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The critical ordering temperature in alloys. II. The existence of a two-phase region

Published online by Cambridge University Press:  24 October 2008

C. E. Easthope
C. E. Easthope
Affiliation:
Trinity College
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By using Bragg and Williams' approximations and Bethe's interaction energies between nearest neighbours, it has been shown that, for alloys of the AB3 type, (1) the maximum value of the critical temperature Tc must occur when c = ½ (2) for all temperatures within the range 0 < T < (Tc)max there is always a concentration range within which the alloy exists in a two-phase state, the one phase being partially ordered and the other disordered. This two-phase region is small at all temperatures and vanishes both at T = 0 and at T = (Tc)max. In particular for T/(Tc)max = 0·82, which is the value of Tc for c = ¼, the limits of the two-phase region are c = 0·245 and c = 0·256. The theoretical phase diagram has therefore qualitatively the shape, given in Fig. 5. For a fixed concentration there is thus a certain temperature range in which a disordered phase and a partially ordered phase exist in equilibrium with one another. The temperature at which the disordered phase completely disappears is that usually given by experimental determinations of the critical temperature. It is somewhat higher than Tc, but its maximum value also occurs when c = ½, so that the discrepancy between theory and experiment for this type of superlattice still remains.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 34 , Issue 1 , January 1938 , pp. 68 - 80
Copyright
Copyright © Cambridge Philosophical Society 1938

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References

* Proc. Cambridge Phil. Soc. 33 (1937), 502.Google Scholar

† Proc. Roy. Soc. 150 (1935), 552.Google Scholar

‡ Proc. Roy. Soc. 145 (1935), 699;Google ScholarProc. Roy. Soc. 151 (1935), 540;Google ScholarProc. Roy. Soc. 152 (1935), 231.Google Scholar These papers are referred to as I, II and III respectively.

* See Fowler, , Statistical mechanics, 2nd ed. (Cambridge, 1936), pp. 791–7.Google Scholar

† Williams, III p. 232, equation (1).

‡ We assume that as the composition is varied the lattice spacings remain unaltered, so that V AA,V BB and V AB are independent of c.

* See Williams, III, equation (14); see also the appendix to this paper.

* See equation (13) below.

* These results will be true, of course, quite irrespective of whether the slope of the curve between P 1 and P 2 is continuous or not.

* See Williams, III p. 238, equation (19).

† Fowler, , Statistical mechanics, 2nd ed. (Cambridge), p. 797.Google Scholar

* The term c(V AA − V BB)+ V BB in (8) has been neglected in these calculations since we are only interested in the intersections of the free energy curves and their relative slopes and not in their absolute values.