Modelling of spectral properties and
population kinetics studies of inertial
fusion and laboratory-astrophysical
plasmas
EMínguez R Florido R Rodríguez J M Gil J G Rubiano
M A Mendoza G Espinosa andPMartel
Abstract
Fundamental research and modelling in plasma atomic physics continue to be essential for
providing basic understanding of many different topics relevant to high-energy-density
plasmas. The Atomic Physics Group at the Institute of Nuclear Fusion has accumulated
experience over the years in developing a collection of computational models and tools for
determining the atomic energy structure, ionization balance and radiative properties of, mainly,
inertial fusion and laser-produced plasmas in a variety of conditions. In this work, we discuss
some of the latest advances and results of our research, with emphasis on inertial fusion and
laboratory-astrophysical applications.
(Some figures may appear in colour only in the online journal)
1.
Introduction
Fundamental research and modelling in plasma atomic
physics continue to be essential for providing both basic
understanding and advancing on many different topics relevant
to high-energy-density systems community [1]. In the most
general scenario, the plasmas are not in local thermodynamic
equilibrium (LTE), which means many atomic properties—
such as cross-section of all relevant collisional and radiative
atomic processes and atomic energy levels—need to be known
to properly describe and diagnose the plasma conditions. The
basis for most non-LTE (NLTE) simulations is the collisional-
radiative (CR) model [2], which describes each ion charge
state in the plasma in terms of a number of atomic levels.
The distribution of atomic populations among the levels is
determined by solving a set of coupled rate equations whose
generation requires calculating all transition rates among the
atomic levels. Once the population distributions have been
obtained, material and radiative properties—such as opacities,
emissivities, specific energies and other quantities of interest
for the equation of state—can be calculated.
In this connection, the Atomic Physics Group at the
Institute of Nuclear Fusion (DENIM) in Spain has accumulated
experience over the years in developing a collection of
computational models and tools for determining the atomic
energy structure, ionization balance and radiative properties
of, mainly, inertial confinement fusion (ICF) and laser-
produced plasmas in a variety of conditions. In this
work, we will focus on some of the latest advances and
results of our research, with emphasis on ICF and, also,
laboratory-astrophysical applications. In the following section
we briefly describe the most recently developed models
and codes, i.e. ABAKO/RAPCAL [3,4] and ATMED [5].
ABAKO/RAPCAL was designed to perform detailed atomic
kinetics and radiative properties calculations over a wide
range of temperature and density, including the coronal,
LTE and non-LTE regimes. Meanwhile, the new average-
atom screened hydrogenic model, ATMED, was aimed for
providing fast estimates and potential in-line hydrodynamical
calculations of emissivities and opacities for plasmas under
LTE conditions. In sections 3 and 4, we report on recent
applications of ABAKO/RAPCAL to two completely different
kind of laboratory-plasma experiments. In the first one,
connected to laboratory astrophysics, radiative properties
calculated from RAPCAL
are
used to characterize
the
radiative
blast waves launched in xenon clusters. In the second one,
relevant for the ICF community, we discuss the use of ABAKO
as part of physics model employed for the spectroscopic
analysis of shock-ignition implosions. Finally, conclusions
are presented.
2.
ABAKO/RAPCAL and ATMED computational
packages
ABAKO is a CR model that can be applied to low-to-high
Z ions for a wide range of laboratory-plasma conditions:
coronal, LTE or NLTE, and optically thin or thick plasmas.
It combines a set of analytical approximations to atomic rates,
which yield substantial savings in computer running time, still
comparing well with more elaborate codes and experimental
data. Autoionizing states are explicitly included in the kinetics
in a fast and efficient way. Radiation transport effects due to
line trapping in
the
plasma are taken into account via geometry-
dependent escape factors. Also, since the atomic kinetics
problem often involves very large sparse matrices, an iterative
method is used to perform the matrix inversion. It has been
shown that ABAKO results compare well with customized
models and simulations of ion population distributions. A
detailed description of the model can be found in [3].
Meanwhile, the RAPCAL [4] code determines several
relevant plasma radiative properties such as the frequency-
dependent opacities and emissivities, mean and multigroup
opacities, source functions, radiative power losses, specific
intensities and plasma transmission. For line transitions,
RAPCAL uses a Voigt profile including natural and Doppler
broadenings, unresolved-transition-array (UTA) corrections,
as well as an approximate formula to account for the electron-
impact width.
In the following two sections, we discuss the use of
ABAKO/RAPCAL for the study and investigation of two com-
pletely different kinds of laboratory-plasma experiments, one
connected to laboratory astrophysics and the other one to ICF.
Furthermore, the ATMED code was aimed for providing
fast calculations of the photon energy dependent opacity as
well as the Rosseland and Planck mean opacities for single
element and mixture hot dense plasmas under LTE. Typical
ATMED running times are of the order of seconds. The
code was developed within the framework of an average
atom model. The needed atomic data are computed using
a relativistic screened hydrogenic model based on a new
set of universal screening constants including j-splitting [5].
These screening constants were obtained from the fit to a
wide database of high-quality atomic energies, ionization
potentials and transition energies taken from
[6,7].
The
opacity calculations take into account the bound-bound,
bound-free, free-free contributions, as well as scattering
processes. The line absorption cross-sections are computed
using a new analytical expression for oscillator strengths
based on relativistic screened-hydrogenic wave functions.
Line shapes include natural, Doppler and electron collisional
10-
1
i—i—i—i—r-
LEDCOP
ATMED
250 500 750 1000 1250 1500 1750 2000
Photon energy (eV)
Figure 1. Frequency-dependent opacity for a Be(99.1%)-Cu(0.9%)
plasma mixture at 250 eV and 0.1 g cirT
3
. A comparison between
ATMED and LEDCOP [9] is shown.
broadenings. To obtain a more realistic value of
the
Rosseland
mean opacity, an additional broadening of the bound-bound
transitions has been included by considering the fluctuations
of the occupations numbers into the atomic shells. ATMED
also provides the plasma equation of state and shock Hugoniot
curves. The model has been successfully tested and, in spite
of its simplicity, it is able to predict the correct order of
magnitude of quantities of interest, so it can be used to model
radiative transport phenomena in hydrodynamic codes and
experiments in an approximate way. A detailed discussion
of many applications of ATMED can be found in [8]. Here,
as a case of illustration, we show in figure 1 the frequency-
dependent opacity corresponding to a Be-Cu plasma mixture.
A calculation like this could be of interest for simulations of
indirect-drive fusion capsules, in which Cu-doped beryllium
has been used as ablator material.
3.
An application to laboratory-astrophysical
plasmas: characterizing radiative blast waves
created in the laboratory
The main goal of laboratory astrophysics is to design and
perform reproducible and well-characterized experiments, and
then use hydrodynamical simulations and scaling arguments
to both explain and predict actual astrophysical phenomena.
Among many others, one particular area of interest within
laboratory astrophysics is that of radiative shocks [10,11],
which are observed around astronomical objects in a wide
variety of forms, e.g. accretion shocks, pulsating stars,
supernovae in their radiative cooling phase, etc.
In this work, we report on atomic kinetics and radiative
properties calculations in connection to radiative blast waves
launched in xenon clusters [12,13]. Ordinarily gases absorb
only a small fraction of the laser energy. However, if atomic
clustered gases are used, the absorption increases dramatically
and a hot (T
e
~ keV), high-energy-density plasma is formed
and, eventually,
a
cylindrical blast wave generated [14]. Using
D.
O
O.
6 8 10 12 14 16
Electron temperature (eV)
50 75 100 125 150
Photon energy (eV)
175 200
Figure 2. (a) Xe charge state distribution as a function of temperature for a mass density of
5
x 10
4
g cm
3
photon free path for a xenon plasma at
10
eV and 5x10
4
g cm
3
.
been added for comparison.
(b) Frequency-dependent
A dashed line indicating the characteristic dimension of the system has
clusters as a target medium, radiative blast waves can be
launched even with laser energies as low as 100 mJ, provided
that the pulse duration is short enough for an efficient energy
coupling into clusters before they explode (which requires sub-
picosecond lasers).
In the experiments here discussed, xenon clusters were
irradiated at average gas densities of 0.16 and 0.25 kg irr
3
with femtosecond laser pulses of 440 mJ and 360 mJ,
respectively. The evolution and morphology of
the
blast waves
were monitored using imaging interferometric and Schlieren
techniques. Time histories of electron density radial profiles
were also determined from interferometric measurements.
ABAKO/RAPCAL was used for a better understanding and
characterization of these radiative blast waves [15,16]. Typical
mass densities and electron temperatures of these xenon
plasmas are expected to be in the range 10~
5
-10~
3
g
cirr
3
and
1-20
eV,
respectively. A first step for the characterization of
the xenon plasmas was to perform a population kinetics study
and compute basic quantities such as the average ionization
and ion population distribution for such plasma conditions.
For the given ranges of temperature and density, and within
the steady-state assumption, it is difficult to know a priori
whether the population distributions of xenon plasmas will be
those characteristic of a coronal, LTE or NLTE regime. This
fact prevents the use of a simplified coronal model as well
as the Saha-Boltzmann (SB) equations, at least until a first
atomic kinetics study has been carried
out
based on collisional-
radiative calculations. At this point, it must be noted that
such xenon atomic kinetics calculations are challenging, since
for the temperatures and densities of interest, xenon will be
only a few times ionized and the calculations will involve
complex ions with a large number of bound electrons. As
the number of bound electrons increases, the number of
energy levels and electronic configurations do dramatically and
some averaging technique must be used to keep the problem
manageable. Here, we took advantage of the proper balance
existing in ABAKO between accuracy and computational cost
to obtain reasonable estimates of average ionization and ion
population distributions. Additionally, we made a careful
selection of electronic configurations to be included following
the criteria discussed in [16]. For each CR calculation, the
model included about ten xenon ions, which yields
a
CR matrix
with ~70000 energy levels. As illustration, in figure 2(a)
the charge state distribution (CSD) of xenon is plotted as a
function of the electron temperature for a fixed mass density
of 5 x 10~
4
gcirr
3
, characteristic of blast waves in xenon
clusters. As seen, population is mainly distributed among four
different charge states for a given temperature value within the
range of interest.
Also,
the conditions to define an optically thin, radiative
regime for laboratory astrophysics experiments relevance to
radiative supernova remnants were formulated in [17]: (a) for
the radiative flux to escape from the blast wave, the mean free
path of the photons, X
ph
, must be larger than a characteristic
size scale of the system, h; and (b) the radiative cooling
time,
T
rad
, must be shorter than the convective transport time,
^conv = h/s, where s is the plasma sound speed. In this
regard, measurements indicate that blast waves in clustered
xenon reached the needed conditions to enter the radiative flux
regime—defined as the situation in which the radiative energy
flux is greater than the material energy flux, thus playing an
important role in the evolution of the blast
wave.
Nevertheless,
a theoretical confirmation of this fact is desirable and here
we show as ABAKO/RAPCAL becomes useful to carry out
this study. The frequency-dependent free path of photons was
computed for a mass density of p = 5 x 10~
4
cm~
3
and an
electron temperature of T
e
=
10
eV (see figure 2(b)), which
are approximated values for the upstream medium according
to measurements and hydrodynamical simulations [12,13].
Characteristic size of the xenon blast waves discussed here
is h «a 0.01 cm, so, as observed in figure 2(b), the upstream
medium can be assumed as optically thin for almost the entire
range of photon energies, although some absorption effects are
expected in the interval from 60 to 100
eV.
We also computed
the radiative cooling time
r
rad
[17] for several temperature and
density values within the range of interest and for all cases
T
rad
was found to be less than
r
conv
«a
1
/zs, which is the
typical convective transport time of the xenon blast waves.
AR
N
a
T
B
(a)
£>
(b)
AR
5
Nr T,
c
J-
c
£>
JV "T,
Figure 3. (a) Schematic illustration of the model used to analyse the data, (b) Simpler geometry with analytical solution. Figure adapted
from [20]. Copyright (2011) by American Physical Society.
Therefore, with the only exception of
a
photon energy interval
of width ~60eV, the theoretical analysis performed using
ABAKO/RAPCAL allows us to classify the xenon blast waves
within the optically thin, radiative regime. This conclusion
agrees with what had been previously suggested based on
experimental data.
The results discussed so far were obtained assuming
steady
state.
Therefore, for them to be meaningful, the validity
of this assumption for the case of study must be checked and
possible time-dependent effects discarded. A key quantity for
this analysis is the time scale of the most frequent atomic
process in the plasma, x
a
[18]. Time-independent models
can be acceptably accurate if the characteristic time i
p
i
asma
for the change in the local plasma conditions is significantly
larger than x
a
, i.e. i
p
i
asma
~S>
r
a
. When this inequality holds
true,
it can be considered that the populations distributions
adjust almost instantaneously to the changing conditions of
the plasma. For the case of blast waves experiments in xenon,
it can be inferred from data that í
p
i
asma
~
1
ns. Moreover,
calculations made with ABAKO for densities from 1(T
4
to
1(T
3
g cirr
3
and temperatures from
1
to 20 eV give x
a
values
which are slightly less than i
p
i
asma
. This means that the results
shown above must be taken with caution and considered only
for the purpose of qualitative analyses. An accurate study
would require full time-dependent calculations.
4.
An application to ICF plasmas: spectroscopic
diagnosis of shock-ignition implosions
ABAKO has been used as part of the physics model
employed for the first spectroscopic analysis of shock-ignition
implosions. The results have been discussed and published
in [19,20]. Here, webriefiy emphasize
the
main
results.
Shock
ignition [21] is an approach to ICF that relies on the low-
adiabat implosion of a thick-wall shell target that creates a
high areal-density fuel assembly, which is subsequently driven
to ignition by a spherically convergent shock wave. Initial
shock-ignition experiments at the OMEGA laser facility were
successfully performed [22]. Furthermore, shock-ignition
has aroused significant interest in the ICF community and
several recent studies have been performed for possible shock-
ignition experiments at the National Ignition Facility [23] in
the United States as well as in the future HiPER [24] and Laser
Megajoule [25] facilities in Europe.
The shock-ignition implosions discussed here and the
associated spectroscopic analysis were performed at the
OMEGA laser facility and led by Mancini from the University
of
Nevada,
Reno. They employed plastic shell targets that had
an internal radius of
387
/xm,
a
wall thickness of
40
/xm and an
outer aluminum coating of
0.1
/xm for sealing purposes. They
were filled with 20 atm of
D
2
and 0.072 atm of Ar, which was
used for spectroscopic
diagnosis.
The observed argon radiation
emission includes K-shell x-ray line transitions that span the
photon energy range from 3000 to 4400 eV.
The model used to analyse the data is schematically
illustrated in figure 3(a). It consists of a uniform sphere to
characterize the state of the implosion core surrounded by a
uniform concentric shell to account for the compressed shell
confining the core. The parameters of the model are the
core radius R, electron temperature T
c
, electron density N
c
,
the compressed shell thickness AR, electron temperature T
s
and electron density JV
S
. On the one hand, the value of R
can be computed from JV
C
assuming mass conservation; this
assumption relies on the approximation that all of the core mass
contributes to the emission of radiation. On the other hand, the
value AR is calculated from the values of R and N
s
, as well as
the fraction
r¡
of
shell
mass that
is
imploded, which is estimated
from hydrodynamic simulations of the implosions [19]. The
uniform model approximation for the spherical shell requires
that the same imploded shell mass that contributes to the
attenuation of the core radiation also contributes to the
self-
emission of the shell. To calculate the emergent intensity
distribution for this model, we integrate the radiation transport
equation for each photon energy along chords parallel to
the direction of observation, and then further integrate all
individual chord contributions to obtain the space-integrated
spectrum for comparison with the experimental spectrum.
The frequency-dependent emissivity and opacity of the
core and shell were calculated with population number
densities from ABAKO. All bound-bound, bound-free and
free-free contributions from the argon-deuterium plasma in
the core and the carbon-hydrogen plasma in the compressed
shell within the spectral range of the measurements were
included. For a given set of plasma temperature and
50
core
emission
shell
transmission
factor
(a)
0.25 2.5
-
attenuated
core
emission
•
shell emission
-final
¡ntensit
(b)
3.4 36 38 4.0
Photon
energy
(keV)
4.2 4.4
LA^-
3.0 3.2
3.4 3.6 3.8 4.0
Photon
energy
(keV)
4.4
Figure 4. The sphere-slab model is used to illustrate the formation of the spectrum. Shell and core parameters values are T
s
= 285 eV,
N
s
= 3.8 x 10
25
cirT
3
, AR =
11
/xm, T
c
= 930
eV,
N
c
= 1.0 x 10
24
cirT
3
, R = 40
/xm.
(a) Core emission before being attenuated as it
goes through the shell and shell transmission factor, (b) Attenuated core emission, shell emission, and final emergent intensity. Figure
adapted from [20]. Copyright (2011) by American Physical Society.
density conditions, ABAKO takes into account all non-
autoionizing and autoionizing states consistent with the
continuum-lowering. Radiation transport effects on level
population kinetics were considered via escape factors [26].
Detailed Stark-broadened line shapes including the ion-
dynamics effect [27-29] were used for parent and satellite
transitions of the argon spectrum. For completeness, natural
and Doppler
line
broadening were included
as
well, but they are
small compared with Stark
broadening.
Additionally,
a
plasma
broadening effect on
the
bound-free emissivity and opacity was
included according to the approximation discussed in [30].
The model geometry shown was integrated numerically
and employed to perform the data analysis. However, to gain
insight into the model characteristics and the formation of
the emission spectrum, we used a simplified model geometry,
which corresponds to the case of a spherical plasma source
(core) seen through a plasma slab (shell)—see figure 3(b).
The advantage of this simplified model geometry is that it has
the same underlying physics of the model in figure 3(a) but
the radiation transport equation can be integrated analytically,
namely,
F"e-<
AR
+
F"
F
v
with
F„
u
= nR
1
1 +
-2K? R
K»R
(1
-2K'R\
F
s
v
= 7tR
2
^(l-e
2 (K^R)
-K'AR\
(1)
(2)
(3)
where v is the photon frequency, F
v
is the final intensity flux
and
e
v
c
,
K
V
C
,
e
s
u
and
K
V
C
stand for the temperature, density, and
photon energy dependent emissivity and opacity of the core
and the shell, respectively.
The first term in equation (1),
F^e^
K
^
AR
,
represents the
radiation emitted by and transported through the core further
attenuated by the transmission through the shell. The various
contributions to F
c
u
within the square brackets emerge after
considering the intensity which results from the integral of the
radiation transport equation computed along a chord inside a
spherical, uniform plasma source and further integrating the
intensity flux along the given line of sight over the surface of
the sphere—details can be found in [31]. The intensity flux
F
c
u
has the small and large optical depth limits given by F
c
u
->
^R
3
c
e
v
c
and F
c
u
-> JTR
2
^ for 2K
V
C
R
C
« 1 and 2K
V
C
R
C
» 1,
respectively. As stated in [31], the physical interpretation
of both limits is clear: in the optically thin case the final
result is the volume integral of the emissivity, and in the high
optical depth limit the result corresponds to a surface radiator
of effective area given by the cross-section of the sphere.
The second term in equation (1), F
s
u
, gives the
self-
emission of the shell. Hence, the emergent intensity includes
both the argon and
the
plastic
emissions.
Because
the
shell
self-
emission decreases with hv mainly as exp(-/zv/7¡), its effect
is mostly noticeable in the low energy side of the observed
photon energy
range.
Figure 4 graphically illustrates
the
details
of the formation of the spectrum according to equation (1) for
a given set of model parameters.
Fitting the observed spectrum by means of a weighted
least-squares minimization procedure yields both core and
shell temperature and density conditions. Furthermore, from
N
c
, R, N
s
and R the areal density pR of the imploded
target can also be obtained. Details of this procedure and
comparisons with hydro-simulations and experimental data
can be found in [20]. From the analysis of
the
space- and time-
integrated spectrum recorded in the referred shock-ignition
experiments, the core and compressed shell temperature and
density conditions values were extracted: T
c
= 930 ±
5%
eV,
JV
C
= 1.0 x 10
24
±
20%
cm"
3
, T
s
= 285 ± 6% and JV
S
=
3.8 x 10
25
±
25%
cirr
3
. Uncertainties in the parameters
were determined from a confidence interval statistical study
that takes into account the correlations between model
parameters [32]. Furthermore, the spectroscopic analysis also
provides an estimate of pR = 0.17 ± 20%gcirr
2
for the
target's areal density at the collapse of the implosion, which
is consistent consistent with the results of early experiments
based on particle diagnostics [22].
5. Conclusions
Fundamental research and modelling in plasma atomic physics
play an important role to understand different phenomena in a