Magnetar X-ray spectra an ultramagnetic QED approach...-Seasons in the Abyss by Slayer (misheard)...

Magnetar X-ray spectra: an ultramagnetic QED approach

by

Alexander Kostenko

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Astronomy & AstrophysicsUniversity of Toronto

Copyright c© 2020 by Alexander Kostenko

Abstract

Magnetar X-ray spectra: an ultramagnetic QED approach

Alexander Kostenko

Doctor of Philosophy

Graduate Department of Astronomy & Astrophysics

University of Toronto

2020

This thesis considers the generation of the bright, broadband and non-thermal

X-ray spectra of magnetars. These spectra are very sensitive to the distribution of e± in

the current-carrying magnetic field tubes; previous approaches considered generating

the non-thermal components by modifying a seed thermal spectrum with resonant

cyclotron upscattering on ultrarelativistic outflows, as well as through bremsstrahlung

and synchrotron emission. The present approach constructs a Monte Carlo model of

this persistent emission, where both “AXP-like” and “SGR-like” spectra are generated

self-consistently as a result of the interaction of a warm magnetospheric plasma layer

and a cold, dense neutron star atmospheric surface layer. This approach is motivated

in part by evidence that the magnetospheric currents are strongly localized. In order

to construct this model, a range of QED rates and cross sections are reviewed and

calculated in the ultramagnetic background, many for the first time. In particular,

the high electron-positron scattering and bremsstrahlung rates as well as the highly

kinematically modified rates of two-photon pair creation and annihilation lead to a

physical situation that differs qualitatively from previous models, including unique

potential observational signatures.

ii

“Close your eyes

And forget your name

Step outside yourself

And let your thoughts reign”

-Seasons in the Abyss by Slayer (misheard)

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Acknowledgements

As is customary, a few thank-yous are in order. First of all, I thank my parents,

Vitaly and Elena, who gave up everything when they moved to Canada in order to give

me a better future, and kept on giving and nurturing all the way up to my defence.

Second, I thank my two Universities (Kharkiv Karazin and UofT) for moulding me

into a capable scientist. Third, I thank my supervisor, Christopher Thompson, for

giving me the privilege of doing meaningful work under his wing in a field that he

basically created, and for being there day and night no matter the question. Fourth, I

thank Massey College for rounding me out as a human and teaching me to look to the

future and my place in it with confidence and optimism. And last but not least, I thank

my love, Mariela Faykoo-Martinez for being a rock of support and understanding in

this incredibly difficult portion of my life and education - I can’t wait to do the same

for you when you defend.

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Contents

1 Introduction 1

1.1 Energy sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Magnetar spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Emission mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Current approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Plan of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 QED Interactions in a Background Magnetic Field 10

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Electrons and Positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Dirac Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Rules for Calculating Matrix Elements . . . . . . . . . . . . . . . . . . . . 17

3 Photon processes: scattering and pair creation 20

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Electron-Photon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Cross Section for B BQ . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 Finite-B Correction to the Cross Section . . . . . . . . . . . . . . . 28

3.2.3 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Electron-Positron Pair Creation . . . . . . . . . . . . . . . . . . . . . . . . 31

v

3.3.1 Single-photon Pair Creation into the Lowest Landau Level . . . . 33

3.3.2 Compton-assisted Pair Creation . . . . . . . . . . . . . . . . . . . 34

3.3.3 Two-photon Pair Creation . . . . . . . . . . . . . . . . . . . . . . . 39

4 e± processes: scattering, annihilation and bremsstrahlung 46

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Two-photon Pair Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Electron-Positron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Scattering of Electrons and Positrons off Heavy Ions . . . . . . . . . . . 56

4.4.1 Quantum Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4.2 Classical Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.5 Relativistic e±-Ion Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . 59

4.5.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5.2 Limiting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5.3 Thermal Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . 66

4.5.4 Validity of the Born Approximation . . . . . . . . . . . . . . . . . 73

4.5.5 Comparison with Radiative Recombination . . . . . . . . . . . . 74

4.5.6 Electron-Positron Bremsstrahlung. . . . . . . . . . . . . . . . . . . 76

5 Self-consistent broadband magnetar X-ray spectra 78

5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Magnetospheric plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Cooling Layer Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4.1 Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4.2 Atmospheric and Magnetospheric Model . . . . . . . . . . . . . . 88

5.4.3 Particle propagation algorithm . . . . . . . . . . . . . . . . . . . . 90

5.4.4 Convergence and data accumulation . . . . . . . . . . . . . . . . 93

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5.1 AXP-like spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5.2 SGR-like spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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6 Conclusions & Future Work 108

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 Future Work and Observational Predictions . . . . . . . . . . . . . . . . . 111

A Electron scattering matrix element 114

B Electron-positron scattering integrals 116

C Ionic Integrals 118

D Integral and differential cross sections 120

D.1 Integral cross sections and accumulation of optical depth . . . . . . . . . 120

D.2 Differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Bibliography 122

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List of Tables

4.1 Gaunt Factor and Related Error for T/m = 10−2,−1.5,...,0.5,1 and B = 100 BQ 68

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List of Figures

3.1 Feynman diagrams for Compton scattering. . . . . . . . . . . . . . . . . 22

3.2 Photon frequency shift by electron scattering. . . . . . . . . . . . . . . . 24

3.3 Differential nonresonant electron scattering cross section dependences. 26

3.4 Integral nonresonant electron scattering cross section dependency. . . . 27

3.5 Analysis of the effects of truncated expansion in intermediate Landau

level on the nonresonant electron scattering cross section. . . . . . . . . 29

3.6 Breakdown of integral nonresonant scattering cross section with respect

to pair creation in the final state. . . . . . . . . . . . . . . . . . . . . . . . 36

3.7 Kinematic range of the initial photon that permits Compton-assisted

pair creation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 Feynman diagrams for two-photon pair creation. . . . . . . . . . . . . . 40

3.9 Direction-averaged two-photon pair creation collision rate. . . . . . . . . 42

3.10 Direction- and frequency-averaged two-photon pair creation collision

rate weighed by target photon distribution. . . . . . . . . . . . . . . . . . 43

4.1 Regions of phase space of the two final-state pair annihilation photons

that result in subsequent one-photon pair creation. . . . . . . . . . . . . 50

4.2 Integral cross section for two-photon annihilation, as a function of the

kinetic energy of the incoming electron and positron as measured in

the center-of-momentum frame. . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Feynman diagrams for electron-positron scattering. . . . . . . . . . . . . 53

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4.4 Feynman diagram for electron-ion scattering. . . . . . . . . . . . . . . . . 57

4.5 Classical and quantum cross sections for the backscattering of a positron

off a proton in a strong magnetic field as functions of initial kinetic

energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.6 Feynman diagrams for bremsstrahlung, evaluated in the approximation

where the ion is infinitely heavy. . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Comparison of the relativistic bremsstrahlung cross section with and

without an ultrastrong background magnetic field. . . . . . . . . . . . . 65

4.8 Gaunt factor as a function of ω/T for a range of temperatures T. . . . . 67

4.9 Comparison of the bremsstrahlung Gaunt factor and the recombination

Gaunt factor for a range of subrelativistic temperatures. . . . . . . . . . 77

5.1 Overview of the combined magnetosphere and cold atmosphere system

under consideration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Sample of frequency grid used for treating two-photon pair creation. . . 88

5.3 Spectral breakdowns for AXP and SGR model spectra. . . . . . . . . . . 96

5.4 Total AXP spectra (without deep cooling E-mode) produced by the model. 97

5.5 Individual components of AXP spectra produced by the model. . . . . . 98

5.6 E-mode contribution for AXPs. . . . . . . . . . . . . . . . . . . . . . . . . 99

5.7 Ratios of the cooling layer blackbody flux and the total pre-redshift

O-mode escaping magnetospheric flux to the seed e± total energy flux. 102

5.8 Annihilation optical depth (analytical and numerical) and pair repro-

duction percentage in magnetosphere. . . . . . . . . . . . . . . . . . . . . 103

5.9 Contribution of electron-positron bremsstrahlung to AXP spectra. . . . 104

5.10 Total SGR spectra produced by the model. . . . . . . . . . . . . . . . . . 106

5.11 E-mode contribution for SGRs. . . . . . . . . . . . . . . . . . . . . . . . . 107

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Chapter 1

Introduction

Formed in the gravitational collapses of stellar cores in supernova explosions, neutron

stars have captivated the scientific community since their discovery some 50 years

ago. In part, this enduring interest is due to their extreme properties, as they feature

some of the current universe’s highest observable densities and magnetic fields. Ob-

servations of neutron stars, which are capable of emitting radiation through a variety

of mechanisms fed by a multitude of energy sources, therefore allow researchers to

probe the extreme limits of physical theories.

1.1 Energy sources

All neutron stars are born with significant amounts of internal heat, which the star

radiates over the course of its lifespan. Hot, newly formed stars cool primarily

through neutrino emission before transitioning to a photon cooling stage (van Riper

1991; Lattimer et al. 1994; Yakovlev et al. 2001; Ofengeim & Yakovlev 2017). The

properties and structure of their photon-emitting atmospheres, particularly with

respect to the effects of the powerful magnetic fields on the energy transfer, have been

considered extensively in prior literature and are now thought to be well understood

(Miller 1992; Shibanov et al. 1992; Pavlov et al. 1994; Ho & Lai 2003; Özel 2001; 2003;

1

Chapter 1. Introduction 2

Potekhin & Chabrier 2003; Ho et al. 2008; Suleimanov et al. 2010; Potekhin et al. 2014;

Bauböck et al. 2019). Radiative transfer dominates energy transport in these low-

density atmospheres, with most of the energy carried by a low-opacity polarization

mode (Ho & Lai 2001; Ho et al. 2007), while electron conduction rises in importance

in the dense envelope and beyond (Hernquist 1985; Heyl & Hernquist 1998; van

Adelsberg & Lai 2006; Potekhin et al. 2007).

Similarly, the conservation of angular momentum during stellar core collapse gives

all neutron stars extremely high initial rotational velocities, which can source simple

rotating magnetic dipole emission (Ng & Kaspi 2011). The combination of rapid

rotation and high magnetic fields also leads to the appearance of magnetospheric

charges and currents. The magnetic field lines anchored at the magnetic poles intersect

the star’s light cylinder and are thus open, leading to particle outflows and torque

even in the absence of dipole emission (Goldreich & Julian 1969; Spitkovsky 2006;

Vink et al. 2011; Timokhin & Arons 2013). In order to sustain these outflows, pair

cascades occur along the polar field lines, causing particle bombardment of the poles

which feeds thermal emission (Ruderman & Sutherland 1975; Kundt & Schaaf 1993;

Zavlin et al. 1995; Becker & Truemper 1997). The various accelerated magnetospheric

particles emit curvature, synchrotron, and inverse Compton radiation (Michel 1991;

Hibschman 2002; Harding et al. 2008; Kisaka & Tanaka 2017).

If a binary system contains a neutron star and a less-evolved companion star,

another emission mechanism opens up. Roche lobe overflow and stellar winds

transfer matter from the companion to the neutron star, which gravitationally captures

it and forms an accretion disk (Ghosh & Lamb 1978). As the matter flows closer to the

star, magnetic pressure begins affecting its dynamics, until it eventually overcomes the

ram pressure and channels the infalling matter along the field lines and towards the

neutron star. Given the powerful magnetic fields in question, this magnetic pressure

domination occurs far out from the star (∼ 108 cm), and so matter accretion columns


with trans-relativistic velocities and temperatures as high as ∼ 108K form above the

neutron star’s polar regions (Basko & Sunyaev 1976; Harding et al. 1984). These

columns are capable of generating and upscattering X-ray radiation (Becker & Wolff

2007; Farinelli et al. 2012; Postnov et al. 2015; West et al. 2017; Wolff et al. 2019), with

the nature of the interaction depending heavily on geometry and the accretion rate

(Harding et al. 1984; Becker et al. 2012). The spectra of a small subset of accreting

pulsars also display cyclotron resonance scattering features (Staubert et al. 2019), the

analysis of which sheds light on the environment close to the neutron star surface,

particularly the magnetic field strength (Wang et al. 1988; Araya & Harding 1996;

Schönherr et al. 2007; 2014; Schwarm et al. 2017a;b).

Finally, the emission of some neutron stars is believed to be powered by the decay

of their strong, non-potential magnetic fields well exceeding BQ ≡ m2/e = 4.4× 1013

G, the field at which the cyclotron energy equals the rest-mass energy for an electron.

Known as magnetars, these objects are rare, with about 20 known to exist (Olausen &

Kaspi 2014; Enoto et al. 2017a). The rotation of these objects, characterized by slow

spin periods P = 2− 12s and spindown rates P ∼ 10−12 − 10−10s s−1 (corresponding

to characteristic ages τc ∼ 104yr) often cannot supply their high persistent X-ray

luminosity LX ∼ 1034 − 1036erg s−1, which can exceed their spindown luminosity by

several orders of magnitude 1. In addition to this persistent emission, some magnetars

also exhibit sporadic bursting behaviour, emitting X-rays and gamma rays at extremely

super-Eddington luminosities and changing the properties of the persistent emission

(Fenimore et al. 1996; Mazets et al. 1999; Ibrahim et al. 2001). Despite the paucity

of detected sources, due to the relatively young ages of active magnetars, some

population synthesis analyses suggest that as many as 20 percent of all supernovae

1This is the case with traditional, persistently bright magnetars, which typically exhibit little fluxvariability outside of bursts. More recently, so-called “transient” magnetars have been discovered,whose X-ray luminosities vary between ∼ 1033 − 1035erg s−1(Ibrahim et al. 2004; Olausen & Kaspi 2014;Alford & Halpern 2016).


give birth to a magnetar (Gill & Heyl 2007).

1.2 Magnetar spectra

Several scenarios have been proposed to explain the properties and emission of

magnetars. These include accretion from a fossil disk (van Paradijs et al. 1995;

Chatterjee et al. 2000; Perna et al. 2000; Alpar 2001; Trümper et al. 2010), quark stars

(Ouyed et al. 2004) and quickly rotating massive white dwarfs (Malheiro et al. 2012).

The analyses presented in this thesis, however, are based on the most widely accepted

model, which posits that the emission is powered by the dissipation of the external

magnetic field’s helicity, a process that converts magnetic energy into plasma energy

(Duncan & Thompson 1992; Thompson & Duncan 1995; 1996). The helicity of this

external field is resupplied when the internal, highly twisted magnetic field shears

the crust during “starquakes” (Thompson & Duncan 2001; Beloborodov & Thompson

2007), a process that is associated with the aforementioned burst activity (Thompson

et al. 2002; 2017). This powerful internal field is assumed to be formed after supernova

core collapse, with convective motions strongly amplifying the seed magnetic field

through dynamo action (Duncan & Thompson 1992; Thompson & Duncan 1993).

The persistent X-ray emission of magnetars often extends with rising intensity

to at least 100 keV (Kuiper et al. 2006; Mereghetti et al. 2006; Olausen & Kaspi

2014). Presently, the combined efforts of several X-ray telescopes (Suzaku, INTEGRAL

NuSTAR, Swift, and RXTE to name a few) are able to probe these spectra from soft

X-rays up to gamma rays, with the highest sensitivity in the 0.2-79 keV range (Morii

et al. 2010; Enoto et al. 2011; Vogel et al. 2014; An et al. 2015; Weng & Gögüs, 2015;

Enoto et al. 2017b). These spectra can be roughly divided into two groups:

1. A blackbody peak that connects to a steep falling intermediate power law

component (photon index of Γ ∼ 2− 4) in the 1-10 keV range, which then breaks to a

hard rising component in the 10-79 keV range (photon index of Γ ∼ 1− 1.5) (e.g. 4U


0142+61, 1E 2259+586,1E 1048.1-5937). The energy flux at the break point is a small

fraction (. 10%) of the blackbody flux.

2. A blackbody peak that connects directly or almost directly to a rising (or flat)

power law component, which sometimes shows a much milder break than in 1 (e.g.

SGR 1806-20, SGR 1900+14, 1E 1841-045). In some cases the 100 keV energy flux is

substantially stronger than the blackbody component.

This spectral classification does not uniquely correlate with the typical anomalous

X-ray pulsar (AXP) and soft gamma-ray repeater (SGR) grouping of magnetars. Nev-

ertheless, AXPs are typically associated with spectra from the first group (Weng &

Gögüs, 2015) with SGRs typically having spectra from the second (Enoto et al. 2017b).

1.3 Emission mechanisms

For about ∼ 106 years after they are formed, isolated neutron stars emit soft thermal

X-ray radiation at temperatures exceeding 105K. The structure of the radiative outer

layers is heavily influenced by the complicated interior physics of the neutron star

(Miller 1992; Shibanov et al. 1992; Pavlov et al. 1994). Powerful magnetic fields signifi-

cantly lower the electromagnetic coupling of the star’s electrons to the polarization

mode that is perpendicular to the magnetic field, greatly increasing that mode’s ability

to carry energy and bringing the photosphere to Thomson depths . 103 (Ventura

1979; Mészáros 1992; Ho & Lai 2001; Potekhin & Chabrier 2003; Ho et al. 2007). The

strong internal magnetic fields also greatly affect the internal temperature profiles of

neutron stars (van Riper 1991; Yakovlev et al. 2001), and can even lead to the formation

of condensed, metallic layers with unique radiative properties (Lai & Salpeter 1997;

Potekhin et al. 1999; van Adelsberg et al. 2005; Harding & Lai 2006).

Numerous analyses have considered the origin of the various non-thermal com-

ponents of persistent magnetar spectra. Resonant cyclotron scattering in particular

has received attention in prior literature owing to its large cross section, allowing


for significant modifications of emergent spectra with astrophysically small amounts

of matter. Given the inhomogeneities of the magnetic field and the e± distributions,

this process can lead to the formation of extended spectral components, rather than

a narrow cyclotron line. Lyutikov & Gavriil (2006) applied the methods developed

by Zheleznyakov (1996) to semi-analytically investigate the effect of resonant scat-

tering on the one-dimensional radiative transfer of thermal surface emission in the

non-relativistic limit. This model, where the resonant cyclotron scattering of thermal

surface emission by the warm magnetospheric plasma generates the nonthermal soft

X-ray component (. 10 keV), was applied by Rea et al. (2008) to fit the X-ray data of

several magnetars.

This approach was subsequently refined to include the detailed, three-dimensional

structure of the magnetosphere (Thompson et al. 2002). Fernández & Thompson (2007)

built a Monte Carlo code to treat the multiple resonant scattering of thermal seed

photons as they escape through a twisted, self-similar magnetosphere. This approach

was updated by Nobili et al. (2008) to include bulk motions of magnetospheric

particles, as well as non-trivial angular and temperature distributions of the seed

thermal photons. Zane et al. (2009) subsequently included the relativistic QED

resonant scattering cross section and used this refined model to fit the same data as

Rea et al. (2008).

Attempts have also been made to explain the persistent hard X-ray emission (& 10

keV) through resonant cyclotron scattering off of particles accelerated to relativistic

energies by the high voltages associated with the magnetic twist (Thompson & Be-

loborodov 2005; Beloborodov & Thompson 2007). Baring & Harding (2007) carried out

a semi-analytical analysis using a collisional integral, finding that relativistic scattering

of thermal seed photons close to the magnetar surface can produce a hard and highly

observer-dependent spectrum. Beloborodov (2009; 2013) developed a model which

posits that the magnetic twist is concentrated on closed polar field lines that extend


far from the star. Discharges along these lines create ultrarelativistic e± outflows that

experience drag from the star’s thermal radiation. When the outflow is still in the

strong-field zone, these upscattered photons are quickly transformed into secondary

e±. Only in the outer portions of the field lines are photons able to escape, producing

a hard, observer-dependent spectrum that typically peaks around 1 MeV.

These hard X-ray components of the persistent spectrum have also been attributed

to synchrotron and bremsstrahlung emission. Heyl & Hernquist (2005a) demonstrated

how powerful MHD waves created by a large-scale disturbance on the stellar surface

can experience a breakdown in response to vacuum polarization, creating a pair

plasma fireball and a subsequent burst of hard X-rays and soft γ-rays. This model

was then applied in Heyl & Hernquist (2005b) to show how a similar breakdown of

weaker MHD waves can seed the hard non-thermal component through synchrotron

emission. Synchrotron-emitting pairs can also be created when positrons accelerated

by electric potentials upscatter thermal surface photons past the threshold for pair

creation (Thompson & Beloborodov 2005). Finally, downward-moving beams of

relativistic, current-carrying charges experience beam instabilities upon entering the

atmosphere, heating a thin transitional surface layer and leading to bremsstrahlung

emission (Thompson & Beloborodov 2005; Beloborodov & Thompson 2007), while

the remaining energy of the beam is deposited at high Thomson optical depths ∼ 102

through collisional beam stopping, feeding thermal emission from the atmospheric

interior (Beloborodov & Thompson 2007; González-Caniulef et al. 2019).

1.4 Current approach

The ultrastrong magnetic fields of magnetars present both challenges and opportu-

nities in the way they modify particle kinematics and the properties of fundamental

QED interactions (Harding & Lai 2006), such that non-magnetic approximations break

down. Close to the surface of a magnetar, the field is so strong that interacting


electrons and positrons may be substantially confined to the lowest Landau state,

resulting in effectively one-dimensional motion. This greatly simplifies the modelling

of e±, and makes the magnetic cross sections of various QED processes tractable and

intuitive. In this thesis we investigate a series of processes occurring near the surface

of the magnetar that may play an important role in the formation of their spectra.

For example, non-resonant scattering in the inner magnetosphere may have a

significant effect on the spectrum, being capable of producing a falling power law

component. e±-ion and electron-positron bremsstrahlung in the cold atmosphere may

contribute to the high-energy rising component. Powerful electron-positron scattering

decreases the annihilation depth of hot positrons entering the atmosphere, producing

a well-expressed annihilation peak outside of the observed range. Pair annihilation

cross sections are suppressed while pair creation cross sections are enhanced.

Additionally, the form of the spectrum produced near the stellar surface strongly

affects the overall plasma state and the physical processes occurring in the upper

magnetosphere, including those mentioned above (e.g. resonant scattering). Therefore,

a good understanding of the processes considered presently provides the opportunity

to explain the entire observed spectral range and to make predictions outside of it -

either by itself or in concert with the upper magnetosphere processes. Motivated by

this, we construct a Monte Carlo model of the persistent X-ray emission of a magnetar.

The photons are created self-consistently as a result of the interaction of a warm

magnetospheric plasma with an underlying cold, dense neutron star atmosphere, and

are subsequently comptonized by the magnetospheric plasma.

1.5 Plan of the Thesis

In Chapter 2 we provide an overview of some basic properties of e± and photon states

in a background magnetic field, in preparation for the subsequent QED evaluations.


We then review the derivation and behaviour of the cross sections of various

relevant electromagnetic interactions between photons, electrons and positrons. Pho-

ton scattering and pair creation are considered in Chapter 3, while pair annihila-

tion, electron-positron and e±-ion scattering and bremsstrahlung, as well as thermal

bremsstrahlung absorption are considered in Chapter 4.

In Chapter 5 we use these cross sections to construct a Monte Carlo model of

the interaction of a warm magnetospheric plasma with a cold, dense neutron star

atmospheric layer. This interaction self-consistently generates persistent X-ray spectra,

which we compare against those of AXPs and SGRs.

Chapter 2

QED Interactions in a Background

Magnetic Field

This chapter draws from Kostenko & Thompson (2018) and Kostenko & Thompson (2019).

2.1 Overview

We now review some basic properties of electron/positron and photon states in a

background magnetic field, in preparation for our evaluation of the various relevant

cross sections. The main choice to be made is of the electron/positron wave function,

in which we follow Sokolov & Ternov (1966) and Melrose & Parle (1983a). Then the

wave function of an electron moving along the magnetic field is connected by a simple

Lorentz transformation to the wave function of an electron at rest. Two considerations

lead us to limit the strength of the background magnetic field to . 103 BQ: (i) vacuum

polarization significantly modifies the photon dispersion relation in stronger magnetic

fields (Adler 1971), and (ii) the decay rate of a photon of energy ω > 2m becomes of

the order of ω, so that a propagating photon state loses meaning.

We adopt natural units (h = c = kB = 1) throughout this Chapter as well as

Chapters 3 and 4, along with the (+ − −−) metric signature. The Dirac gamma

10

Chapter 2. QED Interactions in a Background Magnetic Field 11

matrix convention is the same as that used by Melrose & Parle (1983a),

γ0 =

1 0

0 −1

; γj =

0 σj

−σj 0

, (2.1)

where each 0 and 1 element denotes a 2× 2 matrix, and σj are the usual Pauli matrices.

Landau gauge A = Bxy is chosen for the background vector potential, and we

alternatively use Cartesian coordinates (x, y, z) and spherical coordinates (θ, φ) (with

the axis θ = 0 aligned with z) to describe the wavevectors of interacting particles.

The photon quantum states are divided into a strongly interacting O-mode, whose

cross sections are comparable to non-magnetic values, and the weakly-interacting

E-mode, with cross sections that are suppressed by a factor ∼ (mω/eB)2. The e±

quantum states undergo more noticeable changes. The background magnetic field

results in these states only having two conserved components of generalized momen-

tum - one longitudinal component and one gauge-dependent transverse component.

In the Landau gauge chosen here, this gauge-dependent transverse component is

directed along the y axis, and the wave functions are localized in a narrow strip of

width ∼ λB along the x coordinate. We also take care to follow the approaches of

Sokolov & Ternov (1966) and Melrose & Parle (1983a) to defining these wave functions,

ensuring that the wave functions of e± in motion are obtained from the wave functions

with vanishing pz by a continuous Lorentz transformation (Equation (2.14)). Although

we provide the general form of these wave functions, in the analyses that follow we

will assume that all e± sit in the lowest Landau state, n = 0.

To conclude, an overview of the coordinate space Feynman rules for our situation

is provided, along with several other relations that will be relevant to the subsequent

calculations. Due to the form of the e± wave functions, specifically the non-vanishing

coordinate dependence of the spinors u(σ)n,a (x), v(σ)n,a (x), a transition to the momentum

space Feynman rules is impossible.


2.2 Photons

The two polarization states of photons of frequency ω |e|B/m = (B/BQ)m show a

strong asymmetry in their scattering and emission cross sections (Meszaros & Ventura

1979; Harding & Lai 2006). The ordinary (O) mode interacts much more strongly

with electrons than the extraordinary (E) mode, because a significant component

of its electric vector is directed along the background magnetic field. The O-mode

cross sections for electron scattering and bremsstrahlung emission are comparable

in magnitude to the unmagnetized values. This asymmetry disappears in a narrow

range of propagation directions about the magnetic axis; it also disappears at a critical

electron density where the contributions of vacuum polarization and plasma to the

dielectric tensor nearly cancel (Harding & Lai 2006).

The photon wave function is normalized as

Aµ(xν) =εµ

(2ωL3)1/2 e−ik·x; kµ = ω(1, k). (2.2)

Here, kµ and εµ are the wavevector and polarization 4-vectors, and L3 is a nor-

malization volume. Excepting near a vacuum-plasma resonance, both polarization

modes are highly elliptically polarized. A photon propagating in the direction

k = (kx, ky, kz) = (sin θ cos φ, sin θ sin φ, cos θ) has a unit electric vector parallel to

k× B in the E-mode, and parallel to k× (k× B) in the O-mode, i.e.,

εzO = sin θ; ε±O = εx

O ± iεyO = − cos θe±iφ. (2.3)

For the hard X-rays and gamma rays of interest here, vacuum polarization is the

dominant correction to the dielectric response, and the expressions given for εiO,E

are essentially exact: they are accurate to O(4πn±mec2/B2), where n± is the number

density of electrons and positrons (Meszaros & Ventura 1979).

When Landau resonances are kinematically forbidden, the polarization dependence

of the processes we consider reduces to a dependence on εz. This effectively decouples


the E-mode:

εzE = 0; ε±E = ∓ie±iφ. (2.4)

The coupling to the E-mode is restored when virtual Landau excitations are included

in a matrix element. This introduces terms in each matrix element involving the

ε± polarization components, but with a magnitude suppressed by ∼ mω/eB at

frequencies well below the first Landau resonance. This means that the nonresonant

E-mode cross section is generally suppressed by a factor ∼ (mω/eB)2 compared with

that of the O-mode.

2.3 Electrons and Positrons

Quantum states of an electron or positron of charge q = ∓e in a magnetic field

B = ∇×A = Bz are characterized by two conserved components of the general-

ized momentum: a longitudinal momentum pz, and a gauge-dependent transverse

momentum qA marking a center of gyration x± in the plane perpendicular to B. In

the Landau gauge A = Bxy, the electron wave function is localized in coordinate x,

within a strip of width ∼ λB ≡ (|e|B)−1/2 = (B/BQ)−1/2m−1.

This means, for example, that an electron which absorbs momentum −∆ky by

scattering a photon will see its center of gyration shift by ∆x− = +∆ky/|e|B = λ2B ∆ky;

and that an annihilating pair of positive and negative electrons whose centers of

gyration x± are displaced relative to each other will emit photon(s) carrying net

y-momentum ∆ky = |e|B(x+ − x−). On the other hand, in this particular gauge there

is no conserved x-momentum, meaning that the x-momentum carried by photons in

the final state is constrained only by the conservation of energy. The total cross section

and the kinematic constraints on it are of course always independent of this gauge

choice1.

1In a classical approximation, the translational invariance of the magnetic field implies the con-


The energy levels of the electron or positron are (Berestetskii et al. 1971)

E2 = p2z + m2 + |q|B(2l + 1)− qBσ, (2.5)

where l ≥ 0 is an integer labeling the orbital angular momentum of the mode, and σ

is the eigenvalue of the spin operator:

Σz =

σz 0

0 σz

(2.6)

as evaluated in the particle rest frame (pz = 0). Here, σz is the 2× 2 dimension Pauli

matrix. In the lowest Landau state (E = m), the electron has spin σ = −1 and the

positron σ = +1, as expected from the nonrelativistic expansion of the Dirac equation.

These spin labels can be continuously extended to finite pz, as described in Section

2.4.

The localization of the electron wave function transverse to the magnetic field

depends on the sum of the last two terms on the right-hand side of Equation (2.5),

and so we adopt the simplified notation

E2 = p2z + m2 + p2

n; p2n ≡ 2n|e|B ≡ E2

n0 −m2, (2.7)

where n = l + 12 [1− σ · sgn(q)].

In what follows, we assume that the initial electron or positron sits in the lowest

Landau state, given the short timescale for radiative de-excitation from n > 0. Particles

in all processes are also assumed to carry a small enough kinetic energy to prevent

excitations to n > 0 in the final state, or in resonances.

servation of transverse canonical momentum p+ qA, and one recovers the Lorentz force dp/dt =−q [∂A/∂t + (v ·∇)A] = qv ×B.


2.4 Dirac Spinors

The electron/positron wave functions are written, following Sokolov & Ternov (1966)

and Melrose & Parle (1983a), as

[ψ(σ)∓ (xµ)

]pz,n,a

=

e−ip·x u(σ)n,a (x) (electrons);

eip·x v(σ)n,a (x) (positrons).(2.8)

Here, σ = ±1 labels the spin state, a the center of gyration, and pµ the momentum

4-vector,

pµ = (E, 0, py, pz); py = aqB = sgn(q)a

λ2B

. (2.9)

Under charge conjugation, the sign of pµ reverses, and so the gyration center remains

fixed.

The choice of the positive- and negative-energy spinors u(σ)n,a (x), v(σ)n,a (x), is guided

by the requirement that for finite pz they be continuously related to the spinor of a

particle at rest. Their general form is (Johnson & Lippmann 1949)

C1φn−1(x)

C2φn(x)

C3φn−1(x)

C4φn(x)

, (2.10)

where the φn are harmonic oscillator wave functions,

φn(x− a) =1

L(π12 λB2nn!)

12

Hn

(x− a

λB

)e−(x−a)2/2λ2

B , (2.11)

and Hn is the nth-order Hermite polynomial. For a particle at rest (pz = 0),

C1

C1

C3

C4

=

1√2εEn0(εEn0 + m)

εEn0 + m

0

0

ipn

Dσ=+1 +

0

εEn0 + m

−ipn

0

Dσ=−1

. (2.12)


Here, ε = +1(−1) corresponds to positive (negative) energy states. It is easy to check

that these spinors are eigenstates of the z-component of spin,

∫d3xψΣzψ = D2

σ=+1 − D2σ=−1 = ±1. (2.13)

Hence, Dσ=+1 = 1 (Dσ=−1 = 1) corresponds to a state of spin-up (spin-down). In the

lowest Landau state n = 0, only Dσ=−1 = 1 is available for the positive-energy state

(the function φn is undefined for n = −1).

To obtain the spinors at finite pz, one applies the Lorentz boost parallel to B,

ψ→ exp(−1

2αzρ

)ψ; αz =

0 σz

σz 0

. (2.14)

Here, β ≡ tanh(ρ) is the speed of the boost, so that γ = cosh(ρ). Taking into account

that α2z = I, we have

exp(−1

2αzρ

)= I cosh

(12

ρ

)− αz sinh

(12

ρ

). (2.15)

Applying this transformation to the spinors in Equation (2.12), and dividing by√

γ to compensate for the longitudinal contraction of the wave packet under the boost,

one obtains

u(−1)n,a (x) =

1fn

−ipz pnφn−1

(E + E0n)(E0n + m)φn

−ipn(E + E0n)φn−1

−pz(E0n + m)φn

; u(+1)

n,a (x) =1fn

(E + E0n)(E0n + m)φn−1

−ipz pnφn

pz(E0n + m)φn−1

ipn(E + E0n)φn

(2.16)

for the positive-energy spinors and

v(+1)n,a (x) =

1fn

−pn(E + E0n)φn−1

−ipz(E0n + m)φn

−pz pnφn−1

i(E + E0n)(E0n + m)φn

; v(−1)

n,a (x) =1fn

−ipz(E0n + m)φn−1

−pn(E + E0n)φn

−i(E + E0n)(E0n + m)φn−1

pz pnφn

(2.17)


for the negative-energy spinors. Here, we introduce fn = 2L√

EE0n(E0n + m)(E0n + E).

It is straightforward to check that u(±1)n,a and v(∓1)

n,a are connected by exchanging

σ, pz, E, E0n → −σ,−pz,−E,−E0n.

The wave functions derived by Johnson & Lippmann (1949) do not have this

property of being continuously related by a Lorentz transformation to the wave

function of an electron/positron with vanishing pz. Wave functions equivalent to ours

(up to a trivial phase factor) can alternatively be derived by requiring them to be

eigenstates of an appropriately defined magnetic moment operator (Sokolov & Ternov

1966; Melrose & Parle 1983a).

All the cross sections evaluated presently are independent of the choice of spinor

basis, since they are effectively summed (averaged) over the single admissible spin

state of each outgoing (ingoing) electron or positron (Melrose & Parle 1983a).

2.5 Rules for Calculating Matrix Elements

We complete our review of QED amplitudes in strong magnetic fields by summa-

rizing the Feynman rules as expressed in coordinate space and some computational

procedures.

1. A vertex between photon and electron lines is written as the integral

−ie∫

d4x[ψ(σI)− (x)

]pz,I ,nI ,aI

γµ Aµ(x)[ψ(σi)− (x)

]pz,i,ni,ai

= − ie(2ωL3)1/2

∫d4x e−i(pi±k−pI)·x u(σI)

nI ,aI (x)γµεµu(σi)ni,ai(x). (2.18)

Here, i and I label incoming and internal positive-energy electron states, respectively,

and the photon is either absorbed (wavevector +kµ) or emitted (−kµ). The vertex

between an incoming electron and an internal positron is obtained by substituting

−pI and v(−σI)nI ,aI for pI and u(σI)

nI ,aI .


2. An internal electron line is represented by the propagator in coordinate space,

GF(x′ − x) = −i∫

LdaI

2πλ2B

∫L

dpz,I

2π

∞

∑nI=0

[θ(t′ − t)∑

σI

u(σI)nI ,aI (x

′)u(σI)nI ,aI (x)e

−iEI(t′−t)eipI ·(x′−x)

−θ(t− t′)∑σI

v(σI)nI ,aI (x

′)v(σI)nI ,aI (x)e

iEI(t′−t)e−ipI ·(x′−x)]

. (2.19)

3. An internal photon line is also represented by the propagator in coordinate

space,

Gµνγ (x′ − x) = iηµν

∫ d3q2ω(2π)3

[θ(t′ − t)e−iq·(x′−x) + θ(t− t′)eiq·(x′−x)

], (2.20)

where ηµν is the metric tensor.

4. The combined integral over t and t′ generates a combination of an energy delta

function and an energy denominator:

∓i∫

dt∫

dt′θ[∓(t− t′)] ei(E f +ω f∓EI)t′ e−i(Ei+ωi∓EI)t =2πδ(Ei + ωi − E f −ω f )

Ei + ωi ∓ EI. (2.21)

Here, i and f label incoming and outgoing particles. The contribution of an excited

Landau state to a given term in the matrix element is suppressed by a factor E−1I '

(2n|e|B)−1/2 away from resonance. However, the suppression of the net rate is

generally stronger as the result of a cancellation between S f i[1] and S f i[2].

5. The contraction of the electric polarization vector with γ matrices is

γ0γµεµi = −

0 0 εzi ε−i

0 0 ε+i −εzi

εzi ε−i 0 0

ε+i −εzi 0 0

; (i = O, E). (2.22)

6. The matrix element S f i includes energy and momentum delta functions that,

once squared, are handled according to (e.g. in the case of Compton scattering)[2πδ(pz,i + kz,i − pz, f − kz, f )

]2 → L(2π)δ(pz,i + kz,i − pz, f − kz, f );[2πδ

(ky,i − ky, f −

ai − a f

λ2B

)]2

→ L(2π)δ

(ky,i − ky, f −

ai − a f

λ2B

);

[2πδ(Ei + ωi − E f −ω f )

]2 → T(2π)δ(Ei + ωi − E f −ω f ).


Here, T is the normalization time.

The delta function in py has a term from the change in the guiding center a of

the scattering charge. No delta function in px appears for our choice of background

gauge. For the sake of brevity, such a combination of delta functions will be written

in the following way:

δ(3)f i (E, py, pz). (2.23)

7. Summing over the phase space of a final-state photon involves the integral

∫L3

ω2f dω f dΩ f

(2π)3 , (2.24)

where Ω f is solid angle. For a final-state electron or positron, there is no sum over the

x-component of momentum, hence the integral

|e|B2π

∫Lda f

∫L

dpz, f

2π=∫

Lda f

2πλ2B

∫L

dpz, f

2π. (2.25)

Chapter 3

Photon processes: scattering and pair

creation


3.1 Overview

Close to the surface of a magnetar, the magnetic field is so strong that interacting

electrons and positrons may be substantially confined to the lowest Landau state as

described in Chapter 2. This modifies and complicates the electromagnetic interactions

of photons, electrons and positrons (Harding & Lai 2006). We will now consider a

variety of these interactions to the extent that they are relevant to the generation of

persistent X-ray magnetar emission in the current approach, starting with photon

scattering and pair creation in this Chapter. Pair annihilation, electron-positron and

e±-ion scattering and bremsstrahlung, as well as thermal bremsstrahlung absorption

are considered in Chapter 4.

The results of our calculations are presented in a form that makes them easy to

apply to the analytical and numerical study of ultramagnetic systems, specifically the

inner magnetospheres of magnetars that the current approach considers. The basic

20

Chapter 3. Photon processes: scattering and pair creation 21

approximation made (restricting real and virtual e± to the lowest Landau state) is

shown to be an excellent approximation over a significant range of magnetic field

strengths, (10− 102)BQ, and the dominant finite-B correction to the rates considered is

identified (Equations (3.7) and (3.36)). As discussed in Chapter 2, this approximation

also decouples E-mode photons from e±.

We first review the derivation and behaviour of the electron-photon scattering

cross section, which has received considerable attention in prior literature. The total

cross section for the coupled O-mode photons rises monotonically with photon energy,

starting from values comparable in magnitude to the Thomson cross section and

spiking for ωi & m (Figures 3.3 and 3.4) due to a u-channel pole (corresponding to

final photon energy ω f (1− µ2f ) . 2m). For the first time, we quantify the behaviour of

this pole, which in previously published work only appeared as a term in complicated

infinite-sum formulae.

The rate of single-photon pair creation into the lowest Landau level is determined

(Equation 3.19), based on a detailed balance argument and the single-photon annihila-

tion rate computed by Wunner (1979) and Daugherty & Bussard (1980). This controls

the width of final photon states above the threshold for pair creation. Additionally, it

helps determine the rate of scattering-assisted pair creation,γ + e± → e+ + e− + e±,

for which accurate analytic approximations (Equations (3.29) and (3.30)) and kinematic

constraints (Figure 3.7) are presented.

Finally, we present a simplified expression for the rate of two-photon pair creation

into the lowest Landau state, constructed in terms of invariant quantities involving the

photon 4-momenta (Equation (3.35)). This cross section is greatly enhanced compared

with an unmagnetized vacuum, by a factor of ∼ B/BQ (Figure 3.10).


Figure 3.1: Feynman diagrams for Compton scattering.

3.2 Electron-Photon Scattering

We first consider nonresonant electron-photon scattering, e± + γ→ e± + γ, as modi-

fied by a strong magnetic field (Figure 3.1). Previous QED calculations (Herold 1979;

Melrose & Parle 1983b; Bussard et al. 1986; Daugherty & Harding 1986; Harding &

Daugherty 1991; Gonthier et al. 2000; Baring et al. 2005) have focused on the situation

where the initial photon is energetic enough to excite the scattering charge to a higher

Landau level. We consider the case where Landau resonances are kinematically for-

bidden even for initial photon energies around m. We demonstrate a good agreement

between a truncated formula for the nonresonant scattering cross section and the full

QED result. In this situation, the initial photon energy is restricted by single-photon

pair creation; photons approaching this pair creation threshold have an enhanced

scattering cross section.

The kinematic relation between the initial and final electron states is modified

compared with the vacuum case, because kinetic momentum is conserved only in the

direction parallel to B. (As will be the case throughout this analysis, the labeling of

initial and final particle states is contained in the accompanying Feynman diagrams.)


The photon frequency shift following scattering off an electron at rest (from direction

cosine µi = cos θi to µ f = cos θ f ) is

ω f −ωi =1

1− µ2f

(ωi(µ f − µi)µ f + m−

√ω2

i (µ f − µi)2 + 2mωi(µ f − µi)µ f + m2

). (3.1)

This is derived by invoking the conservation of energy and longitudinal momentum,

ωi + m = ω f + (p2z + m2)1/2 and µiωi = µ f ω f + pz, to obtain the quadratic equation

(ωi + m−ω f )2 = (µiωi − µ f ω f )

2 + m2. (3.2)

Although the outgoing photon frequency depends on three quantities (ωi, µi and µ f ),

the frequency shift depends only on µ f and ωi(µ f − µi) (Figure 3.2). This expression

reduces to ω f ' ωi(1± µi)/(1± µ f ) when ωi|µ f − µi| m, with the upper (lower)

sign corresponding to µ f > µi (µ f < µi).

3.2.1 Cross Section for B BQ

There is a considerable simplification in the electron-photon scattering cross sec-

tion when Landau resonances can be neglected in both intermediate and final elec-

tron/positron lines. The formula for the cross section becomes weakly dependent on

background magnetic field strength B, for the simple reason that an electron begins to

behave like a “bead on a wire.” In the case where the initial electron is at rest, we find

for the differential cross section

12π

dσ

dµ f=

r2e2

ω f

ωi

|F|2m2

E f (E f + m)

|εzi |2|εz

f |2

1− β f µ f, (3.3)

where re = e2/4πm = αem/m is the classical electron radius,~εi, f labels the unit electric

vector of the incoming and outgoing photons, β f = pz, f /E f , and

|F| ≡∣∣∣∣∣ 4m(2m + ωi −ω f )

[2m + ωi(1− µ2i )] [2m−ω f (1− µ2

f )]

∣∣∣∣∣ . (3.4)

The differential cross section is plotted in Figure 3.3. The low-frequency behavior

is consistent with the classical result (Canuto et al. 1971)

12π

dσ

dµ f= r2

e |εzi |2|εz

f |2 = r2

e sin2 θi sin2 θ f . (3.5)


Figure 3.2: Photon frequency shift by electron scattering. Curves correspond to final direction

cosine µ f ranging from −1 to 1 in steps of 0.2.


The high-frequency behavior is more interesting. Even when the initial photon energy

lies well below the first Landau resonance, the scattering cross section spikes (but does

not diverge) at a value of ω f (and therefore ωi) somewhat larger than 2m. This spike

arises from the pole in the matrix element associated with annihilation of the initial

electron with a virtual positron into the final-state photon (Herold 1979; Daugherty &

Harding 1986). It appears at a lower photon frequency than the first Landau resonance

(energy (m2 + 2|e|B)1/2 −m) if B > 4BQ.

The partial cross section for scattering into non-pair-creating states is shown in

Figure 3.4. The high-energy behavior of the total cross section is opposite to the

Klein-Nishina result for scattering in vacuum: the cross section grows as ω rises above

m. Nonetheless, scattering is still suppressed for photons that have a large energy

owing to relativistic motion of the scattering charge along B (Lorentz factor γ 1),

because the sin2 θi factor decreases as ∼ 1/γ2.

The high-frequency dip in the curves shown in Figure 3.4 represents the opening

up of the final-state phase space to pair creation over some range of scattering angles.

A scattered photon can convert directly to an electron-positron pair if ω f sin θ f > 2m,

meaning that the cross section for scattering-assisted pair creation, e± + γ → e± +

e+ + e− can substantially exceed the vacuum value. This phenomenon is examined in

more detail in Section 3.3.

More generally, a high-frequency Klein-Nishina scaling for the scattering cross

section – which is approached in the case of scattering at high-order Landau resonances

in sub-QED magnetic fields (Gonthier et al. 2000) – loses meaning as B rises above

4BQ, because the scattered photon has a high probability of converting to a pair

(Beloborodov & Thompson 2007).


Figure 3.3: Differential nonresonant electron scattering cross section versus initial photon

frequency ωi (top panels) and final photon direction cosine µ f (bottom panels), in the regime

B BQ. Curves in the top panels represent |µ f | = 0, 0.2, 0.4, 0.6. Black curves: µ f > 0; blue

curve: µ f = 0; red curves: µ f < 0. The cross section rises as the scattered photon approaches

the threshold for single-photon pair creation. In the bottom panels, the blue curve marks the

classical limit, and black and red curves are used for clarity. Dotted curve: the final-state

photon rapidly converts to an electron-positron pair.


Figure 3.4: Integral nonresonant electron scattering cross section plotted versus initial photon

frequency ωi for a range of initial direction cosine µi, in the regime B BQ. The high-

frequency peak marks the opening up of pair conversion in the final state (see Figure 3.6).


3.2.2 Finite-B Correction to the Cross Section

A simple correction to the electron-photon scattering cross section representing a finite

magnetic field is available. The overlap of the photon wave function (wavevector kµ)

with a pair of harmonic oscillator wave functions, such as appears in the scattering

matrix, yields a factor e−λ2B(k

2x+k2

y)/4:

∫d3xeik·xφn(x− a)φ0(x− b)e−iay/λ2

B

(e−iby/λ2

B

)∗eipz

(eiqz)∗

=(2π)2

(2nn!)12

δ

(ky −

a− bλ2

B

)δ(kz + p− q)e−λ2

Bk2⊥/4eikx(b+λ2

Bky/2)λnB(−ky + ikx)

n,

(3.6)

as derived by Daugherty & Bussard (1980). Including both photon vertices, the cross

section is multiplied by

σ→ e−λ2B(k

2⊥,i+k2

⊥, f )/2σ; k2

⊥ = k2x + k2

y. (3.7)

In Figure 3.5 we compare the integral cross section derived from Equations (3.3) and

(3.4), with and without this correction, with the full sum over intermediate Landau

states to be found in Herold (1979), Melrose & Parle (1983b), Bussard et al. (1986),

Daugherty & Harding (1986), and Harding & Daugherty (1991). There is very good

agreement for B = 100BQ in both cases, and excellent agreement for B = 4BQ and

10BQ including the correction.

3.2.3 Derivation

Although the electron-photon scattering cross section in a strong magnetic field is

already well covered in the literature, we briefly review its derivation here and in

Appendix A. Various components of the calculation will find use in later sections, and

some of the other processes considered (two-photon pair annihilation and creation)

are related by crossing symmetry. The calculation is based on the rules summarized

in Section 2.5.


Figure 3.5: Comparison of the exact cross section for electron-photon scattering from ni = 0

to n f = 0 (red curves) and the result obtained from truncated expansion in intermediate

Landau level, with the additional correction factor (3.7) included (overlying dotted black

curves). Background magnetic field ranges from B = 4BQ (the threshold value above which

the first Landau resonance is superseded by e± conversion) to 10BQ and 100BQ. Blue curve:

the large-B limit given by Equation (3.3), which coincides very nearly with the uppermost

curve (B = 100BQ).


The cross section is obtained from the integral

σ =L3

T

∫L

da f

2πλ2B

∫L

dpz, f

2π

∫L3

ω2f dω f dΩ f

(2π)3

∣∣S f i[1] + S f i[2]∣∣2. (3.8)

The initial electron is assumed to be at rest, so that the conservation of energy and

longitudinal momentum are given by m + ωi = E f + ω f and kz,i = pz, f + kz, f , leading

to the recoil formula (3.1).

The two terms in the S-matrix correspond to the two diagrams in Figure 3.1. The

first is

S f i[1] = −ie2∫

d4x∫

d4x′[ψ(−1)− (x′)

]pz, f ,n f ,a f

γν Aν(x)∗G f (x′ − x)γµ Aµ(x)[ψ(−1)− (x)

]pz,i ,ni ,ai

,

(3.9)

where ni = n f = 0, and the electron and photon wave functions are given in Secs.

2.2-2.4. The second term in S f i is related to S f i[1] by an interchange of photon labels:

ωi ↔ −ω f and kz,i ↔ −kz, f .

Substituting for the electron propagator from Equation (2.19), and restricting the

sum over intermediate Landau levels to nI = 0, σ = −1/ + 1 (for electrons/positrons),

the first term in S f i becomes

S f i[1] =−ie2

2√ωiω f L3

(L

2π

)22πδ(E f + ω f −m−ωi)

∫dpz,I

∫ daI

λ2B

(I1 I2

m + ωi − EI+

I3 I4

m + ωi + EI

).

(3.10)

Here,

I1 =∫

d3x[u(−1)∗

0,aI(x)]T

γ0γµεµi eiki·xu(−1)

0,ai(x)ei(pi−pI)·x⊥+i(pz,i−pz,I)z (3.11)

and

I2 =∫

d3x′[u(−1)∗

0,a f(x′)

]Tγ0γν

(εν

f eik f ·x′)∗

u(−1)0,aI

(x′)ei(−p f +pI)·x′⊥+i(−pz, f +pz,I)z′ .

(3.12)

The integral I3 is obtained from I1, and I4 from I2, by substituting the negative-energy

wave function v(+1)0,aI

for the positive-energy wave function u(−1)0,aI

, and taking pI → −pI .


These integrals are evaluated in Appendix A, where use is made of the overlap

integral (3.6). We find

S f i[1] =−ie2

2√ωiω f L5 e−ikx, f (a f−ky, f λ2B/2) e−λ2

B(k2i,y+k2

⊥, f )/4 εzi (ε

zf )∗

[2E f (E f + m)]12

ωi(E f + m) + pz, f kz,i

(m + ωi)2 − E2I×

(2π)3δ(3)f i (E, py, pz) (3.13)

where E2I = p2

z,I + m2 = k2z,i + m2. Adding S f i[2] to this and taking λB → 0, one

obtains a factor

F =ωi(E f + m) + pz, f kz,i

(m + ωi)2 −m2 − k2z,i

e−ikx, f a f +−ω f (E f + m)− pz, f kz, f

(m−ω f )2 −m2 − k2z, f

e−ikx, f ai . (3.14)

Substituting kz = µω, and making use of energy-momentum conservation, we obtain

the expression (3.4) that appears in Equation (3.3). The differential cross section

is obtained by substituting S f i[1] + S f i[2] into Equation (3.8) and performing the

pz, f , a f and ω f integrals. The factor (1− β f µ f )−1 comes from integrating over the

combination of the pz and E delta functions.

3.3 Electron-Positron Pair Creation

A strong magnetic field opens up new efficient channels for converting gamma rays

to electron-positron pairs (Erber 1966; Daugherty & Harding 1983; Gonthier et al.

2000; Beloborodov & Thompson 2007). The single-photon channel γ → e+ + e− is

now consistent with conservation of both momentum and energy, because the charges

carry a generalized transverse momentum qA, and because the Lorentz invariance of

the vacuum state is broken. The scattering of a sufficiently energetic gamma ray off an

electron can also mediate pair creation through the single-photon channel, achieving

a high cross section over a narrow range of frequencies.

Here, we are interested in the conversion of a photon (or photons) into a pair

confined to the lowest Landau state. The conversion rate for a single photon is easily

obtained using a detailed balance argument from the cross section for single-photon


annihilation, which was calculated by Wunner (1979) and Daugherty & Bussard

(1980). We next consider the scattering of a photon below the Landau resonance into

a pair-creating state, extending the calculation of Section 3.2. The cross section of this

process is strongly enhanced when the final-state photon resonates with the initial

electron and a virtual positron (a u-channel resonance). The cross section averaged

over frequency significantly exceeds πr2e , and indeed can exceed the cross section for

scattering at the first Landau resonance (where the final-state photon can directly

convert to a pair only if B > 4BQ: Beloborodov & Thompson 2007).

Importantly for the application to magnetars, the cross section for photon collisions,

γ + γ→ e+ + e−, is strongly enhanced compared with the unmagnetized vacuum, by

a factor of ∼ B/BQ. A calculation including the full intermediate Landau level sum

can be found in Kozlenkov & Mitrofanov (1986), and the regime B BQ is addressed

by Thompson (2008). We consider the strong-field regime more fully here. The

kinematic threshold for photon collisions differs significantly from the unmagnetized

case, and the cross section is further enhanced when one of the colliding photons has

a low frequency.

Our main focus is on the conversion of O-mode photons to pairs. E-mode photons

couple weakly to electrons by scattering, with a cross section that is suppressed by a

factor of ∼ (B/BQ)−2 at energy ∼ m. The cross section for a collision with a second

photon is similarly suppressed. High-energy E-mode photons are further depleted by

splitting into two photons (Adler 1971), a process that is kinematically allowed for the

E-mode but forbidden for the O-mode 1.

1Splitting of one photon into two daughters only conserves energy and momentum if the index ofrefraction of one or both of the daughter photons is larger than the index of refraction of the initialphoton. Since nO > nE in magnetic fields both larger and smaller than BQ, only splitting of the E-modeis allowed (e.g. Harding & Lai 2006).


3.3.1 Single-photon Pair Creation into the Lowest Landau Level

The energy of a photon cannot be reduced arbitrarily by a Lorentz boost in the

presence of a background magnetic field, because only a boost parallel to B leaves the

background invariant. The energy is minimized in the frame where k ·B = 0; hence,

the threshold energy for pair creation is

ωmin =2m

sin θ(3.15)

for the ordinary polarization mode.

To obtain the conversion rate, we start with the cross section of the inverse process

of single-photon annihilation of an electron and positron (Wunner 1979; Daugherty &

Bussard 1980),2|pz|

Eσann = 2π2 αem

E2BQ

Be−2(BQ/B)2(E/m)2

. (3.16)

This is evaluated in the center-of-momentum frame, with electron/positron momenta

±pz and kinetic energies (p2z + m2)1/2, corresponding to perpendicular propagation

of the photon. The annihilation rate per electron is suppressed by a factor of ∼ BQ/B

compared with vacuum (where (2|pz|/E)σann = πr2e in the nonrelativistic regime;

Berestetskii et al. 1971), because the e± wave functions are concentrated in an area of

∼ m−2BQ/B transverse to the magnetic field.

Consider now a thermal gas of pairs and photons 2 at uniform temperature T with

densitiesdn±dpz

=|e|B(2π)2 N±;

d2nγ

dωdΩ=

ω2

(2π)3 Nγ, (3.17)

where Nγ = (eω/T − 1)−1, N± = (eE/T + 1)−1. Then detailed balance implies the

following relation for the decay rate Γ± into a pair:

∆ω ∆Ωd2nγ

dωdΩΓ± · (1− N±)2 =

[∆pz

dn±dpz

]2 4|pz|Ep

σann · (1 + Nγ). (3.18)

2This is not indicative of the physical system that will be analysed in Chapter 5. Rather, it is athought experiment that is used to derive cross sections based on thermodynamic arguments. This andother detailed balance approaches presented here are typically verified using QED calculations.


Substituting ω = 2E and the Jacobian factor ∆ω ∆Ω/∆p2z = 4π|pz|/Eω, and boosting

to a general frame gives

Γ±(ω, θ) = 4αemB

BQ

m4

ω2⊥(ω

2⊥ − 4m2)1/2

e−(BQ/2B)(ω⊥/m)2sin θ; ω⊥ = ω sin θ.

(3.19)

This result can be derived alternatively directly from the matrix element,

Γ± =1T

∫L

dpz+

2π

∫L

dpz−2π

∫L

da+2πλ2

B

∫L

da−2πλ2

B|S f i|2, (3.20)

where

S f i = −ie∫

d4x[ψ(−1)− (x)

]pz−,0,a−

γµ Aµ(x)[ψ(+1)+ (x)

]pz+,0,a+

. (3.21)

The contraction of the Dirac spinors with the polarization tensor is easy to work out

when the final-state e+ and e− are both in the lowest Landau state. Making use of the

normalization relation (3.6) and substituting the appropriate wavefunctions gives

S f i = −ieεzm

E(2ω)1/2L7/2 e−λ2Bω2/4 · (2π)3δ3

f i(E, py, pz). (3.22)

for an O-mode photon propagating perpendicular to the background magnetic field

(εz = 1; pz+ = −pz−; E+ = E− = E). Then

Γ± =e2m2

4πE2ωλ2B

e−λ2Bω2/2

∫ da+L

∫ ∞

−∞dpz+ δ(2E+ −ω). (3.23)

This agrees with Equation (3.19) after we substitute E = ω/2.

One observes that the annihilation rate per unit volume is enhanced by a net factor

of ∼ B/BQ compared with a thermal pair plasma in the absence of the magnetic field.

The pair creation rate per photon is therefore enhanced by the same factor, because

the photon phase space does not depend directly on B. The same conclusion applies

for two-photon pair creation (Section 3.3.3).

3.3.2 Compton-assisted Pair Creation

The scattering cross section (3.3), as derived for a stationary target electron or positron,

is strongly peaked when the energy ω f of the final-state photon approaches 2m/(1−


µ2f ). This exceeds the threshold (3.15) for single-photon pair creation, meaning that

electron scattering can effectively mediate pair creation, e± + γ → e± + e+ + e−.

Figures 3.4 and 3.6 show that the cross section peaks strongly for non-pair-converting

final states when µi is small (the initial photon propagates nearly perpendicular to the

magnetic field) but peaks for pair-converting final states when µi → ±1. As we show

here, the cross section can be well approximated analytically in the latter regime.

The initial energy ωi of the photon must be large enough for it to have a chance at

direct pair conversion following scattering. To obtain the minimum ωi as a function

of µi, we first consider the final direction cosine µ f at which the perpendicular energy

s f ω f is maximized. (Throughout this section, we use the shorthand si, f ≡ sin θi, f =

(1− µ2i, f )

1/2.) Differentiating Equation (3.2) with respect to µ f , one finds that this

maximum occurs at µ f = µiωi/(ωi + m) and is equal to

(s f ω f )max =√

s2i ω2

i + 2ωim + m2 −m. (3.24)

Requiring (s f ω f )max > 2m gives the inequality s2i ω2

i + 2ωim > 8m2, which is satisfied

for

ωi >

√1 + 8s2

i − 1

s2i

m. (3.25)

This is shown as the black curve in Figure 3.7; in addition, the process of scattering-

assisted pair creation is interesting only if siωi < 2m. The threshold condition for pair

creation is ωi > 4m in the case of nearly longitudinal propagation.

The pole in the cross section, seen in Equation (3.4), is regulated by taking into

account the width of the final-state photon, ω f → ω f − iΓ±/2, where the decay rate

Γ± is given by Equation (3.19). Then the denominator is replaced by[(ω f −m)2 − E2

I

]2→

[(1− µ2

f )ω2f − 2mω f

]2+ m2Γ2

±. (3.26)

The first term on the right-hand side is minimized for a final-state photon energy and

direction

ω∗f = ωi + m− msi

; µ∗f = µisiωi −m

si(ωi + m)−m, (3.27)


Figure 3.6: Partial cross section for electron scattering into a non-pair-creating final state (black

lines; see also Figure 3.4) and pair-creating final state (red solid lines). Curves are ordered from

left to right by increasing µi = 0, 0.1, ...0.9. Dotted red curves show the integral of the red curve

over frequency starting from low ωi. Dotted green lines show the analytic approximation

obtained by substituting Equation (3.29) for the pole in Equation (3.3), and green squares show

the analytic approximation (3.30) to the frequency-integrated cross section.


Figure 3.7: A photon propagating in direction µi can be scattered into a pair-creating final

state only over a restricted range of initial frequency ωi. The threshold ω f (1− µ2f )

1/2 > 2m

for single-photon pair creation is achievable in the final state only to the right of the black

line (Equation (3.25)); whereas the initial photon is itself below the threshold for pair creation

only to the left of the red line. The range of frequencies where both conditions are satisfied

grows wider as the propagation direction becomes more aligned with B. In magnetar-strength

magnetic fields, scattering-assisted pair creation is possible at frequencies well below the

threshold for resonant excitation to the first Landau level.


corresponding to a final electron speed β f = pz, f /E f = µi. The minimum value of

this term is nonvanishing, excepting when the initial photon approaches the threshold

for one-photon pair creation, siωi → 2m, which also implies the pole condition

s2f ω f → 2m. Near the pole, we set siωi = 2m − ∆ω, and the right-hand side of

Equation (3.26) can be approximated as

4m2

[∆ω + m

(1 + si

si

)2 (µ f − µ∗f

)2]2

+ m2Γ2±. (3.28)

Integrating over µ f , one finds that the pole in Equation (3.3) yields the substitution

1[2m−ω f (1− µ2

f )]2→ πsi

25/2(1 + si)

ω2f /m5/2

|∆ω′| (|∆ω|+ |∆ω′|)1/2 ;

|∆ω′| ≡√

∆ω2 + Γ2±/4, (3.29)

with the other factors evaluated at final frequency and direction (3.27). With this

replacement, we have an accurate formula for the integral cross section in the vicinity

of the divergence; see Figure 3.6.

Fixing the initial photon propagation direction µi, one can next average over ωi by

evaluating the nonresonant factors in Equation (3.3) at ∆ω = 0, to obtain the simple

result∫dωi σ(ωi, µi) = 2π2r2

e m3/2[

Γ±

(ω f =

1 + si

sim, µ f =

µi

1 + si

)]−1/2

. (3.30)

This quantity appears as the green squares in Figure 3.6.

To summarize, the cross section for scattering-assisted pair creation is greatly en-

hanced compared with the vacuum value (∼ αem · r2e , see Berestetskii et al. 1971) owing

to (i) a reduced energy loss to recoil by the scattered photon and (ii) the availability of

rapid pair conversion following scattering. (In other words, no additional vertex need

be included in the diagram to obtain a pair in the final state.)

The frequency-averaged cross section peaks above Thomson, at a value given by

Equation (3.30), which can be estimated as⟨σ⟩(µi) ∼

∫dωiσ(ωi, µi)

2m/si= πr2

e · 10(

B10BQ

)−1/2

. (3.31)


This compares with the optical depth to resonant excitation at the first Landau level,

which is (Gonthier et al. 2000; Beloborodov & Thompson 2007)

⟨σ⟩=

1r

∫dr

2π2e2

mcδ

(ω− |e|B

mc

)∼ 2π2r2

e3αem(B/BQ)

= πr2e ·

30(B/10BQ)

(3.32)

in a dipolar magnetic field with |d ln B/d ln r| ∼ 3. In a magnetar-strength magnetic

field, excitation to the first Landau resonance generally requires relativistic bulk

motion of the scattering charge along the magnetic field.

3.3.3 Two-photon Pair Creation

The collision of two photons to form an electron-positron pair (Figure 3.8) occurs

with a dramatically enhanced cross section in an ultrastrong magnetic field. One

can see this using a detailed balance argument similar to the one given in Section

3.3.1. The thermal equilibrium density of electrons and positrons is enhanced by a

factor of ∼ B/BQ, whereas the density of photons is not. Since the annihilation cross

section is suppressed by one inverse power of B/BQ (Daugherty & Bussard 1980), the

pair-production cross section grows by ∼ B/BQ.

The cross section is evaluated to all orders in the intermediate-state Landau level

by Kozlenkov & Mitrofanov (1986), and the special case of a longitudinal collision

in an ultrastrong magnetic field is derived by Thompson (2008). The result does not

depend on the choice of Dirac spinor basis as long as the spins of the electron and

positron are summed over. Here, we consider the more general behavior of photon

collisions in an ultrastrong magnetic field, where the pair is confined to the lowest

Landau state, and outline the derivation using the spinor basis of Section 2.4.

The kinematic constraints on photon collisions are altered by a magnetic field.

There is always a Lorentz frame in which the total longitudinal momentum of the

photons vanishes, µ1ω1 + µ2ω2 = 0. In this frame, the threshold condition for pair

creation is ω1 + ω2 = 2m. One observes that a photon of energy ω1 slightly less


Figure 3.8: Feynman diagrams for two-photon pair creation.

than 2m is able to collide with a much softer photon if the harder photon moves

approximately perpendicular to the magnetic field: one requires only that ω2 ≥ |µ1|ω1.

This contrasts with the unmagnetized vacuum, where ω2 ≥ m2/ω1, so that generally

ω1 + ω2 > 2m. Boosting along B to an arbitrary frame, the threshold condition is

(ω1 + ω2)2 − (µ1ω1 + µ2ω2)

2 > 4m2 (γ + γ→ e+ + e−). (3.33)

The total cross section is, in the center-of-momentum frame where pz,− = −pz,+ =

pz and E+ = E− = E,

σ =2π

|1− µ12|

(e2

4πm

)2 BBQ

m4

ω1ω2E|εz

1εz2|2|pz|

∣∣∣∣ 4pz

ω1ω2(1− µ1µ2)2 + 4µ1µ2p2z

∣∣∣∣2 . (3.34)

This lines up with the B BQ limit of the result given by Kozlenkov & Mitrofanov

(1986).

The cross section in an arbitrary Lorentz frame is obtained by expressing center-

of-momentum frame quantities µi and ωi in terms of the Lorentz scalars C+ =

(ω1 + ω2)2 − (µ1ω1 + µ2ω2)

2, C⊥i = (1− µ2i )ω

2i (i = 1, 2), and C× = ω1ω2(1− µ1µ2):

σ =32πr2

ek1 · k2

BBQ

√C+ − 4m2

C1/2+

C⊥1C⊥2C2+m6

[C⊥1C⊥2C+ + 4m2(C2× − C⊥1C⊥2)]2

=32πr2

e|1− µ12|

BBQ

√C+ − 4m2

C1/2+ (ω1ω2)3

C2+(1− µ2

1)(1− µ22)m

6

[C+(1− µ21)(1− µ2

2) + 4m2(µ1 − µ2)2]2. (3.35)

Here, µ12 is the relative direction cosine of the two photons, k1 · k2 = ω1ω2(1− µ12),

and µi, ωi are evaluated in an arbitrary frame. The dominant correction to the cross


section at finite B comes from a factor

σ→ e−λ2B(k

2⊥,1+k2

⊥,2)/2 σ; k2⊥ = k2

x + k2y. (3.36)

Expression (3.34) is accurate as long as the colliding photons do not propagate

nearly parallel to B, e.g. as long as (εz)2 & BQ/B. Otherwise, one has (Kozlenkov &

Mitrofanov 1986; Thompson 2008)

β · σ =πr2

e2ω2

BBQ

∣∣∣∣∣ ε+2 ε−1 m2

(ω + |pz|)2 + m2 + 2|e|B −ε+1 ε−2 m2

(ω− |pz|)2 + m2 + 2|e|B

∣∣∣∣∣2

, (3.37)

once again as measured in the center-of-momentum frame where ω1 = ω2 = ω and

β = |pz|/E.

The Lorentz scalar C+ measures how far the photons are above the threshold for

pair creation. For fixed C+ the cross section exhibits a strong dependence on the

energies of the colliding photons, as compared with the Breit-Wheeler cross section as

derived in an unmagnetized vacuum (see Berestetskii et al. 1971):

σ =πr2

e2

(1− β2)

(3− β4) ln

1 + β

1− β− 2β(2− β2)

. (3.38)

In particular, the cross section is significantly enhanced when one of the colliding

photons has a low frequency (Figure 3.9). The enhanced ability of a soft photon to

remove a hard photon propagating perpendicular to B implies that photons close to

the threshold for single-photon pair creation are preferentially removed by interactions

with soft photons. Averaging over a flat frequency spectrum of the target photons

(Figure 3.10) also shows a strong divergence in the averaged cross section as ω1

approaches the threshold for single-photon pair creation.

It is worth summarizing the different dependencies of the various pair creation

channels on the strength of the background magnetic field. Whereas photon collisions

grow more rapid as B grows, the resonant and scattering-assisted channels both get

weaker. In the last case, the weakening is not due to any change in the energy of the

pole, but rather to the increasing width of the scattered photon. In magnetar-strength


Figure 3.9: Average of the photon collision rate over direction µ2 of the target photon, for a

range of photon energy ω1 and for µ1 = 0.


Figure 3.10: Average of the photon collision rate over target photon direction and energy,

weighted by the frequency distribution of target photons, dn/dω ∝ ωα−1. Black curves: µ1 = 0

and α = −2, −1, 0. Blue dotted and dashed curves: range of µ1 and a flat energy spectrum

(α = −1). Blue solid curve: α = −1 and µ1 = 0. The rise at low frequencies and the second

rise near the threshold for single-photon pair creation both reflect the strong inverse frequency

dependence of Equation (3.35), with this involving a low-frequency target photon in the latter

case.


magnetic fields, the single-photon channel is effectively instantaneous when the

photon is above the kinematic threshold, and it will dominate two-body interactions.

Which of the two-body effects most effectively removes hard photons that are below

the single-photon conversion threshold depends on the relative concentration of

photons and pairs, as well as the strength of the magnetic field.

Derivation

The fastest route to a derivation of the cross section for γ+ γ→ e++ e− is to make use

of crossing symmetry and infer the matrix element from that for electron scattering,

γ + e± → γ + e±. The cross section for pair production is then obtained from

σ =L3/T|1− µ12|

∫ Lda+2πλ2

B

∫ Lda−2πλ2

B

∫ Ldpz,+

2π

∫ Ldpz,−2π ∑

σ+,σ−

∣∣S f i∣∣2 . (3.39)

Here, the outgoing electron and positron and the internal electron/positron lines are

all restricted to the lowest Landau level (σ± = ±1). The matrix element for a process

that has a particle φ with 4-momentum pµ in the initial state is the same as the matrix

element of a process that has that the antiparticle φ with 4-momentum −pµ in the

final state (Peskin & Schroeder 1995):

S f i(φ(pµ) + ...→ ...) = S f i(...→ ... + φ(−pµ)). (3.40)

Thus, the initial electron in the scattering process becomes a final positron, and the

final photon a second initial photon. Their momenta are related by

pµi → −pµ

+; kµf → −kµ

2 , (3.41)

along with pµf → pµ

− and kµi → kµ

1 , following the labeling of states shown in Figures

3.1 and 3.8.

Before implementing this procedure, we must write down the scattering matrix

element in a more general frame, where the initial electron is not at rest. One


finds, after integrating over the intermediate-state delta functions in momentum, the

following generalization of Equation (3.13),

S f i[1] =−ie2εz

i εz∗f

2√ωiω f L5 e−ikx, f (a f−λ2Bky, f /2)e−λ2

B(k2⊥,i+k2

⊥, f )/4D(ωi, ω f , pz,i, pz, f )

× (2π)3δ3f i(E, py, pz). (3.42)

Here,

D =pz,i[pz, f (Ei + ωi + m) + pz,I(E f + m)] + (Ei + m)[(E f + m)(Ei + ωi −m) + pz, f pz,I ]

2√

EiE f (Ei + m)(E f + m)[(Ei + ωi)2 − E2I ]

,

(3.43)

and pz,I = pz,i + kz,i, EI =√

p2z,I + m2. The expression reduces to Equation (3.13)

after taking pz,i → 0. Now applying the crossing symmetry and moving into the

center-of-momentum frame of the resulting pair, one finds for the pair-production

matrix element

S f i[1] =−ie2εz

1εz2

2√

ω1ω2EL5 eikx,2(a2+λ2Bky,2/2)e−λ2

B(k2⊥,1+k2

⊥,2)/4 m(ω1µ1 − 2pz,+)

2ω1(µ1 pz,+ − E) + ω21(1− µ2

1)

× (2π)3δ3f i(E, py, pz). (3.44)

(The freedom of choice of background magnetic gauge allows us to set kx = 0 for one

but not both of the colliding photons.) Note also that this matrix element lines up with

the B BQ limit of the two-photon annihilation matrix element from Daugherty &

Bussard (1980), as expected. The second exchange term in the matrix element (Figure

3.8) is obtained by interchanging ε1, kµ1 with ε2, kµ

2 . Substituting S f i[1] + S f i[2] into

Equation (3.39) and setting B BQ gives Equation (3.34).

Chapter 4

e± processes: scattering, annihilation

and bremsstrahlung


4.1 Overview

We continue the analysis started in Chapter 3, now turning our attention to a series

of e± processes: two-photon pair annihilation, Coulomb scattering of electrons and

positrons off ions, relativistic e±-ion bremsstrahlung, and electron-positron scatter-

ing. The complications introduced by performing the full sum over initial, final

and/or intermediate-state Landau levels appear to have inhibited previous attempts

to calculate the last two processes. We also provide an approximate treatment of

electron-positron bremsstrahlung. For the most part, we are able to present results for

differential and total cross sections in terms of compact analytic formulae.

We first consider two-photon annihilation of an e± pair, expressing the cross section

in terms of an integral over the cross section for two-photon pair creation (Equation

(4.7)). The kinematic constraints on reconversion to a pair following two-photon

annihilation are derived (Equations (4.9) and (4.10)), and the resulting suppression

46

Chapter 4. e± processes: scattering, annihilation and bremsstrahlung 47

of pair annihilation is evaluated as a function of the kinetic energy of the colliding

electron and positron (Figure 4.2). Accurate fitting formulae to the net cross section

for pair annihilation are provided (Equations (4.12) and (4.13)).

We then, for the first time, calculate the cross section for the scattering of very

strongly magnetized electrons and positrons. The e± are capable of converting to a

real intermediate-state photon with energy exceeding 2m, which regulates the cross

section through its finite width (the second term in Equation (4.28)) and prevents

the t and s channels from interfering. The calculation of e±-ion backscattering is

generalized to include both relativistic motion and Debye screening (Equation (4.37)),

with the ion treated as immobile.

The cross section for relativistic bremsstrahlung is derived in the Born approxima-

tion, with the ion treated as immobile, and compared with the nonmagnetic result

(Bethe & Heitler 1934). The term in the cross section associated with emission during

backscattering is shown to agree with the expected soft-photon scaling obtained from

the e±-ion backscattering cross section. At low frequencies and particle energies, the

magnetic field suppresses the cross section by about an order of magnitude, with

the suppression becoming stronger at higher energies (Equation (4.47) and Figure

4.7). The free-free emission is then averaged over a one-dimensional thermal electron

distribution. The corresponding Gaunt factor is tabulated in Table 4.1; analytic non-

relativistic approximations to it are derived and found to be suppressed relative to

non-magnetic values by only a factor of 2. This gaunt factor is then used to quantify

thermal bremsstrahlung absorption in the non-relativistic limit.

The uncertainty in the thermal bremsstrahlung emissivity arising from the neglect

of the electron-ion interaction in the spinor wave functions is quantified (Section 4.5.4).

This uncertainty becomes proportionately smaller as the electrons grow more relativis-

tic. The rate of photon emission by radiative recombination is derived and compared

with thermal bremsstrahlung. Free-bound emission is shown to be negligible com-


pared with free-free, except at high frequencies, ω T. To conclude, we provide an

approximate treatment of electron-positron bremsstrahlung, acknowledging that the

uncertainties in the approach merit future consideration. The consequences of these

uncertainties will be demonstrated in Chapter 5 (Figure 5.9).

4.2 Two-photon Pair Annihilation

The cross section for the annihilation of an electron and positron into two photons

is suppressed by a factor of ∼ (B/BQ)−1 in the presence of an ultrastrong magnetic

field. The full sum over intermediate-state Landau levels is presented by Daugherty &

Bussard (1980). The net rate of pair annihilation is further suppressed, as we quantify

here, by the reconversion of one or both of the created photons back to a pair. When a

photon exceeds the energy threshold (3.15), this conversion to a pair is generally very

rapid compared with any further two-particle interactions. The net effect is to reduce

the overall annihilation rate within a gas of electrons and positrons; the strength of

the effect grows rapidly as the colliding pairs become mildly relativistic.

We first present an integral formula relating the annihilation cross section to the

cross section for two-photon pair creation, which takes a very simple form when

B BQ. The annihilation cross section is given by the phase-space integral

σann =

∣∣∣∣ pz,+

E+− pz,−

E−

∣∣∣∣−1 12

∫ L3ω21dω1dΩ1

(2π)3

∫ L3ω22dω2dΩ2

(2π)3

∫ da+L

∫ da−L

L3

T ∑σ+,σ−

|S f i|2. (4.1)

The incoming pair and intermediate e± lines are restricted to the lowest Landau state,

σ± = ±1. Two-photon pair annihilation and creation involve essentially the same

matrix element, with all momenta simply reversed in sign:

|S f i|2ann = |S f i|2cre = S20(pz,−, pz,+, ω1, ω2, µ1, µ2) · (2π)3δ3

f i(E, py, pz). (4.2)

We work in the center-of-momentum frame (pz,− = −pz,+ = pz, E+ = E− = E).

Substituting this into the phase-space integral (3.39) gives a relation between S0 and


the two-photon pair creation cross section (3.34),

S20 =

2πTλ2B|pz|

L8E· |1− µ12|σcre. (4.3)

Combining Equations (4.2) and (4.3) with the integral (4.1) gives for the annihilation

cross section

σann =λ4

B4

∫dµ1dµ2

[ω2

1ω22

|µ1 − µ2|· |1− µ12|σcre

]. (4.4)

This relation can also be obtained by a detailed balance argument. One writes for the

reaction e+ + e− ↔ ω1 + ω2

(1 + Nγ1)(1 + Nγ2)|β+ − β−|d2σann

dµ1dµ2∆µ1∆µ2 · 2

dne+

dpz,+∆pz,+

dne−

dpz,−∆pz,−

= (1− N+)(1− N−)|1− µ12|σcre ·12

d2nγ

dω1dµ1∆ω1∆µ1

d2nγ

dω2dµ2∆ω2∆µ2. (4.5)

The factor of 2 on the left-hand side counts the two signs of pz for electrons and

positrons, and the factor of 12 on the right-hand side takes into account the indistin-

guishability of the two photons produced in an annihilation event. We substitute

dne±

dpz,±=

eB(2π)2 N± =

eB(2π)2

1eE±/T + 1

;

d2nγ

dω1,2dµ1,2=

ω21,2

(2π)2 Nγ1,2 =ω2

1,2

(2π)21

eω1,2/T − 1(4.6)

for a thermal plasma, along with ω1 + ω2 = E+ + E− and the Jacobian factor

∆ω1∆ω2/∆pz,+∆pz,− = |β+ − β−|/|µ1 − µ2|. The factors involving the occupation

numbers N± and Nγ1,2 cancel, and we obtain Equation (4.4).

A more explicit form for this cross section is obtained by substituting Equation

(3.34) for σcre,

σann =2πr2

eB/BQ

β

γ2

∫dµ1dµ2

−|µ1 − µ2|(1− µ21)(1− µ2

2)

µ1µ2[(1− µ1µ2)2 − β2(µ1 − µ2)2]2. (4.7)

Here, β = |pz|/E is the speed of the incoming electron and positron, and

ω1 =2Eµ2

µ2 − µ1; ω2 =

2Eµ1

µ1 − µ2. (4.8)


Figure 4.1: In each panel, the left shading marks the zone where photon 1 (µ1 ≥ 0) is above

threshold for pair conversion. The right shading marks the zone where photon 2 (µ2 ≤ 0) is

above threshold. These zones are excluded from the integral (4.7). Left panel: colliding e±

each have energy γ = 1.3 in the center-of-momentum frame. Right panel: γ = 2.5 > 2; both

photons convert to a pair in the overlapping shaded zone.

Demanding that both of the created photons remain below the threshold for pair

creation, (1− µ2i )

1/2ωi < 2m, restricts the range of µ1, µ2 in the integral (4.7). Since

µ1 · µ2 ≤ 0 is kinematically required in the center-of-momentum frame, we focus on

the quadrant µ1 ≥ 0 and µ2 ≤ 0. Given a value of µ1, consider the threshold for

photon 2 to pair create, ω2(1− µ22)

1/2 > 2m. Substituting Equation (4.8) for ω2, one

obtains the range of µ2 in which two photons survive in the final state,

|µ2| > µth(µ1) ≡µ1(γ

√1 + µ2

1β2γ2 − 1)

1 + γ2µ21

(0 ≤ µ1 ≤ 1). (4.9)

There is a similar bound µ1 > µth(|µ2|) to avoid pair creation of photon 1. The

excluded zones in the space µ1, |µ2| are marked out in Figure 4.1.

The possibility that both photons convert to pairs opens up when the line µ1 = |µ2|

first intersects the curve (4.9), which happens when γ = 2. Then the entire zone

µ1, |µ2| < µmin =

(1− 4

γ2

)1/2

(γ > 2) (4.10)


is excluded. Combining these bounds and setting µmin = 0 for γ < 2 gives the full

integral for final states in which both photons remain below threshold for conversion

to a pair,

σann =8πr2

eB/BQ

β

γ2

∫ 1

µmin

dµ1

µ1

∫ µ1

µth(µ1)

d|µ2||µ2|

(µ1 + |µ2|)(1− µ21)(1− µ2

2)[(1 + µ1|µ2|)2 − β2(µ1 + |µ2|)2

]2 . (4.11)

Just as in the case of two-photon pair creation, the dominant finite-B correction to this

formula comes from the exponential factor (3.36).

Figure 4.2: Net cross section for two-photon annihilation, as a function of the kinetic energy of

the incoming electron and positron as measured in the center-of-momentum frame. Only final

photon states that are below the threshold (3.15) for single-photon pair creation contribute to

the phase-space integral in Equation (4.11). The sharp drop at γ & 2 represents the rapidly

shrinking phase space for non-pair-converting photons.

The evaluation of the integrals in Equation (4.11), as a function of the Lorentz


factor of the incoming electron and positron, is shown in Figure 4.2. The behavior at

low γ lines up qualitatively with the result of Daugherty & Bussard (1980), who found

that the annihilation cross section at rest decreases with increasing B. The decrease

in the cross section beyond γ− 1 ∼ 0.1 is due to the growing restriction on the final

photon phase space from reconversion to a pair. A combined analytic and numerical

fit, valid up to γ ' 2, is

σ =8πr2

eB/BQ

β

γ2

[59− 1.29116β2 − 0.0886β4 − ln(β)

(43+ 0.93634β2 + 4.53916β4

)].

(4.12)

(Here, the leading linear term and the leading term proportional to ln(β) inside the

brackets are derivable analytically in the low-β regime.) The sharp cutoff at higher γ

is fitted as

σ =8πr2

eB/BQ

1γ6

(1 +

2.84γ2 +

82γ6

)(4.13)

for γ & 2, with the leading term representing an analytic fit.

4.3 Electron-Positron scattering

Here we consider electron-positron (Bhabha) scattering in the presence of a magnetic

field,

e−i + e+i → e−f + e+f , (4.14)

with the initial and final particles all confined to the lowest Landau state (Figure

4.3). The particle kinetic energies are assumed to be well above the binding energy of

positronium, which is (Shabad & Usov 2006)

|E±,0| 'α2

em4

(ln

BBQ

)2

m (4.15)

in a strong magnetic field. Although the electron-positron interaction is important

for adiabatic conversion of a photon moving through a curved magnetic field into a


Figure 4.3: Feynman diagrams for electron-positron scattering. The left diagram [1] represents

the scattering (t) channel, and the right diagram [2] represents the annihilation (s) channel.

pair (Shabad & Usov 1985), the virtual photon appearing in the scattering of warm

electrons and positrons is far from the pair creation threshold.

Forward Coulomb scattering features a divergence (Langer 1981) that is resolved

when one takes into account electric field screening (Potekhin & Lai 2007). However,

in the case where all particles are confined to the lowest Landau level, the conservation

of momentum and energy implies that pz−, f = pz−,i, pz+, f = pz+,i. The effects of

forward scattering are therefore uninteresting, and we focus on backscattering of the

electron and positron, corresponding to pz−, f = pz+,i, pz+, f = pz−,i.

The cross section is obtained from the integral

σ =1

|β+ − β−|

∫ da+,i

L

∫ da−, f L2πλ2

B

∫ da+, f L2πλ2

B

∫ dpz−, f L2π

∫ dpz+, f L2π

L3

T

∣∣∣S f i[1] + S f i[2]∣∣∣2,

(4.16)

where β = pz/E. The S-matrix has two terms, corresponding to the two diagrams in

Figure 4.3. For the annihilation diagram, it is

S f i[2] = −ie2∫

d4x∫

d4x′[ψ(−1)− (x′)

]pz−, f ,n=0,a−, f

γµ

[ψ(+1)+ (x′)

]pz+, f ,n=0,a+, f

Gµνγ (x′ − x)

×[ψ(+1)+ (x)

]pz+,i ,n=0,a+,i

γν

[ψ(−1)− (x)

]pz−,i ,n=0,a−,i

. (4.17)


Substituting for the photon propagator from Equation (2.20), this becomes

S f i[2] =ie2

(2π)2 δ(E−,i + E+,i − E−, f − E+, f )∫

d3qI2,µηµν I1,ν

(E+,i + E−,i)2 −ω2 , (4.18)

where

I1,ν =∫

d3x[v(+1)∗

0,a+,i(x)]T

γ0γνu(−1)0,a−,i

(x)ei(p+,i+p−,i)·x⊥+i(pz+,i+pz−,i)ze−iq·x, (4.19)

and

I2,µ =∫

d3x′[u(−1)∗

0,a−, f(x′)

]Tγ0γµv(+1)

0,a+, f(x′)e−i(p+, f +p−, f )·x′⊥−i(pz+, f +pz−, f )z′eiq·x′ . (4.20)

These integrals are evaluated in Appendix B.We work in the center-of-momentum frame, where pz−,i = −pz+,i = pz, β−,i =

−β+,i = β, and E−,i = E+,i = E, giving

S f i[2] = −ie2m2

L4E2

∫dqx

exp[iqx(a+, f + a−, f − a+,i − a−,i)/2− λ2

B(q2x + q2

y)/2]

4E2 − q2x − q2

y(2π)2δ

(3)f i (E, py, pz).

(4.21)

Here the conservation of momentum implies qy = qy,2 ≡ (a+,i − a−,i)/λ2B and qz =

pz+,i + pz−,i = 0.The matrix element for the scattering diagram can be obtained from S f i[2] by

exchanging the momenta of the initial-state positron and the final-state electron,

pµ+,i ↔ −pµ

−, f and a+,i ↔ a−, f ,

S f i[1] = −ie2m2

L4E2

∫dqx

exp[iqx(a+, f + a+,i − a−, f − a−,i)/2− λ2

B(q2x + q2

y)/2]

q2z + q2

y + q2x

(2π)2δ(3)f i (E, py, pz).

(4.22)

Here qz = pz−,i − pz−, f = 2pz (for backscattering) and qy = qy,1 ≡ (a−, f − a−,i)/λ2B.

To evaluate these integrals over qx, we invoke the translational invariance of the

background to set a−,i = 0 and also take λB → 0 in the exponentials. We consider first

the scattering diagram, which has poles at qx = ±i√

4p2z + q2

y,1, so that the exponent

becomes exp[iqxa+,i]. Depending on the sign of a+,i, we close the contour at positive

or negative imaginary qx, yielding

S f i[1] = −ie2m2

2L4E2 exp[−|a+,i|

√4p2

z + q2y,1

] sgn(a+,i)√4p2

z + q2y,1

(2π)3δ(3)f i (E, py, pz). (4.23)


As for the annihilation diagram, we must take into account the finite width of

the intermediate-state photon, which has an energy exceeding 2m. Taking the decay

rate of a photon into a pair confined to the lowest Landau level (Equation 3.19) and

substituting ω = |q| → |q| − iΓ±/2, the integral in Equation (4.21) gives

S f i[2] =e2m2

2L4E2 exp[|a−, f |(iqres

x,R − qresx,I)]sgn(a−, f )

qresx

(2π)3δ(3)f i (E, py, pz). (4.24)

Here qresx = qres

x,R + iqresx,I , where

(qresx,R)

2 =12(4E2 − q2

y,2) +12

√(4E2 − q2

y,2)2 + ω2Γ2

± (4.25)

and

(qresx,I)

2 = −12(4E2 − q2

y,2) +12

√(4E2 − q2

y,2)2 + ω2Γ2

±. (4.26)

Near the pole, one has ω = 2E to lowest order in Γ±.

Substituting Equations (4.23) and (4.24) into Equation (4.16) gives the total cross

section

σ =e4m4

16π|pz|E3

∫da+,i

da−, f

λ2B

da+, f

λ2B

∫dpz−, f dpz+, f δ

(3)f i (E, py, pz)

×

∣∣∣∣∣∣sgn(a−, f )

qresx

exp[|a−, f |(iqres

x,R − qresx,I )]− i

sgn(a+,i)√4p2

z + q2y,1

exp[−|a+,i|

√4p2

z + q2y,1

]∣∣∣∣∣∣2

.

(4.27)

The integrand involves four terms – an annihilation term, a scattering term, and two

cross terms. The cross terms, when integrated over a+, f using the delta function,

yield terms of the form sgn(a+,i)sgn(a−, f ) f (|a+,i|, |a−, f |), and vanish after further

integration over a+,i, a−, f . The remaining terms can be integrated in a straightforward

manner, giving

σ =πα2

emm4

4E2p4z

1 + β2 2EΓ±

π

2+ arcsin

1√1 + Γ2

±/4E2

. (4.28)

This result can be self-consistently applied when the energy of the incoming electron

and positron is too small to permit excitation to the first Landau level, e.g. p2z <


2|e|B = 2(B/BQ)m2. The t-channel dominates at low energy, with a Rutherford-like

scaling in momentum, σ ∝ p−4z . The s-channel begins to dominate when β2 & Γ±/m.

4.4 Scattering of Electrons and Positrons off Heavy Ions

Here we consider the scattering of relativistic electrons and positrons by heavy ions

(Figure 4.4),

Z + e±i → Z + e±f , (4.29)

in a magnetic field B BQ. The ion is treated as a fixed Coulomb field centered at

x = y = z = 0,

A0ion(x) =

Ze4π√

x2 + y2 + z2. (4.30)

We assume that the ion has no gyrational motion in the initial state. Excitation

to a higher Landau level during scattering is suppressed by a factor of ∼ BQ/B

(compare Equations (8) and (9) of Potekhin & Lai (2007) evaluated for final Landau

state N′ = N + 1 = 1 as compared with N′ = N = 0).

4.4.1 Quantum Scattering

We work in the Born approximation, where the deformation of the incoming and

outgoing spinor wave functions by the Coulomb field is ignored. In the case of

electron-proton scattering, this is a good first approximation if the initial kinetic

energy of the electron is much larger than the binding energy of hydrogen in the

strong magnetic field, which is (Harding & Lai 2006)

Ei −m |EH,0| ≈ 0.32(

lnB

α2emBQ

)2

Ryd = 8.7× 10−6(

lnB

α2emBQ

)2

m. (4.31)

A characteristic impact parameter for the incoming particle is ai ∼ 1/pz,i, which

greatly exceeds the localization length λB of the electron or positron in the dimensions

transverse to B. As a result, given our choice of background magnetic gauge, the


Figure 4.4: Feynman diagram for electron-ion scattering. The ion is approximated as infinitely

heavy and is represented by a fixed Coulomb field (gray line).

scattered particle can be assigned a well-defined impact parameter x = ai with respect

to the Coulomb center. Focusing on electron-ion scattering, the matrix element written

in coordinate space is

S f i = −ie∫

d4x[ψ(−1)− (x)

]pz, f ,n=0,a f

γ0A0ion(x)

[ψ(−1)− (x)

]pz,i,n=0,ai

. (4.32)

The spatial portion of this integral is

∫d3x

[u(−1)∗

0,a f(x)]T

A0ion(x) u(−1)

0,ai(x) ei(pi−p f )·x⊥+i(pz,i−pz, f )z

=Ze2

K0[(pz,i − pz, f )b]pz,i pz, f + (Ei + m)(E f + m)

L2[EiE f (Ei + m)(E f + m)]1/2 δ

(a f − ai

λ2B

). (4.33)

As we explain in Appendix C, the delta function in ai arises in the regime ai λB,

meaning that over most impact parameters the scattered particle is only weakly

deflected across the very strong magnetic field.

We therefore focus on the case of backscattering, pz, f = −pz,i. Performing the time

integral gives

S f i = −iZe2

L22πm

EiK0(2pz,iai) δ

(a f − ai

λ2B

)δ(Ei − E f ). (4.34)

The absence of a delta function in pz follows from our assumption of an immobile ion.


The cross section per unit area is

12πai

dσ

dai=

1|βi|

LT

∫L

da f

2πλ2B

∫ 0

−∞L

dpz, f

2π

∣∣S f i∣∣2 =

Z2e4

4π2β2i γ2

iK2

0(2pz,iai), (4.35)

where γi = Ei/m and βi = pz,i/Ei. The integral over impact parameter converges,

giving a total cross section

σ =πZ2r2

e

β4i γ4

i(4.36)

for backscattering of an electron off a heavy positive ion. Here re = αem/m is the

classical electron radius. The cross section for positron-ion scattering, evaluated in the

same Born approximation, is identical to Equation (4.36).

The dependence of σ on the incoming particle momentum is as expected for

relativistic Rutherford backscattering. It also lines up with the B → ∞ limit of the

electron-ion backscattering cross section derived in Bussard (1980).

The effect of Debye screening on Coulomb scattering has previously been computed

in the nonrelativistic regime by Pavlov & Yakovlev (1976), Neugebauer et al. (1996),

and Potekhin & Lai (2007). It is straightforward to include in the present situation,

where relativistic electrons or positrons are confined to the lowest Landau level.

Allowing for a background electron gas of number density ne and temperature T, the

ion’s electric field is reduced by a factor exp(−r/rD), where rD = (T/4πneZ2e2)1/2.

Then, we find

σ =πZ2r2

e

β2i γ2

i

1β2

i γ2i + 1/4m2r2

D. (4.37)

Taking the nonrelativistic limit, this reproduces the results of Pavlov & Yakovlev (1976)

and Neugebauer et al. (1996). The softening of the momentum dependence of the

cross section has an important effect on ion-electron drag when the electrons have a

one-dimensional momentum distribution.


4.4.2 Classical Scattering

There are dramatic differences in the classical backscattering of electrons and positrons

off positive ions that are not evident in the quantum Born calculation. The result for

positron-proton scattering (like charges) is shown in Figure 4.5. As the background

magnetic field B → ∞, the cross section approaches the simple kinematic result

πr2e /(γi − 1)2, which is obtained by treating the light charge as a bead on a wire

and finding the critical impact parameter where the Coulomb repulsion absorbs its

asymptotic kinetic energy. On the other hand, the classical cross section approaches

the low-energy quantum result when γi − 1 & 0.1(B/BQ)2/3.

The case of opposite-charge Coulomb scattering in a magnetic field reveals compli-

cated chaotic motion (Hu et al. 2002). The transfer of Coulomb energy to gyrational

motion allows the classical electron to remain in the neighborhood of the ion for

long intervals, whose precise duration is extremely sensitive to the energy at infinity.

Furthermore, the cross section tends to zero as B→ ∞, because backscattering requires

motion across the magnetic field. This suppression is absent in the quantum case,

because the incident wavepacket can backscatter off an attractive potential.

4.5 Relativistic e±-Ion Bremsstrahlung

We now consider bremsstrahlung (free-free) emission by an electron or positron

interacting with a heavy ion of charge Ze, as modified by the presence of a background

magnetic field (Figure 4.6),

Z + e−i → Z + e−f + γ. (4.38)

In contrast with Pavlov & Panov (1976), we allow for relativistic electron motion but

focus on the case B BQ, where the initial, intermediate, and final electron lines can

be restricted to the lowest Landau level. We then calculate the thermally averaged


Figure 4.5: Classical and quantum cross sections for the backscattering of a positron off a

proton in a strong magnetic field, vs. initial kinetic energy. The positron is confined to the

lowest Landau state, and the proton is approximated as a fixed Coulomb potential. As B rises,

the classical cross section approaches the simple kinematic result πr2e /(γi − 1)2; whereas it

approaches the low-energy quantum result when γi − 1 & 0.1(B/BQ)2/3.


Figure 4.6: Feynman diagrams for bremsstrahlung, evaluated in the approximation where the

ion is infinitely heavy and represented by a fixed Coulomb potential (gray line).

emissivity for a range of temperatures. The error introduced by the adoption of

the Born approximation (the neglect of the electrostatic electron-ion interaction in

the spinor wave functions) is also quantified: it decreases with increasing electron

temperature. We also show that radiative recombination into bound electron-ion states

is suppressed relative to free-free emission by a strong magnetic field.

4.5.1 Derivation

As in Section 4.4, we treat the ion as a fixed Coulomb field centered at x = y = z = 0

and neglect gyrational excitation of the ion in the initial state. The emission of a

photon of energy ' eB/mp has a resonantly enhanced cross section associated with

the transition of the ion to a higher Landau level (e.g. Potekhin & Lai 2007), but

in a super-QED magnetic field this need not exceed the cross section for free-free

emission with the ion remaining in the ground Landau state, being suppressed by a

factor ∼ (B/BQ)−1. For trans-relativistic electrons, the relative importance of resonant

emission is further reduced by the fact that most of the energy in free-free photons is

radiated near ω ∼ m (when there is no Landau transition of the ion), well above the


∼ 10 keV energy of the proton cyclotron line near the surface of a magnetar 1.

Following the same logic outlined in Section 4.4, we assign the incoming electron

an impact parameter ai with respect to the ion. (The calculation for positron-ion

bremsstrahlung gives an identical result.) The first term in the matrix element is

S f i[1] = −ie2∫

d4x∫

d4x′[ψ(−1)− (x′)

]pz, f ,n=0,a f

γν Aν(x)∗G f (x′− x)γ0 A0ion(x)

[ψ(−1)− (x)

]pz,i ,n=0,ai

.

(4.39)

Substituting for the electron propagator from Equation (2.19), this becomes

S f i[1] = −ie2

2π

(L

2ω

)1/2

δ(Ei − E f −ω)∫

dpI

∫ daI

λ2B

(I1 I2

Ei − EI+

I3 I4

Ei + EI

), (4.40)

where

I1 =∫

d3x[u(−1)∗

0,aI(x)]T Ze

4π√

x2 + y2 + z2u(−1)

0,ai(x)ei(pi−pI)·x⊥+i(pz,i−pz,I)z (4.41)

and

I2 =∫

d3x′[u(−1)∗

0,a f(x′)

]Tγ0γν

(εν

f eik·x′)∗

u(−1)0,aI

(x′)ei(−p f +pI)·x′⊥+i(−pz, f +pz,I)z′ . (4.42)

The integral I3 is obtained from I1, and I4 from I2, by replacing u(−1)0,aI

(x′) with the

negative-energy wave function v(+1)0,aI

(x′), and taking pI → −pI .

These integrals are evaluated in Appendix C; substituting into Equation (4.40)

gives

S f i[1] =iZe3π

(2ωL7)1/2 δ(Ei − E f −ω) δ

(a f − ai

λ2B− ky

)(εz)∗e−ikx(ai+a f )/2e−λ2

Bk2⊥/4

×K0[(pz,i − pz, f − kz)ai]F(pz,i, pz, f , kz)

[EiE f (Ei + m)(E f + m)]1/2(E2i − E2

I ). (4.43)

1For subrelativistic electrons, the ratio of resonant to nonresonant bremsstrahlung cross sectionsis ∼ (3π/8)(αemB/BQ)

−1 at ω ' eB/mp, which is not much different from unity when B ∼ 102 BQ.This may be checked by considering the inverse process of free-free absorption, as given by Equations(B1)-(B11) of Potekhin & Lai (2007) and applying Kirchhoff’s law. The resonant contribution to thecross section (B1) for free-free absorption (polarization index α = +1) is averaged over frequency nearω = eB/mp, and compared with the nonresonant contribution (polarization index α = 0). Note thatwhereas both E-mode and O-mode photons have a component of their electric vectors proportional tothe basis vector (εx + iεy)/

√2 (corresponding to α = +1), only the O-mode overlaps with εz (α = 0).


Here

F ≡ pz,i(E f +m)(Ei +E f − 2m)+ pz, f (Ei +m)(Ei +E f + 2m)+ kz[pz,i pz, f +(Ei +m)(E f +m)],

(4.44)

where E2I = p2

z,I + m2 = (pz, f + kz)2 + m2, and E2i − E2

I = (pz,i − pz, f − kz)(pz,i +

pz, f + kz) ≡ ∆p‖(pz,i + pz, f + kz). The absence of a delta function in pz once again

follows from the assumption of an immobile ion. The second term in the matrix

element is related to S f i[1] by substituting pz,i ↔ pz, f and kz → −kz (corresponding

to pz,I = pz, f + kz → pz,i − kz). Summing the two terms, we find

S f i[1] + S f i[2] =iZe3π

(2ωL7)1/2 δ(Ei − E f −ω) δ

(a f − ai

λ2B− ky

)(εz)∗e−ikx(a f +ai)/2e−λ2

Bk2⊥/4

×K0(∆p‖ ai)4m(pz,i + pz, f )[pz, f (Ei + m)− pz,i(E f + m)]

[EiE f (Ei + m)(E f + m)]1/2∆p‖[(pz,i + pz, f )2 − k2z]

. (4.45)

We now take λB → 0 and calculate the differential cross section for a beam of

particles:

12πai

dσ

dai=

1|βi|

LT

∫L3 ω2dωdΩ

(2π)3

∫ Lda f

2πλ2B

∫L

dpz, f

2π

∣∣S f i[1] + S f i[2]∣∣2. (4.46)

Integrating over the cross section of the beam and using the identity∫ ∞

0 tK20(t)dt = 1

2

gives

ωd2σ

dωdΩ= ∑

pz, f

4Z2

π

(e2

4π

)3 |εz|2

|βiβ f |ω2m2(pz,i + pz, f )

2[pz, f (Ei + m)− pz,i(E f + m)]2

(∆p||)4[(pz,i + pz, f )2 − k2z]

2EiE f (Ei + m)(E f + m).

(4.47)

Here we sum over the two energetically permissible values of the final momentum,

pz, f = ±√

E2f −m2.

Note that the integral over impact parameter is convergent, due to the presence of

a strong background magnetic field: the kinetic momentum of the outgoing electron

is directed along the magnetic field. The integral of Equation (4.47) over the solid

angle is elementary but cumbersome; it is used in the numerical evaluation of free-free

emission from a thermal plasma but is not repeated here.


The effect of Debye screening of the ion’s Coulomb field is easy to include in this

calculation, by multiplying the electrostatic potential by exp(−r/rD). However, in

many instances screening has a negligible effect. The dominant contribution in the

integral over impact parameter comes from ai ∼ 1/∆p‖ ∝ ω−1. On the other hand,

the Debye screening length is proportional to the inverse of the plasma frequency.

This means that screening can be ignored as long as ω lies well above the plasma

cutoff, because rD is much larger than the dominant emission impact parameter.

We next compare the integral of Equation (4.47) over solid angle with the for-

mula derived by Bethe & Heitler (1934) for relativistic free-free emission in the Born

approximation and at B = 0:

ωdσ

dω=

Z2

m2

(e2

4π

)3 p f

pi

43− 2EiE f

p2i + p2

f

p2i p2

f+ m2

(εiE f

p3i

+ε f Ei

p3f−

εiε f

pi p f

)

+ Lg

[8EiE f

3pi p f+

ω2(E2i E2

f + p2i p2

f )

p3i p3

f

]+

m2ωLg

2pi p f

[EiE f + p2

i

p3i

εi −EiE f + p2

f

p3f

ε f +2ωEiE f

p2i p2

f

].

(4.48)

Here

εi, f = 2 lnEi, f + pi, f

m; Lg = 2 ln

EiE f + pi p f −m2

mω. (4.49)

Figure 4.7 shows that the magnetic cross section is generally smaller than Equation

(4.48), except near the limiting frequency Ei −m where the final-state electron moves

slowly.

It is possible to improve on the Born approximation for relativistic bremsstrahlung

by correcting the electron wave function for the electrostatic interaction with the ion

(Elwert & Haug 1969; Nozawa et al. 1998; van Hoof et al. 2015). In the absence of a

background magnetic field, this produces the correction factor first derived by Elwert

(1939). Obtaining a similar correction factor for the magnetic case is beyond the scope

of the present investigation. Nonetheless, Equation (4.47) is expected to be a good

approximation as long as the kinetic energy of the final-state electron (or positron)

exceeds the hydrogen binding energy (Equation (4.31)); see Section 4.5.4.


Figure 4.7: Comparison of the relativistic bremsstrahlung cross section for electrons interacting

with protons (Z = 1) in a background magnetic field (integral of Equation (4.47) over solid

angle; black curves) with the relativistic bremsstrahlung cross section in free space (Bethe &

Heitler 1934; blue curves). Sequences of curves correspond to a range of initial kinetic energy

Ei,kin = (γi − 1)m. The cross sections have been multiplied by a factor of Ei,kin for clarity. Each

curve is cut off at the maximum photon frequency ω = Ei,kin.


4.5.2 Limiting Cases

The bremsstrahlung cross section (Equation (4.47)) simplifies in various regimes. At

low emission frequencies, ω Ei,kin = Ei −m, one has

ωd2σ

dωdΩ= |εz|2 Z2αemr2

e(m/Ei)

4

πβ2i (1− βi cos θ)2

[(m/Ei)

2

(1− βi cos θ)2 +1

(1 + βi cos θ)2

].

(4.50)

The two terms on the right-hand side represent emission by forward-scattered and

back-scattered e±. The second term can be obtained from the Coulomb backscattering

cross section (Equation (4.36)) by multiplying by the soft photon factor αem(ω/2π)2(ε ·

p f /k · p f − ε · pi/k · pi)2. When the incident electron moves subrelativistically,

ωd2σ

dωdΩ= |εz|2 Z2αemr2

e2

π|βiβ f |. (4.51)

4.5.3 Thermal Bremsstrahlung

We now evaluate the free-free emission from a thermal electron-proton plasma (Z = 1),

with electrons confined to the lowest Landau level,

d2nγ

dωdt= nenp

⟨|βi|

dσ

dω

⟩. (4.52)

Here the cross section (Equation (4.47)) has been integrated over the solid angle of

the emitted photon; ne, np, and nγ are the number densities of electrons, protons, and

photons, respectively; and⟨|βi|

dσ

dω

⟩=

∫ ∞pz,min

dpz,iβi exp[−γim/T] dσ/dω∫ ∞0 dpz,i exp[−γim/T]

. (4.53)

The cutoff pz,min =√

ω2 + 2mω corresponds to an incident electron of the minimum

energy needed to emit a photon of energy ω. We follow convention and describe this

thermal average in terms of a Gaunt factor g(T, ω), defined as⟨|βi|

dσ

dω

⟩=

8αemr2e

3ω

e−(m+ω)/T

K1(m/T)g(T, ω). (4.54)


Figure 4.8: Gaunt factor as a function of ω/T for a range of temperatures T.


This integral over the momentum of the incoming electron is performed numerically.

The result is shown in Figure 4.8 for a range of temperatures and is tabulated in Table

4.1.

The Gaunt factor can be evaluated analytically in some limiting cases. When

m T ω,

g(T, ω) = ln(

4Tω

)− γEM, (4.55)

where γEM is the Euler-Mascheroni constant. Next, when m ω T, we have

g(T, ω) =

√πTω

. (4.56)

Both of these expressions agree with the nonrelativistic results of Pavlov & Panov

(1976) in the regime B BQ (compare their Equations (38) and (41)). Equations (4.55)

and (4.56) resemble the results for free-free emission from an unmagnetized plasma

with m T 1 Ry, for which g(T, ω) ' 2[ln(4T/ω) − γEM] when ω T, and

g(T, ω) ' 2(πT/ω)1/2 when ω T (e.g. Novikov & Thorne 1973).

We also note that in a non-relativistic thermal background with an emissivity

jω = d2nγ/dωdt, O-mode photons will undergo bremsstrahlung absorption, with an

absorption coefficient

αω = neσω =jωBω

= 16πc3αemσenenig(T, ω)

(sin2 θ +

(ω

ωB

)2)√

2mc2

πkBT

1− exp(− hω

kBT

)ω3 .

(4.57)

Here Bω is the blackbody specific intensity and the formula is written in CGS units

unlike the rest of the Chapter. Owing to the sharp dependency on frequency, it will

primarily have the effect of absorbing low-energy photons.

Table 4.1: Gaunt Factor and Related Error for T/m = 10−2,−1.5,...,0.5,1 and B = 100 BQ

T/m = 10−2 T/m = 10−1.5 T/m = 10−1 T/m = 10−0.5 T/m = 100 T/m = 100.5 T/m = 10

ω/m g δg g δg g δg g δg g δg g δg g δg

0.0100 1.5093 0.7770 2.2798 0.8081 3.1692 0.8184 4.0545 0.8216 4.7988 0.8227 5.3180 0.8230 5.6217 0.8231


Table 4.1 continued

T/m = 10−2 T/m = 10−1.5 T/m = 10−1 T/m = 10−0.5 T/m = 100 T/m = 100.5 T/m = 10


0.0107 1.4678 0.7517 2.2267 0.7817 3.1081 0.7917 3.9889 0.7949 4.7311 0.7959 5.2494 0.7962 5.5529 0.7963

0.0115 1.4270 0.7270 2.1742 0.7561 3.0473 0.7658 3.9235 0.7688 4.6635 0.7698 5.1809 0.7701 5.4840 0.7703

0.0123 1.3870 0.7031 2.1223 0.7313 2.9870 0.7406 3.8583 0.7436 4.5960 0.7446 5.1124 0.7448 5.4151 0.7449

0.0132 1.3477 0.6798 2.0710 0.7071 2.9270 0.7162 3.7933 0.7191 4.5285 0.7200 5.0439 0.7203 5.3463 0.7204

0.0141 1.3092 0.6573 2.0204 0.6837 2.8674 0.6925 3.7284 0.6953 4.4611 0.6962 4.9755 0.6964 5.2774 0.6965

0.0151 1.2715 0.6354 1.9704 0.6610 2.8082 0.6695 3.6638 0.6722 4.3939 0.6731 4.9071 0.6733 5.2086 0.6734

0.0162 1.2345 0.6142 1.9210 0.6390 2.7494 0.6472 3.5994 0.6498 4.3267 0.6506 4.8387 0.6509 5.1398 0.6510

0.0174 1.1982 0.5937 1.8723 0.6176 2.6910 0.6255 3.5352 0.6281 4.2596 0.6289 4.7704 0.6291 5.0710 0.6292

0.0186 1.1628 0.5737 1.8242 0.5969 2.6331 0.6045 3.4713 0.6070 4.1926 0.6078 4.7022 0.6080 5.0023 0.6081

0.0200 1.1281 0.5544 1.7768 0.5768 2.5757 0.5842 3.4076 0.5866 4.1257 0.5873 4.6339 0.5875 4.9335 0.5876

0.0214 1.0941 0.5356 1.7301 0.5573 2.5187 0.5645 3.3441 0.5667 4.0590 0.5675 4.5658 0.5677 4.8648 0.5678

0.0229 1.0609 0.5175 1.6841 0.5384 2.4622 0.5453 3.2809 0.5475 3.9924 0.5483 4.4976 0.5485 4.7961 0.5486

0.0245 1.0284 0.4999 1.6388 0.5201 2.4061 0.5268 3.2180 0.5289 3.9259 0.5296 4.4296 0.5298 4.7275 0.5299

0.0263 0.9967 0.4828 1.5942 0.5023 2.3506 0.5088 3.1554 0.5109 3.8595 0.5116 4.3616 0.5118 4.6588 0.5118

0.0282 0.9657 0.4663 1.5504 0.4852 2.2956 0.4914 3.0930 0.4934 3.7933 0.4941 4.2936 0.4943 4.5902 0.4943

0.0302 0.9355 0.4503 1.5072 0.4685 2.2412 0.4746 3.0310 0.4765 3.7272 0.4771 4.2258 0.4773 4.5217 0.4774

0.0324 0.9059 0.4347 1.4648 0.4524 2.1873 0.4582 2.9693 0.4601 3.6613 0.4607 4.1580 0.4609 4.4532 0.4609

0.0347 0.8771 0.4197 1.4231 0.4367 2.1339 0.4424 2.9079 0.4442 3.5955 0.4448 4.0902 0.4449 4.3847 0.4450

0.0372 0.8490 0.4052 1.3821 0.4216 2.0812 0.4271 2.8469 0.4288 3.5299 0.4293 4.0226 0.4295 4.3162 0.4296

0.0398 0.8216 0.3910 1.3419 0.4069 2.0290 0.4122 2.7862 0.4139 3.4645 0.4144 3.9551 0.4146 4.2478 0.4146

0.0427 0.7949 0.3774 1.3024 0.3927 1.9773 0.3978 2.7258 0.3994 3.3993 0.3999 3.8876 0.4001 4.1795 0.4002

0.0457 0.7688 0.3642 1.2637 0.3790 1.9263 0.3839 2.6659 0.3854 3.3343 0.3859 3.8202 0.3861 4.1112 0.3861

0.0490 0.7434 0.3513 1.2257 0.3656 1.8760 0.3704 2.6063 0.3719 3.2695 0.3724 3.7530 0.3725 4.0430 0.3726

0.0525 0.7187 0.3389 1.1884 0.3527 1.8262 0.3573 2.5472 0.3587 3.2049 0.3592 3.6858 0.3594 3.9748 0.3594


Table 4.1 continued

T/m = 10−2 T/m = 10−1.5 T/m = 10−1 T/m = 10−0.5 T/m = 100 T/m = 100.5 T/m = 10


0.0562 0.6946 0.3269 1.1519 0.3402 1.7771 0.3446 2.4884 0.3460 3.1405 0.3465 3.6188 0.3466 3.9067 0.3467

0.0603 0.6712 0.3153 1.1161 0.3281 1.7286 0.3324 2.4301 0.3337 3.0764 0.3341 3.5519 0.3343 3.8387 0.3343

0.0646 0.6484 0.3040 1.0811 0.3164 1.6808 0.3205 2.3723 0.3218 3.0125 0.3222 3.4851 0.3223 3.7708 0.3224

0.0692 0.6262 0.2931 1.0468 0.3050 1.6336 0.3090 2.3149 0.3102 2.9489 0.3106 3.4185 0.3108 3.7030 0.3108

0.0741 0.6046 0.2825 1.0132 0.2940 1.5872 0.2978 2.2579 0.2991 2.8855 0.2994 3.3521 0.2996 3.6353 0.2996

0.0794 0.5836 0.2723 0.9804 0.2834 1.5414 0.2870 2.2015 0.2882 2.8224 0.2886 3.2858 0.2887 3.5677 0.2887

0.0851 0.5632 0.2624 0.9483 0.2730 1.4963 0.2766 2.1455 0.2777 2.7597 0.2781 3.2197 0.2782 3.5002 0.2782

0.0912 0.5433 0.2528 0.9169 0.2631 1.4519 0.2665 2.0901 0.2676 2.6972 0.2679 3.1537 0.2680 3.4328 0.2680

0.0977 0.5240 0.2435 0.8863 0.2534 1.4082 0.2567 2.0352 0.2577 2.6351 0.2580 3.0880 0.2582 3.3656 0.2582

0.1047 0.5053 0.2344 0.8563 0.2440 1.3653 0.2472 1.9809 0.2482 2.5734 0.2485 3.0225 0.2486 3.2985 0.2486

0.1122 0.4870 0.2257 0.8271 0.2349 1.3230 0.2380 1.9271 0.2389 2.5120 0.2392 2.9572 0.2393 3.2317 0.2394

0.1202 0.4693 0.2173 0.7985 0.2261 1.2815 0.2291 1.8739 0.2300 2.4510 0.2303 2.8922 0.2304 3.1650 0.2304

0.1288 0.4521 0.2091 0.7707 0.2176 1.2407 0.2204 1.8213 0.2213 2.3904 0.2216 2.8275 0.2217 3.0985 0.2217

0.1380 0.4354 0.2012 0.7435 0.2094 1.2007 0.2121 1.7694 0.2129 2.3302 0.2132 2.7630 0.2133 3.0322 0.2133

0.1479 0.4192 0.1935 0.7170 0.2014 1.1614 0.2040 1.7180 0.2048 2.2705 0.2051 2.6989 0.2052 2.9662 0.2052

0.1585 0.4034 0.1860 0.6912 0.1936 1.1229 0.1962 1.6673 0.1970 2.2112 0.1972 2.6351 0.1973 2.9004 0.1973

0.1698 0.3882 0.1788 0.6660 0.1861 1.0851 0.1886 1.6173 0.1893 2.1525 0.1896 2.5716 0.1897 2.8349 0.1897

0.1820 0.3733 0.1719 0.6415 0.1789 1.0481 0.1812 1.5680 0.1820 2.0942 0.1822 2.5085 0.1823 2.7697 0.1823

0.1950 0.3589 0.1651 0.6177 0.1719 1.0119 0.1741 1.5193 0.1748 2.0365 0.1750 2.4458 0.1751 2.7049 0.1751

0.2089 0.3450 0.1586 0.5945 0.1651 0.9764 0.1672 1.4715 0.1679 1.9794 0.1681 2.3836 0.1682 2.6404 0.1682

0.2239 0.3315 0.1523 0.5719 0.1585 0.9418 0.1606 1.4243 0.1612 1.9228 0.1614 2.3218 0.1615 2.5763 0.1615

0.2399 0.3184 0.1462 0.5500 0.1521 0.9079 0.1541 1.3779 0.1548 1.8669 0.1550 2.2605 0.1550 2.5126 0.1550

0.2570 0.3057 0.1403 0.5287 0.1460 0.8748 0.1479 1.3323 0.1485 1.8117 0.1487 2.1998 0.1487 2.4494 0.1488

0.2754 0.2934 0.1346 0.5080 0.1400 0.8424 0.1419 1.2875 0.1424 1.7571 0.1426 2.1396 0.1427 2.3866 0.1427


Table 4.1 continued

T/m = 10−2 T/m = 10−1.5 T/m = 10−1 T/m = 10−0.5 T/m = 100 T/m = 100.5 T/m = 10


0.2951 0.2815 0.1290 0.4879 0.1343 0.8109 0.1360 1.2436 0.1366 1.7033 0.1368 2.0800 0.1368 2.3244 0.1368

0.3162 0.2700 0.1237 0.4684 0.1287 0.7802 0.1304 1.2005 0.1309 1.6502 0.1311 2.0210 0.1312 2.2628 0.1312

0.3388 0.2588 0.1185 0.4495 0.1234 0.7503 0.1250 1.1582 0.1255 1.5979 0.1256 1.9628 0.1257 2.2017 0.1257

0.3631 0.2481 0.1135 0.4312 0.1182 0.7212 0.1197 1.1169 0.1202 1.5464 0.1204 1.9052 0.1204 2.1413 0.1204

0.3890 0.2376 0.1087 0.4134 0.1132 0.6929 0.1147 1.0764 0.1151 1.4958 0.1153 1.8484 0.1153 2.0816 0.1153

0.4169 0.2276 0.1041 0.3963 0.1084 0.6654 0.1098 1.0369 0.1102 1.4461 0.1104 1.7924 0.1104 2.0227 0.1104

0.4467 0.2179 0.0996 0.3797 0.1037 0.6387 0.1051 0.9984 0.1055 1.3973 0.1056 1.7373 0.1057 1.9645 0.1057

0.4786 0.2086 0.0953 0.3637 0.0992 0.6128 0.1005 0.9608 0.1009 1.3495 0.1011 1.6830 0.1011 1.9071 0.1011

0.5129 0.1996 0.0912 0.3483 0.0949 0.5878 0.0962 0.9242 0.0965 1.3027 0.0967 1.6297 0.0967 1.8507 0.0967

0.5495 0.1909 0.0872 0.3334 0.0908 0.5636 0.0920 0.8886 0.0923 1.2570 0.0924 1.5774 0.0925 1.7951 0.0925

0.5888 0.1826 0.0834 0.3191 0.0868 0.5402 0.0879 0.8541 0.0883 1.2123 0.0884 1.5260 0.0884 1.7405 0.0884

0.6310 0.1746 0.0797 0.3053 0.0830 0.5176 0.0841 0.8205 0.0844 1.1687 0.0845 1.4758 0.0845 1.6869 0.0845

0.6761 0.1670 0.0762 0.2920 0.0793 0.4958 0.0803 0.7881 0.0807 1.1262 0.0808 1.4266 0.0808 1.6344 0.0808

0.7244 0.1596 0.0728 0.2793 0.0758 0.4749 0.0768 0.7567 0.0771 1.0850 0.0772 1.3786 0.0772 1.5830 0.0772

0.7762 0.1526 0.0696 0.2672 0.0724 0.4547 0.0734 0.7263 0.0737 1.0449 0.0738 1.3318 0.0738 1.5328 0.0738

0.8318 0.1458 0.0665 0.2555 0.0692 0.4354 0.0701 0.6971 0.0704 1.0060 0.0705 1.2863 0.0705 1.4837 0.0706

0.8913 0.1394 0.0636 0.2444 0.0662 0.4169 0.0670 0.6689 0.0673 0.9684 0.0674 1.2419 0.0674 1.4359 0.0674

0.9550 0.1333 0.0608 0.2337 0.0633 0.3992 0.0641 0.6419 0.0643 0.9320 0.0644 1.1989 0.0644 1.3893 0.0645

1.0233 0.1275 0.0581 0.2236 0.0605 0.3822 0.0613 0.6159 0.0615 0.8969 0.0616 1.1572 0.0616 1.3440 0.0616

1.0965 0.1219 0.0556 0.2139 0.0578 0.3661 0.0586 0.5910 0.0588 0.8631 0.0589 1.1168 0.0589 1.3001 0.0589

1.1749 0.1167 0.0532 0.2048 0.0553 0.3507 0.0561 0.5672 0.0563 0.8305 0.0564 1.0778 0.0564 1.2575 0.0564

1.2589 0.1117 0.0509 0.1960 0.0530 0.3360 0.0536 0.5444 0.0539 0.7993 0.0539 1.0401 0.0540 1.2162 0.0540

1.3490 0.1069 0.0487 0.1878 0.0507 0.3221 0.0514 0.5227 0.0516 0.7693 0.0516 1.0038 0.0517 1.1764 0.0517

1.4454 0.1024 0.0467 0.1800 0.0486 0.3089 0.0492 0.5021 0.0494 0.7407 0.0495 0.9689 0.0495 1.1379 0.0495


Table 4.1 continued

T/m = 10−2 T/m = 10−1.5 T/m = 10−1 T/m = 10−0.5 T/m = 100 T/m = 100.5 T/m = 10


1.5488 0.0982 0.0447 0.1726 0.0466 0.2965 0.0472 0.4825 0.0474 0.7133 0.0474 0.9354 0.0474 1.1008 0.0474

1.6596 0.0942 0.0429 0.1656 0.0447 0.2846 0.0453 0.4638 0.0454 0.6871 0.0455 0.9032 0.0455 1.0652 0.0455

1.7783 0.0905 0.0412 0.1590 0.0429 0.2735 0.0434 0.4462 0.0436 0.6622 0.0437 0.8725 0.0437 1.0309 0.0437

1.9055 0.0869 0.0396 0.1529 0.0412 0.2630 0.0417 0.4295 0.0419 0.6386 0.0420 0.8430 0.0420 0.9980 0.0420

2.0417 0.0836 0.0381 0.1470 0.0396 0.2531 0.0401 0.4137 0.0403 0.6161 0.0404 0.8149 0.0404 0.9665 0.0404

2.1878 0.0805 0.0367 0.1416 0.0382 0.2438 0.0386 0.3988 0.0388 0.5948 0.0389 0.7881 0.0389 0.9363 0.0389

2.3442 0.0776 0.0353 0.1365 0.0368 0.2350 0.0372 0.3848 0.0374 0.5746 0.0374 0.7626 0.0375 0.9074 0.0375

2.5119 0.0748 0.0341 0.1317 0.0355 0.2268 0.0359 0.3716 0.0361 0.5555 0.0361 0.7384 0.0361 0.8799 0.0361

2.6915 0.0723 0.0329 0.1272 0.0343 0.2191 0.0347 0.3592 0.0348 0.5374 0.0349 0.7154 0.0349 0.8536 0.0349

2.8840 0.0699 0.0318 0.1230 0.0331 0.2119 0.0335 0.3475 0.0337 0.5204 0.0337 0.6935 0.0337 0.8286 0.0337

3.0903 0.0677 0.0308 0.1190 0.0321 0.2052 0.0325 0.3366 0.0326 0.5044 0.0326 0.6728 0.0327 0.8048 0.0327

3.3113 0.0656 0.0299 0.1154 0.0311 0.1989 0.0315 0.3263 0.0316 0.4893 0.0316 0.6532 0.0317 0.7821 0.0317

3.5481 0.0636 0.0290 0.1120 0.0301 0.1930 0.0305 0.3167 0.0307 0.4751 0.0307 0.6347 0.0307 0.7606 0.0307

3.8019 0.0618 0.0281 0.1088 0.0293 0.1875 0.0297 0.3078 0.0298 0.4617 0.0298 0.6171 0.0298 0.7401 0.0298

4.0738 0.0601 0.0274 0.1058 0.0285 0.1824 0.0289 0.2994 0.0290 0.4492 0.0290 0.6006 0.0290 0.7207 0.0290

4.3652 0.0586 0.0267 0.1030 0.0277 0.1776 0.0281 0.2915 0.0282 0.4374 0.0283 0.5850 0.0283 0.7024 0.0283

4.6774 0.0571 0.0260 0.1004 0.0270 0.1732 0.0274 0.2842 0.0275 0.4264 0.0275 0.5703 0.0276 0.6850 0.0276

5.0119 0.0557 0.0254 0.0980 0.0264 0.1690 0.0267 0.2774 0.0269 0.4161 0.0269 0.5565 0.0269 0.6685 0.0269

5.3703 0.0545 0.0248 0.0958 0.0258 0.1652 0.0261 0.2710 0.0262 0.4064 0.0263 0.5434 0.0263 0.6529 0.0263

5.7544 0.0533 0.0243 0.0937 0.0252 0.1616 0.0256 0.2651 0.0257 0.3974 0.0257 0.5312 0.0257 0.6382 0.0257

6.1660 0.0522 0.0238 0.0918 0.0247 0.1582 0.0250 0.2595 0.0252 0.3889 0.0252 0.5197 0.0252 0.6243 0.0252

6.6069 0.0512 0.0233 0.0900 0.0242 0.1551 0.0246 0.2543 0.0247 0.3810 0.0247 0.5088 0.0247 0.6111 0.0247

7.0795 0.0502 0.0229 0.0883 0.0238 0.1522 0.0241 0.2495 0.0242 0.3736 0.0242 0.4986 0.0242 0.5987 0.0242

7.5858 0.0493 0.0225 0.0868 0.0234 0.1495 0.0237 0.2450 0.0238 0.3667 0.0238 0.4891 0.0238 0.5870 0.0238


Table 4.1 continued

T/m = 10−2 T/m = 10−1.5 T/m = 10−1 T/m = 10−0.5 T/m = 100 T/m = 100.5 T/m = 10


8.1283 0.0485 0.0221 0.0853 0.0230 0.1470 0.0233 0.2408 0.0234 0.3602 0.0234 0.4801 0.0234 0.5759 0.0234

8.7096 0.0477 0.0217 0.0840 0.0226 0.1447 0.0229 0.2369 0.0230 0.3541 0.0230 0.4717 0.0231 0.5655 0.0231

9.3325 0.0470 0.0214 0.0827 0.0223 0.1425 0.0226 0.2333 0.0227 0.3485 0.0227 0.4638 0.0227 0.5557 0.0227

10.0000 0.0464 0.0211 0.0816 0.0220 0.1405 0.0223 0.2299 0.0224 0.3432 0.0224 0.4564 0.0224 0.5464 0.0224

4.5.4 Validity of the Born Approximation

Equation (4.47) for the cross section, which is obtained in the Born approximation, is

a good first-order approximation only as long as the kinetic energy of the final-state

electron (or positron) exceeds the hydrogen binding energy in a magnetic field (4.31),

E f − m |EH,0|. However, as can be seen from Equation (4.47) and from Figure

4.7, the cross section diverges as β f → 0. Although the thermal average of dσ/dω is

convergent, a significant portion of the integral in Equation (4.53) may be supplied

by low momenta where the electrostatic correction to the electron wave function is

significant.

To quantify this uncertainty, we have separately evaluated the part δg of the mo-

mentum integral in Equation (4.53) supplied by pz,min ≤ pz,i ≤√|EH,0|(2m + |EH,0|).

This estimate of the error in g(ω, T) is included in Table 4.1. As the temperature

becomes more relativistic, the fractional error δg/g drops. The uncertainty is rela-

tively large at subrelativistic temperatures. We expect this result to persist in weaker

magnetic fields even when the emissivity is calculated to all orders in the magnetic

field, as was done by Pavlov & Panov (1976).


4.5.5 Comparison with Radiative Recombination

In an unmagnetized plasma, the radiative recombination of hydrogen has a relatively

large cross section compared with free-free emission when the kinetic energy of the

unbound electron is comparable to the ionization energy. The radiative free-bound

cross section scales as σfb ∼ r2e /αem as compared with ∼ αemr2

e for free-free emission.

Although these scalings with αem persist in the presence of an ultrastrong magnetic

field, as we now show, other factors emerge that suppress free-bound emission relative

to free-free emission. The net result is that free-bound emission is only competitive

for ω > T.

The cross section σfb may be obtained from the photoionization cross section σbf

using the principle of detailed balance. We consider only subrelativistic electrons and

photons of energy ω m interacting with free protons and neutral hydrogen, in

which case (Gnedin et al. 1974)

σbf =8παem

mω

(|EH,0|

ω

)3/2(2 sin2 θ +

ω

2ωBcos2 θ

). (4.58)

Here θ is the propagation angle of the absorbed photon with respect to B and

ωB ≡ eB/m. This cross section represents recombination to the most tightly bound

hydrogen energy level, with angular momentum quantum number s = 0.

To obtain the free-bound cross section, we focus on the simplest case of a partially

ionized hydrogen gas interacting with a blackbody radiation field of a low enough

temperature that cyclotron excitations of the protons can be neglected. The densities

of electrons and photons are

dne

dpz= ne

e−p2z/2mT

(2πmT)1/2 ;d2nγ

dωd(cos θ)=

ω2

(2π)2 Nγ, (4.59)

where Nγ = 1/(eω/T − 1). Detailed balance implies the following relation between

the photoionization and recombination cross sections:

2(1 + Nγ)pz

mdσfb

d(cos θ)np

dne

dpzdpz = σbfnH

d2nγ

dωd(cos θ)dω. (4.60)


The ratio npne/nH, where nH is the density of neutral H atoms, can be obtained

by considering the Boltzmann law, as applied to a two-level system consisting of a

neutral atom and a proton paired with a free electron moving in the momentum range

(pz, pz + dpz):dnp(pz)

nH=

gpge

gHexp

[−|EH,0|+ p2

z/2mT

]. (4.61)

Here the gi label the quantum degeneracies of the various states (gH = gp = 1) and

ge =eBdpz

ne(2π)2 (4.62)

is the differential statistical weight of the electron. Substituting these relations into

Equation (4.61) and integrating over pz, we obtain the relevant Saha ionization relation,

npne

nH=

eB(2π)3/2 (mT)1/2 exp

[−|EH,0|

T

]. (4.63)

Substituting this into Equation (4.60) yields the generalized Milne relation

dσfb

d(cos θ)= σbf

ω2

2eB. (4.64)

As required, this relation is independent of collective properties such as temperature.

Finally, substituting Equation (4.58) gives the differential cross section,

dσfb

d(cos θ)= 4π

r2e

αem

(ωmeB

)( |EH,0|ω

)3/2(2 sin2 θ +

ω

2ωBcos2 θ

). (4.65)

Integrating over the photon direction gives

σfb = 8πr2

eαem

(ω/mB/BQ

)(|EH,0|

ω

)3/2(43+

ω

6ωB

). (4.66)

In the nonrelativistic regime considered here, this cross section is significantly smaller

than that of electron-ionbackscattering, as given by Equation (4.37).

In order to compare the relative importance of recombination emission and

bremsstrahlung, we consider the emission rate from a thermal distribution of electrons,

nenp

⟨|β|dσfb

dω

⟩≡ 2

pω

mσfbnp

dne

dpz

∣∣∣∣pz=pω

dpz

dω

∣∣∣∣pz=pω

, (4.67)


where pω ≡√

2m(ω− |EH,0|), and dpz/dω = m/pω when evaluated at pz = pω.

Defining a recombination Gaunt factor gR(T, ω) similarly to Equation (4.54), one has

gR(T, ω) =3πω2

α2emeB

(|EH,0|

ω

)3/2

exp[|EH,0|

T

] (43+

ω

6ωB

)(ω ≥ |EH,0|). (4.68)

Figure 4.9 plots both this expression and g(T, ω) over a range of frequencies and

temperatures.

4.5.6 Electron-Positron Bremsstrahlung.

To conclude, we briefly consider electron-positron bremsstrahlung. A precise deriva-

tion of this process is outside of the scope of the present analysis. Nevertheless, it is

possible to make a qualitative estimate by assuming that bremsstrahlung processes

scale with their corresponding scattering process. Then,

σbr,± = 4σbr,e−ion,bσe+e−

σCoulomb, (4.69)

where the factor of 4 comes from the fact that the external photon line in electron-

positron bremsstrahlung can be added to both of the particles, and σbr,e−ion,b is the

portion of the electron-ion bremsstrahlung cross section corresponding to backscat-

tering. This formula is easy to verify in the soft photon limit (see Section 4.5.2), and

we assume that it holds at least approximately elsewhere in the parameter space

under consideration. The electron-positron bremsstrahlung absorption coefficient is

enhanced relative to Equation (4.57) in a similar manner.

Therefore, given comparable densities of targets, this electron-positron bremsstrahlung

is significantly enhanced with respect to its ion counterpart. In addition, the presence

of this powerful electron-positron bremsstrahlung in our model further suppresses the

relative importance of resonant e±-ion bremsstrahlung emission in the generation of

the spectrum. Given the relative importance of this process, the approximate treatment

used is sure to have a noticeable effect on the resulting spectra (see Chapter 5).


Figure 4.9: Comparison of the bremsstrahlung Gaunt factor g(T, ω) (black curves) and the

recombination Gaunt factor gR(T, ω) (blue curves) for a range of subrelativistic temperatures

and B = 100BQ. The arrows point in the direction of increasing T. Free-bound emission is

significantly weaker than free-free emission except at high frequencies, where the emission is

exponentially suppressed.

Chapter 5

Self-consistent broadband magnetar

X-ray spectra

This chapter draws from Kostenko & Thompson (2020).

5.1 Overview

Here we construct a self-consistent Monte Carlo model of the persistent X-ray emis-

sion of magnetars (magnetic field 103BQ B BQ). The physical system under

consideration is a hot, dilute, e±-dominated plasma filling the magnetosphere and

positioned above a colder, denser atmospheric cooling layer (see Figure 5.1). Magne-

tospheric pairs diffuse into the cooling layer, which is therefore separated from the

magnetosphere by a pair-loaded transitional layer, with ne− = 2ne+ = 2np (see Section

5.3). The powerful magnetic field restricts e± to one-dimensional motion along the

field, greatly simplifying the numerical treatment. Our model incorporates the various

QED processes investigated in the previous Chapters.

The high cross section for electron-positron scattering greatly decreases the anni-

hilation depth of magnetospheric positrons entering the atmosphere. However, the

differential annihilation cross section (4.7) favours a strong asymmetry in the energies

78

Chapter 5. Self-consistent broadband magnetar X-ray spectra 79

of the produced photons, with most of the energy deposited in a photon with direction

cosine µ ∼ 0. These photons scatter off cold electrons with a resonantly enhanced

cross section (3.3), before escaping and producing a γ-ray annihilation peak with an

extended low-frequency tail. The recoil heated surrounding electrons, together with

the magnetospheric e±, continuously produce bremsstrahlung photons. Collisions

between e+ and e− and between e± and protons contribute comparable amounts to the

bremsstrahlung spectrum. Energy absorbed by the deeper cooling layer is reprocessed

into quasi-blackbody emission.

A large fraction of this emission is carried by the ordinary polarization mode due

to the modest depth of the emitting layer. These O-mode photons then experience

significant non-resonant scattering by the magnetospheric pair plasma. Only after

reaching the top (or sides) of the magnetosphere are photons counted as fully escaped,

at which point they are redshifted and added to the observed spectrum.

This model is motivated in part by the expectation that magnetospheric currents

are strongly concentrated in narrow regions of the magnetar surface, as inferred from

the high temperature and localization of the “hot spot” blackbody emission (Ibrahim

et al. 2001; Woods et al. 2004; Bernardini et al. 2009; Alford & Halpern 2016; Mong &

Ng 2018), and from ab initio calculations of crustal yielding (Thompson et al. 2017).

We conjecture that the combination of a high local magnetic twist energy dissipation

rate, slow annihilations of pairs and rapid collisions of gamma rays in the ultrastrong

magnetic field, along with an enhanced scattering cross section for electrons and

positrons, permits the formation of a quasi-thermal e± plasma in the most intense

current-carrying zones. The densities of this plasma are sufficient for significant

non-resonant scattering, in counterpoint to the much lower densities considered in

resonant scattering approaches (Lyutikov & Gavriil 2006; Fernández & Thompson

2007; Rea et al. 2008; Zane et al. 2009; Beloborodov 2013).

Our model does not self-consistently take into account the emission of thermal


Figure 5.1: Overview of the combined magnetosphere and cold atmosphere system under

consideration. In each panel the shading indicates the relative amounts of deposited and

re-emitted energy, while the red + denote the presence of cold positrons in the atmosphere.

In the processes considered, black denotes the initial state and blue denotes the final state.

The left panel demonstrates the various processes involving photons, namely pair creation,

pair annihilation, non-resonant γ − e± scattering, as well as e±-ion and electron-positron

bremsstrahlung. The right panel demonstrates e±-ion and electron-positron scattering.

E-mode photons due to deep cooling, or deep heating of the magnetar atmosphere by

relativistic e±. This is in counterpoint to analyses of the deep cooling of magnetars,

such as Ho & Lai (2001) and Özel (2001), where the E-mode is the dominant channel

of energy transport from the deep, hot atmospheric layers. But this additional E-mode

component is plausibly present, and can be added to the computed O-mode spectrum

1.

The Chapter is organized as follows. We first provide an overview of the structures

of the magnetosphere and cooling layer in Sections 5.2 and 5.3 respectively. Details of

1In superstrong magnetic fields, mode transfer of X-ray E-mode photons at the plasma-vacuumresonance takes place at depths far below the cooling layer (Lai & Ho 2002). After conversion to theO-mode, the energy flux is trapped, interacting with the surrounding material and generating E-modephotons at the new, lower ambient temperature. Mode conversion therefore decreases the effectivedecoupling depth of the E-mode component but does not result in significant transfer of flux from the∼ 1 keV blackbody to the O-mode (Harding & Lai 2006).


the Monte Carlo model are provided in Section 5.4, with the resulting spectra in the

AXP-like and SGR-like limits presented in Section 5.5. Throughout this chapter, we

adopt CGS units unless specified otherwise.

5.2 Magnetospheric plasma

Beloborodov & Thompson (2007) showed that the strong, decaying magnetic field

of the magnetar leads to the generation of self-induction electric fields that acceler-

ate surface particles and initiate avalanches of pair creation in the magnetosphere.

This results in a quasi-steady state, with the hot, pair-loaded configuration being

continuously replenished with new particles that support the current implied by the

magnetic field. Efficient re-creation of pairs in the magnetosphere contributes to the

pair-loading as well (see Section 5.5 and specifically Figure 5.8). We take this model as

a starting point and make a series of substantial modifications.

To start, the mechanisms of pair generation in the current model are qualitatively

different. In the model considered above, the plasma was replenished by the resonant

upscattering of ∼ 1-10 keV photons off of ultra-relativistic e±. As we will show,

however, the high cross section of Bhabha scattering, Equation (4.28), implies that

positrons diffusing from the magnetosphere into the cooling layer are unable to prop-

agate to significant optical depths prior to annihilation. Therefore, photons produced

in annihilation events can easily escape to the magnetosphere with minimal energy

loss for conversion back into a pair. At the same time, ∼ 100 keV bremsstrahlung

photons generated by the recoil of the energetic annihilation photons off of the cold

background electrons can be upscattered by multiple non-resonant γ− e± scattering

off of trans-relativistic magnetospheric particles to energies capable of pair creation.

This same non-resonant scattering is of interest when considering the spectra of most

active magnetars, since it has the capacity to generate the non-thermal spectral tail


seen in the 1− 10 keV range; in turn, this allows us to potentially diagnose the prop-

erties of the e± magnetosphere, as this channel requires much higher magnetospheric

optical depths than previous models suggest.

Beam instabilities of relativistic e± generated in transient heating events in the

magnetosphere offer both a source of random kinetic energy for the magnetospheric

plasma, as well as a source of deep heating in the magnetar surface, leading to the

generation of thermal E-mode photons that decouple from the e± corona considered

here. The one-dimensional particle-in-cell simulations considered in Beloborodov

& Thompson (2007) could not capture the possibility of three-dimensional oblique

beam-driven modes flattening the e± distribution function.

The lower magnetospheric e± energies considered here, γ ∼ 1, qualitatively change

the way the plasma interacts with the cold atmosphere: rather than propagating to

significant depths and generating pair cascades through interactions with the ions,

the colder plasma will be stopped at lower depths, emitting bremsstrahlung and

annihilating. This shallower propagation requires a careful treatment of the processes

considered in the previous Chapters, noting for example the ∼ (B/BQ)−1 suppressed

annihilation cross section.

We therefore consider a simple, one-dimensional thermal trans-relativistic magne-

tospheric e± distribution, assuming that ne± > |J| /e, such that the current is supplied

by a modest relative e± drift. Efficient electron-positron scattering ensures significant

exchange of momentum between the accelerated electrons and positrons, providing

comparable densities of each at both ends of the magnetic field lines. For the purposes

of the present analysis we assume that the magnetospheric e± plasma behaves as an

isothermal ideal gas at a temperature kBTmag . mc2, implying a thermal scale height

kBTmag/mg ∼ 0.3RNS.

The e± from the magnetosphere propagate into the cold and dense atmosphere,

acting like a beam with a velocity vmag =√

kBTmag/m, energy kBTmag and rest


energy flux F± = nmagT0.5magmc3/

√2π, where Tmag = kBT/mc2 is the normalized

magnetospheric temperature. Assuming that the magnetosphere is a uniform layer

with scale height hmag, which is a good approximation provided that kBTmag/mg &

hmag, this energy flux may be expressed as

F± =τmag√

2πσThmagT0.5

magmc3, (5.1)

where τmag is the magnetospheric Thomson optical depth. The localized “hot spot”

emission seen in many magnetars, meanwhile, suggests that the magnetospheric

width wmag can be as small as a fraction of the height hmag.

Reaching the atmosphere, this beam triggers annihilation and bremsstrahlung

emission. In turn, these processes deposit energy in the atmosphere, feeding its

thermal emission. These emission mechanisms, along with the related processes

they trigger, are what generates the spectrum in our model. Since these mechanisms

deal with O-mode photons, unlike Ho & Lai (2001), most of the relevant processes

occur at τT . 10. Additionally, as will be discussed in Section 5.3, the e± also affect

the structure of the cooling layer itself, generating a pair-loaded upper portion with

comparable densities of cold electrons, positrons and protons. It also bears mentioning

that the values of flux that are inferred from the bolometric output of magnetars are

∼ 1022erg cm−2 s−1, which corresponds to τmag ∼ 1; this suggests that the spectrum

that emerges from the atmospheric cooling layer has the potential to be significantly

comptonized by the trans-relativistic magnetosphere.

5.3 Cooling Layer Structure

Next, we consider the structure equations of our cooling layer - the outer cold atmo-

spheric layer that receives the kinetic and annihilation energy of the magnetospheric

e± beam and re-radiates this energy in the O-mode. These equations will determine

the density and temperature profile of our cooling layer. We consider a plane-parallel


atmosphere composed of hydrogen; typical values of the atmospheric temperature T

are > 106K, high enough to fully ionize the hydrogen. The background magnetic field

B is taken to be collinear to the surface normal z.

As was discussed in Ho & Lai (2001), it is a very good approximation to assume

that the particles on the surfaces of neutron stars obey the ideal gas equation of state

P = nkBT. The radiative transfer equation, meanwhile, is

Frad = − cαR

ddz

(u3

), (5.2)

where Frad is the atmospheric radiative flux, and u = aT4/2 because we are only

considering the electromagnetically coupled O-mode photons. The opacity coefficient

αR is the Rosseland-averaged absorption coefficient of the most dominant process:

1αR

=1

8aT3

∫ ∞

0

∫ 1

−1

1αω

∂uω

∂Tdωdµ; uω =

hπ2c3

ω3

ehω

kBT − 1. (5.3)

In the environments considered, this can either be scattering (αR = nσes) or free-free

absorption, where

αR = 0.1366[

ln(

4e2(1− µ)min

)]−1αemr2

e n2e c3(

kBTh

)−3 ( kBTmc2

)−0.5= 7.12× 10−26 Iµ

n2e,cgs

T3.5cgs

cm−1,

(5.4)

where ne,cgs and Tcgs are the dimensionless CGS values of electron concentration and

temperature, respectively, and e is the mathematical constant. The angular integral

Iµ =[ln(

4e2(1−µ)min

)]−1, being logarithmic, is dropped in subsequent calculations for

simplicity. For the temperatures and densities under consideration, the scattering and

absorption opacities are roughly comparable. Due to this, as well as the observed

insensitivity of the escaping spectrum to the precise details of the model, we will

assume that free-free absorption is the dominant source of opacity.

We also note that the interface of the magnetosphere and the cooling layer will

also become pair loaded owing to the suppression of the annihilation cross section in

a magnetic field (see Equation (4.7)). To show this, we consider the impact of warm

beam positrons on the surface of the cooling layer, seeing how far they propagate


prior to annihilation. We assume that the center-of-momentum speed βCM is trans-

relativistic, such that the s channel in the Bhabha cross section (4.28) dominates and

γCM ∼ 1. Then, for a density of target particles nt, the Bhabha scattering rate is

Γe+e− = 2βCMcσe+e−nt =2π2r2

e cαemB/BQ

nt. (5.5)

Similarly, the rate of annihilation that avoids the re-conversion of a final state photon

into a e± pair is

Γann = 2βCMcσannnt =80πr2

e β2CMc

9B/BQnt. (5.6)

Warm beam positrons that collide with the cooling layer will quickly Bhabha scatter off

of the background cold electrons, becoming cold (and vice versa). However, the warm

electrons from the beam will reheat the positrons back to their initial energy. Since

the cross sections for the cooling and heating reactions are the same, the relative time

a positrons spends as cold and warm is ∆tc/∆tw = Γe+e−(w→ c)/Γe+e−(c→ w) =

ne−,c/ne−,w 1.

When warm, the annihilation rate of positrons is Γwann = (20πr2

e β2wc)/(9B/BQ)ne−,c,

where we have only included annihilation with the cold electrons, taking βCM = βw/2.

Likewise, the annihilation rate of cold positrons is Γwann = (80πr2

e c)/(9B/BQ)[ne−,wβ2w/4+

ne−,cβ2c]. Weighing by the relative time spent in a given state, the average annihilation

rate is

Γann =∆twΓw

ann + ∆tcΓcann

∆tw + ∆tc=

80πr2e c

9B/BQ

[β2

w2

ne−,w + β2cne−,c

]. (5.7)

We approximate the magnetospheric Thomson optical depth as τmag ∼ (kBTw/mg)ne−,wσT

and the cooling layer Thomson optical depth as τT,atm ∼ (kBTc/mpg)ne−,cσT. Given

roughly comparable τT,atm and τmag, which is a reasonable assumption for the range

of parameters under consideration, one sees that the second term in Equation (5.7)

dominates. Furthermore it is clear that the rates for annihilation and reheating are

comparable, such that we can expect positrons to reheat prior to annihilation. There-

fore, considering a hot positron’s mean free path to combined Coulomb scattering


and annihilation, we have

l2ann,sc = lsclann

ne−,w

ne−,c=

β4w

πr2e ne−,c

βw80πr2

e9B/BQ

β2cne−,c

ne−,w

ne−,c, (5.8)

where we have weighed by the relative time that the positron spends in a warm state,

ne−,w/ne−,c. This is the distance the positrons travel on average prior to annihilation,

and therefore we set it to be the physical scale height he+ at which the positrons are

still found below the magnetosphere-cooling layer interface. Setting the corresponding

Thomson optical depth τT,e+ = he+σTne−,c equal to the optical depth of the portion

of the cooling layer under consideration, τT,atm, we obtain an estimate of the optical

depth at which positrons will be found,

τT,e+ = 3.44× 10−1(

B100BQ

) 13 (

Tmag) 1

2(τmag

) 13 , (5.9)

where Tmag ≡ Tw. We also calculated the depth the positrons will diffuse to while

cold, finding that it is smaller than the above result. Therefore, Equation (5.9) describes

the typical depths at which positrons can still be found prior to annihilation. Their

equilibrium concentration ne+,c has to be comparable to the cold electron concentration,

such that the incoming beam is not significantly modified by Bhabha scatterings off of

the background. Demanding quasineutrality, we set ne− = 2ne+ = 2np in this region.

Beyond the cooling layer under consideration, the deeper layers of the neutron star

emit thermally, releasing a spectrum that is primarily in the non-interacting E-mode

(Ho & Lai 2001; van Adelsberg & Lai 2006; Harding & Lai 2006). The source of this

emission is unknown, with possibilities including ultrarelativistic beam instabilities

that deposit half of their energy in the magnetosphere and half deep in the neutron

star crust, or some intrinsic deep heating source, such as the decay of the internal

magnetic field. Additionally, although the precise form of this spectrum is modified by

effects such as resonant ion cyclotron absorption and mode conversion, the emergent

spectrum is fairly close to a blackbody at the deep cooling effective temperature TE,eff,


and we treat it as such. Since this process is decoupled from the cooling layer under

consideration, its only effect in the context of the present model is to add this emergent,

non-interacting thermal spectrum to the model’s escaping O-mode spectrum.

5.4 Monte Carlo Simulation

The probabilistic nature of the QED processes involved makes the Monte Carlo

technique suitable for the analysis of the physical system. We now present the details

of the atmospheric model under consideration, as well as the numerical methods

used.

5.4.1 Grids

For the purposes of the simulation, we construct grids in particle kinetic energy Ek as

well as photon energy hω and direction µ. The energy grids are spaced logarithmically

from 10−4mc2 to 10mc2, with ∼ 40 grid points per decade used for particle kinetic

energy and ∼ 20 for photon energy, while the µ grid is evenly spaced from −1 to 1

with Nµ ∼ 26 grid points, µi = −1 + 2(i− 0.5)/Nµ. A corresponding set of grids with

resolutions ∼ 10 lower is used in certain cases to speed up calculation.

In addition, a separate set of frequency arrays is constructed in order to properly

treat the behaviour of two-photon pair creation when one of the photons is close

to the 2mc2/ sin θ threshold. Here, the frequency interval is broken up into three

sub-intervals: 10−4mc2 to 1mc2, 1mc2 to 2mc2/ sin θ and 2mc2/ sin θ to 10mc2 (if

2mc2/ sin θ > 10mc2, the latter two intervals are merged and cut off at 10mc2), with

each of these sub-intervals having ∼ 40 grid points. The first and third sub-intervals

are spaced logarithmically, while the second is spaced reverse-logarithmically, carefully

sampling the approach to 2mc2/ sin θ (see Figure 5.2).


Figure 5.2: Schematic of a portion of the two-photon pair creation frequency grid that carefully

samples the approach to the 2mc2/ sin θ threshold, as well as a portion of the post-threshold

section.

5.4.2 Atmospheric and Magnetospheric Model

Taking the structure equations from Section 5.3, we consider a fully ionized electron-

proton atmosphere (ne− = np) at equilibrium with a constant Frad. Assuming a power

law scaling of temperature and electron concentration, T = T0zqT and n = n0zqn ,

where z = z/(1 cm), one obtains

T(z) = 1.42× 105 gz K; n(z) = 9.85× 1018F−0.5rad g3.75z3.25g cm−3, (5.10)

where g = g/(1014cm s−2) and Frad = Frad/(1022erg cm−2 s−1) are the normalized

gravitational acceleration and atmospheric radiative flux, respectively. The correspond-

ing Rosseland mean depth is

τR(z) = 1.59× 10−6F−1rad g4z4. (5.11)

The temperature profile of the atmosphere is T(τR) = Teff(3τR/4)0.25, which is the

grey profile with τR → τR − 2/3. Therefore, this solution extends from infinite depths

only up to the boundary of the grey atmosphere zff, where τR(zff) = 2/3, at which

point we transition to an isothermal atmosphere with

T(z) = 3.62× 106F0.25rad K; n(z) = 3.63× 1023F0.3125

rad g0.5 exp [biso(z− zff)] g cm−3,

(5.12)

where biso = (mpg/2kBT)× (1 cm) = 1.67× 10−1F−0.25rad g is the inverse of the normal-

ized thermal scale height and zff = zff/(1 cm). The Thomson optical depth in this


isothermal region is

τT(z) = 1.44F0.5625rad g−0.5 exp (biso(z− zff)) , (5.13)

where we have integrated from z = −∞; however, this regime only continues until

the atmospheric pressure balances the pressure from the hot, dilute magnetosphere,

n(z)kBT(z) = nmagkBTmag. This transition to the magnetospheric beam occurs at

zbeam, where

zbeam = zbeam/(1 cm) = zff + 5.97F0.25rad g−1 ln

(4.69× 10−3F±T0.5

magF−0.5625rad g−0.5

),

(5.14)

where F± = F±/(1022erg cm−2 s−1) is the normalized rest energy flux of the magneto-

spheric beam. Since this is the boundary of our cooling layer, it is prudent to mark

this as the start of our atmosphere z = 0. To implement this change, we must make

the substitution z→ z + zbeam in the model. With this, the temperature dependence is

T(z) =

1.42× 105 g(z + zbeam)K (z > zff − zbeam);

3.62× 106F0.25rad K (z ≤ zff − zbeam),

(5.15)

the concentration dependence is

n(z) =

9.85× 1018F−0.5rad g3.75(z + zbeam)3.25g cm−3 (z > zff − zbeam);

3.63× 1023F0.3125rad g0.5 exp [biso(z + zbeam − zff)] g cm−3 (z ≤ zff − zbeam),

(5.16)

and the Thomson optical depth dependence is

τT(z) =

τT,iso + 1.54× 10−6F−0.5rad g3.75 ((z + zbeam)4.25 − z4.25

ff

)(z > zff − zbeam);

1.44F0.5625rad g−0.5 exp [biso(zbeam − zff)] [exp (bisoz)− 1] (z ≤ zff − zbeam),

(5.17)

where τT,iso ≡ τT(zff− zbeam) is the full optical depth of the isothermal portion. Finally,

recalling Equation (5.9), for τT ≤ τT,e+ we add a background positron concentration

equal to the background proton concentration, and double the background electron

concentration to preserve quasineutrality, such that ne− = 2ne+ = 2np.


Past a sufficient Thomson optical depth τT,atm, any particle or photon in the

physical system under consideration can be assumed to be sufficiently trapped, such

that it is guaranteed to distribute its remaining energy to the background. It is this

region of the atmosphere with τT < τT,atm that is modelled in our code. This region is

broken up into Nz = 30 slices of equal ∆τT, each with its own uniform value of T and

n evaluated at the bottom of the corresponding slice. For the trans-relativistic particle

energies under consideration, τT,atm ∼ 50.

Meanwhile, as discussed in Section 5.2, the magnetosphere is assumed to be

a uniform layer, described by a normalized temperature Tmag, a Thomson optical

depth τT,mag, a physical scale height hmag ∼ RNS/3 ∼ kBTmag/mg and width wmag.

The combined physical system, together with an overview of the relevant physical

processes, is presented in Figure 5.1.

5.4.3 Particle propagation algorithm

We now describe the algorithm for propagating e± and photons through the atmo-

spheric cooling layer and magnetosphere. The e± are restricted to motion along the

magnetic field whereas photons are free to move at arbitrary angles. The background

plasma is labeled by the vertical Thomson depth τT. A description of our treatment of

integral and differential cross sections can be found in Appendix D.

In the cooling layer, we have a particle take a step ∆τT up or down, as appro-

priate, to the boundary of its current slice; the slices are assumed to have uniform

values of T and n. In this step, the various processes all accumulate optical depths

τi = (αi/αT)∆τT/ |µ|, where i = 1, ...N and αi are the respective emission or absorp-

tion coefficients. The propagating e± experience i) Coulomb scattering, ii) e−-ion

bremsstrahlung, iii) electron-positron scattering and bremsstrahlung (for all e+, as

well as for e− where background cold e+ are present) and iv) annihilation (only for

e+ so as to avoid double counting). A photon experiences i) e± scattering and ii)


bremsstrahlung absorption. Single-photon pair creation is assumed to be instanta-

neous, while the ∼ 10 cm scale height of the cooling layer is insufficient for meaningful

two-photon pair creation.

Given that the particle experiences multiple interactions, the probability of process

i occurring once in an interval ∆τT before all the other processes is

Pi =∫ ∆τT

0dτT

αi

αT |µ|exp

[−∑

jτT

αj

αT |µ|

]=

αi

∑j αj

(1− e−∑j τj

). (5.18)

Defining Pi,∞ = limh→∞ Pi = αi/ ∑j αj, one has ∑i Pi,∞ = 1. Therefore, the probability

interval allocated to process i is [∑i−1j=1 Pj,∞, ∑i

j=1 Pj,∞], where ∑0j=1 Pi,∞ = 0.

To determine which process, if any, occurred in a given step, we draw a random

number PR. This identifies a particular process a that corresponds to ∑a−1j=1 Pj,∞ <

PR < ∑aj=1 Pj,∞. If Pa < ∆Pa = PR − ∑a−1

j=1 Pj,∞, no process is triggered, the particle

propagates to the boundary of the current slice and into the next slice (or out of

the model’s cooling layer). If, however, Pa > ∆Pa, the step is truncated so that the

accumulated probability matches ∆Pa,

∆τT,new = ∆τT

− ln[1− ∆Pa

∑j αjαa

]∑j τj

. (5.19)

After the particle takes this truncated step, the process is triggered in accordance with

the procedure described in Appendix D.2.

We adopt a recursive procedure to follow new particles. If the triggered process

involved the start of new particle paths, specifically the bremsstrahlung emission of

photons, annihilation, pair creation, and Bhabha or Compton scattering-triggered

recoil of cold particles, those paths are calculated to completion, at which point we

return to the original particle and continue its propagation as described at the start of

this section.

There are a number of ways in which the propagation of a particle can be termi-

nated in the present analysis. Certain processes, namely annihilation, pair creation and


absorption, imply that the original particle is destroyed. The propagation of an elec-

tron also terminates when its energy drops below the kinetic energy of the cold back-

ground, 0.5kBT(z), while the propagation of a photon terminates 2 when the scattering-

assisted thermal bremsstrahlung absorption cross section τff(z)×max(1, τT(z)) be-

comes greater than 1. Additionally, the propagation of all particles terminates upon

crossing the lower boundary of the cooling layer, τT > τT,atm.

Escape into the magnetosphere is treated differently for e± and photons. When

e± escape, they are assumed to have rejoined the magnetospheric plasma, being

subsequently thermalized; their propagation therefore terminates. Photons, however,

are propagated through the magnetosphere until they pair create, escape, or return to

the cool atmosphere.

The magnetospheric propagation of photons is similar to their cooling layer prop-

agation, albeit with a number of key differences. Because the background e± are

trans-relativistic, effects such as beaming become important, and so their finite ve-

locity must be taken into account in the scattering process. This target e± velocity

distribution is therefore weighted by the e± scattering cross section. Additionally,

thermal bremsstrahlung absorption is ignoreable in the dilute magnetosphere, while

two-photon pair creation is now relevant owing to the large e± scale height (see

Section 5.4.4). Finally, in addition to escaping out the top, photons can also escape out

the sides of the magnetosphere. This is gauged by a random walk approach: when the

sum of the squares of the distances travelled between scattering events ∑ l2i is greater

than the squared magnetospheric width w2mag, the photon is said to have escaped out

the side.

2The photons affected by this cutoff are typically low-energy bremsstrahlung photons, which donot contribute meaningfully to the spectrum (see Figure 5.3). Therefore, although bremsstrahlungabsorption is handled self-consistently in the program, this cutoff does not qualitatively change thespectrum, whereas it greatly reduces the computational time.


5.4.4 Convergence and data accumulation

The physical system under consideration - the atmospheric cooling layer together

with the magnetosphere - is fully described by a set of parameters: F±, Tmag, hmag,

wmag,Frad, TE,eff, g, τT,atm, Nz = 30 and the magnetic field ratio B/BQ. In this system,

we launch equal numbers Ntot ∼ 106 of seed magnetospheric e− and e+ from the

magnetospheric boundary into the cooling layer and propagate them to completion.

The rate of injection with respect to particle velocity v is given by

d2NdAdt

= nmag|v| f (v), (5.20)

where v < 0.

As a result of the magnetospheric e± propagation, the cooling layer absorbs energy

through recoil heating and absorption of soft photons and e±. This energy is assumed

to feed the O-mode emission of the cooling layer Frad. We therefore track the absorbed

energy Eabs following successive e± injections, and emit a corresponding number

of blackbody photons at Teff = (2Frad/σSB)0.25. We also track the ratio RE between

normalized energy in our code and the corresponding vertical flux, calibrating it using

the energy absorbed and re-emitted by the cooling layer:

RE =Frad

Eabs/mc2 . (5.21)

Over the course of the simulation, we track the propagation of photons in the mag-

netosphere. A single photon with frequency ω and directional cosine µ corresponds

to a beam with specific logarithmic intensity

Iln ω =hωRE

mc22π∆µ∆ ln ω(5.22)

where ∆ ln ω and ∆µ are the widths of the corresponding frequency and directional

cosine bins. This corresponds to a photon density n(ω, µ) = Iln ω/(hωc|µ|), since

it is the vertical flux of photons that is conserved in our plane-parallel setup. The


magnetosphere is treated as a single layer, and so we weigh the photon density by

the total residency time in a given (ω, µ) state; e.g. propagation halfway through the

magnetosphere corresponds to a halving of the density. Thus,

ntot(ω, µ) =RE

mc32π∆µ∆ ln ω

∑ hω,µ

hmag, (5.23)

where ∑ hω,µ = ∑ ∆h/|µ| is the total magnetospheric path length travelled by photons

in this bin during the simulation.

The total concentration ntot(ω, µ) is what is used to calculate the two-photon pair

creation rate in the magnetosphere. However, to avoid issues caused by rogue events

occurring at the start of the simulation prior to sufficient statistical accumulation,

the ∑ hω,µ are multiplied by a smoothness factor S2pc = (Ni/N2pc)2, where Ni is the

current number of the seed particle and N2pc ∼ 0.1Ntot, for Ni < N2pc.

After all particles have propagated, we compare the total absorbed energy Eabs to

the total rest energy of all of the seed magnetospheric e±, Erest. Physically, the ratios

of Erest to F± and of Eabs to Frad should be equal. If, after the first iteration, our initial

assumed value of Frad does not satisfy this condition, we set Frad = F±(Eabs/Erest),

update the atmospheric structure accordingly, and launch a new iteration of the

simulation. This is repeated until the value of Frad converges, which typically occurs

over a small number of iterations due to the relative insensitivity of the e± and photon

propagation to the details of the atmospheric model. The converged run’s escaping

spectrum is then augmented by the non-interacting E-mode blackbody spectrum at

TE,eff and redshifted - this combination is taken as the resulting spectrum (see Section

5.5).

This approximate treatment of the effects of bombardment on the structure and

emission of the cooling layer is a key simplification of our model. González-Caniulef

et al. (2019) have recently analyzed the structure of bombarded magnetar atmospheres,

assuming that magnetospheric particles are ultrarelativistic 3 following Beloborodov

3A similar analysis was carried out by Bauböck et al. (2019) for rotationally-powered pulsars and


& Thompson (2007) and Beloborodov (2013). Bombardment by these particles involves

excitation to higher Landau levels, cascades of pair creation and deposition of energy

at high optical depths. In counterpoint, our trans-relativistic particles are unable to

excite Landau resonances, and we find that our energy is primarily deposited at low

optical depths, where bremsstrahlung emission is optically thin.

More work is therefore needed to understand the detailed structure of the cooling

layer in our model, and this is outside the scope of the present analysis. Nevertheless,

we can make some qualitative predictions. For instance, the uppermost layers will most

likely be heated to a temperature significantly higher than TE,eff, but the corresponding

Compton parameter will be small. This layer may be unable to cool effectively by

bremsstrahlung emission, possibly relying instead on Compton cooling from the

deeper, colder thermal emission, similarly to González-Caniulef et al. (2019). These

temperature inhomogeneities will also distort and spread out the thermal spectral

peak, although the comptonization tail produced by the magnetospheric pairs should

dominate the shape of the spectrum at frequencies significantly above the thermal

peak.

5.5 Results

The spectra produced by the present model have several well-defined components,

which can be seen in Figure 5.3 for both AXP-like and SGR-like spectra:

1. An O-mode blackbody peak at Teff sustained by external heating,

2. A power law with a photon index Γ > 2 that connects to the blackbody peak,

produced by multiple non-resonant Compton scattering of O-mode blackbody photons

in the magnetosphere. As Tmag and τmag rise, this component hardens,

3. A hard X-ray/soft γ-ray bremsstrahlung plateau, composed of two broad peaks:

a “first generation” component with hω ∼ kBTmag that is produced directly by the

lower energies, using non-magnetic rates and cross sections.


incoming magnetospheric particles and a “second generation” component that is

produced primarily by particles that are recoil-heated by annihilation photons,

4. A peak at hω ∼ mc2 produced by the annihilation of cooled positrons and

electrons.

The non-interacting, deep cooling E-mode thermal component, here approximated

as a blackbody at TE,eff, is added to the resulting spectrum at the end of the simulation.

Spectra are then redshifted by a factor of 1.3, which is typical of magnetars.

Figure 5.3: Spectral breakdown for the τmag = 1.5, Tmag ≡ kBT/mc2 = 0.2, hmag = 1km,

wmag = 0.2km (AXP-like) case (left) and the τmag = 5.8, Tmag = 0.3, hmag = 1km,

wmag = 3km (SGR-like) case (right) without the deep cooling E-mode component, with

total spectrum (black), comptonized blackbody (red), first-generation bremsstrahlung (green),

second-generation bremsstrahlung (orange) and annihilation (blue). In both cases, B = 10BQ.

5.5.1 AXP-like spectra

In the present model, these spectra correspond to minimal magnetospheric repro-

cessing of the emission produced when the hot e± beam interacts with the cooling

layer. This can either be achieved with τmag . 1, or with a higher tau combined


Figure 5.4: Total AXP spectra (without the deep cooling E-mode component) for a series of

magnetospheric optical depths τmag = 0.15, 0.45, 1.5, 2.25, 3, 3.75, 4.5 and for the temperatures

Tmag = 0.2 (left panel) and Tmag = 0.4 (right panel), for B = 10BQ, hmag = 1km and

wmag = 0.2km. Arrows point in the direction of increasing optical depth.

with a small value of wmag/hmag - values that are consistent with the small “hot spot”

covering fractions inferred for AXPs (Mong & Ng 2018). The results of simulations

for τmag = 0.15, 0.45, 1.5, 2.25, 3, 3.75, 4.5 and for the temperatures Tmag = 0.2 and

Tmag = 0.4, with B = 10BQ, hmag = 1km and wmag = 0.2km, are presented in Figures

5.3-5.7.

These results can be qualitatively understood as follows. When particles from the

hot magnetospheric e± beam interact with the cooling layer, they will immediately be

stopped by rapid electron-positron scattering as described in Section 5.3. Reheating

will propagate them to optical depths comparable to τT,e+ - the optical depth below

the surface where cold positrons are present in our model (recall Equation (5.9)).

Around this depth the beam positrons cool down and annihilate, having transferred

their energy to a background electron. The beam electrons diffuse deeper before

bremsstrahlung emitting and cooling.

For the parameters under consideration τT,e+ ∼ 0.1. Annihilations therefore occur


Figure 5.5: Individual components of the AXP spectra for a series of magnetospheric optical

depths τmag = 0.15, 0.45, 1.5, 2.25, 3, 3.75, 4.5 and for the temperatures Tmag = 0.2 (solid lines)

and Tmag = 0.4 (dashed lines), with B = 10BQ, hmag = 1km and wmag = 0.2km. Arrows

point in the direction of increasing optical depth. Top left: comptonized blackbody. Top right:

second-generation bremsstrahlung. Bottom left: first-generation bremsstrahlung. Bottom right:

annihilation.


Figure 5.6: Comparison of the total spectrum for the τmag = 1.5, Tmag = 0.2 case with and

without a deep cooling E-mode component at kBTE,eff = 0.0011mc2. Note that here the E-mode

contribution is coming only from the “hot spot” area under consideration: if the deep cooling

mechanism is active outside of this region, as is possible for AXPs (Mong & Ng 2018), the

E-mode contribution will be higher.


at very small optical depths. However, as discussed previously, the form of the

annihilation cross section (4.7) favours the production of high-energy photons with

µ ∼ 0. These photons are very likely to recoil-heat the surrounding cold electrons to

potentially significant energies prior to escaping into the magnetosphere.

Both magnetospheric electrons and initially cold electrons heated by magneto-

spheric positrons continuously produce “first generation” bremsstrahlung photons

at energies ∼ kTmag. Meanwhile, the recoil-heated particles will produce a “second

generation” bremsstrahlung component whose shape is independent of the magne-

tospheric temperature. Finally, all of the energy that the cooling layer absorbs is

reprocessed into O-mode blackbody emission at Teff.

After this emission escapes the cooling layer, it is comptonized by the magneto-

sphere, the main effect being the production of the falling power law tail similar to that

observed in AXPs. The comptonization of the bremsstrahlung emission is minimal due

to the comparable energies of the photons and the magnetospheric pairs. Pair creation

primarily depletes the spectrum beyond 100 keV, with the shallow annihilation depths

resulting in the efficient reprocessing of this emission back into magnetospheric pairs.

A non-interacting deep cooling (E-mode) blackbody component at TE,eff is finally

added to the resulting spectrum (see Figure 5.6), being possibly fed by ultrarelativistic

magnetospheric beams, intrinsic deep cooling, or other mechanisms (see Section 5.3).

The ratios of the cooling layer blackbody flux and of the total pre-redshift O-mode

escaping magnetospheric flux to the seed e± total energy flux as functions of τmag are

presented in Figure 5.7. The rapid rise of the first ratio can be primarily attributed to

a higher value of τT,e+ and therefore deeper annihilations that transfer energy from

the annihilation peak to the blackbody peak, as well as to heavier comptonization and

turnaround of emission by the magnetosphere. This latter effect is the primary cause

of the second ratio’s increase with optical depth, and the weak comptonization at low

optical depths explains the observed low-τmag plateau, which is more prevalent for


the lower-Tmag case, as expected.

A comparison of the analytical positron annihilation depth, Equation 5.9, with

the numerical value is presented in the left panel of Figure 5.8. That equation’s

assumption of γCM ∼ 1 begins to break down for the higher-Tmag case, explaining

the worse agreement between the two values in that regime. The efficiency of pair

reproduction in the magnetosphere is presented in the right panel of Figure 5.8. The

high efficiencies in the low-τmag regime can be explained by the minimal reprocessing

of the annihilation peak owing to the low τT,e+ , while the rise in the high-τmag regime

can be attributed to increased availability of mid-energy photons due to increased

comptonization.

We note that a significant portion of the bremsstrahlung emission is due to the

electron-positron bremsstrahlung (see Figure 5.9), with the main effect coming from

enhanced electron emission at τT < τT,e+ . Since our treatment of electron-positron

bremsstrahlung is approximate (see Equation (4.69)), a more careful treatment may

have a significant impact on the resulting spectrum. For example, the spectrum

for τmag = 1.5, Tmag = 0.2 and kBTE,eff = 0.0011mc2 is fairly close to that of the

AXP 4U 0142+61 (Weng & Gögüs, 2015), and the difference is comparable to this

bremsstrahlung uncertainty.

5.5.2 SGR-like spectra

In the present model a more extended and spectrally harder power law appears in the

1-10 keV range if there is significant magnetospheric reprocessing of the cooling layer

emission, with τmag > 1. Such spectra more closely resemble those observed in SGRs,

especially in this band. Here, the surface emission is either heavily comptonized and

escapes, or is directed back towards the cooling layer to be absorbed and re-emitted

as blackbody emission. The value of wmag/hmag is expected to be high given the large

“hot spot” covering fractions inferred for SGRs (Mong & Ng 2018); photons therefore


Figure 5.7: Ratios of the cooling layer blackbody flux (solid) and the total pre-redshift O-mode

escaping magnetospheric flux (open) to the seed e± total energy flux for Tmag = 0.2 (triangles)

and 0.4 (squares).


Figure 5.8: Left panel: numerical results for the average positron annihilation depth (open)

compared with the Equation 5.9 analytical value (solid). Right panel: Ratio of pairs created

in the magnetosphere to the pairs injected into the cooling layer. In both cases Tmag = 0.2 is

denoted with triangles and 0.4 is denoted with squares.

cannot meaningfully escape out the side of the magnetosphere (see Section 5.4.3).

Fixing the other variables, the magnetospheric optical depth can be raised until a

critical value τmag,cr. Past this point the model experiences runaway energy deposition

in multiply scattered photons: successive stages of scattering deposit an increasing

energy fraction in the surface cooling layer, which is further upscattered in the

next iteration. For Tmag = 0.3, 0.5, this critical value is 5.8 and 5.1, respectively, for

B = 10BQ, hmag = 1km and wmag = 3km (Figures 5.3, 5.10 and 5.11).

In both cases, the cooling layer blackbody flux is substantially higher than the seed

e± total energy flux, by factors of 13.09 and 12.88 for the Tmag = 0.3 and 0.5 cases,

respectively. Similarly, the ratio of the total pre-redshift photon energy that escapes

from the magnetosphere to the same total seed e± energy is high, being 26.13 and

26.74 for the Tmag = 0.3 and 0.5 cases, respectively. The efficiency of pair reproduction

in the magnetosphere is also much higher than in the AXP case due to increased

comptonization, being 0.754 and 1.186 for the Tmag = 0.3 and 0.5 cases, respectively.


Figure 5.9: Comparison of the τmag = 1.5, Tmag = 0.2 case with and without electron-positron

bremsstrahlung. As can be seen, this channel nearly doubles the bremsstrahlung emission

that accounts for most of the mid-range spectrum.


In the resulting spectrum, a hard power law produced by magnetospheric scat-

tering connects smoothly to the thermal peak, and the bremsstrahlung/annihilation

components of the atmospheric emission are buried by this power law (see Figure 5.3).

The effect of adding an E-mode component to the spectrum can be seen in Figure 5.11.

The present model is able to approach the nearly-flat SGR-like spectra of magnetars

such as SGR 1806-20 (Esposito et al. 2007; Enoto et al. 2017b), as can be seen in Figure

5.10. We were however unable to obtain the high-energy component that emerges

above 10 keV in spectrally hard magnetars such as 1E 1841-045 (Morii et al. 2010;

Enoto et al. 2017b) or SGR 1900+14 (Mereghetti et al. 2006; Enoto et al. 2017b). This

component might be reproducible by raising the temperature of the magnetospheric

pairs to relativistic values.

Alternatively, a further modification of the model’s escaping spectrum by resonant

scattering is possible. Prior analyses, such as Fernández & Thompson (2007), Baring

& Harding (2007) and Beloborodov (2013) have obtained SGR-like spectra with rising

high-energy components by considering the resonant upscattering of soft X-ray black-

body seed photons of energy ∼ 1 keV. Therefore, replacing this blackbody seed with

the current model’s resulting SGR-like spectra - which are already close in shape to

those observed from SGRs - represents a promising approach (a similar combination of

atmospheric modelling and resonant scattering was recently investigated by Taverna

et al. (2020)).


Figure 5.10: Comparison of total output spectra (without deep cooling E-mode) observed for

τmag = 5.8, Tmag = 0.3 (solid line) and τmag = 5.1, Tmag = 0.5 (dashed line). In both cases

B = 10BQ, hmag = 1km and wmag = 3km. These values of τmag represent the highest possible

magnetospheric comptonization with all other variables being fixed. These spectra do not

include the non-interacting E-mode component.


Figure 5.11: Comparison of the total spectrum for the τmag = 5.1, Tmag = 0.5 case with and

without a deep cooling E-mode component at kBTE,eff = 0.0025mc2. Similarly to Figure 5.6,

the E-mode is only coming from the O-mode emitting “hot spot” region, which is a good

approximation here given the high covering fractions inferred for SGRs (Mong & Ng 2018).

Chapter 6

Conclusions & Future Work

6.1 Conclusions

We have developed a model for the persistent X-ray emission of magnetars - neutron

stars with powerful, non-potential magnetic fields exceeding BQ ≡ m2/e = 4.4× 1013

G. In this model, the emission occurs as a consequence of the interaction of a cold,

dense e−-proton atmospheric cooling layer with an overlying warm magnetospheric

e± plasma. The interaction triggers various QED processes such as bremsstrahlung

emission, electron-positron scattering and pair annihilation, the interplay between

which determines the shape of the model’s spectrum.

Given that the ultrastrong magnetic fields of magnetars have substantial effects

on electromagnetic interactions, we have had to derive the relevant rates and cross

sections. Some of our results are new and some involve considerable simplifications

of preceding treatments appropriate to magnetic fields in the range 103BQ B BQ.

In defining our e± wavefunctions, we follow Sokolov & Ternov (1966) and Melrose &

Parle (1983a) rather than Johnson & Lippmann (1949), such that the wavefunctions of

e± are connected to those at rest by a continuous Lorentz transformation. We note that

in the magnetar-strength magnetic backgrounds considered presently, both real and

virtual e± are essentially confined to the lowest Landau level, which greatly simplifies

108

Chapter 6. Conclusions & Future Work 109

analyses involving these quantum states.

We start by considering the electron-photon scattering cross section, which has

been covered extensively in prior literature but only as complicated infinite sums over

intermediate Landau states, which are not easily implemented in a Monte Carlo code

(Herold 1979; Melrose & Parle 1983b; Bussard et al. 1986; Daugherty & Harding 1986;

Harding & Daugherty 1991; Gonthier et al. 2000; Baring et al. 2005). Our focus is on a

dramatic u-channel pole which emerges below the first Landau resonance, and whose

properties have not been previously examined in detail.

The powerful magnetic background opens up additional channels of pair creation

and greatly modifies existing channels. A single sufficiently energetic photon is

capable of experiencing pair creation, and we quantify this rate by applying detailed

balance arguments to the single-photon annihilation rates presented by Wunner (1979)

and Daugherty & Bussard (1980). The cross section for two-photon pair creation is

shown to be significantly enhanced compared with the unmagnetized value by a factor

of ∼ B/BQ, and is presented in terms of Lorentz scalars for simplicity. We also evaluate

the cross section of the inverse process of two-photon pair annihilation, finding it

to be suppressed by the powerful magnetic field as well as by kinematic constraints

on single-photon pair reconversion. This process distributes energy asymmetrically

between the produced photons, often depositing most of it in a photon with direction

cosine µ ∼ 0. An integral relation between the annihilation and pair creation cross

sections is obtained.

The scattering of electrons and positrons in an ultrastrong magnetic background is

then considered. The cross section is regulated by the decay of an intermediate-state

photon whose energy exceeds the single-photon pair creation threshold. The t-channel

dominates the cross section at low energies where it exhibits a Rutherford-like scaling

in momentum. However, the s-channel begins to dominate at higher energies, giving

the cross section a much shallower scaling with momentum and greatly increasing


its magnitude. To the best of our knowledge, this is the first successful analytical

treatment of electron-positron scattering in an ultramagnetic background. e±-ion

scattering is also considered, and is generalized to include both relativistic motion

and Debye screening.

We then derive the cross section for relativistic bremsstrahlung in the Born approx-

imation, treating the ion as immobile and comparing it with the nonmagnetic result

derived by Bethe & Heitler (1934). This cross section is then thermally averaged to

evaluate thermal bremsstrahlung emission and absorption. The corresponding Gaunt

factor is tabulated for a range of temperatures and frequencies, and analytical approx-

imations in the non-relativistic limit are presented. We also present an approximate

derivation of electron-positron bremsstrahlung using arguments based on the soft

photon limit.

Having armed ourselves with the aforementioned cross sections and rates, we pro-

ceed to construct the magnetar emission model. The structure of the atmospheric cool-

ing layer - with free-free absorption as the dominant source of opacity - is presented.

The transition to an isothermal regime, pressure balance with the magnetosphere,

and pair-loading caused by the suppressed annihilation cross section are taken into

account. The magnetosphere is treated as a uniform layer with constant temperature,

scale height, width and scattering optical depth. The details of the Monte Carlo

procedure - such as the particle propagation algorithm, the treatment of processes

with a distribution of final states and the convergence of the physical system - are

then discussed

The model is capable of producing “AXP-like” spectra with a prominent thermal

blackbody peak, a steep intermediate falling power law component and a rising power

law component that emerges at high photon energies. These spectra correspond to

minimal comptonization of the cooling layer’s radiation by the magnetospheric plasma,

due to a combination of low optical depth and low horizontal width. Therefore,


they clearly demonstrate the interplay between powerful electron-positron scattering,

asymmetric energy distribution in annihilation, resonantly enhanced electron-photon

scattering, and bremsstrahlung emission that takes place in the model. This interplay

is relatively insensitive to the details of the cooling layer’s structure, which serves

to strengthen the presented results. The relative contribution of the deep cooling

E-mode depends on its source, but could be comparable to the O-mode blackbody if

sourced by relativistic magnetospheric particles. In particular, the model can generate

a spectrum that approaches the form of AXP 4U 0142+61 (Weng & Gögüs, 2015), some

of the difference being possibly attributable to the uncertainty in the electron-positron

bremsstrahlung cross section, and also to our assumption of trans-relativistic e± pairs.

“SGR-like” spectra, featuring a blackbody peak that connects directly to a flat

power law component, can also be produced by the model. They are the result of

heavy magnetospheric comptonization of the cooling layer’s radiation combined with

the reprocessing of the comptonized radiation by the cooling layer, and are thus even

less sensitive to the precise structure of the cooling layer. Meanwhile, the deep cooling

E-mode contribution is negligible. These spectra are similar to the spectrum of SGR

1806-20 (Enoto et al. 2017b), but are not quite as hard at the highest photon energies.

6.2 Future Work and Observational Predictions

A significant shortcoming of the present approach is its approximate treatment of

electron-positron bremsstrahlung. Given the high efficiency of electron-positron

scattering, we surmise that the corresponding bremsstrahlung process is similarly

enhanced compared to its e±-ion counterpart. In our model, approximately half of

the bremsstrahlung emission seen in the “AXP-like” spectra is attributable to the

electron-positron channel. A careful QED derivation of this cross section may reveal

unexpected scalings with energy and frequency, and therefore has the potential to

bring the model spectra closer in line with those of real AXPs.


Although we do not expect the changes to alter the overall conclusions of this

analysis, the effects of magnetospheric particle diffusion on the structure and emission

of the cooling layer require detailed, self-consistent modelling similar to the work of

Bauböck et al. (2019) and González-Caniulef et al. (2019). Additionally, more refined

models of the deep cooling E-mode emission can be included (Ho & Lai 2001; van

Adelsberg & Lai 2006; Harding & Lai 2006).

This, together with the assumption that all absorbed energy is re-radiated as a

blackbody at Teff, is a key simplification of our model. Detailed modelling of the

bombarded atmosphere in the vein of Bauböck et al. (2019) and González-Caniulef

et al. (2019) is outside the scope of the present analysis.

The model’s “SGR-like” spectra can be hardened at high photon energies by raising

the temperature of the magnetospheric pairs, or by combining the model with other

models of hard nonthermal magnetar X-ray emission, such as those that consider

resonant cyclotron scattering (Fernández & Thompson 2007; Baring & Harding 2007;

Beloborodov 2013; Taverna et al. 2020) or synchrotron emission (Heyl & Hernquist

2005b; Thompson & Beloborodov 2005).

Non-local collisions between hard (Eγ . 2mc2) photons may be an important

source of charge carriers in parts of the magnetosphere with lower current densities

(e.g. near the poles). This effect may modify current approaches, which assume

that the charges are supplied by a discharge and pair breakdown (Beloborodov &

Thompson 2007; Beloborodov 2013).

We also note that the “AXP-like” spectra generated by this model exhibit clear

annihilation peaks above hω ∼ mc2. These peaks are attributable to the combination

of the asymmetric energy distribution in annihilation events and the high efficiency of

electron-positron scattering, which greatly lowers the annihilation optical depth and

allows the photons to escape easily. To the best of our knowledge, this feature is not

present in any of the other proposed models of hard nonthermal magnetar emission,


and its potential future detection can be used to test the validity of the present model.

Appendix A

Electron scattering matrix element

Here, we evaluate the vertex integrals I1-I4 given by Equations (3.11) and (3.12). Each

involves a contraction of the matrix (2.22) with the spinors (2.16). We include only the

lowest term (nI = 0) in the sum over Landau levels in the Green function (2.19), with

the outgoing electron also being confined to the lowest Landau state. The quantities

which appear in the integrals I1 and I3 are[u(−1)∗

0,aI

]Tγ0γµε

µi u(−1)

0,ai= −εz

ipz,i(EI + m) + pz,I(Ei + m)

2L2[EI(EI + m)Ei(Ei + m)]1/2 φ0(x− ai)φ0(x− aI);

(A.1)[v(−1)∗

0,aI

]Tγ0γµε

µi u(−1)

0,ai= −iεz

ipz,i pz,I + (EI + m)(Ei + m)

2L2[EI(EI + m)Ei(Ei + m)]1/2 φ0(x− ai)φ0(x− aI).

(A.2)

Next, we integrate over x to obtain I1 and I3, making use of the integral formula (3.6):

I1 = −εzi e−λ2

Bk2⊥,i/4 eikx,i(ai+aI)/2 pz,i(EI + m) + pz,I(Ei + m)

2L2[EI(EI + m)Ei(Ei + m)]1/2 (2π)2 δ

(aI − ai

λ2B

+ ky,i

)×δ(pz,i + kz,i − pz,I);

(A.3)

I3 = −iεzi e−λ2

Bk2⊥,i/4 eikx,i(ai+aI)/2 pz,i pz,I + (Ei + m)(EI + m)

2L2[EI(EI + m)Ei(Ei + m)]1/2 (2π)2 δ

(aI − ai

λ2B

+ ky,i

)×δ(pz,i + kz,i + pz,I).

(A.4)

114

Appendix A. Electron scattering matrix element 115

The integrals I2 and I4 are obtained by interchanging Ei, pz,i, ai, and εzi with E f , pz, f ,

a f , and (εzf )∗ in these expressions, with an additional multiplication by −1 for I4. We

can choose kx = 0 in the initial (but not the final) state, given the freedom in the

definition of the background magnetic gauge. Substituting these expressions for I1-I4

into the integral (3.10) gives the expression (3.13).

Appendix B

Electron-positron scattering integrals

Here we evaluate the matrix element for electron-positron scattering in a general lon-

gitudinal reference frame, as given by Equation (4.18). This involves the combination

of integrals I2,µηµν I1,ν. As a first step, note that[v(+1)∗

0,a+,i

]Tγ0γνηµνu(−1)

0,a−,i= i

φ0(x− a+,i)φ0(x− a−,i)

2[E+,iE−,i(E+,i + m)(E−,i + m)]1/2

×

pz+,i[ηµt(E−,i + m)− ηµz pz−,i] + (E+,i + m)[−ηµz(E−,i + m) + ηµt pz−,i]

.

(B.1)

Only the µ = t and z components of I2,µ contribute; these are, respectively,[u(−1)∗

0,a−, f

]Tv(+1)

0,a+, f= −i

pz+, f (E−, f + m) + pz−, f (E+, f + m)

2[E+, f E−, f (E+, f + m)(E−, f + m)]1/2 φ0(x− a+, f )φ0(x− a−, f );

(B.2)

and[u(−1)∗

0,a−, f

]Tγ0γzv(+1)

0,a+, f= i

pz+, f pz−, f + (E+, f + m)(E−, f + m)

2[E+, f E−, f (E+, f + m)(E−, f + m)]1/2 φ0(x− a+, f )φ0(x− a−, f ).

(B.3)

Substituting these results into Equations (4.19) and (4.20), and making use of the

overlap integral in Equation (3.6), we find

I2,µηµν I1,ν =(2π)4eiqx(a+, f +a−, f−a+,i−a−,i)/2e−λ2

B(q2x+q2

y)/2 F(pz−,i, pz+,i, pz−, f , pz+, f )

4L4√

E+,iE−,iE+, f E−, f (E+,i + m)(E−,i + m)(E+, f + m)(E−, f + m)

× δ

(a+,i − a−,i

λ2B

− qy

)δ

(qy −

a+, f − a−, f

λ2B

)δ(pz+,i + pz−,i − qz) δ(qz − pz+, f − pz−, f ),

(B.4)

116

Appendix B. Electron-positron scattering integrals 117

where

F ≡ −[pz+,i pz−,i + (E+,i + m)(E−,i + m)][pz+, f pz−, f + (E+, f + m)(E−, f + m)

]+

[pz+,i(E−,i + m) + pz−,i(E+,i + m)][pz+, f (E−, f + m) + pz−, f (E+, f + m)

].

(B.5)

Substituting Equation (B.4) into Equation (4.18) gives

S f i[2] =−ie2

4L4

∫dqx

eiqx(a+, f +a−, f−a+,i−a−,i)/2e−λ2B(q

2x+q2

y)/2

(E+,i + E−,i)2 − q2

×(2π)2δ

(3)f i (E, py, pz)F(pz−,i, pz+,i, pz−, f , pz+, f )√

E+,iE−,iE+, f E−, f (E+,i + m)(E−,i + m)(E+, f + m)(E−, f + m), (B.6)

where q2 = q2x + q2

y + q2z, qz = pz+,i + pz−,i and qy = (a+,i − a−,i)/λ2

B.

Appendix C

Ionic Integrals

The matrix elements for both e±-ion scattering and e±-ion bremsstrahlung contain

integrals over the ion Coulomb field. The y integral in Equation (4.33) works out to

∫ ∞

−∞dy exp

[iy

ai − a f

λ2B

]1√

x2 + y2 + z2= 2K0

[(ai − a f )

√x2 + z2

λ2B

], (C.1)

where K0 is the modified Bessel function. Taking λB → 0, we see that this quantity

peaks very strongly at ai ≈ a f . It may be expressed in terms of a delta function in a f .

Performing the integral

∫ ∞

−∞da f K0

[(ai − a f )

√x2 + z2

λ2B

]=

πλ2B√

x2 + z2, (C.2)

we have

2K0

[(ai − a f )

√x2 + z2

λ2B

]→ 2πδ

(ai − a f

λ2B

)1√

x2 + z2. (C.3)

Given a typical impact parameter ai ∼ 1/pz,i, we deduce that |a f − ai| ∼ λ2B pz,i, which

is much smaller than the width of the spinor wave functions (δx ∼ λB). Hence, we

may take ai = a f and x = ai in evaluating the x and z integrals, the latter of which

becomes ∫ ∞

−∞dz cos[(pz,i − pz, f )z]

1√a2

i + z2= 2K0[(pz,i − pz, f )ai]. (C.4)

118

Appendix C. Ionic Integrals 119

The Coulomb field is essentially constant over the domain of support of the x integral,

which is easily expressed in terms of the orthogonality relation

∫dx φn f (x− ai)φ0(x− ai) =

δn f ,0

L2 . (C.5)

Along with considerations of energy conservation, this requires the final-state particle

to be confined to the lowest Landau level. The net overlap of the ingoing and outgoing

spinors with the Coulomb potential is presented in Equation (4.33).

The integrals Ii appearing in the bremsstrahlung matrix element (4.40) are evalu-

ated in a similar manner,

I1 =e

4πK0[(pz,i − pz,I)ai]

pz,i pz,I + (Ei + m)(EI + m)

L2[EiEI(Ei + m)(EI + m)]1/2 2πδ

(aI − ai

λ2B

), (C.6)

and

I3 = ie

4πK0[(pz,i + pz,I)ai]

pz,i(EI + m) + pz,I(Ei + m)

L2[EiEI(Ei + m)(EI + m)]1/2 2πδ

(aI − ai

λ2B

). (C.7)

Meanwhile, the two integrals from the bremsstrahlung diagram that involve a real final

photon are unchanged from the electron-photon scattering considered in Appendix A:

I2 = −(εz)∗e−ikx(aI+a f )/2e−λ2Bk2⊥/4 pz,I(E f + m) + pz, f (EI + m)

2L2[EIE f (EI + m)(E f + m)]1/2

× (2π)2δ

(a f − aI

λ2B− ky

)δ(pz,I − pz, f − kz), (C.8)

and

I4 = i(εz)∗e−ikx(aI+a f )/2e−λ2Bk2⊥/4 pz,I pz, f + (EI + m)(E f + m)

2L2[EIE f (EI + m)(E f + m)]1/2

× (2π)2δ

(a f − aI

λ2B− ky

)δ(pz,I + pz, f + kz). (C.9)

Here k2⊥ ≡ k2

x + k2y.

Appendix D

Integral and differential cross sections

Here, we elaborate on the approach used in the present model to treat the integral

cross sections, which are used in optical depth accumulation, as well as differential

cross sections, which are used in determining the results of processes where there is a

distribution of the final state variable(s).

D.1 Integral cross sections and accumulation of optical

depth

As electrons, positrons and photons propagate through the atmosphere and mag-

netosphere, they accumulate optical depths for the various processes considered,

proportional to the product of the path length travelled, the concentration of targets on

this path and the integral cross section of this process, τi = ntσih. For some processes,

the integral cross section can be a simple analytical formula, specifically for e±-ion

scattering, Bhabha scattering, two-photon pair creation and thermal bremsstrahlung

absorption.

The calculation of the integral cross sections for the other processes, however,

involves numerical integration of the differential cross sections with respect to at

120

Appendix D. Integral and differential cross sections 121

least one of the variables (for instance, the differential bremsstrahlung cross section

(4.47) can be integrated analytically over solid angle, while the frequency integral

must be performed numerically). Since a re-calculation of the cross sections for each

combination of parameters would be prohibitively long, we calculate the cross sections

at the grid points of the relevant variable(s), and use the nearest grid point value

during particle propagation.

Additional refinements are also implemented in a series of cases. When there is

a variable whose effect on the cross section we want to sample more precisely than

the relevant grid, we perform a spline interpolation with respect to that variable

while keeping the other variables binned to their respective grids. This occurs for

annihilation and bremsstrahlung emission (where we interpolate over the kinetic

energy of the particle(s)) and Compton scattering (where we interpolate over the

frequency of the scattering photon).

An additional refinement is made for Compton scattering, owing to the cross

section’s resonant behaviour as the the final-state photon’s frequency approaches

2m/(1− µ2f ). For most values of µi, this condition requires that the initial frequency

ωi approach the one-photon pair creation threshold 2m/ sin θi, necessitating a careful

treatment of this region. Taking the |∆ω|−1.5 dependence of the cross section in this

region from Equation (3.29) as a reference point, we use an α = −1.5 power law

analytical interpolation in the immediate vicinity of this threshold.

D.2 Differential cross sections

For many of the processes under consideration, there is no ambiguity as to the final

state. In e±-ion scattering the e± changes direction and loses a small amount of energy

to the ion, while in electron-positron scattering the electron and positron simply

exchange their momenta with each other, so that pz−, f = pz+,i, pz+, f = pz−,i. Similarly,

in both single- and two-photon pair creation the momenta of the created particles are

Appendix D. Integral and differential cross sections 122

uniquely described by the properties of the pair creating photon(s), up to a twofold

degeneracy in the direction of the created particles.

The other processes, however, have a distribution in the possible values of the

final state variable(s) that is set by their differential cross section(s) with respect to

these variable(s). In this case, random numbers are drawn and checked against the

cumulative probability integrals evaluated up to an array of values of the variable

under consideration, with the closest matching array point chosen as the variable

value. For some variables, this probability array can be calculated analytically, while

for other variables numerical integration is required.

In Compton scattering, the post-scattering direction µ f is chosen as the independent

final-state variable, and its probability array must be evaluated numerically. For both

e±-ion and electron-positron bremsstrahlung, there are two independent final-state

variables: the photon frequency ω, whose probability array must also be evaluated

numerically, and the photon direction µ, whose probability array can be determined

analytically at a fixed value of the frequency.

In the case of e± annihilation the independent variables are the directions µ1,2 of the

produced photons, which set the photon frequencies through Equation (4.8). In order

to avoid the appearance of µ-binning related artificial spectral features, a continuous

relation between the randomly drawn probabilities and their corresponding values of

µ1,2 is required. The cumulative probability array of one directional cosine (say µ1)

must be evaluated numerically and then splined with respect to µ1 itself. Then, at a

fixed value of µ1, the cumulative probability with respect to the second directional

cosine µ2 may be evaluated analytically.

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Magnetar X-ray spectra an ultramagnetic QED approach...-Seasons in the Abyss by Slayer (misheard)...

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