Starting from a single resonant rf cavity, disk-loaded travelling(forward or backward) wave accelerating structures'properties, such as the coupling coefficient K in the dispersion relation, group velocity vg, shunt impedance R, wake potential W (longitudinal and transverse), the coupling coefficient β of the coupler cavity and the coupler cavity axis shift δr which is introduced to compensate the asymmetry caused by the coupling apeerture, can be determined by rather simple analytical formulae

By applying the perturbation method, we show in a general way how to determine the coupling coefficient of a waveguide cavity coupling system and as a special example we derive the analytical expressions ofthe coupling coefficient for a waveguide cavity coupling system. The term "cavity" in this paper stands for the cavity in both standing and travelling wave structures.

In this paper, we investigate the dynamical behaviours of a continuous intense charged particle beam in a transport system. It is assumed that fermion particles, such as electron and proton, in the beam follow Fermi-Dirac statistics in the equilibrium state. The halo particles executing stochastic motions due to the envelope oscillation induced nonlinear resonances are investigated. Analytical formulae for the halo current and halo current loss rate on the beam pipe wall are established.

Physically speaking, the delta function like beam-beam nonlinear forces at interaction points act as a sum of delta function nonlinear multipoles. By applying the general theory established by J. Gao (Nucl. Instr. and Meth. A 451 (2000) 545), in this paper we examine analytically the beam-beam interaction limited dynamic apertures and the corresponding beam lifetimes for both the round and the flat beams. Relations between the beam–beam limited beam lifetimes and the beam-beam tune shifts are established, which show clearly why experimentally one has always a maximum beam–beam tune shift, ζy;max, around 0.045 for e+e- circular colliders, and why one can use round beams to double this value approximately. Comparisons with some machine parameters are given.

By considering delta function sextupole, octupole, and decapole perturbations and using di!erence action-angle variable equations, we "nd some useful analytical formulae for the estimation of the dynamic apertures of circular accelerators due to single sextupole, single octupole, single decapole (single 2m pole in general); and their combined e!ects are derived based on the Chirikov criterion of the onset of stochastic motions. Comparisons with numerical simulations are made, and the agreement is quite satisfactory.

nalytical formulae for the coupling coefficient β of a waveguide-traveling wave structure coupling system have been established and compared with experimental results. The dimension of the coupling aperture on the coupler cavity of a traveling wave structure can be determined easily. The relation between the coupling coefficient β and the other parameters. such as aperture dimension, wavelength, group velocity, wall thickness and coupler dimension, is explicitly revealed. These formulae include that for the caes of a waveguide-single cavity coupling system which has been established in ref. [1]

In this paper we consider the wakefields induced in periodic disk-loaded cavities by charged particles of velocities less than that of light. In frequency domain the particle velocity-dependent wakefields can be calculated by generalizing the analytical formulae given in J. Gao (Nucl. Instr. and Meth. A 381 (1996) 174; or LAL/RT 95-01, April 1995). The physical picture of this effect can be drawn so that the frequencies of the excited modes in cavities felt by the particles are increased by a factor of 1/b (b is the normalized particle's velocity). Some examples are given to demonstrate the utility of these formulae in obtaining the quantities, such as loss factors, short-range and long-range wakefields as functions of cavity dimension, bunch length and particle velocity.

Based on the perturbation method, the resonant frequency changes due to apertures on a cavity wall have been investigated, and analytical formulas have been derived and compared with numerical and experimental resykts, The dispersion relation of a peeriodic disk-loaded slow wave structure, which relates the group velocity explicitly to the shapes and sizes of coupling apertures, is established and verified. These explicitly experssed analytical dispersion relations established in this paper will not only serve as useful tools in the design of forward(electrically coupled) and backward wave(magnetically coupled) linear accelerators, but also show a clear picture of relevant physical processes.