# Fast Ion Speed Diffusion Effect on Distributions of Fusion Neutrons
## 3 Calculation Results  
To study the influence of the speed diffusion effect on distributions of emission energies and angles of nuclear fusion products, calculations of energy spectra of neutrons produced in a candidate operating regime of FNS-ST [4] in \({}^{3}\)H(\({}^{2}\)H,n)\({}^{4}\)He and \({}^{2}\)H(\({}^{2}\)H,n)\({}^{3}\)He reactions were performed using the simple analytical anisotropic model (2) similarly to [25], as well as using steady-state numerical solutions of (13) where the speed diffusion effect is included. Parameterisations of differential cross-sections from [40] and [41] were used. Numericalsolutions of the Landau-Boltzmann kinetic Equation (13) were obtained using [39]. The beam and plasma parameters of the FNS-ST candidate operating regime, used herein as a sample case, are similar to those used in simulations reported in [42]. In the modelling of the \({}^{3}\)H(\({}^{2}\)H,n)\({}^{4}\)He reaction, mono-directional injection of monoenergetic deuterons and tritons into Maxwellian deuterium-tritium background plasma was assumed. The injection energy 130 keV, equal electron and ion temperatures 7 keV, and the electron density \(n_{e}=10^{14}\) cm\({}^{-3}\) were taken as input values. Such are the FNS-ST core plasma parameters adopted in [42]. The injection angle value, i.e., the pitch angle of particles injected by the source, was 30\({}^{\circ}\), in other words, the angular dependence factor \(\mathcal{Z}_{1,2}(\theta)\) in (12) and (14) was taken in the form a delta-like peak at the injection angle. In fact, the source function of fast ions originating from neutral beam injection into a toroidal magnetically confined plasma is characterized by a certain angular distribution, rather than a particular injection angle as explained in [43]; however, a narrow distribution around the selected injection angle value was assumed here for simplicity. The fraction of suprathermal tail particles \(A_{1,2}\) in (2) was 2.5%. Figure 1a,b show surface plots of anisotropic distributions of deuteron velocities and triton velocities, correspondingly, calculated for the same conditions. The results of simplified calculations using Formula (2), not taking into account the speed diffusion, are depicted by darker colours, whereas the surfaces depicted by semi-transparent lighter colours refer to the solutions of Equation (13) with the speed diffusion effect. The presence of high-energy tails in the vicinity of the injection angle can be seen. The unit step function in Formula (4) is cutting off these tails above the injection velocities, as the darker surfaces illustrate. The lighter surfaces exhibit the tails extending to the regions above the injection velocities that are different for deuterons and tritons due to the difference of masses. Figure 2 shows contour plots of the same distributions of deuteron velocities and triton velocities as shown in Figure 1. The axes are the parallel and perpendicular projections of velocity \(\mathrm{v}_{\parallel}=\mathrm{v}\cos\vartheta\) and \(\mathrm{v}_{\perp}=\mathrm{v}\sin\vartheta\). Model (2) is shown by dashed lines and model (13) is shown by solid lines. Contour lines of low-energy parts of distributions of thermalized particles are circular. The anisotropy of velocity distributions of high-energy particles can be seen as distortions of contour lines in the vicinity of the injection velocity and injection angle. Figure 1: Surface plots of anisotropic distributions of fuel nuclei for the case of injection energy 130 keV, suprathermal fraction 2.5%, and injection angle 30\({}^{\circ}\) without taking into account the speed diffusion (darker colours) and with the speed diffusion taken into account (lighter colours). (**a**) Deuteron velocity distribution functions; (**b**) triton velocity distribution functions. It should be noted that the speed diffusion is responsible for thermalization of injected fast ions governed by Equation (13). The Maxwellian term was introduced artificially in model (2) to describe the bulk plasma ion population.