Ferromagnetism ofa spin-^12 solid 121fullyalignedvalueM=Nμ,the saturation magnetization. The spontaneous mag-
netizationM( 0 )of the solid (defined as the magnetization in zero applied magnetic
field)is zero.
For theidealparamagneticsolid,M( 0 )willremain zerohoweverfar the tem-
perature is lowered (see again section 3.1.3). However, what happens in any real
substanceisthataphase transition occurs at some temperatureTTTC,drivenbyinterac-
tionsbetween thespins, to an orderedstate. Thesimplest type oforder,but certainly
not the only type, is ferromagnetic ordering, in which the spins all tend to align in
the samedirection,givinga non-zeroM( 0 ). (In allthischapter weignoredomain
formation;M( 0 )refers to the magnetization in a single domain, i.e. in a not too large
sample or in a not too large region of a large sample.) The degree of order in the
ferromagnetic stateischaracterizedbyan order parametermdefinedby
m=M( 0 )/Nμ (11.1)Our main taskistounderstandhowmvaries withtemperature.
The newingredient ofthischapteristoincludeintheargument a treatment of
the interactions between spins. How is the energy of one spin influenced by the spin
state ofalloftheothers? That simple questionhas a complicatedanswer, unless
approximations are made. One extreme approximationisan‘Isingmodel’inwhich
the spins are assumed only to interact with their nearest neighbours. From the present
viewpoint, thatisstilla complicatedproblem eveninprinciple, althougha start can
be made at it usingmethods to be discussed in the next chapter. The other extreme
approximation is to assume that all the other spins are of equal influence, however
distant theyare. Thisiscalledamean fieldapproximation,andwe shallsee thatit
gives a true second-order transition.
The mean field approximation is to assume that the one-particle states of each spin
are, as usual,+μBand−μB,but thevalue ofBischanged. Insteadofbeingequal
toB 0 , the applied magnetic field, the assumption is that we may write
B=B 0 +λM (11.2)whereλis a constant characteristic of the substance. This follows sinceMis the
magnetization ofallthe spins, so the termλMaverages equally over all spins in the
solid.The caseλ=0 recovers theidealparamagnet ofChapter 3. Theferromagnetic
situation corresponds to a large positive value ofλ,a classic case ofpositive feedback
as we shall now see.
11.2.1 The spontaneous magnetization (method 1)
We candirectlygive an expressionfor the magnetization ofthesolid,using the
Boltzmanndistribution withthe energy levelsimpliedby(11.2). Following(3.9) the