Problems and Solutions on Thermodynamics and Statistical Mechanics

(Ann) #1
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.

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