Ramped Bunch Experiment: Theory

Beam Ramping Mechanism

The proposed mechanism for generating a ramped current profile requires an initial beam with an energy chirp in the longitudinal (z) direction, which can be accomplished by injecting the beam into the accelerator cavity with a negative phase offset from the peak of the accelerating field. Since the R56 represents the longitudinal translation per unit of momentum error, a negative R56 acting upon a beam with this sort of chirp will cause particles at the head of the bunch (with higher momentum) to translate backwards, while particles in the tail of the bunch will be translated forward. For the S-Bahn, with an initial chirp of 22 degrees back of crest, the action of the negative compression produces a ramped profile as shown in the following figure.

Simulation Results

The transformation in the above figure, however, is merely a cartoon. Nonlinear optical effects have been neglected. When these effects are included, simulation results using ELEGANT predict that the final phase space is distorted and the ramped profile is lost. The primary cause of the distortion is the 2nd order term T566, which can be eliminated by using sextupoles for chromatic correction. The final phase space without and with the sextupole correction are shown below.

At present, ELEGANT does not include space-charge effects. In order to simulate both space-charge and sextupole corrections, we modified the code for UCLA-PARMELA (which can do either mesh-based or point-to-point space charge calculations) to include sextupole magnets. The results (below) show a broadening of momentum spread and a 4% enhancement in the compression.

Transport Notation

In the absence of interparticle forces, the dynamics of a particle beam in a lattice of magnetic elements is described by the single particle equations of motion, the solutions to which may be expanded using a matrix formalism, which is utilized in various particle tracking codes, including ELEGANT and TRANSPORT. It is customary to attach a curvilinear coordinate system to the design trajectory of the beam and to describe the position of a given particle within the beam by the coordinates (x, y, z) relative to this coordinate system. The variables

form a 6-vector in the trace space (or modified phase space) of the beam, where primes denote derivatives with respect to the path length s along the design trajectory and delta is the fractional momentum error. The matrix transport equation relating the 6-vector X at some initial point s=0 to an arbitrary later point s is given (to third order in powers of the coordinates) by

where superscript noughts indicate initial values at s=0. The longitudinal (i=5) element of this matrix equation describes the transformation of the longitudinal phase space of the beam. The dominant higher-order contributions to this transformation (for a chirped beam with a large energy spread) are those multiplying the momentum error (i.e. chromatic terms), of which T566 and U5666 are the dominant contributors. Consequently, the z transformation to second order is approximated by

The coefficients R56, T566, U5666, ... describe the longitudinal (or temporal) dispersion function of the beam to increasing orders of approximation.

Dispersion in a Dogleg to Second Order

The dogleg lattice may be represented symbolically as

where B represents a bend of angle [theta] and radius [rho] and X is a lattice of focusing elements. With proper use of the focusing elements, the horizontal dispersion and its derivative (R16, R26) may be made to vanish for such a structure. The longitudinal dispersion elements to first and second order are then given by

where Q is the set of values of (i,k) for which Ti6k is nonvanishing. The coefficients ai6k are given by

To eliminate the nonlinear chromatic term T566 and linearize the longitudinal transformation, sextupoles may be inserted into the focusing lattice (X). For optimal effect the sextupoles should be placed in a region of large horizontal dispersion. For a pair of symmetrically placed sextupoles of equal and opposite field strength K2, the T566 can be shown to assume the form

where A is the expression on the right hand side of the equation above for T566 without sextupoles, and B is a complicated algebraic function of the bend radius, bend angle, the drift lengths, and the focal lengths of the quadrupole focusing elements contained in X. The nonlinearity T566 thus varies linearly with sextupole strength and may consequently be manipulated or made to vanish.