Eur. Phys. J. D 23, 201–209 (2003) DOI: 10.1140/epjd/e2003-00032-x T HE EUROPEAN P HYSICAL JOURNAL D Barkas effect, shell correction, screening and correlation in collisional energy-loss straggling of an ion beam P. Sigmund 1 and A. Schinner 2 1 Physics Department, University of Southern Denmark, 5230 Odense M, Denmark 2 Institut f¨ ur Experimentalphysik, Johannes-Kepler-Universit¨at, 4040 Linz-Auhof, Austria Received 24 September 2002 / Received in final form 1st December 2002 Published online 4 February 2003 –c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2003 Abstract. Collisional electronic energy-loss straggling has been treated theoretically on the basis of the binary theory of electronic stopping. In view of the absence of a Bloch correction in straggling the range of validity of the theory includes both the classical and the Born regime. The theory incorporates Barkas effect and projectile screening. Shell correction and electron bunching are added on. In the absence of shell corrections the Barkas effect has a dominating influence on straggling, but much of this is wiped out when the shell correction is included. Weak projectile screening tends to noticeably reduce collisional straggling. Sizable bunching effects are found in particular for heavy ions. Comparisons are made with selected results of the experimental and theoretical literature. PACS. 34.50.Bw Energy loss and stopping power – 52.40.Mj Particle beam interactions in plasmas 1 Introduction In 1915, Bohr [1] predicted that the fluctuation of the energy loss ΔE of a beam of charged particles penetrating matter is characterized by the variance  (ΔE -〈ΔE〉) 2  = NRW (1) with the straggling parameter W given by W = W B =4πZ 2 1 Z 2 e 4 , (2) where 〈ΔE〉 is the mean energy loss over a travelled path- length R, N the number of atoms per volume in the stop- ping material, and Z 1 ,Z 2 the atomic numbers of beam and material atoms, respectively. Equation (1) assumes the stopping medium to be ran- dom and the pathlength R sufficiently small so that the variation with beam energy of the cross-sections responsi- ble for energy loss can be ignored [2,3]. Equation (2) assumes free-Coulomb scattering be- tween beam particles and the electrons of the stopping medium. Bohr showed that unlike the mean energy loss, straggling is rather insensitive to the binding of target electrons. The underlying reason is the fact that the integral W =  T 2 dσ(T ) (3) is dominated by large values of the energy transfer T per collision event even though the differential cross- section dσ(T ) is heavily peaked toward small T . Equation (2), in conjunction with (1), seems to consti- tute one of the most lasting and universally-valid results of the theory of particle penetration. Its range of valid- ity reaches far beyond the initial application area, i.e., α- particles from radioactive sources. Nevertheless there are limitations which, according to common knowledge, may roughly be classified into three groups, 1. at low projectile speed, comparable to electron veloc- ities in the target, W tends to slightly increase above the Bohr value with decreasing velocity before drop- ping toward zero [4], 2. at high speed, relativistic corrections beyond those mentioned already by Bohr [1] become necessary [5] and 3. charge exchange may not be neglected in general as a source of energy-loss straggling [6]. This paper addresses the first group of phenomena which is well separated from the second group because of non-overlapping velocity regimes. Phenomena of the third group need to be considered in comparisons with experiment, but their theoretical study involves different physics [7–11] and will be left out here. Physical effects potentially causing deviations from free-Coulomb scattering at low projectile speed are primarily those that also affect the stopping force, – binding of target electrons, – orbital motion of target electrons, – Barkas effect, – projectile screening by accompanying electrons, – interplay between Born approximation and classical- orbit description of ion-electron scattering.