Statistical Physics, Second Revised and Enlarged Edition

Application to helium-3 93

Liquid

35 Solid

29

0

0. 32 K

T

Gas

P (atm)

Fig. 8. 5 Thephasediagramfor^3 He(notdrawn to scale).

move in order for the liquid to re-establish uniform density. The Landau model ends
up with a gas with the same numberNof particles as atoms, but with theeffective
massofeachparticlebeinglarger than thebare mass. Inliquid^3 Hethiseffective
mass is about 3– 6 times the atomic mass of^3 He, dependingon the applied pressure
and hence on the density.
Thissimplepicture of liquid^3 Heiswellsupportedbythe measuredthermodynamic
properties. The heat capacity follows the general shape of Fig. 8.3, in particular being
linear at low temperatures. But the value ofTTTFdeducedfrom experiment requires,
as suggestedbyLandau, alargeeffective mass. Wegive afullerdiscussion ofthis
topic in Chapter 14 (section 14.2). Similarly the nuclear spin magnetization is found
tofollow theform suggestedin section 8.2for spin magnetization ofafermion gas.
Theseideas alsohelptoexplainthehighlyunusualphasediagram of^3 He,shownin
Fig. 8. 5. There are two points worthy of note, both relevant to statistical physics. The
firstisthat^3 He remainsliquid,atmodest pressures, righttotheabsolute zero. In this
theheliumisotopes are unique. The explanationisthat thebindingenergyofthesolid
is so weak that it can be overcome even by its zero-point vibrations, making the solid
unstable until itisstiffenedupby compression. This ‘quantum manifestation’ occurs
inboth^3 Heand^4 He, showingthat it has nothingto do with fermions or bosons. The
second feature of interest occurs only in^3 He, namely the region of negative slope of
thesolid-liquidequilibriumlinebelow 0.3 K. Itis, therefore, specificallyrelevant to
the fermion system.
An important result of thermodynamics is that such a slope can be related to entropy
andvolume changes usingtheClausius–Clapeyron equation(see e.g. Thermal
PhysicsbyFinn, section 9.4). This states that the slope of the equilibrium line

dP/dT=S/V

whereSandVare, respectively, the entropy and the volume changes which
occur when thephase changetakes place. Usuallywhenasubstance melts,its volume
increases(Vpositive) and its entropyincreases(Spositive). ThereforedP/dT