THE MODEL OF DIFFUSION IN SYSTEMS WITH MULTIWELL POTENTIAL AT JAHN-TELLER IONS
A.Ya.Fishman1, V.B.Vykhodets2
1
Institute of Metallurgy, Ural Department of Russian Academy of Sciences, 101
Amundsen St., 620016 Ekaterinburg, Russia
2
Institute of Metal Physics, Ural Department of Russian Academy of Sciences, 18
S.Kovalevskaya St., 620219 Ekaterinburg, Russia
ABSTRACT
Theoretical investigation of diffusion coefficients of the tracer atoms in systems with
multi-well potential at Jahn-Teller (JT) or pseudo-JT ions is performed. It is shown
that the shape of the potential barrier for a diffusing atom in JT systems can be varied
essentially. On the basis of developed theory the interpretation is given of diffusion
data for carbon and oxygen atoms in -Ti. Measurements of diffusion have shown
that the traditional models of diffusion barriers failed to describe adequately the
relationship between activation energies Q(C) and Q(O) of those atoms. The Q values
for oxygen and carbon atoms were practically the same while carbon atoms cause
sufficiently larger lattice distortions being soluted in -Ti. The reasons of such
disagreement were found with the help of precision experiments on diffusion
coefficient anisotropy, where the different signs of values d(Dzz/Dxx)/dT for oxygen
and carbon were found. These results demonstrated that the activated diffusion states
of carbon atoms are degenerate while for oxygen atoms they are nondegenerate. The
Jahn-Teller effect decreases the diffusion activation energy Q(C) on the value by order
to characteristic Jahn-Teller energy (~1 eV). This explains almost equal values of
Q(C) and Q(O).
INTRODUCTION
The interest to the diffusion phenomena in crystal systems containing ions with
degenerate ground state is connected with the following. The character of the splitting
of the degenerate states in the diffusion process may essentially influence the potential
barriers form and the diffusion coefficient value. This question which is of principle
for understanding the diffusion kinetic is investigated insufficiently. It is important
that diffusion coefficients are rather sensitive to interaction of tracers with impurities
and defects in solid states, to structure transformations, the character of random
deformation fields distribution and thus may give an important information about the
system under investigation.
In our paper [1] the change of diffusion potential barrier due to the degenerate centers
was considered. The vacancy model of anion diffusion in cubic crystal lattice was
choosed as the object for the analysis. The following mechanisms of degenerate
centers influence on diffusion atom potential relief weree investigated: the lowsymmetry crystal fields, the resonance interactions (double exchange), JT interaction.
The Influence Of Degenerate Term Splitting On The Potential Barrier Form Of
The Diffusing Atom
604
The case of diffusion in the lattice with orbitally non-degenerate ions where JT state
(for definiteness the cubic E-term in configuration 3dn1) appears on the cation
closest to the diffusing anion vacancy (see Fig. 1, cation with number k=1) was
considered. If the charge-quadrupole interaction represents the main mechanism of
degenerate term splittting, the splitting energy dependence upon the position of
diffusion atoms may be described as
1/ 2
2
 2 
E (u )   C  q 3   2  (1  u 2  2u ) 3 / 2  3(1  2 u u 2  2u )  4
, (1)
 a 
where a is the distance metal-ligand, u is the relative anion displacement (u = U/a, in
the saddle point u = 1/2), C is the charge-quadrupole interaction parameter. The
corresponding dependence E(u) (in the units C(q2/a3)) is presented on Fig. 2 (curve 1).
It can be seen that the potential barrier caused by the additional JT contribution should
essentially change its form and in particular it should be narrowed.



Fig.1. Crystal environment of diffusing atom.
 - anion,  - cation,  - anion vacancy, e- - excess 3d-electron.
Fig.2. Contribution of degenerate states splitting to potential energy of diffusing atom.
1  the case of non-degenerate 3dn cation states; 2  the case of degenerate 3dn cation
states.
In the case of anion diffusion in a crystal with JT cations (the ground 3dn1 state - Eterm) the analogous calculation has been made for each JT 3d -ion closest to the
hopping path (Fig.1). The typical dependence of the summary free energy splitting is
presented on Fig.2 (curve2). It can be seen that in this case the specific JT contribution
to potential relief may be also rather noticeable. The typical scale of the value E(u)
variation must have the order of value cub/Z (where cub is the cubic crystal field
parameter and Z is the number of nearest to cation ligands), i.e. thousand degrees.
Resonance Interaction In The Complex Of 3d-Mixed Valence Ions
We shall illustrate the influence of the charge transfer effects on the diffusion process
by the example from [1] also for the system with orbitally non-degenerate ions. Then
605
the so called “double exchange” or resonance interaction [2] caused by transfer of an
extra eg-electron takes place mainly between cations with k=1,2 on Fig.1. The energy
levels of this mixed-valence ion pair can be described by the following expressions

 p  2h( E )  4h ( E )
 (1 / 2) t   p  2h( E )  4h ( E )
E1, 2  (1 / 2) t 
2
2
2
1/ 2

1/ 2

,
2
E 3, 4
,
(2)
t  b( E , E )  b( E , E ) ( S ) ,  ( S )  ( S  1 / 2) /( 2S 0  1) ,
p  b( E , E )  b( E , E ) ( S ) ,
here b(E,E) are the transfer integrals of egelectron between state E(1) and E(2)
of cations with k = 1,2 (transfer integral b(E,E) is equal to zero for the present pair
geometry), h(E) and h(E) describe the low-symmetry fields on the ion with 3dn+1
configuration, S is the total pair spin, So is the spin of cation in orbitally nondegenerate state with the 3dn configuration.
The character of behavior of four considered energy levels during the anion diffusion
process is determined by the change of low-symmetry fields (considered earlier) and
transfer integrals t and p. Due to the fact that the transfer effects between cations
through the anion (oxygen, fluorine) depend strongly on the angle of superexchange
bond: transfer integrals t and p take the maximum values for the 180o superexchange
bond and are approximately equal to zero for the 90o bond. The corresponding
expression for bb(u) can be written in the form
b( E, E )  b( E , E ) / 3  (b0 / 4)(a / R)  cos  , cos   u(u  2 ) /(1  u 2  2u) ,
where b is the transfer integral for the 180o bond in the Anderson theory of
superexchange [2],  is the angle of the superexchange bond between cations through
the diffusing anion, R is the cationdiffusing anion length. It is easy to show that in
the vicinity of the saddle-point the parameter  is approximately equal to  = 5 [3].
0
0.2
u , a.u.
0.4 0.6
0.8
1
0
1
F(u) , a.u.
-2
2
3
-4
4
-6
5
-8
Fig.3. Dependence of diffusing potential energy upon transfer integral of extra
electron.
b/(Cq2/a3) = 0.05, 0.1, 0.15, 0.2, 0.25.
606
The dependence of the pair free energy on the anion position at different ratio between
value bo and Cq2/a3 is presented on Fig. 3. It is easy to see that the account of transfer
effects can change essentially the character of Jahn-Teller contribution to the potential
relief. The situation seems to be similiar in the qualitative sence when the anion
diffusion is considered for the system with the orbitally degenerate ground state of
cations in the 3dn configuration. It is caused by the fact that the main contribution to
the energy change due to transfer effects is determined by the resonance interaction of
the same pair of cations (k = 1,2).The character of the corresponding contribution
depends on the electron configuration of 3d-ions, geometry of anion migration and, as
a result, angular dependence of transfer integrals. The value of transfer integral b for
typical distance R = a = 2A in different oxides is equal to 0.10.2 eV [2]. The
corresponding parameters b(E,E) (mainly b(E,E)) may exceed this value more
than four times in accordance with (3). The direct dependence of the considered
contribution to the parameter of diffusion barrier on the states of magnetic subsystem
allows to analyse the influence of magnetic phase transitions on the diffusion kinetics.
Systems With Jahn-Teller Or Pseudo -Jahn-Teller Effect
It was assumed that in crystals with pseudo-JT effect at 3d-ions the additional
contribution to the potential energy of diffusing atom brings about the JT interaction
of two (or more) non-degenerate levels with the corresponding anion displacements.
Then if the crystal potential at 3d-ions occurs to be multi-well one, the same character
is valid for the anion potential energy. The typical changes of the potential barrier for
the diffusing anion while passing from one-well potential to the multi-well potential is
presented on Fig. 4. It was shown that the contribution to the diffusion activation
energy Ea due to multi-well potential at 3d-ions is proportional to the value of oxygen
displacements (caused by JT effect) from equilibrium positions:
E a  V JT u JT ,
(3)
where V(E) is an interaction constant describing the interaction of the degenerate term
with JT active displacements uJT . One can write, that considered effect gives the
value Ea by order of JT stabilization energy, i.e. 1eV.
Fig.4. Influence of Jahn-Teller interaction on the form of potential barrier of diffusing
atom.
607
Thus the presence of the degenerate states in the system can lead to the strong
influence on the potential barrier for the diffusing atom and, as a consequence, on the
diffusion coefficient [1]. The character of degeneracy determines the corresponding
mechanism of this influence. In present work we shall consider the role of degenerate
excited diffusion state in the diffusion properties of the systems -Ti: C,O. On the
basis of developed theory for JT mechanism of the influence on the potential barriers
form the interpretation is given of diffusion data for carbon and oxygen atoms in -Ti.
JAHN-TELLER EFFECT AND DIFFUSION ACTIVATION
ENERGY OF INTERSTITIAL ATOMS IN -TI
Accurate measurements of carbon and oxygen diffusion coefficients in -Ti [4,5] had
shown that traditional models of diffusion barriers failed to describe adequately the
relationship between activation energies of those atoms. Values of Q(C) and Q(O)
nearly coincide (Q (C) = 2.08 eV, Q(O) = 2.13 eV), whereas carbon atoms distort the
lattice to a much greater extent than oxygen atoms do upon dissolution [6].
Estimations founded on some elastic models had showed that difference value Q(C ) Q(O) must be from 0.88 up to 1.24 eV. These estimations were made using Pailing
radius values of interstitial C and O atoms or experimental data on influence of
dissolved interstitial atoms on lattice parameters c and a of -Ti [6]. The clue to
understanding the reasons of such discrepancy was found in interpretation of accurate
experiments concerned with anisotropy of diffusion coefficients Dij in -Ti [4,5].
Those experiments have revealed that d(Dzz/Dxx)/dT had different signs for oxygen
and carbon atoms. The anomalous positive sign for carbon atoms was attributed to a
double-well potential for a diffusing atom at saddle points. In other words, activated
diffusion states of carbon and oxygen atoms were assumed to be degenerate and
nondegenerate respectively [7]. Therefore, it is necessary to revise the nature of the
diffusion activation energy taking into account the Jahn-Teller effect. A modified
model of diffusion potential barriers (calculation of the activation energy) in systems
with degenerate activated diffusion states is proposed. It is shown that in the activated
state the Jahn-Teller effect decreases the diffusion activation energy Q by nearly
characteristic Jahn-Teller energy (~1 eV). This explains almost equal values of Q(C)
and Q(O). Thus, the concept of a degenerate activated state can reasonably account for
all experimental data on diffusion of oxygen and carbon atoms in -Ti.
Model Of Diffusion Barriers And Anisotropy Of Diffusion Coefficients In -Ti
The highest accuracy achieved to date in measuring diffusion coefficients and other
diffusion parameters for the atoms of light elements in solids has been achieved in
investigations of the anisotropy of the diffusion coefficients Dzz/Dxx of tracers 18O [4]
and 14C [5] in the hcp titanium lattice. The most important result of these experiments
is the observation that the sign of d(Dzz/Dxx)/dT is different for oxygen and carbon
(Fig. 5). The temperature dependence of the diffusion anisotropy Dzz/Dxx in the
systems studied arises only in the presence of a deviation of the hcp lattice from the
ideal lattice (for which c/a = (8/3)1/2), in which case two kinds of saddle points for the
migration of interstitial impurity atoms appear in the crystal [8]. The fact that the
oxygen and carbon atoms respond differently to these small deviations (c/a  (8/3)1/2)
shows that the character of their activated states is different. It turned out [7] that the
608
effect is connected with the degeneracy of the energy levels of the impurity carbon
atoms in the activated state. Thus the characteristic structural features of a nonideal
hcp crystal provide a unique opportunity for studying these diffusion states.
For the atomic hopping mechanism realized in a nonideal hcp lattice, the potential
energy profile for an interstitial atom is as shown in Fig. 6. The saddle points S1 and
S2 are located at the centers of the faces of the octahedra and tetrahedra forming the
interstitial sites [8]. One can see that atomic hops between interstitial sites of the same
kind occur with the participation of the saddle points S2, while atomic hops between
interstitial sites of different kinds occur through S1. For the indicated form of the
profile, the hopping frequencies are related as
(4)
TO / TT  OT / OO ,
where the indices 0 and Т denote the octahedral and tetrahedral interstitial sites. Here,
the following temperature dependence was assumed for the hopping frequencies:
ik  ik0 exp{ Eik /( k B T )} ,
(5)
where Eik is the energy parameter of the potential barrier, kB is Boltzmann's constant,
and Т is the absolute temperature.
Fig. 5. Temperature dependence of the anisotropy Dzz/Dxx of the diffusion coefficients
for oxygen and carbon in Ti.
Fig. 6. Potential energy profile for interstitial atoms in a nonideal hcp crystal.
The analysis of the anisotropy of the diffusion coefficients made it possible to connect
the temperature dependence of Dzz/Dxx with the difference E(S1)  E(S2) of the
potential energies of the diffusing atoms at different types of saddle points. The values
obtained for E(S1)  E(S2) were 0.047 eV [4] for oxygen atoms and - 0.208 eV for
carbon atoms [5]. This difference is depicted schematically in Fig. 6. In the case c/a =
(8/3)1/2 the potential energies E(S1) and E(S2) are the same, and the anisotropy of the
diffusion coefficients is temperature-independent.
The packing of the Ti atoms at the saddle points S1 is closer than the packing at S2.
This makes it possible to anticipate which impurity, carbon or oxygen, will have a
positive sign of the parameter E(S1)  E(S2). This result follows from the standard
elastic model of diffusion potential barriers. The difference E(S1)  E(S2) should be
greatest for carbon atoms, since carbon atoms stretch the -Ti lattice while dissolution
609
more strongly than oxygen atoms [6]. Similar results can be obtained by using
different models of the pair interactions Ti-Ti, Ti-C, and Ti-0, if it is assumed that
repulsive forces dominate at distances corresponding to the position of an interstitial
atom at a saddle point.
On this basis, the fact that the signs of the quantities d(Dzz/Dxx)/dT and E(S1)  E(S2)
are different for carbon and oxygen atoms is an unexpected result. The revision of the
viewpoints concerning the nature of the activated state during diffusion of interstitial
atoms at least for carbon atoms is required.
Let us consider this question in greater detail. In an elastic model of the potential
barrier for the diffusion of interstitial atoms it is ordinarily assumed that the energy of
an impurity in the lattice is proportional to the squared difference D - d of the
diameter D of the interstitial atom and the diameter d of the corresponding cavity (or
passage in the case of a saddle point) (see, for example, Refs. [8-9]). As a result, a
decrease in the size of the passage increases the elastic energy of the impurity. If it is
assumed that the change in the parameter d is due to the deviation of the packing of Ti from the ideal packing, then the difference of the energies E(S1) and E(S2) of the
activated states is
(6)
E(S1 )  E(S2 )   2E(S1 )d /( D  d )   2E(S1 )e( A1 )d /( D  d ) ,
where d is the change in the passage diameter d, the corresponding deformation
e(A1) is proportional to {c/a - (8/3)1/2} and is negative.
When the energy state of an interstitial impurity in the position S1 is nondegenerate,
such an effect, which is linear in {c/a - (8/3)1/2}, can be due only to the fully
symmetric e(A1) strains. In Eq. (6) the quantity 2E(S1)d/(D-d), where E(S1)  2eV
[4,5], plays the role of the impurity interaction parameter V(A1) with the indicated
strains e(A1) in the excited state (V(A1) < 0).
A completely different situation takes place in the case of energy degeneracy of
activated states of diffusing atoms. An additional linear interaction of the degenerate
term with incompletely symmetric strain e(E) appears (Fig. 7). This results in splitting
of the impurity level and decrease of the impurity energy. If the hcp lattice deviates
from an ideal lattice, such strains arise as a result of the symmetry change of the
environment of the saddle point by the titanium atoms. This effect occurs at the saddle
point S1, where the three matrix atoms closest to this position form an isosceles
triangle. The nearest-neighbor environment of the saddle point S2 is represented by an
equilateral triangle.
Since only twofold degeneracy (E-term) is possible in positions with trigonal
symmetry, the corresponding decrease of the energy can be described by the
expression
(7)
E(S1 )   (k BT ) ln 2 coshV (E)e(E) /( k BT )  V (E)e(E) ,
where V(E) is an interaction constant describing the interaction of the degenerate term
with the incompletely symmetric strain e(E) and V(E)e(E)> kBT.
As a result, the total effect for E(S1)  E(S2), which is proportional to {c/a - (8/3)1/2}
can be negative:
(8)
E(S1 )  E(S2 )  V (A1 )e(A1 )  V (E)e(E) .
We note that for the particular symmetry of the environment of the saddle point S1
considered here, the strains e(A1) and e(E) are equal:
e(A1) = 2u(A1)/a = e(E) = 2{c/a - (8/3)1/2},
610
 = 21/23-3/2 .
As a result, for condition V(A1)/V(E)< 1 the difference E(S1)  E(S2) is negative
and, correspondingly, d(Dzz/Dxx)/dT is positive.
Fig. 7. The dependence of the potential energy nearly saddle point from JT strain and
the nearest environment of the interstitial atom in a saddle point (S)of hcp lattice and
symmetric displacement of the Ti-atoms nearly saddle point:
u(A1) = (12)-1/2(2y1 + 3x2  y2  3x3 y3) ,
u(A1) = (a/2)e(A1) ,
1 / 2
 3 ( x  x 2  x3 )
;
u( E ' )   1 / 2 1
3 ( y1  y 2  y3 )
(1 / 12 )( 2 x1  x 2  3 y 2  x3  3 y 3 )
u ( E" )  
,
u(E) = (a/2)e(E) .
(
1
/
12
)(
2
y

y

3
x

y

3
x
)
1
2
2
3
3

In summary, the assumption that energy degeneracy is possible in an activated state
can eliminate the previously noted disagreement between the experimental data and
theory. The possible reasons of differences between the activated states of carbon and
oxygen atoms were discussed in [7].
Activation Energies In Model With Degenerate Activated Diffusion States
Let us consider now how the existence of degeneration in exited state is connected
with the value of activation diffusion energy. In crystals with JT effect at interstitial
ions the additional contribution to the potential energy of diffusing atom brings about
the JT interaction of two degenerate exited levels with the corresponding JT
displacements (uJT) or strains (eJT). Then the crystal potential at diffusing atoms
occurs to be multi-well one and the same character is valid for the potential energy
U(eJT) nearly saddle point.
V (E)
0 2
0 2
0
U (eJT )  1 ΩKeJT2  V ( E )eJT  1 ΩK (eJT  eJT
)  1 ΩK (eJT
) , eJT

, (9)
2
2
2
ΩK
where K – is the elastic modulus, describing the interaction of degenerate excited term
with JT strains;  - is the the corresponding atomic volume.
The value - (1/2) K (eJT0 ) 2 represents the JT stabilization energy and e JT0 - JT strain
at minimum of the adiabatic potential. Usually their peculiar order values are equal
correspondingly to (0.1 - 1)eV and (0.01 - 0.1) relative units. Thus influence of orbital
degeneration on the potential relief of the diffusing atom must be essential. Let us
underline, that multi-well form of the potential barrier nearly saddle point takes place
in a plane normal to direction of diffusion migration. In considered case of excited
degenerate state this effect reduces activation energy of diffusion on value
0 2
0
E a  (1/2) K (eJT
)   (1/2)V (E) eJT
.
The JT parameter V(E) can be estimated from the difference of energies E(S1)  E(S2),
determining the anisotropy effect of the carbon diffusion coefficients (8). As a result
for the absolute value of parameter a we have
0
Ea
eJT
.
(10)

E (S1 ) - E (S 2 )
4ζ [c/a - (8/3) 1/2 ]
If take into account the value E(S1)  E(S2) for carbon, one can expect appreciable
reduction of the diffusion activation energy due to JT effect in the case of JT strains
611
0
e JT
 [c / a  (8 / 3)1 / 2 ]  0.046 . For the explanation of the considered divergence
between the experiment and the elastic theory it is enough to have the value e JT0 of the
order of 0.1. This value is entirely reasonable.
SUMMARY AND ACKNOWLEDGMENTS
This investigation has established that there is a substantial difference between the
activated energy states of carbon and oxygen in a-Ti. It was shown that the presence of
the degenerate states in the system can influence strongly the potential barrier of the
diffusing atom and, as a consequence, on the diffusion coefficient. The Jahn-Teller
effect leads to the decreasing of the potential energy of diffusing atoms at saddle
points and explains the decreased value of this energy in saddle points with the closest
packing of the environment. Thus, the concept of a degenerate activated state can
reasonably account for all experimental data on diffusion of oxygen and carbon atoms
in -Ti. Sensitivity of diffusion properties to the orbital states of ions in crystal matrix
allows to use the precision diffusion experiments dealing with atoms of light elements
in solids in order to obtain new fundamental information about the saddle point
properties (structure electronic states).
This work was supported by the Russian Fund for Fundamental Research
under Grant 00-03-32362.
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