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