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This theoretical physics paper presents an improved mathematical framework for describing particle beam behavior in accelerators by incorporating collision effects through the Fokker-Planck equation, which the standard Vlasov equation neglects. The authors derive new equations for beam envelopes and emittances that account for intra-beam scattering, distinguishing between external friction forces and momentum-conserving internal collisions within the beam. The work provides simplified formulas for temperature equilibration in particle beams using an affine mean-flow approximation and clarifies the treatment of longitudinal diffusion.
Why it matters
This improved theoretical description could lead to more accurate predictions of beam behavior in particle accelerators, which is essential for optimizing accelerator performance in applications ranging from high-energy physics experiments to medical radiation therapy and industrial materials processing.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: Beam dynamics calculations based on the collisionless Vlasov equation omit collisional correlations such as intra-beam scattering. For a short-correlated process dominated by small momentum changes, a Fokker-Planck equation can describe their effect on the one-particle distribution. From this equation we derive equations of motion for the beam envelopes and for the rms-emittances. Prescribed friction against an external bath is distinguished from momentum- and energy-conserving self-collisions. For the latter, an affine mean-flow approximation and a scalar relaxation coefficient give simple equations for temperature balancing within the beam. The longitudinal diffusion and temperature are expressed in terms of the physical random momentum rather than the signed effective longitudinal mass. The resulting growth-rate formulas apply to this reduced relaxation model. The original numerical examples are retained for comparison, with the limitations of the storage-ring and numerical-noise applications made explicit. Revised version of: Particle Accelerators~{bf 45}, 229 (1994)
Source: Improved envelope and emittance description of particle beams using the Fokker-Planck approach