Problems and Solutions on Thermodynamics and Statistical Mechanics

Statiaticd Physics 315

where a = po B/kT.

(b) The entropy is

= Nk[ln(e"+e-")-a:tanha:].

(c) The average energy is

• a
  ap

U = --lnz = -NpoBtanh

(d) The average magnetic moment is

• M = Npo tanh (F) PO B.

For a single ion, we have

For the whole system, we have

(AM)2 = N(A~I)~

AM=*

cosh (g).

(e) We see from (b) that the entropy (T is a function of poB/kT. If

the entropy of the spin states is maintained constant, i.e., poB/lcT is kept

constant, then the temperature of the spin states can be lowered by re-

ducing the magnetic field adiabatically. In the crystal, the decrease in the

temperature of the spin states can result in a decrease of the temperature

of the crystal vibrations by 'heat transfer". Therefore, the whole crystal is

cooled by an "adiabatic reduction of the magnetic field". From

we have Tf = K.