Kinetics of electron cooling of positrons in a storage ring
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, NONLINEAR, AND SOFT MATTER PHYSICS
Kinetics of Electron Cooling of Positrons in a Storage Ring L. I. Men’shikov Russian Research Centre Kurchatov Institute, Moscow, 123182 Russia e-mail: [email protected] Received August 14, 2007
Abstract—Kinetic equations are derived for the positron velocity distribution in storage rings with electron cooling. Both drag force and components of the velocity diffusion tensor are calculated. The mechanism of approach to a steady-state positron velocity distribution via electron cooling is discussed. It is shown that the resulting steady-state positron distribution is very close to the electron distribution when the magnetic field is sufficiently strong. PACS numbers: 29.27.Bd DOI: 10.1134/S1063776108060149
1. INTRODUCTION Electron cooling [1] has been successfully used to reduce the phase-space volume of heavy-particle beams with particle mass M m [2, 3], where m is the electron mass (in what follows, the cooled particles are occasionally referred to as M). Electron cooling of positrons (M = m) is a new problem that has arisen in studies of production and properties of antihydrogen and positronium atoms (see [4] for review and [5, 6] for details). To date, cooling theory is well developed for heavy-particle beams, where the contributions of collective effects and binary collisions to the drag force are comparable [3, 4]. However, collective effects play a dominant role in positron cooling, which complicates analysis of cooling kinetics. This may explain why the theory of electron cooling of positrons is not yet fully developed. To this day, only a few papers have been published on the subject [4–9]. In view of the importance of electron cooling of positrons [4–6], the theory is revisited here in order to derive rigorous kinetic equations for positrons and discuss their main implications. Another objective of this paper is to provide a collection of formulas (including concise derivations) that are sufficient for practical calculations of the cooling kinetics of positrons. These formulas are required both for designing positron storage rings and for planning experiments on positrons and positronium atoms. Finally, the third objective is to present some new results in anisotropic plasma physics.
cooling section of length LC ~ 3 m. The electron and positron energies in the laboratory frame are approximately 10 keV, and the electron density n varies between 108 and 109 cm–3. The longitudinal and transverse electron temperatures are T|| ~ 1–10 K and T⊥ ~ 1000 K, respectively. Thus, the electron velocity distribution is anisotropic [10]: dn = f ( v )d v , 3
f ( v ) = G e ( v ⊥ )g ( v || ),
⎛ v 2⊥ ⎞ 1 G e ( v ⊥ ) = ------------2- exp ⎜ – --------⎟ , 2 2π∆ ⊥ ⎝ 2∆ ⊥⎠
(1)
⎛ v 2|| ⎞ 1 g ( v || ) = ---------------- exp ⎜ – ---------2⎟ , 2π∆ || ⎝ 2∆ || ⎠ where ∆⊥ =
T ⊥ /m and ∆|| = T⊥/T|| 1.
T || /m , with (2)
Anisotropy can be combined with magnetization characterized by the condition r H < R || ,
(3)
to produce ultra-cold ion beams with temperatures T ≈ T|| [2, 3, 11]. (Here
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