Statistical Physics, Second Revised and Enlarged Edition

94 Fermi–Diracgases

is usuallypositive. A negativedP/dTcan occurin two situations. Occasionally it
happensthatthesolidcontractsonmelting, icetowaterbeingthecommonestexample.
Inthe case of^3 He,however, thevolume changeis normal.The peculiarity of^3 He
arises entirely from an unusualentropyvariation. Below 0.3 K thesolidentropyis
bigger than the liquid, in other words the solid is more disordered than the liquid!
Thisarisesbecause the entropy of^3 He at theselow temperaturesisdue principally
toits spindisorder. In zero magneticfield,thespinsinthesolid behave precisely
as a spin-^12 solid, as discussed in section 3.1. They are totally disordered, giving an
entropyofNkBln 2 at alltemperatures above afew mK. Thespins orderbytheir own
interactions at around 2 mK. The entropyis illustrated schematicallyin Fig.8. 6 .On
the other hand the Fermi factor keeps the liquid entropy low. As in our discussion of
theheat capacity,onlya smallnumber (oforderkkkBTg(εF))ofthespins arefree to
change their state. The exclusion principle ensures that the others are frozen in, and
atT= 0 zero entropy is achieved by all states below the Fermi level being definitely
full and those above it beingdefinitelyempty: hence Fig.8. 6. The liquid (FDgas)
entropyis not of course limited toNkBln 2, so that at a high enough temperature, it
crosses thesolidcurve andcontinues upward.
Finallywe can note that thisphenomenonis more thanatheorist’s re-creation. It
forms the basis of ‘Pomeranchuk cooling’, a practical method for achieving temper-
atures aslow as 2 mK. Ifliquid^3 Heis precooledto a temperature somewhatbelow
0 .3 K, andisthen convertedto solid bycompression, refrigerationis produced.For
instance if the conversion to solid were isothermal(Tconstant), Fig. 8.6 illustrates
that alarge amount ofentropy would be extractedfrom theheat reservoir. On the
otherhandifthe compression toform solidwere adiabatic(Sconstant), the same
S–Tdiagram shows that the temperature of the^3 He would reduce. With reference to
the phase diagram (Fig. 8.5), the liquid–solid mixture would pass up the anomalous
(negative slope) co-existence curve as the pressureincreases, untilalltheheliumhas
solidified. The entropy scale on Fig. 8.6 is quite large, so that this is an effective
coolingmethod.

Solid

Liquid

2 mK 0. 32 K

T

S

NkkkBln 2

Fig.8. 6 Entropy–temperature curves for liquid and solid^3 He. In the anomalous (= unusual) region below
0.32 K the solid is more disordered than the liquid.