Download Calculation of the Cherenkov Light Yield for High

Calculation of the Cherenkov Light Yield for High-Energy Particle
Cascades
von Ömer Penek
May 2, 2016
Masterarbeit in Physik
vorgelegt der
Fakultät für Mathematik, Informatik und Naturwissenschaften
der RWTH Aachen
angefertigt im
III. Physikalischen Institut, Lehrstuhl für Experimentalphysik III B
Lehr- und Forschungsgebiet Elementarteilchen- und Astroteilchenphysik
Prof. Dr. Christopher Wiebusch
Prof. Dr. Werner Bernreuther
1
Contents
1 Introduction and Motivation
5
2 High-Energy Neutrino Detection
6
2.1
Cosmic Neutrino Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Neutrino Interactions in Matter
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3
Particle Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3.1
Electromagnetic Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3.2
Hadronic Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.4
Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5
The IceCube Neutrino Observatory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Monte Carlo Simulation of Particle Cascades
3.1
12
The GeanT4-Toolkit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1
Initialization Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.2
Action Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3
Simulation of Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4
Data and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4.1
Step Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4.2
Distance of e+ e− Pairs in PeV Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4.3
Number of Steps as Function of the Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Particle Induced Light Yields
4.1
4.2
18
Theoretical Preperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Electrodynamics in Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.1
Fundamentals and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.2
Electrodynamics in Fourier Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3
Particle Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4
Retarded Potentials and Fields of a Moving Point Charge . . . . . . . . . . . . . . . . . . . . . . 27
4.5
Electromagnetic Flux
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5.1
The Connection between Energy Flux and Total Number of Photons
4.5.2
The Master Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5.3
The Master Formula
4.5.4
The Classical Limit and Determination of the Prefactors
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5.5
4.6
Generalization of the Frank-Tamm Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 38
About Cascade Directions and Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2
4.6.1
Cascades from Arbitrary Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.6.2
Cascade Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Calculation Results
5.1
5.2
40
Cherenkov Light Yield Studies and Expectation Validation . . . . . . . . . . . . . . . . . . . . . 40
5.1.1
One Track divided into N Steps ranked together . . . . . . . . . . . . . . . . . . . . . . . 40
5.1.2
Cherenkov Light Yield as Function of the Steplength . . . . . . . . . . . . . . . . . . . . . 41
5.1.3
Cherenkov Light Yield as Function of the Distance . . . . . . . . . . . . . . . . . . . . . . 43
5.1.4
Cherenkov Light Yield of Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1.5
Cherenkov Light Yield of Single and Multiple Scattering . . . . . . . . . . . . . . . . . . . 47
Cherenkov Light Yield of Particle Cascades in Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.1
Cherenkov Light Yield of indpendent Steps . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.2
Cherenkov Light Yield of Particle Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Summary and Outlook
51
3
Eigenständigkeitserklärung:
Ich versichere, dass ich die Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und
Hilfsmittel benutzt sowie Zitate kenntlich gemacht habe.
Aachen, 2. Mai 2016
Ömer Penek
4
Abstract
The measuring and calculation of the Cherenkov light yield in a particle cascade can help to reconstruct the
energy of a cascade inducing particle. The light yield itself is described by the well-known Frank-Tamm formula
which is a measure for the number of photons per unit path length and unit frequency of light of a single
track. However, the theory behind the Frank-Tamm formula uses an infinite track approach using conditions
like ω∆t 1 whereby ω is the frequency of light and ∆t the track length time of the particle. Nevertheless,
simulation studies where the polarization regions of two charged particles may overlap show that the light yield
can increase or decrease depending on the charge configuration [81]. The goal of this thesis is to calculate the
Cherenkov light yield for high energetic particle cascades. As a simplification, the medium is chosen to be ice.
1
Introduction and Motivation
The background of the today quasi well-known cosmic rays has its origin in the balloon experiments of Victor
Hess in 1912 where cosmic radiation was detected and studied the first time. Nowadays the flux of cosmic rays
can be quantitatively described by a broken power law:
dN
∝ E −γ .
dE
(1.1)
Thereby γ is the spectral index describing the strongness of the flux and E is the energy of the cosmic radiation.
One of the best known mediators for cosmic rays are the neutrinos. They can move directly from their sources
to earth without any disturbance of their paths due to the fact that they only weakly interact with matter and
are neutral charged. The discovery of neutrinos go hand in hand with the development of the weak interaction.
In 1911 physicists like Hahn and Meitner measured the β-decay energy spectrum and could show that the
spectrum is continuous. 19 years later Pauli postulated an additional particle to explain the continuitiy of the
β-spectrum. A first principle theoretical approach of the weak force was done by Fermi assuming a four body
interaction with a point-like vertex. Nowadays, it is known that the mediators of the weak force are the W - and
Z-bosons [105]. The IceCube neutrino observatory is focused on the detection of neutrinos using a large neutrino
telescope which is located in the Geographic South Pole. The detection and energy reconstruction of neutrinos
is experienced to be very challenging. Nevertheless, neutrino studies opened a new window in the research
of cosmic sources like supernova remnants, active galactic nuclei or gamma-ray-bursts. Such high energetic
neutrinos can be detected via the Cherenkov radiation using the amount of light emitted by the corresponding
charged lepton which is induced by the neutrino via a charged current reaction. The resulting cascades have
electromagnetic and hadronic character.
The first and main goal of this work is to construct a classical mathematical tool which makes it possible to
calculate the Cherenkov light yield for arbitrary particle systems which take into account high particle densities
and with application to simulations done by Geant4 . The second goal is to study electromagnetic cascade data
relevant for the calculation of the Cherenkov light yield.
5
2
High-Energy Neutrino Detection
In this chapter the principles of high-energy neutrino detection will be presented and discussed. A detector
example is chosen to be the IceCube Neutrino Observatory.
2.1
Cosmic Neutrino Production
Cosmic neutrinos come from pion or kaon decays which are created by interactions of accelerated protons or
nuclei with photons or hadrons. The following reactions explain the creation of e.g. pions:
p+γ
p+p
p+n
→ ∆+ (1232) →

p + π 0
,
n + π +

p + p + π 0
→ ∆++ (1232) →
p + n + π +
(2.1)
,
→ p + ∆0 (1232) → p + p + π − .
(2.2)
(2.3)
Charged pions decay according to π ± → µ± ν and further muons decay according to µ± → e± νν with branching
ratios of approximately ∼ 100 % which lead to the creation of neutrinos. These neutrinos are not influenced by
galactic magnetic fields and move directly to earth without the loss of tracking information.
2.2
Neutrino Interactions in Matter
Neutrinos are uncharged elementary particles which are interacting weakly with matter. From a scientific point
of view this is a disadvantage because the detection of neutrinos becomes rigorous. Nevertheless, this fact has
also its advantage. The solar neutrino flux is about ∼ 1011 cm−2 s−1 and these neutrinos essentially do not
interact with the human body which can be seen as protection mechanism. The detection of neutrinos become
more probable for higher energies. This is caused by the neutrino nucleon cross section, σνN , which increases
with increasing energy. The cross section can be approximated as [110]:
σνN ∼

10 pb
100 pb
E
T eV
E
T eV
E < 10T eV
0.363
.
(2.4)
E > 10 T eV
This cross section is negligibly small compared to the proton-proton cross section which varies from a few
hundred mb to a few b. There are two types of neutrino interactions. The first interaction is called charged
current interaction and the second is called neutral current interaction. This topology has to do with the
mediators of the weak force. The reaction equation for a neutrino with flavor l interacting with a nucleus N is:
W+
νl + N → l + X,
Z
(2.5)
0
νl + N → νl + Y.
6
(2.6)
Thereby the X and Y stand for hadronic cascades. The reaction products are either charged leptons (charged
current interaction, CC) or neutrinos (neutral current interaction, NC). In transparent media like water or ice
the CC reaction is preferred because of the fact that the charged leptons can induce Cherenkov radiation which
can be measured via photomultiplier tubes.
In 1960 Markov gave a solution for the detection principle of neutrinos. Basically he proposed that deep sea
water or ice are suitable sites for neutrino telescopes leading to a large volume for free neutrino targets. The
other reason was that a detector located at in-depth provides a great shielding against secondary particles.
2.3
Particle Cascades
In the following the main two known cascade types will be presented i.e. electromagnetic and hadronic cascades.
In this work only electromagnetic cascades are treated. Nevetheless, by the sake of completeness hadronic
cascades will be discussed briefly.
2.3.1
Electromagnetic Cascades
The carriers of electromagnetic cascades are electrons, positrons and photons. The dominant processes at high
energies are pair production induced by photons and bremsstrahlung induced by electrons or positrons. These
processes occur if a carrier moves a characteristic length scale called the radiation length X0 . It is a material
constant and is measured usually in gcm−2 [109]:
X0−1
4αem NA re2 Z (Z + 1) log 183Z −1/3
=
.
A
(2.7)
Thereby A is the atomic mass number, Z the proton number, αem the fine structure constant, NA the Avogadro
nmber and re the classical electron radius. A typical radiation length in ice simulated by Geant4 is 39.7 cm.
Another length scale for electromagnetic cascades is the Moliére radius which gives the transversal width of the
cascade. It can be described as [109]:
RM =
Es
X0 (Es ≈ 21 M eV ) .
Ec
(2.8)
Thereby Ec is the critical energy. This quantity is really important for the development of showers. It is defined
as the threshold energy where the energy loss due to ionization is equal to the energy loss due to bremsstrahlung.
It is approximately given as [65]:
Ec ≈
610M eV
.
Z + 1.2
The critical energy in ice is about 78.60 MeV (water: 78.60 MeV).
In the following figure a modeling of an electromagnetic shower is illustrated.
7
(2.9)
Figure 2.1: Schematic view of an electromagnetic shower according to Heitler. On the left side the energy
distribution is shown. The quantity R on the right represents the radiation length [17].
As it is realizable an electromagnetic shower is an alternating process of bremsstrahlung and pair production.
The opening angle of pair production can be approximated as [104]:
α∼
1
me c2
∼
.
γ
E
(2.10)
Thus, the higher the energy the smaller the opening angle of pairs.
The number of particles in an electromagnetic cascade has been simulated by Rossi and Greisen in 1941 and
can taken from the following figure.
8
Figure 2.2: Simulation of the number of particles in an electromagnetic cascade according to Rossi and Greisen
(1941)[17, 107]. The number of particles is plotted logarithmically as function of the radiation length. The
energy is given as multiples of the critical energy Ec .
At 20X0 and 106 Ec the number of particles due to Rossi and Greisen can be estimated as ∼ 32000. The
number of particles in a cascade with energy > E can further be approximated as [104]:
N (> E) ∼
Eprimary
.
E
(2.11)
A typical Geant4 simulation of an electron induced cascade is shown in the following figure.
Figure 2.3: Geant4 simulation of an electron induced cascade at 10 GeV. Represented is the xz-distribution,
where z is the shower axis.
2.3.2
Hadronic Cascades
Hadronic cascades occur if nuclear interactions play a crucial role. Hadronic reactions cause e.g. mesons like
pions. The π 0 decays into two photons (branching ratio ∼ 1) leading to an electromagnetic cascade separated
from the hadronic component. The other charged pions π ± decay into muons and neutrinos leading to a
complicated shower development.
9
2.4
Cherenkov Radiation
Cherenkov radiation appears if a charged particle with velocity v moves faster than the speed of light in a
dielectric [101]. Assuming that cn = c0 /n is the velocity of light in a dielectric, c0 the velocity of light in
vacuum and β = v/c0 the Cherenkov condition reads as:
v > cn ⇒ cos θCh =
1
1
=
.
βn
βn
(2.12)
Thereby θCh describes the opening angle of the radiation cone which is illustratively shown in the following
figure.
Figure 2.4: Origin of Cherenkov radiation. On the left column the velocity of the particle is < cn . There is no
resonance and the dipoles in the media neutralize each other. On the right column the Cherenkov condition is
fulfilled and a cone with a defined opening angle is created [65, 108].
The Cherenkov radiation is a polarization effect. If the Cherenkov condition is fulfilled the medium is
polarized and the dipoles have no time to arrange themselves symmetrically to the particle. The dipole field is
not vanishing and the temporal alteration of the dipoles lead to electromagnetic radiation [19, 75, 10, 26]. The
number of photons emitted for a single track is given by the Frank-Tamm formula [102]:
dNγ
dωdL
dNγ
dλdL
=
=
αem
sin2 θCh ,
c0
2παem
z2
sin2 θCh .
λ2
z2
(2.13)
(2.14)
This formula is only valid for ω∆t 1 i.e. for large track length compared to the considered wavelength of
10
light and does not consider high particle densities. The following figure shows the Frank-Tamm formula for one
particle as function of the wavelength λ.
Figure 2.5: Simulation of the classical Frank-Tamm formula for a single charged particle as function of the
wavelength.
2.5
The IceCube Neutrino Observatory
The IceCube Neutrino Observatory is a large neutrino detector with a volume of about ∼ 1 km3 located at the
Geographic South Pole. The following figure shows the detector array.
Figure 2.6: The IceCube Neutrino Detector.
The IceCube detector has about 86 strings and each string has about 60 digital optical modules and is a
suitable site for high-energy neutrino detection.
11
3
Monte Carlo Simulation of Particle Cascades
3.1
The GeanT4-Toolkit
Geant4 is an acronym and stands for geometry and tracking. It is a toolkit which was designed to describe and
simulate the passage of particles through matter and uses C++ language. The project started in december 1994
and its first public release came out on december 1998. The toolkit can be seperated into two main classes. The
first class is the so-called initialization class and consists of two mandatory classes G4VUserDetectorConstruction
and G4VUserPhysicsList [103, 65]. The second class is the so-called action class and consists of one mandatory
class which is called G4VUserPrimaryGeneratorAction. It is possible to add optional actions as well. Some will
be treated in the following chapter in the context of this thesis.
3.1.1
Initialization Classes
G4VUserDetectorConstruction: Detector and Materials
The detector geometry which is used in the simulations is a cylinder with a radius of 20 m and a height of 40
m. The detector material is set as ice and is taken from the NIST database. The defined ice material coincides
with an index of refraction of about 1.33 and thus corresponds to the index of refraction in the south pole where
the IceCube detector is located. The ranges of the x, y and z axes are given as: x, y, z [m] ∈ [−20, 20]. The
alignment of the axes can be taken from the follwoing figure.
Figure 3.1: Detector geometry and the alignment of the axes.
G4VUserPhysicsList: Particles and Processes
The particles which are implemented in the physics list are electrons, positrons, photons and protons. The
standard proceses which can be done by these particles are specified in the classes G4G4EmStandardPhysics
and G4DecayPhysics. The main processes in the standard physics package for electrons, positrons and photons
are given in the following tabular.
12
e−
Multiple Scattering
Electron Ionization
Electron Bremsstrahlung
-
e+
Multiple Scattering
Electron Ionization
Electron Bremsstrahlung
Positron Annihilation
γ
e e conversion
Compton Scattering
Photo-Electric Effect
+ −
Table 1: Standard Physics Processes [103].
The electromagnetic cascades use the G4EmStandardPhysics_option4 class which is designed for simulations
for highest accuracy.
3.1.2
Action Classes
Mandatory Class: G4VUserPrimaryGeneratorAction
This mandatory class is used to specifiy the primary i.e. cascade inducing particle’s properties like particle
type, energy and momentum for instance.
Optional Class: G4UserSteppingAction
The G4UserSteppingAction class is important because the whole needed particle track and step information is defined and called in this class. The information of each particle’s track itself is specified in the
G4VUserTrackInformation which guarantees that every track information is not deleted but saved for every
step. A scattering procedure can be taken from the following figure.
Figure 3.2: Notations and definitions (to read from left to right). The blue line is called Track and the red dots
are called PreStep or PostStep points. The line inbetween two red dots is called an AlongStep [98].
Every information concerning the tracks or the steps can be taken from the simulation. For this work the
following data were simulated:
• Particle charge
• PreStep and PostStep time
• PreStep and postStep positions for every cartesian coordinate (x, y, z)
• β and γ factor
• Unit momentum vector for an along step
13
• Vertex positions where a track is created
• Vertex kinetic energy
3.2
Validation
An experimental validation of the simulated results is very important to become a realistic connection of the
data and the experiment. Electromagnetic cascades which can be simulated in Geant4 “are well validated
in LHC experiments where the gamma energy of interest is below 1 TeV. (..)” (Ivantchenko, 2015, Mail [97]).
Above 1 TeV the Geant4 collaboration talks about theoretical validation which is caused by the following effects:
1. LPM effect in e+ e− bremsstrahlung and gamma conversion,
2. Nuclear form-factor for hadron ionization,
3. Radiative corrections for muon ionization,
4. Muon nuclear reactions.
The maximal energy treated in this work is 1 PeV.
3.3
Simulation of Cascades
The simulation technique of cascades is shown in the following figure.
Figure 3.3: Simulation of Cascades [65].
Thereby a primary particle induces the shower because of its high energy. The created secondary particles
can be charged or neutral.
14
3.4
Data and Results
The simulated data concentrate basically on the carrier of electromagnetic cascades i.e. electrons, positrons and
photons. Important for the light yield is the step time distribution and the distances of the e+ e− pairs.
3.4.1
Step Time Distribution
In the following two figures the step time distributions for the energies 1 GeV and 1 TeV are shown.
Figure 3.4: Step time distribution for an electron induced cascade. Left: 1 GeV. Right: 1 TeV.
Figure 3.5: Step time distribution for a positron induced cascade. Left: 1 GeV. Right: 1 TeV.
It is realizable that the step distributions look absolutely equal and that the average step times are about
0.016 ns.
15
3.4.2
Distance of e+ e− Pairs in PeV Cascades
The distance of an e+ e− pair is defined looking forward to equation (4.86). In the following three figures, the
distances of e+ e− pairs in a 1 PeV cascade are shown. Thereby the left columns contain the distances of the
first steps and the right columns represent the distance with respect to the radiation length X0 . Thereby, only
distances below 2 µm are considered. The histograms can be parameterized as follows:
dstep
∼
dX0
∼
α
Lstep sin
,
α 2
X0 sin
.
2
(3.1)
(3.2)
Thereby dstep stands for the distance of the pairs at the first step, Lstep is the steplength and α the opening
angle. The distance dX0 represents the distance of the pairs with respect to moving a radiation length X0 .
16
Figure 3.6: Distance of e+ e− pairs from electron (top), positron (middle) and photon (bottom) induced cascades
for 1 PeV. Left column: Distance of e+ e− pairs at their first step. Right column: Distance of e+ e− pairs at a
radiation length. Both distance distribution have an upper limit of 2 µm.
3.4.3
Number of Steps as Function of the Energy
The number of steps simulated by Geant4 as function of the energy can be taken from the following chart.
17
Particle
1 GeV
10 GeV
100 GeV
1 TeV
10 TeV
100 TeV
1 PeV
e−
1025
10870
109088
1091460
10906678
109102824
1090940074
e+
1103
10993
109210
1092145
10906393
109116198
1091156311
γ
1107
10945
109134
1090012
10912060
109135999
1091501298
Table 2: Number of steps simulated in Geant4 as function of the energy.
The steps as function of the energy show a linear dependence. Thus, the approximation
NSteps ∼ 1.09 ·
E
M eV
(3.3)
for every carrier can be used as a measure to estimate the number of steps. This relation is important due to
the fact that it determines the amount of Cherenkov photons.
4
Particle Induced Light Yields
The results obtained in this chapter are of huge importance. Therefore a separate part was reserved for it. First
of all the particle sources will be calculated and finally the electromagnetic flux.
4.1
Theoretical Preperations
In the following steps some integration techniques will be presented.
Plane Wave Expansion
Finally, the plane wave expansion method in three dimensions will be described. Assuming that fP ~k is a
function of a vector ~k and P whereby P determines the dependence of f on other parameters e.g. fP ~k =
f ~k, ω then the integral over the ~k-space is invariant under change of the coordinate system i.e.:
ˆ
ˆ
ˆ
ˆ
3
3
~
d kfP k = d kfP (kr , θk , φk ) = d kfP (kρ , kz , φk ) = d3 kfP (kx , ky , kz ) .
3
R3
R3
R3
(4.1)
R3
Thereby ~k = (kr , θk , φk ) stands for spherical, ~k = (kρ , kz , φk ) for cylindrical and ~k = (kx , ky , kz ) for cartesian
coordinates. The integral measures are given by the Jacobian functional determinant:
d3 k = dkr dθk dφk kr2 sin θk = dkρ dkz dφk kρ = dkx dky dkz .
(4.2)
It is important and decisive at this point that the integrals in (4.1) go over the total and infinite ~k-space i.e.
every component of ~k varies between −∞ and ∞. Otherwise the integrals depend strongly on the geometry.
18
~ a position or reference vector in space then ei~kR~
Now, assuming that ~k is a wave vector in Fourier space and R
is called a plane wave. Now the dot product is given as:
~k R
~ cos θ ≡ kR cos θ.
~ = ~k R
(4.3)
According to equation (4.1) the integral does not depend on the coordinate system. Therefore an integral of
type
ˆ
~~
d3 kfP (k) eikR
R3
can always be written as
ˆ
3
d kfP (k) e
~
i~
kR
ˆ
d3 kfP (k) eikR cos θk ,
=
R3
(4.4)
R3
~ i.e. θ = θk correspond with the polar angle of ~k with respect to R
~ expressed
whereby the angle between ~k and R
in spherical coordinates. Hereby, the rotational invariance of spheres is used i.e. it is always possible to find a
~ have the angle θ = θk to each other whereby θ corresponds to the polar
coordinate system such that ~k and R
angle. In principle, (4.4) is only the application of the dot product equation (4.3). It has to be underlined that
fP only depends on the norm of ~k i.e. k. Thus, the plane wave expansion method cannbe useful
o in several integral
~
~
calculations using a spherical symmetry. Generally, a plane wave of the form exp ik R can be expanded as
follows:
e
~
i~
kR
=
∞
X
n
i (2n + 1) jn (kR) Pn
n=0
~k R
~
kR
!
.
(4.5)
Thereby, jn are the n-th order spherical Bessel functions and Pn the n-th order Legendre polynomials.
The Fourier Transformation and its Properties
The Fourier transformation F (H) of a function H ≡ H (~r, t) with restricted p-norm (p ∈ N) i.e. H obeys the
inequation
ˆ
kHkp :=
is defined as
F (H) ≡ H ~k, ω =
p
d3 rdt kH (~r, t)k
1
χr χt
ˆ
1/p
<∞
n o
d3 rdtH (~r, t) exp −i ~k~r − ωt
(4.6)
(4.7)
R4
and its inverse transformation F −1 (H) reads as follows:
F
−1
1
(F (H)) = H (~r, t) =
ψk ψω
ˆ
n o
d3 kdωH ~k, ω exp i ~k~r − ωt .
(4.8)
R4
p
HerebykH (~r, t)k is the absolute value or norm of H to the power of p. The coefficients χr , χt , ψk and ψω
are the weights of the forward and inverse transformations and will be denoted as Fourier factors or Fourier
coefficients from now on. They can be obtained by putting the equations (4.7) and (4.8) into one another which
19
leads to [105, 106]:
H (~r, t)
ˆ
=
1
ψk ψω
=
1
χr χt ψk ψω
R4
4
(2π)
χr χt ψk ψω

n o
n o
d3 rdtH (~r, t) exp −i ~k~r − ωt  exp i ~k~r − ωt
1
χr χt
R4
ˆ
ˆ
n
o
d3 kdω d3 r0 dt0 H (~r0 , t0 ) exp i~k (~r − ~r0 ) + iω (t0 − t)
d3 kdω 
R4
=
ˆ

R4
ˆ
d3 r0 dt0 H (~r0 , t0 ) δ (~r − ~r0 ) δ (t − t0 )
R4
4
≡
(2π)
H (~r, t)
χr χt ψk ψω
!
= H (~r, t) .
(4.9)
Therefore the coefficients obey the following relation:
4
χr χt ψk ψω = (2π) .
(4.10)
At the third step of (4.9) the property
ˆ
d3 r0 dt0 H (~r0 , t0 ) δ (~r − ~r0 ) δ (t − t0 ) = H (~r, t)
(4.11)
R4
was used. Now, all these factors can be chosen freely such that equation (4.10) is fulfilled. If the weights
concerning the ~r and t or ~k and ω transformations should be homogeneous all factors can be chosen equally
i.e. 2π which makes sense. Nevertheless, in the calculations below (see chapter 4.3) they will be left arbitrary
because they play a crucial role while treating fields and sources which are transformed only with respect to
the time coordinate.
A decisive reason for using the Fourier transformation method as it is defined like in equation (4.7) is that
derivation operators in spacetime become factors in Fourier space i.e. the following transitions hold:
∇
∂
∂t
→
i~k,
(4.12)
→
−iω.
(4.13)
This can be proven easily by taking the transform in equation (4.7) using ∂H/∂t or ∇H instead of H and
applying the method of integration by parts. For ∂H/∂t the result is:
ˆ
dt
∂H
∂t
∞
exp {iωt} = [H (t) exp {iωt}]−∞ −iω
|
{z
}
≡0
20
ˆ
dtH (~r, t) exp {iωt} = −iωH (ω) .
(4.14)
Due to the fact that H vanishes for large t the first term in (4.14) vanishes. If the prefactor in the exponential
is −iωt the partial derivative∂/∂t changes over to iω. Thus, the transition is a matter of convention how the
sign in the exponential is defined. Analogously this can be shown for ∇H.
Some Useful Formulae
The following equations can be called master formula for Feynman integrals and are needed in the flux
calculations. They are is given as [85]:
ˆ
d
Eν,µ
=
d
Mν,µ
=
d
d
2 ν
d
kE
d
2 ν−µ+ 2 Γ ν + 2 Γ µ − ν − 2
2
=
π
m
,
2 + m2 )µ
(kE
Γ d2 Γ (µ)
ν
ˆ
ν−µ+ d2 Γ ν + d2 Γ µ − ν − d2
k2
d
ν−µ
m2
,
dd kM 2 M 2 µ = iπ 2 (−1)
Γ (d/2) Γ (µ)
(kM − m )
dd kE
(4.15)
(4.16)
Thereby ν, µ, m and n are generally complex variables, kE a d-dimensional vector in euclidean space and kM a
d-dimensional vector in Minkowski space. The last formula is very important in the context of renormalizability
and renormalized theories [89].
4.2
4.2.1
Electrodynamics in Matter
Fundamentals and Definitions
Microscopic and Macroscopic Field Equations
Electrodynamics in matter can be well described by Maxwell’s equations of motion. They are coupled differ~ = εE
~ and B
~ = µH
~ and are given by [26]:
ential equations of the electromagnetic fields D
~ ≡ ∇D
~ =ρ
divD
~
~ ≡∇×E
~ = − ∂B
rotE
∂t
~ ≡ ∇B
~ = 0,
divB
(4.17)
~
~ ≡∇×H
~ = ~j + ∂ D .
rotH
∂t
(4.18)
~ ≡D
~ (~r, t) and magnetic field B
~ ≡B
~ (~r, t) are spacetime depending vector functions
Hereby, the dielectric field D
and ∇ is the vectorial derivative or nabla operator. The quantities ρ ≡ ρ (~r, t) and ~j ≡ ~j (~r, t) are so-called
sources. Thereby, ρ is the charge density and ~j the current density of the moving charges. The equations (4.17)
and (4.18) convey that the sources of electric fields are charges, magnetic monopoles do not exist and that
temporal variable electric fields induce magnetic fields (and vice versa). Regarding this, the electromagnetic
fields hereof spread in the matter where they are located. The aspect of matter will be described below. It is
possible two divide the sources into three parts, namely the undisturbed, external and induced part such that:
ρ =
~j
ρ0 + ρext + ρind ,
= ~j0 + ~jext + ~jind .
(4.19)
(4.20)
An undisturbed source is a source without external influences and is marked by index 0. An external source
(index ext) causes electromagnetic fields and an induced one (index ind) is caused directly by electromagnetic
21
fields. In this thesis ρ0 , ρind , ~j0 , ~jind are set to zero due to the fact that undisturbed sources are essentially not
present and that the effect by induced sources is negligibly small compared to the external ones. Equations
(4.17) and (4.18) do not contain directly the field response functions or field constants ε and µ. These functions
~ and H
~ as proportionality factors and occur as scalars (generally complex
are coupled directly with the fields E
scalars). Ordinarily, the response functions are not scalars but tensors which can be simplified to scalars if and
only if the considered medium is homogeneous and isotropic. That means the phase of matter is considered
to be more or less ideally gaseous, liquid or solid with cubic symmetry. As a matter of fact the response
functions are the only carriers of the matter (medium) information and therefore really important quantities
[93, 94, 68, 75]. If there is not a medium i.e. electrodynamics in vacuum is taken into account it comes out
that ε = ε0 , µ = µ0 and εµ = ε0 µ0 = 1/c20 . The latter quantity gives the inverse squared of the velocity of
light in vacuum. Assuming that ε and µ are approximately constant they can be represented to leading order
as ε = εr ε0 and µ = µr µ0 due to dimensional reasons. Concerning this the velocity of light cn in the medium
has then to be modified into:
εµ = εr ε0 µr µ0 =
whereby n =
√
1
c2n
⇔ c2n =
n2
,
c20
(4.21)
εr µr is the index of refraction. For a homogeneous and isotropic medium the response functions
ε and µ are approximately complex functions of the frequency i.e. ε ≡ ε (ω). Nevertheless, they vary only
smoothly with the frequency and can be seen as constants to leading order in ω. For ω → 0 they reduce to the
electric and magnetic field constants. As mentioned before ε and µ describe the medium. Therefore they can
be seen as macroscopic quantities. A general problem in electrodynamics is the question whether considering
microscopic or macroscopic field equations. These two scopes have fundamental differences but for both of them
the shape of the Maxwell equations stay the same, fortunately. As the names suggest the difference is a question
of the length scale. Let S ∈ [0, 1] be a dimensionless scale factor, ∆L the minimal variation length (the length
where the variation of the fields is approximately constant) and l the lattice constant. Then
S =1−
∆L
l
(4.22)
can be seen as a measure for the variation of the fields. For 0 < S . 1 the microscopic fields and for S ' 0 the
macroscopic fields dominate. In the latter case the minimal variation length approaches the lattice constant
which means that the fields do not vary within a lattice cell. The variation of the fields is not only the difference
between microscopic and macroscopic fields. The macroscopic ones are obtained by averaging the microscopic
fields with respect to space. The averaging procedure has to be done using a convolution integral. Let E be a
microscopic quantity (H just represents the electromagnetic fields or the sources), δ (~r − ~r0 ) a Dirac sequence
which is by definition located for ~r − ~r0 = ~0. Then
ˆ
hHi (~r, t) =
d3 r0 H (~r0 , t) δ (~r − ~r0 )
(4.23)
defines a macroscopic field [95]. Thereby, l ε λ was considered whereby λ is the wavelength of the light and
in the order of a lattice cell. For → 0 δ (~r − ~r0 ) → δ (~r − ~r0 ) and equation(4.23) becomes hHi (~r, t) ≈ H (~r, t).
22
Now this means the macroscopic fields are related to the microscopic ones. Due to the fact that λ ∼ O (0.1µm)
and therefore the Delta approximation is acceptable if the arguments of E do not influence the limiting process
→ 0. So, for the further calculations the brackets h i will be dropped off.
Gauge Conditions, Fields, Sources and Fluxes
~ and B.
~ Nevertheless, in the most
Maxwell’s equations can be completely uncoupled to wave equations for E
~ which obey e.g. the Coulomb gauge
cases it is practically to introduce electrodynamic potentials Φ and A
~ = 0 or the Lorenz gauge condition
condition ∇A
~ + µε
∇A
∂Φ
= 0.
∂t
(4.24)
The latter equation (4.24) is used in this thesis. These potentials are directly coupled with the electromagnetic
fields according to
~ (~r, t)
E
= −∇Φ −
~ (~r, t)
B
~
= ∇ × A.
~
∂A
,
∂t
(4.25)
(4.26)
Using the Lorenz gauge and the electromagnetic fields (4.25) and (4.26) Maxwell’s equations can be uncoupled
and reduced to two inhomogeneous wave equations:
∂2
Φ
∂t2
2
~ ≡ ∇2 − µε ∂
~
DA
A
∂t2
DΦ ≡
∇2 − µε
ρ
= − ,
ε
(4.27)
= −µ~j.
(4.28)
Thereby, D ≡ ∇2 − µε∂ 2 /∂t2 is the d’Alembert operator and ∇2 is the Laplace operator. Thus, the electromagnetic fields can be obtained by solving these wave equations or just transform these equation to Fourier
space (see 4.1.2). To solve the equations (4.27) and (4.28) a description of the sources is needed. In microscopic
electrodynamics particle densities can be expressed through δ-functions. Assuming that j is the j-th particle’s
index, qj the particle’s charge, ~rj ≡ ~rj (t) the particle’s track function, ~vj ≡ ~vj (t) the particle’s velocity, tj1
the particle’s prestep time, tj2 the particle’s poststep time, ~rj1 ≡ ~rj (tj1 ) the particle’s prestep vector and
~rj2 ≡ ~rj2 (tj2 ) the particle’s poststep vector then the charge and current densities ρj and ~jj are given by [9]:
qj δ (~r − ~rj (t)) Θ (t − tj1 ) Θ (tj2 − t) Θ (~r − ~rj1 ) Θ (~rj2 − ~r) ,
ρj (~r, t)
=
~jj (~r, t)
= qj δ (~r − ~rj (t)) Θ (t − tj1 ) Θ (tj2 − t) Θ (~r − ~rj1 ) Θ (~rj2 − ~r) ≡ ~vj (t) ρj (~r, t) .
(4.29)
(4.30)
The descriptions for the sources are only valid if and only if the Coulomb force FC between two charged particles
is essentially smaller than a reference value Fref which can be set to 1 N . The reference force conforms with a
mass of 100 g experiencing the acceleration of free fall which represents a macroscopic phenomenon. Of course
23
this force must be essentially larger than the Coulomb force given by:
FC =
qQ
,
4πεd2qQ
(4.31)
whereby dqQ is the distance of the particles with charges q and Q. In the case FC Fref the distance of
the particles must be significantly larger than 10−14 m i.e. dqQ 10−14 m. This is the case for particles in
cascades treated in this work. The Coulomb force has to be negligibly small due to the fact that otherwise
the particle attraction and repulsion would play a significant role. These equations can be interpreted as a
particle located at ~rj (t) which moves by approximation rectilinear from ~rj1 = ~rj (tj1 ) to ~rj2 = ~rj2 (tj2 ) i.e. in
the interval ~rj1 ≤ rj (t) ≤ ~rj2 . In a particle cascade the points with index j1 and j2 would be the prestep and
poststep points. The total densities are calculated as a sum of all step points for every particle. The equations
(4.29) and (4.30) can be simplified. This fact is given by the expressions tj2 ≤ t ≤ tj2 and ~rj1 ≤ rj ≤ ~rj2 which
are equivalent to each other therefore it follows that the productΘ (~r − ~rj1 ) Θ (~rj2 − ~r) = 1. Thus, the total
densities yield:
ρ (~r, t)
=
N
X
qj δ (~r − ~rj (t)) Θ (t − tj1 ) Θ (tj2 − t) ,
(4.32)
qj ~vj δ (~r − ~rj (t)) Θ (t − tj1 ) Θ (tj2 − t) ,
(4.33)
j=1
~j (~r, t)
=
N
X
j=1
whereby N is the total number of particles. If a particle moves from −∞ to ∞ the product Θ (t − tj1 ) Θ (tj2 − t)
equals 1 and the densities depend on the δ-functions only. Therefore, this description can be directly transformed
to the model (only δ-functions) and can be seen as a generalization. Furthermore the sources obey the continuity
equation which is an alternative equivalent to the Lorenz gauge, It is given as follows:
∂ρ
∇~j +
= 0.
∂t
(4.34)
Nevertheless, the equations (4.24) and (4.34) are not directly used in the calculations because
Electrodynamic fields support energy. This energy flow can be described by the Poynting vector (named after
John Henry Poynting). It is defined as the cross product of the electric and magnetic field [12]:
~ (~r, t) = 1 E
~ ×B
~ ≡E
~ × H.
~
S
µ
(4.35)
The Poynting vector is a measure for the energy flow of the electromagnetic waves per unit area and time i.e.
the energy flux density. Due to the fact that the electromagnetic fields satisfy the superposition principle the
24
total fields are given by the linear sum of the partial fields, they can be written as:
~ total ≡ E
~
E
=
N
X
~j,
E
(4.36)
~j.
B
(4.37)
j=1
~ total ≡ B
~
B
=
N
X
j=1
Therefore the total Poynting vector yields:
~total ≡ S
~=
S
N
X
~j × H
~ k 6=
E
j,k=1
N
X
~l .
S
(4.38)
l=1
That means the total Poynting vector cannot be written as a sum of partial Poynting vectors. Thus, every field
interacts with each other. The Poynting vector can be used to calculate the energy flux W of the radiation. It
is given as the surface and time integral of the Poynting vector:
ˆ
W
=
ˆ
dt
ˆ
~S
~ (~r, t) =
dA
ˆ
~ (~r, t) ,
dV divS
dt
(4.39)
V
∂V
whereby the Gaussian integral theorem was used. Hereby,∂V is the smooth boundary of the Volume V (In
~ (~r, t) can be simplified applying a
IceCube the volume is parameterised by a cylinder). The divergence of S
vector operation:
~
~
div S=∇
S
=
=
=
=
1 ~
~
∇ E×B
µ
i
1 h~ ~ −E
~ ∇×B
~
B ∇×E
µ
"
!#
~
~
∂E
∂B
1
~
~
~
−B
− E µj + µε
µ
∂t
∂t
1
1 ∂ ~~
~
~
~
~
−
B B + µεE E − µj E
µ
2 ∂t
(4.40)
Thereby, Maxwell’s equations (4.18) for the rotation of the fields were used. Applying the divergence to the
Poynting vector has the advantage that it is independent of the geometry. Therefore the right hand side of
(4.40) can be seen as a coordinate free description. As a matter of fact the spatial dependence ~r of the fields can
thus presume an arbitrary geometrical system. It has to be taken into account that the considered volume V
must be seen as a variable volume. That means if the total volume is finite there are infinite subsets of volumes
as part of the total volume, formally: if MVe is the set of all volumes, v ∈ MVe there exists a SVe ⊆ MVe (a subset
of MVe ) with V ∈ SVe and V ≤ v for all volumes V, v. A variable volume now means that equation(4.39) has to
be find out for every V ∈ SVe and not only for exactly one V ∈ SVe . Therefore∂V represents the boundary of
every volume inside the total volume. Therefore, the flux through arbitrary surfaces is obtained by calculating
W.
25
4.2.2
Electrodynamics in Fourier Space
This chapter is focused on electrodynamics in Fourier space and can be seen as the Fourier description of the
previous chapter.
Application of the Fourier Transform to Electromagnetic Theory
Applying the Fourier transformation to the equations (4.24) (Lorenz gauge), (4.34) (continuity equation),
(4.25) (electric field), (4.26) (magnetic field), (4.27) (wave equation for the scalar field), (4.28) (wave equation
for the vector field), (4.32) (total charge density), (4.33) (total current density) and using the transitions (4.12)
and (4.13) yields:
~ ~k, ω − iµεωΦ ~k, ω = 0,
i~k A
i~k~j ~k, ω − iωρ ~k, ω = 0,
n o
´t
PN
ρ ~k, ω = χr1χt j=1 qj tj1j2 dt exp −i ~k~rj (t) − ωt ,
n o
´ tj2
PN
~k~rj (t) − ωt ,
~j ~k, ω = 1
dt~
v
(t)
exp
−i
q
j
j
j=1
χr χt
tj1
ρ(~
k,ω )
Φ ~k, ω = 1ε k2 −µεω2 ,
~ ~
~ ~k, ω = µ j2 (k,ω) 2 ,
A
k −µεω
~ ~k, ω = −i~kΦ ~k, ω + iω A
~ ~k, ω ,
E
~ ~k, ω = i~k × A
~ ~k, ω .
B
(4.41)
(4.42)
(4.43)
(4.44)
(4.45)
(4.46)
(4.47)
(4.48)
The Fourier transformed gauge condition and the continuity equation lead to a side condition equation which
can be derived as follows:
0 = i~k~j ~k, ω − iωρ ~k, ω ⇒ ~k~vj (t) = ω.
(4.49)
This equation holds only for all tj1 ≤ t ≤ tj2 and gives an important relation between the frequency and the
particle’s velocity. Due to the fact that an along step crossed by a charged particle within the time interval
tj1 ≤ t ≤ tj2 obeys the charge conservation law or the continuity equation the side condition(4.49)
is fulfilled.
~
Nevertheless, it is a side condition i.e. it is a boundary term giving information about which k, ω -states have
to be taken into account.
4.3
Particle Sources
A particle which is indexed by j is moving uniformly with velocity ~vj1 ≡ ~vj and within the time interval
tj1 ≤ t ≤ tj2 if it obeys the following spacetime law:
~rj (t) = ~vj1 (t − tj1 ) + ~rj1 ≡ ~vj (t − tj1 ) + ~rj1 .
26
(4.50)
That means for t = tj1 the particle is located at ~rj (tj1 ) ≡ ~rj1 and for t = tj2 at ~rj (tj2 ) ≡ ~rj2 . Moreover it has
within tj1 to tj2 the constant velocity ~vj or an averaged velocity ~v̄j . Now, as it was derived the charge density
is given by (4.43). For the particle density of the j-th particle it follows for t̄j = (tj1 + tj2 ) /2 ≡ t̄j that:
ρ ~k, ω =
=
=
=
⇒ ρ ~k, ω =
⇒ ~j ~k, ω =
´ tj2
n o
dt exp −i ~k~rj (t) − ωt
n
o
n
o ´
tj2
~k~vj − ω
dt
exp
−i
t
−
t̄
qj exp i~k~vj tj1 − i~k~rj1 − it̄j ~k~vj − ω
j
tj1
n
o exp −i∆t ~k~v −ω /2 −exp i∆t ~k~v −ω /2 {
) }
{ j( j ) }
j(
j
qj exp i~k~vj tj1 − i~k~rj1 − it̄j ~k~vj − ω
−i(~
k~
vj −ω )
o sin ∆t ~k~v −ω /2 n
{ j( j ) }
qj exp −i~k~r t̄j + iω t̄j
(~k~vj −ω)/2
n
o
PN
sin{∆t(~
k~
v −ω )/2}
1
~k~rj t̄j − ω t̄j
q
exp
−i
,
(4.51)
j
j=1
χr χt
(~k~v−ω)/2
o
n
PN
sin{∆t(~
k~
v −ω )/2}
1
.
(4.52)
vj
exp −i ~k~rj t̄j − ω t̄j
j=1 qj ~
~
χr χt
k~
v
−ω
/2
(
)
qj
tj1
The realistic modeling of the current (or charge) density can be simply converted to a classical δ-function
approach using ω∆t 1:
n o
sin ∆t ~k~v − ω /2
→ 2πδ ~k~v − ω .
~k~v − ω /2
(4.53)
Thus, the classical δ-function technique can be verified immediately.
4.4
Retarded Potentials and Fields of a Moving Point Charge
The solutions of the differential equaitons (4.27) and (4.28) are so-called retarded or advanced potentials. They
can be written as a space integral over the sources:
Φ (~r, t)
=
~ (~r, t)
A
=
√
ˆ
r0 , t ∓ µε |~r − ~r0 |
1
3 0ρ ~
d r
,
4πε
|~r − ~r0 |
ˆ
~ r0 , t ∓ √µε |~r − ~r0 |
µ
3 0j ~
d r
.
4π
|~r − ~r0 |
(4.54)
(4.55)
Thereby “−” stands for the retarded and “+” for the advanced solutions. The retarded solutions describe the
delay of the radiation emitted at ~r0 and received at a distance |~r − ~r0 | (i.e. at position ~r) whereas the advanced
solutions are not physical solutions due to the fact that the radiation emitted e.g. at t = 0 would be received
at an earlier time t < 0 which is not possible because of causality. Therefore, only the retarded potentials are
considered in the following steps. The delay can be defined as
δt :=
√
µε |~r − ~r0 | ≡
|~r − ~r0 |
cn
(4.56)
and represents the time which is needed by light in a medium moving a distance of |~r − ~r0 |. Using the one
particle sources (4.29) and (4.30) for ~r = ~x and defining σ = ± the potentials can be calculated with respect to
27
the (~x, t)- and (~x, ω)-space as follows:
Φ (~r, t)
=
=
=
=
ˆ
1
4πε
ˆ
1
4πε
ˆ
1
4πε
ˆ
1
4πε
√
~r0 , t − σ µε |~r − ~r0 |
d r
|~r − ~r0 |
0
ρ (~r , τ )
√
d3 r0 dτ
δ (τ − t + σ µε |~r − ~r0 |)
0
|~r − ~r |
δ
(~r0 − ~r (τ )) Θ (τ − t1 ) Θ (t2 − τ )
√
d3 r0 dτ
δ (τ − t + σ µε |~r − ~r0 |)
0
|~r − ~r |
Θ (τ − t1 ) Θ (t2 − τ )
√
dτ
δ (τ − t + σ µε |~r − ~r (τ )|) .
|~r − ~r (τ )|
3 0ρ
√
√
Now solving the equation τ − t + σ µε |~r − ~r (τ )| = 0 for cn ≡ 1/ µε, ~r (τ ) = ~v (τ − τ1 ) + ~r1 , ~s = ~r − ~r1 + ~v τ1
and σ 2 = 1 leads to:
⇒ τ2
q
2
cn (t − τ ) = σ (~s − ~v τ )
⇒ c2n τ 2 − 2tτ + t2
= s2 − 2~s~v τ + v 2 τ 2
v 2 − c2n − 2τ ~s~v − c2n t + s2 − c2n t2 = 0
~s~v − c2n t
s2 − c2n t2
2
+
= 0
⇒ τ − 2τ 2
v − c2n
v 2 − c2n
~s~v − c2n t
⇒ τ± =
v 2 − c2n
q
1
2
(~s~v − c2n t) − (v 2 − c2n ) (s2 − c2n t2 ) (4.57)
± 2
|v − c2n |
28
This equation implies that the radicand must be positive which leads to an additional side condition. It follows
that:
2
~s~v − c2n t
≥
⇔ (~s~v ) − 2 (~s~v ) c2n t + c4n t2
≥
2
v 2 − c2n
t−
⇔
t−
⇔
t−
⇔
t−
⇔
t−
v 2 s2 − c2n s2 − (~s~v )
2
2
~s~v
t
v2
2
~s~v
v2
2
~s~v
v2
2
~s~v
v2
2
~s~v
v2
2
~s~v
v2
⇔ t2 − 2
⇔
s2 − c2n t2
v 2 s2 − v 2 c2n t2 − c2n s2 + c4n t2
⇔ v 2 c2n t2 − 2 (~s~v ) c2n t ≥
v 2 − c2n 2 (~s~v )
s − 2 2
v 2 c2n
v cn
≥
2
2
v 2 − c2n 2 (~s~v )
(~s~v )
s − 2 2 + 4
2
2
v cn
v cn
v
2 2
2
v − cn 2 (~s~v )
1
1
s
+
−
v 2 c2n
v2
v2
c2n
2
v 2 − c2n 2 (~s~v ) v 2 − c2n
s − 2
v 2 c2n
v
v 2 c2n
i
v 2 − c2n h 2 2
2
s v − (~s~v )
4
2
v cn
≥
≥
≥
≥
v 2 − c2n
2
|~s × ~v | .
v 4 c2n
≥
(4.58)
This condition has to be considered additionally. In total the scalar potentials reads as:
Φ (~r, t) =
1 √
σ µεΘ
4πε
t−
~s~v
v2
2
−
v − c2n
v 4 c2n
!
2
|~s × ~v |
2

 Θ (τ+ − t1 ) Θ (t2 − τ+ ) + Θ (τ− − t1 ) Θ (t2 − τ− )  .
r (τ+ ))~
v
(~
r −~
r (τ− ))~
v
t − τ+ 1 + µε (~r−~
t − τ− 1 + µε t−τ−
t−τ+
(4.59)
The scalar potential shows that it is relatively more difficult to use this derivation way.
Φ (~x, ω)
=
=
=
=
1
χt
ˆ
dtd3 rδ (~r − ~x) Φ (~r, t) exp (iωt)
ˆ
1
ρ (~r0 , t0 )
√
d3 r0 dt0 dt
δ (t0 − t + µε |~x − ~r0 |) exp (iωt)
4πεχt
|~x − ~r0 |
ˆ
q
δ (~r0 − ~r (t0 )) Θ (t0 − t1 ) Θ (t2 − t0 )
√
d3 r0 dt0
exp (iωt0 + iω µε |~x − ~r0 |)
0
4πεχt
|~x − ~r |
ˆt2
q
1
√
dt
exp (iωt + iω µε |~x − ~r (t)|) ,
4πεχt
|~x − ~r (t)|
(4.60)
t1
~ (~x, ω)
⇒A
=
µq
4πχt
ˆt2
dt
~v
√
exp (iωt + iω µε |~x − ~r (t)|) .
|~x − ~r (t)|
t1
29
(4.61)
For the electromagnetic fields in semi Fourier space it follows that:
~
~ (~x, t) = −∇Φ (~x, t) − ∂ A (~x, t) → E
~ (~x, ω) = −∇Φ (~x, ω) + iω A
~ (~x, ω) ,
E
∂t
~ (~x, t) = ∇ × A
~ (~x, t) → B
~ (~x, ω) = ∇ × A
~ (~x, ω) .
B
(4.62)
(4.63)
Because of
~ (~x, t)
R
1~
~x − ~r (t)
=−
≡− R
= er = −eR ,
|~x − ~r (t)|
R (~x, t)
R
1
1
~x − ~r (t)
1 ~
= −
3 = R 3 R = − R 2 er = R 2 eR ,
|~x − ~r (t)|
√
−iω µεeR
eR
√
=
+ 2 exp (iωt + iω µεR) ,
R
R
√
iω µεeR
eR
√
=
+ 2 exp (iωt − iω µεR) ,
R
R
∇ |~x − ~r (t)| =
1
|~x − ~r (t)|
√
exp iωt + iω µε |~x − ~r (t)|
∇
|~x − ~r (t)|
√
exp iωt − iω µε |~x − ~r (t)|
∇
|~x − ~r (t)|
∇
(4.64)
(4.65)
(4.66)
(4.67)
~ = ~r (t) − ~x, it follows that:
whereby eR is the unit vector of R
~ (~x, ω)
E
q
= −
4πεχt
ˆt2
√
−iω µεeR
eR
√
+ 2 exp {iωt + iωσ µεR}
dt
R
R
t1
iωµεq
+
4πεχt
ˆt2
dt
~v
√
exp {iωt + iωσ µεR}
R
t1
=
q
4πεχt
ˆt2
dt
√
iωµε~v + iω µεeR
eR
√
− 2 exp {iωt + iωσ µεR} ,
R
R
(4.68)
√
iωµε~v + iωσ µεeR
eR
√
− 2 exp {iωt + iωσ µεR} ,
R
R
(4.69)
t1
~ σ (~x, ω)
⇒E
=
q
4πεχt
ˆt2
dt
t1
~ (~x, ω)
B
=
µq
4πχt
ˆt2
dteR × ~v
√ iω µε
1
√
−
exp {iωt + iωσ µεR} ,
R2
R
(4.70)
√ iωσ µε
1
√
−
exp {iωt + iωσ µεR} .
R2
R
(4.71)
t1
~ σ (~x, ω)
⇒B
=
µq
4πχt
ˆt2
dteR × ~v
t1
Thereby σ = ± i.e. “−” stands for the retarded and “+” for the advanced solutions. For particles which move
linearly in time and obey the Cherenkov condition this can be underlined by looking at timely constant phases
30
i.e.:
√
iωt + iωσ µεR
=
⇒1
=
const.
√
−σ µεṘ
√
−σ µεeR~v
√
−σ µεv cos θ
| {z }
=
=
≥0
⇒ cos θ
=
⇒σ
=
σ
−√
µεv
−1.
(4.72)
A positive σ would mean that the radiation is emitted backwards which is only possible in meta materials with
complex or negative index of refraction. It is directly recognizable that the fields become complex conjugated
for ω → −ω.
4.5
4.5.1
Electromagnetic Flux
The Connection between Energy Flux and Total Number of Photons
The energy flux dW/dω can be connected with the total number of photons emitted by a single track. Assuming
that Nγ corresponds to the total number of emitted photons and nγ (ω, L) the photon density per unit path
length L of a charged particle and unit frequency of light, the total number photons can be thus written as:
ω=ω
ˆ f
Nγ :=
L=L
ˆ 1
dω
ω=ωi
dL nγ (ω, L) ,
(4.73)
L=L0
whereby ωi,f are the considered initial and final frequencies and ∆L = L1 − L0 the path length of the particle.
To obtain the energy W of the radiation the photon density must be scaled by the photon energy which is ~ω.
This leads to the following relation:
ω=ω
ˆ f
W =
L=L
ˆ 1
dω
ω=ωi
dL ~ωnγ (ω, L) .
(4.74)
L=L0
Both equations can now be connected to each other by using the differential descriptions:
d2 Nγ
2
d W
= nγ (ω, L) dωdL,
= ~ωnγ (ω, L) dωdL,
2
2
⇒
d W
dωdL
(4.75)
= ~ω
d Nγ
d Nγ
≡ Eγ
.
dωdL
dωdL
31
(4.76)
2
(4.77)
Thus, the Cherenkov light yield i.e. the total number of photons per unit step length and unit frequency can
be described as:
d2 Nγ
1 d2 W
1 dNW
=
≡
.
dωdL
Eγ dωdL
ω dL
(4.78)
This relation shows that the radiated energy only has to be normalized with respect to the photon energy ~ω
to get a description of the emitted number of photons. Using dω = −2πc0 /λ2 the number of photons per unit
frequency and unit path length of the particle can be rewritten as:
d2 Nγ
1 d2 W 2πc0
.
=
dλdL
Eγ dωdL λ2
4.5.2
(4.79)
The Master Integral
In this chapter a solution for a special integral will be given. The solution of this integral can be seen as the basis
of the solutions shown in the further chapters and also as a basis for a several number of integrals. Therefore
it makes sense to call this chapter “Master Integral” because one solutions leads to a series of solutions.
Let H be an operator with D ⊆ C defined as:
~ := He−i~kR~ ∧ ∂H = 0, l = (x, y, z) .
H : D → D, H 7→ H R
∂kl
(4.80)
Then the following integral equation holds:
ˆ
I=
1
~~
d k 2
He−ikRj
2
k −m
3
ˆ∞
= −iπ
ˆ1
dk
0
π
= −i mH
2
ˆ2π
d cos θ
−1
0
ˆ1
ˆ2π
d cos θ
−1
32
~~
dϕk 2 δ k 2 − m2 He−ikR
~
−imb
kR
dϕe
0
,
~k
~k
b
k = ~ek = =
k
m
!
.
(4.81)
First of all the exponential should be expanded as superposition of plane waves according to the plane wave
~ = kR cos θ and P0 (cos θ) = 1. Then the integral reads as [2, 1]:
expansion technique with ~k R
´1
´ 2π
k2
dk −1 d cos θ 0 dϕ k2 −m
2 jn (kR) Pn (cos θ) P0 (cos θ)
´∞
P∞ n
2
k2
= 2π n=0 i (2n + 1) 2n+1 δn0 0 dk k2 −m
2 j0 (kR)
´
P∞ n
2
∞
sin{kR}
2
k
= 2π n=0 i (2n + 1) 2n+1
δn0 21 −∞ dk k2 −m
2
kR
l+1
P∞ (−1)l R2l ´ ∞
(k 2 )
dk k2 −m2
l=0 (2l+1)!
−∞
ILHS =
= 2π
´∞
0
P∞
n=0
2
in (2n + 1) 2n+1
δn0 12
(−1)l R2l
l=0 (2l+1)!
P∞
´∞
−∞
l+1
(k 2 )
dk k2 −m2
l 2l
P∞
R
2
1
δn0 21 l=0 (−1)
in (2n + 1) 2n+1
(2l+1)! Ml+1,1
1
1
l 2l
P∞
P∞ n
l
R
2
2
2 l+ 2
= Γ 2πiπ
δn0 21 l=0 (−1)
i (2n + 1) 2n+1
Γ l + 1 + 12 Γ −l − 12
1
(2l+1)! (−1) m
( 2 )Γ(1) n=0
2l 2l
P∞
P∞
R
1
2
2l+1
δn0 l=0 (−1)
l + 12 Γ l + 12 Γ 1 − l + 12 − l+
= πi n=0 in (2n + 1) 2n+1
(2l+1)! m
( 12 )
2l 2l
P∞
P∞
R
2
2l+1
= −iπ n=0 in (2n + 1) 2n+1
δn0 l=0 (−1)
Γ 12 + l Γ 12 − l
(2l+1)! m
2l 2l
P∞
P∞
√ (−1)l 2l √
R
2
2l+1 (2l−1)!!
δn0 l=0 (−1)
π (2l−1)!! π
= −iπ n=0 in (2n + 1) 2n+1
(2l+1)! m
2l
l 2l
P∞
P∞
R
2
2l
= −iπ 2 m n=0 in (2n + 1) 2n+1
δn0 l=0 (−1)
(2l+1)! m
P
∞
2
δn0 j0 (mR)
= −iπ 2 m n=0 in (2n + 1) 2n+1
´
P
1
∞
= −iπ 2 m −1 d cos θ n=0 in (2n + 1) jn (mR) Pn (cos θ)
´ 2π
´1
~
−im~
ek R
.
(translate axes) ⇒ ILHS = −iπ m
2 0 dϕ −1 d cos θe
= 2π
P∞
n=0
(4.82)
Now it was proven that (4.81) is valid.
4.5.3
The Master Formula
The electromagnetic flux is characterized by the Poynting vector and is given by the Fourier description of
(4.39). Assuming that e.g. f~ (~r, t) ∈ R is a real time dependent function it can be expressed in the Fourier
representation as [106]
f~ (~r, t)
=
Re
=
1
2γ
1
γ
ˆ
ˆ
dω f~ (~r, ω) e−iωt
h
i
dω f~ (~r, ω) e−iωt + f~cc (~r, ω) eiωt .
(4.83)
Thereby the index cc indicates the complex conjugation of the function and γ is a prefactor which will be defined
below. Now this equation can be used to calculate the total radiated energy W which is the Poynting flux of
the real electromagnetic fields. It states that [9, 80]:
33
´ ´
~S
~ (~x, t) = dt d3 x∇S
~ (~x, t)
dA
h
i
´
~ (~x, t) × Re B
~ (~x, t)
⇔ W = µ d3 x dt∇ Re E
h
i h
i
´ 3 ´ ´
´
1
~ (ω) e−iωt + E
~ cc (ω) eiωt × B
~ (Ω) e−iΩt + B
~ cc (Ω) eiΩt
= (2γ)
d
x
dt
dω
dΩ∇
E
2
µ
h
´ 3 ´ ´
´
1
~ (ω) × B
~ (Ω) e−it(ω+Ω) + E
~ (ω) × B
~ cc (Ω) eit(Ω−ω)
= (2γ)
d
x
dt
dω
dΩ∇
E
2
µ
i
~ cc (ω) × B
~ (Ω) eit(ω−Ω) + E
~ cc (ω) × B
~ cc (Ω) eit(ω+Ω)
+E
h
´ 3 ´
´
2π
~ (ω) × B
~ (Ω) δ (ω + Ω) + E
~ (ω) × B
~ cc (Ω) δ (ω − Ω)
= (2γ)
d
x
dω
dΩ∇
E
2
µ
i
~ cc (ω) × B
~ (Ω) δ (ω − Ω) + E
~ cc (ω) × B
~ cc (Ω) δ (ω + Ω)
E
h
´ 3 ´
2π
~ (ω) × B
~ (−ω) + E
~ (ω) × B
~ cc (ω)
= (2γ)
d x dω∇ E
2
µ
i
~ cc (ω) × B
~ (ω) + E
~ cc (ω) × B
~ cc (−ω)
E
W =
´
dt
´
1
´
~ (−ω) = B
~ cc (ω) .
Use B
h
i
2π
~ (ω) × B
~ cc (ω) + 2E
~ cc (ω) × B
~ (ω)
⇒= (2γ)
d3 x dω∇ 2E
2
µ
h
i
´ 3 ´
4π
~ (ω) × B
~ cc (ω) + E
~ cc (ω) × B
~ (ω)
= (2γ)
d
x
dω∇
E
2
µ
h
´ 3 ´
4π
~ cc ∇ × E
~ −E
~ ∇×B
~ cc + B
~ ∇×E
~ cc − E
~ cc ∇ × B
~
= (2γ)2 µ d x dω B
h
´ 3 ´
4π
~ cc iω B
~ −E
~ µ~jcc + iωεµE
~ cc
B
d
x
dω
= (2γ)
2
µ
i
~ (−iω) B
~ cc − E
~ cc µ~j − iωµεE
~
+B
h
i
´ 3 ´
4π
~ (ω) ~jcc (ω) − µE
~ cc (ω) ~j (ω)
= (2γ)
d
x
dω
−µ
E
2
µ
´ 3 ´
8π
~ (~x, ω) · ~j (~x, −ω)
= − (2γ)
d x dω E
2 Re
h´
i
3 ~
~j (~x, −ω)
⇒ dW
=
−2πRe
d
x
E
(~
x
,
ω)
·
dω
V
´
´
(4.84)
Thereby the permeability functions µ and ε are assumed to be real and γ is chosen to be unity. Equation
(4.84) can also evaluated for complex permeability functions which leads to additional terms for dW/dω. In the
third line of (4.84) the Fourier description (4.83) was applied and in the 13th line the Maxwell equations for the
rotation of the fields were used. As it is realizable the divergence of the Poynting vector can be represented using
the fields and the sources. Therefore an additional calculation is not needed due to the fact that the sources
and the fields have been already calculated. In [9] equation (4.84) is also called Schwinger’s approach. An
interesting and decisive point in (4.84) is that the parts which contain the quadratic field dependencies vanish.
~ and linearly on ~j which makes it possible to separate fields from
The resulting term depends only linearly on E
other sources like e.g. the influence of bremsstrahlung or the overlapping of other phenomena. Equation (4.84)
can be represented more clear because ~j (~x, −ω) is known according to (??) and contains δ-functions. Now using
34
the master integral (4.81) the resulting energy flow can be written as:
´
= − 2π
χt Re
h´
dW
3 ~
x, ω) · ~j (~x, −ω)
dω = −2π Re V d xE (~
´
P
∞
N
d3 r −∞ dt exp {−iωt} j=1 qj ~vj δ (~r − ~rj (t)) Θ (t
− tj1 ) Θ (tj2 − t)
n
o
´
iωµε~j (~
k,ω )−i~
kρ(~
k,ω )
exp i~k~r
⊗ ψ1k ε d3 k
k2 −µεω 2
h´ P
N
= − 2π
dt j=1 qj ~vj Θ (t − tj1 ) Θ (tj2 − t)
χt Re
n
o
´ 3 iωµε~j (~k,ω)−i~kρ(~k,ω)
1
~
⊗ ψk ε d k
exp ik~rj (t) − iωt Θ (~rj (t) − ~x1 ) Θ (~x2 − ~rj (t))
k2 −µεω 2
⇒ Solution for arbitrary ~x1 , ~x2 can be used to reconstruct local fluxes
⇒ Choose: ~x1 ≤ ~rj (t) ≤ ~x2 ∀t ∈ [tj1 , tj2 ]and ∀j ∈ {1, .., N }⇒ Θ (~rj (t) − ~x1 ) Θ (~x2 − ~rj (t)) ≡ 1.
⇔ Every Particle is included. Focus of interest is the total flux.
h´ P
N
Re
dt j=1 qj ~vj Θ (t − tj1 ) Θ (tj2 − t)
⇒= − 2π
χt
n
o
´
iωµε~j (~
k,ω )−i~
kρ(~
k,ω )
~
⊗ ψ1k ε d3 k
exp
i
k~
r
(t)
−
iωt
j
k2 −µεω 2
i
n
h
o
´ 3 PN
ωµε~
vl ~
vj −~
k~
vj
~k, ω exp −i~k ~rl t̄l − ~rj t̄j + iω t̄l − t̄j
= Re χt ψ−2πi
d
k
C
q
q
2
2
jl
j
l
l=1
k −µεω
k χr χt ε
n
o
´ 3 PN
−2π 2 ψω
→= Re (2π)4 χ ε d k l=1 qj ql ωµε~vl~vj − ~k~vj δ k 2 − m2 Cjl ~k, ω exp −i~k ~rl t̄l − ~rj t̄j + iω t̄l − t̄j
t
´
PN
−2π 2 ψω m
= Re (2π)4 2χ ε dΩ l=1 qj ql (ωµε~vl~vj − m~ek~vj ) Cjl (m~ek , ω) exp −im~ek ~rl t̄l − ~rj t̄j + iω t̄l − t̄j
t
~ β
~
β
Define: d~lj := ~rl t̄l − ~rj t̄j , tlj := t̄l − t̄j , cos {αlj } := βll βjj
n
o
´
PN
−πg 0 m
~lj − ωtlj ,
⇒ dW
=
dΩ
q
q
(ωµε~
v
~
v
−
m~
e
~
v
)
C
(m~
e
,
ω)
cos
m~
e
d
(4.85)
4
j
l
l
j
k
j
jl
k
k
l=1
dω
(2π) ε
o
n
´
PN
~nj
~
ek β
dN
n
ω
dΩ j,l=1 zj zl βj βl cos {αlj } − βnj
ek d~lj − ωtlj clj (−) .
(4.86)
⇒ dωγ = g 0 απem
2ω
βnl cos cn ~
Thereby ~x1 and ~x2 are the spatial boundaries. ~x1 represents the reference or the initial vector of starting to
measure the total radiation and ~x2 represents the observation vector where the measure and addition process of
the radiation is finished. The gauge factor g 0 is defined as −2πψω /2χt . Due to the fact that the total radiation is
of importance for the calculation of the light yield all the emitted photons have to be considered. The equations
(4.84)
and
(4.85) can be denoted as master formulae for the calculation of Cherenkov light yields. The functions
Cjl ~k, ω are pair correlation functions which give information about how the particles with index j and l
influence each other. They are defined as:
Cjl ~k, ω
:=
n
o
n
o
sin ∆tj ~k~vj − ω /2 sin ∆tl ~k~vl − ω /2
n
o
n
o
.
~k~vj − ω /2
~k~vl − ω /2
35
(4.87)
√
Using the extremal boundary condition k = ω µε they can be written as:
o
√
~ek µε~vj − 1 sin ω∆tl ~ek √µε~vl − 1
2
Clj (−) := Cjl (m~ek , ω, ±) :=
≡ Cj Cl ,
√
√
ω ~ek µε~vj − 1 /2
ω ~ek µε~vl − 1 /2
n
o
√
ω∆tj
sin
~ek µε~vj − 1 sin ω∆tl ~ek √µε~vl − 1
2
2
clj (−) :=
≡ cj cl .
√
√
~ek µε~vj − 1
~ek µε~vl − 1
sin
n
ω∆tj
2
(4.88)
(4.89)
As it is recognizable a maximum is reached if and only if ~k~vj = ω and ~k~vj = ω. The correlation functions Cjl
can be divided into a major diagonal and a minor diagonal part according to:
Cjl = Cjl δjl + Cjl pjl .
(4.90)
Thereby pjl is the complementary part of δjl i.e.:
pjl := 1 − δjl =

0,
j=l
1,
j 6= l
.
(4.91)
It can be stated that δjl is the major diagonal part and pjl the minor diagonal part of the double sum elements
occuring in (4.85). The major diagonal part represents the general Frank-Tamm formula and can be solved
analytically.
The Master Formula as a Product of Sums
In the previous chapter the master formula was derived for a chained sum. This can be simplified e.g. for
numerical calculations to reduce the number of calculations from ∼ O N 2 to ∼ O (2N ) by rewriting the master
formula as a product of sums. This can be done as follows:
dW
dω
πg 0 m
(2π)4 ε
´1
´ 2π
hP
N
vj Cj−
j=1 qj ~
exp im~ek ~rj t̄j − iω t̄j
i
PN
⊗ l=1 ql~vl Cl− (ωµε~vl − m~ek ) exp −im~ek ~rl t̄l + iω t̄l
= Re
d cos θ
−1
0
dφ
The Master Formula for Two Particles
For two particles the master formula can be described as follows:
´
n
1
2 2
= g 0 απem
dΩz
β
1
−
cos
θ
2ω
1 1
β1n c11 (β1n cos θ, −)
´
n
dΩz22 β22 1 − cos θ β12n c22 (β1n cos θ, −)
+g 0 απem
2ω
n
o
~n1 +β
~n2 )
´
~
ek (β
0 αem n
+g π2 ω dΩz1 z2 β1 β2 2 cos α12 − βn1 βn2
cos cωn ~ek d~12 − ωt12 c12 (−)
dNγ
dω
(4.92)
To calculate the light yield for two particles equation (4.92) can be used. Thereby the first two lines are the
generalized Frank-Tamm formulae for the two particles and will be derived below.
36
4.5.4
The Classical Limit and Determination of the Prefactors
The determination of the Fourier factors can be done using the extremal condition ∆t → ∞ for one particle i.e.
N = 1 (or for more realistic situations ω ∆t−1 ). This can be reasoned by the fact that the classical FrankTamm formula must be obtained which is a one particle formula at all. For j = l and N = 1 the correlation
function Cjl leads to [84, 92]:
lim Cjl
ω∆t1
lim cjl
ω∆t1
n o
sin2 ∆t ~k~v − ω /2
→ 2π∆tδ ~k~v − ω ,
=
lim 4
n
o2
ω∆t1
~k~v − ω
ω2
π
π
=
lim Cjl
→ ω 2 ∆tδ ~k~v − ω → ω∆tδ ~ek β~n − 1 .
ω∆t1
4
2
2
(4.93)
(4.94)
In the limiting process ω∆t → ∞ the product of the sine functions give usually a product of δ-functions but
nevertheless they would only exist for ω = ~k~vj = ~k~vl and else become zero. Therefore δjl must be implemented
to avoid this problem. Inserting this relation into (4.85) and using (??) and leads to:
n
g 0 απem
2ω
dNγ
=
dω
=
⇒
dNγ
dω
⇒
´
dΩ
PN
j,l=1 zj zl βj βl
dωdL
cos {αlj } −
n 2 2
π
2
g 0 απem
2 ω z β sin θCh 2 ω∆t
~nj
~
ek β
βnj βnl
cos
n
ω
ek d~lj
cn ~
o
− ωtlj clj (−)
´1
g 0 αem z 2 ββn ∆t sin2 θCh β1n
βn >1
d2 Nγ
´ 2π
d cos θ 0 dφδ (βn cos θ − 1)
i
h Θ β1n + 1 − Θ β1n − 1
−1
=
g 0 αcem
z 2 L sin2 θCh
0
(4.95)
=
z 2 sin2 θCh .
g 0 αcem
0
(4.96)
As it is recognizable the master formula yields the classical Frank-Tamm formula with the introduced g 0 -factor
which corresponds with the g 0 -factor in (4.86). To obtain the Frank-Tamm formula the following gauge condition
for the Fourier factors must be applied:
g 0 := g = −
2πψω
≡ 1.
2χt
(4.97)
In the following calculations the g-factor will also appear but in the final calculations it will be set to unity.
This factor shall emphasize that there is an additional factor which could be arbitrary from theoretical point
of view. From a practical point of view this factor must be defined or chosen make comparisons to the classical
Frank-Tamm formula possible.
Approximations for Large Steplengths
~nj ≈ 1. Nevertheless, approxiFor large steplength compared the wavelength of light it can be shown that ~ek β
mations in the master formula have to be treated carefully especially in combination with the highly oscillating
phase factors exp −im~ek ~rl t̄l − ~rj t̄j
. An approximation of ~ek β~nj,l ≈ 1 for both indexes could make the
phase factors vanish. Under certain circumstances like in many body systems this could absolutely falsify the
37
result. The solution for this problem is that the approximation e.g. ~ek β~nj ≈ 1 has to be done with respect to
one index which means that only one function in the pair correlation function Cjl e.g. Cj can be treated as a
δ-function. The other function has to be considered as a test function.
4.5.5
Generalization of the Frank-Tamm Formula
In this subchapter the master formula will be solved for N = 1 analytically. This gives the general Frank-Tamm
formula for arbitrary steplength and arbitrary frequency. Furthermore, this relation is important due to the
fact that it represents the major diagonal part of the master formula. To simplify the expressions the following
integrals can be defined:
ˆ1
d cos θ (βn cos θ − 1)
I (a) :=
a
sin2
ω∆t
2
(βn cos θ − 1)
2
(βn cos θ − 1)
−1
(a = 0, 1)
(4.98)
The master formula reads as:
dW
dω
=
´
π 2 ψω mω4 2 1
q −1
(2π)4 2χt εω 2
=
d cos θ
π 2 ψω mω8π 2
q
(2π)4 2χt εω 2
´ 2π
2 ω∆t ~
2
~n − 1 sin { 2 (~ek βn2−1)}
dφ
µεv
−
1
−
~
e
β
k
0
(~ek β~n −1)
2
π
ψ
mω8π
µεv 2 − 1 I (0) − (2π)4ω2χ εω2 q 2 I (1)
t
2
sin2 θCh I (0, −1) − 2~gn
πεr z αem I (1)
i
h
2~gn 2
= g αcem
z 2 L sin2 θCh 2~βn
π∆t I (0) − πεr z αem I (1) ,
0
i
h
dN
2βn
αem 2
1 dW
2
=
g
z
L
sin
θ
I
(0)
⇒ dωγ = ~ω
Ch
dω
c0
πω∆t
i
h
αem 2
2c0
2
−g c0 z L sin θCh πnωL sin2 θCh I (1) .
=
2~gn 2
2 2
πεr z αem β n
(4.99)
(4.100)
For large steplength compared to the wavelength of visible light the following transitions are valid which were
shown indirectly at equation (4.95):
2βn
I (0)
πω∆t
2c0
I (1)
πnωL sin2 θCh
→
|{z}
1,
(4.101)
→
|{z}
0.
(4.102)
ω∆t1
ω∆t1
1
Defining the operator [f (cos θ)]−1 of a function f as
1
[f (cos θ)]−1 = f (1) − f (−1)
38
(4.103)
the integral solutions I (a) can be represented as follows:
"
I (0)
=
I (1)
=
sin2 ω∆t
ω∆t
2 (βn cos θ − 1)
Si (ω∆t (βn cos θ − 1)) −
2βn
βn (βn cos θ − 1)
−
#1
,
(4.104)
−1
1
1
[Ci (ω∆t (βn cos θ − 1)) − ln (ω∆t (βn cos θ − 1))]−1 .
2βn
(4.105)
Thereby, Si and Ci are defined as the sine- and cosine integrals. Due to the fact that the general Frank-Tamm
formula contains a ∼ ω −1 part the I (0) function dominates in comparison to the I (1) part because it contains
a part which is proportional to ω.
4.6
4.6.1
About Cascade Directions and Interactions
Cascades from Arbitrary Directions
The above formulae can be used to obtain the results for arbitrary oriented cascades i.e it opens a window to
make a cascade orientation reconstruction using the simulation results. This can be done as follows. A cascade
coming from a certain direction can be obtained by using the simulated data and applying an affine matrix
transformation to the particle’s position data. That means two things are needed. Firstly a rotation matrix
M ∈ Rd×d (d is the dimension) which rotates the particle’s position vector ~rj and second a shift vector ~s ∈ Rd
which places the cascade to a definite position. Thus, the following transformation can be applied:
~ j = M~rj + ~s.
R
(4.106)
The rotation matrix in three dimensions could be set as the Euler angle matrices i.e. a product of three rotational
matrices:

1

M= 0
0
0
0

cos ϑ 0
cos ϕ

sin ϕ   0
− sin ϕ cos ϕ
sin ϑ
1
0
− sin ϑ

cos ψ

  − sin ψ
cos ϑ
0
0

sin ψ
0
cos ψ

0 .
1
0
(4.107)
Thereby, the angles ϕ, ϑ and ψ describe the rotations around the conventional x-, y- and z- axes (the rotation
procedure begins first with the z-axis, then the y-axis and finally the x-axis). An experimental determination
~ j (t = 0) i.e. the first interaction point of the cascade
of the shifting vector ~s and the initial cascade position R
inducing particle leads to a system of three linear equations with three independent variables which is absolutely
solvable. This method can also be applied to rotate the detector geometry.
4.6.2
Cascade Interactions
The equation (4.84) can used to differ between two or more different cascades according to:
dW
dω
ˆ
∼
Re
~ (~x, ω) ~j (~x, −ω) .
d3 xE
39
(4.108)
For two different showers the total energy is obtained as:
dW
dω
ˆ
∼
=
∼
Re
h
ih
i
~ 1 (~x, ω) + E
~ 2 (~x, ω) ~j1 (~x, −ω) + ~j2 (~x, −ω)
d3 x E
ˆ
h
i
~ 1~j1cc + E
~ 2~j2cc + E
~ 1~j2cc + E
~ 2~j1cc .
Re
d3 x E
dW
dW
dW
dW
+
+
+
.
dω 1
dω 2
dω 12
dω 21
(4.109)
This equation shows that there are additional nontrivial parts i.e. (dW/dω)12,21 which obviously not vanish at
all. This problem has to be taken into account in an energy reconstruction. A problem would be e.g. for two
cascade inducing particles from the same source having a small temporal or spatial distance to each other. The
additional parts have then to be subtracted from the total energy flow within calculating light yields.
5
Calculation Results
In this chapter the calculation results will be presented. In the first subchapter studies about the light yield are
shown with the goal to understand the light yield of the main processes possibly occuring (like exactly in parallel
moving particles) and definitely occuring in particle cascades (like pair production, single and scattering). The
studies will give a detailed description of how the light yield behave for specific scenarios. The second subchapter
will focus on the calculation of the Cherenkov light yield in particle cascades for energies up to 1 PeV. In the
following chapters always suppressions will be considered due the the fact, that the main focus of research is to
find a deviation to the classical Frank-Tamm formula. The suppression δN is defined as:
δN =
dNγ
dω
dN
− dωγ
MF
F T .
dNγ
dω
(5.1)
FT
Thereby the index F T stands for the classical Frank-Tamm formula and M F for the master formula.
5.1
Cherenkov Light Yield Studies and Expectation Validation
The following simulations are done for β = 1. The cases including two particles are simulated for equal
steplengths i.e. ∆t1 = ∆t2 ≡ ∆t.
5.1.1
One Track divided into N Steps ranked together
The goal of this part is to prove mathematically that the master formula is invariant under dividing a step of
arbitrary length ∆τ into N arbitrary smaller steps ∆t. This is important to test out because the Cherenkov
light yield of one step and one step divided into e.g. N = 2 steps must be the same. Moreover, the result must
be independent of N for sure. For simplifications it is considered that a particle is moving along the z-direction
40
with v = vez starting at (t0 , z0 ) = (0, 0). The following relations must be taken into account:
qj
vj ez,j
∆τj
∆τ
∆t
t̄j
z¯j
= q,
(5.2)
= vez ,
(5.3)
= ∆τ
(5.4)
= N ∆t,
∆τ
=
,
N
∆t
+ (j − 1) ∆t,
=
2
L
=
+ (j − 1) L.
2
(5.5)
(5.6)
(5.7)
(5.8)
Due the fact that the charge, velocity and velocity direction as well as the steplengths do not change it is
sufficient to calculate only the exponential sums. These are:
S :=
X
exp {−imL cos θ (l − j) + iω∆t (l − j)}
=
j,l
N
X
e−il(mL cos θ−ω∆t)
l=1
N
X
eij(mL cos θ−ω∆t) ,
(5.9)
l=1
These sums are geometrical series and are given as (generally) []:
N
X
ejx
=
j=1

ex 1−eN x ,
x 6= 0
N,
x=0
1−ex
.
(5.10)
According to this relation and using ∆τ = N ∆t the sums (5.9) and (??) can be read out as:
S
=
sin2
sin2
ω∆τ
2
ω∆t
2
(βn cos θ − 1)
.
(βn cos θ − 1)
(5.11)
Now putting the equation (5.11) into the master formula for N = 1 and changing the variables ∆τ → ∆t lead
to the master formula for N = 1 which gives the classical result as shown in chapter XYZ. This is reasoned
by the fact that the denominators of equation (5.11) cancel out. The numerators stand for the undivided large
step contribution and is the only term which is left back.
5.1.2
Cherenkov Light Yield as Function of the Steplength
The generalization of the Frank-Tamm formula (4.100) can be used to study e.g. the Cherenkov light yield
as function of the steplength. Especially regions where ω∆t is smaller, equal to or larger than one can be
analyzed. It gives a classical generalization of the Frank-Tamm formula. The following figure shows the light
yield suppression as function of the angular frequency ω in steps of 0.6 nm. The simulation results show that
above 3 µm the radiation suppression becomes smaller than percent. At ∆t = 10−6 the suppression for higher
frequencies does not decrease but becomes constant. This is caused by the fact that the track length corresponds
41
with the minimal wavelength which can be seen as a threshold where the suppression starts to decrease.
Figure 5.1: Cherenkov light yield suppression as function of the track length. The simulations show that the
larger the tracklength is the smaller the suppression which coincides with the classical expectation.
The Cherenkov light yield as function of the track length shows that the light yield of particles with a small
track length is highly suppressed compared to the classical Frank-Tamm formula.
42
5.1.3
Cherenkov Light Yield as Function of the Distance
The Cherenkov light yield of two particles moving in parallel (coherently) e.g. a dipole is useful to analyse the
number of photons as function of the distance. This is really important to know because it solves the answer
to the question at which distance coherently moving particles can be treated as independently. It is sufficient
to simulate the light yield of coherently moving particles for the case where the suppression is negligibly small
(e.g. ∆t = 10−4 ). Only the distance should be in the foreground. In the following figure a schematic view of
the geometry is shown.
Figure 5.2: Schematical view of two particles (represented as vectors) moving in parallel (coherent motion).
The expression (??) can be solved analytically for ω∆t 1. The combined solution using the δjl - and
pjl -part of the master formula is given by [81]:
dNγ
dω
dNγ
dλ
=
=
αem g 2 2
ω
z sin θCh L 1 ± J0
d sin θCh ,
c0
cn
2πg
ω
2 · 2 αem z 2 sin2 θCh L 1 ± J0
d sin θCh .
λ
cn
2·
(5.12)
(5.13)
Thereby J0 is the zeroth order Bessel function, z = q/e and the signs ± consider whether the particles are
opposite charged (q1 = −q2 ) or do have same sign (q1 = q2 ). The results are given as follows.
43
Figure 5.3: Cherenkov light yield of two in parallel moving particles as function of the wavelength. The blue line
emphasizes two times the Frank-Tamm formula, where the particles are assumed to independently moving. Left
figure: Particles have opposite charges. The radiation is suppressed below 1 µm. The black line is multiplied
by a factor of 50 to make it realizable. Right figure: Particles have same charge. For smaller distances the
particles emit fourth time more than only one particle does.
The analysis of two coherently moving particles makes it possible to define cut values. In this work four
cut values are defined. The first one is dmin = λmin . The next cut length is given by dmax = λmax . The
third one is defined as dcut = 1 µm and the final one is defined as dno = 2 µm. At dno the radiation is not
suppressed anymore and the particles can be treated independently. Averagely speaking, the radiation is also
not suppressed for distances at ∼ 1 µm but the addtional cut value dno was taken to become a larger parameter
space.
5.1.4
Cherenkov Light Yield of Pair Production
The considered scenario of pair production is shown in the following figure.
Figure 5.4: Schematical view of a pair production process.
As it is recognizable the coordinate system can be chosen such that the particles e.g. move in z-direction.
This means the velocity vectors can be parameterized as follows:
~v1
=
~v2
=
α
α
+ ey v1 sin ,
2
2
α
α
ez v2 cos − ey v2 sin .
2
2
ez v1 cos
44
(5.14)
(5.15)
Due to the fact that the pair comes from the same vertex the relevant spacetime quantities to allow the
calculation of the light yield are given as:
t̄1 , ~r1 t̄1
t̄2 , ~r2 t̄2
L1
α
L1
α
∆t1
, ez
cos + ey
sin
,
2
2
2
2
2
L2
α
L2
α
∆t2
, ez
cos − ey
sin
.
=
2
2
2
2
2
=
(5.16)
(5.17)
Assuming that the step lengths are equal the distances become:
t21
d~21
= t2 − t1 = 0,
α
= ~r2 t̄2 − ~r1 t̄1 = −ey L sin .
2
(5.18)
(5.19)
The following figure shows the suppression for a pair moving a steplength of ∆t = 10−4 . It is recognizable that
the radiation is absolutely suppressed above E ≤ 10 · Ec which is about 800 MeV.
Figure 5.5: Numerical simulation of a e+ e− pair with a step length of ∆t = 10−4 ns and an opening angle of
about α (E) ≈ me /E.
The next figure shows a pair with a larger step length. The expected suppression starts at 102 · Ec and
coincides with the case for ∆t = 10−4 and E = 10 · Ec .
45
Figure 5.6: Numerical simulation of a e+ e− pair with ∆t = 10−3 ns.
The following figure shows the same simulations for ∆t = 10−3 ns.
46
Figure 5.7: Numerical simulation of a e+ e− pair with ∆t = 10−2 ns.
The simulations of the suppression for pair production shows that as expected the step length and the energy
are antiproportional to each other. The suppression condition can be taken from the pictures above and can
be formulated as follows. If Ec corresponds with a path length of 10−5 ns the radiation of pairs is absolutely
suppressed.
5.1.5
Cherenkov Light Yield of Single and Multiple Scattering
In this part the light yield of a particle will be calculated which is obeying multiple scattering. The following
figure shows the considered situation.
Figure 5.8: Schematic view of multiple (or single) scattering. For the special case α = 0 the result is given by
two times the single steps.
47
Assuming that the first particle moves according to v1 ez the second has to obey e.g. v2 sin αey + v2 cos αez
with α ∈ − π2 , π2 . Furthermore the relevant spacetime quantities are given as follows:
t̄1 , ~r1
t̄2 , ~r2
∆t1 L1
=
,
ez ,
2
2
∆t2
∆t2
, L1 ez + ~v2
.
=
∆t1 +
2
2
According to the master formula the relevant parameters to calculate the light yield are given as:
d~21 = −d~12
=
t21 = −t12
=
T21 = T12
=
~v1~v2
=
L1
∆t2
ez +
~v2 ,
2
2
∆t1
∆t2
+
,
2
2
∆t2
3
∆t1 +
,
2
2
v1 v2 cos α.
The simulation results are shown in the following figure.
48
(5.20)
(5.21)
Figure 5.9: Numerical calculation of the Cherenkov light yield suppression for single and mutliple scattering.
The represented suppressions are valid for the energy regions from Ec to 109 Ec .
The results for single and multiple scattering have been done for all energies between Ec and 109 Ec . The
results stay already the same without large deviations. Thus, this shows that the suppression effect of particles
in the used simulations are negligibly small because the step lengths are about ∼ 10−2 ns. Moreover, the
scattering angle has been simulated using the approach of Molière:
13.6 M eV
α0 =
z
βcp
r
x
x
1 + 0.038 ln
.
X0
X0
(5.22)
Hereby x/X0 represents the thickness of the scattering medium in radiation lengths. It states that (5.22) is
accurate to 11 % or better for 10−3 < x/X0 < 100 [111, 109].
49
5.2
5.2.1
Cherenkov Light Yield of Particle Cascades in Ice
Cherenkov Light Yield of indpendent Steps
The double sum in the master formula can be separated in to a major diagonal and minor diagonal part. In
this chapter the suppressions for the major diagonal part i.e. the generalized Frank-Tamm part will be shown
for 1 TeV and 1 PeV photon induced cascade to illustrate that there is no suppression in the diagonal part of
the master formula.
Figure 5.10: Cherenkov light yield suppression for a 1 TeV cascade (left) and a 1 PeV cascade (right). The
results show that the suppression is negligibly small.
This result is also consistent with electron or positron induced cascades and look exactly the same.
5.2.2
Cherenkov Light Yield of Particle Cascades
The numerical computation of the master formula applied to the Geant4 simulations cannot be solved due
to technical reasons. Nevertheless, due to the fact that the cascade simulations are given, the relevant data
can be created. As shown in the previous chapter the diagonal part of the master formula behaves like the
classical Frank-Tamm formula. Due to the fact that the total number of steps was established already two
counting experiments have been done. Firstly, the number of electrons and positrons with a distance smaller
or equal to dmin , dmax, , dcut , dno with respect to the first step are counted. This is important to get an upper
bound of the maximal step number for light yield suppression. The second counting experiment treats pairs
moving a radiation length where the multiple scattering of the particles is summarized as a rectilinear curve.
Averaging over electron, positron and photon induced cascades (Step number are approximately equal) leads to
the following tabular:
50
Average dL ≤ λmin dL ≤ λmax
1 TeV
2028
5830
10 TeV
20296
57796
100 TeV
203268
581690
1 PeV
2035278
5816432
Average dX0 ≤ λmin dX0 ≤ λmax
1 TeV
2
2
10 TeV
14
42
100 TeV
132
390
1 PeV
1318
3974
dL ≤ dcut dL ≤ dno
6436
12284
63964
122748
642644
1231966
6426694
12317134
dX0 ≤ dcut dX0 ≤ dno
2
8
42
82
432
882
6614
8790
Table 3: Number of Steps obeying the cut conditions for pairs at the first step (top). Number of electrons and
positrons obeying the cut conditions for a radiation length (bottom). The results show the averaged values for
electron, positron and photon induced cascades and represent the number of electron steps plus the number of
positron steps.
The number of steps of the top tabular increases approximately with a factor of ten. Therefore the amount
for every energy scale is approximately the same. For 1 PeV and dL ≤ dno the relative deviation is about 1.1 %
and it becomes smaller the smaller the cut value is. Taking the right tabular and assuming that the electrons
or positron have a steplength of about 1 mm. Then a radiation length corresponds to approximately 400 steps.
That means e.g. taking again 1 PeV and dX0 ≤ dno leads to 0.3 % which is a negligibly small effect. Actually,
the cut valuess were taken were no suppression effect is available. As a matter of fact, the simulation studies
within this chapter show, that the Frank-Tamm formula can be seen as a high-energy physics formula.
6
Summary and Outlook
The goal of this thesis was to calculate the Cherenkov light yield for electromagnetic cascades. It could be
shown that the light yield with energies up to 1 PeV is not suppressed. There are regions where the light yield
is suppressed but comparing these suppressions with the total amount leads effectively to no suppression. It is
astonishing that the Frank-Tamm formula which has a relatively simple mathematical structure can precisely
calculate the light yield of particle cascades which are relatively complicated systems.
Acknowledgement
I would like to thank specially Prof. Dr. Christopher Wiebusch for his patience and his trust in me giving
a really challenging master thesis. Furthermore, I would like to thank Prof. Dr. Werner Bernreuther for the
constructive discussions within the whole work. I also would like to thank Leif Raedel for his support and also
for his efforts in private talks. I thank my family and a special thank goes to my wife Rumy for motivating me
all the time.
51
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Appendix
A Foundations, Bessel Functions and Integration Techniques
A.1 Bessel Functions of the First Kind
Bessel Functions of the First Kind
In this introductory and beginning chapter some functions, integrals
and equations will be presented. First of all it is useful to make familiar with Bessel functions of the first kind
of order n denoted by f (x) = Jν (x) for x ∈ C. These functions obey the following second order differential
equation:
x2
d2 f
df
+x
+ x2 − ν 2 f = 0.
2
dx
dx
(6.1)
A power series ansatz applied to these equations of motion can be used. The main result is given by:
Jν (x) =
∞
X
k=0
k
x 2k+ν
(−1)
.
k!Γ (k + ν + 1) 2
58
(6.2)
This power series is called Bessel functions of the first kind and ν-th order. Thereby Γ (x) represents the gamma
function, a generalization of the factorial function which is restricted to positive integer values. It is defined as:
ˆ∞
duux−1 exp (−u) = (x − 1)!,
Γ (x) =
(6.3)
0
whereby (...)! stands for the factorial function. The other linear independent solution called Bessel functions of
the second kind is given by
Yν (x) =
Jν (x) cos (νπ) − J−ν (x)
.
sin (νπ)
(6.4)
For ν ∈ Z the following equation is valid:
ν
J−ν (x) = (−1) Jν (x) = Jν (−x) .
(6.5)
For even ν, Jν (x) is symmetric and for odd ν antisymmetric under sign change of ν. Furthermore, the gamma
function is used here to generalize the case for arbitrary ν ∈ C. It can be simplified for ν ∈ N0 . Therefore
Γ (k + ν + 1) = (k + ν)! for integer values with k + ν ≥ 0. Bessel functions obey the following useful recurrence
and derivative relations:
2ν
Jν (x) = Jν−1 (x) + Jν+1 (x) ,
x
dJν (x)
2
= Jν−1 (x) − Jν+1 (x) ,
dx
(6.6)
(6.7)
n
1 d
[xν Jν (x)] = xν−n Jν−n (x) ,
x dx
n 1 d
Jν (x)
n Jν+n (x)
= (−1)
.
x dx
xν
xν+n
(6.8)
(6.9)
For large arguments (x ∈ C) i.e.|x| 1 the behavior of the Bessel functions can be approximated as:
r
J±ν (x) ≈
i
2 h
π π
π π
cos x ∓ ν −
Aνn − sin x ∓ ν −
Bνn (n ∈ N, |arg (x)| < π)
πx
2
4
2
4
(6.10)
Thereby the coefficients Aνn and Bνn read as (ξ1 , ξ2 ∈ (−1, 1)):
Aνn
=
n−1
X
aνk + cνn
k=0
Bνn
=
n−1
X
k=0
bνk + dνn
n−1
X
k
1
ν
+ ξ1 |aνn | n > −
,
=
2k
2 4
(2k)!Γ (ν − 2k + 1/2)
k=0 (2x)
n−1
k
X
(−1) Γ (ν + 2k + 3/2)
ν
3
=
+ ξ2 |bνn | n ≥ −
.
2k+1
2 4
(2x)
(2k + 1)!Γ (ν − 2k − 1/2)
(−1) Γ (ν + 2k + 1/2)
(6.11)
(6.12)
k=0
To leading order the coefficients are given by Aν1 ≈ 1 + cν1 and Bν1 ≈ dν1 + 1/ (2x). Bessel functions of the
first kind will accompany this thesis from now on.
The scope of Bessel functions is widespread. One important subject is the so-called Anger-Jacobi identities or
59
Anger-Jacobi expansions. They are given as a series expansion of complex exponentials containing a standard
trigonometric function:
eiaρ cos ϕ
eibρ sin ϕ
∞
X
=
n=−∞
∞
X
=
in Jn (aρ) einϕ ,
(6.13)
Jn (bρ) einϕ .
(6.14)
n=−∞
As it is shown in the equations (6.13) and (6.14) it is possible to separate the ρ-terms from the ϕ-terms. The
result is an unlimited sum which contains Bessel functions as developmental coefficients. This description allows
the usage of a direct integration with respect to ϕ. Using the orthogonal relation
ˆ2π
dϕeinϕ = 2πδn0 = 2πδ0n =
0

2π,
n=0
0,
n 6= 0
(6.15)
it is possible to build the integrals of (6.13) and (6.14) with respect to ϕ directly:
ˆ2π
dϕeiaρ cos ϕ
=
∞
X
in Jn (aρ) 2πδn0 = 2πJ0 (aρ) ,
(6.16)
Jn (bρ) 2πδn0 = 2πJ0 (bρ) .
(6.17)
n=−∞
0
ˆ2π
dϕeibρ sin ϕ
=
∞
X
n=−∞
0
Thereby δmn = δnm is called the Kronecker delta which is a second rank tensor notation of the unity matrix.
In the case of multiplying the two complex exponentials eiaρ cos ϕ and eibρ sin ϕ and building the integral with
respect to ϕ yields:
ˆ2π
dϕe
iaρ cos ϕ ibρ sin ϕ
e
=
∞
X
ˆ2π
n
m,n=−∞
0
=
=
=
=
∞
X
0
in Jn (aρ) Jm (bρ) 2πδ−mn
m,n=−∞
∞
X
2π
2π
2π
n=−∞
∞
X
n=−∞
∞
X
n=−∞
60
dϕei(m+n)ϕ
i Jn (aρ) Jm (bρ)
in Jn (aρ) J−n (bρ)
n
(−i) Jn (aρ) Jn (bρ)
einφ0 Jn (aρ) Jn (bρ) .
(6.18)
n
Hereby einφ0 = (−i) and equations (6.5) and (6.15) were used. Due to the theorem of de Moivre for complex
exponentials i.e.
n
(cos φ0 + i sin φ0 ) = exp (inφ0 ) = cos nφ0 + i sin nφ0
(6.19)
n
there exists a φ0 such that einφ0 = (−i) . A special value for φ0 is given by 3π/2. Now the expression (6.18)
can be simplified using Graf’s addition theorem for products of Bessel functions. Setting x = aρ and y = bρ
this theorem reads as:
Jν
p
x2 + y 2 − 2xy cos φ0
∞
x − y exp (−iφ ) ν/2
X
0
=
Jn+ν (x) Jn (y) exp (inφ0 ) .
x − y exp (iφ0 )
n=−∞
(6.20)
Hereby x, y and ν could be seen as general complex quantities and φ0 as a general real quantity. For general ν
there is a constraint for x and y, namely: |y exp (±iφ)| < |x|, but this constraint can be voided if ν∈ Z which
is the case in (6.18) because ν ≡ 0. Therefore the total result yields:
ˆ2π
ˆ2π
dϕe
iaρ cos ϕ ibρ sin ϕ
e
0
dϕeiaρ cos ϕ+ibρ sin ϕ = 2πJ0
=
p
p
x2 + y 2 = 2πJ0
ρ2 (a2 + b2 ) .
(6.21)
0
Using the above shown procedure complex exponential integrals like
´ 2π
0
dϕ cos ϕeiaρ cos ϕ eibρ sin ϕ and
´ 2π
0
dϕ sin ϕeiaρ cos ϕ eibρ si
can be solved immediately. Nevertheless these integrals can be simplified taking the chain rule:
´ 2π
0
dϕ cos ϕeix cos ϕ eiy sin ϕ =
0
dϕ sin ϕeix cos ϕ eiy sin ϕ =
´ 2π
1 ∂
i ∂x
1 ∂
i ∂y
´ 2π
x2 + y 2 ,
p
∂
x2 + y 2 .
= 2π
i ∂y J0
0
dϕeix cos ϕ eiy sin ϕ =
0
dϕeix cos ϕ eiy sin ϕ
´ 2π
2π ∂
i ∂x J0
p
(6.22)
(6.23)
Thereby the derivative of the Bessel function reads as:
2π ∂
J0
i ∂x
p
x2 + y 2
J1
=
p
2π ∂
J0
x2 + y 2
=
i ∂y
p
x2 + y 2
,
x2 + y 2
p
J1
x2 + y 2
2πiy p
.
x2 + y 2
2πix
p
(6.24)
(6.25)
Equations (6.22) and (6.23) can also be obtained using the complex representations 2i sin ϕ = eiϕ − e−iϕ and
2 cos ϕ = eiϕ + e−iϕ for sine and cosine and after that applying the recurrence relations (6.6) after integration
with respect to ϕ.
A.2 Modified Bessel Functions of the Second Kind
Another important and useful functions for this work are the modified Bessel functions of second kind. They
obey the following differential equation:
x2
d2 f
df
+x
− x2 + ν 2 f = 0.
2
dx
dx
61
(6.26)
The two linear independent solutions Iν (x) and Kν (x) can be presented in terms of Jν (x):
Iν (x)
Kν (x)
= i−ν Jν (ix) ,
π I−ν (x) − Iν (x)
=
.
2
sin (νπ)
(6.27)
(6.28)
The functions Iν (x) are not of further interest due to the fact that only the Jν - and Kν -functions occur directly
as solutions of the electromagnetic fields. Therefore the calculations will be focused on them. It is recognizable
that the modified functions of second kind differ from the ones of first kind by transforming the argument from
x to ix.
A.3 Useful Formulae
Here, some useful formulae will be presented which can help simplifying the solution of integrals. The first
formulae is often called Schwinger parameterization and is given by [87, 88, 89]:
1
1
=
Aν
Γ (ν)
ˆ∞
uν−1 e−uA du.
(6.29)
0
The next formula is called Gauss integral and is given by [59]:
ˆ∞
2
e−ax
+bx+c
dx
p
n
o
 π exp b2 + c ,
a
4a
2πec δ (b) ,
=
−∞
a 6= 0
.
(6.30)
a = 0, Reb = 0
The final formula is given as follows [1, 2]:
ˆ∞
1 2
2
exp − R − uL du =
4u
1
un+1/2
2n+1/2 R2
−n+1/2 √
R2 L2
n−1/2
Kn−1/2
√
R 2 L2
0
r
=
2
n+1/2
L2
R2
!n−1/2
Kn−1/2
√
R 2 L2 .
(6.31)
Thereby A, ν, µ, a, b, c, m and n are generally complex variables, kE a d-dimensional vector in euclidean
space and kM a d-dimensional vector in Minkowski space. The last formula is very important in the context of
renormalizability and renormalized theories. The following formula is called Feynman parameterization and is
useful in the calculations of Feynman integrals:
Γ (ν1 + ... + νn )
1
=
Aν11 ...Aνnn
Γ (ν1 ) ...Γ (νn )
ˆ1
ˆ1
du1 ...
0
dun
0
62
δ (1 − u1 − ... − un ) uν11 −1 ...uνnn −1
ν1 +...+νn
[u1 A1 + ... + un An ]
.
(6.32)
B Delta Function Approach and Applications
B.1 Delta Function
The one dimensional Dirac δ-function (named after P. A. M. Dirac ()) (or δ-distribution) in distributional sense
is defined as:

”∞”, x = 0
δ (x) :=
.
0,
x 6= 0
(6.33)
Thereby the value of ”∞” can be seen symbolically. It is also possible that is a finite value assuming that δ (x)
converges to zero for x 6= 0. (((In combination with (6.48) for d = 1 and k → ∞ the definition (6.33) diverges
for x = 0 and converges for x 6= 0.))) It obeys
ˆ
ˆ
lim
ˆ
δ (y) dy
=
δ (y) dy
=
R
→0
Rr[−,]
lim
→0
−
δ (y) dy = 1,
0,
(6.34)
(6.35)
and has the following defining feature
ˆ
F (x) =
x+
ˆ
F (y) δ (x − y) dy = lim
F (y) δ (x − y) dy
→0
x−
R
(6.36)
i.e. the convolution maps a function to its own. These features make the δ-function to an important tool.
Further properties can be taken from the Appendix. The δ-function is a practical technique to solve integrals
and really important for this work and also for the theory of electrodynamics. It is not only a tool but a decisive
part of the theory. It can be generalized to d-dimensions. Nevertheless, in this work, only the case up to three
dimensions is used. The three dimensional δ-function in cartesian space is given by:
δ (~r − ~r0 ) = δ (x − x0 ) δ (y − y0 ) δ (z − z0 ) .
(6.37)
It can be transformed to cylindrical i.e. (ρ, ϕ, z) and spherical coordinates i.e. (r, θ, ϕ) using the normalization
relation:
ˆ
1
d3 x δ (~r − ~r0 ) = 1.
g
(6.38)
R3
Thereby g is the Jacobian of the geometry. Using
d3 x = dxdydz = r2 drdθdϕ = ρdρdϕdz
63
(6.39)
it states that:

 1 δ (ρ − ρ ) δ (ϕ − ϕ ) δ (z − z ) ,
cylindrical
0
0
0
δ (~r − ~r0 ) = ρ
 1 δ (r − r ) δ (θ − θ ) δ (ϕ − ϕ ) , spherical
0
0
0
r 2 sin θ
(6.40)
This transformations can be seen as a direct transformation from cartesian to cylindrical and spherical space.
Nevertheless, a rotational invariance can help to neglect the angular dependence as it will be shown below. For
test functions f (x) ( f (x) δ (x − y) = (limx→y f (x)) δ (x − y) is well defined) the following relations are valid:
xδ (x)
=
0,
(6.41)
δ(x)
⇒ δ 0 (x) = −
,
x
f (x) δ (x) = f (0) δ (x) ,
n
X
δ (x − xj )
δ (f (x)) =
,
|f 0 (xj )|
j=1
1
b
δ (ax − b) =
δ x−
,
|a|
a
ˆ
ˆ
δ (y + ) − δ (y − )
dδ (y)
dy = lim f (y)
dy
f (y)
→0
dy
2
(6.42)
(6.43)
(6.44)
(6.45)
R
R
ˆ
f (y) δ (n) (x − y) dy
f () − f (−)
= −f 0 (0) ,
2
=
− lim
=
(−1) f (n) (x) .
→0
n
(6.46)
(6.47)
R
B.2 Delta Sequences in Curvilinear Coordinate Systems
There are several ways to define a Dirac delta function. In the context of this thesis it is useful to introduce a
Dirac delta function or delta distribution through a Fourier sequence according to
ˆ
δ~k (~x) :=
dd k 0
d
Vk
0 ⊂Rd
(2π)
exp −i~k 0 ~x
(6.48)
which also satisfies by definition the normalization relation
ˆ
dd xδ~k (~x) ≡ 1
(6.49)
Rd
for all ~k. Thereby Vk0 ⊂ Rd is the considered ~k 0 ∈ Rd×1 -spacevolume, the vector index ~k describes the sequence
index and d is the number of dimension. In this work mostly the cases d = 1, 2, 3 are used for one to three
dimensional Delta function. Depending on the used coordinate system the expression for δ~k (~x) differs. In
64
cartesian, cylindrical and spherical coordinates the dot product ~k~x is given by:
~k~x =
kx x + ky y + kz z,
(6.50)
~k~x =
kρ x cos φ + kρ y sin φ + kz z,
(6.51)
~k~x =
kr cos θ.
(6.52)
Therefore the solution of (6.48) for every coordinate system has its own result because of the different geometry.
The solutions for the three different geometries using the integral measures (4.2) can be derived as follows:
δ~k (~x)
=
=
=
δ~k (x)
=
=
=
=
δ~k (x)
=
=
=
=
=
ˆkx
1
3
(2π)
0
dkx0 e−ikx x
−kx
ˆky
ˆkz
0
dky0 e−iky y
−ky
0
dkz0 e−ikz z
−kz
sin (kx x) sin (ky y) sin (kz z)
πx
πy
πz
δkx (x) δky (y) δkz (z) ,
ˆkρ
1
3
(2π)
ˆ2π
dkρ0 kρ0
(6.53)
ˆkz
0
dφe−ikρ (x cos φ+y sin φ)
0
0
dkz0 e−ikz z
−kz
0
ˆkρ
p
sin (k z)
z
dkρ0 kρ0 2πJ0 kρ0 x2 + y 2 2
z
(2π)
0
p
2 + y2
J
k
x
1
ρ
1
sin (kz z)
p
kρ
2
2
2π
πz
x +y
p
δk ρ
x2 + y 2
p
δkz (z) ,
π x2 + y 2
1
3
ˆk
1
ˆ1
0 02
dk k
3
(2π)
1
0
−1
ˆ1
dk 0 k 02
dk 0 k 0
2
(2π)
0
0
d cos θe−ik r cos θ
−1
0
ˆk
2
0
dφe−ik r cos θ
d cos θ
ˆk
2
(2π)
ˆ2π
(6.54)
sin (k 0 r)
r
0
1 sin (kr) − kr cos (kr)
2π 2
r3
δk (r)
.
2πr2
65
(6.55)
Hereby the techniques presented in 4.1.1 were used especially equation (6.21). Now taking the limit ~k → ∞
~
yields the Dirac delta function:



δ (x) δ (y) δ (z) ,


δ (~x) = lim δ~k (x) = δ(ρ)
πρ δ (z) ,

~
k→∞
~


 δ(r) ,
2πr 2
whereby ρ :=
p
x2 + y 2 and r :=
cartesian
cylindrical
(6.56)
spherical
p
ρ2 + z 2 . The sequences (6.53), (6.54) and (6.55) are called δ-sequences.
They differ essentially from Dirac sequences because of their property to take negative values within their
domain of definition whereas a one dimensional Dirac sequence δk must be positive for all k which is usual for
probability distributions. Therefore a Dirac sequence is always a δ-sequence but all δ-sequences are not Dirac
sequences.
B.3 Dirichlet Kernel
The Dirichlet kernel (named after P. G. L. Dirichlet) is principally the numerical expression of (6.48) and can
be seen as a usual δ-sequence. It is defined as:
Dn (x) =
n
X
Dl (x) =
l=−n
n
X
exp {ilx} =
l=−n
sin {(1/2 + n) x}
sin {(2n + 1) x/2}
=
.
sin {x/2}
sin {x/2}
(6.57)
For n → ∞ it gives the δ-function:
lim Dn (x) = 2πδ (x) .
n→∞
For x = Ω∆tn , n = (2n + 1)
−1
(6.58)
and sufficient large n it can be approximated as:
Dn (Ω∆tn )
≈
∆t∞ D∞ (Ω∆t∞ )
:=
sin (Ω∆t/2)
sin (Ω∆t/2)
⇒ ∆tn Dn (Ω∆tn ) ≈
.
Ω∆tn /2
Ω/2
sin (Ω∆t/2)
.
Ω/2
(6.59)
(6.60)
The approximation (6.59) is valid if and only if n Ω∆t 1. That means the larger n the smaller the error.
Therefore n Ω∆t can be seen as a measure for the error. The latter equation can be derived using the integral:
sin (Ω∆t/2)
∆t
Ω∆t/2
=
∆t
2
ˆ1
dx exp {ixΩ∆t/2}
−1
=
=
≡
∆t eiΩ∆t/2 − e−iΩ∆t/2
2
iΩ∆t/2
∆t 2i sin (Ω∆t/2)
2
iΩ∆t/2
sin (Ω∆t/2)
.
Ω/2
66
(6.61)
The Dirichlet kernel is a really useful tool to approximate integrals.
B.4 Convolution and Approximation of Functions
A convolution integral can be seen as a weighted mean of an integrable function f (x) with respect to an
integrable weight function g (x). Thus, the function f becomes smoother. The convolution integral is defined
ˆ
as:
f (y) g (x − y) dd y.
F (x) :=
(6.62)
Rd
This kind of integrals become more interesting if the weighting function g has certain properties. The function
g is a real and obeys the normalization relation:
ˆ
dd yg (y) = 1.
(6.63)
Rd
Due to the fact that g is real there must exist domains where g ≥ 0 because of (6.63). Otherwise there is no
chance to get the integral normalized. For positive normalized functions g with
ˆb
g (x) dx = 1
(6.64)
a
there exists a class of functions f which obey:
ˆ
x+b
ˆ
f (y) g (x − y) d y =
f (y) g (x − y) dd y.
d
F (x) =
(6.65)
x+a
Rd
This is a significant relation because it allows to restrict the integration space Rd to an essentially smaller
interval K (a, b, x) := (x + a, x + b) and is sometimes useful for e.g. solving integrals or approximate them.
One example for this kind of g is given by the function δ (x) = sin (x/) /πx. This function has the following
property:
ˆ∞
−∞
sin (x/)
=
dx
πx
ˆa
dx
sin (x/)
= 1,
πx
(6.66)
−a
whereby a ≈ 1.926447661. Another one is e.g. the δ-function. The δ-function is a practical technique to solve
integrals and really important for this work and also for the theory of electrodynamics. Nevertheless besides
δ-distributions also δ-sequences play a decisive role. They allow to approximate functions. Again assuming
in the one dimensional case that δk (x − y) ≡ δ:=1/k (x − y) is a Dirac δ-sequence, f (x) a function and I an
67
arbitrary interval I ⊆ Rd (here d = 1) with x − y ∈ I. Then
ˆ
f (y) δ (x − y) dy
f (x) :=
(6.67)
I
approximates the function f with respect to . The generalization to arbitrary dimensions is straight forward
and can be done using the ansatz (6.48). Due to the fact that δ (x) is normalized it is possible to define it as
a normalized function Φ such that:
−τ d Φ
δ (x) =
x
lim −τ d Φ τ
=
→0
x
(6.68)
τ
δ (x) .
(6.69)
Thereby d is the number of dimension and τ an arbitrary but constant number. Using the normalization and
´
the substitution y → x/τ it follows that Rd dd x−dτ Φ xτ ≡ 1. For → 0 the transition f (x) → f (x) holds
but for arbitrary > 0 the function f (x) is approximated by f (x). The approximation can now be done using
(6.68):
ˆ
f (x)
f (y) −τ d Φ
=
x−y
τ
dy
ˆI
f (x − τ z) Φ (z) dz
=
I
=
∞
X
f (j) (x)
j=1
j!
ˆ
jτ
z j Φ (z) dz.
(6.70)
I
Thereby the substitution z = (x − y) /τ was used and f (x − τ z) was expanded into a Taylor series with
respect to τ = 0. The gaussian δ-sequence
δ (x) = √
1
x2
exp − 2
2
2π
(6.71)
leads to Φ (x) = δ (x) and for n ∈ N with I = R it gives:



0,

ˆ

j
z Φ (z) dz = (2n − 1)!!,



I
1,
68
j = 2n + 1
j = 2n
j=0
,
(6.72)
whereby the double factorial is defined as:
x!! :
⇒ (2n − 1)!!
=
=

x (x − 2) (x − 4) ...2,
x even
x (x − 2) (x − 4) ...1,
x odd
,
(2n)!
.
2n n!
Thus,
f (x) =
(6.73)
(6.74)
n
∞
X
f (2n) (x) 2
n!
2
n=0
(6.75)
is a description for f (x). It rapidly converges to f (x) for small and only depends on the even order derivatives
of f . This method allows to calculate approximated functions and depends highly on the chosen δ-sequence.
B.5 Singularities, Discontinuities and Principle Values
The principle value technique can be introduced using a simple singular propagator i.e.
1
,
k + i
(6.76)
whereby has to be considered again as an infinitesimal number. This propagator can be rearranged acoording
to:
k − i
k
k
1
= 2
−i 2
≡ 2
− iπσ () δ (k) .
k + i k − i
k + 2
k + 2
k + 2
Now δ (k) = /π k 2 + 2 is a δ-sequence and σ () the sign of epsilon:



1,


σ () := 0,



−1
(6.77)
>0
=0.
(6.78)
<0
Taking the limit → 0 yields:
1
1
= P V − iπσ () δ (k) =: ∆ (k) .
→0 k + i
k
lim
(6.79)
The abbreviation P V means principle value. It is defined as the following integral value:
 k −

ˆ1
ˆk2
1
1
1
,
P V := lim+ 
+
→0
k
k
k
k0
69
k1 +
(6.80)
whereby (k0 , k2 ) is the integration interval and k1 ≡ 0 the pole of 1/k. The principle value of 1/k is only defined
for k 6= 0 whereas the propagator for k = 0 gives a δ-function. Therefore the function ∆ (k) was introduced:

P V 1 ,
1
k
∆ (k) := P V − iπσ () δ (k) =
−iπσ () δ (k) ,
k
k 6= 0
.
(6.81)
k=0
It has to be taken into account that the δ-function only occurs if a positive or negative limit of is used.
Thereby the limiting process is important. Otherwise i.e. for = 0, 1/k + i = 1/k which leads to the initial
situation. The principle value approach allows to avoid singularities by just shifting the quantity k a little bit
to the complex plane i.e. k → k + i. The shifting to complex plane is important here because of the fact
that the propagator for real k becomes complex and can be thus separated into a real and a complex part.
This is not possible for a pure real shifting i.e. k → k + . The outcoming result for the shifted propagator
is a fundamental feature which can be used for e.g. integration aspects. Nevertheless this approach has to
be applied carefully since the above shown way cannot be applied for every integral. This method helps e.g.
dealing with singularities and discontinuities. A discontinuity of a function D (k) can be seen as the following
difference:
∆D (k) = D (k + i) − D (k − i) .
(6.82)
A special case is given by a singularity at k = 0. Again using the propagator like 1/k with two singularities at
k ± i = 0 yields:
lim+
→0
1
1
−2i
−
= lim+ 2
= −2πiδ (k) .
→0 k + 2
k + i k − i
(6.83)
The two singularities which enclose the contour of integration pinch the singularity at k = 0 for → 0+ and
can be avoided using a residue i.e. the principle value approach. These kind of singularities are called pinch
singularities [112, 87].
B.6 Heaviside Step Function, Sign Function, its Representations and Applications
A step function can be defined variously but a special one is given by the Heaviside step function Θ (t). It is
defined as:


1,


Θ (t) := Θ0 ≡ Θ (0) ,



0,
t>0
t=0,
(6.84)
t<0
whereby Θ0 can be chosen as 0, 1/2 or 1. In this work it is set equal 1 when talking about sources and
particle densities and 0 if it is specially underlined but for integrals it does not matter whether it is set 1
or 0. The step function is used to implement restrictions about the spacetime parameters as functions i.e.
´∞
´b
finite integrals like a f (τ ) dτ can be treated as −∞ f (τ ) Θ (τ − a) Θ (b − τ ) dτ which offers a possibility to use
70
special representations for Θ. A special and useful representation is given by:
−1
Θ (t) =
lim
2πi →0+
ˆ∞
exp {−itκ}
1
dκ =
lim
κ + i
2πi →0+
−∞
ˆ∞
exp {itκ}
dκ.
κ − i
(6.85)
−∞
This integral representation allows to continue analytically definite integrals which are not directly solvable. It
is important that the time integral runs over the whole R-space because otherwise the forward and backward
Fourier transforms would not be inverse to each other. The product Θ (τ − a) Θ (b − τ ) is a simple boxcar
function which can be simplified for integrals to:
β (a, τ, b) := Θ (τ − a) Θ (b − τ ) = Θ (τ − a) − Θ (τ − b) (a < τ < b) .
(6.86)
This relation can be proven by multiplying a test function f (τ ) to it and build the integral:
ˆ
ˆ
b
f (τ ) dτ
∞
f (τ ) Θ (τ − a) Θ (b − τ ) dτ
=
−∞
a
=
F (b) − F (a)
=
F (∞) − F (a) − (F (b) − F (∞))
ˆ ∞
f (τ ) (Θ (τ − a) − Θ (τ − b))
=
−∞
ˆ∞
ˆ∞
dτ f (τ ) −
=
a
dτ f (τ ) .
b
The derivative of the Heaviside function gives the Dirac δ-function which can be shown applying the derivative
operator d/dt to (6.85) for = 0. The sign function can be defined and expressed by the Heaviside function
according to:



1,


σ (x) := Θ (x) − Θ (−x) = β (−x, 0, x) = 0,



−1,
x>0
x=0.
(6.87)
x<0
Finally, a limit representation will be shown which is important for integrals which cannot be solved analytically.
It is given as:
Θ (x) = lim
k→+∞
∞
X
n=0
n
(−1) e−kxn = lim
k→+∞
1
1 + e−kx



1,
x>0


= 1/2, x = 0 .



0,
x<0
(6.88)
Hereby, it can be directly recognized that the notation Θ (0) = 1/2 is taken. As a matter of fact, this definition
does not influence the result of an integral with a Heaviside function because it is just one point. For integrable
71
function this means the following:
ˆ∞
ˆ∞
dτ f (τ )
dτ Θ (τ ) f (τ )
=
−∞
0
(δ < )
=

∞
ˆδ
ˆ−
ˆ
dτ f (τ ) + dτ f (τ )
lim+  dτ f (τ ) −
→0
−∞
ˆδ
⇒
dτ f (τ ) → 0
−δ
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−δ