126 Phase transitionsline upallthespins. Thisrelates to theflatF(m)curves,since theaddition ofan
applied fieldB 0 to Fig. 11.4 (as in (11.6)) tilts the curves somewhat, and can move
theminimum position wellawayfromm=0.
Incidentally,the sameidea oftiltingtheF(m)curves canbe usedintheferro-
magneticregion to describe the influence of an applied magnetic field belowTTTC.
One minimumbecomes oflowerFthan theother, andthewholepicture givesa
good account of hysteresis curves (ofMversusB 0 )inaferromagnetic material.The
interested reader is referred to books on solid-state physics.1 1.3 Real ferromagnetic materials
First a word of warning! We have outlined a simple theory for ferromagnetic
interactionsinaspin-^12 solid.
Note:(i) In many cases the interactions in magnetic materials are antiferromagnetic,
in other words neighbouring spins tend to be oppositely aligned (or to be at a non-
zero angle). Ferromagnetismis not theonly,andindeednot the commonest, situation.
Nevertheless iron and nickel are not unknown materials, either. (ii) As in Chapter 3,
extension to spins other than^12 can readilybemade, withno qualitative changes.
The meanfieldtheoryhas notable successesindescribingthe transitionfromferro-
to para-magnetism. The nature of the spontaneous magnetization, the existence ofa
criticaltemperatureTTTC,theheat capacity anomalyandthe magnetic properties are
all describedwithreasonable accuracy. A true second-order transition(Scontinuous,
but dS/dTdiscontinuous) is predicted by the theory. How does all this compare with
experimentindetail?
Thefirst comment must concern themagnitudeofthe parameterλ. In (11.2)it
seems that we have assumed thatλMis a real magnetic field. And in some cases it
mightwell be so; there willalwaysbe a true magnetic couplingbetween thedipole
moments of adjacent spins. But in the materials we think of as ferromagnets (like iron
for example), the physical origin ofλcomes from far stronger effects, the magnitude
ofλMbeingseveralhundredteslatogiveTTTCabove room temperature. This strong
couplingarises from quantum influences on the electrostatic Coulomb forces between
overlapping electrons on neighbouring atoms. Since electrons are identical spin-^12
fermions, the overlapis strongly dependent on therelative spinofthe two electrons,
and hence the energy of one electron is influenced by the spin of its neighbours. This
gives energy splittings as suggestedby (11.2),but theoriginis not a weakmagnetic
one,butisdue to these muchlarger ‘exchange energies’.
Next let us consider the detailed shapeofthem(T)variation (Fig. 11.3). Although
agreement withexperimentisfair, there are two points ofdetailwhichareincorrect.
Atlow temperatures, the meanfieldapproachsomewhat overestimatesm.Thisis
because it ignores the existence of ‘spin waves’, essentially long wavelength periodic
variationsin magnetization. These are notincludedin our simpletheory, whichis
basedonlong-rangeorderonly,i.e. onthespinsofallrangeshavinganequalinfluence.