- Email: [email protected]
Comment on “dilute Ising models: A simple theory”
Volume 99A, number 5
PHYSICS LETTERS
5 December 1983
COMMENT ON "DILUTE ISING MODELS: A SIMPLE THEORY"
A. ZAGORSKI Institute of Physics, Technical University of Warsaw, Warsaw,Poland Received 20 May 1983 Revised manuscript received 15 September 1983
The work by Boccara, published recently [ 1], needs a short comment. The author claims that he presents "a much less sophisticated method which is conceptually as simple as an ordinary mean field approxirnation but which gives, however, much better results". He considers the Ising triangular lattice with the coordination number n = 3. The crucial point of his method is an identity of the general form:
Suppose that a spin S interacts with n other spins: $1, S 2 .... S n by means of the (reduced) exchange constants K1, K2, ..., Kn, respectively. Its thermodynamical average is then given by the Brillouin function Bs(x), where x=KISf
+...+ K n S Z ~ - K ' S z .
Now we express B s ( x ) by its Fourier transform BS(Y) as follows:
tanh(KlO 1 + K 2 o 2 + K 3 o 3) =AlO 1 +A2o 2 + A 3 o 3 + B a l o 2 o 3,
Bs(X ) = f --
where o i denote spin variables, K i reduced exchange integrals, and A i and B are simple combinations of the functions tarda(K 1 + K 2 + K3). This expression is next appropriately averaged if the distribution of K's or spins is random. The Curie temperature Te is then obtained in a standard way giving the dependence of Tc on the concentration of bonds (or spins). A similar method was developed already ten years ago by Matsudaira [2,3] for cubic lattices with nonmagnetic impurities (site percolation problem). He also calculated different correlations between spins and examined their influence on e.g. the critical temperature. That method was subsequently applied by Ferchmin and coworkers [4] in a series of papers to alloyed thin films. They calculated also Tc for an amorphous ferromagnet [5] and next - the Curie and the compensation temperatures for a disordered model ferrimagnet [6]. In this letter I would like to show, how to extract spin variables from the argument o f the BriUouin function for arbitrary spins, exchange constants and coordination numbers. I hope, this might be useful for those physicists who are involved in Ising systems of any nature. 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
+ o*
exp(21rixy)Bs(y)dY aa n
= f I] exp(21riyKjS~)i~S(y)dy. - ~ ]=1 Exponential functions of the general form exp(qS z) may be written in the form: S exp(qS z) = ~ Pleq I , l=-S where PI denotes the projection operator on the state S z = l of a given spin. It is easy to check that Pl = ( Sz - S)( Sz - S + 1) ... x (s t - t -
1 ) ( s z - t + I ) ... ( s z + S ) I N ,
where N is the same product as in the nominator but with S z replaced by l. PI is then a polynomial of the order 2S with respect to S z. If we repeat this procedure n times then we get:
247
Volume 99A, number 5
Bs(K. s z) +~
: _** f
PHYSICS LETTERS
5 December 1983 1
$1
$2
E 12=-$2 E 11=-$1
Sn
2 n In=-S
For the special case: n = 3, S 1 = S 2 = S 3 = ~, the above formula gives immediately Boccara's result for tanh(K1S ~ + K2S ~ + K 3 S~).
References X Pin exp(27riyK. QBs(Y)dY = ~ PIBs(K" l). 1 Here ! denotes the n-dimensionalvector ( l l , 12, -.., ln), and
P1 = Pla (S~)PI2 (S~) ... Pln(SZ) . Because the Brillouin function is antisymmetric, the polynomial Pl contains only odd products o f spin variables.
248
[1] N. Boccara, Phys. Lett. 94A (1983) 185. [2] N. Matsudaira, J. Phys. Soc. Japan 35 (1973) 1593. [3] N. Matsudaira and S. Takase, J. Phys. Soc. Japan 36 (1974) 305. [4] A. Zag6rski, W. Nazarewicz, A.R. Ferchmin and A. Sukiennicki, Phys. Stat. Sol. 100b (1980) 473, and references therein; Phys. Lett. 71A (1979) 127. [5] W. Nazaxewicz, A. Zag6rski, A.R. Ferchmin and S. Kobe, Phys. Stat. Sol. 106b (1981) K131. [6] A. Zag6rski and W. Nazarewicz, Acta Phys. Polon. A60 (1981) 697.
PHYSICS LETTERS
5 December 1983
COMMENT ON "DILUTE ISING MODELS: A SIMPLE THEORY"
A. ZAGORSKI Institute of Physics, Technical University of Warsaw, Warsaw,Poland Received 20 May 1983 Revised manuscript received 15 September 1983
The work by Boccara, published recently [ 1], needs a short comment. The author claims that he presents "a much less sophisticated method which is conceptually as simple as an ordinary mean field approxirnation but which gives, however, much better results". He considers the Ising triangular lattice with the coordination number n = 3. The crucial point of his method is an identity of the general form:
Suppose that a spin S interacts with n other spins: $1, S 2 .... S n by means of the (reduced) exchange constants K1, K2, ..., Kn, respectively. Its thermodynamical average is then given by the Brillouin function Bs(x), where x=KISf
+...+ K n S Z ~ - K ' S z .
Now we express B s ( x ) by its Fourier transform BS(Y) as follows:
tanh(KlO 1 + K 2 o 2 + K 3 o 3) =AlO 1 +A2o 2 + A 3 o 3 + B a l o 2 o 3,
Bs(X ) = f --
where o i denote spin variables, K i reduced exchange integrals, and A i and B are simple combinations of the functions tarda(K 1 + K 2 + K3). This expression is next appropriately averaged if the distribution of K's or spins is random. The Curie temperature Te is then obtained in a standard way giving the dependence of Tc on the concentration of bonds (or spins). A similar method was developed already ten years ago by Matsudaira [2,3] for cubic lattices with nonmagnetic impurities (site percolation problem). He also calculated different correlations between spins and examined their influence on e.g. the critical temperature. That method was subsequently applied by Ferchmin and coworkers [4] in a series of papers to alloyed thin films. They calculated also Tc for an amorphous ferromagnet [5] and next - the Curie and the compensation temperatures for a disordered model ferrimagnet [6]. In this letter I would like to show, how to extract spin variables from the argument o f the BriUouin function for arbitrary spins, exchange constants and coordination numbers. I hope, this might be useful for those physicists who are involved in Ising systems of any nature. 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
+ o*
exp(21rixy)Bs(y)dY aa n
= f I] exp(21riyKjS~)i~S(y)dy. - ~ ]=1 Exponential functions of the general form exp(qS z) may be written in the form: S exp(qS z) = ~ Pleq I , l=-S where PI denotes the projection operator on the state S z = l of a given spin. It is easy to check that Pl = ( Sz - S)( Sz - S + 1) ... x (s t - t -
1 ) ( s z - t + I ) ... ( s z + S ) I N ,
where N is the same product as in the nominator but with S z replaced by l. PI is then a polynomial of the order 2S with respect to S z. If we repeat this procedure n times then we get:
247
Volume 99A, number 5
Bs(K. s z) +~
: _** f
PHYSICS LETTERS
5 December 1983 1
$1
$2
E 12=-$2 E 11=-$1
Sn
2 n In=-S
For the special case: n = 3, S 1 = S 2 = S 3 = ~, the above formula gives immediately Boccara's result for tanh(K1S ~ + K2S ~ + K 3 S~).
References X Pin exp(27riyK. QBs(Y)dY = ~ PIBs(K" l). 1 Here ! denotes the n-dimensionalvector ( l l , 12, -.., ln), and
P1 = Pla (S~)PI2 (S~) ... Pln(SZ) . Because the Brillouin function is antisymmetric, the polynomial Pl contains only odd products o f spin variables.
248
[1] N. Boccara, Phys. Lett. 94A (1983) 185. [2] N. Matsudaira, J. Phys. Soc. Japan 35 (1973) 1593. [3] N. Matsudaira and S. Takase, J. Phys. Soc. Japan 36 (1974) 305. [4] A. Zag6rski, W. Nazarewicz, A.R. Ferchmin and A. Sukiennicki, Phys. Stat. Sol. 100b (1980) 473, and references therein; Phys. Lett. 71A (1979) 127. [5] W. Nazaxewicz, A. Zag6rski, A.R. Ferchmin and S. Kobe, Phys. Stat. Sol. 106b (1981) K131. [6] A. Zag6rski and W. Nazarewicz, Acta Phys. Polon. A60 (1981) 697.