Differential equations are derived and two computer solutions presented for the case of a relativistic electron beam injected into neutral gas. Processes included are ionization, scattering of electrons on atoms and ions, charge exchange, supply of neutrals to the beam by diffusion, and incoherent radiation by the electrons. A longitudinal electric field is assumed. The main restrictive assumptions are those of uniform densities and hydrodynamic stability. Solutions are given for an 1100-amp, 0.77-Mev, externally-focused beam injected into H2 at pressures of 10−6 and 2×10−5 mm Hg. The beam contracts from an initial radius of 1.73 cm to an equilibrium radius of 7×10−4 cm in 3 and 0.5 msec, respectively. It is found in these cases that neutrality of the beam does not occur until after the radiation-scattering equilibrium, which was first studied by Budker, has occurred. Several difficulties connected with a neutralized beam are discussed. It is shown that a space-charge-limited beam of high phase-space density is the type most suitable for obtaining a small beam diameter in the shortest time. An approximate solution is derived which applies after a specified time near equilibrium and allows calculation of several parameters of interest without the necessity of computer integration.
This corresponds to the classical virial theorem, since in the presence of a magnetic field the right-hand side of (1) is replaced by (a -it) in the quantum case.Perhaps the most useful form of the virial theorem is the case corresponding to no magnetic field, --.It is clear that (5) and (6) do not apply for continuum states. For example, for a free particle (a-p)=(p 2 /E). Alternatively, for a static potential cj>, we consider the expectation value for any stationary state of the anticommutator of £ and H to obtain <0(£-«0)>«l,(that is, 3a-p) = 0) which gives the well-known result 0)=t/E for a free particle. However, for a bound state in the Coulomb field, it follows from (6) thatso that in this case,•e<00> = E»--l.The nonrelativistic limit e{4>)--2{T), T the nonrelativistic kinetic energy operator, follows from (7b); and more generally from (6) the usual limiting form e(r-V) = 2{T) results.
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