Direct Energy, 2018a

292 13.1 Introduction
• Describe the second form of energy E kinetic e in terms of path V . The
resulting energy density is
E kinetic e
V
= c 0 V 5/2 (13.4)
where c 0 is a constant. This step will require the idea of reciprocal
space.
• Write down the Hamiltonian H ( ) ( )
r, V, dV
dr and Lagrangian L r, V,
dV
dr .
• Set up the Euler-Lagrange equation.
( )
∂L
∂V − −→ ∂L
∇·
∂ ( )
dV
dr
â r =0 (13.5)
• Solve the Euler-Lagrange equation for the equation of motion. The
result is
5
2 c 0V 3/2 − ɛ∇ 2 V =0. (13.6)
• Change variables to clean up the equation of motion. The resulting
equation is called the Thomas Fermi equation.
d 2 y
dt 2 = t−1/2 y 3/2 (13.7)
• Voltage and charge density are algebraically related to the quantity
y in the equation above.
To attempt to nd charge density and voltage as a function of position
r from the center of the atom, we will have to make some rather drastic
assumptions. This analysis follows works of Thomas [173] and Fermi [174]
which were originally completed around 1927. This derivation is discussed
by numerous other authors as well [6] [46] [136] [175]. Because of the severe
assumptions made below, the results will not be very accurate. However,
more accurate numerical calculations are based on improved versions of the
techniques established by Thomas and Fermi. We are discussing the most
simplied version of the derivation, but this is the basis of more accurate
approaches.