Statistical Physics, Second Revised and Enlarged Edition

Ferromagnetism ofa spin-^12 solid 123

1

T

m

TTTc

Fig. 11. 3 Variation ofthe magnitudeoftheorder parametermwithtemperatureT.Note thatmvaries
from1atT= 0 to 0 atT=TTTC. It remains zero aboveTTTC.

Themagnitudeofmobtainedbysolving(11.4)isshowninFig. 11.3, plottedasa
functionofT. The order parametergrows rapidlybelowTTTC,reachingalmost unityat
TTTC/3. One of the characteristic features of the mean field approximation is the total
lackoforderwhenT>TTTC;thesolid iseffectively in a paramagnetic stateinspite
of the interactions between spins. However, before comparing with experiment, it is
usefulto rederivethe resultform(T)by another method.

1 1.2.2 The spontaneous magnetization(method 2)

When in doubt, minimize the free energy! That was the burden of Chapter 10, and we
can do that here, as an alternative derivation of the variation of the order parameter
mwithtemperature. To calculateF,wefindbothUandSasfunctions ofm.The
internal energy contribution due to the spins is simply their potential energy in the
magneticfield.Withthe meanfieldassumption (11.2), thisgives

U=−

∫

0

∫∫

BdM

=−B 0 M−λM^2 /2(11.6)

In zero appliedfield(B 0 = 0 )the first term of (11. 6 ) vanishes, and we have as a
functionofm

U=−(λ/ 2 )(Nμm)^2 (11.7)

The entropy contribution from the spins can be worked out directly from (10.3).
Thefractions ofthespinsinthe upper andlower states are givenin terms ofmby
P 2 =( 1 −m)/ 2 andP 1 =( 1 +m)/ 2 respectively.[Check: P 1 +P 2 = 1 ,and