Download NMR in metals, metal particles and metal cluster compounds

Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
NMR in metals, metal particles and metal cluster compounds
J.J. van der Klink a,∗ , H.B. Brom b
a
Institut de Physique Expérimentale, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
b Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, Netherlands
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. NMR theory of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. Orbital and spin magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1. Experimental considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2. Electron spin susceptibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3. Orbital susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4. Nonlinear effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Chemical, or orbital-Knight shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Phenomenological generalized susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1. Nonlocal spin susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2. Knight shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3. Spin–lattice relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4. Indirect spin–spin coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5. Overhauser shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. The Pauli approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5. Spin susceptibility enhancements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1. Local density approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2. Spin fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6. Kramers’ degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1. Time reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2. Shift, hyperfine field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3. Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.4. Metals, superconductors, small particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7. Appendix: second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2. Current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.3. Spin magnetization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.4. Power absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∗ Corresponding author.
E-mail addresses: [email protected] (J.J. van der Klink), [email protected] (H.B. Brom).
0079-6565/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved.
PII: S 0 0 7 9 - 6 5 6 5 ( 9 9 ) 0 0 0 2 0 - 5
91
93
93
93
94
95
96
97
99
99
100
102
103
103
104
106
106
111
113
113
115
115
117
118
118
120
121
122
90
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
3. NMR in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Zero of the shift scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. The reference state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2. Optical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Alkali and noble metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Oscillatory Knight shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Transition metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Structure in the density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6. Strong correlation effects and disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7. Strong exchange: magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.1. Hyperfine fields in ESR and NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.2. NMR of manganese metal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. NMR theory of small particles and clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1. Poisson distribution, electron counting, and charging energies . . . . . . . . . . . . . . . . . . . . .
4.1.2. Statistical distribution functions of the energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Electron density and NMR line width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. Electron density variation due to surface effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2. Statistical distribution of the electron density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3. Other approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Comparison of the different NMR models for the NMR line shape . . . . . . . . . . . . . . . . . . . . . . .
4.3.1. Random matrix theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2. Exponential healing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Relaxation in small metal particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1. Korringa-like description of the relaxation in small particles . . . . . . . . . . . . . . . . . . . . . . .
4.4.2. Relaxation due to discrete energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. NMR in small metal particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Small particles: copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Small particles: silver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Small particles: platinum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1. 195 Pt NMR data analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2. The surface peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3. Effects of chemisorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4. Support effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.5. Pt–Pd Bimetallics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4. Small particles: rhodium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. Confined metal clusters and metal cluster compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1. Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1. Zeolites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2. Metal cluster compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.3. Related cluster compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Physical properties of confined metal particles and clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1. 23 Na of the faujasite-structure zeolite-Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2. Sodalites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3. Si-Na clathrates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.4. Clusters in zeolite supercages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3. Physical properties of metal cluster aggregates, Pt309 as NMR-paradigm . . . . . . . . . . . . . . . . .
124
124
124
125
128
133
134
142
145
146
146
148
152
153
153
154
155
155
157
159
159
159
160
162
162
163
165
165
166
169
169
172
174
176
178
180
182
182
182
183
186
186
186
187
187
188
190
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
6.4. Resonance properties of other cluster compounds and colloids . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1. Pt55 cluster compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2. Crystalline Ni38Pt6 cluster compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3. Osmium cluster compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.4. Semiconductor molecular colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
193
193
194
195
195
196
196
Keywords: Generalized Pauli susceptibility; Knight shift; Nuclear spin-lattice relaxation; Electrons in metals; Hyperfine fields; Metal surfaces;
Mesoscopic systems
1. Introduction
Traditionally resonance methods like ESR and especially NMR have yielded a wealth of information about
the electronic properties of metals [1–3]. Using the position of the resonance line with respect to a well chosen
reference and the relaxation rates, the density of states at the Fermi level and the character (e.g. s or d) of the
various bands could be determined. The last survey of these results appeared two decades ago [3]. Since then
more accurate bandstructure calculations became possible by the development of new algorithms based on density
functional theory in the local density approximation and the availability of powerful computers allowing more
extended basis sets to be handled [4]. Experimentally, especially the progress in photoelectron spectroscopy (PES)
now allows the actual measurements of the dispersion curves of the electron states in k -space with accuracies up
to a few millielectron volts. Also in neutron spectroscopy (inelastic neutron scattering, INS) similar improvements
in the k and ω-dependence of the generalized susceptibility [5] have been achieved. NMR is typically sensitive
in the meV range and below, which is at the lower end of the PES or INS range. Information about dispersion is
only indirectly available as the NMR data depend on the summed contribution of all wave vectors. Still, NMR has
unique possibilities, that cannot be easily matched by other techniques. Especially when the interactions between
the electrons become important or the size of the sample starts to play a role, the changes in the nuclear resonance
properties are pronounced and allow a detailed comparison with theory. For example NMR has played (and is still
playing) a crucial role in the study of high-Tc superconductors. By probing the effect of electron spin excitations at
various places in the unit cell, the antiferromagnetic character of the excitations in the normal state was established
early on, as was the possibility of d-wave pairing [6].
This survey is devoted to considering how the electronic states of bulk metals, small metal particles and the
metal cores in metal cluster compounds can be deduced from NMR. Part of the interest in the properties of small
metal particles and molecular metal cluster compounds comes from catalysis. The catalytic possibilities of small
particles, like those of Pt, Pd or Rh, have been well known for more than a century. The probability of a particular
chemical reaction taking place depends among other factors on the morphology and electronic properties of the
individual particle and the structure of its packing. A better knowledge about these parameters is still required to
optimize the processes. In zeolites and metal cluster compounds the metal particles or cluster cores can be arranged
in lattice structures. This opens the fascinating possibility to build new materials with metal particles as the building
unit. Regarding the electronic properties, one might think that as long as a particle contains a few hundred atoms
or more, deviations from bulk samples can be neglected. This turns out to be an oversimplification. On one hand,
surface effects become increasingly important as the particle shrinks, on the other hand, structure in the electronic
and lattice energy states also start to appear. These aspects are very general, e.g. also in constricted geometries
of nanometre size in electronic devices the conductance becomes quantized [7,8]. Information about the specific
properties of assemblies of small particles is given by Perenboom et al. [9], and by Halperin [10]. Here, we discuss
the more recent NMR experiments in terms of the underlying fundamental physical/chemical properties.
92
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
The usual chemical classification of solids is based on differences in bonding, roughly equivalent to addressing
the question ‘where’ the electrons are. In molecular solids the atoms (e.g. Xe) or molecules (e.g. benzene) retain their
identity and are kept together by the rather weak van der Waals forces, based on induced electric dipoles. Permanent
dipoles give rise to more directional forces, like the hydrogen bonds. Still stronger electrostatic attractions are present
in ionic solids, where ions are formed by the transfer of an electron from one kind of atom to another. Metallic
solids are formed of ions of the same kind, kept together by delocalized electrons. Covalent solids are described
as giant molecules, where each atom has a well-defined number of directional chemical bonds with its neighbors.
Solid state physics on the other hand starts by considering the three-dimensional packing of the constituents, and
relies heavily on Fourier analysis of their periodic spatial structure. Therefore, most of its reasoning takes place in
reciprocal space, the Wigner–Seitz unit cell of which is the first Brillouin zone, and all electrons that appear in the
problem are considered to be delocalized.
The contradiction between the chemical and the physical picture is to a large extent only apparent: the difference
is simply in how an inherently N-electron problem is approximated as a sum of N one-electron problems. It is
usually not possible to do this in a unique way. Very schematically, the chemical viewpoint is an extension of
the one-electron problem of the stationary states of the hydrogen atom, introducing the Aufbau principle. The
physical viewpoint is an extension of the problem of a single particle in a box with rigid walls. In the first case,
the prevailing symmetry is spherical, and the ‘natural’ quantum numbers of the possible one-electron states are
those that characterize the spherical harmonics, l and m. In the second case, the translational symmetry leads to
the quantum numbers associated with plane waves, the components of the wave vector k . But when there are many
electrons, we may obtain a charge distribution that can be decomposed as well in a superposition of spherical
harmonics as in a combination of plane waves. Mathematically, this is illustrated by the Rayleigh expansion of a
plane wave in products of spherical Bessel functions and spherical harmonics.
The existence of a mathematical transformation from plane waves (or, slightly more general, Bloch wave functions) to spherical harmonics (and their extension to directional hybrids) does not of course imply that a given
problem can be equally easily understood in both approaches. Some problems are even hard to understand in either:
e.g. the bonding between a metal surface and an adsorbate, or the transfer of electrons from one phase to another,
as in electrochemistry. A physicist doing NMR sees ‘metallic’ aspects on some adsorbate molecules, because the
spin–lattice relaxation rate of nuclei in the adsorbate follows a temperature-dependence characteristic of metals. A
chemist’s picture is that the adsorbate forms a specific bond with only a few nearby metal atoms. The attempts of
NMR to answer such questions are outside the scope of this article, but have recently been reviewed by van der Klink
[11]. Some books on solid state physics with a chemical flavor, e.g. [12,13], discuss these dualities. If the number
of atoms in the metal core of a cluster compound or in a metal particle is small, it is obvious that a description in
real space is more appropriate than in the reciprocal lattice or wave vector space. When the system grows in size
a band description will become more and more appropriate. This is not only true for the electronic but also for the
magnetic properties.
In Section 2 we develop, using the concept of nonlocal susceptibility, a formal theory of NMR parameters that
are governed by a Fermi contact interaction between nuclear and electronic spins. The basic formalism is valid in
molecules (e.g. oxygen) as well as in solids, but we specialize it for metallic solids. It will be shown among other
things, that due to ferro-or antiferromagnetic interactions the simple Korringa relation has to be modified. In their
more general form, the equations remain rather unwieldy, and in Section 3 some semi-empirical simplifications are
proposed. Those readers not interested in formal development should skip directly to there. Most of Section 3 is
devoted to the NMR of bulk metals, discussing the alkalis, the noble metals and some selected transition metals.
NMR evidence for strong correlation and exchange effects is given for some chosen examples.
By shrinkage of the volume, the surface to volume ratio grows. In addition for a metal the structure in the density
of states around the Fermi energy will no longer be negligible and gaps will appear (quantum size effects). The
consequences for the NMR line width and relaxation rates are the subject of Section 4. A difference between small
metal particles and the metal cores in metal cluster compounds is the presence of the surface ligands in the latter.
It appears that these ligands have a negligible effect on the electronic properties of the inner atoms of the core,
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
93
but have the advantage of allowing a high density packing without the danger of coalescence. In aggregates the
distances between the cores are even so small, that the typical time for electron exchange between the cores will be
of the order of the NMR time scale or shorter. Cluster cores can be arranged as in crystals or be randomly packed.
In Section 4 we also discuss the expression for the NMR line shape that has been derived for the particular case of
random packing with electron exchange.
Experimentally, one of the most striking NMR consequences of decreasing particle size is the large increase in
line width, which often is only very weakly temperature-dependent. The characteristic Korringa relation, which
connects the relaxation rates to the Knight shift and temperature, is usually much less affected by size changes,
although considerable effects have been observed in a few cases. The experimental findings for small particles are
the subject of Section 5, while cluster compounds and confined particles in zeolites and other structures are treated
in Section 6.
The experimental and theoretical work discussed in this survey still continues. In chemistry, cluster compounds
with an even larger variety in number and kind of core atoms are synthesized and distances between cores can
be varied by spacer molecules [14]. We have already mentioned similar developments in zeolites. These materials
which combine aspects of the already metallic cores with those inherent to the regular packing form an intriguing
new way of metal synthesis. Alloying of various elements in the cores of cluster molecules and in small metal
particles is another field, which is still growing. Also for the bulk materials new developments are taking place,
partly inspired by the work on superconducting oxides. New theoretical insights are appearing especially in this
area. We expect NMR to remain an important tool in the analysis of metallic materials because of its relative ease
of operation and its still increasing sensitivity due to the availability of stronger magnetic fields.
2. NMR theory of metals
2.1. Orbital and spin magnetism
2.1.1. Experimental considerations
Three important experimental methods give information about the magnetic properties of a metallic sample. In
susceptometry, one measures the strength of the magnetic dipole moment induced in the sample by an applied field.
In conduction-electron spin resonance (CESR), there are two quantities of interest: the integral of the absorption line
shape, called for short the intensity of the signal, and the resonance frequency ω0 /2π in a given applied field B 0 ,
parametrized by the g-factor: ~ω0 = gµBB · S . For a single ‘free’ electron spin S = 21 we have g = 2.0023 ≈ 2;
but usually in ESR the ‘effective’ spin and the g-factor differ from these values because of spin–orbit coupling. In
this article we (almost) always replace g by the number 2. The third method is of course the study of the NMR shifts
with respect to the resonance frequency expected from the gyromagnetic ratio γ of the nucleus ω0 = γ B0 , which
is a central point of this article. Earlier reviews of the subject are [3,15–18].
For a measurement of the (static uniform) magnetic susceptibility of a metallic body, we need initially a volume
of empty space, in which a field H 0 is present, created by a suitable current distribution that is located completely
outside of the volume; next we bring the body in this volume. The associated field B 0 has two effects: it aligns the
magnetic dipole moments associated with the spin of the conduction electrons in the metal, and it induces a current
density in the electronic charge distribution. As a consequence, the magnetic field H (rr ) changes both inside and
outside the body. At a large enough distance from the body, the sources of the change in the field can be written
as multipoles of the current distribution and of the spin magnetization. For small H 0 the susceptibility of the body
equals the ratio of the magnetic dipole moment per unit volume (called the magnetization M ) to the external field
H 0 . To avoid ambiguities, it is desirable that M is independent of sample size: this can be obtained by choosing
certain well-defined sample geometries, of which a long, thin ellipsoid parallel to H 0 is the simplest. In that case
H 0 + M ) = µ0 (1 + χ )H
H 0,
B in the sample = µ0 (H
(1)
94
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
where µ0 is the vacuum permittivity, and χ the (dimensionless) magnetic susceptibility. We remark that it is really
the field B in the sample that determines the NMR frequencies, not the field H in the sample , which for the geometry chosen
above is equal to H 0 . Such susceptibility corrections are sometimes necessary in high-resolution NMR, but in the
NMR of metals they are usually neglected.
In the susceptometric methods, the two different contributions to the susceptibility (orbital currents and spin
alignment) are measured together. In some favorable cases, the effective-spin susceptibility has been measured separately using a technique to calibrate the intensity of the CESR signal by means of the NMR from the same sample
in the same spectrometer setup. Such experiments show that orbital and spin susceptibility are distinct phenomena,
so that they may be discussed separately. The method is based on the following principle. As a consequence of the
Kramers–Kronig relations, the static susceptibility (of the nuclear and the electron spin magnetism) is proportional
to the integral over the absorption line shape of a magnetic resonance experiment (both for nuclear and for electron
spin resonance). The proportionality factor contains instrument-dependent contributions that are not easy to evaluate. Relative measurements, however, can be performed with good accuracy, by leaving the instrument settings
untouched, except for the magnetic field that is first adjusted to observe one resonance, and then the other. The ratio
of the susceptibilities is then obtained as a ratio of two instrument read-outs in arbitrary units. The nuclear magnetic
moments are only weakly coupled: for temperatures above 1 K they have a Curie–Langevin susceptibility that can be
easily calculated. Next, the electron spin susceptibility is simply obtained from multiplication by the experimentally
observed intensity ratio. Schumacher and Slichter [19] were the first to use this principle for sodium metal.
Langevin derived his result using classical statistical mechanics of an ensemble of N permanent magnetic moments
µ in a magnetic field B . The classical Hamiltonian is
H = H0 − µB
N
X
cos αi ,
(2)
i=1
µ|, and B ≡ |B
B |. The magnitude of the aligned magnetic moment per atom M is found to be
where µ ≡ |µ
kT
µB
−
M = µ coth
kT
µB
(3)
M | is M/,  being the atomic volume. At high temperatures, kT µB,
and the corresponding magnetization |M
the susceptibility χL is
χL =
µ0 µ2
µ0 M
=
,
B
3kT 
(4)
which is easily calculated. The quantum mechanical equivalent, required for the application to the magnetization
of nuclear spins I with gyromagnetic ratio γ is obtained by setting µ2 = (γ ~)2 I (I + 1).
2.1.2. Electron spin susceptibility
The spin susceptibility in metals is usually only slightly temperature-dependent, contrary to the Curie-type spin
susceptibility of isolated paramagnetic centers in insulators. This difference in behavior has first been explained by
Pauli, and is closely related to the exclusion principle, which says that the sets of ‘quantum numbers’ describing
the one-particle states of any two electrons cannot be the same. Consider a uniform gas of electrons in a box. In a
mean-field approximation, each electron has the same electrostatic potential energy (due to the average electric field
of all the other electrons). As a consequence, the electrons cannot have all exactly the same kinetic energy, and the
individual electrons in an N-electron system must have quite a range of energies. The highest energy of the range
is called the Fermi energy Ef (for the present discussion it will be practical to choose the zero of the energy scale at
zero kinetic energy). Although the representation of a metal as a collection of electrons in a box is rather crude, it
captures the essence of this statistical problem. At zero Kelvin, all electrons are in the lowest possible energy state,
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
95
so that all one-electron states with energy above Ef are empty. At practical temperatures kT Ef , so that only
a small band of one-electron energies around Ef has a chance of being partly occupied (e.g. having just a spin-up
electron in it, so that an additional spin-down electron could be added by some other interaction). Only the spins of
electrons in this energy band could be turned over by a magnetic field, and contribute to the equivalent of the Curie
susceptibility of paramagnetic centers. A Curie susceptibility varies as (kT )−1 , and the number of contributing
electrons as kT , the net result being a temperature-independent susceptibility. More precisely, the probability to
find a one-electron state at energy occupied is given by the Fermi–Dirac distribution function f ():
f () =
1
exp{( − ζ )/kT } + 1
(5)
where ζ is the chemical potential that normalizes the number of particles N in the system with energy levels i of
degeneracy di :
N=
∞
X
di f (i ).
(6)
i=1
The Fermi energy of an N-electron system is defined as the zero-temperature limit of the chemical potential
Ef = lim ζ (N, T ).
(7)
T →0
In a metal ζ is hardly temperature-dependent and one usually replaces ζ by Ef in Eq. (5). The degeneracies di are
even numbers, because of the spin. When a field H 0 is applied, the energy of di /2 electrons (the ‘down’ spins)
H 0 |, while that of the other di /2 (the ‘up’ spins) decreases to i − µ0 µB |H
H 0 |. (Here µ0 is
increases to i + µ0 µB |H
the vacuum permeability that connects the fields H 0 and B 0 , and µB is the Bohr magneton, the magnitude of the
elementary magnetic dipole moment associated with the electron). The resulting magnetic moment is
µB (N↑ − N↓ ) = µB
∞
X
di
i=1
2
B 0 |) − f (i + µB |B
B 0 |)) = −µ0 µ2B |H
H 0|
(f (i − µB |B
∞
X
di f 0 (i ).
(8)
i=1
Here the applied field is supposed very small, and f 0 (i ) is the derivative of f () taken in = i . The magnetization
H 0|
is the magnetic moment per volume, and the (zero-field) susceptibility χ the ratio of magnetization and field |H
for vanishing values of the field. Writing the volume V of the sample as N times the volume per electron :
∞
µ0 µ2B X
di f 0 (i ) = µ0 µ2B −1 D(Ef , T ),
χ =−
N
(9)
i=1
where the last equality defines the ‘density of states at the Fermi energy’ at temperature T .
In a metal at practical temperatures kT Ef . In that case, f () is to a good approximation an inverted Heaviside
step function θ at the Fermi energy: f () = 1−θ ( −Ef ), and its derivative is the negative of a Dirac delta-function,
f 0 () = −δ( − Ef ). Then we can write the density of states at the Fermi energy as
D(Ef ) = N −1
∞
X
di δ(i − Ef )
(10)
i=1
and therefore, the susceptibility is temperature-independent.
2.1.3. Orbital susceptibility
We now turn to a brief, qualitative discussion of the orbital susceptibility. The usual expressions contain several
terms, identified as generalizations of the diamagnetic (i.e. negative) Larmor–Langevin susceptibility of free ions;
96
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
of the paramagnetic (i.e. positive) van Vleck susceptibility of free ions, related to mixing of excited states into the
ground state by the applied field; and of the Landau susceptibility of the free-electron gas, that may have either
sign in the general case [20]. While this separation is conceptually convenient, there is only one physically real
orbital susceptibility, formed by the sum of all contributions. This is most easily understood by considering the
induced current distribution as a linear response to the vector potential A 0 associated with the applied field B 0 . The
same field can be described by several different vector potentials; these are said to correspond to different gauges.
Under different gauges, the induced current is expressed as a sum of different terms, while the sum of these terms
is gauge-independent. Therefore, the individual terms in the sum have no real physical meaning. The separation
described above corresponds of course to a ‘suitable’ choice of gauge.
To find expressions for the orbital susceptibility, two different approaches have been used: either we evaluate
the magnetization associated with the current induced by the field (as described above) or we calculate the thermodynamic potential in the presence of the field and take the second derivative with respect to the field (as done
originally by Landau). In the following we only consider the linear-response method (which is valid in small fields,
and does not yield nonlinear effects such as the de Haas–van Alphen (dHvA) oscillations, see Section 2.1.4). Since
these are zero-field calculations the electronic structure can be based on a k -space (Bloch function) description.
While this has the advantage of familiarity, it turns out to be unable to yield the terms corresponding to the Landau
susceptibility. This is because the Bloch description relies on the translational periodicity in the bulk and does not
represent surface effects (unless special measures are taken).
It has already been shown by Teller in 1930 [21] that the Landau diamagnetism of free electrons, represented by
simple plane waves in the Bloch picture, is due to a surface current that gives a finite contribution to the susceptibility
even in the limit of a semi-infinite volume. This is easy to see for a collection of free electrons in a box. Since there is
no internal structure, the system is completely translationally invariant if the box is infinitely large. Because of this
invariance, the magnetization must be uniform; Maxwell’s equations then require zero current density everywhere
inside the box (with possible exception of the surface, which for an infinite box is not well defined). Recently
developed methods [22–24] use Green’s functions to describe the electronic structure. This is more general: it could
be specifically applied to surface situations, and in the context of our article its applicability to clusters is particularly
attractive. For bulk calculations all three types of terms are found.
Another kind of difficulty in a correct formulation of the orbital susceptibility is related to the vector potential A
associated with a homogeneous field B : when the volume tends to infinity, the vector potential must diverge. In an
approach proposed by Luttinger and Stiles [25], the problem is initially formulated in terms of a periodic magnetic
field, and only at the very end as the limit where the period becomes infinitely long.
2.1.4. Nonlinear effects
At low temperatures and/or high applied fields (such that kT is smaller than ~ωc , with ωc the cyclotron frequency
to be defined below), the susceptibility of sufficiently pure specimens shows periodic variations as a function of
field, superimposed on the low-field value: the dHvA oscillations [26]. To explain the basics of the phenomenon,
it is sufficient to consider electrons that (in absence of B 0 ) can be represented by plane waves with wave vector k .
This representation of the electronic eigenstates reflects the translational symmetry in all three directions of space.
When a magnetic field Bz is applied, only the translational symmetry represented by kz is maintained, but kx and
ky are no longer ‘good quantum numbers’. Consider the group of electrons that have all the same component of
their wave vector along the magnetic field, between kz and kz + dkz , and therefore, all the same kinetic energy in
the z-direction. The quantization of their transverse kinetic energy is now replaced by that of a circular motion,
quantized just as a harmonic oscillator,
E(n) = (n + 21 )~ωc
(11)
and the fundamental frequency is the cyclotron frequency ωc related to the field Bz by ωc = eBz /m, where
m is the mass of the electron. The continuous spectrum of transverse kinetic energies has been replaced by the
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
97
discrete spectrum of an harmonic oscillator. To observe this effect, it is of course necessary that the ‘width’ of
approximately kT of the Fermi–Dirac function (Eq. (5)), be smaller than the separation of the cyclotron levels;
otherwise the ‘blurred’ cyclotron energy levels will be essentially indistinguishable from the initial kinetic-energy
levels.
In the absence of electron scattering, the longitudinal kinetic energy of an electron remains unchanged when
the magnetic field is swept. The total transverse energy (the number of electrons in each cyclotron level n times
E(n), summed over n) will vary, and not necessarily quadratic in the field, thus yielding a field-dependent susceptibility. When the uppermost occupied level rises above the Fermi energy, electrons will ‘fall’ into the lower
levels (this is possible because their degeneracy increases with increasing field): at some point in the field sweep
the highest occupied level will actually be below the Fermi energy, and continue to rise towards it. This rising
of the energy of the highest occupied level followed by its emptying causes a sequence of relative minima and
maxima in the total free energy, and therefore, in the susceptibility (which is proportional to the second derivative
with respect to the field). This simplified description makes plausible that the variation in susceptibility is orbital in
character.
There may be a variation in spin susceptibility as well. It turns out that the rising and emptying of the cyclotron
levels described above implies a field-dependent density of states at the Fermi energy, leading to variations in the
spin susceptibility. It is believed that this effect has been seen in Knight shift measurements to be discussed in
Section 3.3.
2.2. Chemical, or orbital-Knight shift
To find the local magnetic field at the site of a nucleus (that we will take as the origin), due to the current
distribution j ind (rr ) induced in the sample by the external field B 0 , we must use the linear-response theory relating
j ind (rr ) to the vector potential A 0 (rr ) associated with B 0 , as mentioned in Section 2.1.3 (Landau’s thermodynamic
approach cannot be used for local quantities). While the Green’s function formulation of this theory is the only
one really suited for actual computation [23,24], we believe that the Bloch-function description is perhaps easier to
grasp. Both methods agree that the Landau-type contributions to the shift are pure susceptibility effects, as expressed
by Eq. (1). In the formulation below, we will find contributions to the shift that correspond to the Larmor–Langevin
and van Vleck terms in the susceptibility.
We start by referring to an extended form of Biot and Savart’s law ([27], Chapter 4) which is the solution of the
differential equation
∇ × ∇ × A ind = µ0j ind ,
given by
A ind (rr ) =
µ0
4π
Z
1
j ind (rr 0 ) 0
drr +
0
4π
sample |rr − r |
(12)
I
surface
B ind (rr 0 ) × n (rr 0 )
dS.
|rr − r 0 |
(13)
The formula is valid for r either inside or outside the sample volume. The first term to the right is the usual
Biot–Savart form, and the second describes the discontinuity at the surface. The n (rr 0 ) is a unit vector normal to the
surface element dS in the point r 0 . For all points r sufficiently far from the surface, we can use the macroscopic
(m)
approximation for B ind (rr 0 ) with r 0 in the surface; for suitable sample shapes the macroscopic B ind is uniform in
the sample, and for the case of Eq. (1) the proportionality constant is χ . Therefore, we have in the origin, r = 0
Z
Z
µ0
χ
1
r 0 × j ind (rr 0 ) 0
r
B ext .
drr 0 + χB
dr
+
∇
B
·
∇
(14)
B ind (0) = ∇ × A ind (0) =
ext
4π sample
4π sample
|rr 0 |
|rr 0 |3
The first term to the right is the field due to the microscopic currents; the two other terms are macroscopic and
describe the (dipolar) demagnetizing field and the overall difference between the macroscopic applied B 0 and the
98
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
macroscopic internal field B in the sample . In the following we only retain the first term; when necessary a macroscopic
correction to the value of B ext can be made to account for the two other terms.
The problem is now to evaluate j ind (rr ) from the, supposedly known, electronic structure in zero field. This can
be done using the methods of second quantization, as shown in Section 2.7. Using Eq. (151) of that section we
obtain
Z
∞ r
µ0
µ0 e2 X
r × j ind (rr )
drr =
−δij f (i ) φi 3 × A ext (rr ) φi
B ind (0) =
4π sample
4π m
|rr |3
|r|
i,j =1
r
I (i , j )
φi 3 × p φj φj A ext (rr ) · p + p · A ext (rr )φi .
+
2m
|r|
(15)
In our application we want A ext (rr ) to represent a homogeneous applied field B 0 . Unless the gauge is carefully
chosen, the vector potential will cause the matrix elements in Eq. (15) to diverge. The way to avoid this problem is
to take initially a periodic field, and let the period go to infinity at the end of the calculation [25]:
A ext (rr ) =
B0 × q
sin(qq · r ) = Aq sin(qq · r ),
|q|2
(16)
where the otherwise arbitrary vector q satisfies q · B 0 = 0, and the right-hand side defines a shorthand Aq ; this
choice of A ext (rr ) gives B ext (rr ) = B 0 cos(qq ·rr ), which has the desired limiting behavior. However, the |q|2 occurring
in Aq may cause problems, unless properly handled.
The way to do this has been indicated in [28]. Let g(x) be some function of x with derivative g 0 (x), and let H0
be the zero-field one-electron Hamiltonian (see Eq. (143)), with eigenfunctions φi and eigenvalues i . We will be
interested in the commutator
2
p
i~
Aq · r )g(qq · r ) =
Aq · r )g(qq · r ) = −
Aq · p )g(qq · r )
, (A
(A
H0 , (A
2m
2m
p · q )(A
Aq · r )g 0 (qq · r ) + (A
Aq · r )g 0 (qq · r )(p
p · q) .
Aq · p ) + (p
(17)
+g(qq · r )(A
The matrix elements of any such commutator are
φi [H0 , X] φj = (i − j ) φi X φj .
(18)
p · q ), as well as the vector
Aq · r ) commutes with (p
Now we choose g(x) = sin(x) − (x/2) cos(x), and use that (A
identity
Aq · p ) − (qq · p )(A
Aq · r )
(qq × Aq ) · (rr × p ) = (qq · r )(A
(19)
to write Eq. (17) in the form
2mi
Aext (rr ) · p + p · A ext (rr )|φi =
Aq · r )g(qq · r )|φi
(j − i ) φj |(A
φj |A
~
B ext (rr ) · r × p |φi − φj 21 R φi ,
+ φj |B
(20)
where the operator R is defined as
Aq · r )(qq · r ) ( sin(qq · r )) (p
p · q)
p · q )(A
Aq · r )(qq · r ) sin(qq · r ) + (A
R = (p
and has a well-defined limit of 0 when |qq | tends to zero, so that we may neglect it in the following.
(21)
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
99
When we insert Eq. (20) into the second term of Eq. (15), the first term on the right of Eq. (20) gives a contribution
∞
X
r
I (i , j )
2mi
Aq · r )g(qq · r )|φi
φi 3 × p φj
(j − i ) φj |(A
2m
~
|r|
i,j =1
∞
r
iX
Aq · r )g(qq · r ) φi
f (i ) φi × p , (A
=
~
|r|3
(22)
i=1
(where we have used the definition of I (i , j ), see Eq. (139)). The commutator
r
r
r
i
Aq · r )g(qq · r ) = 3 × Aq g(qq · r ) + 3 × q (A
Aq · r )g 0 (qq · r )
× p , (A
3
~ |r|
|r|
|r|
=
B ext − (B
B ext · r )rr
(rr · r )B
r × A ext
−
+ R0,
3
3
|r|
2|r|
(23)
where
Aq · r )(qq · r ) sin(qq · r )
R 0 = (A
r ×q
2|r|3
(24)
again has a well-defined limit of 0 when |qq | tends to zero.
The first term on the right of Eq. (23) compensates the first term on the right of Eq. (15). Finally, in the evaluation
of Eq. (15) we are left with the second term in the right-hand side of Eq. (20) and the second term in the right-hand
side of Eq. (23). The induced field in the origin (supposed to be the location of a nucleus) becomes
∞
(rr · r )B
B 0 − (rr · B 0 )rr µ0 e2 X
− f (i ) φi B ind (0) =
φi
4πm
2|r|3
i=1
+
∞
r
µ0 e2 X I (i , j )
φi 3 × p φj φj B 0 · r × p φi
4πm
2m
|r|
(25)
i,j =1
B 0 · B ind (0))/(B
B 0 · B 0 ). The first term is negand the corresponding Knight shift Korb is given by Korb = (B
ative (diamagnetic), the second term positive (paramagnetic). The use of this expression will be discussed in
Section 3.4.
2.3. Phenomenological generalized susceptibility
2.3.1. Nonlocal spin susceptibility
The magnetization (magnetic dipole moment per unit volume) due to the electronic spins is a quantity that can be
defined inside an atom. On that scale, the response to a uniform applied field is nonuniform because of variations
of spin density inside an atom; furthermore we must take into account that due to electron–electron interactions
the ‘effective’ field actually is nonuniform as well. It is, therefore, convenient to introduce a susceptibility that
relates the magnetization in a point r to a field applied in another point r 0 . It will be useful to consider applied
fields that vary harmonically in time, but we will need only slow variations, so that there is no need to take the full
electromagnetic field into account. Let the sample be subjected to a nonuniform field H (rr 0 ) that varies in time as
cos(ωt). We will be interested in the component of M (rr ) colinear with the field, so that we treat the susceptibility
as a scalar function times a unit tensor, and we will not write the tensor explicitly. In the linear approximation,
a harmonic excitation causes a harmonic response, possibly with a phase lag. The susceptibility is defined as a
100
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
complex quantity χ(rr , r 0 ; ω) = χ 0 (rr , r 0 ; ω) − iχ 00 (rr , r 0 ; ω), such that
H (rr 0 ; t) = H (rr 0 ) cos(ωt)
Z
Z
0
0
0
0
H (rr ) drr + sin(ωt)
χ (rr , r ; ω)H
M (rr , t) = cos(ωt)
sample
H (rr 0 ) drr 0 .
χ 00 (rr , r 0 ; ω)H
(26)
sample
We will see that the static uniform susceptibility, the Knight shift, the spin–lattice relaxation rate and the indirect
spin–spin coupling constant can all be expressed in terms of χ(rr , r 0 ; ω) in the limit ω → 0. The static susceptibility
is proportional to χ 0 integrated over both spatial arguments; the Knight shift has an integral over one argument only,
the other being kept fixed at the nuclear site; the coupling constant contains χ 0 with one argument at the site of the
first, and the other at the site of the second nuclear spin; finally the spin–lattice relaxation rate is related to χ 00 with
both arguments at the site of the nucleus.
The response to a position- and time-independent field is given by the static uniform susceptibility χ 0
Z Z
χ 0 = V −1
χ 0 (rr , r 0 ; 0) drr 0 drr .
(27)
sample
The energy absorbed from a nonuniform time-varying field per unit time averaged over a cycle is
Z
Z Z
Z
M
∂M
ω
ω 2π/ω
H (rr 0 ) drr drr 0 .
B (rr ; t) ·
B (rr )χ 00 (rr , r 0 ; ω)H
drr dt =
P (ω) =
2π 0
∂t
2
sample
sample
(28)
In a crystalline solid, the generalized susceptibility is invariant under translation through a vector R α of the underlying
Bravais lattice:
χ (rr , r 0 ; ω) = χ(rr + R α , r 0 + R α ; ω).
(29)
Writing r = ρ + R α and r 0 = ρ 0 + R β , with ρ and ρ 0 in the unit cell at the origin, this gives
ρ , ρ 0 + R β − R α ; ω) = χ(ρ
ρ , ρ 0 + R γ ; ω).
χ (rr , r 0 ; ω) = χ(ρ
(30)
If the crystal consists of N unit cells, there are N different vectors R γ . Sometimes this dependence of χ on lattice
vectors in real space is replaced by a dependence on vectors q α in the first Brillouin zone of the reciprocal lattice
through the Bloch Fourier transform:
ρ , ρ 0 ; q α ; ω) =
χ̃ (ρ
N
X
ρ , ρ 0 + R β ; ω)
exp(iqq α · R β )χ(ρ
(31)
β=1
N −1
N
X
ρ , ρ 0 ; q α ; ω) = χ(ρ
ρ , ρ 0 + R γ ; ω).
exp(−iqq α · R γ )χ̃ (ρ
(32)
α=1
In terms of χ̃ 0 the static uniform susceptibility is
Z Z
ρ , ρ 0 ; 0; 0) dρ
ρ dρ
ρ 0,
χ 0 = −1
χ̃ 0 (ρ
(33)
cell
where  is the volume of the unit cell.
2.3.2. Knight shift
The Knight shift is related to the magnetic interaction between nuclear and electronic spins. Consider the magnetic
dipole µ I associated with a nuclear spin I situated at a point r in a crystal lattice. Let M (rr ) be the electron spin
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
101
magnetization in an arbitrary point r in the crystal. The coupling energy is [29]
2µ0
R)
µ · M (R
3Z I
M (rr ) · µ I ) ((rr − R ) · (rr − R )) − 3 (M
M (rr ) · (rr − R )) ((rr − R ) · µ I )
µ0
(M
drr .
+
4π r 6=R
|rr − R |5
UI S = −
(34)
The r 6= R roughly says that the volume of the nucleus at r should be excluded from the integration. (In actual
applications this is not a problem: in a tight-binding picture s-like wave functions are the only ones that do not go
to zero on the nucleus; but their spherical symmetry integrates to zero net contribution). It should be noted that we
R ) to determine the hyperfine coupling, since the electronic spin–lattice relaxation
can use the thermal average M (R
is much faster than the nuclear spin–lattice relaxation. The first term gives the isotropic part of the shift, usually
called ‘the’ Knight shift K. This term does not exist in a purely classical picture. It is the contact interaction and
occurs when the nuclear and the electronic spin are ‘in the same place’, which is impossible with classical particles.
This part of the interaction energy has the same form as the interaction between a nuclear magnetic moment and
R ) = (2µ0 /3)M
M (R
R ), similar to the expression for a chemical shift. The (isotropic part of)
some ‘external’ field B S (R
K is by convention positive when the field B S is parallel to the uniform applied field B 0 :
Z
Z
R) · B 0
2
2µ0 M (R
2
0
0
0
R) =
ρ , ρ 0 ; 0; 0) dρ
ρ 0 = K(ρ
ρ ),
=
χ (rr , r ; 0) drr =
χ̃ 0 (ρ
K(R
(35)
3 B0 · B0
3
3
where the nucleus under consideration has a relative position ρ in the unit cell.
By definition, the hyperfine field Bhf is simply related to ratio of the Knight shift and the uniform susceptibility
through the dimensionless quantity
ρ)
ρ)
K(ρ
Bhf (ρ
=
.
µ0 µB
χ0
From the expressions for K, Eq. (35), and χ 0 , Eq. (33), we see that
R 0
ρ , ρ 0 ; 0; 0) dρ
ρ0
χ̃ (ρ
2
ρ ) = µ0 µB R R 0
.
Bhf (ρ
ρ , ρ 0 ; 0; 0) dρ
ρ dρ
ρ0
3
χ̃ (ρ
(36)
(37)
The Knight shift is then written as
ρ) =
K(ρ
χ 0
ρ ),
Bhf (ρ
µ0 µB
(38)
but we will see later that this simple definition of ‘the’ hyperfine field is not always physically meaningful, and that
it can be more judicious e.g. to attribute different hyperfine fields to s-like and to d-like electrons.
The second term in Eq. (34) is the usual magnetic dipole–dipole coupling and gives the anisotropy of the Knight
shift in noncubic metals, as can be seen more easily by writing the scalar products in terms of the Cartesian
components
Z R 0
X
χ (rr , r 0 ; 0) drr 0
µI,i Hj
(δij r 2 − 3ri rj ) drr .
(39)
Kdip =
|rr |5
i,j =x,y,z
The (δij r 2 − 3ri rj ) are proportional to the second rank spherical harmonics: therefore, they form a second rank
irreducible tensor, just as the anisotropic part of the chemical shift. If the nucleus is in a site with the symmetry of
one of the five cubic point groups, only the isotropic shift can be nonzero. For the eight point group symmetries
in the orthorhombic, monoclinic and triclinic systems, the anisotropy is determined by two constants and is said
102
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
to be asymmetric; for the remaining 19 point group symmetries of the hexagonal, tetragonal and trigonal systems,
the anisotropy is symmetric around the n-fold (n = 6, 4, 3) rotation, or rotation-inversion axes and is, therefore,
determined by one constant. Note that the dipole–dipole coupling between two well-localized dipoles has axial
symmetry around the line connecting them; but in the present case the electron is not considered as localized, and
therefore, the average interaction can have lower than axial symmetry.
A rather subtle form of this dipolar shift can occur in ionic paramagnetic complexes: as an example we consider
the Cu2+ ion, containing nine d-electrons, surrounded by six anions in an octahedral arrangement ([30], p. 412,
456). The electrostatic interaction with the ‘crystal field’ (as it is called) created by these six ions partially lifts the
degeneracy in energy of the five d-orbitals of the free copper ion, and splits them in one group of two, and another
group of three. It is convenient to consider the missing electron in the copper d-shell as a hole: the hole can be
in either of the two orbitals |x 2 − y 2 i or |3z2 − r 2 i. If both these orbitals were equally occupied, the resulting
‘hole-distribution’ around the copper nucleus would have cubic symmetry, and there would be no spin–dipolar
coupling. As a rule, however (the Jahn–Teller theorem), such orbital degeneracies are not ‘stable’, and the crystal
structure will deform somewhat, to separate |x 2 − y 2 i in energy from |3z2 − r 2 i. The hole will then be on one of
these orbitals only, giving a definitely noncubic distribution, and therefore, an anisotropic Knight shift tensor.
2.3.3. Spin–lattice relaxation
Under the influence of thermal effects (in fact, of the electron spin–lattice relaxation, caused by modulation of
the electron spin–orbit coupling by the lattice vibrations) the transverse part of B S (rr ) fluctuates rapidly (on the
NMR time scale) around an average value of zero. Using BPP theory, this can be treated as ‘scalar relaxation of the
second kind’ ([1], Ch. VIII, Eq. 125) so that the nuclear spin–lattice relaxation rate is given by
Z
2µ0 γI 2 1 +∞
R ; t)M− (R
R ; 0)iexp(−i(ωs − ωI )t) dt,
hM+ (R
(40)
T1−1 =
3
2 −∞
where the subscript I refers to the nucleus, and s to the electron. The integral represents the spectral density of the
transverse fluctuation in the electron spin magnetization on the site R at the difference of the electronic and nuclear
Larmor frequencies. This frequency difference appears because the operators involved are of the type I+ s− and
I− s+ , that flip simultaneously one kind of spin ‘up’ and the other ‘down’. In usual cases the spectral density is flat
between zero and the electronic Larmor frequency: the ‘extreme narrowing’ limit of BPP theory.
In low (zero) field and cubic symmetry we have that
R ; t)M− (R
R ; 0)i = 2hMz (R
R ; t)Mz (R
R ; 0)i.
hM+ (R
(41)
The fluctuation–dissipation theorem says that the spectral density of the correlation function in the right-hand side
of Eq. (41) is proportional to the imaginary part of the susceptibility:
Z
kT 00
1 +∞
R ; t)Mz (R
R ; 0)iexp(−iωt) dt =
R , R ; ω)
χ (R
hMz (R
(42)
2 −∞
µ0 ω
and the expression for the relaxation rate, Eq. (40), becomes
T1−1
= µ0
2γI
3
2
R , R ; ωS − ωI )
χ 00 (R
2kT
= µ0
ωS − ωI
2γI
3
2
2kT N −1
N
X
ρ , ρ ; q α ; ωS − ωI )
χ̃ 00 (ρ
.
ωS − ωI
(43)
α=1
The imaginary part of the susceptibility is an odd function of frequency, and linear for small values of ω. The
right-hand side of Eq. (43) is then frequency-independent and can be evaluated in the limit of vanishing frequencies.
If χ 00 is independent of temperature, then so is the product T1 T . This latter result has been derived by Heitler and
Teller in 1936 [31], well before the discovery of NMR, in a theoretical study of cooling by adiabatic demagnetization.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
103
Note that the sum in the right most member of Eq. (43) represents the average over all vectors q α of the susceptibility
χ̃ 00 . Sometimes a q -dependent hyperfine field Bhf (qq ; ρ ) is defined such that Eq. (43) can be written
(T1 T )−1 = µ0
2γI
3
2
2kN −1
N
X
χ̃ 00 (qq α ; ωS − ωI ) 2
Bhf (qq ; ρ ),
ωS − ωI
(44)
α=1
where χ̃ 00 (qq ; ω) describes the response to a field H cos(ωt) exp(iqq · r ) according to Eq. (26). Of course an actually
applied field must be a combination of exp(iqq · r ) and exp(−iqq · r ).
2.3.4. Indirect spin–spin coupling
The indirect nuclear spin–spin coupling in metals, experimentally detected and theoretically explained by Bloembergen and Rowland [32] (but in the literature more often associated with Ruderman and Kittel [33]) can also be
described in terms of the generalized susceptibility. Basically, the contact interaction between a nuclear spin I 1
situated at R 1 and the local electron spin density propagates through the nonlocal susceptibility to another nuclear
spin I 2 situated at R 2 . The contact field B I (rr ) in point r due to the nuclear spin at R 1 is
B I (rr ) =
2µ0
γ1 ~II 1 δ(rr − R 1 )
3
(45)
and creates an electronic magnetization
M (rr ) =
2µ0
γ1 ~II 1 χ 0 (rr , R 1 ; 0).
3
(46)
R 2 ) has a contact interaction with the nuclear moment γ2 ~II 2 :
In the point r = R 2 the magnetization M (R
2µ0
2µ0
R 2 , R 1 ; 0) = J (R
R 2 , R 1 )II 1 · I 2 ,
γ2 ~II 2 ·
γ1 ~II 1 χ 0 (R
URKBR =
3
3
(47)
which gives the phenomenological equation for the indirect spin–spin coupling, and defines the coupling constant
J , which is very similar to the J encountered in liquid high-resolution NMR. When both spins are in the same unit
cell, at positions ρ 1 and ρ 2 , then the coupling constant is sometimes written as
ρ 1, ρ 2) =
J (ρ
2µ0 ~
3
2
γ1 γ2 N −1
N
X
ρ 1 , ρ 2 ; q α ; 0),
χ̃ 0 (ρ
(48)
α=1
where we use the Bloch Fourier transform according to Eq. (32).
2.3.5. Overhauser shift
The contact interaction is ‘symmetric’ in nuclear and electronic spins, as can be seen more easily by writing the
first term of Eq. (34) as
(c)
UI S = −
2µ0 X
2µ0 X
Rj ) =
R j − r i )(γI ~II j ) · (2µBS i ),
µ Ij · M (R
δ(R
3
3
j
(49)
i,j
where the index j runs over the nuclei and i over the electrons. For the Knight shift, we choose a particular value of
j , and look at the effect of all the i. For the corresponding Overhauser shift in conduction-electron spin resonance
we choose a particular i, and look at the effect of all the j . The Overhauser shift O is thus proportional to the thermal
average hγI ~II i = χnH 0 , with χn the nuclear magnetic susceptibility, just as the Knight shift is proportional to the
electron spin susceptibility. So far as the introduction of a single hyperfine field is meaningful, we will, therefore,
have that O = (χn /χ 0 )K. The interesting thing is that at practical temperatures the nuclei form a magnetically
104
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
noninteracting system with an easily computable Langevin susceptibility, Eq. (4), so that a measurement of O can
be interpreted as a measurement of Bhf .
2.4. The Pauli approximation
The expression for the electron spin susceptibility in a metal that neglects the magnetic interactions between
electrons (the ‘molecular field’, the ‘Stoner enhancement’) is often called the Pauli susceptibility (in a strict sense;
the name is sometimes used as synonym for electron spin susceptibility). We can find the Pauli susceptibility in the
low-frequency limit from the expressions for the spin density and for the absorbed power per unit energy interval
(see Eqs. (161) and (168) in Section 2.7). To indicate the low-frequency limit, we will omit the argument ω in the
following. From a comparison of Eq. (161)
Mz (rr ) =
2µ0 µ2B
∞
X
0
−δij f (i ) + I (i , j )
i,j =1
Z
φi∗ (rr )φj (rr )
sample
H 0 (rr 0 )φi (rr 0 ) drr 0
φj∗ (rr 0 )H
(50)
and the phenomenological definition of the nonlocal susceptibility χ 0 , Eq. (26), we make the identification (subscript
P for the Pauli approximation)
∞
X
χP0 (rr , r 0 ) = 2µ0 µ2B
i,j =1
−δij f 0 (i ) + I (i , j ) φi∗ (rr )φj (rr )φj∗ (rr 0 )φi (rr 0 ),
where the φ are orthonormal in the volume of the sample:
Z
φi∗ (rr )φj (rr ) drr = δij
(51)
(52)
sample
so that with Eq. (35) the Pauli approximation for the Knight shift of a nucleus in position ρ is
∞
X
4
ρ ) = − µ0 µ2B f 0 (i )|φi (ρ
ρ )|2 .
KP (ρ
3
(53)
i=1
By comparing Eq. (168)
∞
X
2P (ω)
2
=
−
2πµ
µ
f 0 (i )δ(j −i −~ω)
lim
0
B
~ω→0 ~ω2
ZZ
i,j =1
sample
B (rr )φj (rr )φj∗ (rr 0 )H
H (rr 0 )φi (rr 0 ) drr drr 0 (54)
φi∗ (rr )B
with the phenomenological definition of the nonlocal χ 00 (Eq. (28)), we can likewise identify
χP00 (rr , r 0 ) = −2π~ωµ0 µ2B
∞
X
f 0 (i )δ(j − i − ~ω)φi∗ (rr )φj (rr )φj∗ (rr 0 )φi (rr 0 ).
(55)
i,j =1
If the sample is a metal, then at practical temperatures the Fermi–Dirac distribution function is an inverted step
function at the Fermi energy, and its derivative is the negative of a Dirac delta-function:
(metal)
−f 0 (i )δ(j − i − ~ω) = f 0 (i )f 0 (j )
2
∞
X
00
0
2 0
∗
0 ρ , ρ ) = 2π~ωµ0 µB f (i )φi (ρ
ρ )φi (ρ
ρ )
χP (ρ
i=1
(56)
(57)
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
105
so that with Eqs. (43) and (53) we immediately find the Korringa [34] relation
2
S(T1 T )−1
P = KP
with the Korringa constant
S=
(2µB )2
.
4π ~kγ 2
(58)
It is sometimes stated that this relation is based on the free-electron approximation, or on a Lindhard type of wave
vector-dependent susceptibility. Our derivation of the Pauli approximation to the nonlocal susceptibility does not
make these approximations (but the Lindhard susceptibility is recovered when in Eq. (50) the wave functions φ are
plane waves and the energy is quadratic in their wave vector). The Korringa relation depends on a low-frequency
approximation for χ 00 (ω), on a nonzero and continuous (but otherwise arbitrary) density of states around Ef , and,
most important, on the neglect of exchange effects in the expression for the susceptibility (while such effects may
perfectly well have been taken into account in the determination of the one-electron energies and wave functions).
As we will see in Section 3.2, even the alkali metals have relaxation rates and Knight shifts that numerically do not
obey Eq. (58) very well.
It is instructive to see the modifications that occur in the temperature dependences when these equations are
applied to an intrinsic semiconductor. At zero temperature, its conduction band is empty, and the unoccupied
one-electron states that are lowest in energy are separated from the highest occupied state in the valence band by
the gap energy Eg . Depending on details of the band structure, the Fermi level lies about halfway in the gap, and at
practical temperatures Eg /2 kT . Therefore, instead of Eq. (56) we have
(i − Ef )
1,
f 0 (i ) ≈ −(kT )−1 f (i ), (semiconductor, i > Ef ).
(59)
f (i ) ≈ exp −
kT
The temperature dependence of the Knight shift is contained in
X
X
− f 0 (i ) = (kT )−1 f (i ) = (kT )−1 Ne (kT ),
i
(60)
i
where Ne (kT ) is the number of electrons excited to the conduction band at temperature T . To calculate this quantity
we must know something about the density of states D() in the conduction band. In a simple case, this is proportional
to ( − Ef + Eg /2)1/2 , so that
Z ∞
Eg
3/2
.
(61)
D()f () d ∝ (kT ) exp −
Ne (kT ) =
2kT
Ef +Eg /2
The temperature dependence of the product (T1 T )−1 is contained in
Z Z ∞
X
Eg
0
−1
0
0
0
.
− f (i )δ(j − i ) = (kT )
D()D( )f ()δ( − ) d d ∝ kT exp −
2kT
Ef +Eg /2
(62)
i,j
Such a temperature dependence is measured in the semiconductor Te [35] below 400 K with Eg = 0.30 eV (see
Fig. 1).
Coming back to the subject of metals, we use that f 0 (i ) ≈ −δ(i − Ef ) and the density of states D(Ef ) (twice
the number of energy levels at the Fermi energy, normalized per electron) to write
χP0 = µ0 µ2B −1 D(Ef )
with
D(Ef ) = 2N −1
∞
X
δ(i − Ef ).
(63)
i=1
The periodicity of the crystal lattice means that the one-electron functions φ(rr ) can be chosen in the Bloch form:
ρ ) exp(ikk · R α ) with ρ restricted to the unit cell in the origin, and R α a vector in the
ρ + R α ) = N −1/2 ϕ(ρ
φ(ρ
106
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 1. T1−1 T −2 as function of the inverse temperature in the semiconductor Te. The dashed line shows the exponential dependence on Eg /2.
(After Selbach et al. [35]. ©1979 American Physical Society).
Bravais lattice of the crystal. The functions ϕ are normalized in the unit cell, but not necessarily orthogonal. The
low-frequency form of the complex Pauli susceptibility becomes:
ρ + R α , ρ 0 + R β ) = 2µ0 µ2B N −2
χP (ρ
∞
X
−δij f 0 (i ) + I (i , j ) − iπ ~ωf 0 (i )f 0 (j )
i,j =1
∗
ρ
ρ
ρ 0 )ϕi (ρ
ρ 0 ) exp
×ϕi (ρ )ϕj (ρ )ϕj∗ (ρ
Rβ − Rα) .
i(kk i − k j ) · (R
(64)
(Do not confuse the imaginary unit i with the summation index i on k and on ). This expression for the complex
susceptibility in the Pauli approximation is useful when we consider the Stoner enhanced susceptibility in Section
2.5.1.
If we restrict the wave vectors k to the first Brillouin zone, then we need a slight change in notation. The indices i
and j will run over these wave vectors, and the one-electron energies as well as the wave functions ϕ will need two
indices: one (i, j ) to indicate the wave vector and another (n, n0 ) as a band index. The Bloch Fourier transformed
susceptibility is
X
X
ρ , ρ 0 ; q α ) = 2πµ0 µ2B N −1 1(qq α + k i − k j )
−δij δnn0 f 0 (i,n ) + I (i,n , j,n0 )
χ̃P (ρ
i,j
0
0
−iπ ~ωf (i,n )f (j,n0 )
n,n0
∗
∗
ρ )ϕj,n0 (ρ
ρ )ϕj,n
ρ 0 )ϕi,n (ρ
ρ 0 ),
× ϕi,n (ρ
0 (ρ
(65)
where N now is the number of unit cells. The 1(qq α + k i − k j ) is a generalized Kronecker symbol that restricts
the summation to wave vectors such that q α + k i − k j = 0. (Note the difference between a Kronecker symbol like
δll 0 , which restricts a summation to terms with l = l 0 , and has no units (no ‘dimension’), and a Dirac delta function
like δ( − i ), which makes an integral over take on the value of the integrand for = i and has the ‘dimension’
of the inverse of its argument. A generalized Kronecker symbol like 1(qq α − q β ) restricts a summation over wave
vectors, but still has no dimension).
2.5. Spin susceptibility enhancements
2.5.1. Local density approximation
As we will see in Section 3.2, measurements in the alkali metals of the Knight shift K, the Overhauser shift
O (which gives the hyperfine field Bhf ) and the calibrated intensity of the conduction electron spin resonance
(CESR) signal (which gives the static spin susceptibility χ 0 ) are consistent with the structure of the equations
given in the Pauli approximation. Also the Heitler–Teller product (more often called Korringa product) T1 T is to a
good approximation temperature-independent. However, the value calculated for the susceptibility from theoretical
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
107
density of states curves does not agree particularly well; neither does the Korringa relation between the square
of the shift and relaxation rate given by Eq. (58). Furthermore in the Pauli approximation the total shift must be
necessarily positive, since the (spin) Knight shift is positive, and the net orbital shift is expected to be either very
small, or dominated by the positive van Vleck contribution. However, in some transition metals (Pt, Pd) the shift is
beyond doubt negative.
These problems are solved in the present approximation, where we use the expressions for susceptibility enhancement as given by density-functional theory. (We use these expressions to show a change in structure of the equations
only; not to perform actual calculations). This enhancement gives at the same time three effects: a Stoner-like
amplification [36] of the static susceptibility; a Moriya-type desenhancement factor [37] in the Korringa relation;
and the Yafet–Jaccarino kind of core-polarization hyperfine fields [38], that may be parallel or antiparallel to the
applied field.
The density-functional theory of the inhomogeneous electron gas [39–42] says that in the local-density approximation the paramagnetic susceptibility can be written in the form of an integral equation:
XZ
0
0
ρ, ρ + Rα) +
ρ , ρ 1 + R β )ν(n(ρ
ρ 1 + R β ))χ (ρ
ρ 1 + R β , ρ 0 + R α ) dρ
ρ 1 , (66)
ρ , ρ + R α ) = χP (ρ
χP (ρ
χ (ρ
cell
β
where ν(n(rr )) is related to a second derivative of the exchange-correlation energy, and is (in the local-density
ρ , ρ 0 + R α ) is the ‘noninapproximation) a function only of the charge density n in the point r . The quantity χP (ρ
0
teracting’ Pauli susceptibility, see Eq. (64). The vectors ρ and ρ 1 are in the unit cell at the origin, and R α and R β
ρ ) in the following, with the
are lattice vectors, as introduced in Eq. (30). For simplicity, ν(n(rr )) will be written ν(ρ
vector ρ in the unit cell at the origin (the charge density having, of course, the periodicity of the lattice).
The equivalent equation for the Bloch Fourier transformed susceptibilities is (compare Eq. (31))
Z
ρ , ρ 0; q α ) +
ρ , ρ 1 ; q α )ν(ρ1 )χ̃(ρ
ρ 1 , ρ 0 ; q α ) dρ
ρ 1.
ρ , ρ 0 ; qEα ) = χ̃P (ρ
χ̃P (ρ
(67)
χ̃ (ρ
cell
The expression for the real part of the susceptibility is directly obtained by adding a prime to all χ in Eq. (66) and
Eq. (67). But for later use we also write it in a slightly different form, using that the susceptibility should obey the
reciprocity relation χ (rr 1 , r 2 ) = χ(rr 2 , r 1 ):
XZ
0
0
0
0
ρ
ρ
R
ρ
R
ρ
ρ 0 + R α − R β , ρ 1 )ν(ρ
ρ 1 )χ 0 (ρ
ρ 1 + R β , ρ ) dρ
ρ 1.
χP0 (ρ
(68)
χ (ρ , + α ) = χP (ρ + α , ) +
cell
β
Once the χ 0 has been found, the χ 00 can be obtained to lowest order in ~ω from
XZ
00
0
00
0
ρ , ρ + R α ) = χP (ρ
ρ, ρ + Rα) + 2
ρ , ρ 1 + R β )ν(ρ
ρ 1 )χ 0 (ρ
ρ 1 + R β , ρ 0 + R α ) dρ
ρ1
χP00 (ρ
χ (ρ
XZ Z
+
cell
β,γ
β
cell
ρ , ρ 1 + R β )ν(ρ
ρ 1 )χP00 (ρ
ρ 1 + R β , ρ 2 + R γ )ν(ρ
ρ 2 )χ 0 (ρ
ρ 2 + R γ , ρ 0 + R α ) dρ
ρ 1 dρ
ρ 2,
χ 0 (ρ
and the Bloch Fourier transform of this equation is
Z
00
0
00
0
ρ , ρ ; q α ) = χ̃P (ρ
ρ, ρ ; q α) + 2
ρ , ρ 1 ; q α )ν(ρ
ρ 1 )χ̃ 0 (ρ
ρ 1 , ρ 0 ; q α ) dρ1
χ̃P00 (ρ
χ̃ (ρ
cell
Z Z
ρ , ρ 1 ; q α )ν(ρ
ρ 1 )χ̃P00 (ρ
ρ 1 , ρ 2 ; q α )ν(ρ
ρ 2 )χ̃ 0 (ρ
ρ 2 , ρ 0 ; q α ) dρ
ρ 1 dρ
ρ 2.
+
χ̃ 0 (ρ
(69)
(70)
cell
In numerical work, the equation can be applied to crystals with several different atoms in the unit cell (e.g.
YBa2 Cu3 O7 , see [43]), but here we will simplify to a system with one atom per unit cell in a cubic structure.
108
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
We will take the Wigner–Seitz cell as the unit cell, approximate it by a sphere, and assume that the spin density and
ρ by 4πρ 2 dρ. The wave functions ϕi (ρ
ρ ) in Eq.
the charge density have spherical symmetry, replacing ρ by ρ, and dρ
m
(64) are developed in spherical harmonics Yl (θ, φ):
ρ ) = (4π)1/2
ϕi (ρ
+l
∞ X
X
l=1 m=−l
Clm (i)ϕl (i , ρ)Ylm (θ, φ),
(71)
ρ )) which can be taken as real, and
where the radial function ϕl (i , ρ) (do not confuse with the total function ϕi (ρ
ρ ).
depends on the index i only through the energy i of the one-electron state described by the wave function ϕi (ρ
The spherical harmonics are orthonormal:
Z 2π Z π
0
Ylm∗ (θ, φ)Ylm0 (θ, φ) sin(θ) dθ dφ = δll 0 δmm0 ,
(72)
φ=0 θ=0
the radial wave functions are normalized as
Z WS
ρ 2 |ϕl (, ρ)|2 dρ = 1,
4π
(73)
0
where WS is the radius of the Wigner–Seitz sphere, and the coefficients are normalized as
+l
∞ X
X
|Clm (i)|2 = 1.
(74)
l=1 m=−l
In the expression for the complex Pauli susceptibility, Eq. (64), we replace the product of two wave functions taken
in the same point ρ by its spherical average:
Z 2π Z π
X
∗
−1
∗
ρ )ϕj (ρ
ρ ) sin(θ ) dθ dφ =
ϕi∗ (ρ
Clm
(i)Clm (j )ϕl (i , ρ)ϕl (j , ρ).
(75)
ϕi (ρ)ϕj (ρ) ≈ (4π)
φ=0 θ =0
lm
Likewise, we will assume that the enhanced susceptibility is given in terms of such spherical averages.
The lm-like partial density of electron states at energy (twice the number of energy levels per atom and per unit
energy interval) Dlm () is related to the set of complex expansion coefficients Clm (i) through Obata’s sum rule
[44]:
2N −1
∞
X
∗
δ( − i )Clm
(i)Cl 0 m0 (i) = δll 0 δmm0 Dlm (),
(76)
i=1
where N is the number of unit cells. Note that on both sides of Eq. (76) the dimension is 1/energy.
Since the spherical harmonics form a complete set, the expansions given in Eqs. (71),(73) and (74) are formally
exact; but of course they will be useful only if it turns out that a small number of l-values is sufficient for a correct
ρ ). These wave functions may be ‘hybridized’ in the sense that they do not need to be purely
representation of the ϕi (ρ
s-like or d-like, or other; only the average value |Clm |2 of the projection constants appears in the expression for the
partial densities of state. This formulation in real space does not require to group the one-electron eigenfunctions
in bands, as is done in the Bloch Fourier transform Eq. (65).
In order to proceed further, we will need, however, some other assumptions in addition to the spherical
averages. Let m(ρ) be a quantity proportional to the magnetization created in point ρ by a uniform field, and
defined as
N Z
X
ρ 0.
χ 0 (ρ, ρ 0 + r α ) dρ
(77)
m(ρ) =
α=1 cell
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Setting −f 0 () = δ( − Ef ) in the expression for χP0 , Eq. (64), and using Eq. (68) we obtain
m(ρ) =
µ0 µ2B
N Z
X
X
Dlm (Ef ) ϕl2 (Ef , ρ) +
α=1 cell
lm
ρ 0 )χ 0 (ρ
ρ0
ρ 0 |)ν(ρ
ϕl2 (Ef , |ρ
ρ |) dρ
ρ
+ R α , |ρ
0
109
!
.
(78)
The contact Knight shift for the nucleus in ρ = 0, therefore, has a direct contribution, proportional to the first term
within parentheses on the right (and nonzero only for l = 0), and a core polarization contribution given by the
second term.
In the following we attempt to decompose the static susceptibility, the Knight shift and the relaxation rate
approximately into sums of lm-like contributions [38,45]. It will be convenient to introduce a quantity
ml (ρ; q α ) =
ϕl2 (Ef , ρ) +
N
X
Z
exp(iqq α · R β )
β=1
cell
ρ 0 |)ν(ρ
ρ 0 )χ 0 (ρ
ρ 0 + R β , |ρ
ρ |) dρ
ρ 0.
ϕl2 (Ef , |ρ
The uniform susceptibility is
Z
Z
X
X
χ 0 = −1 4πρ 2 m(ρ) dρ = µ0 µ2B −1 Dlm (Ef ) 4πρ 2 ml (ρ; 0) dρ =
χ 0lm ,
lm
(79)
(80)
lm
which defines the partial susceptibilities χ 0lm . The enhancement with respect to the Pauli value χ 0P,lm is given by
Z
χ 0lm
1
=
4πρ 2 ml (ρ; 0) dρ =
,
(81)
χ 0P,lm
1 − αl
where the right most expression defines the l-like partial Stoner factor αl .
The Knight shift K(0) is
X
2
Bhf,l (0) X
χ 0lm
=
Klm (0),
K(0) = m(0) =
3
µ0 µB
lm
(82)
lm
where the last equality defines the partial contributions to the shift, and the effective l-like hyperfine field Bhf,l is
defined as
2ml (0; 0)
Bhf,l (0)
R
.
=
µ0 µB
3−1 4πρ 2 ml (ρ; 0) dρ
(83)
This hyperfine field can be nonzero also for l 6= 0, and the sign of ml (ρ; 0) in ρ = 0 (the numerator) may be
opposed to that of its average over the Wigner–Seitz sphere (the denominator), giving a negative hyperfine field.
To separate χ 00 (0, 0), that appears in the expression for the relaxation rate, Eq. (43), into lm-like contributions,
we make the approximation that in the enhancement integrals in Eq. (69) only those terms in the sums give an
important contribution where both arguments of χP00 (rr 1 , r 2 ) are in the same unit cell:
Z
ρ 1 )χ 0 (ρ
ρ 1 , 0) dρ
ρ1
χP00 (0, ρ 1 )ν(ρ
χ 00 (0, 0) ≈ χP00 (0, 0) + 2
cell
XZ Z
ρ 1 )χP00 (ρ
ρ 1 + R β , ρ 2 + R β )ν(ρ
ρ 2 )χ 0 (ρ
ρ 2 + R β , 0) dρ
ρ 1 dρ
ρ 2.
+
χ 0 (0, ρ 1 + R β )ν(ρ
(84)
β
cell
Introducing the spherical-average expression for χP00 (ρ1 , ρ2 )
ρ 1 + R β , ρ 2 + R β ) = χP00 (ρ
ρ 1 , ρ 2 ) = χP00 (ρ1 , ρ2 ) =
χP00 (ρ
X
1
2
π ~ωµ0 µ2B Dlm
(Ef )ϕl2 (Ef , ρ1 )ϕl2 (Ef , ρ2 ), (85)
2
lm
110
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
the relaxation rate can be written as
X
2
S(T1 T )−1 =
kl Klm
,
(86)
lm
where the partial contribution to the Knight shift Klm have been defined in Eq. (82), and where the ‘desenhancement
factor’ kl is given by
kl−1 =
|ml (0, 0)|2
.
P
2
N −1 N
α=1 |ml (0; q α )|
(87)
The kl are commonly called desenhancement factors, but in some cases they actually represent an increase in relaxation rate with respect to the Korringa-like value. It depends on whether the average value of |ml (0; q α )|2 is larger or
smaller than its value for q α = 0. When the most important ml (0; q α ) occurs at zero wave vector, the magnetization
is aligned over many unit cells, and therefore ferromagnetic: the factor kl < 1. In the opposite case, when the biggest
ml (0; q α ) is found well away from q α = 0, the magnetization changes sign over distances comparable to a unit cell
dimension and therefore it represents an antiferromagnetic magnetization: in that case kl > 1.
The desenhancement factor for the homogeneous electron gas has been discussed by Moriya [37]. Because of
ρ ; q ) is independent of ρ , and the hyperfine field is simply Bhf =
the homogeneity there are no indices lm, m(ρ
ρ ) is taken proportional to
2µ0 µB /3 (although different values are usually supplied in an ad-hoc manner). The ν(ρ
ρ ) and the q-dependence of m(ρ
ρ ; q ) becomes of the Lindhard form, i.e. has its maximum at q = 0, and therefore
δ(ρ
a desenhancement is found.
The right-hand sides of Eqs. (80),(82) and (86) are explicitly labeled with a given value of l (and m). Assume for
the sake of argument that in a given system, perhaps a transition metal alloy, we can experimentally vary D00 (Ef ),
without affecting any of the D2m (Ef ); would all quantities labeled with l = 2 remain completely unchanged? This
is only approximately true, as we will show on the example of the uniform susceptibility. From now on, we will no
longer show the argument Ef in the quantities Dlm (Ef ) and ϕl (Ef , ρ). Using Eq. (79) the enhancement factor in
Eq. (81) is written as
Z
Z
4πρ 2 ml (ρ; 0) dρ = 1 + 4πρ 02 ϕl2 (ρ 0 )ν(ρ 0 )m(ρ 0 ) dρ 0
Z
X
2
0
0
4πρ 2 ϕl2 (ρ)ν(ρ)ml 0 (ρ; 0) dρ.
= 1 + µ0 µB Dl m
(88)
l 0 m0
Now approximate the ml (ρ; 0) in the leftmost and right most integrals in Eq. (88) by a function of the form
(1 + Al )ϕl2 (ρ), where the constant Al remains to be determined; Eq. (88) transforms into
Z
Z
X
1 + Al = 4πρ 2 ml (ρ; 0) dρ = 1 + µ0 µ2B Dl 0 m0 (1 + Al 0 ) 4πρ 2 ϕl2 (ρ)ν(ρ)ϕl20 (ρ) dρ
=1+
X
l 0 m0
Dl 0 m0 (1 + Al 0 )νll 0 ,
(89)
l 0 m0
where the last line defines the exchange integrals νll 0 . Only if the diagonal elements of νll 0 are more important than
the off-diagonal ones, νll 0 ≈ νll δll 0 , can we write a Stoner-like expression where the partial Stoner factor αl (see
Eq. (81)) is determined only by l-like quantities:
X
X
Dlm
P
χ0 =
χ 0lm = µ0 µ2B −1
.
(90)
1 − νll m0 Dlm0
lm
lm
In applications of equations for χ 0 , K and T1 T with the structure derived here, it is usual to assume that the νll
and Bhf,l are properties of an atom in its Wigner–Seitz cell, and that, if we put this Wigner–Seitz cell in different
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
111
environments, these intra-atomic exchange integrals and effective hyperfine fields do not vary much: the changes in
environment will only change the densities of states Dlm . In systems that lack translation symmetry (surfaces, small
particles), the different environments are described by different local densities of states. The values of the exchange
constants and hyperfine fields are often obtained once and for all by fitting data obtained in the bulk elemental metal
of interest. (Calculated densities of states for the bulk are believed to be reliable, and are used as input parameters in
the fit). A good example of an important change in Wigner–Seitz cell with experimental conditions is given by the
high pressure experiments on alkalis to be discussed in Section 3.2: here the cell volume changes very considerably,
and the value of the hyperfine field does not remain constant.
2.5.2. Spin fluctuations
In the treatment of Section 2.5.1, the exchange interaction, represented by νll in Eq. (90), has no effect on the
energy of the system when no magnetic field is applied: the system is completely ‘nonmagnetic’. (Of course some
exchange-and-correlation potential has been used to determine the band structure, but that is completely independent
of any magnetic-susceptibility effect). But strictly speaking, once an exchange coupling has been introduced, there
is a magnetic energy in zero applied field associated with the thermal fluctuations of the local spin density (around a
zero average). Simply stated, a thermal fluctuation sets up an internal field, and this field couples to the other modes.
The free energy of the zero-field equilibrium state at a finite temperature now contains a contribution related to
the wave vector-dependent dynamic susceptibility of the system. Therefore, some kind of self-consistent procedure
must be followed to determine this susceptibility. For example, one could start from an enhancement found by a
local-density theory; write a spin–fluctuation term 1F in the free energy using this susceptibility; add a magnetic
field, and determine a new value of the susceptibility (the second derivative of the free energy with respect to the
field). Repeating the procedure should lead to a self-consistent value [46]. It is clear that this program is not easy to
execute, and no local-susceptibility variant exists: all available work is purely thermodynamic, yielding expressions
for the low-frequency wave vector-dependent susceptibility, but not for the corresponding hyperfine fields.
It is plausible that at least a part of the contribution of the spin fluctuations to the susceptibility will be temperature
dependent. In the discussion here we only consider metals at temperatures sufficiently above their magnetic transition
temperatures. Nevertheless, the wave vector-dependence of the strength of the spin fluctuations will reflect the
character of the ordered state: for simplicity we consider only ferro- and simple antiferromagnetic orderings. In a
metal, these orderings are distinguished by saying that the frequency of the fluctuations in a certain wave vector
range becomes very low when the transition temperature is approached from above, and finally these fluctuations
‘freeze out’ (note that this is a dynamic mode, that slows down; which is not the same as creating a one-electron
localized moment). The dynamic susceptibility at these wave vectors is temperature-dependent; but at wave vectors
far away the susceptibility may well have a nearly Pauli-like temperature-independence. In the case of (what at low
temperature will be) ferromagnets, the temperature dependence is near q = 0, whereas in antiferromagnets it is for
wave vectors about halfway into the Brillouin zone, since an antiferromagnetic unit cell has twice the dimensions
of its parent paramagnetic cell. It will be clear that temperature-dependent antiferromagnetic fluctuations can have
an influence on the temperature dependence of the T1 T -product, while at the same time the Knight shift could be
determined by an (almost) temperature-independent long-wavelength susceptibility.
We can give here only the barest of outlines of the application of spin fluctuation theory to nuclear spin relaxation
[47,48]. Consider that the magnetization in a sample fluctuates in time and varies in space: M (rr , t). (We will not
bother about unit cells and atoms). In the paramagnetic phase and in zero applied field:
Z ∞
Z
M (rr , t) dV = 0 =
M (rr , t) dt
(91)
sample
t=0
but the integrals of the squares of these quantities would be zero only if we had rigorously M (rr , t) = 0 at all points
and at all times. More precisely this idea is expressed by the space-time correlation function
M (rr , t)M
M (rr 0 , t 0 )i = FM |rr − r 0 |, |t − t 0 | ,
(92)
hM
112
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
where the h i indicate a thermodynamic averaging, and the absolute-bars in the right-hand side come from the
conditions of translational invariance and of stationarity. In a general case FM is a tensor, but we consider only
one diagonal component (and since our simple system is isotropic, all diagonal components behave the same
way). The double Fourier transform of FM (rr , t) is on one hand related to the q -dependent susceptibility (through
the fluctuation-dissipation theorem and the Kramers–Kronig relations), and on the other hand to our postulated
contribution 1F to the free energy. Let the latter be expressed by an exchange interaction I (rr − r 0 ) (the equivalent
of νll in Eq. (90)):
Z Z
Z Z
M (rr , t) · M (rr 0 , t) drr drr 0 =
I (rr − r 0 )M
I (qq )FM (qq , ω) dqq dω.
(93)
1F = V −1
sample
It should be pointed out that the mechanism postulated by Eq. (93) does not contribute to the temperature dependence
of the susceptibility of a purely local-moment system. A local moment jumps (more or less) randomly between
‘up’ and ‘down’, but its absolute value is temperature-independent (the jump rate need not be), so that Eq. (93)
simply shifts the free energy by a constant amount. Spin–fluctuation theory [46] says that the magnetic moment
per site in the ordered phase of a metallic system can be much smaller than one µB , and that in the paramagnetic
phase the mean-squared moment grows approximately linearly in temperature. However, looking in q -space, there
is a difference between the paramagnetic phases of ferro- and antiferromagnetic systems. In ferromagnetic systems,
the ‘ordering vector’ Q is near Q = 0, while for antiferromagnets it is some finite vector, usually denoted Q A . In
q -space the increase with temperature of FM (qq , ω) in the paramagnetic phase is localized near an ordering vector
of the low-temperature phase.
The self-consistency requirement is now that the dynamical susceptibility calculated from the total free energy
Ft = F0 + FM (qq ; ω) + 1F
(94)
(where F0 is the free energy without any magnetic field effects and FM (qq ; ω) is the magnetic free energy in the presence of an applied field of wave vector q and frequency ω) is the same as that calculated by the fluctuation–dissipation
theorem from FM (qq , ω). This of course is rather formidable, and cannot be done ab initio. The idea has been to
parametrize a simple model, based on the electron gas, and then fit experimental results to parameters of the model.
In the context of NMR, the theory has mainly been used to discuss the difference in temperature dependence of
the T1 T product for the paramagnetic phases of (incipient) ferromagnets and antiferromagnets. Because the range
of q -vectors where χ 00 (qq , ω) changes with temperature is different in the two cases, the integral over χ 00 (qq , ω)/ω,
as in Eq. (44), behaves differently as well. To lowest order in q and ω it has been found ([46], Eqs. 5.2 and 5.6) that
in the paramagnetic phases
Q + q , ω)
κ2
χ 00 (Q
∝
ω
(q/qB )(κ 2 + (q/qB )2 )2
Q + q , ω)
κ2
χ 00 (Q
∝ 2
ω
(κ + (q/qB )2 )2
(ferromagnets Q = 0)
(95)
(antiferromagnets Q = Q A ),
(96)
where qB is the radius of the effective, spherical Brillouin zone qB = (6π 2 /)1/3 with  the volume per magnetic
atom. The dimensionless parameter κ 2 is proportional to the inverse of the enhancement by the spin fluctuations of
the static susceptibility at the ordering wave vector:
κ2 ∝
Q, 0)
χ00 (Q
0
Q, 0)
χ (Q
(97)
and parametrizes the temperature dependence of the problem. It is found in spin fluctuation theory [46] that in the
paramagnetic phases of both ferromagnets and antiferromagnets, sufficiently far from the transition temperature
there is a region where the κ 2 increases approximately linearly with temperature. Note, for antiferromagnets,
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
113
the difference between the static uniform susceptibility (the response to a uniform applied field) and the static
susceptibility at the ordering wave vector (response to an applied field with periodicity Q A ).
To lowest order, the temperature variation of (T1 T )−1 is found by integrating q/qB in Eq. (95) or Eq. (96) over a
unit sphere, and retaining
only the lowest power of κ. The result is that in the paramagnetic phase of antiferromagnets
√
(T1 T )−1 ∝ κ ∝ 1/ T and for ferromagnets (T1 T )−1 ∝ κ 2 ∝ 1/T . In several weak (i.e. low transition temperature)
ferromagnets and antiferromagnets such behavior has been identified in a well chosen temperature region.
2.6. Kramers’ degeneracy
2.6.1. Time reversal symmetry
In the Hartree picture, each energy eigenvalue is associated with two eigenfunctions that have the same orbital
parts, but ‘opposite’ spin parts. In solids, the orbital parts can be represented by Bloch functions. This leads to a very
convenient formulation of the magnetic properties, since the Hamiltonian for the interaction between a magnetic
field and an electron
µB
B · (rr × p + 2ss )
(98)
H=
~
does not connect the two states, and the average value of r × p in each of the states is zero. Therefore, the g-factor
has exactly the free-electron value. By construction, the spin operator s has no matrix elements between states
p can have such elements (they appear e.g. in the expression
belonging to different pairs, but the orbital operator r ×p
for the orbital Knight shift, Eq. (25)).
This two-fold degeneracy is a special result of a theorem due to Kramers: if the Hamiltonian of any system with
an odd number of electrons contains only kinetic-energy and electrostatic terms, then at least two-fold degeneracy
will always exist. This degeneracy is lifted by additional Hamiltonian terms that contain a magnetic field. The
one-electron Hartree Hamiltonian is of course the simplest such case. The general situation is represented by an
‘effective spin 1/2’, in the sense that there is a doublet of states; however, its energy splitting in the presence of an
additional magnetic field does not need to be the same as that of the ‘real spin 1/2’ of a free electron. To emphasize
this, we write the spin in the Pauli matrix notation, where σ = 2ss /~ stands for the collection of the three matrices
0 1
0 −1
1 0
σy = i
σz =
.
(99)
σx =
1 0
1 0
0 −1
If the two degenerate states are represented by two orthonormal wave functions, these wave functions are related by
the ‘time-reversal’ operator K. (The operation actually reverses the direction of all motion, both orbital and spin,
rather than time). In the usual representation where r is diagonal and the σ defined as in Eq. (99), the time-reversal
operator is represented by
K = iσy K0 ,
(100)
where K0 is the ‘complex conjugation’ operator, that converts the spatial part of the wave function into its complex
σ = −σ
σ (iσy ). In the r -representation, p = (~/i)∇, and thus K0p = −p
p K0 .
conjugate. It is easy to verify that (iσy )σ
Therefore, the following anti-commutation rules hold:
σ = −σ
σK
Kσ
(101)
K(rr × p ) = −(rr × p )K.
(102)
Let ψ1 and ψ2 be any two wave functions for the odd-electron system, and let the corresponding ‘time-reversed’
states be Kψ1 and Kψ2 . To avoid ambiguities, we write the corresponding bras and kets as e.g. h(Kψ1 )| and |(Kψ1 )i.
It follows from the usual definition of the scalar product hψ1 |ψ2 i that
hψ1 |ψ2 i = h(Kψ2 )|(Kψ1 )i
(103)
114
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
and therefore also that
σ |ψ2 i = hψ1 |(σ
σ ψ2 )i = h(Kσ
σ ψ2 )|(Kψ1 )i = −h(σ
σ Kψ2 )|(Kψ1 )i = −h(Kψ1 )|σ
σ |(Kψ2 )i∗ .
hψ1 |σ
(104)
Let us now consider two members of a Kramers doublet at energy i , denoted φi,α and φi,β and chosen orthogonal
in the eigenfunction subspace belonging to i such that
φi,α = Kφi,β .
(105)
Choosing ψ1 = ψ2 = φi,α in Eq. (104) it is easy to see that
σ |φi,α i = −hφi,β |σ
σ |φi,β i.
hφi,α |σ
(106)
In a similar way it can be shown that
hφi,α |rr × p |φi,α i = −hφi,β |rr × p |φi,β i.
(107)
In the simplest case, the two functions φi,α and φi,β are pure ‘spin-up’ and ‘spin-down’ functions. The more general
degeneracy that can exist (the ‘mixing’ of the pure spin states) is due to the existence of a spin–orbit coupling
mechanism. A classical image of this coupling, which overestimates it by a factor of two, is as follows. An electron
running at speed v through an electric field E senses a magnetic field B 0 given (to first order in v /c) by
B0 = −
v
×E
c2
(108)
and the magnetic dipole associated with the spin senses this B 0 . The field E is due to the (periodic) electrostatic
potential of the lattice. It is strongest in the core regions of the atoms; and becomes stronger if the core contains
A, see Eq. (143)
more charge (i.e. when the atom is heavier). If an external magnetic field is applied, mvv = p + eA
in Section 2.7. The Hamiltonian becomes
H=
~µB
µB
1
p + eA
A) +
p + eA
A)2 − eV +
σ ).
σ · E × (p
(p
B · (rr × p + ~σ
2
2m
~
2mc
(109)
The term in the spin–orbit coupling containing the magnetic vector potential can be neglected. First, we consider the
σ ·E
E ×p
p as a perturbation on the states |ii of the Hartree Hamiltonian
zero-field case, and take Hso = (~µB /2mc2 )σ
p 2 − eV , resulting in states |i, soi given by
H0 = (1/2m)p
|i, soi = |ii +
X hn|Hso |ii
n
i − n
|ni,
(110)
where now |i, soi is one of the members of the Kramers doublet. Because of the properties discussed below Eq.
(98) we can write the g-factor of this doublet as
X hn|Hso |iihi|B
B|
B · r × p |ni + c.c.
gi |B
B · (rr × p + ~σ
σ )|i, soi = hi|~B
B · σ |ii +
,
= hi, so|B
2
i − n
n
(111)
σ |ni = 0. Here the
where the c.c. stands for complex conjugate, and we have used that hi|rr × p |ii = 0 and hi|σ
summation index n runs over both ‘spin-up’ and ‘spin-down’ functions for n 6= i.
In an electron spin resonance experiment, we induce transitions between the two Kramers conjugate states. This
can only be done if initially the two states are not both full or both empty, that is, if the states are very close to the
Fermi energy. According to the above derivation, different pairs of states at the Fermi energy may have different
g-factors gi . If the electron is scattered (e.g. by phonons) between different states at a rate fast compared to the
corresponding differences in Larmor frequencies, only an average g-value is observed (just as in the case of rapid
chemical exchange in NMR). The intensity of the ESR line is proportional to the number of participating states,
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
115
which is proportional to the density of states at the Fermi energy, and therefore, to the Pauli susceptibility χP . Since
the average g-value in metals is never very different from 2, it is in general not necessary to make a g-correction in
the expression for χP .
2.6.2. Shift, hyperfine field
We now want to see how this degeneracy affects the susceptibility and the Knight shift. For simplicity, we consider
the approximation where the Knight shift can be written as the product of a hyperfine field and a susceptibility. In
this section we look at the hyperfine field; the next one is devoted to susceptibility effects. To discuss the hyperfine
field it is useful to have an expression for the two members of a Kramers doublet that shows explicitly the spatial
and spin parts. It is easily verified that the following φi,α and φi,β are connected by K (note that K2 = −1):
1
0
∗
+ ϕb (rr )
(112)
φi,α = ϕa (rr )
0
1
1
0
+ ϕa∗ (rr )
,
(113)
φi,β = −ϕb (rr )
0
1
where the
1
0
are the eigenvectors of the σz -matrix. The simple Hartree picture is recovered when ϕb (rr ) = 0. We can choose these
B · (rr × p + ~σ
σ ):
wave functions so as to diagonalize the Zeeman Hamiltonian HZ = (µB /~)B
B · (rr × p + ~σ
σ )|φi,α i =
hφi,α |B
B|
g|B
2
B · (rr × p + ~σ
σ )|φi,β i = 0
hφi,α |B
(114)
(115)
and the same with a minus sign when the indices α and β are exchanged. The hyperfine field is proportional to the
matrix elements
B hf (rr ) ∝ hφi,α (r̂r )|σz δ(rr − r̂r )|φi,α (r̂r )i = |ϕa (rr )|2 − |ϕb (rr )|2 .
(116)
B hf (0)| for states at the
In a simplified description, the (direct contact) Knight shift is proportional to the average of |B
Fermi energy, and to χP . The orbital Knight shift will be given by the usual expression, Eq. (25), but using the wave
functions φi,α etc. (This change in orbital Knight shift when the spin–orbit coupling is ‘switched on’ is sometimes
called the spin–orbit Knight shift). As a result of all this, one can expect fairly modest, quantitative, changes in the
NMR quantities when spin–orbit coupling is taken into account.
However, there are two cases where the spin–orbit coupling is believed to modify qualitatively the behavior
expected for the susceptibility in its absence. This is the subject of the next Section 2.6.3.
2.6.3. Susceptibility
For BCS superconductors, one expects spin pairing at low temperatures, and therefore, also a vanishing contact
Knight shift [49], Eq. (35). An example of the same phenomenon in a cuprate superconductor [50] is given in Fig. 2.
Something similar is supposed to occur in the quantum-size regime (see Section 4.1) of small metal particles with an
even number of electrons. Experimentally, such vanishing shifts are not always observed. For experimental reasons,
NMR on superconducting metals must be performed on relatively fine powders. It is thought that in fine powders
and small particles the presence of the surface induces an additional spin–orbit coupling, resulting in one-electron
wave functions of the type of Eqs. (112) and (113). It is then argued that a second-order effect in the spin-only
116
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 2. The normalized Knight shift of 63 Cu in the chains of a cuprate superconductor, as a function of normalized temperature. The line is a fit
to the theory. (After Barrett et al. [50]. ©1990 American Physical Society).
susceptibility restores the susceptibility, and therefore, the Knight shift, to a finite value. In the following we develop
this idea schematically. For simplicity, we assume that in zero applied field only Kramers’ degeneracy is left.
Since the change in χP due to spin–orbit coupling in metals is expected to be small, we set the g = 2 and
HZ = µBB · σ . To second order, the states are modified by the field as
X hφj,α |σz |φi,α i
hφj,β |σz |φi,α i
B )i = |φi,α i + µB Bz
|φj,α i +
|φj,β i
(117)
|φi,α (B
i − j
i − j
j
and the magnetization becomes
X
hφi,α |σz |φi,α if (i + hφi,α |σz |φi,α iµB Bz ) + hφi,β |σz |φi,β if (i + hφi,β |σz |φi,β iµB Bz )
M = − µB
i
+ 2µ2B Bz
X hφi,α |σz |φj,α i2 + hφi,α |σz |φj,β i2 + hφi,β |σz |φj,α i2 + hφi,β |σz |φj,β i2 f (i ) .
j − i
i,j
(118)
Using a closure relation and σz2 = 1, we have the identity:
X
hφi,α |σz |φj,α ihφj,α |σz |φi,α i + hφi,α |σz |φj,β ihφj,β |σz |φi,α i .
1 = hφi,α |σz2 |φi,α i =
(119)
The diagonality hφi,α |σz |φi,β i = 0 makes Eq. (119) reduce to
2 X hφi,α |σz |φj,α i2 + hφi,α |σz |φj,β i2
1 − hφi,α |σz |φi,α i =
(120)
j
j 6=i
(and the same relation with α and β exchanged). Expanding the f (i + hφi,α |σz |φi,α iµB Bz ) up to terms linear in
Bz we obtain for Eq. (118):
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
X
X M
0
hφi,α |σz |φj,α i2 + hφi,α |σz |φj,β i2 + hφi,β |σz |φj,α i2
=
−
2
f
(
)
+
i
2
µB Bz
i
i,j, j 6=i
2 f (i ) − f (j )
+ hφi,β |σz |φj,β i ×
+ f 0 (i ) .
j − i
117
(121)
Here the first sum over i on the right-hand side gives the density of states at the Fermi energy, D(Ef ), compare Eq.
(10). To be able to proceed further with the second sum, we assume that the matrix elements are mainly a function
of the energy difference between states φi and φj , but do not depend (much) on these states themselves. In that case
we introduce a function g(|j − i |) to rewrite Eq. (121) as
X
f (i ) − f (j )
M
0
= D(Ef ) +
g(|j − i |)
+ f (i )
(122)
j − i
µ2B Bz
i,j, j 6=i
2.6.4. Metals, superconductors, small particles
Now we must make a distinction between the cases where there is a gap in the density of states at Ef , as happens
at low temperature in superconductors or in ‘even’ small particles, and the usual situation, where the density of
states is continuous at Ef .
We treat the latter case first, and write
∞
X
(j − i )n−1 (n)
f (i ) − f (j )
0
f (i ),
+ f (i ) = −
j − i
n!
(123)
n=2
where f (n) (i ) is the nth derivative of f (), taken in = i . In Eq. (122) we can change the summation over j in
a summation over energy intervals j − i = 1:
∞
X
X
1X
M
n−1
=
D(E
)
−
g(|1|)(1)
f (n) (i ).
f
n!
µ2B Bz
1
i
n=2
(124)
The nth derivatives are nonzero only in a small region around the Fermi energy, and we can replace the sum over i
by an integral:
Z ∞
X
(n)
f (i ) = D(Ef )
f (n) () d = D(Ef ) f (n−1) (∞) − f (n−1) (0) = 0,
(125)
0
i
which shows that there is no second-order spin susceptibility effect when the density of states is effectively constant
(and nonzero) around the Fermi energy.
The case of a BCS superconductor at zero temperature will only be discussed briefly and qualitatively. The
electrons pair off in the Kramers doublet states, giving a first-order spin susceptibility of zero. The density of
excited states is nonzero only for excitation energies greater than the gap energy, usually denoted as 21. The Fermi
level is in the center of the gap, and therefore, Eq. (122) becomes
X
f (i ) − f (j )
M
=
g(|
−
|)
.
(126)
j
i
j − i
µ2B Bz
i,j, j 6=i
If g(|j − i |) is nonzero for |j − i | ≥ 21 then there will be a second-order contribution to the spin susceptibility,
known as the Ferrell–Anderson effect [51,52]. This is believed to be the reason for the nonvanishing of the Knight
shift in superconducting heavy nontransition metals such as Sn [49].
118
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
In the quantum-size regime of ‘even’ small particles, the equal level spacing model (see Section 4.1) is easiest
to handle. We have the same expression as in the superconductor case, Eq. (126), and now we let 1 be the level
spacing. Then we number the levels such that i = i1 and j = j 1. Slightly rearranging the double sum we have
X
f (i1) − f (j 1)
M
.
(127)
=
2
g(|j
−
i|1)
(j − i)1
µ2B Bz
i,j, j 6=i
Replacing the index j by p = j − i
∞
∞
X
g(p1) X
M
=
2
(f (i1) − f ((i + p)1)) .
p1
µ2B Bz
p=1
i=0
(128)
At low temperatures, where f () is a step function, the sum over i yields p, and thus, reintroducing the full expression
for g(1) defined by Eq. (121) and Eq. (122):
X M
−1
hφi,α |σz |φj,α i2 + hφi,α |σz |φj,β i2 + hφi,β |σz |φj,α i2 + hφi,β |σz |φj,β i2
=
1
µ2B Bz
i,j, j 6=i
X
2
2 (2 − δi )
,
(129)
1 − hφi,α |σz |φi,α i + 1 − hφi,β |σz |φi,β i = 2δi
= 1−1
1
i
where we followed the literature [53] in defining hφi,α |σz |φi,α i = 1 − δi .
2.7. Appendix: second quantization
2.7.1. General
The quantum mechanical wave function that represents a collection of electrons is antisymmetric with respect to
the operation that exchanges the space and spin coordinates of any pair of electrons in the collection (electrons are
fermions). Let a set of orthonormal one-electron wave functions be given by φ1 · · · φn , and let the system contain
N electrons, with N n. Arrange a choice of N (out of n) functions φi in an N × N array where the elements in a
row have constant i, while the arguments of the wave function are the same in any given column, all referring, say,
to the kth electron. An N-electron wave function satisfying the antisymmetry requirement can be represented by
the determinant of this array (called a Slater determinant). If n tends to infinity (as will be implied in all equations
of this Section), we can construct an infinite number of such arrays, and any desired N-electron wave function can
be represented as a suitable linear combination of the determinants of such arrays. Of course it helps for simplicity
to make a suitable choice for the collection φi to start with; but here we will not actually make calculations, and we
need not go into this question.
In many practical problems, we want to determine matrix elements of a many-electron operator that has one
of two relatively simple structures: it can be written as a sum of terms that each contain only operators referring
to one single electron (one-electron operators; e.g. terms in the kinetic energy), or only operators referring to two
single electrons (two-electron operators; e.g. the interelectronic distance occurring in the Coulomb repulsion). It
can be shown that such operators have elements only between determinantal wave functions differing by at most
2 one-electron orbitals. For example, the first array may be constructed choosing all φi with 1 ≤ i ≤ N, and the
second with 3 ≤ i ≤ N + 2. Because of this property, it turns out to be convenient to imagine a ‘creation’ operator
†
bi , that ‘places’ an electron into φi , and its Hermitian conjugate bi , the ‘destruction’ operator, that ‘removes’ an
electron from that orbital. In the example above let the determinant of the first array be denoted 11···N , and that of
the second array by 13···N+2 , then
† †
11···N = b1 b2 bN+1 bN+2 δ3···N +2 .
(130)
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
119
As a further example, we evaluate the matrix element
† b† b
h11···N |bm
2 N+1 bN+2 |13···N+2 i = δm,1 ,
(131)
†
which is zero, unless the index m equals 1. The creation operators bj and annihilation operators bi obey the
anti-commutation rules
†
†
bj bi + bi bj = δij
(132)
† †
† †
b i bj + bj bi = 0
(133)
bi bj + bj bi = 0
(134)
In the following, we consider the interaction of the N -electron system with the magnetic field as a perturbation on
the ground state of the zero-field Hamiltonian H0 . Suppose that, through a suitable choice of one-electron wave
functions φi , the ground state of H0 can be written as a single determinantal wave function, represented by a ket |90 i,
with eigenvalue E0 for H0 . Let other possible arrays that can be formed from the collection of φi be represented by
kets |9e i, with eigenvalues Ee for H0 . Let F and G be two one-electron operators:
F =
n
X
k,l, k6=l
†
Fkl bk bl ,
G=
n
X
i,j, i6=j
†
Gij bi bj
(135)
and suppose that Fkl = Flk∗ and Gij = G∗j i . In perturbation theory we encounter expressions of the type
∞
X
h90 |G|9e ih9e |F |90 i + c.c.
e=1
Ee − E0
=
†
†
n X
∞
n X
X
Fkl Gij h90 |b bj |9e ih9e |b bl |90 i + c.c.
i
k,l=1 i,j =1 e=1
Ee − E0
k
,
(136)
where c.c. stands for complex conjugate. The product of the two matrix elements is nonzero only if k = j and l = i;
in that case Ee − E0 = j − i , where i is the one-electron energy corresponding to an electron ‘annihilated’ by
†
bi , and j corresponds to an electron ‘created’ by bj . Next we use the closure relation
∞
X
|9e ih9e | = 1 − |90 ih90 |
(137)
e=1
to obtain for the perturbation expression, Eq. (136)
n F G h9 |b† b b† b |9 i + c.c.
X
j i ij
0 i j j i
0
i,j =1
j − i
n F G h9 |(1 − b† b )b† b |9 i + c.c.
X
j i ij
0
0
j j i i
=
i,j =1
j − i
.
(138)
†
A matrix element like h90 |bi bi |90 i is interpreted as the probability that the one-electron orbital φi is occupied in
the N -electron state 90 , and this probability is given by the Fermi–Dirac function f (i ). Using, furthermore, the
Hermitian property of the matrices Fkl and Gij we have finally
∞
X
h9e |F |90 ih90 |G|9e i + c.c.
e=1
Ee − E0
=
n
X
I (i , j )Fj i Gij ,
i,j =1
with the definition that I (i , j ) = 0 if i = j and (f (i ) − f (j ))/(j − i ) otherwise.
(139)
120
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
2.7.2. Current density
The current density operator ĵ0 (rr ) for an N -electron system (the index 0 indicates that no magnetic field is
applied in this case) is given by
ĵ0 (rr ) =
N
X
−e
i=1
2m
(p̂ i δ(rr − r̂ i ) + δ(rr − r̂ i )p̂ i ).
(140)
To make a clear distinction between the continuous space variable r and the position operator of the ith electron r̂ i ,
theˆis used here to indicate operator quantities. We will not use this notation in general, however. The structure of
this equation says that electron i does not contribute to the current density in point r , unless its position r̂ coincides
p = −evv , where e is a positive quantity, and the current is
with that point, in which case its contribution is (−e/m)p
opposed to the electron’s motion (this relation between mvv and p is modified by the magnetic field, see below). The
operator ĵ0 (rr ) clearly is a one-electron operator in the sense given above. A general rule for the second-quantization
representation of one-electron operators leads to
ĵ0 (rr ) = −
n
e X
†
hφk |p̂ δ(rr − r̂ ) + δ(rr − r̂ )p̂ |φl ibk bl ,
2m
(141)
k,l=1
where the interesting feature is that the single sum over electrons in Eq. (140) has been replaced by a double sum
over orbitals in Eq. (141). Of course the creation and annihilation operators will ensure that the correct number
of electrons N is used. The one-electron matrix elements in Eq. (141) are easy to evaluate using the explicit
representation p = (~/i)∇:
~ ∗
φk ∇φl − φl ∇φk∗ ,
(142)
hφk |p̂ δ(rr − r̂ ) + δ(rr − r̂ )p̂ |φl i =
i
where the right-hand side is to be evaluated in the point r (but we will not use this expression here).
The Hamiltonian H of a single electron in a magnetic vector potential A and a scalar electric potential V is given
by
1
p + eA
A)2 − eV .
(p
(143)
H=
2m
A, thereby adding a term
From the change in the ‘kinetic energy’ term, we see that now mvv corresponds to p + eA
ĵA (rr ) to the current density operator, Eq. (141):
e2
A (rr )δ(rr − r̂ )
m
and in second quantization for the N-electron system:
ĵA (rr ) = −
(144)
n
ĵA (rr ) = −
X
e2
†
A (rr )
φk∗ (rr )φl (rr )bk bl .
m
(145)
k,l=1
In the Hartree approximation, the Hamiltonian of an N -electron system is a sum of terms like Eq. (143), the potential
V being due to the crystal lattice and the (N − 1) other electrons. Considering A as a perturbation, we write
H = H0 + HA ,
(146)
A2
where, neglecting the terms in A ,
Z
n
e X
†
hφk |p̂ · A (r̂r̂ ) + A (r̂r̂ ) · p̂ |φl ibk bl .
HA = − ĵ0 (rr ) · A (rr ) dV =
2m
k,l=1
Here the notation A (r̂r̂ ) indicates that A now takes the position operator of the electron as its argument.
(147)
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
121
The ket |9A i representing the N-electron system in presence of A can be written as
|9A i = |90 i −
∞
X
h9e |HA |90 i
e=1
Ee − E0
|9e i
(148)
and the current density in the N -electron system in the presence of A is
j (rr ) = h9A |ĵ0 + ĵA |9A i.
(149)
We require j (rr ) to first order in A :
j (rr ) = h90 |ĵA |90 i −
∞
X
h90 |ĵ0 |9e ih9e |HA |90 i + c.c.
.
Ee − E0
(150)
e=1
From the second quantization forms of ĵA (Eq. (145)), HA (Eq. (147)) and ĵ0 (Eq. (141)) and using Eq. (139) we
find
n X
e2
− A (rr )δij f (i )|φi (rr )|2
j (rr ) =
2m
i,j =1
e 2
A(r̂r̂ ) · p̂ + p̂ · A (r̂r̂ )|φi i .
+
I (i , j )hφi |p̂ δ(rr − r̂ ) + δ(rr − r̂ )p̂ |φj ihφj |A
(151)
2m
This completes the derivation of the current density j (rr ) induced by an external vector potential A (rr ). Note that
by definition the sums run over electron states, not over energy levels (the number of states is twice the number of
levels).
2.7.3. Spin magnetization
In this section, it will be useful to have separate spin- and orbital indices to the creation and annihilation operators,
†
e.g. bj ↑ . We will be interested in the representation of the components of the local spin magnetization operator
M̂ (rr ), where r is a continuous space variable (not an electron position operator r̂ ).
M (rr ) =
M̂
N
X
M i δ(rr − r̂r i ).
M̂
(152)
i=1
S i , because
M i = −2µBŜ
The one-electron magnetic moment operator is related to the corresponding spin operator by M̂
spin and magnetic moment have opposite directions. The Bohr magneton µB is the strength of an elementary dipole.
M (rr ) is given by
The second-quantization representation of the irreducible tensor components of M̂
M̂0 = µB
n X
†
†
bk↑ bl↑ − bk↓ bl↓ φk∗ (rr )φl (rr )
(153)
k,l=1
n
√ X
†
bk↑ bl↓ φk∗ (rr )φl (rr )
M̂+1 = −µB 2
(154)
n
√ X
†
bk↓ bl↑ φk∗ (rr )φl (rr ),
M̂−1 = µB 2
(155)
k,l=1
k,l=1
122
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
where the indices ↑ and ↓ refer to magnetic moments parallel and antiparallel, respectively, to the magnetic field.
The relation between Cartesian and irreducible tensor components is
√
√
M̂0 = M̂z ,
M̂+1 = − 21 2(M̂x + iM̂y ).
(156)
M̂−1 = 21 2(M̂x − iM̂y ),
The electron Zeeman Hamiltonian is
Z
n X
†
†
bk↑ bl↑ − bk↓ bl↓ hφk |B0 (r̂r )|φl i,
HZ = − M̂0 (rr )B0 (rr ) dV = −µB
(157)
k,l=1
where the notation B0 (r̂r ) indicates that the argument is the position operator of the electron. In a uniform field this
reduces to
HZ,0 = −µB B0
n X
†
†
bk↑ bk↑ − bk↓ bk↓ .
(158)
k=1
The magnetization Mz (rr ) is given to first order in the field by an expression rather similar to Eq. (150):
Mz (rr ) = h9B |M̂0 (rr )|9B i −
∞
X
h9e |HZ |90 ih90 |M̂0 (rr )|9e i + c.c.
Ee − E0
e=1
,
(159)
where 9B is the ground state in presence of the field, while 90 and 9e are evaluated without the field. In the first
term we must, therefore, distinguish the one-electron energies i↑ and i↓ :
†
†
h9B |bi↑ bj ↑ − bi↓ bj ↓ |9B i = δij f i − µB hφi |B0 (r̂r )|φi i − f i + µB hφi |B0 (r̂r )|φi i
≈ −2µB δij hφi |B0 (r̂r )|φi if 0 (i ),
(160)
where f 0 (i ) is the derivative of the Fermi–Dirac function f (), taken in = i . In the sum over excited states
we take the zero-field values for Ee − E0 , determined by one-electron energies like i and j , and use the general
expression, Eq. (139), so that finally
Mz (rr ) =
2µ0 µ2B
n
X
0
−δij f (i ) + I (i , j )
i,j =1
Z
φi∗ (rr )φj (rr )
sample
H 0 (rr 0 )φi (rr 0 ) drr 0 .
φj∗ (rr 0 )H
(161)
This expression is used in the evaluation of the Knight shift and of the uniform susceptibility in the Pauli approximation, Section 2.4.
2.7.4. Power absorption
To study the average power absorbed by the spin magnetization from a nonuniform alternating field B0 (rr ) cos ωt
we consider the perturbation Hamiltonian
Z
1
H1 (t) = − M̂0 (rr )B0 (rr ) (exp(iωt) + exp(−iωt)) dV
2
n
µB X †
†
bk↑ bl↑ − bk↓ bl↓ hφk |B0 (r̂r )|φl i (exp(iωt) + exp(−iωt))
=−
2
k,l=1
= H1,0 (exp(iωt) + exp(−iωt)) ,
(162)
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
123
where the last line defines the operator H1,0 . Using Fermi’s golden rule, we write the transition probability per unit
time W0e to go, under influence of this perturbation, from the N -electron ground state represented by the ket |90 i
to an excited state represented by |9e i as
W0e (ω) =
2
2π h90 |H1,0 |9e i δ(Ee − E0 + ~ω).
~
(163)
To make the connection between this quantum mechanical result and thermodynamics is not straightforward. Clearly,
if we are sure that the system is initially in the state |90 i then its probability per unit time to go to the state |9e i
by absorption is given by W0e in Eq. (163). But if on the contrary we were sure that initially the system is in
|9e i, then the same expression gives the rate We0 at which it goes to the state |90 i by stimulated emission. So the
thermodynamical average should take into account how likely it is to be initially in each of the states. Since H1,0
is a one-electron operator, this accounting can be done by considering separately processes where a one-electron
state at lower energy is ‘replaced’ by one at higher energy (absorption), and the opposite (stimulated emission). The
total rate of all absorption processes is
W0→ (ω) =
n
X
X
2
π
W0e (ω) = µ2B
f (i ) 1 − f (j ) hφi |B0 (r̂r )|φj i δ(j − i − ~ω)
~
e
(164)
i,j =1
with j > i : in the initial state |φi i is occupied, and |φj i is empty. The total rate of all emission processes is
obtained when initially the higher-energy state is occupied, and the lower-energy one is empty:
W0← (ω) =
n
X
X
2
π
We0 (ω) = µ2B
f (j ) (1 − f (i )) hφj |B0 (r̂r )|φi i δ(j − i − ~ω)
~
e
(165)
i,j =1
again with j > i . The net absorbed power is thus
P (ω) = ~ω (W0→ (ω) − W0← (ω))
n
X
2
f (i ) − f (j ) hφi |B0 (r̂r )|φj i δ(j − i − ~ω),
= (πω)µ2B
~ω > 0.
(166)
i,j =1
For our application we will be interested in the low-frequency limit of P (ω)/~ω2 . We use that
f (i ) − f (j ) δ(j − i − ~ω) = −~ωf 0 (i )δ(j − i − ~ω),
~ω → 0
(167)
and in Section 2.4 we need the limit
n
X
2
2P (ω)
2
=
−2πµ
f 0 (i )δ(j − i − ~ω) hφi |B0 (r̂r )|φj i
B
2
~ω→0 ~ω
lim
i,j =1
=
−2πµ0 µ2B
n
X
0
f (i )δ(j − i − ~ω)
i,j =1
Z Z
sample
B (rr )φj (rr )φj∗ (rr 0 )H
H (rr 0 )φi (rr 0 ) drr drr 0 .
φi∗ (rr )B
(168)
According to this equation, no power is absorbed from a uniform field B 0 (rr ) = B 0 . Clearly, it does not describe
the spin–lattice relaxation process when we switch on a magnetic field; such processes are in fact related to the
evolution in time of W0← (ω) and W0→ (ω).
124
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
3. NMR in metals
3.1. Zero of the shift scale
3.1.1. The reference state
The very first review of NMR in metals [15] already pointed out the importance of the choice of reference
compound to determine the metallic shift. In Section 2.2 we have found the following expression for the orbital
Knight shift (see Eq. (25)):
2
∞
∞
x + y2 lz µ0 e2 X
µ0 e2 X I (i , j )
f (i ) φi (169)
φi 3 φj hφj |lz |φi i.
φi +
Korb = −
4πm
4π m
2m
2|r|3 |r|
i=1
i,j =1
Here the applied field has been taken in the z-direction, and |r|2 = x 2 + y 2 + z2 ; furthermore, lz = (rr × p )z . The
first term is diamagnetic (negative), the second one is paramagnetic, and often called the van Vleck shift, because
it corresponds to a contribution to the magnetic susceptibility of ions that has first been discussed by van Vleck.
(This susceptibility is often called temperature independent paramagnetism, TIP, by magnetochemists).
It is useful to divide the diamagnetic term into two parts. The first diamagnetic contribution is due to the low-lying
(i Ef ) core electronic levels. It is generally assumed, and confirmed by recent ab-initio calculations of chemical
shifts [54], that this term is almost independent of atomic environment, and can be treated as a constant for a given
type of atom. The second diamagnetic contribution is that of valence electrons (i ≈ Ef ). It is usually neglected
compared to the paramagnetic shift which is caused by these same electrons. (Because of the factor I (i , j ) the
core electrons have no effect on paramagnetism). The van Vleck susceptibility χvV is given by
∞
χvV =
2
µ0 e2 −1 X I (i , j ) 
hφi |lz |φj i
4πm
2m
(170)
i,j =1
and the second term in Eq. (169) is often rewritten in terms of this susceptibility and an orbital hyperfine field
defined such that
KvV = χvV
Bhf,orb
.
µ0 µB
It is then usual to define an average value hr −3 i by
P∞
−3
i,j =1 I (i , j )hφi |lz |r| |φj ihφj |lz |φi i
−3
hr i =
2
P∞
i,j =1 I (i , j ) hφi |lz |φj i
(171)
(172)
so that the orbital hyperfine field can be expressed as
Bhf,orb = µ0 µB hr −3 i.
(173)
The ideal ‘reference state’ therefore, is the nucleus with core electrons, rather than the ‘bare’ nucleus. (The latter
was preferred by the nuclear physicists of the early days, who were interested in the gyromagnetic ratio as a nuclear
property, and thought of shifts as a useless nuisance). For the alkalis, silver and some other metals it is possible to
find aqueous solutions of salts with noncoordinating anions, and extrapolate the observed resonance frequency to
infinite dilution to obtain a suitable reference.
For transition metals this usually does not work since even the coordination with water can already be fairly strong.
Since the paramagnetic shift is related to the van Vleck paramagnetism, one might think of correlating the measured
magnetic susceptibility and NMR shift in a series of ionic compounds. If we had some means of independently
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
125
estimating the diamagnetic part of the susceptibility, then we could extrapolate the correlation plot to zero van
Vleck paramagnetism, and find the resonance frequency for the ion with its core electrons only. Unfortunately, the
estimate of the diamagnetic part usually cannot be made with sufficient precision.
A very elegant procedure has been proposed for ions that have paramagnetic groundstates when present as
impurities in simple host crystals, such as Co2+ in MgO ([55], p. 244). The idea is to observe the same ion in a
number of different hosts using a technique called electronic-nuclear double resonance (ENDOR). ENDOR detects
ESR in the presence of an additional continuous frequency-swept decoupling in the NMR frequency range, and
contains information on the electronic as well as the nuclear transitions. The different ligand fields in the different
host crystals will shift both the NMR and the ESR signals from their ‘free ion’ values. One could make a correlation
plot of the observed nuclear and electronic frequencies, and extrapolate to the known ESR frequency at zero van
Vleck susceptibility (corresponding to g ≈ 2): this gives the (unknown) NMR frequency without paramagnetic
contribution. The experiment has never been performed in this complete form, but a simplified version, without
extrapolation, has been used for the 3d5 ions Mn2+ and Fe3+ (it should also be applicable to Cr+ ). In these cases it
is expected on theoretical grounds, and experimentally observed, that the ESR shift (with respect to g = 2) should
be small; therefore, the observed NMR frequency should also be close to the value at zero van Vleck susceptibility.
The case of 55 Mn is discussed at some length in Section 3.7.1, and the result for 57 Fe will be briefly mentioned in
Section 3.1.2.
For ions with nonmagnetic groundstates, a correlation between NMR and optical data can be used. The application
of this method for the determination of the reference gyromagnetic ratio of several transition metal nuclei is the
subject of Section 3.1.2.
3.1.2. Optical methods
Among the first clearly detected chemical shift differences were those for 59 Co in different ‘nonmagnetic’
compounds [56]. For simplicity, we will restrict our discussion mainly to octahedral complexes of d6 ions, like
Co3+ . In an octahedral ligand field the five d-orbitals that are degenerate in energy in the atom are split in one group
of three, and another group of two. For a d6 ion in the ground state, the group of three is fully occupied and the group
of two is empty. The energy splitting often is fairly low, corresponding to optical frequencies in the visible region of
the spectrum. Therefore, the energy denominators in I (i , j ) = (f (i ) − f (j ))/(j − i ) are relatively small, and
the shift according to Eq. (171) is important. It was observed that the 59 Co resonance in a series of such complexes is
shifted to lower field (paramagnetic shift) when the optical transition occurs at longer wavelengths (lower energy):
this correlation is called spectrochemical ordering [57,58]. In a number of Co complexes, the relation between shift
and wavelength is linear, so that extrapolation of the experimental straight line to zero wavelength (infinite energy
difference, therefore I (i , j ) → 0) should yield the zero of the NMR shift scale. This procedure supposes that
the variation in excited-state energy in Eq. (170) is the determining variable, and that the geometric factor hr −3 i
in Eq. (172) is less important. However, this is not the general case, as we will see from data collected in Tables 1
and 2.
In Table 1 the NMR shifts are given for 195 Pt, 103 Rh, 59 Co, 99 Ru and 57 Fe in a series of d6 octahedral or
quasi-octahedral complexes. The data are taken from [59], and given on the shift scales used in that book. So far as
the resonances have been observed, they all go from low field to high field in the order of ligands: F, H2 O, OH, Cl,
Br, ethane-1,2-diamine (abbreviated: en), CN and I, except for the inversion of Br and en for 195 Pt. The data for the
sandwich compounds with cyclopentadienyl (Cp)2 show that the ligand-ordering idea can (in an imprecise way) be
extended to other coordinations. The meaning of the values in the last line will be discussed below. We remark in
passing that on the shift scales used in Table 1, the resonance of Rh metal is at −1370 ppm, and that of Ru metal at
approximately +1500 ppm, refuting the idea that Knight shifts must be ‘unusually’ large. As we will see, the shifts
in the transition metals result from positive and negative contributions, that occasionally may cancel out.
The optical data in Table 2 show that spectrochemical ordering is indeed often observed, although there is a
number of exceptions, the most notable being the position of Cl and Br for 195 Pt and 103 Rh. Closer inspection of
the numerical values also reveals that the linearity between shift and wavelength is usually much worse than for the
126
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Table 1
Metal NMR shifts (ppm) in (quasi)octahedral d6 complexes. Values in parentheses are calculated, but the resonances have not been observed.
The last line gives values of the shift at zero van Vleck contribution (see text)
195 Pt
F6−
(H2 O)6
(OH− )6
Cl−
6
Br −
6
(NH3 )6
(en)3
(CN− )6
I6 −
(Cp)2
zero
103 Rh
59 Co
(11240)
9880
11847
7823
4522
2619
8001
7007
4766
(4070)
3572
667
−1545
−1800
−7000
−2200
99 Ru
57 Fe
15110
16050
8173
7144
0
7820
6600
0
2497
−2400
−8600
−1300
−7300
1532
−2060
Table 2
Wavelength (nm) of the lowest optical ligand-field transition [60]. Where known, the second-lowest transition is given within parentheses. In
spectrochemical ordering the wavelength must decrease monotonically along a column
Pt
Rh
Co
Ru
F6−
(H2 O)6
(OH− )6
Cl−
6
Br −
6
(NH3 )6
317 (275)
469 (360)
398 (310)
606 (402)
529 (386)
(en)3
(CN− )6
215
518 (412)
552 (450)
305 (253)
301 (253)
225
472 (338)
464 (338)
312 (260)
390 (310)
370 (302)
323
379 (353)
444
Fe
323 (270)
Co3+ complexes selected in [57,58]. A better optical parameter in this respect is the ‘internal ligand field strength
parameter’ 6, that can be calculated from the two lowest optical ligand-field transitions by means of intermediate
ligand field theory. We will not attempt to provide a theoretical justification for the use of this parameter, but simply
note its empirical utility [61,62]. An infinite value of 6 corresponds to excited states very far away from the ground
state, i.e. no van Vleck paramagnetism, and a good linear correlation is found between the chemical shifts and the
values of 6 −1 . A weakness of the method is that the slope of the correlation plot does not have a simple interpretation
so that its value cannot be compared between metals.
Values of 6 −1 calculated from the optical data in Table 2 by the method of [61] are given in Table 3. Correlation
plots between data in Table 3 and in Table 1 for 195 Pt, 103 Rh, 59 Co and 99 Ru are shown in Fig. 3. The shift values
Table 3
Values of 1000/6, where 6 is the (dimensionless) internal ligand-field strength parameter, calculated from data in Table 2
F6 −
(H2 O)6
(OH− )6
Br −
6
Br−
6
(NH3 )6
(en)3
(CN− )6
Pt
Rh
Co
Ru
9.754
20.05
18.69
36.58
25.14
27.19
25.32
12.88
16.86
14.58
4.663
16.80
14.68
13.25
12.19
Fe
12.63
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
127
Fig. 3. Correlation plots between the observed metal resonance shift and the inverse of the internal ligand field strength parameter, 1/6, for Pt,
Rh, Co and Ru.
for 6 −1 extrapolated to zero are given in the last line of Table 1 (except the value for 57 Fe, taken from the ENDOR
experiment described in Section 3.1.1). For 103 Rh and 59 Co larger datasets than we use here have been given in
[61,62]; their extrapolations differ from ours by something like 1000 ppm. Our extrapolations for 195 Pt and 99 Ru
should of course be treated with great care, based as they are on a minimal number of experimental points. They
are, however, consistent with the rest of the data: the 195 Pt zero shift is to high field of the (PtI6 )2− resonance, and
the extrapolations for 59 Co and 99 Ru, that have similar shift ranges, are similar. Furthermore the extrapolated value
128
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Table 4
Gyromagnetic ratios γ /2π in MHz/T at the zero of the shift scales chosen by different authors
Reference
195 Pt
103 Rh
59 Co
99 Ru
57 Fe
Table 1
[62]
[61]
[18]
[59]
9.0913
1.3360
1.3387
10.015
1.9464
1.3757
9.094
9.1135
1.340
1.3454
10.026
10.03
10.102
1.9
1.9607
1.3757
1.3785
for 99 Ru found here is nearly the same as that found in [63] by a slightly different parametrization of the same
data.
Table 4 compares a number of proposed reference gyromagnetic ratios, γ /2π, for 195 Pt, 103 Rh, 59 Co, 99 Ru and
57 Fe. (Remember that values quoted here are for the diamagnetic ion, not for the bare nucleus). The conversions
between different authors are based on a field of 2.348720 T to obtain the 1 H resonance in a spherical sample of
water at exactly 100 MHz [64]. The first line (except 57 Fe) is based on extrapolations in Table 1; the values quoted
from [62] for 103 Rh and from [61] for 59 Co are based on more data and should be more precise. With the possible
exception of 99 Ru [65], the reference values proposed here are essentially the same as those in the most widely
used compilation of Knight shift data [18]. Use of shift scales from the chemical literature is not recommended for
Knight shift evaluations.
3.2. Alkali and noble metals
The properties of the alkali metals are often discussed with reference to the free electron gas with the same
electron density. If the volume per electron (and therefore, also per atom) is  and the electron density parameter
rs is defined through  = (4/3)πrs3 then the density of states at the Fermi energy (number of states per atom and
per Joule) in the free electron gas (feg) is
2
rs
,
(174)
Dfeg (Ef ) = 1.869 × 1017
a0
where a0 is the Bohr radius. The susceptibility is
−1
−6 rs
χfeg = 32.65 × 10
a0
(175)
and has no dimension, although it is often referred to as the ‘volume’ susceptibility. The corresponding cgs value
is obtained by dividing the right-hand side by 4π. The cgs susceptibility in emu mol−1 results from a further
multiplication by the molar volume Vm = NA , and therefore, has the dimension of a volume, and units cm3 .
Strictly speaking, there is of course no nuclear magnetic resonance, and therefore, no Knight shift, in a free
electron model. It is nevertheless usual to introduce a Knight shift Kfeg as the product of a susceptibility as obtained
above and the intensity at the nucleus of a Fermi-energy single-electron wave function, |ψ(0)|2 , supposed to be
normalized in the volume 
χ 8π
χ 4π B
feg
feg
hf
= µB Dfeg (Ef )Bhf ,
(176)
|ψ(0)|2 =
Kfeg =
4π
3
4π µ0 µB
where the second line introduces the equivalent hyperfine field Bhf , with units Tesla (T). The corresponding cgs
equations are obtained by replacing the factors χfeg /4π by χ̃cgs /NA , where χ̃cgs is the susceptibility in emu mol−1 ,
and further replacing µ0 by 4π . The resulting hyperfine field is in Gauss (G), and 1 T = 104 G.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
129
In the case of Na, the ‘best’ experimental value for the spin susceptibility at low temperature (4.2 K) is thought
to be (13.63 ± 0.14) × 10−6 [66], and the value found for the hyperfine field by Overhauser shift measurements is
Bhf = 24.76 ± 1.2 T [67]. The molar volume of sodium at low temperatures is Vm = 22.72 cm3 . These ESR-based
values of susceptibility and hyperfine field predict a value for the Knight shift K = (1.09 ± 0.05) × 10−3 , in good
agreement with the accepted value K = 1.07 × 10−3 at 10 K. The value of Bhf corresponds to |ψ(0)|2 = 120−1 ,
which shows clearly how the wave functions that are important for the contact shift pile up near the nucleus.
For rs /a0 = 3.93, appropriate for Na, Eq. (174) gives a density of states Dfeg = 2.89 × 1018 J−1 , or 6.3 states per
atom and per Rydberg (1Ry = 2.1795 × 10−18 J) in good agreement with the value 6.12 Ry−1 from band structure
calculations [68]. Therefore, χfeg = 8.31 × 10−6 is close to the value from the band structure calculation in the
Pauli approximation, but both are clearly different from the experimental χobserved = 13.63 × 10−6 . The Stoner
enhancement factor α (see Eqs. (81) and (90)) defined by
χobserved =
χP
1−α
(177)
is α = 0.407 (the value calculated in [68] is 0.415). Associated with this Stoner enhancement is a disagreement
between the T1 T value calculated from the Korringa relation, 3.29 s K, and the experimental value 5.1 ± 0.3 s K. In
the Shaw–Warren model [69], see also Section 2.5.1, the expected desenhancement factor k(α) for the relaxation
rate is approximately given by
α 2
S
= k(α)K 2 ≈ (1 − α) 1 +
K
(178)
T1 T
4
with the value of the constant S given by S = c(γp /γ )2 , where γ is the gyromagnetic ratio of the nucleus under
consideration, c = 2.63334 × 10−7 s K and γp = 100/2.34870 MHz T−1 (see also Eq. (190) below). This gives
k(0.407) ≈ 0.653, close to the value of 0.64 observed in Na, and shows that susceptibility enhancement is important
even in such a ‘simple’ metal as sodium.
The free electron gas model has a very smooth density of states around the Fermi energy, and the only temperature
effect is thermal expansion, which changes rs /a0 . The same parameter describes pressure effects. The volume
dependence of the susceptibility, Eq. (175), is given by
d ln χfeg
1
=− .
d ln 
3
(179)
Under variation of p or T we can expect from Eq. (176) two extremes of behavior for the Knight shift, written as
d ln χ
d ln |ψ(0)|2
d ln K
=
+1+
.
d ln 
d ln 
d ln 
(180)
If the changes in volume scale at the level of the wave functions, such that |ψ(0)|2 remains constant, then
d ln |ψ(0)|2 /d ln  = −1. In the other extreme, the changes in volume affect the wave function far away from the
nucleus, but |ψ(0)|2 stays constant: d ln |ψ(0)|2 /d ln  = 0. In the free electron gas approximation we have thus
for the volume dependence of the Knight shift
−1/3 ≤
d ln Kfeg
2
≤ .
d ln 
3
(181)
For Na, the change of susceptibility with volume at constant temperature (4.2 K) has been measured by the
CESR method [70] for volume reductions to (p)/(0) = 0.9, and the quantity in Eq. (179) has been found
as −0.34 ± 0.03. (It should be noted, however, that Na is rather the exception: the values for the other alkali metals
are in the range −1 to −2). The 23 Na pressure-dependent NMR data [70–72] in the same range of relative volume
give a slope of +0.13 ± 0.02 for Eq. (180), see Fig. 4, so that d ln |ψ(0)|2 /d ln  ≈ −0.54. At larger volume
130
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 4. Volume dependence of the spin susceptibility in Na metal, measured by the CESR method and of the 23 Na Knight shift. All quantities
relative to room temperature and atmospheric pressure. (After Kushida et al. [73] and Benedek et al. [70]. ©1976, 1958 American Physical
Society).
reduction, see Fig. 5, this slope in the shift data changes sign and becomes approximately −0.1. There are no corresponding data for the susceptibility, but if we continue to use the low-pressure, nearly free-electron-gas value of
−0.34, then d ln |ψ(0)|2 /d ln  ≈ −0.76, closer to the value −1 expected for |ψ(0)|2 constant. If this is correct,
then the NMR data indicate that in the low-pressure range the value of the wave function far away from the nucleus
changes more than the value close to it; but at higher pressure the ‘compression’ becomes more uniform.
A very nice illustration of the equivalence of temperature and pressure variation is given in Fig. 6. Plots like Fig.
5 taken at two different temperatures [72] do not superpose; but if they are reduced to the same lattice parameter (or
to the same rs /a0 ), they superpose to within experimental precision. Note that in Fig. 6 the relative precision within
a temperature run is much better than the precision to which the calibrations of the two runs can be compared.
For Li, the best value for the spin susceptibility is (27.4 ± 0.1) × 10−6 [66], and the hyperfine field from the
Overhauser shift is 5.39 ± 0.2 T [74]. The molar volume is 12.8 cm3 . The Knight shift calculated from these data
is (2.69 ± 0.1) × 10−4 , while the experimental value is 2.6 × 10−4 . These satisfactory agreements hide in fact a
conceptual problem concerning the meaning of the hyperfine fields. Several bandstructure calculations that project
out the partial densities of states of s-, p-and d-symmetries, see e.g. [75], agree that for Li about 3/4 of the density of
states at the Fermi energy corresponds to p-states, that have zero amplitude at the nucleus; for Na the value is about
2/5. A CESR experiment, and therefore, the Overhauser experiment also, sees all electrons at the Fermi energy, as
shown by the good agreement between measured and calculated susceptibilities: the measured hyperfine field is an
average value over all these electrons. This implies that all electron states are hybridized in about the same ratio;
or that all electrons jump rapidly between all states; or both. This raises a delicate question on the principle of a
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
131
Fig. 5. Volume dependence of the relative 23 Na Knight shift in Na metal measured at 295 K in a diamond anvil cell, achieving very large volume
reductions. Two sets of data, taken in slightly different cells. (After Kluthe et al. [72]. ©1996 American Physical Society).
Fig. 6. 23 Na Knight shift in Na metal at two temperatures, as a function of the lattice constant. The vertical shift between the two runs is of the
order of the precision of the absolute shift calibrations, see the error bars, and the difference is probably not significant. (After Kluthe et al. [72].
©1996 American Physical Society).
Knight shift calculation. Should we consider only the s-like susceptibility, its enhancement, and the (high) purely
s-like hyperfine field, or rather the average susceptibility, its enhancement and the (low) average hyperfine field?
The calculated [68] intensity of the Fermi-level wave functions, averaged over all Fermi-level states corresponds
to hyperfine fields of 31 T for Na, and 6.9 T for Li. Both are about 30% higher than the experimental value, but
nevertheless in astonishing agreement. Using also the calculated [68] average susceptibility enhancement we find
shift values 1.38 × 10−3 for Na, and 4.33 × 10−4 for Li, which is within a factor of two of the experimental values.
This reasonable agreement for two metals with rather different band structures makes us prefer, in general, the
‘average sp’-approach over the ‘pure s-type’ models. Of course the pure s-like hyperfine fields calculated in such
models must be higher than the average-sp values: by about a factor of 5/3 for Na, or 4/1 for Li. A disadvantage of
the ‘average sp’-approach is that these hyperfine fields cannot really be considered as ‘atomic’ properties, and can,
strictly speaking, not be compared easily between unit cells in different environments.
For the other alkalis, there are no CESR determinations of susceptibility, and no Overhauser experiments. The
shift and relaxation data for the alkali and the noble metals in Table 5 can be used to find a value for α from Eq.
(178). From this value of α, the calculated value of D(Ef ) and the experimental value of K, the spin susceptibility
χ and the hyperfine field Bhf are calculated as shown in Table 6. The values for Li and Na agree reasonably with
those discussed in the previous paragraphs. For K, Rb and Cs the susceptibility enhancements (1 − α)−1 have
been deduced from dHvA experiments [76]: the values are αK = 0.41, αRb = 0.42 and αCs = 0.43. Estimates for
132
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Table 5
The molar volume Vm [77], the calculated density of states D(Ef ) [68] (except Cs and Au from [75]), the Knight shift K and the product (T1 T )−1
[18] for the alkali and noble metals
Vm (cm3 )
7 Li
12.8
22.7
43.0
52.5
66.5
7.11
10.27
10.21
23 Na
39 K
87 Rb
133 Cs
63 Cu
109 Ag
197 Au
D(Ef ) (Ry−1 )
K
(T1 T )−1 (s−1 K−1 )
6.53
6.13
9.93
12.24
20.43
3.94
3.67
3.99
2.6 × 10−4
1.07 × 10−3
2.5 × 10−3
6.46 × 10−3
1.57 × 10−2
2.38 × 10−3
5.22 × 10−3
1.65 × 10−2
0.023
0.196
0.04
9.09
9.09
0.787
0.111
0.217
Table 6
Values of the Stoner parameter α, the susceptibility χ̃cgs in molar cgs units, the ‘volume’ susceptibility χ in SI units, and the hyperfine field Bhf ,
as deduced from data in Table 5
Li
Na
K
Rb
Cs
Cu
Ag
Au
α
χ̃cgs (µemu mol−1 )
χ (×10−6 )
Bhf (T)
0.476
0.416
0.276
0.526
0.498
0.541
0.562
0.339
29.6
25.0
32.6
61.4
96.7
20.4
19.9
14.3
29.1
13.8
9.52
14.7
18.3
36.1
24.3
17.7
4.9
23.9
42.8
58.8
90.7
65.1
146
642
the spin susceptibility obtained from the measured total susceptibilities and a (large) calculated correction for core
diamagnetism have been given in [18] as χK = 14.6 × 10−6 , χRb = 15.1 × 10−6 and χCs = 19.3 × 10−6 . The
NMR results for αK and χK compare rather poorly with these estimates, but the agreement for Rb and Cs is more
reasonable. The calculations of hyperfine fields in [68] give 45.9, 83.0, 82.9 and 173 T for K, Rb, Cu and Ag (there
are no results for Cs and Au).
The case of Cs is particularly interesting, because of its high compressibility. Applying suitable corrections to
the measured pressure dependence of the total susceptibility yields an estimate for the slope d ln χspin /d ln  of
−2.2 ± 0.4, clearly much larger than the free electron value in Eq. (179). The experiment was performed at 150 K,
and the smallest (p)/(0) was 0.94. The pressure dependence of both K and T1 T have been measured at 4 K
[78]. In the range 0.94 ≤ (p)/(0) ≤ 1 the ratio of the relaxation rate to the square of the Knight shift changes by
only 5%: the change in susceptibility enhancement is only small. Therefore, the volume change of the susceptibility
must be due to a change in the density of states very different from that predicted by the free electron gas model,
Eq. (174). The measured slope d ln K/d ln  is close to −1, so that from d ln χspin /d ln  ≈ −2.2 and Eq. (180) we
find that the hyperfine field changes only slightly with changing volume. This conclusion fits well with an intuitive
idea that the ‘outer reaches’ of the large atoms in the easily compressible Cs can be squeezed while close to the
nucleus the situation does not change much. However, room temperature data for the 133 Cs shift [72] suggest that
for much larger volume reductions, Fig. 7, the hyperfine field scales with volume, i.e. |ψ(0)|2 is constant (there
are no independent susceptibility data for this range of (p)/(0), and we have assumed that there is no change
in the slope d ln χspin /d ln ).
As an example of volume changes due to thermal expansion we discuss the case of silver [79]. Between room temperature and the liquid phase at 1360 K the molar volume increases by nearly 13%. Over this range d ln K/d ln  =
+0.873 ± 0.004; and additional data down to 25 K also fall very well on this plot [80]. This value is outside
the possible range in the free electron gas, Eq. (181). From this value of d ln K/d ln  and Eq. (180) we expect
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
133
Fig. 7. The relative shift of 133 Cs in Cs metal as a function of the relative volume at 295 K. Two sets of data using slightly different diamond
anvil cells. After Kluthe et al. [72]. ©1996 American Physical Society).
−0.127 ≤ d ln χspin /d ln  ≤ +0.873. From an analysis of K 2 T1 T in the solid range it is concluded that the Stoner
susceptibility enhancement is of the order 1.9, and increases by about 5% between room temperature and the melting
point; while measurable, this is a negligible effect. Therefore, the variation is susceptibility must be due to changes
in the density of states that are very different from what is seen in the alkalis, where the slope of d ln χspin /d ln 
is never more positive than the free electron value −0.33.
The Knight shift of 109 Ag in metallic silver plotted as function of molar volume is continuous across the melting
transition, Fig. 8, but the product (T1 T )−1 shows a discontinuous increase in relaxation rate in the liquid, Fig. 9.
While several mechanisms might be suggested, e.g. the effect of atomic motion, it is interesting to speculate that it
might be due to the greater atomic disorder in the liquid. In some simple models [81] such disorder enhances the
dynamic susceptibility (related to the relaxation rate), while it does not affect the static susceptibility (connected to
the shift) by much.
3.3. Oscillatory Knight shifts
As explained in Section 2.1.4, the orbital motion of a conduction electron in directions perpendicular to an
applied magnetic field is quantized in cyclotron orbits. When the applied field is swept slowly, the uppermost
Fig. 8. Relative shift of 109 Ag in Ag metal as a function of volume. Here the experimental parameter is temperature (rather than pressure). The
plot is continuous across the melting transition. (After El-Hanany et al. [79]. ©1974 American Physical Society).
134
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 9. The Korringa ratio (S/T1 T )/K 2 for 109 Ag in Ag metal as a function of volume. The relaxation rate shows a discontinuous increase in
the liquid phase. (After El-Hanany et al. [79]. ©1974 American Physical Society).
occupied cyclotron level rises above the Fermi energy, and deposits its electrons into the next-lower level, that
now becomes the new uppermost occupied level. This emptying and rising of the highest cyclotron level creates a
periodic modulation on the susceptibility, called the de Haas–van Alphen (dHvA) effect. (Actually the periodicity
appears when data are plotted as a function of the inverse of the applied field). The effect is only visible at low
temperatures and in high quality materials, as otherwise the lifetime of an electron in a cyclotron level is too short.
The period (measured in units of T−1 ) of the susceptibility oscillations is related to the geometry of the Fermi surface
in a way that is rather well understood, but has little relevance to our subject. (The Fermi surface is the surface
mapped out in reciprocal space by the wave vectors of all one-electron wave functions at the Fermi energy). The
available interpretation of the amplitude of the oscillations is however only schematic, and that of the periodicity in
the Knight shift even more so.
Periodic variations of the Knight shift with increasing applied field have been observed in tin, cadmium and
aluminum. The samples are made from single crystals, sliced into twenty or so wafers of ≈ 0.3 mm thick, that
are isolated from each other by thin Mylar sheets, and reassembled into a block of roughly a cubic centimeter in
size. The sample temperature is typically 1.2 K, and the applied field ranges from 1 to 2 T. Both susceptibility and
Knight shift oscillations are measured on the same sample. A result for the shift of 113 Cd is shown in Fig. 10. The
amplitude of the oscillating part of the shift is of the order of 30 ppm; the relative amplitude is about one percent.
The Landau susceptibility can have only a very small effect on Knight shifts, because it only can appear in the
macroscopic demagnetizing field (see Section 2.2): therefore, the oscillatory part of the Knight shift indicates the
presence of rather strong oscillations (of the order of one percent) in the spin susceptibility. No serious attempts
have been made to evaluate these numbers. It has been pointed out that not all Fermi-level electrons participate in
the oscillating contribution to the Knight shift, and that those participating may have larger hyperfine fields than the
Fermi-surface average: an amplification by a factor 1.67 has been proposed (for this particular orientation of the
magnetic field with respect to the crystal axes) [82].
3.4. Transition metals
The NMR properties of the transition metals are rather different from those of the simple alkalis in Section 3.2.
Probably the most obvious example is Pt, where the Knight shift is negative for any reasonable choice of the zero
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
135
Fig. 10. Oscillations of the 113 Cd shift in a sliced single crystal of Cd metal as a function of the inverse of the applied field, for a well-chosen
orientation of the field with respect to the crystalline axis and in two field ranges. (After Goodrich et al. [82]. © 1971 American Physical Society).
of the shift scale [83]. This cannot be explained by the equation for the free electron gas, Eq. (176), that consists of
a product of positive factors. However, the expression found for the effective l-like hyperfine fields when exchange
enhancement effects are taken into account, Eq. (83), can, in principle, have either sign. We will assume that the
density of states at the Fermi energy in Pt can be written as a sum of an sp-like and a d-like partial density of
states. This is not quite the same as requiring that individual wave functions have either sp- or d-character: we only
need to have negligible exchange cross-couplings between the sp- and d-like densities of state. In that case, the
susceptibility can be written unambiguously as the sum of an sp-like and a d-like part as in Eq. (90). An effective
sp-like hyperfine field will be associated with the sp-like part in much the same way as we have proposed for the
alkalis: we expect the associated Knight shift to be positive. The effective hyperfine field associated with the d-like
part can be negative, as explained in Eq. (83), and this has been found to be the case for Pt both from the analysis
of experimental data and in calculations [84].
The bulk susceptibility as well as the absolute value of the Knight shift in Pt increase markedly with decreasing
temperature [85]. The plot of shift versus susceptibility, with temperature as an implicit parameter, is a straight line,
the slope of which gives a negative hyperfine field, Fig. 11. The change of susceptibility with temperature is not
thought to be a simple volume effect, as in the thermal expansion of the free electron gas, but rather believed to be
due to a decrease in the effective d-like density of states when the temperature increases. For E near Ef , the density
of states Dd (E) may vary rather strongly with E, contrary to Dsp (E), that should be smooth as for the free electron
Fig. 11. The shift of 195 Pt in Pt metal as a function of susceptibility. The parameter is the temperature, that goes from 295 to 1358 K. The slope
of this plot gives the hyperfine field Bhf,d . (After Shaham et al. [85]. ©1978 American Physical Society).
136
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
density of states. Since the Fermi–Dirac function picks out states within an interval kT around Ef , the effective
d-like density of states that appears in the susceptibility becomes temperature dependent.
The orbital Knight shift in transition metals, given by Eq. (169), can also be important. The susceptibility χvV
contains the function I (i , j ) ∝ f (i ) − f (j ), which is different from zero only when the states i and j are not
both full or both empty. It will be small at the beginning of a transition metal row (where the d-band is nearly empty
and most f () ≈ 0) and also at the end (where the d-band is nearly filled and most f () ≈ 1). This simple idea is
confirmed by the result of relativistic linear response calculations [24] for the 4d transition metals in Fig. 12. The
variation in hr −3 i within the row is found to be monotonous, so that we expect relatively small orbital Knight shifts
at the beginning and the end of a transition metal row, and relatively large ones in between. A full calculation of
Fig. 12. The orbital susceptibility χ̃vV,cgs ; the value of hr −3 i occurring in the approximation for Bhf,orb ; the approximate value of KvV from Eq.
(171); and the exact KvV from Eq. (169) for the 4d transition metals, calculated by a relativistic linear response technique. (After Ebert et al.
[24]).
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
137
KvV confirms this idea, although the exact values in Fig. 12 are up to 20% higher than those obtained using Eq.
(171).
While the shift may contain positive as well as negative contributions, the relaxation rate is always a sum of
positive terms. In the theoretical section we have shown terms proportional to the squares of the sp-like and d-like
Knight shifts introduced above in Eq. (86). There are two other contributions, that we have not discussed [44]. In
a cubic metal, the shift due to spin–dipolar coupling (Eq. (39)) is zero because of symmetry, but the fluctuating
part contributes to the relaxation. Furthermore, the fluctuating part of the van Vleck interaction (Eq. (171)) likewise
gives relaxation. These relaxation rates have only been discussed in the tight-binding approximation, where both
are found to be proportional to (Dd (Ef )Bhf,orb )2 . Strictly speaking, the hyperfine field Bhf,orb that appears in the
relaxation equations is different from that appearing in Eq. (173), but it is usual to neglect this difference. The
dipolar relaxation rate is nearly always much smaller than the orbital rate, and therefore neglected.
A refinement can be made to the relaxation equations given in the theoretical section. In Eq. (86) we have written
sums over m, where m runs from −l to +l. In cubic metals the five values of m corresponding to l = 2 come
together in a triplet t2g and a doublet eg . So instead of
Dd (Ef ) =
2
X
D2m (Ef )
(182)
m=−2
we should write
Dd (Ef ) = Dt2g (Ef ) + Deg (Ef )
(183)
and similarly for the sum of squares
2
X
2
D2m
(Ef )
=3
m=−2
Dt2g (Ef )
3
2
+2
Deg (Ef )
2
2
= Rd Dd2 (Ef ),
(184)
where the last equality defines the ‘orbital reduction factor’ Rd . There is a similar, but slightly differently defined,
reduction factor Rorb that appears in the expression for the orbital relaxation rate. We will not go into the details of
its derivation [44], but mention for reference
Rd =
2Dt22g + 3De2g
Rorb =
(185)
6Dd2
2Dt2g (Dt2g + 6Deg )
9Dd2
.
(186)
In the above paragraphs we have already introduced several approximations in the description of the shift and
relaxation rates in transition metals, the most severe being the introduction of the three densities of states Dsp (Ef ),
Dt2g (Ef ) and Deg (Ef ). The advantage is that these values can be supplied by band structure calculations, and that
the d-like hyperfine field can, sometimes, be found from experiment, as in Fig. 11. We have no reliable means to
calculate the effective Stoner factors αl that appear in Eq. (81) and the desenhancement factors kl in the expression for
the relaxation rate, Eq. (87), are also unknown. We will suppose that kl can be calculated from some l-independent
function of the Stoner parameter k(α), thus kl = k(αl ). A few models exist to derive the relation k(α), all of
them for simple metals [37,69,86,87]. For want of something better they have sometimes been applied to transition
metals as well [85,88,89,90]. Here we will adopt the Shaw–Warren result [69], which can be represented as a simple
polynomial in α, k(α) ≈ (1 − α)(1 + α/4). There is little fundamental justification for doing so, but it leads to a
satisfactory description of e.g. the data for Pt and for Pd. Finally, we will analyze the data in terms of the equations
138
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Table 7
Fitted values of partial contributions to the susceptibility, the Knight shift and the relaxation rate of 195 Pt in platinum metal. The next four rows
give the parameters used: hyperfine fields Bhf , reduction factors R, Stoner parameters α and densities of states D(Ef ). Vm = 9.10 cm3
χ(×10−6 )
K(×10−3 )
(T1 T )−1 (s−1 K−1 )
Bhf (T)
R
α
D(Ef ) (Ry−1 )
χ0
=
µ0 µ2B −1
s
d
vV
Total
Exp.
22.2
7.8
7.2
270
289
−44.2
19.9
−118
0.21
0.77
20.4
13.8
2.0
6.35
110
0.87
242
−34.4
33.4
242
−34.4
33.4
0.4
4.08
Dsp (Ef ) Dd (Ef )
+
1 − αsp
1 − αd
+ χvV = χ 0 sp + χ 0 d + χvV
K = χ 0 sp Bhf,sp + χ 0 d Bhf,d + χvV Bhf,orb

= Ksp + Kd + KvV
µ0 µB
2
S(T1 T )−1 = k(αsp )Ksp
+ k(αd )Kd2 Rd + (µB Dd Bhf,orb )2 Rorb
α
.
k(α) = (1 − α) 1 +
4
(187)
(188)
(189)
(190)
In the expression for the susceptibility a diamagnetic contribution χdia has been left out. As explained in the
theoretical section, the diamagnetism of the conduction electrons only shows up as part of the demagnetizing field
associated with χ 0 , leading to shifts of the order of a few tens of ppm (the numerical value of the diamagnetic
susceptibility). For a derivation of the last term in the expression for the relaxation rate, Eq. (189), see [44].
The experimental, calculated and fitted data for Pt are shown in Table 7. The values for Bhf,s = 270 T, αs =
0.40 and αd = 0.77 have simply been fitted. The orbital susceptibility and hyperfine field can be estimated from
other data. The estimated diamagnetic contribution is not included in the value quoted for the total susceptibility.
The d-like hyperfine field Bhf,d = −118 T is given by the shift versus susceptibility plot. The densities of states
Ds (Ef ) = 4.08 Ry−1 and Dd (Ef ) = 20.4 Ry−1 [91] as well as the reduction factors [75] Rorb = 0.87 and Rd = 0.21
are from bandstructure calculations. The analysis of the 195 Pt shifts is based on the reference value 195 γ /2π =
9.094 MHz T−1 given in the Knight shift compilation of Carter, Bennett and Kahan [18]. This was thought to give
the resonance condition for H2 PtI6 , but more recent values [59] put this resonance at 9.0973 MHz T−1 .
There is a large uncertainty in the reference frequency for 105 Pd. Its NMR has been reported in only one nonmetallic
system, two aqueous solutions of hexachloropalladate [92], with an average ratio of frequency to field of ν/B =
1.9525 MHz T−1 . As we have seen in Section 3.1.2, the zero of the Knight shift scale must be at smaller ν/B than
any resonance in such octahedral complexes. At 4 K, the average of the values of ν/B for the metal given in [93,94]
is 1.8700 MHz T−1 . It follows that the Knight shift K is more positive than −4.2 × 10−2 . The hexachloro complexes
of Rh and Pt, neighbors of Pd in the periodic table, are at +1.5% and +0.67%, respectively, with respect to the (not
very precise) zeroes in Fig. 3, so the Knight shift of Pd is probably between −2.7 × 10−2 and −3.5 × 10−2 . From the
shift-susceptibility correlation plot the low-temperature value of χd = 1.06 × 10−3 , and of Kd = −0.0450 [94]; an
earlier report [93] has χd = 1.01 × 10−3 , and Kd = −0.0438. The experimental value of (T1 T )−1 = 1.35 s−1 K−1 .
Using Eq. (189), the smallest value of Ks that is compatible with these experimental results is Ks = 7.4 × 10−3 ,
with αs = 0. The corresponding hyperfine field is Bhf,s = 520 T, which is probably too large compared to the value
given for Pt in Table 7 (compare the trends in Table 6). If we impose Bhf,s = 200 T, and furthermore assume a 10%
error margin in the relaxation rate, we can obtain the partitioning of the susceptibility, Knight shift and relaxation
rate given in Table 8. The densities of states are from [91] and the reduction factors from [75]. Note the rather large
enhancement of the s-like susceptibility found this way. The value K = −0.0344 gives γ = 1.9366 MHz T−1 as the
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
139
Table 8
Partial contributions to the susceptibility, the Knight shift and the relaxation rate of 105 Pd in palladium metal. The next four rows give the
parameters used: hyperfine fields Bhf , reduction factors R, Stoner parameters α and densities of states D(Ef ). Vm = 8.82 cm3
χ(×10−6 )
K(×10−3 )
(T1 T )−1 (s−1 K−1 )
Bhf (T)
R
α
D(Ef ) (Ry−1 )
s
d
vV
Total
Exp.
45
6.2
0.30
200
1033
−44.1
0.46
−34
0.24
0.900
30.6
28
3.6
0.46
100
0.345
1106
−34.4
1.22
1054
0.75
3.34
1.35
reference gyromagnetic ratio for 105 Pd, so that the chemical shift of the (PdCl6 )2− solutions is 8.2 × 10−3 . It should
be remarked that the spin–lattice relaxation in Pd is perhaps not well described by Eq. (189), based on a local spin
density approximation. Palladium is often considered a ‘nearly ferromagnetic’ metal, for which a description based
on spin fluctuation theory would be more appropriate. The validity of the decomposition in Table 8 is therefore
uncertain.
There is an interesting series of NMR experiments on 195 Pt in platinum alloyed with very small quantities of other
transition metals [95]. In the spectra ‘satellite’ lines appear (see Fig. 13 ), due to 195 Pt in sites close to an alloying
impurity. The difference in shift between a satellite line and the bulk NMR line was taken as a direct measure for
a difference in local susceptibility (the hyperfine fields were taken as site-independent). This spatial distribution of
the susceptibility on Pt sites around the solute is related to the change in the bulk susceptibility of similar (more
concentrated) alloys. To fit the data, an exponential decay of the change in susceptibility with distance from the
impurity was assumed. The characteristic length of the decay is between 0.5 and 0.6 lattice constants (0.392 nm),
depending on the solute. The spatial decay of the local susceptibility corresponds to a decay in the local density of
states at the Fermi energy: the experimental data show that a measurable change in this quantity can extend as far
as the third neighbor Pt shell.
Fig. 13. Sketches of the main (near K = −0.035) and satellite 195 Pt lines in Pt metal slightly alloyed with other transition metals. The assignments
to first, second and third neighbors (numbered arrows) are based on simultaneous analysis of all data. (After Inoue et al. [95]. ©1978 Physical
Society of Japan).
140
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
The considerable NMR literature on concentrated alloys is as a rule not included in this article. An exception is
made here for the system Pt1−x Pdx , because it has also been studied in small-particle form. In the bulk, platinum and
palladium form random alloys at all compositions, and their magnetic susceptibility χ(x), and 195 Pt NMR parameters
195 K(x), 195 T (x) and 195 J (x) have been studied [96–98] (J is the scalar, or Ruderman–Kittel–Bloembergen–Rowland,
1
coupling constant; see Eq. (47)). There are no 105 Pd NMR data. In these random alloys, the platinum sites can have
several different coordinations, and therefore the NMR line broadens with respect to that in the pure metal. To
simplify the discussion we will neglect this broadening, although it is actually quite important in Pt1−x Pdx , and
assume that ‘typical’ values of the x-dependent density of states on platinum sites can be derived from NMR parameters measured at the center of gravity or at the maximum of the 195 Pt NMR line. As a first step, we separate the
experimental shift K into an s-like and a d-like part, neglecting the orbital contribution. To make this separation,
we use the experimental values of the scalar 195 Pt–195 Pt coupling constant Jexp . It is believed that this spin–spin
coupling is (mainly) determined by the s-like density of states, and therefore, should be proportional to Ks :
Kd (x) = Kexp (x) − Ks (x)
(191)
Jexp (x)
Ks (x)
=
,
Ks (0)
J (0)
(192)
where Ks (0) = 7.8 × 10−3 and J (0) = 4.2 kHz are the values in pure platinum. From Eqs. (191) and (192) and
the experimental data, the variations with x of the s-like, d-like and total densities of states on a ‘typical’ platinum
site in Pt1−x Pdx are derived as shown in Fig. 14. An interesting conclusion from this analysis is shown in Fig. 15.
Fig. 14. Smoothed curves of the variation of the partial and total densities of states at the Fermi level D(Ef ) on ‘average’ Pt sites in Pt1−x Pdx
alloys, and comparisons of the experimental 195 Pt NMR parameters with the values (drawn lines) obtained from these curves. (After Tong et al.
[99]. ©1996 American Physical Society).
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
141
Fig. 15. Variation of local susceptibility on ‘average’ Pt and Pd sites in Pt1−x Pdx alloys as deduced from experimental data on alloy susceptibilities
and the density of states curves in Fig 14. (After Tong et al. [99]. ©1996 American Physical Society).
Here the experimental susceptibility χexp (x) is written as the sum of a contribution χPt (x) of typical Pt sites and a
contribution χPd (x) of typical Pd sites:
χexp (x) = (1 − x)χPt (x) + xχPd (x).
(193)
The values of χPt (x) are calculated from Fig. 14, and next the χPd (x) are determined from Eq. (193). As Fig. 15
shows, both on the palladium and on the platinum sites the local susceptibility diminishes rapidly when x decreases
from 1 to 0.8, and more slowly thereafter.
A simplified form of the χ, K, T1 T analysis has been applied to the NMR of Rh metal [100]. Only the density
of 4d-like states at the Fermi energy has been considered, neglecting possible s-like contributions. (There are nine
valence electrons per atom, so that, in principle, we may have a metal without any s-like electron. In Pt or Pd with 10
valence electrons this is impossible). The experimental Knight shift is written as the sum of a rather large positive
orbital shift and a somewhat smaller (in absolute value) negative spin shift. There is a measurable temperature
dependence of the bulk susceptibility and the shift, and the correlation plot yields a core polarization field of −16 T
[101]. This is rather small compared to the value in Pd (−34 T), a neighbor of Rh in the periodic system, and it
has been argued [102] that the actual ranges of the variations in the plot are too small for a valid conclusion. The
authors of [102] propose to take the same value in Rh as found for Pd. We will see their arguments in a more
general discussion of regularities in the hyperfine fields through the transition metal range of the periodic table in
Section 3.7.1. It emerges from that discussion that, although 16 T is almost certainly too small, perhaps 34 T is too
big.
In the case of V metal, measurements of the 51 V NMR shift and of the total susceptibility have been performed
between 20 and 290 K on a high-purity oriented single crystal specimen [103]. This sample form, an unlikely choice
according to simple NMR sensitivity arguments, actually offers many advantages in precision, and furthermore
provides the opportunity to study effects related to sample shape and orientation. Even though the quadrupole
moment of 51 V is rather small, the crystalline perfection and relative purity of a single crystal as compared to a
powder will give a sharper line. More important perhaps is the lower risk of extrinsic causes in the study of rather
small temperature dependent effects. Aligning the magnetic field along a cubic axis of the crystal will diminish the
dipolar line width, and the position of the metal NMR line can be located to within a few ppm.
142
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 16. Frequency shift of the 51 V resonance in a single crystal disk of vanadium as a function of temperature, for two orientations of the
applied field: (a) parallel and (b) perpendicular to the faces of the disk. The resonance frequency is 38.4 MHz. (After Hechtfischer, [103]. ©1976
Springer–Verlag).
Before discussing the components of the shift and the hyperfine fields, we briefly describe a measurement of
total susceptibility, and even of its temperature variation, uniquely from NMR data. Certainly straightforward
susceptometry is easier to perform and more accurate, but it is interesting that this experiment is based on the
demagnetizing fields discussed in Section 2.1.2. The sample was a circular disk of 12 mm diameter and 0.5 mm
thickness. For simplicity, let us assume that because of the skin effect only nuclei near to the two faces of the disk
contribute to the observed NMR signal. It is a general consequence of Maxwell’s equations in the static limit that
at the boundary between two media the perpendicular component of B and the tangential component of H are
continuous (the latter only if no current flows in the surface). If the applied field H 0 is parallel to the faces of the
B 0 . If the field B 0 is applied perpendicular to the
disk, we have H in the sample = H 0 , and B in the sample = (1 + χ)B
H 0 . Therefore, the relative difference of
faces of the disk, we have B in the sample = B 0 and H in the sample = (1 − χ)H
the NMR frequencies for these two orientations of the applied field, see Fig. 16, gives the value of χ . Note that
this is a macroscopic consideration, very different from those in the calculations of contributions to the shift. In the
actual experiment [103] the sample shape was taken into account in a less approximate way then we have done here,
using shape-dependent so-called demagnetization factors. The correct determination of these factors is, however, a
very delicate matter.
The measured temperature dependences (at constant pressure) of the Knight shift K(T ) and of the total susceptibility χ (T ) were corrected for thermal expansion effects to find values at constant volume, Fig. 17. Assuming, as
usual, that this temperature dependence is due to the d-electrons, the core polarization hyperfine field was determined
as −9.0 T. For a discussion of this value, see Section 3.7.1.
It is believed that a nearly perfect compensation of positive s-like and negative d-like spin shifts occurs, so that
the observed Knight shift is equal to Korb . This makes it impossible to test by NMR the spin pairing that according
to BCS theory must occur [49] below the superconducting transition at 5.3 K. To within a few percent relative shift,
no change in 51 V shift is found down to 1 K. Since V is a light metal, the spin–orbit effects of the Ferrell–Anderson
mechanism [51,52], see Eq. (126), should not be operative, and the above compensation is the remaining explanation.
3.5. Structure in the density of states
Metals are good conductors because the wave functions that belong to the continuous DOS around Ef are
delocalized. This delocalized character of the electronic wave functions is a simple consequence of the periodic
repetition of the unit cell in the solid. Quasicrystals are solids that possess an orientational order representing
symmetry axes that are forbidden in classical crystallography, and an associated quasi-periodic long-range order.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
143
Fig. 17. Relative changes in the susceptibility and in the 51 V NMR shift for a V single crystal disk with respect to their values at 293 K, as a
function of temperature. The experimental values have been corrected to obtain values at constant volume. The same sample was used for both
measurements. (After Hechtfischer, [103]. ©1976 Springer–Verlag).
They are thus different from classical crystals by their lack of strict translational order, and different from amorphous
materials by being (in their ideal forms) perfectly ordered. The orientational order associated with the most studied
quasicrystalline alloys is icosahedral or decagonal. Most quasicrystals studied recently are thermodynamically stable
systems (single grain sizes are not larger than a few cubic millimetres). They are frequently ternary alloys, the major
component being Al, and (at least) one of the two others a transition or noble metal. A mainstream of research in the
physics of quasicrystals attempts to decide whether the quasicrystalline order leads intrinsically to properties that
are different from those of crystalline or amorphous materials of similar composition. Quasicrystals usually exist
only for rather narrow composition ranges in the ternary phase diagram; often for adjacent (and equally narrow)
compositions so-called approximant phases are found that have a local order similar to the quasicrystal, but a strict
translational periodicity (with very large unit cells). There is growing evidence that the local atomic rather than the
long-range quasiperiodic order determines the physical properties [104,105].
If seen as transition metal alloys, the low electric and thermal conductivities of quasi-crystals are very unexpected
[105]. Calculations of the density-of-states curve [106,107] in these nonperiodic structures indicate that the Fermi
level lies in a relative minimum of the curve, called the pseudogap, and originates from the strong diffraction
of valence electrons by the lattice (the so-called Hume–Rothery pseudogap [108]). The width of the pseudogap
is between 0.3 and 1 eV. Experimentally, its existence (at least of the part below Ef ) has been inferred from
high-resolution, low-temperature UPS (ultraviolet photoelectron spectroscopy) studies [105]. A more controversial
theoretical result is the prediction of a spiky structure superimposed on the smooth ’valley’ of the pseudogap
[106,107]. The spikes should be a few tens of meV wide, but have not been seen in the UPS experiments. We
discuss below how the electronic density of states and the spike structure at the Fermi level are studied by NMR
[109–111].
The large fraction of Al in most quasicrystal alloys has made 27 Al a frequently studied nucleus. The gyromagnetic
ratio is 1.1103 kHz/G for the aqueous (Al(H2 O)6 )3+ ion at infinite dilution. A few complexes in other solvents exist
that resonate a few tens of ppm to high field of this reference, but it is probably a good approximation for the 27 Al
nucleus in a diamagnetic 2s2 − 2p6 environment. Bulk aluminum is usually considered a free-electron metal. At
300 K the Knight shift has been determined as 1640 ppm [112,113], the product T1 T = 1.85 s K [18]. The Korringa
144
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
constant S 27 = 3.87 ␮ s K and the calculated bulk density of states is 5.39 Ry−1 . Using the method of Table 6 the
spin-only susceptibility becomes χ̃cgs = 17.55 ␮ emu/mol, and the hyperfine field Bhf = 52.2 T.
In bulk fcc Al all atomic positions are magnetically equivalent, and the site symmetry is cubic, so that there are no
static quadrupolar effects: there is a single NMR line and in good-quality (powder) samples its width is determined
by nuclear dipole–dipole couplings that can be suppressed either by MAS or by line-narrowing pulse sequences.
Single-crystal NMR studies are not trivial due to the skin effect, which shields the rf field. For bulk fcc Al one
would expect no variation of the NMR spectrum if a single crystal specimen were oriented at different angles in
the magnetic field. The local environments in quasicrystals are not yet well characterized, but the point symmetry
at the nuclear sites should be less than cubic, so that quadrupolar and anisotropic shift effects should be visible.
In a structural model for the approximant Al80 Cu32 Fe16 with a lattice parameter of 1.23 nm, there are 8 different
Al sites [109]. These studies, also on other samples including the approximants [104], show that even in the (very
large) unit cells there are many inequivalent sites, all having a different quadrupole coupling.
The importance of the static quadrupole interactions raises the question whether dynamic quadrupole effects might
contribute to the observed spin–lattice relaxation and obscure the density of state effects. While the quadrupolar
contribution by conduction electrons to T1 T [114] normally is negligible, the ion cores of the surrounding atoms
might contribute through a phonon-type of mechanism with T1−1 ∝ T 2 at high T . To settle this question the ratio of
the relaxation rates of 63 Cu and 65 Cu at 77 K was studied in AlCuRu and AlCuFe. For Cu the quadrupole mechanism
leads to 65 T1 /63 T1 = (63 Q/65 Q)2 = 1.14 and the hyperfine interaction leads to 65 T1 /63 T1 = (63 γ /65 γ )2 = 0.87.
Values for these ratios are in favor for magnetic relaxation (∼0.87). For 27 Al, only around room temperature
deviations from T1 T constant were observed, which point to the growing importance of phonon contributions
above 77 K. Below 77 K the product T1 T was found to be constant (190 s K). The value, which is the same in
the quasicrystals, in the approximants and in AlRePd [111], is very large compared to the value in bulk Al metal
(1.85 s K). At 77 K the 27 Al shifts are 250 ppm in AlCuRu (3 compositions) and 160 ppm in AlCuFe (quasicrystal
and approximant) [109]. In AlRePd quasicrystals, which have among the lowest conductivities known for this class
of materials, the 27 Al Knight shift is 120 ± 30 ppm, independent of temperature below 100 K [111]. The data
below 77 K clearly show the reduced density of states in the quasicrystals. Fine structures in the DOS of the order
of 10−2 eV, as suggested in the Al-Cu-Ru system by specific heat and thermopower data, could not be confirmed
by NMR studies under hydrostatic pressure up to 2 kbar [110]. However, recently Tang et al. [115] concluded to
sharp features in the pseudogap of i-AlCuRu, i-AlPdRu and i-AlCuFe, and the crystalline approximant phase of
α-AlMnSi. Between 100 and 400 K, the Cu and Al relaxation rates were verified to be magnetic. The 27 Al and
63 Cu relaxation rate is proportional to T 2 , see Fig. 18, and the shift (reported with a precision of the order ±3 ppm)
increases linearly in this temperature range. These nonlinear temperature dependences of the relaxation rate are
seen as a clear signature of sharp features in the DOS at Ef with an estimated width of 20 meV.
Up to now, all NMR work agrees that the density of states (on the Al sites) in these quasicrystals is low, and that
the same holds for the crystalline approximants. Below 80 K, the product T1 T is temperature-independent, very
large compared to the value in pure Al metal, and not very different in quasicrystals of different composition. The
Fig. 18. The inversion recovery curves of the magnetization M ∗ = [M(∞) − M(t)]/[M(∞) − M(0)] vs. tT 2 in i-AlPdRe. The scaling shows
the T 2 -dependence of the relaxation process between 133 K and 673 K. The T 2 -contribution below 400 K is verified to be electronic in origin.
(After Tang et al. [115]. ©1997 American Physical Society).
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
145
hypothesis by Tang et al. [115], that their NMR results between 100 and 400 K, can be explained by the presence of a
sharp cusp at Ef has not yet found confirmation in other NMR work at lower temperature. The temperature variation
of shift and relaxation rate at relatively high temperatures for a system where both these quantities are small to start
with, is perhaps not the most appropriate way to detect by NMR the presence or absence of ’spiky structures’ in
the density of states. One has to suppose that the second-order quadrupole shift is temperature-independent, and
that there are no quadrupolar contributions to the relaxation rate. Because also the work under varying pressure
[110] found no evidence for significant changes in the density of states at Ef , in our opinion, this issue is not yet
completely settled.
3.6. Strong correlation effects and disorder
As has been shown in the foregoing, the enhancement effects seen in the NMR behavior of quite a few metals can
be described reasonably well with Stoner and desenhancement factors, respectively, α and k(α). In the form given
by Eq. (190), 0 < k(α) < 1. An increase of k over its value expected from Eq. (190), and even to values greater than
one is an indication of antiferromagnetic correlations (see the discussion of Eq. (87)), or of the effect of disorder
on the susceptibility [81]. We touch briefly on two examples: expanded liquid alkali metals (for a more complete
review, see [116]) and high-Tc superconductors (many reviews are available; some recent ones are [117–119]).
The effect of thermal expansion on the Knight shift of solid and liquid Ag metal has been shown in Fig. 8.
The liquid alkali metals, particularly Cs, can be expanded to fairly large specific volume by adjusting pressure and
temperature [116]. As we have seen in Fig. 6, the main effect of pressure and temperature on the susceptibility
is to change the density of states; in the simplest cases according to Eq. (174). The variation of the Knight shift
furthermore depends on whether Bhf varies with the atomic volume, e.g. within the limits given by Eq. (181). As
a check on the possible variation of the Stoner parameter α with atomic volume, the desenhancement factor k(α)
(see Eq. (178)) can be used, as was done in our discussion of Fig. 7. There it was concluded that the main effect
of changing the molecular volume at ambient temperature and pressure is a change in the density of states (even
though D(Ef ) is not well approximated by the free electron gas value). The lower panel in Fig. 19 shows that for
relative volumes up to V /V0 ≈ 1.4 the value of k(α) is reasonably constant, and the upper panel gives a reasonably
smooth extrapolation of Fig. 7 up to that value of V /V0 . For still larger expansions, the effects of pressure and
temperature are no longer equivalent for k(α) in Fig. 19, which implies that a gas of independent electrons flying
around in some average electrostatic potential is no longer a good model at such dilutions. At these low densities
Fig. 19. The 133 Cs Knight shift (upper panel) and desenhancement factor k (lower panel) vs. density ρ for expanded liquid caesium. See text for
discussion. (After Warren et al. [120]. ©1989 American Physical Society).
146
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 20. A model of a Fermi surface in high Tc cuprates. The tight-binding two-dimensional Fermi surface is given by the drawn line-the concave
shape arises from diagonal interactions. The dotted lines represent the magnetic Brillouin zone boundary. The intercept of the two lines marks
the center of the hot spots on the Fermi surface near π/a, 0). Because these points can be connected by the wave vector Q = (π/a, π/a), they
are most strongly scattered into each other, after [6]. Only the cold regions are observed as Fermi surfaces in photon electron spectroscopy or
neutron data.
the individual character of the charge carriers starts to show up, because there are not enough electrons available
for effective electrostatic screening. In most models for electron–electron interactions, the lack of screening leads
to antiferromagnetic spin correlations, and therefore, according to the discussion of Eq. (87), the value of k can
increase above 1 (also some models for the susceptibility of a disordered metal give k > 1 [81]). The upturn of
K for the lowest densities (largest V /V0 ) is probably not due to susceptibility enhancement, but to an increasing
D(Ef ), also seen in optical reflectivity experiments [116].
In high Tc superconductors not only the superconducting properties are unusual (d-wave instead of s-wave pairing)
but also the normal state strongly deviates from that of a normal metal. While in ordinary metals the Fermi surface is
a valid concept, the Fermi surface in the cuprate based superconductors seems to exist only in those directions in k
space, where magnetic excitations do not interfere, see Fig. 20 [6]. Even before the peculiarities of the Fermi surface
were well established, the NMR data were successfully described by a semi-phenomenological susceptibility of a
homogeneous spin fluid, containing a mixture of almost localized magnetic and nearly-free electron contributions
[121]. In the rather complex unit cells, localized magnetic correlations develop mainly at the copper sites. The
Knight shifts of the different nuclei in the cell can no longer be described by considering a local susceptibility of
their individual Wigner–Seitz spheres and a corresponding atomic-like hyperfine field. Instead, the Knight shift at
one atomic site is to a large extent determined by the spin state at another site, parametrized through the so-called
transferred hyperfine fields. (This phenomenon is also found in other systems, and is discussed in some more detail
in Section 3.7.2). The variation of the transferred hyperfine field within the unit cell can be described by form factors
in reciprocal space. In the almost two dimensional superconductors, these form factors ‘filter away’ [121–125] the
antiferromagnetic correlations on sites other than Cu. As an example, in YBa2 Cu3 O7 k is smaller than one and
temperature-independent on the Y and O sites, whereas k is larger than one and temperature-dependent on the Cu
sites, see Fig. 21. The temperature dependence of k follows from the increase of the antiferromagnetic correlation
length when lowering T . Also in other models such as the effective one-band description of the copper-oxide plane
by the t − J (or Heisenberg–Hubbard) model [126], similar physics appear.
3.7. Strong exchange: magnetism
3.7.1. Hyperfine fields in ESR and NMR
This section shows how in simple cases the d-like core polarization hyperfine fields in transition metals and their
ions can be considered as ‘atomic’ properties, to a certain extent independent of the environment that we put the
‘atom’ in. (In the next section we will meet some more complicated cases). The hyperfine fields of ‘magnetic’ ions
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
147
Fig. 21. The behavior of (T1 T )−1 for 63 Cu, 89 Y and 17 O in YBa2 Cu3 O7 , after Ref. [123–125]. The Y-relaxation rate (scale at the right) is a
factor 103 to 104 lower than that of O or Cu (scale at the left). The 1/(T1 T )-value of Cu is strongly enhanced compared to its value in other
materials; the enhancement increases with decreasing temperature down to 120 K. The superconducting transition temperature is around 90 K.
diluted in nonmagnetic insulating hosts, typically simple oxides like MgO, are measured by ESR, or for reasons of
sensitivity by ENDOR [55]. The hyperfine fields in nonmagnetically ordered metals can be determined through the
Knight shift in NMR. It should be noted that the values quoted by ESR people for the hyperfine coupling usually
are ‘per unit spin’, that is they give the ESR frequency difference between two adjacent hyperfine transitions.
Metal-NMR people, who are concerned with shifts rather than splittings, find it more natural to quote the hyperfine
field ‘per Bohr magneton’, which is half of the value ‘per unit spin’. Where necessary, the conversion to metal-NMR
convention has been made in this article.
As an example of the use of both techniques, consider 55 Mn. The reference gyromagnetic ratio 55 γ /2π =
10.500 ± 0.007 MHz T−1 , is taken from ENDOR experiments on Mn2+ [127], as described in Section 3.1.1. With
respect to this reference, the shift of the NMR frequency of the aqueous permanganate ion 55 MnO4 2− , which was
one of the first observed manganese resonances [128], is about 0.5%. This large shift may seem surprising, since
in a naive picture the manganese should be in a Mn7+ state, completely devoid of d-electrons. But the deep color
of the complex and the positive magnetic susceptibility show that there are in fact low-lying ligand field states, that
can give a paramagnetic chemical shift by the mechanism of Eq. (171). This situation is general for the tetraoxo
ions of groups 5, 6 and 7 (from V to Re). There are no examples of 3d6 complexes of Mn1+ that have sufficiently
weak ligands to apply the optical correlation method described in Section 3.1.2; it is believed that the 3d6 state can
exist only when stabilized by strong ligands.
There is an interesting difference between the Overhauser-shift CESR method used for the alkali metals, and the
ESR of paramagnetic impurities. The conduction electrons fly through all of the sample during the measurement,
and therefore, see a temperature-dependent average nuclear magnetization, given by the Langevin equation (Eq.
(3)). The well-localized electrons on the paramagnetic impurity on the other hand only see the nuclear spin on their
site. The flipping rate of that nucleus is of the order of its spin–lattice relaxation, slow compared to ESR time scales.
Therefore, each electron sees one well-defined nuclear spin state, but the electrons on different sites see different
spin states: there is a temperature-independent splitting instead of a temperature-dependent shift of the ESR line.
Actual measurements of hyperfine fields are often made using ENDOR, because of its higher precision. The value
so found for Mn2+ in a CaO host [127] is −8.15 T, the minus sign being characteristic of core polarization fields.
Values in the related oxides MgO and SrO differ from this value by less than one percent, but when very different
hosts are chosen, the differences go up to tens of percent. When the ionic configuration is changed to 3d3 , it is found
148
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
that V2+ , Cr3+ and Mn4+ in Al2 O3 have within a few percent the same hyperfine field −9.9 T [55]. For the 3d8
ions Co1+ , Ni2+ and Cu3+ the hyperfine fields are roughly all equal to −12.5 T [55].
Some of these trends are also found in the metals, both in computations and experiments. The calculated core
polarization fields [23] for the 3d neighbor metals V and Cr are −8.5 and −9.1 T, and −13.9 and −16.0 T for the
corresponding 4d pair Nb and Mo. The best experimental value for V metal [103], obtained from the single crystal
data in Fig. 17, is −9.0 T. The values found from NMR in the high-temperature paramagnetic phase of Ni metal
vary between −11.3 and −13.7 T [129,130]. This collection of results suggests that core polarization fields increase
slowly along a row, and rather rapidly in a column of the periodic system. The metal-NMR values increase by
roughly a factor of three from Ni to Pd, and another factor of three from Pd to Pt. The experimental core polarization
field at the end of the 5d transition metal series (Pt) is −118 T, while the calculated value is −1.1 × 102 T [84].
At the end of the 4d series we have Rh and Pd, where the shift-susceptibility correlation plots yield −16 T and
−34 T. This difference is much larger than the above-mentioned calculated difference for Nb and Mo, and it has
been argued that the Rh result is not reliable [102].
3.7.2. NMR of manganese metal
As example of a magnetic metal, we discuss manganese. Actually only one of its four allotropic forms, ␣-Mn,
is antiferromagnetic (below 96 K), the other forms being paramagnetic with large susceptibilities. The cubic A12
structure of ␣-Mn has 58 atoms in the unit cell, on four crystallographically inequivalent sites in the ratios 2 : 8 : 24 : 24
(labeled in that order I, II, III and IV). The cubic A13 structure of ␤-Mn is thermodynamically stable from 1000 to
1368 K, but can be retained to room temperature and below by quenching. There are 20 atoms per unit cell on two
inequivalent sites in the ratio 8 : 12 (labeled I and II). At still higher temperatures are the fcc ␥-Mn and bcc ␦-Mn
structures; finally the melting point is at 1517 K.
We start by considering the high-temperature phases (γ , δ and `, the liquid phase) [131]. Despite the simple
structures, the paramagnetic properties of these phases show several peculiarities. From analysis of the data, see
Table 9, it is found that about one quarter of the susceptibility is of orbital origin. As discussed in Section 3.4,
the orbital susceptibility and orbital shift will be large when the d-band is exactly half filled; and according to
bandstructure calculations this is indeed the case for both the fcc γ and the bcc δ phases [75]. The calculated
densities of states at the Fermi energy are 23.4 Ry−1 for the fcc and 37.4 Ry−1 for the bcc phases, so that the values
for χd mentioned in Table 9 according to Eq. (177) correspond to a Stoner enhancement (1 − α)−1 ≈ 10. The
experimental difference in susceptibility between the phases is smaller than what would be expected from these
DOS calculations, done for a ground state at T = 0 K. At the high temperature of the experiments, the DOS values
are averaged over a large interval kT , and this can change the effective value of the DOS [131].
In all three phases the susceptibility increases with increasing temperature, and the slope of the shift versus
susceptibility correlation plots, Fig. 22, is roughly the same: for ␥-Mn it corresponds to a hyperfine field −22 ± 2 T,
about twice the values from ESR on different Mnn+ ions, mentioned in Section 3.7.1. This suggests the presence of
additional core polarization fields, transferred from neighboring sites. In spin fluctuation theory, this requires that the
coherence of the fluctuations amplifies the hyperfine field more efficiently than it amplifies the bulk susceptibility.
A possible clue is that ␥-Mn can be stabilized at low temperatures by impurities such as Cu, and then becomes
antiferromagnetic below 480 K, with a magnetic moment value of ≈ 2.2µB per atom [46]. In density-functional
Table 9
Spin and orbital susceptibilities, spin and orbital
Table II]
55 Mn
shifts, and the d-like hyperfine field, in fcc (γ ), bcc (δ) and liquid Mn metal. [131,
Phase
T (K)
χ̃d,cgs (␮emu mol−1 )
χ̃orb,cgs (␮emu mol−1 )
Kd
Korb
Bhf,d (T)
γ
δ
Liquid
1400
1450
1525
451
485
504
141
153
146
0.0179
0.0202
0.0198
0.0091
0.0099
0.0095
−22.3
−23.4
−22.0
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
149
Fig. 22. Shift of 55 Mn in the γ , δ and liquid (`) phases of Mn metal, as a function of susceptibility with temperature as parameter. The slopes
give the values of Bhf,d listed in Table 9, and extrapolations of these lines yield the other values. (After El-Hanany et al. [131]. ©1975 American
Physical Society).
theory, we can look at Eq. (78):
m(ρ) =
µ0 µ2B
N Z
X
X
2
Dlm (Ef ) × ϕl (Ef , ρ) +
lm
α=1 cell
!
ρ 0 |)ν(ρ
ρ 0 )χ 0 (ρ
ρ0
ϕl2 (Ef , |ρ
ρ |) dρ
ρ
+ R α , |ρ
0
.
Now m(0) determines the hyperfine field, and the integral over ρ the susceptibility. The proposed ‘transferred’
R α 6= 0) contributes relatively more to m(0) than to the
mechanism requires that the sum over neighboring sites (R
integral over ρ.
In the temperature range where ␤-Mn is thermodynamically stable, only a single resonance is observed. The
behavior of the Knight shift, the relaxation rate, the line width and intensity of this resonance [131] result from
atomic diffusion between the type-I and type-II sites, and it has been derived that at T = 1200 K, KI = +300 ppm,
and KII = −7200 ppm. This shows that the minority sites I are considerably less magnetic than the majority sites
II.
By quenching, ␤-Mn can be obtained at room temperature and below. At low temperature, the system has been
mainly studied by the NQR technique. It has been found that between 2 and 100 K the spin–lattice relaxation rate for
site II increases with the square root of temperature [132]. For site I there is an orbital contribution to the relaxation
rate that increases linearly with T as usual, and a magnetic contribution that increases with the square root of the
temperature [133]. The magnetic relaxation at site II is about 20 times stronger than that at site I, see Fig. 23. This
agrees qualitatively with the shift difference found in the NMR experiment. The T 1/2 dependence agrees with the
Fig. 23. The spin–lattice relaxation rate T1−1 of 55 Mn in the two sites of ␤-Mn metal as a function of temperature. The T 1/2 dependence is due
to antiferromagnetic spin fluctuations in this paramagnetic metal. (After Kohori et al. [132]. ©1993 Physical Society of Japan).
150
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 24. The magnetic neutron scattering rate in ␤-Mn metal as a function of the scattering vector Q measured at zero energy transfer, at T = 7 K
and T = 290 K. The sharp peak at Q = 12 (nm)−1 is due to a MnO impurity. The broad peaks show the antiferromagnetic fluctuations. (After
Nakamura et al. [134]. ©1997 Institute of Physics).
prediction from spin fluctuation theory, see Section 2.5.2. Polarized neutron scattering experiments have also been
performed on quenched ␤-Mn, and a broad peak was found at a wave vector roughly equal to the inverse of twice
the mean interatomic distance, indicating the presence of antiferromagnetic spin fluctuations [134], see Fig. 24.
Finally we mention the internal fields that have been measured on the different sites of the antiferromagnetic
α-phase at low temperatures [135]. From an interpretation of neutron scattering results in terms of a localized
model [136], the magnetic moments on sites I to IV were found to be 1.9, 1.7, 0.6 and 0.2µB . The zero-field NMR
experiment gives the internal fields (in the same order) as 18.8, 14.1, 2.5 and 0.5 T; the fields from ‘one aligned
Bohr magneton’ are therefore, 9.9, 8.3, 4.2 and 2.5 T. The first two values are comparable to the ionic values in
Section 3.7.1. The two other low values perhaps point to an s-like contribution to the hyperfine field, of opposite
sign and with another ratio of hyperfine field to magnetic moment than the d-like contribution. (A superposition of
s-and d-contributions has been assumed on all four sites in the fits of the neutron data [136]). In the paramagnetic
phase of ␣-Mn it is likewise found that the Knight shift in site IV (and probably also in III) is nearly temperature
independent, in sharp contrast to the ‘usual’ temperature-dependent d-like shift for the sites I and II [131].
We see from the above that the magnetism in this ‘magnetic’ metal depends on the crystallographic structure,
and that different sites in a given structure have different magnetic properties. Even in the paramagnetic phases
the Wigner–Seitz cells around individual Mn atoms cannot be considered as independent paramagnetic entities,
and some form of spin fluctuation theory is probably necessary for a correct description of these phases. Another
example of such interatomic effects can be found in dilute alloys of Ag in Pt or Pd [137,138]. In these alloys,
the 109 Ag shift is negative: when extrapolated to zero Ag concentration, the value in Pd is 109 K = −0.0146, and
in Pt 109 K = −0.0046. The extrapolated values of the spin–lattice relaxation rate, however, are very nearly the
same, Fig. 25. While Eq. (187) to Eq. (190) can describe situations where the shifts are identical but the relaxation
rates different, the opposite is almost impossible. It is therefore unlikely that the negative shifts of 109 Ag are due
to a local density of d-like states on the Ag sites, with an associated intra-atomic negative core polarization field.
Instead, one must think of an interatomic exchange enhancement effect: the high local exchange enhancement of
the susceptibility on the host atoms (Pt or Pd) happens to create a negative s-like spin density on the neighboring
Ag atom. Something similar occurs in the interstitial regions between the atoms in pure platinum: from local spin
density calculations that explicitly include the presence of a magnetic field [84] it is found that the spin densities in
the interstitial regions are negative, see Fig. 26. The relaxation rate of the 109 Ag nuclei is much less affected by these
interatomic exchange effects than the shift. This can be roughly understood from the difference in the corresponding
nonlocal susceptibilities as defined by Eq. (26). According to Eq. (35) the shift is proportional to the integral of
χ 0 (0, r ), and the point r where the field is applied runs over all space, and therefore, also over the neighboring Pd
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
151
Fig. 25. Shift and spin–lattice relaxation rate of 109 Ag in Agx Pt1−x and Agx Pd1−x alloys. Note that near x = 0 the relaxation rates are the same,
and the shifts are different in the two hosts. (After Ebert et al. [138]. ©1984 Institute of Physics. Additional data from Narath et al. [137]).
152
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 26. Contour plot of the calculated spin density in one half of a five-layer slab of Pt(0 0 1) in the presence of a (very large) magnetic field
(235 T). In the hatched interstitial regions the spin density is negative. (After Weinert et al. [84]. ©1983 American Physical Society).
sites (0 is the Ag site). The shift of 109 Ag is mainly due to the field applied at the Pd sites. The relaxation rate is
proportional to χ 00 (0, 0), and now 0 is the only point where the field is applied. Therefore, the interatomic exchange
comes in only as a kind of second-order effect: the field applied at the Ag site induces a magnetization on the Pd
site, and this magnetization reacts back on the Ag. Since the induced magnetization is expected to be smaller than
that directly created by an applied field, the relaxation rate is smaller than would be expected from the observed
shift. The 109 Ag relaxation rate is, therefore, simply determined by the local densities of state, as in Eq. (189), that
happen to be rather similar in both hosts.
4. NMR theory of small particles and clusters
From simple free electron considerations for a small metallic particle containing N electrons one expects the
spacing between energy levels to be of the order Ef /N, where Ef is the Fermi energy of the corresponding bulk
solid as defined in Section 2.1.2. At temperatures much larger than this level spacing the system will behave
like a bulk metal, but with decreasing temperature the gaps between the levels can no longer be neglected and
deviations from bulk behavior are expected. Clear observation of such a crossover from the bulk to the quantum size
regime has been the goal of many experiments on assemblies of small metal particles since a few decades [9,10].
Such assemblies (required for sufficient experimental sensitivity) always showed substantial size distributions of
the particles, blurring the results. But even in a sample of nominally identical particles, the precise energy level
structure is expected to differ between particles due to differences in the individual boundary conditions because of
surface effects and randomness in the packing [139]. The energy gap then becomes a pseudogap and to predict the
thermodynamic behavior one has to take recourse to statistical theories for the energy level distributions [140], such
as the Random Matrix Theory [141–144]. The predictions depend on the possibility of electron exchange between
the particles. If electron exchange is possible at the relevant time scale of the experiment, the grand canonical
ensemble is required, while under conditions of negligible exchange the canonical ensemble applies. As realized a
few years ago, not only the spread in the density of states, but also multiple scattering can be very important for the
thermodynamic properties [145]. Below we discuss these concepts in detail.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
153
Fig. 27. Nearest-neighbor level spacing distribution P (x) for the Poisson (complete random sequence), orthogonal, unitary, and symplectic
ensembles, as function of the level spacing normalized by the average splitting x = 1/δ.
4.1. Energy levels
4.1.1. Poisson distribution, electron counting, and charging energies
Kubo considered an ensemble of noninteracting particles and assumed a random distribution of the energy levels.
If the particles have the same average level splitting δ, each interval in energy has the same probability of containing
a level [9]. The probability of finding a level spacing of 1 is given by a Poisson distribution, see Fig. 27
P (1) =
1 −1/δ
,
e
δ
(194)
where P (1)d1 is the probability of finding the nearest level in the interval (1, 1 + d1). The highest probability
is found for δ = 0. This means that for a random arrangement, the levels seem to attract each other, leading to
accidental degeneracy. The average distance δ between two spin degenerate levels at the Fermi energy is related to
the density of states (twice the number of energy levels per unit energy interval and per atom) D(Ef ) at the Fermi
level by [10,139],
δ=
2
N D(Ef )
(195)
with N the number of electrons in the particle; for the free electron model D(Ef ) is given by Eq. (174). The second
important aspect introduced by Kubo is that in the canonical ensemble an even or odd number of electrons per
particle [139,140] makes a difference. A particle with an odd number of electrons has at T = 0 K the highest
occupied level filled with one electron with spin up or down. For T δ/k this spin behaves like a free spin and the
susceptibility obeys Curie’s law. In the even case the spins are paired in the ground state. Therefore, the susceptibility
is zero at 0 K. Under electron exchange odd–even effects will be less pronounced. In Fig. 28 this effect is illustrated
for the canonical and grand-canonical case for a particle with equal level spacing.
Also charging energies [146] become important if particle sizes become small. The electrostatic energy of a
sphere with radius R and one extra electron is U1 = e2 /(8π 0 R). This quantity becomes very large for small R
(830 K for R = 10 nm). Kubo concluded that the number of electrons in a small particle is strictly fixed and that the
particle will be neutral for kT U1 . However, these considerations do not take into account that an electron will be
attracted by a neutral conductor by image forces, from which it even follows that a neutral conductor has a positive
electron affinity [146]. This implies that a system of small metal particles is not necessarily an ensemble of neutral
154
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 28. The normalized spin susceptibility χ/χP as a function of temperature for a particle with an equal level spacing between the levels. Near
kT /δ = 0 the two upper curves represent particles with an odd number of electrons, the two lower curves represent even particles. From top
to bottom the ensembles are the canonical, the grand canonical, the grand canonical and the canonical ensembles. (After Denton et al. [140].
©1973 American Physical Society).
particles. Arguments based on random matrix theory lead to a similar conclusion [147]: the charging energies can
become irrelevant due to the effects of the distribution in the splitting between energy levels.
4.1.2. Statistical distribution functions of the energy levels
Highly symmetric particles or particles with complete random energy levels are not very realistic. What can we
say about the energy levels around the Fermi level of an ensemble of particles with slightly different boundary
conditions? To calculate these levels one would have to incorporate in the Hamiltonian all effects which are of the
order of the level spacing at the Fermi level, which increases when the volume V of the particles decreases. Due
to the randomness, it is impossible to calculate the energy levels up to the Fermi energy exactly. The electronic
energies are the eigenvalues of a fixed Hamiltonian but with random boundary conditions. The boundary conditions
may be incorporated into the random matrix by a fictitious potential [144,148]. This means that an ensemble of
small metal particles is represented by an ensemble of random matrices and that the distribution function of the
energy level spacings is that of an average over particles. The symmetry of the Hamiltonian is supposed to be
common to all particles. The possible symmetries can be classified in terms of the orthogonal (integer total angular
momentum), symplectic (half integer total angular momentum) and unitary groups [142,143]. The unitary group
has no time inversion symmetry. With a few reasonable assumptions [9] it follows that each pair of eigenvalues
for all ensembles shows level ‘repulsion’. The (nearest-neighbor) level spacing distributions in terms of x = 1/δ
(compare Eq. (194)) are [148]:
Poisson :
P (x) = exp(−x)
1
1
πx exp − πx 2
2
4
64 2
218 4
symplectic : P (x) = 6 3 x exp − x
9π
3 π
4
32
unitary : P (x) = 2 x 2 exp − x 2
π
π
orthogonal :
P (x) =
(196)
(197)
(198)
(199)
These functions are shown graphically in Fig. 27 . Here P (x)dx represents the fraction of particles in the ensemble
that has a level spacing 1 between xδ and (x + dx)δ. The average value of x is 1.
In the case of negligible spin–orbit coupling and no magnetic field only two-fold degenerate states exist, and the
orthogonal ensemble must be used. When the number of electron is odd and the spin–orbit coupling becomes so
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
155
large that it mixes the levels originally determined by the orthogonal ensemble, the symplectic ensemble should be
applicable (see also Eqs. (108) and (129)). Since a magnetic field breaks the time reversal symmetry, the unitary
ensemble must be applied in strong magnetic fields [149,150]. For a particle with a radius of R = 1 nm, like e.g.
the metal core of the Pt309 compound discussed in Section 6, the crossover field will be about 65 T, almost an order
of magnitude higher than the static NMR fields commonly used.
4.2. Electron density and NMR line width
4.2.1. Electron density variation due to surface effects
In s-like metals like Li, Na, Cu only the contribution of the contact term to the Knight shift is important and it is
proportional to the spin susceptibility. In small particles without interparticle hopping of electrons the temperature
dependence of the susceptibility, and therefore the Knight shift, is different for particles with an even or with an
odd number of electrons. With decreasing T , the Knight shift of the particles with an even number of electrons
should go to zero (because all spins are paired off) and the shift of the odd particles should increase (because their
susceptibility is Curie like), see Fig. 28. At high temperatures, much larger then the average level spacing, the
odd–even effects will disappear and the susceptibility will behave like that of the bulk material. There is, however,
another reason why the NMR of small particles is different from that of the bulk material. Small particles have a
high surface to volume ratio. At the surface the periodic potential felt by the conduction electrons is interrupted.
This results in a oscillatory behavior of the probability density of the electrons near the surface (Bardeen–Friedel
oscillations). The Knight shift is proportional to the electronic spin density and is therefore expected to oscillate
near the surface. These oscillations enlarge the NMR line width for small metal particles with respect to the bulk,
even at high temperatures.
For transition metals like platinum, the spatial dependence of the Knight shift will be different for the 6s and 5d
conduction electrons. Slichter et al. [151] estimated the contribution from the s-electrons, Ks , from the bulk value
of the Knight shift, Ksb , using the spin density in the noninteracting electron picture. The d-electron contribution is
treated as in the bulk with a correction for surface effects. In this model the particles are simplified by spheres with
a spherical square well potential. The boundary conditions are that of an infinite square well of which the effective
radius is somewhat larger (0.15 nm) than that of the particle. The calculation is now that of solving the Schrödinger
equation for free electrons in a spherical square well, which is an exactly solvable problem. The solutions for the
wave function 9R,E (r) and energy, E, are [152]
r
~2 Znk 2
2 jn [rZnk /R]
,
E=
,
(200)
9R,E (r) =
3 jn−1 Znk
2me
R
where jn (x) is the nth spherical Bessel function, Znk is the kth zero of jn (x), me is the mass of the electron and
9R,E (r) has been normalized per unit volume. Due to the spherical symmetry each energy level has a degeneracy of
2n + 1. The Knight shift due to the 6s electrons is obtained via a scaling function S(r, R) proportional to |9R,E (r)|2 |
taking the energies and the (2n + 1)-fold degeneracy of the levels into account
Ks (r, R) = S(r, R)Ksb .
(201)
Deviations from spherical boundary conditions remove the high degeneracy by mixing up the order of filling states,
thereby reducing the spin density oscillations. This feature can be incorporated in the model by artificial broadening
of the energy levels. A simple way of introducing this broadening is to regard the temperature as a parameter. An
artificially high temperature causes more of the states near the Fermi energy to contribute in the Knight shift, which
is equivalent to the mixing of states that an explicit level broadening would introduce. The 5d electron contribution
is much more difficult to handle in a rigorous way. It is assumed that the local 5d electron density at Ef is reduced
at the surface and heals back exponentially to the bulk value when moving away from the surface.
Kd (r, R) = [1 − ξ d(r, R)]Kdb ,
(202)
156
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
where ξ is the fractional reduction in the local density of states of the 5d electrons at the surface and
1,
r≥R
d(r, R) =
exp[(r − R)/λ], r ≤ R
(203)
with λ as the characteristic healing length and R is slightly smaller (0.05 nm) than the radius of the particle. The
density of states is much larger for the 5d electrons than for the 6s electrons. At the surface it is suggested that the
6s states hybridize with the 5d states [153]. This hybridization leads to an enhanced contribution of the 6s electrons
to the Knight shift at the surface and should fall off rapidly by going away from the surface. In the model this is
included by an enhancement factor η that falls off exponentially with the 5d healing length. Then Eq. (201) will
change into
Ks (r, R) = [1 + ηd(r, R)]S(r, R)Ksb .
(204)
The total Knight shift is then K(r, R) = Ks (r, R) + Kd (r, R). Use of these expressions gives for the line shape of
a particle with radius R
Z R+0.5
B0
2π
B0
,R =
r 2δ
−
dr.
(205)
L
ν0
ν0
γ [1 + K(r, R)]
0
Here ν0 /B0 is the position in the line (in units of MHz T−1 ) and γ /2π is the gyromagnetic ratio in the same units.
For the line shape only the Knight shift is included, other effects like chemical shifts or homogeneous broadening
are neglected since they are expected to be much smaller. Fig. 29 shows the Knight shift as a function of the radial
Fig. 29. Knight shift as a function of the radial position in a particle of 1.0 and 2.0 nm for different numbers of electrons in the particle. The
smooth line is for the exponential healing for both s and d electrons, see [154] for comparison.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
157
Fig. 30. Line shape for a sphere in the exponential healing model for different values of the radius of the sphere, with d-electrons only.
position in a spherical particle of 1.0 and 2.0 nm for a different number of electrons per particle. The straight line
indicates the case when only the exponential healing is included for both s and d-electrons. From Fig. 29 it can be
clearly seen that the number of electrons has a large influence on the position and amplitude of the oscillations in
the total Knight shift. For larger particles R > 5.0 nm, the influence of the oscillations in the Knight shift due to
the s-electrons is negligible and the spatial dependence of the Knight shift is given accurately by the exponential
healing effect only. The line shape can be calculated using the spatial dependent Knight shift and Eq. (205) (see
Fig. 30).
4.2.2. Statistical distribution of the electron density
As mentioned before, in the canonical ensemble at temperatures (much) lower than the average level spacing
different properties can be expected for particles with even or odd number of electrons. Due to electron tunneling
(grand canonical ensemble) the odd–even differences will disappear, but new quantum features that affect the
electron density will appear and will influence the NMR data. When the inelastic mean free path exceeds particle
sizes not only the variation in the energy density of states, but also multiple scattering leading to local electron
density variations can be very important for the thermodynamic properties [145]. In an ensemble of small metal
particles with slightly different boundary conditions (e.g. due to random packing) together with electron exchange
[155,156] the electron wave function will exhibit strong local variations. The local susceptibility χ(r) is proportional
to the wave function at point r and therefore also varies from point to point. In this limit the NMR-line shape
has been calculated using the zero dimensional supersymmetric σ model [155,156] and also by random matrix
theory [157]. In these models the changes in the electron density due to the presence of healing lengths in the
charge density, electron–electron interactions, and interactions with magnetic impurities are neglected. The spin
susceptibility can then be expressed through the local density of states D(Ef , ra ), see Section 2.4. At T = 0 K (cf.
Eq. (9))
χ (ra ) = µ0 µ2B −1 D(Ef , ra ).
(206)
158
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
The NMR-line shape can be written as
X
δ(ω − ω0 − 1ω(ra )),
I (ω) = W
(207)
a
where W is a weight factor and ω0 = γ B0 and the sum is over all nuclear spins. In a bulk metal the shift is the
same for all spins and the NMR line is narrow. In calculating the NMR-line shape for small metallic particles it is
assumed that the nuclear spins are located in a system of macroscopically identical particles. At the same time, the
particles experience random perturbations so that the level distribution will differ from particle to particle although
it is assumed that the volume of all the particles is the same. To calculate the NMR-line shape, I (ω/ωbulk ) has to
be averaged over this disorder. At T = 0 K I (ω/ωbulk ) scales with the distribution function for the local density of
states. This distribution function is calculated in [155,156] for a model in which there is a finite probability for an
interchange of electrons between the particle and its environment. Due to this connection with its surrounding, the
energy levels of the particle are no longer infinitely sharp but have a finite width described by a parameter γ . The
calculation of the normalized line shape gives [155,156]
α 1/2
1
1
1
α
x+
× 2cosh α + x + −
sinh α
(208)
x −3/2 exp −
I (x) =
8π
2
x
x
α
with x = 1ω/1ωbulk . The dimensionless parameter α is proportional to the ratio of the lifetime broadening γ and
the mean level spacing δ:
α=
γπ
.
δ
(209)
The function I (x) given in Eq. (208) is drawn in Fig. 31 for several values of α. The limit α → ∞ corresponds to
particles of such large sizes that the discreteness of the levels is not felt. The normalized line shape or local density
of states becomes I (x) = δ(x − 1), i.e. the shape is very narrow with a line positioned at the Knight shift of the
bulk metal. In the other extreme α 1 the discreteness of the levels becomes very important. The function I (x) is
very broad and asymmetric and its maximum moves to zero when α → 0. To account for finite temperatures, the
width parameter in Eq. (209) is written as
γ (T ) = γ (0) + cT .
(210)
Fig. 31. The normalized line shape I (x) vs. x with x = D(Ef , r)/D(Ef )bulk = 1ω/1ωbulk for different values of α, where α is a measure for
the tunnel rate of the electrons.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
159
Fig. 32. Sketch of the change in line profile (i.e. the signal intensity as function of the Knight shift) for the bulk sites when U is increased from
0 to U = 2V with W = 1V and η = 0.05V. (After Pastawski et al., [158]. ©1997 American Physical Society).
4.2.3. Other approaches
The mesoscopic solutions [155–157] describe the variations in the local density of states, but neglect the difference
in binding of surface and bulk atoms. Starting with a simple tight binding model for the electronic structure,
this limitation [158] might be repaired as follows. Using the second-quantization formalism (see Section 2.7) the
Hamiltonian is written as
H=
M
M
M X
X
X
†
†
†
Ei bi bi +
V (bj bi + bi bj ),
i=1
(211)
j >i i=1
where Ei is the energy of the s-electron state centered at site i of a cube; V is the kinetic energy involved in hopping
between nearest-neighbor sites i and j . To represent shape and crystal inhomogeneities the site energies are taken
between −W/2 and W/2 (Anderson disorder). Surface sites, which are defined as those sites which being at the
cube’s surface also have an additional energy shift U , are introduced by hand. All other sites are bulk sites. Tunneling
among particles broadens the line inhomogeneously with the highest broadening (0s ) at particular surface sites. The
characteristic level broadening for all sites is taken equal to η0 = 0s /M. Finite temperature effects are reproduced
by an additional broadening, similar to Eq. (210), η = η0 +kT . Resorting to a matrix continued fractions calculation
of the Green’s function, the line position and shape are found to have universal scaling behavior with respect to the
variation of the thermal energy or the mean level spacing — the same result as found by the previously discussed
mesoscopic methods. For very small particles surface states become relevant and the line shape starts to deviate
from the universal scaling function. This broadening effect is shown in Fig. 32 for a cube of 7 × 7 × 7 sites having
49 surface (one side of the cube) and 294 bulk sites.
4.3. Comparison of the different NMR models for the NMR line shape
The NMR line shape arises from different electron densities at different sites. In the mesoscopic model, the
difference is due to multiple scattering within a small particle. In the exponential-healing model the difference is
empirically described as a surface effect. Recently these two sources for the line width were compared [159], and
below we summarize the findings and apply this approach to spherical particles.
4.3.1. Random matrix theory
Let us consider N randomly packed metallic particles with volume V separated from each other by tunnel barriers.
The shape and electronic properties changes from particle to particle, but the volume V is the same. Each particle
160
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
contains on the average M electrons. The system is, therefore, an (N × M) electron system. The energy level
distribution is quasi continuous, but each corresponding eigenfunction is supposed to be well localized in one of the
N particles. Therefore, this model is referred to as the tunnelling/localization model. The particles are numbered
by n and the energy levels within a particle by k. It is assumed that the probability distribution of the energy of an
electron in a level (n,k) is given by a Lorentzian L(E − Ekn ) with width γ /2 independent of n and k.
γ 2 −1
γ
2
.
(212)
E +
L(E) =
2π
2
The charge distribution |ψkn (r)|2 is not supposed to be uniform. To calculate the thermodynamic properties there
are two steps involved [139]. The first step is to average over an ensemble of particles which are all completely
identical to particle n. The total number of electrons is supposed to be so large and the temperature so low that the
chemical potential ζ (see Section 2.1.2) is temperature independent. Due to electron tunneling or hopping ζ is equal
in all the particles. The second step takes into account the differences in energy levels for different particles. The
thermodynamic average for the susceptibility of particle n, χ n , is given by
Z
2µ0 µ2B X
∂f
dE
(213)
L(Ekn − E)
χn = −
V
∂E
k
and the second step is the average over particles n
X
χ̄ = N −1 χ n .
(214)
n
The second moment of the susceptibility is defined by
X
1χ 2 = N −1 (χ n − χ̄)2 .
(215)
n
Using the same methods the average Knight shift (or center of gravity of the line) becomes
K = 23 χ .
(216)
The expression for the second moment of the resonance lines not only depends on 1χ 2 , but also on the two level
correlation function and the variations in |ψkn (r)|2 within the particles. From random matrix theory explicit forms
of the level correlation functions can be obtained. The second moment of the Knight shift becomes
1K 2 = 49 1χ 2 + 49 (πα)−1 χ̄ 2 A = K̄ 2 (πα)−1 (g(α) + A).
(217)
The α in Eq. (217) is a levelwidth parameter similar to Eq. (209), and A has the value of 2/β with β = 1, 2, 4 for
the orthogonal, unitary and symplectic ensembles [157]. The first term of Eq. (217) comes from the variation in
susceptibility due to the difference in energy levels around Ef between particles: the stronger the level repulsion the
less variation in χ, see Fig. 33. The second term comes from the variation of the wave functions within the particles.
At high temperatures, this means large values of α, the first contribution vanishes, and the variation of the wave
function dominates. At low temperatures, i.e. small values of α, the influences of both terms are of equal size.
4.3.2. Exponential healing
The exponential healing model focuses on surface effects. For comparison with the multiple scattering model let
us consider an infinite slab of thickness 2d. The Knight shift varies with distance x from the center of the slab as
x−d
K(x)
= 1 − η exp
(218)
Kb
λ
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
161
Fig. 33. The weight function g(α) as function of α (after [159]) for the orthogonal, unitary, and symplectic ensemble.
with η the fractional reduction of the local states at the surface, λ the healing length and Kb the bulk Knight shift.
All these parameters are supposed to be independent of the slab thickness. In this model the NMR intensity as
function of the Knight shift is proportional to dx/dK times the number of atoms at position x (which for a slab is
x independent). This gives the following normalized line shape
(λ/d)(1 − (K/Kb ))−1 K0 < K < Kd
(219)
I (K) =
0
K < K0 ∨ K > Kd
with
K0 = (1 − η)Kb ,
−d
Kd = 1 − η exp
Kb .
λ
(220)
K0 is the Knight shift at the surface and Kd is the Knight shift at the center of the slab. The first moment of this
NMR line is
Z
λ
K = I (K)K dK = Kb − (Kd − K0 ).
(221)
d
The second moment is given by
1K 2 = K 2 − K̄ 2 = Kb − K̄
K−
1
2
(Kd + K0 ) .
(222)
In Eq. (222) the first factor on the right-hand side goes to zero for a thick slab and the second factor goes to zero
for a very thin slab, see Fig. 34. In the multiple scattering model α increases as the square of a linear dimension in
the low temperature limit. In Fig. 34 the second moment of a slab and a sphere in the exponential healing model
are plotted as function of half the thickness of a slab or the radius of a sphere. The third line is the second moment
for the unitary ensemble in the multiple scattering model with the assumption that the parameter α is proportional
to the surface of the particle. Because the proportionality constant is not known, in Fig. 34 the second moment of
the Knight shift from the tunneling model is scaled in the x-direction (particle radius).
For large diameters both models give a second moment that goes to zero, corresponding to the narrow bulk line
at the bulk Knight shift position. For small diameters the two models give very different results. For the exponential
healing model the second moment goes to zero and the line width becomes very small at the chemical position, see
Fig. 34. In the multiple scattering model the first moment or center of gravity of the line remains at the bulk Knight
shift position, although for small α (small particles) there is a large intensity at the chemical position.
162
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 34. Radius dependence of the second moment of the NMR line for a slab and sphere in the exponential healing model and for the
tunneling/localization model for the unitary ensemble.
4.4. Relaxation in small metal particles
4.4.1. Korringa-like description of the relaxation in small particles
To describe the nuclear spin lattice relaxation rates in small platinum particles, one can start from the equations
for the susceptibility, the Knight shift, and the relaxation rate in bulk Pt, given in Eq. (187) to Eq. (190). The new
assumption made [90] is that the equations are still valid for small metal particles, but with a local density of states
being site dependent due to the healing effect, and all other parameters at the values given in Table 7. In Fig. 35 the
Knight shift and the relaxation rate are plotted as function of the local s-and d-density of states. From this figure it
is clear that nuclei with the same total Knight shift (at the same frequency in the NMR line) can have very different
Fig. 35. Knight shift in percent and S(T1 T )−1 , in units of (10−6 ), of 195 Pt as function of the local s and d density of states at the Fermi level.
(After Bucher et al. [90]. ©1988 American Physical Society).
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
163
relaxation rates. By using the Knight shift and the relaxation rate at that particular position in the NMR line it is
possible to find with the use of Fig. 35 the s-and d-density of states and to see differences between the bulk and
surface parts of the NMR line shape.
4.4.2. Relaxation due to discrete energy levels
Till now we formulated the general relation between the dynamic susceptibility and the spin–lattice relaxation
rate. To discuss the effects of discrete levels, the authors of [160] have used the Hebel–Slichter derivation [161] of
T1−1 . That result can be written as
S(T1 T )−1 =
(µB Bhf )2 1 X
f (i ) 1 − f (j ) δ(j − i − ~ω)
kT
N2
(223)
i,j
with N defined in Eq. (6) and Bhf in Eq. (36).
In a bulk metal we have, with the usual approximation ~ω kT ,
f (i ) 1−f (j ) δ(j −i − ~ω) ≈ f (i ) (1 − f (i )) δ(j − i − ~ω) = −kTf 0 (i )δ(j − i − ~ω) (224)
and next the use of Eq. (56) leads to the Korringa relation given in Eq. (58). The sum over f 0 (i ) is related to the
density of states at the Fermi energy D(Ef ) through Eq. (63). These equations remain valid in small metal particles
with thermal broadening exceeding the splitting of the levels around the Fermi energy, but if the energy levels
are very sharp (no thermal broadening) it is usually impossible to satisfy the energy conservation expressed by
δ(i − j − ~ω) and the relaxation rate should fall to zero. Thermal broadening of discrete levels can be introduced
in Eq. (223) in two different ways.
In the model of Ref. [160] the thermal effects are introduced as a relief of the requirement of exact energy
conservation; but the probabilities of occupation of the relevant levels are computed from their sharply defined
discrete energies. The Fermi energy is supposed to be halfway between two levels n and n + 1, and at low enough
temperatures one has
f (i ) = 1,
i ≤ n,
f (j ) = 0,
j ≥ n + 1.
(225)
Only the terms with i = n and j = n + 1 are retained in the double sum of Eq. (223). The effects of thermal
broadening are modeled by writing
δ(n+1 − n − ~ω) ≈
γ /2
1
,
π (γ /2)2 + (1 − ~ω)2
(226)
where γ = γ (T ) is a temperature-dependent broadening parameter and 1 = n+1 − n . For the small particle the
number of electrons N that appears in Eq. (223) can be expressed through Eq. (195). Setting 1 − ~ω ≈ 1 one then
finds in the low-temperature limit defined by Eq. (225)
T1−1 (T , 1)
−1
T1,bulk
=
γ /2δ
δ
,
kT (γ /2δ)2 + (1/δ)2
(227)
where T1 (T , 1) is the relaxation time at temperature T of all nuclei in a particle with a splitting 1 between the
highest occupied and lowest empty levels. One expects the ratio given in Eq. (227) to be smaller than one, and to
fall to zero when T → 0. This then requires that γ (T ) goes to zero faster than linear in the temperature. In [160]
γ (T ) was chosen to depend on the level splitting and on the temperature as
−1
.
(228)
γ (T , 1) = γ0 (T ) exp
kT
164
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Since T1,bulk T is independent of T , the temperature variation of T1 (T , 1) is due to γ (T , 1) appearing in Eq. (227).
On physical grounds, we expect that both T1−1 (T , 1) and γ (T , 1) increase with temperature (for a given particle
with a given 1). This amounts to the condition
21 < γ (1, T ).
(229)
This condition cannot be met for 1 ≈ 0, but such particles are not frequent in e.g. an orthogonal distribution, Eq.
(197). Let us consider only the 95% of the particles that have 1 ≥ δ/4; then the temperature region where Eq. (227)
is valid is determined by
kT <
δ
.
4 ln(2γ0 (T )/δ)
(230)
The restriction in the double sum of Eq. (223) requires that the temperature be low enough that γ0 (T )/δ > 0.64.
Another possible approach is to assume that the levels are broadened into a structured, but continuous density
of states, and that the probability of occupation is a continuous function of energy; and furthermore that energy
is conserved exactly. Then the only difference with the case of bulk metals is that now the density of states at the
Fermi energy depends on the temperature T and the level distribution parameter 1. We start from the definition of
the density of states in Eq. (63):
D(Ef ) = 2N
−1
∞
X
δ(i − Ef ).
(231)
i=1
Now broaden the δ-functions into Lorentzians similar to Eq. (226) (all with the same width for simplicity), and
replace N by using Eq. (195):
Dparticle (Ef ) = Dbulk (Ef )δ
∞
X
1
i=1
π
(γ /2)2
γ /2
.
+ (Ef − i )2
(232)
As an approximate way to perform the sum over i in Eq. (232) for a particle that has a level splitting 1 around the
Fermi energy we can write in the equal level spacing approximation
∞
πγ X
Dparticle (Ef )
4
γ
=
tanh
=1
.
Dbulk (Ef )
π γ 2 + (2k − 1)2 12
21
(233)
k=1
The right most equality is exact for a system with all levels equally spaced at interval 1, the Fermi energy halfway
between two levels, and infinitely many levels below Ef . The spin–lattice relaxation rate is now given by
T1−1 (T , 1)
−1
T1,bulk
= tanh2
πγ 21
.
(234)
This equation has so far not been applied to experimental data.
The expression for the magnetization recoveries M(t) after saturation via the usual saturation pulse trains is a
superposition of individual recovery curves with characteristic times T1 (T , 1), given by Eq. (227) or Eq. (234):
Z
−t
P (1) d1,
(235)
M(t) = M0 1 − exp
T1 (T , 1)
where P (1) is the spacing distribution function, see Eqs. (196)–(199).
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
165
Fig. 36. 63 Cu NMR spectrum of a supported copper sample with average particle diameter 3.4 nm, taken at 1.5 K. The left marker gives the
position of the bulk copper resonance, the right marker the reference position. (After Tunstall et al. [165]. ©1994 Institute of Physics).
5. NMR in small metal particles
5.1. Small particles: copper
Because 63 Cu is a very convenient NMR nucleus, there have been quite a few experiments on small copper
particles (early papers are [162,163]). In most of these, the particles were created by evaporation of the metal, and
imbedded in simultaneously evaporated SiO. Here we will discuss a series of experiments on particles made by the
colloidal route, using poly(ethylenimine) as a protecting polymer [164,165]. The particles were deposited on silica,
and the resulting samples were red in color. When these samples were allowed to become oxidized (by exposure
to air), those containing smaller-sized particles turned blue, and those with larger sizes dark green. These oxidized
products also showed clear EPR signals. The absence of these blue/green colors, and of detectable EPR in the
samples used for the NMR study was taken as an indication of the absence of oxides. This is of importance, because
in the earlier work [163] it was thought that the observed very large width of the 63 Cu NMR line could be related
to unidentified paramagnetic impurities, such as surface oxides. The particle size distribution was determined from
TEM micrographs. The mean volume-weighted particle diameter hdiw for the smallest-particle sample was 3.4 nm;
the largest particles studied had hdiw = 12 nm. The mean particle sizes were also determined from the broadening
of X-ray diffraction lines (XLBA): these results consistently gave smaller values than the TEM analysis.
The NMR experiments were performed by the point-by-point field-swept spin echo method, at ω0 /2π =
143.3 MHz (B0 = 12.7 T), and at temperatures between 1.2 and 4.2 K. The full-width at half-maximum (FWHM)
δω at 1.5 K was δω/ω0 = 4.5 × 10−3 for the smallest particles, and δω/ω0 = 2.1 × 10−3 for the largest. An
earlier report [164] found that the line width is not simply proportional to the applied field, thereby suggesting the
possibility of quadrupolar broadening. However, in later work it was found that there is an important instrumental
factor to the observed line width, and that the nonlinearity is within this instrumental error. The Knight shift of bulk
copper is 2.38 × 10−3 , so these are really large widths, see Fig. 36. For samples with hdiw < 8 nm the shift of
the maximum of the line varies with temperature (in the range studied), and is less than the bulk Knight shift, Fig.
37. The spin–lattice relaxation rate has not been determined. The shift of the maximum of the line towards K = 0
when the temperature decreases as well as the slightly asymmetric line shape would make these data interesting
candidates for an analysis in terms of the multiple scattering models.
The authors [165] have preferred the classical odd–even analysis in the canonical ensemble, but taking into account
nonlinearities. In this description, a given particle gives rise to a single ‘infinitely sharp’ resonance (mesoscopic or
surface effects are neglected), but different sizes of particle have different susceptibilities, and therefore different
resonance frequencies; furthermore odd and even particles are treated separately. For both classes of particles, the
susceptibility is taken as nonlinear. For the odd ones, this is simply done by assuming that at these high fields and
166
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 37. Shift at the peak of spectra as in Fig 36 as function of temperature, for samples with typical diameters 3.4 nm (bottom), 4.4 nm (middle)
and 8.0 nm (top). Relative shift, with respect to the bulk metal; estimated error ±0.05 in these units. (After Tunstall et al. [165]. ©1994 Institute
of Physics).
low temperatures the lone spin is fully aligned, and uniformly distributed over all sites in the particle. From the
experimental data for bulk copper we have estimated in Table 6 that the hyperfine field ‘per fully aligned Bohr
magneton’ is approximately 65 T; one than finds that the Knight shift for all the nuclei in e.g. a 1001-atom particle
is (65/1001)/12.7 ≈ 0.0051, since the applied field is 12.7 T. For the even particles, the differential susceptibility
∂M/∂H is related to the two-level correlation function R2 (1 , 2 ). This quantity is defined as follows: in the ensemble
of particles under consideration, a fraction R2 (1 , 2 )d1 d2 has an energy level in the interval (1 , 1 + d1 ) and a
second, distinct level in the interval (2 , 2 + d2 ), irrespective of the indices of the levels (i.e. irrespective of how
many other levels there might be in between). In random matrix theory, it is found that R2 is of the form R2 (x) with
x = |1 − 2 |/δ, where δ is the mean level spacing. At very low temperatures, where only a few levels are partially
occupied (i.e. have an occupation probability different from 0 and from 1), an expression for the even-particle
susceptibility has been obtained [140] by counting explicitly all thermally accessible configurations. The result is
2µ0 µ2B
(π kT )2 00 2µB B0
2µB B0
∂M
R2
+
+
·
·
·
,
(236)
=
R
2
∂H even
δ
δ
δ
6δ 2
where R200 (x) is the second derivative of R2 (x). To obtain the magnetization of the sample at 12.7 T, one integrates
the differential susceptibility for B0 going from 0 to 12.7 T.
An alternative interpretation of the 63 Cu NMR has been presented as well [165]. This is based as usual [163]
on the assumption that the resonance frequency of nuclei in the odd particles is strongly dependent on the average
level spacing, and that, therefore, the result of a size distribution will be to smear this signal out over a very large
frequency range, making it in fact unobservable. At low temperatures the susceptibility of even particles should drop
to zero; that the maximum of the NMR spectrum does not correspond to zero shift is then interpreted as evidence
for spin–orbit coupling effects, see Eq. (129) in Section 2.6.4. It should be pointed out, however, that numerical
fits of data for copper particles to such spin–orbit models for the Knight shift agree rather poorly. Copper is not a
particularly heavy metal, and there is no Ferrell–Anderson effect (see Section 2.6.4) in the low-temperature shift of
a cuprate superconductor, see Fig. 2.
5.2. Small particles: silver
The number of NMR studies on small silver particles is quite limited. This is rather surprising when one thinks
of the advantage that 109 Ag has over 63 Cu in being a spin 1/2, thereby avoiding possible problems with quadrupole
broadening of the signals. It would seem, however, that the reactivity of silver makes the preparation of small
particles of this metal rather difficult. Three groups have reported 109 Ag NMR [166–168] in systems of supported
particles: but two of them found it impossible to detect signals in samples where all particles were smaller than
50 nm. The sample with the smallest particles in the third report [168] had an average particle diameter determined
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
167
Fig. 38. Line widths (FWHM) in units of the bulk metal shift for 109 Ag in small silver particles as a function of particle diameter. The lines are
simple guides to the eye, showing slopes d −1 (drawn) and d −2 (dashed). Data at 20 K. (After Bercier et al. [168]. ©1993 Institute of Physics).
from X-ray line broadening analysis (XLBA) of 13 nm. These samples were prepared by methods similar to those
used for the copper particles described above, with the important difference that the protecting agents for the colloids
were surfactants rather than polymers. Other samples were prepared by impregnation of the supports with solutions
of silver salts, followed by reduction. The supports were alumina, titania and silica. There is very little information
on the particle size distributions, because of contrast problems in TEM. The available TEM pictures suggest that
nearly all samples contained at least some large particles of the order of 50 nm.
The 109 Ag NMR in all samples (as said, with average diameters determined by XLBA of 13 nm and more) showed
a symmetric line, detectable by Fourier transform methods but clearly broadened with respect to a bulk signal. The
line position and the spin–lattice relaxation time were those of bulk silver (K = 5.215 × 10−3 at room temperature
with respect to the ion at infinite dilution, T1 T = 8.7 ± 0.2 s K; compare also Table 5). Most experiments were
performed at 20 K, but occasionally temperatures down to 15 and up to 400 K were used.
For quantitative analysis of the linebroadening, the observed signals were fitted to a sum of two Lorentzians,
centered at the same frequency, but with different width. While this was done simply for convenience, it was also
found this way that the line shape for different samples could be very different. This is thought to be related to
details in the particle size distributions (e.g. bimodality), and makes it very difficult to characterize ‘the’ line width
by a single number. Two different measures were used: the first simply the FWHM of the fitted line; the other the
ratio (M23 /M4 )1/2 , with M2 and M4 the second and fourth moment for a truncated Lorentzian (it is well known that
the moments of a full Lorentzian diverge). If the two measures gave a very different ranking for the same spectrum
in the particle diameter versus line width ordering, the data on that sample were discarded. Nevertheless, there
remained considerable scatter in the plot of line width vs. particle diameter d, although it is pretty sure that the
variation is somewhere between proportional to d −1 and d −2 , see Fig. 38.
Given the particle sizes and the temperature at which the experiments are performed, the quantum size effects
can be excluded as the reason for the particle-size dependent linebroadening. The proposed explanation is related to
one of the first discussions of size effects in the NMR of metals, concerning the broadening of the 207 Pb NMR line
in filaments of lead obtained by the impregnation of porous glass [169]. Two samples with filament diameters of
10 and 30 nm were studied, and the observed broadening was ascribed to surface-induced spatial variations in the
density of Fermi-level electrons. The associated charge oscillations in the free electron gas have been mentioned
168
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
by Bardeen [170], and go asymptotically as (2kf x)−2 cos(2kf x) where x is the distance from the surface and kf
the Fermi wave vector. The asymptotic variation of the susceptibility goes as (2kf x)−1 sin(2kf x), and therefore
decays slower than the charge perturbation [171]. The free electron gas has no crystal lattice, and therefore it is
not easy to compare its characteristic distance, the inverse of the Fermi wave vector, with a distance in a crystal
lattice (for silver, (2kf )−1 = 0.04 nm). Anyway, it has been proposed [168] that these Bardeen-Friedel oscillations
make that in the surface region some sites have higher and other sites have lower local susceptibilities than the
average, thereby leading to NMR line broadening. Strictly speaking of course the decaying oscillation cannot give
a perfectly symmetric broadening, but there remains a clear difference with the ‘exponential-healing’ model and its
exponential decay that creates a one-sided broadening.
A very coarse estimate of the broadening can be made on basis of the free electron model. The calculation of
the position-dependent susceptibility [171] (with the free electron gas parameter set for silver) indicates that the
relative shift of the surface atoms with respect to the bulk might be as large as ±0.15, of the order of the line width
of the smallest sample at low temperatures. It is then calculated that the local susceptibility oscillations decay to
an amplitude that corresponds to the bulk line width over a distance of 10 atomic layers. In the largest-size sample
studied, this region contains about one-tenth of all atoms; in the smallest-size sample it is about half of all atoms.
This model suggests that for samples with average particle diameters above 5.7 nm (20 atomic layers) the line
width should decrease as d −1 . The scatter in the data of Fig. 5 is then attributed to differences in the particle size
distributions.
There is a remarkable variation of the observed line width with temperature, Fig. 39. To a good approximation,
the line shapes observed at 20 and 80 K can be brought into coincidence by a scaling of the shift axis (and, of
course, of the amplitude). The scaling factor varies somewhat from one sample to the next, but is about 1.6, clearly
Fig. 39. Line widths (FWHM) in units of the bulk metal shift for 109 Ag in small silver particles as a function of temperature. Typical particle
diameter 1.3 nm (top) and 8.0 nm (bottom). (After Bercier et al. [168]. ©1993 Institute of Physics).
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
169
different from the factor 4 that one would expect for broadening by paramagnetic impurities in the dilute limit. (The
line shapes scale linearly with applied field between 4 and 8 T). When the temperature is increased further, the line
width decreases more rapidly. The temperature dependence does not have the characteristics of thermally activated
motional narrowing, e.g. by self diffusion. The proposed explanation is increasing vibrational motion of the surface
of the particles. The motion diminishes the sharpness of the boundary of the electron gas, and thus diminishes the
amplitude of the Bardeen oscillations in the model of [171]. A more dynamic image of this process can be obtained
by considering the local hyperfine field on some arbitrarily selected nucleus inside the particle. When the surface
vibrates, the Bardeen oscillations and therefore the local field on this nucleus vary in time. When the variation is
rapid compared with the total range of NMR frequencies of the nucleus during the course of one vibration period,
there will be ‘motional narrowing’ of the NMR signal, but this time not due to motion of the nucleus itself, but of
the surface of the particle.
5.3. Small particles: platinum
5.3.1. 195 Pt NMR data analysis
The relation between particle size distribution and NMR spectrum has been modeled in the ‘NMR layer model’
[172]. Here we describe a simple version; a more general one has been discussed in Section 4.2. We start by
constructing size histograms from TEM micrographs. To do this, it is customary to consider the images on the
micrograph as circles with diameter d equal to that of effectively spherical metal particles that caused the image.
Furthermore, this diameter is converted to the total number NT of atoms in the particle by (for the fcc structure,
four atoms per unit cell):
√ 2π d 3
π 2 d 3
=
,
(237)
NT =
3 a
6
2r
where a is the (bulk) lattice parameter, and r the hard-sphere radius. For platinum, a = 0.392 and 2r = 0.277 nm.
While both hypotheses seem reasonable for particles containing several hundreds of atoms, a justification for
their use in very small particles is lacking, since both the electron-beam/sample interaction and the transfer of
information in the microscope are nonlinear processes. A further experimental problem is caused by the presence of
a strong granularity on the micrographs, in addition to the image of the particle. It is general practice to ignore these
complications, and to accept some uncertainty in the true diameter of the smallest particles. In the extreme, Eq. (237)
says that a single atom will yield an image with d = 2.2r. In a hard-sphere model for both zeolite and platinum, the
biggest fcc particle that fits into the supercage (cage diameter 1.3 nm) contains 31 atoms [173]; its image, according
to Eq. (237) has a diameter of 0.96 nm. Comparison of electron micrographs for untreated and platinum-loaded
zeolites under our experimental conditions [174] shows that the smallest features that can be reasonably attributed
to platinum particles have d ≈ 0.5 nm. For oxide-supported particles contrast is often a problem, and the lower
limit is closer to 1 nm.
For the NMR layer model, the atoms in a small-particle sample are divided into groups belonging to different
atomic layers: the surface layer, the subsurface layer and so on. To find the fraction of atoms in each group, the
particle size histograms obtained by electron microscopy are interpreted in terms of fcc cubo-octahedra. The smallest
such particle contains 13 atoms; the next-larger one 55. To convert a histogram into layer statistics, we use Eq. (237)
to find an average NT for each class of the histogram; next we determine the corresponding (noninteger) number of
cubo-octahedral layers m from [172]:
NT =
10 3
11
m − 5m2 + m − 1
3
3
(238)
and the number of surface atoms NS from
NS = 10m2 − 20m + 12.
(239)
170
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 40. Particle size distributions and layer statistics. (a,b) size histograms for two samples of Pt particles on titania TiO2 . (c) distribution of
atoms over the layers of the NMR layer model. Layer 0 is the surface. After Bucher et al. [172].
The number of atoms in subsurface layers is obtained by replacing m in Eq. (239) by m − 1, m − 2 and so on. It
should be noted that the highest dispersion possible in this model is 12/13 for a 13-atom particle.
We assume that the different sites in a given layer, Fig. 40, of a cubo-octahedral particle are sufficiently similar
that the resonance frequencies of all nuclei in the same layer are relatively close to each other on the scale of the total
spectrum width. The superposition of NMR signals from a given layer we will call a ‘peak’: its (inhomogeneous)
width is supposed to be of the order of a MHz. A similar assumption is of course the basis for the correlation
between low-field NMR intensity and particle size, already found in the early work of Slichter’s group [153]. For
convenience, these peaks are taken to be Gaussians, completely characterized by the position of their maximum in
the spectrum, by their width and by their integral. The integral must be proportional to the relative number of atoms
in the corresponding layer, given in Fig. 40. For the position of the maximum as a function of layer number, we
impose an exponential decay (see Eq. (241) below), similar to the behavior in the vicinity of an impurity in very
dilute alloys (Fig. 13 in Section 3.4). Finally, the width of the Gaussian is considered a freely fittable parameter,
but for not too different samples (like the case of Fig. 41) it is assumed to be sample-independent. The maximum
of the peak corresponding to the nth layer is assumed to occur at a Knight shift Kn (K0 is the Knight shift on the
surface, K∞ that in the infinite solid) obeying the relation
n
,
(240)
Kn − K∞ = (K0 − K∞ ) exp −
m
where the dimensionless constant m represents the ‘healing length’ for the Knight shift (compare also Eq. (202)),
expressed in units of a layer thickness (0.23 nm). According to this assumption and to the data in Fig. 40c, the NMR
spectrum of Fig. 41b should consist mainly of a superposition of three Gaussian peaks with relative areas 0.60, 0.29
and 0.09. The spectrum of Fig. 41a contains these same Gaussians (having the same positions in the spectrum and
the same widths), but now with relative areas 0.36, 0.25 and 0.17, and several more Gaussian peaks. Fits according
to this principle are shown in Fig. 41c and Fig. 41d. They correspond to K0 = 0 and m = 1.35 (the corresponding
‘healing length’ is 1.35 nm × 0.23 nm) in Eq. (240). The agreement between fitted and experimental spectra is
sufficient to demonstrate the usefulness of the NMR layer model. The fitted subsurface (n = 1) peak of the clean
sample falls approximately halfway between the surface and bulk resonances. This is in very good agreement with a
five-layer slab calculation [84] and shows that more than half of the spectrum contains information from the surface
region.
The NMR layer model simply considers the layer-to-layer variation of the NMR shift. It is perhaps more reasonable
to look instead at the local density of states at Ef (LDOS). A further refinement is to characterize each layer by two
quantities instead of one: the average LDOS and the width of the distribution of LDOS values on the different sites
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
171
Fig. 41. 195 Pt NMR spectrum and layer statistics. (a,b) point-by-point spectra under clean-surface conditions for the samples in Fig 40. (c,d) fits
by a superposition of Gaussians that represent the NMR line of a given layer. After Bucher et al. [172].
that make up the layer. The necessary additional experimental information can be obtained from the spin–lattice
relaxation data in the following way. It is easily seen from Fig. 35 in Section 4.4 that at a single resonance frequency
(fixed value of K) one might find signals from nuclei with many different combinations of s-and d-like LDOS. Each
such combination would give rise to the same K, but a different T1 . Therefore, generally, the spin–lattice relaxation
curves measured at a certain resonance frequency could be nonexponential. The spin–lattice relaxation mechanism
by conduction electrons requires that such nonexponential decay curves obey ‘time-temperature scaling’, because
the spin–lattice relaxation rate T1−1 for each individual nucleus is proportional to temperature T , so that T1 T = C.
In a relaxation experiment at a given resonance position and temperature, we measure a series of recovered signal
amplitudes Ai as a function of the relaxation interval τi between the initial saturation pulses and the inspection
pulse(s). When the Ai are normalized by the fully relaxed amplitude, the Bloch equation for a single site is
τi
(241)
1 − Ai = exp −
T1
and using the Heitler–Teller–Korringa relation, one has
τi T
.
1 − Ai = exp −
C
(242)
If the relaxation curve is a sum of N different exponentials (corresponding to different sites), there are N different
constants C; but a collection of curves taken at different values of T will collapse into a single curve when plotted
as normalized Ai versus τi T , as shown in Fig. 42 [175]. It is usually impossible to determine a value of N from
the experimental data. We find that most relaxation curves can be described by a sum of two exponentials, with
temperature-independent amplitude ratios. While it is clear that in such a case nuclei in at least two different
environments resonate at the frequency under consideration, it is of course impossible to demonstrate that there are
not more than two environments. Therefore, one should be very cautious in interpreting the amplitude ratio of the
two exponential decays as a ratio of ‘site occupations’.
To arrive at an estimate for the widths and for the average values of the LDOS-distribution on the surface sites,
we consider the relaxation curve measured at the frequency of the maximum in the spectrum as representative of
the surface sites: we take the range of their spin–lattice relaxation times as lying between the ‘fast’ and the ‘slow’
172
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 42. Time-temperature scaling for nonexponential relaxation curves in a series of 195 Pt saturation-recovery experiments. The time points τ
are multiplied by the temperature of the experiment (see key to symbols) and the individual equilibrium signal amplitudes are scaled to the same
value. The sample is Pt/TiO2 of dispersion 0.6 under several hydrogen coverages (0.1, 0.5 and 1.0 monolayer). The squares in c) show data at
110 K for another Pt/TiO2 catalyst of dispersion 0.36. (After Tong et al. [175]. ©1994 American Chemical Society).
values of the double-exponential fit. We use Eqs. (188) and (189) to obtain the LDOS corresponding to the fast and
the slow relaxation values. The range between the LDOSs of the slowly and the rapidly relaxing nuclei is now taken
as a measure of the width of the LDOS distribution over all surface sites. To obtain the average values, the ‘fast’
and the ‘slow’ LDOS are weighted by the corresponding fractions of the double-exponential fits. The main purpose
of such slightly complicated fitting procedures is the study of the change in surface properties after chemisorption.
The surface-LDOS is a quantity that appears in some models for the reactivity of a metal surface. These applications
are outside the scope of the present article, but have been reviewed in [11].
5.3.2. The surface peak
The ‘NMR layer model’ can obviously be used to determine the dispersion (the fraction of metal atoms in the
surface) of a sample. However, two experimental considerations are important. First, very small particles (say,
below 25 atoms) will be not metallic in the NMR sense: they have no Knight shift and no Heitler–Teller–Korringa
spin–lattice relaxation. Since all other spin–lattice relaxation mechanisms are less efficient, the T1 of such particles
will be very long, and in practice their NMR signal will be unobservable due to saturation. Second, the 195 Pt NMR
spectra are obtained by measuring the amplitude of spin-echoes, created by a pair of rf pulses. The echo amplitude
depends on the pulse spacing through an (effective) transverse relaxation time T2 , which varies across the spectrum.
Usually, spectra are plotted without correcting for this effect, but in quantitative work this must of course be done.
When zeolite-Y is loaded with platinum by generally accepted methods of preparation, the 195 Pt NMR signal
usually indicates the presence of particles larger than the supercage, see Fig. 43. Electron microscopy of 40 nm thick
slices prepared by ultra-microtomy has shown that these particles are inside the zeolite matrix, and the histograms
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
173
Fig. 43. Point-by-point 195 Pt NMR spectra of six Pt/NaY samples under clean-surface conditions. Most of these samples have detectable intensity
in the high field region of the spectrum, indicating the presence of particles larger than the supercage. After Tong et al. [174].
Fig. 44. Particle size histograms for the samples in Fig 43 from electron micrographs of ultramicrotome slices. These particles are all inside the
zeolite, but most do not fit in an undamaged supercage. After Tong et al. [174].
are given in Fig. 44. From Figs. 43 and 44, and with the Gaussian peaks fitted in Fig. 41, we ‘predict’ the spectra
in Fig. 45. The samples contained between 0.5 and 1 platinum atoms per supercage, so that the fraction of cages
damaged by the growth of the big particles was at most a few percent, and the overall zeolite structure was retained.
As remarked above, the NMR cannot exclude the simultaneous presence of much smaller, ‘nonmetallic’ clusters
in undamaged supercages; but if a large fraction of Pt would have gone undetected, the remaining signal should
be measurably weaker than expected on the basis of the known platinum content. Six samples were studied with
some variation in the parameters of their preparation. There was no clear correlation between the parameters of the
preparation and the particle size [174].
Yu and Halperin [176] have observed a 195 Pt surface resonance from platinum particles initially prepared on
silica gel; afterwards the carrier was removed with a solution of sodium hydroxide to form a ‘self-supported’
174
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 45. Predicted spectra for the samples in Fig 43a (dashed curve) and Fig 43f (full curve). The agreement is sufficient to indicate that the
particles seen by TEM and by NMR are the same (see text). After Tong et al. [174].
powder sample. Its average particle diameter determined by TEM was 4 nm; assuming a log-normal distribution
this corresponds to a dispersion of 0.31. The NMR samples were extensively washed with water, dried, and left
exposed to the atmosphere. A signal detected at 1.089 G kHz−1 was attributed to surface platinums; its relative
area indicated a dispersion of 0.47. The Illinois group [153] has identified the signal at this position in their
samples as due to H2 Pt(OH)6 or a similar compound, formed by interaction with the atmosphere. It disappears after
oxidation/reduction treatment, which moves the peak position to around 1.10 G kHz−1 . For the ‘self-supported’
sample, no surface-NMR signal was found after hydrogen treatment.
5.3.3. Effects of chemisorption
It has been shown in the first papers from the Slichter group [153] that the shape of the 195 Pt NMR spectrum of
a platinum catalyst varies strongly with surface conditions. In Fig. 46 two sets of spectra are shown: one set from
the Illinois group, and one set from Lausanne. The two catalysts have been prepared by different methods, and the
spectra have been taken in different laboratories. But their particle sizes, as measured by TEM, are very similar, and
Fig. 46. Point-by-point 195 Pt NMR spectra of Pt/Al2 O3 by the Illinois group (panels a and b, after Rhodes et al. [153]; ©1982 American Physical
Society) and of Pt/TiO2 from the Lausanne group (panels c and d, after Bucher et al. [172]). Clean surface conditions in (a) and (c), saturation
hydrogen coverage in (b) and (d). The dispersions of the two samples measured by electron microscopy are very similar.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
175
Fig. 47. Spin–lattice relaxation rate as a function of temperature for 195 Pt in the surface peak of the spectrum. The straight lines indicate metallic
behavior. (a) Pt/TiO2 with (circles) and without (triangles) hydrogen. (After Bucher et al. [90]. ©1988 American Physical Society). (b) Pt/Al2 O3
covered with CO. (After Ansermet et al. [177]).
so are the spectra for particles with or without chemisorbed hydrogen. Note that the sample Pt-46 has a dispersion of
0.46 measured by hydrogen chemisorption, but 0.61 according to electron microscopy and 0.52–0.62 measured from
the clean-surface NMR. In the interpretation of these spectra it is assumed that the geometry (i.e. the layer statistics
in the NMR layer model) of the particles does not change upon chemisorption, but that the surface-LDOS and the
healing length are affected. It has been stated [151] that the changes in the 195 Pt spectrum upon chemisorption of
CO are similar to those for hydrogen covering; also the 195 Pt T1 in the surface region of the spectrum changes in a
similar way for both adsorbates. The product T1 T is temperature-independent, see Fig. 47, showing that even after
the adsorption of hydrogen or carbon monoxide the platinum surface has metallic character. It is found that T1 T
increases after adsorption, indicating a drop in the local density of states at the Fermi energy on the surface sites.
In the interpretation of the spectra in Fig. 46 it is assumed that the geometry (i.e. the layer statistics in the NMR
layer model) of the particles does not change upon chemisorption, but that the surface-shift and the healing length
(represented by K0 and m in Eq. (240)) are affected. Repeating a fit as in Fig. 41 for hydrogen-covered samples, we
find the new value of m. From spin–lattice relaxation curves measured at the position of the Gaussians, we derive
the layer-by-layer variation of the LDOS. The result of such an analysis [172] is shown in Fig. 48. The increase
Fig. 48. The LDOS variation with layer number for Pt/TiO2 with clean and hydrogen-covered surfaces. Upon hydrogen chemisorption the LDOS
on the surface (layer 0) drops, and the healing length increases. After Bucher et al. [172].
176
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 49. Comparison of the effect of hydrogen and of oxygen chemisorption on 195 Pt NMR spectra of Pt/TiO2 . Data taken at 80 K, except the
open symbols that are scaled (at the maximum of the spectrum) from data at 20 K. The changes in the spectra for hydrogen-covered samples are
the effect of an increasing healing length, as in Fig 48, that moves all Gaussians to lower field. Under hydrogen, the spin–lattice curves obey
time-temperature scaling, as in Fig 42, but under partial oxygen coverage as in panels (f) and (g) this is not the case, so that some platinum must
be in a nonmetallic environment. (After Tong et al.: [175], © 1994 American Chemical Society; and [179], ©1995 Institute of Physics).
in ‘healing length’ upon hydrogen chemisorption can be interpreted as saying that the hydrogen forms a bond not
only with surface platinums, but also with subsurface atoms and beyond. A drop in surface-LDOS upon hydrogen
chemisorption has also been found in calculations for hydrogen chemisorption on Pd [178].
The effect of oxygen chemisorption on the 195 Pt spectrum is less drastic [179], see Fig. 49, but the behavior of
the spin–lattice relaxation is more complicated. For Pt/oxide samples, at intermediate oxygen coverages, a fraction
of the 195 Pt surface signal no longer has a temperature-independent T1 T ; this means that on some surface sites
the density of Fermi-energy electrons is very low. In principle, when T1 T is temperature-dependent one expects
to observe temperature-dependent features in the spectrum as well, but these have not been detected. The reason
for this is not clear, but the same discrepancy has experimentally been found in the 195 Pt NMR of other systems,
see Section 6.3: it seems that spin–lattice relaxation is a more sensitive monitor of ‘nonmetallic’ behavior than the
spectral shape. At the lowest coverage studied (0.16 atoms of oxygen per surface atom of platinum) approximately
three nonmetallic Pt sites are found for each oxygen atom; at a coverage of 0.40 this number is roughly halved and
at 0.75 coverage it is zero (i.e. at this coverage all surface platinum atoms again have a Korringa-type relaxation).
Under UHV conditions the saturation coverage on the Pt(1 1 1) surface is one oxygen atom per four platinum
atoms [180]. For a coverage of three oxygen atoms per four platinums, which is the saturation value in Fig. 49,
oxygen–oxygen direct bonding becomes important, and restores the metallic character of the underlying platinum
surface. At the lower coverages, the oxygen–oxygen interaction is mainly indirect, the platinum-oxygen interaction
is rather localized, and the NMR properties of atoms in the deeper layers are essentially unaffected. However, the
metallic surface sites have their LDOS modified by the presence of oxygen [179].
5.3.4. Support effects
The 195 Pt NMR of small platinum particles on classic oxide supports shows that the clean-surface-LDOS is largely
independent of the support (silica, alumina, titania) and of the method of preparation (impregnation, ion exchange,
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
177
Fig. 50. 195 Pt relaxation curves for platinum particles in three different types of zeolite. The data for NaX have been taken at several temperatures
between 80 and 250 K; the others are at 80 K only. Data for three samples in NaY plotted together. Similarly, data for two samples in NaHY
plotted together. The small black dots with error bars represent a double-exponential fit that has been used for a quantitative analysis of these
data. (After Tong et al. [183]. ©1997 American Chemical Society).
deposition of colloids). At a given resonance position, one always finds the same relaxation rate, independent of
particle size or support. The shape of the spectrum is related to the sample dispersion. The same is true for particles
protected in films of PVP. (However, samples prepared in conditions of strong metal-support interaction behave
differently [181]).
For platinum particles in zeolites, the 195 Pt NMR behavior is more complicated [182]. The low-temperature
data, that indicate a transition towards nonmetallic character, are discussed in Section 6.2. Experiments at 80 K
and above show that the spin–lattice relaxation at fixed resonance position is still independent of particle size for a
given zeolite; but at constant dispersion it is different in different zeolites. As remarked in Section 5.3.1, most of the
particles in these samples do not fit inside an undamaged supercage. It is found that there is a systematic variation
of the spin–lattice relaxation in the surface region of the 195 Pt NMR spectrum with the framework acidity of the
encaging zeolite. Results in zeolites of three different acidities (NaX, NaY and partially hydrogen-exchanged NaY)
are shown in Fig. 50. There are three samples with different dispersions in NaY, and two dispersions in NaHY.
For a given zeolite, all data have been fitted together. Data for NaY and NaHY are at one temperature only (80 K);
those for NaX have been obtained at different temperatures between 80 and 250 K. Roughly speaking, the data
represented by the circles are a ‘faster’ relaxation than those given by the squares, and the triangles are in between.
An increase in zeolite acidity decreases the metal surface-LDOS [183]. It has been shown that this has interesting
consequences for the chemisorption strength of e.g. carbon monoxide [184].
Recently, the 195 Pt NMR of commercial fuel cell electrode material has been observed [185,186], see Fig. 51.
This consists of platinum supported on carbon black, and pressed into graphitized-carbon cloth. Because of the
conducting nature of the carrier, one might expect to see differences with respect to NMR of particles supported on
oxides; furthermore, if an electrolyte is present in the NMR sample, the electric double layer at the metal/electrolyte
interface might influence the 195 Pt surface signal. The as-received material shows the NMR peak at 1.089 G kHz−1
characteristic of platinum particles exposed to the atmosphere [153]. For oxide-supported catalysts, it disappears
after thermal treatment under oxygen and hydrogen. The electrocatalysts are cleaned by electrochemical methods,
where the sample is used as the working electrode in a three-electrode cell. In the first experiments, the as-received
material was subjected to extensive potential cycling; but later it was found that results are more reproducible
when the potential is held fixed within the double-layer region (250 mV) until the reduction current falls below mA
values, and can no longer be measured (this takes several hours). After this cleaning procedure, the fuel cell electrode
material, together with some of the electrolyte (0.5 M H2 SO4 ) is transferred into an NMR ampoule. Three samples
with different particle sizes have been studied, Fig. 51. The platinum loadings and average particle diameters
178
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 51. 195 Pt NMR spectra of Pt/graphite fuel cell electrode material with Pt particles of different sizes. The pair a,b is on the same vertical scale,
and the pair c,d also. (a) 2.5 nm particles, as-received material. (b) as a., but after electrochemical cleaning. (c) 8.8 nm particles, electrochemically
cleaned. (d) 2.0 nm particles, electrochemically cleaned. ([(a) after Tong et al. [185], ©1997 American Chemical Society; (b)-(d) after Rice et al.
[186]).
were provided by the manufacturer of the material. From the hydrogen adsorption/desorption profiles in cyclic
voltammetry experiments, the total quantity of adsorbed hydrogen was determined; together with the value of the
platinum loading this yields a ratio for (atom H)/(total atoms Pt), the dispersion. An independent value of dispersion
can be obtained from the average diameters, assuming the particles to be cubo-octahedra. Reasonable agreement
between these values and the intensities in the 195 Pt NMR spectra was obtained by considering the spectral region
below 1.11 G kHz−1 in Fig. 51 as due to surface platinums. (No correction for T2 effects was applied). A remarkable
difference in the spectra of oxide- and carbon-supported platinum is especially clear for the 2.5 nm sample: the
fuel-cell material shows much less intensity at the bulk resonance position, 1.138 G kHz−1 . A similar difference is
seen in the spectrum for the 2.0 nm sample. In terms of the NMR layer model, this means that the healing length
is larger in the carbon-supported material. It is not yet clear whether this is related to the conducting nature of the
carrier, or to the presence of the electrolyte: comparisons between ‘wet’ and ‘dry’ samples are needed.
5.3.5. Pt–Pd Bimetallics
The NMR lines in bulk alloys are considerably broader than those in pure metals; this simply reflects the distribution of atomic environments, and thus of Ef -LDOS, in an alloy. The lines in high-dispersion bimetallic catalysts
have additional broadening according to the NMR layer model. Furthermore, if segregation occurs, the average
composition of each layer may be different. Therefore, the interpretation of 195 Pt NMR in Pt1−x Pdx particles is
only qualitative. The samples [99] were prepared as colloids, protected by a PVP polymer film. Layer statistics
according to the NMR layer model, Eq. (239), for samples with x =0, 0.2 and 0.8 are shown in Fig. 52 . The
metal/polymer films were loaded in glass tubes closed with simple stoppers.
The NMR spectrum and spin–lattice relaxation times of the pure platinum polymer-protected particles are practically the same as those in clean-surface oxide-supported catalysts of similar dispersion. This implies that the
interaction of the polymer with the surface platinums is weak and/or restricted to a small number of sites. The
spectrum predicted using the layer distribution from Fig. 52, and the Gaussians from Fig. 41 shows qualitative
agreement with the observed spectrum for x = 0 (Fig. 53). No metal-NMR data are available for pure palladium
catalysts. From magnetic susceptibility studies it has been found that the susceptibility of surface Pd atoms is less
than that of the bulk [187]. A similar conclusion has been reached for Pd cluster molecules [188]. This means that
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
179
Fig. 52. Layer statistics for three samples of Pt1−x Pdx in PVP film. The line is calculated for a sample of particles with 2.35 nm diameter. For
NMR purposes, the three size distributions are identical. (After Tong et al. [99]. ©1996 American Chemical Society).
Fig. 53. 195 Pt spectra (a,b,c) and spin–lattice relaxation time (d,e,f) for Pt1−x Pdx particles with closely matching size distributions. The differences
are, therefore, due to alloy formation. The right most arrows in a,b,c indicate the peak position of the bulk spectrum. The left most arrows divide
the observed intensity in two halves. For x = 0 the dashed line is the spectrum ‘predicted’ from the size distribution. The full curve in the
relaxation panels represents the data for Pt/TiO2 catalysts. The filled circles are data for the bulk. Note that the edges of the spectra and the
extrapolation of the relaxation data follow the bulk behavior. (After Tong et al., [99]. ©1996 American Chemical Society).
180
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
in Pd particles the Ef -LDOS is lower on surface sites than on bulk sites, just as in Pt. Therefore, we assume that the
195 Pt NMR spectra of Pt Pd
0.2 0.8 and Pt0.8 Pd0.2 particles can also be interpreted with an exponential-healing model,
Eq. (240).
In 195 Pt NMR spectra of catalysts, the nuclei in a bulk-Pt-like environment (those that have roughly two layers
of platinum atoms around them) resonate in the region 1.12–1.14 G kHz−1 . It is seen that the fraction of such nuclei
in Fig. 53 decreases with increasing Pd content. As far as 195 Pt NMR goes, the size distributions are completely
characterized by the data in Fig. 52. Since the site statistics of layers 0–4 are virtually identical, and the absolute
values for the deeper layers are very small, the effective size distributions are all the same (and very close to that
for a monodisperse sample of 2.35 nm particles). Therefore, the differences in the NMR spectra cannot be due to
differences in size distribution and they must be an effect of the alloying. The right most arrows in Fig. 53a–c
give the (average) resonance position in the corresponding bulk materials. It is seen that the high-field edges of
the spectra follow these positions very well, as expected from Eq. (240) for layers with n ≥ 2m. Furthermore,
the nuclear spin–lattice relaxation rates at the high-field edges tend toward the corresponding bulk values, see Fig.
53d–f. This shows that on the scale of 1–2 healing lengths, the composition of the interior of the particles is to a good
approximation that of the overall formula (and therefore, the same holds for the surface composition). The products
T1 T are essentially temperature independent in all points of the spectra for x = 0 and x = 0.2. In the surface
region of the spectrum for x = 0.8, the T1 is comparatively shorter at low temperature, indicating an increase of the
effective density of states. The order of magnitude of T1 T in the surface region of all three spectra is compatible
with the existence of atoms in a metallic environment.
The catalytic activity of similar polymer-protected bimetallics has been found to vary strongly with composition
[189]. In the bulk alloys the Ef -LDOS on both Pt and Pd sites varies rapidly with composition around x = 0.8 (see
Fig. 15). The 195 Pt NMR spectra and relaxation rates at the high-field end of Fig. 53 show that the interior of the
alloy particles is bulk-like. It is supposed, but not proven, that on the surface of the alloy particles the Ef -LDOS
changes rapidly with composition as well, and that this explains the variation in catalytic activity.
5.4. Small particles: rhodium
Just as for small platinum particles, the main motivation for 103 Rh NMR is the study of supported metal catalysts
(although in both cases the metal loadings used in NMR far exceed what is usual in catalysis). Little work has
been published so far [100], and we will have to refer in part to preliminary data from the Lausanne group. It
seems certain that the NMR behavior of small rhodium particles is quite unlike that of platinum particles: the
exponential-healing model does not apply. The 103 Rh NMR spectra of small rhodium particles on different oxide
supports (titania, alumina) or protected by a poly(N -vinyl-2-pyrrolidone) (PVP) film are rather symmetric and to a
good approximation centered at the resonance position of the bulk metal. When the particles are smaller, the line is
broader both to low and to high field, and no clear ‘surface region’ emerges in the spectra, see Fig. 54. Chemisorption
of a monolayer of hydrogen has a measurable, but small effect: it mainly shifts the whole line slightly upfield, see
Fig. 55. Preliminary relaxation data taken at 80 K indicate that T1 is somewhat shorter than in the bulk, even at the
‘bulk position’ in the spectrum.
In calculations for platinum [84] and palladium [178] it has been found that the Fermi-level electrons have a
lower density near the surface than in the bulk of the metal. This is the basis of the exponential-healing model for
the 195 Pt spectra. But a recent calculation for a rhodium slab finds that the local density of states at the Fermi energy
on surface sites is similar to or slightly higher than that of the bulk [190]. This probably is the reason why no surface
region can be identified in the NMR spectra. In another calculation [191] the effects of hydrogen chemisorption on
the Ef -LDOS on the fcc (1 1 1) surfaces of rhodium and palladium were compared. For Pd the earlier result [178]
that hydrogen causes a further drop in the LDOS was confirmed; but for Rh the effect on the surface Ef -LDOS was
found to be rather small, see Fig. 56. Assuming that Pt behaves as Pd, this figure can explain that the chemisorption
of hydrogen, which has a rather dramatic effect in the case of 195 Pt NMR as shown in Fig. 46, does not change the
103 Rh spectrum by much. Remarkably, the shift of the 1 H NMR line for chemisorbed hydrogen is much larger on
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
181
Fig. 54. 103 Rh point-by-point NMR spectra in three oxide-supported samples. Rh/TiO2 , with some of the largest particles over 6 nm diameter,
loading 10%; Rh/Al2 O3 with a fairly narrow distribution around 2.5 nm, loading 20%; Rh/Al2 O3 with a fairly narrow distribution around 1.5 nm,
loading 4%. Note that elimination of the large particles broadens the spectrum in a rather symmetric fashion, contrary to what is always seen
for small 195 Pt particles.
Rh than it is on Pt or Pd [11], which may correspond to the higher D(Ef ) values in the hydrogen layer on Rh in
Fig. 56.
It has been estimated [101] that the spin susceptibility of bulk rhodium is roughly one-tenth of that of palladium;
but also that the orbital susceptibility of Rh is twice that of Pd. In the exponential-healing model for 195 Pt NMR,
the site-to-site variation of the orbital susceptibility is neglected. The rhodium data suggest that such a neglect is
not justified in small-particle 103 Rh NMR. To simplify, we assume that the s-like contributions to the shift and
relaxation rate can be neglected [100]. The site-to-site variation of the relaxation rate is then determined by the
variation in Dd (Ef ). Experimentally, the change of T1 across the spectrum is not very large, and therefore the
site-to-site variation of the shift must be mainly determined by variations in the orbital susceptibility, Eq. (170).
This susceptibility depends on both occupied an unoccupied ranges of the local density of states, and apparently
varies as much among surface sites as between the surface layer and the deeper ones, so that no clear surface
Fig. 55. 103 Rh point-by-point NMR spectra for Rh/Al2 O3 -20% with clean and hydrogen covered surfaces. The changes are very small compared
to those in Fig. 46.
182
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 56. Layer resolved density of states (DOS) for hydrogen-covered seven-layer metal slabs of palladium and rhodium. Top: DOS on the
hydrogen layer; middle: on the metal surface layer; bottom: on the metal subsurface layer. (After Löber et al. [191]. ©1997 American Physical
Society).
region appears in the spectrum. The experimental fact that the spin–lattice relaxation rate at the ‘bulk position’ in
the spectrum is measured to be slightly faster than the bulk value then indicates that there are also 103 Rh nuclei
in the surface that resonate at this position in the spectrum. The calculations mentioned before suggest that the
surface-LDOS is slightly higher than that of the bulk, and therefore, the relaxation time is expected to be shorter.
6. Confined metal clusters and metal cluster compounds
In the previous section NMR experiments on small metal particles formed on substrates were central. One of
the (unwanted but inevitable) features of these particles is the presence of a size distribution. Chemistry offers
other routes to a large variety of clusters and solid state structures, that are better defined in number of atoms
[14,192–196]. Two particular examples of these systems are the so-called metal cluster compounds with cores,
that might exceed 561 metal atoms in size, and the alkali-metal atoms that are confined in cages (as in sodalites,
clathrates and zeolites). The regular arrangement of the filled ␤-cages in the zeolites and the random organization
of many of the metal cluster molecules in aggregates are new forms of solids that can be well studied by resonance
techniques.
6.1. Materials
6.1.1. Zeolites
Zeolites are aluminosilicates. A space-filling framework is built from (approximately) tetrahedral (SiO2 ) and
(AlO2 ) elementary units linked through the oxygens. The simplest structural element constructed from these tetrahedra is the sodalite cage, a truncated octahedron of about 0.9 nm diameter, with Si and Al atoms on its vertices,
see Fig. 57. Different types of zeolites are obtained by linking the sodalite cages indirectly (through oxygen atoms)
in different ways, usually so as to create large voids called the supercages in between them. These supercages
are interconnected by rather large pores or ‘channels’. If the sodalite cages are linked directly (through the metal
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
183
Fig. 57. Sodalite cage. The cage is the building block of zeolites. Alternating silicon and aluminum atoms at the vertices are interconnected by
oxygen atoms.
atoms), the resulting structure is called a sodalite (after a naturally occurring mineral of that name). Unless the
framework consists of pure silica, it will bear a negative charge. This is compensated by the presence of counterions
at well-defined sites in the structure. A very common counterion is Na+ . In addition to these indispensable counterions, additional ionic constituents, e.g. NaBr, and crystal water may be present, but they can usually be removed
by thermal treatment. The counterions can be changed by ion-exchange procedures. Such ion exchange is the usual
first step in the preparation of zeolite-encaged small particles of catalytically active transition metals, Pt, Pd, Rh.
Most methods of preparation of such catalysts do damage to the zeolite framework, and the particles are larger than
the volume of a supercage.
In this subsection, we will be concerned with alkali metal clusters, that are believed to leave the framework
intact. The oldest known such cluster is (Na4 )3+ , a group of four sodium ions with an additional electron, that gives
rise to a very characteristic 13-line ESR spectrum [197]. For the formation of these clusters in the sodalite cages
(and sometimes in the supercages) various treatments are used. Conceptually the simplest is the treatment with a
vapor of sodium atoms: the atom, equivalent to an ion plus an electron, enters the structure and teams up with three
sodium counterions to form the tetrasodium cluster. Larger units may be formed in the supercages; if they have
singlet electronic states their presence is not detected by ESR, but they might be seen by NMR. The least ambiguous
experimental results are those obtained with sodalite structures, where only one type of cavity is present. The filling
of the cages in zeolites and sodalites can be seen as the formation of sodium-or potassium based ‘cluster crystals’
which depending on the overlap of the wave functions might exhibit magnetism or metallicity [198].
6.1.2. Metal cluster compounds
In a metal cluster compound (MCM) a core or network of metal atoms is surrounded by nonmetallic ligands. One
variety (type A) of MCMs consist of a metal core in a cuboctahedral arrangement with a magic number of atoms
given by Eq. (238) with integer m: NT = 13, 55, 147, 309, . . . , see Figs. 58 and 59, and hence have a uniform
(monodisperse) size.
The different magic numbers are generated by adding successive new shells of atoms to the cluster core. The
cluster cores in these compounds are chemically stabilized by organic molecules bonded to the metal surface atoms.
184
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 58. Pt core of the metal cluster compound Pt309. The phen∗ ligands (right figure) are attached to the black Pt atoms, while oxygens are
bonded to the dark gray sites.
These ligand molecules isolate the particles from each other; without them the particles would coalesce into bulk
metal. The great majority of the type A compounds form noncrystalline solids, that is the cluster molecules are
packed densely, but with only short-range order, like in a glass. The chemical bonds of the ligands with the metal
atoms at the surface of the cluster core will influence the surface atoms and possibly also the atoms inside the
metal core, so that these influences need to be assessed. Many different metal cluster compounds are synthesized
with metal cores of various metals like Pt, Au, Pd, Ni, Fe and with different ligands attached to the metal cores.
For an overview the reader is referred to references [194,199]. In this section we focus on measurements on two
metal cluster compounds synthesized by Schmid and coworkers, Pt309 Phen∗36 O30 and Pt55 [As(t-Bu)3 ]12 Cl20 [194],
respectively, abbreviated as Pt309 and Pt55.
There is a second class (type B) of metal cluster compounds, that is especially successful to obtain monometallic
(e.g. osmium, nickel, platinum) or bimetallic cores (e.g Cu/Fe, Pt/Ni) organized in a crystalline lattice [199]. For
example, in [HNi38 Pt6 (CO)48 ]5− [192], the metal cluster framework is a penta-anion. The crystal contains two
crystallographically independent anionic clusters in the unit cell: one in the (0,0,1/2) and one in the (1/2,0,0)
position. Because the monoclinic cell is pseudotetragonal (β = 91.1◦ is close to 90◦ ; a = 3.093 nm and b =
3.188 nm have almost the same length) the anions are packed in a quasi-tetragonal (body centered) fashion in a
cell having a 0 ≈ b0 ≈ 2.23 nm and c = 1.90 nm. The structure of the Ni38 Pt6 anion, which is also present in the
analogous compound (TEA)5 [HNi38 Pt6 (CO)48 ], is shown in Fig. 60. It is not a perfect octahedron as the faces of
the outer nickel octahedron are bent slightly outwards. This is because the platinum atoms are slightly larger than
Fig. 59. Pt core of the metal cluster compound Pt55 and the As(t-Bu)3 ligand. Most of the outer Pt atoms are bonded to either As(t-Bu)3 ligands
or Cl atoms.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
185
Fig. 60. Schematic drawing of the metal core of the metal cluster compound [Pt6 Ni38 H(CO)48 ]5− [N(C2 H5 )4 ]+
5 . The dark spheres are Pt atoms
and the light spheres are Ni atoms.
the nickel atoms. X-Ray studies [192] reveal that the platinum-platinum distance is 0.2719 nm (in bulk platinum
0.277 nm). The nickel-platinum distances vary between 0.2510 nm and 0.2764 nm, and the nickel-nickel distances
between 0.2393 nm and 0.2762 nm (in bulk nickel 0.249 nm). Hence there is only one platinum position.
The synthesis of the metal cluster compounds of type A is performed by reduction of a metal compound to metal
atoms. When small metal particles are synthesized by reduction of a salt with hydrogen (i.e. H2 PtCl6 ), there will be a
size distribution in the resulting particles. If metal cations in solution are reduced to metal atoms almost immediately
polycrystalline metallic precipitates are formed. To stop this quick crystal growth and end up with mono-crystalline
metallic particles, the particles must be trapped by appropriate ligand molecules. The concentration of ligand
molecules plays a key role in this process. When it is too high, mononuclear complexes are formed, and when it
is too low one obtains bulk metal. The choice of the ligands also controls the solubility of the ligand-stabilized
clusters. Another important aspect is the formation of full-shell clusters. They are more stable (lower energy) than
the nonfull-shell clusters. Full-shell cores consist of a cubic or hexagonal closed packed structure with a closed
outer geometry and have magic numbers of atoms, 13, 55, 147, 309, . . . . To synthesize a specific full-shell cluster
not only the ligand concentration but also the ligand size and geometry, the ligand strength, and the kinetic and
thermodynamic parameters are important [194]. Although the formation of ligand stabilized clusters depends on
many parameters, it is possible to control these parameters and synthesize ligand stabilized clusters having all the
same magic number of metal atoms.
The two shell clusters M55 L12 Clx , M =Au, Pt, Rh, Ru; L =PR3 , AsR3 ; R =alkyl group; x = 6, 20 are synthesized
in the above way. Metal salts are reduced in solution in the presence of appropriate ligands L. However, as these
ligands have an organic nature, the reduction is carried out in organic solvents and with B2 H6 as a reducer instead
of H2 . B2 H6 also binds excess of ligands as H3 B-L adducts, so that the concentration of L is kept low. For instance,
if (C6 H5 )3 PAuCl is reduced in benzene or toluene solution by diborane, Au55 [(C6 H5 )3 P]12 can be isolated as an
air-stable brown-colored solid in about 20% yield. It is soluble in dichloromethane and in THF (tetrahydrofuran).
Other two-shell clusters with Pt (like Pt55), Rh or Ru are extremely air sensitive and are available only in very
low yields. The larger metal cluster compound Pt309 is synthesized by reducing Pt(II)acetate by hydrogen in the
186
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
presence of phen∗ (phen∗ is a 1,10-phenanthroline derivative). By subsequent cautious oxidation, the four shell
cluster Pt309 is formed as a water-soluble, black powder. Recently nanocrystallites of various sizes of Au cores
up to 3.5 nm have been synthesized by passivation by straight-chain alkylthiolate molecules [195] and the Au55
clusters can now also be made in crystalline form [14].
Mixed Ni-Pt metal clusters (type B) are synthesized by an oxidative condensation reaction without incorporation of the oxidizing agent. The reaction of [Ni6 (CO)12 ]2− with PtCl2 or K2 PtCl4 in acetonitrile results in
the synthesis of cluster anions having the general formula [H6−n Ni38 Pt6 (CO)48 ]n− with n = 4, 5 [192,200] or
[H4−n Ni9 Pt3 (CO)21 ]n− with n = 3, 4.
6.1.3. Related cluster compounds
Crystals containing very large metal cluster anions of [Al77 R20 ]2− where R is N(SiMe3 )2 can be formed [201].
These clusters are prepared by slowly evaporating a solution of aluminum iodide in a solvent mixture of toluene
and diethylether added to a solution of base-free LiN(SiMe3 )2 [201]. They are more metal-like than the even
larger slat-like structures such as Cu146 Se73 [193]. In the latter compounds the metal atoms form a network with
metal-metal bonds, that differ from those in simple bulk metals. We will not cover this subject here. NMR and µSR
have also be proven to be valuable tools in another interesting field of cluster physics, i.e. mesoscopic magnetic
molecular clusters, like Mn12 O12 acetate [202]. This topic will not be discussed either.
6.2. Physical properties of confined metal particles and clusters
The intracrystaline channels and pores in open structures like zeolites (aluminosilicates) [203] are not only of
practical use as sieves or in catalysis, but are also used as host for clusters of guest molecules in the so-called
supercages. For a review of the preparation of such systems, see [204]. The uniform pore size helps to narrow
the size distribution. The smaller sodalite cages are often filled with one alkali atom only. The distances between
the filled cavities can be small enough to couple the resultant alkali-clusters inside the cavities in such a unique
way, that new physics appears [205]. Below we summarize the main results obtained in K and Na clusters in the
faujasite-structure of zeolite-X and -Y, and in sodalites. We also give some results of the clusters formed in the
supercages.
We have seen above that the various routes to cluster molecules lead to a packing of metal cluster cores that can
be random or regular. The physical properties are expected to show the typical signatures of mesoscopic physics,
the subject of Section 4. NMR is very suited as experimental technique as the line width is sensitive for the electron
density and the relaxation rates for the energy splittings. We will use Pt309 as paradigm to illustrate these aspects.
6.2.1. 23 Na of the faujasite-structure zeolite-Y
Several groups have studied the 23 Na of the faujasite-structure zeolite-Y. The Y indicates a relatively high Si/Al
ratio in the framework; the unit cell composition for the dehydrated form is typically close to Na56 (AlO2 )56 (SiO2 )136 ,
abbreviated (Na+ )56 Y56− . There are eight sodalite cages (called β-cages to distinguish them from the α-supercage)
in a faujasite unit cell. In theory a nominal sodium metal concentration of 8 per unit cell is sufficient to place an Na3+
4
unit in each sodalite (three Na atoms initially there and the fourth from the absorbed atom) and form a perfect cluster
56− . Inevitable slight uncertainties in sample composition are particularly important at
crystal: (Na+ )32 (Na3+
4 )8 Y
low Na concentrations. In the analysis of their ESR data Woodall et al. [198] solved most of the discrepancies
+
56− ,
found in the past [197,206–208]. At low doping first Na3+
4 paramagnetic clusters are formed. For Na8 (Na )56 Y
3+
75% of these clusters reside in the sodalite cages. The ESR susceptibility shows one spin per Na4 cluster, and
the interactions between the clusters are sufficiently strong to wash out the hyperfine splitting pattern. Powder
neutron diffraction does not show the presence of a second cluster. Electrons released above or below this filling
fraction seem to be strongly antiferromagnetically coupled into a singlet state. Incomplete filling together with
this interaction explains the controversial data observed in the past. The metallic Knight shifts observed in some
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
187
Fig. 61. Inverse paramagnetic shift of the main 27 Al resonance in sodium-electro-sodalite as function of temperature. The straight line is a
Curie–Weiss fit with 2 = −178 ± 8 K to the data, and is indicative of the strong antiferromagnetic interaction in the paramagnetic phase. (After
Srdanov et al., [211]. ©1998 American Physical Society).
experiments [206] are most likely explained by the formation of aggregates. Similar features are to be found in the
sodalites, discussed below.
6.2.2. Sodalites
Essentially the same Na3+
4 cluster has been identified in sodalites (the so-called black sodalites [205]). The ideal
−
composition of a unit cell of dehydrated sodalite with one tetrasodium cluster per cavity is (Na3+
4 )2 (SiO2 )6 (AlO2 )6 .
3+
The Na4 ionic clusters form a bcc lattice with a band gap of 0.7 eV optical gap [209]). In samples with composition
close (73% of cavities occupied) to the ideal one, 27 Al and 29 Si MAS-NMR found signals shifted by 70 ppm to low
field with respect to the resonances in unloaded sodalite. As under MAS only isotropic shifts survive, the interaction
must be an isotropic transferred hyperfine interaction [210]. The wave function, which has a predominant s-character,
is centered in the sodalite cage and extends beyond the cage boundaries. The 27 Al and 23 Si NMR data, and also
the ESR and susceptibility measurements show that the strong exchange coupling between the unpaired electrons
leads to an antiferromagnetic transition around 48 K [211], see Fig. 61. Although LDA bandstructure calculations
allow for a metallic band [209] of the Na4 clusters in the wide aluminosilicate bandgap, more refined self consistent
LSDA calculations show the system to change from a metal to an antiferromagnetic insulator when spin ordering
is allowed. Correlation effects are apparently strong enough that metallic behavior is absent. The absence of the
sodium resonance in these cubic sodalites (there is only one crystallographic Na site, of which the resonance is
expected to be shifted by 0.1%), might be due to electron mediated spin–lattice relaxation.
6.2.3. Si-Na clathrates
Si-Na clathrates (inclusion compounds) are built of Si polyhedra containing 20–28 atoms. The crystallographic
structure is similar to zeolites [212–214]. They also resemble the alkali-doped C60 [215,214], of which some
show a insulator-metal transition upon increase of the alkali content, and superconductivity in the metallic phase
(Rb3 C60 has a superconducting transition around 30 K). Silicon clathrates codoped with Ba: Na2 Ba6 Si46 indeed
show superconductivity in the sp3 covalent network of silicon [216,217], see Fig. 62.
The NMR of 29 Si, 23 Na, 137 Ba, and 135 Ba indicate that the electron density in the compound is strongly site
dependent; all sites show Korringa behavior, see Fig. 62. Gryko et al. [218] measured the temperature and frequency
dependence of 23 Na in Si136 Nax , Si46 Nax and Si46 (Na,Ba)x with 7 ≤ x ≤ 9. Typical shifts are of the order of
0.1% (compared to 0.113% in Na-metal) and are interpreted as Knight shifts due to delocalized electrons. In Na
compounds without Ba, the Knight shift has an activated behavior that contrasts with the T -independent K-values in
normal metals. The T -dependence might be due to structure in the density of states at the Fermi level, as confirmed
by band structure calculations [219], which show that the material is a narrow band semiconductor or semimetal.
ESR and susceptibility data in Si136 Nax with 1.3 < x < 22 and Si46 Na8 [215] give a nice indication of the
metal–insulator transition. Due to the formation of an impurity band of clusters of neighboring Na atoms the ESR
lines that are well resolved at low doping disappear at higher doping.
188
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 62. (a) Temperature dependence of the resistivity of a pressed powder of (Na,Ba)x Si46 near the critical temperature. (After Kawaji et al.
[216]. ©1995 American Physical Society). (b) The temperature dependence of the nuclear relaxation rates of 29 Si, 23 Na, and 137 Ba for three
different Si-sites. The solid line corresponds to constant T1 T . (After Shimizu et al. [217]. ©1996 American Physical Society).
6.2.4. Clusters in zeolite supercages
In a highly loaded (15.8% by weight according to atomic absorption spectroscopy) sample of platinum in
zeolite-X of reasonably high dispersion (0.65 according to TEM), the 195 Pt spin–lattice relaxation at the position 1.10 G kHz−1 was studied between 22 and 250 K [182]. From 250 K down to 80 K the relaxation curves can
very well be normalized by time/temperature scaling (for this method of data treatment, see Eq. (242)). Below 80 K
only a temperature-dependent fraction of the experimental relaxation curve can still be normalized this way. At
22 K, only about 1/3 of the nuclei resonating at this position still show the ‘metallic’ spin–lattice relaxation. As
mentioned before, there is no detectable change in the NMR spectrum when the nonmetallic relaxation behavior
sets in. This is not well understood, but has been experimentally observed in the 195 Pt NMR of other systems as
well, see Section 6.3. Samples of this dispersion show virtually no signal at the bulk resonance position, since in
terms of the NMR layer model (see Section 5.3.1) less than 1% of the atoms is below layer 2 (the sub-subsurface
layer). The scaling behavior of the relaxation could be studied below 83 K at one spectral position, 1.091 G kHz−1 ,
where only surface atoms should resonate (the layer model supposes that the maximum of the surface peak is at
1.10 G kHz−1 , but there is some signal from the subsurface layer at this frequency), and another, 1.110 G kHz−1 ,
where the contribution from the subsurface layer should be more important. At both frequencies, roughly the same
1/3 of the signal has metallic relaxation at 22 K. This suggests that there are two classes of particles: those that have
at 22 K metallic relaxation on all sites, and those that have no metallic relaxation, see Fig. 63. It is then reasonable
Fig. 63. Time-temperature-scaled relaxation data for three clean-surface PtNaY-samples, taken at spectral position 1.100 G Hz−1 . The full curve
in all panels is a double exponential fit to the 80 K data, see also Fig. 43 and Fig. 50. (After Tong et al. [182]. ©1995 American Physical Society).
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
189
to think that the metallic particles must be the larger ones, and from the size distribution it is calculated that their
diameter is 1.6 nm or more. Similar results were found for samples in zeolite-Y with dispersions between 0.56 and
0.77; but the relaxation of oxide-supported platinum follows time/temperature scaling down to 20 K. The difference
between relaxation behavior in oxide-supported and in zeolite-encaged particles is probably related to the fact that
particles sit on the surface of the oxide, but are enclosed in the matrix of the zeolite. Although most of the particles
in the zeolite do not fit in an undamaged supercage, they may still fit rather snugly inside the cavity created by
the local partial collapse of the aluminosilicate network. On the other hand, one might argue that the results on
oxide-supported platinum are the unusual ones, not those on zeolite-encaged particles. Indeed, at some point during
the lowering of the temperature one might expect to start to see the effect of the discreteness of the electronic levels.
Therefore, one has to find a mechanism that could wash out the level discreteness in the oxide samples, but not in
the zeolite samples.
For the 195 Pt the same vibrations of the particle surface may be important that explain the temperature dependence
of the 109 Ag line width in small silver particles (Section 5.2). In particular, a nearly-free supported particle can
execute ‘breathing’ vibrations, whereby the volume of the particle varies in time. This is a phonon-like motion, and
the electron distribution can adjust very rapidly on this timescale, so as to be at any moment in equilibrium with the
instantaneous configuration of the nuclei. As a simplification one might consider the particle as an elastic continuum
to describe the vibrations, and as a free electron gas inside this vibrating membrane to calculate an effective density
of states. If the instantaneous volume per electron in the free electron gas is  and if as usual (see Section 3.2) rs
is defined through  = (4/3)πrs3 then the instantaneous Fermi energy is Ef = c(rs /a0 )−2 with c = 8.03 × 10−18
J. On the other hand, the usual estimate of the level splitting in a particle containing N electrons is (see Eq. (195))
δ = 2/ND(Ef ). If under influence of the vibration the radius of the particle varies between
R + 21 1R
and
R − 21 1R,
the electron density parameter will vary between
rs + 21 1rs
and
rs − 21 1rs ,
with
1R
1rs
=
rs
R
and
R = N 1/3 rs .
It is expected that washing-out of the level structure will occur if the variation in Fermi energy that corresponds to
the variation in rs is larger than the level splitting δ:
Ef (rs − 21 1rs ) − Ef (rs + 21 1rs ) ≥
2
ND(Ef )
(243)
which leads to the condition 1R/rs ≥ (2/3)N −2/3 . For N = 100, the requirement is that the amplitude of the
surface motion is about 3% of an atomic radius; in bulk gold the atomic rms displacements are estimated to satisfy
the requirement above 100 K; and it is well known that the amplitude of thermal motion in a surface is larger than
that in the bulk. Such a vibration mechanism could, therefore, explain the absence of discrete-level effects above
20 K in particles with essentially ‘free’ surfaces, such as those anchored on oxides. On the other hand, one could
suppose that ‘encaged’ particles are mechanically clamped, so that a higher temperature is needed to have a sufficient
amplitude of these breathing modes to average out the discrete level structure.
The loss of metallic behavior due to quantum size effects have also been observed for other metals: K-clusters
in the supercages show ferromagnetic properties [220], and the ESR intensity of Cs-clusters in zeolite-X follows
Curie–Weiss behavior [221].
190
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
6.3. Physical properties of metal cluster aggregates, Pt309 as NMR-paradigm
The availability of various (nm) diameters of the same element makes metal cluster compounds of type A
particularly suited to demonstrate the difference between the properties of nanometer-scale metallic particles and
those of the bulk. Due to the presence of the ligand shells high densities of metal particles are possible without
danger of coalescence, enabling NMR experiments with good sensitivity even though the resonance lines are very
broad. These aggregates of randomly packed cluster molecules form a new kind of solid, where quantum size effects
and mesoscopic density variations dominate the physical properties at low temperatures [160].
Although the ligands are beneficial for the separation of the metal cores, they will influence the properties of
the core-surface atoms and possibly also of deeper lying atoms. Mössbauer and susceptibility data show that the
ligands only affect the surface atoms and that the inner core atoms behave as small pieces of metal [199,222–224].
For these inner cores, quantum size effects were nicely demonstrated to play an important role in a series of Pd
clusters of nm dimension. By combining susceptibility and specific heat data below 4 K the effect of odd–even
numbers of electrons could be demonstrated to be the origin of the upturn in the susceptibility below kT /δ and
in the electronic part of the specific heat to give rise to a T 2 instead of a T term [225]. It is also shown that
although due to the confinement in the metal cluster cores below 20 K intracore phonons are no longer excited,
ligand and intercluster vibrations give rise to a large contribution to the specific heat and are responsible for the
thermal equilibrium within the electronic spin system [225–230]. Because of the irregular packing and the presence
of an insulating ligand layer, charge transport in type A clusters proceeds by hopping, and has many features in
common with other disordered systems, such as cermets, conductor/nonconductor composites and doped polymers
[231]. The time scale of the intercluster electron exchange can be estimated from dielectric measurements [232]
on Au55; the typical hop between two nearby Au55-clusters appears to be of the order of 10−10 s (for Pt309 the
time scale will be somewhat shorter due to the higher DC-conductivity values). Extrapolating these data to low
temperatures shows that the electronic inter- as well as intracluster transitions should involve a sufficient number
of different cluster cores on the time scale of the NMR relaxation and line width experiment to provide a source
for the broadening of the energy levels. Relaxation measurements of 195 Pt in aggregates of cluster molecules
with the same Pt309 core, but larger ligands, have confirmed the importance of broadening due to intercluster
processes.
The mesoscopic multiple scattering model for the NMR spectrum of small metal particles predicts a temperaturedependent broadening of the line profile (see Section 4.2) and the hypothesis of discrete energy levels, broadened
by lifetime effects due to interparticle electron jumps, leads to a lack of time/temperature scaling in the spin–lattice
relaxation (see Section 4.4). The dimensions of the metal core (2.1 nm diameter) and the packing of the cluster
molecules in the compound [194,199] Pt309 Phen∗36 O30 (abbreviated as Pt309) are well suited to test these predictions.
For this 309 atom particle the mean energy splitting δ at Ef can be estimated from the simple relation δ ∼ Ef /N, with
N the number of free electrons in the enclosed volume, or by using the bulk density of states per atom (1.55 × 10−4
states/K atom); the resulting values are, respectively, 60 K and 40 K [10,233].
In Fig. 64 representative data for the line shape are shown for a number of temperatures. Like in other studies
on Pt particles [90,151,234], the width is seen to be very broad and T -independent. The MHz width of these line
profiles are in sharp contrast to those found in bulk platinum metal (about 30 kHz in 9.4 T) or in simple chemical
compounds (a few kHz) and are typical for small metal particles. The chemical compounds listed in Table 1 have
their resonances at B/ν < 0.109923 T MHz−1 (H2 PtI6 ). The zero of the shift scale is at 0.109963 T MHz−1 . At low
temperatures, the bulk Pt resonance occurs at 0.11388 T MHz−1 ; it moves to 0.11323 T MHz−1 at room temperature.
The Fig. 65 shows the spin–lattice relaxation at 85.55 MHz in 9.4 T as function of temperature for the cluster
compound Pt309 for temperatures higher than 80 K, i.e. higher than the expected average level spacing. The value
found here and in Fig. 47 (for a clean catalyst surface) agree within 20%. Because the line width is much broader
than the bandwidth of the π/2 pulse, the frequency at which the relaxation has been measured must be specified.
The frequency dependence of T1 at 80 K is shown in Fig. 66. At all scanned frequencies above 65 K the relaxation is
single-exponential and T1 T is constant. Close to the reference position (at 0.109 T MHz−1 ) a second Pt line with a
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
191
Fig. 64. The line intensity of the cluster compound Pt309 for different temperatures measured by frequency and field sweeps at B=9.4 T. Drawn
and dashed lines are fits discussed in the text. The possible influence of circuit retuning needed during these sweeps is checked by performing a
field sweep at 67.65 MHz and 70 K. When plotted vs. B/ν all data coincide. Intensity calibration shows that more than 90% of the 195 Pt nuclei
participate in the signal.
Fig. 65. Relaxation time for Pt309 in 9.4 T as function of the inverse temperature at B/ν = 0.110 T MHz−1 (65 K < T < 165 K). The solid line
shows T1 T =80.6 ms K; for bulk platinum at room temperature K=-2.92% and T1 T = 33 ms K.
Fig. 66. Relaxation time for Pt309 as function of B/ν at 80 K. The solid line is a fit based on the full Korringa relation [90], see text.
192
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
much longer relaxation time appears, which is due to Pt-atoms that are chemically strongly bound to ligand groups
[233,235]. Here we neglect this second resonance.
As explained in Section 4.4, the exponential-healing models [90,151] analyze small-particle NMR data using
Eq. (187)–(190), assuming that only the densities of states vary from site to site. From the combination of Knight
shift and relaxation time, the s-and d-like density of states can be computed. The single exponentiality of the
magnetization recoveries means that at different frequencies only one combination of s-and d-density of states is
involved. The solid line in Fig. 66 is a fit that takes only a site dependence of the d-part of the Knight shift into
account [90], while the s-part is uniform. (An equally good fit can be obtained by including a healing length for
the s-electron density too [151]). The constant value of Ds (Ef ) taken in the fit is 0.28 states eV−1 at.−1 , close to
the bulk value of 0.30 states eV−1 at.−1 (see Table 7). The bulk value of Dd (Ef ) (reached far from the surface) is 5
times larger than Ds (Ef ).
In the calculation of the line shape L(ν) as a function of ν the Knight shift K has to be weighted by the number
of nuclei having that particular shift. This can be done in different ways. For the NMR layer model, similar to
exponential healing, results of calculations have been illustrated in Fig. 41. The multiple scattering model gives
in the case of the unitary ensemble a simple zero-Kelvin expression for L(ν), Eq. (208), which depends only
on one parameter α, which is proportional to the ratio of the level broadening γ and the mean level splitting δ
[155,156]; for the orthogonal ensemble only small changes are expected. (Going from the unitary to the orthogonal
ensemble changes the second moment of the line shape at most by a factor of two [157]). With increasing T the
resonance profile is predicted to shift and narrow to the bulk position and shape. Although the measured line shape
can be well fitted by the mesoscopic expression (the drawn line in Fig. 64), the expected temperature dependence
is absent. Above 65 K, where according to Fig. 65 the Korringa relation holds, the Pt309 data show that the line
shape is still broad. Calculating L(ν) with the exponential-healing parameters that were derived from the frequency
dependence of the relaxation rates in this Korringa regime, L(ν) is well reproduced [90,151,233] (dashed line in
Fig. 64). In this temperature range, the densities of states are smooth functions of energy, and the surface effects are
better represented by exponential healing than by multiple scattering. Incorporating additional surface effects in the
multiple scattering model by considering the different binding situation, as in Eq. (211), shows [158] that the latter
effect may indeed overshadow the other mesoscopic features. The conclusion is that at temperatures higher than
65 K the spin–lattice relaxation rate and Knight shift of the Pt-cores behave as in bulk Pt, but with a reduced and
site dependent d-density of states. Up to this point, the Pt309 results are similar to those on other small Pt particles
discussed in Section 5.3 (but compare the discussion of platinum particles in zeolites in Section 6.2).
These common features become different below 65 K. Fig. 67 shows the recovery of the nuclear magnetization
as function of the time τ times the temperature T , for Pt309 at various temperatures down to 5 K. By scaling the
time as in Eq. (242), the recovery curves fall on top of each other in the Korringa regime (T > 65 K; the 80 K data
are given as reference). Non-exponentiality is observed to start below about 50 K and becomes strongly pronounced
below 10 K. In contrast to Korringa behavior, the scaled curves are now all temperature specific. While at 10 K
equilibrium could be reached in an hour, at 5 K it is expected that the magnetization needs at least three orders
of magnitude more for its return to the thermal equilibrium value M0 . Therefore, at 5 K, M0 is determined from
equilibrium data taken at 10 K by assuming that M0 will obey the Curie law. For an explanation of the loss of
time-temperature scaling in Fig. 67 at 15 K and below we note that the expected average level splitting in the Pt-core
and the observed cross-over temperature are around 40 K. When thermal broadening is no longer sufficient to have
a quasi-continuous density of states at Ef , the differences in energy splitting become manifest, similar to what is
seen in Fig. 63. The fits in Fig. 67 are based on a generalization of Eq. (227), written as
T1−1 (T , 1) = A(T )
γ (T , 1)
γ 2 (T , 1) + 412
,
(244)
where A(T ) is a slightly temperature-dependent proportionality constant, independent of 1, and γ (T , 1) is of the
form given in Eq. (228). The lines in Fig. 67 are of the form given by Eq. (235):
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
193
Fig. 67. Recovery of the magnetization as function of τ times temperature for the cluster compound Pt309 and fits with Eq. (245) for the
orthogonal level distribution. (After Fritschij et al. [160]. ©1999 American Physical Society).
Z
M(t) =
M0 1 − exp
−t
T1 (T , 1)
P (1) d1
(245)
with P (1) the orthogonal distribution function given by Eq. (197) [160]. As the error bars in the fitting parameters
are rather large due to the exponential dependences, various parameter sets work equally well.
These results show that in aggregates of Pt309 clusters the nuclear spin–lattice relaxation rate obeys the same
Korringa relation as in bulk metal down to 65 K. Around 15 K the recovery curves of M(t) no longer follow
time-temperature scaling. This temperature is indeed below the estimated average energy gap at Ef for these
clusters. The nonexponential recoveries show that within the Pt cluster assembly the energy splittings vary from
core to core. Local variations of the electron densities in the Pt cores enable the observation of particles with small
and large energy splittings at the same Knight shift (electron exchange allows a grand-canonical treatment of the
ensemble). The distribution of the splittings is well described by the orthogonal ensemble if the thermal broadening
of the energy levels is taken into account. In contrast to the relaxation behavior the overall line shape is found to be
insensitive to the spiky structure of the density of states in a given particle: the spectra are temperature-independent.
This finding is not well explained yet.
6.4. Resonance properties of other cluster compounds and colloids
We selected Pt309 as paradigm for what can be expected for the physics in a typical cluster molecule of sufficient
size, as it is monodisperse, well characterized by a variety of techniques and NMR has contributed a unique element
to its understanding. Below we discuss other cluster compounds which are analyzed by NMR. In some of these the
chemistry leads to some ambiguity, in others the cluster cores are too small to be in the mesoscopic regime. We also
mention the results obtained on a semiconducting core to illustrate that the composition of the cores does not need
to be necessarily of one element but can be that of an alloy or a semiconductor. In the CdSe example given, the core
is covered by covalently bonded organic ligands, a technique that is often used for larger particles (colloids) and
leads to some size distribution.
6.4.1. Pt55 cluster compounds
The metal core of Pt55 [As(t-Bu)3 ]12 Cl20 , see Fig. 59, has a diameter of 1.2 nm. The mean level spacing estimated
for such a small particle is about 150 K. Fig. 68 shows the NMR signal at 300 K and 30 K corrected for T2 effects and
194
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 68. Pt55 line shape at 30 and 300 K. The drawn line is a Gaussian fit to the data, the dashed line is the predicted line shape for the
tunneling/localization model [155–157]. The experimental line shape is T -independent.
the predicted line shapes for the two models. (It is obvious that Pt55 is too small to be described by a spherical particle
with a continuous reduced Knight shift at the surface. Also surface effects due to the ligands should play a very
important role). Although both models fit the line shape reasonably well, the absence of a temperature dependence
of the line shape and position is in disagreement with the predictions of the multiple scattering model. Calibration
measurements show that only about 20% of the Pt nuclei contribute to the final signal, even at room temperature.
This result is not yet explained. At 150 K the relaxation recovery at 85.5 MHz is extremely long, T1 T = 2 × 103 s K,
a factor 104 longer than in Pt309 at the same temperature and frequency.
6.4.2. Crystalline Ni38Pt6 cluster compounds
The compound containing the ion [HNi38 Pt6 (CO)48 ]5− , abbreviated Pt6Ni38, is a nonmagnetic material [222,223].
The core consists of 44 atoms, which makes it the largest metal atom core that has been realized in a single crystal.
Besides 38 Ni-atoms the core contains six crystallographically equivalent Pt atoms (see Fig. 60). There are three
different Ni positions when looking at the isolated metal core, but after including the attachment of the CO ligands
there are six different Ni sites. Since the Pt sites are equivalent one would expect that surface effects can only shift
the Pt-NMR line position, but broadening should be excluded. For the mesoscopic multiple scattering model the line
broadening mechanism will still work as long as the distance between the Pt6Ni38 cores does not exclude electron
tunneling (slight variations in core properties e.g. due to ligand effects or packing still have to be present). The samples
indeed show a moderate DC-conductivity σDC around room temperature of σDC = 10−2 exp(−3×103 /T )(cm)−1
[231].
In the NMR signal less than half of the 195 Pt nuclei participate, which indicates interchange of surface nickel and
inner core platinum atoms. The large line width (see Fig. 69, ) can be explained by the mesoscopic charge density
variations in and between the metal cores (about 1.7 nm apart), but more likely by the variation of the charge density
and/or the presence of internal magnetic fields due to the interchange effects [236]. The relaxation rates in Pt6Ni38
are larger than in the comparable compound Pt55 (at 20 K the stretched exponential magnetization recovery reaches
its 1/e-value in 2.5 s). The temperature dependence (T1 T is constant above 10 K) and field dependence of T1 are
typical for tunneling between two level systems [236].
Line shapes in Pt38 (CO)44 H2 [N(PPh3 )2 ]2 and [Pt26 (CO)32 H2 ][PPh4 ]2 [237] resemble those of other Pt-particles
of similar core size. Relaxation times were non-Korringa like and are likely influenced by the presence of paramagnetic spins, e.g. generated by a partial oxidation.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
195
Fig. 69. 195 Pt NMR line width of the Ni38Pt6 metal cluster compound at 20 K, measured with a frequency and field sweep.
When compared to Pt309, these other metal cluster compounds all show similar line shape features. The interpretation of the relaxation rates is less unambiguous due to the presence of paramagnetic or magnetic centers which
is not under control. As the chemistry of cluster molecules is still leading to new compounds, one might expect to
follow the mesoscopic NMR effects as function of core size of the same element. Depending on the compounds
made, also other topics might become in reach. For example, if superconducting cores could be manufactured,
correlation and size effects in superconductivity are also accessible.
6.4.3. Osmium cluster compounds
Carbon-centered metal-carbonyl clusters, which are formed by a range of transition metals, are composed of
discrete clusters of metal atoms enveloped by ligands [238]. Cluster geometries include close packed structures
of up to more than 40 metal atoms. Precise measurement of the chemical shift [239] of the central carbon gives
valuable information about the valence state of the ion. In some of these clusters hydrogen is present as an interstitial
[240]. This will have an influence on those physical properties which depend on the number of the electrons in the
cluster, such as the shape of the ESR line. In H2 Os10 C(CO)24 [241] a symmetrical almost exactly Lorentzian line
shape was observed in the X-band below 100 K. Above this temperature the response is asymmetric and resembles
a Dysonian form, associated with electrical conducting samples thicker than the microwave skin depth. Hydrogen
motion has been suggested as origin. Very peculiar for this even-electron molecular cluster is the occurrence of
paramagnetism below 100 K, seen both in the ESR and susceptibility data. Such a behavior is also seen in other
even-electron clusters like those of Pd. The presence of hydrogen in the Pd-core might play a crucial role, but it
cannot easily be detected by a direct measurement. Variation of the H-content from cluster to cluster will turn some
of them even and others odd in electron count.
6.4.4. Semiconductor molecular colloids
Quantum confinement is responsible for the colors of nanometer sized cadmium selenide crystallites. The larger
crystallites can absorb lower energy photons and so appear red, whereas the smallest absorb only higher energy
quanta and so appear yellow [242]. Thayer et al. [243] prepared three samples of CdSe coated by covalently attached
organic ligands and performed 77 Se NMR. The sizes varied from 1.2, via 1.5–1.8 to 3.0–3.5 nm diameters. As in
the other small metal particles discussed so far, these NMR studies show the line width to gradually shift away from
the bulk position (taken as 0 ppm) with lower particle sizes, see Fig. 70.
The variety and sizes of metal cluster compounds are still expanding [14,195,196]. The new compounds help in
our understanding of what happens when particle sizes become smaller and smaller as needed in the information
technology. They might also be of direct relevance in this field as entities, that can be addressed directly. Here we
196
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
Fig. 70. Sketch of the 77 Se NMR spectra of three sizes of CdSe-R (R =phenyl, butyl) clusters in pyridine, and bulk CdSe. On the horizontal axis
the frequency shift is given in parts per million. (After Thayer et al. [243]. ©1988 American Physical Society).
have shown that the effects of surfaces on the electron densities in the particles are crucial for an explanation of the
experimental resonance data. The predicted mesoscopic effects in the line shape due to variations in the electron
density are absent. So far, only in the temperature dependence of the relaxation rate indications for the importance
of level statistics and density variations are found.
Acknowledgements
We thank the following colleagues for kindly allowing us to use material from their publications: Detlef Brinkmann,
Michael Duncan, Hubert Ebert, Uri El-Hanany, Günther Engelhardt, Arthur Freeman, R.G. Goodrich, Dieter Hechtfischer, O. Kanert, T. Kohara, B. Mühlschlegel, Horacio Pastawski, Charles Slichter, Yuye Tong, David Tunstall,
Bill Warren, Yue Wu, S. Yamanaka. Special thanks are due to Jos de Jongh and Franco Fritschij for their contribution
to the chapters on metal cluster molecules. The NMR group in Lausanne is partially funded by the Swiss National
Science Foundation, recently under Grant 20-53637.98.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
A. Abragam, Principles of Nuclear Magnetism, Oxford University Press, Oxford, 1961/1986.
C.P. Slichter, Principles of Magnetic Resonance, Springer, Berlin, 1990.
J. Winter, Magnetic Resonance in Metals, Oxford University Press, Oxford, 1970.
P. Fulde, Electron Correlations in Molecules and Solids, Springer, Berlin, 1993.
A.W. Lovesey, Condensed Matter Physics. Dynamic Correlations, Benjamin/Cummings, Menlo Park, CA, 1980.
D. Pines, Z. Phys. B 103 (1997) 129.
C.W.J. Beenakker, H. van Houten, Solid State Phys. 44 (1991) 1.
Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, Oxford, 1997.
J.A.A.J. Perenboom, P. Wyder, F. Meier, Physics Reports 78 (1981) 173.
W.P. Halperin, Rev. Mod. Phys. 58 (1986) 533.
J.J. van der Klink, Adv. Cat. 44 (1999) 1.
P.A. Cox, The Electronic Structure and Chemistry of Solids, Oxford University Press, Oxford, 1987.
J.K. Burdett, Chemical Bonding in Solids, Oxford University Press, Oxford, 1995.
G. Schmid, R. Pugin, J.-O. Malm, J.-O. Bovin, , Eur. J. Inorg. Chem. (1998) 813.
W.D. Knight, Solid State Phys. 2 (1956) 93.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
197
T.J. Rowland, Prog. Mat. Sci. 9 (1961) 1.
L.E. Drain, Metallurgical Rev. 119 (1967) 195.
G.C. Carter, L.H. Bennett, D.J. Kahan, Prog. Mat. Sci. 20 (1977) 1.
R.T. Schumacher, C.P. Slichter, Phys. Rev. 101 (1956) 58.
R. Peierls, ZS. f. Phys. 80 (1932) 763.
E. Teller, ZS. f. Phys. 67 (1930) 311.
J. Benkowitsch, H. Winter, J. Phys. F: Met. Phys. 13 (1983) 991.
H. Ebert, H. Winter, J. Voitländer, J. Phys. F: Met. Phys. 16 (1986) 1133.
H. Ebert, M. Battocletti, M. Deng, H. Freyer, J. Voitländer, , J. Computational Chemistry, in press.
J.M. Luttinger, P.J. Stiles, J. Phys. Chem. Solids 17 (1961) 284.
D. Shoenberg, Magnetic Oscillations in Metals, Cambridge University Press, Cambridge, 1984.
J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941.
K.H. Oh, B.N. Harmon, S.H. Liu, S.K. Sinha, Phys. Rev. B 14 (1976) 1283.
B. Bleaney, in: A.J. Freeman, R.B. Frankel (Eds.), Hyperfine Interactions, Academic Press, New York, 1967.
A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Oxford University Press, Oxford, 1970 (Dover reprint
1986).
W. Heitler, E. Teller, Proc. Roy. Soc. A 155 (1936) 629.
N. Bloembergen, T.J. Rowland, Phys. Rev. 97 (1955) 1679.
M. Ruderman, C. Kittel, Phys. Rev. 96 (1954) 99.
J. Korringa, Physica 16 (1950) 601.
H.S. Selbach, O. Kanert, D. Wolf, Phys. Rev. B 19 (1979) 4435.
E.C. Stoner, Proc. Roy. Soc. A 165 (1938) 372.
T. Moriya, J. Phys. Soc. Jpn. 18 (1963) 516.
Y. Yafet, V. Jaccarino, Phys. Rev. A 133 (1964) 1630.
S.H. Vosko, J.P. Perdew, Can. J. Phys. 53 (1975) 1385.
O. Gunnarson, J. Phys. F: Met. Phys. 6 (1976) 587.
J.F. Janak, Phys. Rev. B 16 (1977) 255.
A.R. Williams, U. von Barth, in: S. Lundquist, N.H. March (Eds.), Theory of the Inhomogeneous Electron Gas, Plenum Press, New York,
1983.
W. Götz, H. Winter, J. Phys.: Condens. Matter 4 (1992) 6253.
Y. Obata, J. Phys. Soc. Jpn. 18 (1963) 1020.
J.J. van der Klink, J. Phys.: Condens. Matter 8 (1996) 1845.
T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism, Springer, Berlin, 1985.
T. Moriya, K. Ueda, Solid State Commun. 15 (1974) 169.
A. Ishigaki, T. Moriya, J. Phys. Soc. Jpn. 65 (1996) 3402.
D.E. MacLaughlin, Solid State Phys. 31 (1976) 2.
S.E. Barrett, D.J. Durand, C.H. Pennington, C.P. Slichter, T.A. Friedmann, Phys. Rev. B 41 (1990) 6283.
R.A. Ferrell, Phys. Rev. Lett. 3 (1959) 262.
P.W. Anderson, Phys. Rev. Lett. 3 (1959) 325.
J. Sone, J. Phys. Soc. Jpn. 42 (1977) 1457.
G. Schreckenbach, T. Ziegler, Int. J. Quantum Chem. 60 (1996) 753.
S. Geschwind, in: A.J. Freeman, R.B. Frankel (Eds.), Hyperfine Interactions, Academic Press, New York, 1967.
W.G. Proctor, F.C. Yu, Phys. Rev. 81 (1951) 20.
J.S. Griffith, L.E. Orgel, Trans. Faraday Soc. 53 (1957) 601.
R. Freeman, G.R. Murray, R.E. Richards, Proc. Roy. Soc. A 242 (1957) 455.
Joan Mason (Ed.), Multinuclear NMR, Plenum Press, New York, 1987.
A.B.P. Lever, Inorganic Electronic Spectroscopy, Elsevier, Amsterdam, 1984.
R. Bramley, M. Brorson, A.M. Sargeson, C.E. Schäffer, J. Am. Chem. Soc. 107 (1985) 2780.
M.C. Read, J. Glaser, M. Sandström, J. Chem. Soc. Dalton Trans. 233 (1992).
N. Juranic, J. Magn. Res. 71 (1987) 144.
E.R. Cohen, B.N. Taylor, Europhysics News 18 (1987) 65.
A. Burgstaller, J. Voitländer, H. Ebert, J. Phys.: Condens. Matter 6 (1994) 8335.
C. Oliver, A. Myers, J. Phys.: Condens. Matter 1 (1989) 9457.
Ch. Ryter, Phys. Lett. 4 (1963) 69.
V.L. Moruzzi, J.F. Janak, A.R. Williams, Calculated Electronic Properties of Metals, Pergamon Press, New York, 1978.
R.W. Shaw Jr., W.W. Warren Jr., Phys. Rev. B 3 (1971) 1562.
G.B. Benedek, T. Kushida, J. Phys. Chem. Solids 5 (1958) 241.
R. Bertani, M. Mali, J. Roos, D. Brinkmann, J. Phys.: Condens. Matter 2 (1990) 7911.
198
[72]
[73]
[74]
[75]
[76]
[77]
[78]
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
[93]
[94]
[95]
[96]
[97]
[98]
[99]
[100]
[101]
[102]
[103]
[104]
[105]
[106]
[107]
[108]
[109]
[110]
[111]
[112]
[113]
[114]
[115]
[116]
[117]
[118]
[119]
[120]
[121]
[122]
[123]
[124]
[125]
[126]
[127]
[128]
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
S. Kluthe, R. Markendorf, M. Mali, J. Roos, D. Brinkmann, Phys. Rev. B 53 (1996) 11369.
T. Kushida, J.C. Murphy, M. Hanabusa, Phys. Rev. B 13 (1976) 5136.
Ch. Ryter, Phys. Rev. Lett. 5 (1960) 10.
D.A. Papaconstantopoulos, Handbook of the Bandstructure of Elemental Solids, Plenum Press, New York, 1986.
B. Knecht, J. Low Temp. Phys. 21 (1975) 619.
W.B. Pearson, Handbook of the Lattice Spacings of Metals and Alloys, Pergamon Press, New York, 1967.
H.T. Weaver, A. Narath, Phys. Rev. B 1 (1970) 973.
U. El-Hanany, M. Shaham, D. Zamir, Phys. Rev. B 10 (1974) 2343.
J.-J. Bercier, Thesis EPFL, Lausanne, 1993.
B.S. Shastry, E. Abrahams, Phys. Rev. Lett. 72 (1994) 1933.
R.G. Goodrich, S.A. Khan, J.M. Reynolds, Phys. Rev. B 3 (1971) 2379.
A.M. Clogston, V. Jaccarino, Y. Yafet, Phys. Rev. A 134 (1964) 650.
M. Weinert, A.J. Freeman, Phys. Rev. B 28 (1983) 6262.
M. Shaham, U. El-Hanany, D. Zamir, Phys. Rev. B 17 (1978) 3513.
A. Narath, H.T. Weaver, Phys. Rev. 175 (1968) 373.
P. Bhattacharyya, K.N. Pathak, K.S. Singwi, Phys. Rev. B 3 (1971) 1568.
F.Y. Fradin, D.D. Koelling, A.J. Freeman, T.J. Watson-Yang, Phys. Rev. B 12 (1975) 5570.
L.M. Noskova, E.V. Rozenfeld, Y.P. Irkhin, Sov. Phys. Solid State 26 (1984) 1686.
J.P. Bucher, J.J. van der Klink, Phys. Rev. B 38 (1988) 11038.
A.H. MacDonald, J.M. Daams, S.H. Vosko, D.D. Koelling, Phys. Rev. B 23 (1981) 6377.
M.A. Fedotov, V.A. Likhobolov, Bull. Acad. Sci. USSR; Div. Chem. Sci. 33 (1984) 1751.
J.A. Seitchik, A.C. Gossard, V. Jaccarino, Phys. Rev. 136 (1964) A1119.
M. Takigawa, H. Yasuoka, J. Phys. Soc. Jpn. 51 (1982) 787.
N. Inoue, T. Sugawara, J. Phys. Soc. Jpn. 45 (1978) 450.
W. Treutmann, Z. Angew. Phys. 30 (1970) 5.
S. Kobayashi, H. Launois, P. Lederer, C. Froidevaux, W. Treutmann, E. Vogt, Solid State Commun. 6 (1968) 265.
A. Narath, H.T. Weaver, Solid State Commun. 6 (1968) 413.
Y.Y. Tong, T. Yonezawa, N. Toshima, J.J. van der Klink, J. Phys. Chem. 100 (1996) 730.
P.A. Vuissoz, T. Yonezawa, D. Yang, J. Kiwi, J.J. van der Klink, Chem. Phys. Lett. 264 (1997) 366.
J.A. Seitchik, V. Jaccarino, J.H. Wernick, Phys. Rev. 138 (1965) A148.
A. Narath, H.T. Weaver, Phys. Rev. B 3 (1971) 616.
D. Hechtfischer, Z. Physik B 23 (1976) 255.
F. Hippert, L. Kandel, Y. Calvayrac, D. Dubost, Phys. Rev. Lett. 69 (1992) 2086.
Z.M. Stadnik, D. Purdie, M. Garnier, Y. Baer, A.-P. Tsai, A. Inoue, K. Edagawa, S. Takeuchi, K.H.J. Buschow, Phys. Rev. B 55 (1997)
10938.
G. Trambly de Laissardière, T. Fujiwara, Phys. Rev. B 50 (1994) 5999.
M. Krajci, M. Windisch, J. Hafner, G. Kresse, M. Mihalkovic, Phys. Rev. B 51 (1995) 17355.
W. Hume-Rothery, G.V. Raynor, The structure of Metals and Alloys, Institute of Metals, London, 1954.
A. Shastri, F. Borsa, D.R. Torgeson, J.E. Shield, A.I. Goldman, Phys. Rev. B 50 (1994) 15651.
A. Shastri, D.B. Baker, M.S. Conradi, F. Borsa, D.R. Torgeson, Phys. Rev. B 52 (1995) 12681.
J.L. Gavilano, B. Ambrosini, P. Vonlanthen, M.A. Chernikov, H.R. Ott, Phys. Rev. Lett. 79 (1997) 3058.
M. Mehring, H. Raber, Solid State Commun. 13 (1973) 1637.
E.R. Andrew, W.S. Hinshaw, R.S. Tiffen, J. Magn. Res. 15 (1974) 191.
Y. Obata, J. Phys. Soc. Jpn. 19 (1964) 2348.
X.-P. Tang, E.A. Hill, S.K. Wonnell, S.J. Poon, Y. Wu, Phys. Rev. Lett. 79 (1997) 1070.
W.W. Warren Jr., J. Phys.: Condens. Matter 5 (1993) B211.
K. Asayama, Y. Kitaoka, G.-Q. Zheng, K. Ishida, Progr. NMR Spectrosc. 28 (1996) 221.
C. Berthier, M.H. Julien, M. Horvatić, Y. Berthier, , J. Phys. I (France) 6 (1996) 2205.
H.B. Brom, Proceedings of the International School of Physics ‘Enrico Fermi’, vol. 136, 1998, p. 77.
W.W. Warren Jr., G.F. Brennet, U. El-Hanany, Phys. Rev. B 39 (1989) 4038.
A.J. Millis, H. Monien, D. Pines, Phys. Rev. B 42 (1990) 167.
F. Mila, T.M. Rice, Physica C 157 (1989) 561.
P.C. Hammel, M. Takigawa, R. Heffner, Z. Fisk, K.C. Ott, Phys. Rev. Lett. 63 (1989) 1992.
H. Alloul, P. Mendels, P. Monod, Phys. Rev. Lett. 61 (1988) 746.
H. Alloul, T. Ohno, P. Mendels, Phys. Rev. Lett. 63 (1989) 1700.
B. Sriram Shastry, Phys. Rev. Lett. 63 (1989) 1289.
W.B. Mims, G.E. Devlin, S. Geschwind, V. Jaccarino, Phys. Lett. A 24 (1967) 481.
W.G. Proctor, F.C. Yu, Phys. Rev. 77 (1950) 716.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
[129]
[130]
[131]
[132]
[133]
[134]
[135]
[136]
[137]
[138]
[139]
[140]
[141]
[142]
[143]
[144]
[145]
[146]
[147]
[148]
[149]
[150]
[151]
[152]
[153]
[154]
[155]
[156]
[157]
[158]
[159]
[160]
[161]
[162]
[163]
[164]
[165]
[166]
[167]
[168]
[169]
[170]
[171]
[172]
[173]
[174]
[175]
[176]
[177]
[178]
[179]
[180]
[181]
[182]
[183]
[184]
[185]
199
P.J. Segransan, Y. Chabre, W.G. Clark, J. Phys. F 8 (1978) 1513.
M. Shaham, J. Barak, U. El-Hanany, W.W. Warren Jr., Phys. Rev. B 22 (1980) 5400.
U. El-Hanany, W.W. Warren Jr., Phys. Rev. B 12 (1975) 861.
Y. Kohori, Y. Noguchi, T. Kohara, J. Phys. Soc. Jpn. 62 (1993) 447.
M. Katayama, S. Akimoto, K. Asayama, J. Phys. Soc. Jpn. 42 (1977) 97.
H. Nakamura, K. Yoshimoto, M. Shiga, M. Nishi, K. Kakurai, J. Phys.: Condens. Matter 9 (1997) 4701.
H. Yamagata, K. Asayama, J. Phys. Soc. Jpn. 29 (1970) 1385.
T. Yamada, N. Kunitomi, Y. Nakai, D.E. Cox, G. Shirane, J. Phys. Soc. Jpn. 28 (1970) 615.
A. Narath, J. Appl. Phys. 39 (1968) 553.
H. Ebert, J. Abart, J. Voitländer, J. Phys. F: Met. Phys. 14 (1984) 749.
R. Kubo, J. Phys. Soc. Jpn. 17 (1962) 975.
R. Denton, B. Mühlschlegel, D.J. Scalapino, Phys. Rev. B 7 (1973) 3589.
E.P. Wigner, Proc. Cambridge Phil. Soc. 47 (1951) 790.
F.J. Dyson, J. Math. Phys. 3 (1962) 140.
F.J. Dyson, M.L. Mehta, J. Math. Phys. 4 (1963) 701.
L.P. Gorkov, G.M. Eliashberg, Sov. Phys. JETP 21 (1965) 940.
B.L. Altshuler, in: W.P. Kirk, M.A. Reed (Eds.), Nanostructures and Mesoscopic Systems, Elsevier, Amsterdam, 1991.
M.P.J. van Staveren, H.B. Brom, L.J. de Jongh, G. Schmid, Z. Phys. D 12 (1989) 451.
E. Cuevas, M. Ortuño, J. Ruiz, Phys. Rev. Lett. 71 (1993) 1871.
M.L. Mehta, Random Matrices, Academic Press, New York, 1991.
K.B. Efetov, Physica D 83 (1995) 151.
C.W.J. Beenakker, Rev. Mod. Phys. 69 (1997) 731.
C.D. Makowka, C.P. Slichter, J.H. Sinfelt, Phys. Rev. B 31 (1985) 5663.
L.I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, New York, 1968, pp. 83–88.
H.E. Rhodes, P.-K. Wang, H.T. Stokes, C.P. Slichter, J.H. Sinfelt, Phys. Rev. B 26 (1982) 3559.
H.T. Stokes, H.E. Rhodes, P.-K. Wang, C.P. Slichter, J.H. Sinfelt, Phys. Rev. B 26 (1982) 3575.
K.B. Efetov, V.N. Prigodin, Mod. Phys. Lett. B 7 (1993) 981.
K.B. Efetov, V.N. Prigodin, Phys. Rev. Lett. 70 (1993) 1315.
C. Beenakker, Phys. Rev. B 50 (1994) 15170.
H.M. Pastawski, J.A. Gascôn, Phys. Rev. B 56 (1997) 4887.
J.J. van der Klink, J. Phys.: Condens. Matter 7 (1995) 2183.
F.C. Fritschij, H.B. Brom, L.J. de Jongh, G. Schmid, Phys. Rev. Lett. 82 (1999) 2167.
L.C. Hebel, C.P. Slichter, Phys. Rev. 113 (1959) 1504.
S. Kobayashi, T. Takahashi, W. Sasaki, J. Phys. Soc. Jpn. 32 (1972) 1234.
P. Yee, W.D. Knight, Phys. Rev. B 11 (1975) 3261.
M.J. Williams, P.P. Edwards, D.P. Tunstall, Faraday Discuss. 92 (1991) 199.
D.P. Tunstall, P.P. Edwards, J. Todd, M.J. Williams, J. Phys.: Condens. Matter 6 (1994) 1791.
J.K. Plischke, A.J. Benesi, M.A. Vannice, J. Phys. Chem. 96 (1992) 3799.
V.M. Mastikhin, I.L. Mudrakovsky, S.N. Goncharova, B.S.Balzhinimaev, S.P. Noskova, V.I. Zaikovsky, React. Kinet. Catal. Lett. 48 (1992)
425.
J.J. Bercier, M. Jirousek, M. Graetzel, J.J. van der Klink, J. Phys.: Condens. Matter 5 (1993) L571.
R.J. Charles, W.A. Harrison, Phys. Rev. Lett. 11 (1963) 75.
J. Bardeen, Phys. Rev. 49 (1936) 653.
R.L. Kautz, B.B. Schwartz, Phys. Rev. B 14 (1976) 2017.
J.P. Bucher, J. Buttet, J.J. van der Klink, M. Graetzel, Surf. Sci. 214 (1989) 347.
A. Kleine, P.L. Ryder, Catal. Today 3 (1988) 103.
Y.Y. Tong, J.J. van der Klink, G. Clugnet, A.J. Renouprez, D. Laub, P.A. Buffat, Surf. Sci. 292 (1993) 276.
Y.Y. Tong, J.J. van der Klink, J. Phys. Chem. 98 (1994) 11011.
I. Yu, W.P. Halperin, Phys. Rev. B 47 (1993) 15830.
J. Ph. Ansermet, C.P. Slichter, J.H. Sinfelt, Progr. NMR Spectrosc. 22 (1990) 401.
S.G. Louie, Phys. Rev. Lett. 42 (1979) 476.
Y.Y. Tong, J.J. van der Klink, J. Phys.: Condens. Matter 7 (1995) 2447.
A.G. Luntz, M.D. Williams, D.S. Bethune, J. Chem. Phys. 89 (1988) 4381.
J.P. Bucher, J.J. van der Klink, M. Graetzel, J. Phys. Chem. 94 (1990) 1209.
Y.Y. Tong, D. Laub, G. Schulz-Ekloff, A.J. Renouprez, J.J. van der Klink, Phys. Rev. B 52 (1995) 8407.
Y.Y. Tong, J.J. Billy, A.J. Renouprez, J.J. van der Klink, J. Am. Chem. Soc. 119 (1997) 3929.
Y.Y. Tong, P. Meriaudeau, A.J. Renouprez, J.J. van der Klink, J. Phys. Chem. B 101 (1997) 10155.
Y.Y. Tong, C. Belrose, A. Wieckowski, E. Oldfield, J. Am. Chem. Soc. 119 (1997) 11709.
200
[186]
[187]
[188]
[189]
[190]
[191]
[192]
[193]
[194]
[195]
[196]
[197]
[198]
[199]
[200]
[201]
[202]
[203]
[204]
[205]
[206]
[207]
[208]
[209]
[210]
[211]
[212]
[213]
[214]
[215]
[216]
[217]
[218]
[219]
[220]
[221]
[222]
[223]
[224]
[225]
[226]
[227]
[228]
[229]
[230]
[231]
[232]
[233]
[234]
[235]
[236]
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
C. Rice, Y.Y. Tong, E. Oldfield, A. Wieckowski, Electrochim. Acta 43 (1998) 2825.
S. Ladas, R.A. Dalla Betta, M. Boudart, J. Catal. 53 (1978) 356.
D.A. van Leeuwen, J.M. van Ruitenbeek, G. Schmid, L.J. de Jongh, Phys. Lett. A 170 (1992) 325.
N. Toshima, T. Yonezawa, K. Kushihashi, J. Chem. Soc. Faraday Trans. 89 (1993) 2537.
A. Eichler, J. Hafner, J. Furthmüller, G. Kresse, Surf. Sci. 346 (1996) 300.
R. Löber, D. Hennig, Phys. Rev. B 55 (1997) 4761.
A. Ceriotti, F. Demartin, G. Longoni, M. Manassero, G. Piva, M. Sansoni, Angew. Chem. Ed. Engl. 24 (1985) 697.
H. Krautscheid, D. Fenske, G. Baum, M. Semmelmann, Angew. Chem. 105 (1993) 1364.
G. Schmid, Clusters and colloids: from theory to applications, in: G. Schmid (Ed.), Structure and Bonding, vol. 62, VCH, Weinheim,
1985, p. 51.
R.L. Whetten, J.T. Khoury, M.M. Alvarez, S. Murthy, I. Veznar, Z.L. Wang, P.W. Stephens, C.L. Cleveland, W.D. Luedtke, U. Landman,
Adv. Mater. 8 (1996) 428.
A. Müller, S.Q.N. Shah, H. Bögge, M. Schmidtmann, Nature 397 (1999) 48.
H. Nakayama, D.D. Klug, C.I. Ratcliffe, J.A. Ripmeester, J. Am. Chem. Soc. 116 (1994) 9777.
L.J. Woodall, P.A. Anderson, A.R. Armstrong, P.P. Edwards, J. Chem. Soc. Dalton Trans. (1996) 719.
L.J. de Jongh (Ed.), Physics and Chemistry of Metal Cluster Compounds, Kluwer Academic Publishers, Dordrecht, 1994.
A. Ceriotti, R. Pergola, L. Garlaschelli, Physics and Chemistry of Metal Cluster Compounds, Kluwer Academic Publishers, Dordrecht,
1994.
A. Ecker, E. Weckert, H. Schnöckel, Nature 387 (1997) 397.
A. Lasciafari, Z.H. Jang, F. Borsa, P. Caretta, D. Gatteschi, Phys. Rev. Lett. 81 (1998) 3773.
J. Klinowski, Progr. NMR Spectrosc. 16 (1984) 237.
J.B. Nagy, I. Hannus, I. Kiricsi, in: J.H. Fendler (Ed.), Nanoparticles and Nanostructured Films, Wiley-VCH, Weinheim, 1998.
O.F. Sankey, A.A. Demkov, T. Lenovsky, Phys. Rev. B 57 (1998) 15129.
P.J. Grobet, G. van Gorp, L.R.M. Martens, P.A. Jacobs, Proc. Coll. Ampere 23 (1986) 356.
P.A. Anderson, P.P. Edwards, J. Am. Chem. Soc. 114 (1992) 10608.
E. Trescos, F. Rachdi, L.C. de Ménorval, F. Fajula, T. Numes, G. Feio, J. Phys. Chem. 97 (1993) 11855.
N.P. Blake, V.I. Srdanov, G.D. Stucky, H. Metiu, J. Chem. Phys. 104 (1996) 8721.
G. Engelhardt, M. Feuerstein, P. Sieger, D. Markgraber, G. Stucky, V. Srdanov, J. Chem. Soc. Chem. Commun. (1996) 729.
V.I. Srdanov, G.D. Stucky, E. Lippmaa, G. Engelhardt, Phys. Rev. Lett. 80 (1998) 2449.
C. Cros, M. Pouchard, P. Hagemuller, Compt. Rend. 260 (1965) 4764.
J.S. Kasper, P. Hagemuller, M. Pouchard, C. Cros, Science 150 (1965) 1714.
A.A. Demkov, O.F. Sankey, K.E. Schmidt, G.B. Adams, M.O’Keeffe, Phys. Rev. B 50 (1994) 17001.
S.B. Roy, K.E. Sim, A.D. Caplin, Phil. Mag. B 6 (1992) 1445.
H. Kawaji, H. Horie, S. Yamanaka, M. Ishikawa, Phys. Rev. Lett. 74 (1995) 1427.
F. Shimizu, Y. Maniwa, K. Kume, H. Kawaji, S. Yamanaka, M. Ishikawa, Phys. Rev. B 54 (1996) 13242.
J. Gryko, P.F. MacMillan, R.F. Marzke, A.P. Dodokin, A.A. Demkov, O.F. Sankey, Phys. Rev. B 57 (1998) 4172.
G.B. Adams, M. O’Keeffe, A.A. Demkov, O.F. Sankey, Y-M. Huang, Phys. Rev. B 49 (1994) 8048.
Y. Nozue, T. Kodaira, S. Ohwashi, T. Goto, O. Terasaki, Phys. Rev. B. 48 (1993) 12253.
K.W. Blazey, K.A. Müller, F. Blatter, E. Schumacher, Europhys. Lett. 4 (1987) 857.
D. van Leeuwen, J. van Ruitenbeek, Phys. Rev. Lett. 73 (1994) 1423.
David A. van Leeuwen, J.M. van Ruitenbeek, L.J. de Jongh, A. Ceriotti, G. Pacchioni, O.D. Häberlen, N. Rösch, Phys. Rev. Lett. 73
(1994) 1432.
F.M. Mulder, T.A. Stegink, R.C. Thiel, L.J. de Jongh, G. Schmid, Nature 367 (1994) 716.
Y. Volokitin, J. Sinzig, L.J. de Jongh, G. Schmid, M.N. Vargaftik, I.I. Moiseev, Nature 384 (1996) 621.
H.H.A. Smit, P.R. Nugteren, R.C. Thiel, L.J. de Jongh, Physica B 153 (1988) 33.
J. Baak, H.B. Brom, L.J. de Jongh, G. Schmid, Z. Phys. D 26 (1993) S30.
H.B. Brom, J. Baak, in: L.J. de Jongh (Ed.), Physics and Chemistry of Metal Cluster Compounds, Kluwer Academic Publishers, Dordrecht,
1994, p. 211.
Y. Volokitin, J. Sinzig, G. Schmid, H. Bonnemann, L.J. de Jongh, Z. Phys. D 40 (1997) 136.
Y. Volokitin, C. Bernard, G. Schmid, I.I. Moiseev, L.J. de Jongh, Czech. J. Phys 46 (Suppl. S4) 2373 (1996).
M.P.J. van Staveren, H.B. Brom, L.J. de Jongh, Phys. Rep. 208 (1991) 1.
J.A. Reedijk, H.B. Brom, L.J. de Jongh, G. Schmid, Phys. Rev. B 57 (1998) R15116.
H.B. Brom, D. van der Putten, L.J. de Jongh, Physics and Chemistry of Metal Cluster Compounds, Kluwer Academic Publishers, Dordrecht,
1994, p. 227.
I. Yu, W.P. Halperin, J. Low Temp. Phys. 45 (1981) 189.
D. van der Putten, H.B. Brom, L.J. de Jongh, G. Schmid, in: P. Jena (Ed.), Physics and Chemistry of Finite Systems: From Clusters to
Crystals, Kluwer Academic Publishers, Dordrecht, 1992, p. 1007.
H.B. Brom, J.J. van der Klink, F.C. Fritschij, L.J. de Jongh, R. della Pergola, A. Ceriotti, Z. Phys. D 40 (1997) 559.
J.J. van der Klink, H.B. Brom / Progress in Nuclear Magnetic Resonance Spectroscopy 36 (2000) 89–201
[237]
[238]
[239]
[240]
[241]
[242]
[243]
B.J. Pronk, H.B. Brom, A. Ceriotti, G. Longoni, Solid State. Commun. 64 (1987) 7.
P. Chini, J. Organomet. Chem. 200 (1980) 37.
J.D. Harris, T. Hughbanks, J. Am. Chem. Soc. 119 (1997) 9449.
E.C. Constable, B.F.G. Johnson, J. Lewis, G.N. Pain, M.J. Taylor, J. Chem. Soc. Chem. Commun. (1982) 754.
R.E. Benfield, J. Phys. Chem. 91 (1987) 2712.
M.A. Reed, Scientific American, January 1993, 98.
A.M. Thayer, M.L. Steigerwald, T.M. Duncan, D.C. Douglas, Phys. Rev. Lett. 60 (1988) 2673.
201