The radial distribution of plasma which results from field ionization of excited atoms injected into simple mirror machines is calculated and compared with measurements made in the Phoenix machine. The excited atoms are assumed to have a level population distribution given by An−3 where n is the principal quantum number. The fraction of the atoms ionized per unit interval along their path is obtained by calculating the quantum mechanical barrier penetration probability at each point. Finite Larmor orbit diameter, precessional drift around the field axis and finite atom beam dimensions are taken into account in obtaining the radial distribution of plasma that results from this method of injection. The plasma distribution rises to a maximum on the axis in contrast to the distribution obtained by assuming that ionization occurs immediately a critical field is reached. Good agreement with experiment is obtained both for the absolute density and for radial distributions. The fraction ionized per unit interval at the axis is calculated for the Alice and Ogra II experiments without their stabilising fields; this fraction is approximately proportional to the square root of the radial field gradient coefficient. However, the fraction ionized per unit interval is also calculated for an artificial flat-topped field shape and the results show that even in the absence of field gradient, useful trapping can be obtained. Calculations are also made of the effect of pre-ionization on the radial extent of the plasma.
An anisotropic equilibrium for a high-β theta-pinch