120 Phase transitions
T (and P)G
A B
2 '
1 '
1
2
Fig. 11. 1 The Gibbs free energyGaround a first-order phase transition. In region A (ofTandP)phase 1 is
stable, whereasinregionBphase2isstable.Atthetransition,jumpsthereforeoccurinthefirstderivativesof
G.Supercoolingoccurs when curve 2 ′is followed (an unstable situation) below the transition; superheating
corresponds to curve 1′.
Aunifyingconcept for the understandingof all such transitions is that of an ‘order
parameter’. At high temperatures, the substance is disordered and the order parame-
teris zero. As onelowers the temperature throughthe transition, theorder parameter
suddenlystarts togrow from zero, attainingits maximum value bythe time the
absolute zero is reached, a state of perfect order. For liquid^4 He and also for an
idealbosongas (ofcoldatoms, say), theorder parameteristhe superfluid den-
sity. For superconductivity it is the so-called energy gap. For ferromagnetism it
isthe spontaneous magnetization ofthesubstance,i.e. the magnetizationin zero
appliedfield.Andintheorderingofabinaryalloysystem (suchasbeta-brass, CuZn)
with two types of lattice site, the order parameter characterizes which atoms are on
whichsite.
Inthe sense that the onset oforderinthese systemsissudden, the transitions
are often called ‘order–disorder transitions’. The sudden onset of order implies a
co-operative effect. Inboson gases, the co-operationisforcedbythe(friendly) BE
statistics. Butintheother cases, the co-operation arisesfrominteractionsbetween
the particles of the system. The importance of this chapter is that we can see how to
apply statisticalmethodstoatleast one situation where the particles arenotweakly
interacting.Inthe next section, we shalltakethe transitionfrom paramagnetism to
ferromagnetism in a spin-^12 solid as model example for these ideas.
1 1.2 Ferromagnetism of a spin-^12 solid
In section 3.1, wediscussedtheproperties ofaspin-^12 solid,anidealparamagnetic
material, in which the spins are weakly interacting. In particular ((3.9) and Fig. 3.8),
we workedout the magnetizationMofthesolidas thespinsline upin a magneticfield
B.The result was thatifB=0,thenM= 0 ;butifBkkkBT/μthenMtends to the