MODEL OF PHASE EQUILIBRIUM IN SYSTEMS WITH COMPETING JAHN-TELLER
INTERACTIONS
A.Ya. Fishman1, M.A. Ivanov2, N.C. Tkachev3
1
Institute of Metallurgy, Ural Department of Russian Academy of Sciences,
Amundsen St. 101, 620016 Ekaterinburg, Russia
2
Institute of Metal Physics, Ukrainian National Academy of Sciences,
Vernadsky St. 36, Kiev-142, 03680 Ukraine
3
Institute of High Temperature Electro-Chemistry, Ural Department of Russian Academy of
Sciences, S.Kovalevskaya St. 18, 620218 Ekaterinburg, Russia
The decomposition phase diagrams of systems with cooperative Jahn-Teller effect are investigated.
It is found, that the Jahn-Teller mechanism of interaction between ions is capable to describe the
main topologically various types of immiscibility phase diagrams in quasi-binary systems.
Introduction
Phase diagrams of mixed condensed systems with the limited solubility of various components
differ by large topological variety [1,2]. However, until now there are several problems with their
microscopic description and especially with identification of mechanisms, which are charged with
the characteristic peculiarities of the diagrams. In this connection the tasks related to modeling of
phase equilibriums of systems with essential cooperative interactions appear to be actual. The
crystal systems with the Jahn-Teller (JT) centers are an ideal object for such evaluation. JT
mechanism of inter-particle interaction answers for the phase transformations into the phase of JT
glass [3,4], structural phase transitions (SPT) with ferro- and antiferro- distortion ordering [5] and
transitions of spin re-orientation type [6,7]. It is also possible to consider as established, that the
effects of stabilization connected to splitting of degenerate states by the field of environment
charges, result in the decomposition of the mixed condensed systems (see, for example, [8,9,10]).
In the submitted work the research of the immiscibility phenomenon in the degenerate crystal
systems is continued [11,12]: the opportunity of the description of the topologically various
decomposition phase diagrams within the framework of simple microscopic model which is taking
into account only JT interactions is considered.
The model of phase transformations offered by J. Kanamori [5], is used for the analysis at one of
three types of transitions: structural, spin-re-orientation and disintegration. In such model the
anharmonic interactions are considered as rather weak in comparison with JT interactions and the
structural phase transformations of both the first and the second order can be described. In systems
with the ferro-type of JT ordering the anharmonic interactions determine the gradual changes of the
order parameter at temperature of the phase transition of the first sort. If one select by appropriate
way JT constant for initial compound, it is possible to describe full spectrum of immiscibility phase
1 - 41
diagram without any adjustable parameters. We have confirmed the efficiency of such approach at
the analysis of experimental data for crystals Mn3-xAlxO4 and Mn3-xCrxO4 [13].
In this work the basic attention is focused on relatively weak inter-particle interactions, which
nevertheless can strongly influence the topology of the immiscibility phase diagrams. Quasi-binary
crystal systems with substitution of JT ions by orbital non-degenerate ions or by the others JT ions
are considered. As a result it is established, that JT mechanism of interpartial interaction is able to
respond for significant variety of the decomposition phase diagrams, at least, in systems of quasibinary type. The recognized results can be used for the qualitative analysis of the experimental
phase diagrams of immiscibility for the various mixed condensed systems without dependence from
presence of JT effect.
Structural phase transitions (cooperative JT effect)
Let us restrict ourselves by consideration of JT ferroelastic with the doublet-type of degenerate ions.
In this case Kanamori proposed the simple model for the description of structure phase transitions,
caused by the cooperative Jahn-Teller effect [5]





H  E0   VE eE  hE , s  V2 (eE2   eE2  ) U E , s  VeE  hE , s  2V2eE eE U E , s ,
s




  1 0
 0 1 / 2
U E  
, U E  

,
 0 1
1/ 2 0 
1 0 2

E0  N C JT
eE  eE2   V3eE eE2   eE2   ,
2

(1)
where N is the number of elementary cells in crystal; V2 and V3 are the parameters of anharmonic
interaction of the doublet term with uniform JT deformations eE and eE; hE,s and hE,s are the
components of 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 occur when anharmonic interactions are taken into
account. The cubic crystals with the spinel structure containing JT ions of the iron group (Mn3+,
Cu2+, ets.) in octa-sites, can serve as an example of the considered ferroelastics.
In theory of the cooperative Jahn-Teller effect the deformation eE plays the role of the order
parameter, which has to be determined by the equation [6, 14]:
V  2V e  (V e  h  V e 2 )  E (e, h) 
E
2
2
,
e  c JT  E
th
 E
0
E (e, h)
 k BT 
 C JT  

(2)

1/ 2
2


E (e, h)   VE e  hE  V2e 2  hE2  , e  eE  .


Here сJT is the concentration of JT ions, straight horizontal line over the expression means the
1 - 42
configuration averaging over 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 the dispersion  are presented in Fig. 1.
The structural 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 are 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
C
0
JT
Ω
, e0  c JT
VE
C0 Ω
(3)
,
JT
where TD is the temperature of the structure phase transition, e0 is the distortion describing JT
deformation at T  0K, the quantity p << 1.
Fig. 1. 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.
Decomposition of degenerate systems
Let us consider now the phase transition such as the decomposition of degenerate system and
analyze the nature 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 the separation into phases with higher and lower concentrations of
JT ions. The inter-coupling between structural phase transformations and phase transitions of
separation type always take place. This interconnection appears in JT crystals due to the fact that
the system has a tendency to lower its free energy as a result of maximum possible splitting of
degenerate states in both cases. The possibility of the microscopic description of different types of
1 - 43
the immiscibility phase diagrams within the simplest JT subsystem models with structural phase
transitions (SPT) of the first and second orders was shown in [10-13].
Model of mixture. Substitution of JT ions by orbital nondegenerate ions. The model of mixture in
which free energy is determined by splitting of degenerate levels, and the configuration entropy
corresponds to a random distribution of JT ions over all sub-lattices is stated below. The
generalization of the Kanamori model on the case of quasi-binary system AX JT B1 X JT (XJT is the
concentration of JT-ions) presents no difficulties. In such approximation multi-component solid
solutions in which the replacement occurs only over the JT-ions sub-lattice can be considered. In
particular, this situation can take place in crystals where JT-centers are substituted by orbital nondegenerate ions. In this case, the free energy of a quasi-binary system per structural unit can be
written in the form:

 E (e, h)  
e2
 ,
, FJT   k BT ln 2cosh 
2
 k BT  

Fid  T xJT ln xJT  1  xJT ln 1  xJT  .
0
F  xJT FJT  Fid  C JT
(5)
Expressions (13), (15) allow to derive 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 [15,16,17]:

 x

F
 FJT
 a  b  FJT  xJT
 T ln  JT  ,
 xJT
xJT
1  xJT 
a  F  1  xJT    FJT  xJT 1  xJT 
 FJT
 T ln xJT .
xJT
(6)
(7)
The standard condition of equilibrium for the calculation of the concentrations of components
(binodals) in the coexisting phases I and II can be used:
aI = aII ,
bI = bII .
The boundaries of the absolute instability region are determined by the equation [1]:
2F
2
xJT


0 .
 xJT
(8)
The corresponding equilibrium phase diagram (binodal curve) is shown in Fig. 2 for the case of
SPT of the second order. The critical point of mixing is accomplished at temperature kBTcrit = 0/3
(0  VEe0) and concentration xcrit = 1/3.
The results of computation of the phase boundaries (binodal lines) for the cooperative JT systems
with the first order SPT are shown in Fig. 3. In contrast to the system considered above, the critical
point of mixing coincides with the SPT temperature of the pure compound with JT ions. Such phase
1 - 44
diagram is in good qualitative agreement with typical fragments of the experimental diagrams for
the immiscible JT systems (see, for example [18]).
Therefore, it is shown that the model of degenerate system with the simplest substitutions allows to
describe a separation of the mixed system into phases with higher and lower concentrations of
degenerate centers in all temperature range below Tcrit.
Fig. 2. Phase diagram of the quasi-binary JT
systems with SPT of second order.
Fig. 3. Phase diagram of the quasi-binary JT
systems with SPT of first order.
Bold curve is the binodal and dashed one is the
spinodal .
It is possible to show, that the class of the immiscibility phase diagrams described by JT model of
inter-particle interaction can be extended significantly. It follows, in particular, from the analysis of
phase equilibriums of crystalline systems with two types of JT ions.
Quasi-binary systems with substitution of one JT-ion into another JT-ion. Now let us consider
another mixed system where one type of JT-ions is replaced by another one, i.e., by ions with the
mismatched parameters of Hamiltonian (1). For the sake of simplicity we can restrict ourselves by
varying only one parameter characterizing “new” JT-ions, namely, the parameter of anharmonicity
denoting it as p2 . Then, pure components or boundary compositions of JT system with X = 0 and X
= 1 have at least different temperatures of structural phase transitions:


0
tD  tD
1  6  p1  pX  (1  X ) p2 2 ,
t D  k BTD /V E e0 ,
(9)
0
where t D
 1 is the temperature of structural transition of the second order in the system where the
anharmonicity is negligible. When the signs of p and p2 coincide the character of JT-deformation in
pure components is one and the same. In contrast, when the signs at p and p2 are opposite
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subsequently the competing anisotropy springs up and as a result the phase transformation of the socalled spin-re-orientation type holds with the formation of the “oblique” phase [6].
The free energy and the chemical potentials of mixed ions in such systems with the chaotic atomic
distribution can be described quite analogously to (5)- (7), as follows:

 z  pyz 2 
  ln X  ,
  ln cosh 


k BT
t



a
z  e / e0 ,

 z  p2 yz 2 
  ln 1  X  ,
  ln cosh 


k BT
t



b

  ln cosh

k BT


ex


t
(10)
 z  p2 yz 2 
  ln  X
cosh 
 z  pyz 2 






t
F  X a  (1  X ) b , y  cos(3 ), tg 

1  X  ,
2ezz  exx  e yy
,
3 ( exx  e yy )
where the value of y defines the character of resulting JT–deformation in crystal. In addition, the
parameter y, in sharp contrast from the above-illustrated cases may be not equal to 1.
Corresponding equations for evaluation of the equilibrium values of the order parameters z and y
have the form:
1  3 p1 zy  X
 (1  X )
1  3 pzy  2( pz ) 2
E ( z, y)
1  3 p 2 zy  2( p 2 z ) 2
e( z , y )
 E ( z, y) 
tanh 

 t

 e( z , y ) 
tanh 
  0,
 t 
(11)
 E ( z , y )  (1  X ) p 2
 e( z , y ) 
p1 
tanh 
tanh 

0 ,
 t

 t 
E ( z, y)
e( z , y )
Xp

E ( z , y )  z 1  2 pyz  ( pz ) 2

1/ 2

, e( z , y )  z 1  2 p 2 yz  ( p 2 z ) 2

1/ 2
.
The boundaries of phase equilibriums are determined by standard equilibrium conditions — by the
equalities of chemical potentials of atoms in coexisting phases.
The typical phase diagrams of such mixed JT systems is presented in Fig. 4-8. The systems are
considered with one type of anharmonic coupling (Fig. 4) and competing anharmonic couplings
(Fig. 5-8).
Systems with the same type of deformation on mixed JT –ions. It is naturally, that in this case the
simplest portrait of phase diagram with the restricted solubility region has occurred. It can be
1 - 46
readily seen from fig. 4 that two-phase region is the “cigar”-like at high temperatures and the
miscibility gap at low temperatures. Somewhat different (from fig. 4) configuration of phase
diagram occurs when the structural transition temperatures at boundary concentrations with X = 0
and X = 1 are mismatched drastically (in order of magnitude and more). Here within the decreasing
of temperature of structural transition of one of the components «cigar» becomes thicker at the
beginning with further overlapping by rising miscibility gap. At the asymptotic limit TD(0)/TD(1) 
0 this diagram corresponds to the one of JT-system substituting by non-degenerate ions (as was
already shown by Fig. 3).
Fig. 4. 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.
Immiscibility in systems with competing interactions. One of the specific features of mixed JT
systems with competing anharmonic interactions is the formation of the so-called “oblique” phase
[6,7]. Let us consider for simplicity solution with two different types of JT-ions with just opposite
signs of anharmonicity parameters p2 =  p. In that case with random distribution of JT-ions over
the cation sub-lattice the curve of phase transitions of the first-order has parabolic form and the top
of the parabola (tс =1, Xс = 1/2) is the critical point where the second-order structural transition
takes place. In addition, in the temperature region below the critical one t  tс =1, i.e., in “ferrodistorted” phase with y = -1 (at X < 1/2) or y = 1 (at X > 1/2) the intermediate (by type of distortion)
«oblique» phase occurs with y.
From figs. 5-8 one can see, as the topology of phase diagram is re-constructed with the increasing
of the anharmonicity parameter, in particular, its immiscibility regions grow up much. 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
1 - 47
dotted curves in diagram bottoms show boundaries of an angular phase. Single-phase regions of
orbital ordering from the left side of diagrams correspond to values of y =-1 and from the right
correspond to y = 1. Two-phase regions of “cigar”-like form at the top are presented by the
equilibriums like “para”-“ferro” (cubic and tetragonal phases). At the bottom of the diagrams twophase regions are presented mainly (5, 6, 8 – fully, 7 – in part) with the equilibriums between
“ferro”-distorted states having opposite orientations of tetragonal deformations, i.e., with y =-1 and
y = 1. The «oblique» phase of solid solution with the chaotic distribution of JT-ions always lies
inside of the latter two-phase region. In other words, at the equilibrium state decomposition of the
solution is much more preferable with respect to the formation of «oblique» phase. One can expect,
that in systems where the equilibrium cannot be realized due to kinetic reasons, the decomposition
tendency must lead to the formation of complexes enriched by one or another component relating to
concentration interval.
Fig. 5. The phase diagrams of the quasi-binary
JT system with competing anisotropy
(p = p1 = p2 = 0.2).
Fig. 6. The phase diagrams of the quasi-binary
JT system with competing anisotropy
(p = p1 = p2 = 0.25)
Phase diagram with p = 0.3 (Fig. 7) illustrates the specific features of the transformation from one
topology to another, namely, from “aseotropic” to “eutectic” character. It can be easily seen that in
comparatively narrow temperature region near tc = 1 new two-phase («ferro»- «ferro») and
moreover three-phase equilibriums («para» - « ferro » - « ferro ») occur in both sides of the diagram
(x  1/2 и x  1/2). Additional points of three-phase equilibriums appear either above or below tc =
1 respectively.
1 - 48
Fig. 7. The phase diagrams of the quasi-binary JT system with competing anisotropy
(p = p1 = p2 = 0.3).
Fig. 8. The phase diagrams of the quasibinary JT system with competing
anisotropy (p = p1 = p2 = 0.35).
Conclusion
It was shown that the phenomenon of restricted solubility is the specific feature of the degenerate
systems and the JT model taking into account the cooperative inter-ionic interactions allows us to
clear up and describe broad class of the phase diagrams with immiscibility. Topological
peculiarities of the phase diagrams of the decomposition in JT crystal systems are determined by the
1 - 49
anisotropy anharmonic interactions. Besides, the so-called “oblique” phases in systems with
competing anisotropy could be corresponded only to the meta-stable state of the solid solution due
to its decomposition into the phases with the opposite directions of dominated anisotropy turns out
to be much more preferable thermodynamically.
It is obvious, that the asymmetrical model of JT system with the competing anisotropy (p1  p2 )
allows for the description of all the types of the immiscibility phase diagrams for quasi-binary
systems [19]. We hope, that the developed microscopic theory can serve as the basic model for
calculations of phase equilibriums of composite systems with various types of the degenerate states.
The work was supported by Russian Foundation for Basic Research (grant 02-03-32877).
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