1. INTRODUCTION
The production of fast ions by short laser pulses was considered a
long time ago (Caruso & Gratton, 1971). The
idea was based on the concept that a substantial amount of energy can be
transferred to the electrons of a small amount of matter by short laser
pulses before quenching of the laser light absorption, caused by target
expansion. The energy is ultimately transferred to the ions by
quasi-neutral electrostatic coupling during the overall plasma
expansion.
In this approach, by a proper target dimensioning, a high energy per
nucleon can be in principle obtained by limiting the amount of heated
matter (Controlled Amount of Matter mode, (CAM)). As shown in the
following, energy per nucleon appropriate for different applications can
be achieved by this method.
The CAM mode was more recently reconsidered in connection with the ICF
approach based on the injected entropy method, a sort of assisted spark
ignition formation technique (Caruso & Strangio,
2001).
To activate the CAM mode high contrast laser pulses are needed.
Actually, to get high energy per nucleon, the target to be irradiated can
be quite tiny, depending on the laser energy available. This means that
any kind of long prepulse has to be kept to a very low level, to avoid
target destruction before the arrival of the main pulse. This matter will
be addressed below by a proper dimensioning of the target and laser pulse
by using a simple model.
The OAM mode (Hatchett et al.,
2000; Clark et al., 2000; Roth et al., 2002; Hegelich et al., 2002; Allen et al., 2003; Matsukado et al., 2003) was activated in
experiments performed with sub-picosecond PW pulses (several hundred
Joules of energy). Multi-MeV protons attributed to target contamination
were detected as emitted from the rear and the lateral sides of the
target. The acceleration process was attributed to the electric field
existing in the negative charge sheath formed, at the rear and lateral
sides of the target, by fast electrons generated in the laser-matter
interaction zone (Target Normal Sheath Acceleration (TNSA)). The
electrons, in transit through the target, are trapped by the charge
separation electric field over a distance of the order of
λDh = [Th
/(4πnh
e2)]1/2, where
Th is the hot electrons temperature in energy
units and nh their local density. In such a
sheath electric fields on the order of E ≈
(Th
/e)/λDh exist.
The concentration of high specific energy in the proton component of
ions was attributed to the highest charge-to-mass number ratios (Z/A)
featuring protons. The same mechanism is also active in any ionic
acceleration mechanism based on quasi-neutral collisionless plasma EW
(Allen & Andrews, 1970; Caruso et al., 1970; Pearlman & Morse, 1978; Denavit, 1979; Gitomer et
al., 1986), since in this case, an electrostatic field couples
the ions to the electrons (E ≈
(Th
/e)/(vs /t),
where vs is the ion-sound velocity and
t the time). Ionic acceleration by this mechanism has been also
considered to explain fast ion production by ultrashort pulses (Wilks et al., 2001).
In this paper both mechanisms (TNSA and EW) for the OAM mode are
reviewed. It will be shown that the most consistent scenario for fast ion
production is the one in which part of the acceleration process occurs in
rarefaction shocks (Bezzerides et al.,
1978) embedded in a two-temperature isothermal EW; the rest of the
acceleration is provided by the EW itself. The structure of the
two-temperature isothermal EW will be discussed with special attention to
the energetic of the process (that includes consideration of the heat flow
in isothermal structures due to infinite thermal conductivity). It is
shown that the position of the discontinuities in the EW is determined by
local thermodynamic quantities only. Finally a simple estimate for the
amount of fast ions produced is presented.
2. ESCAPING ELECTRONS
The interaction of high intensity radiation with solid targets results
in the production of high energy electrons beamed in the forward direction
and with a spectrum having a logarithmic slope temperature
Th equal to the relativistic oscillation
energy in the electromagnetic field (Hatchett et al., 2000), that is,
Th ≈ 1MeV ×
(I19
λμm2)1/2, where
I19 is the light flux density (in units of
1019 W/cm2) and λμm
is the light wavelength (in μm). Of these electrons a (usually) small
number escapes the target, the rest being trapped in it by the electric
field due to the positive target charge. The fraction of escaping
electrons is easily evaluated for fast electrons distributed according to
a Boltzmann distribution as (Ne
/Th)exp(−K/Th).
Ne is the total number of electrons produced
by the laser pulse and K is the kinetic energy. Electrons
escaping from the target will be the ones with K >
Km = α(eQ/a) where
Q is the total (positive) charge on the target, e the
unit atomic charge, a the typical radius of the charged region,
and α a form factor depending on the charge distribution and on the
initial position of the escaping electron. For instance, if Q is
distributed uniformly in a thin disk of radius a, α = 2 for
an electron positioned at the center of the charge surface and α =
4/π = 1.273 for an electron at the disk boundary. The fraction of
escaping electrons is f =
exp(−Km
/Th), whereas Q =
fNe e. From these considerations it
follows that
The number ε can be expressed as ε =
λD /a where
λD =
[a3Th
/(αe2Ne)]1/2
is the Debye length. Eq. (1) is represented in Fig.
1. The loss of electrons (target charging) will be neglected
throughout this paper because, in situations of interest here, ε =
0.003 ÷ 0.03 so that the energy associated to the escaping
electrons are much smaller than that of the trapped ones.
Fraction of fast electrons escaping a target as function of the ratio
between the Debye length λD to the charged region
size (a).
3. THE CAM MODE-TARGET AND FOCAL SPOT
DIMENSIONING
In the CAM mode the laser light energy is first transferred with
efficiency ηt to the target creating a single
high temperature (Th) electronic population.
Energy is ultimately transferred to the ions by electrostatic coupling in
a quasi-neutral, isothermal expansion.
It is assumed that the laser light energy can be transferred to the
target before the density ρ becomes less than some critical value,
ρt, when the target becomes transparent.
Some degree of beam collimation may be necessary for certain
applications. This can be obtained by exploding thin flat targets since
the pressure gradient is for these cases essentially normal to the target
surface. For this reason in the following treatment we will highlight the
case when short laser pulses are used to irradiate thin disks. To provide
a dimensioning for this case, we assign the available laser energy
(Elaser) and the required energy per nucleon
(Enucl). The ionic velocity to be achieved
will be
where Mo = 1.660 ×
10−24 g is the nucleon mass. The initial target aspect
ratio (q = Ro
/Δo, radius/thickness) is chosen such that
transparency to the laser light begins for a plasma thickness of the order
of the target diameter, that is when three-dimensional (3D) expansion is
also effective. From this it follows that q =
Ro /Δo =
ρo /ρt, where
ρo is the initial target density. The appropriate
pulse duration is tpulse ≈
Ro /Vi. From
this model simple, useful formula for target and pulse preliminary
dimensioning can be easily deduced (Caruso &
Strangio, 2001) under the assumption that
ρt = ρc is the critical
density.
For the target it is found that
where σ is the target mass density per unit surface and λ the
light wavelength.
The corresponding formulae for the laser pulse are
where tpulse is the pulse duration and
I is the light power density at the target surface. For pulse
energies in the kJ range and ionic energies in the MeV range the numerical
coefficients in Eqs. (2) and (3) frame properly the target dimensioning,
due to the weak dependences from these quantities. A fast dependence on
Enucl can be noted for
tpulse and I, this last being
independent from Elaser. Numerical examples
are reported in Table 1. To properly activate
the CAM mode it is necessary to use a pulse with an adequate contrast.
More precisely the ablation due to any pedestal forerunning the main pulse
has to be much smaller than the σ given by Eq. (3). This matter has
been recently investigated by Strangio et al. (2004).
4. THE TNSA ACCELERATION MECHANISM
The TNSA mechanism assumes that a negative sheath of thickness
λDh = (Th
/4πnh
e2)1/2 is formed at the rear and
lateral target sharp surfaces, due to the fast electrons trapped in the
target by self-consistent electric field. In the previous formula
nh is the hot electrons density. In what
follows the structure of this sheath is described and the motion of a test
particle in the structure is studied. It is assumed that the ions with
charge Z are at rest with a uniform density
nio and occupy the space for x ≤
0 (one-dimensional (1D) calculations, see Fig.
2). Two groups of electrons are considered: hot electrons
(temperature Th, density
nho for x → −∞),
and cold electrons (temperature Tc, density
nco for x → −∞).
The electronic density distributions are assumed to be of the hydrostatic
type, nh = nho
exp(eV/Th) and
nc = nco
exp(eV/Tc), where V is
the electrostatic potential.
The double layer structure created at a sharp plasma surface by fast
electrons. For x [lE ] 0, where an ionic background exists, a
positive sheath is formed (ionic excess). For x>0, where ions
are absent, hot electrons create a negative sheath.
The following dimensionless quantities will be used: h =
nh /nho,
c = nc
/nco, f =
nho /Znio,
θ = Th
/Tc and the normalized potential φ =
eV/Th, so that
The Poisson equation reads
where the coordinate x is measured in units of
λD = (Th
/4πZnio
e2)1/2, the Debye length with
temperature Th and density
Znio.
Neutrality is assumed for x → −∞, so that
for h = c = 1 the second derivative in the first of the
Eq. (7) results identically zero. The electric field must be continuous at
x = 0 (no surface charge) and zero at ±∞ (global
neutrality for the double layer). Eq. (7) can be easily integrated once
and the proper conditions for the electric field imposed.
From the continuity of the electric field at x = 0 it follows
that the dimensionless potential must be
The approximation is valid for θ >> 1, a situation usually
verified (as that f << 1). The corresponding dimensioned
value is VB ≈
−(Tc /e)(fθ
+ 1). In the following we shall consider cases for fθ ≈
1. In these cases VB ≈
2Tc /e.
The scale of variation of φ deep in the region x < 0
(that positively charged) is easily obtained from Eq. (8) by using the
approximation for φ small, dφ/dx ≈
[f + (1 −
f)θ]1/2φ ≈
θ1/2φ. Since the scale for x is
λD the distance of variation for φ is
λDc = λD
/θ1/2 = (Tc
/4πZnio
e2)1/2. The treatment for the region
x > 0 (negatively charged) can be simplified by neglecting the
contribution of the cold electrons that is the second term in the r.h.s.
of Eq. (9) for θ >> 1. The solution is
The dimensioned scale of variation is
where λDh = [Th
/(4πnho
e2)]1/2. For x →
∞ the potential φ → −∞ with a slow logarithmic
divergence so that, to evaluate the voltage drop across the sheath, a
cutoff must be introduced at a distance where the breakdown occurs for the
1D model. This occurs at a distance a, the smaller between the
target size and the laser spot radius. The voltage drop is then
.
The hot electrons density decreases much faster as
A qualitative plot of the potential φ is represented in Fig. 2 to illustrate the double layer structure. A
test ion of charge Zt and mass
Mt inserted at position
xo in the positive sheath is accelerated
toward the plasma surface at x = 0 (see Fig.
2). The extraction time can be evaluated by assuming a simplified
form of the potential for φ << 1, namely φ ≈
φB
exp(x/λDc). Whence the extraction
time (for fθ ≈ 1)
For large, negative z the approximation
holds, whereas for small, negative z the approximation
can be used. Since g(−0.5) ≈ 1.34, for a test particle
located within the positive sheath textr ≈
1/ωhybr, whereas for
xo → −∞ the extraction time
is exponentially long.
The transit time of a test ion through the negative sheath can be
estimated as
and the ratio between the extraction time and the transit time is
textr
/ttrans ≈ f1/2
<< 1. This result shows that an EW is driven in the plasma in a time
shorter than the acceleration one. This implies that the expansion wave
ought to be considered in the process right from the beginning.
5. THE EW IN A TWO-TEMPERATURE SYSTEM
The EW in a collisionless, quasi-neutral plasma has been considered by
several authors (Pearlman & Morse, 1978;
Bezzerides et al., 1978; Denavit, 1979; Gitomer et
al., 1986) also in contexts different from laser plasma
interaction (Allen & Andrews, 1970).
Important papers on the ionic acceleration by EW have been published in
the context of the studies on the plasma plume expansion toward the
focusing optics in experiments on laser light interaction with solid
targets. The ion acceleration mechanisms by EW in ultra-short pulse
interaction with solid targets have been studied by numerical simulations
(Wilks et al., 2001).
The EW in a two-temperature quasi-neutral plasma will now be
reconsidered analytically. Attention will be paid to the energetic of the
process and to the formation of gas-dynamic discontinuities. With regard
to the last topic, it will be shown that the position of the
discontinuities in a self-similar EW depends on local thermodynamic
quantities, the inner structure of the discontinuity being not relevant in
the context.
The notation used in the previous section is also adopted in the EW
treatment. Since ion motion is now taken into account, the following
additional definitions and equations will be used. The normalized ionic
density i = ni
/nio will be used in the neutrality
condition i ≈ fh + (1 − f)c,
in the continuity equation di/dt =
−i(∂v/∂x), and in the
momentum equation, iM(dv/dt) =
−ieZ(∂V/∂x). In these
equations, v is the flow (ionic) velocity and
d/dt = ∂/∂t +
v(∂/∂x). The electronic densities
h and c are given by Eqs. (6). Eliminating the potential
V, the quasi-neutrality and momentum equations become
In Eq. (17), the pressure p is measured in units of
nio
Mvso2 and
vso is the sound velocity evaluated at a
temperature Tc.
For a complete description of the process going on in EW, the system
of Eqs. (15) and (16) must be implemented by the energy equation. This is
always true, also in single-temperature EWs. Actually, in this case, since
the temperature is constant through the EW, the internal energy per unit
mass is also constant, so that the flow kinetic energy must be due to a
flow of heat. In other words, the isothermal expansion wave is driven by
an infinite conductivity thermal flow, the electric field being a vector
to transfer energy from electrons to the ions, in a sort of
thermo-electric acceleration process. The heat flux must be calculated by
the energy balance equation, in a way reminiscent of the well-known case
of the hydrodynamics of infinitely conductive fluids, where the
Ampère law is used to evaluate the current.
The energy equation is
where the heat flux Fh and the specific
energy are measured in units of nio
Mvso3 and
vso2 respectively. The pure number
δ accounts for relativistic effects (Cox &
Giuli, 1968). Limiting cases are: non relativistic δ = 1;
extreme relativistic δ = 2.
Eq. (15) reveals that for θ >> 1 an abrupt change of regime
occurs when i ≈ f. As easily seen, for i
< f the approximation i ≈ fh is very
good, that is cold electrons are expelled from the regions where
i < f (see Eq. (15)). How sharp the transition is can
be appreciated from Fig. 3a. In Fig. 3b a pictorial representation of the situation is
displayed.
(a) Normalized hot electrons density versus normalized ionic
density in a quasi-neutrality regime. For large electronic temperature
(θ = Th
/Tc >> 1) the dependence shows a
sharp change in behavior at i ≈ f =
nho /Znio
(b) For large electronic temperature (θ =
Th /Tc
>> 1) hot electrons expel cold electrons from the region where
ni Z [lE ]
nho.
An additional interesting feature that results from quasi-neutrality
and large temperature ratio regards the normalized sound velocity
vs2 = dp/di
(the normalization unit is vso2 =
ZTc /M).
When the density i decreases below f (where
i ≈ fh), the value of
vs2 pass suddenly from a value on
the order of unity to the value φ (see Fig.
4). This means that if the gradient ∇p =
vs2(∇i)i=f is evaluated, at this transition, a sudden increase of
∇p occurs and the low-density plasma is strongly
accelerated producing a high-energy jet of ions.
Normalized squared sound velocity
(vs2 =
dp/di) as function of i. The normalization
unit is vso2 =
ZTc /M. For large electronic
temperature (θ = Th
/Tc >> 1) the dependence shows a
sharp change in behavior for i ≈ f =
nho /Znio,
with a transition from the value θ for small i to a value of
the order of unity for larger i.
The solution for a self-similarity EW wave can be obtained by standard
methods (Landau & Lifshitz, 1959). We assume
that the EW starts at x = 0 and propagates towards −∞
into the plasma that occupy the region x < 0 (The region
x > 0 is initially empty). The result is
Both vs and v are measured in
units of
.
Trying a continuous solution everywhere, Eq. (20) is integrated with the
boundary condition v = 0 for i = 1 and Eq. (22) with
Fh = 0 for i = 0. In the limiting
case
.
From this it follows that at the left boundary of a conventional
single-temperature isothermal EW, the normalized heat flux
Fho = θ3/2 must be
supplied.
In the general case, non-physical and multi-valued solutions along the
variable ξ may result. This occurs where dξ = 0, that is,
from Eqs. (20) and (21), where
d(ivs) = 0. This happens for
The quantities h+ and h−
are real and distinct when 1 − 10θ + θ2 ≥ 0.
The equation 1 − 10θ + θ2 = 0 gives the
threshold temperature ratio
.
The other root is (by symmetry) 1/θthr <
1 and will be not considered. This result is not dependent on f.
The usual well known, well behaving solutions for f = 0 and
f = 1 (single temperature) correspond to displace the
multi-valued quantities region to h → 0 or to h
→ ∞, respectively.
The behavior of the density i is shown for θ =
θthr and for θ >
θthr (see Fig. 5). To
restore a physical behavior a discontinuity must be introduced
(rarefaction shock). At the interface flux of mass, momentum, and energy
must be conserved. Another condition is that the discontinuity must be
consistent with the self-similar character of the solution. For this
reason to the shock must be attributed a constant value of the similarity
variable ξs, its velocity. On both sides of the
shock, the solutions Eqs. (20), (21), and (22) must be matched. For this
reason the flow velocity relative to the shock moving with the velocity
ξs is v −
ξs = vs that is, a
local thermodynamic quantity.
The two-temperature EW displays, for any value of f, a
multi-valued behavior for all the quantities when the temperature ratio
exceeds
.
In the shock treatment the indexes 1 and 2 will be used respectively
for the inflow (left side) and for the outflow (right side). The equations
for conservation of the mass and momentum fluxes read
Since all the thermodynamic quantities in Eqs. (24) and (25) are
functions of h (see Eqs. (15) and (19)), the previous equations
allow us to find numerically the corresponding values of
h1 and h2. The shock position
ξs, and the velocity v2 follow
from
In Eq. (26) the flow velocity v1 is obtained by
integrating Eq. (20) from h = 1 (where v = 0) to the
known value of h1. The quantities
vs1 and vs2 are
local functions of h1 and h2. A
complete description of the flow is obtained by evaluating the velocity
for ξ > ξs. Eq. (20) is integrated from
h2 where the velocity is v2.
Conservation of energy is used to find the heat fluxes entering and
leaving the shock:
The enthalpy is w = ε + p/i and
the value of Fh2 is obtained by integrating
Eq. (22) starting from h = 0 where
Fh = 0, to h =
h2. Upstream the flux is evaluated by integrating Eq.
(22) starting from h1, where
Fh = Fh1. In
Fig. 6 a typical case of rarefaction shock
treatment is presented.
Expansion shock positioning for the case θ = 100, f =
0.01. It is noted the substantial drop in heat flux across the shock and
the corresponding increase in flow velocity. Downstream to the shock the
flow is practically that of a isothermal EW at the single temperature
Th.
Across the discontinuity hot electron density decreases
(h2 < h1) and the same occurs
for the voltage V:
Typically a double layer as described in Fig.
7 will be formed. Associated to this potential drop are the
increase in flow velocity (v2 >
v1) and the decrease in heat flux
(Fh2 < Fh2).
The ion acceleration in the double layer appears as due to a potential
drop powered the heat flow, in a sort of thermoelectric process.
A double layer is formed at the expansion shock in a two temperature
plasma. Quasi-neutrality will be violated within the
structure.
The flow downstream of point 2 is practically a single temperature
(Th) EW. Typical features are the exponential
dependences of i and Fh, the linear
dependence for the flow velocity v and the constant value of
.
This part of the flow represents a low density, quasi-neutral jet formed
by energetic ions. The relevant dependences for the main quantities
are:
Numerically, for a wide range of the parameters θ and f,
it is found:
In Eq. (35) H2 is the dimensioned version of the
heat flux Fh2 driving the single-temperature,
isothermal EW for ξ > ξs. The quantity
Ni represents the number of fast ions per
unit surface in this region. Strictly the time t ought to be
interpreted as the laser pulse duration, since the light pulse is the
source of energy that maintains the hot electrons temperature at the value
Th.
6. CONCLUSIONS
The process of fast ion generation by high power short laser pulses
interacting with a solid target has been discussed in the context of
different regimes.
Fast ions can be generated in the CAM mode by putting a limit on the
amount of heated matter by proper target design. Energy is transmitted to
the target before light absorption quenching starts due to target
expansion. It is found that in this regime the required power density on
the target and the laser pulse duration depend critically on the assigned
energy per nucleon.
To activate this fast ion generation mode high contrast short laser
pulses are required. This is the case because the thin target to be used
in this mode could not survive the ablation induced by any long prepulse
forerunning the energetic short pulse used to explode the target.
The TNSA scheme has been analyzed and the structure of the double
layer at the target surface involved in the process has been studied in
the framework of a two electronic temperature model (ions cold at rest).
This structure is composed by a positive layer of thickness
λDc = (Tc
/4πZnio
e2)1/2 where
Tc is the cold electron temperature and
nio the density of ions with charge
Z. In the framework of the 1D model used in this paper, the fast
electrons form a negative cloud on the surface of the target that creates
a logarithmically diverging potential. Clearly a cut-off is needed to
evaluate the voltage drop (ΔV) in this negative region. The
natural candidate is the minimum (a) between the laser spot and
the target size. The result is
where λDh = [Th
/(4πnho
e2)]1/2.
Th and nho are
the fast electrons temperature and the density. The scale of variation of
the hot electrons density is
.
A test particle placed inside the positive sheath of the double layer
is extracted from this sheath in a time much shorter than its transit time
through the negative sheath. This means that an expansion wave starts
during the ion acceleration process in the double layer, and the picture
of a static double layer accelerating ions to energies of the order
ZeΔV is not fully consistent.
The acceleration process by a two-temperature isothermal EW has also
been considered. All the ion acceleration mechanism appears as a
thermoelectric process driven by the heat flow carried by the hot
electrons. It is found that essentially the ion acceleration occurs in two
steps.
The first step is the transit through a rarefaction discontinuity
embedded within the EW, after a modest acceleration in the high-density
part of the EW. The introduction of these discontinuities is required
because multi-valued solutions in space occur for high temperature ratios
(Th /Tc).
The positioning of this shock within the EW requires the evaluation of
local thermodynamic quantities, the structure of the discontinuity being
not involved. The rarefaction discontinuity has the nature of a double
layer, where the jump of the accelerating potential is maintained by the
absorption of the heat flux flowing through the entire EW structure.
The second step takes place in the EW part located downstream of the
rarefaction discontinuity. This part of the EW is practically
single-temperature (Th) and represents a low
density, high temperature jet emitted by the high-density region.
All the relevant parameters of the ion jet follow from the EW solution
without any need for the details of the rarefaction shock structure. Here
the number of fast ions contained per unit surface in the jet is:
The time t in Eq. (37) ought to be interpreted as the laser
pulse duration, since the light pulse is the source of energy that
maintains the hot electron temperature at the value
Th.
