INSTITUTE OF PHYSICS PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY
Nucl. Fusion 43 (2003) 362–368
NUCLEAR FUSION
PII: S0029-5515(03)61692-3
Fundamental issues in fast ignition
physics: from relativistic electron
generation to proton driven ignition
A. Macchi1,a , A. Antonicci1,2 , S. Atzeni3 , D. Batani4 , F. Califano1 ,
F. Cornolti1 , J.J. Honrubia2 , T.V. Lisseikina1 , F. Pegoraro1 and
M. Temporal2,5
1
Istituto Nazionale per la Fisica della Materia (INFM), sezione A, Dipartimento di Fisica
‘Enrico Fermi’, Università di Pisa, Via Buonarroti 2, I-56100 Pisa, Italy
2
ETSII, Universidad Politecnica de Madrid, Spain
3
Dipartimento di Energetica, Università di Roma ‘La Sapienza’ and INFM, Roma, Italy
4
Dipartimento di Fisica, Università di Milano ‘Bicocca’ and INFM, Milano, Italy
5
ETSII, Universidad de Castilla-La Mancha, Spain
E-mail: macchi@df.unipi.it
Received 14 October 2002, accepted for publication 31 March 2003
Published 2 May 2003
Online at stacks.iop.org/NF/43/362
Abstract
In recent years, several schemes for laser-driven fast ignition (FI) of inertial confinement fusion targets have been
proposed. In all schemes, a key element is the conversion of the energy of a Petawatt laser pulse into a beam of
strongly relativistic electrons and the transport of the latter into a dense plasma or a solid target. The electron beam
may either drive ignition directly or be used to accelerate a proton beam which is in turn used to ignite. Both ignition
scenarios involve a number of physical processes which are widely unexplored and challenging for theory and
simulation. In this contribution, we present theoretical and numerical investigations of several fundamental issues
of relevance to FI from the stage of electron generation and transport to that of proton energy deposition, including
electron beam instabilities, electron transport in solid-density plasma, proton transport in the coronal plasma, and
requirements for proton beam driven ignition.
PACS numbers: 52.57.Kk, 52.57.-z, 52.50.Gj, 52.50.Jm
1. Introduction
An essential element of fast ignition (FI) of inertial
confinement fusion (ICF) targets [1] is the conversion of the
energy of a Petawatt laser pulse into a beam of strongly
relativistic electrons. In the original proposal [2], the electron
beam is generated by direct interaction of the laser pulse with
the coronal plasma and reaches the precompressed fuel core
after propagating in the dense plasma region. The electron
beam is expected to carry a current of several megaampere,
and its transport properties are thus affected by self-generation
of very high electric and magnetic fields, beam instabilities,
and inefficient charge neutralization due to low conductivity
of background electrons. Efficiency of energy transport is also
a key factor in alternative FI schemes [3].
Electric field generation by relativistic electrons penetrating in solid targets is also important, as an accelerating
a Author to whom correspondence should be addressed.
0029-5515/03/050362+07$30.00
© 2003 IAEA, Vienna
mechanism of hydrogen ions (i.e. protons), which may be
present in the target material either as a component or as
impurities. Energetic proton emission from solid targets has
been observed by several groups (see, e.g. [4] and references
therein). The proton beam may be intense enough to provide an alternative driver for FI, as suggested in a more recent
proposal [5, 6].
On the route from fast electron generation to ignition,
many physical processes are at play and the resulting scenario
is therefore quite complex. One needs, for example, to
understand the basic physics of processes such as generation
of electron jets and related beam instabilities, or to develop
simulation models to support test-bed experiments. The final
goal is, of course, to estimate the energy of the driving
laser pulse for a given ignition scheme. This requires to
collect and integrate the information provided by basic theory,
experiments and simulation as much as possible.
In this paper, we present several investigations of physical
processes relevant to FI which have been performed within
Printed in the UK
362
Fundamental issues in fast ignition physics
our collaboration project. We first investigate (see section 2)
the role of collisionless instabilities which may lead to the
filamentation of the fast electron current in the overdense
plasma. Next, in section 3, we discuss models for electron
transport in solid targets, and compare simulation results with
recent experimental data. Finally, in section 4, we study proton
beam requirements for FI of precompressed deuterium–tritium
(DT) fuels.
2. Fast electron filaments: Weibel or surface
instabilities?
2.1. Three-dimensional fluid simulation of the Weibel current
filamentation instability
The basic process of relativistic (or ‘fast’) electron generation
by high-intensity laser interaction with an overdense plasma is
mostly studied with particle-in-cell (PIC) simulations. Many
of such simulation studies have evidenced a filamentary
structure of the fast electron current penetrating into the
plasma, with filaments originating immediately after the
laser–plasma interaction region either in two-dimensional
[7, 8] or three-dimensional [9, 10] geometry. The origin
of current filaments has been often attributed to a beam
instability of the Weibel type, driven by the magnetic
repulsion between the fast electron current and the oppositely
directed ‘return’ current of ‘cold’ background electrons.
This current filamentation (CF) instability has been studied
extensively analytically and numerically in two dimensions
([11, 12] and references therein) assuming a plasma which is
homogeneous along the direction of the currents. Thus, in
order to address the relevance of this instability in the threedimensional regime of laser interaction with highly overdense,
inhomogeneous plasmas, a broader basic investigation is
required. In the following, we address three-dimensional
effects on the evolution of the CF instability by performing
three-dimensional relativistic, collisionless fluid simulations.
In particular, we discuss the coupling with longitudinal modes
which play an essential role in three dimensions. We stress
that our study is directly relevant to the interpretation of
PIC simulations of laser interaction with ‘coronal’ plasmas of
moderate density, where collective effects dominate. The role
of Weibel-like or filamentation instabilities in the dynamics of
fast electron transport at high densities (see section 3) should
deserve a separate analysis, taking into account, e.g. thermal
and collisional effects on the ‘cold’ beam (see, e.g. [13, 14]
where the analysis is limited to purely transverse instabilities).
The simulations are initialized assuming two oppositely
directed electron beams with zero net current, i.e. n1 v1 =
−n2 v2 where n1,2 and v1,2 are the beam densities and velocities,
respectively, and n1 + n2 = n0 where n0 is the total electron
density. A highly asymmetrical configuration (n1 /n2 = 19 ,
v1 = 0.95c, v2 = −0.105 56c) was simulated. This
corresponds to fast electrons with 1 MeV energy carrying,
for n0 = 1028 m−3 (typical of PIC simulations), an energy
flux equal to 4.6 × 1022 W m−2 . A white random noise of
amplitude 10−6 times the initial values is added as a seed
for the instability. The dimensions of the numerical grid were
64 × 64 × 128 with a spatial resolution of 0.5d
e , where
de = c/ωp is the electron skin depth and ωp = n0 e2 /0 me is
the plasma frequency.
Figure 1. Three-dimensional fluid simulation of the Weibel CF
instability. Isosurfaces of the component of the vector potential
along the beams direction at t = 60ωp−1 .
In figure 1 the component of the vector potential along the
beam direction (Az ) at t = 60ωp−1 10 fs for n0 = 1028 m−3
is shown. It is convenient to show Az because the magnetic
field generated by the instability lies predominantly in the
plane perpendicular to the beams, i.e. Bz Bx , By . We
thus see that the development of the Weibel instability in three
dimensions generates magnetic structures with typical length
scales comparable to a few times de both in the direction of
the beams and in the perpendicular plane, in contrast to what
is generally observed in PIC simulations where the filaments
extend along the beam direction for hundreds of de .
Furthermore, as observed also at later times before
entering the nonlinear regime, there is no inverse cascade in
the (x, y) plane (as well as in the (x, z) and (y, z) planes),
i.e. no coalescence of small structures into larger ones. These
results are different from the results obtained in purely twodimensional simulations in the plane perpendicular to the beam
direction, as for example, in [15]. This is not surprising
since inverse cascade processes are typical of two-dimensional
systems.
Our numerical results are consistent with the observation
that in this parameter range, the wave vector of the most
unstable mode is not strictly perpendicular to the beam
direction. In particular, by solving the full dispersion relation
[12] for the simulation parameters, the maximum growth rate is
found at k 0.8 and k⊥ 2.0 (figure 2). The curve becomes
flat for larger k⊥ ; however, at large k⊥ values the growth
rate must decrease due to thermal and collisional effects not
considered here. Furthermore, the growth rate curve becomes
nearly flat for smaller values of k in the range 0 k 0.5
and decreases very rapidly for larger values of k > 1. These
results are quite typical of the relativistic case and do not
depend qualitatively on the particular parameter choice (see
figure 2 in [12]). For the parameters of the simulation reported,
363
A. Macchi et al
(a)
(b)
Figure 2. Growth rate γ (normalized to ωp ) of the
three-dimensional Weibel CF instability vs the longitudinal (k ) and
transverse (k⊥ ) components of the wavevector (normalized to c/ωp ),
calculated analytically from a two-fluid, cold plasma model. Left
frame: γ vs k⊥ for k = 0. Right frame: γ vs k for
k⊥ = 0.6, 1, 2, 5. Parameters n1 and v1 are the normalized density
and velocity of the ‘fast’ beam.
the analytical maximum growth rate is γth 0.2ωp , close to
the value observed in simulations γnum = 0.18ωp .
2.2. Impact of surface instabilities on fast electron
filamentation
An alternative picture for the generation of fast electron
filaments is based on the effect of surface deformations at
the laser–plasma interface. It was suggested that a ‘smooth’
density deformation wide as the laser spot, as that produced
during the hole boring process, leads to high absorption and to
collimation of fast electrons due to a geometrical ‘funnel’ effect
[16]. Due to the appearance of smaller scale deformations
at higher intensities, several electron filaments, each of them
correlated with a small-scale deformation, may appear [17].
Plasma surface rippling for picosecond pulses may be due
in principle to hydrodynamic instabilities of the Rayleigh–
Taylor type, because of the strong acceleration produced by
the intense radiation pressure. However, experiments of laser–
solid interaction with shorter (<100 fs) pulses [18] suggest that
surface rippling may occur much more rapidly than the typical
scales of ion motion, and must be therefore due to electronic
mechanisms.
The search for electron surface instabilities has suggested
a more direct correlation between the plasma dynamics at
the surface and the origin of fast electron filaments. Twodimensional PIC simulations with immobile ions showed
the generation of a standing, ‘period-doubling’ oscillation
at the plasma surface due to the onset of laser-stimulated
parametric instabilities [19]. In the low-intensity regime
the process is understood as the parametric excitation of
counterpropagating surface waves (two-surface wave decay)
[20]. Figure 3 shows results for a laser pulse of intensity
1023 W m−2 and wavelength 0.25 µm impinging on a stepboundary plasma with initial density 8 × 1026 m−3 = 5nc , nc
being the cut-off density for laser propagation. Phase space
364
Figure 3. Two-dimensional PIC simulations of the interaction of a
1023 W m−2 , 0.25 µm laser pulse with an overdense plasma (density
ne = 5nc ). (a) Electron density contours showing the onset of
surface ripples due to parametric instabilities. (b) Phase space
projections showing longitudinal electron acceleration correlated
with ripple locations [19].
projections show that fast electrons are accelerated mostly near
the maxima of the standing surface oscillations, leading to an
‘imprint’ for a filamentary structure.
The apparent relation between fast electron jets and
surface instabilities needs to be further investigated. However,
it is attractive since it would imply that the filaments are
generated already at the plasma surface (i.e. over a very narrow
region, in contrast to the ‘Weibel’ picture which is based on
homogeneous, counterpropagating currents in the bulk of the
plasma), that their location is correlated with that of density
‘ripples’ at the plasma surface, and, at least when the dispersion
relation of surface modes is similar to that of linear surface
waves, that their transverse dimensions scale approximately
with the laser wavelength. All these features have been
observed in several simulations.
3. Modelling electron transport in solid targets
3.1. Experimental indications of electric inhibition
The study of fast electron transport in solid targets provides a
preliminary test bed for the stage of energy transport into the
core in the FI scheme. In a solid target, the ‘cold’ background
electrons, which provide the neutralizing return current, are
expected to be very collisional. Assuming that a simple Ohm’s
law holds for the background,
jb = σ E,
(1)
for low values of the conductivity σ a strong electric field E
must be generated. This field has a slowing effect for fast
Fundamental issues in fast ignition physics
(a)
(b)
Figure 4. Kα yield from foam plastic targets with constant mass
thickness ρL. The deviation from a constant behaviour vs ρ
indicates that the penetration range of fast electrons is not due to
collisions only, but is affected by self-generated fields [22].
electrons, whose penetration should then be inhibited in a
low-σ material.
The importance of self-generated electric fields was
studied by comparing results on the range of penetration for
different type of materials, e.g. metallic and dielectric targets
[21]. The experiment used multilayer targets all of which had
the same metal coating on the front side (i.e. the side irradiated
by the laser pulse) to ensure that the parameters of the fast
electron source were the same in all targets at a given value
of the laser intensity. The penetration of fast electrons was
diagnosed by Kα emission from fluorescent layers on the rear
side of the target. It was shown that a purely collisional (MonteCarlo) modelling of fast electron penetration was unable to
explain the differences found between targets with a central
layer made of metal or plastic, respectively, giving a first
indication of transport inhibition by self-generated fields.
The use of foam targets, having a controllable average
density which is lower than that of a solid target, was effective
in giving a further evidence of electric field inhibition [22]. For
purely collisional transport the penetration depth R and the Kα
yield would be proportional to (ρ)−1 and to ρL, respectively,
where ρ and L are the target density and thickness. Figure 4
shows the Kα yield from two fluorescent materials (Mo, Pd),
plotted against areal density from foams and normal density
plastics. As we can see there is an approximate power law
scaling with a slope 0.52 for both Pd and Mo. Since the areal
density ρL is the same for all targets, the reduction of Kα yield
when target density is decreased gives experimental evidence
of inhibition of fast electron penetration, with respect to what
can be calculated with collisional models only, and clearly
indicates the role of electric inhibition.
3.2. Hybrid simulations of fast electron transport with the
PENELOPE code
The study of fast electron transport in solid-density materials
(including plasmas, solid and dielectrics) cannot be presently
performed with PIC codes, since these require too large
computing resources for dense, cold plasmas, especially when
collisions or ionization processes are included. This motivated
the development of hybrid models integrating a kinetic, MonteCarlo description of fast electrons with the inclusion of selfgenerated magnetic fields [13, 23–25]. In all these codes, the
Figure 5. Simulations of fast electron transport in solid targets with
the hybrid MC-fluid code derived from the PENELOPE code:
experimental and calculated Kα yield vs thickness for Al (a) and
plastic targets (b).
background is modelled by Ohm’s law (1), the displacement
current is neglected and the local charge neutrality is assumed
everywhere (∇ · j = 0). This last assumption implies that
the electric field E is purely inductive, i.e. purely electrostatic
fields are neglected. The electric field is computed from Ohm’s
law ‘a posteriori’. Thus, the equation for the electric and
magnetic fields read
1
1
j
∇ ×B ,
−jf +
E= b =
σ
σ
µ0
(2)
∂t B = −∇ × E,
where jf is the fast electron current, evaluated from the particle
Monte-Carlo model. If, in addition, magnetic diffusion is
neglected [23], the electric field is opposite to the fast current
(E = −jf /σ ) and increases for low σ .
To develop an improved tool to analyse fast electron
transport experiments, the three-dimensional MC electron–
photon transport code PENELOPE [26] has been adapted by
including electron injection, field generation and diagnostics
[25]. The fields are evaluated using equations (2) in the
assumption of cylindrical symmetry. The code has been used
to simulate the experiments of [21]. A total energy of 5 J
delivered in 350 fs (fwhm) into a focal spot diameter of 10 µm,
which corresponds to a laser intensity of 1.8 × 1023 W m−2 ,
and a conversion efficiency of 25% were assumed. Electrons
propagated through a transport layer of Al or CH, followed by
two fluor layers of Mo and Pd. The results for Kα yield are
shown in figure 5. The mean energy of electrons comes out
from fitting the numerical results to the experimental points.
For Al targets, the calculations performed with fields not
included or included, give a value for the mean energy of
500 keV and 560 keV, respectively. Thus, the effect of selfgenerated fields is significant even in conducting materials.
365
A. Macchi et al
(a)
3.3. A preliminary study of electrostatic effects
Results from models assuming local charge neutrality show a
good agreement with experiments for conducting materials.
In the case of dielectrics, the results are based on a
phenomenological estimation of the resistivity and also in
several approximations to calculate the fields [13]. The
accuracy of these approximations in dielectrics has not been
addressed yet. In general, the assumption of local charge
quasi-neutrality is reasonable for a good conductor, but is more
questionable for low conductivity materials. In the latter case,
electrostatic fields generated from charge separation may play
a relevant role.
An hybrid transport code which includes electrostatic
effects is under development. In this code, the net charge
density is evaluated and Poisson’s equation is used to find
the electrostatic fields. In addition, the existence of surface
charges at the boundaries of the numerical box is allowed. Here
we report preliminary results in a one-dimensional geometry,
which implies that the electrostatic field is overestimated;
however, results are indicative at least as far as the penetration
distance of fast electrons is less than the laser spot radius.
In figure 7, simulation results for the propagation of
200 keV electrons into an Al isothermal plasma with a temperature of 100 eV are shown at the time t = 100 fs from
the simulation start. The flux of fast electrons is constant
in time (‘flat-top’ pulse) and corresponds to an intensity of
3.6 × 1021 W m−2 . Both the field profiles and the phase space
distributions evolve slowly and look very similar at following
times, as far as the injected fast electron current is constant. It
(a)
40
In the case of plastic targets, electric inhibition is greater as
consequence of its high resistivity at low temperatures. The
resistivity of plastic has been taken from the heuristic model
of [24].
The pinching effect of self-generated magnetic fields on
the fast electron beam can be seen by comparing the timeintegrated Kα distribution in the azimuthal plane (integrated
in time and in the direction of propagation) with and without
fields, shown in figure 6 for the Al target of 50 µm. If fields are
not taken into account, electrons propagate with a cone angle
of 30˚ (i.e. diameter of beam at the fluor layer of, roughly,
60 µm). When fields are turned on, the collimation of the
electron beam is enhanced.
The last issue considered has been the relative importance
of fields and collisions in the balance of energy deposition. The
comparison has been done in terms of the energy absorbed
by joule heating due to return current and by collisions of
fast electrons. The two cases of Al and plastic targets of
50 µm have been compared. In the simulations, open boundary
conditions at both sides of the simulation box were used, i.e.
electrons were allowed to escape the target. In the Al target,
26% of the fast electron energy is transferred to the plasma by
joule heating, 23% is deposited by collisions of fast electrons
and the remainder is leaked out. In the plastic target, these
fractions are 54% and 15%, respectively. It is worth pointing
out that these values depend on the boundary conditions used
at the front and rear side of targets (i.e. if refluxing effects
are taken or not into account), and to a less extent on the
distribution in energy of the injected electrons. However, also
when fast electrons are confined to the target by reflecting
boundary conditions, the ratio between energy deposition by
joule heating and by collisions is larger in plastic than in Al.
We conclude that fields are important in both cases, but play a
prominent role in non-conducting materials, as expected.
8
8
0
2.1x10
8
1.1x10
-20
y (µm)
20
3.1x10
7
-40
1.4x10
-40
(b)
-20
0
20
40
(b)
40
x (µm)
8
y (µm)
0 20
1.6x10
8
1.1x10
7
-20
6.1x10
7
-40
1.1x10
-40
-20
0
x (µm)
20
40
Figure 6. PENELOPE simulations: Kα distribution in the azimuthal
plane for Al targets with fields turned on (a) and off (b), showing
enhanced beam collimation in the first case.
366
Figure 7. One-dimensional simulations of electrostatic stopping of
a 200 keV, 3.6 × 1021 W m−2 fast electron beam injected into an Al
plasma with a 200 eV temperature. (a) Profiles of fast electron
density (——), electric field (——), and analytical predictions
of [27] (· · · · · ·). (b) (x − px ) phase space of fast electrons.
Broadening in momentum space is due to Coulomb collisions.
Fundamental issues in fast ignition physics
4. Requirements for proton beam driven ignition
A few years ago, beam requirements for FI were evaluated
assuming that the hot spot is created by pulses of particles
with assigned range and constant stopping power [28]. For
optimally focused pulses, the minimum beam energy and
power required to ignite a DT sphere with density ρ are,
respectively Eopt = 140/ρ̂ 1.85 kJ, and Popt = 2.6/ρ̂ PW,
where ρ̂ = ρ/(105 kg m−3 ). The optimal focal radius is
ropt ≈ 60/ρ̂ µm. The recent proposal of FI by laser accelerated
proton beams [5] has motivated new studies, because (i) laserinduced proton sources have wide energy spectra; (ii) the
associated velocity dispersion may give rise to substantial
power spread at the dense fuel, located at some distance d
from the source; (iii) proton stopping power in a dense target
depends on proton energy and plasma density and temperature.
Proton beam requirements for DT FI have then been
studied [29,30] by an analytical model and by two-dimensional
numerical radiation-hydro-nuclear simulations, performed
with the DUED code. In this code, proton–plasma interaction
is dealt with by the model of [31], which applies binary
collision theory to a plasma of arbitrary ionization and
degeneration, and also account in a simplified way for ion
correlation. The simulations assume that the beam is optimally
focused and the proton velocity distribution is exponential
or Maxwellian. Indeed, details of the proton spectrum vary
between experiments, but most of the spectrum is reasonably
fitted by an exponential law.
We have first considered the ignition of homogeneously
precompressed DT spheres, and have determined the proton
energy required as a function of fuel density, proton beam
temperature Tp and source-to-fuel distance d. Results for fuel
density ρ = 4×105 kg m−3 are shown in figure 8. Simulations
for other values of the density show that the optimal Tp is in
any case about 5 MeV. For d = 1–4 mm, density in the range
2 × 105 kg m−3 ρ 8 × 105 kg m−3 and optimal Tp , the
ignition energy is reasonably fitted by
Eig∗ =
90(d/1 mm)0.7
kJ.
ρ̂ 1.3
(3)
A simple analytical model [29] shows similar trends and helps
in acquiring insight into the relevant physics. An important
60
E*ig (kJ)
is found that most of the fast electrons injected into the plasma
are kept near the surface by a capacitor-like electrostatic field
within a few microns at the time shown (t = 100 fs), and only
a small fraction propagates deeper into the target. The electrostatic field is mostly generated by surface charges, while charge
neutralization is very well satisfied in the plasma volume.
In figure 7 the fast electron density profile is compared
with the analytical solution given in [27] for an homogeneous,
isothermal plasma with efficient charge neutralization. The
analytical modelling of [27] assumes that the fast electron
distribution is confined by the electrostatic field and the
density profile is normalized to the total number of electrons
injected into the target. The numerical results are very
different from the analytical predictions. One reason for this
discrepancy is that most of the injected electrons are actually
reflected from the surface field and go back to the surface, so
the total number of electrons inside the target is much smaller
than the total number of electrons injected at that time.
d = 4 mm
40
d = 2 mm
20
0
d = 1 mm
0
5
10
15
20
25
Tp (MeV)
Figure 8. Minimum proton beam energy Eig∗ for ignition of a DT
sphere (ρ = 4 × 105 kg m−3 ), radius (R = 100 µm) vs average
proton energy Tp and source-fuel distance d [29].
result is that the values of Eig∗ in equation (3) for d = 3 mm
are higher by a factor about 4 than those estimated in earlier
work [5]. Also, it should be observed that the scaling of Eig∗
with density in equation (3) is weaker than found in [28].
With the guidance provided by the above results, we have
then studied ignition and burn of fuel configurations generated
by capsule implosion [30]. In this case, we have first simulated
(by the one-dimensional radiation-hydrocode SARA [32])
the implosion of a capsule driven by a time-shaped thermal
radiation pulse. The subsequent stage of proton irradiation
and ignition has then been studied by the two-dimensional code
DUED. The simulations show that the ignition requirements
are similar to those for precompressed homogeneous fuels and
exhibit the same trends as functions of d and Tp . Frames of a
two-dimensional simulation of the ignition of a capsule with
3.5 mg of DT fuel, imploded after absorbing 635 kJ of thermal
x-rays with peak temperature of 210 eV, are shown in figure 9.
The compressed fuel has peak density of 6.7 × 105 kg m−3
and average density of 3.2 × 105 kg m−3 . Ignition requires a
26 kJ proton beam (with Tp = 3 MeV, and d = 4 mm). The
fuel releases about 490 MJ of fusion energy. It is interesting to
observe that, despite the larger than expected requirements for
ignition, a target based on standard indirect drive implosion
(with 30% of the generated x-rays absorbed by the fusion
capsule), followed by proton driven ignition is estimated to
achieve an energy gain of about 200 at a driver energy of about
2.5 MJ.
The simulations also show rather complex features related
to the time-dependence of the proton beam spectrum at the
target, and the time- and space-dependence of fuel temperature
and density [30]. It is found that only a portion of the proton
spectrum actually contributes to the generation of the ignition
hot spot. By appropriately shaping the proton spectrum, the
ignition energy could then be lowered by a factor about 2.
Beam parameters can be made less demanding by placing
the source very close to the fuel, as foreseen, e.g. by conically
guided spherical capsules [33]. To understand relevant
issues, we have performed a preliminary set of simulations,
concerning the ignition of a sector of capsule, induced by a
proton beam generated at very small distance d from the sphere
centre [30]. These simulations confirm the substantial decrease
of the required ignition energy (e.g. Eig = 12–14 kJ for beams
with temperature Tp = 3–5 MeV, and d = 0.5 mm), without
burn degradation. The feasibility of the high-convergence
367
A. Macchi et al
300 µm
ρ (g/cm3)
1000
500
0
t = 105 ps
t = 145 ps
t = 155 ps
Ti (keV)
100
10
1
0.1
Figure 9. Two-dimensional simulation of proton beam driven
ignition of radiatively imploded fuel. The frames show
two-dimensional density maps (upper row) and temperature maps
(lower row) at selected times. The igniting proton beam is released,
at time t = 0, by a source located at distance d = 4 mm from the fuel
centre (on the right-hand side of the frames), in a burst of negligible
duration. It delivers an energy of 26 kJ onto a focal spot with radius
of 10 µm; the average proton kinetic energy is Tp = 3 MeV [30].
implosion of a sphere sector thus becomes a key, and so far
an unexplored issue.
Acknowledgments
This work was supported partly by INFM through the PAIS
project GENTE and the Supercomputing Initiative, and by the
Italian Ministry of University and Research (MIUR) through
the project ‘Generation of fast electron and ion beams by
superintense laser irradiation’. S.A. gratefully acknowledges
two travel grants by the European Science Foundation FEMTO
programme.
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