Stationary solution of the Fokker-Planck equation for linearly coupled motion in an electron storage ring
Effects upon longitudinal bunch shape in a storage ring due to linear and nonlinear potential can be calculated by finding the stationary solution to the Fokker-Planck equation for the particle distribution. Effects upon transverse bunch shape of a stored electron beam due to photon emissions and damping can be calculated by this method. It has been found that this method can also be used for a case in which the transverse modes of oscillation are coupled to the energy deviation delta. Examples of lattice elements which produce linear coupling between these oscillations are skew quadrupole magnets and solenoid magnets. For the linearly coupled case the stationary solution has been found to be given by exp (..sigma sigma..A/sub ij/ x/sub i/x/sub j/) with x/sub i/ the canonical variables (x,p/sub x/, y, p/sub y/, delta, p/sub delta/) and A /sub ij/ some constants. The solution for the values of A /sub ij/'s will be described in this report. It will be shown that this solution can be expressed in a compact form. For simple cases, this form of solution leads directly to analytic expressions for the values of A /sub ij/'s and the bunch shape can be calculated by integrating the distribution function over some of the coordinates; for the more complex cases, it can be conveniently adapted as an algorithm for numerical evaluation. 16 refs.
- Research Organization:
- Stanford Linear Accelerator Center, Menlo Park, CA (USA)
- DOE Contract Number:
- AC03-76SF00515
- OSTI ID:
- 6925388
- Report Number(s):
- SLAC-PEP-NOTE-188; PEP-130; ON: DE88014676
- 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
ANALYTICAL SOLUTION
BEAM BUNCHING
BEAM DYNAMICS
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
DISTRIBUTION FUNCTIONS
EQUATIONS
EQUATIONS OF MOTION
FOKKER-PLANCK EQUATION
FUNCTIONS
PARTIAL DIFFERENTIAL EQUATIONS
PEP STORAGE RINGS
STORAGE RINGS