Statistical Physics, Second Revised and Enlarged Edition

Aspin-^12 solid 35

Withoutlabouringthepoint,itisnoaccident that the workingsubstance ofarefrigera-
tor is also a possible candidate for thermometry – the refrigerant must haveSa strong
function ofT,i.e.it mustbethermally active (‘doing something’)inthat temperature
range!

Magnetization We characterize the magnetic properties bythe magnetic moment
Mof the whole assembly ofNspins. Interactions will be neglected, at any rate until
wediscussferromagnetism muchlaterinthebook,sothatBmaybetaken as the
externallyapplied field. In the lower state, each spin has a componentμaligned
to theBfield, whereas each spin in the upper state has a component –μ.And the
numbers ofspinsinthe two statesinthermalequilibrium at temperatureTaregiven
bythe Boltzmann distribution (3.2). Hence we have

M=n 0 (μ)+n 1 (−μ)

=Nμ

[exp(μB/kkkBT)−exp(−μB/kkkBT)]

[exp(μB/kkkBT)+exp(−μB/kkkBT)]

=Nμtanh(μB/kkkBT) (3.9)

This expression is plotted in Fig. 3.8, and one can note:

1. 
   

   At high enoughB,or low enoughT,the magnetization saturates at the valueNμ,
   anobvious resultsince then alltheNspinshave alignedwiththefield.

   
   


2. 
   

   For a solid with a higher spin (like the spin-^32 Cu nuclei) the shape of the curve is
   superficially similar, although it is not a simple tanh function.
   3 .Inthe weaklyparamagneticlimit, i.e. μB/kkkBT  1, atlowfieldsorhigh
   temperatures, then the curve is essentiallylinear and we have:

   
   

M=Nμ(μB/kkkBT) (3.10)

12

M

Curie's law region

NNN Saturation limit

B/kkkBT

Fig. 3.8Themagnetization ofaspin-^12 solidas a(universal) function ofμB/kkkBT.