THE SIMULATION OF THE PROPERTIES OF METAL, OXIDE
AND OTHER MATERIALS WITH DIFFERENT TYPES OF
ENERGY OR ORIENTATIONAL DEGENERACY ON THE BASE
OF JAHN-TELLER SYSTEM BEHAVIOUR
B.S.Tsukerblat
Ber-Sheva University, ISRAEL
L.D. Falkovskaya, A.Ya. Fishman, V.Ya. Mitrofanov
Institute of Metallurgy of UB RAS, Ekaterinburg, RUSSIA
M.A. Ivanov
Institute of Metal Physics of NAN, Kiev, UKRAINE
V.I. Tsidilkovski
Institute of High-Temperature Electrochemistry of UB RAS,
Ekaterinburg, RUSSIA
1. Introduction. There is a large class of systems containing centers,
whose ground state is degenerate, quasi-degenerate or orientationally
degenerate. It is enough to mention centers with charge transfer between
nearest to non-isovalent substitution ions, splitting (dumbell-like)
configurations, off-center ions, impurity pairs consisting of substitution
and interstitial ions in high symmetry lattices, systems with orbital
degeneracy etc. Distinctive feature of all these centers is multy-well
profile of their potential energy. The physicochemical properties of such
systems are determined by splitting of energy levels in equivalent
potential wells due to the tunneling effects, external fields, random
crystal fields and molecular fields of various types. Our interest to these
systems is stipulated by the following circumstances.
1) The anomalously large contribution of such centers to different
properties of condensed systems and the unusual character of these
properties in comparison with the non-degenerate systems.
2) The strong dependence of properties upon the distribution of random
crystal fields removing degeneracy on the considered ions. On one hand
these fields decrease the susceptibility of the system to external fields
and on the other hand they can form the properties of the system e.g. the
low temperature - heat capacity. In the diluted degenerate systems one of
the main sources of such fields can be just the multy-well potential
centers.
233
Jahn-Teller (JT) ions occupy the special position among the multy-well
potential centers [1,2,3,4,5,[6],7,8]. The microscopic nature of multy-well
potential is known for the JT systems (unlike other ones) and the
exhibitions of multy-well potential in different properties (magnetic,
thermodynamic, spectral and other) are very diverse. The JT centers can
have simultaneously spin and orbital degeneracy, the dipole moment,
that is why as a degenerate subsystem they make available the high
susceptibility of crystal to the various external actions. Besides, JT states
are extended enough widely, e.g., in oxides with cubic lattice one of the
3d ion charge state can be orbitally degenerate. Here, we’ll try to
illustrate the peculiarity of structural, thermodynamic, magnetic, spectral
and other properties of degenerate systems on the example of JT ions or
molecules in crystals.
2. Jahn-Teller states - simple example. The JT theorem claims that for
orbitally degenerate ions and molecules the symmetric configuration of
environment is unstable with respect to deformations. As a result the
adiabatic potential of such ions or molecules (systems with pseudodegeneracy) turns out to be of multy-well type (see Fig. 1).
Fig. 1. Three-well
potential at JT ions
with cubic E-term
in the ground state
in the absence of
tunnelling splitting.
This potential W() can be described by the following function of JT
strains e(E) = ezz – (exx + eyy)/2 and e(E) = 3(exx – eyy)/2:
W() = W0cos3, ctg = e(E)/e(E) .
(1)
234
The characteristic displacements of nearest to JT center anions in states
corresponding to minimums of adiabatic potential (  -2/3, 0, 2/3)
are presented on Fig. 2.
X
Y
JT
JT
Z
Fig. 2. The displacement
of nearest to JT center
anions at minimums of
adiabatic potential.
JT
3. The behavior of the dilute systems with degenerate centers in
random fields. The low-symmetry crystal fields may be the dominant
mechanism of removing degeneracy at the JT ion. In this case the
character of random field distribution function determines the
peculiarities of low-temperature behaviour of thermodynamic, magnetic
and other properties of JT systems. The distinctive property of these
functions is their many-component character. For centers with cubic Eterm we have the following Hamiltonian H of interaction with random
fields, described by the two-component distribution function
f (hE , hE ) [9],


H   hE , s U E , s  hE , s U E , s ,
s



f (hE , hE )    hE  hE , s  hE  hE , s ,
(2)
s
where h ,s (   ) are the components of random field at the JT
center with index s, U,s are orbital operators. The explicit form of these
functions may be sometimes obtained if the character of random field
source and the law of low-symmetry field decrease with distance from a
235
source are given. For example, the random field distribution function of
deformational nature f(h,h) for a small concentrations of these field
sources xi << 1 has the following form
f hE , hE   f (h) 


3 / 2
N JT
 hE2  hE2  Γ 2
,
2
   xi VE
i
 1 1  
,
xi  1  
(3)
where NJT is the number of JT ions in crystal, VE is the parameter of
vibronic interaction of JT ion with JT active deformations,  is the
volume of the unit cell,  is the Poisson coefficient. At high
concentrations of random field sources xi ~ 1 we have for f(h,h)
f hE , hE   f (h) 
N JT


 
exp  hE2  hE2 / Δ 2 ;
(4)
where the dispersion  is equal to the random field value at the nearest
to a source of such field position, h  (h,h).
Density of states, heat capacity. The important property of degenerate
system is density of states for the excitation energies g(E):
g ( E )  

  E  Ei, excited (h)  E0 (h) f (h)dh ,
 k


(5)
where Ei,exited(h,h) and E0(h,h) are the energies of excited and
ground states of JT center in random fields correspondingly. We
consider the case when the dispersion  of random fields is larger than
the parameter tun, describing tunnelling between the minimums of
adiabatic potential. Then approximately in some region of energies E <
 constant density of states g(E) ~ NJT / takes place. As a result the
linear temperature dependence of heat capacity occurs in the crystals
with JT centers
CJT ~ Tg(E=0) ~ NJTT/ ,
kBT <  .
(6)
Because of the fact that at small concentration of random field sources xi
<< 1 dispersion  ~ xii, the heat capacity CJT is proportional to
NJTT(xii)-1. In the case when just the degenerate centers are the source
236
of random field, the density of states and heat capacity in corresponding
regions of energies and temperatures become independent upon xJT.
However the width of the region, where this effect takes place, depends
linearly upon xJT.
Similarly the modification of the other thermodynamic property
behaviour of crystals with the degenerate centers (elastic modules,
thermal expansion coefficient, magnetic and dielectric susceptibility)
always takes place at temperatures k BT   () .
These results have rather common character and can be generalized
easily on the case of concrete degenerate systems. So, the first
examination of obtained results was performed for the analysis of heat
capacity of the inert gases crystals with molecular impurities (for
example, Kr: 14N2) having the identical system of lowest excited states
[10]. Other exotic object for application of obtained results is weakly
doped fullerites AxC60 (x<<1), where the considered peculiarities of low
temperature properties can be also observed [11].
The dipole centers R3+-OH in oxide proton conductors АIIВIV1-xRIIIxO3-y
can serve one more example for illustration [12]. Protons are attached to
the oxygen ions forming OH centers, which themselves possess dipole
moment and form more complex dipoles with dopant ions R3+. Four
equivalent potential minima for protons exist at every oxygen and
respectively six minima for OH centers near the dopant in the
perovskite lattice. Both types of multy-well states without regard for
interactions between different “dipoles” and their interactions with other
imperfections are orientationally degenerate. At low temperatures, when
hydrogen migration between oxygen atoms is impossible practically, it is
enough to consider the four-level states of R3+-OH center. These
tunnelling states of hydrogen have symmetry A1+E+B1 of point group
C4v and energy values:
E(A1) = tun,
E(E) = 0 ,
E(B1) = tun ,
where tun is the tunneling splitting parameter. Temperature
dependencies of the contribution of the indicated centers to the heat
capacity С and dielectric susceptibility  are shown in a Fig. 3-4.
We used the relative temperature scale with t = kBT/tun. It is seen, that
237
heat capacity С is a linear function of temperature in low-temperature
region kBT < . For this system the value -1/2N(OH)exp{-(tun/)2}/
plays the role of density of states g(0). As a result the coefficient at
linear temperature term is maximum, when /tun = 2.
Fig. 3. Calculated
temperature
dependences of heat
capacity C.
/tun= 0(1), 0.5(2),
1(3), 5(4).
Fig. 4. Calculated
temperature dependences of
dielectric susceptibility
 .
/tun= 0(1),0.5(2),1(3),4(4)
It is important to underline, that completely similar results take place for
a heat capacity and magnetiс susceptibility of dilute JT systems with
zircon structure and JT ions Tm3+, Tb3+, Dy3+[8]. In this case tetragonal
crystal field is the analog of tunnel splitting parameter for hydrogen and
ensures the identical systematisation of terms in four-level system.
4. Dilute systems with cooperative interactions between degenerate
centers. In JT systems with xJT << 1 interaction between JT ions is
stipulated mainly by interchanging through phonons. It decreases with
238
distance as R3, and its angular dependence has alternating character. As
a result, when this interaction is dominating, the state of lowtemperature phase of a system can not be characterized by the longrange order parameter different from zero. Nevertheless, the lowtemperature phase of a glass type can take place in such system. The
analysis of thermodynamic properties of a system with interacting JT
ions has shown that at low temperatures new type of state - JT glass may
occur.
The thermodynamic functions of JT subsystem can be retrieved by a
method of virial expansion at temperatures exceeding transition
temperature Tg to a low-temperature phase of JT glass. In the case of JT
ions, whose ground state is the orbital doublet, the following expressions
take place [13] for transition temperature Tg, heat capacity C,
generalized susceptibility  (magnetic or dielectric susceptibility and
elastic constant-module, describing interaction of degenerate term with
strains)
Tg ~ x JT VE2 / s 2 ,
(7)


 B g 2
 N JT
k BT


p2
C ~ N JT k B Tg / T ,    0 1  Tg / T ,  0   N JT 0 ,
k BT

0

C JT




where  is the density of matrix, s is the velocity of a sound, p0 is the
electrical dipole moment of the degenerated center, CJT0 is the elastic
module of a matrix in the absence of interaction between JT ions and
strains. It is obvious, that the value of Tg/T is the parameter of virial
expansion and the phase transition into the low-temperature glass phase
at temperature Tg is possible. As a result quite definite frozen
configuration of strains takes place at the degenerate centers (at given
random field distribution in a system) for T < Tg [13,14]. However, the
average value of strains in crystal remains equal to zero. Experimentally
239
such states were observed in mixed JT systems with zircon structure
(TmxY1-xVO4, TmxY1-xPO4) [15,16].
Thus, JT model enables to describe the low-temperature
thermodynamics of degenerate system both at a dominating role of
random fields, and in the case of cooperative interaction between the
degenerate states.
5. Peculiarities of dilute magnetic systems with JT ions. Magnetic
anisotropy and magnetostriction. In comparison with the nonmagnetic crystals splitting of orbitally degenerate state in magnetics
depends essentially on the direction of exchange fields at JT ion. E.g. the
Hamiltonian of magnetic anisotropy in view of three lowest states of JT
ion with cubic E-term can be written as





2
H anis  DS ( S  1 / 2) 3n z2  1 U E , s  3 n x2  n 2y U E , s ,
3
0 
0 
 1 0
0 0
(8)




U E , s   0 1 / 2 0  , U E , s  0 1 / 2
0 ,
 0
0 0  1 / 2
0 1 / 2
where D is the single-ion anisotropy constant, S is spin, n is the unit
vector of magnetization.
The role of the anisotropy energy is played by the splitting energy of
three lowest vibronic states by the spin-orbit interaction in the second
order. As a result the angular dependence of the free energy of magnetic
anisotropy Fanis has quite non-traditional form at temperature kBT < D
(see also Fig. 5)
n x2  1 / 3


Fanis   2 DS ( S  1 / 2) Max n z2  1 / 3
n 2  1 / 3
 z

(9)
Such unusual angle dependence leads to stepwise change of Fanis
(anisotropy of “light axis”-type) with the magnetization direction (see
Fig. 5).
240
Fig. 5. The light axes
of JT subsystem
The anomalously large value of this contribution to the magnetic
crystallographic anisotropy energy is caused by the fact that the
constants of single-ion anisotropy for orbitally degenerate centers are
proportional to  or 2cub, where cub is the cubic crystal field
parameter. For usual non-degenerate ions the anisotropy constants are
proportional to 4cub3 in cubic crystals.
This peculiarity of the behavior of JT subsystem free energy leads to the
appearance of oblique phases in systems with competing anisotropy (due
to the JT term proportional to ni2 in contrast to ni4 in magnetic anisotropy
energy of matrix).
The magnetostriction properties of these systems are also original. At a
given direction of magnetization the spin-orbit interaction removes the
degeneracy from the directions of local JT deformations which were
earlier equivalent and this leads to the magnetostriction distortion of the
crystal. The JT ions contribution to the magnetostriction constants in the
absence of random fields is equal to xJTeJT, where eJT is the characteristic
local JT deformation (eJT ~ 10-1-10-2).
Only tetragonal deformations can take place at the centers under
consideration and each of the three energy levels turns to be the lowest
in the definite region of magnetization angles. The transfer of
magnetization through the boundary of these regions is accompanied at
241
low temperatures by the change of the type of tetragonal deformation at
JT ions, that is the JT ion contribution to magnetostriction. Such effects
were observed in the system YIG: Mn3+ [17].
The low-symmetry random fields which are always present in real
systems remove partly or absolutely the degeneracy at JT center. If
random field dispersion   D, then the resulting contribution of JT
centers to magnetoelastic characteristics diminishes D/ times (when
kBT < ). However, even in this case the values of magnetostriction
constants largely surpass the corresponding values which are typical for
orbitally non-degenerate ions.
Behavior of reorientable centers containing a 3d-ion of mixed valence
in cubic magnets. The indicated results (besides JT systems) have the
direct relation to magnetic properties of cubic magnets with mixed
valence (MV) centers stipulated by non-isovalent substitutions or
vacancies (see, for example, [18])). Such centers of several 3d-ions with
the charge, localized on them, arise in the case of non-isovalent
substitutions and in the presence of anion or cation vacancies. The MV
complexes in addition to traditional properties of JT centers have
peculiarities stipulated by reorientation of an extra charge (electron or
hole) between ions of a complex, when the degeneracy is taken out at
the expense of external perturbations or cooperative interactions.
Fig. 6. The trigonal MV center
near anion vacancy
V - vacancy, C - cation, А anion, h - extra hole
First of all it concerns to occurrence of essential electrical dipole
moment at MV center. The example of MV center with С3v symmetry is
presented in Fig. 6 for the case of cubic crystal lattice (in particular for
242
spinel structure). It consists of three cations (3d-ions) with localized t2g hole on them.
In the case of small hopping integrals b of an excess charge
(b<Dor ) reorientation of MV center (the transfer of an extra
charge) is similar to tunneling of JT ion between minimums of adiabatic
potential. The similar systems are characterized by anomalously strong
contribution to constants of magnetic anisotropy and magnetostriction,
nonconventional angular dependence of the free energy of a magnetic
anisotropy and effects, connected with competing anisotropy [19]. An
example of the typical phase diagram (relative temperature concentration) for systems, where the competition of a magnetic
anisotropy of a matrix and MV centers takes place is presented in Fig. 7.
Fig. 7. Magnetic
phase diagram of
a system with
mixed
valence
centers Cr4+-Cr3+
(Cr2+-Cr3+).
Continuous and
dashed
curves
represent the lines
of first - and
second - order
phase transitions,
respectively.
7. Resonance properties of degenerate systems. As a rule the
character of resonance spectrums of degenerate centers is determined by
the dominating mechanism of splitting of the ground and lowest excited
states. Such spectrums are very sensitive to any quantitative and
qualitative changes of the system state. As a result the degenerate center
spectrums can be of the convenient source of information about the main
crystal matrix.
243
JT ions or mixed-valence centers of the 3d group have a substantial
influence on the different resonance spectrums (ultrasonic attenuation,
ESR, FMR, NMR and other) of crystals. Let us consider the distinctive
features of the observed NMR spectrum of such centers in magnetic
cubic crystals. In this case the hyperfine fields at the nuclei of
degenerate centers have a substantial anisotropic component and
therefore depend strongly on the state of the orbital subsystem. In the
case of doubly degenerate JT ions with octahedral (Mn3+, Cu2+) or
tetrahedral coordination (Fe2+) the Hamiltonian of hyperfine interaction
Hhf has the following form [20, 21].
(10)
A
H hf  A1IS   2 U E 3I z S z  IS   3U Eε I x S x  I y S y .
2



Here S and I are the electron and nuclear spins of JT ion, A1 and A2 are
parameters of the isotropic and anisotropic hyperfine interactions. We
are omitting rather small quadrupole interactions to streamline
equations. Then according to the expression (10) the NMR frequency of
the JT ion is equal to

  c  d 2 nz2  c  d  2 nx2  c  d  2 n 2y
1/ 2 ,
(11)
C  A1S , d  A2S  U E , d  A2S  U E  3U E  / 2,
where ... means the quantum-statistical or quantum-mechanical
expectation value. It is easy to see that the spectral distribution of NMR
frequencies strongly depends on the quantum-mechanical average values
of orbital operators <UE> and <UE> or in other words on the dominant
mechanism of the orbital degeneracy removal (one-ion anisotropy field,
random and local low-symmetry crystal fields). It is obvious that in
contrast to pure spin 3d ions with the resonance frequencies depending
on one parameter of the isotropic hyperfine interaction A1, the NMR
spectrum of JT ions is highly anisotropic. The anisotropy factor is
determined by the ratio A2 / A1 which can be of the order of unity or
greater.
In the case when the main contribution to the splitting of the orbital
doublet brings in the one-ion anisotropy, the NMR spectrum at low
244
temperatures contains only one line. The position of this line depends on
the direction of magnetization. The character of the NMR spectrum is
drastically changed when random crystal fields bring in the dominant
contribution to the splitting of the orbital doublet. At  / D > 1and low
temperatures (kBT < ) the distribution of orientations of pseudo-spin
moments <UE> and <UE> becomes practically uniform and the
 

absorption band of width A2 S 1  3 n x2 n 2y  n x2 n z2  n 2y n z2 1 / 2 thus
arises. The maximal width of the NMR spectrum takes place when the
direction of magnetizations is parallel to [100]-axes. In this case the
absorption peaks in the static limit of the JT effect correspond to the
frequencies [20-22]
1 = A1 + A2S,
2 = A1 - A2/2S .
These frequencies determine the high and low boundaries of the
absorption band, if the parameters of hyperfine interactions satisfy the
conditions: 2A2/A1 > 1 or A1/A2 < -1. In the opposite case the
absorption band begins from  = 0 and its high-frequency boundary
corresponds to Max(1, 2).
The
typical
NMR
spectral
distribution
Ni0.35Zn0.09Cu0.12Fe2.29O4 [22] is given in Fig. 8 (A,B).
for
Cu2+
in
Fig. 8A. Experimental NMR spectrum of Cu2+ ions in nickel-zinc ferrite
1-63Cu2+, 2 - 65Cu2+, 3 – integral spectrum.
245
Fig. 8B. Theoretical spectral distribution of NMR for JT ions with
ground cubic E-term in the case kBТ/<<1 [22]
1–dynamic JT effect (A1=260, A2=-120 MHz); 2– static JT effect
(A1=260, A2=-120 MHz); 3 – static JT effect (A1=-70, A2=-310 MHz)
The typical NMR spectrums of Mn3+ ions in ferrites [20,21] are
demonstrated in Fig. 9.
Fig. 9. The NMR spectrum of Mn3+ ions in ferrites MnFe2O4 (a) and
Mn0.82Zn0.18Fe2O4 (b) for the magnetization orientations along [001],
[111], [110]-axes correspondingly.
246
The comparison of the theory and observed low-temperature spectrums
of the considered systems has shown that these spectrums are
characterized by the structure (angular and frequency dependences)
which is typical for ions with the static Jahn-Teller effect in crystals with
large random fields (  / D > 1). While the anisotropy factor for Cu2+ is
large (A2/A1>1/2) the situation for Mn3+ is quite opposite (A2/A1<1).
The effect of the NMR spectrum contraction must be observed in
considered systems with increasing temperature [21]. Firstly, the such
phenomenon was locked in for Mn3+ ions in a lithium ferrite [23] (see
Fig. 10).
Fig. 10. NMR spectrum contraction of Mn3+ ions in Li0.5Fe3.5O4
monocrystal at 77 and 300К.
a – H0 = 0, b – H0 = 1.5 kOe
The splitting of the lowest lying vibronic states of Mn3+ ions in octasites
Ei (i = x, y, z) is determined primarily by the low-symmetry crystal
field of the lithium sublattice. Thus the resultant low-temperature NMR
spectrum is a superposition of several anisotropic spectra of Mn3+ ions in
crystallographically different octasites. At high temperatures, kBT > Ei,
the anisotropic components of the hyperfine fields are determined by the
difference between the populations of the split states; they tend to zero
as temperature is raised. As a result, a significant contraction of the
NMR absorption takes place. A similar effect observed in the ESR of
Cu2+ ions has been labeled as a “dynamic spectral contraction’’ (Ref. [ 5],
247
for example). We note that the effect discussed here may also occur in a
long list of systems with JT ions, whose orbital degeneracy is lifted by
the field of anisotropy or random crystal fields.
Quite different mechanism of nonuniform broadening of the NMR
spectrum was proposed for orbitally degenerate centers in crystal with
mixed-valence ions [24]. A distinctive feature of Jahn-Teller states of
such clusters is seen when degeneracy is lifted: the lifting of degeneracy
gives rise to a nonuniform distribution of the net charge which
accompanies a characteristic deformation of the system. In random
crystal fields, the NMR spectrum of the cluster ions spreads out into a
band with a width of the order of the isotropic hyperfine interaction
parameter and is distinct from the conventional mechanisms, which stem
from anisotropic hyperfine interactions at JT ions [21]. The proposed
mechanism of nonuniform broadening of the spectra of nuclear
transitions is typical for a broad class of coordination compounds,
including high-TC superconducting oxides and doped fullerites. Spectra
features like those discussed here have been observed in the NMR
spectra of mixed-valence chromium ions in chromium chalcogenide
spinels [25].
8. Cooperative systems with JT ions. Since the microscopic nature of
the structure phase transitions for similar systems is known they may be
an ideal model for investigation of phenomenon accompanying such
phase transitions. Let us, for example, consider the problem of
suppression of the structure phase transitions by random crystal fields
and decomposition of mixed systems with degenerate centers.
Structural phase transitions (cooperative JT effect). We shall confine
the consideration to JT ferroelastic with doubly degenerate ions. In this
case the simple model was proposed by Kanamori [6] for the description
of structure phase transitions, caused by the cooperative Jahn-Teller
effect
(12)


2
2

 V e  hE , s  V2 (eE  eE ) U E , s  

H  E0    E E
,

Ve

h

2
V
e
e
U


s 
E
E , s
2 E E
E , s



248




1 0 2

E0  N  C JT
eE  eE2   V3eE eE2   eE2   ,
2

  1 0
 0 1 / 2
U E  
, U E  

,
 0 1
1 / 2 0 
where N is the number of elementary cells in crystal; V2 and V3 are the
parameters of anharmonic interaction of the doubly degenerate term with
uniform JT deformations eE and eE ; hE,s and hE,s are the components
of the two-dimensional random field at the JT ion, index s labels JT ions.
In the absence of anharmonic interactions the model describes the
structure phase transitions of the second order, while transitions of the
first order are described when anharmonic interactions are taken into
account. The cubic crystals with spinel structure, containing JT ions of
the iron group (Mn3+, Cu2+, ets.) in octasites, can serve as an example of
considered ferroelastics. The analogous model may be utilized for the
rare-earth compounds with the zircon structure (for example, TmVO4).
As a rule, the structure phase transitions of the first order occur in the
cubic crystals while of the second order - in compounds with zircon
structure.
In the theory of the cooperative Jahn-Teller effect the role of the order
parameter is played by the deformation eE, which is defined by the
equation [6, 26]
V  2V e  (V e  hE  V2e2 )  E (e, h) 
,
e  cJT  E 0 2  E
th
E (e, h)
 k BT 
 C JT  


(13)
1/ 2
2


E (e, h)   VE e  hE  V2e2  hE2 


, e  eE  .
Here сJT is the concentration of JT ions, the line over the expression
means the configuration averaging with random field distribution
function f(h) (3)-(4), for simplicity the quantity V3 is set to zero. The
typical temperature dependencies of the order parameter for different
values of dispersion  are presented in Fig. 11.
249
The phase transition can be described analytically when the anharmonic
parameter is small. Then, in the absence of random fields, the phase
transitions are nearly of the second order and characterized by the
following parameters
TD  (VE e0 )(1  6 p 2 ),
eE (TD )  6 pe0 ,
eE (T  0)  (1  2 p )e0sign( p) ,
p 
V2
0
C Ω
, e0  cJT
JT
VE
C0 Ω
(14)
,
JT
where TD is the temperature of the structure phase transition, e0 is the
parameter, describing JT deformation at T  0K, the quantity p << 1.
Fig. 11. Temperature dependencies of the order parameter.
The following dimensionless variables and values of parameter p are
used: x = eE/e0, t = kBT/(VEe0), /(VEe0)= 0.4 (1), 0.8(2), 0.9(3),
1.1(4); p = -0.2.
Random crystal fields and dilution of JT systems by nondegenerate ions
reduce the temperature of phase transition and value of the order
250
parameter in the low-symmetry phase. The phase transition temperature
TD is equal to zero, when the critical value of the dispersion arrives at
crit, which has the same order as the quantity VEe0
(13)
1/ 3

TD
 
 A 1 


  crit 
,  crit 



2 
VE e0 1  2p 2 , A3 
.
2
3 (3)
The critical concentration of JT ions may be determined in the
framework of this approach if the dependence of the dispersion as a
function of сJT is defined, e.g,:
2(сJT)= 2(0) + (1 - сJT)12.
Such calculations have been performed for the solid crystal solution
Tm1-xLuxVO4, where the critical concentration of JT ions Tm3+ is
situated in the interval 0.18 < xcrit< 0.28 (x = 1 - сJT) [27]. The model
shows good agreement between theoretical and experimental results [28].
Cooperative JT model can be used also for the analysis of the structure
phase transitions in quasi two-dimensional crystal La2MeO4- (Me = Cu,
Co). The interest to the question icreased after the discovery of HTSC.
The phase transition from the high-temperature tetragonal phase to the
low-temperature octahedral phase occurs due to rotation of the oxygen
octahedron around the [110] axis and antiphase rotations of oxygen
octahedrons in the neighboring cells [29-30]. For the description of such
phase transitions a model of multi-well potential centers was developed
on the basis of a theory of the pseudo Jahn-Teller effect [31]. In this
model the expressions for the order parameter and temperature of the
structure transition TD are similar to (11)-(13). If the main contribution
to the dispersion is made by one type of random field source, e.g. Sr
impurities, the corresponding expressions for TD are transformed to the
form (see also Fig. 9)
TD  TD ( x  0) 1  2/π x/xcrit , TD   , (14)


2
TD 
TD ( x  0)1  x/xcrit 1/ 2 , TD   .
π
where the critical concentration xсrit = {2TD(x = 0)/()}2. The
comparison of the theoretical dependence TD(x) with experimental data
is shown in Fig. 12. The resulting, without any fitting parameters,
251
theoretical curves are in good agreement with experimental data.
Fig. 12. Concentration dependence of the transition temperature from
tetragonal phase to orthorhombic phase in La2-xSrxCuO4-.
Decomposition of degenerate systems. Let us consider now the phase
transition such as decomposition of degenerate systems and analyze the
origin of immiscibility (or rather weak miscibility) of different
components in crystals with the cooperative JT effect. In such systems
the substitution of JT ions by orbitally nondegenerate ions can lead to a
separation into phases with a higher and lower concentrations of JT ions.
The connection between structural phase transformations and phase
transitions of separation type always takes place. This connection
appears in JT crystals due to the fact that in both cases the system has a
tendency to lower its free energy as a result of maximum possible
splitting of degenerate states. The possibility of the microscopic
description of different types of immiscibility phase diagrams using
simplest JT subsystem models with structural phase transitions (SPT) of
the first and second orders was shown in [32,33,34].
Model of mixture. Phase instability regions in quasi-binary system. The
model of a mixture in which free energy is determined by splitting of
252
degenerate levels, and the configuration entropy corresponds to a
random distribution of JT ions in the mixture is considered. In this case,
the free energy of a quasi-binary system per structural unit can be
written in the form:
(15)
F

 E (e, h)  
e2
 ,
, FJT   k BT ln 2cosh 
2
k
T
 B 

Fid  T xJT ln xJT  1  xJT ln 1  xJT  .
0
xJT FJT  Fid  C JT
Expressions (13), (15) allow to calculate the chemical potentials a (of
components with JT ions), b (of component with orbitally
nondegenerate ions), and the exchange chemical potential  = a -b of
the quasi-binary system under investigation [35,36,37]:

 x

F
 FJT
 a  b  FJT  xJT
 T ln  JT  , (16)
 xJT
xJT
1  xJT 
 a  F  1  xJT    FJT  xJT 1  xJT 
 FJT
 T ln xJT . (17)
xJT
One can use the standard condition of equilibrium for the calculation of
the concentrations of components (binodals) in the coexisting phases:
aI = aII ,
bI = bII ,
where indexes I and II correspond to the phases with tetragonal and
cubic symmetry respectively. The boundaries of the region of absolute
instability of the solution are determined by the condition [38]:
2F
2
xJT


0 .
 xJT
(18)
The corresponding equilibrium phase diagram (binodal) and the spinodal
curve are shown in Fig. 13 for the case of SPT of the second order. The
critical point of mixing is achieved at temperature kBTcrit = 0/3 (0 
VEe0) and concentration xcrit = 1/3.
The results of calculation of the phase boundaries (binodal lines) for the
cooperative JT systems with the first order SPT are shown in Fig. 14. In
contrast to the system considered above, the critical point of mixing
253
coincides with the SPT temperature of the pure compound with JT ions
[39]. Such phase diagram is in good qualitative agreement with the
typical fragment of the experimental diagram for the stratified JT
systems (see, for example [39]).
Thus it is shown that the model of degenerate system with substitutions
allows to describe a separation of the mixed system into phases with a
higher and lower concentrations of degenerate centers in all temperature
range below Tcrit.
Fig. 13. Phase diagram of the quasi- Fig. 14. Phase diagram of the quasibinary JT systems with SPT of
binary JT systems with SPT of first
second order.
order.
Bold curve is the binodal and
dashed is the spinodal one.
It is possible to show, that the class of the immiscibility phase diagrams
described by JT model of interparticle interaction can be essentially extended. It
follows, in particular, from the analysis of phase equilibriums of crystalline
systems with two types of JT ions. The typical phase diagrams of such
mixed JT systems is presented in Fig. 15-16. The systems are considered
with one type of anharmonic coupling (Fig. 15) and competing
anharmonic couplings (Fig. 16).
254
It is obvious, that the asymmetrical model of JT system with a
competing anisotropy (p1  p2 ) allows to describe the basic types of
the immiscibility phase diagrams for quasi-binary systems [40].
Fig. 15. The phase diagram of
the quasi-binary JT system
without competing anisotropy
(p1 = 0.3, p2 = 0.05).
The dashed line in the top of
the diagram shows a line of
structural phase changes in a
system with a random
distribution of two types of JT
ions. The dotted line in the
bottom of the diagram shows
the curve of absolute
(spinodal) instability of a
system.
Fig. 16(a,b). The phase diagrams of the quasi-binary JT system with a
competing anisotropy (p = p1 = p2 = 0.25 (a), 0.35 (b)).
The solid lines correspond to phase boundaries. The dashed curve in the
diagram top shows a line of structural phase transitions in a system with
a random distribution of two types of JT ions. The dotted curves in
diagram bottoms show boundaries of an angular phase.
255
8. Conclusion. We hope, that the developed microscopic theory of wide
class of JT system properties (frequently unusual) can become a model
basis for calculations of phase equilibriums, structural, thermodynamic,
magnetic and spectral properties of composite systems with various
types of the degenerate states. For today there are enough examples
confirming such point of view.
The work was supported by Russian Foundation for Basic Research
(grant 02-03-32877).
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