1. INTRODUCTION
Optical properties of plasmas are a powerful tool for plasma
diagnostics, which is a key issue to understand beam plasma interaction
phenomena with intense laser and particle beams (Deutsch, 2004; Hasegawa et
al., 2003; Kato et al.,
2004; Baldina, 2004). As it is known,
optical properties depend strongly on the level populations in the plasma.
For this reason, it is necessary to calculate the populations with high
accuracy. As level populations depend mainly on the atomic properties of
the ions immersed in the plasma, the first stage is to obtain the atomic
magnitudes for a rich set of ionic configurations. Once they have been
obtained, the level populations can be calculated through the
Saha-Boltzmann equation for LTE conditions or solving the rate equations
for non-LTE conditions, wherein the transition rates, which also depend on
the atomic properties, appear explicitly.
On the other hand, for optically thick non-LTE plasmas, the rate
equations and the radiation transfer equation are coupled, and they should
be resolved in a simultaneous way, under these conditions the reabsorption
of the radiation in the plasma influences considerably in the level
populations.
Up to now the models proposed by us (Mìnguez et al.,
1998, 2003) to
calculate the level populations had not taken into account the
reabsorption of the radiation. However, our participation on experiments
carried out at LULI (Leboucher-Dalimier et
al., 2001), in which aluminum thick plasmas was obtained,
which led us to include this effect into our calculations through two
different ways. As there are improvements in our previous models, the next
section is devoted to explain them briefly in order to show the
achievements reached and to clarify our starting point. In the third
section, we present the models proposed to include the radiation in the
calculation of level populations. The first one is based on the solution
of the radiation transfer equation and the second one is based on the
escape factor formalism. This work will allow us to analyze the influence
of the reabsorption on the level populations and the influence of them on
the specific intensity, and also to check the accuracy of the escape
factors formalism for uniform plasmas. Finally, in the fourth section, the
results and the discussion are presented.
2. REVIEW OF THE PREVIOUS MODELS
As we said in the previous section, up to now the models that we had
developed to calculate level populations and opacities had not been taken
into account the reabsorption of the radiation in the plasma. These models
were grouped into two codes named ANALOP and JIMENA.
JIMENA code (Mìnguez et al.,
1998) was developed to perform calculations of absorption and
emission coefficients and mean and multi-frequential opacities under LTE
conditions. It uses analytical expressions for the cross sections of the
radiative processes and a Vogt profile for the line shape (Doppler,
collision, natural, and UTA). It works under the assumption of the average
atom and the atomic data are obtained into this context using the
Ion-Sphere Self-consistent Model. However, JIMENA is also able to obtain
level populations for detailed configuration using the Argo-Huebner
method.
The ANALOP code (Mìnguez et al.,
2003) started from JIMENA code and it has gradually increased its
calculation capacity. Now it is able to calculate emission and absorption
coefficients in plasmas both in LTE and non-LTE conditions, and also mean
and multi-frequential opacities in LTE. The cross sections of the
radiative processes are calculated using relativistic quantum calculations
and it uses a Vogt profile for the line shape. ANALOP obtains the level
populations both at LTE (Saha-Boltzman) and non-LTE (rate equations)
conditions for optically thin plasmas. Finally, the atomic data are
calculated in the code using analytical potentials which, as they avoid
the iterations of the self-consistent potentials (Martel et al., 1995; Gil
et al., 2002; Rodrìguez et
al., 2002; Rodriguez et. al.,
2002), allow us to handle with a high number of ionic
configurations (detailed configuration) into the Independent Particle
Model (IPM) context.
In the following section it is presented an improvement to ANALOP
code. Due to the modular structure of this code it results very simple to
couple it this new improvement.
3. MODELS PROPOSED TO INCLUDE REABSORPTION
OF THE RADIATION
In order to introduce the reabsorption of the radiation in the
calculation of level populations in thick plasmas in non-LTE conditions,
we have developed two models, named LTNEP and M3R-EF in the following.
This section is devoted to explaining them.
In the LTNEP model, 1D plasma divided into N cells is
considered; each cell has density and temperature. The profiles of density
and temperature are provided by hydrodynamics calculations as an input of
the model. Then, the atomic kinetics and the radiation transfer equation
are solved self-consistently for the whole plasma. The rates equations are
solved in the Collisional-Radiative Steady State (CRSS) model. The atomic
processes included in the rates equation are:
- Spontaneous emission.
- Resonant emission and stimulated emission.
- Photoionization and radiative recombination.
- Collisional excitation and dexexcitation.
- Collisional ionization and three-body recombination.
- Dielectronic recombination.
The forward rates coefficients are obtained with widely used formulas
while inverse rate coefficients are obtained from detailed balance
relations. The atomic data required for the calculations are provided to
the model as an input either by our models or by others. Once radiation
dependent populations have been obtained for each cell, they are used as
input of line shape Code Pim Pam Poum (Calisti et
al., 1990). With this code we obtain the opacity and emissive
Stark profiles for each line involved in the spectral range. The source
function is then obtained from line opacity calculated with Pim Pam Poum
code and bound-free opacity calculated with hydrogenic formulas. Finally,
the specific intensity is determined solving the transfer equation with
the known source function.
The other model, M3R-EF, modelizes the reabsorption of the radiation
through the escape factor formulism. The escape factor θ denotes the
mean probability that a photon emitted anywhere in the source travels
directly to the surface of the source in any direction and escapes. In
this work, we have assumed a uniform distribution of emitting atoms and
isotropic emission and a slab geometry. Under these assumptions, the
escape factor can be written as (Mancini et
al., 1987)
with
where ν is the central line frequency corresponding to electronic
transition from the upper level u to the lower level l,
and τ0 is the optical thickness.
Escape factors are frequently used in plasma spectroscopy as an
approximate way to account for the effects of reabsorption, which can be
very important, especially for resonance line transitions. Escape factors;
θ enter the calculations in two ways. First, they enter the atomic
physics calculations of excited-state populations; as a result there is a
reduction in the Einstein spontaneous emission coefficient,
Au→l which is written as
θAu→l. Second, they appear
in the determination of the total emergent line intensity. This
modification circumvents the need to perform a simultaneous calculation of
radiation transport and atomic physics.
For a line of frequency ν, the optical thickness for this
frequency is given by
with κν the opacity and D the slab width.
The opacity is given by
with pl the fractional population of
level l, flu the absorption
oscillator strength, and Ni total ion
density. We can see that the opacity is dependent on populations and
therefore the escape factor, too. So, given plasma with electron density
Ne and temperature T, if we include
the new spontaneous emission coefficient
Au→l* in the rate equations,
this set of equations becomes non-linear, and we have to use an iterative
procedure to find the level populations pj.
The rate equations are solved in the Collisional-Radiative Steady State
(CRSS) model and the atomic processes included are the following:
- Spontaneous emission.
- Radiative recombination.
- Collisional excitation and dexexcitation.
- Collisional ionization and three-body recombination.
- Dielectronic recombination.
As it happens, in the LTNEP model, the atomic data required in M3R-EP
model can also be provided either by JIMENA or ANALOP codes or by
others.
4. RESULTS
In this work, we have focused our results and discussion on aluminum
plasmas because they were obtained in the experiments at LULI.
First of all, we are interested in verifying that the reabsorption of
the radiation in plasmas introduces perceptive changes in the level
populations. With this purpose, we have plotted in Figure 1 the level populations for aluminum uniform
plasma calculated assuming optically thin plasma, and optically thick
plasma (M3R-EF and LTNEP models). It was considered in the calculations
eight ionization states and 326 energy levels.
Uniform Al plasma. Plasma conditions: 1023
cm−3 and 300 eV. Plasma length: 100μ.
As can be seen from the figure, there are relevant discrepancies
between optically thin and thick plasma calculations. This result was
obtained under other plasma conditions, and therefore, as in non-LTE
conditions plasma radiation plays an important role; it should be included
in population calculations. On the other hand, from the figure we can
observe that a good agreement is reached between the calculations made
with M3R-EF and LTNEP.
Due to the specific intensity depends on the populations, we also
analyzed that influence. In Figure 2, we show
the quotient of the population for each level involved in the calculations
by the degeneracy of the level for the three most ionized aluminum ions at
the same conditions of Figure 1. The ground
configuration of each ion is the first one being the remaining excited
configurations. The same qualitative conclusions that are in Figure 1 can be obtained.
Population for each level of the three most ionized ions for a uniform
Al plasma. Plasma conditions: 1023 cm−3 and
300 eV. Plasma length: 100μ.
In this case, for the Lyman series, we obtain that the ratio of the
level populations calculated assuming optically thick and thin plasma,
Pthick
/Pthin is equal to 10 for the ground
states and 102 for the excited states while for the Helium
series is equal to 1 and 10, respectively. The source function for a given
frequency is given by
with εν and κν being the
emissive and the opacity respectively for that frequency. Denoting as
Pu and Pl, the
populations of the upper and the lower level of the transition,
respectively, calculated as the quotient of the population by the
degeneracy, the source function can be expressed as
Taking into account the results of the ratios of populations shown
above,
Sνthick/Sνthin
≈ 10. According to the relation between the specific intensity and the
source function the last result implies that the specific intensity
calculated for the optical thick plasma is ten times greater than for the
optical thin plasma. It can be observed in Figure
3. This figure also shows the good agreement between both models,
M3R-EF and LTNEP, for the specific intensity. The same results were
obtained for the same plasma at a temperature of 400 eV instead of 300
eV.
Specific intensity for uniform optically thick and thin Al plasma for
the Lyman and Helium series. Plasma conditions: 1023
cm−3 and 300 eV. Plasma length: 100μ.
According to these results, we have verified that the escape factor
formulism is a good alternative to those methods based on the resolution
of the rate equations, coupled to the radiation transport equation for
uniform plasmas. However, for non-uniform plasmas, this formalism is no
longer accurate. This result is expected since the escape factor formalism
does not couple cells with different plasma conditions. In general, we
have observed that this fact leads to a decrease of the mean ionization
which affects mainly the Lyman series and the He-α line, as it can be
observed from Figure 4. The density and
temperature profile was obtained using hydrodynamics simulations.
Specific intensity of Lyα and Heα lines for non uniform Al
plasma. Plasma conditions: 1023 cm−3 and 300
eV. Plasma length: 100μ.
4. CONCLUSIONS
It has been shown, the relevance of the reabsorption of the radiation
in plasmas under non-LTE conditions in the calculation of the level
populations, and in the specific intensity as well. This fact led us to
improve ANALOP code to include it in two models: M3R-EF based on the
escape factor formalism and LTNEP which solves self-consistently the rates
equations, and the radiative transfer equation. For uniform plasmas, the
escape factor formalism is a useful and accurate alternative to solve the
problem of the reabsorption of the radiation, avoiding the self-consistent
procedures. However, for non-uniform plasmas, the context of the escape
factors the cells of the plasma with different temperature and density
remain uncoupled, the results are no longer reliable and we have to use
LTNEP model. Finally, it is worth noting that ANALOP code is able to work
either M3R-EF or LTNEP.
ACKNOWLEDGMENTS
This work was partially supported by the program Keep-in-Touch of the
European Union and by the project UNI2003/22 of the ULPGC.
