THE LOWER BOUNDARY OF THE TYPICAL ANGLE OF SCATTERING OF IONS

Key et al. (Reference Key, Freeman, Hatchett, Mackinnon, Patel, Snavely and Stephens2006) estimated the typical angle of scattering of ions in the material of the protective membrane by using the Molière theory. According to this theory, after passage of the completely ionized ions of element with the atomic number Z through the uniform foil consisting of atoms of one element with the atomic number Z _f and the atomic mass A _f, the angular distribution of the ions is determined by the parameter

(1)

where t _f is the surface density of the foil or, in other words, its thickness measured in g/cm², p and v are the ion momentum and the ion velocity corresponding to its average kinetic energy ɛ in the foil, and the parameter B that is the root of the equation B − ln B = b, where

(2)

β is the ratio of v to the velocity of light, α = ZZ _fe ² /(ħ v), e is the absolute value of the electron charge, and ħ is the Planck constant (Bethe, Reference Bethe1953; Bichsel, Reference Bichsel and Gray1963, Reference Bichsel, Attix and Roesch1968; Key et al., Reference Key, Freeman, Hatchett, Mackinnon, Patel, Snavely and Stephens2006; Marion & Zimmerman, Reference Marion and Zimmerman1967; Molière, Reference Molière1948). According to Bichsel (Reference Bichsel and Gray1963, Reference Bichsel, Attix and Roesch1968), the Molière theory is applicable at B ≥ 4.5 (see also Bethe, Reference Bethe1953; Key et al., Reference Key, Freeman, Hatchett, Mackinnon, Patel, Snavely and Stephens2006; Marion & Zimmerman, Reference Marion and Zimmerman1967).

Key et al. (Reference Key, Freeman, Hatchett, Mackinnon, Patel, Snavely and Stephens2006) used the value as the typical angle of scattering of the ions. About 62 to 63% of the scattered ions will have the angle between the vector of the velocity and the initial direction of motion in the range from zero to γ₀, while about 73 to 75% of the scattered ions will have this angle in the range from zero to 1.2γ₀ (Bichsel, Reference Bichsel and Gray1963, Reference Bichsel, Attix and Roesch1968).

The Molière theory describes the situation when the stage of ionization of the material of the foil corresponds to the normal conditions (Bethe, Reference Bethe1953; Bichsel, Reference Bichsel and Gray1963, Reference Bichsel, Attix and Roesch1968; Marion & Zimmerman, Reference Marion and Zimmerman1967; Molière, Reference Molière1948). In fast ignition scenarios under consideration, the laser-accelerated ions will scatter either in a two-layer obstacle, consisting of the non-ionized and ionized material of the membrane, or, if the membrane is sufficiently thin, in the ionized material of the membrane (Barriga-Carrasco et al., Reference Barriga-Carrasco, Maynard and Kurilenkov2004; Maynard & Barriga-Carrasco, Reference Maynard and Barriga-Carrasco2005; Ramis & Ramírez, Reference Ramis and Ramírez2004). Scattering in the ionized material is stronger than that in the non-ionized one with the same surface density. This can be demonstrated by using the Thomas-Fermi theory. According to this theory, ionization of the atom or an increase in the ionization stage of the isolated positive ion reduces the negative charge density ρ_e, corresponding to the bound electrons, in any space region (see, e. g., March, Reference March, Lundqvist and March1983). A decrease in ρ_e results in reduction of screening of the positive electric charge of the nucleus and, thereby, enhancing of the electric field scattering the moving charged particles. When the ions are placed in the plasma, the electric fields of the nuclei are also screened by the free electrons, but such screening is weaker than that by the bound electrons. Note that in the Molière theory the atoms of the scattering foil are being described within the framework of the Thomas-Fermi theory (Bethe, Reference Bethe1953; Molière, Reference Molière1948). Note also that Maynard and Barriga-Carrasco (Reference Maynard and Barriga-Carrasco2005) presented the results of the computer simulation of the effects related to density and temperature of the foil scattering protons.

Thus, when the whole membrane or its layer is ionized and t _f is known, γ₀ can serve as the lower boundary of the typical angle of scattering of the ions in the material of the membrane.

When calculating t _f, it is convenient to use the formula t _f = k _ex ρ_m l _m, where k _ex is a coefficient taking into account a possible decrease in the surface density of the ionized material due to its transversal expansion, ρ_m is the density of the membrane, and l _m is its initial thickness. Since the expansion of the membrane material will occur mainly in the direction that is perpendicular to the membrane surface, k _ex will probably not be less than 0.5 to 0.7 even if the special measures to reduce it are taken and in some cases will be close to unity (see Barriga-Carrasco et al., Reference Barriga-Carrasco, Maynard and Kurilenkov2004; Maynard & Barriga-Carrasco, Reference Maynard and Barriga-Carrasco2005; Pert, Reference Pert1979; Ramis & Ramírez, Reference Ramis and Ramírez2004; Taylor, Reference Taylor1987).

REQUIREMENTS ON COMPOSITION AND MINIMUM THICKNESS OF THE GOLD PROTECTIVE MEMBRANE OF THE TARGET WITHOUT CONE WITH THE EQUIMOLAR D-T FUEL

The protective membrane of the target without cone will be influenced by the compressing radiation (Albright et al., Reference Albright, Schmitt, Fernández, Cragg, Tregillis, Yin and Hegelich2008; Barriga-Carrasco et al., Reference Barriga-Carrasco, Maynard and Kurilenkov2004; Fernández et al., Reference Fernández, Albright, Flippo, Hegelich, Kwan, Schmitt and Yin2008, Reference Fernández, Honrubia, Albright, Flippo, Gautier, Hegelich, Schmitt, Temporal and Yin2009; Honrubia et al., Reference Honrubia, Fernández, Temporal, Hegelich and Meyer-Ter-Vehn2009; Maynard & Barriga-Carrasco, Reference Maynard and Barriga-Carrasco2005; Ramis & Ramírez, Reference Ramis and Ramírez2004; Roth et al., Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pegoraro, Bulanov, Campbell, Perry and Powell2001). Making such a membrane of one or several low-Z elements or such elements and small amount of the high-Z element(s) is inexpedient. The main reasons are the following. The influence of the compressing radiation on such membrane would result in formation of a rather large plasma cloud (see, e. g., Eidmann et al., Reference Eidmann, Földes, Löwer, Massen, Sigel, Tsakiris, Witkowski, Nishimura, Kato, Endo, Shiraga, Takagi and Nakai1995) and the necessity to prevent damage of the fuel capsule by this cloud. Also the motion of the non-evaporated part of the membrane toward the ion source under the influence of the ablative pressure would result in the necessity to place the ion source at the relatively long, on the order of 1 mm, distance d ₁ from the membrane (see also Borghesi et al., Reference Borghesi, Fuchs, Bulanov, Mackinnon, Patel and Roth2006; Ramis & Ramírez, Reference Ramis and Ramírez2004). When using at least some of the methods of acceleration of the ions, this would cause the strong deterioration of focusing of the ions on the ion source (see, e. g., Gus'kov, Reference Gus'kov2001; Shmatov, Reference Shmatov2003).

Thus, the protective membrane being influenced by the compressing radiation should be made of the high-Z element(s). According to Borghesi et al. (Reference Borghesi, Fuchs, Bulanov, Mackinnon, Patel and Roth2006) and Roth et al. (Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pegoraro, Bulanov, Campbell, Perry and Powell2001), if the material of the membrane has the usual solid-state density, l _m should be on the order of 10 µm (here and below the target parameters, which can depend on the composition of the fuel, correspond to the equimolar D-T fuel). For example, Roth et al. (Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pegoraro, Bulanov, Campbell, Perry and Powell2001) described the gold membrane with l _m = 30 µm. Barriga-Carrasco et al. (Reference Barriga-Carrasco, Maynard and Kurilenkov2004) and Maynard and Barriga-Carrasco (Reference Maynard and Barriga-Carrasco2005) analyzed operation of the gold membranes with l _m from 1.5 to 46.5 µm. At l _m = 3 µm, the membrane side that faces the proton source would be heated to the temperature of about 190 eV and, as a result, d ₁ must be at least of 7 cm (Barriga-Carrasco et al., Reference Barriga-Carrasco, Maynard and Kurilenkov2004). Such values of d ₁ seem to be unacceptable, because they impose the very hard requirements on the smallness of spreads of the directions and absolute values of the initial ion velocities (see also Atzeni et al., Reference Atzeni, Temporal and Honrubia2002; Barriga-Carrasco et al., Reference Barriga-Carrasco, Maynard and Kurilenkov2004; Gus'kov, Reference Gus'kov2001; Maynard & Barriga-Carrasco, Reference Maynard and Barriga-Carrasco2005; Shmatov, Reference Shmatov2003).

The requirements on the composition and minimum thickness of the protective membrane of the indirect compression target without cone seem to be close to those on these parameters of the layer confining radiation in hohlraum of this target. The results of computer simulation of operation of the walls of such hohlraums confirm that at least when the maximum compressing radiation temperature T ^max_rad is about 300 eV, l _m of the gold membrane should be on the order of 10 µm. For example, Ramis and Ramírez (Reference Ramis and Ramírez2004) have described the situation when T ^max_rad = 329 eV and evaporation of 30% of the material of the gold wall with the initial thickness of 40 µm occurs. Note that many of the parameters describing operation of the membrane, made of the low-Z element(s) or containing large amount of the low-Z elements, would be close to those describing operation of the ablators of the indirect compression targets.

EXAMPLES OF THE TYPICAL ANGLES OF CARBON ION SCATTERING AND THE IMPORTANCE OF THIS EFFECT

The examples of γ₀ for scattering of ¹²C⁺⁶ ions with 100 MeV ≤ ɛ ≤ 825 MeV in the gold membranes with ρ_m = 19.32 g/cm³ (see, e. g., Berdonosov, Reference Berdonosov and Prokhorov1990) and k _exl _m = 10, 20, 30, and 40 µm are presented in Table 1. The chosen range of ɛ coincides approximately with the range of carbon ion kinetic energies in fast ignition scenarios considered by Albright et al. (Reference Albright, Schmitt, Fernández, Cragg, Tregillis, Yin and Hegelich2008), Fernández et al. (Reference Fernández, Albright, Flippo, Hegelich, Kwan, Schmitt and Yin2008, Reference Fernández, Honrubia, Albright, Flippo, Gautier, Hegelich, Schmitt, Temporal and Yin2009), Honrubia et al. (Reference Honrubia, Fernández, Temporal, Hegelich and Meyer-Ter-Vehn2009), and Shmatov (Reference Shmatov2003, Reference Shmatov2008, Reference Shmatov2011).

Table 1.

Some values of γ₀[rad] for scattering of ¹²C ⁺⁶ ions in gold

At γ₀ ≪ 1 rad, the typical displacement Δ of the point of hit of the ion on hot spot due to scattering is about γ₀d (Key et al., Reference Key, Freeman, Hatchett, Mackinnon, Patel, Snavely and Stephens2006).

According to Atzeni and Tabak (Reference Atzeni and Tabak2005), the sum E _hs of the kinetic energies of the ions hitting the hot spot should obey the condition

(3)

where E _hs^opt [kJ] = 140(ρ /100 g/cm³)^−1.85, ρ is the density of the fuel at the stage of heating the hot spot, R is the range of the ion in the hot spot, R ₀ = 1.2 g/cm², r _hs is the hot-spot radius,

(4)

(see also Atzeni, Reference Atzeni1999; Shmatov, Reference Shmatov2011; Tabak et al., Reference Tabak, Hinkel, Atzeni, Campbell and Tanaka2006). Here it is assumed that the spread of the ion ranges is small, the hot spot is transversally uniform, and the ion stopping power in the hot spot is uniform.

The data from Table 1 and Eqs. (3) and (4) yield that in the scenarios with the transversally uniform hotspot, the importance of scattering of carbon ions in the protective membrane of the indirect compression target without cone at the realistic l _m, ɛ, and d depends strongly on ρ or, in some situations, on both ρ and the parameters of ion beam before its passage through the membrane.

Let us consider the examples with two values of ρ. Albright et al. (Reference Albright, Schmitt, Fernández, Cragg, Tregillis, Yin and Hegelich2008) and Fernández et al. (Reference Fernández, Albright, Flippo, Hegelich, Kwan, Schmitt and Yin2008, Reference Fernández, Honrubia, Albright, Flippo, Gautier, Hegelich, Schmitt, Temporal and Yin2009) described the scenarios with a relatively low ρ of 150 g/cm³, the fuel capsule radius r _c = 730 µm, the external and internal radii of the D-T ice in the fuel capsule of 580 µm and 330 µm, respectively, and bombardment of the compressed fuel by two counter-propagating beams of carbon ions with the energy spread of ±10%. The highest capsule gain corresponded to the average ion kinetic energy of about 440 MeV (Fernández et al., Reference Fernández, Honrubia, Albright, Flippo, Gautier, Hegelich, Schmitt, Temporal and Yin2009). The radius r _b^main of the main, i. e., high-density, region of the blob of the compressed fuel in the scenarios under consideration is about 60 µm. Let us assume that the thermal radiation compressing the fuel capsule is generated by the laser beams, the internal radius R ^int_h of the hohlraum equals 2.56r _c ≈ 0.187 cm (Tabak et al., Reference Tabak, Hinkel, Atzeni, Campbell and Tanaka2006), the internal surface of the protective membrane coincides with the internal surface of the hohlraum wall, and

(5)

Using Eqs. (4) and (5) and the data from Table 1, we obtain r _opt (ρ = 150 g/cm³) ≈ 40 μm

(6)

(7)

According to Eq. (3), Eq. (6) describes a relatively weak scattering of the ions in the protective membrane, while the importance of scattering described by Eq. (7) depends on the transversal size of the ion source and the distribution of the directions of the ion velocities before passage of the ions through the membrane. This distribution is determined by the spread of the directions of the initial velocities of the ions accelerated in the regions with the small transversal sizes and, in some situations, by the measures undertaken to focus the ions (see also Borghesi et al., Reference Borghesi, Fuchs, Bulanov, Mackinnon, Patel and Roth2006; Gus'kov, Reference Gus'kov2001; Key et al., Reference Key, Freeman, Hatchett, Mackinnon, Patel, Snavely and Stephens2006; Roth et al., Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pegoraro, Bulanov, Campbell, Perry and Powell2001; Shmatov, Reference Shmatov2003).

Ramis and Ramírez (Reference Ramis and Ramírez2004) described the scenario with ρ = 400 g/cm³, the D-T fuel mass of 0.8 mg, that corresponds to r _b^main≈78 μm, R _h^int = 0.28 cm and heating of the hot spot by the laser-accelerated protons passing through the 40-μm-thick gold hohlraum wall. Let us consider the scenario that is almost the same but the hot spot is heated by laser-accelerated carbon ions with ɛ ≈ 500 or 825 MeV.

Assuming that d ≈ R _h^int − R _b^main ≈ 0.27 cm, k _ex ≈ 1 and using the data from Table 1 and Eq. (4), that yields r _opt(ρ = 400 g/cm³) ≈ 16 μm, we obtain

(8)

(9)

Eqs. (8) and (9) describe the important scattering of the ions in the protective membrane (see Eq. (3)).

Shaping the bombarded region of the compressed fuel can provide a significant decrease in the minimum acceptable value of E _hs compared with that corresponding to the transversally uniform bombarded region (Atzeni & Tabak, Reference Atzeni and Tabak2005; Temporal et al., Reference Temporal, Ramis, Honrubia and Atzeni2009). In the effective scenarios of such a kind, the difference between the typical outer and inner radii of the ring-shaped bombarded region should probably be about 10 µm (see, e.g., an example presented by Temporal et al. (Reference Temporal, Ramis, Honrubia and Atzeni2009)). This corresponds to the acceptable values of Δ for a few microns. The data from Table 1 and Eqs. (1), (2), (6)–(9) show that for the indirect compression targets without cones with heating the compressed fuel by carbon ions, the realization of such values of Δ will be difficult or even impossible.

SOME MEASURES TO MINIMIZE THE TYPICAL ANGLE OF SCATTERING OF IONS

In some situations the optimization of the ignition scenario can include the measures to minimize the typical angle of scattering of ions.

A decrease in γ₀ with increasing ɛ (see Eqs. (1), (2), (8), (9), and Table 1) can be one of the factors determining the optimum typical kinetic energies of the laser-accelerated ions. Note that other factors that can result in the expedience of acceleration of the ions to the relatively high kinetic energies were considered by Albright et al. (Reference Albright, Schmitt, Fernández, Cragg, Tregillis, Yin and Hegelich2008), Fernández et al. (Reference Fernández, Albright, Flippo, Hegelich, Kwan, Schmitt and Yin2008, Reference Fernández, Honrubia, Albright, Flippo, Gautier, Hegelich, Schmitt, Temporal and Yin2009), Honrubia et al. (Reference Honrubia, Fernández, Temporal, Hegelich and Meyer-Ter-Vehn2009), and Shmatov (Reference Shmatov2011).

The minimization of the typical angle of scattering of the ions by means of the optimization of composition and/or density of the protective membrane can also be expedient (see Callahan-Miller & Tabak, Reference Callahan-Miller and Tabak2000; Rosen & Hammer, Reference Rosen and Hammer2005; Wilkens et al., Reference Wilkens, Nikroo, Wall and Wall2007). For example, Wilkens et al. (Reference Wilkens, Nikroo, Wall and Wall2007) describe the hohlraum wall, the main region of which has the thickness of about 7 µm, and consists of 185 pairs of 30-nm-thick layers of uranium and 8.2-nm-thick layers of gold. The average composition of this “cocktail” wall region corresponds to the formula U_0.75Au_0.25 (Wilkens et al., Reference Wilkens, Nikroo, Wall and Wall2007). The sum of the thicknesses of the uranium and gold layers in this region are about 5.55 µm and 1.52 µm, respectively. Using Eqs. (1) and (2), it is possible to show that scattering of ¹²C⁺⁶ ions with ɛ on the order of 100 MeV in the 5.55-μm-thick layer of uranium is being described by approximately the same value of γ ₀ as scattering of such ions in the 5.79-μm-thick layer of gold. Thus, scattering of carbon ions in the “cocktail” layer under consideration is equivalent to that in the 7.3-μm-thick layer of gold. Note that for such a gold layer γ₀ (ɛ = 500 MeV) ≈ 4.3 × 10⁻³ rad.

A decrease in the typical angle of scattering of the ions can also be achieved by means of increasing the atomic number of the element the ions of which heat the hot spot. For example, Eqs. (1) and (2) yield that for scattering of the ⁶⁴Z ⁺³⁰ ion with ɛ = 3900 MeV in the gold membrane with k _exl _m = 40 μm, γ₀ ≈ 7.37 × 10⁻³ rad. The chosen value of ɛ corresponds to R ≈ 1.2 g/cm² at ρ = 500 g/cm³ and the electron temperature T _e = 12 keV (Shmatov, Reference Shmatov2008; here and below R is calculated according to the model of Bychenkov et al. (Reference Bychenkov, Rozmus, Maksimchuk, Umstadter and Capjack2001)). These values of R,ρ, T _e, and k _exl _m also correspond to ɛ ≈ 143 MeV, γ₀ ≈ 3.93 × 10⁻² rad for the ¹²C⁺⁶ ions (see also Shmatov, Reference Shmatov2008). However, it should be emphasized that the effective acceleration of the ions of elements with the relatively high Z, for example, with Z ≥ 20, may be accompanied by the serious technical difficulties (see Albright et al., Reference Albright, Schmitt, Fernández, Cragg, Tregillis, Yin and Hegelich2008; Bychenkov et al., Reference Bychenkov, Rozmus, Maksimchuk, Umstadter and Capjack2001; Borghesi et al., Reference Borghesi, Fuchs, Bulanov, Mackinnon, Patel and Roth2006; Fernández et al., Reference Fernández, Albright, Flippo, Hegelich, Kwan, Schmitt and Yin2008; Honrubia et al., Reference Honrubia, Fernández, Temporal, Hegelich and Meyer-Ter-Vehn2009; Shmatov, Reference Shmatov2008).