Direct Energy, 2018a

300 13.3 Derivation of the Lagrangian
electrons varies as a function of the spatial frequency of charges in a crystal
or other material, and this is the idea represented by functions of wave
vector such as ρ ch
( −→k
). We are trying to solve for charge density ρ ch (r).
We expect that electrons are more likely to be found at certain distances
r from the center of the atom than at other distances. However, there is
no pattern to the charge density as a function of wave vector, ρ ch
( −→k
).
Assume that ρ ch is roughly constant with respect to | −→ k | up to some level.
With some more work, this assumption will allow us to solve for charge
density ρ ch (r).
The kinetic energy of a single electron is given by
E kinetic e
= 1 e − 2 m|−→ v | 2 (13.31)
where m is the mass of the electron. We can write this energy in terms
of momentum, −→ M = m −→ v . (Note that momentum −→ M and generalized momentum
M are dierent and have dierent units.)
E kinetic e
= |−→ M| 2
(13.32)
e − 2m
We do not know how the energy varies as a function of position r. Instead,
we can write the energy as a function of the crystal momentum −→ M crystal
or the wave vector −→ k , and we know something about the variation of these
quantities. Crystal momentum is equal to the wave vector scaled by the
Planckconstant.
−→
M crystal = −→ k (13.33)
It has the units of momentum kg·m
s , and it was introduced in Sec. 6.4.2.
The kinetic energy of one electron as a function of the crystal momentum
is given by
( −→M
) 2 (
E crystal | −→ ) 2
k |
kinetic e
=
=
e − 2m 2m . (13.34)
A vector in reciprocal space is represented Eq. 13.30, and Eq. 13.34 can
be simplied because we are assuming spherical symmetry ˜θ = ˜φ =0.
The magnitude of the wave vector becomes | −→ k | =˜r, and we can write the
energy as
E kinetic e
= 2˜r 2
e − 2m . (13.35)
Just as each electron has its own momentum m| −→ v |, each electron has its
own crystal momentum | −→ k |. However, we know some information about