Subatomic Physics

2.5. Beam Optics 23
Figure 2.8: Rectangular dipole magnet. The optical analog is a prism, shown at the right.
The force is normal to the trajectory. For the normal component of the force,
Newton’s law, F = dp/dt, and Eq. (1.7) give
so that with Eq. (2.21) the radius of curvature becomes (6)
Fn = pv
, (2.22)
ρ
ρ = pc
. (2.15)
|q|B
Problems arise when a beam should be focused. Figure 2.8 makes it clear that
an ordinary (dipole) magnet bends particles only in one plane and that focusing can
be achieved only in this plane. No magnetic lens with properties analogous to that
of an optical focusing lens can be designed, and this fact stymied physicists for many
years. A solution was finally found in 1950 by Christofilos and independently by
Courant, Livingston, and Snyder in 1952. (7) The basic idea of the so-called strong
focusing can be explained simply by referring to Fig. 2.7: If focusing and defocusing
elements of equal focal lengths are alternated, a net focusing effect occurs. In beam
transport systems, strong focusing is most often achieved with quadrupole magnets.
A cross section through such a magnet is shown in Fig. 2.9. It consists of four
poles; the field in the center vanishes and the magnitude of the field increases from
6Equation (2.15) is given in Gaussian units, where the unit for B is 1 G, and the unit of
potential is 1 stat V = 300V. To compute ρ for a particle with unit charge (|q| = e), express pc in
eV; then Eq. (2.15) yields
B(Gauss) × ρ(cm) = V
. (2.23a)
300
As an example, consider an electron with a kinetic energy of 1 MeV; pc follows from Eq. (1.2) as
pc =(E 2 kin +2Ekinmc 2 ) 1/2 =1.42 × 10 6 eV.
V then is 1.42 × 10 6 VandBρ =4.7 × 10 3 G cm. Equation (2.15) can also be rewritten in mks
units, where the unit of B is
1T(Tesla)=1Wb(Weber)m −2 =10 4 G.
7 E. D. Courant, M. S. Livingston, and H. S. Snyder, Phys. Rev. 88, 1190 (1952).