Volume 26A, number 5 PHYSICS LETTERS 29 January 1968

Although the experimental results at present similar to those observed by Litton and Reynolds available are not sufficient to give a detailed ex- [3] in CdS. planation for this negative resistance effect (and in particular to distinguish between the theory of References

i.

3.

Smith and that of Steele et al. ), it seems reason- able to conclude that an injected plasma of elec- trons and holes is present in the AgCl crystal.

The observations reported here might shed

R. W. Smith, Phys. Rev. 105 (1957) 900. G. A. Marlor and J. Woods, Proc. Phys. Eoc. 81 (1963) 1013. C. W. Litton and D. C. Reynolds, Phys. Rev. 133 (1964) A536.

some light onto the experimental results of Leh-

feldt [6] and Jung [+I], who found light-induced low resistivity and electroluminescence in AgCl at low temperatures. These results might be ex- plained with light-induced double injection effects,

4.

5.

6. 7.

M. C.Steele, K. Ando and M. A. Lampert, J. Phys.Soc. Japan 17 (1962) 1729. M. A. Lampert, Phys. Rev. 125 (1962) 126 and Proc. IRE 50 (1962) 1781. W. Lehfeldt, Nachr. Ges. Wiss. Giittingen 263 (1933). L. Jung, Z. Phys. 146 (1956) 479.

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DETERMINATION OF CRITICAL BEHAVIOUR IN LATTICE STATISTICS FROM SERIES EXPANSIONS

A. J. GUTTMANN, B. W. NINHAM and C. J. THOMPSON Applied Mathematics Department,

The University of New South Wales, Kensington, N. S. W., Australia

Received 23 October 1967

A simple method is proposed for determining the critical point (P) and given the critical exponent (Y) from terms of a series expansion for a restricted class of functions. The advantage of the method is that under certain reasonable assumptions it provides precise estimates for errors.

The class of functions considered are those of and we hope to be able to report on this at a the form later date.

f (xl = (1 - ~L?c)-~~(~) , (1)

where h(x) is analytic in and on the circle I& 6 1 in the complex z-plane, except at px = - 1 where it has a singularity weaker than a simple pole. The problem is to determine p and y from terms of a series expansion forf (X).

Our method rests on the simple observation that for functions of the form (l), the coefficients am of (lx)m in

h,(x) = (1 + IJ.x? (1 - &Y(x) (2)

As examples we consider the high temperature susceptibility x for the square and simple cubic Ising model lattices. The variable x in this case is tanh (J/LT) (with the usual notation) and in asserting that x has the form (1) we are assum- ing that the closest singularity to the origin is at lyxl = 1, the ferromagnetic singularity factors (both assumptions are commonly made in the literature [ 1,2]) and that the singularity at 1-1 x = - 1 is weaker than a simple pole [3]. The only functions for the Ising model which we believe have the form (1) are the x’s for the loose packed lattices. We believe however that our method can be generalised to cover a wider class of functions

forn =O, 1,2, . . . . oscillate in sign and decrease in magnitude for sufficiently large m. This is proved for n = 0 by using the asymptotic form for the coefficients of ho(x) and the result for n = 1, 2, follows simply by induction.

We consider first x for the square lattice for which fifteen terms of the series are known [4]. Our procedure is to obtain, for a range of values for p and y, the values of n for which at least the last six terms of h,(x) oscillate in sign and de- crease in magnitude. The results are shown in fig. 1 as a contour map of n values in the ~.l-l-y plane. For n > 2 the & series are not sufficiently regular to be able to draw reasonable conclusions. Also, if one demands more regularity, for example

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Volume 26A. number 5 PHYSICS LETTERS 29 January 1968 ~___

ff = 2, 7 = 1.7497 +Z 0.0048. c1-l = 0.414 2136 f i 0.000 080.

In the present case we know that n = 4 + 1 [l], so that from fig. 1 we obtain y = 1.7499 * 0.0002 for r = 6 (II = 2). Similarly we obtain for Y = 8 (% = 1) y = 1.7497 f 0.0004 and for Y = 13 (fi= 0) y = 1.7500 f 0.0045, which support very strongly the suggestion that y = f [l].

1.746

3-

Fig. 1. Contour map of n values in the /.L-1-Y plane for the high temperature susceptibility series of the two

We next consider x for the simple cubic lattice for which eleven terms of the series are known. We have scanned a range of values of 1-1 and T as indicated above and the resulting contour map is very similar to that shown in fig. 1. In parti- cular for K = 1 we obtain y = 1.249 8 f 0.073, ,u-l = 0.218 15 * 0.000 14 and for ff = 2, y = 1.248 6 i 0.001 6, p-l = 0.218 166 i 0.000030. Also, if we assume that ‘)-’ = {, the commonly assumed value, we find that n-I=0.21814365~0.00000035forr=5(~=2)~ p-1 = 0.218 143 9 f 0.000 008 9 for ‘Y = 8 (II = 1). and n-1 = 0.218 133 8 f 0.000 037 5 for Y = 10 (ff = 0); which differ somewhat from the Pade estimate [l] of p-1 = 0.218 156 f 0.000 006.

dimensional square lattice Ising model.

requiring that at least the last eight terms of the hnseries oscillate and decrease, one obtains no n = 2 values, and requiring that the last thirteen terms oscillate and decrease, one obtains no n = 1 or 2 values. The prescription one adopts then is that the maximum value of n, denoted by Z(r), where ‘Y is the number of terms of the hn series one requires to oscillate and decrease, corresponds to the correct p and y. As can be seen from the figure this prescription is satis- fied by a range of values lying within a narrow region (for n > 0). For S = 1 we obtain y = 1.756 + 0.018, p-1 = 0.414 32 + 0.0031, and for

Two of us (B. W. N. and C. J. T. ) are grateful to the Commonwealth Governement for the award of Queen Elizabeth II fellowships and the other for the award of a Commonwealth Post Graduate Award. We would also like to thank Professor J. M. Blatt for his comments on the manuscript.

References 1.

2.

3.

4.

M. E. Fisher. Boulder Lectures 1964 (University of Colorado Press. Boulder. Colorado, 1965). C. Domb, Proc. Conf. on critical phenomena, Was- hington D.C., 1965 (N.B.S. Miscellaneous Publication 273,1966). M. E. Fisher and M. F. Sykes. Physica 28 (1962) 919. 939. M. F. Sykes. J. Math. Phys. 2 (1961) 52,63.

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