Comparison of analytical and Monte Carlo calculations of multi-photon effects in bremsstrahlung emission by high-energy electrons A. Mangiarotti a,b , P. Sona c,d,⇑ , S. Ballestrero e,f , U.I. Uggerhøj g , K.K. Andersen g a Laboratório de Instrumentação e Física Experimental de Partículas, Coimbra, Portugal b Departamento de Física, Faculdade de Ciências e Tecnologia da Universidade de Coimbra, Coimbra, Portugal c Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, Polo Scientifico, Via G. Sansone 1, 50019 Sesto Fiorentino, Italy d INFN, Sezione di Firenze, Italy e Department of Physics University of Johannesburg, Johannesburg, South Africa f CERN PH/ADT, Geneve, Switzerland g Department of Physics and Astronomy, University of Aarhus, Aarhus, Denmark article info Article history: Received 30 April 2012 Received in revised form 5 July 2012 Available online 25 July 2012 Keywords: Multi-photon emission Discrete-bremsstrahlung LPM effect Pair production Monte Carlo abstract Approximate analytical calculations of multi-photon effects in the spectrum of total radiated energy by high-energy electrons crossing thin targets are compared to the results of Monte Carlo type simulations. The limits of validity of the analytical expressions found in the literature are established. The separate contributions to spectral distortion of electromagnetic processes other than bremsstrahlung are also studied in detail. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction The theory of bremsstrahlung emission by high-energy elec- trons or positrons penetrating matter has been developed to an ever increasing degree of accuracy starting with the seminal paper by Bethe and Heitler [1]. The main goal of the theory is, of course, to provide as accurate expressions as possible of the differential cross section for radiating a single photon per unit energy interval [2–4]. These expressions will then undergo a validation process when the appropriate experiments are performed. However, the experiment cannot approach too closely the ideal conditions of an infinitely thin target, owing to the unavoidable background gen- erated by several materials present along the beam line (windows for the vacuum tubes, section of the beam line in plain air, etc.). Truly enough, one can perform a no-target run and subtract the re- sult from the main data, but even this remedy has severe limita- tions in the presence of sizable background since the simple subtraction does not restore the correct original shape of the target related spectrum [5]. In other words, it is in practice always neces- sary to use targets where the probability of emission of more than one photon by a single impinging electron is non negligible. As a consequence, one usually measures the spectrum of the total radi- ated energy (in the following indicated by K) which, generally speaking, differs from that predicted by the ‘‘single-photon’’ theory (as given by the above mentioned cross section) in that the high- energy section of the spectrum is enriched at the expense of the low-energy one. The comparison of the experimental data with the ‘‘single-photon’’ theory needs therefore an intermediate step implying either an analytical approach to provide a suitable for- mula which takes into account the multi-photon emission in some approximation or the powerful Monte Carlo (MC) approach with its capability of including not only the multi-photon emission but also many different mechanisms of spectral distortion (e.g. the pair cre- ation by radiated photons). Surprisingly, only two groups have, to our knowledge, tackled the problem via an analytical approach [6– 8]. 1 Baier and Katkov [6], starting from the Landau [9] solution of the kinetic equation concerning the energy loss distribution of charged 0168-583X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2012.07.018 ⇑ Corresponding author at: Dipartimento di Fisica e Astronomia, Polo Scientifico, Via G. Sansone 1, 50019 Sesto Fiorentino, FI, Italy. Tel.: +39 055 4572271; fax: +39 055 4572351. E-mail address: pietro.sona@fi.infn.it (P. Sona). 1 Actually, a closely related problem, namely the calculation of the electron straggling function in targets of thickness ranging from a few percent of radiation length to several radiation lengths, has been amply considered both via analytical and numerical studies in the older literature (see Ref. [2] for a review). These investiga- tions were mainly focused on the hard section of the radiated spectra having as a main goal the intense photoproduction of particles and as such are not explicitly considered in this work which is aiming at an accurate description of the whole bremsstrahlung spectrum. Nuclear Instruments and Methods in Physics Research B 289 (2012) 5–17 Contents lists available at SciVerse ScienceDirect Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb