Gaussian fluctuations of a molecular field

Volume 76A, number 5,6

PHYSICS LETTERS

14 April 1980

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GAUSSIAN FLUCTUATIONS OF A MOLECULAR FIELD ~

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Z. ONYSZKIEWICZ

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Magnetism Theory Division, Institute of Physics, A. Mickiewicz University, Pozna,i, Poland Received 3 January 1980

Within the high density expansion method a new way of calculating gaussian molecular field fluctuations is established. With the help of the approximation proposed the magnetization, the mean squares of molecular field fluctuations and the free energy for Ising ferromagnets are calculated.

Within the high density expansion method the mean squares of gaussian fluctuations of a molecular field were calculated in several papers [1—6].The present letter is concerned with the calculation of the mean square of gaussian fluctuations of a molecular field within the high density expansion method, in a new way, different from how it has been performed before. We consider an ideal Ising ferromagnet with spin Sat each lattice site. The Ising hamiltonian is J,-

=

2

1~ss~1

(1)

.

In the approximation which includes gaussian fluctuations of the molecular field the following expression for the relative magnetization [3] is obtained: (2) where the bar denotes average over a gaussian distribution: L lnl( 3~)=

—~-—

f exp(—~x 2)L E’~1(~,+~yx)~ (3) a° Ish[(S÷~)(y+&vx2)1(4) aY°

t

sh[~(y+6yx)]

i

=

~

J1~’= ~
~ = (kB T) 1,

z is the number of spins interacting with a given spin. Eq. (3) implies that &y denotes a mean fluctuation of the molecular field and is determined by the formula 2=N~~ J(k),

(5)

(~Y) where J(k) is the Fourier transform of the effective interaction. From refs. [1,2,6] it follows that: J(k)~3J(k)[I —f3J(k)L”(y)]~ ,

(6)

where J(k) ~Jff~ exp[ik(f—f’)]

(7)

.

On the other hand, in refs. [3—5],when calculating the series J(k)=1

)+

~

i-....

(8)

and it is found that ~ Work sponsored by the Institute of Low Temperature and Structural Research, Polish Academy of Science, Wroclaw, Poland.

J(k) = ~J(k) [1

—

~J(k) L “(p)]

1

,

(9)

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Volume 76A, number 5,6

PHYSICS LETTERS

where ~3J(k)

f~IF~=L”(y).

We see that eq. (9) is in fact only a simplification of eq. (6). In both approximations (6) and (9), the calculation introduces some shortcomings into the theory which we specify as follows: (i) Namely, the approximations (6) and (7) used

( 1,0

i H

14 April 1980

for a flat Ising model lead to a zero Curie temperature = 0. This is inconsistent with the exact result obtamed by Onsager [7]. (ii) Moreover, in approximation (6) for an sc lattice we obtain numerically a discontinuous phase transition from the ferromagnetic to the paramagnetic state. This result is contradictory to the results following from the high temperature expansion and the renormalization group approach. (iii) Moreover, within approximation (9) we obtain a nonmonotonic dependence 3y = ~v(T) for T ~ T~ in an sc lattice. (iv) In addition within approximation (9) no expression exists for the free energy, F, which could satisfy the two stationary conditions, i.e., aF/oy 0, aF/o(8y)=0. (10) In connection to this we propose a new approximation to calculate ~y. We assume the following self-consistent renormalization of an interaction line: J(k)=fl=÷~~

+

0,5

Fig. 1. Relative magnetization u = 2(S/) in the approximation (11) versus relative temperature t = T/TMFA, where T~4FA is the Curie temperature in the moleculax field approximation. (The Curie temperature in approximation (11) is t~= 0.856 and t~= 0.799 for z 6 and z = 4, respectively),

~0

~•

~ [~J(k)12L~(y).(ll)

~

~

C

Fig. 2. Mean fluctuation of the molecular field ay (in relative units) plotted against the relative temperature t = ~ where T~~IFA is the Curie temperature in the molecular field approximation.

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Volume 76A, number 5,6

PHYSICS LEUERS

The renormalization (11) is performed with the use of the simplest possible correlation function which involves a gaussian fluctuation of the molecular field. Within our approximation it is easy to construct a free energy satisfying the conditions (10): 2 ~3lLIy) + ~(f33zJ2)~~y4, (12) F/N ~JZ(S/) —

14 April 1980

molecular field approximation. Hence, our approximation yields the best result known up to date for four-dimensional Ising models in the whole range of temperatures. The author wishes to thank Professor H. Cofta for his helpful discussions and for a critical reading of the manuscript.

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and the obvious condition F FMFA for ~iy-+ 0, where FMFA is the free energy in the molecular field approximation. On applying the approximation proposed we come to the results which are presented in figs. I and 2. We see that our results do not have the defects known from the —~

previous works [1—3] The approximation (11) at high temperatures gives results similar to the ones obtained by the Bethe—Peierls methods and in the low temperature region it gives results similar to those obtained within the .

References

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[11G. Horwitz and H.B. Callen, Ploys. Rev, 124 (1961) 1757. 121 B. Muhlschlegel and H. Zittartz, Z. Phys. 175 (1963) 553. [31 Yu.A. Izyumov, F.A. Kassan-Ogly and Yu.N. Skryabin, J. de Phys. 325 (1971) C1-87. Phys. Lett. 57A (1976) 480. [51 Z. Onyszkiewicz, Phys. Lett. 68A (1978) 113. [6] Z. Onyszkiewicz and H. Cofta, Acta Phys. Pol. 56A (1980). [71 L. Onsager, Phys. Rev. 65 (1944) 117.

[41 Z. Onyszkiewicz,

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