Summary 41
Forbothsubstances, the one-particle energystates are well describedbyquantum
mechanics.
A spin-^12 particlehasjust two quantum energylevels,dependent on any applied
magneticfield(Zeeman effect).
A harmonic oscillator has an infinite number of equally-spaced energy states.
The partitionfunctionZ(T)is readilyfoundinbothcases. It canbe expressed
as a zero-pointfactor (whichdependsonthe energyzerobut not on temperature)
multiplied by a thermal factor (which is independent of the energy zero but does
dependon temperature).
The thermal properties are dependent on the ratio of two energyscales. One derives
from the one-particle energy structure, the other iskkkBT,the ‘thermal energy scale’.
The entropyrises from zero at low temperature to approximatelyNkkkBlnGathigh
temperature, whereGis the number of one-particle states accessed at the high
temperature. For the spin-^12 solidG= 2 , independent of temperature; whereas for
the oscillatorsGisproportionaltoT.
Cooling byadiabatic demagnetisation of a paramagnetic material is readily
understood from the properties of a spin-^12 solid.
Harmonic oscillators provideamodel(theEinsteinmodel)for understandingsome
aspects of the thermal properties of solids, even though the atoms of a solid are
hardly ‘weakly-interacting’. We shallreturn to this questionlaterinChapter 9.