Transverse Quadrupole Wake Field Effects in High Intensity Linacs*

Transverse quadrupole wake fields exist whenever the beam is not round, and therefore in an alternating gradient focusing system these fields cannot be eliminated. As a result, these fields represent a potential limitation on high intensity linac performance. In this note we calculate the magnitude of quadrupole wake field effects for the SLAC linac. (Submitted to Particle Accelerators) *Work supported by the Department of Energy, contract DE-AC03-7GSF00515. 2INTRODUCTION As an intense bunch of charged particles travels down a linear accelerator, it interacts with the surrounding accelerating structure and generates wake electromagnetic fields behind it. If the beam intensity is high and therefore the wake field is strong, the particle distribution within the bunch may be significantly affected and the effective beam emittance will grow. This effect may impose an important limit on the bunch intensity for the linear colliding beam devices now being proposed since the luminosity of these devices depends critically on maintaining a beam size of the order of 1 pm at the collision point. As we report in this note, the inclusion of the quadrupole wake effects in calculations for the SLAC Linear Collider (SLC)' indicates that these effects may be important for other high intensity linear accelerators as well. The wake field generated by an intense particle bunch in a cylindrically symmetric accelerator pipe can most conveniently be described by decomposing it into several components, each giving rise to different beam dynamical effects. The first component is the longitudinal wake field; it consists mainly of a longitudinal electric field which is uniform across the transverse cross section of the accelerator pipe and varies longitudinally along the pipe. -The longitudinal wake field gives rise to a variation in particle energy within the length of the bunch. The second component of the wake field is the transverse dipole wake generated by an off-axis motion (i.e., the dipole moment) of the bunch; it deflects the tail portions of the bunch. -3The third component of the wake field is a transverse quadrupole wake caused by the bunch having a transverse spatial distribution that is not round (i.e., having a finite quadrupole moment). The effect of the quadrupole wake is to cause the transverse distribution in the tail portions of the bunch to change without affecting the beam center of a properly centered beam. The longitudinal wake and the transverse dipole wake have been extensively studied.' In this paper we will study the beam dynamical effects caused by .he quadrupole wake field component. lJe first present some analysis assuming a simplified accelerator lattice model. The results are then applied to the presently proposed SLAC linac collider. A numerical tracking program has been written to test these results. . The properties of the quadrupole wake fields are taken from the results of Ref. 2. The longitudinal wake field effects can be minimized by adjusting the phase of the radio-frequency accelerating field relative to the arrival time of the particle bunch. The effects of the transverse dipole fields can in principle be eliminated by proper alignment of the accelerator pipe, and by accurate bunch injection. In contrast to the dipole wake component, the quadrupole wake field exists even if the particle bunch is perfectly centered on the pipe axis. This observation follows from the fact that the quadrupole wake field is generated by the quadrupole moment rather than the off-axis dipole moment of the particle distribution. Thus the quadrupole wake field represents a potential fundamental limiting influence on the transverse beam dynamics in the -4linac and as such needs to be studied. This is especially the case if the linac lattice is of the alternating gradient type so that the transverse spatial distribution of the beam is in general elliptical. QUADRUPOLE WAKE FIELD A quadrupole wake is generated if the transverse beam distribution posesses a quadrupole moment relative to the accelerator pipe axis. In that case, two types of quadrupole moments may exist: 41 = (1) Qt = 2 where x and y refer to the transverse coordinates defined by the focusing scheme of the accelerator, and < > means averaging over the transverse beam dimensions. . Consider a sl'ce of charge travelling down the linac. Let the charge slice have quadrupole moments 91 and 42. The wake field generated by this charge slice at a point z following the slice gives rise to a transverse Lorentz force given by e[l + i x ii] = etW(z) [Q, (xit ~$1 + Q,(yji: + x~)] (2) where W(z) is a certain quadrupole wake function characteristic of the accelerator pipe structure, i and 3 are the unit vectors in.the x and y directions. The electromagnetic force generated by 41 resembles that of a quadrupole focusing magnet, while the force generated by Q2 resembles that of a skew quadrupole magnet. For simplicity, in the following we assume the transverse beam shape is an upright ellipse for which 42 = 0 and therefore there are no skew quadrupolar fields.