Energy spectra from “beam-target” nuclear reactions in magnetic fusion devices

Abstract Analytical expressions for the energy distribution of fusion products created in “beam-target” fusion reactions in a strong magnetic field are derived.

In controlled fusion experiments, the largest fusion reaction rates [1] and the most easily measured fusion spectra are achieved in plasmas with non-thermal populations. Measurements of these spectra can provide information about the "tail" (non-thermal part) of the ion distribution function that may be difficult to obtain using other diagnostic techniques. For example, recent measurements of the energy distribution of 15 MeV protons produced in the d(3He, p)a fusion reaction during fast wave (ICRF) heating in the PLT tokamak provided the first experimental evidence that the 3He ions accelerated by the waves are anisotropic in velocity space [2]. Expressions relating the temperature of thermal reactants to the width of the fusion spectrum are well known [3,4], but only one analytical treatment relating the distribution of non-thermal reactants to the energy distribution of fusion reaction products has been published [5] and this work contains errors [6]. Other workers have studied the fusion spectra produced by non-thermal plasmas numerically [7] but this approach, though accurate, is tedious and provides little physical insight into the factors affecting the fusion product energy distribution. This paper extends previous analytical work [5] by considering explicitly the effect of a strong magnetic field on the fusion spectra produced in "beam-target" reactions and by carrying the calculation to higher order in the ratio of beam energy to fusion energy. Under some conditions, the derived expressions permit a measured spectrum to be related directly to the beam-ion distribution that produced it.
For a reaction 2(1, 3)4 where particles 1, 2, 3, 4 are the projectile, target, and products, respectively, the energy of particle 3 using non-relativistic kinematics is [3] m4 ~ 2m3m4 where Q is the fusion energy, K = lmlm2o2/(m 1 + m2) is the relative kinetic energy, v = v 1 -v 2 is the relative velocity, V= (m 1 Vl + m2v2)/(ml + m2) is the center-of-mass (cm) velocity, and 0 is the angle between V and the cm velocity of particle 3. Our objective is to deduce the energy distribution of product 3, F( E 3 ), given the reactant distributions fl(vl ) and f2 (v2) and the reaction probability o v (v, 0). We assume that the fusion cross section is isotropic in angle, i.e., o = o(v), which is valid for the d(t, n)a and d(3He, p)a fusion reactions below the resonance energy [8,9]. In laboratory plasmas, the fusion energy Q is several orders of magnitude larger than the reactant kinetic or cm energy so we order the equation in half-integer powers of the small parameter c = m3VZ/Q and neglect terms that enter in order e3/2. For most cases of interest o(v) is a rapidly increasing function of the relative velocity so that the weighting factor ovfl(vl)f2(v2) is largest when v is maximized. For ion distributions fl and f2 with comparable velocities, such as in a thermonuclear plasma, v is largest when the reactants collide head on. This implies that most reactions occur with the cm velocity V= 0, which from eq. (1) implies that the resulting fusion-product energy distribution is peaked about E3=m4( Q + g)/(m3+ m4) [3]. Here, however, we consider the special case of "beam-target" reactions for which Under these conditions, it is only the distribution function of the beam and the cross section that determine the fusion-product energy distribution. The general expression for the fusion-product energy distribution function F(E 3) becomes otherwise.
Due to a particle's fast gyromotion in a magnetically confined plasma, the particle distribution function is independent of direction in the plane perpendicular to the magnetic field, f(v ±, vii ) = f(v±, vii ). We denote the angle made by the magnetic field vector and the beam particle velocity by X~ and the angle between the field and the velocity vector of the fusion product by X3. The cm angle 0 can be reexpressed in terms of the laboratory angle, between the projectile velocity vl and the product velocity v 3 according to cos 0 --cos *~1 -(k o sin ,)2 _ ko sin 2 *, where The next step is to carry out the angular integration in eq. (2). Define a reduced fusion-product energy distribution function F(E). For the special case of a monoenergetic beam, fl(vl)= 8(v-Vb)g(vl/Vl).

IE-E I<AE,
where and m4 The analysis for an anisotropic beam is similar. We consider a monoenergetic beam that is unidirectional in pitch angle X but isotropic in gyroangle % g(~)dff = 8( X -x1)dxdq0. Reexpressing cos 0 (eq. (3)) in terms of X1, X3, and ¢p and proceeding as in the isotropic case yields the fusion-product energy distribution produced by an anisotropic monoenergetic beam, The accuracy of eq. (7) was tested by integrating it numerically over sin X1 dxa to recover the isotropic distribution eq. (6). The final result for the fusion-product energy distribution function is found by taking the weighted average of reduced distribution functions F(E3) = fdElo(Ea)E~fl(E~)P(E3,Ea). (8) The reduced distribution functions eqs. (7) and (6) are plotted in figs. la and lb. The physical origin of the twin lobes ( fig. la) for a perpendicular beam distribution viewed in the perpendicular plane is that the probability of a reaction is constant over a gyro period but more of the orbit forms an angle 0 near I cos 01 = 1 than near sin 0 = 0 ( fig. lc). Similar results for the fusion spectra from a monoenergetic beam were obtained by Lehner (figs. 2 and 4 of ref. [5]).
If the beam distribution function fl(E1) decreases rapidly with energy [e.g., if fl(E1) is a Boltzmann distribution], then the weighting factor o(E 1) E 1 fl(E1) in eq. (8) peaks strongly for some energy E 1 = Epeak and the fusion-product energy distribution function is approximately As predicted by eq. (9), spectra from realistic beam distributions look like smoothed monoenergetic spectra ( fig. 2). The expressions, eqs. (6)-(9), have found application in analysis of 15 MeV proton spectra measured during fast wave (ICRF) heating of 3He ions in the PLT tokamak [2]. In those experiments, spectra similar to the one plotted in fig. la were observed with a detector oriented perpendicular to the field (X3 = 7r/2), which was taken as evidence that _> 90% of the 3He ions accelerated to -200 keV by the ICRF was  Fig. 2. Spectrum of protons ermtted at an angle X3 = ~r/3 with respect to the magnetic field in d(3He, p)a fusion reactions between anisotropic 3He ions and a cold deuterium plasma for various angles X] of the 3He ions with respect to the field (eq. (7)). A distribution in energy of file (EHe) 0~ exp( --EHe/50 keV) is assumed for the 3He ions and an analytical fit [10] is employed for the cross section o in eq. (8).
1~. W.. Heidbrink / Energy spectra from beam-target nuclear reactions anisotropic in velocity space. Although the data from this experiment provided unambiguous evidence of beam anisotropy, the information was insufficient to establish the direction of 3He anisotropy. In fig. 2, possible spectra from a similar experiment with a detector oriented at X3 = rr/3 are plotted. The great variation in the relative amplitude of the two peaks as a function of the angle of 3He velocity X1 (fig-2) indicates that, with properly oriented detectors, spectral measurements can diagnose both the direction of fast beam ions and their degree of anisotropy.