Angular distribution of Bremsstrahlung photons and of positrons for calculations of terrestrial gamma-ray flashes and positron beams

Highlights

• Analytical results on doubly differential cross-sections for typical TGF parameters

• Distribution of emission angles and energies for Bremsstrahlung photons

• Distribution of emission angles and energies for positrons in pair production

• C++ code with the analytical cross section results is provided.

Abstract

Within thunderstorms electrons can gain energies of up to hundred(s) of MeV. These electrons can create X-rays and gamma-rays as Bremsstrahlung when they collide with air molecules. Here we calculate the distribution of angles between incident electrons and emitted photons as a function of electron and photon energy. We derive these doubly differential cross-sections by integrating analytically over the triply differential cross-sections derived by Bethe and Heitler; this is appropriate for light atoms like nitrogen and oxygen (Z = 7, 8) if the energy of incident and emitted electron is larger than 1 keV. We compare our results with the approximations and cross section used by other authors. We also discuss some simplifying limit cases, and we derive some simple approximation for the most probable scattering angle.

We also provide cross sections for the production of electron positron pairs from energetic photons when they interact with air molecules. This process is related to the Bremsstrahlung process by some physical symmetry. Therefore the results above can be transferred to predictions on the angles between incident photon and emitted positron, again as a function of photon and positron energy. We present the distribution of angles and again a simple approximation for the most probable scattering angle.

Our results are given as analytical expressions as well as in the form of a C++ code that can directly be implemented into Monte Carlo codes.

Introduction

Terrestrial gamma ray flashes (TGFs) were first observed above thunderclouds by the Burst and Transient Source Experiment (BATSE) (Fishman et al., 1994). It was soon understood that these energetic photons were generated by the Bremsstrahlung process when energetic electrons collide with air molecules (Fishman et al., 1994, Torii et al., 2004); these electrons were accelerated by some mechanism within the thunderstorm. Since then, measurements of TGF's were extended and largely refined by the Reuven Ramaty Energy Solar Spectroscopic Imager (RHESSI) (Cummer et al., 2005, Smith et al., 2005, Grefenstette et al., 2009, Smith et al., 2010), by the Fermi Gamma-ray Space Telescope (Briggs et al., 2010), by the Astrorivelatore Gamma a Immagini Leggero (AGILE) satellite which recently measured TGFs with quantum energies of up to 100 MeV (Marisaldi et al., 2010, Tavani et al., 2011), and by the Gamma-Ray Observation of Winter Thunderclouds (GROWTH) experiment (Tsuchiya et al., 2011).

Hard radiation was also measured from approaching lightning leaders (Moore et al., 2001, Dwyer et al., 2005a); and there are also a number of laboratory experiments where very energetic photons were generated during the streamer–leader stage of discharges in open air (Stankevich and Kalinin, 1967, Dwyer et al., 2005b, Kostyrya et al., 2006, Dwyer et al., 2008b, Nguyen et al., 2008, Rahman et al., 2008, Rep'ev and Repin, 2008, Nguyen et al., 2010, March and Montanyà, 2010, Shao et al., 2011).

Next to gamma-ray flashes, flashes of energetic electrons have been detected above thunderstorms (Dwyer et al., 2008b); they are distinguished from gamma-ray flashes by their dispersion and their location relative to the cloud — as charged particles in sufficiently thin air follow the geomagnetic field lines. In December 2009 NASA's Fermi satellite detected a substantial amount of positrons within these electron beams (Briggs et al., 2011). It is now generally assumed that these positrons come from electron positron pairs that are generated when gamma-rays collide with air molecules.

Two different mechanisms for creating large amounts of energetic electrons in thunderclouds are presently discussed in the literature. The older suggestion is a relativistic run-away process in a rather homogeneous electric field inside the cloud (Wilson, 1925, Gurevich, 1961, Gurevich et al., 1992, Gurevich and Zybin, 2001, Dwyer, 2003, Dwyer, 2007, Milikh and Roussel-Dupré, 2010).

More recently, research focuses on electron acceleration in the streamer–leader process with its strong local field enhancement (Moss et al., 2006, Li et al., 2007, Chanrion and Neubert, 2008, Li et al., 2009, Carlson et al., 2010, Celestin and Pasko, 2011, Li et al., 2010).

Whatever the mechanism of electron acceleration in thunderstorms is, ultimately one needs to calculate the energy spectrum and angular distribution of the emitted Bremsstrahlung photons. As the electrons at the source form a rather directed beam pointing against the direction of the local field, the electron energy distribution together with the angles and energies of the emitted photons determine the photon energy spectrum measured by some remote detector. The energy resolved photon scattering angles are determined by so-called doubly differential cross-sections that resolve simultaneously energy ħω and scattering angle Θ_i of the photons for given energy E_i of the incident electrons. The data is required for scattering on the light elements nitrogen and oxygen with atomic numbers Z = 7 and Z = 8, while much research in the past has focused on metals with large atomic numbers Z. The energy range up to 1 GeV is relevant for TGF's; we here will provide data valid for energies above 1 keV.

As illustrated by Fig. 1, the full scattering problem is characterized by three angles. The two additional angles Θ_f and Φ determine the direction of the scattered electron relative to the incident electron and the emitted photon. The full angular and energy dependence of this process is determined by so-called triply differential cross-sections. A main result of the present paper is the analytical integration over the angles Θ_f and Φ to determine the doubly differential cross-sections relevant for TGF's.

As the cross-sections for the production of electron positron pairs from photons in the field of some nucleus are related by some physical symmetry to the Bremsstrahlung process, we study these processes as well; we provide doubly differential cross-sections for scattering angle Θ₊ and energy E₊ of the emitted positrons for given incident photon energy ħω and atomic number Z.

With the doubly differential cross sections for Bremsstrahlung and pair production a feedback model can be constructed tracing Bremsstrahlung photons and positrons as a possible explanation of TGFs (Dwyer, 2012).

Our present understanding of Bremsstrahlung and pair production was largely developed in the first half of the 20th century. It was first calculated by Bethe and Heitler (1934). Important older reviews are by Heitler (1944), by Hough (1948), and by Koch and Motz (1959). We also used some recent text books (Greiner and Reinhardt, 1995, Peskin and Schroeder, 1995); together with Heitler (1944) and Hough (1948), they provide a good introduction into the quantum field theoretical description of Bremsstrahlung and pair production. The calculation of these two processes is related through some physical symmetry as will be explained in Section 3.

As drawn in Fig. 1, when an electron scatters at a nucleus, a photon with frequency ω can be emitted. The geometry of this process is described by the three angles Θ_i, Θ_f and Φ. Cross sections can be total or differential. Total cross sections determine whether a collision takes place for given incident electron energy, singly differential cross sections give additional information on the photon energy or on the angle between incident electron and emitted photon, and doubly differential cross sections contain both. Triply differential cross sections additionally contain the angle at which the electron is scattered. As two angles are required to characterize the direction of the scattered electron, one could argue that this cross section should actually be called quadruply differential, but the standard terminology for the process is triply differential.

Koch and Motz (1959) review many different expressions for different limiting cases, but without derivations. Moreover, some experimental results are discussed and compared with the presented theory. Bethe and Heitler (1934), Heitler (1944), Hough (1948), Koch and Motz (1959), Peskin and Schroeder (1995), Greiner and Reinhardt (1995) use the Born approximation to derive and describe Bremsstrahlung and pair production cross sections.

Several years later new ansatzes were made to describe Bremsstrahlung. Elwert and Haug (1969) use approximate Sommerfeld–Maue eigenfunctions to derive cross sections for Bremsstrahlung under the assumption of a pure Coulomb field. They derive a triply differential cross section and beyond that also numerically a doubly differential cross section. Furthermore they compare with results obtained by using the Born approximation. They show that there is a small discrepancy for high atomic numbers between the Bethe–Heitler theory and experimental data, and they provide a correcting factor to fit the Bethe–Heitler approximation better to experimental data for large Z. However, they only investigate properties of Bremsstrahlung for Z = 13 (aluminum) and Z = 79 (gold).

Tseng and Pratt (1971) and Fink and Pratt (1973) use exact numerical calculations using Coulomb screened potentials and Furry–Sommerfeld–Maue wave functions, respectively. They investigate Bremsstrahlung and pair production for Z = 13 and for Z = 79 and show that their results with more accurate wave functions do not fit with the Bethe–Heitler cross section exactly. This is not surprising as the Bethe–Heitler approximation is developed for low atomic numbers Z and for Z dependent electron energies as discussed in Section 2.2.

Shaffer et al. (1996) review the Bethe–Heitler and the Elwert–Haug theory. They discuss that the Bethe–Heitler approach is good for small atomic numbers and give a limit of Z > 29 for experiments to deviate from theory. For Z < 29 the theory of Bethe and Heitler, however, is stated to be in good agreement with experiments for energies above the keV range. They calculate triply differential cross sections using partial-wave and multipole expansions in a screened potential numerically for Z = 47 (silver) and Z = 79 and compare their results with experimental data. Actually their results are close to the Elwert–Haug theory which fits the experimental data better than their theory.

Shaffer and Pratt (1997) also discuss the theory of Elwert and Haug (1969) and compare it with the Bethe–Heitler theory and, additionally, with the Bethe–Heitler results multiplied with the Elwert factor and with the exact partial wave method. They show that all theories agree within a factor 10 in the keV energy range, and that the Elwert–Haug theory fits the exact partial wave method best. However, they only investigate Bremsstrahlung for atomic nuclei with Z = 47, 53 (iodine), 60 (neodymium), 68 (erbium) and 79, but not for small atomic numbers Z = 7 and 8 as relevant in air. In summary, Elwert and Haug (1969), Tseng and Pratt (1971), Fink and Pratt (1973), Shaffer et al. (1996) and Shaffer and Pratt (1997) calculate cross sections for Bremsstrahlung and pair production for atomic numbers Z = 13 and Z > 47 numerically, but not analytically, and they do not provide any formula or data which can be used to simulate discharges in air.

The EEDL database consists mainly of experimental data which have been adjusted to nuclear model calculations. For the low energy range Geant4 takes over this data and gives a fit formula. The singly differential cross section related to ω which is used in the Geant4 toolkit is valid in an energy range from 1 keV to 10 GeV and taken from Seltzer and Berger (1985). The singly differential cross section related to Θ_i is based on the doubly differential cross section by (Tsai, 1974, Tsai, 1977) and valid for very high energies, i.e., well above (1–10) MeV. But in the preimplemented cross sections of Geant4 the dependence on the photon energy is neglected in this case so that it is actually a singly differential cross section describing Θ_i.

Table 1 gives an overview of the available literature and data for total or singly, doubly or triply differential Bremsstrahlung cross sections; parameterized angles or photon energies are given, as well as the different energy ranges of the incident electron. Furthermore, the table shows the atomic number Z investigated and includes some further remarks.

For calculating the angularly resolved photon energy spectrum of TGF's, we need a doubly differential cross section resolving both energy and emission angle of the photons; we need it in the energy range between 1 keV and 1 GeV for the small atomic numbers Z = 7 and 8. Therefore most of the literature reviewed here is not applicable. However, the Bethe–Heitler approximation is valid for atomic numbers Z < 29 and for electron energies above 1 keV (Shaffer et al., 1996). How the range of validity depends on the atomic number Z is discussed in Section 2.2. We therefore will use the triply differential cross section derived by Bethe and Heitler (1934) to determine the correct doubly differential cross.

Carlson et al., 2009, Carlson et al., 2010 use Geant 4, a library of software tools with a preimplemented database to simulate the production of Terrestrial Gamma-Ray Flashes. But Geant 4 does not supply an energy resolved angular distribution as it does not contain a doubly differential cross section, parameterizing both energy and emission angle of the Bremsstrahlung photons (see Table 1). Furthermore, it is designed for high electron energies. It also includes the Landau–Pomeranchuk–Migdal (LPM) (Landau and Pomeranchuk, 1953) effect and dielectric suppression (Ter-Mikaelian, 1954) which do not contribute in the keV and MeV range. We will briefly discuss the cross sections and effects implemented in Geant 4 in Appendix D.

Lehtinen has suggested a doubly differential cross section in his PhD thesis (Lehtinen, 2000) that is also used by Xu et al. (2012). Lehtinen's ansatz is a heuristic approach based on factorization into two factors. The first factor is the singly differential cross section of Bethe and Heitler (1934) that resolves only electron and photon energies, but no angles. The second factor is due to Jackson (1975, p. 712 et seq.), it depends on the variable (1 − β²) [(1 − βcosΘ_i)² + (cosΘ_i − β)²] / (1 − βcosΘ_i)⁴, where β = |v_i|/c measures the incident electron velocity on the relativistic scale. However, this factor derived in Jackson (1975, p. 712 et seq.) is calculated in the classical and not quantum mechanical case, and it is valid only if the photon energy is much smaller than the total energy of the incident electron. We will compare this ansatz with our results in Appendix E.

Dwyer (2007) chooses to use the triply differential cross section by Bethe and Heitler (1934), but with an additional form factor parameterizing the structure of the nucleus (Koch and Motz, 1959). We will show in Appendix F that this form factor, however, does not contribute for energies above 1 keV. This cross section depends on all three angles as shown in Fig. 1. If one is only interested in the angle Θ_i between incident electron and emitted Bremsstrahlung photon, the angles Θ_f and Φ have to be integrated out — either numerically, or the analytical results derived in the present paper can be used.

In chapter 2 we introduce the triply differential cross section derived by Bethe and Heitler. Then we integrate over the two angles Θ_f and Φ to obtain the doubly differential cross section which gives a correlation between the energy of the emitted photon and its direction relative to the incident electron. Furthermore, we investigate the limit of very small or very large angles and of high photon energies; this also serves as a consistency check for the correct integration of the full expression.

In Section 3, we perform the same calculations for pair production, i.e., when an incident photon interacts with an atomic nucleus and creates a positron electron pair. As we explain, this process is actually related by some physical symmetry to Bremsstrahlung, therefore results can be transferred from Bremsstrahlung to pair production. We get a doubly differential cross section for energy and emission angle of the created positron relative to direction and energy of the incident photon.

The physical interpretation and implications of our analytical results is discussed in Section 4. Energies and emission angles of the created photons and positrons are described in the particular case of scattering on nitrogen nuclei. For electron energies below 100 keV, the emission of Bremsstrahlung photons in different directions varies typically by not more than an order of magnitude, while for higher electron energies the photons are mainly emitted in forward direction. For this case, we derive an analytical approximation for the most likely emission angle of Bremsstrahlung photons and positrons for given particle energies.

In Section 5, we will briefly summarize the results of our calculations.

Details of our calculations can be found in Appendix A The residual theorem to calculate integrals with trigonometric functions, Appendix B The doubly differential cross section for, Appendix C The doubly differential cross section for, Appendix D Discussion of Geant 4, Appendix E Comparison with, Appendix F Contribution of the atomic form factor, Appendix G Contribution of the integrals, Appendix H Conservation of energy and momentum, Appendix I Approximation for Θ. Beyond that we provide a C++ code. The C++ code can be downloaded directly from the website of the journal.