Aspin-^12 solid 29T
2 3 SNkkkBln 20 Fig.3. 4 The variation of entropywith temperature for a spin-^12 solid. Note the transition fromperfect
order(S=0) towards total disorder (S=NkkkBln 2).3 .1.2 Magnetic solidsActuallythe two-state situationisquite a common one. It occursbecause ofthe
existence of spin states in atoms. If a single atom has an angular momentum quantum
number of^12 ,then whenit experiences a magneticfieldB,the atomic energylevels
splitinto two. Onelevelgoes upbyμBandtheothergoesdownbyμB,whereμ
isthez-component of the magnetic moment associated with the angular momentum.
Thisisthe Zeeman effect (muchlovedby spectroscopists, since thissplitting of2μB
canbeobservedbya varietyoftechniques). In thelowest (ground) state one can
imagine the atomic moment to be aligned parallel to the field, whereas in the upper
(unstable) stateitis antiparallel.
Therearetwooriginsofthesemagneticmomentsinsolids. Oneisfromtheelectrons
in the atom,and the other from the nucleus. An electronic effect can come both from
theorbitalmotion andalsofrom theintrinsicspinoftheelectron. The precise value
ofμdepends on the details of the atomic origin of the moment, and need not concern
us in this discussion, except to note that in order of magnitude it will equal theBohr
magnetonμB(=e/ 2 m= 0. 93 × 10 −^23 JT−^1 ).For simplicitywe shall infuture refer
to the angular momentum as ‘spin’, regardless of its actual orbital or intrinsic origin.
The second cause is nuclear spin. Here even more the magnitude ofμdepends on the
specific nucleus,but the muchsmaller order ofmagnitudeisthenuclear magneton
μN(=e/ 2 MMMP= 5 .0 5 × 10 −^27 JT−^1 ).
We now consider thethermalproperties ofasolidwhose atomshave spin^12 .To
whateverproperties thesolidwouldexhibitwithout the 2 μBsplitting, we must add
aspin contribution. And in many situations this contribution is precisely described
bythe two-state modelofthe previous section. In orderfor our modelto apply, all
we requireisthat thelocalizedspins are weakly interactingandidentical.Thisisa
good approximation if (but only if) the spins are sufficiently far apart that theB-field
experiencedby eachspinarises almost entirelyfrom an externally appliedfield, rather
thanfrom theinfluence ofits neighbours. In the usualterminologythis means that