The Effect of an Under-Dense Plasma Density Gradient on the   Backstreaming Ion Mechanism
  The space charge limited emission of ions from a target in the focus of an intense relativistic electron beam is studied analytically for the case of a spatially varying target density profile. In particular, the emission in the presence of an under-dense plasma shelf in contact with the solid density target dramatically differs from the case of an abrupt solid-vacuum boundary. It is found that an under-dense gradient scale length several times that of the beam radius at the focus reduces the emission by at least an order of magnitude over that to be expected from a solid-vacuum boundary. 
George J. Caporaso