Accelerator Physics/Physics of linear accelerators (Focusing on longitudinal dynamics)/Important concepts and definitions/Transit Time Factor

Transit time factor, , characterizes the energy gain of a particle passing through an acceleration gap. The energy change of the particle is given by where is the charge of the particle, the maximum voltage difference between the gap, the initial phase, defined as the phase of the oscillating field at , compared to the crest. The expression of is given by Derivation Consider a relativistic particle passing through an acceleration gap, and ignore the velocity change of the particle during this acceleration, such that the time is related with its longitudinal position by . The energy change is given by the integration: where the second integration vanishes because the odd-function of . Property as a function of is shown in the figure to the right. To maximize the acceleration efficiency the length of the gap needs to be chose wisely to let be as close to unity as possible. This can be done by choosing to be , for example. thumb|427x427px However, if is too small, sparks will appear in the acceleration device as the gradient increases. There is little to be gained by reducing it to less than, say . More complicated but more realistic model The equations above assumed a uniform and constant electric field in the derivation. Realistically, the field is a function of , where is the radius of the particle trajectory w.r.t. the center of acceleration gap, the longitudinal position, and the time. The modified transit time factor is then where is the zeroth order modified Bessel function, the zeroth order Bessel function, , and the drift-tube bore radius. Derivation of the above model Let's consider the general expression for The integration in the simple model becomes Define the axial RF voltage as , and the transit time factor as , the expression of is then the same as the first equation of this first section's first equation. Now since the integration is only effective in the field region between to , the limits can actually be expanded to infinity, such that where is the wave number. Noticing that the integral has a form of the Fourier cosine integral, the Fourier transform can be calculated by expanding this expression to off-axis regions, then This expression has to satisfy the wave equation, which is given by in cylindrical coordinates. Noticing the azimuthal symmetry of the system, and applying expression of , the wave equation becomes where