194 AppendixE
(b)Anegative temperatureinaspinsystem can neverthelessbe reached;indeedit
isaprerequisite for a laser or maser. Look upand write a brief account of the
methods used.
(c) Explainwhynegative temperatures cannotbe contemplatedinthe assemblyof
harmonic oscillators.
4 The energylevelsofalocalizedparticle are 0,ε,2ε.Themiddlelevelisdoubly
degenerate (i.e. two quantum stateshave this same energy)andtheotherlevels are
singly degenerate. Write down and simplify the partition function. Hence compare
thethermalproperties(U,C,S) an assemblyofthese particles withthe properties
of the spin-^12 solid.
5 Explain why iron is not used as the coolant for adiabatic demagnetization.
Chapter 4
1 Consider waves trapped in, (a) a one-dimensional, and (b) a two-dimensional box.
In eachcase,deriveg(k)δkandcompare your results with(4.4). Findthe energy
dependence ofthedensityofstatesinεfor agas ofparticles ofmassM, comparing
with(4.9).
Chapter 5
1 Repeat question 1 of Chapter 1 for a model assemblyof four particles with the
same energy states and withU= 6 εas before, for the cases when the particles are,
(a)gaseousbosons, and(b)gaseousfermions. Compare the resultsfor thethree
cases.
2 (Not an easy problem.) As a game – which has no physical basis – work out the
statistics ofagas obeying‘intermediate statistics’. The one-particle states ofthe
gas maybe occupied by0, 1, 2,...p−1,pparticles, so that the two physical cases
arep=1(FD)andpinfinite (BE). Obtain expressions fort{(ni)}andfor the
thermaldistribution, valid for anyspecifiedvalue ofp.Checkthat thetwolimits
give the correct results. (Hint:If you get stuck, see if your library hasMolecular
Physics,vol. 5, p. 525 (1962) in which all is revealed!)
Chapter 6
1 Check the derivation of (6.8) from (6.7), including the value of the constantC.
2 UsingtheintegralsofAppendixC,verifythe statedexpressionsfor the mean and
the rms speeds in section 6.2.
3 What is the distribution function invvxfor an MB gas? Thisisdefinedsuchthat
n(vvx)dvvxisthe number ofparticles withx-component ofvelocitybetweenvvxand
vvx+dvvx.
4 In an experiment to studytheMBdistribution, gas molecules are observedafter
theyhave escapedthrougha small holeinthe oven. Show that the rms speedof
the escaped molecules equals 2vT(i.e. higher than the rms speed inside the oven).
(Hint:Do question 3first.)
5 Find the constant in equation ( 6 .10).