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Hot electron transport modelling in fast ignition relevant targets with non-Spitzer resistivity

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2010 J. Phys.: Conf. Ser. 244 022031 (http://iopscience.iop.org/1742-6596/244/2/022031) View the table of contents for this issue, or go to the journal homepage for more

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The Sixth International Conference on Inertial Fusion Sciences and Applications IOP Publishing Journal of Physics: Conference Series 244 (2010) 022031 doi:10.1088/1742-6596/244/2/022031

Hot electron transport modelling in fast ignition relevant targets with non-Spitzer resistivity D A Chapman1 , S J Hughes2 , D J Hoarty1 , and D J R Swatton1 1 2

Plasma Physics Department, AWE, Aldermaston, Reading, Berkshire, RG7 4PR, UK Computational Physics Group, AWE, Aldermaston, Reading, Berkshire, RG7 4PR, UK

E-mail: [email protected] Abstract. The simple Lee–More model for electrical resistivity is implemented in the hybrid fast electron transport code THOR. The model is shown to reproduce experimental data across a wide range of temperatures using a small number of parameters. The effect of this model on the heating of simple Al targets by a short–pulse laser is studied and compared to the predictions of the classical Spitzer–H¨ arm resistivity. The model is then used in simulations of hot electron transport experiments using buried layer targets.

1. Introduction Fast ignition (FI) Inertial Confinement Fusion schemes, and many laser–driven High Energy Density Physics experiments, rely on the efficient propagation of an energetic electron beam through high density, partially ionised (and potentially degenerate) matter [1]. The transport of the beam energy is strongly affected by the resistivity of the target, which dictates the heating due to the return current via Ohm’s law, E ≈ −η(T )J. In this work the resistivity model of Lee and More [2] is used in simulations of fast electron transport in FI relevant targets using the relativistic explicit hybrid transport code THOR. The code is based on the work of Davies et al. [3], i.e. the transport of the hot electron population is modelled through interaction with a cold fluid background of electrons and ions on an Eulerian grid. The code is also computationally inexpensive, scales well on parallel systems compared to many PIC or VFP schemes, and can treat both 2D and 3D Cartesian and axisymmetric geometries. 2. Lee–More resistivity model The classical Spitzer scaling of electrical resistivity [4] is known to break down in the low temperature/high density conditions produced during short–pulse laser–matter interactions. In low temperature regions of the target the resistivity is then significantly overestimated (see Figure 1a), leading to far larger return current heating and magnetic fields than may be expected. Consequently, the Spitzer resistivity is often capped, ηmax = ηSH (100 eV), to prevent unphysical behaviour in transport calculations. However, it may be naive to assume that such an approach is sufficient for describing the evolution of solid density targets initially at room temperature. The resistivity model of Lee and More (L–M) is a practical alternative to rigorous calculations [5] which adequately captures the effects of strong coupling and degeneracy, is material dependent and easily implemented. The resistivity is calculated from an effective collision frequency,

c 2010 IOP Publishing Ltd

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The Sixth International Conference on Inertial Fusion Sciences and Applications IOP Publishing Journal of Physics: Conference Series 244 (2010) 022031 doi:10.1088/1742-6596/244/2/022031 2 ηLM = νeff /A(βµ) ε0 ωpe , appropriate to the temperature regime. A summary of the L–M model used in this work is given below

¡ ¢ -1 -1 -1 νeff = min νmax , ( νhot + νcold ) , F2 (−βµ) 4 A(βµ) = , 2 3 (1 + e−βµ )F1/2 (−βµ) Z ∗ e4 ne ln(Λ) , νhot = f (Z ∗ ) 3(2π)3/2 m2e ε20 vT3 1 4 (v + vT4 )1/4 , νmax = r0 F T ν , νcold = max 50 Tm

(1) (2) (3) (4) (5)

where f (Z ∗ ) ≈ (1 + 3.3/Z ∗ ) is an empirical factor to account for electron–electron collisions [6], vT, F is the thermal/Fermi speed, r0 is the mean ion sphere radius, Tm is the melting temperature of the material, βµ is the normalised chemical potential and Fj is the Fermi–Dirac integral of order j. A good fit to experimental data (blue curve) [7] can be obtained using the L–M model (red curve) with a set of adjustable parameters αj , that scale the collision frequency in the cold, saturation, and plasma regimes respectively. It is clear that in the low temperature regime the capped Spitzer model (dashed black curve) is inadequate to describe the resistivity.

(a)

(b)

(c)

Figure 1: (Colour online) (a): temperature scaling of resistivity models. (b–c): comparisons of the background temperature (b), and magnetic field (c) for the optimised Lee–More model (top half) and capped Spitzer (bottom half) models in initially cold (T0 = 0.025 eV) material. To gauge the effect of non–Spitzer resistivity on the heating and field structure a number of simulations of simple Al blocks have been performed using THOR. The electron source is created near the surface with Gaussian spatial and temporal profiles (to represent the laser pulse), and a Maxwell–Boltzmann distribution of energies. The distribution is characterised by a temperature, given by a suitable scaling on the laser irradiance, and by an angular spread, which is also Gaussian with a width of θ1/2 = 15◦ . The energy of the source is 10 J, which is determined by the absorbed intensity fabs I = 0.2 × 1019 Wcm-2 , spot size σ ≈ 20 µm, and pulse length τ = 0.5 ps. The resulting heating profiles at t = 2 ps through the simulations are similar for both models near the surface, but begin to diverge in the low temperature region. The average temperature for z & 25 µm is then greater by a factor of ∼ 2 for the capped Spitzer 2


model (Figure 1b). Consequently, the structure of the B–field, which is primarily generated near the surface, is largely unaffected, although the lobe extends further into the target for the Spitzer model (Figure 1c). However, this difference is too small to make a noticeable impact on the collimation of the electron beam for these conditions. Furthermore, changing the laser wavelength from 1ω (λ = 1.054 µm for Nd:Glass laser) to 2ω (λ = 527 µm), and thereby decreasing the hot electron temperature, gives greater heating near the surface due to increased field stopping and drag whilst the heating at large depths is not affected. As expected, if the initial temperature is high (T0 ∼ 100 eV) then the heating and field growth converge to the predictions of the Spitzer model. Whilst this result suggests that the Spitzer model is not unreasonable for hot electron transport modelling, it should be noted that the seemingly good agreement with the L–M model seen here is most likely due to the relatively high energy and deposition rate of the source; the target is rapidly heated to the temperature regime in which the models give similar results. Greater differences may therefore be expected if the energy of the source were substantially reduced, i.e. for ultra–short–pulse laser or X–ray driven targets. 3. Modelling fast electron heating in buried layer targets Recent AWE experiments [8] have measured the heating of thin (∼ 0.2 µm) Al foils buried in plastic targets using 1ω and 2ω p–polarised light from the CPA arm (τ = 0.5 ps) of the HELEN laser (see Figure 2a). These experiments are of interest to fast ignition as an understanding of the deposition rate of the electron beam energy in dense materials is critical to meeting ignition criteria and modelling isochoric heating experiments. A number of THOR simulations using L–M resistivity have been performed to model the range of buried layer depths and laser intensities used in the experiment; zAl = 2 − 20 µm and I = 6 × 1017 Wcm-2 (2ω), 6 × 1018 Wcm-2 (1ω), 1 × 1019 Wcm-2 (2ω). In the experiments the laser pre–pulse was mitigated to ensure isochoric heating by using a plasma mirror for the 1ω shots, and from the conversion crystal for the 2ω shots. This is modelled in the simulations by reducing the energy in the laser pulse from 50 J (a conservative estimate for HELEN) to 40 J for 1ω and 25 J for 2ω. For Gaussian spatial and temporal intensity profiles the characteristic radius of the laser spot is then given by σ = 5.47(EJ /I18 τps )1/2 µm. Beg’s law, Thot = 100(I17 λ2µ )1/3 keV, is used for the hot electron temperature (since a0 ≤ 2.2 for all intensities considered) and estimates of the absorption and divergence are made from experimental data [9, 10].

(a)

(b)

Figure 2: (Colour online) (a): Schematic representation of the experimental set up. (b): comparisons of the experimental data (dashed curves) and THOR calculations (solid curves).

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Figure 2b shows the calculated temperatures in the buried Al layers at the end of the laser pulse (solid curves) and the experimental data (dashed curves) for comparison. The average temperature has been calculated by integrating over the thickness of the buried layer and out to a cutoff radius corresponding to the point at which the temperature reaches the detector threshold (T ≈ 200 eV). The shape of the heating profiles from the high intensity runs (red and green curves) are broadly consistent with the trends observed in the data up to z ≈ 12 µm, showing a gradual decrease in temperature. However the rapid drop–off in temperature beyond this point is not observed. The high intensity 2ω run (green curve) also produces greater heating near the surface and reduced heating at depth compared to the 1ω run. Whilst this is consistent with simple estimates of the Ohmic heating rate; T˙ ∝ (I/λ)4/3 , where the current density is estimated as Jhot = efabs I/ kB Thot , the data does not support this. A sharp drop–off in temperature is seen for the low intensity 2ω run (blue curve), although this occurs slightly earlier in the target and the temperatures are significantly lower than observed. This could suggest that the absorption of the low intensity pulse may have been considerably larger (∼ 60%) than has been considered in this work. Improved agreement with the experiment may be achieved via a semi–integrated approach to modelling the electron source, e.g. taking the distributions of electron energy and angular spread from PIC or direct Vlasov simulations, and by better modelling the detector. 4. Conclusions The Lee–More resistivity model has been implemented in the hot electron transport code THOR, and the subsequent effects on hot–electron heating have been studied with a number of simulations of simple Al targets. It is shown that a simple capped Spitzer model overpredicts the heating at large depths (50 − 100 µm) by a factor of ∼ 2 compared to the L–M model, although the structure and peak value of the resulting magnetic field are not strongly affected. This result is largely independent of the hot electron temperature, as well as the initial temperature of the simulations. Initial simulations of buried layer targets heated by short–pulse produced hot electron beams have also been performed using this improved model. Whilst the results show that the heating profiles calculated by THOR broadly agree with experimental observations, the characteristic temperature drop–off for depths of z ≈ 15 µm is not well reproduced, particularly for runs with high intensity. Future work in this area will focus on refining the resistivity model and using integrated modelling including radiation–hydrodynamics and full kinetic simulations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Tabak M et al. 1994 Physics of Plasmas 1 1626–1634 Lee Y T and More R M 1984 Physics of Fluids 27 1273–1286 Davies J R 1997 Phyical Review E 56 Cohen R S et al. 1950 Physical Review 80 230–238 Ichimaru S 2004 Statistical Plasma Physics Volume II: Condensed Plasmas (Westview Press) Atzeni S and Meyer-Ter-Vehn J 2004 The Physics of Inertial Confinement Fusion (Oxford Press) Milchberg H M et al. 1988 Physical Review Letters 61 2364–2369 Hoarty D J et al. 2009 IFSA Plenary 3.0.2 Ping Y et al. 2009 Physical Review Letters Green J S et al. 2008 Physics Review Letters 100 015003–1–4

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