Following these leads, the *n*i*t*E vs. *T*i values required for ignition (*Q*pB = 1), as *T*i/Te 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 re-evaluated 11B(p,3a) fusion reaction cross sections,19 the accuracy of which was more recently further refined,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 parameters9 are not included in the calculation.

Here, *Q*pB = 1 is defined where the fusion *a* self-heating power exactly replaces the externally applied heating power that is required in the first place to bring the plasma to ignition. This definition is appropriate since p-11B produces 100% of its fusion energy in the form of near-3-MeV a particles, which can in principle be well confined in a tokamak or another magnetic configuration of sufficient size and field. Note that Lawson’s net energy gain model18 requires that , where Q is the plasma fusion energy gain and is the fusion energy conversion efficiency 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-11B fusion ignition could in theory be achieved if the Lawson triple product (*n*i *t*E* Ti*) can reach ~1024 m-3 s keV. This is however over two orders of magnitude higher than the ITER D-T fusion goal of ~6x1021 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 confinement 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 *Ñ**P*i/Zi, 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/Cs) 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-11B 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 confinement 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 *Ñ**P*i/Zi, 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/Cs) 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-11B 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:

Figure 2 shows the variations of this p-11B reaction rate, as a function of *T*i, and as *V*d/Cs = 0, 1, 2, 3, where *C*s = [(Ti + Te)/MpB]0.5. It is seen 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 ~102 keV) could raise the p-11B fusion reaction rate to ~2x10-22 m3/s or ~6x10-22 m3/s, respectively, from the ~1x10-22 m3/s level in a static plasma. The results show that the maximum reaction rates below 300keV would nearly double to ~7x10-22 m3/s as *V*d is increased to 2*C*s; a large increase to a value of 7.5x10-22 m3/s would be obtained when *T*i is reduced toward 50keV, if *V*d is increased to 3*C*s.

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/Te = 4 are assumed. Here *Q*pB is defined as the fusion power replacing part of or all the externally applied heating power, which is used to reach a specific *Q*pB value in the first place. It is seen that the minimum triple product (*n*i E Ti) required to obtain these *Q*pB values would be lowered to ~1.4x, ~7x, ~24x 1022 m-3 s keV, respectively. A theoretical possibility of a driven or a sustained p-11B 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 1021 m-3 s keV.22