Page 7 - Fusion reactivity of the pB11 plasma revisited
P. 7
Nucl. Fusion 59 (2019) 076018
situation is reversed, and the protons become faster than alphas. The electrons are much faster than alphas in the whole range of energies. The Borons are universally slower than alpha particles.
In the steady state, the alpha-particle distribution function satisfies the equation:
1 ∂ v2î(D +D +D )∂fα +m vÄDαp+DαB +Dαeäf ó v2∂v αp αB αe∂v α T Te α
S.V. Putvinski et al
Now we can turn to solving equation (B.4). We note that the loss rate ∝ 1/τ is small in comparison with slowing down rate ∝ 1/τs by collisional processes and it can be neglected. For the highly suprathermal alpha-particles one can drop the diffusive term in this equation and retain only dynamic fric- tion. This yields a corrected slowing-down distribution in the familiar form:
− fα τα (v)
+ Sα2δ(v−v1)=0. 4πv1
Sτ α s
4π(v3 +v3∆(v)) ∗
(B.4) Here v1 is defined by equation mαv21/2 = Eα, and Sα(v1) is
alpha particle source per unit volume with initial velocity v1. The second-to-last term in the l.h.s. of equation (B.4) describes alpha-particle losses (without this term they would be accumulated in the system). One can anticipate develop- ment of techniques that would allow removal of only slowed- down alphas, so that most of energy released in them would go to plasma heating, and only a small fraction of the fusion
yield would be lost by the removal of slow alphas.
Because of low density and high energy of energetic alpha particles we have not included collisions between energetic
alpha-particles in equation (B.4).
Using equations of appendix A, we find the following
expression for the alpha-particle diffusivity for the fast alpha particles:
This expression describes diffusion of alpha particles with velocities exceeding the ion thermal velocity, i.e. the alphas with the energy exceeding roughly 1 MeV (the superscript ‘f’ means ‘fast’.
When the alphas slow down below this energy limit, their further diffusion is described by the expression:
fα =
where correction factor ∆(v) is
(B.10)
(B.11)
nβZβ β mβ
2
v3
v3 +(3√π/4)v3
D(f)= αp. (B.5) αp A2 m3v3
∆(v)=
nβZβ β mβ
Tβ .
2
4πΛe4Z2 n T
At v ≫ vT we have ∆ (v) → 1 and equation (B.10) becomes equation (B.1). The correction for finite thermal ion velocity is important only for protons.
Figure B1 shows correction factor, ∆(v), and ratio of corrected distribution function, equation (B.10), to the one given by equation (B.1). The calculations are done for T = 300 keV, Te = 150 keV, initial alpha particle energy 4 MeV, and Boron concentration nB/ni = 0.15. One can see that the effect of finite proton velocity is significant. Note, however, that the power transferred by the alphas to protons and electrons contains an integration of the alpha distribution
2
function with the weight v , so that the effect of the lower
alpha velocities is suppressed, and the net heating power of either electrons or ions is still determined by v/v1 in the range of 0.7–1, where the correction is modest. The net effect is a roughly ten percent increase of the proton and electron heating (a favorable effect).
Expression (B.10) should be considered as a Green func- tion and the slowing down distribution can be evaluated by integrating distribution function (B.10) with the energy spec- trum of the alpha particle source in the laboratory frame. The energy spectrum of the source can be evaluated using the alpha particle source presented in [9], Snet (E), with account of the gain by alpha particles as an additional kinetic energy from colliding protons and Borons which are assumed at the moment to be Maxwellian with temperatures T. This kinetic addition is sizeable at the high energy where cross section has its peak. Using relative velocity, u, and velocity of CM of col- liding particles, v, the source spectrum in the lab frame can be defined as:
(B.12)
We have normalized all velocities on their thermal veloci-
ties. If Snet is normalized as Snet dE = 1 then Slab is also nor- malized on 1. The average energy of alpha particle accounting for finite temperature of reagents will be
√24... D(s) = 8 πΛZαe np mp .
αp 3A2 m2 2T αp
(B.6)
αp
One can combine equations (B.5) and (B.6) into one equa- tion that covers both the case of fast and slow alphas. The interpolation formula is:
4πΛe4Zα2 npT 1
Dαp= A2m3v3 v3+(3√π/4)v3 . (B.7)
α p th,p
The thermal velocity of protons is vTp = 2T/mp. Note also that v/vTp = 1 corresponds to the alpha-particle energy of 4T. A similar diffusion coefficient can be adopted for Borons,
Slab (E)
although correction for finite Boron velocity is small as noted 4 ´∞ Snet(E−1Tv2−1Tu2)exp{−v2−u2}v2σ(u)u3dudv
1/20 33
above =π ´ .
4 π Λ e 4 Z α2 Z B2 n B T 1
√ .(B.8)
∞ exp(−u2)σ(u)u3du 0
DαB=
Diffusion on the electrons occurs with a coefficient (see
A2Am3v3 v3+(3 π/4)v3 αBp TB
´
appendix A)
8√πΛZα2 e4ne ... me
T T ´∞0 exp−u2σ(u)u5du
Dαe =η(Te) 3A2m2 2T . αpe
(B.9)
⟨E⟩lab=⟨E⟩net+2+3´∞ 2 3 . (B.13) 0 exp(−u )σ(u)u du
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