Photon dose estimation from ultraintense laser–solid interactions and shielding calculation with Monte Carlo simulation

Radiation Physics and Chemistry 131 (2017) 13–21

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Photon dose estimation from ultraintense laser–solid interactions and shielding calculation with Monte Carlo simulation

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Bo Yanga,b, Rui Qiua,b, , JunLi Lia,b, Wei Lua,b, Zhen Wuc, Chunyan Lic a b c

Department of Engineering Physics, Tsinghua University, Beijing 100084, People's Republic of China Key Laboratory of Particle & Radiation Imaging, Ministry of Education, Beijing 100084, People's Republic of China Nuctech Company Limited, Beijing 100084, People's Republic of China

A R T I C L E I N F O

A BS T RAC T

Keywords: High-intensity laser Solid targets Photon dose Shielding Monte Carlo simulation

When a strong laser beam irradiates a solid target, a hot plasma is produced and high-energy electrons are usually generated (the so-called “hot electrons”). These energetic electrons subsequently generate hard X-rays in the solid target through the Bremsstrahlung process. To date, only limited studies have been conducted on this laser-induced radiological protection issue. In this study, extensive literature reviews on the physics and properties of hot electrons have been conducted. On the basis of these information, the photon dose generated by the interaction between hot electrons and a solid target was simulated with the Monte Carlo code FLUKA. With some reasonable assumptions, the calculated dose can be regarded as the upper boundary of the experimental results over the laser intensity ranging from 1019 to 1021 W/cm2. Furthermore, an equation to estimate the photon dose generated from ultraintense laser–solid interactions based on the normalized laser intensity is derived. The shielding effects of common materials including concrete and lead were also studied for the laser-driven X-ray source. The dose transmission curves and tenth-value layers (TVLs) in concrete and lead were calculated through Monte Carlo simulations. These results could be used to perform a preliminary and fast radiation safety assessment for the X-rays generated from ultraintense laser–solid interactions.

1. Introduction A hot plasma is produced when an ultraintense laser beam focuses on a solid target. Laser–plasma interactions subsequently accelerate the electrons in the plasma to relativistic energies through various physical processes such as J×B heating (Kruer and Estabrook, 1985; Pukhov and Meyer-ter-Vehn, 1998), resonance absorption (Forslund et al., 1977), vacuum heating (Brunel, 1987), and skin-layer heating (Bauer and Mulser, 2007). The generated hot electrons have a Maxwellian-type distribution characterized by an electron temperature T (Wilks and William, 1997; Beg et al., 1997; Haines et al., 2009). The interaction of these relativistic electrons with atoms of a solid target produces X-rays through the Bremsstrahlung process. In the last decade, many research groups (Guo et al., 2001; Magill et al., 2003; Chen et al., 2004; Courtois et al., 2011, 2013) have studied the generation of laser-driven Bremsstrahlung X-rays and some applications such as flash radiography and transmutation of long-lived nuclides. At the same time, an ionizing radiation hazard produced from the interaction between the high-intensity laser and a solid target has been observed. Measurements show that this type of ionizing radiation can ⁎

produce significant radiation dose. Borne et al. (2002) reported that photon dose varied between 0.7 and 73 mSv near the chamber at a laser intensity of ~1019 W/cm2 for 150 laser shots with energies ranging from 1 to 20 J on solid targets such as Au, Al, and Teflon. Clarke et al. (2006) measured photon doses of up to 43 mSv at 1 m per shot at a laser intensity of ~4×1020 W/cm2 with ~230 J energy on 1mm-thick gold target at the Vulcan Petawatt (PW) laser. Henderson et al. (2014) measured photon doses of up to 70 mSv at 1 m outside the target chamber at a laser intensity of ~2×1021 W/cm2 with ~510 J total energy on 2-mm gold targets at the Texas PW laser. However, the radiation protection research on this type of ionizing radiation is very limited. The photon dose generated by ultraintense laser–solid interactions was studied by Hayashi et al. (2006). They proposed an equation to estimate the photon dose as a function of the electron temperature, and assumed that the relationship between the electron temperature and the laser intensity is given by Wilks’ scaling (Wilks and William, 1997). Recently, Liang et al. (2015) have compared the equation results with a series of measurements at Stanford Linear Accelerator Center National Accelerator Laboratory's Matter in Extreme Conditions (MEC) facility and Lawrence Livermore National Laboratory (LLNL)’s Titan laser system. They found that by

Corresponding author at: Department of Engineering Physics, Tsinghua University, Beijing 100084, People's Republic of China. E-mail address: [email protected] (R. Qiu).

http://dx.doi.org/10.1016/j.radphyschem.2016.10.010 Received 10 May 2016; Received in revised form 10 October 2016; Accepted 13 October 2016 Available online 14 October 2016 0969-806X/ © 2016 Elsevier Ltd. All rights reserved.

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Maxwellian distribution, f (E) =2(E/π)1/2 exp(−E/T)/(T3/2). It can be seen that regardless of the distribution used, the exponentially decreasing feature is well known, and T is the key parameter to characterize the exponentially decreasing slope of the electron spectrum. Previous studies have shown that T is mainly dependent on the normalized laser intensity (Iλ2), which is the product of the laser intensity (I) and the square of the laser wavelength (λ) (for common amplifying mediums, Nd: glass and Ti: sapphire laser, λ values are 1.054 and 0.8 µm, respectively). Various scaling laws to determine the relationship between T and Iλ2 have been derived in the past two decades, such as Wilks’ scaling [Eq. (1)] (Wilks and William, 1997), Beg's scaling [Eq. (2)] (Beg et al., 1997), and Haines’ scaling [Eq. (3)] (Haines et al., 2009). Wilks’ scaling was derived on the basis of the ponderomotive force theory. Beg's scaling was obtained with fits to the measurement data at intensities between 1017 and 1019 W/cm2. Haines's scaling has been recently derived applying the energy and momentum conservation laws:

using Wilks’ scaling to determine the electron temperature, the equation results would overestimate the photon dose at least an order of magnitude for laser intensities > 1019 W/cm2. For mitigating the X-ray hazards at high-intensity laser facilities to an acceptable level, Allott et al. (2000) used mass attenuation coefficients of lead (0.0497 cm2/g)and concrete (0.0228 cm2/g) for the photon energy of 10 MeV, and calculated the shielding requirements for the peak intensity of 1021 W/cm2 at the PW laser. Qiu et al. (2014) calculated the dose transmission curves in several typical shielding materials for electron temperatures of 0.4 and 1 MeV. Although the laser-driven X-ray hazard imposes new constraints on radiation safety at high-intensity laser facilities, a comprehensive shielding calculation has not been still found for the laser intensity range of 1019−1021 W/cm2. In this study, extensive literature on the physics and properties of hot electrons has been reviewed, including the laser-to-electron energy conversion efficiency, hot electron spectrum, and electron divergence. On the basis of these information, the photon dose generated from ultraintense laser–solid interactions was simulated with the Monte Carlo (MC) code FLUKA (Ferrari et al., 2005). The variations in photon dose with the electron temperature, target material and thickness, and hot electron divergence were studied. The shielding effects of common materials including concrete and lead were also studied for the laserdriven X-ray source. The dose transmission curves and tenth-value layers (TVLs) in concrete and lead were calculated through MC simulations for electron temperatures of 1−10 MeV, which covered the laser intensity range of 1019−1021 W/cm2.

T = 0.511 × [(Iλ 2 /1.37 + 1)1/2 −1],

(1)

T = 0.215 × (Iλ 2 )1/3 ,

(2) 1/2

T = 0.511 × {[(2Iλ 2 /1.37)1/2 + 1] −1},

(3)

where T is the electron temperature in MeV, I is the laser intensity in units of 1018 W/cm2 and λ is the laser wavelength in μm. H. Chen et al. (2009) compared these scaling laws with recent experimental results at laser intensities above 1019 W/cm2, as shown in Fig. 1. The experimental results show the electron temperature increases as (Iλ2)0.34, which agrees well with Beg's scaling (Iλ2)0.33. It can also be seen that Haines’ scaling is in close agreement with the experimental scaling over the relevant intensity range. However, the electron temperature determined by Wilks’ scaling is higher than the experimental data at laser intensities ≥1019 W/cm2 μm2. As the laser intensity reaches up to 1021 W/cm2 μm2, Wilks’ scaling is approximately five to six times higher than the experimental scaling. The numerical simulations indicate that Wilks’ scaling overestimates the electron temperature because the plasma electron density profile steepens by photon pressure at intensities > 1020 W/cm2 (Chrisman et al., 2008). Hence, in this study, Haines’ scaling is adopted to determine the electron temperature T.

2. Source term As discussed above, Bremsstrahlung photons are generated by hot electrons, which are accelerated due to the laser–plasma interactions. To estimate the photon dose and carry out radiation protection studies, it is essential to understand the properties of hot electrons, and three key factors that describe the electron source term are discussed in the following sections: 1. Laser-to-electron energy conversion efficiency. 2. Hot electron spectrum. 3. Hot electron divergence. 2.1. Laser-to-electron energy conversion efficiency

2.3. Hot electron divergence Laser-to-electron energy conversion efficiency is used to characterize the hot electron yield, which represents the fraction of laser energy on the target converted to the total energy of hot electrons. Although the data are not always conclusive and consistent, many experiments show that 10−50% of laser energy is converted to hot electrons over the laser intensity range of 1018–1020 W/cm2 (Key et al., 1998; Hatchett et al., 2000; Yasuike et al., 2001). On the basis of some experimental results, Hayashi et al. (2006) used the conversion efficiencies of 30%, 40%, and 50% for laser intensities of 3×1019, 1020, and 3×1020 W/cm2, respectively. Recent experimental results show an enhanced conversion efficiency at intensities above 1020 W/cm2, reaching 80% for 45° incidence, and 60% for the near-normal incidence (Ping et al., 2008).

Hot electrons usually have a large divergence. Increasing the electron divergence increases the isotropic dose and decreases the

2.2. Hot electron spectrum The energy spectrum of electrons generated from an ultraintense laser hitting a solid target can be described as the relativistic Maxwellian distribution (Phillips et al., 1999; Yabuuchi et al., 2007; Tanimoto et al., 2009): f (E)=E2 exp(−E/T)/(2T3), where E is the electron energy and T is the electron temperature. Other distributions (Hatchett et al., 2000; Pennington et al., 2000; Ewald et al., 2003; Mordovanakis et al., 2010) are sometimes used to fit the experimental data, such as the Boltzmann distribution, f (E)=exp(−E/T)/T or the

Fig. 1. Comparison of various temperature scaling laws and experimental data of T.

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conversion coefficients (ICRP-74, 1997; Pelliccioni, 2000), provided in the FLUKA code. On the basis of the normal incidence model presented above, the electron divergence was added in the simulation to study the effect of hot electron divergence on photon dose. The electron divergence angle (θdiv) was assumed to have an energy-dependent distribution given by Eq. (4). Here, hot electrons were injected into 1-mm Au target with a randomly chosen angle up to a maximum off-axis angle θdiv, and the generated dose was recorded at 1 m downstream from the target. We shall refer to this model involving the hot electron divergence as the electron divergence model.

forward dose. To describe the divergence angle of hot electrons generated from ultraintense laser–solid interactions, a simple electron divergence model has been used in MC simulations, and the experimental results have been successfully reproduced (Stephens et al., 2004; C.D. Chen et al., 2009). This model is based on the classical ejection angle of a single electron in the laser field where the electrons’ ejection angles depend on their energy (Moore et al., 1995). The ejection angle of electrons in this model is given by

θ = tan−1{[2/(γ − 1)]1/2 },

(4)

where θ is the ejection angle with respect to the laser forward direction and γ is the relativistic mass increase factor (Miller, 1952). The model predicts that low-energy electrons are almost injected into a hemisphere, and high-energy electrons tend to concentrate along the laser forward direction, which is consistent with the measurements showing that high-energy electrons are better collimated than low-energy electrons (Cowan et al., 1999; Santala et al., 2000; Schwoerer et al., 2001; Clarke et al., 2006).

3.2. Shielding calculation For the shielding calculation, hot electrons were normally injected into 1-mm Au target, and a series of slabs of different thicknesses were located downstream from the target as shown in Fig. 2. The slab materials were lead (11.36 g/cm3) and concrete (composition by weight: 2.2% H, 0.3% C, 57.5% O, 1.5% Na, 0.1% Mg, 2.0% Al, 30.5% Si, 1.0% K, 4.3% Ca, and 0.6% Fe; ρ=2.35 g/cm3). Totally, 14 thicknesses in the range of 30 cm were used for lead and 19 thicknesses in the range of 300 cm were used for concrete. The distance from the target to the front surface of concrete was fixed at 200 cm, which is referred to the shielding layout of Vulcan (Clarke et al., 2006) and Texas (Henderson et al., 2014). For lead, this distance was adjusted to 100 cm referring to the shielding layout of the 100 TW laser in the Laboratory for the Use of Intense Lasers (LULI) at Ecole Polytechnique (Borne et al., 2002). The detector was an air sphere with a radius of 10 cm in which the transmitted dose was recorded. The detector was located 30 cm from the outer part of slab, as shown in Fig. 2, because 30 cm is a typical distance used in shielding calculation, as suggested by NCRP 147 (NCRP-147, 2004). To improve the efficiency of MC simulation, variance reduction techniques were used, such as Leading Particle Biasing that can avoid the geometrical increase of the number of particles in an electromagnetic shower and region importance sampling where each region importance is adjusted to maintain the number of particles approximately constant throughout the shielding material. The application of variance reduction techniques together with a sufficient number of transport histories gives a statistical uncertainty of < 3%.

3. MC Simulation The photon dose generated from the interaction between hot electrons and a solid target was studied by MC simulations. The simulations were performed with the MC code FLUKA and included two parts. In the first part, the variations in photon dose with the electron temperature, target material and thickness, and hot electron divergence were studied. In the second part, dose transmission curves and TVLs in common shielding materials, concrete and lead, were calculated. 3.1. Dose calculation For the dose calculation, it is initially assumed that hot electrons were normally injected into the target, which gives a conservative estimation of the forward dose of photons. The electron spectrum was described as the relativistic Maxwellian distribution. The electron temperature T was set to be 0.5−10 MeV, which covered the laser intensity range 1019−1021 W/cm2. The target materials included gold (Au), copper (Cu), aluminum (Al), and polyethylene (PE), and the target thickness was set to be 0.02−6 mm, similar to that used in the previous experiments (Guo et al., 2001; Borne et al., 2002; Magill et al., 2003; Chen et al., 2004; Clarke et al., 2006; Courtois et al., 2011, 2013; Henderson et al., 2014). The photon flux was recorded by a sphere detector with a radius of 3 cm, which was located 1 m downstream from the target, as shown in Fig. 2. The effective dose was estimated online, folding the particle flux with built-in flux-to-dose equivalent

4. Results and discussion 4.1. Photon dose versus hot electron temperature To study the variation in photon dose with electron temperature, the photon dose generated from the interaction between high-energy electrons with the relativistic Maxwellian distribution and a gold target was calculated for electron temperatures of 0.5–10 MeV. The target thickness here was set to be the optimized thickness 1 mm corresponding to the maximum dose (see Section 4.2). The simulation results can be fitted by the following equation, as shown in Fig. 3:

Hx = 0.144 × T 1.77,

(5)

where Hx is the photon dose at 1 m per electron energy (mSv/J) and T is the electron temperature in MeV. Fig. 3 also plots the results estimated with the dose equation given by Hayashi et al. (2006). It was obtained by integration over the relativistic Maxwellian distribution of an empirical formula given for monoenergetic electrons of linear accelerators. It is shown that the FLUKA simulation results agree with the equation results within a factor of 2. It is important to note that according to the equation results, the photon dose is proportional to T2 in the range < 3 MeV and to T in the range > 3 MeV. This is different from the FLUKA simulation results indicating that the photon dose is proportional to T1.77. This

Fig. 2. Sketch of the geometry in the simulation. Detector a is used to measure the forward dose of photons directly from the target (without shielding in the dashed frame). Detector b is used to measure the transmitted dose after the shielding material. The distance from the target to the front surface of the shielding is adjusted to 100 cm, if the shielding material is lead. Figure is not drawn to scale.

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Fig. 5. Plots of photon dose obtained by the electron divergence model (cycles) and the normal incidence model (squares).

Fig. 3. Photon dose versus electron temperature T. The squares are estimated using the Monte Carlo code FLUKA in this study. The solid line is calculated with the dose equation given by Hayashi et al. (2006).

4.2. Photon dose versus target thickness and material difference may be caused by the uncertainty of the empirical formula given for monoenergetic electrons of linear accelerators. It is shown in the IAEA 188 report (Swanson, 1979) that the empirical formula is consistent with the experimental result of photon dose within a factor of 2.

Fig. 4(a)−(c) shows the variation in photon dose with target thickness and material for electron temperatures of 1, 3, and 10 MeV, respectively. The target materials here include Au, Cu, Al, and PE, and the target thickness ranges from 0.2 to 6 mm. The simulation results show some similar trends for electron temperatures of 1−10 MeV. For high-Z targets such as Au, the variation

Fig. 4. Photon dose versus target material thickness for T=1 MeV (a), 3 MeV (b), and 10 MeV (c).

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in photon dose with target thickness is small, and the photon dose agrees within a factor of 3 over the target thickness ranging from 0.02 to 6 mm. The maximum dose is reached for a target thickness of about 1 mm. For Cu, the photon dose increases sixfold over the target thickness ranging from 0.02 to 1 mm, and seems saturated when the target thickness is larger than 1 mm. For low-Z targets, the photon dose increases approximately 30-fold over the target thickness ranging from 0.02 to 6 mm for Al or 0.1–6 mm for PE. For thin targets, the photon dose significantly increases with atomic number Z. The difference of photon dose between Au and PE is nearly two orders of magnitude at a target thickness of 0.1 mm. On the contrary, photon doses agree within a factor of 2 for different target materials (Au, Cu, Al, and PE) when the target thickness increases to 6 mm. These simulation results are consistent with the observations made in the previous experiments with various target materials and thicknesses (Clarke et al., 2006; Sawada et al., 2015). Fig. 6. θdiv/θ1/2 and θdiv versus electron energy. The electron divergence θdiv determined by Eq. (4) is shown as a dashed line corresponding to the right axis. The Bremsstrahlung divergence θ1/2 is given by θ1/2 [deg] =100/E [MeV], where E is the electron energy (Swanson, 1979).

4.3. Effect of hot electron divergence on photon dose Fig. 5 plots the photon dose obtained by the electron divergence model. Here, the photon dose can be fitted by the following equations, as shown in Fig. 5. The photon dose in Section 4.1, obtained by the normal incidence model, is also plotted for comparison.

Hdiv = 0.091 × T 1.52 (0.5 MeV < T < 2 MeV ), Hdiv = 0.113 × T 1.28(2 MeV ≤ T < 10 MeV ),

(6)

where Hdiv is the photon dose at 1 m per electron energy (mSv/J) and T is the electron temperature in MeV. Because high-energy electrons are better collimated than lowenergy electrons in ultraintense laser–solid interactions, as discussed in Section 2.3, one could think that the difference in photon dose between the two models would decrease as the electron temperature increases. However, the simulation results show that the difference in photon dose between the two models gradually increases with the electron temperature. The explanation of this variation could refer to the results given by Hayashi et al. (2006), who found that the effect of the electron divergence on the forward photon dose in fact depended on the ratio of the electron divergence θdiv and the Bremsstrahlung divergence θ1/21 A plot of the ratio θdiv/θ1/2 against electron energy based on the hot electron divergence given by Eq. (4) is plotted in Fig. 6. It is evident from the figure that although θdiv is reduced with the increasing electron energy, the ratio θdiv/θ1/2 is approximately proportional to the square root of electron energy. This explains why the influence of hot electron divergence on photon dose gradually increases with the electron temperature.

Fig. 7. Comparison of the calculated dose and the experimental results. Different symbols represent the experimental results obtained at different laser facilities. The solid line represents the photon dose estimated with Eq. (5) and Haines’ scaling. The dashed line represents the photon dose estimated with Eq. (6) and Haines’ scaling.

4.4. Comparison of the calculated dose and the experimental results Experimental results of photon dose obtained at the different laser facilities, LULI (Borne et al., 2002; Lefebvre et al., 2003), Vulcan (Edwards et al., 2002; Clarke et al., 2006), Titan (Bauer et al., 2011), Texas (Henderson et al., 2014), and OMEGA Extended Performance (EP) (Courtois et al., 2011, 2013) and MEC (Liang et al., 2015), are presented in Fig. 7. In these experiments, the laser intensity ranges from 5×1018 to 2×1021 W/cm2, and in most cases, high-Z targets are used with thicknesses of 1–2 mm. A large dispersion of the experimental data is observed at the same normalized laser intensity. It may be caused by the variation in the experimental parameters (e.g., fluctuations of density profile of the plasma created by the prepluse and fluctuations of the laser focal spot leading to variation on the focused intensity on target). It should also be pointed out that thin or Fig. 8. Plots of dose transmission curves in concrete for electron beams of 6 and 10 MeV: calculated by FLUKA (squares and cycles) and Nelson and LaRiviere (1994) (solid and dashed lines).

1 The angle θ1/2 represents the Bremsstrahlung divergence generated by monoenergetic electrons and the high-Z target, which is given by θ1/2 [deg] =100/E [MeV], where E is the electron energy (Swanson, 1979).

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Fig. 9. Dose transmission curves in concrete (a) and lead (b) for T=1−10 MeV.

Fig. 10. The first and subsequent TVLs in concrete (a) and lead (b) for T=1−10 MeV.

upper boundary of the experimental results over the relevant intensity range. It is also shown that the difference in the calculated dose between the normal incidence model and the electron divergence model is much smaller than the large dispersion of the experimental results. The comparison indicates that the calculation method used in this study is applicable to estimate the photon dose generated from ultraintense laser–solid interactions for radiation protection purposes. Here, using Haines’ scaling to determine the electron temperature and the assumption made concerning η(Iλ2), an equation to estimate the photon dose as a function of the normalized laser intensity is obtained from Eq. (5), given by

low-Z targets (2.5-μm PE, 10-μm Al, 15-μm Ni, 5-μm Au) were used in the experiments at MEC. The photon dose is calculated using Eqs. (5) and (6) on the basis of the normal incidence model and the electron divergence model, respectively. In addition, two assumptions are made to correlate the calculation results with the experimental results. Assumption 1:. Haines’ scaling is used to determine the electron temperature T, as discussed in Section 2.2. Assumption 2:. On the basis of some experimental results in Section 2.1, the laser-to-electron energy conversion efficiency η(Iλ2) is assumed to be 30%, 50%, and 60% for 5×1018 W/cm2 μm2 < Iλ2 < 5×1019 W/ cm2 μm2, 5×1019 W/cm2 μm2 < Iλ2 < 5×1020 W/cm2 μm2, and 5×1020 W/cm2 μm2 < Iλ2 < 2×1021 W/cm2 μm2, respectively. On the basis of the above assumptions, Fig. 7 presents the comparison of the calculated dose and the experimental results. It is evident from the figure that the calculation results almost cover the

1/2

1.77

Hx′ = 0.044 × {[1 + (2Iλ 2 /1.37)1/2 ] −1} Hx’

× η(Iλ 2 ),

(7) 2

is the photon dose at 1 m per laser energy (mSv/J) and Iλ is where the normalized laser intensity in units of 1018 W/cm2 μm2. For 5×1018 W/cm2 μm2 < Iλ2 < 5×1019 W/cm2 μm2, 5×1019 W/cm2 μm2 18

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< Iλ2 < 5×1020 W/cm2 μm2, and 5×1020 W/cm2 μm2 < Iλ2 2×1021 W/cm2 μm2, η (Iλ2) is 30%, 50%, and 60%, respectively.

electron temperatures of 1−10 MeV. It is also shown that the first TVL is always larger than the second. This could be due to the build-up factor (Chilton et al., 1984) that occurs for broad beams, or the fact that the high-energy secondary particles induce a shower at inner layers of the shielding material when the electron temperature is high enough. Moreover, for the low electron temperature, such as, T=1 MeV, except for the first TVL value, the subsequent TVL value gradually increases with the shielding thickness, and reaches an equilibrium value very deep in concrete. Fig. 11(a) and (b) shows the photon spectra at different depths in concrete at T=1 and 10 MeV, respectively. For the case of T=1 MeV, it is shown that the photon spectrum seems much less energetic than that at T=10 MeV, and is consecutively hardened throughout the shielding material as the shielding thickness increases. In contrast to concrete, the TVLs in lead show less correlation with the electron temperature, and range from 5.0 to 5.5 cm. Fig. 12(a) and (b) shows the photon spectra at different depths in lead at T=1 and 10 MeV, respectively. It can be seen that the photon spectrum in lead is rapidly modified according to the shape of the attenuation coefficient curve for lead (NIST Standard Reference Databases, 1996). Here, the radiation field is dominated by photons near the Compton minimum, and the attenuation is controlled by the attenuation coefficient near that energy. In addition, for T=1 and 2 MeV, the first TVL is smaller than the subsequent TVL because of the effective attenuation of lowenergy photons in lead. For T=10 MeV, except for the first TVL, the subsequent TVLs are smaller than those at lower electron temperatures. This could be attributed to the fact that the attenuation

<

4.5. Verification of MC methodology in transmission calculation To verify the applied MC methodology in transmission calculation in the shielding material, a comparison was made with the transmission curves calculated by Nelson and LaRiviere (1994) and by FLUKA. Nelson et al. calculated dose transmission curves in ordinary concrete for monoenergetic electron beams of 6, 10, and 25 MeV, which have been validated through comparison with the measurement data. Here, the simulation parameters set in FLUKA were identical to the parameters used by Nelson et al., including source, target, geometry, and scoring. Fig. 8 shows a plot of the transmission curves calculated by FLUKA and Nelson et al. for electron beams of 6 and 10 MeV. It is evident from the figure that a rather good agreement could be obtained between the literature results and the results calculated by FLUKA. 4.6. Dose transmission curve and TVL Fig. 9(a) and (b) shows families of dose transmission curves in concrete and lead for electron temperatures of 1−10 MeV. Furthermore, the first and subsequent TVLs in concrete and lead are derived as shown in Fig. 10(a) and (b). For concrete, it is shown that the TVL values increase with the electron temperature as expected, and range from 32 to 57 cm for

Fig. 11. Photon spectra at various depths in concrete for T=1 MeV (a) and 10 MeV (b).

Fig. 12. Photon spectra at various depths in lead for T=1 MeV (a) and 10 MeV (b).

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coefficient in lead is significantly increased when the photon energy is > 10 MeV, because of the pair production process. 5. Conclusions In this study, the photon dose generated from laser–solid interactions was simulated with the MC code FLUKA. The variations in photon dose with several key parameters such as electron temperature, target material and thickness, and hot electron divergence, were studied. Comparison of experimental results with the literature data shows that the calculation method used in this study is applicable to estimate the photon dose generated from ultraintense laser–solid interactions for radiation protection purposes, and an equation to estimate the photon dose as the function of the normalized laser intensity is given. The shielding effects of common materials including concrete and lead were also studied for the laser-driven X-ray source. The dose transmission curves and TVLs in lead and concrete were calculated through MC simulations. The results of this study could be used to perform a preliminary and fast safety assessment for the X-rays generated from ultraintense laser–solid interactions. It should be pointed out that the development of high-intensity laser facilities continues to be rapid, and multi-PW or multi-10 PW facilities will appear online within the next few years. For those facilities, the electron energy can be accelerated up to a few hundred MeV or GeV, and particular attention should be given to the neutron hazards. It is well known that the high-energy neutrons cannot be effectively attenuated, and they usually dominate the particle environment outside thick shielding. Acknowledgments This study was supported by the National Natural Science Foundation of China [Grant Nos. 11275110, 11375103], the Subject of National Science and Technology Major Project of China [Grant No. 2013ZX06002001-007], and the Collaborative Innovation Center of Public Safety. References Allott, R., Wright, P., Danson, C., Edwards, C., Neely, D., Norreys, P., Rodkiss, D., Wyborn, B., 2000. Vulcan Petawatt Upgrade: the radiological perspective. Central Laser Facility Annual Report 1999/2000 pp. 177–179. Bauer, D., Mulser, P., 2007. Vacuum heating versus skin layer absorption of intense femtosecond laser pulses. Phys. Plasmas 14, 023301. Bauer, J., Liu, J. C., Prinz, A. A., Tran, H., Xia, Z., 2011. High intensity laser induced radiation measurements at LLNL. SLAC Radiation Physics Note RP-11-11 Beg, F.N., Bell, A.R., Dangor, A.E., Danson, C.N., Fews, A.P., Glinsky, M.E., Hammel, B.A., Lee, P., Norreys, P.A., Tatarakis, M., 1997. A study of picosecond laser–solid interactions up to 1019 Wcm−2. Phys. Plasmas 4, 447–457. Borne, F., Delacroix, D., Gelé, J.M., Massé, D., Amiranoff, F., 2002. Radiation protection for an ultra-high intensity laser. Radiat. Prot. Dosim. 102, 61–70. Brunel, F., 1987. Not-so-resonant, resonant absorption. Phys. Rev. Lett. 59, 52–55. Chen, C.D., Patel, P.K., Hey, D.S., Mackinnon, A.J., Key, M.H., Akli, K.U., Bartal, T., Beg, F.N., Chawla, S., Chen, H., Higginson, D.P., Link, A., Ma, T.Y., MacPhee, A.G., Stephens, R.B., Van Woerkom, L.D., Westover, B., Porkolab, M., Freeman, R.R., 2009. Bremsstrahlung and Kα fluorescence measurements for inferring conversion efficiencies into fast ignition relevant hot electrons. Phys. Plasmas 16, 082705. Chen, H., Wilks, S.C., Kruer, W.L., Patel, P.K., Shepherd, R., 2009. Hot electron energy distributions from ultraintense laser solid interactions. Phys. Plasmas 16, 020705. Chen, L.M., Forget, P., Fourmaux, S., Kieffer, J.C., Krol, A., Chamberlain, C.C., Hou, B.X., Nees, J., Mourou, G., 2004. Study of hard x-ray emission from intense femtosecond Ti: sapphire laser-solid target interactions. Phys. Plasmas 11, 4439–4445. Chilton, A.B., Shultis, J.K., Faw, R.E., 1984. Principles of Radiation Shielding. PrenticeHall, Inc, Englewood Cliffs, NJ. Chrisman, B., Sentoku, Y., Kemp, A.J., 2008. Intensity scaling of hot electron energy coupling in cone-guided fast ignition. Phys. Plasmas 15, 056309. Clarke, R.J., Neely, D., Edwards, R.D., Wright, P.N.M., Ledingham, K.W.D., Heathcote, R., McKenna, P., Danson, C.N., Brummitt, P.A., Collier, J.L., Hawkes, S.J., Hernandez-Gomez, C., Holligan, P., Hutchinson, M.H.R., Kidd, A.K., Lester, W.J., Neville, D.R., Norreys, P.A., Pepler, D.A., Winstone, T.B., Wyatt, R.W.W., Wyborn, B.E., Hatton, P.E., 2006. Radiological characterisation of photon radiation from ultra-high-intensity laser–plasma and nuclear interactions. J. Radiol. Prot. 26, 277–286. Courtois, C., Edwards, R., La Fontaine, A.C., Aedy, C., Barbotin, M., Bazzoli, S., Biddle,

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