Order–disorder transformations in alloys 127In practice spin wavesdoexist, andtheyare seeninthelow-temperature regionboth
inmand also as a small additional heat capacity.
Theother region ofinterestisclose toTTTC. Here the meanfieldtheory gives too small
avalue ofm.The transitionis somewhat more catastrophicthan wehavepredicted.
The theory very close toTTTCsuggestsm∝(TTTC−T)^1 /^2 ,whereas reality is closer to
m∝(TTTC−T)^1 /^3 .The studyofsuch‘criticalexponents’is a very searching way of
understandingphase transitions.
Correspondingly, the thermal properties show deviations from Fig. 11. 5 , and we do
nothave a true second-order transition. Theheat capacityshows alambdasingularity,
differingfrom the theoryin two respects. Firstly(as withm)it is more sharpjust below
the transition, giving a much higher value asTTTCis approached. And secondly there
isanadditionalheat capacityjust above the transition (theotherhalfoftheλ-shape).
These effects are thought to arise from the existence of short-range order. Above
TTTC, althoughm=0 identically, i.e. there is no long-range order, we still expect
neighbouringspins tointeract andperhapsform transientislandsoflocalorder. (We
have alreadynoted the verylarge fluctuations possible here.) The paucityof the mean
field approach is that, as we have seen, it treats all spins equally, whereas in reality
theinteraction withcloser neighbours will be stronger than thatfromdistant ones. So
treatments based on, say, an Ising model will describe these details more accurately.
But that wouldalso takeusfarbeyondthe scope ofthischapter!
Finally(compare section 11.2.5) we can note that there is further evidence of the
influence of short-range order in the magnetic properties aboveTTTC. Until one is close
to the transition, theform of(11.10), the Curie–Weisslaw,isfoundto agree well
withexperiment. However, theparameter‘TTTC’whichentersitisfoundtobe not the
same as the actual ferromagnetic transition temperature, but rather a temperature a
little higher (e.g. about 375◦Cinnickel, comparedwithaTTTCof^358 ◦C). Close to the
transitionit seems thatM/B 0 is betterdescribed as(T−TTTC)−^4 /^3 , another example
of mean field theory not getting the critical exponent quite right.
11.4 ORDER–DISORDER TRANSFORMATIONSIN ALLOYS
Before leavingthe topic, we maynote that the same type of mean field theorycan
be constructed for many other transitions. Here we just consider one other example,
that ofthephase transition whichoccursinanalloysuchasbeta-brass CuZn.
This alloycontains equal numbers of Cu and Zn atoms. At high temperatures,
the atoms lie on a body-centred cubic lattice in an entirely random arrangement.
But below a transition temperature (about 4 60 ◦C), the atoms start toform an ordered
arrangement, with one type of atom at the corners of each unit cell and the other type at
the centre ofthecell.(Abody-centredcubic structure canbethoughtofas consisting
oftwointerpenetratingsimplecubicsub-lattices, analogous to a two-dimensional
centred square lattice, illustrated in Fig. 11.6.)
The transformation canbedescribedbythe meanfieldapproximation, withmuch
the same success andfailure asintheferromagneticsituation. Thelong-rangeorder