2. PLASMA BLOCK GENERATION MECHANISM

Problems appeared from the discovery of few rare observations of anomalies at the interaction of laser pulses of TW power and ps duration in contrast to the usually measured relativistic effects. Zhang et al. (Reference Zhang, He, Chen, Li, Zhang, Wong Lang, Li, Feng, Zhang, Tang and Zhang1998) was applying very clean laser pulses where the prepulses were suppressed by a factor 10⁸ (contrast ratio) until 50 ps before the main pulse arrived leading to measured very low X-ray emission in contrast to the usual experiments. Measuring the energy of the fast ions with such clean pulses, Badziak et al. (Reference Badziak, Kozlov, Makowksi, Parys, Ryc, Wolowski, Woryna and Vankov.1999) arrived at much lower ion energies than usual and the number of the fast ions did not change by varying the laser power. This led to the explanation that relativistic self-focusing was suppressed and a predominant nonlinear (ponderomotive) force acceleration occurred resulting in directed plasma blocks from the skin layer (Hora et al., Reference Hora, Badziak, Boody, Höpfl, Jungwirth, Kralikova, Krasa, Laska, Parys, Perina, Pfeifer and Rohlena2002, Reference Hora2007; Hora, Reference Hora2003).

The resulting plasma blocks with ion current densities above 10¹¹ A/cm² could then be directly applied to the side-on ignition of uncompressed DT as calculated by Chu (Reference Chu1972) and Bobin (Reference Bobin, Schwarz and Hora1974) in similar way as a modified fast ignition scheme using 5 MeV intense electron beams (Nuckolls et al., Reference Nuckolls and Wood2002) for ignition of DT at low compression of only about 10 times higher density than the solid state. In both cases, with the ion currents of the blocks or the electrons, the crucial problem of the necessary high thresholds E _t* for the energy flux density E* may be reached as calculated from hydrodynamics (Chu, Reference Chu1972; Bobin, Reference Bobin, Schwarz and Hora1974). This threshold even may be reduced by a factor up to 20 due to interpenetration processes of the particles (Hora, Reference Hora1983). Another relaxing correction of the threshold E _t* is possible by later discovered effects. One of these phenomena is the stopping power of the alpha particles from the fusion reaction in the target by collective effects. After reviewing the background of the mentioned collective effect, the results of the hydrodynamic theory are reported.

3. EMISSION OF BREMSSTRAHLUNG

The whole analysis of Chu (Reference Chu1972) included the emission of bremsstrahlung from the interaction region at varying the incident energy flux density E* given in J/cm² or ergs/cm². The key process of the ignition could be seen in diagrams of the generated temperature T in the interaction area depending on time t (see Fig. 2 of Chu (Reference Chu1972)). Ignition was seen when the resulting curves did not decrease. The limiting curve going into constant T(t) for large t defined the threshold E _t* for this side-on ignition. Irradiating solid DT fusion fuel arrived at the value 4.3 × 10⁸ J/cm² which was out of any range to ignite solid DT by laser or energetic particle beams. Interestingly, the evaluation of very detailed computations (Storm et al., Reference Storm, Lindl, Campbell, Bernat, Coleman, Emmett, Hogan, Horst, Krupke and Lowdermilk1988) of spark ignition for spherical laser fusion (Hora et al., Reference Hora, Azechi, Kitagawa, Mima, Murakami, Nakai, Nishihara, Takabe, Yamanaka, Yamanaka and Yamanaka1998) resulted in similar or one order higher E*-values for generating the detonation front from the central spark produced by volume ignition for moving into the very high density and modest temperature outside fuel mantle.

Since the nonlinear force driven plasma block generation resulted in space charge neutral directed plasma of modest temperatures and directed ion energy up to 100 keV with ion current densities above 10¹¹ Amps/cm², it became possible to reach the conditions for side-on ignition of uncompressed DT with PW-ps laser pulses. Figure 1 shows the results without inhibition factor (Ghorannevis et al., Reference Ghoranneviss, Malekynia, Hora, Miley and He2008) with E* as parameter for the ignited cases. The lower set of curves is for the same conditions of Chu (Reference Chu1972), while the higher set refers to the later discussion. The lowest curve shows how T merges into a constant value for high t showing that the threshold for ignition is as known from Chu

(1)

Fig. 1. Side-on ignition characteristics following Chu (Reference Chu1972) for laser ignition of solid state DT by nonlinear force driven plasma blocks. Lower set of curves to reproduce the results of Chu (Reference Chu1972), upper set with later discussed collective effect.

For lower values of E*, the curves decayed on time corresponding to no ignition. The code could be used to compute the total amount of emitted bremsstrahlung per generated fusion energy. This factor was plotted in Figure 2 for decaying curves of input energy flux density. The factor was one for the ignition threshold given in Eq. (1) and then increasing for lower E*.

Fig. 2. Dependence of Input energy and bremsstrahlung loss with Collective effect for unignited case DT . Summary of the stopping length R for alpha particles depending on the plasma temperature T = T _i = T _e and a set of plasma densities including the liquid – close to the solid-state for deuterium tritium DT at 5 × 10²² cm⁻³ for different binary interaction models collected by Stepanek (Reference Stepanek, Schwarz, Hora, Lubin and Yaakobi1981) showing the strong difference to the collective model (Ray & Hora, Reference Ray and Hora1977). Factor of bremsstrahlung loss per generated fusion energy for side-on ignition of solid DT from the computation following Chu (Reference Chu1972) for input energy flux densities E* in the non-ignition range below the ignition threshold to demonstrate the then dominating losses by bremsstrahlung.

This factor of unity of Figure 2 could alternatively be used to determine the ignition condition of using the plots of Figure 1 for stationary plasma temperature.

4. COLLECTIVE EFFECTS

The deposition of energy by alpha particles that are produced by DT laser driven fusion is important in calculations of the ignition and development of the inertial confinement fusion. The plasma is reheated by the collisions of the high energy alpha particles generated in the fusion reactions. The problem of reheat is involved with the very complex question, what penetration depth for the MeV alpha particles in the high density plasma is to be taken. Ray and Hora (Reference Ray and Hora1977) calculated it on the basis of a modification of the well known Bethe-Bloch formalism by Gabor (Reference Gabor1952) and showed an increase nuclear fusion yields of inertially confined DT plasma due to reheat.

The basic physics responsible for slowing down of alpha particles is energy exchange through Coulomb interaction. This is a similar process for the treatment of the stopping power of alpha particles in plasmas following the Bethe-Bloch theory with several modifications by further authors as reviewed by Stepanek (Reference Stepanek, Schwarz, Hora, Lubin and Yaakobi1981) where binary collisions between the alphas, protons or other charged particles from nuclear reactions with electrons are considered. A visible discrepancy appeared with the measurements by Kerns et al. (Reference Kerns, Rogers and Clark1972) where an electron beam of 2 MeV energy and 0.5 MA current of 2 mm diameter was hitting deuterated polyethylene CD₂. The penetration depth of the electrons was measured by changing the thickness d of the CD_2, and the saturation of the emission of fusion neutrons at d = 3 mm was a proof for the very much shorter stopping length of the electrons than binary collision theories predicted. The energy loss rate of a energetic alpha particle to the DT plasma results from alpha-electron and alpha-ion interactions. The alpha-electron interaction is given by the Rutherford scattering cross section, and the alpha-ion interaction is given by the combination of the cross section of Coulomb and elastic nuclear processes.

An explanation of the value d was obtained (Bagge et al., Reference Bagge and Hora1974) when Bagge's theory of the stopping of cosmic rays was applied where the interaction of the charged energetic particles was to be taken by the whole electron cloud in a Debye sphere for the electrons as discussed before by Gabor (Reference Gabor1933, Reference Gabor1952). This collective model in contrast to the binary stopping power was treated by Ray and Hora (Reference Ray and Hora1976, Reference Ray and Hora1977) based on kinetic theory using the Fokker-Planck equation and quantum electrodynamics.

The extreme discrepancy between binary and the collective theory of the stopping length of 3.5 MeV alpha particles in solid density DT is shown in Figure 3 following Ray and Hora (Reference Ray and Hora1977). For plasma temperatures above 0.1 keV binary Bethe-Bloch theory arrives at a stopping lengths R _BB given by the Winterberg approximation used by Chu (Reference Chu1972).

(2)

Fig. 3. Temperature dependence of the stopping length R (range) for alpha particles of 2.89 MeV in a fusion plasma of solid state density with binary electron collisions (Fokker-Planck F.P. collisions and quantum electrodynamic (Pauli) cut-off) and collisions with the electron collective in a Debye sphere (Ray & Hora, Reference Ray and Hora1977) corresponding to the summary by Stepanek (Reference Stepanek, Schwarz, Hora, Lubin and Yaakobi1981, Fig. 6; Malekynia et al., Reference Malekynia, Hora, Ghoranneviss and Miley2009, Fig. 1) where corrections to the binary collision theory with screened Debye potential according to the Gabor (Reference Gabor1952) collective effect are included.

In strong contrast to this binary stopping theory, the collective stopping length is nearly constant (Ray & Hora, Reference Ray and Hora1977) for higher temperatures. The more precise expression taken from Fig. 3 is

(3)

taking into account the very slight decrease of R at higher temperatures T (Ray & Hora, Reference Ray and Hora1977). Where Ei was given by Hora and Ray (Reference Hora and Ray1978) and E ₀ is the initial energy of alpha particle. For temperatures less than 0.1 keV, the later extensions of the Bethe-Bloch theory and the collective theory resulted in nearly the same stopping lengths, see Figure 6 of Stepanek (Reference Stepanek, Schwarz, Hora, Lubin and Yaakobi1981). This result of the much shorter stopping length of the reaction products in laser fusion was the reason of the strong reheats in laser irradiated fusion pellets for DT at fully detailed inclusion of the adiabatic expansion dynamics of the spherical plasmas leading to the discovery of the volume ignition (Hora & Ray, Reference Hora and Ray1978 Eq. (3) can be approximated by

(3a)

The Gabor theory (Gabor, Reference Gabor1952) of the stopping power of alpha particles for collisions with the whole collective of the electrons in the Debye sphere in contrast to the binary collisions with electrons following from the Bethe-Bloch theory, needed some closer consideration in view of results of volume ignition of spherically compressed pellets for fusion energy. As shown in Figure 3, the stopping lengths of both theories are not very much different for plasma temperatures up to about 100 eV (Stepanek, Reference Stepanek, Schwarz, Hora, Lubin and Yaakobi1981), while the collective effect arrives at very different values for volume ignition at higher temperatures (Ray & Hora, Reference Ray and Hora1977), see Figure 2 of Malekynia et al. (Reference Malekynia, Hora, Ghoranneviss and Miley2009). This resulted in higher gains for DT fusion at volume ignition or the sub-ignition burn with spherical irradiation using the collective effect in agreement with measurements (Hora et al., Reference Ghoranneviss, Malekynia, Hora, Miley and He1998). The experiments by Kerns et al. (Reference Kerns, Rogers and Clark1972) were a direct proof of the much shorter stopping length at very high intensity particle interaction which could immediately and convincingly be explained by the Gabor collective model (Bagge et al., Reference Bagge and Hora1974).

The problems with the stopping power were observed by the experiments of Hoffmann et al. (Reference Hoffmann, Weyrich, Wahl, Gardes, Bimbot and Fleurier1990) directly showing that the ranges of particle beams in laser produced plasmas are drastically reduced compared with the binary stopping theories. The complexity of the theories is well known (Stepanek, Reference Stepanek, Schwarz, Hora, Lubin and Yaakobi1981; Deutsch, 1984; Deutsch et al., Reference Deutsch and Popoff2007) but the experiments of Kerns et al. (Reference Kerns, Rogers and Clark1972) with the exact agreement to Gabor's results (Bagge et al., Reference Bagge and Hora1974) and the measurements by Hoffmann et al. (Reference Hoffmann, Weyrich, Wahl, Gardes, Bimbot and Fleurier1990) may be a basis for using the collective effect.

It was evaluated, what differences in the computations with the collective effect and without inhibition factor will appear with respect to the comparison of the new results (Hora et al., Reference Hora, Malekynia, Ghiranneviss, Miley and He2008; Malekynia et al., Reference Malekynia, Hora, Ghoranneviss and Miley2009) in comparison with the stopping power in the temperature range below 100 eV where Chu (Reference Chu1972) used the then not valid (see Fig. 3) Winterberg Eq. (2).

5. REVISED HYDRODYNAMIC COMPUTATIONS WITH THE COLLECTIVE MODEL

In order to see the importance of the collective effect of the stopping power in the hydrodynamic equations, we are comparing the results with the initial computations performed by Chu (Reference Chu1972). The correction by inclusion of the reduction of the thermal conductivity expressed by the inhibition factor was presented before (Ghoranneviss et al., Reference Ghoranneviss, Malekynia, Hora, Miley and He2008), which discussion is separated from the present considerations.

The hydrodynamic equations used are the same as presented before (Ghoranneviss et al., Reference Ghoranneviss, Malekynia, Hora, Miley and He2008, Hora et al., Reference Hora, Malekynia, Ghiranneviss, Miley and He2008) where for the reason of comparison with the results of Chu (Reference Chu1972) no essential changes were taken into account. The only differences are that the bremsstrahlung is based on the electron temperature T _e working with Eq. (15) from Chu (Reference Chu1972) with the maximum at x = 0, thus,

(4)

Eq. (4) is a little different from Eq. (20) of Chu (Reference Chu1972) where T _i = T _e is assumed while the following computation with the collective stopping has to be performed for general temperatures. The α particles are assumed to deposit their energy in the plasma. They have a mean free path at solid state density DT according to Figure 3 for the Ray-Hora case of collective effects given by Eq. (2). The action of the stopping with the collective effect is expressed by the temperature T from Eq. (2). For the calculation of the collective effect we added a term to right-hand of Eq. (3). Thus

(5)

where P is the thermonuclear heating rate per unit time obtained from the burn rate and the fractional alpha particle deposition including the loss Y of burned fuel.

(6)

(7)

E_α = 3.5 MeV and f is the fraction of alpha particle energy absorbed by electrons or ions, which has been given by

(8)

For the equations after Eq. (4), the temperature of the electrons and of the ions were used to be equal to T, as used in Eq. (2) for the following numerical evaluations. These relations had to be repeated from before because we are using the exact stopping length from the collective theory (3) for comparison with the approximation (3a) in the following numerical evaluations.

6. NUMERICAL RESULTS WITH INCLUSION OF THE COLLECTIVE EFFECT

The following numerical evaluations are reported where the development of the temperature with the time t is used in order to compare the ignition condition with the results of Chu given in his Figure 2 (Chu, Reference Chu1972). The here presented Figure 4 shows a lower set of curves that are very similar to that of Chu where all his conditions are the same with exception of the use of the temperature T _e instead of the ion temperature T _i because of dependency of the bremsstrahlung and thermal conductivity to T _e. Figure 1 also contains the new results with inclusion of the collective effect leading to higher temperatures as expected from the shorter stopping length using the same irradiation energy flux densities as in the cases of Chu for ignition given as E* in erg/cm². Chu found that ignition happens at E* = 4.3 × 10⁸ J/cm² for the case without collective effect as seen here too from the time dependence of the curve continuing to be constant on time. For higher E*, the curves for the temperature are still increasing in time. It was evident that the evaluation was to some extend difficult where the more precise threshold values are to be found because the characteristic curves in Figure 1 had to be followed up for long times to check whether there is a very slight increase or decrease on time. Figure 4 shows the result for longer times than reported before and for a higher resolution of the ordinate. It turned out that the convincing result for ignition at inclusion of the collective effect without the inhibition factor is at E* at 2.5 × 10⁸ J/cm² and the exact threshold is at a little lower E* value.

Fig. 4. Results as in Figure 1 showing ignition of solid deuterium tritium with the collective effect being reduced by about a factor 4 to E* = 1.05 × 10⁸ J/cm².

The results of Figure 1 and 4 including the collective effect were using the linear approximation (Eq. (2)) Malekynia et al. (Reference Malekynia, Hora, Ghoranneviss and Miley2009). We checked what differences appear if the exact formulation (3) is used (Fig. 5). The differences to the linear approximation of the slightly bent curve for the collective stopping is very small (Fig. 3) and are very close to the various different derivations of stopping ranges summarized by Stepanek (Reference Stepanek, Schwarz, Hora, Lubin and Yaakobi1981, Fig. 6; Malekynia et al., Reference Malekynia, Hora, Ghoranneviss and Miley2009, Fig. 1) where corrections to the binary collision theory with screened Debye potential according to the Gabor (Reference Gabor1952) collective effect are included. A difference appears when instead of the linear approximation, the exact stopping lengths were now used as shown in Figure 5. The calculations with the exact stopping lengths of Eq. (3) arrive at little higher temperature than with the linear approximation. Although the differences are not very strong it is remarkable, what the very minor difference between the approximation and the exact stopping length is leading to. For lower E* values than the threshold, the difference of the results with approximated and exact stopping length are seen to be stronger. This may be due to the fact that in the range much below the threshold, the losses by emission of bremsstrahlung is much stronger against the generation of fusion energy.

Fig. 5. Max Ion Temperature without Inhibition Factor For The Unignited Case Side-on ignition characteristics for solid DT for E* as parameter. The lowest of each of the three plots is without collective effect, the upper very close lines show the use of the approximation of the collective stopping, Eq. (2) while the highest lines uses the un-approximated values from Figure 3.

It is important to note that in all these results, that of Chu, that with collective effect only (Malekynia et al., Reference Malekynia, Hora, Ghoranneviss and Miley2009 and the here presented results), and with collective effect and inhibition factor, the plasma temperature with ignition of DT is always higher than 4 keV. This is the well known limit above which the fusion energy only can be higher than the bremsstrahlung. This is understandable because the side-on ignition is generated in interaction fronts of the fusion flame (according to the terminology of Bobin, Reference Bobin, Schwarz and Hora1974) where the involved volume is so thin that neither re-absorption of bremsstrahlung nor additional heating in the plasma by the reaction products (re-heat) can have an effect. This is different for the larger volume effects with the spherical laser compression scheme, where re-absorption and re-heat can lead to optimum ignition conditions at much lower temperatures than 4 keV, even down to 500 eV (Miley et al., Reference Miley, Hora, Osman, Evans and Toups2005).

Based on the here elaborated results, the side-on ignition of proton-11 boron (HB11) fusion was studied and reported separately (Hora et al., Reference Hora, Miley, Azizi, Malekynia, Ghoranneviss and He2009a, Reference Hora, Miley, Ghoranneviss, Malekynia and Azizi2009b).

It should be realized that the new direction for laser driven fusion energy by side-on ignition with nonlinear force driven plasma blocks is just at the beginning to be explored (Hora, Reference Hora, Miley, Azizi, Malekynia, Ghoranneviss and He2009; Hora et al., Reference Hora, Miley, Azizi, Malekynia, Ghoranneviss and He2009a, Reference Hora, Miley, Ghoranneviss, Malekynia and Azizi2009b) and general difficulties may appear at further studies when other methods than hydrodynamics are used. The anomalous generation of plasma blocks by interaction of petawatt-picosecond laser pulses permits side-on ignition of uncompressed solid fusion fuel. After application of the Chu-model for deuterium-tritium and including of the inhibition factor for thermal conduction (Ghoranneviss et al., Reference Ghoranneviss, Malekynia, Hora, Miley and He2008; Zhou et al., Reference Zhou, He and Yu2008), the studies for the reduced stopping lengths for alpha particles are extended here continuing the preceding considerations (Malekynia et al., Reference Malekynia, Hora, Ghoranneviss and Miley2009) including a detailed description of the hydrodynamic properties with application of the generally rather complex theory of the stopping power. The separate elaboration of the collective effect compared with the inhibition factor was needed for a detailed understanding in view of the complexity of the stopping power theory, after first results of the combination of the collective stopping with the inhibition effects were summarized (Hora et al., Reference Hora, Malekynia, Ghiranneviss, Miley and He2008) and the influence by losses of bremsstrahlung at non-ignition conditions below the threshold were understood. The fact has to be underlined that the ignition temperatures for all the considered fuel are above the energy loss by bremsstrahlung as known since the computations by Chu for DT and as it was repeatedly confirmed in the present treatments. This emission was based on collective effect stopping power of alpha particles.

The side-on ignition is a process in a shock front with energy production from the ions and the generated alpha particles. These conditions are determined by thin volume geometry of the side-on ignition where the advantages of the volume ignition at spherical laser fusion are not possible. The problem is that the here reported hydrodynamic analysis is based on one dimensional plane and infinitely extended geometry. In reality there is a lateral spread of the interacting laser beam though this is of a radius in the dimension of thousand laser wave lengths. A number of these geometries have been evaluated by Bobin (Reference Bobin, Schwarz and Hora1974) and a first reference to these results may confirm that the better conditions for the nonlinear force driven block ignition of HB11 by several orders of magnitudes in contrast to the spherical laser compression scheme may sustain the further analysis (Hora et al., Reference Hora, Miley, Azizi, Malekynia, Ghoranneviss and He2009a, Reference Hora, Miley, Ghoranneviss, Malekynia and Azizi2009b).

The here elaborated result about the stopping length of alpha particles from the DT fusion reaction relates to a kind of a delta function or at least of a layer with highly reduced thickness. We know from Figure 3 and related theories based on Gabor's (Reference Gabor1933; Reference Gabor1952) collective model that the stopping length of alphas of about 3 MeV energy in plasmas of any subrelativistic temperature is in the range between 0.001 and 0.008 cm to explain the validity of the one-dimensional hydrodynamic analysis beginning with Chu (1971) and Bobin (Reference Bobin, Schwarz and Hora1974). This is in essential contrast to the three-dimensional problems of the electron beam ignition with the Nuckolls-Wood (Reference Nuckolls and Wood2002) method. The three-dimensional discussion of Nuckolls and Wood is then based on the volumetric (see page 4 of Nuckolls et al., Reference Nuckolls and Wood2002) ρR criterion where ρ is the density of plasma uniformly compressed in a sphere of radius R. This criterion was derived from the numerically calculated optimum fusion gains G (Hora et al., Reference Hora and Pfirsch1970; Hora & Pfirsch, Reference Hora, Pfirsch, Schwarz and Hora1972) giving the fusion energy per input laser energy E ₀ into a spherical volume with radius R of fusion fuel of density ρ per solid state density ρ_s

(9)

where E_BE is the break even energy with the value 6 MJ for DT. The first expression of (9) was formulated in 1970 (Hora et al., Reference Hora and Pfirsch1970) and the second, resulting from E ₀ ~ ρR ³, was first published by Kidder (Reference Kidder1974). This is the result of volume burn at spherical uniform compression according to the self-similarity model (Hora, Reference Hora1981, Section 5) only at optimum temperatures (of 11.5 keV for DT) and is valid only up to gains G < 8 for DT (Hora et al., Reference Hora, Azechi, Kitagawa, Mima, Murakami, Nakai, Nishihara, Takabe, Yamanaka, Yamanaka and Yamanaka1998). For higher gains, volume ignition was discovered (Hora et al., Reference Hora and Ray1978), confirmed by Wheeler modes (Kirkpatrick et al., 1971), where other gain formulas were derived for higher gains (Hora, 1987; Betti, 2009) apart from the detailed numerical diagrams (1998). Under these special conditions, the ρR formula (9) can be used for three-dimensional geometry only. The side-on ignition by nonlinear force driven plasma blocks with the generation of shock fronts as fusion flames (Hora, Reference Hora2009) is a one-dimensional problem and not that of three dimensions as in the case of electron beams (Nuckolls et al., Reference Nuckolls and Wood2002) with few mm stopping lengths of the electrons as measured (Kerns et al., Reference Kerns, Rogers and Clark1972).