Properties ofan ideal Fermi–Dirac gas 87Three remarksabout the Fermienergy follow.- We can now quantify the need for FD statistics in the systems mentioned earlier, by
substitutinginto (8.5). It is useful in so doingto calculate the ‘Fermi temperature
TTTF’, defined by:
μ( 0 )=kkkBTTTFFor example, if we are interested in electrons, (8. 5 ) can be writtenTTTF=4. 1 × 10 −^15 (N/V)^2 /^3For electrons at metallic densities (say6× 1023 electronsinamolarvolume of
9 × 10 −^6 m^3 ) this gives a Fermi temperature of about 70 000 K. Therefore, at
ambient temperatures we are alwaysinthelimitTTTTF,so that the systemis
dominatedbyFD statistics (see againFig. 8.1). For electronsin a semiconductor,
the Fermi temperature will be lower and there are some situations in which MB
statistics are adequate. Finally we may note that thedense-free electrons existing
inatypical white-dwarf star have Fermi temperatures of around 1 09 – 1010 K. Since
the internal temperature of such a star is (only!) a mere 1 07 K, this is again a highly
degenerate fermiongas. We return to this topic in Chapter 15.
2 .The calculation ofμ(0) can be directlyrelated to the degeneracyparameterA
of (8.1). In fact substituting (8. 5 ), together with the definition above ofTTTF,we
obtain
A=( 8 / 3√
π)(√√
TTTF/T)^3 /^2 ≈ 1 .50(TTTF/T)^3 /^2
This result explicitly demonstrates that the degeneracy conditionA>1iseffec-
tively equivalent toT<TTTF.AndcorrespondinglytheclassicallimitA 1 isthe
same asTTTTF.
3. There is a temperature variation toμ. In fact in the classical regionμdiverges
to minusinfinity as−TlnT,a result derivable from the methods of Chapter 6.
However,inthedegenerateregion(TTTTF)thevariationissmall,andcanoftenbe
neglected. It is still determined by (8.4), the number restriction. The temperature
enters onlythroughthe Fermifunctionf(ε). Butinthedegenerate region (see
again curves 1 and2ofFig. 8.1), the variation off(ε)isonlyasubtleblurringin
an energy range of orderkkkBTaroundμ.In fact ifg(ε)were a constant independent
ofε,there would bevirtuallyno variation ofμwithTinthedegenerate region. As
Tis raised, the fillingof states above the Fermi energywould be compensated by
the emptying of those below. It is only wheng(ε)varies withεthat a small shift in
μoccurs, to allowfor theimbalancein numbers ofstates above andbelowμ.For
atwo-dimensional electrongas (exercise!),g(ε)isaconstantandμ(T)=μ(0)
to a high degree of accuracy. For the usual three-dimensional gas we have a rising