Statistical Physics, Second Revised and Enlarged Edition

Aspin-^12 solid 31

T

TTT 2 TTT 1

(Final temperature) (Precooling temperature)

S C

NkkkB ln 2

Low field,B 2

High field,

B 1

B

A

Fig. 3. 5 Coolingbyadiabatic magnetization. The twoS−Tcurvesgive the behaviour in a highmag-
netic field,B 1 ,and a low field,B 2 , AB represents the precoolingprocess (removal of entropyatahigh
temperature), and BC the adiabatic demagnetization to the final temperatureTTT 2.

The theory of the method is easily followed from anS−Tdiagram as shown
in Fig. 3.5. The figure shows two entropy curves, one in the highest available field
(B 1 ),andoneinalowfield(B 2 ). As notedabove,Sisafunction onlyofB/T,
so the only difference between the two curves is the scaling along theT-axis.The
cooling methoduses twodistinct steps. Firstlythesolid is magnetizedat thelowest
available precoolingtemperature,T 1 .This is illustrated in Fig. 3.5 bythe step AB,
which represents an isothermal magnetization. In the second step, the system is
then thermally isolatedandthemagneticfield graduallyreduced,givingan adiabatic
reversible (i.e. isentropic) process. In other words the systemgoes from B to C on
the diagram, and its temperature reduces fromT 1 toTTT 2 .(Actually the spin-^12 solid
is a verysuitable workingsubstanceforarefrigerator – the requirements are thatS
should varywithTand with another controlparameter. Here theparameter is the
applied magnetic field; in a more common fluid refrigerator it is the pressure.)
Itisinterestingto consider the two processes microscopically.Intheisothermal
magnetization leg(AB), the spins tend to align in the strongmagnetic field sinceμBis
comparable tokkkBT.Inthis process, the population ofthelower state (energy−μB,
i.e.withμparalleltoB)grows at the expense ofthe upper state (energy +μB).
The population becomes (somewhat) ordered, and the entropy decreases. During this
processheat ofmagnetizationis evolvedfrom the system, andinfact the precooling
refrigeratorhas tobe veryactive to maintainthe temperature atT 1 .In the second
adiabatic step (BC), the situation is different. The spins are no longer in contact with
aheatbath,sothe occupation numbers cannot change. AsdiscussedinChapter 2,S
constantmeanst({nnj})constant whichmeans{nnj}constant.Hence atCthe occupation
numbersn0,1are just the same as they were atB. However, the energy level spacing
has reduced(from 2μB 1 to 2μB 2 )andso the temperature musthave alsodecreased
in the same ratio, in order to keepμB/kkkBTconstant. Therefore the coolingobeys the