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A study of electron recombination using highly ionizing particles in the ArgoNeuT Liquid Argon TPC

TL;DR: The dependence of recombination on the track angle with respect to the electric field direction is much weaker than the predictions of the Jaffe columnar theory and by theoretical-computational simulations as mentioned in this paper.
Abstract: Electron recombination in highly ionizing stopping protons and deuterons is studied in the ArgoNeuT detector. The data are well modeled by either a Birks model or a modified form of the Box model. The dependence of recombination on the track angle with respect to the electric field direction is much weaker than the predictions of the Jaffe columnar theory and by theoretical-computational simulations.

Summary (3 min read)

1. Introduction

  • Liquid Argon Time Projection Chambers (LAr TPCs) offer excellent calorimetry and mm-scale position resolution in a large volume and are increasingly favored for the next generation of neutrino detectors.
  • Particles in the MeV - few GeV energy range of interest for neutrino detectors are usually contained within the TPC and can be well characterized by range, calorimetry and decay topology.
  • Calorimetric reconstruction requires calibration to correct for detector specific effects such as liquid argon impurities and electronics response and calibration for charge loss due to electron-ion recombination.
  • The data were taken during an exposure of 1.35x1020 protons on target in the Fermilab NuMI neutrino beam[1].

2. Recombination Models

  • Electrons emitted by ionization are thermalized by interactions with the surrounding medium after which time they may recombine with nearby ions.
  • In the columnar model of Jaffe[3], published in 1913, recombination depends on the collective electron and ion charge density from multiple ionization interactions in a cylindrical volume surrounding the particle trajectory.
  • The relative importance of these two theories for liquid argon can be estimated by comparing the average electron-ion distance with the average ion-ion distance.
  • These studies favor a columnar theory approach and imply that collective effects are important.
  • (2.2) In this equation, s is a dimensionless variable that characterizes the time dependent overlap of the electron and ion distributions.

2.1 Theoretical application

  • The Birks and Box model equations do not provide a consistent global description of all data, which is not remarkable considering the assumptions made in their development.
  • They do however provide good agreement with data in some regimes, for example radioactive source data where dE/dx is fixed and the electric field is varied.
  • A technical difficulty arises when applying a Birks model correction to highly ionizing particles.
  • The recombination factor using the Birks model with ICARUS parameters is shown by the blue curve with an electric field of 0.5 kV/cm and liquid argon density of 1.383 g/cm3.
  • One can adjust the canonical Box model β value to 0.30 cm/MeV to match the blue curve at dE/dx = 7 MeV/cm and achieve good qualitative agreement to very high values of stopping power (solid red curve).

3. Calorimetric Reconstruction in the ArgoNeuT Detector

  • The ArgoNeuT detector is described in reference [10].
  • All data were taken with an electric field of 0.481 kV/cm.
  • Clusters in each wire plane, or view, are matched to form three-dimensional (3D) tracks comprised of a set of space points.
  • A calorimetric measurement of the stopping power, (dE/dx)calo, is then found for each space point by applying a recombination correction using equation 2.9.
  • The rms difference between (dE/dx)calo and (dE/dx)hyp is found for each value of ∆.

4. Stopping Particle Identification

  • Unlike the situation in a test beam where the incident particle type and energy is known, a wide variety of particles are produced by the neutrino beam in a wide range of energies.
  • The authors introduce a particle identification technique using this feature that is intended to minimize potential sources of selection bias.
  • The energy lost by ionization in the first step is found by multiplying the stopping power as calculated from the BetheBloch equation[11] by the step length.
  • Table 1 shows the power law parameterization for particles of interest in this analysis.
  • If, for example, there is an overall scale error in the detector calibration, the mean values of the Gaussian peaks will be shifted from the expected values of A. Likewise, use of an incorrect recombination correction will shift the peaks and possibly broaden the PIDA distributions.

5. Data Selection

  • There are several sources of protons that will stop in the detector.
  • Fully contained protons produced by neutrino interactions in the detector are an obvious source.
  • Tracks entering the detector are not rejected.
  • The shaded histograms in figure 4 show the PIDA distributions of these samples.
  • The pion and deuteron Gaussian distributions are extrapolated into the proton selection region to estimate the proton selection purity (≈95%) and efficiency (≈95%).

6. Detector Simulation

  • The LArSoft simulation is based on GEANT4.
  • One track travels parallel to the wire plane (φ = 90◦) and the other is inclined relative to the wire plane.
  • Ionization electrons arrive at the collection plane with a larger spread in time for the inclined track case than for the parallel track case.
  • Hits and tracks are then reconstructed using the same calorimetry code that is used to analyze the data.
  • The authors next confirm that this calibration correction does not introduce an artificial angular dependence in the recombination analysis.

7. Recombination Simulation

  • Jaskolski and Wojcik provided a theoretical-computational approach to the columnar model in [16] to test the validity of the theoretical premises.
  • Positive ions are uniformly distributed along a line representing the track trajectory.
  • The simulation results are insensitive to the assumptions used for the initial electron distributions because the magnitude of the initial state conditions is small compared to conditions after thermalization.
  • The poorer agreement at low dE/dx can be rectified by introducing a factor that accounts for energy loss due to δ -rays which are not included in the model.
  • Studies show that the recombination factor becomes independent of ymax for tracks at φ = 40◦ when ymax ≥ 7000 nm.

8.1 Proton Sample

  • The Birks and Box recombination parameters are found by fitting histograms of dQ/dx vs (dE/dx)hyp.
  • The error on (dE/dx)hyp is found by propagating the 0.1 cm residual range error (Section 3) using equation 4.1.
  • The recombination fits in each angle bin are shown in figure 10.
  • Both the Birks and modified Box model equations provide a good representation of the data in the range 2 MeV/cm < dE/dx < 24 MeV/cm.

8.2 Deuteron Sample

  • Recombination at higher stopping power can in principle be studied with deuterons.
  • The small number of tracks in their sample limits its usefulness however.
  • The deuteron sample is subjected to the stopping point fitting algorithm described above with (dE/dx)hyp calculated using the deuteron hypothesis.
  • The red points and curve are the data and modified Box fit parameterization for protons in the same φ bin.
  • The authors conclude that deuterons are indeed present in this sample and that the recombination fits found above are applicable to dE/dx = 35 MeV/cm.

9. Discussion

  • The significant disagreement between published data and theory belies that statement however.
  • A calculation of the Debye length using the measured drift electron lifetimes in the ArgoNeuT detector results in a screening length of 400 - 600 nm.
  • This is four times larger than the Onsager length [2] and should therefore not be a significant contribution.
  • This result can be understood by noting that the electron thermalization distance in liquid argon is ∼2500 nm[4], so the electrons that are assumed to escape recombination at ymax = 2000 nm have rather high kinetic energies.
  • This may be due to the presence of positive and negative charges that are left from different ionization events.

10. Conclusions

  • The angular dependence is significantly weaker than that predicted by the Jaffe columnar theory and by a recombination simulation.
  • As noted above, this is not the sole example of disagreement with recombination models.
  • The authors have presented two possibilities to explain this discrepancy.
  • Both possibilities have a common theme - that impurities in the argon may affect the micro-physics of electron recombination.

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source: https://doi.org/10.7892/boris.58426 | downloaded: 10.8.2022
Preprint typeset in JINST style - HYPER VERSION
A study of electron recombination using highly
ionizing particles in the ArgoNeuT Liquid Argon TPC
R. Acciarri
a
, C. Adams
b
, J. Asaadi
c
, B. Baller
a,
, T. Bolton
d
, C. Bromberg
e
,
F. Cavanna
b, f
, E. Church
b
, D. Edmunds
e
, A. Ereditato
g
, S. Farooq
d
, B. Fleming
b
,
H. Greenlee
a
, G. Horton-Smith
d
, C. James
a
, E. Klein
b
, K. Lang
h
, P. Laurens
e
,
D. McKee
d
, R. Mehdiyev
h
, B. Page
e
, O. Palamara
b,i
, K. Partyka
b
, G. Rameika
a
,
B. Rebel
a
, M. Soderberg
a,c
, J. Spitz
b
, A.M. Szelc
b
, M. Weber
g
, M. Wojcik
j
, T. Yang
a
,
G.P. Zeller
a
a
Fermi National Accelerator Laboratory, Batavia, IL 60510 USA
b
Yale University, New Haven, CT 06520 USA
c
Syracuse University, Syracuse, NY 13244 USA
d
Kansas State University, Manhattan, KS 66506 USA
e
Michigan State University, East Lansing, MI 48824 USA
f
Universitá dell’Aquila e INFN, L’Aquila, Italy
g
University of Bern, Bern, Switzerland
h
The University of Texas at Austin, Austin, TX 78712 USA
i
INFN - Laboratori Nazionali del Gran Sasso, Assergi, Italy
j
Lodz University of Technology, Lodz, Poland
E-mail:baller@fnal.gov
ABSTRACT: Electron recombination in highly ionizing stopping protons and deuterons is studied
in the ArgoNeuT detector. The data are well modeled by either a Birks model or a modified form
of the Box model. The dependence of recombination on the track angle with respect to the electric
field direction is much weaker than the predictions of the Jaffe columnar theory and by theoretical-
computational simulations.
KEYWORDS: Time projection chambers; Noble-liquid detectors; Data analysis.
Corresponding author.
arXiv:1306.1712v1 [physics.ins-det] 7 Jun 2013

Contents
1. Introduction 1
2. Recombination Models 2
2.1 Theoretical application 3
3. Calorimetric Reconstruction in the ArgoNeuT Detector 5
4. Stopping Particle Identification 5
5. Data Selection 7
6. Detector Simulation 10
7. Recombination Simulation 11
8. Analysis and Results 12
8.1 Proton Sample 12
8.2 Deuteron Sample 15
9. Discussion 15
10. Conclusions 16
1. Introduction
Liquid Argon Time Projection Chambers (LAr TPCs) offer excellent calorimetry and mm-scale po-
sition resolution in a large volume and are increasingly favored for the next generation of neutrino
detectors. Particles in the MeV - few GeV energy range of interest for neutrino detectors are usually
contained within the TPC and can be well characterized by range, calorimetry and decay topology.
Calorimetric reconstruction requires calibration to correct for detector specific effects such as liq-
uid argon impurities and electronics response and calibration for charge loss due to electron-ion
recombination.
In this paper, we report on a study of the recombination of electron-ion pairs produced by
ionizing tracks in a sample of stopping protons and deuterons in the ArgoNeuT detector with a par-
ticular emphasis on the angular dependence. The data were taken during an exposure of 1.35x10
20
protons on target in the Fermilab NuMI neutrino beam[1].
1

2. Recombination Models
Electrons emitted by ionization are thermalized by interactions with the surrounding medium after
which time they may recombine with nearby ions. The Onsager geminate theory[2] presumes that
the dominant recombination process is the re-attachment of the electron to the parent ion under the
influence of the Coulomb field of the pair. In the columnar model of Jaffe[3], published in 1913,
recombination depends on the collective electron and ion charge density from multiple ionization
interactions in a cylindrical volume surrounding the particle trajectory.
The relative importance of these two theories for liquid argon can be estimated by comparing
the average electron-ion distance with the average ion-ion distance. The average separation dis-
tance between ions, r
ion
, is W
ion
/(dE/dx) where W
ion
= 23.6 eV is the energy required to ionize an
argon atom. For the range of particle stopping power, dE/dx, in this analysis, r
ion
varies between
10 and 50 nm. Transient conductivity measurements of ionization produced by charged particles
show that electrons reach thermal energies in 1 - 2 ns [4] in liquid argon during which time the elec-
trons are estimated to travel O(10
3
) nm. As a result, the probability of geminate recombination
should be small since the electron-ion separation distance is much larger than the ion-ion separation
distance. A simulation of thermalization and recombination in [5] estimates that the probability of
geminate recombination is 10
3
. These studies favor a columnar theory approach and imply that
collective effects are important.
In the Jaffe theory, electrons and ions are assumed to have a Gaussian spatial distribution
around the particle trajectory during the entire recombination phase. The spatial distribution and
the charge mobility, µ, are assumed to be equal for electrons and ions. With these assumptions,
Jaffe found that the fraction that survive recombination is
R
J
=
1 +
αN
o
8πD
r
π
z
0
S(z
0
)
1
where z
0
=
b
2
µ
2
E
2
sin
2
φ
2D
2
, (2.1)
where α is a recombination coefficient, N
o
is the number of ion-electron pairs per unit length, D is
the diffusion coefficient, E is the electric field and b = r
o
p
4/π where r
o
is the average ion-electron
separation distance after thermalization. The variable φ is the angle between the electric field and
the particle direction. Equation 2.1 is now referred to as Birks law [6] which was developed to
model the scintillation light yield of a particle as a function of the stopping power. The angular
dependence inherent in the dimensionless variable z
0
may be modified by the S(z
0
) factor,
S(z
0
) =
1
π
Z
0
e
s
ds
p
s(1 +s/z
0
)
. (2.2)
In this equation, s is a dimensionless variable that characterizes the time dependent overlap of the
electron and ion distributions. Numerical integration of equation 2.2 shows that S(z
0
) approaches
one when z
0
exceeds 10. To estimate z
0
for this analysis, we assume that electrons are thermalized
and use the Einstein-Smoulchowski relation D/µ E
thermal
0.01 eV for liquid argon. We set E
to a low electric field typically used in a LAr TPC (0.5 kV/cm), set r
o
= 2500 nm[5] and find that
z
0
is typically > 40 for the range of angles in this analysis. The variation in S(z
0
) is less than 1% in
this range, allowing the use of a more compact form
2

R
J
= [1 + k
c
(dE/dx)/(E sinφ)]
1
, (2.3)
where k
c
is a constant that is assumed to be specific to liquid argon.
Thomas and Imel [7] noted that electron diffusion and ion mobility are negligible in liquid ar-
gon during recombination. After dropping these terms in the Jaffe diffusion equations and applying
“Box model” boundary conditions they find
R
Box
=
1
ξ
ln(α + ξ), where ξ = k
Box
N
o
/4a
2
µE . (2.4)
The quantity N
o
/4a
2
represents the charge density in a microscopic box of size a. The variable
α is explicitly equal to one in the canonical form of the model. The model was developed to
parameterize the electric field dependence of recombination using radioactive sources where the
charge density in the box is a fixed quantity. The similarity between the Box model and the Jaffe
theory becomes clear by performing a series expansion of equation 2.4, ln(1 + ξ)/ξ = 1/(1 +
ξ /2...). The Jaffe and Box recombination k factors are related by a multiplicative factor when ξ <
1.
The electric field is held constant in this analysis but the charge density varies along a track.
We therefore associate the charge density with dE/dx and re-cast ξ in the form
ξ = β(dE/dx), (2.5)
where β is a parameter found by performing a recombination fit. There is no explicit mention of
angular dependence in the Box model. It is generally accepted that E can be replaced by E sinφ .
2.1 Theoretical application
Ideally, a theoretical understanding of recombination would result in a prescription for calorimet-
ric reconstruction of particles in a liquid argon TPC. The Birks and Box model equations do not
provide a consistent global description of all data, which is not remarkable considering the as-
sumptions made in their development. They do however provide good agreement with data in
some regimes, for example radioactive source data where dE/dx is fixed and the electric field is
varied. ICARUS[
8] was the first experiment to study recombination with variable dE/dx and vari-
able E in the range of interest for neutrino liquid argon TPCs. The ICARUS data are well described
by a Birks form similar to equation 2.3:
R
ICARUS
=
A
B
1 + k
B
·(dE/dx)/E
(2.6)
with A
B
= 0.800 ± 0.003 and k
B
= 0.0486 ± 0.0006 (kV/cm) (g/cm
2
)/MeV. Note that in the high
electric field limit, R
ICARUS
0.8 whereas the canonical box model would predict R
Box
1. The
recombination factor approaches 0 for the canonical forms of both models. This expected behavior
at small electric field is supported by measurements with relativistic heavy ions (R 0.003) but
not with low energy electrons (R 0.35)[9].
A technical difficulty arises when applying a Birks model correction to highly ionizing parti-
cles. Calorimetric reconstruction of particle tracks relies on the equation
3

dE/dx = (dQ/dx)/(R W
ion
), (2.7)
to determine the stopping power given a measured value of the charge deposited per unit length,
dQ/dx, along the particle trajectory. The result, using the Birks model form for R is
dE/dx =
dQ/dx
A
B
/W
ion
k
B
·(dQ/dx)/E
. (2.8)
Spurious values of dE/dx result when the denominator approaches zero (at large values of dQ/dx).
In contrast, the inverse Box equation has an exponential form that does not suffer from this malady:
dE/dx = (exp(βW
ion
·(dQ/dx)) α)/β . (2.9)
In summary, the Birks model provides a consistent description of recombination over a limited
range of dE/dx but there are technical difficulties applying it to calorimetric reconstruction at high
ionization. The Box model has no technical difficulties but has inadequate behavior at low dE/dx.
This deficiency in the Box model can be corrected by allowing α < 1 as illustrated in figure
1. The recombination factor using the Birks model with ICARUS parameters is shown by the blue
curve with an electric field of 0.5 kV/cm and liquid argon density of 1.383 g/cm
3
. One can adjust
the canonical Box model β value to 0.30 cm/MeV to match the blue curve at dE/dx = 7 MeV/cm
and achieve good qualitative agreement to very high values of stopping power (solid red curve).
The significant disagreement at low dE/dx can be eliminated by setting α = 0.93 while keeping
β = 0.30 cm/MeV (dotted red curve). In this report we will refer to this case as a “modified Box”
model.
Figure 1. Qualitative comparison of the recombination factor for the Birks equation using ICARUS param-
eters (blue curve), the “canonical” Box model with α = 1 and β = 0.30 (red curve) and a “modified Box"
equation with α = 0.93 and β = 0.30 cm/MeV (dotted red curve).
In the analysis of ArgoNeuT data, we perform fits to the Birks model recombination param-
eters A
Argo
and k
Argo
in different angular bins, and to the modified Box model recombination pa-
rameters α and β in different angular bins to test the validity of the φ dependence predicted by the
columnar theory.
4

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References
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Journal ArticleDOI
S. Agostinelli1, John Allison2, K. Amako3, J. Apostolakis4, Henrique Araujo5, P. Arce4, Makoto Asai6, D. Axen4, S. Banerjee7, G. Barrand, F. Behner4, Lorenzo Bellagamba8, J. Boudreau9, L. Broglia10, A. Brunengo8, H. Burkhardt4, Stephane Chauvie, J. Chuma11, R. Chytracek4, Gene Cooperman12, G. Cosmo4, P. V. Degtyarenko13, Andrea Dell'Acqua4, G. Depaola14, D. Dietrich15, R. Enami, A. Feliciello, C. Ferguson16, H. Fesefeldt4, Gunter Folger4, Franca Foppiano, Alessandra Forti2, S. Garelli, S. Gianì4, R. Giannitrapani17, D. Gibin4, J. J. Gomez Y Cadenas4, I. González4, G. Gracia Abril4, G. Greeniaus18, Walter Greiner15, Vladimir Grichine, A. Grossheim4, Susanna Guatelli, P. Gumplinger11, R. Hamatsu19, K. Hashimoto, H. Hasui, A. Heikkinen20, A. S. Howard5, Vladimir Ivanchenko4, A. Johnson6, F.W. Jones11, J. Kallenbach, Naoko Kanaya4, M. Kawabata, Y. Kawabata, M. Kawaguti, S.R. Kelner21, Paul R. C. Kent22, A. Kimura23, T. Kodama24, R. P. Kokoulin21, M. Kossov13, Hisaya Kurashige25, E. Lamanna26, Tapio Lampén20, V. Lara4, Veronique Lefebure4, F. Lei16, M. Liendl4, W. S. Lockman, Francesco Longo27, S. Magni, M. Maire, E. Medernach4, K. Minamimoto24, P. Mora de Freitas, Yoshiyuki Morita3, K. Murakami3, M. Nagamatu24, R. Nartallo28, Petteri Nieminen28, T. Nishimura, K. Ohtsubo, M. Okamura, S. W. O'Neale29, Y. Oohata19, K. Paech15, J Perl6, Andreas Pfeiffer4, Maria Grazia Pia, F. Ranjard4, A.M. Rybin, S.S Sadilov4, E. Di Salvo8, Giovanni Santin27, Takashi Sasaki3, N. Savvas2, Y. Sawada, Stefan Scherer15, S. Sei24, V. Sirotenko4, David J. Smith6, N. Starkov, H. Stoecker15, J. Sulkimo20, M. Takahata23, Satoshi Tanaka30, E. Tcherniaev4, E. Safai Tehrani6, M. Tropeano1, P. Truscott31, H. Uno24, L. Urbán, P. Urban32, M. Verderi, A. Walkden2, W. Wander33, H. Weber15, J.P. Wellisch4, Torre Wenaus34, D.C. Williams, Douglas Wright6, T. Yamada24, H. Yoshida24, D. Zschiesche15 
TL;DR: The Gelfant 4 toolkit as discussed by the authors is a toolkit for simulating the passage of particles through matter, including a complete range of functionality including tracking, geometry, physics models and hits.
Abstract: G eant 4 is a toolkit for simulating the passage of particles through matter. It includes a complete range of functionality including tracking, geometry, physics models and hits. The physics processes offered cover a comprehensive range, including electromagnetic, hadronic and optical processes, a large set of long-lived particles, materials and elements, over a wide energy range starting, in some cases, from 250 eV and extending in others to the TeV energy range. It has been designed and constructed to expose the physics models utilised, to handle complex geometries, and to enable its easy adaptation for optimal use in different sets of applications. The toolkit is the result of a worldwide collaboration of physicists and software engineers. It has been created exploiting software engineering and object-oriented technology and implemented in the C++ programming language. It has been used in applications in particle physics, nuclear physics, accelerator design, space engineering and medical physics.

18,904 citations

DOI
01 Jan 2005
TL;DR: The 2005 version of the Fluka particle transport code is described in this article, where the basic notions, modular structure of the system, and an installation and beginner's guide are described.
Abstract: This report describes the 2005 version of the Fluka particle transport code. The first part introduces the basic notions, describes the modular structure of the system, and contains an installation and beginner's guide. The second part complements this initial information with details about the various components of Fluka and how to use them. It concludes with a detailed history and bibliography.

2,271 citations

ReportDOI
14 Dec 2005
TL;DR: The 2005 version of the Fluka particle transport code is described in this article, where the basic notions, modular structure of the system, and an installation and beginner's guide are described.
Abstract: This report describes the 2005 version of the Fluka particle transport code. The first part introduces the basic notions, describes the modular structure of the system, and contains an installation and beginner's guide. The second part complements this initial information with details about the various components of Fluka and how to use them. It concludes with a detailed history and bibliography.

1,896 citations

Journal ArticleDOI
Lars Onsager1
TL;DR: In the absence of forces other than the Coulomb attraction, the probability of escape equals the reciprocal of the Boltzmann factor as discussed by the authors, which is the proper procedure whenever the Langevin factor equals unity, as in gases at high pressures.
Abstract: The probability that a pair of ions of given initial separation will recombine with each other is computed from the laws of Brownian motion, which is the proper procedure whenever the Langevin factor equals unity, as in gases at high pressures. In the absence of forces other than the Coulomb attraction, the probability of escape equals the reciprocal of the Boltzmann factor. This result includes the correlation between temperature and pressure coefficients of the ionization by light particles previously predicted by Compton, Bennett and Stearns, if one allows their basic hypothesis about the laws which govern the initial separation of the ions. The effect of an electric field is to increase the fraction of escaping ions by a factor which in the incipient stage of the effect is proportional to the field intensity and independent of the initial distance, although it depends on the orientation of an ion pair. The predicted increase of the ionization current is a little more than one percent for every 100 volts/cm, which accounts for the observed effects of fields exceeding 1500 volts/cm. A reasonable amount of columnar recombination would help to explain the proportionately greater effects of weak fields. The inferred initial separations of the ions are apparently compatible with present knowledge of electron scattering and attachment.

1,598 citations

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Frequently Asked Questions (2)
Q1. What are the contributions in this paper?

Electron recombination in highly ionizing stopping protons and deuterons is studied in the ArgoNeuT detector. 

The authors have presented two possibilities to explain this discrepancy. Both possibilities have a common theme - that impurities in the argon may affect the micro-physics of electron recombination.