Polarization correction in the theory of energy losses by charged particles
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Polarization Correction in the Theory of Energy Losses by Charged Particles D. N. Makarov and V. I. Matveev Lomonosov Northern (Arctic) Federal University, Arkhangelsk, 163002 Russia email: [email protected] Received September 27, 2014
Abstract—A method for finding the polarization (Barkas) correction in the theory of energy losses by charged particles in collisions with multielectron atoms is proposed. The Barkas correction is presented in a simple analytical form. We make comparisons with experimental data and show that applying the Barkas correction improves the agreement between theory and experiment. DOI: 10.1134/S1063776115040111
1. INTRODUCTION At present, the ionization energy losses in colli sions of heavy ions with atoms of matter have been adequately studied for the range of ion velocities v va [1], where va ~ 1 is the characteristic velocity of an electron in an atom. The Bethe–Bloch formula with standard corrections to it [1] is commonly used in this range of collision velocities (here and everywhere below, unless stated otherwise, atomic units are used): 2
S = 4π ZN (L 2 a v
Bethe
+ ΔL
Bloch
+ ΔL
Shell
+ ΔL
Barkas
), (1)
where Z and v are the projectile charge and velocity, Na is the number of electrons in the target, the quantity LBethe = ln(2v2/I) was calculated by Bethe [2] in the lowest order of the perturbation theory, I is the mean ionization potential of the target, ΔLBloch = –Reψ(1 + iZ/v) + ψ(1) is the Bloch correction [3], ψ(x) is the logarithmic derivative of the Γ function, ΔLShell is the shell correction (see, e.g., [4]), and ΔLBarkas is the Bar kas correction [5]. It is the investigation of the Barkas correction that this paper is devoted to. The necessity of introducing the Barkas correction in the Bethe– Bloch theory (it is often called the polarization correc tion) appeared as a result of the experimental detec tion [6] of a difference by several percent between the ranges of π+ and π– mesons of the same energy in pho tographic emulsion. The physical essence of the emer gence of this correction lies in the fact that when a charged particle interacts with an atom, the electron shells of the atom are elongated toward the charged particle if the particle charge is positive, but they are elongated in the opposite direction if the particle charge is negative; thus, the electron shells are elon gated differently, depending on the sign of the particle charge. There exists a quantum solution of this prob lem in the second order of the perturbation theory [7]
in the range of its validity, i.e., at Z/v 1, where Z is the ion charge and v is the ion velocity, and this cor rection makes a small contribution to Eq. (1). How ever, if the energy losses of not light but heavy charged particles are considered, then, as it became clear later, this correction can give large values [1, 8, 11] and a nonperturbative consideration becomes necessary. There is no quantummechanical, exact solution of this problem at present either. Indeed, the wave func t
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