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The transverse electromagnetic interaction of an intense azimuthally uniform beam of particles with itself, including the effect of a resistive vacuum tank, is investigated theoretically. It is shown that a beam of particles all having velocity ν is unstable against the development of transverse waves having a phase velocity close to (1 − v/n)ν, where v is the number of transverse free betatron oscillations occurring in one revolution and n is a positive integer greater than v. The growth rate for the instability is proportional to N/σ½, where N is the number of particles in the beam and σ is the conductivity of the surface material of the vacuum tank. Stabilizing mechanisms are examined by means of the Vlasov equation and it is shown that a spread in the quantity (n−v)ν, evaluated for particles in the unperturbed beam, will prevent the instability. A criterion for the spread required is shown, in the limit of walls of high conductivity, to depend upon the beam intensity and energy as well as upon certain geometrical properties of the accelerator, but not upon the conductivity. Numerical examples covering a range of particle accelerators are presented, and suggest that the theory is in agreement with the coherent beam behavior recently observed in a number of accelerators.

It is now easy to see that a linear relation exists also between energy differences in the two configurations. If we add to the interactions Vij a constant term -EQ (EO might be chosen to be the energy of the ground state in the ^3/2/7/2 configuration), it will contribute to all levels of the configuration ^3/2/7/2 the same energy -EQ, and to all levels of the dziifi^ configuration the same energy -3Eo. Thus we obtain the following relations, where £^=£(^3/2/7/2/) and E/^E{d zl if v J):

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