New integral representation for the spin S Ising ferromagnet

Printed in Great Britain

Pergamon Press.

Solid State Communications, Vol. 25, pp. 853-854, 1978.

NEW INTEGRAL REPRESENTATION FOR THE SPIN S ISING FERROMAGNET B. Frank and 0. Mitran Department of Physics, Sir George Williams Campus, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal, Quebec H3GlM8, Canada (Received 17 October 1977 by R. Barrie)

A recently discovered integral representation for the spin l/2 Ising ferromagnet is generalized to the case of spin S. with the identifications

AN INTEGRAL representation for the magnetization in the case of the Ising model has been derived previously by the authors [ 1,2] in two different ways. One method [ 1 ] uses spectral representations for the relevant time correlation functions, while a simpler method [2] uses a known formula for the thermal average of the zcomponent of spin at site f,63; interms of hyperbolic functions. In this note, the second method is generalized to yield an integral representation for the Ising ferromagnet for general spin S, involving an expansion in semiinvariant averages [ 3,4] . We consider the Hamiltonian

y = 7r/[P(2S + l)]

or

n/P

for the first or second term respectively of (2), and a = x&/3. The result is cSj> = G

I& exp (ih&exp mm

(iO,t)>F(t; 6)

(4)

where

-(5)

(1) which describes an Ising system in an applied nonuniform magnetic field represented by the ht, and in which the only restriction on the exchange integral Jzp coupling the sites f and g is Jr, = 0. Using the method of Suzuki [ 51, the general formula for spin S

In terms of cumulants [3,4], (4) may be written _ (- l)nt2*‘Mf h,t+ c lntl n=o(2n + l)!

W;, = ;IdtF(t;@sin 0

u;)

= (S + f )(coth (S + t)x$-- 3 (~0th ix,>

(2)

p)“tl”~~n

x exp

where

i

xf = P(Oz+h,)

@-=

l/kT)

and

or * c Jf4& P

is easily derived. This equation is the analog of the paramagnetic result for free ions, in terms of the Brillouin functions [6]. Now all operators in (2) commute, and so one may make use of the integral formula [7]

pl

(2nY

1

(6)

where the M% are the semi-invariant (or cumulant) averages, with the 0, as the relevant statistical variables. This result reduces to that of [l] and [2] for the case S = l/2. IPM, (n > 1) are set equal to zero, (6) can be shown, again using (3) to reduce to the molecular field result [6]: a;) = (S+f)coth

@(S+t)&)-f

coth@&}

(7)

where I%$, =c

J,,S,Z+h,.

m

s

dt sin (at) [ cloth (yt) - 1] = g coth lf 0 0

- 1

a

(3)

In tfe general case, (6) may serve as a starting point for the treatment of the critical properties of the spin S Ising system, in a manner similar to that used in [ 11. 853

854

NEW INTEGRAL

REPRESENTATION

FOR THE SPIN S ISING FERROMAGNET

Vol. 25, No. 11

Acknowledgements - The authors wish to thank the National Research Council of Canada for financial support (grant number A3 15 1).

REFERENCES 1.

FRANK B. & MITRAN O.,J. Phys. ClO, 2641 (1977).

2.

FRANK B. & MITRAN O., Solid State Commun. 24,343

3.

KIRKWOOD J.G.,J.

4.

HORWITZ G. & CALLEN H.,Phys. Rev. 124,1757

5.

SUZUKI M.,Phys. Lett. 19,267

6.

MORRISH A.H., The Physical Binciples of Magnetism, p. 262 Wiley, New York (1965).

7.

GRADSHTEYN I.S. & RYZHIK I.W., Table of Integrals Series and Products. p. 507, No. 3.987. Academic Press, London (1965).

(1977).

Chem. Phys. 6,70 (1938). (1961).

(1965).