Direct Energy, 2018a

13 THOMAS FERMI ANALYSIS 303
We know something about the wave vectors of lled states in reciprocal
space. At T =0K, the lowest states are lled, and all others are empty,
and they are lled up to a radius of k f . The volume of a sphere of radius
k f is given by 4 3 πk3 f , and this represents the number of lled k states per
volume of reciprocal space. We can therefore simplify the expression above.
ρ ch = −2q · 4
3 πk3 f ·
1
(2π) 3 (13.50)
ρ ch = −q
3π 2 k3 f (13.51)
( ) −3π
2 1/3
k f = ρ ch (13.52)
q
We want to write E kinetic e
as a function of generalized path V .Wecan
V
now achieve this task by combining Eqs. 13.47 and 13.52.
E kinetic e
V
( )
= −32 −3π
2 2/3
10mq ρ ch ρ ch (13.53)
q
E kinetic e
V
= −32
10mq
( −3π
2
q
) 2/3
ρ 5/3
ch
(13.54)
Electrical energy is the product of charge and voltage. More specically,
from Eq. 2.8, it is given by
E = 1 QV. (13.55)
2
Electrical energy density is then given by
E
V = 1 2 ρ chV. (13.56)
Use Eq. 13.56 to relate ρ ch and V .
E kinetic e
V
= 1 2 ρ chV = −32
10mq
( −3π
2
q
) 2/3
ρ 5/3
ch
(13.57)
We have now related the generalized path and the generalized potential.
ρ ch =
V = −32
5mq
(
−5mq
3 2 ·
( −3π
2
q
) 2/3
ρ 2/3
ch
(13.58)
( ) )
−3π
2 −2/3 3/2
V 3/2 (13.59)
q