Statistical Physics, Second Revised and Enlarged Edition

34 Two examples

interactionsleft, evenifthese are onlymagneticdipole–dipoleforcesbetween the
Nspins. As remarked earlier, although our treatment neglects interactions between
particlessomeinteractionsareessential,otherwisethermalequilibriumcouldneverbe
reached:hence ‘weakly interacting’. We mayturn offthe externallyappliedmagnetic
field(B 0 say), but some residual interactions must remain. If we characterize these
byaneffectivefield (Bint), then theBentering equations suchas (3.7) should bethe
appropriate sum ofB 0 andBint.Bearinginmindthe randomdirection ofBint,the
correct expression is

B=(B^20 +B^2 int)^1 /^2 (3.8)

The energy levels obtained from (3.8) are plotted in Fig. 3.7.
The upshot ofallthisonthelowestfinaltemperatureis:

1 .Theabsolute zerois unattainable; the temperature reachedin zerofinalapplied
fieldisT 1 (Bint/B 1 ),a non-zero number.

2. Another way ofexpressing this resultistoobserve that thespins willorderin zero
   appliedfieldat arounda temperatureTTTint=μBint/kkkB.Referringback to Fig. 3.5,
   the entropy does not stay atNkkkBln2asTis lowered past this value, but falls to
   zero to achieveS= 0 atT=0(yet another statement ofthethirdlaw).


3. Since the coolingmethod is useful onlyaroundμB/kkkBT∼ 1 ,the lowest attainable
   practical temperature is aroundTTTint. For CMN this is about 1 mK, a very low value
   for an electronicspinsystem;hence theimportance ofthis particular salt. For Cu
   spins in thepure metal,TTTintis less than 0. 1 μK,one of the reasons for its choice.

3 .1.4 Magnetization and thermometry

Beforeleavingthe topicofthespin-^12 solidwediscussits magnetic properties. These
maybe readily derivedfrom theBoltzmanndistribution, andtheygive a convenient
method for measuring the low temperatures reached by adiabatic demagnetization.

Applied field B 0

Bint

2 Bint 0

e=–B

e e = +B

Fig. 3. 7 The energies ofaspin-^12 solidas afunction ofappliedmagneticfieldB 0.