# In-Line Target Production for Laser IFE
## 2 Modeling Results
### FST-Layering Method for Classical High Gain Targets
Therefore, the TIP criterion can be written in the following type (\(\tau_{\text{rol}}\) is the time of pure target rolling):  
\[\tau_{\text{form}}\leq\tau_{\text{res}}=\tau_{\text{rol}}. \tag{2}\]  
Thus, determination of the rolling conditions is one of the main problem, which influences the choice of the FST-LM operation including simplifying the physics design and modifying the specifications. First of all we should to determine the spiral angles \(\alpha\) for realizing the pure target rolling (R&R-mode). Consider a cylindrical spiral of radius \(R_{0}\). In each point of the spiral the tangent makes a constant angle \(\alpha\) with the horizontal. Let a spherical target of radius \(R\) and mass \(m\) is moving with a velocity \(V\) (t) along the spiral LC. Then the system of equations in accordance with [PERSON]'s second law of motion has the following form:  
\[dV/dt=gsin\alpha-F_{f}/m, \tag{3}\]  
\[(J/R)d\omega/dt=F_{f}-k_{r}, \tag{4}\]  
where \(g\) is the free fall acceleration, \(J\) and \(\omega\) are the moment of the target inertia and the angular velocity of its rotation, \(F_{f}\) is the friction force, \(N\) is the reaction of support from the LC wall, and \(k_{r}\) is the rolling friction. If the target rolls without sliding the linear and angular velocities are related by equation of rolling  
\[V=\omega R. \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(D_{1}\) (\(\mu m\)) & \(\Delta_{I}\) (\(\mu m\)) & \(D_{0}\) (\(\mu m\)) & \(\rho_{I}\) (\(g/cm^{3}\)) \\ \hline D–T (vapor) & 0 & 3000 & 3000 & 0.3-\(10^{-3}\) \\ \hline D–T (solid) & 3000 & 190 & 3380 & 0.25 \\ \hline CH (DT)\({}_{64}\) & 3380 & 261 & 3902 & 0.25775 \\ \hline CH layer & 3902 & 1 & 3904 & 1.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: CHGT-2 specifications. Excluding the friction force from Equation (3), we obtain the rolling equation:  
\[(1+\zeta)\mathrm{dV/dt}=\mathrm{gsin}\alpha-\mathrm{k_{r}N/m},\,\zeta=\mathrm{J/(mR ^{2})}. \tag{6}\]  
Besides, there is the conditional relation that should be maintained for the accelerated target to be in pure rolling:  
\[\mathrm{F_{f}}\leq\mathrm{F_{max}}=\mathrm{k_{s}N},\,\mathrm{N}=\mathrm{mcos} \alpha[g^{2}+(V^{4}\mathrm{cos}^{2}\alpha)/R^{2}]^{0.5}, \tag{7}\]  
where \(\mathrm{k_{s}}\) is the sliding friction. The second component under the root is the square of the centripetal acceleration. Solving together Equations (6) and (7), we obtain the working range \(\Delta\alpha\), which is given by two inequalities:  
\[\mathrm{k_{r}}=\mathrm{tg}\alpha_{\mathrm{min}}<\mathrm{tg}\alpha<\mathrm{tg }\alpha_{\mathrm{max}}=\{(1+\zeta)\mathrm{k_{s}}-\mathrm{k_{r}}\}/\zeta. \tag{8}\]  
The next step is modeling of the CHGT-2 layering. We used a simulation code based on solving the [PERSON]'s problem [25; 26]. Using the data in Table 2 and the data on the fuel heat capacity and conductivity [19], we got the following values:  
* \(\mathrm{T_{in}}=\mathrm{T_{s}}\sim 35\) K: \(\mathrm{\tau_{form}}=22.45\) s for D\({}_{2}\) fuel and \(\mathrm{\tau_{form}}=28.52\) s for D\({-}\)T fuel,
* \(\mathrm{T_{in}}=\mathrm{T_{d}}\sim 28\) K: \(\mathrm{\tau_{form}}=12.05\) s for D\({}_{2}\) fuel and \(\mathrm{\tau_{form}}=14.25\) s for D\({-}\)T fuel,  
where \(\mathrm{T_{s}}\) is the temperature of fuel separation into the liquid and gaseous phases, and \(\mathrm{T_{d}}\) is the depressurization temperature at which the excess gas is removed from the SC. Note that depending on the shell strength \(\mathrm{T_{d}}\) can be \(\sim\mathrm{T_{s}}\). The final step is the computation a set of the optimization parameters related to the LC geometry to maintain the process of the CHGT-2 fabrication by the FST layering method. Using (8), we have found that the CHGT-2 can be fabricated in the FST-LM by using a double-spiral LC that is confirmed by the experimental modeling with pilot LC mockups. The obtained results are presented below. 1.