Self-consistent in internal and external parameters model of indactively coupled rf discharge at low-pressure

A model of a low-pressure ICRF discharge is considered as a nonlinear eigenvalue problem with a parameter for a system including electron balance equations and Maxwell's equations with mixed boundary conditions. The problem is considered in a cylindrical coordinate system for a plasmatron with solenoid coil. Maxwell's equations are considered in the transformed form to a system of elliptic equations in the squares of the modulus of electric and magnetic strengths. The coefficients of ambipolar diffusion, the ionization frequency and the frequency of elastic collisions of electrons with atoms and ions are assumed to be functions of the reduced electric field $E(r)/N$, they are calculated by the BOLSIG+ program using the LXCAT cross-section database. It is shown that the spectral parameter of the problem is the electric field strength at the discharge boundary $E_R$, and the free parameter is the value of the electron concentration in the center of the plasma bunch $n_{e0}$. A condition for the existence of a nontrivial solution of the system is obtained as a nonlinear function of the smallest eigenvalue of the auxiliary linear Sturm-Liouville problem versus the value of the electric field strength at the discharge boundary. This approach makes it possible to find not only the self-consistent distribution of the electron concentration, electric, and magnetic fields in the discharge, but also to relate the value of $ n_ {e0} $ (an internal parameter) with the inductor current $ I_ {ind} $ (an external parameter). A program in Python has been developed to solve the system of boundary value problems. The equations of the system were discretized by the finite differences. The system of difference equations was solved using the Seidel-type iteration. The results of calculating the dependences of $ n_ {e0} $, electric and magnetic fields on $ I_ {ind} $ for an IC RF discharge in a discharge chamber $ 2{}4 $ cm in diameter at a pressure of $ 60 $ Pa and a generator frequency of $ 13{,}56 $ MHz are presented.