Aneutronic Fusion in a Degenerate Plasma

Aneutronic Fusion in a Degenerate Plasma

S. Son and N. J. Fisch, Princeton Plasma Physics Laboratory, Princeton University.

Princeton Plasma Physics Laboratory report PPPL-4003, UC-70, September 2004. Prepared for the U.S. Department of Energy under Contract DE-AC02-76CH03073.

Key words: aneutronic, fusion, degeneracy, stopping, bremsstrahlung, proton, boron, helium, deuterium. PACS: 52.27.Gr, 52.27.-z, 52.25.Mq.

Abstract

In a Fermi-degenerate plasma, the electronic stopping of a slow ion is smaller than that given by the classical formula, because some transitions between the electron states are forbidden. The bremsstrahlung losses are then smaller, so that the nuclear burning of an aneutronic fuel is more efficient. Consequently, there occurs a parameter regime in which self-burning is possible. Practical obstacles in this regime that must be overcome before net energy can be realized include the compression of the fuel to an ultra dense state and the creation of a hot spot.

1. Introduction

A fusion reaction in which no neutrons are produced is safer and cleaner than a D-T fusion reactor. Proton plus boron-11 giving three alpha particles at 2.7 MeV, and deuterium plus helium-3 giving a 14.7 MeV proton and a 3.6 MeV alpha particle, are the most promising reactions for this purpose. However, the cross-section for these fuels is appreciable only when the ion temperature exceeds 100 keV for P-B-11, and 50 keV for D-He-3. P-B-11 is the cleanest, but it needs a high temperature for burning. D-He-3 requires only a moderate temperature, but it has two disadvantages. One is the production of neutrons from the D-D and D-T reactions. This can be partially overcome with a high-He-3 and low-D fuel mixture, which then must be burned above 100 keV. The other drawback is the scarcity of He-3 on Earth.

For P-B-11, Dawson pointed out that the bremsstrahlung power at a temperature of 200 keV is greater than the fusion power, which makes self-burning unlikely. To avoid the bremsstrahlung losses, the electron temperature must be much lower than the ion temperature, but not too low, because the fusion byproducts should be preferentially stopped by the ions. In view of this consideration, the electron temperature must be in a narrow range around 100 keV to retain the possibility of self-burning.

For inertial confinement fusion using P-B-11, there have been several theoretical attempts to generate a detonation wave. Martinez-Val and Eliezer showed that compressed fuel can be burned by an expanding ion fusion-burning wave preceded by an electron-conduction heat detonation wave. A large gap between an electron temperature of about 80 keV and an ion temperature of about 200 keV might then be achievable. Leon suggested that the bremsstrahlung, the stopping power of an alpha particle, and the ion-electron collision rate are all reduced due to the electron degeneracy, facilitating the detonation wave. However, the feasibility of self-burning even with this temperature differential still remains unclear. We show here that, without consideration of the practicality, for a particular parameter regime, Fermi degeneracy plays an important role in reducing the alpha particle stopping, ion-electron collisions and the bremsstrahlung, so that self-burning is possible. The optimal regimes are characterized by an electron temperature much lower than the 80 keV suggested by Eliezer and Martinez-Val.

The feasibility of inertial confinement fusion in D-He-3 is greater if the fusion reactivity is higher. In D-He-3 fusion, a 14 MeV proton transfers its energy mainly to the electrons, and so the electron temperature is the same as or higher than the ion temperature. Because of the bremsstrahlung losses, the concentration of deuterium has a minimum value, below which burning is impossible. By reducing this minimum value, we can achieve the minimum number of neutrons. Honda pointed out that, due to the nuclear elastic scattering, there will be more energy transfer to the ions from a 14 MeV proton, which though still smaller than to the electrons, improves the fusion reactivity. We show that, due to the degeneracy, the proton can be stopped mainly by He-3 through the nuclear elastic scattering, and the fuel might be burned at low electron temperature and high ion temperature.

Accessing a regime in which the ion temperature greatly exceeds the electron temperature is always useful for achieving controlled fusion. For example, in magnetically confined fusion, the low electron temperature reduces the requirement on confining plasma pressure, and the reactivity can be improved by channeling the alpha energy in the D-T reaction. In a tokamak, this might be accomplished through rf waves. Here, the regime of hot ions and cold electrons affords the possibility of achieving ignition altogether in aneutronic fuel.

The paper is organized as follows. In Sec. 2, the stopping power formula in an electron degenerate plasma is presented. In Sec. 3, the regime of self-burning is identified using the formula in Sec. 2. In Sec. 4, the density-radius equation for fuel burning is solved to find the appropriate pellet dimension and the constraint on the laser or ion beam power. In Sec. 5, we discuss the implications and limitations of these results. In Sec. 6, we summarize our main results.

2. Electronic Stopping Power

The electronic stopping power in an electron degenerate metal has been intensively studied both theoretically and experimentally. In a fully degenerate plasma, when the velocity of an ion is smaller than the electron Fermi velocity, the electronic stopping power becomes almost independent of the density and proportional to the ion velocity. Equation (1) of the report gives the rate of energy loss as a numerical coefficient times the square of the ion charge, times the square of the electron mass, times the fourth power of the elementary charge, divided by the ion mass and the cube of the reduced Planck constant, all multiplied by the ion energy and by a slowly varying function of the coupling parameter. That parameter is the square of the elementary charge divided by pi times the reduced Planck constant times the Fermi velocity, and the function is the standard logarithmic form given by Nagy and co-workers.

The above formula is valid if the ion velocity is much less than the Fermi velocity and the mean interelectron spacing parameter is much less than one. The collisions occur between the ion and the fastest electrons rather than, as in a weakly-coupled hot plasma, between the ion and the thermal electrons. The collisional cross-section decreases as the inverse fourth power of the Fermi velocity. This strong dependence of the cross-section on the Fermi velocity just suffices to cancel the effect of the greater electron density, the greater energy loss per collision, and the greater relative velocity of the colliding particles. The stopping frequency then is independent of the electron density.

The ion-electron collision frequency (Equation 2) is 3.47 times ten to the thirteenth per second, multiplied by the square of the ion charge divided by the nucleus mass in units of the proton mass; the coupling function takes the value 2 when the density is ten to the twenty-eighth per cubic centimetre. For further details, see Appendix A.

3. Regimes of Self-burning

3.1 P-B-11

When the electron temperature is below the ion temperature, the ion kinetic energy is drained into the electrons. The rate of the energy drain from the ions to the electrons is three halves of the ion-electron collision frequency times the ion density times the ion temperature, with the collision frequency given by Eq. (2) and with the ion temperature assumed much greater than the electron temperature. The rate of the fusion energy production is the product of the two fusing densities, the Maxwellian-averaged reactivity, and the energy released per fusion.

We now consider P-B-11 fuel. Equation (3) gives the ratio of the ion-to-electron drain power to the fusion power as a function of the boron-to-proton ratio, the electron density, the reactivity and the ion temperature. For the fuel to be burned, firstly, that ratio must be less than one, and secondly, the fusion product must be stopped mainly by the ions and not by the electrons.

Note that Eq. (3), as a function of the boron-to-proton ratio, has a minimum when that ratio is about 0.3. With a ratio of 0.3 and an ion temperature of 200 keV, we note that the drain power equals the fusion power when the electron density is 6.69 times ten to the twenty-eighth per cubic centimetre. Thus any electron density above that value satisfies the first requirement. We used the recent reaction activity data of Nevins and Swain, a reactivity of 2.5 times ten to the minus sixteenth cubic centimetres per second, which is less than the activity given by the old data by 37.5 percent.

For the second requirement, with a ratio of 0.3, the ion stopping frequency of the alpha particle is 7.32 times ten to the thirteenth per second, scaled by the density ratio and by the three-halves power of the ratio of the initial alpha energy of 2.7 MeV to the instantaneous energy, with a Coulomb logarithm of 5 and the classical collision formula. Equation (4) integrates the fraction of the energy transfer from the alpha particle to the ions over the slowing-down history. When the electron density equals the threshold value, 85 percent of the alpha particle energy goes to ions; at twice the threshold density, the fraction is 92 percent. Therefore, the second requirement is satisfied automatically when the density exceeds the threshold.

The electron temperature is determined from the balance between the energy input from the ions and the losses from the bremsstrahlung (Equation 5). For twice the threshold density and a boron-to-proton ratio of 0.3, the ion-to-electron power is 9.3 times ten to the forty-seventh electron volts per cubic centimetre per second. Using the classical bremsstrahlung formula (Equation 6), which scales as the electron density times the square root of the temperature times the sum over ion species of the density times the squared charge, with a relativistic correction factor, we obtain an electron temperature of about 27 keV.

The above analysis shows that, in principle, P-B-11 can be burned with an ion temperature of 200 keV and an electron temperature of 27 keV, with the optimized fuel concentration of 0.3 overcoming the bremsstrahlung losses. The density of the system is slightly more than 3.8 times ten to the fifth grams per cubic centimetre, and the Fermi energy is 95 keV. Since the electron temperature is below the Fermi energy, the electrons are still degenerate.

3.2 D-He-3

In D-He-3 inertial confinement fusion, the electron temperature can generally be no less than the ion temperature, since a 14 MeV proton is mainly stopped by electrons. For example, if the deuterium-to-helium ratio is 0.1 to assure a low neutron level, then the ratio of bremsstrahlung to fusion power is larger than one at all temperatures, and a self-sustaining burn is not possible. However, nuclear elastic collisions do channel energy from a 14 MeV proton to He-3, improving the chance for a self-sustaining burn. Nonetheless, since most of the energy still goes from the proton to the electrons, the electron temperature still cannot be much lower than the ion temperature. Thus, it appears, at first sight, necessary to increase the deuterium fraction for burning. We show, however, that, in an ultra dense plasma at ten to the fifth grams per cubic centimetre, the nuclear elastic collisions can transfer the proton energy mainly to He-3, and so the electron temperature might be lower than the ion temperature as in the P-B-11 case, thereby achieving the self-sustaining burn condition.

If Coulomb stopping of the proton by the ions is ignored, the energy loss of the proton to the ions (Equation 7) is the nuclear-elastic-collision cross-section times the proton velocity times the fraction of proton energy lost per collision. Equation (8) then gives the fraction of the energy deposited in the electrons as an integral over the slowing-down history of the ion-electron collision term divided by the sum of that term and the nuclear elastic term, starting from 14.7 MeV.

Using nuclear elastic collision data, roughly, the electron fraction is one divided by one plus the helium density scaled by 3.47 times ten to the twenty-eighth. For a density of three times ten to the fifth grams per cubic centimetre and a deuterium-to-helium ratio of 0.1, 35 percent goes to electrons. The alpha particle also transfers less than 10 percent of its energy to the electrons. Overall, 70 percent of the energy from the fusion products goes to the ions.

For that fuel ratio, an ion temperature of 70 keV and the density given above, we note that the ion-to-electron power is 2.94 times ten to the forty-seventh and the fusion power is 5.79 times ten to the forty-seventh electron volts per cubic centimetre per second, using a reactivity of ten to the minus sixteenth cubic centimetres per second, and that the bremsstrahlung is 2.6 times ten to the forty-fifth times the square root of the electron temperature in electron volts, assumed non-relativistic. We note also that seven tenths of the fusion power divided by the ion-to-electron power is 1.38, greater than one, and so the fuel burns. By balancing the ion-to-electron power plus three tenths of the fusion power against the bremsstrahlung, we find an electron temperature of 32 keV. The plasma is still partially degenerate since the Fermi energy is 90 keV.

4. Reactor Prospects

To find the pellet dimension and total power, we solve the density-radius equation in P-B-11 with a boron-to-proton ratio of 0.3 and a density of 3.8 times ten to the fifth grams per cubic centimetre. Equation (9) gives the burn-up rate as 1.2 times ten to the thirteenth times a factor linear in the remaining fuel fraction, and integrates to an exponential decay of that fraction with an e-folding time of the inverse of 1.2 times ten to the thirteenth seconds.

For total burn-up, the confinement time, the pellet radius divided by the sound speed, must be larger than ten to the minus thirteenth seconds. Assuming the sound speed is set by the Fermi energy and the mass density, the pellet radius must be larger than ten to the minus fourth centimetres. The electron degeneracy energy is three times ten to the ninth joules per gram. As an example, for a radius of ten to the minus third centimetres, by putting in 4.78 MJ, we get 88 MJ out, and so the gain is 18.31.

For D-He-3 with a deuterium-to-helium ratio of 0.1 and a density of three times ten to the fifth grams per cubic centimetre, the pellet dimension and the energy input characteristic are almost the same as with the P-B-11 analysis. The gain is 15.

The feasibility as a reactor for either of these fuels is low because the gain is smaller than 20, and more than 200 is usually required. We note that the gain can be as large as 1000 in D-T fuel. The creation of a hot spot for fast ignition might be a way of improving the gain substantially. This might be done by using a small D-T pellet inside the aneutronic fuel, or a fission-fusion hybrid concept. Because of the ultra dense condition, a difficult practical requirement for the uniformity of the laser or particle beam in compression must be met.

5. Discussion

As shown in Sec. 3, the radiation losses can be overcome sufficiently for self-sustained burning. Rider pointed out that the fusion power of an aneutronic plasma is substantially smaller than the minimum recirculating power needed to maintain the non-equilibrium condition of high ion and low electron temperature, which diminishes the prospect for utilizing aneutronic fuel. However, his derivation is under the assumption that the two-body effects are proportional to the volume integral of the squared density. In an ultra dense plasma, we showed the stopping power is not proportional to the electron density, breaking that assumption, and thus avoiding the negative conclusion by Rider.

The practicality as a reactor is likely small, as discussed in Sec. 4, unless the gain can be made larger. In this respect, note that certain assumptions made here might be too pessimistic. One is that we assumed total electron degeneracy in the calculation of the ion-electron collision frequency. In a partially degenerate plasma, that frequency has the tendency to decrease as a function of the electron temperature, but the detailed result of the stopping power in a partially degenerate plasma has not been incorporated into our calculation.

To see why, in a partially degenerate plasma, the slowing down might be smaller yet, consider a plasma in which electrons can be assumed to be classical. When the ion velocity is much less than the thermal electron velocity, the ion energy loss comes solely from collisions of the ion with the electrons slower than the ion. For an isotropic velocity distribution, faster electrons do not drag the ion because of a well-known cancellation. On the other hand, if the ion velocity is much less than the Fermi velocity and the electrons are completely degenerate, then the drag on the ion comes mainly from the electrons at the Fermi velocity. The force from those electrons does not cancel, in contrast to the classical limit. This is because, due to the lack of the asymmetry of the electron-hole transition probability, the drag force of electrons on an ion is not exactly an inverse-square law: it depends on the direction relative to the ion velocity. The cancellation, however, occurs only for inverse-square forces. Electrons slower than the Fermi velocity minus the ion velocity do not drag the ion, because these electrons do not collide with the ion due to the lack of available holes.

Consider now a case where the ion velocity is much less than the thermal electron velocity, which is in turn much less than the Fermi velocity. The electrons slower than the ion still do not drag the ions, because no hole is available. The drag by the electrons near the Fermi velocity is greatly reduced compared to the case of complete degeneracy, since the transition probability asymmetry is not very sharp, but has instead the scale of the thermal electron velocity. Its effect can be roughly estimated, and it might imply that the ion-electron collision frequency must be reduced by the ratio of the thermal electron velocity to the Fermi velocity compared to Eq. (3). When the temperature approaches the Fermi energy, mainly electrons slower than the ion contribute to the stopping. The stopping frequency is then proportional to the classical ion-electron collision frequency at the Fermi temperature, multiplied by a Fermi-function factor built from the chemical potential.

The above rough considerations seem to imply, if speculatively, that if the ion velocity is much less than the thermal electron velocity, which is much less than the Fermi velocity, the ion-electron collision frequency is reduced further as a function of the electron temperature in the window between the ion kinetic energy scale and the Fermi energy. This is especially true for P-B-11, because the ratio of thermal ion to thermal electron velocity is about 0.1 with an electron temperature of 10 keV and an ion temperature of 200 keV. In D-He-3, the 14 MeV proton velocity is too large to have such a separation. But in P-B-11, we might speculate that there will be a big reduction of the stopping frequency for an appropriate electron temperature.

Secondly, the bremsstrahlung is also reduced. When the Fermi energy greatly exceeds the electron temperature, not all electrons collide with the ions, since many of the electron-hole transitions are forbidden. The estimate, using the classical derivation of the bremsstrahlung, shows that the total loss will be reduced from the classical formula by the three-halves power of the ratio of temperature to Fermi energy. If the bremsstrahlung is reduced too much so that the electrons begin to heat up, we can put some high-Z impurity into the fuel so that we can fine tune the bremsstrahlung to balance with the ion-electron energy transfer at the optimal electron temperature.

Thirdly, at such a high density as ten to the twenty-ninth per cubic centimetre, the plasma frequency corresponds to 10 keV, and a significant fraction of the energy radiated will be re-absorbed, given the fact that the electron temperature is a few tens of keV. The Compton heating of the electrons also turns out to be significant.

The above considerations tell us that the severe condition imposed for self-burning in Sec. 3 can be eased by the further reduction in the ion-electron collision frequency, the bremsstrahlung losses and the re-absorption. However, this is all quite speculative; an estimation of how much it will help remains to be seen. In particular, the bremsstrahlung and the stopping power should be taken into account in the full context of the partial degeneracy.

As a warning, we note the following. In the stopping power estimation of Sec. 2, we assume a density of ten to the twenty-eighth per cubic centimetre at zero electron temperature, but for ten to the twenty-ninth per cubic centimetre we note a 10 percent increase in the electron stopping compared with Eq. (2). We also note that the relativistic effect should be taken into account in the calculation of the stopping power and the bremsstrahlung, because the Fermi energy is 20 percent of the electron mass energy. We estimate that, in the bremsstrahlung, the partial degeneracy is a much more important effect than the relativistic one. But, in the stopping power reduction, the relativistic effect might be as important as the partial degeneracy.

Therefore, we propose that the full time evolution of the fuel burning should be obtained with the relativistic effect, the partial degeneracy, the local field correction and the other effects mentioned taken into account. While this is beyond the scope of the present manuscript, it is clear to the extent that these effects tend to reduce the coupling of the electrons, and it will be even easier to maintain disparate ion and electron temperatures and hence greater activity.

6. Summary

In this paper, we identified a possible ignition regime for P-B-11 and D-He-3, in which the density exceeds ten to the fifth grams per cubic centimetre, the ion temperature is about 100 keV, and the electron temperature is 30 keV. The degeneracy of the electrons reduces the stopping power and the bremsstrahlung losses, which facilitates self-sustained burning. It is mainly the reduction in the stopping power of the electrons that enables such a large differential between ion and electron temperature. While the power requirements suggest that this regime is still impractical for inertial confinement fusion, the regime may be practical should the present assumptions turn out to be wrong concerning the electron stopping in a partially degenerate plasma, the Compton heating, the reduction of the bremsstrahlung, relativistic effects, or the re-absorption of the radiation. Some arguments are given suggesting that these assumptions may in fact overstate the stopping by electrons.

The authors thank R. Kulsrud, G. Hammett and S. Cohen for useful discussions. This work was supported by the U.S. DOE under contract AC02-76CH0-3073.

Appendix A: Stopping Power in a Degenerate Plasma

The appendix derives the stopping formula from the dielectric response of the degenerate electron gas, following Lindhard. The total field generated by a test particle travelling at constant velocity is the bare field divided by the dielectric function (Equation 10). The polarisation field at the position of the test particle is obtained by Fourier transform (Equation 11), and reduces to an integral over wavevector of the imaginary part of the dielectric function weighted by the projection of the wavevector on the velocity. The stopping power, the energy loss of the particle per unit length by the drag, is the charge times that polarisation field (Equation 12).

The longitudinal dielectric function in a completely degenerate plasma (Equation 13) is one plus three times the squared plasma frequency divided by the squared wavevector and squared Fermi velocity, multiplied by the Lindhard function (Equation 14), whose real and imaginary parts are given in the small-damping limit (Equation 15) in the two dimensionless variables formed from the wavevector scaled by twice the Fermi wavevector and the frequency scaled by the wavevector times the Fermi velocity. Substituting into the stopping integral gives the standard form (Equation 16): four pi times the squared ion charge, the fourth power of the elementary charge, the electron density and a stopping number, divided by the electron mass and the squared velocity, with the stopping number given as a double integral (Equation 17).

In the limit where the ion velocity is much less than the Fermi velocity, the imaginary part of the Lindhard function becomes linear in frequency below the cutoff and vanishes above it (Equation 18), and the stopping power reduces to a form proportional to the ion velocity and almost independent of the electron density (Equations 19 and 20). If the real part is set to its zero-argument value and the coupling parameter is small, the coefficient reduces to minus the logarithm of that parameter, the result obtained by Fermi. The ion-electron collision frequency then follows from the stopping power divided by the kinetic energy and the velocity.

(The reference list and the original equations are in the complete report at osti.gov/servlets/purl/835895. On this site, the case for hydrogen-boron as the clean fusion fuel is at /library/stm-8dfcd2d56d, the first measurement of proton-boron-11 fusion in a magnetically confined plasma is at /library/stm-9e2a22de0f, the dense plasma focus route to the same fuel is at /library/stm-05100e66da, and the Defense Intelligence Reference Document on aneutronic fusion propulsion is at /library/stm-b5e092d030.)