# Fast Ion Speed Diffusion Effect on Distributions of Fusion Neutrons
## 2 Modelling Techniques
Modified S- and L-algorithms were published in [27] on the basis of the same geometric technique as in [26]. This method still retains the generality of [25]. Detailed descriptions of the computations of neutron spectra are explained in the corresponding references. The compact analytical anisotropic model, used in [25] as a sample case of fuel nuclei velocity distributions, that does not take into account the speed diffusion effect, is as follows:  
\[f_{1,2}(\mathrm{v},\theta)=(1-A_{1,2})f_{1,2}^{(\mathrm{M})}(\mathrm{v})+A_{1,2}\sum_{n=0}^{\infty}\phi_{n}^{(1,2)}(\mathrm{v})P_{n}(\cos\theta) \tag{2}\]  
where the second term describes populations of suprathermal particles of species \"1\" and \"2\" with energetic tail fractions \(0\leq A_{1,2}\leq 1\), the term  
\[f_{1,2}^{(\mathrm{M})}(\mathrm{v}_{1,2})=\frac{1}{\pi^{3/2}}\bigg{(}\frac{m_{ 1,2}}{2T_{1,2}}\bigg{)}^{3/2}\exp\left(-\frac{m_{1,2}\mathrm{v}_{1,2}^{2}}{2T_ {1,2}}\right) \tag{3}\]  
is the Maxwellian distribution of bulk thermalized particles, \(P_{n}(\cos\theta)\) are Legendre polynomials,  
\[\phi_{n}^{(1,2)}(\mathrm{v})=\frac{\kappa_{1,2}Z_{1,2}^{(\mathrm{b})}}{V_{3}^ {3}}(2n+1)\mathcal{Z}_{n}^{(1,2)}\frac{H(\mathrm{v}_{inj1,2}-\mathrm{v})}{b_{1,2}(\mathrm{v})}e^{-\frac{\pi(n+1)}{L_{1}2T_{\mathrm{v}}}\int\limits_{\mathrm{ v}}^{\mathrm{inj1,2}}\frac{c_{1,2}(\vec{\gamma})}{b_{1,2}(\vec{\gamma})}d\vec{ \upsilon}} \tag{4}\]Dimensionless functions \(b_{1,2}(\mathrm{v})\) and \(c_{1,2}(\mathrm{v})\) in (4) are given by  
\[b_{1,2}(\mathrm{v})=Z_{1,2}^{(\mathrm{b})}\left(1+\frac{\mathrm{v}^{3}}{V_{c1,2} ^{3}}\right) \tag{5}\]  
\[c_{1,2}(\mathrm{v})=\frac{\epsilon_{1,2}Z^{(\mathrm{eff})}}{2}\frac{v_{Te}}{ \mathrm{v}}+\frac{2\epsilon_{1,2}}{3\sqrt{\pi}} \tag{6}\]  
where  
\[Z^{(\mathrm{eff})}=\frac{1}{n_{e}}\sum_{i=1}^{N_{i}}n_{i}Z_{i}^{2} \tag{7}\]  
The value  
\[V_{c1,2}=\left(\frac{3\sqrt{\pi}}{4}Z_{1,2}^{(\mathrm{b})}\right)^{1/3} \epsilon_{1,2}\mathrm{v}_{Te} \tag{8}\]  
referred to as critical velocity is proportional to the electron thermal velocity \(\mathrm{v}_{Te}=\sqrt{2T_{e}/m_{e}}\) determined by the electron temperature \(T_{e}\). The parameters in (8) are dimensionless quantities given by  
\[\varepsilon_{1,2}=\left(\frac{m_{e}}{m_{1,2}}\right)^{1/3} \tag{9}\]  
\[Z_{1,2}^{(\mathrm{b})}=\frac{m_{1,2}}{n_{e}}\sum_{i=1}^{N}\frac{n_{i}Z_{i}^{2 }}{m_{i}} \tag{10}\]  
where \(m_{e}\) is the electron mass, \(n_{e}\) is the electron density, the summation is over the \(N\) species of the background plasma ions with masses \(m_{i}\), electric charge numbers \(Z_{i}\) and densities \(n_{i}\). The values  
\[f_{1,2}(\mathrm{v},\theta)=(1-A_{1,2})f_{1,2}^{(\mathrm{M})}(\mathrm{v})+A_{1,2}\sum_{n=0}^{\infty}\phi_{n}^{(1,2)}(\mathrm{v})P_{n}(\cos\theta) \tag{2}\]0  
are normalizing constants. Finally, coefficients  
\[f_{1,2}(\mathrm{v},\theta)=(1-A_{1,2})f_{1,2}^{(\mathrm{M})}(\mathrm{v})+A_{1,2}\sum_{n=0}^{\infty}\phi_{n}^{(1,2)}(\mathrm{v})P_{n}(\cos\theta) \tag{2}\]1  
are dimensionless quantities determined by the unity-normalized angular dependence factor \(\mathcal{Z}_{1,2}(\theta)\) of the monoenergetic fast particle source in the plasma, \(H(\mathrm{v}_{inj1,2}-\mathrm{v})\) is the unit step function, and \(\mathrm{v}_{inj1,2}\) are the injection velocities of fast particles.