Page 8 - Fusion reactivity of the pB11 plasma revisited

Nucl. Fusion 59 (2019) 076018
The three alphas get additional 3T/2 from Doppler shift (second term) and energy of relative motion weighted by cross section.
Figure B2 compares the alpha particle spectrum at T = 325 keV with the one based on the data presented in [9] (T = 0).
To get the slowing down distribution function we inte- grated equation (B.10) with the spectrum equation (B.12),
S.V. Putvinski et al
The ion velocity is small compared to the electron velocity. The energy transfer to the ion can then be evalu- ated for the resting ion. We use a technique described in [22]. It is based on the fact that the main contribution to the electron scattering on the Coulomb center comes from the large impact parameter and small scattering angles. For large impact parameters the electron deflection can be found perturbatively. The momentum pi acquired by the ion is
(C.6) where v is the electron velocity and a is impact parameter. Accordingly, the energy acquired by the ion with atomic mass
fα =
ˆ ∞ 2
Slab (E)τsdE 4π(v3 +v3∆(v)).
2Ze2 pi = av
mv /2
∗
Appendix C. Electron–ion energy exchange in the case of relativistic electrons
This appendix is concerned with the energy exchange rate between the heavy non-relativistic ions and sub-relativ- istic electrons. The electron temperature anticipated for the pB11 reactors is Te = 150 keV, so that a part of an electron Maxwellian distribution occupies energies exceeding 500 keV. The significance of this effect has been pointed out to us by Barnes (private communication).
®􏰀´˙
f (p) = nC(Te)exp − Te (C.1) 4πΛZ2e4 ˆ ∞ 4πp2
v = pc/􏰀p2 + m2c2. (C.8)
Integrating by the impact parameters and electron distribution function, we find the rate of the ion energy increase:
where p is a momentum and C(Te) is a normalization factor = Amp 0 v f ( p) dp, (C.9) determined from the following condition:
ˆ where Λ is the Coulomb logarithm. Then, noting that for
A is
∆W = 2Z2e4 . Amp a2 v2
(C.7)
The velocity v is related to the electron momentum by:
The relativistic Maxwellian momentum distribution is:
2Z2e4 ˆ 2πda ˆ ∞ 4πp2 m2c4+p2c2−mc2 W= Amp a 0 v f(p)dp
fd3p=n. (C.2)
2
The last term mc in the expon􏰁ent of equation (C.1) is intro-
duced to make the distribution function matching both non- relativistic and ultra-relativistic limits.
In normalized variables
p Te
􏰁p=mc,Te=mc2 (C.3)
the normalization constant becomes:
Maxwellian ions T˙ i = (2/3) W˙ , and using expression (C.1) for the distribution function, we find after some algebra:
Ä 2ä 2nmc 4πΛZ2e4 1+2T􏰁e+2T􏰁e
(E)
νei =3M· 22 ·´∞ ß√1+p2−1TM .
(mc) 0exp−􏰁Te 􏰁p2d􏰁p (C.10)
In the non-relativistic limit the exchange rate (we will use a notation ν(E)∗ for it) coincides with well-known results pre-
ñ®􏰀´ô ˆ∞ 2 −1
sented in Braginski’s review [18]. In Braginski’s notation we have ν(E)∗ = 2 (m /m τ ), see equations (4.5) and (4.8) in
− 3 2 1 + 􏰁p − 1 C=(mc)4π􏰁pexp−􏰁 d􏰁p
ei eie
ei
0 Te
= (mc)−3C􏰁 ÄT􏰁eä . (C.4)
[18]. The ratio ν(E)/ν(E)∗ is used in equation (9) in section 2.2. ei ei
References
[1] Wittenberg L.J., Santarius J.F. and Kulcinski G.L. 1986 Fusion Technol. 10 1
[2] Binderbauer M.W. et al 2015 Phys. Plasmas 22 056110 [3] Gota H. et al 2017 Nucl. Fusion 57 116021
[4] Dawson J.M. 1981 Fusion vol 1, ed E. Teller (New York:
Academic) p 453
[5] Nevins W.M. 1998 J. Fusion Energy 17 25
[6] Nevins W.M. and Swain R. 2000 Nucl. Fusion 40 865
[7] Stave S. et al 2011 Phys. Lett. B 696 26
[8] Spraker M.C. et al 2012 J. Fusion Energy 31 357
[9] Sikora M.H. and Weller H.R. 2016 J. Fusion Energy 35 538
[10] Moreau D.C. 1977 Nucl. Fusion 17 13
We are going to evaluate the energy exchange rate between the ions and electrons which is described by the equation
˙ (E)
Ti=−νei (Ti−Te).
(C.5)
The first term comes from the dynamic friction of the ions versus electrons, whereas the second one comes from the ion diffusion produced by random electron ‘kicks’. The structure of this equation is consistent with the detailed balance prin-
ciple. To find the parameter ν(E) it is sufficient to assess one of ei
the two processes (dynamic friction and scattering). Below we evaluate the second term.
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