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179 10 EPR Spectroscopy 10.1 Introduction to EPR Spectroscopy Electron paramagnetic resonance (EPR) shares the theoretical description and many experimental concepts with NMR spectroscopy. From a methodological point of view, the main difference is the much larger magnetic moment of electron spins, which exceeds the one of protons by a factor of 660. Therefore EPR has a higher sensitivity at given spin concentration, can observe spin-spin interactions over longer distances, and is sensitive to molecular motion at shorter time scales. Resonance frequencies are in the microwave (mw) range (see Table 10.1). Typical pulse lengths and signal observation times are also shorter by about a factor of 500. Spectrometer technology is thus more complicated and may differ considerably between the different mw bands. In this lecture course, we shall refer to the most common band, the X band, unless noted otherwise. Table 10.1: Microwave frequency bands used in EPR spectroscopy (band definition by Radio Society of Great Britain).. Band Frequency range Typical EPR Frequency [GHz] L S X Ku Q V W D 1-2 2-4 8-12 12-18 30-50 50-75 75-110 110-170 1.5 3.0 9.5 17 36 70 95 140 a. Assuming Typical wavelength [mm] Typicala EPR Field B 0 [mT] 200 100 30 17 8 4 3 2 54 110 340 600 1280 2500 3390 5000 g g e = 2.002319 From an application point of view almost all pure substances contain magnetic nuclei and are thus accessible to NMR spectroscopy, while only few pure substances contain unpaired electrons and are thus accessible to EPR spectroscopy. This is because chemical binding is based on electron pair formation with spin cancellation. Most stable compounds are 180 thus diamagnetic. The occurence of an EPR signal is thus often related to enhanced reactivity, as for instance in transition metal complexes and free radicals. Therefore, EPR is particularly well suited for studies on synthetic and biological catalysts and for studies of radical-induced degradation processes in organisms and materials. As an example consider the enzyme methyl-coenzyme M reductase (MCR), which catalyzes methane formation from methyl-coenzyme M (methyl-CoM) and coenzyme B. Biochemical studies do not reveal whether MCR catalyzes the reaction by attacking the thioether sulfur atom of methyl-CoM or the carbon atom of the CH3S group. This question was addressed by studying the reaction of MCR with the inhibitor 3-bromopropane sulfonate (BPS), which results in a stable paramagnetic species with a single electron in a d 2 2 orbital x –y on the nickel ion. By using BPS 13C labeled at the methyl group and the two-dimensional HYSCORE experiment at X band (Section 10.4.4.4), it was possible to detect hyperfine coupling between the unpaired electron and the methyl carbon atom (Figure 10.1). The couplings to the protons of BPS were obtained from Q-band HYSCORE and electron nuclear double resonance (ENDOR) spectra. From this information a model for the binding mode could be proposed. C A 30 O – H OOC 20 ?2 / MHz HN H2NOC CH3 N H3C – OOC N COO COO – N H E 10 N Nin+ gz – 0 -30 H? -20 -10 O O F430 H2N a'147 Gln COO – D 0 ?1 / MHz 10 20 30 N N H?2 * H? 30 Ni SO3H?1 N N 20 B Br g b a SO3- BPS ?2 / MHz 10 0 -30 OGln? ‘147 -20 -10 0 ?1 / MHz 10 20 30 Figure 10.1: Determination of the binding mode of the inhibitor 3-bromopropane sulfonate (BPS) to the active center F430 of methyl-coenzyme M reductase (MCR). A) Structure of the active center. B) Structure of the inhibitor. C) X-band HYSCORE spectrum for the product obtained from F430 and BPS 13C-labelled at the -position. The left quadrant contains the 13C peaks. D) Simulated HYSCORE spectrum. E) Structural model for BPS bound to F430 (from Hinderberger D. et al, Angew. Chem. Int. Ed. 2006, 45, 3602-3607). Some defects in inorganic materials, such as semiconductors, are also paramagnetic. These defects determine the electronic and optical properties of these materials. EPR 181 spectroscopy provides more detailed information on their electronic and spatial structure than electrical or optical measurements. Furthermore, species with unpaired electrons are often observed as reaction intermediates, in particular in one-electron transfer processes. Such processes are the basis of energetics of living cells and occur in many redox reactions. Shortlived intermediates can be studied with special transient EPR experiments (photosynthetic reaction centers and light-sensitive proteins) or after spin trapping (Section 10.5.5). For instance, neurodegeneration during Alzheimers disease is related to an inflammatory process that is in turn characterized by abnormally high levels of free radicals. EPR spectra recorded in the presence of the spin trap N-tert-butyl--phenylnitrone (PBN) showed that toxic forms of amyloid- peptide lead to a formation of radicals. Such radical formation is not observed in the absence of these peptides or in the presence of a peptide that is rendered nontoxic by replacement of the methionine residue at position 35 by norleucine (Figure 10.2). This established the role of residue Met35 in the neurodegenerative process, the probable involvement of the sulfur atom of this residue in the chemistry of radical formation, and a potential protective effect of spin traps against Alzheimer’s disease. Meanwhile PBN and similar compounds are discussed as potential drugs. A B C D E Figure 10.2: Demonstration of radical formation by amyloid- peptide using spin trap EPR. A) Spin trap N-tert-butyl--phenylnitrone (PBN). B-E) EPR spectra for B) a control solution without A-peptide, C) with A 1-42 added, D) as before, but with residue Met35 replaced by a norleucine residue, E) with A 25-35 added. Residues 1-24 and 36-42 appear to be not essential for radical formation (from: S. Varadarajan et al., J. Struct. Biol. 2000, 130, 184-208. For such applications EPR is better suited than NMR, mainly because of the low concentrations involved and the higher sensitivity of EPR. Unless paramagnetic centers relax very fast, they also cause strong broadening of NMR lines of nuclei in their vicinity. Active centers of paramagnetic metalloproteins thus cannot be characterized by NMR. On the other hand, the structure of domains remote from the active center are not accessible to EPR, but can be studied by NMR. The two spectroscopies are thus often complementary. 182 A C B b C2 C -SH 180° y MTSSL O S S O x N O D E VIIA R49 N O b C -S S Z q Y f L210 W258 Q47 IXB R204 X V254 Figure 10.3: Structural model of the dimer of the Na+/H+ antiporter of Escherichia coli determined from the x-ray structure of the monomer by EPR distance measurements between spin-labeled residues. A) A selected residue is mutated to a cystein and the methanethiosulfonate spin label (MTSSL) is attached. B) For a homodimer with a C2 symmetry axis, four degrees of freedom (x, y, , ) determine the relative position of the two moieties. C) Nine distances were measured between spin labels in the two moieties of the dimer were measured (red lines). The dashed line marks a distance that was too long for the applied technique. The eight known distances overdetermine the structure. D) Interaction between -sheets in the two moieties contributes most to binding in the dimer. E) Another contribution to binding results from contacts between two helices (original publication: Hilger D. et al, Biophys. J. 2007, 93, 3675-3683). EPR spectroscopy can also complement NMR spectroscopy on diamagnetic systems by providing access to larger distances between sites and to dynamics on shorter time scales. For this purpose, stable free radicals have to be introduced in the system. EPR is then used as a probe technique. Usually, nitroxides (Section 10.5.1) are introduced as non-covalently attached spin probes or covalently bound spin labels. For proteins site-directed spin labeling (SDSL) have been developed. This technique is based on mutation of an amino acid near the site of interest to a cystein and covalent attachment of a thiol-specific spin marker. As is the case with NMR, structure determination of biomacromolecules by EPR does not require crystallization. For instance, the dimer of the Na+/H+ antiporter NhaA of Escherichia coli is weakly bound and does not survive crystallization. A structural model could be determined from the xray structure of the monomer and measurements of nine distances between spin-labeled residues in the two moieties of the dimer (Figure 10.3). If the two monomers in a homodimer are considered as rigid bodies, their relative position is fixed by only four degrees of freedom. Hence, the problem is overdetermined. The structural model shows that, to a large extent, binding between the monomers is due to interactions of two -sheets and to a lesser extent by 183 hydrophobic interactions between two helices. The -sheet binding motif is common for soluble proteins, but was found here for the first time for a membrane protein. Later biochemical experiments confirmed the importance of hydrogen-bonding residues in the sheets for dimer formation. 10.2 Differences between EPR and NMR spectroscopy The basics of the description of spin systems and spin dynamics are the same for EPR and NMR. Many of the experimental concepts from NMR can also be applied. In the remaining lectures we mostly discuss concepts that are particular to EPR. Such concepts result from the following differences. First, the larger magnetic moment implies frequencies in the mw range (Table 10.1) that pose more difficult problems for spectrometer technology. Larger excitation bandwidths are required, which implies pulses with lengths in the nanosecond rather than microsecond range. For these reasons pulse techniques were developed later in EPR than in NMR. The typical number of pulses in an experiment is smaller in EPR and pulse shape, phase, and frequency cannot be varied so easily. Experimental schemes are thus less complex in EPR compared to NMR. Due to the larger magnetic moment of the electron spin, typical hyperfine couplings between an electron and a nuclear spin are by at least two orders of magnitude larger than typical couplings between two nuclear spins. At the same time, nuclear Zeeman frequencies are smaller than in NMR as external magnetic fields are usually smaller. Thus, hyperfine couplings are often of the same order of magnitude as the nuclear Zeeman interaction and lead to mixing of nuclear spin states. The usual spectroscopic selection rules do no longer apply. Formally forbidden transitions can be excited by a single pulse or even by continuous irradiation. This allows for detection of nuclear frequencies in electron spin echo envelope modulation (ESEEM) experiments (Section 10.4.4). The pulse sequences consist of only mw pulses that are far off-resonant for the nuclear spins. Such experiments have no equivalent in NMR. Generally, the spin dynamics is more complex in EPR compared to NMR. In addition, the larger ratio between couplings and resonance frequencies than in NMR makes relaxation rates faster compared to the resonance frequencies. As line widths are proportional to the transversal relaxation rate, this leads to a larger ratio between line widths 184 and resonance frequencies, i.e. to lower resolution. To some extent this lower resolution is compensated by a larger spread of magnetic parameters. The equivalent to the chemical shift in NMR is the relative deviation (g-ge)/ge of the electron g value from the ge value for the free electron. This deviation is typically three orders of magnitude larger than chemical shifts (ppt instead of ppm). Because of the larger frequeny shifts and the larger couplings, EPR spectra extend over a broader frequency range relative to the mean resonance frequency. This requires a larger relative excitation bandwidth. However, the relative excitation bandwidths of pulses in the radio frequency (rf) and mw range are almost the same, if the same pulse power is available. It is thus not usually possible in EPR spectroscopy to excite the whole spectrum by a single pulse. This eliminates the sensitivity advantage of pulse Fourier transform experiments, so that continuous-wave (cw) EPR is still the method of choice for recording basic EPR spectra (Section 10.7.1). Furthermore, cw techniques are applicable even if the transverse relaxation time T2 is shorter than the shortest available pulses. Therefore, cw EPR is applicable in a broader temperature range than pulsed EPR. NMR spectroscopy relies strongly on the concept of separation of interactions by multipulse sequences and 2D spectroscopy. These concepts can be transferred to pulse EPR. By using them, resolution can be improved tremendously and more information can be obtained. Therefore both cw and pulse EPR are commonly applied. Even in pulsed EPR the external magnetic field is varied to overcome the limitations in excitation bandwidth. Another difference arises from the much larger change of the EPR resonance frequency associated with a change of the orientation of the molecule in the magnetic field. Anisotropies of EPR frequencies are on the order of tens of Megahertz to several Gigahertz. This matches the range of inverse rotational correlations times of molecules in fluid solution or soft matter. Hence, orientations of the molecule exchange on the time scale of cw EPR experiments. The effect on the spectral line shape can be treated in the same way as chemical-exchange phenomena in NMR were treated in Chapter 3. The rotational correlation time of the molecule can then be inferered from the line shape (Section 10.5.2). Like NMR, EPR spectroscopy can be applied to fluid or solid samples. As the coupling between two electrons at the same distance is by a factor (e/H)2 = 436’000 larger than 185 between two protons, the concentration of paramagnetic centers needs to be smaller to avoid excessive line broadening by spin-spin interaction. Best resolution and sensitivity for liquid samples is achieved at concentrations of of about 10-3-10-5 M. In crystalline samples the paramagnetic centers are often diluted by doping the paramagnetic compound into an isomorphous diamagnetic compound. Typical dilutions are 1:100 to 1:10000. If isomorphous diamagnetic substitution is impossible, solid-state measurements are performed in frozen solution. Aggregation of the dissolved paramagnetic compound on freezing is avoided by using glass-forming solvents. A selection of pure glas-forming solvents is given in Table 10.2. For amphiphilic compounds mixtures of polar and apolar solvents have to be used. Such glassforming solvents are, for instance, isopentane/isopropanol in a 8/2 mixture, toluene/acetone in a 1/1 mixture or with an excess of toluene, and toluene/methanol in a 1/1 mixture or with an excess of toluene. For biomacromolecules, water/glycerol mixtures are often used. Table 10.2: Glass-forming solvents with their melting temperatures Tm and glass transition temperatures Tg. Solvent Tm/K Tg/K 2-methylbutane 113.3 68.2 2-methyltetrahydrofuran 137 91 ethanol 155.7 97.2 methanol 175.2 102.6 1-propanol 146.6 109 toluene 178 117.2 ethylene glycol 255.6 154.2 di-1,2-n-butyl phthalate 238 179 glycerol 291.2 190.9 o-terphenyl 329.3 246 poly(styrene) 513 373 186 Another important difference between NMR and EPR is the appearance of the spectra. For technical reasons (see Section 10.7.1), best sensitivity and resolution in EPR is achieved with cw experiments at fixed mw frequency. The magnetic field is swept and at the same time modulated with a typical frequency of 100 kHz. Phase-sensitive detection of the signal modulation at this frequency yields the derivative of the absorption spectrum (Fig. 10.4). Although the shape of a derivative Lorentzian absorption line looks like the shape of a dispersion line, the derivative absorption line has a smaller linewidth. G= 2 . h DV 3T2 g µB DB0 DV C A B B0 DBpp Figure 10.4: Detection of the derivative of the absorption spectrum by field modulation with modulation amplitude B0. The amplitude of the reflected mw V is measured (A) providing a derivative lineshape (B). To avoid artificial line broadening the condition B 0 3B pp has to be fulfilled. The peak-to-peak linewidth Bpp = in field-swept spectra scales as 1/g (roughly with B0) if the transversal relaxation time T2 does not depend on g (C). 10.3 Interactions and EPR Hamiltonians 10.3.1 Classification of electron spin systems To avoid excessive relaxational line broadening, EPR spectra are usually measured in a concentration regime where interactions between paramagnetic centers are negligible. An exception are systems where the distance between two (or more) paramagnetic centers is relatively well defined and is measured by EPR (Section 10.3.6). An isolated paramagnetic center can contain a single unpaired electron (electron spin S=1/2, Section 10.3.3) or several tightly coupled electron spins (group spin S>1/2, Section 10.3.4). The latter situation is 187 encountered in high-spin states of transition metal complexes or triplet states of organic molecules. If the zero-field splitting in such a high-spin system exceeds the mw frequency, transitions can be induced only with pairs of levels with magnetic quantum numbers m S of the electron group spin S. Such a pair of levels is a Kramers doublet, meaning that the levels are degenerate in the absence of an external magnetic field and are split by the field. The Kramers doublet can be treated as an effective spin S’=1/2 (Section 10.3.5). A B Figure 10.5: Topology of spin systems. A) System consisting of seven nuclear spins. All magnetic moments and hence all 21 spin-spin couplings are comparable. B) System consisting of one electron spin with a large magnetic moment and six nuclear spins with much smaller moments. Only the six hyperfine couplings between the electron spin and the nuclear spins are significant. In most paramagnetic centers the electron spin is coupled to nuclear spins. Because of the presence of one spin that has a much higher magnetic moment than all the other spins, such electron-nuclear spin systems have a different topology compared to the nuclear spin systems observed in NMR. In typical nuclear spin systems all spins exhibit significant couplings to several other spins (Fig. 10.5A), while in typical electron-nuclear spin systems the nuclearnuclear couplings are negligible. Hence, the electron spin couples to many nuclei, but each nucleus couples significantly only to the one electron spin (Fig. 10.5B). The spectrum of the nuclear spins is thus easier to analyze and better resolved. It can be observed by ENDOR (Section 10.4.3) or ESEEM (Section 10.4.4) experiments. Furthermore, spin evolution can be described by considering only subsystems consisting of the electron spin and one nuclear spin, since the matrix representation of the spin Hamiltonian factorizes into Hamiltonians of twospin systems. This is an important simplification, since two-body problems can be solved exactly, while many-body problems cannot. 188 10.3.2 Units of magnetic parameters in EPR spectroscopy In NMR spectroscopy magnetic parameters are generally given in frequency units or are dimensionless. In EPR spectroscopy, magnetic field units are also used. Furthermore, wavenumbers are sometimes given for zero-field splittings and large hyperfine couplings that may be resolved in optical spectra. Frequency units and wavenumbers are energy units that can be interconverted without further knowledge (1 MHz = 3.335641·10-5 cm-1). In contrast, the conversion between magnetic field units and frequencies is not trivial, as it depends on the g value, and the g value often cannot directly be read off the spectrum. The conversion may thus require a full analysis of the spectrum. If the g value is known with sufficient precision, we have B 0 = h g B . [10.1] In particular, for g g e (organic radicals), 1 MHz corresponds to 0.0356828 mT or 1 mT to about 28 MHz. The cgs unit Gauss (G) is often used to specify the magnetic field. In fact it is a unit of magnetic induction (SI unit Tesla (T)), with 10000 G = 1 T. In both NMR and EPR the magnetic induction is usually referred to as the magnetic field. More precisely, the magnetic field unit is A m-1 in SI units or Oersted in cgs units (1 A m-1 = 4·10-3 Oersted). If the relative permeability of a material is unity, the magnetic induction in Gauss is the same as the magnetic field in Oersted. This is a good approximation for diamagnetic and paramagnetic substances, but not for ferromagnetic materials. Finally, when discussing spin Hamiltonians, we use angular frequency units for energies. Angular frequencies are related to frequencies by =2, and to energies E by = E h . 10.3.3 One electron spin S=1/2 coupled to nuclear spins The spin Hamiltonian in the absence of mw or rf irradiation (static Hamiltonian) is given by ˆ0 = H ˆ EZ + H ˆ HF + H ˆ NZ + H ˆ NQ , H [10.2] 189 ˆ EZ , the hyperfine interaction H ˆ HF , the where the terms are the electron Zeeman interaction H ˆ NZ , and the nuclear quadrupole interaction H ˆ NQ . In the nuclear Zeeman interaction H following they are discussed in turn. 10.3.3.1 The electron Zeeman interaction This interaction is formally equivalent to the (nuclear) Zeeman interaction treated in Sections 5.1 and 5.4. It is given by B T ˆ EZ = H ------B 0 gSˆ , h where – 24 B = e = 9.27400899 37 10 – 34 h = h 2 = 1.05457159682 10 J·T-1 [10.3] is the J·s Planck’s quantum of action, Bohr T B0 magneton, the transpose (T) of the magnetic field vector B 0 = B 0x B 0y B 0z , g the g tensor, and Sˆ the electron spin vector operator Sˆ x Sˆ y Sˆ z . In the PAS of the g tensor, the magnetic field vector can be written as B 0 = B 0 sin cos sin sin cos , [10.4] where and are polar angles that characterize the direction of this vector in the PAS. Expansion of Eq. [10.3] in this coordinate system yields B ˆ EZ = ------B 0 g x sin cos Ŝ x + g y sin sin Ŝ y + g z cos Ŝ z . H h [10.5] For an anisotropic g tensor,1 the direction of the effective field depends on the orientation of the molecule in the magnetic field. In other words, the quantization axis of the electron spin is not necessarily parallel to the magnetic field and can have different directions for different orientations of the molecule. The effective g value is thus given by g eff = 1 2 2 2 g x sin2 cos2 + g y sin2 sin2 + g z cos2 . [10.6] Strictly speaking, g does not have the transformation properties of a tensor, as it connects two independent spaces (spin space and laboratory space). Therefore it is sometimes called an interaction matrix. In this lecture, we call all interaction matrices tensors, as most magnetic resonance literature does. 190 In the high-field approximation, where the electron Zeeman interaction dominates over all the other interactions, its contribution to the energy of the spin states is proportional to m S g eff , where mS is the magnetic quantum number of the electron spin. The resonance field at this orientation of the molecule with respect to the magnetic field is then h mw B 0 res = --------------- . g eff B [10.7] The deviation of the g value of bound electrons from the ge value of the free electron is mostly due to spin-orbit coupling. As orbital angular momentum is quenched for nondegenerate ground states, spin-orbit coupling is due to admixture of excited states to the ground state by the orbital angluar momentum operator l̂ . This admixture is usually small and can thus be treated by perturbation theory. To second order the g tensor is given by g ij = g e ij + 2 k k m0 m l̂ ki 0 0 l̂ kj m ---------------------------------------------- = g e ij + 2 k k ij , E0 – Em [10.8] k where indices i and j run over the Cartesian directions x, y, and z and ij is the Kronecker delta. The outer sum with index k runs over all atoms in the molecules, where k is the spinorbit coupling constant for the kth atom and l̂ ki and l̂ kj are Cartesian components of the orbital angular momentum operator of this atom. As spin-orbit coupling is a relativistic effect, k increases with the mass of the atomic nucleus. The inner sum with index m runs over the molecular orbitals with index 0 designating the singly occupied molecular orbital (SOMO), where the unpaired electron resides in the ground state. Note that this index can be both negative (occupied orbitals) and positive (empty orbitals). Accordingly, the energy difference E 0 – E m in the denominator can also be positive or negative. Spin-orbit coupling with empty p-, d-, or f-orbitals thus leads to negative deviations of g from ge (high-field shifts), while spinorbit coupling with occupied orbitals leads to positive deviations (low-field shifts). The latter case is more often encountered. Spin density in s orbitals does not contribute to deviations of g from ge. The g tensor is thus a complicated function of the frontier orbitals of the molecule. It is used as fingerprint information for the class of paramagnetic centers. Since light elements (first and second period) have a small spin-orbit coupling constant, deviations of g from ge are only 191 –3 of the order of 10 10 –2 for organic radicals, unless there exist near degenerate frontier orbitals ( E 0 – E m 0 ). Typical cases of orbital degeneracy are the hydroxyl radical ·OH, alkoxy radicals RO·, and thiyl radicals RS· in the gas phase. In condensed phase, degeneracy is lifted by interaction with neighboring molecules. For these radicals, the g tensor thus strongly depends on intermolecular interactions. For first row transition metal ions deviations of g from –2 –1 ge are of the order of 10 10 . They are dominated by contributions from orbitals on the transition metal ion, so that we find g = g e 1 + 2 m0 m l̂ ki 0 0 l̂ kj m ---------------------------------------------- = g e 1 + 2 , E0 – Em [10.9] where is the spin-orbit coupling constant of the transition metal. 10.3.3.2 The hyperfine interaction More detailed information on the electronic and spatial structure of the paramagnetic center can be obtained from the hyperfine couplings. The hyperfine interaction term of the spin Hamiltonian is given by ˆ HF = H ˆ T A ˆI k , k S [10.10] k where k runs over all magnetic nuclei (Ik >0) in the molecule that have hyperfine couplings larger than the linewidths in the spectrum under discussion. The ˆI k are nuclear spin vector operators for these nuclei and the A k are hyperfine tensors. When analyzing liquid-state EPR spectra, hyperfine couplings exceeding 3 MHz have to be considered. In solid-state EPR spectra, such couplings are usually resolved only when they exceed 15 MHz. When analyzing ENDOR, ESEEM, or hyperfine sublevel correlation (HYSCORE) spectra the resolution limit is at 0.2-0.5 MHz. Within the high-field approximation the contribution of the hyperfine coupling to the energies of the spin states is m S m I A eff , where Aeff is an effective hyperfine coupling. Hyperfine coupling of the electron spin to a nuclear spin comes about by through-space dipole-dipole coupling of the two magnetic moments and by the Fermi contact interaction. The Fermi contact interaction is due to the non-zero probability to find the electron spin at the 192 same point in space as the nuclear spin. This happens only when the unpaired electron is in an T s orbital. The Fermi contact interaction leads to a purely isotropic coupling a Sˆ ˆI with iso 2 0 2 a iso = s --- -----g e B g n n 0 0 , 3h [10.11] where s is the spin density in the s orbital under consideration, gn is the nuclear g value and – 27 n = n = 5.05078317 20 10 0 0 2 J·T-1 the nuclear magneton ( g n n = n h ). The factor is the probability to find the electron at this nucleus in the ground state with wavefunction 0 .1 The magnetic moment of the electron is fully characterized by the ge value of the free electron, as there is no orbital angular momentum in s orbitals. Unpaired electrons in p-, d, and f- orbitals do not contribute to Fermi contact interaction, as these orbitals have a node at the nucleus. However, due to the non-spherical symmetry of these orbitals, dipole-dipole coupling between the magnetic moments of the electron and nuclear spin does not average. Spin density in such orbitals thus gives rise to purely anisotropic couplings.2 In general, the matrix elements Tij of the total anisotropic hyperfine coupling tensor T of a given nucleus are computed from the ground state wavefunction by 2 0 3r i r j – ij r T ij = ---------- g e B g n n 0 -------------------------- 0 . 5 4h r [10.12] Quantum chemistry programs such as ORCA, ADF, or Gaussian can compute T as well as aiso. A special situation applies to protons, alkali metals and earth alkaline metals, which have no significant spin densities in p-, d-, or f-orbitals. In this case, the anisotropic contribution can only arise from through-space dipole-dipole coupling to other centers of spin density. In a point-dipole approximation the hyperfine tensor is then given by 1 Tabulated in: J. R. Morton, K. F. Preston, J. Magn. Reson. 1978, 30, 577. 2 For given spin densities these couplings can be computed from parameters given in: J. R. Morton, K. F. Preston, J. Magn. Reson. 1978, 30, 577. 193 T 0 3n j n j – 1 T k = ---------- g e B g n n j ---------------------- , 3 4h Rj [10.13] jk where the sum runs over all centers with significant spin density j (summed over all orbitals at this center), the Rj are distances between the nucleus under consideration and the centers of spin density, and the n j are unit vectors along the direction from the considered nucleus to the center of spin density. For protons in transition metal complexes it is often a good approximation to consider spin density only at the central ion. The distance R from the proton to the central ion can then be directly inferred from the anisotropy of the hyperfine coupling. Admixed orbital angular momentum also contributes to dipole-dipole coupling. This can be considered by a simple correction, giving the dipole-dipole hyperfine tensor gT A dd = ------- . ge [10.14] We thus obtain for the total hyperfine tensor A k in Eq. [10.10] gT k A k = a iso k 1 + --------- . ge [10.15] Note that the product gT k may have an isotropic part, although T k is purely anisotropic. This pseudocontact contribution depends on the relative orientation of the g tensor and the spinonly dipole-dipole hyperfine tensor T k . The hyperfine couplings can be used to map the SOMO in terms of a linear combination of atomic orbitals. For catalytic species this can provide insight into reactivity and thus into the mechanism of catalysis. For organic compounds that form reasonably stable radical anions or cations the SOMO of the radical anion corresponds to the LUMO of the parent compound and the SOMO of the radical cation to the HOMO of the parent compound. The contributions discussed so far can be understood in a single-electron representation. A further contribution arises from correlation between electrons in the molecule. Assume that the pz orbital on a carbon atom contributes to the SOMO, so that the spin state of the electron is preferred in that orbital (Fig. 10.6). Electrons in other orbitals on the same atom will then also have a slight preference for the state, as electrons with the same 194 spin “tend to avoid each other” and thus have less electrostatic repulsion.1 In particular, this means that the spin configuration in Fig. 10.6A is slightly more preferable than the one in Fig. 10.6B. According to the Pauli principle the two electrons that share the bond orbital of the CH bond must have antiparallel spin. Thus, the electron in the s orbital of the hydrogen atom that is bound to the spin-carrying carbon atom has a slight preference for the state. This corresponds to a negative isotropic hyperfine coupling of the directly bound proton, which is induced by the positive hyperfine coupling of the adjacent carbon atom. The effect is termed ‘spin polarization’, although it is entirely different from the polarization of electron spin transitions in an external magnetic field. A B pz pz C H C s H s Figure 10.6: Spin polarization on an adjacent hydrogen atom ( proton) induced by spin density in a pz orbital on carbon. The spin configuration in A) is slightly favored with respect to the one in B). A B c H H C Figure 10.7: Isotropic hyperfine coupling of a next-neighbor hydrogen atom ( proton) induced by hyperconjugation. The overlap of the carbon pz orbital and the hydrogen s orbital depends on dihedral angle . Spin polarization is observed both in radicals, where the spin density is confined to a single pz orbital, and in radicals, where the spin density is delocalized in a system. In 1 This preference for electrons on the same atom to have parallel spin is also the basis of Hund’s rule. 195 radicals spin polarization is the only contribution to the isotropic hyperfine coupling of a proton that is directly bound to a carbon atom of the conjugated system. The isotropic hyperfine coupling of such an proton can thus be predicted by the McConnell equation A iso H = Q H , [10.16] where is the spin density at the adjacent carbon atom and QH is a parameter of the order of – 2.5 mT, which slightly depends on the structure of the system. The isotropic hyperfine coupling of protons in radicals is caused by spin delocalization from the spin-carrying pz orbital to the s orbitals on the hydrogen atoms. These orbitals are sufficiently close in space to allow for overlap. The extent of this hyperconjugation depends on the dihedral angle between the pz orbital lobes and the C-H bond. When the C group rotates freely with respect to the spin-carrying orbital, an average hyperfine coupling is observed. Unlike spin polarization, hyperconjugation does not depend on electron correlation, but can rather be seen as spin density transfer. 10.3.3.3 Nuclear Zeeman interaction The nuclear Zeeman interaction can be neglected in the analysis of EPR spectra, unless it is of the same order of magnitude as a resolved hyperfine coupling of the same nucleus. In analysis of ENDOR, ESEEM, and HYSCORE spectra, the nuclear Zeeman interaction is used to assign hyperfine couplings to elements (isotopes). This information is missing in EPR spectra. Chemical shift information cannot be obtained from EPR, ENDOR, ESEEM, or HYSCORE spectra, as chemical shifts are smaller than the linewidths for nuclear spins with significant hyperfine couplings. Chemical shift is thus neglected in the nuclear Zeeman Hamiltonian, which is given by n n T ˆ ˆ NZ = – g---------H -B 0 I . h [10.17] In the laboratory frame this simplifies to ˆ NZ = I , H I z [10.18] 196 with I = -nB0. Within the high-field approximation the nuclear Zeeman interaction contributes an energy mI I to the spin states. At S-band to Q-band frequencies the nuclear Zeeman frequency is often comparable to hyperfine couplings or smaller. This leads to state mixing by anisotropic hyperfine interactions, which is the basis of ESEEM and HYSCORE experiments. At W-band frequencies and above this phenomenon is observed only for a few low- nuclei with large hyperfine couplings (large spin densities), such as for nitrogen atoms that are directly coordinated to a transition metal ion. 10.3.3.4 Nuclear quadrupole interaction The nuclear quadrupole interaction (Section 5.9) does not contribute to EPR spectra unless it is of the same order of magnitude as the hyperfine coupling of the same nucleus and the hyperfine coupling of this nucleus is resolved. Usually this happens only for heavy elements from the third row of the periodic table onwards. In analyzing ENDOR, ESEEM, and HYSCORE spectra of nuclei with spin Ik>1/2 the nuclear quadrupole interaction has to be considered. In an EPR context, the nuclear quadrupole Hamiltonian is often written as ˆ NQ = ˆI T PIˆ , H [10.19] where the principal values of the nuclear quadrupole tensor are given by 2 e qQ P z = ---------------------------- , 2I 2I – 1 h [10.20] P x – P y = P z , [10.21] Px + Py + Pz = 0 . [10.22] and The principal values are thus directly related to the field gradient components Vxx, Vyy, and Vzz defined in Section 5.9. Within the high-field approximation the contribution of the nuclear 2 quadrupole interactions to the energies of the spin states is proportional to m I P eff , where Peff is an effective nuclear quadrupole interaction. To first order, the transition energy for a pair of 197 levels m I thus does not depend on the nuclear quadrupole interaction. In particular, this applies to the allowed transition m I = 1 2 – 1 2 . 10.3.4 One electron group spin S>1/2 coupled to nuclear spins If m electrons are distributed indegenerate orbitals, each orbital is first singly occupied (Fig. 10.8A, Hund’s rule). This is because electron pairing in the same orbital would lead to increased Coulomb repulsion. In the presence of a ligand field that removes orbital degeneracy, pairing is preferred if the energy difference between the upper and lower set of orbitals exceeds electron pairing energy (Fig. 10.8B,C). Depending on the strength of the ligand field, several electrons may thus be unpaired (high-spin state) or paired as far as possible (low-spin state). A B C eg eg d t2g t2g Figure 10.8: High-spin and low-spin configurations for a d5 transition ion, such as Fe3+. A) In the absnece of a ligand field all unpaired electrons in the degenerate d orbitals have parallel spin (high spin S= 5/2). B) In a weak ligand field, where the splitting between the eg and t2g levels is smaller than the electron pairing energy, all spins are parallel (high spin, S= 5/2). C) In a strong ligand field, where the level splitting exceeds the electron pairing energy, all electrons occupy the set of lowest-lying orbitals, where they pair (low spin, S= 1/2). In high spin states of transition metal ions all the unpaired electrons reside mainly in orbitals on the same atom. They are so close in space that they couple strongly and cannot be excited separately from each other. Hence, it is appropriate to describe them as a single group spin S = m/2. For instance, Fe3+ with a 3d5 configuration in weak ligand fields assumes an S = 5/2 high-spin state, while in strong ligand fields it assumes an S = 1/2 low-spin state. On first sight one might expect that the distribution of five unpaired electrons on all five 3d orbitals is spherically symmetric so that no equivalent to the nuclear quadrupole 198 interaction would result. However, spin-orbit coupling breaks this symmetry. Electron-group spins S > 1/2 thus have an additional interaction that is formally analogous to the nuclear quadrupole interaction. This zero-field splitting1 contribution to the spin Hamiltonian is written as ˆ ZF = Sˆ T DSˆ , H [10.23] where D is the traceless zero-field splitting tensor. In the principal axes frame of this tensor, the Hamiltonian can be written as PAS 2 2 2 ˆ ZF H = D x Sˆ x + D y Sˆ y + D z Sˆ z 2 1 2 2 = D Sˆ z – --- S S + 1 + E Sˆ x – Sˆ y 3 . [10.24] with D = 3Dz/2 and E = (Dx-Dy)/2. By convention, Dz is the principal value with the largest magnitude, so that E cannot exceed D/3. For axial symmetry, E = 0. In cubic symmetry, quadrupolar zero-field splitting vanishes (D= E = 0). In that case and for S > 3/2, the much smaller hexadecapolar contribution may become observable. For transition metal ions, g shifts with respect to ge (Eq. [10.9]) and zero-field splitting are related, 2 D = . [10.25] Within the high-field approximation the contribution of the zero-field splitting to the energy 2 levels is proportional to m S D eff . Thus the transition m S = 1 2 – 1 2 that is allowed in EPR spectra of half-integer group spins S is not affected to first order by zero-field splitting. 10.3.5 Effective spin S’=1/2 in a high-spin system S>1/2 If the zero-field splitting is much larger than the mw quantum and the electron Zeeman interaction at accessible magnetic fields, only part of the transitions of a spin S > 1/2 are accessible for EPR. In conventional EPR at X band, the limit is at about 20 GHz. For integer group spins S, none of the transitions is accessible unless the zero-field splitting tensor has at 1 The term of the Hamiltonian is sometimes called fine structure term instead of zero-field splitting. 199 least axial symmetry. Such integer-spin systems are termed EPR silent, although they are paramagnetic and can be studied at higher fields and frequencies. For half-integer group spins there is at least one pair of low-lying levels that are degenerate in the absence of a magnetic field and are split in its presence. Such a Kramers doublet can be described as an effective spin S’ = 1/2. For instance, for high-spin Fe3+ (S = 5/2) in non-cubic symmetry, EPR transitions can be observed only within the three Kramers doublets, but not between them (Fig. 10.9). The spin Hamiltonian for each Kramers doublet is equivalent to the spin Hamiltonian of a spin S = 1/2 (Section 10.3.3), except that the effective g values depend on the zero-field splitting parameters D and E and are not strictly field-independent. g = 9.67 g = 4.3 g = 0.6 0 B0 Figure 10.9: Kramers doublets with effective spin S’=1/2 for high-spin Fe3+ (3d5) with E/D = 1/3. The mw quantum (vertical bars) is too small to excite transitions between different Kramers doublets. The three Kramers doublets have different effective g values. 10.3.6 A pair of weakly coupled electron spins 10.3.6.1 Exchange coupling Consider two electron spins S1 = 1/2 and S2 = 1/2 as individual paramagnetic centers. At distances up to at least 15 Å there is still significant overlap of the two SOMOs, so that the two electrons can exchange. This exchange interaction leads to a splitting between the singlet state (group spin S = 0) and the triplet state (group spin S = 1) of the coupled system. If this splitting 200 is so small that transitions between the singlet and triplet state can be excited in an EPR experiment, it is more convenient to treat the system in terms of two individual spins that are coupled to each other. Typically at distances larger than 5-10 Å the individual spin treatment is applied. The energy difference between the singlet and triplet state is the exchange integral J = – 2e 2 1 r 1 2 r 2 1 r 2 2 r 1 - dr 1 dr 2 , ----------------------------------------------------------------------r1 – r2 [10.26] where 1 and 2 are the wavefunctions of the two unpaired electrons.1 For positive J, the singlet state is lower in energy, i.e., the orbital overlap is bonding and the interaction is antiferromagnetic. Negative J correspond to a lower-lying triplet state, i.e., antibonding orbital overlap and a ferromagnetic interaction. The exchange term of the spin Hamiltonian is given by ˆ EX = JSˆ T1 Sˆ 2 = J Sˆ 1x Sˆ 2x + Sˆ 1y Sˆ 2y + Sˆ 1z Sˆ 2z . H [10.27] This term describes a scalar (purely isotropic) coupling. Anisotropic contributions to the exchange coupling may occur for species involving heavy elements, but can be neglected for organic radicals. Within the high-field approximation the contribution of the exchange interaction to energy levels is proportional to mS,1mS,2J. To a good approximation the exchange interaction decays exponentially with distance r between the paramagnetic centers. The decay rate depends on the conductivity of the medium between the centers and on the presence or absence of a conjugated network of bonds between them. If conjugation is weak and the matrix isolating, exchange coupling is much smaller than the dipole-dipole coupling through space at distances longer than 15 Å. 1 There exist different conventions for the sign of J and the factor 2 may be missing. One should always ascertain which convention a certain author is adhering to. 201 10.3.6.2 Dipole-dipole coupling The dipole-dipole coupling between two electron spins is analogous to that between two nuclear spins (Section 5.8). For transition metal and rare earth ions the g anisotropy is so large that the two magnetic moments are not parallel to the magnetic field. The interaction energy according to Eq. [5.74] is then parametrized by three angles 1, 2, and (Fig. 10.10) and is given by 0 1 E = – ------ 1 2 ---3- 2 cos 1 cos 2 – sin 1 sin 2 cos . 4 r [10.28] The dipole-dipole coupling term of the spin Hamiltonian assumes the form 1 0 3 2 ˆ dd = Sˆ T1 DSˆ 2 = --H -3 ---------- g 1 g 2 B Sˆ 1 Sˆ 2 – ---2- Sˆ 1 r Sˆ 2 r . r 4h r B0 [10.29] ® µ2 ® µ1 f q1 q2 ® r Figure 10.10: Coupling between two magnetic moments 1 and 2 in a general orientation with respect to the external magnetic field B 0 and definition of the angles 1, 2, and that parametrize this interaction. Note that these angles implicitly depend on the orientation of the molecule in the magnetic field. In the special case of two parallel magnetic moments, which is a good approximation for two organic radicals, the dipole-dipole coupling tensor D has the principal values -dd, dd, 2dd with 1 0 2 dd = ---3- ---------- g 1 g 2 B . r 4h [10.30] The orientation of the molecule can then be characterized by a single angle between the magnetic field axis and the spin-spin vector. The orientation-dependent dipolar splitting thus assumes the form d = 3 cos2 – 1 dd . [10.31] 202 For organic radicals, g 1 g 2 g e is a good approximation, so that the dipole-dipole coupling dd=dd/2 in frequency units is given by 52.04 MHz dd = ---------------------------- . 3 –3 r nm [10.32] With typical transversal relaxation times T2 of electron spins of a few microseconds, dipole-dipole couplings can be measured down to about 100 kHz in favourable cases, corresponding to distances of 8 nm. For soluble proteins and membrane proteins reconstituted into detergent micelles this upper distance limit may reduce to about 6 nm and for membrane proteins reconstituted into liposomes to about 5 nm. These limits are comparable to the diameter of protein molecules. Within the high-field approximation the contribution of the dipole-dipole interaction to energy levels is proportional to mS,1mS,2 dd,eff. This is the same dependence on magnetic quantum numbers as for the exchange coupling. The two contributions to electron-electron coupling thus cannot be separated by spin manipulation. a b ·O=O · O c N ·O=O · O N ·O=O · O N Figure 10.11: Transversal and longitudinal relaxation of the electron spin of a nitroxide molecule by a collsion with triplet dioxygen. (a) The molecules are separated and diffuse towards each other. For example, both unpaired electrons of the oxygen molecule are in the state, while the unpaired electron of the nitroxide molecule is in the state. (b) The molceules collide, their orbitals overlap, and the three unpaired electrons are no longer distinguishable. (c) The molecules have separated again. With a probability of 2/3 the nitroxide molecule is now left with one of the unpaired electrons that originally belonged to the oxygen molecule (and vice versa). The electron spin of the nitroxide thus has lost phase memory and has changed its spin state. 10.4 Measurement of hyperfine couplings 10.4.1 cw and echo-detected EPR spectroscopy Hyperfine couplings manifest in the spectra of both electron and nuclear spins. Large hyperfine couplings (>20...50 MHz or 0.7...2 mT at g = ge) can often be determined with 203 sufficient precision from EPR spectra even in the solid state. For small radicals in solutions with low viscosity, precision of an EPR measurement may even be sufficient for hyperfine couplings as small as 3 MHz. In these situations, cw EPR or field-swept echo-detected EPR is the method of choice, as it is more sensitive and technically easier than measurement of the nuclear spectrum. To obtain utmost hyperfine resolution, line broadening due to couplings between electron spins has to be avoided. In the solid state this requires concentrations of 1 mmol L-1 or less. . c) Figure 10.12: Line broadening due to collisonal exchange. 1,8-Dimethyl naphthaline radical anion at a concentration of a) 10-3 M, b) 10-5 M. c) Wurster’s blue without (top) and with (bottom) oxygen in the solution. In the liquid state linewidths are smaller. Furthermore, exchange broadening can also arise due to collisions between paramagnetic molecules. During such a collision the orbitals overlap, the unpaired electrons of both molecules become indistinguishable and may be exchanged when the molecules separate again (Fig. 10.11). This leads to sudden changes in resonance frequency and thus to phase relaxation. The transversal relaxation time T2 is shortened and lines are broadened. In exceptional cases, concentrations down to 200 mol L-1 may be required to avoid such broadening (Fig. 10.12A). Exchange broadening is also caused by dissolved oxygen, as dioxygen has a triplet ground state and is thus paramagnetic. The effect is often tolerable for measurements in aqueous solution, but highly detrimental for measurements in unpolar solvents, where oxygen solubility is much higher (Fig. 10.12B). 204 . 6 H 5 H 2H(1.743 mT) 3 2H(0.625 mT) H H 375 2 H 4 1H(0.204 mT) 380 385 B0 (mT) Figure 10.13: Simulated EPR spectrum of the phenyl radical (bottom) and schematic drawing of how the splitting pattern arises (top). The hyperfine splitting pattern of radicals in solution is generated in the same way as discussed in Section 7.1 for J-coupled spectra of weakly-coupled nuclear spins in NMR, except that only the spectrum of one of the spins (the electron spin is observed). Furthermore, the number of nuclear spins with significant hyperfine coupling to the electron spin is usually larger than the corresponding number in J-coupled NMR spectra. As a particularly simple case, the EPR spectrum of the phenyl radical is shown in Fig. 10.13. This radical can be generated for instance by photolysis of phenylbromide in solution. Due to the low natural abundance of 13C, carbon hyperfine splittings are apparent only in weak satellite lines that do not concern us here. The splitting pattern is thus entirely due to proton hyperfine couplings. Among the protons, the ones in positions 2 and 6 are magnetically equivalent with a coupling of 1.743 mT, the ones in positions 3 and 5 with a coupling of 0.625 mT and the proton in position 4 has a 205 coupling of 0.204 mT. The resulting spectrum, a triplet of triplets of doublets has 18 lines. As the pattern is clearly apparent, the hyperfine couplings can be read off directly. Figure 10.14: Experimental EPR spectrum of the paracyclophan free radical. Such a simple analysis may no longer be possible if the subpatterns overlap. This happens almost with certainty when the number of hyperfine coupled nuclei increases, as the number of lines in the EPR spectrum increases multiplicatively n EPR = 2k i Ii + 1 , [10.33] i where index i runs over the groups of equivalent nuclei, the ki are the numbers of nuclei within each group, and the Ii their nuclear spin quantum numbers. For instance, for the free cyclophan radical in solution with 4 equivalent aromatic protons, 9 equivalent exo, and 9 equivalent endo protons, 5 9 9 = 405 lines arise (Fig. 10.14). Coordination with a K+ ion (nuclear spin I = 3/2 for 39K, quadruplet splitting) removes the equivalence between the two moieties of the molecule, so that each group of equivalent protons splits into two inequivalent subgroups. The number of lines increases to 3 3 5 5 5 5 4 = 22500 . Spectra like this are difficult to analyse. Due to the special topology of electron-nuclear spin systems (Fig. 10.5B), the nuclear spectrum in the liquid state has a much smaller number of lines 206 n NMR liq = 2S + 1 , [10.34] i where i again runs over the groups of equivalent nuclei. For instance, the nuclear spectra of phenyl radicals, uncoordinated paracyclophan radicals, and K+-coordinated paracyclophane radicals have only 6, 6, and 14 lines, respectively. In the solid state, additional splitting due to nuclear quadrupole couplings leads to a number of lines n NMR sol = 2Ii 2S + 1 . [10.35] i Among our examples, this would change the number of lines only for K+-coordinated paracyclophane radicals to 18. hnmw/g^mB A hnmw/gymB C q = 90° q = 0° 300 hnmw/gxmB hnmw/gzmB hnmw/g||mB 320 300 340 B D 320 hnmw/gzmB 340 hnmw/gymB hnmw/g^mB hnmw/g||mB hnmw/gxmB 300 320 B0 (mT) 340 300 320 340 B0 (mT) Figure 10.15: Typical S = 1/2 EPR spectra in the absence of hyperfine couplings. The frequency dispersion is caused only by a g tensor with axial symmetry (A,B) or orthorhombic symmetry (C,D). The low-field and high-field edges of the absorption spectra (A,C) correspond to selection of a small subset of orientations near the x or z axis of the g tensor PAS. The first derivative lineshape, as detected in cw EPR spectroscopy, is dominated by the singularities that arise at principal axis directions. 207 The nuclear spin spectra cannot be measured with an NMR spectrometer, as the hyperfine splittings exceed the bandwidth of NMR probeheads by orders of magnitude. Furthermore, detection at the low NMR frequency insted of the high EPR frequency would lead to a drastic loss in sensitivity. For these reasons spectra of hyperfine coupled nuclei are detected by EPR-based methods such as ENDOR or ESEEM. The dispersion of electron spin resonance frequencies due to g anisotropy for powders or frozen solutions causes a substantial drop in hyperfine resolution compared to the liquid state. Even for transtion metal ions with only moderate g anisotropy, such as V(IV), Cu(II), or Co(II) the spectrum in the absence of any hyperfine couplings extends over tens of mT (Fig. 10.15). Except for the singularities corresponding to principal axis directions, the first derivative of the absorption lineshape is barely detectable. If hyperfine anisotropy is much smaller than g ansiotropy or if the PAS of the g and hyperfine tensors coincide, observable hyperfine splittings thus correspond to the principal axis directions of the g tensor. If g and hyperfine anisotropy are of the same order of magnitude and the PAS are non-coincident, spectrum analysis requires lineshape simulations. hfi at g|| hfi at g^ A mI = |+3/2ñ L L Cu L mI = |+1/2ñ L B hnmw/g||mB mI = |-1/2ñ mI = |-3/2ñ 0.3 g||,A|| 0.32 B0 (T) 0.34 C A|| hnmw/g^mB 0.3 0.32 0.34 B0 (T) Figure 10.16: Hyperfine splittings in solid state EPR spectra of powders, glasses or frozen solutions. (simulations for a square planar copper(II) complex with four equal ligands L). A) Each nuclear spin state gives rise to a separate axial powder pattern. B) The absorption spectrum, as detected by fieldswept echo-detected EPR is the sum of the four separate patterns. C) The derivative of the absorption spectrum, as detected by cw EPR, exhibits hyperfine splitting along g||. The inset shows a stereoview of the complex with the unique axis of the g and copper hyperfine tensor. 208 A simple case is illustrated in Fig. 10.16 on the example of a square planar copper complex with four equal ligands L. This species has a C4 symmetry axis. A Cn symmetry axis with n 3 implies an axial g tensor with its unique axis coinciding with the symmetry axis. As the SOMO also has C4 point symmetry at the Cu2+ ion, the copper hyperfine tensor is also axial and has the same unique axis. For each magnetic quantum number mI = -3/2, -1/2, +1/2, and +3/2 the two tensors for the nuclear Zeeman and copper hyperfine interaction add to a total axial tensor that describes the anisotropy of the resonance frequency.1 Each nuclear spin state thus gives rise to a separate axial powder pattern (Fig. 10.16A). The edges of the patterns are shifted with respect to each other by multiples of the hyperfine coupling A|| (low-field edge) and A (high-field edge). The small splitting A is usually unresolved in the EPR spectrum (Fig. 10.16B). The parameters g||, g , and A|| can be directly read off the cw EPR spectrum (Fig. 10.16C). 10.4.2 Nuclear spin spectra Allowed transitions in nuclear spin spectra involve a change of the magnetic quantum number of the nuclear spin mI by unity while the magnetic quantum number of the electron spin mS remains constant, i.e., they are of the type m I m I + 1 . If the high-field approximation applies to both the electron and nuclear spin, the angular frequencies of such transitions are given by = I + m S A eff + 2m I + 1 P eff . [10.36] For simplicity, the following considerations are restricted to a system consisting of a nuclear spin I=1/2 coupled to an electron spin S=1/2. This system has two nuclear transitions with the angular frequencies A eff = I ---------- . 2 [10.37] Depending on the sign of the gyromagnetic ratio of the nuclear spin, the sign of the hyperfine coupling, and the relative magnitudes of the two interactions, the transition frequencies can be 1 The argument applies strictly only in frequency domain, but the qualitative conclusions are true also in field domain. 209 either negative or positive and they can have either the same or a different sign. The absolute sign of the frequency can only be detected if the nuclear transitions are directly excited with circularly polarized radiofrequency (rf) irradiation. As linearly polarized irradiation is used this information is lost. The relative sign can be detected in experiments that correlate the two transitions, such as in the hyperfine sublevel correlation (HYSCORE) experiment (Section 10.4.2.3). The sign uncertainty complicates assignment and interpretation of the spectra. This is first discussed for the case of isotropic hyperfine coupling. The same considerations apply to spectra of single crystals. 10.4.2.1 Isotropic case Consider the case of the phenyl radical whose EPR spectrum is shown in Fig. 10.13. At X-band frequencies the nuclear Zeeman frequency I=I/2 is about -15 MHz. The hyperfine couplings are A1/2 = 5.7 MHz for the para proton H4, A2/2 = 17.5 MHz for the meta protons H3 and H5, and A3/2 = 48.8 MHz for the ortho protons H2 and H6. The couplings of the para and meta protons are in the weak coupling regime A 2 I (weak coupling) [10.38] while the couplings of the ortho protons are in the strong coupling regime A 2 I (strong coupling) . [10.39] In the weak coupling case the doublet is centered at |I| and split by |A|/2(Fig. 10.17A,B)1 In the strong coupling case, one of the frequencies has opposite sign, corresponding to precession of the magnetization with an opposite sense of rotation. As the sign is lost, the corresponding line is “mirrored” to positive frequencies. The doublet is thus centered at |A|/4|A|/2 on an angular frequency scale) and split by 2|I|. When different nuclear isotopes contribute to the spectrum, transitions of strongly coupled low- nuclei, e.g. 14 N, may thus overlap with transitions of weakly coupled high- nuclei, e.g. 1H. This problem is common at X-band 1 In the literature, A is often used as an angular frequency in equations but as a frequency in spectra or tables. Beware! 210 frequencies and below. The overlap may be eliminated by going to higher frequencies or by HYSCORE. 10.4.2.2 Anisotropic case In the regime of very weak coupling A « I , the high-field approximation is valid for the nuclear spins. The nuclear spins are then quantized along the direction of the external magnetic field. The hyperfine interaction perpendicular to this direction is truncated, and Eq. [10.37] applies with Aeff = Az being the hyperfine coupling along the external field direction. A A1 wnuclear w(1H) 0 B A2 wnuclear w(1H) 0 A3 C 2w( H) 1 w(1H) 0 A3/2 wnuclear D w( H) 1 0 wnuclear Figure 10.17: Schematic nuclear spectrum of the phenyl radical. A) Subspectrum of the para proton (weak coupling). B) Subspectrum of the meta protons (weak coupling). C) Subspectrum of the ortho protons (strong coupling). The line at (relative) negative frequencies is “mirrored” to positive frequencies. D) Total spectrum. If the nuclear Zeeman and hyperfine interaction are of the same order of magnitude, the hyperfine field at the nucleus perpendicular to the external field (terms A Sˆ z ˆI x and A Sˆ z ˆI y zx zy of the Hamiltonian) can no longer be neglected. Instead of being non-secular, as in the highfield approximation, these terms are now pseudosecular. Together they contribute an effective interaction B = 2 2 A zx + A zy in the xy plane of the laboratory frame. The corresponding field 211 at the electron spin (terms A xz Sˆ x ˆI z and A yz Sˆ y ˆI z of the Hamiltonian) is usually still negligible as are the remaining off-diagonal terms, since the high-field approximation still applies to the electron spin with its much larger Zeeman interaction. Thus, the truncated Hamiltonian for the S=1/2, I=1/2 spin system becomes ˆ 0 = Sˆ z + ˆI z + ASˆ z ˆI z + BSˆ z ˆI x . H S I [10.40] A new x direction was chosen for the nuclear spin frame to simplify the Hamiltonian.1 The Hamiltonian is written in a singly rotating frame, where the electron spin space rotates with the mw frequency while the nuclear spin space is fixed with respect to the laboratory frame. Hence, S is a resonance offset, while I is the total nuclear Zeeman frequency. z wa a A/2 hb ha -A/2 b wb wI 2h b -B/2 B/2 a x Figure 10.18: Local fields on the nucleus in the mS=1/2 () and mS=-1/2 () states. The effective field directions for the two states differ; they are tilted by angles and with respect to the external field direction z. When divided by the gyromagnetic ratio n, the Hamiltonian terms containing I spin operators can be interpreted as local fields at the nuclear spin (Fig. 10.12). The pseudosecular contribution causes a tilt of the effective field away from the external field direction z. As the pseudosecular contribution has different signs for the and states of the electron spin, the effective field axis ist tilted by angles and in different directions. As the effective field component along z is once the sum and once the difference of the nuclear Zeeman field and the secular hyperfine field, the magnitudes of and also differ. The quantization axis of the nuclear spin thus is no longer well defined and mI is no longer a good quantum number. 1 This is convenient, unless the nuclear spins are directly irradiated by rf. 212 This has two important consequences. First, an mw pulse that excites the electron spin will also have an influence on nuclear spin state, as the direction of the local field at the nucleus is changed. To some extent the pulse thus excites forbidden transitions in which both mS and mI change. This is the basis of ESEEM experiments (Section 10.4.4). Second, Eq. [10.37] for the nuclear frequencies does no longer apply. The two nuclear frequencies are now given by 2 = 2 + A +B ------- I 2 4 = 2 B – A + ------- , I 2 4 [10.41] and 2 [10.42] as can be inferred by vector addition (Fig. 10.18). The doublet is now centered at a frequency that is slightly higher than I while the splitting is slightly smaller than A. For A = 2 I the hyperfine and nuclear Zeeman field along z cancel exactly for one of the two electron spin states, either or . In this situation of exact cancellation the effective field is within the xy plane, the nuclear frequency is B/2, and the transition moments of ‘allowed’ and ‘forbidden’ transitions are nearly the same. For quadrupolar nuclei (I>1/2) with small hyperfine anisotropy the nucleus at exact cancellation experiences a near zero-field situation. Narrow lines at the zero-field nuclear quadrupole resonance (NQR) frequencies are then observed. In macroscopically disordered systems, such as powders, glasses, and frozen solutions, A and B depend on the relative orientation of the hyperfine tensor PAS and the magnetic field. For a hyperfine tensor with axial symmetry and principal values (aiso+2T, aiso-T, aiso-T), the orientation dependence is fully described by the angle between the unique axis of the hyperfine tensor and the magnetic field, A = a iso + T 3 cos2 – 1 , [10.43] B = 3T sin cos . [10.44] 213 Powder patterns computed from Eq. [10.41-10.44] are shown for weak coupling, exact cancellation, and strong coupling in Fig. A, B, and C, respectively. wa wb A wI 0 B wI 0 B/2 wb 0 wa C aiso/2 Figure 10.19: Simulated anisotropic nuclear frequency spectra for an S=1/2, I=1/2 system with weak coupling (A), at exact cancellation (B), and with strong coupling (C).The same sign is assumed for aiso and I. 10.4.2.3 Hyperfine sublevel correlation (HYSCORE) The frequencies of a pair of transitions of the same nucleus in the electron spin and manifold can be correlated in a 2D experiment by using a 180 mw pulse for mixing. To keep with usual notation in EPR literature flip angles are given in radians from now on. A 180 pulse is thus a pulse and a 90 pulse a /2 pulse. The correlation peaks created by mixing with the pulse appear in the quadrant ( 1 0 , 2 0 ) for weakly coupled nuclei, while they appear in the quadrant ( 1 0 , 2 0 ) for strongly coupled nuclei. A schematic spectrum for the phenyl radical is shown in Fig. 10.20 and can be compared to the EPR spectrum in Fig. 10.13 and the one-dimensional nuclear spectrum in Fig. 10.17. For macroscopically disordered systems with anisotropic hyperfine coupling, curved ridges result. The anisotropy of the hyperfine coupling can be extracted from their curvature and their shift with respect to the antidiagonal at I, even if only part of the correlation ridge can be observed (Fig. 10.21). 214 1 2n ( H ) n2 A2 A1 A3/2 -n(1H) 0 n2 n(1H) Figure 10.20: Schematic HYSCORE spectrum for the phenyl radical. The two doublets of the weakly coupled para and meta protons are separated from the doublet of the strongly coupled ortho protons. wI l na go ia tid an a T+ is o 2 is o I 9T Iw I 32 +a 2T Figure 10.21: Schematic HYSCORE spectrum for a macroscopically disordered system with ansiotropic hyperfine coupling.The dashed line is the I antidiagonal. 215 10.4.3 ENDOR spectroscopy 10.4.3.1 Davies ENDOR The polarization of electron spin transitions is at least 660 times larger than the one of nuclear spin transitions. In ENDOR experiments, part of this polarization is transferred to the nuclear transitions, so that these transitions can be detected with much higher sensitivity. The detection is also performed on electron spin transitions. Because of the higher energy of mw photons compared to rf photons, this results in an additional sensitivity gain. For liquid samples, ENDOR is best performed as a cw experiment. As cw ENDOR involves simulataneous detection and strong mw and rf irradiation, it is a technically demanding experiment. It is often easier to analyze the more complicated EPR spectra or to perform a pulsed ENDOR experiment on a frozen solution, which yields additional information on the anisotropy of the hyperfine couplings. For solid samples, pulsed ENDOR experiments have replaced cw ENDOR almost completely, as they are technically less demanding and work over a broader range of temperatures. Therefore, only pulsed ENDOR is discussed in this lecture course. p p p/2 t T t m.w. 0 1 p r.f. 2 Figure 10.22: Pulse sequence of the Davies ENDOR experiment. The influence of a rf pulse with variable frequency on echo inversion is observed. The echo is formed and inverted by selective mw pulses with an excitation bandwidth that is smaller than the hyperfine coupling. Level populations at points 0 (thermal equilibrium), 1 (after inversion), and 2 (after the rf pulse) are shown in Fig. 10.23. The conceptually most simple ENDOR experiment is Davies ENDOR (Fig. 10.22). In this experiment one of the two electron spin transitions of a hyperfine multiplet is selectively inverted by an mw pulse. The influence of this inversion pulse on level populations for the S = 1/2, I = 1/2 system is shown in Fig. A,B. The pulse creates a state of two-spin order. An 216 ideally selective mw pulse creates polarization with opposite sign on the two individual nuclear spin transitions. This polarization is just as large as the thermal equilibrium polarization of the electron spin transitions. C RF A |aañ |abñ B D MW |bañ |bbñ E RF Figure 10.23: Polarization transfers in the Davies ENDOR experiment for an S = 1/2, I = 1/2 system. A) Thermal equilibrium populations corresponding to point 0 in the pulse sequence shown in Fig. 10.22. Only the electron transitions are significantly polarized. B) The selective mw pulse has generated a state of two-spin order, where both nuclear transitions are oppositely polarized (point 1 in the pulse sequence). C-E) Situations corresponding to point 2 in the pulse sequence for different frequencies of the selective rf pulse. C) Situation after an rf pulse that is resonant with the transition. Both electron transitions are saturated, i.e. devoid of polarization. D) Situation after a non-resonant rf pulse. The originally excited electron transition is still inverted. E) Situation after an rf pulse that is resonant with the transition. Both electron transitions are saturated. The rf pulse is also transition-selective. It inverts either the transition (Fig. 10.23C) or the transition (Fig. 10.23E) or it is off-resonant (Fig. 10.23D). In the cases where the rf pulse is resonant with one of the nuclear transitions, both electron transitions become saturated. In this situation the subsequent detection echo subsequence, consisting of selective mw /2 and pulses on transition , does not give rise to an echo (dotted zero line in Fig. 10.22). For an off-resonant rf pulse the transition is still inverted and a negative echo is observed (solid signal line). A Davies ENDOR spectrum of a single crystal of a copper(II) complex diluted into the corresponding nickel(II) complex is shown in Fig. 10.24A. The lines are due to protons with 217 small hyperfine couplings (arrows) and directly coordinated nitrogens with large hyperfine couplings. Note that unlike integral NMR line intensities, ENDOR line intensities are not proportional to the number of observed nuclei. This is because the rf field amplitude is not constant over such a broad frequency range. Furthermore the rf field at the nucleus is enhanced by the hyperfine coupling. This hyperfine enhancement results from the non-resonant adiabatic motion of the magnetic moment of the electron spin induced by the rf field. A C N HN HN Cu B 5 N 10 nrf (MHz) NH N H 20 15 Figure 10.24: Davies ENDOR spectra of [(2,2’-bipyridylamine)(diethylenetriamine)Cu]2+ diluted into a single crystal of the corresponding isomorphous Ni(II) complex obtained at a temperature of 10 K. A) Lengths of the mw pulses t(1) = 400 ns, t = 200 ns, t(2) = 400 ns, both proton and nitrogen transitions are observed, arrows indicate proton transitions; B) Lengths of the mw pulses t(1) = 20 ns, t= 100 ns, t(2) = 200 ns, most of the proton transitions are suppressed. C) Structure of the complex (adapted from: C. Gemperle, A. Schweiger, Chem. Rev. 1991, 91, 1481-1505). Polarization transfer in Davies ENDOR crucially depends on the selective excitation of one electron spin transition. The excitation bandwidth of the mw pulses thus has to be smaller than the hyperfine coupling. With decreasing hyperfine coupling excitation thus has to be more and more selective, so that the experiment looses sensitivity. Davies ENDOR is thus best suited to detect nuclei with relatively large hyperfine couplings (5-50 MHz). The suppression of signals with small hyperfine couplings can be described by a selectivity parameter 1 A eff t S = ------------------ , 2 [10.45] 218 1 where Aeff is the hyperfine splitting and t the length of the first mw pulse. Maximum absolute ENDOR intensity Vmax is obtained for S = 2 2 . As a function of S, the absolute ENDOR intensity is given by 2 S V S = V max --------------------- . 2S + 1 2 [10.46] Eq. [10.46] describes the hyperfine contrast selectivity that can be used to edit spectra. By 1 shortening the mw pulse length t from 400 to 20 ns in the example shown in Fig. 10.24 ENDOR lines from the weakly coupled protons are strongly suppressed. Ghom A B nmw Ginh G > 2/T2 W 0 W Figure 10.25: Effect of selective irradiation of an inhomogeneously broadened spectral line. A) The inhomogeneously broadened line is a superposition of lines of spin packets with a smaller homogeneous linewidth determined by T2 and different resonance offsets. B) The spin packets can be individually excited, so that a hole is burnt into the spectral line. The minimum width of the hole for very weak irradiation is the homogeneous linewidth 2/T2. For pulsed irradiation, the width of the whole is determined by the maximum of 2/T2 and the excitation bandwidth of the pulse. The suppression effect can be understood by considering the polarization changes in an inhomogeneously broadened EPR line. Such a line with width inh consists of many superimposed lines of spin packets with different resonance frequency (Fig. 10.25). Each of these lines has a homogeneous linewidth hom = 2 T 2 . With weak continuous mw irradiation it is possible to saturate just one spin packet and burn a hole with width hom into the line. An mw pulse affects spin packets within its excitation bandwidth. Sensitivity of pulsed EPR experiments increases with the number of excited spin packets. A resonant rf pulse shifts half of the hole (half of the electron spin polarization) to a transition whose resonance frequency differs by the hyperfine splitting Aeff. This shift occurs towards higher or lower frequencies, depending on which of the transitions or was inverted by the mw pulse. Thus half of the whole remains at zero 219 resonance offset, while a quarter is shifted by Aeff and another quarter by -Aeff (Fig. 10.26). This shift of the hole can only occur if Aeff is larger than the width of the hole. Aeff B A Aeff W G » 2p/tp W Figure 10.26: An on-resonant rf pulse in Davies ENDOR shifts intensity from a hole burnt into the inhomogeneous EPR line to side holes. A) Situation before the rf pulse (point 1 in Fig. 10.22). B) Situation after the rf pulse (point 2). 10.4.3.2 Mims ENDOR The inversion of spin packets can be performed in a more controlled way by splitting the mw pulse in two /2 pulses that are separated by an interpulse delay . This can be seen by computing the effect of such a subsequence (/2)x--(/2)x on an inhomogeneously broadened line with product operator formalism. For brevity, only the electron spin S is ˆ 0 = Sˆ z . The experiment starts at considered so that the rotating-frame Hamiltonian is H S thermal equlibrium with density operator ̂ = – Sˆ z .1 Application of an ideal mw /2 pulse eq along x, ̂ eq 2 Sˆ x ̂ 1 , [10.47] results in ̂ 1 = Sˆ y . [10.48] ˆ 0 the density operator is given by After subsequent evolution for time under H ̂ 2 = cos S Ŝ y – sin S Sˆ x . 1 [10.49] The negative sign is a consequence of the negative electron charge that leads to preferential population of states of electron spins with their spin antiparallel to the external field. 220 The following /2 pulse along x leaves the second term on the right-hand side of Eq. [10.49] unaffected, while the first term is converted to negative polarization ̂ 3 = cos S Ŝ z – sin S Sˆ x . [10.50] . The second term on the right-hand side of Eq. [10.50] decays by transversal relaxation or can be removed by a phase cycle. The first term corresponds to a polarization grid in the inhomogeneously broadened EPR line whose resolution depends on interpulse delay , but not on the excitation bandwidth (Fig. 10.27). The excitation bandwidth determines the envelope of the grating. It is thus possible to create a finely spaced grid to which many spin packets contribute. DW 1/t Figure 10.27: Polarization grating in an inhomogeneous line created by a (/2)x--(/2)x subsequence (numerical simulation). Unlike in the product operator treatment, the limited excitation bandwidth of the mw pulses is considered in this simulation. The polarization grid can be detected by applying another mw /2 pulse. This creates an FID signal of the oscillatory grating. As the Fourier transform of a cosine function is a Dirac delta function, the FID signal corresponding to the first term on the right-hand side of Eq. [10.50] is confined at time . It is a stimulated echo signal. The width of the echo is determined by the excitation bandwidth of the /2 pulses and is thus comparable to the pulse length. In the Mims ENDOR experiment the effect of an rf pulse with variable frequency on the intensity of the stimulated echo is observed (Fig. 10.28). An on-resonant rf pulse shifts two quarters of the polarization grating by Aeff and -Aeff, respectively, just as the rf pulse in Davies ENDOR shifts two quarters of the burnt hole (compare Fig. 10.26). The unshifted half of the grating and the two shifted quarters interfer. For A eff = 2k + 1 with k = 0 1 interference is destructive. The stimulated echo is canceled. For an on-resonant rf pulse and A eff = 2k , the shifted gratings interfer constructively and the stimulated echo is nearly 221 unaffected. This situation cannot be distinguished from the one for an off-resonant rf pulse. The Mims ENDOR experiment thus features blind spots at A eff = 2k .. p/2 p/2 t m.w. p/2 T ^ s ^ s 0 1 ^ s ^ s 2 3 t p r.f. Figure 10.28: Pulse sequence for the Mims ENDOR experiment. The intensity of the stimulated echo is observed as a function of the frequency of the rf pulse. The density operators ̂ 1 ̂ 3 are given in Eq. [10.47-10.50] with ̂ 0 = ̂ eq . The ENDOR efficiency 1 F ENDOR = --- 1 – cos A eff 4 [10.51] quantifies this blindspot behavior. The total intensity of the stimulated echo scales with e – 2 T 2 . By considering this and Eq. [10.51], it can be shown that very small hyperfine couplings are detected with highest sensitivity at = T2. To safely detect all large couplings, Mims ENDOR spectra have to be recorded at different and added. This averages the blind spots. 10.4.4 ESEEM spectroscopy and the HYSCORE experiment In NMR spectroscopy spin echo signals usually decay smoothly as a function of the interpulse delays. In EPR spectroscopy, the decay is often superimposed by modulations with nuclear frequencies. This electron spin echo envelope modulation (ESEEM) effect allows for a detection of nuclear spectra without directly exciting the nuclear spins. The indirect excitation of the nuclear spins arises from the tilt of the effective field at the nucleus with respect to the external field (Fig. 10.18). Excitation of coherence on formally forbidden transitions can be understood by transforming the mw Hamiltonian to the eigensystem of the static Hamiltonian given in Eq. [10.40]. 222 10.4.4.1 Transition moments for allowed and forbidden transitions ˆ 0 of the S=1/2, I=1/2 spin system is given in a In Eq. [10.40] the static Hamiltonian H system where the z axes of both the electron and nuclear spin frame are parallel to the external magnetic field and perpendicular to the linearly polarized mw field. In this frame the oscillatory mw Hamiltonian, which describes the excitation, is given by ˆ 1 = Sˆ x , H 1 [10.52] with 1 = g eff x B B 1 h . The effective g value geff,x pertains to the x direction of the mw field. The spin transitions are, however, defined between eigenstates of the spin system. To compute the excitation strengths for these transitions, Eq. [10.52] thus has to be transformed to ˆ 0 . The necessary transformations can be read off directly from Fig. 10.18. the eigenframe of H To bring the effective field axis to the z axis, the subspace of the electron spin S has to be rotated clockwise about the y axis of the nuclear frame by angle . This is described by a unitary transformation U = exp – i Sˆ ˆI y , [10.53] –B = arc tan ------------------- . 2 I + A [10.54] where is defined as The polarization operator Sˆ is given by Sˆ = Eˆ 2 + Sˆ z . The negative sign of the argument of the arcustangens arises since a clockwise rotation is opposite to a rotation in mathematical sense. A clockwise rotation is required since the y axis of a right-handed frame points into the paper plane in Fig. 10.18. Likewise, the subspace corresponding to the state of S has to be rotated anticlockwise about the y axis of the nuclear frame by angle . This is described by a unitary transformation U = exp – i Sˆ ˆI y with [10.55] 223 B = arc tan ------------------- , 2 I – A [10.56] and the polarization operator Sˆ = Eˆ 2 – Sˆ z . In this frame the static Hamiltonian for the weak coupling case1 takes the form ˆ 0 = Sˆ z + Sˆ ˆI z + Sˆ ˆI z . H S [10.57] Since Sˆ ˆI y and Sˆ ˆI y commute, the transformation from Eq. [10.40] to Eq. [10.57] can be described by a single unitary transformation U EB = U U = U U = exp – i Sˆ ˆI y + Sˆ ˆI y . [10.58] ˆ 0 is more simple with The transformation of Eq. [10.52] to the eigensystem of H Cartesian operators. By substituting the polarization operators, UEB takes the form U EB = exp – i 2Ŝ z ˆI y + Î y , [10.59] – + = ------------------- , = ------------------- . 2 2 [10.60] with Since [ˆI y,Sˆ x] = 0 , the second term in the argument on the right-hand side of Eq. [10.59] has ˆ 1 . In the eigenbasis, the oscillatory Hamiltonian takes the form no effect on H ˆ 1EB = U H ˆ ˆ ˆ ˆ † H EB 1 U EB = cos 1 S x + sin 1 S y I y . [10.61] The first term on the right hand side of Eq. [10.61] describes excitation of the allowed transitions. The second term is more easily interpreted when the product operator is written in terms of ladder operators, - + + + - 1 + Sˆ y ˆI y = --- Sˆ ˆI + Sˆ ˆI – Sˆ ˆI – Sˆ ˆI . 4 1 [10.62] In the strong coupling case, the sign of either the or the term changes. Which sign changes depends on the relative signs of I and A. 224 The terms on the right hand side of Eq. [10.62] drive the forbidden electron-nuclear transitions (first two terms) and (last two terms) as indicated in Fig. 10.29A. A 1 aa wa ab 2 B w13 w24 w cos h 2 w14 4 w+ sin h 2 bb wb 3 w23 Ws ba Figure 10.29: Level scheme and schematic EPR spectrum of the S=1/2, I=1/2 system. A) Level scheme with allowed electron (green), forbidden electron-nuclear (red), and nuclear (blue) transitions. B) Spectrum with allowe (green) and forbidden (red) transitions, where - = – and + = + The transition moment of allowed transitions thus scales with cos, while the transition moment of forbidden transitions scales with sin. Spectral intensities are proportional to the transition probability, which is the square of the transition moment. This is because the transition moment applies to both excitation and detection. The EPR spectrum of the S=1/2, I=1/2 system with anisotropic hyperfine interaction thus has the appearance shown in Fig. 10.29B. The hyperfine splitting is given by the difference frequency A eff = - = – [10.63] and the splitting of the forbidden transitions by the sum frequency + = + . [10.64] 10.4.4.2 Two-pulse ESEEM Two-pulse ESEEM is observed by measuring the amplitude of a (/2)--()- echo as a function of the interpulse delay . The first /2 pulse excites coherence with amplitude cos on the two allowed transitions and coherence with amplitude sin on the two forbidden transitions. The coherence evolves during the first interpulse delay , defocuses due to the distribution of resonance offsets S, and decays with the transversal relaxation times T2a or T2f of the respective transitions. The pulse inverts the phase of all coherences and thus leads to a 225 refocusing of magnetization after another delay . Those pathways, where the coherence remains on the same transition during the transfer by the pulse, contribute two-pulse echoes that smoothly decay with the respective relaxation rates. B A |aañ |abñ |bañ |bbñ |aañ |abñ |bañ |bbñ áaa| áab| h ába| cos2 h -sin cos h sin h cos 2 h cos h sin h 2 cos h -sin h sin h sin2 h cos h ább| Figure 10.30: Branching of magnetization (coherence transfers) during the pulse in the two-pulse ESEEM experiment. A) Transfer of coherence on the allowed transition to the two allowed and two forbidden transitions combined with a phase inversion. B) Transfer of coherence on the forbidden transition to the two allowed and two forbidden transitions combined with a phase inversion. However, the pulse also changes the nuclear spin states with a probability sin2, and this corresponds to a transfer of coherence between the different transitions. This branching of magnetization is indicated in a density matrix representation in Fig. 10.30. For instance, coherence on the allowed transition is merely phase-inverted with probability cos2and transferred with probability sin()cos() to the forbidden transition (Fig. 10.30A). In contrast, coherence on the forbbiden transition is phase-inverted only with the small probability sin2and transferred to the forbidden transition with the larger probability cos2 (Fig. 10.30B). All transferred coherences also experience phase inversion. Thus they also form echoes. These coherence transfer echoes oscillate with the difference of the transition frequencies before and after the pulse. From Fig. 10.29 and Eqs. [10.63] and [10.64] it can be verified that the possible difference frequencies are , , , and . Each of the basic nuclear frequencies and appears in two coherence transfer pathways, and each of the combination frequencies and appears in only one pathway. Furthermore, the product of the excitation probability (transition moment for excited coherence), transfer probability (branching factor), and detection probability (transition moment for detected coherence) is the 226 same for each pathway, it takes the value sin2 cos2 = sin2(2)/4. The amplitude of all nuclear modulations thus scales with the modulation depth parameter B I 2 k = sin2 2 = -------------- . [10.65] With proper bookeeping of the coherence transfer pathways, the two-pulse ESEEM formula k V 2p = 1 – --- 2 – 2 cos – 2 cos + cos - + cos + 4 [10.66] results. A schematic spectrum for the weak coupling case is shown in Fig. 10.31. Note that he combination frequencies are close to A and 2I, but are slightly shifted towards lower and higher frequencies, respectively. The phase information (positive basic frequency peaks and negative combination peaks) is often hard to access, as spectrometer dead time prevents observation of the signal at short values. Usually phase correction fails and magnitude spectra are displayed, where all peaks are positive. 2wI A wI Figure 10.31: Schematic two-pulse ESEEM spectrum for a weakly coupled S=1/2, I=1/2 system. 10.4.4.3 Three-pulse ESEEM Nuclear frequencies are measured by two-pulse ESEEM as differences between the frequencies of electron spin transitions. Hence, the linewidth in the spectra is twice the homogeneous EPR linewidth, which is much larger than the natural linewidth for nuclear 227 transitions. The resolution can thus be drastically improved by observing the evolution of coherence on nuclear transitions. B A |aañ |abñ |bañ áaa| |bbñ |bbñ i cosh 2 - i sinh 2 áab| 2i sinh - i cosh 2 ába| ább| |aañ |abñ |bañ i sinh 2 - i sinh 2 i cosh 2 i cosh 2 Figure 10.32: Magnetization branching during the second /2 pulse in a mw pulse subsequence 2 – – 2 . A) Coherence on the allowed electron spin transition is transferred to nuclear coherences with probability sin. B) Coherence on the forbidden transition is transferred to nuclear coherences with probability cos. A single, ideal mw pulse does not excite nuclear coherence. However, an mw pulse subsequence (/2)--(/2) does generate such coherence due to magnetization branching during the second /2 pulse (Fig. 10.32). This nuclear coherence evolves during delay T and is then transferred to observable electron coherence by another /2 pulse. After another delay , the dispersion of electron spin resonance offsets S is refocused. The coherence transfer pathways that involve nuclear coherence contribute signals that oscillate with frequencies and as a function of time T. Coherence transfer pathways that do not involve nuclear coherence give rise to a smoothly decaying stimulated echo. The total echo signal as a function of the fixed interpulse delay and the variable delay T, neglecting relaxation, is given by k V 3p T = 1 – --- 1 – cos 1 – cos T + 4 + 1 – cos 1 – cos T + . [10.67] No oscillation with combination frequencies is observed. This simplifies spectra and is an additional advantage of three-pulse ESEEM compared to two-pulse ESEEM. However, the amplitude of the nuclear oscillation now depends on the nuclear frequency in the other electron spin state and on the fixed delay . This leads to blind spots, which are a disadvantage of three- 228 pulse ESEEM. To avoid suppression of peaks, the experiment thus has to be performed at several values and the magnitude spectra have to be added. p/2 p/2 t p/2 t T Figure 10.33: Pulse sequence for three-pulse ESEEM. The echo amplitude is observed as a function of interpulse delay T for fixed delays . 10.4.4.4 HYSCORE Hyperfine sublevel correlation spectra, as introduced in Section 10.4.2.3, can be measured by a two-dimensional extension of the three-pulse ESEEM experiment. For that purpose, an mw pulse is introduced at a variable delay t1 after the second /2 pulse (Fig. 10.34). This pulse transfers nuclear coherence between the two manifolds corresponding to the electron spin states and . Thus it correlates frequency of a given nucleus with frequency of the same nucleus and vice versa. The new frequency is measured by introducing another variable delay t2 between the pulse and the final /2 pulse that reconverts the nuclear coherence to observable electron coherence. p p/2 p/2 t p/2 t1 t2 t Figure 10.34: Pulse sequence of the two-dimensional HYSCORE experiment. Echo amplitude is observed as a function of the two variable delays t1 and t2 for fixed delays . Forbidden transitions during the pulse convert some nuclear coherence to electron polarization and vice versa. This leads to axial peaks at 1 = 0 and 2 = 0. These peaks are unwanted, as they do not contain correlation information. They can be removed by baseline correction of the time-domain data before Fourier transformation. Because of its limited excitation bandwidth, the nominal pulse does not fully invert the electron spin for large resonance offsets S. This leads to diagonal peaks 1 =2 = and 1 = 2 = . These 229 diagonal peaks cannot reliably be removed by data processing and may obscure cross peaks for small hyperfine couplings. Their contribution scales with the ratio between the excitation bandwidths of the /2 pulses and the pulse. Therefore, it is advantageous to use the full mw power for the pulse and less power for the /2 pulses, as indicated in Fig. 10.34. The modulation in the HYSCORE experiment corresponding to the cross peaks is described by k V 4p t 1 t 2 = --- sin ---------- sin --------- V t 1 t 2 + V t 1 t 2 2 2 2 [10.68] with V V t 1 t 2 = cos2 cos t 1 + t 2 + + --- – sin2 cos t 1 – t 2 + - --- ,[10.69] 2 2 t 1 t 2 = cos2 cos t 1 + t 2 + + --- – sin2 cos t 1 – t 2 + - --- .[10.70] 2 2 In this representation with unsigned nuclear frequencies, the weak coupling case corresponds to cos2 sin2 and the strong coupling case to sin2 cos2 . In the former case, the cross peaks are thus stronger in the quadrant (1>0, 2>0) and in the latter case in the quadrant (1>0, 2<0). The blind spot behavior of the HYSCORE experiment is described by the two factors sin 2 and sin 2 in Eq. [10.68]. A cross peak is thus suppressed when one of the two correlated nuclear peaks is suppressed in three-pulse ESEEM. Therefore it is even more important in HYSCORE to add several magnitude spectra obtained at different delays . 10.5 Spin probes, spin labels, and spin traps Structure and dynamics of complex biological systems or synthetic materials are often difficult to characterize by techniques that derive their signal from the whole sample. The problems are a lack of contrast between different components, insufficient resolution, and ambiguities in signal assignment. The situation simplifies considerably with techniques that detect signals only from a probe molecule that is attached to a certain site of interest in the system. EPR spectroscopy can be used as such a probe technique for diamagnetic systems. The 230 spectroscopic response of the spin probe is interpreted in terms of structure and dynamics of the system. Ideal probes do not perturb the system under investigation, can be directed at will to sites of interest, and give a spectroscopic response that strongly depends on their environment. The EPR spin probe technique is particularly suitable for soft organic matter, as it does not rely on long-range order and as the spectra of typical spin probes are very sensitive to molecular dynamics on time scales between 10 ps and 1 s. This time range corresponds to motion on length scales longer than a chemical bond, but smaller than colloidal dimensions that start at about 100 nm. On these length scales supramolecular interactions determine the self-organization, stability, and functionality of soft matter systems. Distances between spin probes can be measured on length scales between about 0.8 and 8 nm. O R N H N O 1 O R 2 N O R N O 3 O H3 C S S O N O 5 6 4 N O Figure 10.35: Structures of the most common spin probes and labels. R is a functional group used for dierctiong the probe or attaching the label to the site of interest. 1 4-oxo-2,2,6,6-tetramethyl-piperidin. 2 2,2,6,6-tetramethyl-piperidin-1-oxyls, TEMPO derivatives. 3 2,2,5,5-tetramethyl-pyrrolydine-1oxyls, PROXYL derivatives. 4 2,2,5,5-tetramethyl-pyrroline-1-oxyls. 5 4,4,-dimethyl-3oxazolidinyloxy probes, DOXYL derivatives. 6 Methanethiosulfonate spin label, MTSSL. The most widely applied class of spin probes are nitroxides (Fig. 10.35). They are originally derived from compound 1, 4-oxo-2,2,6,6-tetramethyl-piperidin, which is the basis compound of hindered amine light stabilizers. This compound is formed in a one-pot reaction from acetone and ammonia and is easily oxidized to TEMPON, 4-oxo-2,2,6,6-tetramethyl-piperidin-1-oxyl (compound 2 with -R being =O). The carbonyl group can be converted to other functional groups (e.g. OH, -NH2, -N+(CH3)3). Such groups direct site attachment via hydrogen bonds or electrostatic interactions (spin probes in a strict sense). Reactive groups R can be used for covalent attachment to the site of interest (spin labels). Somewhat smaller probes or labels are obtained by reducing the ring size to 231 PROXYL derivatives 3 or 3,4-dehydro-PROXYL derivatives 4. The most important label is the methanethiosulfonate spin label (MTSSL) 6 that attaches with high selectivity under very mild conditions to cysteine residues in proteins. The size of the MTSSL-derived sidechain is similar to the one of an aromatic amino acid residue. The SDSL technique for proteins involves mutation of an amino acid at the site of interest to cystein and subsequent reaction with a thiol-selective spin labels such as MTSSL, iodoacetamido-PROXYL, or maleimido-PROXYL. For distance measurements two selected sites are labeled. Synthesis of DOXYL derivatives 5 is more difficult and proceeds with lower yields. Their advantage is a very rigid attachment to alkyl chains that allows to probe motion and liquid crystalline ordering of surfactant molecules, lipids, and steroids in more detail. The unpaired electron in nitroxides is stabilized by delocalization over the N-O bond and by the steric hindrance due to the four methyl groups that prevents dimerization. Derivatives of TEMPO, PROXYL, and 3,4-dehydro-PROXYL are usually stable up to temperatures of 130 C. They are degraded by strong acids (pH 1 and below) and slowly decompose in solution at pH 2 and below. Nitroxides are reduced to the corresponding hydroxylamines by ascorbic acid and may be reduced by some thiols. PROXYL derivatives are less susceptible to this reduction than TEMPO derivatives. The hydroxylamines are reoxidized by air. 14 14 A( N) A( N) z y N . O x R H Figure 10.36: Molecular frame (PAS of the g tensor) for nitroxides and dependence of the hyperfine coupling on orientation of the magnetic field with respect to this frame. 14N 232 10.5.1 Nitroxide spectra with and without orientational averaging The unpaired electron in nitroxides is distributed mainly in a orbital along the N-O bond that is made up of the pz orbitals of the nitrogen and oxygen atom. There is also some spin density in the s orbitals of these two atoms, but few delocalization to the neighboring carbon atoms. The deviation of the g value from ge is due to spin orbit coupling in excited states that involve the lone pair orbitals on oxygen. This deviation is maximum in the x direction of the molecular frame along the N-O bond (Fig. 10.36), intermediate in the y direction, and almost negligible in the z direction. Typical principal g values are gxx = 2.0090, gyy=2.0060, and gzz= 2.0024. In contrast, the hyperfine coupling to 14N is strongest if the external field is along the z direction, i.e. parallel to the lobes of the pz orbital on nitrogen. Differences between the hyperfine coupling in the x and y direction are minute, so that an axially symmetric hyperfine tensor can be assumed. Typical hyperfine couplings are Axx= Ayy = 18 MHz, Azz= 96 MHz. All magnetic parameters differ slightly between the different classes of nitroxides. The z axes of the hyperfine and g tensor are nearly coincident. Hyperfine couplings to the protons of the methyl groups in the 2- and 6-positions may be resolved in the limit of fast motion (see Section 10.5.2) in the absence of oxygen. In other cases the proton couplings are unresolved and lead to line broadening. Narrower lines are then obtained with deuterated nitroxides. E +1 0 -1 mS +1 0 -1 mI +1/2 hnmw -1/2 -1 0 +1 B0 Figure 10.37: Schematic energy level scheme and cw EPR spectrum of a nitroxide radical. Only the electron Zeeman and 14N hyperfine interactions are considered. The EPR spectrum of a nitroxide in a single orientation or in the regime of fast orientational averaging is schematically shown in Fig. 10.37. The energy levels of the unpaired electron result from electron Zeeman splitting in the external field (magnetic quantum number 233 mS) and hyperfine coupling to the 14N nuclear spin (product of quantum numbers mS and mI). For the allowed transitions mI is constant, so that a hyperfine triplet results. As the hyperfine coupling is positive, the transition with the highest frequency corresponds to mI= +1. In a field sweep this transition is detected at lowest field. When Fig. 10.37 is interpreted as a spectrum in the regime of fast orientational averaging, the center line is at a field corresponding to the isotropic g value giso= (gxx+gyy+gzz)/3 and the splitting between two outer lines is 2Aiso= 2(Axx+Ayy+Azz)/3. If the Figure is interpreted as a single-crystal spectrum, the center line is positioned at the g value along the magnetic field direction in the molecular frame and the splitting is determined by the hyperfine coupling along this direction. The powder spectrum is the superposition of the spectra at all orientations with proper weighting. Each of the lines with different 14N magnetic quantum number mI can be considered separately (Fig. 10.38A). The variation of the resonance field is minimum for the mI= 0 transition, where the hyperfine contribution vanishes. It is maximum for the mI= -1 transition, where the hyperfine contribution and electron Zeeman contribution have the same sign, and intermediate for the mI= +1 transition, where the two contributions have different sign. A 2Azz B mI=+1 mI=0 mI=-1 0.34 0.345 B0 (T) 0.35 0.34 0.345 0.35 B0 (T) Figure 10.38: Nitroxide spectrum for a macroscopically disordered system at X band frequencies. A) Absorption subspectra of the three transitions with different 14N nuclear magnetic quantum number mI. B) Total absorption spectrum, resulting from the sum of the three subspectra (top) and its derivative (bottom) that corresponds to the cw EPR spectrum. The total absorption spectrum is the sum of the two subspectra (Fig. 10.38B). It can be measured by echo-detected field-swept EPR. In cw EPR the derivative of this absorption spectrum is observed. It has a positive peak at the low-field edge that corresponds to the edge of the mI= +1 line and a negative peak at the high-field edge that corresponds to the edge of the mI= -1 line. The outer extrema splitting between these two peaks is twice the hyperfine 234 coupling along the molecular z orientation 2Azz. The central line with a quasi-dispersive shape results mainly from the mI= 0 transition. 10.5.2 Nitroxide spectra for incomplete orientational averaging The effect of rotational motion of the nitroxide on the spectrum can be understood in terms of a multi-site exchange between different orientations. The exchange rate is related to the rotational diffusion rate R, which in turn is given by R= 1/6r, where r is the rotational correlation time. The basic concepts are the same as for two-site chemical exchange (Sections 3.1 and 3.2). In the limit of slow exchange, the spectrum is lifetime broadened (Figure 3.8). This effect is unresolved in nitroxide spectra. If the motion becomes faster, exchange between orientations in addition to the line broadening leads to partial averaging of the two resonance frequencies (Fig. 3.6). As a consequence the outer extrema splitting 2A’zz reduces (Fig. 10.39). This effect becomes observable at rotational correlation times shorter than about 1 s. Fast regime Slow tumbling 2Azz' » 3.0 mT tr tr 10 ps 4 ns 32 ps 10 ns 100 ps 32 ns 316 ps 200 ps 1 ns 316 ns 3 ns 1 µs 2Azz' » 6.8 mT Figure 10.39: Dependence of the nitroxide cw EPR spectrum on the rotational correlation time r. 235 For a symmetric two-site exchange a normalized exchange rate constant of 1 2 2 corresponds to coalescence, i.e. a collapse of the doublet into a single line, and maximum exchange broadening (Fig. 3.6). For the multi-site orientational exchange in nitroxides such coalescence is observed at a rotational correlation time of about 3.5 ns, where the appearance of the spectrum changes from a powder like lineshape (right column in Fig. 10.39, slow tumbling) to a triplet of Lorentzian broadened lines (left column, fast regime). In the fast regime the outer extrema splitting 2A’zz is only slightly larger than twice the isotropic hyperfine coupling (Fig. 10.40). The outer extrema splitting depends most strongly on r at coalescence, where it is 2A' zz 5 mT. Dynamic processes are often studied as a function of temperature. They can then be characterized by the temperature T5mT (or T50G), where 2A' zz = 5 mT is attained. 2Azz' (mT) 7 6 5 fast regime slow tumbling 4 3 -10 -9 -8 -7 log(tr /s) -6 Figure 10.40: Dependence of the outer extrema splitting 2A’zz on the rotational correlation time (semilogarithmic plot). For thermally activated processes, 1/r is expected to have an Arrhenius dependence on temperature, ln 1 r = A + E A RT , [10.71] where EA is the activation energy. The rotational correlation times at different temperatures can be determined by lineshape fitting. These considerations assume isotropic Brownian rotational diffusion, which is often a good approximation for small, almost spherical spin probes like TEMPO (compound 2, R=H). Different lineshapes are observed for more complicated dynamic processes. However, only in simple cases it is possible to derive a model for the motion from the spectral lineshape. 236 Lineshapes predicted from molecular dynamics simulations are often in good agreement with experimental lineshapes. In the fast regime, the widths of the three lines are different. This can be understood from Fig. 3.9. The mean square difference 1 – 2 2 of the resonance frequencies of exchanging orientations is smallest for the narrow central line with mI= 0 and largest for the broad high-field line with mI= -1. Therefore, the high-field line is broader also in the case of fast orientational exchange. Only in the fast limit, where other contributions to line broadening dominate, equal widths and amplitudes are observed for the three lines. This limit is attained at rotational correlation times of 10 ps or shorter. In the fast regime r can be obtained from analyzing the linewidths B according to Kivelson theory.1 According to this theory, the ratio of the line width of one of the outer lines to the line width of the central line is given by T 2–1 m I ------------------- = 1 + Bm I + Cm I2 , T 2–1 0 [10.72] 4 B = – ------ bB 0 T 2 0 r 15 [10.73] 1 C = --- b 2 T 2 0 r , 8 [10.74] where and with the hyperfine anisotropy parameter A xx + A yy 4 b = ------ A zz – ------------------------3 2 [10.75] and the electron Zeeman anisotropy parameter 2 B g xx + g yy = ------------- g zz – ---------------------- . h 2 1 D. Kivelson, J. Chem. Phys. 1960, 33, 1094-1106. Theory of ESR Linewidths of Free Radicals. [10.76] 237 The relaxation time T2(0) for the central line can be computed from the corresponding peak-topeak linewidth in field domain B pp 0 as h T 2 0 = ------------------------------------------------ . 3g iso B B pp 0 [10.77] Thus, Eqs. [10.72-10.74] can be solved for the only remaining unknown r. In practice, ratios of peak-to-peak line amplitudes I(mI) are analyzed rather than linewidth ratios, as they can be measured with higher precision. The linewidth ratio is related to the amplitude ratio by T 2–1 m I ------------------- = T 2–1 0 I mI ------------- , I0 [10.78] since the integral intensity of the absorption line is the same for each of the three transitions. The rotational correlation time can thus be determined by, e.g., 3 b 4B 0 r = ---------- --- – ---------------b 8 15 iso B ---------------- B pp 0 h –1 g I0 ------------- – 1 , I –1 [10.79] where B0 is the peak-to-peak linewidth of the central line. Similar formulas can be obtained by using different pairs of line amplitudes or all three line amplitudes. It is good practice to test whether several of these formulas give similar values for r. If this is not the case, the motion cannot be characterized by a single rotational correlation time. 10.5.3 Dependence of magnetic parameters on polarity and hydrogen bonds The g and hyperfine tensor of a nitroxide depend on the distribution of the unpaired electron between the orbitals on the nitrogen and oxygen atom. This distribution in turn is influenced by the polarity of the environment, as can be understood from the mesomeric structures shown in Fig. 10.41. Localization of the unpaired electron at the nitrogen atom corresponds to a charge-separated structure that is stabilized by a polar environment. Therefore more spin density N on the nitrogen and less spin density O on the oxygen atom is found in such a polar environment. N + O - N O 238 Figure 10.41: Mesomeric structures of a nitroxide with the unpaired electron localized on the nitrogen atom (left) and the oxygen atom (right). The isotropic 14N hyperfine coupling Aiso and the coupling along the lobes of the pz orbital (Azz) increase with increasing N and thus with increasing polarity. The deviation of the g value from ge changes in the opposite way, as it results mainly from spin-orbit coupling in an excited state that involves alone pair localized on oxygen. The strongest change is observed for gxx, as is expected when the lone pair orbitals have mainly py character.1 The g value shift gxx= gxx-ge is influenced not only by changes in the spin density distribution, but also by changes in the energy difference between the SOMO and the lone pair orbital. Hydrogen bonding lowers the energy of the lone pair and thus increases this energy difference. This corresponds to a blue shift of the n-* transition in optical spectra and to an increase of the denominator in Eq. [10.8]. Thus hydrogen bonding leads to a smaller g shift gxx at same polarity. Hydrogen bonding also leads to some delocalization of the lone pair orbitals, which leads to a minor further decrease of gxx. By measuring both Azz and gxx with high precision with high-field EPR, it is possible to separate the contributions due to polarity and hydrogen bonding. 10.5.4 Accessibility measurements The influence of Heisenberg exchange of unpaired electrons between paramagnetic species on relaxation times (see Section 10.4.1, Fig. 10.12) can be used to estimate the local concentration of a paramagnetic quencher near a nitroxide spin label. An ubiquitous paramagnetic quencher is triplet oxygen, which is much better soluble in apolar environments, such as lipid bilayers, than in polar environments. Water-soluble quenchers are transition metal complexes such as the electroneutral Ni(II) complex of ethylendiaminediacetic acid (EDDA) or the negatively charged Cr(III) complex of oxalate [Cr(C2O4)3]3-. At very high local concentrations such quenchers cause line broadening. At lower concentration their influence on relaxation times can be quantified easily and precisely by measuring saturation curves (see Section 2.7.2). 1 Fo a full discussion, see: T. Kawamura, S. Matsunami, T. Yonezawa, Bull. Chem. Soc. Japan 1967, 40, 1111-1115. Solvent Effects on the g-Value of Di-t-butyl Nitric Oxide. 239 The change in relaxtion rates is purely through collisions (Fig. 10.11) if the longitudinal relaxation time of the quencher is much shorter than the lifetime of the collisional encounter complex. In this regime it is equal to the exchange rate Wex, 1 1 ------ = ------ = W ex = k ex C , T 1 T 2 [10.80] where kex ist the exchange rate constant and C the local quencher concentration.1 In the following we assume the case of insignificant exchange broadening, W ex « 1 T 2 . Up to a constant factor , Wex can then be determined from saturation curves. For this, the peak-topeak amplitude A of the first derivative central line of the nitroxide spectrum (mI= 0) is measured as a function of mw power P. The data are described by the function I P A P = ---------------------------------------------------------- 1 1 + 2 – 1 P P 1 2 [10.81] with the power-independent amplitude factor I, the power at half saturation P1/2, and the inhomogeneity parameter as adjustable parameters. The inhomogeneity parameter takes the value = 1.5 in the homogeneous limit, which is assumed from here on, and = 0.5 in the inhomogeneous limit. More precisely, P1/2 is the incident mw power where A is reduced to half of its unsaturated value. This is given by 3 4–1 P 1 2 = ------------------------ , e2 2 T 1 T 2 [10.82] where = B 1 P is the conversion efficiency of the mw resonator. For small quencher concentrations exchange broadening is insignificant and T2 is the same in the presence and absence of the quencher. The change in P1/2 due to addition of the quencher is then given by P 1 2 1 3 4 – 1 W ex = ------------------------------- . e2 2 T 2 [10.83] The term ‘local’ refers to the characteristic diffusion length of the quencher on the time scale given by Wex. The factorization into kex and C may be ill-defined when comparing labels attached at different sites of a protein, but Wex remains a meaningful parameter. 240 To eliminate the dependencies on T2 and , a dimensionless accessibility parameter is defined as1 P 1 2 B pp 0 = -------------------------------------------------- = W ex , P 1 2 B pp 0 ref [10.84] where the denominator is obtained from measurements on a reference substance such as dilute diphenylpicrylhydrazyl powder in KCl. 10.5.5 Spin trapping In many processes that involve free radicals, these radicals react very fast, so that their lifetimes are extremely short and their steady-state concentrations extremely low. Such radicals can be detected indirectly by trapping them with a diamagnetic compound. The reaction product is a stable free radical whose magnetic parameters provide some information on the original, short-living radical. The most important class of diamagnetic spin traps are nitrones, such as -phenyl-t-butyl nitrone (PBN) or 5,5-dimethyl-1-pyrroline N-oxide (DMPO). Free radicals add to the carbon of nitrones and a nitroxide is formed (Fig. 10.42). The hydrogen bound to the carbon makes nitroxides generated by such reactions less stable than the nitroxides used as spin probes. The lifetime of spin trap products in solution typically ranges between a few seconds and a few hours. This is usually sufficient to accumulate steadystate concentrations that are detectable by cw EPR H a N PBN N O O +R· DMPO R +R· a N O · R N · H O Figure 10.42: Reaction of the spin traps -phenyl-t-butyl nitrone (PBN) or 5,5-dimethyl-1-pyrroline N-oxide (DMPO) with free radicals R· 1 C. Altenbach, W. Froncisz, R. Hemker, H. Mchaourab, W. L. Hubbell, Biophys. J. 2005, 89, 2103-2112. Accessibility of Nitroxide Side Chains: Absolute Heisenberg Exchange Rates from Power Saturation EPR. 241 . Figure 10.43: cw EPR spectrum observed by UV irradiation ( < 380 nm) of a dispersion of ZnO in heptane in the presence of oxygen and the spin trap DMPO (from C. Cheng, R. P. Veregin, J. R. Harbour, M. I. Hair, S. L. Issler, J. Tromp, J. Phys. Chem. 1989, 93, 2607-2609. Photochemistry of ZnO in Heptane: Detection by Oxygen Uptake and Spin Trapping. The hyperfine coupling to the -proton is usually resolved, so that a six-line spectrum results. In some cases, further couplings may be resolved. For instance, the spectrum of the DMPO adduct of the superoxide anion radical O2 (Fig. ) has twelve lines with couplings AN = 12.9 G, AH= 6.3 G, AH= 1.6 G. The coupling to one of the two -protons is not fully resolved. Spin trapping has its main field of application in studies on degeneration processes in living cells. Such processes involve reactive oxygen species, most of which are radicals. Furthermore spin-trapping is used in studies on hetrerogeneous catalysts that promote radical formation. The radical adducts of spin traps are mainly identified via the N and -1H 14 hyperfine couplings. The isotropic g value of the adduct also depends on the type of original radical, but is less characteristic. An internet database for identification of radical adducts of several widely used spin traps is maintained by the NIH (http://tools.niehs.nih.gov/stdb/ index.cfm). 242 10.6 Distance measurements by EPR techniques The high sensitivity of EPR spectroscopy and the larger magnetic moment of the electron spin allow for the measurement of longer distances between spins than is possible with NMR. In conjunction with SDSL this allows for studying structure and structural dynamics of biomacromolecules and their complexes, even if they cannot be crystallized and have molecular weights larger than 100 kDa. The main current application field are large membrane proteins. Similar techniques can also be applied to synthetic polymers or to soft matter that forms self-organized structures based on supramolecular interactions. Distance measurements are based on the inverse cube dependence of the magnetic dipole interaction on distance (Section 5.8). The data can be analyzed with high precison ( 0.5 Å for distances up to 40 Å) if the spin pairs are sufficiently diluted and exchange coupling (10.3.6.1) is negligible. On length scales between 0.8 and 8 nm, where such measurements are possible, spins are sufficiently diluted at concentrations of 200 M or less. Exchange couplings are usually negligble at distances beyond 1.5 nm. Shorter distances than 1.5 nm can be determined with a precision of about 2 Å. 10.6.1 cw EPR 10.6.1.1 Analysis of dipolar line broadening The peaks in a well resolved first derivative absorption spectrum of a nitroxide in the rigid limit have a typical width of 0.4 mT corresponding to 11.2 MHz. An additional broadening with about half this width can be safely detected. According to Eq. [10.32] this corresponds to a spin-spin distance of up to 21 Å that still leads to recognizable dipolar broadening. If the spin label environment is homogeneous and very similar for both labels, the peak width in cw EPR spectra is determined by unresolved hyperfine couplings of methyl protons in the label. The limit can then be shifted to distances of about 25 Å by using a deuterated label. If the environment is heterogeneous or different for the two labels, differences in hydrogen bonding and polarity may cause additional broadening due to a distribution of the 14N hyperfine coupling and g value (Section 10.5.3). The upper limit for 243 elucidating accurate distances then decreases to about 16 Å and cannot be extended by deuteration. The distance can be determined with the highest precision if the cw EPR spectra both in the absence and presence of the dipole-dipole interaction are available. Hence, the spectrum of the doubly-labeled macromolecule and the two spectra of the corresponding singly-labeled macromolecules have to be measured. The spectrum of the doubly labeled molecule can then be fitted by convolution of the sum of the spectra of the isolated paramagnetic centers with a dipolar broadening function (Fig. 10.44). The dipolar broadening function in turn is simulated by superimposing Pake patterns for a Gaussian distribution of distances. Fit parameters are the mean distance r and width r of the Gaussian peak. Integrate isolated centers exp. dipolar broadened trial distance distribution B0 Convolution FT B0 Multiply inverse FT derivative sr árñ r Simulate/ fit dipolar spectrum sim. dipolar broadened B0 FT B0 exp. dipolar broadened Figure 10.44: Distance determination from a dipolar broadened cw EPR spectrum by fitting of a convolution of the spectrum of isolated paramagnetic centers with a dipolar broadening function. The convolution method assumes that the macromolecule is completely doubly labelled. This assumption can be relaxed by fitting a superposition of the dipolar broadened and unbroadened lineshapes. If the spectrum of the isolated centers is unknown, a simulated spectrum can be used instead. Both these variations of the method cause a loss in accuracy and a decrease in the upper distance limit. 10.6.1.2 Intensity of half-field transitions The convolution technique neglects the influence of exchange coupling between the electron spins on the line shape. This is a good approximation at the upper end of the distance range (1.5... 2 nm), but a poor approximation at the lower end (0.8... 1.2 nm). The influence of the isotropic exchange coupling is eliminated in a technique that relies on mixing of the states 244 of the two spins by the anisotropic dipole-dipole coupling. The two types of electron-electron coupling can be distinguished, since the anisotropic coupling contributes off-diagonal terms that connect the and states of the S1= 1/2, S2= 1/2 two-spin system ( Eˆ and Fˆ terms in the dipolar alphabet, see Eq. [5.80] and Fig. ), while the exchange coupling does not. Therefore, only the dipole-dipole coupling mixes theses levels and leads to a non-zero transition probability of the double quantum transition. At constant field, this transition would be observed at the sum of the two electron Zeeman frequencies, i.e., at twice the electron Zeeman frequency of the nitroxide radical. Since cw EPR experiments are performed at constant mw frequency and variable field, the double-quantum transition is observed at half the field of the usual single-quantum spectrum. |bbñ SQ ^^ C,D SQ DQ ^^ C,D ^ B |bañ ZQ ^^ C,D ^ ^ E F SQ |abñ SQ ^^ C,D |aañ Figure 10.45: Energy level scheme for a system consisting of two electron spins S1= 1/2 and S2= 1/2 with assignement of the terms of the dipolar alphabet and of zero-quantum (ZQ), single-quantum (SQ), and doubl;e-quantum (DQ) transitions. The intensity of the half-field transition depends on the ratio between the dipole-dipole coupling and the electron Zeeman interaction. The distance can thus be determined from the integral intensities V(DQ) of the double-quantum (DQ) transition and V(SQ) the four single quantum (SQ) transitions.1 Numerical computations provide the approximate relation2 V DQ 9.1 GHz 2 Å 6 ------------------ = 19.5 + 10.9g -------------------- ----- mw r 6 V SQ [10.85] that is valid up to X band frequencies. The approach neglects anisotropic contributions of the exchange coupling, as is appropriate for distances larger than about 6 Å. For distances larger 1 S. S. Eaton, G. R. Eaton, J. Am. Chem. Soc. 1982, 104, 5002-5003. Measurement of Spin-Spin Distances friom the Intensity of the EPR Half-Field Transition. 2 R. E. Coffman, A. Pezeshk, J. Magn. Reson. 1986, 70, 21-33. Analytical Considerations of Eaton’s Formula for the Interpsin Distance between Unpaired Electrons in ESR. 245 than about 12 Å the intensity of the half-field transition at X-band frequencies is too weak for this kind of analysis. 10.6.2 DEER At distances longer than 18... 20 Å the dipole-dipole coupling needs to be separated from other interactions that lead to broadening of EPR lines. The experimentally most robust technique for such separation is the double electron electron resonance (DEER) experiment, which is also known als pulse electron electron double resonance (PELDOR). By employing two mw frequencies this experiment excites each of the two coupled spins separately. It is thus akin to a heteronuclear rather than a homonuclear NMR experiment. The separate excitation of two electron spins is possible even if they both belong to nitroxide radicals, as the excitation bandwidth of mw pulses is significantly smaller than the width of the nitroxide spectrum. The two frequencies thus select different orientations or different spin states of the 14N nucleus in the two radicals (Fig. 10.46). pump (n2) observer (n1) B B0 A N . . O O N excitation bands B0 schematic spectra Figure 10.46: Selective excitation of two radicals by observer and pump pulses in a doubly nitroxide labeled macromolecule. The radicals have different orientations with respect to the external magnetic field or different 14N nuclear spin states and thus different resonance frequncies. The subsequence for the observer spin A at frequency 1 is a refocused echo sequence (Fig. 10.47). It consists of a part (/2)-1-()-1 that creates a first echo and a part 2-()-2 that refocuses the signal once again and creates a second echo. This second echo is observed. As the observer pulses excite only the A spin, they refocus all interactions that are linear in operators of this spin. These are the electron Zeeman interaction, the hyperfine interaction of the A spin, and also the dipole-dipole interaction between the A spin and the pumped B spin. 246 Only the dipole-dipole interaction is reintroduced by the pump pulse at frequency 2, which exclusively excites the B spin. This pump pulse inverts the state of the B spin and thus leads to a change of the resonance frequency of the observer spin A by the dipole-dipole splitting d() given in Eq. [10.31]. This frequency change is induced at a variable delay t with respect to the first observer echo, while delays 1 and 2 are fixed. The echo signal is thus modulated with dd as a function of time t. From dd the distance can be determined using Eq. [10.30] or [10.32]. The modulation is not damped by relaxation, as the total duration of the experiment is constant. Nevertheless distance resolution and the upper distance limit depend on electron spin relaxation, as the maximum observation time tmax of the dipolar modulation cannot be much longer than T2. Therefore, the experiment is performed at temperatures of about 50 K, where transversal electron spin relaxation of nitroxides asymptotically approaches its low-temperature maximum. At this temperature transversal relaxation is mainly driven by fluctuating hyperfine fields from weakly coupled protons. A decrease of proton concentration by deuterium exchange extends the upper distance limit and improves resolution. p p/2 n1 p t1 t2 p t n2 Figure 10.47: Pulse sequence of the DEER experiment consisting of the refocused echo subsequence at frequency 1 that excites exclusively observer spins A and a pump pulse at frequency 2 that excites exclusively pumped spins B. Delays 1 and 2 are fixed, while delay t is varied. The echo amplitude oscillates with the dipolar splitting d() as a function of delay t. As the pump pulse has limited excitation bandwidth and inverts only a fraction of the B spins, only a fraction of the echo intensity is modulated. The remaining fraction 1- of the echo amplitude is independent of t for an isolated pair. The DEER signal F(t,) for an isolated spin pair is thus given by F t = F 0 1 – 1 – cos d t . [10.86] The form factor F(t) for a macroscopically disordered sample is obtained by powder averaging 247 2 Ft = V t sin d . [10.87] 0 A O O N (CH2 )5CH3 (CH2) 5CH3 O N N O H3C(H2 C) 5 H3 C(H2C)5 H3CO(H2 C) 5 1 V(t)/V(0) (CH2 )5 OCH3 N O O 1 B 0.9 C background 0.8 l 0.9 correction 0.7 0.8 0.6 0 5 10 0 t (µs) 5 10 t (µs) ier n ur tio Fo a m for ns a tr Tikhonov regularization ndd E I(n) P(r) D -1 0 n (MHz) 1 3 4 r (nm) 5 6 Figure 10.48: DEER distance measurement on a rigid biradical. A) Structure of the biradical. B) Normalized echo amplitude as a function of delay t. The read line is a fit of the background function B(t) according to Eq. [10.88]. C) Form factor F(t) obtained by dividing the normalized echo amplitude by the background function. The red line is the theoretical form factor corresponding to the extracted distance distribution P(r). D) Dipolar spectrum (Pake pattern) obtained by subtracting the constant part from the form factor F(t), multiplication with a Hamming window, zero filling, and Fourier transformation. The red line is the theoretical spectrum corresponding to the extracted distance distribution P(r). E) Distance distribution P(r) obtained by Tikhonov regularization. Interaction of the A spin with spins in neighboring molecules modifies the signal, as a fraction of theses spins is also inverted by the pump pulse. If the neighboring spins are homogeneously distributed in space, the background factor B(t) assumes the form 248 2g 2 B2 0 N A B t = exp – ---------------------------------- ct , 9 3h [10.88] where the average modulation depth is the fraction of spins excited by the pump pulse, g is an average g value, and c is the total concentration of spins. The total DEER signal V t = F t B t [10.89] for a well-defined distance between the spins has an appearance as shown in Fig. 10.48B. The form factor for the isolated pair can thus be obtained by fitting the background function and dividing the DEER signal V(t) by this function (Fig. 10.48B). The dipolar spectrum is computed by Fourier transformation after removing the constant component 1- of the form factor (Fig. 10.48C). From the singularity of the Pake patern, dd can be determined and the distance calculated according to Eq. [10.32]. Due to flexibility of the macromolecule and conformational degerees of freedom of the label, the spin-spin distance is not sharply defined but rather distributed. Extraction of this distribution of distances P(r) from the form factor F(t) is an ill-posed problem. In ill-posed problems small errors in the input data F(t) may result in large errors of the output data P(r). Errors in the input data come from noise or incomplete background correction and are unavoidable. Therefore the solution has to be stabilized by additional constraints. Constraints can be derived from the properties of the distribution to be non-negative (P(r)>0 for all distances r) and smooth. Smoothness corresponds to a small square norm of the second derivative = 2 d Pr dr 2 . [10.90] Furthermore, the mean square deviation of the simulated form factor S(t) from the experimental form factor F(t), = St – Ft 2 , [10.91] has to be small (goodness of the fit). The relative weight of these two criteria for a good solution of the problem is experessed by a regularization parameter . The best solution for the distance distribution is obtained by minimizing 249 G = + [10.92] This solution can be directly computed by matrix algebra with the Tikhonov regularization algorithm. The result of Tikhonov regularization depends on the choice of the regularization parameter . A good compromise between undersmoothing (too small ) and oversmoothing (too large ) can be found by plotting log vs. log . Because of its typical shape this plot is termed L curve (Fig. 10.49). In the steep branch of the curve at small an increase in leads to a strong decrease in the square norm (much smoother distribution), but to only a small increase in the mean square deviation of the simulated from the experimental data. In the flat branch at large , a further increase of causes a strong increase in the mean square deviation (deterioration of the fit), but to only a small improvement in smoothness. The best compromise corresponds to the corner of the L (arrow in Fig. 10.49). -8 a= 10 -5 log h -12 a= 10 -1 -16 -20 a= 103 -5 -4.5 -4 log r Figure 10.49: L curve for the distance distribution of the biradical in Fig. 10.48. The optimum regularization parameter = 0.1 corresponds to the corner of the L (red). The distance distribution obtained with this regularization parameter is shown in Fig. 10.48E. Determination of the distance distribution rather than only the mean distance is particularly important for systems that simultaneously attain two or more conformational states. This situation is encountered in studies of protein function or folding. 250 10.7 Technical considerations 10.7.1 The cw EPR experiment The general setup of an EPR spectrometer is similar to the one of an NMR spectrometer discussed in Chapter 2.5 (Fig. 10.50). The mw source in modern spectrometers is a Gunn diode whose frequency can be tuned over a range of typically 1 GHz. The NMR coil and capacitance are substituted by a cavity resonator (Fig. 10.51A) or, for special experiments, a loop-gap resonator. The dimensions of cavity resonators are comparable to the wavelength of the mw (Table 10.1). The standing wave in the resonator has spatial regions with a large magnetic and a small electric component of the alternating field (Fig. 10.51B). This is where the sample has to be placed, as resonant absorption of the magnetic component provides the signal, while non-resonant absorption of the electric component by the electric dipoles of molecules in the sample leads to mw losses and sample heating. Therefore sample size must be much smaller than the wavelength. Typical outer diameters of sample tubes for cw EPR are 4 mm at X band, 1.6 mm at Q band, and 0.9 mm at W band. Samples that contain highly polar solvents, such as water, have high dielectric losses and thus need to be confined to a region where alternating electric fields are very small. They are thus measured in capillaries or flat cells. Typical sample amounts are 150 mg/150 l at X band and 1 mg/1 l at W band for nonlossy samples and by about a factor 2 to 5 less for aqueuos samples. It is possible to routinely detect micromolar concentrations with a microliter of sample at X band, corresponding to 1 paramagnetic centers. bias microwave source reference phase arm f m.w. diode PSD 1 circulator Signal 11 pmol of or 6 10 3 attenuator 2 resonator magnet modulation coils modulation generator Figure 10.50: Schematic drawing of a cw EPR spectrometer. 251 The mw from the source (power up to Pmw = 200 mW at X band) is attenuated to a level that does not cause saturation of the electron spin transitions (see Section 2.7.2). With typical attenuations between 20 and 40 dB the power incident on the sample is about 2 mW to 20 W. This power reaches the cavity resonator through a circulator, so that no power can directly pass on to the mw diode that functions as the detector (Fig. 10.50). The mw is transmitted via a waveguide that is coupled to the resonator by an iris with a tuning screw (Fig. 10.51). For best sensitivity the spectrometer has to be tuned and matched. A B B1 B1 Figure 10.51: Cavity resonator used in cw EPR experiments. A) Rectangular cavity with iris tuner. The mw is fed from the left front side by a rectangular waveguide. B) Distribution of the mw field. Solid lines with arrows marked B1 correspopnd to the magnetic field component, while dots and crosses correspond to the electric field component.The central plane of the resonator is a nodal plane of the electric field. Tuning means that the frequency of the Gunn diode is adjusted to the resonance frequency of the cavity resonator. For that the mw absorption of the cavity is measured as a function of frequency and displayed as a tuning picture (Fig. 10.52). The dip is moved to the center of the picture by adjusting the center frequency of the sweep. Maximum sensitivity is achieved if the impedance R0 of the transmission line (50 ) is matched to the impedance R of the cavity resonator. Due to the non-resonant mw losses, R depends on the dielectric properties of the sample. Tuning and matching thus have to be repeated for each sample. A matched cavity does not reflect any mw power if the electron spins are off resonance (Fig. 10.52). This means that zero mw power is transmitted through 252 ports 2 and 3 of the circulator towards the detecting mw diode (Fig. 10.50). Matching is achieved by adjusting the iris tuning screw. This changes the coupling n between the transmission line and the resonator until 2 R0 n = R , [10.93] a condition that is termed critical coupling. The loaded quality factor, which characterizes the resonant enhancement of the mw field in the cavity, is then given by 2 resonator L Q L = -------------------------------- , 2 R0 n + R [10.94] where L is the inductivity of the cavity. The amplitude of the magnetic field component B1 becomes 2Q L P mw ----------------------------------- , 0 V c resonator B1 = [10.95] –3 where Vc is the effective volume of the resonator, which scales roughly as resonator . For the same QL, less mw power Pmw is thus required at higher frequencies to achieve the same amplitude of the alternating magnetic field. More power is needed if the sample is lossy (large R), corresponding to low QL. . signal voltage after m.w. diode total reflection Dn -3 dB DVrefl -2 -1 no reflection 0 1 2 n-nresonator (MHz) Figure 10.52: Idealized tuning picture of a matched (solid line) cavity resonator with a loaded quality factor QL = 10000 at X band frequencies. The dashed line corresponds to resonant mw absorption by the electron spins.The EPR signal Vrefl is due to the power reflected in this situation The bandwidth of the resonator (Fig. 10.52) is given by 253 resonator = ---------------------- . QL [10.96] In cw EPR the bandwith of a high-quality cavity resonator is always sufficient. For pulsed EPR, bandwidths of 30 to 200 MHz are required. This is achieved by intentionally spoiling resonator quality (increasing R) or by overcoupling (increasing n beyond critical coupling). With the spectrometer tuned and matched, the measurement is performed by keeping the mw frequency constant and sweeping the magnetic field through resonance of the electron spins. On resonance, the spin systems absorbs mw energy and dissipates the energy through longitudinal relaxation. This corresponds to an increase in cavity impedance R and thus to a violation of the critical coupling condition, Eq. [10.93]. The cavity is no longer matched and mw power is reflected (Fig. 10.52). This reflected power is transmitted through ports 2 and 3 of the circulator to the detecting mw diode (Fig. 10.50). The change in the voltage incident at the mw diode is given by V refl = C B 0 Q L , [10.97] where ’’(B0) is the field-dependent imaginary part of magnetic susceptibility arising from the electron spins, is the filling factor of the resonator and C is an apparative constant. The function ’’(B0) is the EPR absorption spectrum. The filling factor is the ratio of the integrals of the mw magnetic field amplitude over the sample and the whole resonator, B 1 dV sample = -------------------------------- . B1 dV [10.98] resonator Sensitivity of the cw EPR experiment can thus be increased by optimizing either the quality factor QL (cavity resonators) or the filling factor (loop-gap resonators, dielectric ring resonators). The first strategy works better for samples with low dielectric losses that are available in sufficient quantities (100 mg). The second strategy is preferable for lossy samples or small sample quantities. Dielectric resonators combine high QL and high . They are only used for special applications or pulse EPR as the dielectric rings used for concentrating the mw field have background signals. 254 The reflected mw power is detected by a mw diode, which can manufactured with a low intrinsic noise figure. However, mw diodes have two drawbacks. First, they are not sensitive at very low incident powers (Fig. 10.53). Second, they detect mw power over a very broad frequency range, thus also collecting thermal noise from this whole frequency range. These problems require two additions to the spectrometer. To achieve best sensitivity and linearity of the output signal with respect to incident voltage, the diode has to be biased to its operating point (Fig. 10.53). This is done by transmitting a small fraction of the mw power directly from the source to the diode via the reference arm (Fig. 10.50). The operating point is attained by adjusting the bias attenuator. The output current with the sample off resonance should correspond to the horizontal dotted line in Fig. 10.53 (typically 200 A). The mw transmitted through the reference arms must have the same phase as the mw reflected from the resonator to interfere constructively at the diode output current input. This is achieved by adjusting the mw phase in the reference arm. operating point incident voltage Figure 10.53: Characteristic curve of an mw diode for cw EPR detection (solid line). Highest sensitivity and linear behavior is obtained near the operating point (full circle). The diode is damaged at too high incident power (full diamond). In modern spectrometers, the diode is protected from damage by limiting the input power. The characteristic curve of a limiter-protected diode is shown as a dashed line. The detection band can be narrowed and thus much of the noise excluded by imposing a modulation on the signal. This can be done most easily by modulating the external magnetic field at a low frequency that can pass the mw. diode (typically 100 kHz). Such modulation leads to an oscillation of the reflected mw with the same frequency (Fig. 10.4). For sufficiently low field modulation amplitudes B0 the amplitude of the voltage oscillation is proportional to the derivative d dB 0 , i.e., the derivative of the absorption spectrum. This amplitude is 255 measured by a phase-sensitive detector (PSD). As a result cw EPR spectra are derivative absorption spectra. With increasing modulation amplitude the signal increases, while noise remains the same. This applies until the modulation amplitude reaches the peak-to-peak linewidth Bpp. With this modulation amplitude detection is most sensitive. However, if the modulation amplitude exceeds Bpp/3, the line is artificially broadened. Therefore, the modulation amplitude is adjusted to about one third of the width of the most narrow line in the spectrum. In rare cases natural linewidths can be smaller than 100 kHz, corresponding to 36 mG or 3.6 T on a magnetic field axis. In this case the modulation frequency has to be decreased to avoid line broadening. Finally, saturation of the electron spin transitions (see Section 2.7.2) may also cause line broadening. Such power broadening is avoided in the linear regime where the signal amplitude increases with the square root of the mw power, V P mw . In this regime the amplitude increases by a factor of two when decreasing attenuation by 6 dB ( 10 0.6 = 4 4 = 2 ). phase shifter m.w. source circulator 1 * MPFU mixer f reference arm ** S r.f. source power amplifier main attenuator 2 receiver amplifier video amplifier ELDOR 2nd m.w. source ENDOR 3 magnet r.f. coils r.f. PFU r.f. amplifier resonator video signal (MHz range) ** * MPFU m.w. pulse forming unit S protection switch or power limiter Figure 10.54: Schematic drawing of a pulse EPR spectrometer. 10.7.2 Pulsed EPR experiments In a pulsed EPR spectrometer the pathways for excitation power, signal and reference mw are similar to the ones in a cw spectrometer (Fig. 10.54). The mw pulses are created in an mw pulse forming unit (MPFU), where the mw power is divided into several channels. In the 256 most flexible setup each channel has its own attenuator, phase shifter, and PIN diode switch (Fig. 10.55A). This allows for independent adjustment of power, phase, and timing of the pulses. The PIN diode switches have rise and fall times of about 3-4 ns, so that a pulse with a length of 32 ns is not quite rectangular (Fig. 10.55B). The power from all channels is recombined at the output of the MPFU. Modern commercial spectrometers have one preset four-channel MPFU with fixed quadrature phases (+x, +y, -x, -y). The power of the four channels cannot be adjusted independently. This MPFU provides best performance in phase cycling and ease in experiment setup. For full flexibility, additional MPFUs with adjustable phase and amplitude in each channel have to be installed A MPFU B attn. phase j j j j t=0 PIN diode S+áxñ S+áyñ tp S-áxñ S-áyñ 32 ns Figure 10.55: MW pulse formation. A) MW pulse forming unit (MPFU) with four channels whose amplitudes and phases can be adjusted independently. B) Typical pulse shape in EPR with the rise and fall times of the pulse-forming switch being comparable to the pulse length. Much larger B1 fields are required for pulse EPR than for cw EPR. Therefore, the mw excitation power is amplified in a traveling wave tube ( P mw 1 kW) or solid-state power amplifier ( P mw 10 W). A precision attenuator after this amplifier allows for adjusting pulse power. The pulses enter the resonator via the circulator. Critical coupling of the resonator is not required in pulsed EPR, since excitation and detection are separated in time. In fact, resonators are usually overcoupled on purpose to increase bandwidth (see Eqs. [10.94,10.96]). For instance, a pulse with a length tp= 12 ns has an excitation bandwidth of about 1/tp = 83 MHz and thus requires QL < 115 at a frequency of 9.6 GHz. Such overcoupling leads to substantial power reflection. Reflected mw power in the range of tens to hundreds of Watt would destroy the sensitive receiver amplifier. Therefeore, the amplifier is protected by a switch or a power limiter. During detection the protection switch is open. The signal is amplified by about 20 dB by the mw receiver amplifier. This amplifier has a very low noise figure. The signal is then fed to a mixer, where a video signal at the difference frequency between the reference mw and the 257 signal mw is generated. This frequency subtraction corresponds to detection in the rotating frame. Sign information on the difference frequency is lost, unless two mixers with a reference phase difference of 90 are used. Such quadrature detection with a doubly-balanced mixer is optionally available in modern EPR spectrometers. It should be used only when necessary, as the splitting of the signal to two mixers decreases signal-to-noise ratio compared to singlechannel detection. The video signal is passed through a low-pass filter, amplified, and detected by a fast digitizer with a typical time resolution between 1 and 8 ns. Modern digitizers allow for signal accumulation at repetiton rates up to 1 MHz. In echo experiments, the digitized signal is integrated over a certain time range. This improves signal-to-noise ratio. The width of the integration gate puts an upper limit on the frequencies that can be detected in the signal. Optimum sensitivity is achieved by matching detection bandwidth to excitation bandwidth. This corresponds to an integration gate that is approximately as wide as the full width at half height of the echo, which is in turn given by the length of the longest excitation pulse (window with length tintg,SN in Fig. 10.56). For recording field-swept echo-detected EPR spectra the detection bandwidth should be much smaller than the width of the most narrow peaks in the absorption spectrum. This is achieved by integrating over the whole echo. Typical widths for the integration gate in this mode of operation (tintg,res in Fig. 10.56) are 100 to 200 ns. signal voltage (a.u.) t' echo td FID tintg,SN tp tintg,res 0 100 200 300 400 t (ns) Figure 10.56: Signal trace in a two-pulse echo experiment starting immediately after the falling edge of the pulse (simulation for an infinetely broad inhomogeneous line). Pulse lengths of 16 and 32 ns were assumed for the /2 and pulse, respectively. 258 The theoretically expected signal trace after a two-pulse sequence with an interpulse delay ’ measured between the falling edge of the /2 pulse and the rising edge of the pulse is shown in Fig. 10.56. The simulation assumes that the EPR line is inhomogeneously broadened and is much broader than the excitation bandwidth, as is often the case. For such a line, the FID length tFID is determined by the length of the pulse. The echo appears at a time slightly longer than ’, as the effective evolution time is approximately the delay between the centers of the two mw pulses. In practice, only the echo but not the FID can be detected, as the signal with a power of a few microwatts is superimposed by stronger residual power from the highpower pulses within dead time td. For a residual power of 100 W to decay to the level of 1 W a time ln(108)tring has to pass, where tring is the ringing time of the system. This time cannot be shorter than the ring down time of the resonator tring,res= QL/(mw). For QL= 100 at 9.6 GHz this corresponds to a dead time of approximately 60 ns. Any reflection of power in the mw bridge increases tring and proportionally td. The decrease in QL that allows for a larger excitation bandwidth leads to a loss in sensitivity compared to cw EPR. This loss is only partially compensated by the contribution of more spin packets to the signal. Furthermore, according to Eq. [10.95] this decrease also leads to a smaller mw field amplitude B1 at given power Pmw. These unwanted effects can be compensated by a different resonator design than in cw EPR. While cw EPR resonators are optimized by maximizing QL, pulse EPR resonators are optimized for high filling factors (Eq. [10.98]) and for effective volumes Vc that match available sample volume as closely as possible. Both criteria require a confinement of the magnetic component of the mw field to dimensions that are much smaller than the wavelength in air. There exists the additional constraint that the electric field component in the sample should be as small as possible to avoid sample heating. The dimensions of an mw resonators can be reduced by filling it with a material with much larger dielectric constant than the one of air (= 1). The mw wavelength is inversely proportional to . Furthermore, if a hole is drilled in the dielectric medium, the electric component is mainly condined in the dielectric ring and the magnetic component in empty space. Such a dielectric ring (Fig. 10.57A) within a cylindical cavity resonator combines a large filling factor , small effective volume, and high loaded quality factor QL at critical coupling. When critically coupled it allows for cw EPR experiments with good sensitivity and 259 when overcoupled for pulse EPR experiments with good power conversion. Compared to an optimized cavity resonator without dielectric ring it has somewhat lower cw EPR sensitivity. For some cw EPR applications and pulse EPR at very low temperatures it is not suitable, as dielectric materials generally have some background signal due to transition metal impurities. Usually sapphire ( 10 ) is employed. A B B1 B1 loop B0 gap Figure 10.57: Resonator designs used for pulsed EPR. A) Dielectric ring resonator, consisting of a ring of material with a high dielectric constant that is placed inside an mw shield or outer cylindrical resonator. The sample is placed in the center hole. B) Loop-gap resonator consisting of a metal ring with a vertical slit. The sample is placed in the center hole. More flexibility in scaling of the critical volume is achieved with lumped circuit resonators, which can be pictured as LC resonance circuits with the inductivity L and the capacitance C in the same place. More strictly defined the travel time of the electric signal between the elements of a lumped circuit is negligible. The most simple lumped circuit resonator is the loop-gap resonator, which is a massive metal ring with a slit that has a constant width (Fig. 10.57B). The slit (gap) can be considered as the capacitance C while the metal piece is a one-turn coil, i.e., the inductivity L. Accordingly, the electric component of the mw field is concentrated across the gap, while the magnetic component is particularly strong inside the ring (loop), where the sample is placed. The dimension of a loop-gap resonator can be adjusted without changing its frequency, as L and C can be varied independently by changing the radius of the loop and the width of the gap, respectively. An antenna is used to couple mw into such a loop-gap resonator. The coupling n can be changed by shifting the antenna vertically (along the B1 direction) with respect to the resonator. Such antenna coupling can also be employed for the dielectric ring resonator. 260