# Neutral Beam Coupling with Plasma in a Compact Fusion Neutron Source
## 3 BTOR Software and LNB Computational Methods
The neutral beam intensity attenuation along the beam track is calculated as:  
\[\frac{dI}{dx}=-n_{e}\sigma_{\mathrm{s}}I, \tag{2}\]  
where \(n_{e}\) is the plasma target density, and \(\sigma_{\mathrm{s}}\) is the effective beam stopping cross-section due to ionization:  
\[\sigma_{\mathrm{s}}=<\sigma_{ie}|\mathrm{v}-\mathrm{V_{b}}|>/V_{\mathrm{b}}+ \frac{n_{p}}{n_{e}}(\sigma_{ip}+\ \sigma_{\mathrm{cx}})+\frac{n_{z}}{n_{e}}\sigma_{z}\approx< \sigma_{ie}\
u>/V_{\mathrm{b}}+\ \sigma_{ip}+\ \sigma_{\mathrm{cx}}+\frac{n_{z}}{n_{e}}\sigma_{iz}. \tag{3}\]  
Here, \(\sigma_{\mathrm{ie}}\), \(\sigma_{\mathrm{ip}}\), \(\sigma_{\mathrm{iz}}\) are ionization cross sections of atoms in collisions with electrons, protons, and impurities of charge Z, respectively, \(\sigma_{\mathrm{cx}}\) is the charge exchange cross section on hydrogen ions, and brackets mean averaging over the Maxwell distribution of electron velocities. Graphs of the beam stopping cross section in the range of FNS-ST plasma parameters are shown in Figure 8. Since the neutral beam energy for FNS-ST is E\({}_{\mathrm{b}}\)\(\sim\)100-140 keV, the NB is produced from positive ions. After neutralization and dissociation of molecular ions, the beam carries three energy fractions that are shown by violet bars in Figure 8a. It can be seen that for the low-energy fraction E\({}_{1/3}\), the stopping cross-section can be up to three times higher than that for the full energy fraction (E\({}_{\mathrm{full}}\)), which leads to the ionization and capture of low-energy beam fractions (E\({}_{1/2}\) and E\({}_{1/3}\)) at the plasma periphery and reduces the total CD output from NBI. The velocity of fast ions released is decomposed into two components with respect to the magnetic field line: parallel velocity which enables the ion circulation around the torus main axis, and transverse velocity, i.e., orthogonal component associated with the Larmor rotation around the field line. The toroidal current in the plasma is created by the parallel ion velocity. The collisions cause fast ion momentum transfer to plasma electrons and ions. The classical solution for fast ion deceleration due to Coulomb collisions [6] is applied to calculate the fast ions thermalization time (\(\tau_{\mathrm{s}}\)) as a function of its initial energy E\({}_{\mathrm{b}}\), the critical energy E\({}_{\mathrm{c}}\), and the value \(\tau_{\mathrm{se}}\), which is the characteristic Spitzer time:  
\[\tau_{\mathrm{s}}=\frac{\tau_{se}}{3}\ln\left[1+\left(\frac{E_{\mathrm{b}}}{ E_{\mathrm{c}}}\right)^{3/2}\right]; \tag{4}\]  
\[E_{\mathrm{c}}=14.8\left(A_{\mathrm{b}}/A_{i}^{2}/3\right)\cdot T_{e}, \tag{5}\]  
\[\tau_{\mathrm{S}_{e}}\ \propto\ \left(A_{\mathrm{b}}T_{e}^{3/2}/n_{e}\right), \tag{6}\]  
Here, \(A_{b}\) and \(A_{i}\) are the atom mass of fast ions and plasma ions, respectively, and \(\propto\) is means direct proportionality. \(E_{\mathrm{c}}\) is the value of the critical beam energy, which defines the level when the electron and ion heating rates are equal. If the beam ion energy is higher, it loses its energy mainly due to collisions with electrons, with a slight change in momentum, while the scattering angles of the ions are small. If the ion energy is below the critical level, deceleration occurs mainly due to collisions with plasma ions, and in this case the ion is deflected by significant angles. From Equation (5), the highest value of the critical energy E\({}_{\rm c}\) corresponds to the maximum mass ratio of the beam ions to background ions, and directly increases with plasma temperature. For a deuterium beam in a deuterium plasma with T\({}_{\rm e}\) = 5 keV, the critical energy is \(\sim\)93 keV (E\({}_{\rm c}\approx 19\times\) T\({}_{\rm e}\)). Fast ion distribution in space and angle can be obtained from a kinetic equation solution with a beam source and a collisional operator, accounting for fast ion deceleration and pitch angle scattering.