Abstract
Small-angle multiple intrabeam scattering (IBS) emittance growth rates are normally expressed through integrals, which require a numeric evaluation at various locations of the accelerator lattice. In this paper, I demonstrate that the IBS growth rates can be presented in closed-form expressions with the help of the so-called symmetric elliptic integral. This integral can be evaluated numerically by a very efficient recursive method by employing the duplication theorem. Several examples of IBS rates for a smooth-lattice approximation, equal transverse temperatures and plasma temperature relaxation are given.
Highlights
This paper presents the results, previously obtained by Bjorken and Mtingwa [1], as closed-form analytic expressions
The intrabeam scattering (IBS) formulas, I am proposing in this paper, require evaluating the symmetric elliptic integral, with its variables cycled, 3 times at each point of the accelerator lattice
Some of the IBS rates for special cases are expressed by the following combination of elliptic integrals: x; y; z 2xRD y; z; x yRD z; x; y zRD x; y; z: (8)
Summary
This paper presents the results, previously obtained by Bjorken and Mtingwa [1], as closed-form analytic expressions. There have been attempts in the past [1,2] to express the intrabeam scattering (IBS) rates through Legendre’s incomplete elliptic integrals. Rational operations and square roots are required. Such a numerical method gives, in my opinion, the main advantage for expressing the IBS rates through this integral. The IBS formulas, I am proposing in this paper, require evaluating the symmetric elliptic integral, with its variables cycled, 3 times at each point of the accelerator lattice. Some of the IBS rates for special cases are expressed by the following combination of elliptic integrals: x; y; z 2xRD y; z; x yRD z; x; y zRD x; y; z:. Eq (3.6) in B-M [1] paper can be written as follows:
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