30 Two examples
thesubstanceis paramagnetic(alignment ofpermanent momentsinanappliedfield)
rather than ferromagnetic or antiferromagnetic (spontaneous alignment of permanent
moments). Another profitable way ofstating the requirementisthat the energylevels
ε 0 andε 1 do not themselvesdependon the occupation numbersn 0 andn 1 ,but onlyon
the applied field. This is a good approximation for most nuclear spin systems, since
μNis small. It applies onlytoalimitedrange ofelectronicspin systems, notably
dilutedparamagneticsalts suchas cerium magnesium nitrate (CMN) whichcontains
only a few spins (Ce^3 +ions) separated by a lot of non-magnetic padding (all the other
atoms,includingthe water ofcrystallization).
For this type of ideal paramagnetic substance the hard work is alreadydone. (A
treatment of ferromagnetism appears later, in Chapter 11.) We may use the results of
section 3.1.1 todetermine the contribution ofthespins to thethermalproperties ofthe
solid. The energydifferenceεbetween the two states is equal to 2μB,and hence the
characteristic temperatureθequals 2 μB/kkkB. The numberNis the number of spin-^12
particlesinthesolid(i.e. muchless than the number ofatomsinthe case ofCMN).
Note the following:
1 .Thethermalproperties ofthespins,dependent onlyonθ/T,are thusfor agiven
system universal functions ofB/T.We shall use this result forSin particular in
the next section.
2 .Typical values ofθin strongmagnetic fields are a few degrees K for an electronic
spin system, but a few mK for a nuclear spin system. As we shall see later in the
book,there arefew other thermalcontributions at suchlow temperatures, so that
the spins in fact form the major thermal bath in the solid.
- We are restricting our discussion to a spin-^12 solid, one which has just two spin
states. The treatmentfor ahigher spinfollows verysimilarlines, withqualitatively
identical results. The major difference is that, since a spinIhas(2I+1)possible
states, the high temperature entropy isNkkkBln( 2 I+ 1 )rather thanNkkkBln 2. The
generalform ofallthefunctionsissimilar, withagainacharacteristic temperature
θwhich relates to the Zeeman splitting. - Theheat capacity (Fig. 3.3)is worthyofnote. Heat capacities ofthis sort are called
Schottkyanomalies. The word‘anomalies’is usedbecause ofthe potentialupset
to an unsuspecting experimenter. As the range of measurement is reduced below
room temperature, thelargelattice contribution to a typicalsolid heat capacity
reduces rapidly(often asT^3 ) towards zero. Imagine the consternation whena
furtherreductionofTsees anincreasingT−^2 contribution starting to come in!
However, thisis preciselywhathappens as spinsinthe systembecome orderedas
θis reached.
3 .1.3 Coolingby adiabatic demagnetization
The paramagnetic solid can form the basis of a method of cooling, and one which is of
greatimportanceinphysics, sinceitistheonlyworkablemethodinthesub-millikelvin
region.