Statistical Physics, Second Revised and Enlarged Edition

92 Fermi–Diracgases

g↓() g↑()

F



B

B

+B

• B

Fig. 8. 4 Calculatingthemagnetization ofaspin-^12 FDgas atT=0. Theleft-handhalfofthefiguregives

the densityof states for spin-down fermions, whereas the right-hand half is for spin-up fermions. The full

curve shows the densities of states whenB= 0 ,the dashed curves when a fieldBis applied. Onlya small

number of the spins close to the Fermi energy(indicated bythe shaded region) are able to realign whenB

is applied.

has its spin reversed (i.e. changed by 2μ), giving a total magnetization of

M=μ^2 Bg(εF) (8.13)

This expression is independent ofTin the degenerate region, since the argument of

Fig. 8.4 is unaffected by the slight blurring of the Fermi surface. Equation (8.13) is

againacharacteristic FD result. Sinceg(εF)= 3 N/ 2 εFfor theidealgas, we see that

the magnetization is reduced from the Boltzmann value, (8.12), by the usual Fermi

factor(kkkBT/εF). Againthe smallmagnitudeandthelackofT-dependence ofthe

magnetization are wellconfirmedbyexperiment, asisthedependence ong(εF)in

particular metals.

8.3 Application tohelium-3

Helium-3 is a spin-^12 atom, having a nuclear spin of^12 .Therefore^3 He gas is a fermion
gas. However, as explained in Chapter 6, it never exists with a density high enough
for FD statistics tobe more than a smallcorrection to the MB treatment.But thatis
not theendofthe^3 He story, since it turns out that FD statistics can usefully be applied
toliquid^3 He.
Againthisidea shouldcome as a surprise! There are verystronginteractions
between molecules in a liquid, and the atoms in liquid^3 He are no exception. One of
theachievements ofLandau was todevelop thetheory ofa Fermiliquid,andto show
thatfor manypurposesit canbe treatedas a weakly interactinggas. However, the
particles of the gas are not^3 He atoms,but are^3 He ‘quasiparticles’, entities which can
bevisualizedasbare^3 He atomsdressedwiththeinteractions ofother nearby atoms.
After all, if one atom in the liquid is moved, then a whole lot of other atoms must