the term S
rad
ð Þ
n
describing the inter-state transitions stimulated by the photons. These
two features of our analysis could only be reconciled for the case of the so-called
“transparent” media, where the effective mean free path of a photon to absorption by
a neutral/ion, ℓ abs , is longer than the characteristic scale-length, L, (e.g. for the case
of radiation in the divertor volume, it could be the poloidal width of the divertor). An
opposite case, where ℓ abs < L, is called the “opaque” or “optically thick” one.
Since the photon absorption rate is proportional to the density of available
absorbers, and absorption process in edge plasma largely has a resonant nature
(recall the relation ħω 0 % ΔE nk ), edge plasma can be transparent for the radiation
corresponding to some lines and opaque for the other ones (the latter case is often
referred to as radiation “trapping”). Because the density of hydrogen atoms (hydrogen molecules are dissociated rather quickly due to electron impact) is the largest
among the radiating species in the edge plasma, the line radiation related to atomic
hydrogen is the first candidate for being trapped. Moreover, since the majority of
atomic hydrogen is in the ground state, the most strongly “trapped” hydrogen lines
are Ly α and Ly β , related, respectively, to the transitions n ¼ 2 ! k ¼ 1 and
n ¼ 3 ! k ¼ 1 (we will see that the absorption of other lines in Lyman series is
weaker due to reduction of corresponding oscillator strengths). It was shown
(e.g. see [43, 44]) that the trapping effects for Ly α and Ly β lines become important
already for current tokamaks and they are expected to be much more pronounced in
the future tokamak reactors (e.g. ITER [45]).
As one could notice, we have stated that the frequency of the emitted photon ω 0 is
only approximately equal to ΔE kn /ħ. The reason for this is the so-called line
“broadening”, which results in the fact that emitted photons have some frequency
distribution, described by the “line shape” function, a(ω) (localized around ω 0 and
having characteristic width Δω ( ω 0 ), such that
R
a(ω)dω ¼ 1. There are few reasons
for line broadening in edge plasma. First, there is a “natural” broadening of the line,
Γ, caused by the finite time of the radiation emission, which corresponds to the decay
rate of the excited state determined by the Einstein coefficients (2.2). However, in
practice Δω is, in most cases, much larger than Γ. In edge plasma both Δω and the
shape of the function a(ω) are largely determined by (i) Doppler broadening related
to the shift of the frequency of the radiation emitted by moving particles, so that
Δω D ~ ω 0 (V th /c), where V th is the particle thermal speed; (ii) Stark broadening due to
the micro-electric fields E micro ∼ en
2=3
e , causing a change in the energy of the
quantum states and yielding, for a hydrogen atom, Δω S ~ (ħ/m e e)E micro ; (iii) Zeeman
effect that results, in the presence of a strong magnetic field, in splitting the quantum
states; and, finally, (iv) the so-called motional Stark effects related to the effective
electric field, E B ~ (V N /c)B. (e. g. see [46, 47], and the references therein).
For the case where the line width is only determined by the decay rate, the
effective cross-section of absorption of a resonant photon, σ abs (ω 0 ), is proportional
to the square of the photon wavelength, i.e. σ abs (ω 0 ) ~ (c/ω 0 )
2 . We note that for the
edge plasma conditions, ω 0 ) ω pe
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4πn e e 2 =m e
p
, where ω pe is the Langmuir
frequency, so one can neglect plasma effects in the dispersion of the line radiation
and take ω 0 ¼ k 0 c, where k 0 is the photon wavenumber. However, broadening of the
28
2 Atomic Physics Relevant to Fusion Plasmas
rad
ð Þ
n
describing the inter-state transitions stimulated by the photons. These
two features of our analysis could only be reconciled for the case of the so-called
“transparent” media, where the effective mean free path of a photon to absorption by
a neutral/ion, ℓ abs , is longer than the characteristic scale-length, L, (e.g. for the case
of radiation in the divertor volume, it could be the poloidal width of the divertor). An
opposite case, where ℓ abs < L, is called the “opaque” or “optically thick” one.
Since the photon absorption rate is proportional to the density of available
absorbers, and absorption process in edge plasma largely has a resonant nature
(recall the relation ħω 0 % ΔE nk ), edge plasma can be transparent for the radiation
corresponding to some lines and opaque for the other ones (the latter case is often
referred to as radiation “trapping”). Because the density of hydrogen atoms (hydrogen molecules are dissociated rather quickly due to electron impact) is the largest
among the radiating species in the edge plasma, the line radiation related to atomic
hydrogen is the first candidate for being trapped. Moreover, since the majority of
atomic hydrogen is in the ground state, the most strongly “trapped” hydrogen lines
are Ly α and Ly β , related, respectively, to the transitions n ¼ 2 ! k ¼ 1 and
n ¼ 3 ! k ¼ 1 (we will see that the absorption of other lines in Lyman series is
weaker due to reduction of corresponding oscillator strengths). It was shown
(e.g. see [43, 44]) that the trapping effects for Ly α and Ly β lines become important
already for current tokamaks and they are expected to be much more pronounced in
the future tokamak reactors (e.g. ITER [45]).
As one could notice, we have stated that the frequency of the emitted photon ω 0 is
only approximately equal to ΔE kn /ħ. The reason for this is the so-called line
“broadening”, which results in the fact that emitted photons have some frequency
distribution, described by the “line shape” function, a(ω) (localized around ω 0 and
having characteristic width Δω ( ω 0 ), such that
R
a(ω)dω ¼ 1. There are few reasons
for line broadening in edge plasma. First, there is a “natural” broadening of the line,
Γ, caused by the finite time of the radiation emission, which corresponds to the decay
rate of the excited state determined by the Einstein coefficients (2.2). However, in
practice Δω is, in most cases, much larger than Γ. In edge plasma both Δω and the
shape of the function a(ω) are largely determined by (i) Doppler broadening related
to the shift of the frequency of the radiation emitted by moving particles, so that
Δω D ~ ω 0 (V th /c), where V th is the particle thermal speed; (ii) Stark broadening due to
the micro-electric fields E micro ∼ en
2=3
e , causing a change in the energy of the
quantum states and yielding, for a hydrogen atom, Δω S ~ (ħ/m e e)E micro ; (iii) Zeeman
effect that results, in the presence of a strong magnetic field, in splitting the quantum
states; and, finally, (iv) the so-called motional Stark effects related to the effective
electric field, E B ~ (V N /c)B. (e. g. see [46, 47], and the references therein).
For the case where the line width is only determined by the decay rate, the
effective cross-section of absorption of a resonant photon, σ abs (ω 0 ), is proportional
to the square of the photon wavelength, i.e. σ abs (ω 0 ) ~ (c/ω 0 )
2 . We note that for the
edge plasma conditions, ω 0 ) ω pe
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4πn e e 2 =m e
p
, where ω pe is the Langmuir
frequency, so one can neglect plasma effects in the dispersion of the line radiation
and take ω 0 ¼ k 0 c, where k 0 is the photon wavenumber. However, broadening of the
28
2 Atomic Physics Relevant to Fusion Plasmas