196 AppendixE
that theradiation maybedescribedasaphotongas. (a) Explaintheoriginofthe
T^4 factor. (b) Compare your theoretical value ofσwith the experimental value,
σ= 5. 67 × 10 −^8 Wm−^2 K−^4.
4 Avacuuminsulatedflaskhas theshapeofasphere (witha narrow neck!) ofvolume
5 × 10 −^3 m^3 .The flask is filled with liquid nitrogen (boiling point 77 K, latent heat
1 .7 0 × 108 Jm−^3 ). (a) Estimate theholdtime oftheflask(important practicallyif
theflaskis usedas a ‘coldtrap’), assumingthat the outer surface oftheflaskisat
3 00 K and that both surfaces behave as perfect black bodies. (b) How in practice
is sucha(Thermos)flaskmade more efficient? Explain.
5 Wien’s law for black-bodyradiation (used for thermometrybycolour) states that
λmaxT=constant, whereλmaxrefers to the maximum inu(λ),the energy den-
sityper unit wavelength.The experimentalvalue ofWien’s constantisabout
0.002 9 K m. Calculate the theoretical value of the constant as follows:
(a) Re-express Planck’s law (9.12) in terms of wavelengthλinstead of frequency
ν.Explainwhythe peaksinu(ν)andinu(λ)do not occur at the samephoton
states.
(b) Usingtheusualtypeofdimensionlessvariabley=ch/λkkkBTobtainanequation
forymax,they-value correspondingtoλmax.
(c) Solve the equation numerically or graphically. (Hint:thesolutionoccursaty
just less than 5 .) Hence compute Wien’s constant.
(d)Show that the maximumin (9.12) occurs atay-valuejustless than 3, as
mentioned in the text.
6 Startingfrom (9.14)for thethermalenergy ofasolid: (a) Show that athightem-
peratures, theclassicalresultCV= 3 NkkkBis recovered.(Hint:onlysmallvalues
ofyare involved, so expand the exponential.) (b) Show that the DebyeT^3 follows
atlow temperatures. (c) At evenlower temperatures, (9.14) will becomeinvalid
becauseitisbasedon thedensityofstates approximation. Estimate the temperature
below which (9.14) will severely overestimateU(and henceCV)for a sample of
size, (i) 1 cm, and(ii)1μm. The speedofsoundinatypicalsolid is 4000 m s−^1.
Chapter 1 0
1 (a) Verifythe derivations of (10.4) and (10.5).
(b) (Harder) Find the analogous expression forSfor ‘intermediate statistics’(see
question 2 for Chapter 5), and check that it has the correct limits.
2 Consider the formation of case 2 vacancies(often called Frenkel defects)as intro-
duced in section 10.3. The defect occurs when an atom leaves a normal site and
moves to aninterstitialsite. Suppose thecrystalhasNnormalsites andN 1 inter-
stitial sites (these numbers will be the same within a small multiple which depends
on the crystal structure). The energy of formation of the defect is. Following the
methodofsection 10.3, show that the numbernofvacancies at temperatureTis
n=(NN 1 )^1 /^2 exp(−/ 2 kkkBT)