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SPECTRAL DISTRIBUTION OF NMR IN MIXED
VALENCE COMPLEXES. ELECTRIC-DIPOLE ABSORPTION
L.D. Falkovskaya, A.Ya. Fishman, V.Ya. Mitrofanov, B.S. Tsukerblat 
Institute of Metallurgy, Ural Division of the Russian Academy of Sciences,
Ekaterinburg, Russia

Ben-Gurion University of the Negev, Beer-Sheva, Israel
In this study we reveal the specific character of the frequency spectral
distribution of NMR in mixed valence (MV) complexes in magneto-ordered crystals
with cubic structure. Chalcogenide chromium spinels can be exemplified in wich Cr2+
or Cr4+ ions exist in the octahedral sub-lattice along with Cr3+ ions. It was shown that
due to the presence of essential dipole momentum and unquenched orbital angular
momentum in such MV centers one can expect a considerable amplification of NMR
signal induced by a high intensity electric-dipole transition between components of
E-term .
INTRODUCTION
In crystals with small concentration of non-isovalent substitutions, vacancies
in cation or anion sub-lattices, the extra electrons or holes are localized by the
Coulomb interactions near the lattice defects [1-3]. The extra electrons can be
delocalized over the metal ions in the vicinity of the defect giving rise to a mixedvalence clusters containing metal ions in different oxidation degrees like manganese
and chromium dimers, Mn4+ - Mn3+, Cr4+ - Cr3+.
One of the constituent ions of these MV complexes, either dn or dn1, usually
has orbitally degenerate ground state so that the MV clusters entire can possess some
peculiarities that are intrinsic to the isolated Jahn-Teller (JT) systems. Low symmetry
local fields created by the excess charge remove this degeneracy. Nevertheless the
ground state of the whole MV complex can be degenerate due to excess electron
transfer among 3d-ions of the complex. Due to the external perturbation or
cooperative interactions the orbital degeneracy proves to be removed and the system
is expected to possess some new properties resulting from the redistribution or
reorientation of the density of the excess electron. This originate a dipole moment of a
MV cluster essentially depending upon the character of its orbital state. Close
interrelation between magnetic and electric properties of these compounds can be
understood on the basis of the model of the double exchange [4,5].
Being extremely sensitive to the local fields the nuclear magnetic resonance
(NMR) provides an important information regarding the peculiarities of the magnetic
states and gives an opportunity to analyze the local charge distribution in MV centers.
The lack of the theory of NMR spectral distributions in systems containing transition
metal MV clusters in magnets essentially complicates the interpretation of
observable spectrums. To a large extent this is connected with the absence of adequate
models of electronic structure of the impurity complexes. This task is just more
complicated for systems in which some types of defects in cation and anion sublattices take place. Such situation is typical for NMR spectrums of 55Mn [6-8] in
(LaMn)1-2xO3 crystals containing MV clusters of manganese ions. The corresponding
NMR spectrum of 55Mn [6] represents a series of non- uniformly broadened lines. A
drastic amplification of the NMR signal was observed in the whole frequency range
with the increase of the defect concentration. Till now these results have not received
3 - 21
any adequate interpretation in terms of current theories.
The present paper is devoted to the theoretical treatment of the peculiarities of
the spectral distribution in NMR spectra closely related to the essential dipole
momentum of MV complexes. As an example we consider MV centers in the
octahedral positions of the crystals with spinel structure. Analysis of MNR spectra
was carried out for the case of orbitally degenerate ground state of MV complex (Eterm). In this case the main mechanisms determining splitting of the lowest E-term are
the spin-orbit interaction and random crystal fields. The peculiarities of NMR spectra
of such centers arise from the existence of the anomalously strong direct interaction of
the type of A1(I) between orbital  and nuclear I momenta of 3d-ion in one of two
possible charge configurations, either 3dn or 3dn1 [9,10]. We show that due to the
essential dipole momentum of MV center the intensity of NMR transitions under
consideration can be significantly amplified.
HAMILTONIAN OF MIXED VALENCE CENTER. COUPLING BETWEEN
ORBITAL AND NUCLEAR SUB-SYSTEMS
Let us consider a MV center consisting of a triad of exchange-coupled 3d-ions
and an excess t2g-hole (or t2g-electron) delocalized over these ions. Such center in a
cubic crystal can appear in the presence of cation or anion vacancies or nonisovalent
substitutions. For the sake of definiteness we shall consider a MV center consisting of
a cluster with a point defect in cation sublattice of crystal with spinel structure and
t2g-hole coupled to this system (see Fig. 1). The basis set of the system involves three
localized states corresponding to three 3d n 1 configuration of the magnetic ions each
containing the excess hole. The chalcogenide chromium spinels where along with
Cr3+ (3d3) ions also Cr4+(3d2) ions exist in octa-positions [9, 10] could be mentioned
as an example.
Fig. 1. Mixed valence center of
formed by three cations in
octahedral spinel sub-lattice with
the delocalized excess t2g-hole
PD - point defect in cation
sublattice, A - anion, C - cation,
p – extra hole.
It is supposed that the lowest energy state for the ions with 3d n 1
configuration is two-fold degenerate with respect to orbital quantum number (trigonal
E-term). The ground electronic state of two other magnetic ions with configuration
3dn is assumed to be orbitally non-degenerate.
We shall represent the Hamiltonian of a triad by
(1)
H  H res   H k ; H res    bk ,l ck cl ,
k
k  l  ,
where Hres is the Hamiltonian describing the transfer of a t2g – hole in a triad (double
exchange), and Hk is the Hamiltonian of the cluster when a t2g hole is localized at
center к (к = 1,2,3), c+k and cl are the creation and annihilation operators for t2g
holes whose spin projection is  at centers k and l in orbital states  and ,
3 - 22
correspondingly, bkl is the transfer integral for a t2g hole between the mentioned
states.
Let us confine our treatment to the states with the maximum projection of the
total spin of the triad along the quantization axis. Since the maximum energy gain is
achieved when the spin of the dn triad and the moving t2g hole are parallel the ground
state of the system is contained precisely in the mentioned group of the levels. Six
corresponding lowest states of a triad (MV center) are analyzed. The low-lying group
of the levels arise from three localized states dn-1-dn-dn, dn- dn-1-dn, dn-dn-dn-1,each state
being double degenerate and corresponds to localization of the excess t2g hole on one
of the constituent ions. The delocalized states
transform accordingly to the
irreducible representations of the symmetry group C3v of the whole system: А1, А2 and
2E. Leaving aside spin-orbit interaction one can find the following expressions for the
energy levels [11]:
E ( A1 )  h0  b1  2b2 ,
E ( A2 )  h0  b1  2b2 ,
(2)
E  ( E )  b1 / 2  [(h0  b2 ) 2  (3 / 2b1 ) 2 ]1 / 2 .
In this case the energies are counted of from the ground ferromagnetic state of
the magnetic material with an excess t2g-hole; the energy levels E(E) belong to the
repeated E representations, h0 is the parameter of tetragonal crystal field caused by the
source of excess charge. The transfer integrals are equal to [11]


b1   2b1 ,2  b1 ,2  b1 ,2 / 3 ,


b2   b1 ,2  b1 ,2  2b1 ,2 / 3 .
(3)
Further on we shall focus on the situation when a doublet state turns out to be
the lowest state of the triad (this is valid when h 0>0 and b1/b2  1/2 [11]). The main
interactions determining splitting of the ground Е–term, and hence also properties of
MV centers are the spin-orbit interaction and low symmetry fields. They can be
represented as:
H el  qS    h    h  ,
(4)
where orbital operators are defined in space trigonal triad basis
 1 0
0 1
 0 i
,    
,    
,
1
1 0
  i 0
   
0
(5)
 is the spin-orbit interaction parameter, q is the reduction parameter for the orbital
angular momentum in the ground triad state comparatively to that in a free 3dn-1 ion,
index  enumerates the projection on the trigonal symmetry axis

S  3S max mx m y n z  mx mz n y  m y mz n x

,
(6)
Smax is the maximum projection of system total spin, m is the unit vector along the
trigonal axis of cluster, n is the unit vector parallel to magnetization, h и h are
components of low-symmetry field acting on the JT cluster. The low-symmetry fields
can include both the random crystal fields and magnetic anisotropy fields caused, for
example, by the second order effects of spin-orbit interaction [11,12].
The functions of the constituent ions can be involved with different weights in
the wave functions of the lowest (0) and excited (ex) complex states with the
maximum total spin projection
3 - 23
3
3
k 1
k 1
0   [ C (0, k)  k ()  C (0, k)  k ()], ex   [ C (ex,)k  k ()  C (ex,)k  k ()],
(7)
where functions k() are the antisymmetrized products of wave functions of three
ions with t2g hole on a site к (к = 1,2,3) in one of states E ( = +, ).
While describing the hyperfine interactions in impurity complex let us for
simplicity confine ourselves (like in [10]) by taking into account of only isotropic
term for ions with configuration 3dn
(8)
H hf (3d n )  A0 (IS) ,
n-1
For state 3d we shall take into account the isotropic hyperfine interaction and
anisotropic direct interaction of the orbital angular momentum with nuclear spin
H hf (3d n 1 )  A0 (IS)  A1  I ,
(9)
'
where I is the nuclear spin of 3d ion, S , S, A'0, A0 are spins and parameters of isotropic
hyperfine interaction in configurations 3dn and 3dn-1,  is the orbital operator defined
in the trigonal basis of an isolated center.
It is convenient to transform the Hamiltonian of hyperfine interaction using the
electronic functions 0 and ex diagonalizing Hamiltonian (4) as basis functions.
Respectively, under this rotation the orbital operators , ,  are transformed into
the new orbital operators  ,  and . As a result the Hamiltonian doublet state of
the triad that includes spin-orbit interaction, low-symmetry fields and hyperfine
interaction takes the following form


3
3




H  E el      k I kz  A1    U k*  n k D*k I k   n k D k  I k  n k β k I kz   k .c. ,





k 1
 k 1 



Eel  2 qS 2  h 2
, h 2  h2  h2 ,
(10)
2
2
 a (k )  b1 (k )

 a (k )  b2 (k )

 2k   1
 b1 (k )  a1 (k )       2
 b2 (k )  a 2 (k )     
2
2




 a (k )  b1 (k )
  a (k )  b2 (k )

 2 1
 b1 (k )  a1 (k )      2
 b2 (k )  a 2 (k )    nn k  ,
2
2



where Eel is the energy splitting of ground E-term of MV center, k are NMR
frequencies of transitions associated with the ions k = 1, 2, 3, belonging to MV
center. One can see that direct interaction mixing the orbital and nuclear excitations
on the MV center takes place. Such interactions can induce a new electric-dipole
mechanism of NMR absorption, a new contribution to NMR signal amplification and
occurrence of the “hybrid” orbital-nuclear states in the region of crossing of
corresponding excitation energies.
The following notations where used in expressions (10):
a1 (k )  A0 S    A0 S  A0 S Wk , a2 (k )  A1Wk ,
b1 (k )  A0 S    A0 S  A0 S Vk , b2 (k )  A1Vk ,
Wk  C0, k
2

Vk  Cex
,k
2
 *
 C0, k
2
Wk  C0, k
2

, Wk  Cex
,k
2
,

 Cex
,k
2
 *
 C0, k
2

 Cex
,k
,
2
  C

U k  C0, k Cex
,k
0, k Cex, k ,
3 - 24
,
(11)
y
y
nk  3  mkz mk , mkx mkz , mkx mk  ,  ki  H ki k , i  x, y, z


 a (k )  b1 (k )
H ki   1
 b1 (k )  a1 (k )  
2


 a (k )  b2 (k )
 n i   2
 b2 (k )  a 2 (k )  
2


 

 n ik ,

  i y   x  z 
 i x   y  z 
z 2
1


k
k
k
k
k
k
k
 , Dy  
 , Dz 
Dkx  
.
k
k
2
2
2
z
z
2 1 k
2 1 k
 
 
where mk is a unit vector directed along local trigonal axis of triad ion with number k.
The obtained expressions allow one to proceed to the analysis of spectral
distribution for frequencies of nuclear transitions in 3d ions of MV centers.
DIPOLE MOMENTUM OF MV CENTER. COEFFICIENT OF ELECTRICDIPOLE ABSORPTION OF ELECTROMAGNETIC RADIATION
The coefficient of electric-dipole absorption in the triad under consideration
can be presented in the following form

(12)
  N0
Im  ( ),  ( )   P / P    ,
cn
where P is dipole momentum of the MV complex, N0 is the number of MV centers, с
is the light velocity, n is medium refraction coefficient. For the centers with double
degenerate ground state under investigation such electric dipole momentum is
described by the following expressions
P  2 p0 Ax cos   Ay sin    A*x cos   A*y sin  
,
  x cos    y sin  

Ax 

 i y   x z 
2 1   z2

 
, Ay 

i x   y z 
2 1   z2

, Az 
1   z2
,
2
(13)
 i  Vi Eel , V x  2h , V y  2h , V z  2qS  .
Here indexes  and  = x, y, z enumerate tetragonal coordinate axes,  = 0 (z),
2  /3 (x), -2  /3 (y). Parameter p0 characterizes the absolute value of the dipole
momentum, which can be attained in MV center,
p0  xq0 R0 , x  b2  h0 [b2  h0 2  3b1 / 22 ]1 / 2 ,
(14)
where q0 is the value of the excess charge, R0 is the distance between the source of
excess charge and nearest 3d-ions of cluster .
After some transformations the expression for coefficient of electric-dipole
absorption, eq. (12), can be presented as:

*     
(15)
  4 N 0 p02
Im K        K
,

cn

2
K  Ax сos cos    A y

2
sin  sin    Ax Ay cos  sin   
 A y Ax sin  cos   .
Green-functions that are contained in expression (15) can be obtained in the
second order of perturbation theory with respect to the Hamiltonian of hyperfine
interaction. In this case one can obtain for absorption coefficient the following
3 - 25
expression that is accurate in the region far from crossing of orbital and nuclear
excitation energies:
(16)
 K *
 3 4 I z   U 2 n D 2 
K
k k
k k




2
2
k

  16 N 0 p0
A1    Im

 .
2
2
2
2
cn

  Eel   k 1
k  
   Eel 





The following notations are adopted here
E
     1 th el ,
2 2T

2
 I kz    1 th k , U k  1 (1  x 2 )(1   2z ) .
2 2T
9
(17)
The NMR frequencies k (k = 1,2,3) of the triad can be presented in the form
2
2


 2k    x (1   2z ) cos(   k )      2 A1  z (1  x 2 )     
 3





 2 2 A1  z (1  x 2 )       x (1   2z ) cos(   k )    (nn k ) ,
 3


  1 ( A0 S  2 A0 S ) ,   2 ( A0 S  A0 S ) ,  k  2k / 3 , ctg  h / h .
3
3
(18)
As long as we are interested in the absorption coefficient within the
range of NMR frequencies which are considerably smaller than electron
energies
 k  E el ,
(19)
we
can
neglect
in
comparison
with
Eel
in
the

square brackets of expression (16). Then it turns out to be that
K  K * Eel2  2 Re K / Eel2 ,
Re K 
x
 2z  2x   2y
4(1   2z )
cos  cos   
(20)
 2z  2y   2x
4(1   2z )
2qS
2h cos 
 2h sin 
, y
, z
.
Eel
Eel
Eel
sin  sin   
h 2 sin 2
sin(     ) ,
2 E2
el
Taking into consideration the well-known formula
2Ωk
Im
  (Ω k   ),
(21)
(Ω 2k  ω 2)
one can obtain the following expression for the absorption coefficient in the range of
NMR frequencies:
  32 N 0
Re K
 2 2 2
2 3
2
p0 A1    2
U k  n k D k  I kz  ( k   )
2
cn
Eel
k 1
(22)
When one summarizes contributions to the absorption from different MV
centers it is necessary to consider not only the MV clusters with trigonal axis [-111] as
represented in Fig. 1, but also the remaining three types of clusters in the lattice with
the following trigonal symmetry axes: [1-11], [-1-1-1] and [11-1].
In the case of systems where random crystal fields introduce the dominating
contribution to low symmetry fields removing degeneracy , expression (22) should be
averaged over different configurations of the random fields, h, h , using for
example the Gaussian distribution function
3 - 26
g (h,  ) 
1
2
exp( h ), h 2  h2  h2 .
2
2
(23)


where  is the dispersion of random crystal fields.
RESULTS OF THE CALCULATIONS THE NMR SPECTRAL
DISTRIBUTION
1) First let us consider NMR spectrum of MV centers in the case when
external magnetic field n is directed along tetragonal crystal axis [001]. Since four
space diagonals of a cube make the same angle with cubic axis [001] all four types of
clusters give the same contribution to the absorption coefficient. The typical
frequency dependence of NMR absorption was shown in Fig. 2. The following values
of parameters were used:
x  0.5,     0 , A0 S / 3qS max  A0 S  / 3qS max  0.02,
A1  0.2 3qS max , k B T / 3qS max  0.1 ,
 / 3qS max  0.1.
One can see, that the values of all energy parameters are given in the relative
units , i.e. presented as the ratio of these parameters to the maximum value of spinorbit splitting 3qSmax). It can be easily cleared up that the absorption maximum at
the frequency /3qSmax = 0.047 in Fig. 2 is related to the ions k = 1,2, while the
peak at frequency /3qSmax = 0.069 is related to the ion 3. While integrating over
the orientation of the random fields the main contribution to the absorption coefficient
comes from the range of values h  . Fig. 2 represents the results for the case of
relatively weak random fields /3qSmax = 0.1, therefore the absorption peaks are
observed at the frequencies 1,2/3qSmax = 0.047 and 3/3qSmax = 0.069.
Fig. 2. Frequency dependence of NMR spectral distribution on MV centers at electrodipole absorption mechanism. Magnetic field is directed along axis of [001]-type.
2) Let’s now analyze NMR spectral distribution in the case when magnetic
field n is directed along axis [110].
3 - 27
Frequency dependence of the absorption coefficient in this case is presented in
Fig. 3 for the following set of parameters:
x  0.5,     0, A0 S / 3qS max  0.01, A0 S  / 3qS max  0.02,
A1 / 3qS max  0.2,
k B T / 3qS max  0.1,  / 3qS max  0.1.
Fig. 3. Frequency dependence of NMR spectral distribution of MV centers, electricdipole absorption mechanism is involved. Magnetic field is parallel to axis of [110]type. a. MV centers with trigonal axes [-1-1-1] and [11-1],
b. MV centers with axes [-111] and [1-11].
3 - 28
At such direction of magnetization MV clusters are divided into two groups clusters with the trigonal axes [-111] and [1-11], which are perpendicular to the
selected direction of magnetization, and clusters with the axes [-1-1-1] and [11-1],
which are not perpendicular to magnetization. For clusters of the first type the
combination
  mx m y n z  mx mz n y  m y mz n x
(24)
is equal to zero so that spin-orbit interaction does not contribute to the splitting of
degenerate ground cluster state. As a result NMR frequencies depend only upon
random fields
 k  2 A0 S   A0 S  2( A0 S  A0 S ) x cos(   k )  
1
.
3
(25)
After configuration averaging the spectral distribution of NMR for clusters of
this type does not depend upon number k and it is characterized by a broad absorption
band centered at frequency  = (2A0S + A0S)/3.
The second group of clusters for which  ()1/2, gives weak peaks at
frequencies /3qSmax = 0.06 and /3qSmax = 0.072. If the value of random field
is h  , the first frequency coincides with the NMR frequency for the constituent
ions with numbers k = 1 and 2 while the second one – for the ion with number k = 3.
One can see that the intensity of absorption by the clusters of the first type is three
orders larger than that for the second type of clusters. Analytical estimations show
that at relatively weak random fields the ratio of the corresponding intensities turns
out to be of the order of ()4. Thus one can expect an appreciable amplification of
the NMR signals in the fields that are perpendicular to trigonal axis of MV clusters.
CONCLUSION
1. The existence of electric-dipole contribution to the NMR spectral
distribution of MV centers was determined. It was shown that due to the unquenched
orbital angular momentum an essential enhancement of NMR signal could be
achieved that is expected to be promising for applications. The effect is attained by
means of high intensity electro-dipole transitions between split components of MV
center E-term and of mixing orbital and nuclear states by hyperfine interaction. The
comparison of electric-dipole, eq. (16), and magnetic-dipole (see for example [9,10])
contributions shows that the role of the first mechanism can be rather important in a
wide range of low-symmetry fields values. The magnetic-dipole contribution of MV
centers to the absorption of electromagnetic radiation in the range of NMR
frequencies can be described by

2  Im N g 2       2S ' Ng 2  a a  
 md   B
0 l

0 0

0
cn

,
(26)
where the first term is related to the excitation of the orbital sub-system, while the
second one describes the absorption caused by the interaction of nuclear sub-system
with spin waves at k = 0. The contribution of the direct absorption in the nuclear sub-
3 - 29
system was dropped from equation (26) because it is (N / B)2 times less than the
terms that have been taken into consideration.
The first term in md is (p0/B)2  10 6 times less than the corresponding
electric-dipole contribution. The ratio of intensity of electromagnetic radiation
absorption described by the second term in md ,eq.(26), to the intensity of electricdipole absorption in the range of NMR frequencies turns out to be of the order of
p 02 A12 E el2  В2 A02  k20 , where k=0 is the energy of magnons with k = 0.
If one accepts the following values of parameters entering this ratio,A1  10A0,
Eel  10-100 см-1, k=0  0.1 см-1, then the electric-dipole contribution turns out to be
considerably larger or at least comparable with magnetic-dipole one.
2. The spectral distributions for electric-dipole and magnetic-dipole
contributions to the absorption of electromagnetic radiation in NMR are found to be
appreciably different. This result allows understand the appearance of additional
peaks in NMR spectrums of systems under investigation in the case of comparable
contributions of mentioned mechanisms.
This work was supported by the Russian Foundation for Basic Research, grant
02-03-32877 and Special Federal Program “Integration”, project B 0035.
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