(including both radiative and three-body recombination). Accordingly, the expressions in the first and second braces on the right-hand side of Eq. (2.15) are called the
hydrogen ionization, K
H
ion , and electron-ion recombination (EIR, which includes
both radiative and three-body recombination processes), K
H
rec , rate constants. As a
result, we can re-write Eq. (2.15) as follows
d H
½
dt
¼ ÀK
H
ion n e H
½ þ K
H
rec n e H
þ
½ þS
transp
ð
Þ
n
:
ð2:16Þ
The CRM is widely used in the modeling of gas discharge, fusion, and astrophysical plasmas (e.g. see [12, 13, 23, 25, 27–31] and the references therein). It is
much simpler than the time-dependent rate Eq. (2.12) whereas allowing for the
impact of excited states on both the ionization and recombination processes. We
notice that since the population of excited states is established in a competition of
electron-induced transitions and spontaneous decays, both rate constants K
H
ion and
K
H
rec depend on the electron density and temperature. However, in optically thick
plasma, re-absorption of resonance photons (in fusion plasmas they are usually Ly α
and Ly β ) can alter the population of the excited states (recall terms S
rad
ð Þ
n
in
Eq. (2.13)) and, therefore, the ionization and recombination rate constants.
We will see below that because of high ionization rates of the excited states, the
multistep processes including excitation and quenching of the excited hydrogen
levels result, at sufficiently high plasma density, in a significant increase of the
effective hydrogen ionization rate constant. Moreover, even the constants of elastic
processes involving excited states can depend on n. In particular, the rate K
n
ð Þ
cx of the
so-called charge exchange process, H(n) + H
+
! H
+ + H(n), increases with
increasing n and is proportional, although approximately, to n
4 [32]. Such elastic
collisions play a vital role in plasma-neutral momentum exchange and divertor
detachment physics (e.g. see [33] and the references therein). Therefore, correct
assessment of the impact of excited states on overall momentum exchange is
important.
A crude estimate of the effective resonance charge exchange rate constant, based
on a simple averaging of corresponding cross-sections over relative population of
excited states, similar to that of the ionization rate constant, K cx ¼
P
n K
n
ð Þ
cx H n
½ = H
½ ,
demonstrates a significant increase of K cx in comparison to the cross-section
involving hydrogen in the ground state, K
1
ð Þ
cx [34]. If this simple estimate held, it
would have an important impact on both hydrogen transport and plasma-neutral
coupling described with both Monte Carlo codes and fluid models (e.g. see [35]).
However, more thorough consideration shows that such a simplified description of
the contribution of the excited states to hydrogen transport and plasma-neutral
momentum exchange is incorrect [36].
For a proper consideration of ion-neutral momentum exchange, we need to
consider the distribution functions of protons, f i v
!
, and atoms in all excited states,
f
H
n
ð Þ v
!
. Then, since the charge-exchange cross-section of excited states is much
24
2 Atomic Physics Relevant to Fusion Plasmas
hydrogen ionization, K
H
ion , and electron-ion recombination (EIR, which includes
both radiative and three-body recombination processes), K
H
rec , rate constants. As a
result, we can re-write Eq. (2.15) as follows
d H
½
dt
¼ ÀK
H
ion n e H
½ þ K
H
rec n e H
þ
½ þS
transp
ð
Þ
n
:
ð2:16Þ
The CRM is widely used in the modeling of gas discharge, fusion, and astrophysical plasmas (e.g. see [12, 13, 23, 25, 27–31] and the references therein). It is
much simpler than the time-dependent rate Eq. (2.12) whereas allowing for the
impact of excited states on both the ionization and recombination processes. We
notice that since the population of excited states is established in a competition of
electron-induced transitions and spontaneous decays, both rate constants K
H
ion and
K
H
rec depend on the electron density and temperature. However, in optically thick
plasma, re-absorption of resonance photons (in fusion plasmas they are usually Ly α
and Ly β ) can alter the population of the excited states (recall terms S
rad
ð Þ
n
in
Eq. (2.13)) and, therefore, the ionization and recombination rate constants.
We will see below that because of high ionization rates of the excited states, the
multistep processes including excitation and quenching of the excited hydrogen
levels result, at sufficiently high plasma density, in a significant increase of the
effective hydrogen ionization rate constant. Moreover, even the constants of elastic
processes involving excited states can depend on n. In particular, the rate K
n
ð Þ
cx of the
so-called charge exchange process, H(n) + H
+
! H
+ + H(n), increases with
increasing n and is proportional, although approximately, to n
4 [32]. Such elastic
collisions play a vital role in plasma-neutral momentum exchange and divertor
detachment physics (e.g. see [33] and the references therein). Therefore, correct
assessment of the impact of excited states on overall momentum exchange is
important.
A crude estimate of the effective resonance charge exchange rate constant, based
on a simple averaging of corresponding cross-sections over relative population of
excited states, similar to that of the ionization rate constant, K cx ¼
P
n K
n
ð Þ
cx H n
½ = H
½ ,
demonstrates a significant increase of K cx in comparison to the cross-section
involving hydrogen in the ground state, K
1
ð Þ
cx [34]. If this simple estimate held, it
would have an important impact on both hydrogen transport and plasma-neutral
coupling described with both Monte Carlo codes and fluid models (e.g. see [35]).
However, more thorough consideration shows that such a simplified description of
the contribution of the excited states to hydrogen transport and plasma-neutral
momentum exchange is incorrect [36].
For a proper consideration of ion-neutral momentum exchange, we need to
consider the distribution functions of protons, f i v
!
, and atoms in all excited states,
f
H
n
ð Þ v
!
. Then, since the charge-exchange cross-section of excited states is much
24
2 Atomic Physics Relevant to Fusion Plasmas