Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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182 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.<br />
expect th<strong>at</strong> if we take the limit of an infinite cluster we might be able to get<br />
the exact solution, as long as we do take limits in the appropri<strong>at</strong>e order.<br />
In our case we have short range (nearest neighbor) interactions only. So we<br />
would expect rapid convergence with cluster size. But th<strong>at</strong> is not necessarily<br />
true, since near a phase transition the correl<strong>at</strong>ion length becomes infinitely large.<br />
The cluster we consider here consists of a central <strong>at</strong>om and its q neighbors.<br />
This is the approach taken by Bethe. Because of symmetry we assume th<strong>at</strong> all<br />
neighboring sites are equivalent. We label the central spin 0 and the neighbors<br />
1,2,..,q. The energy of this cluster is<br />
Ec(σ0, · · · , σq) = −Jσ0<br />
q<br />
σi − h<br />
i=1<br />
q<br />
i=0<br />
σi − h ′<br />
q<br />
σi + f(h ′ ) (8.109)<br />
The last term represents the interactions with the environment. We assume th<strong>at</strong><br />
the environment is in some average thermodynamic st<strong>at</strong>e and th<strong>at</strong> the effects of<br />
the environment on the outer spins are noticeable through an effective magnetic<br />
field h ′ acting on the surface of the cluster. The average field due to the outside<br />
acts only on the <strong>at</strong>oms th<strong>at</strong> are in contact with the outside. Our result can be<br />
generalized to larger and non-symmetric clusters, which makes the m<strong>at</strong>hem<strong>at</strong>ics<br />
much more complic<strong>at</strong>ed. For example, it would introduce effective fields th<strong>at</strong><br />
are site specific. In our case we use the same effective field h ′ on all neighbors,<br />
since the cluster is symmetric.<br />
For this cluster we now have to impose transl<strong>at</strong>ional symmetry, and we need<br />
〈σi〉 = 〈σ0〉. This gives an equ<strong>at</strong>ion which determines the value of h ′ . We did<br />
already make one choice here, and we assumed th<strong>at</strong> the effective field on the<br />
central <strong>at</strong>om is zero. Th<strong>at</strong> makes sense. If we make the cluster larger, however,<br />
we have to introduce an effective field for each shell around the center, and<br />
we get an equ<strong>at</strong>ion equ<strong>at</strong>ion magnetic moments for each shell, and can again<br />
determine all fields. Only the central effective field can be set equal to zero! An<br />
individual <strong>at</strong>om in the nearest neighbor shell is not <strong>at</strong> a symmetric position with<br />
respect to the surface of the cluster, and hence not equivalent to the central cell.<br />
It therefore needs an effective field.<br />
Next, we need to calcul<strong>at</strong>e the partition function. In order to do so, we drop<br />
the term f(h ′ ) in the energy, since it will give a factor in the partition function<br />
th<strong>at</strong> does not depend on the values of σ and hence does not influence the spin<br />
averages. We do need it <strong>at</strong> the end again, however, to calcul<strong>at</strong>e the energy. The<br />
partition function for the cluster is<br />
Zc = <br />
{σ0,···,σq}<br />
i=1<br />
e −βEc(σ0,···,σq)<br />
(8.110)<br />
The deriv<strong>at</strong>ion is a bit tedious. First we introduce the energy formula and<br />
separ<strong>at</strong>e the sum on the central spin from the others.<br />
Zc = <br />
{σ1,···,σq}<br />
<br />
e βJσ0<br />
σ0<br />
q<br />
i=1 σi e βh<br />
q<br />
i=0 σi e βh ′ q<br />
i=1 σi (8.111)