arX

iv:1

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4879

v4 [

phys

ics.

ao-p

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3

Angular distribution of Bremsstrahlung photons

and of positrons for calculations of

terrestrial gamma-ray flashes and positron beams

Christoph Kohna and Ute Eberta,b

a CWI, P.O.Box 94079, 1090GB Amsterdam, The Netherlands, andb Eindhoven University of Technology, P.O.Box 513, 5600MB Eindhoven, The

Netherlands.

Abstract

Within thunderstorms electrons can gain energies of up to hundred(s) ofMeV. These electrons can create X-rays and gamma-rays as Bremsstrahlungwhen they collide with air molecules. Here we calculate the distributionof angles between incident electrons and emitted photons as a function ofelectron and photon energy. We derive these doubly differential cross-sectionsby integrating analytically over the triply differential cross-sections derivedby Bethe and Heitler; this is appropriate for light atoms like nitrogen andoxygen (Z=7,8) if the energy of incident and emitted electron is larger than1 keV. We compare our results with the approximations and cross sectionused by other authors. We also discuss some simplifying limit cases, and wederive some simple approximation for the most probable scattering angle.

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

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

Keywords: Bremsstrahlung, pair production, analytical integration of

Email address: koehn@cwi.nl, ebert@cwi.nl (Christoph Kohna and Ute Eberta,b)

Preprint submitted to Elsevier October 29, 2018

Bethe-Heitler equation, distribution of scattering angles

1. Introduction

1.1. Flashes of gamma-rays, electrons and positrons above thunderclouds

Terrestrial gamma ray flashes (TGFs) were first observed above thunder-clouds by the Burst and Transient Source Experiment (BATSE) (Fishmanet al., 1994). It was soon understood that these energetic photons weregenerated by the Bremsstrahlung process when energetic electrons collidewith air molecules (Fishman et al., 1994; Torii et al., 2004); these electronswere accelerated by some mechanism within the thunderstorm. Since then,measurements of TGF’s were extended and largely refined by the Reuven Ra-maty Energy Solar Spectroscopic Imager (RHESSI) (Cummer et al., 2005;Smith et al., 2005, Grefenstette et al., 2009, Smith et al., 2010), by theFermi Gamma-ray Space Telescope (Briggs et al., 2010), by the Astroriv-elatore Gamma a Immagini Leggero (AGILE) satellite which recently mea-sured 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 Thun-derclouds (GROWTH) experiment (Tsuchiya et al., 2011).

Hard radiation was also measured from approaching lightning leaders(Moore et al., 2001; Dwyer et al., 2005); and there are also a number oflaboratory experiments where very energetic photons were generated duringthe streamer-leader stage of discharges in open air (Stankevich and Kalinin,1967; Dwyer et al., 2005b; Kostyrya et al., 2006; Dwyer et al., 2008b; Nguyenet al., 2008; Rahmen et al., 2008; Rep’ev and Repin, 2008; Nguyen et al.,2010; March and Montanya, 2010; Shao et al., 2011).

Next to gamma-ray flashes, flashes of energetic electrons have been de-tected above thunderstorms (Dwyer et al., 2008b); they are distinguishedfrom gamma-ray flashes by their dispersion and their location relative tothe cloud - as charged particles in sufficiently thin air follow the geomagneticfield lines. In December 2009 NASA’s Fermi satellite detected a substan-tial amount of positrons within these electron beams (Briggs et al., 2011). Itis now generally assumed that these positrons come from electron positronpairs that are generated when gamma-rays collide with air molecules.

Two different mechanisms for creating large amounts of energetic elec-trons in thunderclouds are presently discussed in the literature. The oldersuggestion is a relativistic run-away process in a rather homogeneous electric

2

field inside the cloud (Wilson, 1925; Gurevich, 1961; Gurevich et al., 1992;Gurevich, 2001 Dwyer, 2003, 2007; Milikh and Roussel-Dupre, 2010).

More recently research focusses on electron acceleration in the streamer-leader process with its strong local field enhancement (Moss et al., 2006; Liet al., 2007; Chanrion and Neubert, 2008; Li et al., 2009; Carlson et al., 2010;Celestin and Pasko, 2011; Li et al., 2012).

1.2. The need for doubly differential cross-sections

Whatever the mechanism of electron acceleration in thunderstorms is, ul-timately one needs to calculate the energy spectrum and angular distributionof the emitted Bremsstrahlung photons. As the electrons at the source form arather directed beam pointing against the direction of the local field, the elec-tron energy distribution together with the angles and energies of the emittedphotons determine the photon energy spectrum measured by some remotedetector. The energy resolved photon scattering angles are determined byso-called doubly differential cross-sections that resolve simultaneously energy~ω and scattering angle Θi of the photons for given energy Ei of the incidentelectrons. The data is required for scattering on the light elements nitrogenand oxygen with atomic numbers Z = 7 and Z = 8, while much research inthe past has focussed on metals with large atomic numbers Z. The energyrange up to 1 GeV is relevant for TGF’s; we here will provide data valid forenergies above 1 keV.

As illustrated by Fig. 1, the full scattering problem is characterized bythree angles. The two additional angles Θf and Φ determine the directionof the scattered electron relative to the incident electron and the emittedphoton. The full angular and energy dependence of this process is determinedby so-called triply differential cross-sections. A main result of the presentpaper is the analytical integration over the angles Θf and Φ to determinethe doubly differential cross-sections relevant for TGF’s.

As the cross-sections for the production of electron positron pairs fromphotons in the field of some nucleus are related by some physical symmetryto the Bremsstrahlung process, we study these processes as well; we providedoubly differential cross-sections for scattering angle Θ+ and energy E+ ofthe emitted positrons for given incident photon energy ~ω and atomic num-ber Z.With the doubly differential cross sections for Bremsstrahlung and pair pro-duction a feedback model can be constructed tracing Bremsstrahlung photonsand positrons as a possible explanation of TGFs (Dwyer, 2012).

3

Figure 1: Parametrization of the Bremsstrahlung process: Momenta of incident electronpi, scattered electron pf and emitted photon ~k form the angles Θi = ∢(pi,k) andΘf = ∢(pf ,k), and Φ is the angle between the planes spanned by the vector pairs (pi,k)and (pf ,k). The scattering nucleus has atomic number Z.

1.3. Available cross-sections for Bremsstrahlung

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

As drawn in Fig. 1, when an electron scatters at a nucleus, a photon withfrequency ω can be emitted. The geometry of this process is described by thethree angles Θi, Θf and Φ. Cross sections can be total or differential. Totalcross sections determine whether a collision takes place for given incidentelectron energy, singly differential cross sections give additional informationon the photon energy or on the angle between incident electron and emittedphoton, and doubly differential cross sections contain both. Triply differentialcross sections additionally contain the angle at which the electron is scattered.As two angles are required to characterize the direction of the scatteredelectron, one could argue that this cross section should actually be calledquadruply differential, but the standard terminology for the process is triply

4

differential.Koch and Motz (1959) review many different expressions for different

limiting cases, but without derivations. Moreover, some experimental resultsare discussed and compared with the presented theory. Bethe and Heitler(1934), Heitler (1944), Hough (1948), Koch and Motz (1959), Peskin andSchroeder (1995), Greiner and Reinhardt (1955) use the Born approximationto 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 eigenfunctionsto derive cross sections for Bremsstrahlung under the assumption of a pureCoulomb field. They derive a triply differential cross section and beyondthat also numerically a doubly differential cross section. Furthermore theycompare with results obtained by using the Born approximation. They showthat there is a small discrepancy for high atomic numbers between the Bethe-Heitler theory and experimental data, and they provide a correcting factorto fit the Bethe-Heitler approximation better to experimental data for largeZ. 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 numericalcalculations using Coulomb screened potentials and Furry-Sommerfeld-Mauewave functions, respectively. They investigate Bremsstrahlung and pair pro-duction for Z = 13 and for Z = 79 and show that their results with moreaccurate wave functions do not fit with the Bethe Heitler cross section ex-actly. This is not surprising as the Bethe-Heitler approximation is developedfor low atomic numbers Z and for Z dependent electron energies as discussedin section 2.2.

Shaffer et al. (1996) review the Bethe Heitler and the Elwert Haug the-ory. They discuss that the Bethe Heitler approach is good for small atomicnumbers 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 goodagreement with experiments for energies above the keV range. They calculatetriply differential cross sections using partial-wave and multipole expansionsin a screened potential numerically for Z = 47 (silver) and Z = 79 and com-pare their results with experimental data. Actually their results are close tothe Elwert Haug theory which fits the experimental data better than theirtheory.

Shaffer and Pratt (1997) also discuss the theory of Elwert and Haug(1969) and compare it with the Bethe Heitler theory and, additionally, with

5

the Bethe Heitler results multiplied with the Elwert factor and with the exactpartial wave method. They show that all theories agree within a factor 10in the keV energy range, and that the Elwert-Haug theory fits the exactpartial wave method best. However, they only investigate Bremsstrahlungfor 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. Insummary, Elwert and Haug (1969), Tseng and Pratt (1971), Fink and Pratt(1973), Shaffer et al. (1996) and Shaffer and Pratt (1997) calculate crosssections for Bremsstrahlung and pair production for atomic numbers Z = 13and Z > 47 numerically, but not analytically, and they do not provide anyformula or data which can be used to simulate discharges in air.

The EEDL database consists mainly of experimental data which havebeen adjusted to nuclear model calculations. For the low energy range Geant4takes over this data and gives a fit formula. The singly differential crosssection related to ω which is used in the Geant4 toolkit is valid in an energyrange from 1 keV to 10 GeV and taken from Seltzer and Berger (1985). Thesingly differential cross section related to Θi is based on the doubly differentialcross 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 Geant4the dependence on the photon energy is neglected in this case so that it isactually a singly differential cross section describing Θi.

Table 1 gives an overview of the available literature and data for totalor singly, doubly or triply differential Bremsstrahlung cross sections; param-eterized angles or photon energies are given, as well as the different energyranges of the incident electron. Furthermore, the table shows the atomicnumber 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 emis-sion angle of the photons; we need it in the energy range between 1 keVand 1 GeV for the small atomic numbers Z = 7 and 8. Therefore most ofthe literature reviewed here is not applicable. However, the Bethe-Heitlerapproximation is valid for atomic numbers Z < 29 and for electron energiesabove 1 keV (Shaffer et al., 1996). How the range of validity depends on theatomic number Z is discussed in section 2.2. We therefore will use the triplydifferential cross section derived by Bethe and Heitler (1934) to determinethe correct doubly differential cross.

6

Data/Paper Information Energy range Atomic Number Z RemarksBethe and ω 1 keV - 1 GeV 7,8 energy range depends on Z

Heitler (1934,1944) ω,Θi,Θf ,ΦTotal different lower depends on the

Koch and ω bounds, no used formulaeMotz (1959) ω,Θi,Φ upper bounds

ω,Θi,Θf ,ΦAiginger (1966) ω,Θi 180, 380 keV 79, Al2O3 experimentalElwert and ω,Θi keV range 13,79Haug (1969) ω,Θi,Θf ,Φ

Penczynski and ω,Θi (300± 10) keV 82 experimentalWehner (1970)Tseng and ω,Θi,Θf ,Φ keV, MeV range 13,79Pratt (1971)Fink and ω keV, MeV range 6,13,79,92 also for pair production

Pratt (1973) ω,Θi,Θf ,ΦTsai (1974,1977) ω,Θi > few 10 MeV all

Seltzer and ω 1 keV - 10 GeV Z=6,13,29,47,74,92Berger (1985)EEDL (1991) Total 5 eV - 1 TeV all see (Cullen et al., 1991)Nackel (1994) ω,Θi keV 6,29,47,79 only twodimensional description

Schaffer ω,Θi,Φ keV range 6,13,29,47,74,92et al. (1996)Schaffer and ω,Θi,Θf ,Φ keV range 47,53,60,68,79Pratt (1997) ω,Θi

Lehtinen (2000) ω,Θi 1 keV - 1 GeV 7,8 Simple product ansatz forangular and frequency part

Total 5 eV - 1 TeV all based on EEDLGeant 4 (2003) ω 1 keV - 10 GeV 6,13,29,47,74,92 based on Seltzer and Berger (1985)

Θi > few 10 MeV all based on Tsai (1974, 1977)

Table 1: Available data for Bremsstrahlung cross sections. Besides the available information on total or singly, doubly or triplydifferential cross-sections, the range of validity of the incident electron energy and of the atomic number is given. If not statedotherwise, these are theoretical expressions.

7

I

1.4. Bremsstrahlung data used by other TGF researchers

Carlson et al. (2009, 2010) use Geant 4, a library of sotware tools with apreimplemented database to simulate the production of Terrestrial Gamma-Ray Flashes. But Geant 4 does not supply an energy resolved angular distri-bution as it does not contain a doubly differential cross section, parameteriz-ing both energy and emission angle of the Bremsstrahlung photons (see Table1). Furthermore, it is designed for high electron energies. It also includes theLandau-Pomeranchuk-Migdal (LPM) (Landau and Pomeranchuk, 1953) ef-fect and dielectric suppression (Ter-Mikaelian, 1954) which do not contributein the keV and MeV range. We will briefly discuss the cross sections andeffects implemented in Geant 4 in Appendix D.

Lehtinen has suggested a doubly differential cross section in his PhDthesis (Lehtlinen, 2000) that is also used by Xu et al. (2012). Lehtinen’sansatz is a heuristic approach based on factorization into two factors. Thefirst factor is the singly differential cross section of Bethe and Heitler (1934)that resolves only electron and photon energies, but no angles. The secondfactor is due to Jackson (1975, p. 712 et seq.), it depends on the variable(1 − β2) [(1 − β cosΘi)

2 + (cosΘi − β)2] / (1 − β cosΘi)4, where β = |vi|/c

measures the incident electron velocity on the relativistic scale. However,this factor derived in Jackson (1975, p. 712 et seq.) is calculated in theclassical and not quantum mechanical case, and it is valid only if the photonenergy is much smaller than the total energy of the incident electron. Wewill compare this ansatz with our results in Appendix E.

Dwyer (2007) chooses to use the triply differential cross section by Betheand Heitler (1934), but with an additional form factor parameterizing thestructure of the nucleus (Koch and Motz, 1959). We will show in Appendix Fthat 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 oneis only interested in the angle Θi between incident electron and emittedBremsstrahlung photon, the angles Θf and Φ have to be integrated out —either numerically, or the analytical results derived in the present paper canbe used.

1.5. Organization of the paper

In chapter 2 we introduce the triply differential cross section derived byBethe and Heitler Then we integrate over the two angles Θf and Φ to ob-

8

tain the doubly differential cross section which gives a correlation betweenthe energy of the emitted photon and its direction relative to the incidentelectron. Furthermore, we investigate the limit of very small or very largeangles and of high photon energies; this also serves as a consistency checkfor the correct integration of the full expression.

In chapter 3 we perform the same calculations for pair production, i.e.,when an incident photon interacts with an atomic nucleus and creates apositron electron pair. As we explain, this process is actually related by somephysical symmetry to Bremsstrahlung, therefore results can be transferredfrom Bremsstrahlung to pair production. We get a doubly differential crosssection for energy and emission angle of the created positron relative todirection and energy of the incident photon.

The physical interpretation and implications of our analytical results isdiscussed in chapter 4. Energies and emission angles of the created photonsand positrons are described in the particular case of scattering on nitrogennuclei. For electron energies below 100 keV, the emission of Bremsstrahlungphotons in different directions varies typically by not more than an order ofmagnitude, while for higher electron energies the photons are mainly emittedin forward direction. For this case, we derive an analytical approximationfor the most likely emission angle of Bremsstrahlung photons and positronsfor given particle energies.

In chapter 5 we will briefly summarize the results of our calculations.Details of our calculations can be found in Appendix A - Appendix I.

Beyond that we provide a C++ code The C++ code can be downloadeddirectly from the website of the journal.

9

2. Bremsstrahlung

2.1. Definition of the process

If an electron with momentum pi approaches the nucleus of an atom,it can change its direction due to Coulomb interaction with the nucleus;the electron acceleration creates a Bremsstrahlung photon with momentumk that can be emitted at an angle Θi relative to the initial direction ofthe electron. The new direction of the electron forms an angle Θf with thedirection of the photon. The angle Φ is the angle between the planes spannedby the vector pairs (pi,k) and (pf ,k). This process is shown in figure 1. Avirtual photon (allowed by Heisenberg’s uncertainty principle) transfers amomentum q between the electron and the nucleus. Therefore both energyand momentum are conserved in the scattering process.

The corresponding triply differential cross section was derived by Betheand Heitler (1934):

d4σ =Z2α3

fine~2

(2π)2|pf ||pi|

dω

ω

dΩidΩfdΦ

|q|4 ×

×[

p2f sin

2Θf

(Ef − c|pf | cosΘf)2(

4E2i − c2q2

)

+p2i sin

2Θi

(Ei − c|pi| cosΘi)2(

4E2f − c2q2

)

+ 2~2ω2p2i sin

2Θi + p2f sin

2Θf

(Ef − c|pf | cosΘf )(Ei − c|pi| cosΘi)

− 2|pi||pf | sinΘi sinΘf cosΦ

(Ef − c|pf | cosΘf)(Ei − c|pi| cosΘi)

(

2E2i + 2E2

f − c2q2)

]

.(1)

Here Z is the atomic number of the nulceus, αfine ≈ 1/137 is the fine struc-ture constant, h ≈ 6.63·10−34 Js is Planck’s constant, ~ = h/2π and c ≈ 3·108m/s is the speed of light. The kinetic energy Ekin,i/f of the electron in theinitial and final state is related to its total energy and momentum as

Ei/f = Ekin,i/f +mec2 =

√

m2ec

4 + p2i/fc

2 (2)

where me ≈ 9.1 · 10−31 kg is the electron mass. The conservation of energyimplies

Ef = Ei − ~ω (3)

10

which determines Ef as a function ofEi and ~ω. The directions of the emittedphoton with energy ~ω and of the scattered electron are parameterized bythe three angles (see Fig. 1)

Θi = ∢(pi,k), (4)

Θf = ∢(pf ,k), (5)

Φ = Angle between the planes (pi,k) and (pf ,k). (6)

The differentials are

dΩi = sinΘi dΘi, (7)

dΩf = sinΘf dΘf . (8)

Furthermore one can get an expression for the absolute value of the virtualphoton q with the help of the momenta, the photon energy ~ω and the angles(4) - (6). Its value is

− q2 = −|pi|2 − |pf |2 −(

~

cω

)2

+ 2|pi|~

cω cosΘi − 2|pf |

~

cω cosΘf

+ 2|pi||pf |(cosΘf cosΘi + sinΘf sinΘi cosΦ). (9)

2.2. Validity of the cross sections of Bethe and Heitler

The cross sections of Bethe and Heitler (1) are valid if the Born approxi-mation (Bethe and Heitler, 1934) holds

v ≫ Zc

137(10)

For nitrogen with Z = 7 and for oxygen with Z = 8, this holds for electronvelocities |vZ=7| ≫ 15 · 106 m/s and |vZ=8| ≫ 18 · 106 m/s; this is equivalentto a kinetic energy of

Ekin =mec

2

√

1− v2

c2

−mec2 ≫

670 eV, Z = 7875 eV, Z = 8

. (11)

This means that incident electron energies above 1 keV can be treated withEq. 1, for lower energies, one cannot calculate with free electron waves any-more, but has to use Coulomb waves (Heitler, 1944; Greiner und Reinhardt,1995); in this case one cannot derive cross sections like (1) analytically any

11

more. Thus the Bethe Heitler cross section and our results must not be usedfor energies of the electron in the initial and final state smaller than 1 keV.However, for higher energies of the electron in the initial and final state,the approximation by Bethe and Heitler becomes more accurate; thus thisapproximation is better if Ekin ≥ 10 keV.

2.3. Integration over Φ

The easiest way is to integrate over the angle Φ between the scatteringplanes first (see Fig. 1) . For this purpose it is useful to redefine somequantities in the following way; therefore (1) can be written much moresimply:

α := 2|pi||pf | sinΘi sinΘf , (12)

β := −p2i − p2

f −(

~

cω

)2

− 2~

cω|pf | cosΘf + 2

~

cω|pi| cosΘi

+ 2|pi||pf | cosΘi cosΘf (13)

A :=Z2α3

fine

(2π)2|pf ||pi|

~2

ω, (14)

a1 :=

( |pf |2c2 sin2Θf

(Ef − |pf |c cosΘf)2+

|pi|2c2 sin2Θi

(Ei − |pi|c cosΘi)2

)

· A, (15)

a2 :=

(

− 2|pi||pf |c2 sin Θi sin Θf

(Ei − |pi|c cosΘi)(Ef − |pf |c cosΘf)

)

· A, (16)

a3 :=

(

4E2i |pf |2 sin2Θf

(Ef − |pf |c cosΘf)2+

4E2f |pi|2 sin2Θi

(Ei − |pi|c cosΘi)2

+2~2ω2(|pi|2 sin Θi + |pf |2 sin2Θf)

(Ei − |pi|c cosΘi)(Ef − |pf |c cosΘf )

)

· A, (17)

a4 :=

(

−|pi||pf | sinΘi sinΘf (4E

2i + 4E2

f)

(Ei − |pi|c cosΘi)(Ef − |pf |c cosΘf)

)

· A. (18)

With (12) - (18), Eq. (1) can be written as:

d4σ

dωΩidΩfdΦ=

a1α cosΦ + β

+a2 cosΦ

α cosΦ + β

+a3

(α cosΦ + β)2+

a4 cosΦ

(α cosΦ + β)2; (19)

12

thus the integration over Φ simply reads

d3σ

dωdΩidΩf=

2π∫

0

dΦ

[

a1α cosΦ + β

+a2 cosΦ

α cos Φ + β

+a3

(α cosΦ + β)2+

a4 cosΦ

(α cosΦ + β)2

]

(20)

where ai, i ∈ 1, . . . , 4, α and β still depend on Θf and Θi. These integralscan be calculated with the help of the residue theorem which is reviewedbriefly in Appendix A. If R(x, y) : R2 → R is a rational function withoutpoles on the unit circle x2 + y2 = 1, then

2π∫

0

R(cosΦ, sinΦ)dΦ = 2πi∑

|z|<1

Res(f, z) (21)

where f is a complex function which is defined as

f(z) :=1

izR

(

1

2

(

z +1

z

)

,1

2i

(

z − 1

z

))

. (22)

The residuum of a pole zj of order n is defined as

Res(f, zj) =1

(n− 1)!limz→zj

dn−1

dzn−1

[

(z − zj)f(z)]

. (23)

To integrate the functions in Eq. (20), we write

R1(x, y) :=a1

αx+ β, (24)

R2(x, y) :=a2x

αx+ β, (25)

R3(x, y) :=a3

(αx+ β)2, (26)

R4(x, y) :=a4x

(αx+ β)2; (27)

13

and get from (22)

f1(z) =2a1

i(αz2 + 2βz + α), (28)

f2(z) =a2z

2 + a2zi(αz2 + 2βz + α)

, (29)

f3(z) =4a3z

i (αz2 + 2βz + α)2, (30)

f4(z) =2a4(z

2 + 1)

i (αz2 + 2βz + α)2. (31)

The poles of the functions fi(z) in (28) - (31) are given by

z1,2 = −β

α±

√

(

β

α

)2

− 1. (32)

For f1,2 poles are of order one and for f3,4 of order two. In addition f2 has apole at

z3 = 0. (33)

According to (21) one needs poles with |zi| < 1. For z3 it is quite clear that|z3| = 0 < 1. As the angles Θi and Θf are between 0 and π, the expressionα > 0 in Eq. (12). Furthermore cosΘf > −1, cosΘi < 1 and |pi| > ~/c ω.

14

Hence

β (34)

= −p2i − p2

f −(

~

cω

)2

− 2~

cω|pf | cosΘf + 2

~

cω|pi| cosΘi

+ 2|pi||pf | cosΘi cosΘf (35)

= −p2i − p2

f −(

~

cω

)2

+ 2~

cω|pi| cosΘi

+ 2|pf | cosΘf

(

−~cω + |pi| cosΘi

)

(36)

< −p2i − p2

f −(

~

cω

)2

+ 2~

cω|pi|+ 2|pf | cosΘf

(

−~cω + |pi|

)

(37)

< −p2i − p2

f −(

~

cω

)2

+ 2~

cω|pi|+ 2|pf |

(

−~cω + |pi|

)

(38)

= −p2i − p2

f −(

~

cω

)2

+ 2~

cω|pi| − 2|pf |

~

cω + 2|pf ||pi| (39)

= −(

|pi| − |pf | −~

cω

)2

< 0 (40)

Therefore β/α in Eq. (32) is a negative real number. Furthermore sinΘi < 1and sinΘf < 1. Thus

−β − α (41)

= p2i + p2

f +

(

~

cω

)2

+ 2~

cω|pf | cosΘf − 2

~

cω|pi| cosΘi

− 2|pi||pf | cosΘi cosΘf − 2|pi||pf | sinΘi sin Θf (42)

> p2i + p2

f +

(

~

cω

)2

− 2~

cω|pf | − 2

~

cω|pi| − 4|pi||pf | (43)

> p2i + p2

f +

(

~

cω

)2

> 0 (44)

⇒ −β

α> 1 (45)

15

It follows immediately that |z1| > 1 and |z2| < 1. For all residua one obtains

Res(f1, z2) = −a1i

1√

β2 − α2, (46)

Res(f2, z2) =a2β

αi

1√

β2 − α2, (47)

Res(f2, z3) =a2αi

, (48)

Res(f3, z2) = −a3βi

1

(√

β2 − α2)3, (49)

Res(f4, z2) = a4α1

(√

β2 − α2)3. (50)

With the knowledge of these residua and using (21), the integral in (20) canbe calculated elementarily

d3σ

dωdΩidΩf=

2πa2α

+2π

√

β2 − α2

[

−a1 +a2β

α

− a3β

|β2 − α2| +a4α

|β2 − α2|

]

. (51)

2.4. Integration over Θf

After having obtained an expression for the “triply” 1 differential crosssection, there is still the integration over Θf left. This calculation is mainlystraight forward, but rather tedious. Using expression (51), it is

d2σ

dωdΩi

=

π∫

0

dΘf

[

2πa2α

+2π

√

β2 − α2

(

−a1 +a2β

α

− a3β

|β2 − α2| +a4α

|β2 − α2|

)]

sinΘf . (52)

1 Here “triply” really means the dependence on the photon frequency and two angles.

16

Let’s now consider the first integral of (52). If one inserts (12) and (16), itbecomes

π∫

0

dΘf2πa2α

sinΘf = − 2πAc2

Ei − cpi cosΘi

π∫

0

dΘfsin Θf

Ef − cpf cosΘf

(53)

= − 2πAc2

Ei − cpi cosΘi

+1∫

−1

dx1

Ef − cpfx(54)

where the substitution x := cosΘf was made in the second step. (54) israther simple and yields

π∫

0

dΘf2πa2α

sin Θf = − 2πAc

(Ei − cpi cosΘi)pfln

(

Ef + pfc

Ef − pfc

)

. (55)

This was a quite simple calculation. All the other integrals can be calculatedsimilarly, but with more effort. As another example let’s consider the lastintegral. Before inserting (12), (13) and (18) one can define for simplicity

∆1 := −p2i − p2

f −(

~

cω

)2

+ 2~

cω|pi| cosΘi, (56)

∆2 := −2~cω|pf |+ 2|pi||pf | cosΘi. (57)

The expression β from Eq. (13) is then

β = ∆1 +∆2 cosΘf . (58)

Thus the regularly appearing term β2 − α2 can be written as

β2 − α2 = (∆22 + 4p2i p

2f sin

2Θi) cos2Θf + 2∆1∆2 cosΘf

+ (∆21 − 4p2i p

2f sin

2Θi) (59)

= 21 cos

2Θf + 2∆1∆2 cosΘf +22 (60)

where the definitions

21 := ∆2

2 + 4p2i p2f sin

2Θi, (61)

22 := ∆2

1 − 4p2i p2f sin

2Θi (62)

17

have been introduced.By using (12), (18) and (60), the last integral of (51) becomes

π∫

0

dΘf2πa4α

√

(β2 − α2)3sinΘf = −

16πAp2i p2f sin

2Θi(E2i + E2

f)

Ei − cpi cosΘi

×π∫

0

dΘfsin2Θf

√

(21 cosΘ

2f + 2∆1∆2 cosΘf +

22)

3(Ef − cpf cosΘf)sinΘf

(63)

= −16πAp2i p

2f sin

2 Θi(E2i + E2

f )

Ei − cpi cosΘi

×+1∫

−1

dx1− x2

√

(21x

2 + 2∆1∆2x+22)

3(Ef − cpfx)(64)

where x = cosΘf has been substituted again.This integration can be performed elementarily by finding the indefinite

integral

21x

2 + 2∆1∆2x + 22

√

(21x

2 + 2∆1∆2x + 22)

3(∆21∆

22 −

21

22)(

21E

2f

+ 2∆1∆2Efpf c + 22pf c)

×

×(

−41Efx +

42pf c + 2∆

21∆

22(Efx − pf c) + ∆1∆2

22(Ef + pf cx)

− 21(

22(Efx − pf c)∆1∆2(Ef + pf cx))

)

+E2

f − p2f c2

√

(21E

2f

+ 2∆1∆2Ef pf c + 22pf c)3

ln(

(Ef − pf cx)(21Efx +

22pf c

+ ∆1∆2(Ef + pf cx) +√

21x

2 + 2∆1∆2x + 22 ×

×√

21E

2f

+ 2∆1∆2Ef pf c + 22pf c)

)

, (65)

by inserting +1 and −1 as upper and lower limit, using (61) and (62) and

18

simplifying. The integral in (63) is then finally

π∫

0

dΘf

2πa4α√

(β2 − α2)3sinΘf −

16πAp2i p2f sin2 Θi(E

2i + E2

f )

Ei − cpi cos Θi

×

×

−2(∆2pf c + ∆1Ef )

(−∆22 + ∆2

1 − 4p2ip2fsin2 Θi)((∆2Ef + ∆1pf c)2 + 4m2c4p2

ip2fsin2 Θi)

+m2c4

√

((∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi)

3×

× ln

(

(

(Ef − cpf )(4p2i p

2f sin

2Θi(−Ef − pf c) + (∆1 − ∆2)((∆2Ef + ∆1pf c)

−√

21E

2f

+ 2∆1∆2Ef pf c + 22pf c)

)(

(Ef + cpf )(4p2i p

2f sin

2Θi(+Ef − pf c)

+ (∆1 + ∆2)((∆2Ef + ∆1pf c) −√

21E

2f

+ 2∆1∆2Efpf c + 22pf c)

)

−1)]

.

(66)

All the other integrals can be calculated similarly where one always has tosubstitute x = cosΘf . With this technique the whole doubly differentialcross section finally becomes

d2σ(Ei, ω,Θi)

dωdΩi=

6∑

j=1

Ij (67)

19

with the following contributions:

I1 =2πA

√

∆22 + 4p2

ip2fsin2 Θi

ln

∆22 + 4p2i p

2f sin2 Θi −

√

∆22 + 4p2

ip2fsin2 Θi(∆1 + ∆2) + ∆1∆2

−∆22 − 4p2

ip2fsin2 Θi −

√

∆22 + 4p2

ip2fsin2 Θi(∆1 − ∆2) + ∆1∆2

×

1 +c∆2

pf (Ei − cpi cos Θi)−

p2i c2 sin2 Θi

(Ei − cpi cos Θi)2

−2~2ω2pf∆2

c(Ei − cpi cos Θi)(∆22 + 4p2

ip2fsin2 Θi)

, (68)

I2 = −2πAc

pf (Ei − cpi cosΘi)ln

(

Ef + pf c

Ef − pf c

)

, (69)

I3 =2πA

√

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi

× ln

(

(

(Ef + pf c)(4p2i p

2f sin

2Θi(Ef − pf c) + (∆1 + ∆2)((∆2Ef + ∆1pf c)

−√

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi))

)(

(Ef − pf c)(4p2i p

2f sin

2Θi(−Ef − pf c)

+ (∆1 − ∆2)((∆2Ef + ∆1pf c) −√

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi))

)

−1)

×

−(∆2

2 + 4p2i p2f sin2 Θi)(E

3f + Efp2f c2) + pf c(2(∆2

1 − 4p2i p2f sin2 Θi)Ef pf c + ∆1∆2(3E

2f + p2f c2))

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi

−c(∆2Ef + ∆1pf c)

pf (Ei − cpi cosΘi)

−4E2

i p2f (2(∆2Ef + ∆1pf c)2 − 4m2c4p2i p

2f sin2 Θi)(∆1Ef + ∆2pf c)

((∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi)

2

+8p2i p

2fm2c4 sin2 Θi(E

2i + E2

f ) − 2~2ω2p2i sin2 Θipf c(∆2Ef + ∆1pf c) + 2~2ω2pfm2c3(∆2Ef + ∆1pf c)

(Ei − cpi cosΘi)((∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi)

,

(70)

I4 = −4πApf c(∆2Ef + ∆1pf c)

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi

−16πE2

i p2fA(∆2Ef + ∆1pf c)2

((∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi)

2, (71)

I5 =4πA

(−∆22 + ∆2

1 − 4p2ip2fsin2 Θi)((∆2Ef + ∆1pf c)2 + 4m2c4p2

ip2fsin2 Θi)

×

[

~2ω2p2f

Ei − cpi cos Θi

×Ef [2∆2

2(∆22 − ∆2

1) + 8p2i p2f sin2 Θi(∆

22 + ∆2

1)] + pf c[2∆1∆2(∆22 − ∆2

1) − 16∆1∆2p2i p

2f sin2 Θi]

∆22 + 4p2

ip2fsin2 Θi

+2~2ω2p2i sin2 Θi(2∆1∆2pf c + 2∆2

2Ef + 8p2i p2f sin2 ΘiEf )

Ei − cpi cos Θi

+2E2

i p2f 2(∆2

2 − ∆21)(∆2Ef + ∆1pf c)2 + 8p2i p

2f sin2 Θi[(∆

21 + ∆2

2)(E2f + p2f c2) + 4∆1∆2Efpf c]

((∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi)

+8p2i p

2f sin2 Θi(E

2i + E2

f )(∆2pf c + ∆1Ef )

Ei − cpi cosΘi

]

, (72)

I6 =16πE2

f p2i sin2 ΘiA

(Ei − cpi cos Θi)2(−∆2

2 + ∆21 − 4p2

ip2fsin2 Θi)

. (73)

Eq. (67) depends explicitly on Ei, ω and Θi while Ef and pf are functionsof Ei and ω through (2) and (3). (67) is the final result of the integrationof (1) over Φ and Θf with the help of the residue theorem and some basiccalculations. Now this result can be used both as input for Monte Carlocode and for discussing some basic properties of the behaviour of produced

20

Bremsstrahlung photons.Actually (67) is also valid for Θi = 0, as will be shown in the next section,

but the simple way just to set Θi = 0 in (67) will fail, especially for numericalpurposes, because the logarithmic part in (68) tends to ln(0/0) for Θi → 0and so fails for numerical applications. Thus we need an additional expressionfor Θi = 0 which has to be consistent with (67).

2.5. Special limits: Θi = 0, π and ~ω → Ekin,i

For some special cases the integration of (1) over Φ and Θf is easier. Thisinformation can also be used to verify (67) by checking consistency and usethem for Monte Carlo codes.

2.5.1. Θi = 0 or Θi = π

If one is only interested in forward and backward scattering, one can setΘi = 0 or Θi = π before integrating; then (1) becomes

d4σ

dωdΩidΩfdΦ=

Z2α3fine~

2

(2π)2|pf ||pi|

1

ω

1

|q|4

×(

p2f sin

2Θf

(Ef − c|pf | cosΘf )2(

4E2i − c2q2

)

+ 2 ~2ω2

p2f sin

2Θf

(Ef − c|pf | cosΘf)(Ei ∓ c|pi| cosΘi)

)

(74)

where the momentum q of the virtual photon can be written as

− q2 = −p2i − p2

f −(

~

cω

)2

− 2~

c|pf | cosΘf ± 2

~

cω|pi| ± 2|pi||pf | cosΘf .

(75)

Here the upper sign corresponds to Θi = 0 and the lower one to Θi = π.As (74) and (75) do not depend on Φ at all, the Φ integration simply

21

gives a factor of 2π, and (74) becomes

d3σ

dωdΩidΩf

=Z2α3

fine~2

2π

|pf ||pi|

1

ω

1

|q|4

×(

p2f sin

2Θf

(Ef − c|pf | cosΘf)2(

4E2i − c2q2

)

+ 2 ~2ω2

p2f sin

2Θf

(Ef − c|pf | cosΘf )(Ei ∓ c|pi| cosΘi)

)

. (76)

Finally this expression has to be integrated over Θf in order to obtain thedoubly differential cross section. Similarly to the total integration of (52) itis convenient to define

∆1 := −(

pi ∓~

cω

)2

− p2f , (77)

∆2 := −2~cωpf ± 2pipf (78)

where ∆1,2 = ∆1,2(Θi = 0, π), j ∈ 1, 2 , with definitions (56) and (57). Eq.(75) can then be rewritten as

− q2 = ∆1 + ∆2 cosΘf (79)

and

d2σ

dωdΩi

=Z2α3

fine~2

2π

|pf ||pi|

1

ω

π∫

0

dΘf

[

|pf |2c2 sin2Θf

(Ef − |pf |c cosΘf)2(∆1 + ∆2 cosΘf)

+4E2

i |pf |2 sin2Θf

(Ef − |pf |c cosΘf)2(∆1 + ∆2 cosΘf)2

+2~2ω2|pf |2 sin2Θf

(Ei ∓ c|pi|)(Ef − |pf |c cosΘf)(∆1 + ∆2 cosΘf)2

]

sin Θf

(80)

22

where the integration is rather elementary and can be performed by substi-tuting x = cosΘf again. Thus (80) yields

d2σ

dωdΩi

(Ei, ω,Θi = 0, π) =Z2α3~2

2π

|pf ||pi|

1

ω

[

− 2|pf |c∆2Ef + ∆1|pf |c

+|pf |2c2(−∆2

1 + ∆22)

∆2(∆2Ef + ∆1|pf |c)2ln

(

∆1 + ∆2

∆1 − ∆2

)

+2∆1Ef |pf |c+ ∆2(E

2f + |pf |2c2)

(∆2Ef + ∆1|pf |c)2ln

(

Ef + |pf |cEf − |pf |c

)

− 16E2i |pf |2

(∆2Ef + ∆1|pf |c)2− 4~2|pf |2ω2

(∆2Ef + ∆1|pf |c)(Ei ∓ c|pi|)∆2

− 8E2i |pf |2(∆1Ef + ∆2|pf |c)(∆2Ef + ∆1|pf |c)3

ln

(

(∆1 − ∆2)(Ef − |pf |c)(∆1 + ∆2)(Ef + |pf |c)

)

+2~2|pf |2ω2(2∆1∆2Ef + ∆2

1|pf |c+ ∆22|pf |c)

(∆2Ef + ∆1|pf |c)2(Ei ∓ c|pi|)∆22

ln

(

∆1 + ∆2

∆1 − ∆2

)

+2~2|pf |ω2(E2

f − c2|pf |2)(∆2Ef + ∆1|pf |c)2(Ei ∓ c|pi|)c

ln

(

Ef − |pf |cEf + |pf |c

)

]

. (81)

This expression is much simpler than (67), but only valid for Θi = 0 orΘi = π. Actually this expression has also been obtained by calculating thelimit Θi → 0 or Θi → π in (67); hence the consistency check is succesful.Details can be found in Appendix B.

2.5.2. ~ω → Ekin,i

The other case which can be investigated easily is when almost all kineticenergy of the incident electron is transferred to the emitted photon, i.e.,

Ef = Ei − ~ω = Ekin,i +mec2 − ~ω

~ω→Ekin,i−−−−−−→ mec2 (82)

and

|pf | =

√

E2f

c2−m2

ec2

~ω→Ekin,i−−−−−−→√

m2ec

4

c2−m2

ec2 ≡ 0 (83)

23

and consequently from Eq. (9)

− q2 ~ω→Ekin,i−−−−−−→ −p2i −

(

~

cω

)2

+ 2~

cω|pi| cosΘi =: δ (84)

⇒ q4 ~ω→Ekin,i−−−−−−→ δ2. (85)

With these limits it follows for the triply differential cross section (1)

d4σ

dωdΩidΩfdΦ

~ω→Ekin,i−−−−−−→Z2α3

fine~2

(2π)2|pf ||pi|

1

ω

1

δ2

[ |pi|2 sin2Θi

(Ei − c|pi| cosΘi)2

× (4E2f + δc2) + 2 ~

2ω2 |pi|2 sin2 Θi

(Ei − c|pi| cosΘi)Ef

]

.

(86)

Actually (86) depends neither on Φ, nor on Θf . Therefore

2π∫

0

dΦ

π∫

0

sinΘidΘi = 4π (87)

which leads to a very simple expression for the doubly differential cross sec-tion

d2σ

dωdΩi

~ω→Ekin,i−−−−−−→ Z2α3~2

π

|pf ||pi|

1

ω

1

δ2

[ |pi|2 sin2Θi

(Ei − c|pi| cosΘi)2(4E2

f + δc2)

+ 2 ~2ω2 |pi|2 sin2Θi

(Ei − c|pi| cosΘi)Ef

]

. (88)

Although taking the limit ~ω → Ekin,i contradicts Eq. (11) as the energy ofthe emitted electron should be larger than 1 keV (11), (88) can be used fortwo purposes.

As (88) can be obtained, as well, by taking the limit |pf | → 0 in (67),the complicated expression (67) is checked for consistency analytically. Forfurther details the reader is referred to Appendix C. Furthermore we willsee in section 4.1.5 that the most probable scattering angle does not dependon the photon energy for Ekin,i ≥ 1 MeV. Therefore this cross section can beused for calculating the most probable scattering angle in this energy range.

24

3. Pair production

Pairs of electrons and positrons can be produced if a photon interactswith the nucleus of an atom. This process is related by some symmetry tothe production of Bremsstrahlung photons. Bremsstrahlung occurs when anelectron is affected by the nucleus of an atom, scattered and then emits aphoton. So there are three real particles involved: incident electron, scat-tered electron and emitted photon. As the photon has no antiparticle one canchange the time direction of the photon. For antimatter it is well known thatantiparticles can be interpreted as the corresponding particles moving backin time. So one can substitute the incident electron by an positron movingforward in time. Thus by substituting emitted photon by incident photonand incident electron by emitted positron (due to time reversal and changingits charge) it is possible to describe pair production from Bremsstrahlung.Thus the emitted photon in the Bremsstrahlung process has to be substitutedby the incident photon from the nucleus and the incident electron by the pro-duced positron. With these two replacements one gets the differential crosssection for pair production (Heitler, 1944; Greiner and Reinhardt, 1995)

d4σ =Z2α3

finec2

(2π)2~|p+||p−|

dE+

ω3

dΩ+dΩ−dΦ

|q|4 ×

×[

− p2− sin2Θ−

(E− − c|p−| cosΘ−)2(

4E2+ − c2q2

)

− p2+ sin2Θ+

(E+ − c|p+| cosΘ+)2(

4E2− − c2q2

)

+ 2~2ω2 p2+ sin2Θ+ + p2

− sin2Θ−

(E+ − c|p+| cosΘ+)(E− − c|p−| cosΘ−)

+ 2|p+||p−| sinΘ+ sin Θ− cos Φ

(E+ − c|p+| cosΘ+)(E− − c|p−| cosΘ−)

(

2E2+ + 2E2

− − c2q2)

]

,

(89)

where Z, αfine, h, ~ and c are the same parameters as in Eq. (1). ω is thefrequency of the incident photon, E± and p± are the total energy and themomentum of the positron/electron with

E± =√

p2±c

2 +m2ec

4. (90)

Similarly to (1) there are three angles, Θ± between the direction of the photonand the positron/electron direction, Θ+ = ∢(p+,k),Θ− = ∢(p−,k), and Φ

25

is the angle between the scattering planes (p+,k) and (p−,k). The absolutevalue of the momentum of the virtual photon is

− q2 = −|p+|2 − |p−|2 −(

~

cω

)2

+ 2|p+|~

cω cosΘ+ + 2|p−|

~

cω cosΘ−

− 2|p+||p−|(cosΘ+ cosΘ− + sinΘ+ sinΘ− cos Φ). (91)

Algebraically one obtains (89) from (1) by replacing

Ef → E−, (92)

Ei → −E+, (93)

pi → −p+, (94)

pf → p−, (95)

ω → −ω, (96)

Θi → π −Θ+, (97)

Θf → Θ−, (98)

Φ → Φ− π (99)

where the quantities on the left hand side are for Bremsstrahlung, and thoseon the right hand side for pair production. At the end one has to multiplywith an additional factor to get the correct prefactor. With all the mentionedsubstitutions it is

d4σbrems ←~3ω2

|p+|2c2dω

dE+d4σpair, (100)

Because of this symmetry the results for pair production follow easily fromthose for Bremsstrahlung.

The direction of the positron relative to the incident photon is given byintegrating (89) over Φ and Θ−. But this is the same exercise as to integrate(1) over Φ and Θf . Because of the symmetry between Bremsstrahlung andpair production one can take (67) and substitute (92) - (99) to obtain adoubly differential cross section

d2σ(E+, ω,Θ+)

dE+dΩ+=

6∑

j=1

Ij (101)

26

with the following contributions

I1 =2πA

√

(∆(p)2 )2 + 4p2

+p2−

sin2 Θi

× ln

(∆(p)2 )2 + 4p2+p2

−sin2 Θi −

√

(∆(p)2 )2 + 4p2+p2

−sin2 Θi(∆

(p)1 + ∆

(p)2 ) + ∆

(p)1 ∆

(p)2

−(∆(p)2 )2 − 4p2+p2

−sin2 Θi −

√

(∆(p)2 )2 + 4p2+p2

−sin2 Θi(∆

(p)1 − ∆

(p)2 ) + ∆

(p)1 ∆

(p)2

×

−1 −c∆

(p)2

p−(E+ − cp+ cos Θi)+

p2+c2 sin2 Θi

(E+ − cp+ cosΘi)2

−2~2ω2p−∆

(p)2

c(E+ − cp+ cosΘi)((∆(p)2 )2 + 4p2+p2

−sin2 Θi)

,

(102)

I2 =2πAc

p−(E+ − cp+ cos Θi)ln

(

E− + p−c

E− − p−c

)

, (103)

I3 =2πA

√

(∆(p)2 E− + ∆

(p)1 p−c)2 + 4m2c4p2+p2

−sin2 Θi

× ln

(

(

(E− + p−c)(4p2+p

2−

sin2Θi(E− − p−c) + (∆

(p)1 + ∆

(p)2 )((∆

(p)2 E− + ∆

(p)1 p−c)

−

√

(∆(p)2 E− + ∆

(p)1 p−c)2 + 4m2c4p2+p2

−sin2 Θi))

)(

(E− − p−c)(4p2+p

2−

sin2Θi(−E− − p−c)

+ (∆(p)1 − ∆

(p)2 )((∆

(p)2 E− + ∆

(p)1 p−c) −

√

(∆(p)2 E− + ∆

(p)1 p−c)2 + 4m2c4p2+p2

−sin2 Θi))

)

−1)

×

c(∆(p)2 E− + ∆

(p)1 p−c)

p−(E+ − cp+ cos Θi)

+[

((∆(p)2 )

2+ 4p

2+p

2−

sin2Θi)(E

3−

+ E−p−c) + p−c(2((∆(p)1 )

2− 4p

2+p

2−

sin2Θi)E−p−c

+ ∆(p)1 ∆

(p)2 (3E

2−

+ p2−

c2))][

(∆(p)2 E− + ∆

(p)1 p−c)

2+ 4m

2c4p2+p

2−

sin2Θi

]

−1

+[

− 8p2+p

2−m

2c4sin

2Θi(E

2+ + E

2−) − 2~

2ω2p2+ sin

2Θip−c(∆

(p)2 E− + ∆

(p)1 p−c)

+ 2~2ω2p−m

2c3(∆

(p)2 E− + ∆

(p)1 p−c)

][

(E+ − cp+ cos Θi)((∆(p)2 E− + ∆

(p)1 p−c)

2+ 4m

2c4p2+p

2−

sin2Θi)

]

−1

+4E2

+p2−(2(∆

(p)2 E− + ∆

(p)1 p−c)2 − 4m2c4p2+p2

−sin2 Θi)(∆

(p)1 E− + ∆

(p)2 p−c)

((∆(p)2 E− + ∆

(p)1 p−c)2 + 4m2c4p2+p2

−sin2 Θi)

2

, (104)

I4 =4πAp−c(∆

(p)2 E− + ∆

(p)1 p−c)

(∆(p)2 E− + ∆

(p)1 p−c)2 + 4m2c4p2+p2

−sin2 Θi

+16πE2

+p2−A(∆

(p)2 E− + ∆

(p)1 p−c)2

((∆(p)2 E− + ∆

(p)1 p−c)2 + 4m2c4p2+p2

−sin2 Θi)

2, (105)

I5 =4πA

(−(∆(p)2 )2 + (∆

(p)1 )2 − 4p2+p2

−sin2 Θi)((∆

(p)2 E− + ∆

(p)1 p−c)2 + 4m2c4p2+p2

−sin2 Θi)

×

[

~2ω2p2

−

E+cp+ cosΘi

[

E−[2(∆(p)2 )

2((∆

(p)2 )

2− (∆

(p)1 )

2) + 8p

2+p

2−

sin2Θi((∆

(p)2 )

2+ (∆

(p)1 )

2)]

+ p−c[2∆(p)1 ∆

(p)2 ((∆

(p)2 )

2− (∆

(p)1 )

2) − 16∆

(p)1 ∆

(p)2 p

2+p

2−

sin2Θi]

][

(∆(p)2 )

2+ 4p

2+p

2−

sin2Θi

]

−1

+2~2ω2p2+ sin2 Θi(2∆

(p)1 ∆

(p)2 p−c + 2(∆

(p)2 )2E− + 8p2+p2

−sin2 ΘiE−)

E+ − cp+ cos Θi

−[

2E2+p

2−2((∆

(p)2 )

2− (∆

(p)1 )

2)(∆

(p)2 E− + ∆

(p)1 p−c)

2+ 8p

2+p

2−

sin2Θi[((∆

(p)1 )

2+ (∆

(p)2 )

2)(E

2−

+ p2−c2)

+ 4∆(p)1 ∆

(p)2 E−p−c]

][

(∆(p)2 E− + ∆

(p)1 p−c)

2+ 4m

2c4p2+p

2−

sin2Θi

]

−1

−8p2+p2

−sin2 Θi(E

2+ + E2

−)(∆

(p)2 p−c + ∆

(p)1 E−)

E+ − cp+ cosΘi

, (106)

I6 = −16πE2

−p2+ sin2 ΘiA

(E+ − cp+ cos Θi)2(−(∆

(p)2 )2 + (∆

(p)1 )2 − 4p2+p2

−sin2 Θ+)

(107)

27

with

A =Z2α3

finec2

(2π)2~

|p+||p−|ω3

, (108)

and with ∆(p)1 ,∆

(p)2 defined as

∆(p)1 := −|p+|2 − |p−|2 −

(

~

cω

)

+ 2~

cω|p+| cosΘ+, (109)

∆(p)2 := 2

~

cω|pi| − 2|p+||p−| cosΘ+. (110)

28

0.01

0.1

1

10

100

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

ω/Z

2 d2 σ(

Eki

n,i,ω

,Θi)/

(dω

dΩi)

[mb/

rad]

Θi

Experimental results

Result of integration 0.1

1

10

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

ω/Z

2 d2 σ(

Eki

n,i,ω

,Θi)/

(dω

dΩi)

[mb/

rad]

Θi

Experimental results

Result of integration

a) Ekin,i = 180 keV, ~ω = 50 keV b) Ekin,i = 380 keV, ~ω = 100 keV

Figure 2: ω/Z2 · d2σ/(dωdΩi)(Ekin,i, ω,Θi) for Bremsstrahlung as a function of the scat-tering angle Θi between emitted photon and incident electron for gold Z = 79 where 1 mb= 10−31 m2. The energies are a) Ekin,i = 180 keV, ~ω = 50 keV and b) Ekin,i = 380 keV,~ω = 100 keV. The solid lines shows our result (67); the dotted lines show experimentalvalues (Aiginiger, 1966).

4. Discussion

4.1. Bremsstrahlung

4.1.1. Comparison with experiments

If electrons are scattered at nuclei, they can produce hard Bremsstrahlungphotons with frequency ω and direction Θi relative to the direction of theelectrons.

Figure 2 compares our equation (67) with experimental results for gold(Z = 79) for different electron and photon energies (Aiginger, 1966). ForZ = 79 the minimal electron energy (11) for the Born approximation to bevalid, is Ekin,i,f = 115 keV. Figure 2 shows that the cross sections agreeoverall in size for Ekin,i = 180 keV, ~ω = 50 keV and for Ekin,i = 380 keV,~ω = 100 keV. However, for the first case, the energy of the electron in thefinal state is Ekin,f = 130 keV ≈ 115 keV, thus close to the velocity limit.Therefore there is a larger deviation, especially for small angles, than for thesecond case where a very good agreement can be observed.

4.1.2. Angular distribution of Bremsstrahlung

Figure 3 shows the doubly differential cross section (67) for Bremsstrah-lung for several electron and photon energies. At first, the probability for

29

1e-51

1e-50

1e-49

1e-48

1e-47

1e-46

1e-45

1e-44

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

−hω/Ekin,i=1%

−hω/Ekin,i=10%

−hω/Ekin,i=50%

−hω/Ekin,i=99%

1e-54

1e-53

1e-52

1e-51

1e-50

1e-49

1e-48

1e-47

1e-46

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 πd2 σ(

Eki

n,ω

,Θi)/

(dω

dΩi)

[m2 s/

rad]

Θi

−hω/Ekin,i=1%

−hω/Ekin,i=10%

−hω/Ekin,i=50%

−hω/Ekin,i=99%

a) Ekin,i = 10 keV b) Ekin,i = 150 keV

1e-56 1e-55 1e-54 1e-53 1e-52 1e-51 1e-50 1e-49 1e-48 1e-47 1e-46 1e-45

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

−hω/Ekin,i=1%−hω/Ekin,i=10%

−hω/Ekin,i=50%

−hω/Ekin,i=99% 1e-64 1e-62 1e-60 1e-58 1e-56 1e-54 1e-52 1e-50 1e-48 1e-46 1e-44 1e-42

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

−hω/Ekin,i=1%

−hω/Ekin,i=10%

−hω/Ekin,i=50%−hω/Ekin,i=99%

c) Ekin,i = 1 MeV d) Ekin,i = 100 MeV

Figure 3: The doubly differential cross section d2σ/(dωdΩi)(Ekin,i, ω,Θi) forBremsstrahlung (Z = 7) versus the scattering angle Θi between emitted photon andincident electron. The electron energies are a) Ekin,i = 10 keV, b) Ekin,i = 150 keV, c)Ekin,i = 1 MeV and d) Ekin,i = 100 MeV. In each plot the the photon energy ~ω amountsto 1%, 10%, 50% and 95% of the kinetic energy of the incident electron.

30

generating photons decreases with increasing photon energy for fixed electronenergy. This can be understood easily by applying (88). As can be seen there,the doubly differential cross section grows linearly in the momentum of theelectron in the final state which is equivalent to

d2σ

dωdΩi∼ |pf | (111)

for high photon energies. So, if all kinetic energy is transferred from theelectron onto the photon, the final momentum |pf | vanishes, and thus

d2σ

dωdΩi→ 0. (112)

For nonrelativistic electron and photon energies the scattering angle tends tobe mainly equally distributed, i.e. the photons do not have a preference for aparticular direction. When the photon energy increases, photons are mainlyemitted in forward direction, but the ratio between forward and backwardscattering is at least three orders of magnitude lower than for a relativisticelectron. This case belongs to the classical case where the velocity is smallcompared to the speed of light. Namely, it is v/c|Ekin,i=10keV ≈ 0.20 andnon-relativistic equations will be enough to describe these phenomena. Inthe relativistic case (v/c|Ekin,i=1MeV ≈ 0.94 and v/c|Ekin,i=100MeV ≈ 0.99999)the differential cross section becomes more and more anisotropic. Forwardscattering is preferred to backward scattering although the maximal crosssection does not lie precisely at Θi = 0 as can be seen in figure 3 c). Butthe more the electron energy increases the more the maximum wanders tosmaller angles, for example, it seems in figure 3 d) that the maximal emissionis indeed for Θi=0. As mentioned in section 2.4 and in Appendix B, formula(68) cannot be evaluated directly at Θi = 0. However, for this purpose, wederived (81) which is valid for Θi = 0 and Θi = π. Figure 4 shows again (67)for two relativistic electron and different photon energies but for a smallerrange of angles. It shows in more detail that the angle of maximal scatteringis small, but not 0.

Figure 5 shows the ratio between the cross section for backward scat-tering and forward scattering. It can be clearly seen that the tendency forbackward scattering decreases for increasing electron energy. The lower theelectron energy becomes, the more forward and backward scattering becomesimilar and in general, the scattering tends to be isotropic. Only for ratios

31

1e-52

1e-51

1e-50

1e-49

1e-48

1e-47

1e-46

1e-45

0 5π/180 10π/180 15π/180 20π/180

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

−hω/Ekin,i=1%

−hω/Ekin,i=10%

−hω/Ekin,i=50%

−hω/Ekin,i=99% 1e-52

1e-50

1e-48

1e-46

1e-44

1e-42

1e-40

1e-38

0 0.5π/180 1π/180 1.5π/180 2π/180

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

−hω/Ekin,i=50%

−hω/Ekin,i=99%

a) Ekin,i = 1 MeV b) Ekin,i = 100 MeV

Figure 4: The doubly differential cross section d2σ/(dωdΩi) as in figure 3 for a smallerangular range

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

100000 1e+06 1e+07 1e+08d2 σ(E

kin,

i,ω,Θ

i=18

0 °)

)/d2 σ(

Eki

n,i,ω

,Θi,m

ax)

Ekin,i [eV]

−hω/Ekin,i=50%

−hω/Ekin,i=10%

−hω/Ekin,i=1%

−hω/Ekin,i=99%

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100000 1e+06 1e+07 1e+08d2 σ(E

kin,

i,ω,Θ

i=18

0 °)

)/d2 σ(

Eki

n,i,ω

,Θi,m

ax)

Ekin,i [eV]

−hω/Ekin,i=1%, 10%, 50%

−hω/Ekin,i=99%

a) b)

Figure 5: The ratio between the doubly differential cross section for backward scattering(Θi = 180) d2σ(Ekin,i, ω,Θi = 180)/(dωdΩi) and the maximum of this cross sectionmax

(

d2σ(Ekin,i, ω,Θi)/(dωdΩi))

vs. the kinetic energy of the incident electron for differ-ent ratios between photon and electron energies in a a) linear and b) logarithmic scale forZ = 7 .

32

a) b)

Figure 6: a) In the rest frame of the electron where the nucleus is moving instead radiationis emitted isotropically with a small-angle deflection. b) If, however, one transforms thesituation into the rest frame of the nucleus where the electron is moving, most of theradiation is emitted in forward direction relative to the direction of the electron.

between photon energies and electron energies close to 1, forward scatteringis preferred for the whole range of energies, but still decreases with increasingelectron energies.

In energetic electron avalanches electrons scatter frequently which leadsto a large velocity dispersion. It depends on the direction of the appliedelectric field whether electrons move forward or whether their directions aredistributed arbitrarily. If so, however, this implies that photons will notnecessarily move in a preferred direction, but in the direction of the inci-dent electron. Their motion and thus change of direction depend on photonprocesses, such as Compton scattering.

4.1.3. Relativistic transformation

The tendency of forward scattering in the case of relativistic incidentelectrons can be understood by applying the laws of relativistic transforma-tions. Imagine a non-quantum field theoretical description of Bremsstrahlung(Jackson, 1975, p. 712 et seq.). If one regards an inertial system in which theincident particle is at rest (Fig. 6 a) ), radiation is emitted isotropically witha small-angle deflection. If the physical laws for this process are relativisti-cally transformed into the laboratory system where the nucleus is at rest andthe electron moving, most of the radiation is emitted in forward directionrelative to the electron direction (Fig. 6 b) ). Because this transformation isvalid for a non-quantum field theoretical, relativistic electron, it must also betrue for a relativistic quantum theoretical description, therefore we see thatthe forward scattering of photons can simply be explained as a result of the

33

1e-55

1e-54

1e-53

1e-52

1e-51

1e-50

1e-49

1e-48

1e-47

1e-46

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

−hω/Ekin,i=1%

−hω/Ekin,i=10%

−hω/Ekin,i=50%

−hω/Ekin,i=99%

1e-55

1e-54

1e-53

1e-52

1e-51

1e-50

1e-49

1e-48

1e-47

1e-46

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

−hω/Ekin,i=1%

−hω/Ekin,i=10%

−hω/Ekin,i=50%

−hω/Ekin,i=99%

a) Ekin,i = 400 keV b) Ekin,i = mec2 ≈ 511 keV

Figure 7: The doubly differential cross section d2σ/(dωdΩi)(Ekin,i, ω,Θi) forBremsstrahlung (Z = 7) versus the scattering angle Θi between emitted photon andincident electron. The electron energies are a) Ekin,i = 400 keV, b) Ekin,i ≈ 511 keV.In each plot the the photon energy ~ω amounts to 1%, 10%, 50% and 95% of the kineticenergy of the incident electron.

relativistic transformation.The forward scattering can moreover be understood by using the conser-

vation laws of energy and momentum. They predict that photons have to bescattered in forward direction if electron and photon energy are high. Theinterested reader is referred to Appendix H.

Although figure 3 shows that the maxima of the doubly differential crosssection form with increasing electron energy, it is difficult to determine inthese plots when these maxima really start to be generated clearly. Figure7 shows the doubly differential cross section in dependence of the incidentelectron energy for Ekin,i = 400 keV and Ekin,i ≈ 511 keV when the kineticenergy is equal to the rest energy. For 400 keV and for ~ω/Ekin,i = 0.01 thecross section for forward scattering is already two orders of magnitude largerthan for backward scattering, but a clear maximum cannot be seen. How-ever, for the same kinetic energy, but for ~ω/Ekin,i = 0.95 there is alreadya clear maximum formed. But if the kinetic energy grows up to 511 keVwhich is equal to the rest energy of the electron, there is even a maximumfor ~ω = 0.01Ekin,i. This can be expected due to the relativistic transforma-tion. If Ekin,i ≪ mec

2, then the photon emission is relatively isotropic. Butif the kinetic energy is approximately equal to the rest energy of the electron,

34

relativistic laws are valid. Especially for Ekin,i = mec2

v

c=

√3

2≈ 87%, (113)

therefore the electron has to be treated relativistically and clear maximaclose to Θi = 0 form for every possible photon energy.

4.1.4. Dependence on the energy of the emitted photon

Figure 3 shows that for both slow and relativistic electrons the doublydifferentical cross section also varies with the photon energy for fixed electronenergies. For fixed electron energy, lower photon energies are more likely.Moreover, photons are more likely for certain angles. They are more likelyfor lowly energetic electrons in the limit Θi → 180 and for highly energeticelectrons in the limit Θi → 0. Figure 8 shows the doubly differential crosssection in another way. Now the photon energy is fixed and the electronenergy differs within one plot. For all cases it is more likely that low energeticelectrons create photons than relativistic electrons do, in the limit Θi → 180.But for small angles, i.e. for forward emission of photons, the probabilityrapidly increases for relativistic electrons and exceeds the probability at smallelectron energies.

4.1.5. The most probable scattering angle

Figure 3 also shows that the angle for which maximal scattering takesplace, is rather independent of the photon energy. Hence, one can use (88)to determine a formula for that scattering angle. Actually this derivationleads to a quartic equation which can, however, be approximated for smallangles, i.e. Θi . 20, through a quadratic equation. The reader is referredto Appendix I for the detailed calculation. The solution of the quadraticequation reads

Θi =

√

√

√

√

√

− δ0~ω(4E2

f + δ0c2)− 2δ0~ωEf

(Ei − c|pi|)

2 |pi|c

[

4E2f + δ0c2 +

2~2ω2

Ef(Ei − c|pi|)

]

− |pi|δ0c− ~ωEf

c|pi|δ0(114)

with

δ0 := −|pi|2 −(

~

cω

)2

+ 2~

cω|pi|, (115)

35

1e-50

1e-50

1e-48

1e-46

1e-44

1e-42

1e-40

1e-38

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

Ekin,i=150 keV

Ekin,i=1 MeV

Ekin,i=100 MeV

1e-52

1e-51

1e-50

1e-49

1e-48

1e-47

1e-46

1e-45

1e-44

1e-43

1e-42

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 πd2 σ(

Eki

n,ω

,Θi)/

(dω

dΩi)

[m2 s/

rad]

Θi

Ekin,i=150 keV

Ekin,i=1 MeV

Ekin,i=100 MeV

a) ~ω = 50 eV b) ~ω = 500 eV

1e-56

1e-54

1e-52

1e-50

1e-48

1e-46

1e-44

1e-42

1e-40

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

Ekin,i=1 MeV

Ekin,i=100 MeV 1e-56

1e-54

1e-52

1e-50

1e-48

1e-46

1e-44

1e-42

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

Ekin,i=1 MeV

Ekin,i=100 MeV

c) ~ω = 500 keV d) ~ω = 950 keV

Figure 8: The doubly differential cross section d2σ/(dωdΩi)(Ekin,i, ω,Θi) forBremsstrahlung (Z = 7) vs. the scattering angle Θi for several electron and photonenergies. In each panel the photon energy ~ω is fixed and the cross section is plotted forvarious kinetic energies of the incident electron.

36

0

10

20

30

40

50

100000 1e+06 1e+07 1e+08

Θi [

°]

Ekin,i [eV]

99.99 %25 %1 %

Θi (Ekin,i) from quartic equationΘi (Ekin,i) from quadratic approximation

0.01

0.1

1

10

100000 1e+06 1e+07 1e+08

Θi [

°]

Ekin,i [eV]

Θi (Ekin,i) from quadratic approximation

Θi (Ekin,i) from quartic equation

99.99 %25 %1 %

a) b)

Figure 9: Θi for maximal scattering vs. incident electron energy in a a) semilog and b)loglog plot for Z = 7 . Besides (114) for ~ω = 0.9999Ekin,i, the exact solution of thequartic equation and various data for different ~ω/Ekin,i are shown.

~ω → Ekin,i, e.g. ~ω = 0.9999Ekin,i and

pi =

√

Ekin,i

(

Ekin,i

c2+ 2me

)

. (116)

Figure 9 shows (114) and manually extracted values for Θi for different pho-ton energies. It shows much better than figure 3 that Θi is rather independentof the photon energy for relativistic electron energies. Besides (114), the so-lution of the quartic equation, is shown. Moreover, it shows that (114) givesa good approximation for those angles Θi for which scattering is maximal.Actually, we see that the exact solution describes the angle for maximal scat-tering better, especially for low energies, but for high energies both curvesfit very well.

By inserting Ekin,i = ~ω/0.9999 into (114) one obtains a formula whichrelates the photon energy to the most probable scattering angle.

4.2. Pair production

4.2.1. Basic properties of pair production

We now proceed from Bremsstrahlung to pair production. One photonwith energy ~ω creates two particles, namely an electron and a positron,both with rest energy mec

2. Therefore

Ekin,− + Ekin,+ = ~ω − 2mec2 (117)

37

follows for the kinetic energies of these two particles. Thus the photon energyhas to be ~ω ≥ 2mec

2 ≈ 1.022 MeV for pair production and the kineticenergy of the particles is bounded as Ekin,± ≤ ~ω − 2mec

2. Figure 10 showsthe doubly differential cross section (101) for different photon and positronenergies. Forward scattering is dominant, there is almost no case now ofmore isotropic scattering. This results from the fact that almost all positronenergies in Figure 10 are relativistic. For very highly energetic photons, e.g.50 MeV and 100 MeV, and thus relativistic positron energies in Fig. 10there are clear maxima for forward scattering. For energies ~ω < 50 MeV ,however, the maxima are > 5 , but forward scattering is still preferred. Pairproduction is symmetric in positron and electron energy. Thus for the singlydifferential cross section

dσ

dE+

(E+, ω) =

π∫

0

d2σ(E+, ω,Θ+)

dE+dΩ+

sin Θ+dΘ+ (118)

the probability of the creation of a positron with a given energy is as largeas the probability to create an electron with this energy:

dσ

dE+

∣

∣

∣

∣ E+~ω−2mec2

=dσ

dE+

∣

∣

∣

∣

1−E+

~ω−2mec2

. (119)

Figure 11 shows the doubly differential cross section (101) for fixedpositron and different photon energies. Again positrons which are gener-ated with high velocities predominantly scatter forward while this tendencyvanishes if the positron energy is very low. This can be traced back to the rel-ativistic behaviour again. If a positron is very energetic, it has to be treatedrelativistically and the relativistic transformation leads to forward scattering(this is the same explanation as for Bremsstrahlung). We also see that thecreation of positrons is more likely for highly energetic photons.

38

1e-22

1e-21

1e-20

1e-19

1e-18

1e-17

1e-16

1e-15

1e-14

1e-13

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

+,ω

,Θ+)/

(dE

+dΩ

+)

[J-1

m2 /r

ad]

Θ+

E+/(−hω-2mec2)=10 %

E+/(−hω-2mec2)=30 %

E+/(−hω-2mec2)=50 %E+/(−hω-2mec2)=95 % 1e-24

1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 πd2 σ(

Eki

n,+,ω

,Θ+)/

(dE

+dΩ

+)

[J-1

m2 /r

ad]

Θ+

E+/(−hω-2mec2)=10 %

E+/(−hω-2mec2)=30 %

E+/(−hω-2mec2)=50 %

E+/(−hω-2mec2)=95 %

a) ~ω = 5 MeV b) ~ω = 10 MeV

1e-26

1e-24

1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

+,ω

,Θ+)/

(dE

+dΩ

+)

[J-1

m2 /r

ad]

Θ+

E+/(−hω-2mec2)=10 %

E+/(−hω-2mec2)=30 %

E+/(−hω-2mec2)=50 %

E+/(−hω-2mec2)=95 %

1e-28

1e-26

1e-24

1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

+,ω

,Θ+)/

(dE

+dΩ

+)

[J-1

m2 /r

ad]

Θ+

E+/(−hω-2mec2)=10 %

E+/(−hω-2mec2)=30 %

E+/(−hω-2mec2)=50 %E+/(−hω-2mec2)=95 %

c) ~ω = 50 MeV d) ~ω = 100 MeV

Figure 10: Doubly differential cross section d2σ(E+, ω,Θ+)/(dE+dΩ+) as a function ofthe angle Θ+ between incident photon and created positron for Z = 7: The cross sectionis shown for fixed photon energies a) ~ω = 5 MeV, b) ~ω = 10 MeV, c) ~ω = 50 MeV andd) ~ω = 100 MeV. In each panel different positron energies E+ relative to the availablephoton energy ~ω − 2mec

2 are plotted.

39

1e-20

1e-19

1e-18

1e-17

1e-16

1e-15

1e-14

1e-13

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

+,ω

,Θ+)/

(dE

+dΩ

+)

[J-1

m2 /r

ad]

Θ+

−hω=100 MeV

−hω=50 MeV−hω=10 MeV

−hω= 5 MeV

1e-22

1e-21

1e-20

1e-19

1e-18

1e-17

1e-16

1e-15

1e-14

1e-13

1e-12

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

+,ω

,Θ+)/

(dE

+dΩ

+)

[J-1

m2 /r

ad]

Θ+

−hω=100 MeV−hω=50 MeV

−hω=10 MeV

−hω= 5 MeV

a) Ekin,+ = 150 keV b) Ekin,+ = 1 MeV

Figure 11: Doubly differential cross section d2σ(E+, ω,Θ+)/(dE+dΩ+) as a function ofthe angle Θ+ between incident photon and created positron for Z = 7: The cross sectionis shown for fixed positron energies a) Ekin,+ = 150 keV and b) Ekin,+ = 1 MeV. In eachpanel curves for the photon energies ~ω = 5 MeV, 10 MeV, 50 MeV and 100 MeV areincluded.

4.2.2. The most probable scattering angle

As for Bremsstrahlung one can get a simple formula for the preferreddirection. Performing the same calculation as for Bremsstrahlung one obtains

Θ+ =

[

(

− δ(p)0

~ω(−4E2

− − δ(p)0 c2)− 2δ

(p)0 ~ω

E−

(E+ − c|p+|))

×(

2|p+|c

[

−4E2− − δ

(p)0 c2 +

2~2ω2

E−

(E+ − c|p+|)]

− |p+|δ(p)0 c

− ~ω

E−c|p+|δ(p)0

)−1]

12

(120)

with

δ(p) := −|p+|2 −(

~

cω

)2

+ 2|p+|~

cω (121)

and

E+ −mec2

~ω − 2mec2≈ 1. (122)

Figure 12 shows that (120) is a good approximation for Θ+ for high photon

40

0

5

10

15

20

25

30

35

40

45

50

1e+06 1e+07 1e+08

Θ+ [°

]

−hω [eV]

99 %75 %50 %25 %10 %1 %Θ+

0.0001

0.001

0.01

0.1

1

10

1e+06 1e+07 1e+08

Θ+ [°

]

−hω [eV]

99 %75 %50 %25 %10 %1 %Θ+

a) b)

Figure 12: Θ+ for maximal scattering vs. incident photon energy in a a) semilog and b)loglog plot for Z = 7 . Besides (120) for Ekin,+/(~ω − 2mec

2) = 0.9999 various data fordifferent Ekin,+/(~ω − 2mec

2) are shown.

energies and high ratios between photon and positron energy. The smallerthe ratio between photon and positron energy, however, is, the worse (120)becomes for low photon energies. If the photon energy is larger than 50 MeV,relativistic positrons are created; therefore forward scattering takes place andΘ+ can be calculated with (120).

41

5. Conclusion

We have reviewed literature relevant for Bremsstrahlung in Terrestrialgamma-ray flashes (TGFs) (Bethe and Heitler, 1934; Heitler, 1944; Elwertand Haug, 1969; Seltzer and Berger, 1985; Shaffer et al., 1996; Agostinelleet al., 2003). Focussing on atomic numbers Z = 7 (nitrogen) and Z = 8(oxygen) and an energy range of 1 keV to 1 GeV, no good parametrizationof an energy resolved angular distribution in the form of doubly differentialcross section is available. The theory of Bethe and Heitler covers this energyrange for Z = 7, 8, but it parameterizes the direction of the scattered elec-tron as well; therefore we integrated their triply differential cross section toobtain the correct energy resolved angular distribution for Bremsstrahlungand pair production. Other authors (Lehtinen, 2000; Dwyer, 2007; Carlsonet al., 2009, 2010) used different approaches, as discussed in the introduction.They use singly or triply differential cross sections which do not give a directrelation between the photon energy and the direction of the photon relativeto the motion of the electron. As positrons are created within a thundercloudas well (Briggs et al., 2010), we used a symmetry between the production ofBremsstrahlung and the creation of an electron-positron pair both in the fieldof a nucleus to obtain a cross section which relates the energy of the createdpositron with its direction.

We have seen that emitted Bremsstrahlung photons are mainly releasedin forward direction if the electron which interacts with the nucleus has sucha high energy that it has to be treated relativistically. For lower energiesscattering tends to be more isotropic. For the case that almost all kineticenergy of the incident electron is transformed into photon energy, we derivedan approximation for the most probable photon emission angle as a functionof the incident electron energy and of the photon energy. The expressionis valid for all ratios of photon over electron energy if the electron motionis relativistic. So, when photons have been created within a thundercloudor discharge, they are mainly scattered in forward direction as long as theelectrons move relativistically, i.e. if their kinetic energy is at least as largeas their rest energy.

Similar results hold for pair production. Next to the doubly differen-tial cross section we derived a simple approximation for the most probablepositron emission angle for the case that the photon energy is larger than10 MeV (for ratios between the kinetic energy of the positron and availablephoton energy down to 25%) or than 100 MeV (for ratios lower than 25%).

42

We have seen that for very highly energetic photons that almost all positronsare scattered in forward direction. If, however, the photon energy decreases,the probability of forward scattering decreases as well. Instead the maxi-mal cross section can be found at Θ+ ≈ 90 for low ratios between E+ and~ω − 2mec

2 and is, beyond that, symmetric to this angle.Our analytical results for the doubly differential cross-sections for Brems-

strahlung and pair production are also supplied in the form of two functionswritten in C++. In this form the functions can be implemented into MonteCarlo codes simulating energetic processes like the production of gamma-raysor electron positron pairs in thunderstorms.

Acknowledgements: We dedicate this article to the memory of Davis D.Sentman. He was a great inspiration and discussion partner in the summer of2011 when C.K. worked on this project, and he was an invaluable colleagueand co-organizer for U.E. for many years. We acknowledge fruitful and moti-vating discussions with Brant Carlsson. C.K. acknowledges financial supportby STW-project 10757, where Stichting Technische Wetenschappen (STW)is part of The Netherlands’ Organization for Scientific Research NWO.

43

Appendix A. The residual theorem to calculate integrals with trigono-metric functions

In this appendix the method how to calculate integrals of the form

2π∫

0

R(cosΦ, sin Φ)dΦ (A.1)

shall be discussed where R(x, y) : R2 \ x, y ∈ R|y = ±√1− x2 → R is a

rational function without poles on the unit circle x2 + y2 = 1. But beforeexplaining this method let’s briefly review some general facts about residua.

Appendix A.1. The residual theorem

Let f : C ⊃ I → C, z 7→ f(z), be a holomorphic function and Γ : [a, b]→C, t 7→ Γ(t), a closed curve in the complex plane. Then one can calculatecomplex curve integrals via

∫

Γ

f(z)dz = 2πi∑

j

Res(f, zj) (A.2)

where the sum has to be taken over all poles zj of f and complex curveintegrals are defined as

∫

Γ

f(z)dz :=

b∫

a

f(Γ(t)) · dΓdt

(t)dt. (A.3)

The residuum of a pole zj can be calculated via

Res(f, zj) =1

(n− 1)!limz→zj

dn−1

dzn−1

[

(z − zj)f(z)]

(A.4)

where n denotes the order of the pole.

Appendix A.2. Integral with trigonometric functions

With the help of the (A.2) one can simply perform the integration of(A.1). For that purpose define

f(z) :=1

izR

(

1

2

(

z +1

z

)

,1

2i

(

z − 1

z

))

(A.5)

44

and choose the unit circle

Γ(t) = eit, t ∈ [0, 2π] (A.6)

as closed curve; hence (A.3) becomes

∫

Γ

f(z)dz =

2π∫

0

1

ieitR

(

1

2

(

eit + e−it)

,1

2i

(

eit − e−it)

)

ieitdt (A.7)

=

2π∫

0

R

(

1

2

(

eit + e−it)

,1

2i

(

eit − e−it)

)

dt (A.8)

=

2π∫

0

R(cos t, sin t)dt (A.9)

where the identities cos t = 1/2 (eit + e−it) and sin t = 1/(2i) (eit − e−it) wereused in the last step.

Finally with (A.2) and (A.9) one gets a simple formula to calculate (A.1):

2π∫

0

R(cosΦ, sinΦ)dΦ = 2πi∑

|z|<1

Res(f, z) (A.10)

with f being defined in (A.5).

Appendix B. The doubly differential cross section for Θi = 0 andΘi = π

In order to get (81) from (67) it is rather straight forward to set Θi = 0or Θi = π. Especially it is

∆1(Θi = 0, π) = ∆1, (B.1)

∆2(Θi = 0, π) = ∆2. (B.2)

But there is one case which should be considered a bit more thoroughly.This regards the logarithm in (68). For Θi = π it is ∆2(Θ = π) = ∆2 =

45

−2pf(~/c ω + pi) < 0; thus |∆2| = −∆2 and

ln

(

∆22+4p2i p

2fsin2 Θi−

√∆2

2+4p2i p2fsin2 Θi(∆1+∆2)+∆1∆2

−∆22−4p2i p

2fsin2 Θi−

√∆2

2+4p2i p2fsin2 Θi(∆1−∆2)+∆1∆2

)

√

∆22 + 4p2i p

2f sin

2Θi

∣

∣

∣

∣

∣

∣

∣

∣

Θi=π

(B.3)

=1

|∆2|ln

(

∆22 − |∆2|(∆1 + ∆2) + ∆1∆2

−∆22 − |∆2|(∆1 − ∆2) + ∆1∆2

)

(B.4)

= − 1

∆2

ln

(

∆1 + ∆2

∆1 − ∆2

)

(B.5)

which is a very simple calculation. However, for Θi = 0 it is ∆2(Θi = 0) =∆2 = −2pf(~/c ω− pi) which can be both negative or positive depending onvalues of pi and ~/c ω If ∆2 < 0 then equations (B.3) - (B.5) are valid again.If ∆2 > 0, however, it follows for the argument of the logarithm

∆22 + 4p2i p

2f sin

2Θi −√

∆22 + 4p2i p

2f sin

2Θi(∆1 +∆2) + ∆1∆2

−∆22 − 4p2i p

2f sin

2Θi −√

∆22 + 4p2i p

2f sin

2Θi(∆1 −∆2) + ∆1∆2

∣

∣

∣

∣

∣

∣

Θi=0

(B.6)

=∆2

2 − ∆2(∆1 + ∆2) + ∆1∆2

−∆22 − ∆2(∆1 − ∆2) + ∆1∆2

=0

0. (B.7)

Hence it is necessary to use the rule of L’Hopital:

limΘi→0

∆22 + 4p2i p

2f sin

2Θi −√

∆22 + 4p2i p

2f sin

2Θi(∆1 +∆2) + ∆1∆2

−∆22 − 4p2i p

2f sin

2Θi −√

∆22 + 4p2i p

2f sin

2Θi(∆1 −∆2) + ∆1∆2

(B.8)

= limΘi→0

8p2i p2f sinΘi cosΘi − ∆1+∆2

2√

∆22+4p2i p

2fsin2 Θi

· 8p2i p2f sin Θi cosΘi

−8p2i p2f sinΘi cosΘi − ∆1−∆2

2√

∆22+4p2i p

2fsin2 Θi

· 8p2i p2f sinΘi cosΘi

(B.9)

= ln

(

∆1 − ∆2

∆1 + ∆2

)

. (B.10)

46

With (B.10) the whole limit yields

ln

(

∆22+4p2i p

2fsin2 Θi−

√∆2

2+4p2i p2fsin2 Θi(∆1+∆2)+∆1∆2

−∆22−4p2i p

2fsin2 Θi−

√∆2

2+4p2i p2fsin2 Θi(∆1−∆2)+∆1∆2

)

√

∆22 + 4p2i p

2f sin

2Θi

∣

∣

∣

∣

∣

∣

∣

∣

Θi=0

(B.11)

=1

∆2

ln

(

∆1 − ∆2

∆1 + ∆2

)

= − 1

∆2

ln

(

∆1 + ∆2

∆1 − ∆2

)

(B.12)

which is and has to be identical with (B.5). So in both cases, ∆2 > 0 and∆2 < 0, (B.5,B.12) are generated by setting Θi = 0, π; therefore one doesnot have to distinguish between the these cases in (81).

But it is of importance to mention that due to (B.7) one can get numericalproblems if one only implements (67) and wants to calculate the doublydifferential cross section for Θi = 0. Thus it is useful to distinguish forΘi 6= 0 and Θi = 0 and to use (81) instead for the latter case.

For the rest of limiting forward and/or backward scattering it is, however,straight forward to insert Θi = 0, π and thus can deduce (81) from (67) withthe additional help of (B.10).

Appendix C. The doubly differential cross section for ~ω → Ekin,i

There are three contributions from (67) which lead to (88) in the limit~ω → Ekin,i ⇔ |pf | → 0:

ι1 =16πE2

f p2i sin2 ΘiA

(Ei − cpi cos Θi)2(∆2

1 − ∆22 − 4p2

ip2fsin2 Θi)

, (C.1)

ι2 = −2πAp2i c

2 sin2 Θi

(Ei − cpi cos Θi)2

1√

∆22 + 4p2

ip2fsin2 Θi

× ln

∆22 + 4p2i p

2f sin2 Θi −

√

∆22 + 4p2

ip2fsin2 Θi(∆1 + ∆2) + ∆1∆2

−∆22 − 4p2

ip2fsin2 Θi −

√

∆22 + 4p2

ip2fsin2 Θi(∆1 − ∆2) + ∆1∆2

, (C.2)

ι3 = −4π~

2ω2p2i sin2 ΘiA

Ei − cpi cos Θi

−2∆1∆2pf c + 2∆2

2Ef + 8p2i p2f sin2 ΘiEf

(−∆22 + ∆2

1 − 4p2ip2fsin2 Θi)((∆2Ef + ∆1pf c)2 + 4m2c4p2

ip2fsin2 Θi)

+pf c(∆2Ef + ∆1pf c)

√

((∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi)

3

× ln

(

(

(Ef + pf c)(4p2i p

2f sin

2Θi(Ef − pf c) + (∆1 + ∆2)((∆2Ef + ∆1pf c)

−√

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi))

)(

(Ef − pf c)(4p2i p

2f sin

2Θi(−Ef − pf c)

+ (∆1 − ∆2)((∆2Ef + ∆1pf c) −√

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi))

)

−1)]

(C.3)

47

while all other integrals which appear in (67) cancel each other (which willbe shown in an example later). It can be verified easily that

limpf→0

∆1 = δ, (C.4)

limpf→0

∆2 = 0 with ∆2 ∼ |pf | (C.5)

according to definitions (56), (57) and (84). With these limits the behaviorof ι1 for small pf can be calculated in a straight forward way:

limpf→0

ι1 =16πAE2

fp2i sin

2Θi

(Ei − cpi cosΘi)2δ2. (C.6)

For (C.2) and (C.3), however, there is more effort to be invested. As it is√

∆22 + 4p2i p

2f sin

2Θi → 0 and ln

(

(

∆22+4p2ip

2f sin

2Θi−√

∆22 + 4p2i p

2f sin

2Θi

× (∆1 + ∆2) + ∆1∆2

)

/(

− ∆22 − 4p2i p

2f sin

2Θi −√

∆22 + 4p2i p

2f sin

2Θi(∆1

− ∆2) + ∆1∆2

)

)

→ 0 for pf → 0, one has to use the rule of L’Hopital.

If one rewrites

∆2 = Ψpf (C.7)

with

Ψ := −2~cω + 2pi cosΘi (C.8)

this rule leads to

limpf→0

1√

∆22 + 4p2i p

2f sin

2Θi

×

× ln

∆22 + 4p2i p

2f sin

2Θi −√

∆22 + 4p2i p

2f sin

2Θi(∆1 +∆2) + ∆1∆2

−∆22 − 4p2i p

2f sin

2Θi −√

∆22 + 4p2i p

2f sin

2Θi(∆1 −∆2) + ∆1∆2

= −2δ; (C.9)

48

thus

limpf→0

ι2 =4πAp2i c

2 sin2Θi

(Ei − cpi cosΘi)2δ. (C.10)

The limit of (C.3) can also be calculated by using (C.7). It is

ι3 = −4π~

2ω2p2i sin2 ΘiA

Ei − cpi cosΘi

−p2f (2∆1Ψc + 2Ψ2Ef + 8p2i sin2 ΘiEf )

p2f(−∆2

2 + ∆21 − 4p2

isin2 Θi)((ΨEf + ∆1c)2 + 4m2c4p2

isin2 Θi)

×p2f c(ΨEf + ∆1c)

p2f((ΨEf + ∆1c)2 + 4m2c4p2

isin2 Θi)

×1

pf

√

(ΨEf + ∆1c)2 + 4m2c4p2isin2 Θi)

× ln

(

(

(Ef + pf c)(4p2i p

2f sin

2Θi(Ef − pf c) + (∆1 + Ψpf )(pf (ΨEf + ∆1c)

− pf

√

(ΨEf + ∆1c)2 + 4m2c4p2isin2 Θi))

)(

(Ef − pf c)(4p2i p

2f sin

2Θi(−Ef − pf c)

+ (∆1 − Ψpf )(pf (ΨEf + ∆1c) − pf

√

(ΨEf + ∆1c)2 + 4m2c4p2isin2 Θi))

)

−1)]

(C.11)

While pf can simply be reduced in the fractions, one has to use the rule ofL’Hopital again for the logarithmic part because it ispf√

(ΨEf +∆1c)2 + 4m2c4p2i sin2Θi → 0 and the logarithm→ 0 for pf → 0.

Its limit is

limpf→0

1

pf

√

(ΨEf + ∆1c)2 + 4m2c4p2isin2 Θi

× ln

(

(

(Ef + pf c)(4p2i p

2f sin

2Θi(Ef − pf c) + (∆1 + Ψpf )(pf (ΨEf + ∆1c)

− pf

√

(ΨEf + ∆1c)2 + 4m2c4p2isin2 Θi))

)(

(Ef − pf c)(4p2i p

2f sin

2Θi(−Ef − pf c)

+ (∆1 − Ψpf )(pf (ΨEf + ∆1c) − pf

√

(ΨEf + ∆1c)2 + 4m2c4p2isin2 Θi))

)

−1)]

= −2

Ef∆1

∣

∣

∣

∣

∣

pf→0

= −2

Ef δ; (C.12)

thus the whole limit yields after some further calculations

limpf→0

ι3 =8π~2ω2p2i sin

2ΘiA

(Ei − cpi cosΘi)δ2Ef. (C.13)

Finally, if one inserts (14), the sum of (C.6), (C.10) and (C.13) leads to (88).All other terms which appear in (67) vanish. For this purpose one should

regroup all terms according to their origin. As an example let’s consider the

three contributions which have arisen fromπ∫

Θf=0

2π∫

Φ=0

a2 cosΦα cosΦ+β

dΦdΩf . For this

49

integral it follows

π∫

Θf=0

2π∫

Φ=0

a2 cos Φ

α cos Φ + βdΦdΩf

= −2πAc

(Ei − cpi cos Θi)pfln

(

Ef + pf c

Ef − pf c

)

−2πAc2

Ei − cpi cos Θi

−

∆2√

∆22 + 4p2

ip2fsin2 Θipf c

× ln

∆22 + 4p2i p

2f sin2 Θi −

√

∆22 + 4p2

ip2fsin2 Θi(∆1 + ∆2) + ∆1∆2

−∆22 − 4p2

ip2fsin2 Θi −

√

∆22 + 4p2

ip2fsin2 Θi(∆1 − ∆2) + ∆1∆2

−∆2Ef + ∆1pf c

pf c√

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θ2

i

× ln

(

(

(Ef + pf c)(4p2i p

2f sin

2Θi(Ef − pf c) + (∆1 + ∆2)((∆2Ef + ∆1pf c)

−√

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi))

)(

(Ef − pf c)(4p2i p

2f sin

2Θi(−Ef − pf c)

+ (∆1 − ∆2)((∆2Ef + ∆1pf c) −√

(∆2Ef + ∆1pf c)2 + 4m2c4p2ip2fsin2 Θi))

)

−1)]

~ω→Ekin,i−−−−−−−−−→

2πAc

Ei − cpi cos Θi

[

−2c

Ef

+ Ψ

(

−2

δ

)

+(

ΨEf + δc) 2

Ef δ

]

= 0 (C.14)

where we have used (C.9), (C.12) and

limpf→0

1

pfln

(

Ef + pfc

Ef − pfc

)

=2c

Ef(C.15)

in the limiting step. Of course, this term has to vanish because a2 ∼ pf , butthe concrete calculation after having integrated over Φ and Θf is much morecomplicated. Therefore we have just given an example here. Similarly, allother terms cancel so that only the limits of ιi, i ∈ 1, 2, 3, stay.

Appendix D. Discussion of Geant 4

As mentioned in the introduction, preimplemented cross sections for Brems-strahlung can be found in the Geant 4 software library (Agostinelli et al.,2003; geant4.cern.ch). Geant 4 contains data for the total cross section σ, thesingly differential cross section dσ/dω and a singly differential cross sectiondσ/dΩi depending on Θi, but not on ω.

The singly differential cross section dσ/dω by Bethe and Heitler is appro-priate for small Z; it is (Bethe and Heitler, 1934; Heitler, 1944)

dσ

dω(Ei, ω) = χ0

1

ω

pfpi

(χ1 + Lχ2) (D.1)

50

with

χ0 =Z2r20137

, (D.2)

L = ln

(

p2i + pipf − Ei·~ωc2

p2i − pipf − Ei·~ωc2

)

, (D.3)

ǫ0 = 2 ln

(

Ei + cpimec2

)

, (D.4)

ǫ = 2 ln

(

Ef + cpfmec2

)

, (D.5)

χ1 =4

3− 2

EiEf

c2p2f + p2ip2i p

2f

+m2ec

2

(

ǫ0Ef

cp3i+

ǫEi

cp3f− ǫ0ǫ

pipf

)

, (D.6)

χ2 =8

3

EiEf

c2pipf+

(

~

cω

)21

p3i p3f

(

E2i E

2f

c4+ p2i p

2f

)

+m2

ec~ω

2pipf

(

EiEf

c2+ p2i

p3iǫ0 −

EiEf

c2+ p2f

p3fǫ+ 2

~

c3ωEiEf

p2fp2i

)

(D.7)

with the quantities as described in section 2.1.Geant 4 uses a fit formula which is appropriate for large Z; it is (Agostinelli

et al., 2003)

dσ

dω(Ei, ω) =

S(

~ωEkin,i

, ω)

Cω(D.8)

where C is a constant which is not specified in the Geant 4 documentation,nor in the source code. S is defined as

S

(

~ω

Ekin,i, ω

)

=

1 + al~ω

Ekin,i+ bl

(

~ωEkin,i

)2

, Ekin,i < 1 MeV

1− ah~ωEiF1 + bh

(

~ωEi

)2

F2, Ekin,i ≥ 1 MeV(D.9)

where Ekin,i = Ei − mec2 is the kinetic energy of the incident electron. F1

and F2 are defined as

F1 =

F0(42.392− 7.796δ + 1.961δ2 − F ), δ ≤ 1F0(42.24− 8.368 ln(δ + 0.952)− F ), δ > 1

, (D.10)

F2 =

F0(41.734− 6.484δ + 1.250δ2 − F ), δ ≤ 1F0(42.24− 8.368 ln(δ + 0.952)− F ), δ > 1

(D.11)

51

with F = 4 ln(Z)−0.55(ln(Z))2, F0 = 1/(42.392−F ) and δ = 136mec2ǫ/(Z1/3Ei(1−

ǫ)) where ǫ = ~ω/Ei is the ratio between the photon energy and the totalenergy of the incident electron.

ah, bh, al and bl in (D.9) are defined as

ah = 1 +ah1u

+ah2u2

+ah3u3

, (D.12)

bh = 0.75 +bh1u

+bh2u2

+bh3u3

, (D.13)

al = al0 + al1u+ al2u2, (D.14)

bl = bl0 + bl1u+ bl2u2, (D.15)

with u = ln (Ekin,i/(mec2)). The ahi, bhi, ali, bli are directly defined in the

Geant 4 source code as

ahj = ahj,0[Z(Z + 1)]13

(

ahj,1 + [Z(Z + 1)]13ahj,2

)

, j ∈ 1, 2, 3,(D.16)

bhj = bhj,0[Z(Z + 1)]13

(

bhj,1 + [Z(Z + 1)]13 bhj,2

)

, j ∈ 1, 2, 3,(D.17)

alj = alj,0[Z(Z + 1)]13

(

alj,1 + [Z(Z + 1)]13alj,2

)

, j ∈ 1, 2, 3, (D.18)

blj = blj,0[Z(Z + 1)]13

(

blj,1 + [Z(Z + 1)]13 blj,2

)

, j ∈ 1, 2, 3, (D.19)

where all the coefficients are also defined in the source code:

(ah)i,j =

4.67733 −0.619012 −0.020225−7.34101 1.00462 −0.03209852.93119 −0.403761 0.0125153

, (D.20)

(bh)i,j =

4.23071 −6.10995 −0.0195531−7.12527 0.969160 −0.02742552.69925 −0.363283 −0.00955316

, (D.21)

(al)i,j =

−2.05398 0.0238815 0.000525483−0.0769748 −0.0691499 0.002224530.0406463 −0.0101281 0.000340919

, (D.22)

(bl)i,j =

1.04133 −0.00943291 −0.0004547580.119253 0.0407467 −0.00130718−0.0159391 0.00727752 −0.000194405

, (D.23)

52

0

5e-48

1e-47

1.5e-47

2e-47

2.5e-47

10000 100000 1e+06 1e+07 1e+08

dσ/d

ω (

Eki

n,i,ω

) [m

2 s]

Ekin,i [eV]

Bethe - Heitler

Geant 4 0

2e-49

4e-49

6e-49

8e-49

1e-48

1.2e-48

1.4e-48

1.6e-48

100000 1e+06 1e+07 1e+08

dσ/d

ω (

Eki

n,i,ω

) [m

2 s]

Ekin,i [eV]

Bethe - Heitler

Geant 4

a) ~ω = 10 keV b) ~ω = 100 keV

0

2e-50

4e-50

6e-50

8e-50

1e-49

1.2e-49

1.4e-49

1e+06 1e+07 1e+08

dσ/d

ω (

Eki

n,i,ω

) [m

2 s]

Ekin,i [eV]

Bethe - Heitler

Geant 4

c) ~ω = 1 MeV

Figure D.13: The singly differential cross sections (D.1) and (D.8) as a function of thekinetic energy Ekin,i of the incident electron for Z = 7 (nitrogen) and for fixed photonenergy a) ~ω = 10 keV, b) ~ω = 100 keV and c) ~ω = 1 MeV.

with i ∈ 1, 2, 3 and j ∈ 0, 1, 2. In (D.20) - (D.23) the first index denotescolumns, the second one denotes rows.

Figure D.13 compares the Bethe Heitler cross section (D.1) with that ofGeant 4 (D.8) where we have chosen C = 1028 for all energies in such a waythat the orders of magnitude of (D.1) and (D.8) agree with each other. Itshows that (using exactly the values provided in the source code of Geant4) that there is a good quantitative and qualitative agreement for electronenergies of ≈ 1 MeV and ≈ 10 MeV. But above and below that, both crosssections certainly differ.

That is because Geant 4 was developed for high energy energy physicsin particle accelerators and thus for high atomic numbers. Thus the cross

53

0.85

0.9

0.95

1

1.05

1.1

1000 10000 100000 1e+06 1e+07 1e+08 1e+09

S(E

γ)

Eγ [eV]

Ekin,i=2 keV, 100 keV,1 MeV, 10 MeV

Ekin,i=1 GeV

0.85

0.9

0.95

1

1.05

1.1

1000 10000 100000 1e+06 1e+07 1e+08 1e+09

S(E

γ)

Eγ [eV]

Ekin,i=2 keV, 100 keV,1 MeV, 10 MeV

Ekin,i=1 GeV

a) ne ≈ 2 · 1025 m−3 b) n ≈ 1024 m−3

Figure D.14: The dielectric factor S (D.24) vs. the photon energy for different electronenergies for a) ne ≈ 2 · 1025 m−3 and b) ne ≈ 1024 m−3

sections used in Geant 4 are not appropriate to describe the production ofBremsstrahlung photons in air. The Bethe - Heitler theory for the energyrange we consider, is used for small atomic numbers.

Geant 4 also includes dielectric suppression, i.e. the suppression of theemission of lowly energetic photons because of their interaction with theelectrons of the background medium (Ter-Mikaelian, 1954), and the Landau-Pomeranchuk-Migdal (LPM) effect (Landau and Pomeranchuk, 1953), i.e.the suppression of photon production due to the multiple scattering of elec-trons.

The influence of the dielectric effect can be estimated by

S(~ω) =(~ω)2

(~ω)2 +~2E2

i nee2

m3ec

2ǫ0

(D.24)

where ne is the density of free electrons. For densities between 1020 m−3

and 1025 m−3, S is almost 1. Figure D.14 shows (D.24) for different photonenergies, electron energies and densities. Dielectric suppression has a verysmall effect when Ekin,i ≈ 1 GeV; thus it can be neglected.

The LPM effect is not important, either. The LPM threshold energy is≈ 1019 eV (Bertou et al., 2000); this is much higher than typical energies ofelectrons in the atmosphere.

The preimplemented cross sections used in Geant 4 are supposed to beused for high electron energies & 1 MeV and high atomic numbers Z. In

54

the case of TGFs it is necessary to treat electron energies in the keV andMeV range and small atomic numbers where the LPM effect and dieletricsuppression are not significant.

Appendix E. Comparison with Lehtinen (2000)

Figure E.15 shows the comparison of (67) and the doubly differential crosssection used by Lehtinen. Lehtinen uses a product ansatz for the angularand the frequency part; here the angular part is a non-quantum mechanicalexpression taken from (Jackson, 1975, p. 712 et seq.). This cross section isonly valid if ~ω ≪ Ei. There is a good agreement for low ratios betweenphoton and electron energy, but a large deviation for larger ratios. Thereforethis cross section is not appropriate for high ratios needed to obtain photonswith energies up to several tens of MeV to determine the high energy tail ofthe TGF spectrum where almost all electron energy is converted into photonenergy.

Appendix F. Contribution of the atomic form factor

Dwyer (2007) uses the triply differential cross section by Bethe and Heitler(1934), but with an additional form factor F (q) parameterizing the structureof the nucleus (Koch and Motz, 1959). F is defined as

F (q) := − 1

Ze

∫

d3r(r)e−i~q·r (F.1)

where Z is the atomic number and the charge density

(r) = Zeδ(r)− Ze

4πa2re−

ra (F.2)

with a = 111λ/Z−1/3 where λ/ = λ/(2π) is the reduced Compton wave lengthof the electron. The delta function describes the nucleus itself and the Debyeterm describes the electrons of the shell. Performing the Fourier transforma-tion in (F.1) gives

F (q) =q2

q2 + ~2

a2

(F.3)

with q as in Eq. (9). We calculated the value of F (q) for different angles,electron and photon energies [a) Ekin,i = 100 keV, ~ω = 10 keV, Θf =

55

1e-45

1e-44

1e-43

1e-42

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

Product ansatz (Lehtinen)

Result of integration

1e-52

1e-51

1e-50

1e-49

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 πd2 σ(

Eki

n,ω

,Θi)/

(dω

dΩi)

[m2 s/

rad]

Θi

Product ansatz (Lehtinen)

Result of integration

a) Ekin,i = 150 keV, ~ωEkin,i

= 10−5 b) Ekin,i = 150 keV, ~ωEkin,i

= 0.9

1e-47

1e-46

1e-45

1e-44

1e-43

1e-42

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

Product ansatz (Lehtinen)

Result of integration 1e-54

1e-53

1e-52

1e-51

1e-50

1e-49

0 π/9 2π/9 3π/9 4π/9 5π/9 6π/9 7π/9 8π/9 π

d2 σ(E

kin,

ω,Θ

i)/(d

ωdΩ

i) [m

2 s/ra

d]

Θi

Product ansatz (Lehtinen)

Result of integration

a) Ekin,i = 1 MeV, ~ωEkin,i

= 10−5 b) Ekin,i = 1 MeV, ~ωEkin,i

= 0.9

Figure E.15: Comparison of the product ansatz from Lehtinen (2000) with our result (67)of the integration of (1) for different electron energies (Z = 7): doubly differential crosssection versus the scattering angle Θi between incident electron and emitted photon. Theratio between the photon energy ~ω and the kinetic electron energy Ekin,i is fixed to0.001% and 90%. The Born approximation (11) is valid in all cases.

56

1e-47

1e-46

1e-45

1e-44

1e-43

1e-42

1e-41

1e-40

0 20 40 60 80 100 120 140 160 180

[m2 s

]

Θi [°]

I6(Ekin,i,ω,Θi)

I3(Ekin,i,ω,Θi)

d2σ(Ekin,i,ω,Θi)/(dωdΩi)

1e-50

1e-49

1e-48

1e-47

0 20 40 60 80 100 120 140 160 180

[m2 s

]

Θi [°]

I6(Ekin,i,ω,Θi)

I3(Ekin,i,ω,Θi)

d2σ(Ekin,i,ω,Θi)/(dωdΩi)

a) Ekin,i = 100 keV, ~ω = 1 keV b) Ekin,i = 100 keV, ~ω = 95 keV

1e-52

1e-50

1e-48

1e-46

1e-44

1e-42

1e-40

0 20 40 60 80 100 120 140 160 180

[m2 s

]

Θi [°]

I6(Ekin,i,ω,Θi)

I3(Ekin,i,ω,Θi)

d2σ(Ekin,i,ω,Θi)/(dωdΩi)

1e-60

1e-58

1e-56

1e-54

1e-52

1e-50

1e-48

1e-46

0 20 40 60 80 100 120 140 160 180

[m2 s

]

Θi [°]

I6(Ekin,i,ω,Θi)

I3(Ekin,i,ω,Θi)

d2σ(Ekin,i,ω,Θi)/(dωdΩi)

c) Ekin,i = 10 MeV, ~ω = 100 keV d) Ekin,i = 10 MeV, ~ω = 9.5 MeV

Figure G.16: Contribution of (68) - (73) to (67) in a semilog plot for different electron andphoton energies (Z = 7).

37,Φ = 87; b) Ekin,i = 100 keV, ~ω = 80 keV, Θf = 62,Φ = 43; c)Ekin,i = 10 MeV, ~ω = 1 MeV, Θf = 12,Φ = 31 and d) Ekin,i = 50 MeV,~ω = 10 MeV, Θf = 52,Φ = 90]. In all these cases the atomic form factoris 1. Hence, it can be neglected. As it makes the integration over Φ and Θf

more complicated, it is useful not to use F (q).

Appendix G. Contribution of the integrals

As equation (67) is rather complicated, it is interesting to see which termshave the most important contribution. Figure G.16 shows the contributionof all parts to the final result in a logarithmic scale while Fig. G.17 showsthe same in a linear scale. In all cases, i.e. low and high electron energiesand low and high ratios between ~ω and Ekin,i, the main contribution comesfrom (70). It is important to state that not all contributions can be seen in

57

-8e-42

-6e-42

-4e-42

-2e-42

0

2e-42

4e-42

6e-42

8e-42

0 20 40 60 80 100 120 140 160 180

[m2 s

]

Θi [°]

I6(Ekin,i,ω,Θi)

I5(Ekin,i,ω,Θi)

d2σ(Ekin,i,ω,Θi)/(dωdΩi)

-1.5e-48

-1e-48

-5e-49

0

5e-49

1e-48

1.5e-48

0 20 40 60 80 100 120 140 160 180

[m2 s

]

Θi [°]

I6(Ekin,i,ω,Θi)

I5(Ekin,i,ω,Θi)

I3(Ekin,i,ω,Θi)d2σ(Ekin,i,ω,Θi)/(dωdΩi)

a) Ekin,i = 100 keV, ~ω = 1 keV b) Ekin,i = 100 keV, ~ω = 95 keV

-8e-40

-6e-40

-4e-40

-2e-40

0

2e-40

4e-40

6e-40

8e-40

0 20 40 60 80 100 120 140 160 180

[m2 s

]

Θi [°]

I6(Ekin,i,ω,Θi)

I5(Ekin,i,ω,Θi)

d2σ(Ekin,i,ω,Θi)/(dωdΩi)

c) Ekin,i = 10 MeV, ~ω = 100 keV

Figure G.17: Contribution of (68) - (73) to (67) in a linear plot for different electron andphoton energies (Z = 7).

58

figure G.16 because some of the terms have negative values which, however,are shown in figure G.17. So one might think that for Ekin,i = 100 keV and~ω = 1 keV, equation (73) has the largest contribution, but as figure G.17shows, (72) has nearly the same absolute value, but opposite sign; thereforethey cancel. Thus the third integral (70) is the most important one. Thesame holds for other electron energies and ratios between ~ω and Ekin,i. Weconclude that (70) is the dominant contribution for all relevant parametervalues.

Appendix H. Conservation of energy and momentum

One can also gain information on the scattering angle Θi for high electronenergies from the conservation of energy and momentum,

Ei + Eq = Ef + ~ω, (H.1)

pi + q = pf + ~k (H.2)

where Ei,f and pi,f are the energy and the momentum of the electron in theinitial and final state. ~k is the momentum of the photon which is relatedto its energy ~ω through

~|k| = ~

cω, (H.3)

and Eq and q are the energy and the momentum of the virtual photon be-tween electron and nucleus. q changes the momentum of the nucleus. Butthe contribution to the kinetic energy can be neglected; thus Eq ≡ 0 and

Ei = Ef + ~ω, (H.4)

pi − ~k = pf − q. (H.5)

Squaring (H.5) and using pi · k = |pi||k| cos∢(pi,k) = pi k cosΘi, the angleΘi is:

cosΘi =(pf − q)2 − p2i − ~2k2

−2~pik. (H.6)

By using (H.4) and the relativistic energy-momentum relation (2) we get anexpression for the momentum of the electron in the final state

pf =

√

p2i + ~2k2 − 2~k√

p2i +m2ec

2 (H.7)

59

which leads to

cosΘi =2~ωEi

c+ 2q

√

(Ei − ~ω)2 −m2ec

4 cos∢(pf ,q)− cq2

2~ωpi. (H.8)

Although this is an analytical expression for the scattering angle Θi oneshould take into account that it depends on the vector q of the virtual photonwhich is not known in forehand. Thus, depending on q, only a statisticalstatement can be made about Θi.

It is, however, possible to investigate the limit of (H.8) for high electronenergies which yields

limEi→∞

cosΘi = 1 +cq

~ωcos∢(pf ,q) (H.9)

As Θi ∈ R⇔ cosΘi ∈ [−1,+1] and c, q, ~ω > 0 we can conclude that

cos∢(pf ,q) ≤ 0 (H.10)

Especially for | cq~ω

cos∢(pf ,q)| ≪ 1,Θi ≈ 0, i.e., the photon is mainly emittedin forward direction.

If, additionally, the photon energy ~ω also increases more and more (forhigh electron energies) it is

lim~ω→∞

(

1 +cq

~ωcos∢(pf ,q)

)

= 1; (H.11)

thus

limEi,~ω→∞

Θi = 0. (H.12)

Hence, we conclude from simple considerations about energy and momentumconservation that the photon is mainly scattered in forward direction if theenergies of electron and photon are both very high.

60

Appendix I. Approximation for Θi

In order to obtain (114) we calculate the derivative of (88) after Θi:

∂

∂Θi

(

d2σ

dω sinΘidΘi

)

=Z2α3

fine~2

π

|pf ||pi|ω

[

4~

cω|pi| sinΘi

δ3(Θi)×

×(

sin2Θi

(Ei − c|pi| cosΘi)2(4E2

f + δ(Θi)c2) +

2~2ω2

Ef

sin2Θi

Ei − c|pi| cosΘi

)

+1

δ2(Θi)

(

2 sinΘi cosΘi(Ei − c|pi| cosΘi)− 2c|pi| sin3Θi

(Ei − c|pi| cosΘi)3

× (4E2f + δ(Θi)c

2)− 2~cω|pi| sin3Θi

(Ei − c|pi| cosΘi)2

+2~2ω2

Ef

2 sinΘi cosΘi(Ei − c|pi| cosΘi)− c|pi| sin3Θi

(Ei − c|pi| cosΘi)2

)]

(I.1)

with definition (84) for δ. In order to calculate the extrema one has to setequation (I.1) equal to zero:

0 = 4~

cω|pi|

[

sin2Θi

(Ei − c|pi| cosΘi)2(4E2

f + δ(Θi)c2)

+2~2ω2

Ef

sin2Θi

Ei − c|pi| cosΘi

]

(Ei − c|pi| cosΘi)3

+ δ(Θi)[

2(Ei cosΘi − c|pi|)(4E2f + δ(Θi)c

2)

− 2~cω|pi| sin2Θi(Ei − c|pi| cosΘi)

+2~2ω2

Ef

(

2 cosΘi(Ei − c|pi| cosΘi)2

− c|pi| sin2Θi(Ei − c|pi| cosΘi))]

. (I.2)

As δ(Θi) ∼ cosΘi, expression (I.2) is quartic in cosΘi; therefore (I.2) can besolved analytically in principle, but the solution will be long and complicated.Figure 3 also shows that the angles for maximal scattering are very small forrelativistic electrons, therefore one can approximate cosΘi ≈ 1 and sinΘi ≈Θi. This leads to

δ(Θi) ≈ −|pi|2 −(

~

cω

)2

+ 2~

cω|pi| = δ(Θi = 0) =: δ0 (I.3)

61

and

0 = 4~

cω|pi|

[

Θ2i (4E

2f + δ0c

2) +2~2ω2

Ef

Θ2i (Ei − c|pi|)

]

+ δ0

[

2(4E2f + δ0c

2)− 2~cω|pi|Θ2i +

2~2ω2

Ef

(

2(Ei − c|pi|)− c|pi|Θ2i

)

]

(I.4)

with solution

Θi =

√

√

√

√

√

− δ0~ω(4E2

f + δ0c2)− 2δ0~ωEf

(Ei − c|pi|)

2 |pi|c

[

4E2f + δ0c2 +

2~2ω2

Ef(Ei − c|pi|)

]

− |pi|δ0c− ~ωEf

c|pi|δ0(I.5)

62

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69

Vitae:Christoph Kohn studied physics in Kiel and Hamburg, Germany from

2005 till 2010. After having finished his diploma thesis on six-dimensional su-pergravity, he started his PhD studies at CWI Amsterdam, The Netherlands.

Ute Ebert studied physics at the University of Heidelberg, Germany,and she wrote her PhD thesis at the University of Essen, Germany. Aftera postdoc period at the University of Leiden, Netherlands, she became staffmember at CWI Amsterdam, Netherlands. Since 2002 she leads a researchgroup at CWI and is part time professor at Eindhoven University of Technol-ogy. The research of her group concentrates on transient electrical discharges,both in plasma technology and in atmospheric electricity.

• Analytical results on doubly differential cross-sections for typical TGFparameters

• Distribution of emission angles and energies for Bremsstrahlung pho-tons

• Distribution of emission angles and energies for positrons in pair pro-duction

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

70