Page 5 - Fusion reactivity of the pB11 plasma revisited

Nucl. Fusion 59 (2019) 076018
dnp = 4πfp (v) v2dv. (A.3)
The normalization has the same form for the other comp- onents as well.
To find the modification of the proton distribution with respect to Maxwellian, we have to find the diffusion coeffi- cients on the Maxwellian ions for the tail ions; we have also to find contributions from the slowing-down alpha-particles. To do the latter (the most difficult part of the problem), we have to find distribution of the non-thermal alphas and expressions for D∗pα and Fp∗α.
To evaluate the coefficients in equation (3) we use Landau collision integral as presented in a review paper by Trubnikov
[19]. The particle flux in the velocity space, jρ/σ (v), is (see equation (11.8) in [19]): i
224ˆÅ′′′ã
jρ/σ = 2πΛZρZσe Uik fρ vk ∂fσ − fσ vk ∂fρ d3v′.
i A m2 A v′ ∂v′ A v ∂v ρpσρ
(A.4) The test (field) particles are marked by ρ and σ, respectively, and their velocities by v and v′; we depart from the traditional notation to save the marker ‘α’ to denote the quantities related
to alpha particles.
In the isotropic case the flux is collinear to the velocity
vector,
S.V. Putvinski et al I= 4v′/3, v<v′ . (A.11)
4v′4/3v3, v > v′
We are now prepared to evaluate the coefficients in equa-
tion (3). Using equations (A.9) and (A.11) we find that
Dρσ =
This general expression is manageable for the arbitrary veloci- ties of the test particles, but for the first assessment we will use sometime either an appropriate limiting case, or an extrapola- tion between the two limiting cases.
Analogously, equations (A.4) and (A.6) yield:
2πΛZ2Z2e4 4 ï 1 ˆ v ˆ ∞ ò ρ σ 2π v′4fσ (v′)dv′ + v′fσ (v′)dv′ .
A2m2 3 v3
ρp 0 v (A.12)
ïˆ
2 π Λ Z ρ2 Z σ2 e 4 4 1 v
′ 3 ∂ f σ ( v ′ ) ′ Fρσ=−A2ρm2p 32πv20v∂v′dv
+v
v
∂v′
dv′ .
ˆ ∞ ∂fσ (v′) ò
(A.13)
jρ/σ (v) = jρ/σ (v) vi ; jρ/σ (v) = jρ/σ (v) vi iviv
If the field particles all have velocity higher than velocity of the test particle, then both integrals are obviously zero. In other words, only those field particles that are slower than the test particles contribute to the dynamic force (A.13).
Another representation of the dynamic force can be obtained by the integration by part in equation (A.13). It yields:
(A.5) ∂f + 1 ∂ Äv2􏰀 jρ/σ(v)ä=S(v). (A.6)
Fρσ = 4πΛZρ2Zσ2e4 4π ˆ v v′2fσ (v′)dv′. AAm2 v2
This representation shows even more clearly that only those of the field particles whose velocity is slower than that of the test particles contribute to the drag force. Another interesting point is that for the Maxwellian distribution of the field par- ticles one can reduce equation (A.12) to a more compact form:
and the kinetic equation becomes:
(A.14)
∂t v2 ∂v σ
Then, inspecting equations (A.4) and (A.6), one sees that the
tensor
Uik=δik−uiuk; u=v−v′ u u3
(A.7) enters equation (A.4) only in the form of convolutions Uikvivk
4πΛZρ2Zσ2e4 4πTσ ˆ v ′2 ′ ′ A2 A m3 v3 v fσ (v ) dv .
ρσp 0
Dρσ =
In some of the equations below this diffusion coefficient enters
and Uikviv′k. Simple algebra shows that
2 ′2 ′ 2
ρσp 0 (A.15) in the combination v3D. This combination is proportional to
Uikviv′k =Uikvivk = v v −(v·v)
(A.8)
which yields the following expression for the diffusion coefficient
σ
2πΛZ2Z2 e4 ˆ ρ σ
v′2sin2θ A2m2 u3
fσ(v′)d3v′
At this point we note that the integration over v′ in equa-
tion (A.9) can be represented as
ˆ v′2sin2θfσ′ (v′)d3v′ = 2πˆ ∞ fα′ (v′)I(v,v′)dv′, (A.10)
Dpσ=
where θ is an angle between v and v′.
(A.9)
u
3
ˆ v v′2f (v′)dv′, (A.16)
0
where subscript β refers to the Maxwellian distribution of the particle of the type β. If the velocity of the test particle is much smaller than the thermal velocity of the field particle,
ρp
v ≪ vTσ ≡ »2Tσ/mσ
then the integral (A.16) is approximately equal to
u3 0
Conversely, for
􏰂v3/3􏰁 nσ(mσ/2πTσ)3/2. v ≫ vTσ ≡ »2Tσ/mβ,
(A.17) (A.18) (A.19)
where I is the Rosenbluth–McDonald–Judd (‘electrostatic’) integral [20]:
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