4. MARCO DE REFERENCIA
4.2 MARCO TEÓRICO
4.2.2 Placas bipolares.
Theoritical Estimates on the Collisional Ionization Rates
By the laser plasma interaction on the front side, large amounts of hot electrons are accelerated which penetrate through the target and reach the back surface. The hot electrons exit the target rear surface and only the most energetic escape the target completely, while the majority will be drawn back by space charge effects and hit the target rear surface again. Recent experiments by Mackinnon et al. [42] even suggest extensive oscillations in the target. Furthermore these electrons draw colder return currents in the opposite direction. All these electrons can and will ionize atoms they encounter, however mostly within the target. To analyze the ion spectra it is important to know the contribution of collisional ionization to the back surface ions, since only those are accelerated. The ionization within the target is a more complex problem which is difficult to address experimentally but is part of the studies performed by PIC-codes.
Electron-atom collisions can be divided into soft or distant collisions with a large impact parameter and hard or close collisions with small impact parameter. The Mott theory [70] accounts for hard collisions as between two electrons but not for soft collisions [71]. As was shown by Bethe, soft collisions essentially take place by dipole interaction between the incident electron and the target electron [72]. A combined model valid for energies from ionization the ionization threshold to the keV range was developed by Kim in [73] and extended to the high energy MeV-range by Tikhonchuk in [54]. This model is used to estimate the amount of collisional ionization for the relevant range of parameters.
3.3. COLLISIONAL IONIZATION 33
3.3.1
Collisional Ionization by the hot electron component:
Analytical Estimate
From the TNSA-model of Chapter 2 we know that we have to expect a hot electron component with a temperature of∼2 MeV at a density of∼1019 cm−3. The cross sec-
tions for ionization by electron impact are well known. An estimate of the contribution of this electron component to the overall ionization balance can be obtained from [54]:
(3.10) Wcol ≈4πa2bnevet U2 H UkkBTe ln µ kBTe Uk ¶ .
whereabis the Bohr radius,ve is the electron velocity andUkandUH are the ionization potentials of the ionized species and hydrogen, respectively. The results are collected in Sect. 3.5 where they are compared to the field ionization contributions as well as to the numerical results calculated by the FLY-code.
3.3.2
Collisional Ionization by cold electrons in return cur-
rents : Analytical Estimate
The hot electron component drives return currents in the target in order to stay below the Alfv´en-limit. Measurements by Gremillet et al. [74] and simulations by Ruhl [75] suggest temperatures on the order of tens of eV for those return currents. The contribution to the ionization balance return current heating can easily be estimated. The return current with a temperature kBTeret∼50eV , which must balance the hot electron flow, i.e.
(3.11) nrete veret ' nehotvhote ⇒ nrete ∼ n
hot e vehot vret e ∼ nhote s Thot e Tret e .
While the ionization rates due to a 50 eV electron component at 100 times higher density are clearly much higher than those due to the hot component, one has to take into account the interaction times as well. The hot component can ionize the surface atoms on the rear for as long as hot electrons are supplied from the laser interaction region, i.e. for the duration of the laser pulse. The cold electrons however are pushed into the target by the quasistatic electric field on the back surface as soon as it gains some strength. When this field is strong enough to start ionizing carbon it is on the order of several GV/m, which is still more than three orders of magnitude below its maximum. While the cold electrons will be pushed back into the target, the surface ions will be accelerated outwards. The resulting overlap time of a 50eV electron and a
0.1 1 10 100 1E-9
1E-8 1E-7 1E-6
Region of "warm" electron population (50 eV) C1+ion
x
[m
]
Time [fs]
0.1 1 10 100 1E-9 1E-8 1E-7 1E-6 0.0 2.0 4.0 6.0 8.0 10.0x
[m
]
Time [fs]
E
le
c
tr
ic
F
ie
ld
[G
V
/m
]
Efield at rear surface
Figure 3.2: In the presence of a comparatively weak field of ∼ 109V/m, a cold electron component is pushed back into the target, while C1+-ions on the surface are accelerated outward. The mutual overlap time is less than 2 fs. The field is calculated by selfconsistently solving Poisson’s equation using our 1D-kinetic code (Chapter 8).
C1+-ion both initially at rest in presence of this comparatively low field of ∼109V/m
is less than 2fs as can be seen in Fig. 3.2. A higher collisional ionization frequency due to lower temperatures is therefore immediately compensated for by a decreasing interaction time.
3.4
Numerical Model of Collisional Ionization and
Recombination Processes using the FLY-code
In addition to the simple analytical estimates the numerical FLY-code [76] can be used to simulate collisional ionization and recombination with time-dependent electron den- sity ne(t), and ion density ni(t), and an equally time-dependent electron temperatureThot(t).
For a model case of a typically measured C4+-spectrum there is no significant transfer
to other charge states due to collisional ionization or recombination. The simulation starts with solid state density at the target rear surface, ni in the adjacent half space
3.5. COMPARISON OF IONIZATION MECHANISMS 35