We investigate the development of electromagnetic cascades in ice using a
GEANT Monte Carlo simulation. We examine the Cherenkov pulse that is generated
by the charge excess that develops and propagates with the shower. This study
is important for the RICE experiment at the South Pole, as well as any test
beam experiment which seeks to measure coherent Cherenkov radiation from an
electromagnetic shower.Comment: 8 pages, 6 figure

Besson, David Z.

McKay, Douglas W.

Razzaque, Soebur

Seunarine, Surujhdeo

English

arXiv

Soebur Razzaque

Crossref

Signal characteristics from electromagnetic cascades in ice

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02
Signal Characteristics from
Electromagnetic Cascades in Ice
Soebur Razzaque, Surujhdeo Seunarine, David Z. Besson, and
Douglas W. McKay
Department of Physics and Astronomy
University of Kansas, Lawrence, Kansas 66045-2151
Abstract. We investigate the development of electromagnetic cascades in ice using a
GEANT Monte Carlo simulation. We examine the Cherenkov pulse that is generated
by the charge excess that develops and propagates with the shower. This study is
important for the RICE experiment at the South Pole, as well as any test beam exper-
iment which seeks to measure coherent Cherenkov radiation from an electromagnetic
shower.
I INTRODUCTION
Ultra high energy neutrinos can travel a long distance without scattering. By
detecting the sources of these particles, we can probe the physics of the standard
model (and beyond) at energies un-obtainable at current accelerators. An ultra
high energy neutrino can interact via a charge current interaction, giving most of
its energy to the secondary electron, which can then initiate an electromagnetic
cascade or shower. Askaryan [1] predicted a negative charge imbalance in the
cascade which gives rise to coherent Cherenkov radiation at radio frequencies. The
predicted flux of ultra high energy neutrinos is small and model dependent [2–4].
An experiment to detect ultra high energy neutrinos using radio antennas requires
a large volume because of the small flux. A dense, radio-transparent target is
needed for the small shower size needed for coherence and small signal attenuaton.
Antarctic ice is suitable for this purpose. A detailed analysis of such an experiment
was done [5] which concluded that a modest array of antennas can detect many
events per year. The Radio Ice Cherenkov Experiment (RICE) at the South Pole [6]
is a prototype designed to detect neutrinos with energy ≥ PeV using this method.
A reliable Monte Carlo simulation tool is needed to study the shower development,
Cherenkov radiation, detector, and data acquisiton system. One can also test the
idea of coherent Cherenkov emission at accelerator facilities by dumping bunches of
electrons or photons in a dense target like sand or salt or any other suitable medium
[7]. Zas, Halzen and Stanev (ZHS) [8] developed a Monte Carlo simulation to
study electromagnetic showers and Cherenkov emission in ice. Buniy and Ralston
[9] have also developed a method to estimate the coherent Cherenkov signal by
parametrizing the cascades.
We have developed a GEANT Monte Carlo simulation primarily to study the
coherent Cherenkov emmission in ice but it can easily be modified for other similar
studies with different materials. GEANT [10] is a well known and widely used
detector simulation package in particle physics. It allows access to all details of the
simulation such as controls of various processes, definition of target and detector
media, and a complete history of all events simulated. It can be used to simulate
all dominant processes in the 10 keV − 10 TeV energy range. Cross sections of
electromagnetic processes are reproduced in GEANT within a few percent up to a
hundred GeV . We use GEANT to simulate electromagnetic cascades inside mate-
rials from which we extract track information. We take this track information and
determine the resulting radio pulse using standard electrodynamics calculations.
We also calculate other shower parameters using the track and energy information
from GEANT.
II SHOWER DESCRIPTION
When a high energy electron (e−) or photon (γ) hits a material target, an elec-
tromagnetic cascade is created inside the material. Bremmstrahlung and pair pro-
duction are the dominant high energy processes at the beginning of the shower
development. Due to bremmstrahlung, an e− loses 1/e of it’s energy on average
over a distance X0, the radiation length. The secondary γ can then produce an
e+ e− pair. The number of particles thus grows exponentially. e− and e+ lose
energy due to ionization as they travel inside the material. After reaching a critical
energy(Ec), when the energy loss due to bremmstrahlung is equal the energy loss
due to ionization, e−, e+ lose their energy mostly due to ionization and the cascade
eventually stops. A rough estimate of the critical energy is Ec ≈ 605/Z MeV which
can be obtained by equating the radiation and the ionization loss formulae. Z is
the atomic number of the medium.
All models of an electromagnetic cascade, including a simplified model developed
by Heitler [11], and a more realistic model developed by Carlson and Oppenheimer
[12], show similar basic features which include linear scaling of the track lengths
and logarithmic scaling of the maximum position with primary energy.
The cascade is concentrated near an axis along the direction of the primary.
Multiple Couloumb scattering is responsible for the transverse spread of the cas-
cade. A quantity called Moliere radius (RM) is determined by the average angular
deflection per radiation length at the critical energy (Ec) and is used to estimate
the transverse spread. One can also look at the fraction of energy that escapes
transverse to the shower axis [13,14] to get a good estimate of RM . About 90% of
the primary energy is contained inside a tube of radius RM along the shower axis.
A relationship between the Moliere radius, the critical energy and the radiation
length is RM = X0Es/Ec where Es ≈ 21.2 MeV is the scale energy.
Low energy processes like Compton, Moller and Bhabha scatterings and positron
annihilation build up a net charge (more e− than e+) in the cascade, as atomic
electrons in the target medium are swept up into the forward moving shower.
III ELECTROMAGNETIC PULSE
High energy e− and e+ in the cascade can travel faster than the speed of light in
the medium and give rise to Cherenkov radiation. The electric field due to a single
charge moving uniformly from position ~r1 to ~r2 = ~r1+~vδt in the Fraunhoffer limit,
is given by
R~E(ω) =
µr√
2π
(
e
c2
)
eiω
R
c eiω(t1−n
~β·~r1)~vT
(eiωδt(1−n̂·
~βn) − 1)
1− n̂ · ~βn
(1)
where µr is the relative permeability and n is the refractive index of the medium.
The condition 1− n̂ · ~βn = 0 defines the Cherenkov angle θc as cos θc = 1/nβ. At
or very close to the Cherenkov angle or at low frequency, the equation (1) can be
reduced to the form
R~E(ω) =
µriω√
2π
(
e
c2
)
eiω
R
c eiω(t1−n
~β·~r1)~vT δt. (2)
See Fig. 1 for a description of various quantities in equations (1, 2).
R
Shower axis
r
1
t
1
r
2
t
2
v
v
T n̂
observer
FIGURE 1. Set-up to calculate electromagnetic pulse from a single track.
IV MONTE CARLO RESULTS
Since on average, the electrons lose 1/e of their energy in each radiation length
due to bremmstrahlung, one can fit an exponential to the energy loss data generated
by the Monte Carlo and find a value for the radiation length (X0). This will serve
as an internal consistency check to ensure that we are tracking all the particles
along with their energies. We generated 500 showers of 50 GeV electrons and
recorded the energy loss due to bremsstrahlung. A fit to this gives a radiation
length of 41.5± 3.2 cm. Given the molecular composition, GEANT also calculates
the medium’s radiation length. For ice it is 38.8 cm, which is in agreement with
the value we calculated.
Table I GEANT consistency checks
Parameters Iron Lead Ice
X0 (cm) (from ref [15]) 1.76 0.56 39.05
RM (cm) 2.1 1.6 13
Ec = X0Es/RM (MeV ) 17.77 7.42 63.7
Ec = 605/Z (MeV ) 23.3 7.4 83.8
To calculate the Moliere radius (RM) we construct an imaginary cylinder centered
on the shower axis (up to the physical length of the shower). We add the energy
(U), of all the tracks that leave the cylinder without re-entering. By varying the
radius of the cylinder we arrive at RM which is the radius of the cylinder when the
fraction U/E is equal to 0.1, E being the initial energy of the cascade. We have
checked RM for lead, iron and ice (see table I). One can also calculate the critical
energy (Ec) using two different formulae quoted earlier.
Shower depth (t) is measured in units of radiation length (X0) as t = x/X0.
We have simulated 30 GeV electron-induced cascades in iron. The longitudinal
profile was obtained by adding the number of particles with total energy greater
than 1.5 MeV crossing planes at every half radiation length perpendicular to the
shower axis as shown in Fig. 2a. The number of particles agrees reasonably with
EGS4 simulation of the same shower [15]. The Greisen-Rossi distribution for total
number of particles (electrons and positrons) is shown by the dashed line. A least
squares fit to the GEANT data with Greisen-Rossi distribution gives about 90%
confidence level.
We used the same method to study the longitudinal profile of showers in ice.
Simulation shows the correct scaling behavior of the number of particles with the
initial energy of the shower. The position of the shower maximum also shows
the correct logarithmic scaling with the initial energy. The charge excess (∆Q) is
defined as ∆Q = N(e)−N(p)
N(e)+N(p)
where N(e) and N(p) are the number of electrons and
positrons respectively as functions of shower depth. According to our simulations,
the net charge imbalance is 15%− 18% at the shower max. Note that there is no
direct experimental data on this.
A comparison of 100 GeV shower (averaged over 20 showers) with 611 KeV
threshold from GEANT to the same from ZHS Monte Carlo (Fig. 2b) shows about
a 25%− 35% discrepancy for the total number of particles at the shower max. The
discrepancy between the two simulations remains the same at higher energies.
In Fig. 3, we show the charge excess in a 500 GeV shower from GEANT with
kinetic energy threshold 1.5 MeV . This is an average of 20 showers and the charge
excess is broken down to different energy bins. It shows that most of the contribu-
tions to charge excess come from the energy range 1.5 to 30 MeV . The Cherenkov
pulse from a cascade is proportional to the total track length (energy deposition) of
the cascade. Our results show the correct scaling behavior of the track lengths with
the initial energy (Eo). If we consider all processes to be elastic except ionization,
then an upper bound for the total track length is L = Eo/
(
dE
dx
)avg
ion
where (dE
dx
)avgion is
the average ionization energy loss per unit length. To determine (dE
dx
) we generated
500 of 1 GeV tracks and kept a record of the rate at which energy was lost due to
ionization. Fig.4a shows the average (dE
dx
) for the 500 tracks. It can be seen that the
average value matches theoretical Bethe-Bloch result reasonably well. The average
ionization loss in the relativistic rise region is approximately 1.9 MeV/cm.
V PULSE CALCULATION
To calculate the electric pulse from the cascade, we added up the contributions (1)
from all charged tracks. There are two assumptions we made to evaluate pulse (1)
and (2) from GEANT track information. First, we assume an azimuthal symmetry
0
10
20
30
40
50
60
70
80
90
100
0 2 4 6 8 10 12 14 16 18 20
N
u
m
b
e
r 
o
f 
p
a
rt
ic
le
s 
(e
+
p
)
Depth in radiation length
Eth = 1.5 MeV
e+p (PDG)
Greisen
e+p (GEANT)
e+p (ZHS)
-20
0
20
40
60
80
100
120
140
160
180
0 2 4 6 8 10 12 14 16 18 20
N
u
m
b
e
r 
o
f 
p
a
rt
ic
le
s
Depth in radiation length
Eo = 100 GeV
GEANT (e+p)
ZHS (e+p)
GEANT (e-p)
ZHS (e-p)
FIGURE 2. (a) The longitudinal profile of a 30 GeV cascade in iron using GEANT (average
of 30 showers) and comparison with the same produced by EGS4 (from particle data book) and
by ZHS Monte Carlos. The error bars correspond to standard error. (b) Comparisons between
the profiles of a 100 GeV cascade (averaged over 20 showers each) from GEANT and from ZHS
Monte Carlos. The total energy threshold is 0.611 MeV in both cases.
about the shower axis, which allows us to evaluate pulse equations in 2-dimensions.
This is a good approximation as long as we have many tracks in the shower. A
typical 500 GeV shower has thousands of tracks and the number goes up as we
increase the energy. We checked this approximation on a shower-by-shower basis
and did not find any noticeable difference between the pulses evaluated in the x−z
plane and in the y− z plane (z - is the shower axis). Second, we evaluate the times
t1 and δt in (1) and (2) assuming that the particles travel at the speed of light. This
is a good approximation if the frequency is not too high as can be seen from (1).
In Fig. 4b, we calculate pulses1 from 1 TeV and 500 GeV showers at 1 GHz and
300 MHz frequencies. The pulse height at the Cherenkov peak (≈ 55.8o) scales
with the energy of the shower, and the Gaussian half width is inversely proportional
to the frequency which is analogous to a slit diffraction pattern.
VI CONCLUSION
We find distinctive, coherent, Cherenkov radio frequency emission from simu-
lated electromagnetic cascades in ice with energies in the 100 GeV to 1 TeV range.
The multi-purpose, detector simulation package GEANT provides a suitable mod-
elleing of electromagnetic shower details for this purpose, with the flexibility to
include hadronic cascades in future studies. This work also serves as an indepen-
dent treatment of the problem, which has been studied previously [8]. The GEANT
1) we use different normalization here for the field, which requiring that one multiply (1) and (2)
by 2
√
2π
2 4 6 8 10 12 14 16 18 20
Depth
2.5
5
7.5
10
12.5
15
e -p
FIGURE 3. Excess charge plot for a 500 GeV cascade (averaged over 20 showers) in ice using
GEANT. The excess charge is shown in energy bins (from bottom to top) 1.5-5 MeV (white),
5-30 MeV (light gray), 30-100 MeV (dark gray) and above 100 MeV (black).
generated showers are qualitatively in agreement with those in [8]. The showers
differ in several details; however our far-field calculation of the pulse from a shower
gives the same result up to a GHz frequency as that of [8] when both are applied
to the same shower simulation. This has been checked explicitly by calculating
the field using our field code and the track informtions from ZHS code with the
direct field output from ZHS code for the same shower. The differences between a
GEANT and a ZHS shower of initial energy 100 GeV is summarized in Table II.
The origin of differences between the two Monte Carlos is under study.
Our direct calculation of Moliere radius, critical energy and shower population
as a function of shower depth using GEANT all agree within a few percent with
measurements or with published EGS4 simulations for iron and lead [15]. The
extension to ice is straightforward and we expect the same level of reliability here.
Our results agree with the theoretically expected [16] linear scaling of num-
ber of particles at shower maximum and logarithimic scaling of depth at shower
maximum with energy. These are not straightforward results, since the theoret-
ical predictions are based on inclusion of bremmstrahlung and pair production
only, while the physics at the shower maximum is not dominated by these pro-
cesses. It includes important contributions from Compton, Bhabha and Moller
processes as well, which give rise to charge imbalance and, consequently, coherent
radio Cherenkov field emission.
0
1
2
3
4
5
6
0.1 1 10 100 1000
(d
E
/d
x)
 io
n
iz
a
tio
n
 (
M
e
V
/c
m
)
Energy (MeV)
Bethe-Bloch
GEANT average
GEANT data
0
1e-08
2e-08
3e-08
4e-08
5e-08
6e-08
30 35 40 45 50 55 60 65 70 75 80
R
 x
 E
(n
u
) 
in
 v
o
lts
/M
H
z
Angle in degrees
500 GeV 300 MHz
500 GeV 1 GHz
1TeV 300 MHz
1 TeV 1 GHz
FIGURE 4. (a) (dE/dx)ionization calculated from GEANT output. The average is over 500
electron tracks with 0.1 MeV kinetic energy threshold. We also show theoretical Bethe-Bloch
curve. (b) Electromagnetic pulses from cascades in ice generated by GEANT. Pulse height scales
linearly and pulse width scales inversely with primary energy. Fraunhoffer limit has been taken
to calculate field.
Table II Comparisons between GEANT and ZHS Monte Carlos
(averaged over 20 showers each)
Quantity GEANT ZHS
Primary Energy (GeV ) 100 100
Total Energy Threshold (MeV ) 0.611 0.611
Total absolute track length (meter) 398.8± 4.7 642
Total projected (e + p) track length (meter) 374.4± 4.3 518.7
(sum of electron and positron track lengths
projected along the shower axis)
Total projected (e− p) track length (meter) 70.0± 8.4 131.2
(difference of electron and positron track lengths
projected along the shower axis)
Position of the shower max. (radiation length) 7 7
Number of particles (e + p) at shower max. 111± 26 155± 45
Excess electrons (e− p) at shower max. 20± 11 37± 14
Fractional charge excess at the shower max ∼ 18% ∼ 24%
Cherenkov peak at 1 GHz (V olts/MHz) 7.46× 10−9 1.08× 10−8
Acknowledgement Thanks to J. Alvarez-Muñiz, T. Bolton, R. Buniy, G.
Frichter, F. Halzen, J. Ralston, D. Seckel and E. Zas for help and advice at various
stages. This work is supported in part by the NSF, the DOE, the University of
Kansas General Research Fund, the RESEARCH CORPORATION and the facili-
ties of the Kansas Institute for Theoretical and Computational Science.
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5. G. Frichter, D. W. McKay, and J. P. Ralston. Phys. Rev., D53:1684, 1996.
6. RICE Collaboration. ICRC proceedings, 1997.
7. D. Saltzberg et al. hep-ex/0011001, 2000.
8. E. Zas, F. Halzen, and T. Stanev. Phys. Rev., D45:362, 1992.
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