Page 6 - Fusion reactivity of the pB11 plasma revisited

Nucl. Fusion 59 (2019) 076018
one can extend integration in the integral (A.16) to infinity and
obtain the following expression for this integral:
nσ /4π. (A.20) One can combine both limits by a simple interpolation:
ˆv′2′′∼nσ v3
v fσ(v)dv =4πv3+(3√π/4)v3 . (A.21)
0 Tσ
We could use an exact value of the integral, but approximation (A.21) has already an accuracy of better than 2%–3% for the relevant velocities.
For the collisions of the suprathermal protons with thermal protons, Borons, and alphas, one has v ≫ v′, and we obtain the following expressions for the diffusivities:
S.V. Putvinski et al
Figure B1. Correction factor ∆ (v) and ratio of corrected function given by equation (B.10) to the conventional slowing down distribution, equation (B.1). Ti = 300 keV, Te = 150 keV, nB/ni = 0.15.
D = î 4πΛe4Tnp
pp m3 v3 + (3√π/4) v3
ó
(A.22) (A.23)
p Tp
4πΛZB2 e4TnB DpB = A m3v3
.
In equation (A.22) we retain a term accounting for the finite thermal velocity of thermal protons (see definition of the thermal velocity in equation (A.17)). For the proton collisions with the Borons and thermal alpha particles, the corresponding term (accounting for the thermal spread of the Borons and alphas, equation (A.21)) is much smaller and we neglect it.
For collisions with the electrons, the situation is opposite, v′ ≫ v, and we get from equation (A.15):
Dpe =η(Te)2πΛe4 24πˆ ∞v′fe(v′)dv′ m2p 3 0
bp
= η (Te) 8√πΛe4ne ... me . (A.24) 3m2 2T
We have introduced here a correction factor η (Te) related to relativistic effects in the electron distribution functions and in electron–ion collision rates and evaluated in appendix C. For our reference temperature of Te = 150 keV it is η ≈ 1.1.
Appendix B. Distribution of high energy alpha particles
Usually the distribution function of high energy alpha par- ticles is evaluated assuming that they are born with the same initial energy (as in DT plasmas) and that their velocity is much higher than ion velocity but lower than electron velocity as in DT plasmas. The slowing down distribution of alpha par- ticles in this approximation is usually expressed as:
Sα τs
fα0 = 4π(v3 +v3) (B.1)
∗
at v < v1 and fα0 = 0 at v > v1, where v1 is the birth velocity of the alpha particles (see for example [21]). Here Sα is source density, τs is slowing down time of alphas on electrons
pe
Figure B2. Comparison of energy spectrum of alpha particle source fitted to the data presented in in [9] (solid line) with the source corrected for finite ion temperature, T = 325 keV (dashed line).
and
3 m T3/2
τs=√αe, (B.2) 1/2 Λ e4Z2 n
42πme e αe
Ç√ å1/3Å ã
3 π Λ i m e 􏰀 Z β2 n β v∗=4Λene βmβ
2 T e 1 / 2
me .(B.3)
In pB11 plasmas with three alpha particles born in a single reaction event their source spectrum is broad [7–9]. In addi- tion, at ion temperature Ti ∼ 200–400 keV proton velocity can be comparable with velocity of alpha particle with energy of few MeV. Thus equation (B.1) must be corrected.
The just born alpha-particles have a velocity higher than the proton velocity, but at the energy of E1 ∼ 1 MeV the
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