propagation direction in the solid angle Ω
!
. Then, assuming: i) steady-state approximation for I ω r
!
, Ω
!
and ii) that the absorption and emission line shapes are
described by the same isotropic function a(ω), we arrive at the following equation:
4π
ħω
c
! Á ∇I ω ¼ a ω
ð Þ A u,d D u
½ þ I ω B u,d D u
½ À I ω B d,u D d
½
ð
Þ ,
ð2:24Þ
where [D (. . .) ] are the densities of atoms in the high and low energy states; c
! is the
photon velocity vector; B d, u , B u, d , and A u, d are the Einstein coefficients
B u,d ¼ A u,d
4π
3 c
2
ħω 3 , B d,u ¼
g u
g d
B u,d ,
ð2:25Þ
g (. . .) is the statistical weight of the state (. . .), and A u, d can be found from quantum
mechanics (e.g. for hydrogen atom it is determined by Eq. (2.2)).
The terms in the brackets on the right-hand side of Eq. (2.24) describe, respectively, spontaneous and induced emission of the photons and photon absorption. For
edge plasmas, induced emission is small and can be neglected. By integrating the
photon absorption term we find the source for the population of the “u” quantum
state caused by absorption of the radiation by atoms in the “d” quantum state,
S
rad
ð Þ
u
¼ 4π
ð Þ
À1 R
a ω
ð ÞI ω dωdΩ
!
B d,u D d
½ , which enters in Eq. (2.13) for the population
of excited states.
We note that in edge plasmas, the atom densities and the line shape functions
depend on the spatial coordinate. This makes it extremely difficult to find reliable
estimates of radiation transport [12, 50]. In addition, the radiation trapping modifies
the rates of the atomic processes (recall Eq. (2.13) and, therefore, the population
densities appearing in Eq. (2.24). Thus, we see that both radiation transport and the
dynamics of the population of the excited states of atomic hydrogen become
coupled. Moreover, since the emission of a photon can happen in one region and
its absorption in another one, the synergistic effects of radiation transport and atomic
processes appear to be non-local.
As a result, except very crude models that will be discussed later, realistic
solutions of these complex coupled problems can only be found numerically.
At this moment, the most advanced numerical package capable of treating both
the atomic physics and radiation transport effects in complex edge plasma geometry
is built into the EIRENE Monte Carlo code (e.g. see references [40, 52, 53, 77]).
Another multi-dimensional code which was used for the radiation transport modeling in edge plasma is Cretin [51, 54–56]. The results of the simulations performed
with both EIRENE and Cretin show a reasonably good agreement.
Modeling of the JET, Alcator-C-Mod, and ITER plasmas with EIRENE shows
that radiation transport plays a crucial role in hydrogen ionization processes in
optically thick (large product of the hydrogen atom density and the spatial scale
length) devices. For example, while in JET, the radiation-stimulated ionization
contributes only 10–20% to the total ionization source, in more optically thick
2.3 Line Radiation Transport in Edge Plasma
31
!
. Then, assuming: i) steady-state approximation for I ω r
!
, Ω
!
and ii) that the absorption and emission line shapes are
described by the same isotropic function a(ω), we arrive at the following equation:
4π
ħω
c
! Á ∇I ω ¼ a ω
ð Þ A u,d D u
½ þ I ω B u,d D u
½ À I ω B d,u D d
½
ð
Þ ,
ð2:24Þ
where [D (. . .) ] are the densities of atoms in the high and low energy states; c
! is the
photon velocity vector; B d, u , B u, d , and A u, d are the Einstein coefficients
B u,d ¼ A u,d
4π
3 c
2
ħω 3 , B d,u ¼
g u
g d
B u,d ,
ð2:25Þ
g (. . .) is the statistical weight of the state (. . .), and A u, d can be found from quantum
mechanics (e.g. for hydrogen atom it is determined by Eq. (2.2)).
The terms in the brackets on the right-hand side of Eq. (2.24) describe, respectively, spontaneous and induced emission of the photons and photon absorption. For
edge plasmas, induced emission is small and can be neglected. By integrating the
photon absorption term we find the source for the population of the “u” quantum
state caused by absorption of the radiation by atoms in the “d” quantum state,
S
rad
ð Þ
u
¼ 4π
ð Þ
À1 R
a ω
ð ÞI ω dωdΩ
!
B d,u D d
½ , which enters in Eq. (2.13) for the population
of excited states.
We note that in edge plasmas, the atom densities and the line shape functions
depend on the spatial coordinate. This makes it extremely difficult to find reliable
estimates of radiation transport [12, 50]. In addition, the radiation trapping modifies
the rates of the atomic processes (recall Eq. (2.13) and, therefore, the population
densities appearing in Eq. (2.24). Thus, we see that both radiation transport and the
dynamics of the population of the excited states of atomic hydrogen become
coupled. Moreover, since the emission of a photon can happen in one region and
its absorption in another one, the synergistic effects of radiation transport and atomic
processes appear to be non-local.
As a result, except very crude models that will be discussed later, realistic
solutions of these complex coupled problems can only be found numerically.
At this moment, the most advanced numerical package capable of treating both
the atomic physics and radiation transport effects in complex edge plasma geometry
is built into the EIRENE Monte Carlo code (e.g. see references [40, 52, 53, 77]).
Another multi-dimensional code which was used for the radiation transport modeling in edge plasma is Cretin [51, 54–56]. The results of the simulations performed
with both EIRENE and Cretin show a reasonably good agreement.
Modeling of the JET, Alcator-C-Mod, and ITER plasmas with EIRENE shows
that radiation transport plays a crucial role in hydrogen ionization processes in
optically thick (large product of the hydrogen atom density and the spatial scale
length) devices. For example, while in JET, the radiation-stimulated ionization
contributes only 10–20% to the total ionization source, in more optically thick
2.3 Line Radiation Transport in Edge Plasma
31