Statistical Physics, Second Revised and Enlarged Edition

128 Phase transitions

A A A A A

A

B

B

B

B

B

B

B

B

B

B

(a) (b)

B

B

A A A A

A A A A

A

A A

A

A

A

A

A

B

A

A A A A A

Fig. 11.6Lattice sitesin a crystal.Thesketches show two (equivalent) types oflattice site, A andB, (a)

in a squarelatticein two-dimensional,and(b)inathree-dimensionalbody centredcubiclattice, as occurs

inbeta-brass.

parametermcaninthissituationbedefinedas:m=PA−PB.HerePAistheproportion
ofCu atoms on A sitesinthelattice (whichisalso theproportion ofZn atoms on
Bsites), andPBis the proportion of Cu atoms on B sites. At high temperature, we
expectPA=PB=^12 ,so thatm= 0. Butinthefullyorderedstate we would have
m=+1or−1, dependingon whether Cu or Zn occupies the A sites (another case of
a ‘spontaneously broken symmetry’!).
Whether the transition proceedsdependsontherelativebondenergy between
like and unlike nearest neighbours. If it is favourable for neighbours to be
unlike, we may expect a transition to occur. WritingVVVXYas a bond energy
between neighbouringXandY atoms, we require V = VVVCuCu+VVVZnZn−
2 VVVCuZn > 0. This parameter V plays the role of λ in the ferromagnetic
situation. The mean field approximation in the present case is to assume that the
number ofbondsofeachtypeisdeterminedpurely bythe parameterm, assuming
that the positions on the sub-lattices have no short-range correlations. It then follows
(see Exercise 11.3) that the lattice internal energy can be written asU= constant
− 2 NVm^2 (compare (11.7)). Thewholeproblem can thenbesolvedasin section 11.2.
The graphs of section 11.2 may then be applied to this situation also. The ordering
(i.e.m(T))is easilyfollowedbythe appearancebelowTTTCofextralinesin an X-ray
powder photograph.Theeffects on thethermalproperties are also readilymeasured,
and there is an anomaly inCnear the transition, associated with the entropy changes.
Asbefore, the anomalyin practiceindicates alambda transition, rather than thesimple
second-order transition of Fig. 11.5. Again, since the physical interaction is based on
nearest neighbour interactions, it is not surprising that the mean field approach, with
its neglect ofshort-range order,does not containthefulltruth.