# Fast Ion Speed Diffusion Effect on Distributions of Fusion Neutrons
## 2 Modelling Techniques
The model used here to take into account the speed diffusion effect is based on numerical solutions of the Landau-Boltzmann kinetic equation for the distribution functions of fuel nuclei \(n_{1,2}(\mathbf{r})f_{1,2}(\mathbf{v})\) [cm\({}^{-6}\)s\({}^{3}\)]  
\[f_{1,2}(\mathrm{v},\theta)=(1-A_{1,2})f_{1,2}^{(\mathrm{M})}(\mathrm{v})+A_{1,2}\sum_{n=0}^{\infty}\phi_{n}^{(1,2)}(\mathrm{v})P_{n}(\cos\theta) \tag{2}\]2  
where \(S_{1,2}\) is the monoenergetic fast particle source  
\[f_{1,2}(\mathrm{v},\theta)=(1-A_{1,2})f_{1,2}^{(\mathrm{M})}(\mathrm{v})+A_{1,2}\sum_{n=0}^{\infty}\phi_{n}^{(1,2)}(\mathrm{v})P_{n}(\cos\theta) \tag{2}\]3  
with \(\delta(\mathrm{v}-\mathrm{v}_{inj1,2})\) being the Dirac delta function and \(S_{1,2}^{inj}\) [cm\({}^{-3}\)s\({}^{-1}\)] being the source rate, i.e., the number of injected particles of species \"1\" or \"2\" per unit volume per unit time. Fast particle lifetime is denoted by \(\tau_{1,2}\) to enable a simple simulation of losses. Maxwellian background plasma is assumed, and the Landau collision term for fast ion species \"1\" or \"2\" is  
\[ \begin{array}{c}C_{1,2}=\frac{\mathrm{v}_{c1,2}^{3}}{\tau_{s1,2}}\frac{1}{\mathrm{ v}^{2}}\Big{(}\frac{\partial}{\partial v}\left(\mathrm{v}_{c1,2}^{2}\frac{a_{1,2} (\mathrm{v})}{2v}\frac{\partial(n_{1,2}f_{1,2})}{\partial v}+b_{1,2}(\mathrm{v })(n_{1,2}f_{1,2})\right)\\ +\frac{\mathrm{c}_{1,2}(\mathrm{v})}{\mathrm{v}_{c1,2}}\frac{1}{\mathrm{sin} }\frac{\partial}{\partial v}\left(\mathrm{sin}\,\vartheta^{\frac{\partial}{ \partial}(n_{1,2}f_{1,2})}\right)\Big{)}\end{array}  \tag{15}\]  
where  
\[\mathrm{v}_{c1,2}=\varepsilon_{1,2}\mathrm{v}_{T_{c}} \tag{16}\]  
and  
\[\tau_{s1,2}=\left(\frac{m_{1,2}}{Z_{1,2}e\omega_{pe}}\right)^{2}\frac{\mathrm{ v}_{c1,2}^{3}}{\Lambda m_{e}} \tag{17}\]  
with \(Z_{1,2}\) being the electric charge numbers of the injected particles, \(\Lambda\) being the Coulomb logarithm, and  
\[\omega_{pe}=\sqrt{\frac{4\pi n_{e}e^{2}}{m_{e}}} \tag{18}\]  
being the electron plasma frequency with \(e\) designating the elementary charge. The terms containing \(a_{1,2}(\mathrm{v})\) and \(c_{1,2}(\mathrm{v})\) are associated with the diffusion tensor in velocity space. These terms are responsible for the speed diffusion and pitch angle scattering, respectively. The term with \(b_{1,2}(\mathrm{v})\) is associated with the dynamic friction force and describes the slowing-down process. Expressions (5) and (6) given above are simplified. Complete formulae for \(a_{1,2}(\mathrm{v})\), \(b_{1,2}(\mathrm{v})\), and \(c_{1,2}(\mathrm{v})\) are given in [39], as well as the details of obtaining numerical steady state solutions of (13). It is worth mentioning that an analytical approach is also possible. In brief, for an isotropic case, such an approach can be described as follows. The steady state implies zero time derivative in Equation (13). The isotropy implies no angular dependence, i.e., zero angular derivative in the term with \(c_{1,2}(\mathrm{v})\). Neglecting the speed diffusion, i.e., neglecting the term with \(a_{1,2}(\mathrm{v})\) bearing in mind the small parameter \(\varepsilon_{1,2}\), and assuming an infinite \(\tau_{1,2}\), i.e., no losses of particles for simplicity, a steady state isotropic solution of (13) can be readily obtained in the form of the so-called classical slowing-down distribution  
\[n_{1,2}f_{1,2}(\mathrm{v})=\frac{S_{1,2}^{inj}\tau_{s1,2}}{4\pi v_{c1,2}^{3}} \frac{1}{b_{1,2}(\mathrm{v})}H(\mathrm{v}_{inj1,2}-\mathrm{v}) \tag{19}\]  
where the unit step function is obviously cutting off the distribution tail above \(\mathrm{v}_{inj1,2}\). The most straightforward way to obtain a more sophisticated analytical solution taking into account the speed diffusion can be demonstrated for the case when all background plasma species are in thermal equilibrium with equal temperatures \(T\). Since the collision term nullifies in thermal equilibrium, the Maxwellian distribution function \(\exp\bigl{(}-m_{1,2}v^{2}/2T\bigr{)}\) is to be a partial solution of the homogeneous differential equation corresponding to (13).