Transverse Periodic Beam Loading Effects in a Storage Ring

CBP 840 TRANSVERSE PERIODIC BEAM LOADING EFFECTS IN A STORAGE RING ∗ J. R. Thompson † , J. M. Byrd, LBNL, One Cyclotron Road, Berkeley, California 94720, USA Abstract Uneven beam ﬁll patterns in storage rings, such as gaps in the ﬁll patterns, leads to periodic, or transient loading of the modes of the RF cavities. We show that an analo- gous effect can occur in the loading of a dipole cavity mode when the beam passes off the electrical center of the cavity mode. Although this effect is small, it results in a variation of the transverse offset of the beam along the bunch train. For ultralow emittance beams, such as optimized third gen- eration light sources and damping rings, this effect results in a larger projected emittance of the beam compared with the single bunch emittance. The effect is particularly strong for the case when a strong dipole mode has been purposely added to the ring, such as a deﬂecting, or “crab” cavity. We derive an approximate analytic solution for the variation of the beam-induced deﬂecting voltage along the bunch train. ﬂected, perturbing the beam orbit. If the beam is uni- form, all bunches are deﬂected equally and therefore have the same orbit. However, if the storage ring beam ﬁll pat- tern contains a gap, the beam-induced deﬂecting ﬁelds will evolve during the gap and the head and tail of the bunch train will receive unequal deﬂections and follow different orbits. As we show later, this effect is negligibly small except for a few, but important, cases. It is important to note that this effect is not an instability but a steady-state situation that slowly evolves with the beam current. The deﬂecting modes also give rise to a number of coherent instabilities, particularly multibunch instabilities. For the purposes of this study, we ignore these effects and assume that they are damped by an active feedback system. This paper provides an analytical calculation of the scale of this effect. One of situations where transverse periodic beam load- ing is nontrivial is the case of ultralow vertical emittance rings such as damping rings for linear colliders and the next wave of ring-based synchrotron light sources such as NSLS-II, PEPX, and PETRA3. In this case, the small dif- ference in vertical position along the bunch train can be a substantial fraction of the small vertical beam size. Another case where the effect is non-negligible is when a large cav- ity deﬂecting mode has been intentionally added to the ring for the purpose of exciting head-tail transverse oscillations of the bunch. For example, in two-ring colliders, there is a reduction of the luminosity from the geometrical overlap of beams colliding with a crossing angle. Several schemes have been proposed to use deﬂecting cavities to tilt the beams at the interaction point[7, 8, 9]. Another example is a scheme to use deﬂecting cavities to excite a head-tail vertical oscillation larger than the beam divergence[10] and use the vertical head-tail correlation of the emitted photon pulse to shorten its time duration. INTRODUCTION Periodic, or “transient”, beam loading of the accelerat- ing mode of a storage ring RF system has presented one of the challenges to operating high current electron storage rings, particularly in the case of the “ﬂavor”–factories (B and Φ factories) and for third generation light sources with harmonic RF systems. These rings are typically operated with a gap in the ﬁll pattern ranging from a few to about 20%. During the gap, the accelerating voltage and phase evolve such that there is a signiﬁcant difference in the syn- chronous phase of the beam from the head to tail of the bunch train. This synchronous phase transient can create a number of problems for ring operation and most rings are limited to operating with a total transient of 20 degrees of RF phase or less. There has been a number of studies of periodic beam loading over the past decade[1, 2, 3, 4, 5, 6]. It has been shown that the effect is proportional to the average beam current, the size of the gap, and the total R/Q of the RF system. The effect does not explicitly depend on the quality factor, or Q of the RF system and occurs for both normal and superconducting cavities. Although the effect is usu- ally referred to as transient beam loading, it is important to note that effect is steady-state and only slowly evolves as the beam current changes. In this paper, we introduce the analogous effect of pe- riodic beam loading in the deﬂecting mode of an RF cav- ity. In this situation, a beam passes through a cavity with a transverse offset from the electrical center of the dipole mode, exciting the mode. Subsequent bunches are de- ∗ Supported † jrt45@cornell.edu TRANSVERSE TRANSIENTS When a bunch traverses a cavity excited in the TM110 (i.e. deﬂecting) mode parallel to the cavity axis but with some transverse offset from the electrical center of the mode, xoff, it is excited and deﬂects subsequent bunches. The wake voltage excited by the bunch passage can be rep- resented by a complex voltage phasor and is given by V = −j2k ⊥ qx of f by the US-DOE under Contract DE-AC02-05CH11231 where k ⊥ = πω r R/2Q is the transverse loss parameter, Q is the mode quality factor, R/Q is the geometric trans- verse shunt impedance, and q is the charge of the bunch. As each subsequent bunch passes it will experience a deﬂec- tion based on the voltage excited by each of the previous