Energy deposition of fast α particles in a fully ionized deuterium-tritium plasma

Volume 78A, number 3

PHYSICS LETTERS

ENERGY DEPOSITION

OF FAST a PARTICLES

IN A FULLY IONIZED DEUTERIUM-TRITIUM D.C. KHANDEKAR

4 August 1980

PLASMA

and D.C. SAHM

Theoretical Reactor Physics Section, Reactor Group, Bhabha Atomic Research Centre,

Bombay400 085, India

Received 12 February 1980

An (Yparticle transport equation has been solved analytically in one-dimensional spherical geometry when the plasma properties are uniform in space and time. The results agree well with earlier results.

1. Introduction. The problem of energy deposition of (IIparticles in a plasma is of considerable significance in the context of laser induced fusion. As such this has been the subject of many earlier investigations [l-6]. The fast charged particles lose their energy in two ways, by Coulomb interactions and by scattering with ions and neutrals. The most dominant process, however, is the small angle Coulomb scattering, characterized by low energy transfer and high cross section. Thus the particles lose their energy more or less continuously along the trajectory just as they would if they were influenced by a viscous drag. Recently Antal and Lee [5 ] have derived equations for the energy and mass transport of energetic charged particles using the conservation laws of density and energy in phase space. They have replaced the large number of small angle scattering collisions by a net frictional force or the “Coulomb drag force”. The resulting transport equation resembles the one encountered in neutron transport problems and can be solved by well known methods such as S, methods used by these authors. In this paper we obtain an analytical solution to the transport equation in one-dimensional spherical geometry when the plasma properties are assumed to be constant in space and time. Further, it is also assumed that the small angle scattering is the only mechanism for the energy transfer from Q particles to the plasma. For the sake of comparison we treat the

same model problem treated by Antal and Lee [6]. 2. Formulation. Following Antal and Lee the equation for kinetic energy of the charged particles can be written as

~+vr’(“~)fv”*(u~)=s~+$(U.u)IL, (1) where $ (r, u, t) dr du is the kinetic energy at time t carried by the charged particles with their position coordinates lying between r and r + dr and velocities lying between u and u t do. ois the acceleration of the particles due to Coulomb drag force. S,(r, u, t) is the source of charged particles and no collision terms on the right hand side of eq. (1) appear as we have neglected the large angle and nuclear scattering events. Assuming spherical symmetry, eq. (2) reduces to

=S&,v,P,t)-y,

(2)

2d

where p =rw/rv, O
u=a(v)u/v,

a>O,

-lQ/.l
O~VQV,.

condition

on IJJis given by

Vu,t;

~E(--1,0).

(3)

(4) 259

Volume 78A, number 3

PHYSICS LETTERS

The boundary condition of eq. (4) results from the fact that the spherical surface being convex is non-reentrant. We further assume that, at f = 0, $ vanishes everywhere. The detailed form of the acceleration (I has been given by Evans [7]. In the present work we have used the expression for a in the form used by Antal and Lee [6] in which (I is a function of u only. 3. Analytical solution to the transport equation. Now we apply the following transformation to both the dependent variable $ and the independent variables r, u, p: $(r, u, c1, t) = f(r, u, P,

.$=r(l -p2), -(R2

77=rb,

j+)

.v=

t)/a(u) ,

(6)

- t2)112
(7)

0~vsv,=p-$

0

can then be written down as S~(r,~,u,t)=So6(u-u0),

(8)

0

(11)

where u. is the velocity corresponding to an energy of 3.5 MeV. If this source distribution is substituted in eq. (lo), the integrations over u’, t’ and n’ can be easily performed to obtain the result f(L 9, V, t) = SOB(t - b, + b) X [fI(V,-

V)-B(Vo-V-(R2-~2)1/2-~)],

(12)

where 0(x)= 1,

(5)

O
4 August 1980

X>O,

=o,

XGO,

is the Heaviside unit function and b, = b(uo). The transforming relations (5)-(8) can now be utilised to write $ as a function of r, p, V and t. The resulting function can be written as: J/(r,P, K t)=a

e(t-b@(g)

- e(g-R1)l,

(13)

to obtain the equation where

(9) Eq. (9) can be easily solved using the standard Fourier transform techniques. The detailed derivation has been given by Khandekar and Sahni [8]. The solution is given by the relation f(L q, V, t) =

f

-ea

du’ .y dt’ j! -(p-p)*‘2

0

x 6 [t - t’ - b(u’) where

b(u)

= J;

dr)‘tS&, rl’, u’, 1’)

+ b(u)]

6 [Tj’ - ?j + V(u’)

(10) -k(u)]),

“0

b, =

s

du/a,

,’

(14)

R1=rp+

[R2-r2(1

is given by &(RJ)=j

dt$dr~duV,.*(uJ/) 0

du/a(u) and 6 (x) is the Dirac delta T

(15)

1

dtS~ju2du[u~ilr(R,~,u,f)l,

=R2[

260

--~~)]l’~.

5. Evaluation of leakage and absorption fractions for the model problem. The total energy $s (R , T) that leaks out of the sphere of radius R up to time T

function. 4. Model problem. We apply the result of eq. (10) to a model problem studied by Antal and Lee [6] where the source of the charged particles was assumed to be isotropic and uniform in a sphere of radius R. Further these particles are born with a fixed energy of 3.5 MeV. The sphere was assumed to contain an equal number of deuterium and tritium ions and was assumed to be fully ionized. The source distribution

g-i”+,

U

0

0

0

while the energy absorbed by the plasma particles of type i is given by

(16) where ai is the partial drag due to the plasma compo-

PHYSICS LETTERS

Volume 78A, number 3

nent of type i particles as defmed in ref. [6]. We now define the two fractions, namely the leakage fraction F, and the absorption fraction Fai, by the relation F, = $,I&

(17)

Fai = J/ails 3 S=r

dtj-drj-duSE,

(18)

(19)

i

where the index i runs over plasma constituents. In the context of laser fusion one is particularly interested in knowing the fraction of total deposited energy which is deposited to the plasma constituent ions of type i. This is defined ai Pai = F,iIF, .

C(T, LJ)= 1 - bl/T.

The velocity u3 is defined as follows. Let u2 be the solution of the equation

g(u2) = 2R,

g(O) 2 2R,

u2=0,

otherwise,

b&)=T,

b,(O)>T,

u;=o,

otherwise ,

u3 = max(u2, vi>.

(28)

It can now be easily seen that as T + a, F,(R, T) and F,(R)

T) tend to their asymptotic

F;(R)==?_

F;(R)=

$l(u)C(T,u)du,

(21)

4u8R v3 T)=3

i u;R3

u;B(u)C(T,u)du,

(22)

$”

A (u) , (29)

4uoR “2

FQ(R,T)=l-7

3 T)=-_f u20R3 vz

Fai(Ro,

Tlim_ F,(R,

T)= Lj u;R3

udu;B(u), (30) duuB(u),

(31)

02

Similar expressions can be found for the other limiting case i.e. (an infinite sphere) R + = and are given by

v3

F;(T) = R?_ F, (R , T) j” u;R3

values given by:

FQ(R,T)=+j

F$(R)=iFm

T)=L

(27)

Then u3 is given by

over r and p. The resulting expressions for these fractions can be written as

F,(R,

(26)

cw

These fractions can be easily obtained after explicitly substituting the value of S from eq. (18) and J/ from eq. (13) and then performing the integration

F,i(R,

(25)

and let u; be the solution of

while the total absorption fraction is given by Fa= CFaip

4 August 1980

uB(u) C(T, u) du ,

= 0,

(32)

(23)

v3

F~i(T)=~~mF~(R,T)=~~o

where

~%C(T,u)du, uo II;

A(u) = 1 - g2/4R2,

B(u)=Z@ 3

(33)

-@g+$g3 (2ia,b)

Table 1 Range g(O) and slowing down time B1(0) for variousNpl and Tpl. Npl =

g(O) otm) 61(O) (PSI --D

10z3 cmW3

NPr = 102’ cmm3

TPr = 1 keV

TPr= 10 keV

TPr = 1 keV

TPr = 10 keV

435.4 1.0158 X 10-r

6152.1 1.0827

0.2112 4.3759 x 10-s

1.3069 2.1992 x 10-4

261

PHYSICS LETTERS

Volume 18A, number 3 Table 2 The values of F;(R)

4 August 1980

for ions (deuterium and tritium) and F;(R) for various Npl and Tpl. The numbers in brackets indicate Npl = 102’ cmw3

Npl = 1O23 cmw3

R

Tpl = 1 keV

F;(R).

Tpl = 10 keV

Tpl = 1 keV

Tpl = 10 keV

-_ 0.1

0.8924 (0.0029)

0.9099 (0.0541)

0.8998 (0.0088)

0.9160 (0.0829)

0.5

0.5303 (0.0060)

0.6000 (0.0894)

0.5546 (0.0181)

0.6192 (0.1284)

1.0

0.2960 (0.0195)

0.3469 (0.1655)

0.2865 (0.0479)

0.3603 (0.2177)

3.0

0.1017 (0.0293)

0.1202 (0.2109) .__.___

0.1079 (0.0636)

0.1251 (0.2686)

Table 3 The values of F:(T)

for ions (deuterium and tritium) andFi(T)

T

for various Npl and Tpl. The values in brackets indicate I??(T).

Npl = 1O23 crns3

Npl = 102’ crns3

Tpl = 1 keV

7’pl= 10 keV

TpL = 1 keV

Tpl = 10 keV

0.1

0.3071 (0.0034)

0.2006 (0.0573)

0.2586 (0.0108)

0.1806 (0.0868)

0.5

0.08726 (0.0082)

0.7009 (0.1034)

0.8157 (0.0234)

0.6637 (0.1439)

1.0

0.9942 (0.0323)

0.9965 (0.2256)

0.9994 (0.0709)

0.9965 ((I.285 1)

-

F,‘cT?=iF_Fa(R,T)=$/e

uC(T,u)du.

(34)

uo vh _me fractions F;(R), F;(R) = F$(R)/F,(R), F;(T) = F$(T)/F,(T) and F:(T) have been tabulated for two sets of plasma densitiesiVpl = 1O23 cmW3 and 1O27 cm-3 and two sets of plasma temperatures Tpl = 1 keV and 10 keV (see table 1). The radius R has been measured in terms of the range g(0) while the time T has been measured in terms of the slowing down time b l(O). Our asymptotic results for T + CQ presented in table 2 agree well with those of Antal and Lee [5] which they have obtained by solving the time dependent equation numerically by the S, method. This is understandable because they have used a time-step of the order of lo-10 s or greater whereas the slowing down time bl(0) is of the order of picoseconds. The time dependence of the results in the

262

limit R + m for the same source is explicitly displayed in table 3. Results with other source distributions have also been obtained and agree well with the earlier published results [ 1 A].

References [l] E.G. Corman, W.E. Lowe, G. Cooper and A. Winslow, Nucl. Fusion 15 (1975) 377. [2] R. Cooper and F. Evans, Phys. Fluids 18 (1975) 332. [ 31 H. Tsuji, M. Katsurai, T. Sekiguchi and N. Nakano, Nucl. Fusion 16 (1976) 287. [4] E. Greenspan and D. Shavarts, Nucl. Fusion 16 (1976) 295. [5] M.J. Antal andC.E. Lee, J. Comput. Phys. 20 (1976) 298. [6] M.J. Antal and C.E. Lee, Nucl. Sci. Eng. 64 (1977) 379. [7] F. Evans, Phys. Fluids 16 (1973) 1011. [ 81 D.C. Khandekar and D.C. Sahni, Phys Fluids, to be pub hshed.