0 ¼ À H n
½ δ n
X n max
k¼1, k6 ¼n
n e K
e
ð Þ
n!k þ n e K
ion
ð Þ
n!cont þ
X nÀ1
k¼1
ν
rad
ð Þ
n!k þ K
n
ð Þ
cx n i
(
)
þ
X n max
k>n
ν
rad
ð Þ
k!n H k
½ δ k þ
X n max
k¼1, k6 ¼n
n e K
e
ð Þ
k!n H k
½ δ k þ K
n
ð Þ
cx n i H n
½ ,
ð2:19Þ
and the following equation for the evolution of the function f
H v
!
:
df
H v
!
dt
¼ ÀK
H
ion n e f
H v
!
À K
H
cx n i f
H v
!
À H
½ f i v
!
n
o
,
ð2:20Þ
where the first term on the right-hand side of Eq. (2.20) describes just the effective
ionization rate constant, recall Eq. (2.16), and
K
H
cx ¼ K
1
ð Þ
cx þ
X n max
k¼2
n
À1
i
n e K
e
ð Þ
k!1 þ ν
rad
ð Þ
k!1
δ k H k
½ = H
½
ð
Þ ,
ð2:21Þ
is the effective charge exchange rate constant including the contribution of the
excited states. As we see, the structure of the expression (2.21) is very different
from the effective charge exchange cross-section accounting for the contribution of
excited states K cx ¼
P
n K
n
ð Þ
cx H n
½ = H
½ suggested in Ref. [34].
By adopting Grad expansion of both the neutral and ion distribution functions
(see Ch. 6 for details), it is possible to extend the analysis of the role of excited states
to a very general hydrogen-ion elastic collision operator (including the chargeexchange one). However, it goes beyond our simple demonstration of the possible
extension of the CRM to the evaluation of an impact of the excited states on elastic
collisions and neutral transport. Calculations performed in [37] have shown that in a
contrast to the results from [34], K
H
cx exceeds K
1
ð Þ
cx by only 10–15% and an impact of
the excited states on neutral hydrogen transport is not very important. This is because
of: i) the comparability of the magnitude of K
1
ð Þ
cx to the electron impact excitation rate
constant of the hydrogen atom, and ii) fast transition from excited to the ground state.
However, the conclusion of the importance of electron exchange recombination of
an impurity ion in the course of the interactions with excited hydrogen atoms [34]
largely holds because hydrogen in the ground state does not undergo such a
“resonance” charge exchange with impurity.
The application of the CRM to impurity atoms/ions and molecules results in
equations more complex than Eqs. (2.13) and (2.15). This is because (i) the number
of states, which should be considered within CRM for each individual atom/ion,
increases; (ii) few ionization states of the same kind of atom/ion can exist for given
plasma density and electron temperature; (iii) the population of excited states of
impurity atom/ion in edge plasma can be affected by charge-exchange process
involving hydrogen atoms (e.g. A
Z + 1 + H ! A
Z (n) + H
+
, where A
Z and A
Z (n) are
26
2 Atomic Physics Relevant to Fusion Plasmas
½ δ n
X n max
k¼1, k6 ¼n
n e K
e
ð Þ
n!k þ n e K
ion
ð Þ
n!cont þ
X nÀ1
k¼1
ν
rad
ð Þ
n!k þ K
n
ð Þ
cx n i
(
)
þ
X n max
k>n
ν
rad
ð Þ
k!n H k
½ δ k þ
X n max
k¼1, k6 ¼n
n e K
e
ð Þ
k!n H k
½ δ k þ K
n
ð Þ
cx n i H n
½ ,
ð2:19Þ
and the following equation for the evolution of the function f
H v
!
:
df
H v
!
dt
¼ ÀK
H
ion n e f
H v
!
À K
H
cx n i f
H v
!
À H
½ f i v
!
n
o
,
ð2:20Þ
where the first term on the right-hand side of Eq. (2.20) describes just the effective
ionization rate constant, recall Eq. (2.16), and
K
H
cx ¼ K
1
ð Þ
cx þ
X n max
k¼2
n
À1
i
n e K
e
ð Þ
k!1 þ ν
rad
ð Þ
k!1
δ k H k
½ = H
½
ð
Þ ,
ð2:21Þ
is the effective charge exchange rate constant including the contribution of the
excited states. As we see, the structure of the expression (2.21) is very different
from the effective charge exchange cross-section accounting for the contribution of
excited states K cx ¼
P
n K
n
ð Þ
cx H n
½ = H
½ suggested in Ref. [34].
By adopting Grad expansion of both the neutral and ion distribution functions
(see Ch. 6 for details), it is possible to extend the analysis of the role of excited states
to a very general hydrogen-ion elastic collision operator (including the chargeexchange one). However, it goes beyond our simple demonstration of the possible
extension of the CRM to the evaluation of an impact of the excited states on elastic
collisions and neutral transport. Calculations performed in [37] have shown that in a
contrast to the results from [34], K
H
cx exceeds K
1
ð Þ
cx by only 10–15% and an impact of
the excited states on neutral hydrogen transport is not very important. This is because
of: i) the comparability of the magnitude of K
1
ð Þ
cx to the electron impact excitation rate
constant of the hydrogen atom, and ii) fast transition from excited to the ground state.
However, the conclusion of the importance of electron exchange recombination of
an impurity ion in the course of the interactions with excited hydrogen atoms [34]
largely holds because hydrogen in the ground state does not undergo such a
“resonance” charge exchange with impurity.
The application of the CRM to impurity atoms/ions and molecules results in
equations more complex than Eqs. (2.13) and (2.15). This is because (i) the number
of states, which should be considered within CRM for each individual atom/ion,
increases; (ii) few ionization states of the same kind of atom/ion can exist for given
plasma density and electron temperature; (iii) the population of excited states of
impurity atom/ion in edge plasma can be affected by charge-exchange process
involving hydrogen atoms (e.g. A
Z + 1 + H ! A
Z (n) + H
+
, where A
Z and A
Z (n) are
26
2 Atomic Physics Relevant to Fusion Plasmas