# Fast Ion Speed Diffusion Effect on Distributions of Fusion Neutrons
## 2 Modelling Techniques  
At an early stage of nuclear fusion research, possible approaches to calculating the distributions of fusion neutrons were discussed in [28], where deuterium-deuterium fusion reactions were considered and the formula for the fusion neutron spectrum was given for the case when the interacting fuel nuclei are identical and their distributions are Maxwellian, without taking into account the anisotropy of the differential cross section. Some other special cases were also considered, namely, for the generalized Maxwellian distribution with different values of perpendicular and parallel temperatures and for monoenergetic distributions. According to the available data for the \({}^{3}\)H(\({}^{2}\)H,n)\({}^{4}\)He reaction, the differential cross-section exhibits a rather weak angular dependence over a wide energy range, whereas for the \({}^{2}\)H(\({}^{2}\)H,n)\({}^{3}\)He reaction, the anisotropy of the differential cross-section is quite distinct. For some particular cases, a simplified angular dependence of the differential cross-section of the deuterium-deuterium fusion reaction was assumed in [28]. Later, in [29], a method to calculate the energy spectra of neutrons was proposed, based on chain rule differentiation of the total cross-section with respect to the emission angle cosine, which, in turn, is related to the kinetic energy of the neutron owing to energy and momentum conservation. Estimations of fusion neutron spectra in a beam-Maxwell plasma reported in [30] were based on [28; 29]. Suprathermal ions, other than those injected by heating beams, may influence the neutron spectra, e.g., so-called knocked-on deuterons and tritons formed due to close elastic collisions with fusion-born \(\alpha\)-particles, as discussed in [31; 32]. Neutron spectra obtained using Monte Carlo modelling techniques were described in [33], and afterwards in other studies of magnetic and inertial confinement fusion, such as [34; 35]. For a number of particular cases, analytical expressions were considered, such as [36; 37]. However, the mentioned publications [28; 29; 30; 31; 32; 33; 34; 35; 36; 37] do not contain general formulae. For the purpose of clarifying the effect of the speed diffusion of fast ions on neutron spectra, straightforward analytical results of a general form for double differential fusion reactivities \(\frac{d^{2}R_{12}}{d^{2}\omega d\Omega_{3}}\) with respect to neutron energy and laboratory emission angle obtained in [25] have been used in this work. For reactions between two colliding fuel nuclei (species \"1\" and \"2\") and two product particles (species \"3\" and \"4\"), the general methods in [25] are called the S- and the L-algorithms, enabling calculations of distributions of fusion products for arbitrary anisotropic distributions of fuel nuclei velocities, also taking into account the angular anisotropy of differential cross-sections of fusion reactions. Either of these equivalent methods are suitable. Species \"3\" means neutrons herein. Although the S- and the L-algorithm differ mathematically, both are 5-fold integrals essentially based on the use of the relation  
\[(\mathbf{v}_{3}-\mathbf{V})^{2}=\frac{2m_{4}}{Mm_{3}}\left(\frac{\mu\
u^{2}}{ 2}+q_{f}\right) \tag{1}\]  
following from the energy and momentum conservation laws with \(M\) and \(\mu\) being the sum of the fuel particle masses \(m_{1}\) and \(m_{2}\) and their reduced mass, respectively, \(\mathbf{v}_{3}\) being the laboratory velocity of product \"3\", \(\mathbf{V}\) being the centre of mass velocity, \(\upsilon\) designating the relative velocity of the fuel particles, and \(q_{f}\) being the energy released in an elementary fusion reaction due to the mass defect. Up-to-date values of particle masses can be found in [38]. Anisotropic distributions of fuel nuclei velocities in fusion plasma subject to neutral beam injection or radiofrequency heating were described, for example, in [39] and the references therein. This anisotropy, in turn, leads to an anisotropy of the distributions of nuclear fusion products.