8 Fermi–Dirac gases
We return now to the main stream of the book, and to the basic statistical properties
of idealgases as introduced in Chapter 5. Of the three types of statistics we have so
far discussed only the classical limit, corresponding to Maxwell–Boltzmann gases.
Inthe next two chapters so-called‘quantum statistics’isdiscussed,thatis statistics
where the antisymmetric (Fermi–Dirac) or thesymmetric (Bose–Einstein) nature of
the wavefunction plays a significant role.
Theimportance ofthe FD or BE natureis most readily seen when we consider
the state ofthegas atT =0. At theother extreme, wehave alreadynotedthat
MB statistics is a high-temperature approximation, corresponding to the degeneracy
parameterA(defined by (6.6a)) being1. AsTapproaches zero, andthereforeA
becomesinfinitely large, the quantumlimitisobvious. For BE statistics,inwhich
any number of particles can occupy a state, theT=0 state is for allNparticles to
occupy the groundstate. Thisgivesasituation oflowest energy andofzero entropy,
i.e. ofperfect order. In contrast to the‘friendly’bosons, the ‘unfriendly’fermions
operate an exclusion principle. Therefore theT=0 state for FD statistics is with the
Nparticles neatlyandseparately packedinto theNstates oflowest energy, giving a
large zero-point energy,but again zero entropy(because ofthelackofambiguityin
the arrangement).
The conditionA 1 for validityofthe MB approximation contains otherindi-
cations about whenquantum statistics should be used. First, the value ofTneeded
tomakeA= 1 gives a ‘scale temperature’ below which quantum effects will dom-
inate,givingestimates ofthecharacteristic temperatures (or equivalent energies) to
be introduced in the next two chapters. Second, if we considerkkkBTasathermal
energy scale, we can workoutathermalmomentum scaleandhence (usingp=h/λ)
a‘thermalde Broglie wavelength’,λ,tocharacterise thequantumproperties ofa
typical gas molecule. A little rearrangement shows thatA 1 thentranslatesinto
λ(V/N)^1 /^3 ,whichisthe averagedistancebetween gas particles. So we can see
that quantum effectsbecomeimportant when thede Broglie wavelengthsofnearby
particles overlap, a pretty idea.
Inthischapter we nowdiscuss the properties ofFDidealgases. The BE caseis
treatedinChapter 9.
83