Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 20.2-3

POSTLUDE: SYMMETRY AND THE CONCEPTUAL FOUNDATIONS OF THERMOSTATISTICS

Physics — Statistical Mechanics · mean-field-heisenberg-ferromagnet

Problem

For the Heisenberg ferromagnetic model the Hamiltonian is

\mathscr{H} = -\sum_{i,j} J_{ij}\mathbf{S}_i\cdot\mathbf{S}_j - (\mu_B B_e)\sum_j S_{jz}

where

\mu_B

is the Bohr magneton and

B_e

is the magnitude of the external field, which is assumed to be directed along the

z

axis. The

z

-components of

S_j

are quantized, taking the permitted values

S_{jz} = -S, -S+1, \ldots, S-1, S

. Show that for

S = \frac{1}{2}

the mean field theory is identical to the mean field theory for the two-state Ising model if

2S

is associated with

\sigma

and if a suitable change of scale is made in the exchange interaction parameter

J_{ij}

. Are corresponding changes of scale required for the

S = 1

case (recall Problem 20.2-1), and if so, what is the transformation?