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Chemical and magnetic ordering in nonstoichiometric solid solutions
Journal of Magnetism and Magnetic Materials 81 (1989) 173-178 North-Holland, Amsterdam
CJ3EMICAL AND MAGNETIC
173
ORDERING IN NONSTOICHIOMETRIC
SOLID SOLUTIONS
Ludwik DABROWSKI Znsfitute of Nuclear Chemistry and Technology, Dorodna 16, 03-195 Warsaw, Poland Received 2 April 1989
A method of statistical description of chemical and magnetic ordering in multicomponent nonstoichiometric solid solutions in arbitrary temperature range has been worked out. The system Hamiltonian adopted for this purpose has a form of perturbation series by multiparticle and multispin interaction potentials. The character of magnetic interactions has been assumed according to the Ising model. The method takes into account multiparticle and multispin correlations as well as the mutual dependence of atomic and magnetic ordering. The free energy has been expressed overtly by the probability of occurrence of appropriate chemical and magnetic clusters in the solution.
1. Introduction Local atomic ordering in solid solutions causes local fluctuations of the electron structure parameters. This phenomenon has been known for a long time and is experimentally confirmed. In particular, the effective magnetic field of Mbssbauer nuclei changes considerably depending on the presence of diamagnetic atoms in the first and the second coordination shells of magnetic atoms [1,2]. From the theoretical point of view, the problem seems to be very complicated and, therefore, rather rarely undertaken. The only approach is to make significant simplifying assumptions. Our proposal presented in this paper is based upon the following assumptions: Magnetic interactions regard localized spins and are of the Ising type; the related Hamiltonian being adopted as a perturbation series product of the spin operators with the exchange constants dependent on the local atomic configuration. Chemical interactions, in turn, involve an arbitrary number of irreducible multiparticle interaction potentials. In both cases, from the electron structure viewpoint this constitutes an averaged field approximation. On the other hand, in atomic and magnetic structure all local fluctuations as well as their mutual correlations have been taken into account. The crystalline lattice is considered as an ideal and infinite one; each atom can occupy only one lattice site. As far as the methodology is concerned, the present work constitutes a development of the theoretical approach reviewed in our previous paper [3], current approach can be used for description of ordering in both stoichiometric and nonstoichiometric solid solutions.
2. Formulation of the problem Let us consider a canonical ensemble. Using an identical transformation Z=exp(S/k)[exp(-S/k)Z]
of the partition function
=exp(S/k)(Z),
(1)
where S is the entropy and k the Boltzmarm constant for the internal energy, we obtain: U= -kT
In(Z).
0304-8853/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
174
L. Dqbrowski
/ Ordering in nonstoichiometric
solid solutions
The expression of the above type for the internal energy of a solid solution was first introduced by Khachaturyan [4]. It is known that when approaching the thermodynamic limit the system permits, at one given temperature, existence of the configuration with the same energy only and in this case averaging in the formulae (1) and (2) has the mean$g of thermodynamic averaging. Let us introduce the operator X according to the following definition:
R= (2-
1)/@+
l),
where 2 = exp( - fi/kT)
(3)
and I? denotes the Hamiltonian
of the system. Having in mind the identity:
l/@+l)=;(l-R),
(4)
the expression (3) can be rewritten as follows:
where Xi = AZ/(2 + AZ) and AZ = 2 - (2). Basing upon the definition of AZ given in expression (5) and applying formulae (3) and (4) we obtain the dependence between Xi and X: if-
,. x1=
(2) - (&) + (ri,ri)
(6)
1-2(R)+X(X)+(X1R)-qq~
Resulting from expression (5) the internal energy (2) can be presented in the following form: U= -k,[ln(l
+ R-
3X1 + Xi_?) - ln(1 -R--X1
+ X1X)].
(7)
Since Xi can be expressed in terms of X on the basis of eq. (6), the internal energy (7) can be given as a perturbation series with X as its parameter. Using the iteration method in the first approximation we obtain: qil”‘= (X-
(Q/(1
-2(X)
+ J?(R)).
(8)
The method is in this case fully applicable since (X1,> -K (2).
3. Free energy Let us consider the Hamiltonian A=
of a multicomponent
solid solution in the following form:
-cc~~~4+~cc(~.~~~~++~“)~~~+ ij pv i B + +
aFk
,,c.,
(-.$:;;;“gq
...
. * * 3; + vy..p)e;q
. . . ek” + . . . )
(9)
where yiP= g#V’h, $‘ and e!’ are the z th component of the spin and concentration operator, respectively, while J.Y..:,’ and I$Y.:‘kh are the magnetic and chemical interaction potentials. The subscripts i, j, k are the numerals of the lattice sites while II, v, X denote types of atoms in the solution.
L Dqbrowski
/ Ordering in nonstoichiometric
solid solutions
175
The first term of the Hamiltonian (9) regards interactions with the external magnetic field h, the second two particle interactions, the remaining terms regard modification of interactions in dependence on the local environment. The following equations occur:
where Rro = exp( y,%i/kT)
- 1,
Rr,Y.‘.:‘&,
T$?.:‘,’ = exp( - V$Y.:‘;‘/kT)
p = exp( J~‘I.:‘k?!$‘S~ - - - SpA/kT) - 1,
- 1.
The @’ operators are the pseudoconcentration relationships:
operators introduced in ref. [5]; they fulfil the following
$P = &S$;-, w
(11)
where Si are the subsequent projections of the spin operator along the quantization axis. Eq. (10) can be obtained basing upon the properties of the concentration and pseudoconcentration operators, as it has been presented in detail in refs. [3,5]. Since all the terms of Hamiltonian (9) are commulative, the 2 operator defined by expression (3) can be presented as an infinite product of all subscripts of the type i, IL, w. Executing the multiplication operations Z can, in turn, be presented in the form of an infinite series of all possible products of the concentration and pseudoconcentration operators:
The expression denoting ZT. 1:&o five of them have the following form: Zfto = exp( yrSl/kT)
- 1,
=
JJ,an
Z$
=
result directly from transformation
- 1 - Z;“’ - Zr,
exp[ (J$Y’S$?~ - f$’ + y,!?$ + y,?S;)/kT] exp[ ( - J$iA - v::f” - qLA - V?)/kT]
- 1 - ZcU - Zi’,o - Zr
‘J --
m
n
- Z!‘.” - ZKc, tJ,o
- 1 - Z!$’ - Z,!” - Zkr.
The expressions presented indicate obvious algorithm of constructing ZVY;: .k,oQ...p* A
(10). The first
Zy=exp(-v:‘/kT)-1,
Zzu = exp[ ( - F$” + yj’Sl)/kT]
Z!V
p coefficients
the next coefficients:
(13)
176
L Dqbrowski
/ Ordering in nonstoichiometric
solid solutions
The sum symmetrical in respect of all type subscripts i, p, w must be created for all possible expressions and
-V’j;:::;/kT
J:;:::;2,,n..., M_c_
P
4
r
for 2 I p I n, 1 I q I n, 1 I r I m. The argument created in such a way is then exponentiated the result unity and all possible combinations
and from
ZPY::’.k,wD...p A ‘J --
s
I
for which s + t -C n + m are subtracted. An analogous structure as 2 in the expression Regarding the equations:
(12) has also gfl raised to any arbitrary
power.
where T,(T,r;
“’
lJ
A=
RiJ!!Y;;;i
[ exp( - vyY:.k’/kT)] p=
n - 1,
[exp( ~~?.~.~~“S$!$” *. . S;)/kT]
n - 1,
we obtain: p
=
c cc^“cz;“‘““Q”
1+
ip x
[
z!r,:$“’ 'J
A+
+ ... +
-$ .J ’ q...kpv...X
w
c _qpy, w
1
A(?;, + . . . + . . . .
(15)
This indicates that the expression (15) can be obtained from the formula (12) if in (12) all exponential terms are raised to the power n. Let us decompose the 2 operator, defined by expression (3) into a power series in respect to 2”. The result is as follows: R=1+2
E
(-l)“.?.
n=l
Taking into account relationship (15) for the J? operator we obtain the following expression:
The overt form of the coefficients x~y:.’ A ‘J
--
..k,‘d..‘~,
P
4
(16)
L. Dqbrowski / Orderingin nonstoichiometric solid solutions
appearing in expression (17) is relatively easy to obtain. summing-up the series comprising the coefficients Z’ljl.‘,,’ ‘I -_c
177
It should be noted that they result from
A
,d2...p
P
4
having the same p and q, and different n. As a result of this operation we obtain: X[o = th( y/‘S:/2kT), Xzw = th[ (-
X&” = th( - V$f’/2kT),
I$’ + y/‘S;)/2kT]
- Xtw - X$‘.
(18)
Generally speaking, any arbitrary coefficient of the form qJY::&c ___p can be obtained from a corresponding coefficient 2;“. 1:‘,$,o p if a formal change Z+Xand exp(...)-l+th[(...)/2] is executed. The obtained expressions for the above coefficients adopt limited values within the whole temperature range. In our final account, we have thus obtained development of the internal energy into a series in respect of <@ejv * - * 2;) and (e/‘“?jvo * * * ?ip), i.e. in respect of corresponding probabilities of occurrence of respective chemical and magnetic clusters pr.1.:: iA and P.Y .‘.‘,A,D. . p. The above manner of presentation of the series (17) is equivalent yo the commonly applied diagram technique, being, however much simpler and more comprehensive. Including the multiparticle and multispin potentials into the presented considerations changes, of course, the system entropy. This is caused by consequent change of the values of multiparticle and multispin correlation functions. Nevertheless, the general expressions determining the total configuration entropy given in ref. [3] remain valid and have the following form: s=
C(s;+s;), R
09)
where
1 ,
P/J:::&, p and G~::&n chemical and magnetic clusters, respectively. while pr. 1:iA, gr.I:iA,
...p are the probabilities
and correlation
functions of
4. Discussion In the present paper, a simple algorithm of construction of internal energy of a solid solution with omission of a widely applied diagram technique has been proposed. As it has been shown in ref. [6],. inclusion of multiparticle interactions into account while using this technique causes the necessity of graphically distinquishing the terms connected with reducible and irreducible interaction potentials. In our case, extension of the Hamiltonian on the multispin interactions complicates the diagram technique to the degree when its application becomes impractical. Besides, employing transformation (3) results in
178
L. Dqbrowski
/ Ordering in nonstoichiometric
solid solutions
obtaining the internal energy in the form of a series whose development coefficients always adopt finite values and do not require renormalization. Thus, in comparison with ref. [3], we achieve the following advantages. First, while calculating the internal energy the number of terms taken into account is not restricted a priori and can be successively increased during computations. Secondly, it is not necessary to solve an additional complicated system of equations determinin g the renormalization coefficients. Third, and probably the most important advantage is the fact that the knowledge of an asymptotic solution for ultralow temperatures is no longer required. Therefore, the proposed method seems to be much more versatile and, at the same time, simpler than the previous one, presented in ref. [3]. In the case of strongly nonstoichiometric solutions when one of the components predominates, convergence of the series (17) is superior to that for stoichiometric solutions.
Acknowledgements The author is thankful to Professor J. Leciejewicz for his valuable comments and suggestions. This work was sponsored by the Central Research Programme number 01.09.
References [l] [2] [3] [4] [5] [6]
G.K. Wertheim, V. Jaccarin o, J.H. Wemick and D.N.E. Buchman, Phys. Rev. Lett. 12 (1964) 25. T. Moriya, H. Ino and F.E. Fujita, J. Phys. See. Japan 24 (1968) 60. L. Dgbrowski, J. Magn. Magn. Mat. 73 (1988) 184. A.G. Khachaturyan, Progr. Mater. Sci. 22 (1978) 1. L. Dgbrowski, J. Magn. Magn. Mat. 62 (1986) 283. L. Dgbrowski, Phys. Stat. Sol. (b) 135 (1986) 61.
CJ3EMICAL AND MAGNETIC
173
ORDERING IN NONSTOICHIOMETRIC
SOLID SOLUTIONS
Ludwik DABROWSKI Znsfitute of Nuclear Chemistry and Technology, Dorodna 16, 03-195 Warsaw, Poland Received 2 April 1989
A method of statistical description of chemical and magnetic ordering in multicomponent nonstoichiometric solid solutions in arbitrary temperature range has been worked out. The system Hamiltonian adopted for this purpose has a form of perturbation series by multiparticle and multispin interaction potentials. The character of magnetic interactions has been assumed according to the Ising model. The method takes into account multiparticle and multispin correlations as well as the mutual dependence of atomic and magnetic ordering. The free energy has been expressed overtly by the probability of occurrence of appropriate chemical and magnetic clusters in the solution.
1. Introduction Local atomic ordering in solid solutions causes local fluctuations of the electron structure parameters. This phenomenon has been known for a long time and is experimentally confirmed. In particular, the effective magnetic field of Mbssbauer nuclei changes considerably depending on the presence of diamagnetic atoms in the first and the second coordination shells of magnetic atoms [1,2]. From the theoretical point of view, the problem seems to be very complicated and, therefore, rather rarely undertaken. The only approach is to make significant simplifying assumptions. Our proposal presented in this paper is based upon the following assumptions: Magnetic interactions regard localized spins and are of the Ising type; the related Hamiltonian being adopted as a perturbation series product of the spin operators with the exchange constants dependent on the local atomic configuration. Chemical interactions, in turn, involve an arbitrary number of irreducible multiparticle interaction potentials. In both cases, from the electron structure viewpoint this constitutes an averaged field approximation. On the other hand, in atomic and magnetic structure all local fluctuations as well as their mutual correlations have been taken into account. The crystalline lattice is considered as an ideal and infinite one; each atom can occupy only one lattice site. As far as the methodology is concerned, the present work constitutes a development of the theoretical approach reviewed in our previous paper [3], current approach can be used for description of ordering in both stoichiometric and nonstoichiometric solid solutions.
2. Formulation of the problem Let us consider a canonical ensemble. Using an identical transformation Z=exp(S/k)[exp(-S/k)Z]
of the partition function
=exp(S/k)(Z),
(1)
where S is the entropy and k the Boltzmarm constant for the internal energy, we obtain: U= -kT
In(Z).
0304-8853/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
174
L. Dqbrowski
/ Ordering in nonstoichiometric
solid solutions
The expression of the above type for the internal energy of a solid solution was first introduced by Khachaturyan [4]. It is known that when approaching the thermodynamic limit the system permits, at one given temperature, existence of the configuration with the same energy only and in this case averaging in the formulae (1) and (2) has the mean$g of thermodynamic averaging. Let us introduce the operator X according to the following definition:
R= (2-
1)/@+
l),
where 2 = exp( - fi/kT)
(3)
and I? denotes the Hamiltonian
of the system. Having in mind the identity:
l/@+l)=;(l-R),
(4)
the expression (3) can be rewritten as follows:
where Xi = AZ/(2 + AZ) and AZ = 2 - (2). Basing upon the definition of AZ given in expression (5) and applying formulae (3) and (4) we obtain the dependence between Xi and X: if-
,. x1=
(2) - (&) + (ri,ri)
(6)
1-2(R)+X(X)+(X1R)-qq~
Resulting from expression (5) the internal energy (2) can be presented in the following form: U= -k,[ln(l
+ R-
3X1 + Xi_?) - ln(1 -R--X1
+ X1X)].
(7)
Since Xi can be expressed in terms of X on the basis of eq. (6), the internal energy (7) can be given as a perturbation series with X as its parameter. Using the iteration method in the first approximation we obtain: qil”‘= (X-
(Q/(1
-2(X)
+ J?(R)).
(8)
The method is in this case fully applicable since (X1,> -K (2).
3. Free energy Let us consider the Hamiltonian A=
of a multicomponent
solid solution in the following form:
-cc~~~4+~cc(~.~~~~++~“)~~~+ ij pv i B + +
aFk
,,c.,
(-.$:;;;“gq
...
. * * 3; + vy..p)e;q
. . . ek” + . . . )
(9)
where yiP= g#V’h, $‘ and e!’ are the z th component of the spin and concentration operator, respectively, while J.Y..:,’ and I$Y.:‘kh are the magnetic and chemical interaction potentials. The subscripts i, j, k are the numerals of the lattice sites while II, v, X denote types of atoms in the solution.
L Dqbrowski
/ Ordering in nonstoichiometric
solid solutions
175
The first term of the Hamiltonian (9) regards interactions with the external magnetic field h, the second two particle interactions, the remaining terms regard modification of interactions in dependence on the local environment. The following equations occur:
where Rro = exp( y,%i/kT)
- 1,
Rr,Y.‘.:‘&,
T$?.:‘,’ = exp( - V$Y.:‘;‘/kT)
p = exp( J~‘I.:‘k?!$‘S~ - - - SpA/kT) - 1,
- 1.
The @’ operators are the pseudoconcentration relationships:
operators introduced in ref. [5]; they fulfil the following
$P = &S$;-, w
(11)
where Si are the subsequent projections of the spin operator along the quantization axis. Eq. (10) can be obtained basing upon the properties of the concentration and pseudoconcentration operators, as it has been presented in detail in refs. [3,5]. Since all the terms of Hamiltonian (9) are commulative, the 2 operator defined by expression (3) can be presented as an infinite product of all subscripts of the type i, IL, w. Executing the multiplication operations Z can, in turn, be presented in the form of an infinite series of all possible products of the concentration and pseudoconcentration operators:
The expression denoting ZT. 1:&o five of them have the following form: Zfto = exp( yrSl/kT)
- 1,
=
JJ,an
Z$
=
result directly from transformation
- 1 - Z;“’ - Zr,
exp[ (J$Y’S$?~ - f$’ + y,!?$ + y,?S;)/kT] exp[ ( - J$iA - v::f” - qLA - V?)/kT]
- 1 - ZcU - Zi’,o - Zr
‘J --
m
n
- Z!‘.” - ZKc, tJ,o
- 1 - Z!$’ - Z,!” - Zkr.
The expressions presented indicate obvious algorithm of constructing ZVY;: .k,oQ...p* A
(10). The first
Zy=exp(-v:‘/kT)-1,
Zzu = exp[ ( - F$” + yj’Sl)/kT]
Z!V
p coefficients
the next coefficients:
(13)
176
L Dqbrowski
/ Ordering in nonstoichiometric
solid solutions
The sum symmetrical in respect of all type subscripts i, p, w must be created for all possible expressions and
-V’j;:::;/kT
J:;:::;2,,n..., M_c_
P
4
r
for 2 I p I n, 1 I q I n, 1 I r I m. The argument created in such a way is then exponentiated the result unity and all possible combinations
and from
ZPY::’.k,wD...p A ‘J --
s
I
for which s + t -C n + m are subtracted. An analogous structure as 2 in the expression Regarding the equations:
(12) has also gfl raised to any arbitrary
power.
where T,(T,r;
“’
lJ
A=
RiJ!!Y;;;i
[ exp( - vyY:.k’/kT)] p=
n - 1,
[exp( ~~?.~.~~“S$!$” *. . S;)/kT]
n - 1,
we obtain: p
=
c cc^“cz;“‘““Q”
1+
ip x
[
z!r,:$“’ 'J
A+
+ ... +
-$ .J ’ q...kpv...X
w
c _qpy, w
1
A(?;, + . . . + . . . .
(15)
This indicates that the expression (15) can be obtained from the formula (12) if in (12) all exponential terms are raised to the power n. Let us decompose the 2 operator, defined by expression (3) into a power series in respect to 2”. The result is as follows: R=1+2
E
(-l)“.?.
n=l
Taking into account relationship (15) for the J? operator we obtain the following expression:
The overt form of the coefficients x~y:.’ A ‘J
--
..k,‘d..‘~,
P
4
(16)
L. Dqbrowski / Orderingin nonstoichiometric solid solutions
appearing in expression (17) is relatively easy to obtain. summing-up the series comprising the coefficients Z’ljl.‘,,’ ‘I -_c
177
It should be noted that they result from
A
,d2...p
P
4
having the same p and q, and different n. As a result of this operation we obtain: X[o = th( y/‘S:/2kT), Xzw = th[ (-
X&” = th( - V$f’/2kT),
I$’ + y/‘S;)/2kT]
- Xtw - X$‘.
(18)
Generally speaking, any arbitrary coefficient of the form qJY::&c ___p can be obtained from a corresponding coefficient 2;“. 1:‘,$,o p if a formal change Z+Xand exp(...)-l+th[(...)/2] is executed. The obtained expressions for the above coefficients adopt limited values within the whole temperature range. In our final account, we have thus obtained development of the internal energy into a series in respect of <@ejv * - * 2;) and (e/‘“?jvo * * * ?ip), i.e. in respect of corresponding probabilities of occurrence of respective chemical and magnetic clusters pr.1.:: iA and P.Y .‘.‘,A,D. . p. The above manner of presentation of the series (17) is equivalent yo the commonly applied diagram technique, being, however much simpler and more comprehensive. Including the multiparticle and multispin potentials into the presented considerations changes, of course, the system entropy. This is caused by consequent change of the values of multiparticle and multispin correlation functions. Nevertheless, the general expressions determining the total configuration entropy given in ref. [3] remain valid and have the following form: s=
C(s;+s;), R
09)
where
1 ,
P/J:::&, p and G~::&n chemical and magnetic clusters, respectively. while pr. 1:iA, gr.I:iA,
...p are the probabilities
and correlation
functions of
4. Discussion In the present paper, a simple algorithm of construction of internal energy of a solid solution with omission of a widely applied diagram technique has been proposed. As it has been shown in ref. [6],. inclusion of multiparticle interactions into account while using this technique causes the necessity of graphically distinquishing the terms connected with reducible and irreducible interaction potentials. In our case, extension of the Hamiltonian on the multispin interactions complicates the diagram technique to the degree when its application becomes impractical. Besides, employing transformation (3) results in
178
L. Dqbrowski
/ Ordering in nonstoichiometric
solid solutions
obtaining the internal energy in the form of a series whose development coefficients always adopt finite values and do not require renormalization. Thus, in comparison with ref. [3], we achieve the following advantages. First, while calculating the internal energy the number of terms taken into account is not restricted a priori and can be successively increased during computations. Secondly, it is not necessary to solve an additional complicated system of equations determinin g the renormalization coefficients. Third, and probably the most important advantage is the fact that the knowledge of an asymptotic solution for ultralow temperatures is no longer required. Therefore, the proposed method seems to be much more versatile and, at the same time, simpler than the previous one, presented in ref. [3]. In the case of strongly nonstoichiometric solutions when one of the components predominates, convergence of the series (17) is superior to that for stoichiometric solutions.
Acknowledgements The author is thankful to Professor J. Leciejewicz for his valuable comments and suggestions. This work was sponsored by the Central Research Programme number 01.09.
References [l] [2] [3] [4] [5] [6]
G.K. Wertheim, V. Jaccarin o, J.H. Wemick and D.N.E. Buchman, Phys. Rev. Lett. 12 (1964) 25. T. Moriya, H. Ino and F.E. Fujita, J. Phys. See. Japan 24 (1968) 60. L. Dgbrowski, J. Magn. Magn. Mat. 73 (1988) 184. A.G. Khachaturyan, Progr. Mater. Sci. 22 (1978) 1. L. Dgbrowski, J. Magn. Magn. Mat. 62 (1986) 283. L. Dgbrowski, Phys. Stat. Sol. (b) 135 (1986) 61.