Abstract
Electron cooling is a well-established method to improve the phase space quality of ion beams in storage rings. In the common rest frame of the ion and the electron beam, the ion is subjected to a drag force and it experiences a loss or a gain of energy which eventually reduces the energy spread of the ion beam. A calculation of this process is complicated as the electron velocity distribution is anisotropic and the cooling process takes place in a magnetic field which guides the electrons. In this paper the cooling force is calculated in a model of binary collisions (BC) between ions and magnetized electrons, in which the Coulomb interaction is treated up to second order as a perturbation to the helical motion of the electrons. The calculations are done with the help of an improved BC theory which is uniformly valid for any strength of the magnetic field and where the second-order two-body forces are treated in the interaction in Fourier space without specifying the interaction potential. The cooling force is explicitly calculated for a regularized and screened potential which is both of finite range and less singular than the Coulomb interaction at the origin. Closed expressions are derived for monochromatic electron beams, which are folded with the velocity distributions of the electrons and ions. The resulting cooling force is evaluated for anisotropic Maxwell velocity distributions of the electrons and ions.
Highlights
In most experiments with particle beams a high phase space density is desired
The perturbative binary collisions (BC) model as well as the nonperturbative classical trajectory Monte Carlo (CTMC) which are based on the full equations of motion in the presence of a magnetic field exhibit a much more intricate structure, in particular at small vi?, the formation of two maxima of the parallel force Fk at parallel and transverse electron thermal velocities
In this paper we presented and discussed analytic expressions for calculating the cooling force on ions in a model of binary collisions (BC) between ions and magnetized electrons within second-order perturbative treatment
Summary
In most experiments with particle beams a high phase space density is desired. In electron cooling of ion beams [1] this is achieved by mixing the ion beam with a comoving electron beam which has a very small longitudinal momentum spread. [21,22,23,24,25], we here consider the (macroscopic) cooling forces which are obtained by integrating the binary force of an individual electron-ion interaction with respect to the impact parameter and the velocity distribution function of electrons. The resulting expressions involve all cyclotron harmonics of the electrons’ helical motion, and are valid for any interaction potential and any strength of the magnetic field and anisotropy of the velocity distribution of the electron beam. The regularization parameter and the screening length involved in the interaction potential are specified and discussed in Appendices B and C
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