# Focusing

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## Focusing

in particle accelerators, the process of controlling the transverse motion—that is, motion perpendicular to the orbit—of charged particles. In most cases this process can be studied independently of phase stability, which provides stability of particle motion in the longitudinal direction—that is, along the orbit.

Depending on the type of accelerator being considered, the distance the particles travel ranges from a few meters to hundreds of thousands of kilometers. Because particles are not injected into an accelerator at exactly the same position, angle, and energy, they deviate from the equilibrium orbit. In the course of acceleration the deviations may increase owing to collisions with the molecules of the residual gas in the accelerator chamber and to imperfections in the magnet and accelerating systems. The Coulomb repulsion between particles may also lead to deflection from the equilibrium orbit. Sufficiently strong focusing must be provided so that the aggregate effect of the factors mentioned above does not result in the particles’ colliding with the walls of the accelerator chamber. Other conditions being equal, the focusing force determines the maximum number of particles accelerated.

Magnetic focusing is most common. It is provided by a certain configuration of the magnetic field and depends primarily on the field index n (B ~ rn, where B is the magnetic induction and r is the radius as measured from the orbit’s center of curvature). In an axisymmetric magnetic field (as in a cyclotron, betatron, or the early electron and proton synchrotrons) what is called weak, or constant-gradient, focusing is achieved when 0 < n < 1. Strong, or alternating-gradient, focusing is accomplished in an azimuthally periodic magnetic field—for example, when n = n0 sin (Nφ), where φ is the azimuth and N is the number of periods per orbit. The permissible values of n0 ≫ 1 depend on N. In practice sector-type focusing (n > 0) and defocusing (n < 0) magnets are used together with straight sections without a magnetic field. Typical arrangements are FODO and FDODFO, where F represents a focusing magnet, D a defocusing magnet, and O a straight section. In linear accelerators alternating-gradient focusing is carried out by means of magnetic quadrupole lenses. The introduction of strong focusing made possible a substantial reduction in the transverse dimensions of accelerator chambers—that is, a reduction in the weight of the magnets and, consequently, in the cost of the accelerators.

Electric field focusing is employed only for low-energy heavy particles in cyclotrons and in linear accelerators. The principles of such focusing do not differ from those used in electron optics. (See.)

M. S. RABINOVICH

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