1. INTRODUCTION
The problem of inertial fusion energy (IFE) production is addressed
by many research groups. Its attractiveness has increased recently with
the development of high power laser [e.g., National Ignition
Faciltiy (NIF) (USA), MEGAJOULE (France)] and accelerator
[e.g., Heavy Ion Synchrotron (SIS) (Germany), Terawatt Accumulator
(TWAC) (Russia)] facilities. The estimated parameters of these
machines would allow researchers to achieve high plasma parameters,
and, therefore, make a good step forward toward the investigation of
IFE targets. The laser Phelix constructed at present at Gessellschaft
für Schwerionenforschung (GSI, Germany) provides the unique
capability to investigate the physics of high energy density in matter
using both laser and ion beams.
The investigation of such complex systems requires a combination of
experimental studies using high-resolution diagnostics methods and
advanced numerical techniques. One of the basic physical processes is
soft X-ray transport in hohlraums. The reliable modeling of radiation
transport requires taking into account various parameters, such as
radiation intensity, spectrum, and angular and spatial flux
distribution, as well as optical and thermodynamical properties of the
medium. The selection of an adequate approximation for description of
radiation transport in these conditions becomes crucial.
In most of the systems of interest both for IFE research and current
and near-future experimental investigations of high energy density in
matter, optical properties of the medium and thermal conditions change
both from one region to another and in time. Therefore, a
self-consistent combination of different approximations of radiation
transport is required.
2. THE KINETIC EQUATION OF RADIATION
TRANSPORT
Let us consider in more detail the process of radiation transport in
a hohlraum, be it an inertial confinement fusion (ICF) or an
experimental target. The optical properties of the medium are known to
vary strongly within the problem. The radiation is absorbed practically
completely in a surface layer of the optically thick casing, heating
the material. The material, under the impact of radiation, changes its
internal and kinetic energy and starts expanding, and, therefore, its
optical properties change as well (the optical depth of the surface
layer, heated and expanded, decreases). In the optically thin interior
of the casing (vacuum or filled with low-density and/or low-Z
material), on the contrary, radiation penetrates through the whole region
with only a small fraction of its energy lost (absorbed and scattered) on
the way. Therefore, it is necessary to account for the long-range effects
of X-ray interaction with the medium. The radiation diffusion
approximation, which implies local equilibration of radiation field and
material thermal properties, cannot account for remote energy sources.
So, to adequately describe the physics, the kinetic equation of
radiation transport should be applied in this case:
where t is time, c is the light velocity,
Iν(r, ω,t) is
spectral radiation intensity at point r in the direction
ω, εν(r,
ω,t) is the equilibrium radiation flux at point
r in the direction ω,
κν(r,t) is the spectral radiation
attenuation coefficient.
Assuming that the surface radiates according to the Lambert law,
Iν(ω) = const, we get
for the radiation flux into half-space
The radiation intensity as a function of frequency is, according to
the Planck law
where h is Planck's constant, k is
Boltzmann's constant, and T is the effective
temperature.
Neglecting the interaction of radiation with the medium inside the
hohlraum, the transport equation can be written in the integral form:
Here t is time, c is the light velocity,
P, Q are the points on the boundary surface
S of the vacuum region, r(P,Q) is
the distance between these points, n is the inner normal to
the surface S, Jν+,
Jν− are one-way fluxes along
the inner and the outer normals to S, respectively,
qν is the normal component of the total
radiation flux, and S(P) is the part of the surface
S visible from the point P.
One of the specific features of Eq. (4) is that it can have a
discontinuous solution integrable over the surface, depicting the
geometry of the system or the discontinuities of the boundary
conditions (Babaev et al., 1995).
There are clear physical reasons why such solutions occur. The
boundary of the system may be “spotted,” with sharp
interfaces between illuminated and shaded regions due to the system and
source geometry. The other reason is the discontinuity of the boundary
conditions, that is, a strong local change of absorbing and reflecting
properties of the walls, for example, for different wall materials, or
in case of slits or holes in the surface.
For a numerical solution, the surface is approximated by a set of
patches, discretization over the frequency range is made, and
J+(ν), J−(ν) are
substituted by
Jνk+,
Jνk−:
where Nν is the number of frequency
intervals. So, Eq. (4) is replaced by the following set of
equations:
and Eqs. (5) and (6) by
The relation between
Jνk+ and
Jνk− is
In the case of the “grey” (single-group) approximation,
the equations take the form
where qi(t) ∼
T4. The coefficients
aij are the view factors between the
surface elements i and j. The accuracy of the
numerical solution of Eqs. (8) is determined by the accuracy of the
view factor calculation, which is often more complicated than the
solution of the radiation transport problem itself. The view factor for
isotropic surface sources is determined by the formula (see, e.g.,
Siegel & Howell, 1972)
where Si,
Sj are the areas of the surface elements
i, j; αi,
αj are the angles between the normals to the
surfaces at certain points of these elements and the straight line
connecting these points; and rij is the
distance between these surface points. Integration is done over the
visible fractions of the surfaces Si,
Sj. Thus the defined view factor is
symmetric: aij =
aji. It is evident also, that
[sum ]jaij =
Si. Source anisotropy may easily be
accounted for by introducing the weighting function
w(θi) in the view factor integral to
determine nonuniform input of different directions into the radiation
intensity integrated over the solid angle:
The efficient method of view factor calculation for axisymmetrical
two-dimensional (2D) geometries is described by Vasina (1997) and Vasina and Chekshin (1998). Equations (8) are solved by the
“saldo” method using view factors (see, e.g., Abzaev et al., 2000; Vasina & Vatulin, 2001).
3. TREATING OF DISCONTINUITIES OF BOUNDARY
CONDITIONS AND SOLUTION WITH THE VIEW FACTOR APPROACH
The capability of a numerical model to reproduce such peculiarities
of real problems as various kinds of discontinuities is very important
in cases when high accuracy of radiation transport simulation is
crucial. Two problems having analytical solutions illustrate how they
are handled using the view factor approach.
3.1. Radiation transport in a spherical cavity with
three types of boundary conditions
There is a time-independent radiation source with the flux
q1 at the fraction of the spherical surface
S1; the surface S2 is an
absolute diffuse reflector, q2 = 0; the surface
S3 is an absolute absorber, q3
= J+. This stationary problem has an analytical
solution. The temperatures T1,
T2, and T3 depend only on the
value of the source flux and the ratios of the surface areas, rather
than on the shape and place of S1,
S2, and S3. For
S1 = S3, and
q1 = Q1 < 0 the stationary
temperatures are
The numerical solution precisely follows the discontinuities of the
boundary conditions (see Table 1).
The deviation of the calculated temperatures from the analytical ones
for the different surface grids
3.2. Cooling of a sphere—time-dependent solution
and radiation intensity at the boundary as a function of angle
The problem of cooling of a sphere uniformly filled with radiation
corresponding to effective temperature T0 at the
initial moment has an analytical solution (V.A. Karepov, pers. comm.).
The temperature at the boundary of the sphere is
where c is the light velocity, t is the time, and
D is the diameter. This problem has an intrinsically
three-dimensional (3D) character and cannot be adequately modeled in
the 2D approximation. Table 2 shows the
accuracy of the calculation of the 3D view factors and temperature at
the surface (the errors do not grow in time).
Numerical “non-sphericity” as a function of the number of
surface patches
The radiation intensity and flux are, in principle, functions of the
solid angle ω into which a point radiates. The angular
dependence of radiation field parameters not only influences
redistribution of integral fluxes between different surface patches,
but also affects local interaction of radiation illuminating the walls
at different angles.
The radiation flux indicatrix at the surface of the sphere is at each
moment of time a step function:
where ϑ is the angle of the arc at the maximum cross section
of the sphere from the considered point to the point from which
radiation reaches the sphere boundary, 0 ≤ ϑ ≤ π. The
numerical solution is also a step function, and the accuracy of it is
determined by the angular size of the mesh cell at the sphere boundary
(no smearing occurs).
4. CONCLUSION
The hohlraum systems are typical in ICF research and experimental
investigation of X-ray–matter interaction. Modeling of such
systems is quite complicated and requires a thorough account of various
features characteristic of these problems: the system geometry (shading
by occluding bodies, etc.), the discontinuities of initial and boundary
conditions, and angular anisotropy of absorbing and reflecting
properties of the wall material. The multigroup kinetic approximation
of radiation transport allows us to reliably model such problems. The
optimal numerical implementation of this model is the integral
transport equation solved with the view factor method.
This approach was successfully used to simulate plasma conditions for
the proposed experiments at the Phelix + SIS complex (GSI, Germany;
Vasina & Vatulin, 2000). The preparatory
experimental study in this direction was started recently with the
nhelix laser at GSI Plasma Physics group (M. Roth & T. Schlegel,
pers. comm.).
The method has been applied to modeling of experiments on
X-ray–matter interactions carried out during several years on the
ISKRA-5 laser (All-Russian Scientific Research Institute of
Experimental Physics (VNIIEF), Russia) and has proven its reliability
(see, e.g., Abzaev et al., 2000).
Coupled to the kinetic + hydro description of the wall dynamics, it
represents a powerful tool for integrated investigation of the systems
with strongly inhomogeneous optical properties.
