v
! Á ∇f i=e
v
! , z
Æ e
v
! Â B
!
Mc
Á ∇ v
! f i=e
v
! , z
Ç
e
M
dφðzÞ
dz
e
!
z Á ∇ v
! f i=e
v
! , z
¼ 0,
ð4:6Þ
where f i=e v
! , z
is the ion/electron velocity distribution function and e
!
z is the unit
vector along the z-coordinate. A similar equation can be used for the electrons.
Incorporating the solutions of the ion and electron kinetic equations into the Poisson
equation, we can find all the necessary information. However, in practice, to find the
constraint similar to Eqs. (4.4) and (4.5), we do not need to have the full solution of
these equations. As we have found in the course of the derivation of Eq. (4.4), the
Bohm-Chodura constraint comes from the asymptotic behavior of the solution of the
Poisson equation, recall Eq. (4.3), at z ! À1 where the electrostatic potential is
low. Therefore, instead of solving the complex nonlinear system of the kinetic and
Poisson equations, we can consider their linearized versions. As a result, we come to
the problem similar to that of finding the dielectric constant of the plasma, ε ω, k
!
.
Usually, it is determined as a function of the frequency, ω, and the wavenumber, k
!
,
which characterize the plasma wave. However, in our case, we are looking for the
conditions of the formation of a stationary evanescent electrostatic potential that
links the sheath region with the plasma interior. Therefore, we should consider the
plasma dielectric constant with zero frequency and the wavenumber having the
imaginary part ensuring that φ(z ! À1) ! 0.
Following [4] we take f i=e v
!
, z
¼ f
À1
ð
Þ
i=e
v
!
þ f
1
ð Þ
i=e v
!
, z
, where f
1
ð Þ
i
v
!
, z
is
a small correction. Moreover, by analogy with Eq. (4.3), we will assume and justify
a posteriori that φ z
ð Þ ¼ φ 0 exp z b k z
and f
1
ð Þ
i=e v
!
, z
¼ e f
1
ð Þ
i=e v
!
exp z b k z
, where
b k z is an adjustable parameter playing the role of the wavenumber of the evanescent
wave, which can be found from the solution of the Poisson equation. For a “smooth”
transition from the sheath to z ! À1 we need Re b k z
> 0.
We notice that f
À1
ð
Þ
i=e
v
!
, being the solution of the stationary kinetic equation
with φ(z) ¼ 0, should be expressed in terms of the integrals of motion, which gives:
f
ðÀ1Þ
i=e ðv
!
Þ F i=e ðv k , ε ⊥ Þ where ε ⊥ ¼ v
! 2
⊥ =2 whereas v
!
⊥ and v k are the velocity
components perpendicular and parallel to the magnetic field. Then, from [4] we have
b k
2
z ¼ À
X
i=e
4πe
2
m i=e
X 1
j¼À1
* J
2
j
À i cosðαÞ
b kzv⊥
ΩB i=e
jΩ B i=e À i sinðαÞ b k z v k
jΩ B i=e
∂F i=e
∂ε ⊥
À i sinðαÞ b k z
∂F i=e
∂v k
!+
, ð4:7Þ
where J j (x) are the Bessel functions and h. . .i ¼
R
(. . .) dε ⊥ dv k . We notice that the
normalization of F i/e (v k , ε ⊥ ) assumes hF i/e (v k , ε ⊥ )i ¼ n sh .
4 Sheath Physics
77
! Á ∇f i=e
v
! , z
Æ e
v
! Â B
!
Mc
Á ∇ v
! f i=e
v
! , z
Ç
e
M
dφðzÞ
dz
e
!
z Á ∇ v
! f i=e
v
! , z
¼ 0,
ð4:6Þ
where f i=e v
! , z
is the ion/electron velocity distribution function and e
!
z is the unit
vector along the z-coordinate. A similar equation can be used for the electrons.
Incorporating the solutions of the ion and electron kinetic equations into the Poisson
equation, we can find all the necessary information. However, in practice, to find the
constraint similar to Eqs. (4.4) and (4.5), we do not need to have the full solution of
these equations. As we have found in the course of the derivation of Eq. (4.4), the
Bohm-Chodura constraint comes from the asymptotic behavior of the solution of the
Poisson equation, recall Eq. (4.3), at z ! À1 where the electrostatic potential is
low. Therefore, instead of solving the complex nonlinear system of the kinetic and
Poisson equations, we can consider their linearized versions. As a result, we come to
the problem similar to that of finding the dielectric constant of the plasma, ε ω, k
!
.
Usually, it is determined as a function of the frequency, ω, and the wavenumber, k
!
,
which characterize the plasma wave. However, in our case, we are looking for the
conditions of the formation of a stationary evanescent electrostatic potential that
links the sheath region with the plasma interior. Therefore, we should consider the
plasma dielectric constant with zero frequency and the wavenumber having the
imaginary part ensuring that φ(z ! À1) ! 0.
Following [4] we take f i=e v
!
, z
¼ f
À1
ð
Þ
i=e
v
!
þ f
1
ð Þ
i=e v
!
, z
, where f
1
ð Þ
i
v
!
, z
is
a small correction. Moreover, by analogy with Eq. (4.3), we will assume and justify
a posteriori that φ z
ð Þ ¼ φ 0 exp z b k z
and f
1
ð Þ
i=e v
!
, z
¼ e f
1
ð Þ
i=e v
!
exp z b k z
, where
b k z is an adjustable parameter playing the role of the wavenumber of the evanescent
wave, which can be found from the solution of the Poisson equation. For a “smooth”
transition from the sheath to z ! À1 we need Re b k z
> 0.
We notice that f
À1
ð
Þ
i=e
v
!
, being the solution of the stationary kinetic equation
with φ(z) ¼ 0, should be expressed in terms of the integrals of motion, which gives:
f
ðÀ1Þ
i=e ðv
!
Þ F i=e ðv k , ε ⊥ Þ where ε ⊥ ¼ v
! 2
⊥ =2 whereas v
!
⊥ and v k are the velocity
components perpendicular and parallel to the magnetic field. Then, from [4] we have
b k
2
z ¼ À
X
i=e
4πe
2
m i=e
X 1
j¼À1
* J
2
j
À i cosðαÞ
b kzv⊥
ΩB i=e
jΩ B i=e À i sinðαÞ b k z v k
jΩ B i=e
∂F i=e
∂ε ⊥
À i sinðαÞ b k z
∂F i=e
∂v k
!+
, ð4:7Þ
where J j (x) are the Bessel functions and h. . .i ¼
R
(. . .) dε ⊥ dv k . We notice that the
normalization of F i/e (v k , ε ⊥ ) assumes hF i/e (v k , ε ⊥ )i ¼ n sh .
4 Sheath Physics
77