New Theory of Single Bunch Stability in a Linac with Quadrupole Displacements

The analytical treatment previously described [1] has been extended to include the important effect of magnetic quadrupole transverse displacements, the chromatic variation of the magnetic focusing, the energy spread along the bunch and possible microwave quadrupoles, the last two in relation to BNS damping. Both, the longitudinal and transverse equations of motion are solved, the second by using the perturbation method with partial expansions developed for this theory. The localized nature of the quadrupole displacements is preserved by using thin lenses and the superposition principle for the kick effects. The causality principle applied to the downstream beam oscillations due to the kicks is introduced via Heaviside functions. The treatment presented [2] provides formulae for the tuneshift in the bunch and first-order solutions for the transverse beam off-sets within the bunch. It presents a break-through in the recent efforts [3] to solve the problem of the bunch stability theoretically, with realistic beam and linac models. 1 EQUATIONS OF MOTION The equations of motion for the longitudinal and transverse (vertical) plane in a linac with longitudinal and transverse wakefields is given in the form of two semicoupled partial and linear integro-differential equations [4] as: ∂γ(s, z) ∂s = eU m0c cos (kRF z − Φ̄RF ) − (1) −C ∫ z 0 ρ(z∗)[WL0 + WL1 − WL0 lB (z − z∗)]dz∗ ∂x(s, z) ∂s2 − K(s)[1 + ∆k(z)]x = WT0C γ0 ∫ z 0 ρ(z∗)(z − z∗)x(s, z∗)dz∗ + +K(s)[1 + ∆k(z)]xQ(s) (2) The initial conditions are: x(0, z) = 0 (3) ∂x ∂s (s = 0) = 0 (4) The independent variables s and z represent the distance along the linac and the coordinate inside the bunch. z is zero at the head and equal to the bunch length lB at the tail of the truncated bunch. The unknowns γ(s, z) and x are the energy Lorentz factor as well as the vertical transverse displacement along the bunch at a given linac position. A piecewise constant energy of the bunch along the different linac sectors is assumed so that no acceleration term proportional to ∂x ∂s appears in the equation of motion. In addition, a linear variation of the wakefield level along the bunch in both planes is assumed. While WL0 and WL1 represent the longitudinal wakefield (WL) at the head and tail of the bunch, WT0 stands for the transverse wakefield (WT ) at the tail. We use a 4-th order Chebyshev expansion of a normalized Gaussian charge distribution in the range of ±2σz given by: ρ(z) ' 75 46lB [ 1 20 ( 4z lB − 2 )4 − 41 100 ( 4z lB − 2 )2 + 1 ] (5) The quantity ∆k(z) in Eq. (2) represents a variation of the focusing force inside the bunch which is caused by the energy dependent focusing (chromatic effect) as well as by the application of RF quadrupoles in order to reduce the emittance blowup caused by the presence of wakefields. The constant C is defined as C = 4π 0reN where 0 is the permittivity of free space, N the number of particles in the bunch and re the classical electron radius. The function xQ(s) represents the actual quadrupole misalignments as function of the position s. This function is either random or given by recurrence relations representing a trajectory correction [2]. 2 SOLUTION OF THE LONGITUDINAL EQUATION Eq. (2) can be solved in a straightforward way by simple integration w.r.t the independent variable s. This yields γ(s, z) = γ0 + eUs m0c cos (kRF z − Φ̄RF ) − −4π 0reNsR(z) (6)