Statistical Physics, Second Revised and Enlarged Edition

Ferromagnetism ofa spin-^12 solid 125

T

S,C

S =NkkkBln 2

C

S

TTTc

Fig.11. 5 Entropyand heat capacityin the mean field approximation. Full curve, entropy. Dotted curve,

heat capacity(vertical scale×0.6).

T-dependences ofSandCin our mean field approximation are shown in Fig.11.5.
The entropy curve shows the sudden onset of order asTis reduced throughTTTC.
AboveTTTC,S=NkkkBln2andthespins are completely disordered. Correspondingly,
Cvanishes aboveTTTC, but shows a considerable anomaly over the temperature region
belowTTTCinwhichSischanging.

1 1.2.4 The paramagnetic region

Notwithstanding the zero heat capacity in the paramagnetic (T>TTTC)regime, we
have alreadyremarkedupon the veryshallowF(m)curves close to the transition.
Therefore, it is not surprising that the magnetic properties are affected aboveTTTC.
Althoughthespontaneousmagnetizationis zero, the response to an appliedfield is
increasedbytheferromagneticinteractions.
The magnetization in this region can be worked out readily as in Chapter 3.
Equation (11.3) canbesimplified(sinceB 0 andM arebothsmall)by replacing
tanhxbyxtogive

M=Nμ[μ(B 0 +λM)/kkkBT]

and using the definition (11. 5 )ofTTTCthis rearranges to

M

B 0

=

Nμ^2

kkkB(T−TTTC)

(11.10)

This should be compared with (3.10) for the ideal paramagnet. The new feature

isthe−TTTCinthedenominator, whichensures that the ‘paramagnetic’ susceptibility

does notfollow Curie’slaw,but the so-calledCurie–Weisslaw. Agraphofinverse

susceptibility againstTremains linear, but does not pass through the origin. Since

TTTC >0the susceptibilityis everywhereincreasedbytheinteractions, andinfact

diverges as the transition is approached – even a small applied field is enoughto