26 BELL SYSTEM TECHNICAL JOURNAL 



The energy of interaction, / — the positive ordinate of Fig. 14 — can 

 be estimated from the value of the Curie temperature, 6, in a manner 

 suggested by Stoner.^* 



Let 2 J be the difference in the energy of interaction between two atoms when 

 their moments are respectively parallel and antiparallel. The total energy of 

 these two atoms is therefore 



2£ = 2£o ± / 



where £o is the energy of an isolated atom. The negative sign applies when the 

 spins are parallel, the positive when they are antiparallel. Imagine a crystal 

 in which each atom of moment ma is surrounded at equal distances by z other 

 atoms of which x have their spins parallel and y antiparallel. Then turning one 

 atom from the parallel to antiparallel position produces a change of {y — x) in 

 the number of parallel pairs and (x — y) in the number of antiparallel pairs and, 

 therefore, requires an energy 



£ = 2Jix - y). ■ (5) 



Since in each atom the moment must be parallel or antiparallel to the field, the 

 magnetization of the material as a whole will depend on the average value of 



X — y: 



Ilh = {x- y)lz. (6) 



According to Boltzmann's equation an atom will have the following probabilities 

 of being parallel and antiparallel 



Pp=: l/[l+exp(-€/^r)] 



Pa = exp (- elkT)/\:i + exp (- e/kT^. 



Since all atoms behave in the same way on the average x and y must be zPp and 

 zPo- Hence we have 



7//o = {x - y)/z = Pj,- P„ = tanh (e/2^r) 

 or using (5) and (6) 



Comparing this with the modified Weiss equation, Eq. (4), 



I , ^ haNI ^ ,Ilh 

 -rr = tanh , „ = tanh -^rj^ 

 Jo kT Tje 



we have J in terms of the molecular field constant or the Curie temperature: 



J = ijaNIoIz = kelz. 



For iron, z = 8, J = kd/S = 1.8 X 10"" erg or 0.01 electron volt. 



This derivation indicates that J is proportional to 0, and that the constant of 

 proportionality depends on the number of nearest neighbors. The number of 

 neighbors has not been taken into account in the following discussion of Fig. 14. 



The interaction curve is substantiated in a quaHtative manner by 

 the observed variation of the Curie points of the iron-nickel alloys. ^^ 



" E. C. Stoner, Phil. Mag., 10, 27-48 (1930). Stoner's original work appears to 

 have been in error by a factor of two; the modified treatment given here is due to 

 W. Shockley and follows closely the method employed in dealing with order and 

 disorder in alloys (see e.g. Eqs. 1.11, 1.12, 2.2 and 2.16 in the article by F. C. Nix 

 and W. Shockley, Rev. Mod. Phys. 10, 1-71 (1938)). 



1* Summarized by J. S. Marsh, Alloys of Iron and Nickel, v. 1, pp. 45 and 142, 

 McGraw-Hill, New York (1938). 



