Transient particle distribution for linearly coupled motion in an electron storage ring
The stationary distribution function for an electron bunch in a storage ring has been studied in the presence of radiation damping, quantum excitation and linear coupling between the transverse modes of oscillation, x and y, and energy collision, delta. This distribution, /psi//sub s/, has been found to be given in a form exp (/Sigma Sigma/ A/sub ij/ x/sub i/x/sub j/) with x/sub i/ denoting the coordinates (x, p/sub x/, y, p/sub y/, delta, p/sup delta/) and A/sub ij/'s some constants. This distribution function can be obtained by finding the time-independent solution of the Fokker-Planck equation. Under certain conditions, e.g., during injection or the presence of perturbations, the particle distribution function, /psi/, is nonstationary and is different from /psi//sub s/. After injection is terminated or the perturbations are removed, /psi/ leaves its initial form /psi//sub o/, and migrates toward its final form, /psi//sub s/. This transient behavior of /psi/ can be obtained by solving the time-dependent Fokker-Planck equation which satisfies the boundary condition /psi/ = /psi//sub o/ at t = 0. It will be shown in this report that the time-dependent particle distribution function can be expressed in a similar form as /psi//sub s/ except that the A/sub ij/'s are functions of time. A solution for A/sub ij/(t) will be described, and an example will be presented as an illustration. The result could be applied to find the quantum loss rate due to an aperture limit. Smooth approximation will be used in the analysis. 7 refs.
- Research Organization:
- Stanford Linear Accelerator Center, Menlo Park, CA (USA)
- DOE Contract Number:
- AC03-76SF00515
- OSTI ID:
- 6955591
- Report Number(s):
- SLAC-PEP-NOTE-189; PEP-131; ON: DE88014834
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
430200 -- Particle Accelerators-- Beam Dynamics
Field Calculations
& Ion Optics
430400* -- Particle Accelerators-- Storage Rings
BEAM BUNCHING
BEAM DYNAMICS
DIFFERENTIAL EQUATIONS
DISTRIBUTION FUNCTIONS
EQUATIONS
EQUATIONS OF MOTION
FOKKER-PLANCK EQUATION
FUNCTIONS
MATRIX ELEMENTS
PARTIAL DIFFERENTIAL EQUATIONS
STORAGE RINGS
WAVE FUNCTIONS