Hohlraum physics
Ref.:1. J. Lindl, Phys. of Plasmas 2, 3933-4024 (1995) and 11,
339 (2004).
2. Laser Plasma Interactions 5, Edited by M.B.Hooper, Scottisch
Uni. Summer School, IOP, 45 (1994)
For ICF two principal approaches are pursued to delivering the
required energy to drive the implosion. Direct drive applies the
laser energy close to the critical density surface. The energy is
then transported to the ablation surface via electron transport.
In indirect drive an X-ray pulse is produced by focusing laser
beams at the inner walls of a high Z hohlraum producing an
optical thick, high temperature wall. The wall equilibriates with
the radiation field, which comes close to thermodynamic
equilibrium and becomes isotropic and Planckian. With these
properties it is relatively easy to obtain drive symmetry on a
capsule with no small scalelengths drive perturbations. In
addition, a higher mass ablation rate leads to lower growth rates
for hydrodynamic instabilities (ablative stabilisation).
X-ray drive has the disadvantages, as energy is lost to the walls
of the hohlraum.
Black body radiation
The Planckian or back body spectral intensity, the
power/bandwidth/solid angle, is
where p is the Planckian radiation energy density/bandwidth.
The peak of Ip or p occurs at hmax = 2.8kT, (and at ~ 5kT for
I). The black body spectral intensity is isotropic. The one-sided
flux Fp is
where  is the Stefan-Boltzmann constant. In units useful for
ICF, xWcm-2eV-4. ICF needs radiation temperatures
in the range 200 – 300 eV leading to an inward flux onto a
capsule T4 of xxWcm-2 giving ablation
pressures in the range of 100Mbar. This is the order of
magnitude of ablation pressure required.
The Planckian radiation energy density p = 4T4/c. At 200
eV it is xJ/cm3.
Laser absorption and conversion to X-rays
The laser light can propagate up to critical (electron) density nc
for its wavelength.
nc= 1021/2 cm-3
where  is the laser wavelengths in microns.
For densities close to critical, the absorption coefficient due to
inverse Bremsstrahlung absorption becomes high.
Collisional absorption: linear inverse Bremsstrahlung
The mechanism called inverse Bremsstrahlung is due to
electron-ion collisions in the plasma. It is so called because it
can be seen as the reverse process of classical bremsstrahlung, in
which a free electron is slowed down in the Coulomb field of an
ion and emits a photon. Instead, in inverse bremsstrahlung
absorption the electron, while oscillating in the field of the
radiation absorbs a photon during a collision with an ion. In the
two-fluid model, collisions are taken into account as an
additional term, eineve, in the force equation [Kruer, 1988]
where ei is the ion-electron collision frequency. It can be shown
that, if the ions are considered as a fixed, neutralising
background, the dispersion relation for the light wave in the
plasma becomes ( in the condition ei /<<1)
o
~ 2 = k2 c 2 + ~ 2pe (1 - i ~ei )
Consequently, the light waves are damped with damping rate :

 2pe
( 2pe  k 2c 2 )
1
2
 ei
In simple words this means that, while oscillating in the electric
field, the electrons loose their oscillation energy in collisions
withthe ions. The rate of energy lost from the light wave must
balance the rate at which the oscillatory energy of the electrons
is randomised by the electron-ion scattering with a frequency ei.
If the problem is considered from a spatial point of view ( real
and k complex), a spatial rate of damping can be deduced [Max,
1981]:
 
ki  ei
c 
2
pe
2
1
2
 ne 
1 
 nc 
Simple considerations can be used to deduce the scaling of ei
[Max, 1981]: for an electron of thermal velocity vte colliding

with background
ions of number density ni, the collision
frequency can be expressed as ei ~ nivte, where the cross
section is ~b2. The distance of closest approach b can be
obtained by balancing the potential and kinetic energy:
Ze 2 m e v 2te

b
2
Therefore,
2Ze 2 2
ne Z 2

 ei  n i  2  v te  3
me v te 
T2
e
Consequently, inverse bremsstrahlung is more efficient at
densities
 close to critical (the densest layer to which the radiation
can penetrate) and in low temperature plasmas. A rigorous
calculation of the absorption coefficient for a thermal
distribution of ions and a Maxwellian distribution of electrons,
gives:
16 
Zne2e 6 ln 
kib  2 

 3 c m KT 32  2 1n n 12
e
e
e
c

where  is the ratio of the maximum and minimum impact
parameter of the collisions, usually expressed as [Kruer,1988]:

vte
1
 pe max[ Ze 2 Te ,h 2 (meTe ) 2 ]
In particular ,

 ei  3 10 ln 
6
ne [cm 3 ]Z
3
2
e [eV ]
T

An uniform underdense plasma of length L and density ne
causes a fractional absorption a = 1-exp(-kiL) of the energy of a

pulse propagating
through it.
From the scaling inverse Bremsstrahlung is most efficient for
short laser wavelengths, high Z targets and for high densities.
Measurements of plasma absorption at different laser
wavelengths and pulse lengths and material were carried out by
Garban- Labaune et al. 1982.
From the point of view of ICF, with L ~ mm, high Z plasma, T ~
keV, short laser wavelength ~ 0.3µm and with laser fluxes in the
range of 1014 -1015 W/cm2 the absorption is about 80-90%.
X-ray conversion efficiency
The conversion of laser light to x-rays can be written as
Ix-ray = x-ray kib Iinc
where x-ray is the conversion efficiency of the absorbed flux to
x-rays. The phenomena determining the values of x-ray are
complex- they involve an accurate description of atomic physics
and radiation transfer as well as the laser absorption processes.
The measured x-ray conversion efficiency for gold is
X-ray albedo
The x-ray albedo is defined as
Radiation temperature
The radiation temperature in a hohlraum is determined by a
balance of sources and sinks.
The x-ray source is PL where PL is the laser power, the hole loss
is T4Ah, and the wall loss is (1-)T4Aw, where Ah and Aw are
the areas of the hole and wall respectively, and T is the effective
brightness temperature of the hohlraum. In balance,
Scaling experiments have been performed using 1ns constant
power (square) laser pulses. The radiation temperature TR was
measured for 1.6x2.5mm hohlraums. The data fits a simple
power balance model with x-ray conversion efficiency 
between 60 and 70%.
In general the temperature is determined by the wall loss 1 -  ~
1/TR0.5t0.5. For longer times or higher temperatures the wall loss
drops. For Nova, t ~ 1ns, T ~ 260eV and 1 -  = 0.2; in contrast
for NIF, t ~ 10ns, T ~ 300eV and 1 -  = 0.11.
Ablation pressure
Imagine wavefront moving left to right, taking in material at
velocity va and expelling at vb. From eqn's of conservation of
mass, momentum and energy:
with c = 3.3x10-3-2µm gcm-3 and assuming  =5/3
Pb = 14 -2/3 (Iinc/1014)2/3 Mbar
More accurate models make the constant 10 instead of 14.
The pressure behind the shock is Pb + bvb2 and is approximately
2 Pb.
The ablation pressure scaling with laser wavelength is
This model work well for laser light with a critical density well
below solid density, i.e.  = 0.35µm. It demonstrates that the
ablation pressure increases as c increases, that is as the
wavelength decreases. For x-ray drive, critical density is at a
much higher density.
A useful model for x-ray drive gives
Pa = 0.5T4/kT/m)
Mass ablation rate
The mass ablation rate
= v ~ Iinc1/32/3
Soft x-ray drive characterization
Laser absorption and the resultant x-ray production in or near the
walls of the hohlraum initiates the process of generating
radiation drive. This radiation is reabsorbed in all protions of the
wall (which in turn reradiates), creating an approximately
thermal flux distribution. The capsule is thus bathed in this
blackbody, approximately isotropic flux distribution. The
radiation field is not a perfect blackbody for several reasons, not
the least of which are the significant laser entrance holes. The
field is however not far from blackbody.
X-rays are absorbed on the surface of the capsule according to
the opacity of material as it is heated to a few hundred eV.
Typical value of opacity (for cold latter) for a material like
plastic result in penetration depths of several microns for 200
eV radiation. The penetration becomes greater as the material
heats up. Very simple estimates can be made of the opacity that
give for the typical case of plastic capsule: ~ 3()1/3(TkeV)-2.5. In
contrast to x-ray absorption, optical radiation is absorbed near
the critical surface for the laser wavelength and energy is
deposited much less than solid density.
X-ray (or optical) ablation provides the rocket-like force that
drives the implosion. In the x-ray case, the ablation pressure can
be calculated from the radiation hydrodynamic theory for
subsonic heat waves. The hohlraum is radiating at the rate of
T4, energy is absorbed according to opacity and drives a
radiation dominated ablation wave. Using simple
approximations for the heat capacities, the pressure from both
the “rocket-like” force and the direct heat pressure can be
approximated for plastic as
Pa ~ 8TR3.5 (Mbar)
Where the radiation temperature is in 102eV. Ablation pressures
of about 100 Mbar can be generated for a 200eV hohlraum drive.
The ablation pressure drives a “rocket-like” acceleration. The
implosion velocity can be written
where m0 is the initial mass of the shell and
the areal mas
ablation rate. From dimensional consideration the ablation
velocity can be approximated
It has been shown from computer simulations and experiments
that the quantity Pa/ Cs (where Cs is the isothermal sound
speed) is nearly constant over significant ranges of interesting
implosion parameters and in the case of a thin shell is ~ 1.5. The
sound speed can be approximated as
and vi can be rewritten as
Soft x-ray drives – symmetry considerations
Tuning of the symmetry to achieve as close as possible to
spherical illumination of the capsule, is one of the most crucial
steps in achieving high performance implosions. The localized
laser deposition and the entrance holes create departures from
spherical symmetry that must be compensated by tuning. Plasma
filling of the hohlraum, and specific details of the laser
deposition create a complicated environment that cannot be
calculated and optimum conditions can only be found by
experiment. Two cones composed of several laser beams each
create something close to a two rings of illumination on the inner
wall of the hohlraum. Moving these rings in and out with respect
to the plane of the capsule has a large effect on the low order
modes.
The symmetry conditions for particular tuning parameters are
not time invariant. The blow-off from the walls and the resulting
hohlraum filling plasma change the configuration of the drive on
the capsule. There are three principle reasons for temporal
variations of drive symmetry:
 Heating of the walls and the resultant change in albedo
 Movement of laser deposition regions due to blowoff from
walls-spot motion
 Refraction of the laser beams by filling plasma