7 BEAM DYNAMICS R. H. Helm, G. A. Loew, and W. K. H. Panofsky, Editor This chapter deals with the general features of particle motion1 through the standard sectors of the two-mile accelerator. The special problems associated with the injector and positron source are described in Chapters 8 and 16, respectively. The approach in this chapter is to identify those general features of particle motion which do not depend on details of design of the disk-loaded waveguide configuration. Emphasis is, therefore, placed on general formula- tion rather than detailed computation. The first portion of the chapter deals with vacuum trajectories of single particles and thus ignores any effects that depend on beam intensity. The analysis is first made assuming no external focusing. Then the magnetic lens system now employed in the accelerator is taken into account. For this pur- pose the matrix formalism applying to beam transport problems is briefly reviewed; not only are orbits for the ideal accelerator derived, but the effects of misalignments and other perturbations are also studied. The second part of the chapter deals with those phenomena resulting from the influence of the electron current upon individual particle behavior. The dominant item in this category is the beam breakup phenomenon which is discussed first in terms of a general asymptotic theory identifying the phenomena involved and is then examined through more detailed numerical computations. The chapter concludes with a summary of experimental observations taken to date on the beam breakup phenomena. 7-1 The "ideal" linear accelerator (WKHP) Consider an ideal accelerator having cylindrical symmetry about the z axis. Let total differentiation with respect to z and referred to the moving electron be denoted by a prime ('); let y be the energy of the electron in 163 164 R. H. Helm, G. A. Loew, and W. K. H. Panofsky units of the rest energy m. Equating the velocity of light c to unity yields /-^m (7-D where Ez is the axial component of the electric field. In this chapter, (3 = (1 — l/y2)1/2 will be used for the velocity and the usual notation will be adopted for the components of electric and magnetic fields. It will be assumed that y and y' are given functions of z, resulting from the integration of Eq. (7-1) in a given accelerating field Ez. The radial equation of motion is2'3 «±l* (7-2) In the absence of external focusing, the electric and magnetic terms in Eq. (7-2) almost cancel for a traveling-wave accelerator in which the phase velocity of the wave matches the particle velocity; in the relativistic limit (/?« 1) the right-hand side of Eq. (7.2) becomes small. Thus in the usual description all radial forces are neglected and the integral of Eq. (7-2) becomes I (7-3) where 00 is the slope drjdz at an arbitrary starting point and the subscript zero identifies the values of the other variables at that point. If the energy gain is uniform, y = y'z, where y' is constant; furthermore if the energy is sufficient to make /? « 1, Eq. (7-3) becomes / -7 \ (7-4) where z0 is defined by y'z0 = y0. These logarithmic orbits are a simple conse- quence of the transverse momentum being a constant while the longitudinal momentum increases linearly; such a momentum relationship implies r' = 00z0/z from which Eq. (7-4) follows. Equation (7-4) can be interpreted in terms of an " effective length " L given by L = z0 In (-) (7-5) which would be the length given by r = r0 + 60L, i.e., the length over which a corresponding radial excursion in the absence of acceleration would occur (Fig. 7-1). The quantity L will also be recognized as the "contracted" length of the accelerator as seen from a frame of reference moving with the electron. The right-hand side of Eq. (7-2) does not vanish exactly in the relativistic limit. To illustrate this, the "paraxial" equation will be formed, carrying terms linear in r only. With this approximation, the transverse field compon- ents can be expressed in terms of the longitudinal electric field as follows. Beam dynamics 165 I AXIS Figure 7-1 Orbit geometry. In the absence of space-charge effects, Therefore, remembering that Ez is circularly symmetric and thus has no first- order dependence on r, one obtains er 8EZ mr dy' '=~~~ = (7-6) Similarly, from which it follows that er dEz mr dy' (7-7) Hence the general first-order paraxial equation of motions is*: (7-8) since — = ~d~z = is the total derivative with respect to z. As y becomes large the last term * Note that (E + p X B)^ =0, i.e., the Lorentz force has no azimuthal component. 166 R. H. Helm, G. A. Loew, and W. K. H. Panofsky vanishes (corresponding to cancelation of electric and magnetic forces in the relativistitic limit); the differential equation then becomes, with (3 = 1, (yrj + ^y" = o = {yr' + r- y'J - ^ (7-9) Integrating from a point denoted by the subscript zero to a final point, where the initial point is assumed to be in field-free space, one finds 1 rr r yr' - (yr') = -\ y' dr - - / 0 J ± r0 2. or yr' - (yr')0 = ±{-roy + K? - /)} (7-10) where y' is a mean value of the rate of energy gain y'. Hence the radial momen- tum differs from being a constant of the motion by the two terms on the right-hand side of Eq. (7-10). The first term —^r0y' represents the converging lens effect at the beginning of the accelerating region. The effective thin-lens focal length /0 at entry is simply 1 I/ /- = + 57 (M1) The second term gives an alternating focusing-defocusing action due to the fluctuations in y' about the mean value of y'. The result is a net " strong focusing" action the strength of which can easily be computed by conven- tional strong focusing theory. Note that even if the term in y' — y' is negligible within the accelerator section, the exit fringe field still contributes a radial impulse + \ r y' and is thus equivalent to a diverging lens of focal length 1 I/ 7 = -27 (M2) Thus the field-free gaps between sections are equivalent to weak, alternating gradient doublets. These effects will not be analyzed further here, since the strength of the radial forces discussed in this section for the parameters of the SLAC accelerator is small compared with the action of the external lenses which will be discussed in the next section. 7-2 External focusing (RHH) In real accelerators the transverse position and quality of the beam are affected by numerous small perturbations, such as stray magnetic fields, misalignments, RF asymmetry effects, scattering by residual gas, and in some cases a transverse instability ("beam breakup") resulting from electromag- netic interaction with the accelerator structure. In addition, the finite phase space volume of the injected beam imposes limits on the distance the beam Beam dynamics 167 can be transported before filling the entire radial aperture, even in an ideal accelerator. External focusing is a practical means of containing the initial phase space which has the additional advantage of suppressing the various perturbing effects to a considerable extent. A major disadvantage arises from the beam deflection caused by misalignment of focusing lenses, which, therefore, must be aligned to very tight tolerances. In the present section the principles of linac focusing are discussed, with special emphasis being given to problems affecting very long accelerators, and to the design of the focusing system for SLAC. Phase space The general definition of phase space volume which will be employed in this chapter is 1/6 = JJJJJJ dx dpx dy dpy dt where (x, px), (y, Py), and (t, y) are the conjugate coordinate pairs appropriate to a Hamiltonian system in which the independent variable is the longitudinal coordinate, z, and in which there is no scalar potential.* According to Liou- ville's theorem, the coordinates of a given set of particles are contained in a volume which is invariant provided that only nondissipative forces act on the particles. In beam transport (i.e., the motion of streams of charged particles through a complex accelerating, steering, and focusing system) it often happens that one or more components of the motion are decoupled from other motions, so that phase volumes are conserved in certain subspaces; e.g., the projected phase plane areas ux = JJ dx dpx uy = JJ dy dpy ut = JJ dt dy (7-14) might each be conserved. Frequently the coupling between different components is weak enough to permit the use of perturbation calculations in which such subspace projections of the phase volume are conserved as a first approxi- mation. Because of the conservation, whether exact or approximate, the concept of phase space emittance of a beam source is useful as a figure of merit. For example, a small transverse emittance implies that the beam may be focused in such a way as to have simultaneously a moderate size and a very small transverse momentum, so that it may consequently be transmitted for long distances without further focusing. Small transverse emittance also implies * If a scalar electric potential were present, the longitudinal canonical momentum could not be equated with the energy y. 168 R. H. Helm, G. A. Loew, and W. K. H. Panofsky that the beam may be focused to very small size without excessive angular spread—a desirable property for allowing a small target size for physics experiments. The concept of phase space is also useful in specifying the properties of a beam transport system.