Statistical Physics, Second Revised and Enlarged Edition

36 Two examples

This resultmayreadily be verifiedbyexpandingthe exponentialsin (3.9),bearing

in mind that the exponents are small. Hence in this limit the magnetic susceptibility,

essentiallyM/B,is proportionalto 1/T.ThisisCurie’slaw.

Thereforewehaveadirect 1 /Tthermometer. For an electronic paramagnetlike CMN,

the magnetization is usuallymeasured directlyfrom the total magnetization of the

solid. The usual method is to measure the mutual inductance between two coils,

woundover the CMN,givinga constant reading(whichcanbesubtracted)anda

componentproportional to 1/T. This maybe used as a thermometer from 10 K or

higher to around 1 mK, where interactions take over and spoil the simple story. Below

about 20 mK,however, a nuclear system (often Pt nuclei)becomes useful. However,

the value ofμis so small that the direct method is impractical – one would measure

the effect of minute magnetic impurities with electronic spins. Therefore a resonance

methodis used(pulsedNMR) whichsingles out the particular energy levelsplitting

of interest bymeans of radio frequencyphotons of the correct frequency. The strength

of the NMR signal is directly proportional to 1/T.

3 .2 Localized harmonic oscillators

Asecond example which can be solved with little computational difficulty is that
ofan assemblyofNlocalizedharmonic oscillators. Suppose the oscillators are
eachfree to movein onedimension onlyandthat theyeachhave the same clas-
sical frequencyν.They are therefore identical localized particles, and the results
ofChapter 2 maybe usedtodescribethe equilibrium properties ofthe assemblyat
temperatureT.
First we need the result from quantum mechanics for the states of one particle.
For a simple harmonic oscillator there is an infinite number of possible states (((j=
0, 1, 2, 3,...)whose energies aregiven by

εεj=

(

j+

1

2

)

hv (3.11)

(Mostbooks on quantum mechanics, e.g.Chapter 4 ofDavies andBetts’book,

Quantum Mechanics,in this series,include a discussion of the states of a harmonic

oscillator,if you are notfamiliar withthe problem.)

3 .2.1 The thermal properties

There is an infinite number of statesgiven by(3.11), and therefore an infinite number

of terms in the partition functionZ.Nevertheless the even spacing of the energy levels