The Focusing of Charged Particles by a Spherical Condenser

Abstract

The paths of charged particles traversing a portion of an ideal spherical condenser are worked out. The section of the condenser considered is bounded by two rays, enclosing an angle $\ensuremath{\Phi}$, from the common center of curvature, $O$, of the equipotential surfaces. It is shown that a group of particles, homogeneous in energy, leaving a point $P$ on a normal to one of these boundaries and entering the condenser along this normal as a diverging bundle, will be brought to a focus at a point $Q$ lying on the line $\mathrm{PO}$ extended, if the proper potential is applied to the condenser. This permits the whole condenser gap to be used as a focusing energy analyzer, or monochromator, of very large useful aperture. The velocity dispersion and reduced velocity dispersion are calculated for the most general case, and are found to take the same simple form as do the corresponding expressions for the limited homogeneous magnetic field spectrograph. The expressions for the reduced velocity dispersion are identical in the two cases. Compensation for edge effect is discussed. The relativistic modification of the theory required for high speed particles is discussed and results are presented which indicate that the simple theory of the electrostatic spectrograph may be inadequate even for fairly low values of $\frac{v}{c}$. It is suggested that this difficulty may be avoided by the choice of suitable instrument parameters.An analyzer is described which has a useful aperture of 0.210 steradians, a theoretical reduced dispersion of 1010, and which requires a total focusing potential of 0.315 $E$, where $E$ is the particle energy in equivalent volts. The operation of the analyzer in focusing electrons accelerated by a field designed to furnish an equivalent point source is described.