Statistical Physics, Second Revised and Enlarged Edition

188 AppendixC

Hence

4 III 02 =

∫∞

−∞

∫∫ ∫∞

−∞

∫∫

exp[−b(x^2 +y^2 )]dxdy

=

∫∞

0

∫∫

exp(−br^2 ) 2 πrdr

= 2 πI 1

=π/b

Thus

III 0 =

1

2

(π/b)^1 /^2 .(C.4)

The three equations (C.2), (C.3) and (C.4) between them enable one to evaluate

anyofthe requiredintegrals.

2 FERMI–DIRACINTEGRALS

In Chapter 8, we gave several results for the properties of an ideal FD gas in the
limitTTTTF.Thecalculations require the (approximate) evaluation ofintegralsof
the form:

I=

∫∞

0

∫∫

[dF(ε)/dε]]]f(ε)dε (C.5)

The functionF(ε)is chosen to suit the property required, andf(ε)is the FD
distribution, equation (8.2). Integration byparts of equation (C.5)gives

I=−F( 0 )−

∫∞

0

∫∫

F(ε)[ddf(ε)/d(ε)]dε (C. 6 )

where we have usedf( 0 )=1andf(∞)=0. Usually one can choose the (‘user-
defined’)function so thatF( 0 )=0, so we shall ignore the first term of equation (C. 6 ).
Thefunction (−ddf/dε)isaninterestingoneinthelimitkkkBTμ.Itis zero except
within aboutkkkBTof the Fermi energy, and in fact behaves like a ‘delta-function’with
anonzero width.(AtT=0itbecomesidenticalto thedeltafunction.) Therefore
equation (C. 6 ) is evaluated byexpandingthe functionF(ε)asaTaylor series about
μ, since only its properties close toε=μare relevant. The result is

I=F(μ)+(π^2 / 6 )(kkkBT)^2 F′′(μ)+··· (C.7)

Note that:(i)the first term, the value ofFat the Fermi level, is the only one to survive
atT=0; thatisthedelta-function property, (ii)thereisnofirstderivative term since
(−ddf/dε)issymmetricalabout the Fermilevel;andhence no termlinearinT,and