Slowing-down of Charged Particles in a Plasma

P/2532 France Slowing-down of Charged Particles in a Plasma By G. Boulègue, P. Chanson, R. Combe, M. Félix and P. Strasman We shall investigate the case in which the slowingdown of an incident particle Р г can be explained by binary collisions with the constituents of the plasma, collective oscillations entering only in a weak or negligible manner. 1 EPRESSION FOR THE SLOWING-DOWN (NON-RELATIVISTIC CALCULATION) The particle Р г (mass m v velocity v v kinetic energy E 1} charge Z x e) moves in an ensemble of elements P 2 (mass, charge Z 2 e) undergoing Maxwellian agitation (temperature T 2 ). At a given moment the velocity of one of these particles P 2 is F 2. We designate by u the velocity of Р г relative to P 2 : u = Vj - V 2 When a collision occurs the trajectory of Р г is deviated by an angle a in the coordinate system of P 2. One supposes that the function oc(u, a) is known, a being the " collision parameter " (distance of P 2 to the initial relative velocity vector). After the collision the relative velocity becomes u/, with Let us call dn 2 ) the density of particles P 2 whose speed is between F 2 and F 2 + dv 2 in absolute value and whose velocity vector is in the small element of solid angle ( 1, SI + dsi). At the zero instant all the elements P 2 having velocity V 2 which should interact with P x during the unit time following, with collision parameter a and azimuth ф (with respect to any origin, e.g., the plane Vx, u) are contained in a volume иааааф. Their average number is then The momentum conservation in the collision (l)-(2) allows one to write, after a simple calculation: = F-, 2 F/ 2 = - ( F-, 2 Fo 2 -\ ^ u 2 1 (1 cos OL) sin в sin a cos, being the angle between the vectors "V^ and u. ш ф is obtained by multiplying by dn 2 ) Letting I (1 cos oc)ada = a{u), jo R is the interaction radius of the particles, a is a function of u, obtained from the laws of scattering. Evaluating the integrals over ф and a we obtain: v. = dn 2 ) - uo(u) by collisions with particles (2) having velocities V 2. Let us express dn 2 ) as follows: n 2 ) - «^(F 2 ) represents the Maxwellian distribution. Here n 2 is the total number of particles P 2 per unit volume, в being the angle (V^, V 2 ). One obtains after integration over the azimuth : n = 71-2 т л 4- n. ( ( l J JO<e<n This represents, but for a numerical coefficient, the mean loss of energy of a particle (1) per unit of time The mean loss of energy per unit time is Original language: French. * Laboratoire Central de l'armement. 242

SLOWING-DOWN OF CHARGED PARTICLES 243 and per unit length : It is convenient to use as variables и and F 2. finds: One Taking into account ^(F 2 ) = ^4 2 F 2 2 exp ( y 2 2 V 2 2 ) A 2 = (2/л)ЦкТ 2 /т 2 )-''' and y 2 2 = /2kT 2, one arriyes dz - " nn 2 f " *(u)i(u)du, Uo The calculation of dejdz is therefore reduced to the evaluation of a single integral, the function a(u) being assumed known. I(u) can be transformed in various ways of which the most useful for us is the following: \du Г si I _ 2y 2 2 -± sinh (2y 2 W^ u 2 a(u)i(u)du = exp ( y 2 2 u 2 ) sinh 2y 2 exp (- 1 exp ( - sinh (2y 2 2 F 1^) -^ 1 ^(T(w) exp ( Vfv-x COULOMB INTERACTION LIMITED BY THE DEBYE LENGTH (the subscript e refers to the electrons), one deduces, setting Let m be the reduced mass defined by one knows that Шл Wo = ^ i + that x 9 9 = У2 dz from which, exp l Е -т«^ w x kt G = I exp ( m^x^m^) sinh /3: Letting D be the Debye length (1) d Чп (1 +«т 1

244 SESSION A-5 P/2532 G. BOULÈGUE et al. and 2kT 2 D If E 1 /kt 2 is large, a good approximation is obtained by calculating the term in d/dx for a mean value : nu»/* kt 2 m x TIME OF EQUIPARTITION We now assume that an ensemble Р г is slowed down by an ensemble P 2. The particles (1) have a Maxwellian distribution of velocities at temperature T v Per cm 3 and per second, the particles (1) lose energy due to interaction with particles (2) that is given by: Ф being the error function Ф{х) = 2; - x 2 )dx. From this G^VTrexp -_ V m-, Ф A x is the coefficient of the Maxwellian distribution for the particles (1). dejdz is expressed by an integral over u, W lt 2 by a double integral. These calculations give 7\ T 2 (m, and u 5 a(u) exp du. te E-L If we assume a Coulomb interaction limited to the Debye length, it is necessary to calculate This is practically the formula arrived at by the elementary considerations. On the other hand, for arbitrary values of E x \kt 2y the evaluation of G may be rather long; a series expansion of sinh fix is convenient. Some results for a particular case will be given later. It is interesting to investigate the case of slowingdown of a heavy particle by electrons (fn 2 f small), when EJkT 2 is small. One then finds (putting y = x 2 ) : + x 2 y 2 )dy 2k exp ^-+^ du. This integral is derived from that in the previous equation by replacing a(u) by its value given in (1). Letting and g = 2^( ^ 4 \ /i/i/i /мл т л J(x) =2 cos х~ г Í xr 1 cos xdx Jy.- 1 + 2 sin xr 1 I ^ л;" 1 sin x dx one has w b2 = In and finally = 2[cos ÍHH- Ci^- 1 + sin ^-1 Sbr- 1 ], (2) and further: ^ / m \i/2 dz ~ 2 x^/ \3 /гт 2 the function /(g) having been defined in Eq. (2). When g is large (> 10) one has One notes that EJife is zero for E x = Ъ Г 2 \2 } С = In у is the Euler constant: С = 0.5772... which is completely reasonable, " С rapidly becomes negligible in comparison to In g.

SLOWING-DOWN OF CHARGED PARTICLES 245 Figure 1. Loss of energy of a tritium nucleus in a plasma of 6 Li-D at a temperature of 5 kev, as a fraction of triton energy Figure 2. Loss of energy of a tritium nucleus in a plasma of 6 Li-D at a temperature of 2 kev, as a function of triton energy 500 1000 2000 (feev)

246 SESSION A-5 P/2532 G. BOULÈGUE et ai. Let us assume, in particular, that the temperatures T lf T 2 and T e are of the same order of magnitude T. From the equations Txr 3,dT 1 3 jdt 2 W = n k ^ = n k one easily finds dt-, 3 kn^f and from an analogous equation for dtjdt one has t eq is a time for equipartition : Numerically, 3 kn 1 n 2 4.81 10 8 К 3.27 10 14 dt As-IZJ 2& LJ-]j f bt ht \ /Í1 j /vi 2 \ ' 7~ + ) J{g) Ш г Ш 2 J{g) ' Let us compare these results with those of Spitzer : 2 4леМ= 2ЬЛШ ' For large values of g and Л, we have /(g) ^ 2 In g - 1.16 = 2 In - 1.16 + 2 In Л. Our value for the equipartition time approaches that of Spitzer if g is large, but it diners considerably for average or small values of g. We feel, therefore, that our formula is more general than that of Spitzer, which it approaches for large values of Л. APPLICATIONS We shall apply the formulae of a previous section to the problems of the slowing-down of tritium nuclei by a plasma composed of 6 Li nuclei, deuterons and electrons at various temperatures, the densities being n u = 0.617 xlo 23 /cm 3 Njy = 0.617 n e =2.468. Slowing-down by the Electrons It is especially for this case that our formulae are applicable since the collisions are very frequent and each has only a small effect ; the slowing-down is thus quasi-continuous and our expressions can give a good approximation. When the energy Е г of the triton approaches the thermal energy, the slowing-down factor loses much of its importance and approaches zero for E x = ( ) kt. Slowing-down by the Ions It is precisely in this zone that the slowing-down by the ions becomes predominant. It is certain that in this case our formulae no longer have the same validity, due to the marked discontinuity of the slowing down. Nevertheless, it seems reasonable that they can represent the average slowing-down. Figures 1 and 2 show the results of calculations for kt 5 kev and kt = 2 kev. Probability of the Reaction D (t, n) 4 He The probability that such a reaction takes place during the slowing-down of the triton, calculated for the energy range 1 Mev to AkT, is expressed by : (- dejdz) J а(е г ) is the reaction cross section. /гг(кеу) 50 20 1051 p 2.86 10-3 1.77 x 10-2 2.50 x 10-2 3.45 x 10-2 5.38 x 10-2 One finds: Without being completely negligible, this probability stays small. We conclude that the majority of the tritons will be thermalized before producing a thermonuclear reaction. REFERENCES 1. D. Pines and D. Bohm, Phys. Rev., 85, 338 (1952). 2. L. Spitzer, Physics of Fully Ionized Gases, p. 72, Eqs. (5-14), p. 80, Eqs. (5-31), Interscience Publishers, Inc., New York (1956).