STATISTICAL MECHANICS OF ORDERING ALLOYS.
VIBRATIONAL CONTRIBUTIONS IN MEAN FIELD AND
QUASICHEMICAL APPROXIMATIONS
Edward Brook-Levinson, Vladimir Manov
AMTR Scientific Ltd.,
11 Hana Senesh St., Petah Tikva, 49507 Israel
A new statistical-mechanical approach to ordering phenomena in binary alloys with point
defects has been developed by the authors [1] base on the statistical method of
conditional distributions [2, 3]. This approach differs from the widely used lattice
theories (e.g., the cluster variation method so called CVM [4]) by allowing for the
contribution of lattice vibrations and point defects. It is reached by introducing the
hierarchy of partition distribution functions (DFs) defining the probability density of
finding atoms at any point within crystalline cell, e.g., the Wigner-Seitz cell. Integrating
one-particle atomic DF over the cell volume gives the probability of finding the atom in
the cell while CVM operates with the probabilities of finding atoms at lattice sites.
Therefore, the approach presented provides more information on the microscopic
structure and macroscopic properties of a real crystal as compared to the known theories.
To apply the general approach to calculating the ordering crystal properties two
approximations must be introduced. The first approximation relates to the interatomic
correlation described in [1]. It results in a system of integral equations for mean force
potentials (MFPs). The second approximation corresponds to expansion of the mean
force potential in series of the atom displacement from the lattice site (e.g., harmonic
approximation).
The structure of this paper is as follows. In the first section the mean field approximation
(MFA) is considered. The quasi chemical approximation is developed in section 2. The
third section comprises solution of integral equation for MFPs by expanding the MFPs in
series of the atomic deviations.
1. The Mean Field Approximation
1.1. Theory
Although the general approach is applicable to a binary crystal with vacancies [1], it is
known that the vacancy fraction m0 is small even at melting point (m0~10-3÷10-4).
Moreover it is small for the temperature range corresponding to the ordered phase which
is much below the melting temperature. Therefore neglecting the vacancy contribution
we obtain the following expression for the specific free energy of the ordering crystal
f    1 ln{  [
 (Q 
 1, 2 K  0 , A, B
( )
K
()
()
/ PK(  ) ) PK ] }
(QA(1) PA( 2) ) /(QA( 2) PA(1) )  (QB(1) PB( 2) ) /(QB( 2) PB(1) )
3 - 145
(1)
(2)
where Eq. (2) defines the equilibrium order parameter
 P()A- - P()A = P()B - P()B
In the absence of the vacancies the numbers of cells and atoms coincide, therefore the cell
volume and atomic volumes are identical as well as the component and atomic fractions,
i.e., w=v, mA=nA, mB=nB. Accordingly
 P   1,   1,2
K  A, B
( )
K
(3)
P()K = mK + (2)K , P()K = mK - (1)K
K= A, B
(4)
We consider now the Mean Field Approximation in which the binary distribution
functions are presented by product of one particle distribution functions
Fi (  ) j ( ) (iK(  ) , jK( ) ) = Fi (  ) (iK(  ) ) F j ( ) ( jK( ) )
(5)
By definition
()
i (  ) i (  ) ,l (  ) ( iK
) 
 dl
(  ) ( )
(  ) ( )
()
i (  )  ( iK
,lT ) *Fi (  )i l (  ) ( iK
,lT ) / Fi (  ) ( iK
)
( )
T  A ,B w (  )
l
so the equation for MFP takes the form
i (  ) i (  ) ,l ( ) (iK(  ) ) 
  dl  
( )
T  A , B w ( )
i( )
 (iK(  ) , lT( ) ) *Fl  (lT( ) )
( )
(6)
l
To solve the equation we use the fact that one-particle function has a sharp maximum at
the lattice site in the crystalline state which allows using the Laplace method [2} to
solving Eq. (6). With the accuracy to the main asymptotic term we obtain
i (  ) i (  ) ,l ( ) (iK(  ) ) 
 P 
T  A, B
T
( )
i( )
 (iK(  ) , l T ).
Integrating the equation we arrive at
 i  ,l  (iK(  ) ) =  PT [ (iK(  ) , l T( ) )  (1 / 2) (i(K ) , l T( ) )]
( )
( )
(7)
T  A, B
The integration constant in Eq. (7) is chosen so that the limit of (7) is the result of the
lattice theory in the case of the static lattice.
The total MFP is obtained in nearest neighbor (NN) approximation by summing eq. (7)
over the first coordination sphere:
3 - 146
i  (iK(  ) ) 
( )
 Z  P [ (i  , l
T  A, B
( )
K
T T
( )
T
( )
( )
)  (1 / 2) (i K , l T )]
(8),
where ZT is the NN number in the  type lattice for a cell from the  type lattice.
With the MPF known we can determine to the same approximation the normalization
constant in Eqs. (1) and (2):
( )
Q K(  )  exp{   i (  ) (i K )}( 2 /  ) 3 / 2 [ K(  ) ]1 / 2
(9)
i  (iK(  ) )  (1 / 2)
( )
 Z  P (i
T  A, B
 K(  ) 

( )
Kq
T T
 (

q x, y , z
( )
K
2
q x, y , z
( )
(10)
, lT )
( )
/ q 2 ) i (  ) (i K )
(11)
Note that the MFP derivatives with respect to different coordinates are different which is
explained by different numbers of atoms surrounding a cell of a given type. In disordered
state =0, P()K equals the component fractions. As a result the derivatives dependence on
the directions disappears.
The obtained results correspond to the harmonic approximation of the dynamic theory of
lattice. In fact, the second derivatives of the interatomic potential are included in Eq. (1)
for the free energy through the normalization constant Q()K. It is also confirmed (see
Section 3) by the expression for the atomic mean square displacement (MSD):
( ) 2
( )
 (U Kq
)  PK(  ) (  Kq
)
(12)
Finally the free energy can be written in the form
f   1


 P  [ln P 
( )
1, 2
K  A, B
( )
K
( )
K
 (1 / 2) ln[( 2 /  ) 3 /  K(  ) ] ] 
 (1 / 2)[ Z (n A AA  nB BB )  (n A nB   (1) ( 2) 2 [Z  Z ((11))  Z ((22)) ])W ]
where  KT   (i
Z Z
( )

( )
K
Z
 1, 2
,l
( )
( )
( )
T
),
W   AA   BB  2 AB
is
the
(13),
mixing
energy,
(   1,2) is the total NN number for a cell of any type.
Differentiating Eq. (12) with respect to  we arrive to the equation we arrive at the
equation for determination of the equilibrium order parameter dependence on the
temperature (the extremum condition)
3 - 147
ln[( PA(1) PB( 2) ) /( PA( 2) PB(1) )]   [Z  Z ((11))  Z ((22)) ])W 
(1 / 2) ln[  A(1) B( 2) ) /( A( 2) B(1) )  f /( (1) ( 2) )
(14),
where
f   (1 / 2)  (  )
 1, 2
 P   (1 / 
K  A, B
( )
K
q x, y , z
( )
Kq
( )
) Kq
/ .
The difference between Eqs (12)-(13) and the results of Gorsky-Bragg-William theory
consists of the additional terms depending on  K(  ) . These values present the contribution
of the lattice vibrations in harmonic approximation.
Above the order-disorder transition point the order parameter equals zero. Expanding
both sides of Eq. (13) in power series of  and keeping the linear terms at 0 we obtain
the equation for the temperature of the second kind order-disorder phase transition
~
 [ Z  Z ((11))  Z ((22)) ])W  1 /( n A n A )  f 
(15)
where
~
f   (1 / 2)  ( A /  A   B /  B )  (1 / 24)
 1, 2
 K  (1 / 3) i i  (i
( )
( )
K
)
 0
n
K  A, B
K
~
~
/  K2 [( Z (1)  K ) 2 / ( 2)  (Z ( 2)  K ) 2 / (1) ]
 (1 / 6) Z Kn ,  Kn  (n A  KA  nB  KB ) ,
~
Z (  )  Z (())   (  ) Z ,  K  [ KA   KB ] ,  is the Laplace operator.
We would like to emphasize that the obtained equations are valid for any cubic lattice
and can be readily extended to any number of coordination spheres.
1.2. Results
Alongside with the values of the interatomic potential at the lattice points (i.e., the mixing
energies for different coordination spheres) the above equations include the potential
derivatives at the same points. To allow for the vibration contributions we need detailed
information on the interatomic potentials for wide range of interatomic separation. We
select, as an example, rare-gas crystals since their interatomic potentials are presented
with the good accuracy by the Lennard-Jones 6-12 potentials for these solids:
(m)
m)
 AA
  (AA
/  AA  4(rm12  rm6 ) ,
(m)
(m)
m)
 AB
  BA
  (AB
/  AA  4 1 / 2 {[(1   ) / 2rm ]12  [(1   ) / 2rm ]6 }
(m)
m)
 BB
  (BB
/  AA  4 {[  / 2rm ]12  [  / 2rm ]6 } .
3 - 148
Here the interaction energies and interatomic separation rm are converted into
dimensionless forms using potential hole depth, AA, and the effective size of an A atom,
AA, m is the number of the coordination sphere,  =BB/AA,  =BB/AA, =-1=kBT/AA,
is the dimensionless temperature, r1=R=RAA/AA the nearest neighbor distance, kB, the
Boltzmann constant. Herewith we use the known rule for calculating AB and AB [5] i.e.,
AB =AABB)1/2, AB =AA+BB)/2.
The results of the calculations by the formulas of Sec. 1.1 are presented in Figs 1-3.
The 2D surfaces (Fig. 1) correspond to a binary alloy with fcc lattice at stoichometric
composition (nA=(1)=0.25). They characterize the effect of the potential parameters 
and  on the order-disorder phase transition temperature, c. We can conclude that c is
mainly defined by the value of  while the dependence on  is weak.
a)
b)
Fig.1. The order-disorder phase transition temperature c as a function of the interaction
potential parameters  and  for a binary alloy with the fcc lattice at stoichometric
composition ((nA=(1)=0.25); a) R=1.08; b) R=1.12
c becomes negative at ≤ 0.8 which indicates that the ordered phase does not exist in
this range of the parameter. The calculation shows the mean square displacement (MSD)
for B atom (BB≤0.8 AA, while nB=3 nA) also becomes negative for this range of ,
which corresponds to arising of imaginary frequencies in the dynamics theory of lattice
[6]. It evidences the loss of the lattice stability.
The pronounced minimum of c at  1.0 is explained by the mixing energy tending to
zero and by degeneration of the binary crystal into the pure one.
c grows monotonously with growing  at >1. The MSDs (Eq. (12)) becomes compared
with unity which implies localization of atoms within crystalline cells is impossible.
Hence the crystalline state of the system cannot exist.
3 - 149
It is important to underline that the calculation through Eq. (15) yield the finite values of
c for any and . Moreover, the situation is similar for a wide range of the
thermodynamic parameters: c is finite for those values at which the crystalline state of
the system does not exist (over the melting curve, in the decomposition range, etc.). At
the same time using the dynamic characteristics (Eq. (12)) makes it possible even within
the MFA to conclude about the alloy stability which is important advantage of the present
theory compared to the lattice approach.
We conclude that the ordered binary crystal can exist within limited range of interatomic
parameters only with prevailing effect of the effective atomic size ratio.
The growth of the NN distance, R, decreases the transition temperature and narrows the
existence range of the ordered state (Fig. 1b). It correlates with decrease in the mixing
energy and growth of MSD.
Fig. 2 shows the long range order parameter  as a function of temperature for argonkrypton alloy (with argon as A species, =1.049, =1.582) at nA=0.25 with fcc lattice
calculated by Eq. (14). The dependence of h is not one valued. The system achieves the
ordered state a sharp leap. Therefore we have here the first kind order transition allowing
existence of both ordered and disordered phases, one of them (with higher free energy)
being in the metastable state.
Fig. 2. The long-range order parameter as a function of temperature for argon-Krypton
alloy at nA=0.25 with the fcc lattice for different NN distance values, R: 1-1.12, 2-1.10, 31.08,4-1.06
a)
b
)
3 - 150
Fig. 3. The free energy as a finction of the atomic fraction and the normalized order
parameter, a) – =17, b) -  =10
Fig. 3 presents the free energy as a function of the atomic fraction and the normalized
order parameter  =/max where max is the maximum value of  for a given nA. The
disordered state is stable of =10 while the ordered one characterized by local free energy
minimum around nA =0.25, 0.75 is metastable. At =17 (Fig. 3b) the free energy has its
absolute minimum for =1 (-1), the closer nA being to (1) (stoichiometric composition)
the deeper the free energy minimum.
To conclude the section we note that the developed approach allows even in the simplest
MFA to determine the boundaries of the order-disorder phase transition existence for
binary alloy with component individual properties taken into account. This allows to
indicate the limits of applicability of the lattice theories.
2. Quasichemical approximation
2.1. Theory
The MFA although describing the long range order gives no information about the short
range order which in fact leads to ordering in a real alloy. Roughly speaking, the ordering
occurs when the alloy energy lowers with A atom surrounded preferably by B atoms and
vice versa. In this section we consider the Quasichemical Approximation (QCA) allowing
for the short range order effect. We also compare the results with those of the MFA.
The system of integral equations for MFPs in the QCA approximation takes the form:
exp{  i (  ) , j ( ) (i K(  ) )} 
 dj
x
w
( )
L
exp{  [(i
( )
K
P 
L  0 , A, B
( )
L
/QL( ) x
, j L( ) )   j ( ) ( j L( ) )   j ( ) ,i (  ) ( j L( ) )]}
(15)
j ( )
Calculating again the integrals in Eq. (15) by the Laplace method, and setting P0(  )  0
with accuracy to the main asymptotic term the gives
 i (  ) , j ( ) (i K(  ) )   PL( ) K KL /  j ( ) ,i (  ) ( j L( ) )
(16)
L  A, B
Q
( )
L
  j ( ) ( j
 L( ) 
 
( )
Lq
q x, y , z
where
 j
( )
,i (  )
( )
L
)}(2 /  ) 3 / 2 [ L( ) ]1 / 2

 (
q x, y , z
2
(17)
( )
/ q 2 ) j ( ) ( j L )
 j  ( j L( ) )  exp{   j  ( j L( ) )} ,
( )
( )
( j L( ) )  exp{   j ( ) ,i (  ) ( j L( ) )} .
3 - 151
(18)
K KL  exp{ (iK(  ) , j L( ) )} ,
Substituting i K(  )  i K(  ) in Eq. (18) yields the closed system of equations for unknown
values of  j ( ) ,i (  ) ( j L( ) )   L,  . The system can be solved only for the following value
combinations:
 A, A,  PA( ) K AA  PB( ) K AB  ,
 B, B,  PA( ) K BA /   PB( ) K BB
(19)
where
    A, / B,  K AB~ /( 2PB( ) K BB )
~  PB( )  PA( )  ( K  1) P  {( PB( )  PA( )  ( K  1) P ) 2  4KPA( ) PB( ) }1 / 2
P11  P22  0, P12  P21  , K  K AA K BB /( K AB K BA ) .
Differentiating Eq. (19) with respect to i K(  ) allowing for (18), (19), we come to
( )
 Kq

~q ~ ~
~ ~
( ) ~ q
( )
( )
{
(
P
b

0
.
5
b
C

)
/(
P

0
.
5
C
 A KA
KB K 
A
K  )  D Kq }
j ( )  i (  )
(20)
where
~
~
~
~
( )
q
q
DKq
  0.5C K~ PA( ) (a~KA
 a~KB
) 2 /( PA( )  0.5C K~ ) 2 , C A  1 / K , C B  1
a~ q  ( / q)(i (  ) , j ( ) ), a~ q  ( 2 / q 2 )(i (  ) , j ( ) ) .
KA
K
L
Remind, the value of 
KA
( )
Kq
L
( )
defines the MSD (Eq. (12)). Since DKq
>0, it always lowers
( )
, and, hence, leads
 Kq
( )
E Kq
K
the increase of <U2>. Then the value of
~q
~q ~ ~
~
( )
  DKq
/  ( PA( ) bKA
 0.5bKB
C K  ) /( PA( )  0.5C K~ ) , can be considered
j ( ) i (  )
to
j ( )  i (  )
( )
as an alloy stability criterion and referred to as destabilization factor. If EKq
 1 , then
( )
( )
( )
( )
 Kq
 0 , and  U Kq
  , and the alloy is unstable, while at EKq
 0 ,  Kq
 ,
( )
 U Kq
 0 , and the alloy stability is provided.
( )
Note, DKq
is proportional to the squared difference of the interatomic potential first
q
q 2
 a~KB
) , and from this point of view a pure crystal is absolutely stable
derivatives (a~KA
while the binary alloy can be stable when the difference is small, i.e. at close values of
the component potential parameters ( and  not far deviating from unity). This
conclusion is in accordance with both experimental data [7] and the results of our
calculations in MFA (the preceding section).
3 - 152
Write down the final expression for the binary alloy free energy, corresponding to the
above approximations,
f 
  {P  (ln P 


( )
( )
A
1, 2

Z 


m 1, 2
( )
( )m
 (1 / 2) ln[( 2 /  ) 3  A(  ) ]  PB(  ) (ln PB(  )  (1 / 2) ln[( 2 /  ) 3  B(  ) ] 
( )
A
m ~m
m
m
m
( PA(  ) (ln[ K AB
  /( 2 PB(  ) K BB
)]  (1 / 2) ln[ PB(  ) K BB
(2 PB(  ) / ~ 
 1])}
(22)
Eq. (2) for the equilibrium long range order parameter now takes the form
ln( PA(1) / PA( 2) )  (1   Z m ) ln( PB(1) / PB( 2) ) 
m
 (1 / 2) ln  
( 2)
A
where Z m 
(1)
B
/( 
(1)
A
Z 


( m ) 1, 2
( )
( )m
( 2)
B
 (Z 

m 1, 2
(1)
( )m
ln ~1m  Z ((2)) m ln ~2m ) 
)] (23)
, (=1,2) is the total neighbor number on the mth coordination
m
sphere, ~
is defined by Eq. (18) for each sphere separately.
The derived equations for the free energy and order parameter of an ordering alloy with a
cubic lattice allow for any number of coordination spheres. In distinct to the known
results of the lattice theories, Eqs. (21), (22) include additional terms depending on  K(  )
and reflecting contribution of the atomic vibrations.
m
With ~
known, we can find the vacancy equilibrium fractions:
P0(1)  Q0(1) [ PA(1) Q A( 2) /(Q A(1) PA( 2) )]
( 2)
exp{ Pw}
P0( 2)  Q0( 2) [ PA( 2) Q A(1) /(Q A( 2) PA(1) )] exp{ Pw}
(1 )
Taking into account Eqs. (16)-(18),
{P 


P0( )  [ PA( ) / PA( ) ( A( ) /  A( ) )]1 / 2 exp{ Pw}
( )
A
m 1/ 2
/( K AA
) 
( m ) 1, 2
m
 0.5~m /( K BB
K m )1 / 2 )
Z (( )) m
( )
~ m / K m ]Z(( ))m ( ) / 2
[
P


0
.
5

 A

 1, 2
Where   2;1 at   1;2 .
Te pressure P in the above approximations can be written as
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Pw  1  (  / 6)
   Z 


( )
m 1, 2
K , L  A, B
( )
( )m
m
PK( ) PL(  ) K KL
 KLm rm /( Km, Lm,  ) .
 1, 2
3.2. Results
We illustrate the obtained results using again the binary rare gas alloy with fcc lattice.
The configuration part of the free energy by Eq. (22) for krypton-argon alloy with
nA=0.25, =17, R=1,12 as a function of the long-range order parameter is presented in
Fig. 4. The ordered state characterized by the free energy minimum at   0 is observed
only with five or more coordination spheres taken into account. The NN approximation
does not describe the phenomenon.
Fig. 4. The free energy F*=[f()-f(0)] for Krypton-Argon alloy (nA=0.25, =17,
R=1,12) as a function of the long range order parameter with different number of
coordinate spheres, m, taken into account, 1 – m=1, 2 -3, 3 -5, 4 – 4, 5 -2, 6 – 10, 7 – 20.
Fig. 5 presents the order parameter temperature dependence for the same alloy at m=5.
The curves correspond qualitatively to the MFA results (Fig. 2). They show the ordering
is implemented by the first kind phase transition. Comparing the temperature values at
given  (Fig. 5b) we can conclude that QCA gives significantly lower values that the
MFA. The difference decreases with growing .
a)
b)
Fig. 5. a) the order parameter temperature dependence for the Krypton-Argon alloy at
m=5, R: 1-1.12, 2-1.10, 3-1.08, 4-1.06, 5 – 1.04, 6 – 1.02, 7-1.00; b)temperature values at
given  in the MFA (1) and QCA (2) at R=1.12.
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Te correlation analysis will be given in the next publication. Here we only note that the
interatomic correlations in the nearly ordered state tend to zero so the atoms become
statistically independent. As a consequence, the QCA results approach the MFA ones at
.
The free energy as a function of atomic fraction and normalized order parameter,, is
presented in Fig. 6. It follows the ordered state is implemented for alloys with nearly
stoichiometric composition.
Fig. 6. The free energy as a function of the atomic fraction and the normalized order
parameter at =17.
( )
The destabilization factor, E Kq
, as a function of nA and  is shown in Fig. 7 for two argon
atoms from two sublattices at NN distance. At nA (the free energy minimum on Fig. 6)
this state of the atoms is the most unstable while the destabilization factor decreases
monotonously for two NN argon atoms from the 2nd sublattice at growing .
a)
b)
Fig. 7. The destabilization factor as a function of nA and  for two Argon atoms at the NN
separation: a) =1,=2; b) -=2,=2.
The destabilization factor behavior as a function of nA and  is similar to that of the free
energy. Therefore the factor can be used as an additional criterion of stability of one or
another state of the alloy at varying its thermodynamic parameters allowing physical
interpretation of the alloy transition from one equilibrium state into another.
Te free energy minimum at   0 implies the opportunity for the system to be at
equilibrium but does not explain the way how the equilibrium is achieved. Assuming the
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alloy atomic structure changes through the component diffusion via vacancy mechanism
then it is obvious the vacancies must be taken into account.
The developed approach makes it possible to calculate in self-consistent manner both
equilibrium vacancy concentrations (Eq. (24)) and the diffusion coefficients.
Fig. 8 presents the pressure and vacancy concentrations, lnm0, as functions of the NN
distance and inverse temperature at   0 , while Fig. 9 shows its dependence on the
order parameter. The values of P0(  ) sharply drop with the temperature lowering and
pressure growth, the difference between P0(1) and P0( 2 ) is becoming more significant, as
increases, and achieving 103-104 at   1 .
Fig. 8. The pressure and vacancy concentrations, ln(m0), as functions of the NN distance,
R, and inverse temperature b, at   0 , nA=0.25, 1- =0.7, 2- =1.0.
Thus the equilibrium vacancy fraction is extremely low at   2 . This is the reason why
the ordered state of the system cannot be achieved during a finite time interval implying
it is implemented via vacancy mechanism. One of the methods to achieve ordering
avoiding the equilibrium line is vacancy quenching by rapid cooling a disordered solid
solution.
Fig. 9. The pressure and vacancy concentration dependence on the order parameter  at
nA=0.25, 1- =2, 2- =5, 3- =8, 4- =11, 5- =14, 6- =17.
Generally, the QCA as compared to MFA considerably narrows the existence range of
the Cu3Au-like ordered alloys implying both the interatomic potential parameter and
atomic fraction values.
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4. Harmonic approximation
4.1. Theory
Microscopic density inhomogeneity which is characteristic for a crystal and related to the
atomic localization around lattice sites predetermines the form of the one particle DF.
The DF should have a maximum at the lattice site while the corresponding total MFP
should have a minimum. As a consequence, we can expand all the MFPs in Eq. (15) in
series of the atomic deviations from its lattice sites, (Ui). Restricting ourselves to
quadratic terms, which corresponds to harmonic approximation (HA) we obtain

 
 i  (i K(  ) )   iK(0)   iK(1)  U i  (1 / 2) iK( 2)  U iU i
( )
( )
( )
( )

 
 i  , j  (iK(  ) )   iK(0,)j    iK(1,)j   U i  (1 / 2) iK( 2, )j   U iU i
( )
( )
( )
( )
( )
( )
( )
( )


 
 
 
0)
(1)
j (1)
jj ( 2 )
(i K(  ) , j L( ) )   (KL
  iKL
 U i   KL
 U j  (1 / 2)[ iiKL( 2)  U iU i  2 ijKL( 2)  U iU j   KL
 U jU j ]
(24)
where the expansion coefficients are the corresponding potential derivatives, with
account
 iK( 2) 
( )

j ( ) i ( )
 iK( 2, )j 
( )
( )
and  iK( ()1)  0 , as the total MFP has its minimum at the lattice site (Ui=0).
Hence, in the HA the system (15) takes the form

 
exp{   [ iK( ()0)   iK( ()1)  U i  (1 / 2) iK( ()2)  U iU i } 
 
 (1 / 2) iiKL( 2)  U iU i   Lj((0) )   Lj ((0) ),i (  ) ]}x
 dj
x
w
( )
L
P 
L  0 , A, B
( )
L

 
(1)
( 2)
exp{  [  KL
 U i  (1 / 2) KL
: U jU j ]}
j (  ) ,i (  )
j (  ) ,i (  )
j ( )
where
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
(1)
/QL( ) x exp{   [ (KL0 )   iKL
U i 
(25)

(1)
j (1)
ji ( 2 )
L (1)
 KL

U
( ) (  )   KL   KL
j   j (  ) ,i (  ) ,
j ,i
( 2)
jj ( 2 )
 KL
  KL
  Lj((2) )   Lj ((2) ),i (  ) ,
j (  ) ,i (  )
QL( ) 
 dj
( )
L
w(j )
 
exp{  [ Lj((0) )  (1 / 2) Lj((0) )  U jU j ]}
Differentiating Eq. (25) with respect to Ui and setting Ui=0, we can arrive to a closed
system of equations for determination of the MFPs and its derivatives at the lattice sites.
Omitting here the calculation details we write down the one particle DF in the HA from
which the expression for MSD follows,

 
1/ 2
Fi (K ) (U i )  PK(  ) [  /( 2 )] 3 / 2  iK( ()2) exp{(  / 2) iK( ()2 ) : U iU i }
 
  

 U iK(  ) U iK(  )   dU iK(  ) U iK(  ) U iK(  ) Fi (K ) (U iK(  ) )  (3 /  ) PK(  ) [ iK( ()2) ]1
(26)
V
The system of equations following from Eq. (25) was solved numerically by the iteration
method with Eq. (19) taken as an initial approximation.
4.2. Results.
Fig. 10 presents the free energy configuration part versus the long range order parameter
for argon-krypton alloy (=17, =15), calculated in HA. Comparing it with the results of
the main Laplacian term approximation of the preceding section shows the more detailed
allowing for vibration contributions only raises slightly the phase transition temperature,
and, from the quantitative point of view, one can confine the calculation to Eq. (22).
However, the free energy local minimum at =0 implies the instability of the disordered
state, and thus the qualitative distinction of the two calculations. Note also the phase
transition for the system under investigation was observed at rather low temperature
(=1/0.06) when the vibration effects are small.
a)
b)
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Fig. 10. F*=[f()-f(0] as a function of  in the QCA (1) and HA (2); a) =17, b) =15.
The distinction between the QCA and HA is most clearly demonstrated by Figs. 11 and
12 presenting the values of  K( )ii (i  1,2,3) (i=1,2,3) and the A and B atoms MSDs for the
two sublattices. In accordance with the lattice symmetry  K(1)11   K(1) 22   K(1)33   K(1) and
 K( 2)11   K( 2)33   K( 2) 22 , and, therefore, the MSD from the second type lattice site in the
Y-axis direction differs from these in X and Z directions.
Fig. 11. K()ii as a function of at b=17 in the QCA (1,2,3) and HA (4,5,6); 1,2,4,5-=2,
3,6- 1; 1,4 – i=1,2, 5-2, 3, 6-3.
Tending  (U A( 2) ) 2  and  (U B(1) ) 2  to zero at  growing is explained by the factor
PK(  ) in Eq. (26) and related to the fact the probability of finding and A(B) at the 2 nd (1st)
type lattice site is small at 1. For the same reason a heavier B atom MSD (nA=0.75) is
larger as compared to that of a lighter A atom (nA =0.25) at the disordered state (=0).
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Fig. 12. Statistical mean square displacement as a function of , in QCA (a) and HA (b);
1 -5-=1, 2, 3-2, 1-i=1, 2-2, 3-3
Such a behavior relates to the fact the MSD according to its definition is the value
averaged over the ensemble of all the system microscopic states with the weight factor
PK(  ) defining relative contributions of one or another state. However the MSD of a given
atom (the dynamic MSD) should not explicitly depend on the number of the same atoms
in a given sublattice. In other words, observing mentally the given atom behavior in the
given cell we deal with the realized “atom-cell”. Therefore to determine the K atom
dynamical MSD at a  type lattice site, we have to set PK(  ) =1. Then the MSDs will be
exactly proportional to the values of  K(  ) (Fig. 11).
Note the system of equations derived in the HA has no stable solutions at 1.141.10 with
any initial approximation which implies the Cu3Au type structures with the fcc lattices
can exist within only limited pressure range, the alloy composition being close to the
stoichiometric one and the interatomic potential parameters, 8, 8, to the unity.
The developed formalism allows with the initial (disordered) and final (ordered) alloy
structure known, calculating the corresponding free energy values, and thus to estimate
the possibility of existence of one or another state of a system. Factually however, the
ordering process can go through some stages, every stage being characterized by its own
ordered structure. Thus, at investigating binary ordering metal alloys by numerical
simulations, 27 ordered structures for bcc and 18 for hcp lattices have been discovered
[8].
Our attempt to apply the HA to ordering process in bcc lattice was unsuccessful since the
iteration procedure for the lattice showed no convergence. This confirms the bcc structure
of the Lennard-Jones atoms is not stable. It is in accordance with the experimental
observations the rare gas solids have the fcc lattice.
Conclusion
In the framework of the developed formalism thermodynamic properties of a binary
ordering crystal have been calculated in the different correlation approximations extended
to the vibration effects and vacancies.
In the MFA the phase transition temperature as a function of concentration, pressure and
interatomic potential parameters have been obtained. It is shown the approach allows to
determine the range of the order-disorder phase transition existence for binary alloys with
the component individual properties taken into account, thus setting the validity limits of
the lattice theories.
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In the QCA the equilibrium vacancy concentrations and long range order parameter have
been calculated. The first type order-disorder phase transition has been found for the rare
gas binary alloys with the fcc lattice with more that five coordination spheres taken into
account.
Generally, the QCA as compared to the MFA, considerably narrows the existence range
of the Cu3Au-like ordered alloys with respect to both the interatonic potential parameters
and atomic fraction values.
On the basis of analysis of vibration contributions the destabilization factor has been
introduced. The factor can be used as an additional criterion of stability of one or another
state of the alloy at varying its thermodynamic parameters. It allows physical
interpretation of the alloy transition from one equilibrium state to another.
The approach allows consequently improving the accuracy of the results. For example, at
calculating the vibration contributions we can take into account the second term of the
Laplace asymptotics, or the anharmonic terms at expanding the MFPs in series of the
atomic deviations. Although we have restricted ourselves to pair cluster approximation,
the higher-order clusters can be also included into consideration.
REFERENCES
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of the composition, structure and properties of metals, oxides, composites,
nano and amorphous materials”
2. Rott L.A. Statistical Theory of Molecular Systems, Nauka Publishers,
Moscow,
3. Brook-Levinson E.T. 1989 Statistical Theory of Real crystals (Minsk: High
School Publishers) (in Russian)
4. Kikuchi R. Chem.Phys. 60 1071 1974.
5. Hirschfelder J.O., Curtiss Ch.F., and bird R.B. Molecular Theory of Gases
and Liquid.,(N.Y., London) 1954
6. Born M., Kun H. dynamical theory of Crystal Lattice, Oxford: Clarendon
Press, 1954
7. Khachaturyan A.G. Theory of Phase Transformations and structure of
Solid Mixtures, Moscow, Nauka Publishers, 1974
8. Hauck J., Henkel D., Mika K. Z. Pys. B 71 187 1988
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