VbvAstE-001

where ˜ λ C =
m ec
is the Compton radius.
At introducing of the new variable
we can transform the Laplacian:
ϕ(r) = χ(r)
r , (10.15)
∆ϕ(r) = 1 r
d 2 χ(r)
dr 2 . (10.16)
As (Eq.(10.12))
χ(r) = 3 m ec 2
Yβr , (10.17)
8 e
the differential equation can be rewritten:
where
L =
d 2 χ(r)
dr 2
= χ(r)
L 2 , (10.18)
( ) 1/2 9π Yβ
˜ λ
32 αξ 3 C , (10.19)
α = 1 is the fine structure constant.
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With taking in to account the boundary condition, this differential equation has
the solution:
χ(r) = C · exp
(
− r L
)
. (10.20)
Thus, the equation of equilibrium of the electron gas inside a cell (Eq.(10.10)) obtains
the form:
Ze
r · e−r/L = 3 8 mec2 βY . (10.21)
10.2 The Thomas-Fermi screening
Let us consider the case when an ion is placed at the center of a cell, the external shells
don’t permit the plasma electron to approach to the nucleus on the distances much
smaller than the Bohr radius. The electron moving is non-relativistic in this case. At
that ξ → 0, the kinetic energy of the electron
E kin = 3 8 mec2 ξY → 3 EF , (10.22)
5
and the screening length
√
EF
L → . (10.23)
6πe 2 n e
Thus, we get the Thomas-Fermi screening in the case of the non-relativistic motion of
an electron.
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