Thermal conduction via the transport equation with applications to laser-fusion

Abstract

We study the problem of heat conduction in a pellet of deuterium and tritium heated by a laser. In particular, we examine the heat flux between the critical surface and the pellet surface. Because of the smallness of this region compared with an electron mean free path, the problem is treated by transport theory using the Boltzmann equation. Two distinct problems are considered. In the first, only ion-electron collisions are allowed and the resulting linear transport equation is solved analytically. Numerical evaluation of the results illustrates the shortcomings of continuum theory and shows the general structure of the temperature profile across the conduction region. A review of previous work in this area is given. To account for the electron-electron collisions the full non-linear Boltzmann equation is employed with the BGK scattering model. In this approach we allow both electron-electron and ion-electron scattering and by various transformations are able to cast the problem into the form of three coupled, non-linear integral equations. These are solved numerically and values of the heat flux as a function of optical thickness of the conduction zone are obtained. The results show the importance of a rigorous transport treatment of conduction and highlight the limitation of previous attempts to solve this problem. It is not claimed that the theory presented here can be directly applied to the laser-fusion problem because the effects of non-thermal electron distributions and strong electric and magnetic fields have been neglected. Nevertheless, the general conclusions are believed to be true regardless of the physical application.