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Inertial fusion features in degenerate plasmas

Published online by Cambridge University Press:  07 June 2005

PABLO T. LEÓN ,
SHALOM ELIEZER ,
MIREIA PIERA  and
JOSÉ M. MARTÍNEZ-VAL
PABLO T. LEÓN
Affiliation:
Nuclear Fusion Institute (IFN), Madrid, Spain
SHALOM ELIEZER
Affiliation:
Nuclear Fusion Institute (IFN), Madrid, Spain
MIREIA PIERA
Affiliation:
Nuclear Fusion Institute (IFN), Madrid, Spain
JOSÉ M. MARTÍNEZ-VAL
Affiliation:
Nuclear Fusion Institute (IFN), Madrid, Spain
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Abstract

Very high plasma densities can be obtained at the end of the implosion phase in inertial fusion targets, particularly in the so-called fast-ignition scheme (Tabak et al., 1994; Mulser & Bauer, 2004), where a central hot spark is not needed at all. By properly tailoring the fuel compression stage, degenerate states can be reached (Azechi et al., 1991; Nakai et al., 1991; McCory, 1998). In that case, most of the relevant energy transfer mechanisms involving electrons are affected (Honrubia & Tikhonchuk, 2004; Bibi & Matte, 2004; Bibi et al., 2004). For instance, bremsstrahlung emission is highly suppressed (Eliezer et al., 2003). In fact, a low ignition-temperature regime appears at very high plasma densities, due to radiation leakage reduction (León et al., 2001). Stopping power and ion-electron coulomb collisions are also changed in this case, which are important mechanisms to trigger ignition by the incoming fast jet, and to launch the fusion wave from the igniting region into the colder, degenerate plasma. All these points are reviewed in this paper. Although degenerate states would not be easy to obtain by target implosion, they present a very interesting upper limit that deserves more attention in order to complete the understanding on the different domains for inertial confinement fusion.

Keywords

DegenerateFast-IgnitionInertial fusion
Type
Research Article
Information
Laser and Particle Beams , Volume 23 , Issue 2 , June 2005 , pp. 193 - 198
Copyright
2005 Cambridge University Press

1. INTRODUCTION

In inertial confinement fusion, high density and temperature plasmas are created during a small time interval, to obtain energy from fusion reactions, one of the most used approaches is the “hot spark.” In this approach, the isobaric or isochoric model (Meyer-ter-Vehn, 1992; Kidder, 1974), during the compression phase, a high temperature region is generated in the target, and from this high temperature region, the fusion burning wave is launched to the rest of the low temperature pre-compressed plasma. The hot spot generation during the compression phase means the appearance of instability problems, and it is difficult to obtain high gains.

In the past, trying to solve the problems related to the “hot spark” technique, a new idea emerged, where the compression phase was separated from the generation of the hot spot. This new technique was called “fast ignition,” and the principle characteristic was the separation of the two phases (Tabak et al., 1994). During the first compression phase, high densities are reached in the plasma by a nearly isentropic compression, without a hot spot generation in the plasma. Once the plasmas is compressed, and in a second phase, a small region of the target is heated (ignitor), and from the ignitor the fusion burning wave is launched to the rest of the pre-compressed plasma. Different heating methods were studied (electrons, light ions, heavy ions, plasma jets, etc.) (Roth et al., 2001; Martìnez-Val & Piera, 1997; Piriz & Sanchez, 1998; Azechi et al., 1991; Key, 2001; Kodama et al., 2001; Velarde et al., 2003; Deutsch et al., 1997a, 1997b, 2004). The main advantages of this method consist in the optimization of the compression phase, minimizing the energy needed for obtaining high densities in the plasma increasing the gain considerably, and with fewer problems in the compression phase.

In the case where high density and low temperatures are reached in the plasma in the fast ignition approach, the plasma electrons can be degenerate (León et al., 2001). The equations that predict the behavior of these plasmas are different from the classical ones. For example, due to the Pauli's exclusion principle, the Bremsstrahlung emission in degenerate plasmas is strongly reduced as compared to the classical values (Eliezer et al., 2003). This reduction follows a decrease in the ignition temperature of the plasma. To demonstrate this assertion, the program FINE (fast ignition nodal energy) was developed. In the program, the equations defined are valid for the degenerate and classical region, taking into account the possibility that the plasmas pass from a degenerate state to a classical state during the heating process. The results of the program are only valid to study the possibility of ignition of plasmas in the fast ignition concept, not the burn up phase of the target. Ions were the choice for ignitor heating until ignition conditions are reached. As will be seen in the conclusions, the ignition temperature is reduced in degenerate plasmas, but the amount of energy needed to obtain these very high compression targets reduces significantly the gain of the pellet.

2. EQUATIONS THAT DEFINES THE FINE PROGRAM

The FINE program is defined to solve the nodal energy conservation equation. In the fast ignition concept, two nodes were studied. The first one, defining the ignitor, the second one, defining the volume annexed to the ignitor. Figure 1 depicts these two nodes in the pre-compressed target.

Fast Ignition in the two nodal concept.

The size of the ignitor is defined by the mean free path of the external heating ions in the pre-compressed plasma. The fuel is a DT stequiometric mixture, and the projectiles are deuterons, as they are part of the fuel composition. The node 2 size is defined by the 3.5 MeV alpha particles (coming from DT fusion reactions) mean free path. The equations defining the energy conservation law solved by the FINE program are:

Node 1: Ignitor.

Node 2: Volume annexed to the ignitor.

The equations of energy conservation, for both nodes, were divided into two species, electrons and ions. In the fast ignition regime, the temperatures of both species are going to be different, so two energy equations are needed. The radiation temperature is not included in the analysis. Only the Bremsstrahlung emission is taken into account in the electron energy equation. The reabsorption of the Bremsstrahlung emission in nodes 1 and 2 by Compton and by inverse Bremsstrahlung in the pre-compressed target is not important for ignition calculations, only for the study of the burning wave in the target (that is not the goal of the FINE program).

In the ignitor equations, the following terms were taken into account. Initially, the variation of electron energy is due to the external ion beam heating, Pign (the term ηd is the amount of the total energy of the projectile that is deposited in the electrons, the rest is deposited in the ions, all the energy is deposited in the ignitor region by definition of ignitor). In the program, the maximum beam powers were delimited by the energy of the projectile. For a given energy of the deuterons, and taking as the maximum limit of the density of the projectiles, the one given by the solid state density (in the case of Deuterium, 0.15 g/cm3 were taken), the maximum beam power is fixed. This maximum beam power was the choice for the analysis. If a higher power of the external ion beam (deuterons) is needed to reach ignition, a higher energy of the deuterons is needed. This higher energy supposes a change in the ignitor size, due to the dependence of the stopping power with the projectile energy. The amount of the total projectile energy that goes to electrons or ions depends on the stopping power equation of the external ions in both species. For degenerate or classical states, a different equation exists. The stopping power of the ions by the electrons is reduced if the plasmas are degenerate, due to the Pauli's exclusion principle (Skupsky, 1977; Deutsch et al., 1997a). The size of the ignitor strongly depends on the degeneracy of the plasma, and it is a very important variable in the ignition calculations. The ignitor size is defined as a cylinder, with a length equal to the mean free path of the projectiles, and a radius equal to the radius of the beam. Once the heating of the ignitor starts, and the temperatures of both species are different (due to the different amount of energy that goes from the projectiles to electrons and ions), the Pie (power from ion to electron) term appears. This equation is different, depending on the plasma state (classical or degenerate). The electrons, while increasing their temperature, emit Bremsstrahlung radiation (Pb). The Bremsstrahlung emission is much smaller in the degenerate state of the plasma that in the classical one (Eliezer et al., 2003). The next term in the energy conservation equation, Pfα, is the energy released by the alpha particles in electrons and ions (η and (1 − η) fractions), multiplied by the fraction of energy deposited by the alpha particles in node 1 or ignitor, and not in node 2 (factor f). Due to the difference in the electron temperature between node 1 and 2, and the pressure difference, the terms Phe (heat conduction that also depends strongly on the state of the plasmas, degenerate or classical) (Brysk, 1974), Pme, and Pmi (mechanical losses of plasma) appears. In the ion equation, the term f represents the fraction of neutron energy, Pfn (14 MeV DT fusion neutrons) that deposit their energy in node 1. Due to the high densities needed for reaching degenerate plasma, the fraction of neutron energy absorbed is much higher than in classical plasmas. This energy absorbed contributes also (with the minimizing of Bremsstrahlung emission) to the decrease in ignition temperature in degenerate plasmas.

The equations defined for node 2 are similar as the ones defined for node 1, with the contribution of the energy released by node 1 that is deposited in node 2. This energy is the principle energy gain of this node that permits the reaching of ignition conditions, and the launch of the fusion burning wave to the rest of the pre-compressed plasma. The above equations are solved, to obtain the ignition temperature, the gain for a given pre-compressed pellet, and a defined external ion beam.

3. RESULTS OF THE FINE PROGRAM: IGNITION TEMPERATURE AND GAIN

The equations defined above were solved for different plasma initial densities and temperatures (in this case, DT fuel has been chosen, but the analysis can be easily done with different fuels, as pB11) (Eliezer & Martìnez-Val, 1998; Son & Fisch, 2004), using the advantages of low degenerate emission in degenerate plasmas. In all cases, we depart from well degenerate plasma. The ignition temperature for the fast ignition concept was calculated as follows. Initially, the minimum external ion beam power has to be determined for each case. As was noted before, this power depends linearly on the energy of the projectile, so the higher the power needed, the higher the energy of the projectile used. Different projectile energies suppose different ignitor sizes. If the power of the external ion beam (or the energy of the external ion) is not high enough, equilibrium is reached between the power leakage from the ignitor, and the power gained in the ignitor by the external beam before ignition temperature is reached, so it is not possible to launch the fusion burning wave. Also, it is possible that the equilibrium is not reached, by the time needed for the ignitor to reach the ignition temperature is much higher than the confinement time, so it is not possible to launch the fusion burning wave before target dismantling. Once the minimum power for ignition is determined, the second step is to obtain which minimum ion temperature is needed for ignition (that is, going to be different than the electron temperature, as was explained before). The external beam is switched off once the ion temperature reaches a certain value. If the fusion burning wave is not launched, the ion temperature is below the ignition temperature. If the fusion burning wave is launched, the ion temperature is above the ignition temperature. With this iterative scheme, the minimum ion temperature capable of launching the burning wave to the rest of the pre-compressed plasma is determined. This minimum ion temperature is called the ignition temperature of the fast ignition scheme.

The gain is defined as the ratio of the energy obtained from fusion and the energy introduced in the pellet (compression energy and energy of the external ion beam). In the definition of the energy introduced in the pellet, the energy finally introduced is computed, not the total energy, taking into account the losses in the compression phase or the external ion beam generation. DT plasmas with different densities, initially well degenerate after compression, were analyzed. For a high density plasma (ρ = 10,000 g/cm3), Figure 2 defines the power evolution of the different energy exchange mechanisms in the plasma.

Energy exchange mechanisms in the node 1 or ignitor.

In Figure 2, Pf stands for fusion power (total) released in the ignitor, Pfalfa is the fusion power of alpha particles released in the ignitor, and Pfneut is the fusion power of neutrons released in the ignitor (it can be seen that this energy is higher than the energy released by the alpha particles). This is due to the high density, that permits to recover part of the energy from neutrons (80% of the total energy released by fusion), not as in classical cases. Pign stands for the power released by the ignitor, Pb is the power of Bremsstrahlung emitted, Pie is the energy interchanged among electrons and ions, Phe is the power of heat conduction (that goes to node 2), and Pme&i is the mechanical losses to the node 2, due to expansion. From Figure 2, it can be seen that the ignitor beam is heating the ignitor for 4 picoseconds. After this moment, the energy from fusion is larger than the energy losses from node 1, and the fusion burning wave is launched without additional external heating. The energy of the deuterons projectiles heating the ignitor is 10 MeV, the area power is 4.2120 W/cm2, the radius of the beam is 1.25−3 cm, and the ignition temperature is 3 keV. For this plasma density, the Fermi energy is 6.53 keV. This means that the heating phase, until the ions of the ignitor reaches the ignition temperature, occurs in degenerate plasma. All the equations defining energy exchanges of the electrons must to be defined for degenerate plasmas. This is not the case for smaller densities, as will be seen later.

The analysis of the ignition temperature and the gain was computed for different plasma densities. In each case, a 1 mg DT fuel pellet was analyzed. The following tables show the results of the calculations.

Results from high degeneracy calculations

Results from high degeneracy calculations

The first important result is the ignition temperature. In the classical limit, where the ignition temperature is defined as the plasma temperature to which the fusion power equals the Bremsstrahlung emission, the value is 4 keV independently of the density. In the case of degenerate plasma, Bremsstrahlung emission is much smaller, so the ignition temperature decreases notably (1.5 keV for 100.000 g/cm3) (Eliezer et al., 2003). The neutron energy absorbed by the ignitor region contributes to the decrease of the ignition temperature. For example, the neutron energy contribution is taken artificially as zero, the new ignition temperature for the 100.000 g/cm3 case is 2 keV. So, as it was suggested, degeneracy in high density plasmas permits an important decrease of ignition temperature. The problem is, to reach this very high densities, the energy introduced in the pellet is high. Figure 3 shows the electron compression energy (not taking into account the energy losses during compression phase from the driver, etc.) for different plasma densities and temperatures. In high degeneracy plasmas, the compression energy is independent of the temperature, and only depends on the density (Pauli's exclusion principle). If the electron temperature increases, and goes above the Fermi energy, the classical results are obtained, with a linear dependence of the temperature. For example, the compression of 1 mg of a well degenerate plasma of 10,000 g/cm3 density is higher than the compression of 1 mg of a classical plasma (100 g/cm3) to 1 keV. The high energy of compression needed (much higher than the energy needed for heating the ignitor with the external ion beam in all cases) gives a poor result in gain, as compared with smaller densities (e.g., 1,000 g/cm3).

Electron Compression Energy.

In the case of smaller densities (100, 1,000 g/cm3), the time the plasma remains degenerate during the heating period is quite small, since the ignition temperature is higher than the Fermi energy, and the specific heat of degenerate plasmas is much smaller than the classical ones (a small amount of energy in a degenerate plasma suppose a high increase in electron temperature). That is, the reason the ignition temperature is quite similar to the classical value.

4. CONCLUSIONS

The optimization of the compression phase in fast ignition inertial fusion, to obtain low temperature and high density plasmas, can lead to degenerate plasma. The equations that govern these plasmas are different than the classical ones. The decrease in Bremsstrahlung emission permits the decrease in ignition temperature, for high density plasmas. This assumption was demonstrated. The high energy needed to obtain this high density degenerate plasmas decrease the gain as compared to the results obtained in more moderated densities.

ACKNOWLEDGMENT

Many thanks to Dr. D. Hoffmann for bringing to our attention some very important research in the fast ignition field.

Footnotes

This paper was presented at the 28th ECLIM Conference in Rome, Italy.

References

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Figure 0

Fast Ignition in the two nodal concept.

Figure 1

Energy exchange mechanisms in the node 1 or ignitor.

Figure 2

Results from high degeneracy calculations

Figure 3

Results from high degeneracy calculations

Figure 4

Electron Compression Energy.