Statistical Physics, Second Revised and Enlarged Edition

32 Two examples

simplelaw

TTT 2 =T 1 (B 2 /B 1 ) (3.7)

Thisisanice exampleoftheimportance ofentropy.Inthefirstlegwe reduce the
entropyof the solid at a high temperature byan amountS,say,byextractinga
quantityT 1 Sofheat. Theadiabaticlegthentransferstheentropyreductiontoalower
temperature, andthespins can absorbheat ofatleastTTT 2 Sfrom their surroundings
before warmingup again to the startingpoint. (More accuratelythe amount of cooling
available from the spins equals the area between the entropy curve and theS-axis
between pointsCandA,i.e.

∫A

C

∫∫

TdS).

Some experimental details We shall refer to two implementations of the cooling
method. One is the use of CMN to cool to about 2 mK, and the second is the use of
Cu nucleito coolinto theμKregion.
The necessarystartingconditions are easyto work out. To obtain a significant
entropy reductionSunder the starting conditions, one requiresμB 1 to be of the
same order ofmagnitudeaskkkBT 1 .In the case ofelectronic moments (CMN), this
requirement is comfortablyattained with a precoolingtemperatureT 1 ofaround1K
and an applied fieldB 1 of 1T. For nuclear moments (Cu) the starting conditions are
more stringentbyafactor ofover 1000. Theyare marginallymetinmodern techniques
using almost the highest available magnetic field – typically a 7T superconducting
magnet, anda precooling refrigerator operating at around10 mK. Even so onlyafew
per cent ofthe copper spin entropyis removed.
The essential ingredients are indicated in Fig. 3.6. For CMN the precooling stage is
usually a pumpedhelium cryostat (reaching about 1 K with^4 He, or 0.3 K with^3 He).

Precooling stage

(temperature TTT 1 )

Superconducting

solenoid

(Maximum magnetic

field B 1 )

Thermal link

Coolant

Heat switch

Vacuum

insulation

(^4) He bath
Fig. 3. 6 A schematic apparatus for coolingbyadiabatic demagnetization.