The fundamental difficulty with nonthermal ions is apparent when one compares the fusion cross section (_fusion  1 barn for p-¹¹B at E_cm = 580 keV, where E_cm is the energy in the center-of-mass frame) to the effective cross section for many small-angle Coulomb scattering events that combine to produce a scattering angle of 90° rms in an incident beam (_eff  60 barns for protons scattering on ¹¹B at E_cm = 580 keV). Highly nonthermal systems, like the colliding beam reactor proposed by Rostoker et al. would relax to local thermal equilibrium before a significant amount of fusion power could be produced. Alternatively, the nonthermal ion distribution could be maintained by cycling sufficient power through the system.

With this possibility in mind, it is instructive to examine the operating point presented by Rostoker et al. (n_p = 4 × 10¹⁵/cm³, n_B = 1 × 10¹³/cm³, T_p = 25 keV, and T_e = 20 keV), where n_p is the proton density, n_B is the boron density, T_p is the temperature of the protons, and T_e is the temperature of the electrons. We consider the problem in the frame-of-reference of the protons. Maintaining E_cm = 580 keV would require that the boron beam velocity be u_B  1.1 × 10⁹ cm/s. Coulomb collisions with the boron beam would heat the proton distribution at a rate (5) of 1/2m_Bu_B²n_B^B/p  500 W/cm³, where m_B = 11 AMU is the boron mass and ^i/j is the energy transfer rate from collisions between particles of species "i" and "j." As Rostoker et al. suggest, heat would be removed from the protons by collisions with the (colder) electrons at a rate of 3/2n_p(T_p  T_c) ^p/c  350 W/cm³. The relatively small net heating of the proton distribution could be eliminated if T_e were reduced to 18.6 keV.

The largest term in the electron power balance is direct heating by the boron beam, which would proceed at the rate of 1/2m_Bu_B²n_B^B/e  2.0 kW/cm³. Maintaining the electrons at T_e  18.6 keV would require that P_electron  2.5 kW/cm³ be extracted from the electron distribution. While bremsstrahlung could provide only 44 W/cm³ of this electron cooling, experience suggests that it would not be difficult to find other channels for electron heat loss. The 2.5 kW/cm³ extracted from the electrons must be balanced by ion heating.

The fusion power (P_fusion) calculated by Rostoker et al. at the operating point is P_fusion = n_pn_BY_{p-¹¹Bp-¹¹B}v  45 W/cm³, where Y_p-^11B = 8.7 MeV is the energy yield per fusion event (and v is the velocity of the proton and boron nuclei). The difference between P_electron and P_fusion sets a lower limit on the external ion heating power. Thus, the fusion gain (which is normally defined as the ratio of the fusion power to the external heating power), Q  P_fusion/P_heat  P_fusion/(P_electron  P_fusion)  0.02 would be much lower than the value of 2.7 stated by Rostoker et al. (1).

Highly nonthermal fusion schemes generally suffer from an unattractive power balance (4), and so magnetic fusion research has focused on systems that are in local thermal equilibrium. In such systems, high fusion gain (Q >> 1) remains a distinct possibility.

W. M. Nevins
Lawrence Livermore National Laboratory,
7000 East Avenue,
Livermore, CA 94550, USA

REFERENCES AND NOTES

1. N. Rostoker, M. W. Binderbauer, H. J. Monkhorst, Science 278, 1419 (1997) [Abstract/Free Full Text] .
2. For example, J. R. McNally, Nuclear Fusion 11, 187 (1971).
3. For example, M. Tuszewski, ibid. 28, 2033 (1988).
4. For example, T. H. Rider, Phys. Plasmas 4, 1039 (1997).
5. D. L. Book, NRL Plasma Formulary (Naval Research Laboratory, Washington, DC, 1987), p. 31.

Recognizing the unfavorable power balance of a thermal proton-boron (p-¹¹B) plasma, N. Rostoker et al. (1) propose restricting the proton energy relative to the boron to be near the resonance in the fusion cross section at E₀±E = (580 ± 140) keV. Although this beam-beam configuration would avoid the large power required (2) in the migma (3) to maintain the proton distribution against self-collisions, a large power input would nevertheless be required to replace the directed energy lost to frictional heating of the proton and boron beams. The classical formula (4) for this power density is where ln   15 (or larger) is the Coulomb logarithm and Z_B is the atomic number of boron, 5. Because the fusion power density is P_fusion = (8.7 MeV) n_pn_B, if we set the reactivity equal to its value at the resonance, enhanced by a factor of 1.6 for spin polarization,  = 1.3 × 10²¹ m³ s¹, then we find the ratio to be P_fusion/P_fric  0.12, which shows that the proposal for a reactor with net electrical power output is unrealistic.

The power balance would be at least another factor of three less favorable than this estimate because the coupling of the ions through the electrons would be stronger than the direct coupling if T_e < E₀/15 = 40 keV. If the electron coupling were decreased by raising T_e, T_p must also rise. Rostoker et al. suggest that the T_p has only to be less than 140 keV, but this is not sufficiently cool. In order for protons with velocity ±  to lie within the resonance, T must be _BT_p  (E)²/2E₀ = 17 keV. Rostoker et al. published a similar unworkable reactor design earlier (5) with _BT_p = 200 keV.

Next to power balance, a serious problem of the CBFR is equilibrium. The plasma volume envisaged would be a long, thin cylindrical shell with thickness/radius r/r  0.08. Such a configuration would not be in axial equilibrium because the tension of the field lines curving around the shell at the ends would provide a powerful compressive force. For 2-dimensional equilibria, r/r  0.5 (6), so the highly localized profiles required to prevent radial particle and energy losses would not be maintained.

A. Carlson
Max Planck Institute for Plasma Physics,
85748, Garching, Germany

REFERENCES

1. N. Rostoker, M. W. Binderbauer, H. J. Monkhorst, Science 278, 1419 (1997) .
2. T. H. Rider, thesis, Massachusetts Institute of Technology (1995).
3. N. Rostoker, et al., Phys. Rev. Lett. 70, 1818 (1993) [CrossRef] [ISI] [Medline].
4. B. A. Trubnikov, in Reviews of Plasma Physics, M. A. Leontovich, Ed. (Consultants Bureau, New York, 1965), vol. 1, pp. 105-204.
5. N. Rostoker, M. Bindebauer, H. J. Monkhurst, Comments Plasma Phys. Controlled Fusion 18, 11 (1997).
6. W. T. Armstrong, R. K. Linford, J. Lipson, D. A. Platt, E. G. Sherwood, Phys. Fluids 24, 2068 (1981) [CrossRef].

Response: The main point of the comment by Nevins is that direct heating of electrons from Coulomb scattering by the boron beam (2.0 kW/cm³) is the largest term in the electron power balance, and that we neglected it in our article (1). The electrons would have to be maintained at a low T in order to cool the proton beam. The problem is with the calculation of the electron T, which is not detailed in our article or in the comment by W. M. Nevins. The calculation of heating of electrons by boron scattering does give a large result. However, electrons lose energy to the boron ions because their velocity would be higher and there would be a collisional drag effect. This energy loss would cancel the energy gain. The electron temperature T_e is defined by where n_e is the electron density, _e is the average electron velocity, f_e(, ) is the electron distribution function, and N_e is the line density of electrons, which is constant. The calculation of dT_e/dt involves the Fokker-Planck equation and is complicated. If it is done in a straightforward way with the above definition, the result will be as Nevins describes. However, if we note that in a steady state dV_e/dt = 0 and calculate only d²/dt, the cancellation takes place, and electrons would only be heated by the protons. Similar considerations show that protons are cooled by electrons and boron.

We have revisited ideas such as "the use of p-¹¹B as a fuel." However, we have developed new and systematic calculations and the conclusions are not the same as they were with the use of that proposed fuel.

Carlson employs a classical generic formula for the power density required to overcome the friction between proton and boron beams. This formula is inadequate for the Colliding Beam Fusion Reactor. The magnetic field is important, and it is distinguished by its absence in this formula. The complete formula can be derived by taking the appropriate moment of the Vlasov/Fokker-Planck equation. First, the equilibrium conservation of momentum equation is with i = (1, 2) for protons and boron. Here n_i is the particle density; e_i is the charge; m_i is the mass; T_i is the temperature; E_r and B_z are the electric and magnetic field, respectively; and V_i is the velocity in the azimuthal direction. The kinetic equation obtained from the same moment, but including the collision operator, is with a similar equation for ₂. The dot product of ₁ with the above equations yields an energy (power) equation and the magnetic field seems to vanish. The power is The first term is precisely the expression for power density employed by Carlson. The second term involves the radial velocity V_ir and is positive definite. It can be estimated with some approximations; V_ir << V_i and n_iV_ir = D_i(n_i/_r), where D_i is the diffusion coefficient; D₁  a₁²/t₁₂, where a₁ = V₁/₁ is the gyro-radius and ₁ = e₁|B_z|/m₁c. Similar expressions obtain for boron. Although V_ir may be neglected as compared with V_i, it may not be neglected in the second term. This term can be estimated from the equilibrium equation. One can see that the magnetic field that previously seemed to vanish has returned. The second term is positive definite and the magnitude n₁m₁(a₁/L₁)(V₁²/t₁₂) is similar to that of the first term. L₁ is the scale length of the equilibrium, that is (1/L₁) = (1/n₁)|dn₁/dr|. To determine this power quantitatively requires a considerable amount of work (2). It requires a new development in classical transport theory because earlier studies assume a_i << L₁, which would not be the case in the CBFR. This calculation should also include electrons and the fusion products. The result is that P_fric would be tolerable.

Concerning the resonance, we have made detailed calculations. If the beam temperatures are less than one-half of the half width of the resonance, the reactivity should be greater than one-half of the maximum reactivity for zero temperature beams. The result stated by Carlson that the beam temperature must be less than 17 keV seems to contradict this. However, no results for are given for comparison.

The equilibrium calculations to which Carlson refers are not appropriate for the CBFR. Axial equilibrium requires an axial T, or the FRC will contract in the axial direction. It has been observed experimentally that FRCs have long axial equilibria. We have previously considered long, thin cylindrical shell models because they simplify many calculations.

Norman Rostoker
Michl W. Binderbauer
Department of Physics and Astronomy,
University of California,
Irvine, CA 92697-4575, USA
Hendrik J. Monkhorst
Department of Physics,
University of Florida,
Gainesville, FL 32611-8435, USA

REFERENCES

1. N. Rostoker, M. W. Binderbauer, H. J. Monkhorst, Science 278, 1419 (1997) .
2. ___, in preparation.