Statistical Physics, Second Revised and Enlarged Edition

102 Bose–Einsteingases

Liquid^4 He remainsfluidto thelowest temperatures atpressuresbelow about
25 atm, for reasons discussed in section 8.3. Its phase diagram is sketched in Fig. 9.4.
The outstandingfeatureistheexistence ofaliquid–liquidphase transition at around
2Kfrom an ordinaryliquid(HeI) athightemperature to an extraordinary‘superfluid’
phase (HeII) at low temperatures. The transition is evidenced by a heat capacity
anomalyatthe transition temperature, commonlycalledthelambda-pointbecause
of theresemblance of theC−Tcurve to the Greekletterλ.The curveisshownin
Fig. 9. 5 , with the correspondingS−Tcurve in Fig. 9.6. The nature of the singularity
is‘logarithmic’,inthat althoughCCPisinfinite,itsintegralacross the transitionis
finite so there is no latent heat associated with the transition. The entropycurve is
continuous, but it does have a vertical tangent at the transition. The explanation of the
veryflat solid–liquid line on thephasediagram can again (asin^3 He)beunderstood
from the Clausius–Clapeyron equation. Below the lambda-point, both liquid (because
of superfluid ordering) and solid(^4 He has zero spin, unlike^3 He) are highly ordered
andhave virtuallyzero entropy. Hence the entropy differencebetween thephasesis
almost zero so that thephase equilibrium line has almost zero slope.

Solid

Liquid

He I

Liquid

He II

123 4 5

PP (atm)^25

Gas

–line

0

TTT(K)

Fig.9. 4 Thephasediagram of^4 He.

T

TTT

C

Fig. 9. 5 The heat capacityof liquid^4 He as a function ofT,showingthe lambda anomaly.