U n (R) remains the same for all isotopologues of the hydrogen molecule, from the
definition of E rot and ω v we find their dependences on the reduced mass of the
nucleon: E rot / 1= e
M , ω v / 1=
ffiffiffiffi ffi
e
M
p
, and for the H 2 molecule, ħω v ¼ 0.54 eV
[4]. The vibrational energy quanta for different isotopologues of the hydrogen
molecule are different but the energy terms U n (R) are the same. Therefore, the
same dissociation energy of 4.48 eV results in different numbers of the available
vibrational quantum states, which can be estimated as v max /
ffiffiffiffi ffi
e
M
p
(e.g. the H 2
molecule has v max ¼ 14).
In Fig. 2.3, in addition to the molecular terms for H 2 , are also shown the terms for
the positive molecular ion, H
þ
ð Þ
2 , and the negative one, H
À
ð Þ
2 . The latter is metastable
with a relatively short natural life-time. However, it plays a crucial role in both the
excitation of vibrational levels by electron impact H 2 v
ð Þ þ e ! H
À
ð Þ
2
! H 2 v
0
ð Þ þ e
and dissociative attachment H 2 v
ð Þ þ e ! H
À
ð Þ
2
! H þ H
À (e.g. see [12, 17, 18] and
the references therein).
2.2 Collisional-Radiative Model
As we already mentioned, excited states of atoms and molecules in edge plasmas
play important roles in virtually all atomic physics-related processes. Here, using the
example of a hydrogen atom, we consider the basic physics of the CRM which is the
main approach for a quantitative description of the processes the excited states can
be involved in.
The population of excited states (for the simplest case of hydrogen-like ions, they
correspond to n > 1) in edge plasma, depending on the plasma parameters, is
determined by an interplay of electron impact excitation, electron transition from
the continuum, radioactive decay, re-absorption of resonance photons, etc. We
notice that for the energies relevant for the edge plasmas, the impact excitation of
electronic and vibrational states by atoms and ions is usually negligible. Therefore,
we start with a short review of electron-involved processes and their contributions to
the rate equations governing the population of the excited states.
The binary interactions of particles, including electrons and heavy particles
(atoms, molecules, and ions), are usually described by the effective cross-section,
σ, of the particular process. In a very general case, σ depends on the relative speed of
the interacting particles. Taking into account the large mass difference between the
electrons and nuclei, for the case of electron interactions with atoms, molecules, and
ions, the relative speed of the interacting particles is virtually equal to the electron
speed. As a result, the cross-section of such interaction only depends on the electron
kinetic energy, E e . The cross-sections of many processes relevant for the edge
plasmas are available from both quantum mechanical calculations and experiments
(e.g. see [19–21] and the references therein). We notice that the cross-sections of the
forward and reverse processes are related by the principle of detailed equilibrium
(e.g. see [22]), so there is no need to calculate them separately.
2.2 Collisional-Radiative Model
19
definition of E rot and ω v we find their dependences on the reduced mass of the
nucleon: E rot / 1= e
M , ω v / 1=
ffiffiffiffi ffi
e
M
p
, and for the H 2 molecule, ħω v ¼ 0.54 eV
[4]. The vibrational energy quanta for different isotopologues of the hydrogen
molecule are different but the energy terms U n (R) are the same. Therefore, the
same dissociation energy of 4.48 eV results in different numbers of the available
vibrational quantum states, which can be estimated as v max /
ffiffiffiffi ffi
e
M
p
(e.g. the H 2
molecule has v max ¼ 14).
In Fig. 2.3, in addition to the molecular terms for H 2 , are also shown the terms for
the positive molecular ion, H
þ
ð Þ
2 , and the negative one, H
À
ð Þ
2 . The latter is metastable
with a relatively short natural life-time. However, it plays a crucial role in both the
excitation of vibrational levels by electron impact H 2 v
ð Þ þ e ! H
À
ð Þ
2
! H 2 v
0
ð Þ þ e
and dissociative attachment H 2 v
ð Þ þ e ! H
À
ð Þ
2
! H þ H
À (e.g. see [12, 17, 18] and
the references therein).
2.2 Collisional-Radiative Model
As we already mentioned, excited states of atoms and molecules in edge plasmas
play important roles in virtually all atomic physics-related processes. Here, using the
example of a hydrogen atom, we consider the basic physics of the CRM which is the
main approach for a quantitative description of the processes the excited states can
be involved in.
The population of excited states (for the simplest case of hydrogen-like ions, they
correspond to n > 1) in edge plasma, depending on the plasma parameters, is
determined by an interplay of electron impact excitation, electron transition from
the continuum, radioactive decay, re-absorption of resonance photons, etc. We
notice that for the energies relevant for the edge plasmas, the impact excitation of
electronic and vibrational states by atoms and ions is usually negligible. Therefore,
we start with a short review of electron-involved processes and their contributions to
the rate equations governing the population of the excited states.
The binary interactions of particles, including electrons and heavy particles
(atoms, molecules, and ions), are usually described by the effective cross-section,
σ, of the particular process. In a very general case, σ depends on the relative speed of
the interacting particles. Taking into account the large mass difference between the
electrons and nuclei, for the case of electron interactions with atoms, molecules, and
ions, the relative speed of the interacting particles is virtually equal to the electron
speed. As a result, the cross-section of such interaction only depends on the electron
kinetic energy, E e . The cross-sections of many processes relevant for the edge
plasmas are available from both quantum mechanical calculations and experiments
(e.g. see [19–21] and the references therein). We notice that the cross-sections of the
forward and reverse processes are related by the principle of detailed equilibrium
(e.g. see [22]), so there is no need to calculate them separately.
2.2 Collisional-Radiative Model
19