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Magnetic ordering in the Ising model
Journal of Magnetism and Magnetic Materials 53 (1986) 313-317 North-Holland, Amsterdam
313
MAGNETIC ORDERING IN THE ISING M O D E L L. D 4 B R O W S K I Institute of Nuclear Research and Nuclear Energy, 72, Boulevard Lenin, 1184 Sofia, Bulgaria Received 18 July 1985
The general theory of atomic ordering in binary alloys is adapted to the case of magnetic interaction with localized spins of i The problem is solved using the idea of short- and long-range ordering entropy. The method of high-temperature expansion ~. is employed to calculate the free energy making use of the diagram technique.
1. Introduction
A variety of methods have been developed for solving the problem of the magnetic interactions [1,2]; work on this problem is still in progress. Some of these methods, the Ising model in particular, have drawn on from the theory of atomic ordering in alloys [3]. The ideas of short-range ordering in alloys [4-6] have also been based on the familiar Bethe-Peierls-Weiss method [7]. This paper presents an analogous approach to solving the problem which is based on the modern results of the theory of binary alloys [8]. The method outlined herein makes it possible to avoid certain restrictions of earlier methods. Such is in particular the limitation of the magnetic interaction radius. The method is also insensitive to the crystallographic structure of the magnetic material. 2. Formulation of the problem The Ising model Hamiltonian in an external magnetic field h is of the form: *=
-
+
+
(1)
i,j
where Y = gBh c/~j, c = 1/N.
N is the number of lattice sites, while the remaining notations are common. In the case under consideration the summation indices i, j cover all crystal lattice sites. Following certain manipulations [9] the free energy is obtained as F = <.Ig'> - K T S +
aF,
(2)
where:(./g') is the average of (1), S is the entropy of the system, K, T are the Boltzmann constant and the absolute temperature, aF=
-KT
ln{exp(-A.,~/KT));
a . ~ = . , * ' - G,'e').
The average of (1) can be represented as <*>
=
+
+
/j
0304-8853/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
L, Dqbrowski / Magnetic ordering in the Ising model
314
= - ½E~,
[s:,s:, +
~e,, +
~(s., +
(3)
s./) ] ,
{/
where S:,, S:j are the average values of S_., and S.~. ~ij = is the static pair-correlation function, AS_i, AS__~ are the local spin functions at the respective points. In case of a binary alloy the entropy S is composed of the long-range entropy S e expressed as [9]: S,.= - £ [ c , In c, +(1 - c,) ln(1 - c,)]
(4)
i
and the short-range entropy S, h [8]:
+ [ c i ( 1, - c , ] -
,
+[(1- c,)(1-c/)+
-- ~ In 1
fgii] ln(1 + k
(1 - c,)(1 - c,)
,
(s) where c, is the concentration of component A and ~¢o = ( A c , A c / ) is the static pair-correlation function. The binary alloy problem reduces to the Ising model through introducing a concentration operator: ~; = S:, + ½,
(6)
the average values of which are ci. In our case the expression
(7)
corresponds to the concentration c,. Having in mind (6) we get
.%=
(AS.,AS.,) =,
(8)
where f¢~j is the spin correlation function. The expressions obtained (6) and (7) make it possible to find the entropies St. and S.h in the case under consideration, using eqs (4) and (5). We finally get Sc = - E [ ( ½ -
S--i)ln(½- S.i) + (½ + S_i)In(½+ S_-i)]
(9)
i
and
s,,,=-½2{[(½+~i s~,)(-~+ s.,)+ v,] In(1+ (~+ s:,)(~+ s.,) ~¢,j ) +[(~-s:,)(~+<)-~e,,lln 1 (½-s_,)(-~+s.,) + [ ( ½ + S_.,)(½-S./)-f¢,i]
ln(1
+ [ ( ½ - S_.,)(½- S_j) + (g,/] ln(1 +
(~+ s,)(-~- s,)
-))
(~- s,)(~- s,)
(10)
L. Dqbrowski / Magnetic ordering in the lsing model
315
3. Calculation of the fluctuation potential by means of traditional diagram techniques The high-temperature expansion method will be used for calculating A F: AF
~_. M,, ( A . ~ )
KT-
.=1 (-KT)"n''.
(11)
where M,, is the nth cumulant,
+ Y(& + &-
s,)]
= It is convenient to represent the cumulants by means of the diagram technique. According to [10,11] these are constructed in the following way: Each diagram is composed of vertices and bonds. The vertices symbolize independent summation subscripts, connected with the bonds. Single bonds symbolize the expressions B,j, while n-fold bonds symbolize the nth powers of these same expressions. The preceding factors are the same as in [10,11], namely they are equal to n!/g, where n is the order (the number of bonds) of a diagram, while g is the number of identical transformations of that diagram. On increasing the order of the cumulants, the number of diagrams grows rapidly. If the higher-order correlation functions are neglected and account is taken only of the expressions linear with respect to the pair-correlation function, the number of diagrams is reduced considerably. For example, all disconnected diagrams, such as the second-order ones ~ , make a vanishing contribution to the free energy. Diagrams with the same number of vertices and bonds, such as -',-, have to be discarded too, since their contribution is always vanishing. This is due to the fact that summation in this type of diagrams is limited to a volume of the order of the interaction radius, while the summation for the other types spreads over the entire crystal. The total number of diagrams can be reduced considerably through applying the method described in the following section.
4. Simplified diagram technique Simplifying the diagram technique is achieved through summing up same-type diagrams of different order up to and including infinite order. Same-type diagrams are those having the same number of vertices and which are connected in the same manner. They differ from each other only in the number of bonds connecting the vertices. It is convenient at an intermediate stage to pass based on eq. (6) from the spin operator ~ , to the pseudo-concentration operator ~i. Now, Ad~' assumes the form
/J -
'
(12)
O"
where
a=
+ s., + s , ) .
L. Dqbrowski / Magnetic ordering in the lsing model
316
The b~ operators obey (~),, = { a~, 1,
nU:O. n=0
(13)
Having in mind (13) we get after certain manipulations
(<,)"
= ~,~,(2~ + ~a + - ~ -
<&~,>)"
+(a, + e,- 2a,~/)(v- ½- (g.,~,))" +(1 - e , - e , +
(14)
e,e,)(fa+ I - ( < , L , ) ) "
Turning back from the operator ~, to the operator S:, we get (U,/)" = [ S:,S--.i + ½( S:, + S:/) + ¼](2y + ¼- ( S : , 4 i ) ) "
+
+ a-¼-
•
^
^
ii
<4,&>)" The contribution of all two-vertex diagrams to the free energy can be expressed by a general formula
1
1
I ! ( _ K T ) ~,
~
_+ 2!(- KT) 2 •
1 O+ 3!(-KT)
3
q~--------O + - . .
(/,,)" <<;>
=
~-" ]~-~tKT n=l #
(16)
n! =½Y'(Q"> t)
where Qij = [~,S:, + ½(S_., +
S:,)+~]{exp[(a~,,/KT)(2y+I2+¼-(S:i~,))]-I}
+ ( 3 - 2S:,S:.,){exp[(J~JKT)(fa + ¼-(S:,Z/))]
- 1}
,^ + [4A:,- ~(S_-, + ~,)+¼]{exp[(/,,/Kr)(ga+¼-<~:,~,>)]-l}.
It can be shown in the same way that the contribution of the three-vertex diagrams is equal to: 1
2!(-
KT ) 2
~
+
1
3!( -
O---"---O
KT ) 2 ~
~. + . . . .
½ y~ (Q,/Q/h >,
(17)
iak
where Q~j and Q/k are the same expressions as those entering (16). This result can be generalized for an arbitrary number of vertices and represented as a new type of diagrams. The graphical picture of the new diagrams is identical with that of the lowest-order diagram in the given sum. In other words, the new diagrams contain single bonds only. The vertices again symbolize the independent summation indices, while the bonds symbolize the expressions Qi/which change as a result of the summing. Inasmuch as the n! factor [cf. eqs (16) and (17)] is already incorporated into the new diagrams, the new factor is n! times smaller than the old one; namely, it is equal to 1/g.
L. Dqbrowski / Magnetic ordering in the 1sing model
317
5. Conclusion T h e m e t h o d p r e s e n t e d here is c o n v e n i e n t for o b t a i n i n g n u m e r i c a l results b y m e a n s of c o m p u t i n g techniques. F r o m the m i n i m u m - f r e e - e n e r g y c o n d i t i o n we o b t a i n a system of e q u a t i o n s whose solution yields values for t h e r m o d y n a m i c a l l y a v e r a g e d spin at a given p o i n t , as well as v a l u e s for the correlation functions. It can easily be verified that setting A F = 0 a n d fCij = 0 we get the t e m p e r a t u r e d e p e n d e n c e of the average spin in the form of the familiar Brillouin f u n c t i o n B1/2. Expressing the free energy as an explicit function of the p a i r c o r r e l a t i o n s constitutes an a d d i t i o n a l interest f r o m the v i e w p o i n t of investigating the p a i r - c o r r e l a t i o n functions on b o t h sides of the critical temperature.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
S.V. Vonsovskii, Magnetism (Nauka, Moscow, 1971). C. Domb and M.S. Green, Phase Transitions and Critical Phenomena (Academic Press, London and New York, 1974). L.S. Stilbans, Zh. Eksp. Teor. Fiz. 9 (1939) 432. [in Russian] P.R. Weiss, Phys. Rev. 74 (1948) 1493. H.H. Bethe and H.H. Wills, Prec. R. Soc. A150 (1935) 552. H.H. Bethe, Prec. Roy. Soc. A154 (1935) 207. R. Peierls, Prec. Camb. Phil. Soc. 32 (1936) 477. L. Da,browski, Phys. Stat. Sol. (b)128 (1985) 371. A.G. Khachaturyan, Prog. Mat. Sci. 22 (1978) 1. G. Horvitz and H. Callen, Phys. Rev. 124 (1961) 1757. D.A. Badalian and A.G. Khachaturyan, Fiz. Tverd. Tela 12 (1970) 439.
313
MAGNETIC ORDERING IN THE ISING M O D E L L. D 4 B R O W S K I Institute of Nuclear Research and Nuclear Energy, 72, Boulevard Lenin, 1184 Sofia, Bulgaria Received 18 July 1985
The general theory of atomic ordering in binary alloys is adapted to the case of magnetic interaction with localized spins of i The problem is solved using the idea of short- and long-range ordering entropy. The method of high-temperature expansion ~. is employed to calculate the free energy making use of the diagram technique.
1. Introduction
A variety of methods have been developed for solving the problem of the magnetic interactions [1,2]; work on this problem is still in progress. Some of these methods, the Ising model in particular, have drawn on from the theory of atomic ordering in alloys [3]. The ideas of short-range ordering in alloys [4-6] have also been based on the familiar Bethe-Peierls-Weiss method [7]. This paper presents an analogous approach to solving the problem which is based on the modern results of the theory of binary alloys [8]. The method outlined herein makes it possible to avoid certain restrictions of earlier methods. Such is in particular the limitation of the magnetic interaction radius. The method is also insensitive to the crystallographic structure of the magnetic material. 2. Formulation of the problem The Ising model Hamiltonian in an external magnetic field h is of the form: *=
-
+
+
(1)
i,j
where Y = gBh c/~j, c = 1/N.
N is the number of lattice sites, while the remaining notations are common. In the case under consideration the summation indices i, j cover all crystal lattice sites. Following certain manipulations [9] the free energy is obtained as F = <.Ig'> - K T S +
aF,
(2)
where:(./g') is the average of (1), S is the entropy of the system, K, T are the Boltzmann constant and the absolute temperature, aF=
-KT
ln{exp(-A.,~/KT));
a . ~ = . , * ' - G,'e').
The average of (1) can be represented as <*>
=
+
+
/j
0304-8853/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
L, Dqbrowski / Magnetic ordering in the Ising model
314
= - ½E~,
[s:,s:, +
~e,, +
~(s., +
(3)
s./) ] ,
{/
where S:,, S:j are the average values of S_., and S.~. ~ij =
(4)
i
and the short-range entropy S, h [8]:
+ [ c i ( 1, - c , ] -
,
+[(1- c,)(1-c/)+
-- ~ In 1
fgii] ln(1 + k
(1 - c,)(1 - c,)
,
(s) where c, is the concentration of component A and ~¢o = ( A c , A c / ) is the static pair-correlation function. The binary alloy problem reduces to the Ising model through introducing a concentration operator: ~; = S:, + ½,
(6)
the average values of which are ci. In our case the expression
(7)
corresponds to the concentration c,. Having in mind (6) we get
.%=
(AS.,AS.,) =
(8)
where f¢~j is the spin correlation function. The expressions obtained (6) and (7) make it possible to find the entropies St. and S.h in the case under consideration, using eqs (4) and (5). We finally get Sc = - E [ ( ½ -
S--i)ln(½- S.i) + (½ + S_i)In(½+ S_-i)]
(9)
i
and
s,,,=-½2{[(½+~i s~,)(-~+ s.,)+ v,] In(1+ (~+ s:,)(~+ s.,) ~¢,j ) +[(~-s:,)(~+<)-~e,,lln 1 (½-s_,)(-~+s.,) + [ ( ½ + S_.,)(½-S./)-f¢,i]
ln(1
+ [ ( ½ - S_.,)(½- S_j) + (g,/] ln(1 +
(~+ s,)(-~- s,)
-))
(~- s,)(~- s,)
(10)
L. Dqbrowski / Magnetic ordering in the lsing model
315
3. Calculation of the fluctuation potential by means of traditional diagram techniques The high-temperature expansion method will be used for calculating A F: AF
~_. M,, ( A . ~ )
KT-
.=1 (-KT)"n''.
(11)
where M,, is the nth cumulant,
+ Y(& + &-
s,)]
= It is convenient to represent the cumulants by means of the diagram technique. According to [10,11] these are constructed in the following way: Each diagram is composed of vertices and bonds. The vertices symbolize independent summation subscripts, connected with the bonds. Single bonds symbolize the expressions B,j, while n-fold bonds symbolize the nth powers of these same expressions. The preceding factors are the same as in [10,11], namely they are equal to n!/g, where n is the order (the number of bonds) of a diagram, while g is the number of identical transformations of that diagram. On increasing the order of the cumulants, the number of diagrams grows rapidly. If the higher-order correlation functions are neglected and account is taken only of the expressions linear with respect to the pair-correlation function, the number of diagrams is reduced considerably. For example, all disconnected diagrams, such as the second-order ones ~ , make a vanishing contribution to the free energy. Diagrams with the same number of vertices and bonds, such as -',-, have to be discarded too, since their contribution is always vanishing. This is due to the fact that summation in this type of diagrams is limited to a volume of the order of the interaction radius, while the summation for the other types spreads over the entire crystal. The total number of diagrams can be reduced considerably through applying the method described in the following section.
4. Simplified diagram technique Simplifying the diagram technique is achieved through summing up same-type diagrams of different order up to and including infinite order. Same-type diagrams are those having the same number of vertices and which are connected in the same manner. They differ from each other only in the number of bonds connecting the vertices. It is convenient at an intermediate stage to pass based on eq. (6) from the spin operator ~ , to the pseudo-concentration operator ~i. Now, Ad~' assumes the form
/J -
'
(12)
O"
where
a=
+ s., + s , ) .
L. Dqbrowski / Magnetic ordering in the lsing model
316
The b~ operators obey (~),, = { a~, 1,
nU:O. n=0
(13)
Having in mind (13) we get after certain manipulations
(<,)"
= ~,~,(2~ + ~a + - ~ -
<&~,>)"
+(a, + e,- 2a,~/)(v- ½- (g.,~,))" +(1 - e , - e , +
(14)
e,e,)(fa+ I - ( < , L , ) ) "
Turning back from the operator ~, to the operator S:, we get (U,/)" = [ S:,S--.i + ½( S:, + S:/) + ¼](2y + ¼- ( S : , 4 i ) ) "
+
+ a-¼-
•
^
^
ii
<4,&>)" The contribution of all two-vertex diagrams to the free energy can be expressed by a general formula
1
1
I ! ( _ K T ) ~,
~
_+ 2!(- KT) 2 •
1 O+ 3!(-KT)
3
q~--------O + - . .
(/,,)" <<;>
=
~-" ]~-~tKT n=l #
(16)
n! =½Y'(Q"> t)
where Qij = [~,S:, + ½(S_., +
S:,)+~]{exp[(a~,,/KT)(2y+I2+¼-(S:i~,))]-I}
+ ( 3 - 2S:,S:.,){exp[(J~JKT)(fa + ¼-(S:,Z/))]
- 1}
,^ + [4A:,- ~(S_-, + ~,)+¼]{exp[(/,,/Kr)(ga+¼-<~:,~,>)]-l}.
It can be shown in the same way that the contribution of the three-vertex diagrams is equal to: 1
2!(-
KT ) 2
~
+
1
3!( -
O---"---O
KT ) 2 ~
~. + . . . .
½ y~ (Q,/Q/h >,
(17)
iak
where Q~j and Q/k are the same expressions as those entering (16). This result can be generalized for an arbitrary number of vertices and represented as a new type of diagrams. The graphical picture of the new diagrams is identical with that of the lowest-order diagram in the given sum. In other words, the new diagrams contain single bonds only. The vertices again symbolize the independent summation indices, while the bonds symbolize the expressions Qi/which change as a result of the summing. Inasmuch as the n! factor [cf. eqs (16) and (17)] is already incorporated into the new diagrams, the new factor is n! times smaller than the old one; namely, it is equal to 1/g.
L. Dqbrowski / Magnetic ordering in the 1sing model
317
5. Conclusion T h e m e t h o d p r e s e n t e d here is c o n v e n i e n t for o b t a i n i n g n u m e r i c a l results b y m e a n s of c o m p u t i n g techniques. F r o m the m i n i m u m - f r e e - e n e r g y c o n d i t i o n we o b t a i n a system of e q u a t i o n s whose solution yields values for t h e r m o d y n a m i c a l l y a v e r a g e d spin at a given p o i n t , as well as v a l u e s for the correlation functions. It can easily be verified that setting A F = 0 a n d fCij = 0 we get the t e m p e r a t u r e d e p e n d e n c e of the average spin in the form of the familiar Brillouin f u n c t i o n B1/2. Expressing the free energy as an explicit function of the p a i r c o r r e l a t i o n s constitutes an a d d i t i o n a l interest f r o m the v i e w p o i n t of investigating the p a i r - c o r r e l a t i o n functions on b o t h sides of the critical temperature.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
S.V. Vonsovskii, Magnetism (Nauka, Moscow, 1971). C. Domb and M.S. Green, Phase Transitions and Critical Phenomena (Academic Press, London and New York, 1974). L.S. Stilbans, Zh. Eksp. Teor. Fiz. 9 (1939) 432. [in Russian] P.R. Weiss, Phys. Rev. 74 (1948) 1493. H.H. Bethe and H.H. Wills, Prec. R. Soc. A150 (1935) 552. H.H. Bethe, Prec. Roy. Soc. A154 (1935) 207. R. Peierls, Prec. Camb. Phil. Soc. 32 (1936) 477. L. Da,browski, Phys. Stat. Sol. (b)128 (1985) 371. A.G. Khachaturyan, Prog. Mat. Sci. 22 (1978) 1. G. Horvitz and H. Callen, Phys. Rev. 124 (1961) 1757. D.A. Badalian and A.G. Khachaturyan, Fiz. Tverd. Tela 12 (1970) 439.