Diffraction Radiation from a Charged Particle Moving through a Penetrable Sphere

The formal exact solution to the problem of the radiation of a charged particle traveling with a constant velocity through a dielectric sphere is obtained. The electromagnetic field may be expressed in terms of an infinite number of spherical normal modes. It is found that radiation may be emitted for arbitrarily small particle velocities. For larger spheres (i.e., $ka\gg 1$ when $k$ is the wave number and $a$ is the radius of the sphere) the radiated field is predominantly Čerenkov-type radiation when the velocity of the particle is above the Čerenkov threshold velocity. For small spheres ($ka\gg 1$) and low particle velocities the radiation is shown to be mainly of transition type. Numerical results are presented to illustrate the behavior of the spectra of the various lower-order radiative modes as the velocity of the particle is varied.