NUMERICAL CALCULATIONS OF THE ELECTRON COOLING DRAG FORCE IN A MAGNETIC FIELD

Abstract The longitudinal drag force that the electrons in an electron cooler exert on a circulating ion beam has been measured at many electron-cooling installations and also at CRYRING. Although different theoretical models have been used for calculation of this drag force, the discrepancy between theory and experiment have sometimes been quite big due to the theoretical difficulty in treating the interaction between charged particles in a magnetic field. We here present the beginning of an attempt to numerically calculate the energy loss suffered by ions in binary collisions with electrons in the presence of a finite magnetic field. Results for the longitudinal drag force are presented for relative velocities between ions and electrons and magnetic fields that are relevant for electron cooling at CRYRING and similar storage rings. THE PROBLEM The magnitude of the drag force that an ion experiences when it moves through the electron beam of an electron cooler has been calculated, with or without the inclusion of a magnetic field, using a number of different approaches such as binary collisions [1], dielectric theory [2] or molecular-dynamics simulations In our case, the electron is more or less strongly bound to a field line and it can approach the ion several times during successive cyclotron orbits. For large impact parameters this is conventionally regarded as the ions colliding with an electron "disk", whose charge is smeared out over the cyclotron orbits and which has a very small transverse motion, leading to an increased drag force. Indeed, a drag force larger than what has been calculated from theories that do not take the magnetic field into account has been observed at several cooler installations. Another feature of magnetized collisions, in contrast to classical Rutherford scattering, is that negative and positive ions have different drag forces. This effect that also has been observed experimentally Here we present the beginning of an attempt to get accurate numerical values for the electron-cooling drag force in the presence of a finite magnetic field using a binarycollision approach. The aim is to calculate the energy loss for heavy ions passing through an electron gas that has the anisotropic temperature and the magnetization that are characteristic of an electron cooler. Outside the scope of this investigation, however, is the inclusion of plasma effects or electron-electron interactions. THE METHOD The method we use to calculate the energy loss of an ion colliding with an electron is simply to numerically integrate the classical equations of motion of the two particles. We define the magnetic field to be in the positive z direction, and the terms 'longitudinal' and 'transverse' refer to the direction of the field. The initial condition is that the electron is performing gyro motion on a circle around the origin in the plane z = 0 with velocity v e,0 . The ion starts far below that plane and moves toward it with a velocity v i,0 directed along the z axis. It scatters against the electron and continues until it no longer is influenced by the force of the electron, whereupon the energy loss suffered by the ion is computed. The energy loss thus gives the longitudinal drag force for an ion that has an initial velocity that is purely longitudinal. This energy loss or drag force is then integrated over all impact parameters of the ion and over all phases of the electron gyro motion (or, equivalently, over an impact-parameter plane assuming that the electron always starts in the same (x, y) coordinate). The drag force is finally normalized to an electron density of 1 × 10 14 m −3 . The force acting between the ion and the electron is a screened Coulomb force, and the Debye length is used as the screening length. Since the Debye length depends on the electron density n e and the electron temperature T , there are two additional choices to be made here. We have used an electron density of 1 × 10 13 m −3 , which is typical for CRYRING, and as the electron temperature we have taken kT e = m e v 2 e,0 /2. However, for a few different combinations of v e,0 and v i,0 other values of λ D were also used in order to test how sensitive the results are to the choice of λ D , see below. The calculations were performed for a magnetic field of 0.1 T, but the results can easily be transformed to other fields. It is readily seen that the equations of motion ar

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