ANALYSIS OF DYNAMICS OF INTENSIVE ELECTRON BEAM IN DISK- LOADED WAVEGUIDE WITH VARIABLE PHASE VELOCITY

The results of electron dynamics numeral simulation in an unhomogeneous disk-loaded waveguide, which is used in the S-band linac, with average power of an accelerated beam of 10 kW, are presented. Taking into account the self-fields of beam radiation, two approaches are considered: the first method is an estimative based on the power diffusion equation; the second one is based on selfconsistent equations of field excitation and particles motion. The self-consistent approach showed the presence of substantial phase slipping of particles in the homogeneous part of the rf structure, conditioned by the reactive beam loading. INTRODUCTION The low-energy rf linac for technology applications for which we study beam dynamics consists of the elements showed in Fig. 1. Figure 1: A linac outline: electron gun (1), magnetic lenses (2), magnetic screen (3), solenoid (4), accelerating section (5), quadrupole lenses (6), beamline (7). The accelerating section is a piece-wise unhomogeneous disk-loaded waveguide, which consists of a buncher conjugated with a homogeneous constantimpedance accelerating part of this section. The buncher includes 15 cells with variable phase velocity, β, as shown in Tab.1. Table 1: Structure Parameters Cell Number Iris Radius, cm Length, cm β 1 – 6 2.165 0 – 10.3 0.6998 7 2.165 10.3 – 12.3 0.7494 8 2.11 12.3 – 14.6 0.8494 9 – 13 2.035 14.6 – 26.6 0.8994 14 2.035 26.6 – 29.1 0.9244 15 1.715 29.1 – 31.7 0.9741 16 – 101 1.5 31.7 – 263.5 1 The regular accelerating part consists of 86 cells with phase velocity equals to the velocity of light, c. Phase advance per cell is π/2 at the operating frequency 2797 MHz. A 80 kV diode electron gun provides beam current up to 1.5 A per pulse. The focusing system includes 2 lenses, a solenoid and one quadrupole duplet. THE POWER DIFFUSION TECHNIQUE The equation of power diffusion often used to calculate fields induced by an ultrarelativistic beam [1,2] has the form 2 r r r dP P IE dz α + = (1) where α is the attenuation constant, Pr is the power induced by a beam, I is the beam current averaged over an RF period, Er is the longitudinal component of the induced electric field. To use the well-known code PARMELA that calculates motion of particles at given fields it is required to make some assumptions, which ensue from the power diffusion approximation: i) the beam is a sequence of point bunches; ii) beam particles move synchronously with the induced wave at the maximum of decelerating field. By using ( ) 2 r r ser P E R z = (where Rser is the serial impedance) the induced field can be found from Eq.(1). The total accelerating field of a fundamental space harmonic is given by ( ) ( ) ( ) ( ) 0 0 0 , cos , tot r E z t E z z t E z φ = − , (2) where Е0(z) is the field from an rf source; t0 is the entry time of a particle into a accelerating section. The phase of a particle with respect to wave phase is defined as 0 0 0 ( , ) ( , ) ( ) z ph z t t z t dz z φ ω ω ν Λ ′ ′ = − ∫ , (3) where νph is the wave phase velocity, 0 ( , ) t z t Λ is the particle Lagrangian time, ω is the angular frequency. To analyze phase motion of a beam, we use the phase of the first Fourier harmonic of current, ( ) z ψ , which is defined from the following expression: