1549380323-Statistical Mechanics Theory and Molecular Simulation

430 Quantum ideal gases

=

V

π^3

∫∞

0

1

s^5

[

sin(kFs)−

s

lF

cos(kFs)

] 2

, (11.5.68)

wherekF= 2πnF/L. If we now introduce the change of variablesx=kFs, we find
that the expression separates into a density-dependent part and a purely numerical
factor in the form of an integral:

Ex=−

V

π^3

k^4 F

∫∞

0

dx

(sinx−xcosx)^2

x^5

. (11.5.69)

Even without performing the remaining integral overx, we can see thatEx ∼k^4 F
and, therefore,Ex∼ρ^4 /^3. However, the integral turns out to be straightforward to
perform, despite its foreboding appearance. The trick (Parr andYang, 1989) is to let
y= sinx/x. Then, it can be shown that

dy

dx

=−

sinx−xcosx

x^2

d^2 y

dx^2

=−

2

x

dy

dx

−y. (11.5.70)

Finally,
∫∞

0

dx

(sinx−xcosx)^2

x^5

=

∫∞

0

dx

(sinx−xcosx)

x^2

(sinx−xcosx)

x^3

=

∫∞

0

dx

(

dy

dx

)(

1

x

dy

dx

)

=−

1

2

∫∞

0

dx

(

d^2 y

dx^2

+y

)(

dy

dx

)

=−

1

4

∫∞

0

dx

d

dx

[

y^2 +

(

dy

dx

) 2 ]

=−

1

4

[

y^2 +

(

dy

dx

) 2 ]∣∣

∣

∣

∣

∞

0

. (11.5.71)

Bothyand dy/dxvanish atx=∞. In addition, by L’Hˆopital’s rule, dy/dxvanishes
atx= 0. Thus, only the sinx/xterm does not vanish atx= 0, and the result of the
integral is simply 1/4. Using the definitions ofkFandnF, we ultimately find that

Ex=CxV ρ^4 /^3 , (11.5.72)

which is the desired result.

11.5.5 Thermodynamics at low temperature

At low but finite temperature, the Fermi–Dirac distribution appearsas the dashed line
in Fig. 11.1, which shows that small excitations above the Fermi surface are possible