Dierences in the energy loss of protons and positive muons in solids J.E. Vald es a , P. Vargas a , N.R. Arista b, * a Departamento de F  õsica, Universidad de Santiago de Chile, Av. Ecuador 3493, Santiago-2, Chile b Divisi on Colisiones At omicas, Instituto Balseiro and Centro At omico Bariloche, RA-8400, S. C. de Bariloche, Argentina Received 3 May 2000; received in revised form 13 September 2000 Abstract We evaluate the magnitude of the dierence in the electronic energy loss of positive muons and protons in solids in the low-energy range using both the dielectric formalism and the non-linear transport cross-section (TCS) method. The eects of mass increase at low energies, showing stopping dierences between 5% and 1% for velocities in the range 0:2 < v < 1 a.u. We ®nd that nuclear stopping eects must also be taken into account for a consistent analysis in the low-energy range. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 34.50.Bw; 34.50.Fa Keywords: Muon energy loss; Stopping power; Isotopic eects 1. Introduction One of the most usual assumptions in the treatment of the energy loss of protons or heavier ions in matter is that the ratio of the electron to the ion mass may be neglected for practical purposes when calculating the electronic stopping power at intermediate or large velocities, relative to the Bohr velocity v 0 . Theoretical calculations based on this assumption predict that the energy loss of dierent ionized particles depends on the particle velocity but not on its mass. Therefore, it is ex- pected that the energy loss of protons, deuterons or positive muons in the same material and with the same speed would be equal. Several measurements with protons and deute- rons show no evidence of isotopic eects [1,2], and also measurements in the low energy range (of a few keV/u) for protons and deuterons showed no dierences in the electronic energy loss within the limits of experimental precision [3±6]. Recent theoretical estimations of the expected dierences were made both for high energies [7] and for intermediate and low energies [8]. These estimations show that the magnitudes of the dif- ferences are very small, but they increase at low energies. In the case of positive muons, it was Nuclear Instruments and Methods in Physics Research B 174 (2001) 9±15 www.elsevier.nl/locate/nimb * Corresponding author. Tel.: +54-2944-445100; fax: +54- 2944-445299. E-mail address: arista@cab.cnea.gov.ar (N.R. Arista). 0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 0 ) 0 0 5 1 0 - 3