Direct Energy, 2018a

13 THOMAS FERMI ANALYSIS 301
the wave vector | −→ k | of the electrons in the atom.At T =0K, electrons
occupy the lowest allowed energy states.Energy states are occupied up to
some highest occupied state called the Fermi energy E f .While electrical
engineers use the term Fermi energy, chemists sometimes use the term
chemical potential μ chem . The lowest energy states, are occupied while the
higher ones are empty.Similarly, wave vectors are occupied up to some
highest occupied wave vector called the Fermi wave vector k f .
| −→ k | =
{
lled state
˜r k f
(13.36)
The Fermi energy and the Fermi wave vector are related by
E f = 2 kf
2
2m . (13.37)
We use the idea of reciprocal space to write an expression for the kinetic
energy of the electrons per unit volume [136, p.49].The kinetic energy
due to any one electron as a function of position in reciprocal space is
given by Eq.13.35.Note that at each value of | −→ k | =˜r, the electron has
a dierent kinetic energy.To nd the kinetic energy per unit volume due
to all electrons, we integrate over all | −→ k | =˜r in spherical coordinates that
are occupied by electrons, and then we divide by the volume occupied in
−→ k space.
E kinetic e
V
1
=
vol.occupied in k space
· ´lled
( Ekinetic e
) ( )
e
k levels −
e − volume
d (vol.all k space)
(13.38)
The number of electrons per unit volume is given by
( ) e
−
volume
= −ρ ch
. (13.39)
q
The volume occupied in reciprocal space is 4 3 πk3 f , the volume of a sphere
of radius k f .
E kinetic e
V
= 1
4
3 πk3 f
ˆ
( 2˜r 2
·
lled k levels 2m
)(
−ρch
A dierential element of the volume is expressed as
q
)
d (vol.all k space)
(13.40)
d 3 ∣ ∣∣ −→ k
∣ ∣∣ =˜r 2 sin ˜θ d˜r d˜θ d˜φ. (13.41)