Toroidal plasma conditions where the p-11B fusion Lawson criterion could be eased

We examine the theoretical conditions in which the Lawson ignition criterion for p- 11 B fusion in a magnetized toroidal plasma can be reduced substantially. It is determined that a velocity differential between the protons and the boron ions of the order of the plasma sound speed (Mach number of 1 or 2 at a plasma temperature of ~10 2 keV) could raise the p- 11 B fusion reaction rate to ~2x10 -22 m 3 /s or ~6x10 -22 m^3/s, respectively, from the ~1x10 -22 m 3 /s level in a static plasma. The Lawson triple product (n i τ E T i ) required for ignition can thereby be reduced to as low as ~10 23 m -3 s keV, which is one order of magnitude above the ITER requirement for D-T burn. Since order-unity Mach numbers in velocity differentials between deuterons and impurity carbon ions have been maintained in tokamak plasmas under excellent connement conditions, similar levels of velocity differentials between protons and minority boron-11 ions could in principle be maintained also. A theoretical possibility of achieving p- 11 B fusion ignition in a toroidal plasma of ~10 2 keV in ion temperature is hereby presented. Similar p- 13 C plasmas, for example, will introduce a possibility of measuring the CNO fusion chain reaction rates in a laboratory.


Introduction
Neutron-and tritium-free fusion reactions, such as the CNO chain, 1-2 have powered the stars. The aneutronic p-11 B fusion reaction has lured the imagination of fusion energy researchers of magnetic con nement systems since the 70's. [3][4][5] Research and development toward this clean fusion energy from a magnetically con ned plasma, however, due to its very high required plasma temperatures (greater thañ 10 2 keV in the case of p-11 B), is an area of research imbued with great challenges. Overcoming them would create opportunities to change and possibly ease the engineering and technology requirements of a power producing fusion reactor. There is no guarantee that a practical solution for p-11 B fusion plasma exists or could be found. But if so, it would create a realistic chance to achieve the great goal of a universally available, inexhaustible, and environmentally benign power source. In this paper we introduce analysis showing that this challenge could in theory be met by creating a velocity differential between the protons and the boron ions that is of the order of the plasma sound speed.
In recent years however, the theoretical possibility of net energy gain from a magnetized p-11 B fusion plasma has remained elusive, 6 and has been considered by many to be theoretically impossible. The basis for this view has been known to include the following: 1. Bremsstrahlung radiation from a plasma increases strongly with the electron temperature, density and the effective ion charge Z eff . [7][8] As the plasma temperature is raised beyond 50 keV, relativistic effects would strongly enhance this radiation. 9 The presence of a magnetic eld can change the orbit of the electrons as they encounter Coulomb interactions with the ions. The magnetic eld can substantially change the bremsstrahlung radiation when the Larmor radius r B of an electron is smaller than the characteristic impact parameter r s of close Coulomb collisions. 10 According to analysis in [11], magnetic elds would reduce the bremsstrahlung radiation power. When r B is larger than r s , the effect of magnetic eld on bremsstrahlung radiation becomes negligible. For the magnetic con ned plasma discussed in this paper, r B is much larger than r s . So the traditional calculation for the bremsstrahlung without magnetic eld will be applied in this paper. Where the electron and ion temperatures are close to each other, the Bremsstrahlung radiation loss power density at a few hundreds of keV in plasma electron and ion temperatures would exceed the local power density produced by p-11 B fusion. 6 2. In the case of the tokamak, its energy con nement time 12 scales unfavorably with the total plasma heating power P (µP -0.69 ) required to reach such high temperatures. To maintain a plasma temperature at ~10 2 keV, a tokamak would require increased physical parameters (radii, magnetic eld, plasma current, etc.) to compensate for the degradation in energy con nement when the heating power (including fusion self-heating) is raised. However, sustained fusion plasma burn would become impossible under large increases in con nement when fusion ash would accumulate, dilute the fuel concentration, and discontinue fusion burn. 13 More recently, however, ideas were introduced that would mitigate certain aspects of these di culties, such as: 1. It was observed in tokamaks under intense Neutral Beam Injection (NBI) heating, [14][15][16] that during the so-called energetic or hot-ion mode, the central ion temperatures T i (0) were as high as 35keV while maintaining a ratio to the central electron temperature T i (0) / T e (0) higher than a factor of 3. This created a plasma condition where the Bremsstrahlung radiation loss power density could be reduced while maintaining the fusion power density. 6,17 2. By increasing the proton fraction x p of the plasma ions, reducing the effective charge (Z eff ) of the plasma, the Bremsstrahlung radiation loss would be reduced furthe,r 6,17 without reducing the fusion power substantially.

Method And Results
Following these leads, the n i t E vs. T i values required for ignition (Q pB = 1), as T i /T e and x p are varied from 2 to 8 and 0.5 to 0.9, respectively, and are calculated using the common approaches. 6,18 Recent reevaluated 11 B(p,3a) fusion reaction cross sections, 19 the accuracy of which was more recently further re ned, 15 are included. The kinetic effects that may cause an increase of the number of protons at higher energies (with respect to a pure Maxwellian distribution), leading to a net effect of approximately 30% increase of the fusion yield for the same global plasma parameters 9 are not included in the calculation.
Here, Q pB = 1 is de ned where the fusion a self-heating power exactly replaces the externally applied heating power that is required in the rst place to bring the plasma to ignition. This de nition is appropriate since p-11 B produces 100% of its fusion energy in the form of near-3-MeV a particles, which can in principle be well con ned in a tokamak or another magnetic con guration of su cient size and eld. Note that Lawson's net energy gain model 18 requires that , where Q is the plasma fusion energy gain and is the fusion energy conversion e ciency back to a fusion plasma in a power reactor. It is seen that Q pB = 1 corresponds to the requirement of high > 0.5. Here one could rely on some form of direct conversion of the plasma kinetic energy to electricity. 21 Figure 1 shows that under these conditions, p-11 B fusion ignition could in theory be achieved if the Lawson triple product (n i t E Ti) can reach ~10 24 m -3 s keV. This is however over two orders of magnitude higher than the ITER D-T fusion goal of ~6x10 21 m -3 s keV. 22 More recently, toroidal rotation velocities of ions of different charges (Ne-X, C-VI) were measured and compared with the calculated velocities of the main deuteron ions in a "Quiescent Double-Barrier" plasma with an excellent energy con nement time on the DIII-D tokamak. 23 Similar results of plasmas with a strong "Internal Transport Barrier" were found on the JET tokamak. [24][25] The impurity velocities were found to be proportional to ÑP i /Z i , where P i is the pressure and Z i the charge state of the species i. This is consistent with the theoretical neoclassical transport model of a tokamak plasma. 26 A substantial difference, V d , in the toroidal velocities of the impurity ions and the deuterons was observed, the corresponding differences in Mach numbers (M = V d /C s ) being near order-unity in magnitude. This points to a possibility of maintaining a substantial velocity differential between the protons and the boron ions in a future high-temperature p-11 B plasma.
We assume that these ions on the average rotate with a velocity differential V d . We further assume that these ions can be approximately described by a Maxwellian velocity distribution with a temperature T, written as: More recently, toroidal rotation velocities of ions of different charges (Ne-X, C-VI) were measured and compared with the calculated velocities of the main deuteron ions in a "Quiescent Double-Barrier" plasma with an excellent energy con nement time on the DIII-D tokamak. 23 Similar results of plasmas with a strong "Internal Transport Barrier" were found on the JET tokamak. [24][25] The impurity velocities were found to be proportional to ÑP i /Z i , where P i is the pressure and Z i the charge state of the species i. This is consistent with the theoretical neoclassical transport model of a tokamak plasma. 26 A substantial difference, V d , in the toroidal velocities of the impurity ions and the deuterons was observed, the corresponding differences in Mach numbers (M = V d /C s ) being near order-unity in magnitude. This points to a possibility of maintaining a substantial velocity differential between the protons and the boron ions in a future high-temperature p-11 B plasma.
We assume that these ions on the average rotate with a velocity differential V d . We further assume that these ions can be approximately described by a Maxwellian velocity distribution with a temperature T, written as: In the mass-centered coordinates the fusion reaction rates can be written as: A pronounced effect can also be seen in the corresponding Lawson criterion, 18 as shown in Figure 3.
The calculated n iE vs. T i values for Q pB = 0.2, 1, 2 with T i up to 300 keV are provided in Figure 3, where x p = 0.9 and T i /T e = 4 are assumed. Here Q pB is de ned as the fusion power replacing part of or all the externally applied heating power, which is used to reach a speci c Q pB value in the rst place. It is seen that the minimum triple product (n i E T i ) required to obtain these Q pB values would be lowered to ~1.4x, 7x, ~24x 10 22 m -3 s keV, respectively. A theoretical possibility of a driven or a sustained p-11 B plasma fusion burn via high V d is hereby indicated, reducing the Q pB = 1 requirement to about one order of magnitude above the ITER Phase I target of ~6x 10 21 m -3 s keV. 22 Discussion Another possible method of producing such large V d values are introduced through injection of a compact laser plasma accelerator (CLAPA) 27 proton stream into a toroidal plasma. When an ultraintense laser beam is incident on a plastic or metal foil, an intense proton beam of up to 150 pC with an energy spectrum of an exponential pro le from 100 keV to 10MeV and a divergence angle of tens degrees can be generated. 28 A velocity differential between the protons and the heavier ions (such as boron or carbon) in a toroidal plasma can be created during a CLAPA pulse. Although the time-duration of the proton bunch is as short as 100fs at the point of injection, it will lengthen to nanosecond or more due to the energy spread, depending on the ight path length.
Once integrated with a magnetic con nement con guration, CLAPA can reliably deliver intense proton beams of tens of pC, with such as 1% energy spread of different energies up to 10MeV, and good spatial uniformity. When injected into a p-11 B plasma of su cient temperature, density and beam path-lengths, the theoretical effects of large V d on p-11 B plasma fusion reaction rates can in principle be measured.
It is noted that ion acceleration driven by ultra-intense laser pulses has been an active eld of research for years because it delivers ion bunches of multiple-MeV energy with ultrashort duration, originating from μm-scale spots. Near the source, such bunches can be ~10 10 times denser than classically accelerated ion bunches. 29 This technical feature is of great importance not only in high energy density physics studies, but also in magnetic fusion plasma studies by enabling poloidal magnetic and electric eld measurement in 2D pro les. 30 An implication of creating a large V d in a proton and modest-Z ion plasma of ~10 2 keV in temperature would be a rst laboratory experiment on the aneutronic CNO fusion chain 1-2 reaction rates. By replacing 11 B with some of the stable isotopes in this reaction chain ( 12 C, 13 C, 14 N, or 15 N), a fusion device could be so arranged as to measure the corresponding fusion reaction rates, albeit at minute yet detectable levels.
Estimates of these Q p13C values (of the order 0.0001) for the ranges of V d and (n i Ei T i ) values considered in Figure 3 are shown in Figure 4. Although the values estimated could be uncertain at the present time by an order of magnitude, the potential contribution of such a laboratory test to the eld of aneutronic fusion physics in the stars would be large, and therefore deserves consideration.
It is of importance to point out that this theory assumes the maintenance of the protons and the 11 B ions that are not by themselves in thermal equilibrium. It has been calculated 31 that in this case the externally applied recirculating power density needed to maintain this condition would be higher than the fusion power density so produced. However, a combination of particle, momentum, and energy input into and loss from such a "toroidal magnetically con ned high-temperature plasma of multiple ion species" can lead to a physical system in which apparently non-equilibrium components are maintained as part of a more complex macroscopic uid equilibrium. 36 A surprising feature of such a plasma equilibrium is a theoretical possibility of substantial spatial separation of a relativistic electron uid from the thermal electron uid and the two thermal ion uids of different charge-to-mass ratios and ow velocities. The co-location requirement assumed in [31] is no longer appropriate, leaving open the question of the required external power to maintain this aparent absence of thermal equilibrium.
It is further necessary to point out that the four-uid equilibrium model contains a freedom to choose as input the spatial distributions of temperatures, densities, and velocities of these two ion uids. The stability of near sonic or supersonic velocities and velocity differentials in such an equilibrium becomes an important open question that must be analyzed during the next stage of this theoretical work.

Conclusion
This work establishes a theoretical basis for lowering the Lawson criterion of triple product (n i E T i ) for p- 11 B magnetic fusion plasma burn to about one order of magnitude above the ITER goal, by assuming x p = 0.9, T i /T e = 4, and V d = 2C s . This result indicates that a net energy gain via p-11 B magnetic plasma fusion is in theory possible. A scienti c bonus of this nding would be a possibility to measure the CNO fusion chain reaction rates in a suitably enabled toroidal fusion experimental device in a laboratory. This new experimental test could be enhanced by the application of the CLAPA technology. The development of techniques and magnetic con gurations that maintain a high, stable and stationary V d /C s value while reducing the recirculating power substantially is therefore of great importance to the goal of realizing a practical energy gain in a p-11 B magnetic fusion plasma.