2024-182 CDS-Library

C. Amsler1, H. Breuker2, M. Bumbar3,4, M. Cerwenka1,4, G. Costantini5,6, R.
Ferragut7,8, M. Fleck1††thanks: Present address: University College London,
Department of Physics & Astronomy, London, United Kingdom., M. Giammarchi8, A.
Gligorova1††thanks: Present address: University of Vienna, Faculty of Physics,
Vienna, Austria.††thanks: Electronic mail<EMAIL_ADDRESS>G.
Gosta5,6, E. D. Hunter1††thanks: Present address: Experimental Physics
Department, CERN, Geneva, Switzerland.,, C. Killian1, B. Kolbinger1††thanks:
Present address: DATA TECHNOLOGY, Vienna, Austria., V. Kraxberger1,4, N.
Kuroda9, M. Lackner1, M. Leali5,6, G. Maero8,10, V. Mascagna5,6, Y. Matsuda9,
S. Migliorati5,6, D. J. Murtagh1, A. Nanda1,4, L. Nowak3,4, S. Rheinfrank1, M.
Romé8,10, M. C. Simon1, M. Tajima11††thanks: Present address: Japan
Synchrotron Radiation Research Institute, Hyogo, Japan., V. Toso8,10, S.
Ulmer2, M. van Beuzekom12, L. Venturelli5,6, A. Weiser1,4, E. Widmann1, Y.
Yamazaki2, J. Zmeskal1

(The ASACUSA-Cusp Collaboration)

1Stefan Meyer Institute for Subatomic Physics, Austrian Academy of Sciences,
Vienna, Austria
2Ulmer Fundamental Symmetries Laboratory, RIKEN, Saitama, Japan
3Experimental Physics Department, CERN, Geneva, Switzerland
4University of Vienna, Vienna Doctoral School in Physics, Vienna, Austria
5Dipartimento di Ingegneria dell’Informazione, Università degli Studi di
Brescia, Brescia, Italy
6INFN sez. Pavia, Pavia, Italy
7L-NESS and Department of Physics, Politecnico di Milano, Como, Italy
8INFN sez. Milano, Milan, Italy
9Institute of Physics, Graduate School of Arts and Sciences, University of
Tokyo, Tokyo, Japan
10Dipartimento di Fisica, Università degli Studi di Milano, Milan, Italy
11RIKEN Nishina Center for Accelerator-Based Science, Saitama, Japan
12Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands

The mechanism of antiproton-nucleus annihilation at rest is not fully
understood, despite substantial previous experimental and theoretical work. In
this study we used slow extracted, sub-keV antiprotons from the ASACUSA
apparatus at CERN to measure the charged particle multiplicities and their
energy deposits from antiproton annihilations at rest on three different
nuclei: carbon, molybdenum and gold. The results are compared with predictions
from different models in the simulation tools Geant4 and FLUKA. A model that
accounts for all the observed features is still missing, as well as
measurements at low energies, to validate such models.

Keywords: antiproton annihilation, nucleus, Monte Carlo simulations,
annihilation products, low energy, nuclear reaction

##  1 Introduction

Antiproton ($\mathrm{\bar{p}}$) annihilation on a nucleus (A) is a particular
nuclear reaction in which a relatively large amount of energy is released.
Part of this energy is transferred to the nucleus with reduced linear and
angular momentum, compared to collisions with protons or heavy ions.
Consequently, the decay mechanism of nuclei heated by stopped and energetic
antiprotons allows the observation of some effects characteristic of heavy-ion
reactions, such as multifragment disintegration, without nuclear compression
[1, 2, 3]. Our understanding of the mechanism of antiproton annihilation in
nuclear matter is based on observables such as multiplicity, energy and
angular distributions of secondary particles. Various models, including the
Intranuclear Cascade (INC) model [4, 5], the statistical multifragmentation
model [6] or the Lanzhou quantum molecular-dynamics approach [7] describe the
nuclear dynamics and decay mechanism induced by antiprotons with varying
degrees of success. A great amount of complexity in the antiproton-nucleus
annihilation comes from the rich variety of possible reaction channels
triggered by the primary annihilation mesons.

Annihilation at rest most of the time occurs on a single nucleon in the
peripheral nuclear region, where the nucleon density is less than 10% of the
central density [5]. For $\mathrm{\bar{p}p}$ annihilation, the total energy of
1877 MeV is converted into an average of 3.0$\pm$0.2 charged pions
($\pi^{\pm}$) and 2.0$\pm$0.2 neutral pions ($\pi^{0}$), with an average
kinetic energy of 230 MeV [8, 9]. Kaons and $\eta$ mesons are produced in
about 6% and 7% of all annihilations, respectively. When the annihilation
occurs on a neutron, on average 1.07$\pm$0.04 $\pi^{+}$ and 2.07$\pm$0.04
$\pi^{-}$ are produced [10]. Absorption or scattering (final state
interactions, FSI) of the emitted mesons may follow, with or without break-up
of the nucleus, which depends upon the solid angle under which the remaining
nucleons are seen from the annihilation point [11], and which may modify the
detected meson distribution [12]. While many annihilation channels have been
measured for antiproton-nucleon ($\mathrm{\bar{p}}$N) annihilation at rest,
for $\mathrm{\bar{p}}$A the branching ratios have been hitherto reported only
for $\mathrm{{}^{3}He}$ and $\mathrm{{}^{4}He}$ [13, 10, 14], and the total
prong multiplicity has been measured for a very limited number of nuclei [15].
Furthermore, even for $\mathrm{\bar{p}N}$ annihilation the current models do
not perform satisfactorily in describing all the aspects of the measured data.

The bulk of $\mathrm{\bar{p}A}$ data originates from LEAR (Low Energy
Antiproton Ring) experiments, taken about three decades ago by degrading
$\sim$20 MeV antiprotons in a number of moderators, and ultimately stopping
them in thick targets. In these measurements only light prongs (charged
particles) were detected (helium and lithium ions being the heaviest), in
certain energy intervals and within narrow solid angles [3, 16, 1, 17].
Features such as particle yields and momentum spectra of the emitted pions,
protons (p), deuterons (d) and tritons (t) with energies above 10 MeV were
reported, as well as the energy dissipation (energy transfer to the nucleons
and mean excitation energies [3]). The residual nuclei have been investigated
through measurements of their radioactivity after $\mathrm{\bar{p}}$
irradiation, i.e. of their gamma spectra [18, 19, 20, 21, 22, 23], and the
neutron density distributions were deduced through nuclear spectroscopy
analysis of the antiproton annihilation residues [24]. The experimental
results have been most extensively compared to the INC, which models the
$\mathrm{\bar{p}}$A annihilation through the antiproton annihilation on a
single nucleon, generating a number of pions which then cascade through the
nucleus in a sequence of interactions with the rest of the nucleons. Good
agreement with data was found on the absolute pion and proton yields, however
the momentum distribution of pions and protons emerging from the annihilation
were described with less accuracy [25]. For the residue distribution, the
evaporation model provides a satisfactory description [26].

Nevertheless, some characteristics of $\mathrm{\bar{p}}$A annihilation at
rest, such as hadronization and total final product multiplicities remain
unknown. The production of fragments with short ranges has also not been
studied and there is no experimental evidence for heavier nuclei than
$\mathrm{{}^{8}He}$ in the energy distributions [17]. The charge and mass
distribution of the highly ionizing nuclear fragments carry information on the
energy deposited during the intranuclear cascade.

Besides its relevance for nuclear physics studies, the antiproton-nucleus
annihilation at rest is one of the key processes for experiments at the
Antiproton Decelerator (AD) at CERN, that study antimatter in laboratory,
detecting it through annihilation [27, 28, 29, 30, 31, 32]. Monte Carlo
simulations are performed to assess the efficiency of tagging antihydrogen
events, relying on the physics assumptions implemented either in Geant4, such
as the CHiral Invariant Phase Space event generator (CHIPS) [33, 34, 35],
Fritiof-Precompound (FTFP) [36, 37, 38] and the newly added Intranuclear
Cascade de Liège (INCL) model [39], or in FLUKA [40, 41, 42]. The low energy
annihilation models used in these software packages are, however, based on
hadronic high energy interactions (CHIPS, FTFP) or were developed for medical
physics applications (FLUKA), and were extrapolated to low energies in spite
of the fact that the low-energy annihilation mechanism is still not well
understood [43]. Recent studies have been performed using 5 MeV pulsed
antiproton beam from the AD, degrading it to energies of $\sim$100 keV in a
broad distribution. Antiprotons annihilating at rest in Al, Cu, Ag, Au and Si
were measured with emulsion and silicon detectors, as reported in [44, 45,
46]. The results revealed significant discrepancies in the prong multiplicity
between measured data and the Geant4 models, with differences ranging from 30%
to a factor of 4.

In this work, we present new measurements of $\mathrm{\bar{p}}$A annihilation
at rest using a slow extracted, monoenergetic beam of 150 eV, accelerated to
1.15 keV by the electric potential applied on the target. The antiprotons
annihilated on thin targets, providing a crucial comparison with the models in
which the stopping effects and production of secondaries due to the particle
propagation can be neglected. In addition, this allows the emergence and
detection of the heavy nuclei produced in the annihilation.

##  2 Experimental setup

The ASACUSA-Cusp experiment at CERN, the main purpose of which is to measure
the ground-state hyperfine splitting of antihydrogen [47, 48], is also suited
for studies with a continuous (DC) beam of sub-keV antiprotons. The apparatus
was supplied with short pulses of 5.3 MeV antiprotons ($\sim
3\cdot~{}\mathrm{10^{7}}$ $\mathrm{\bar{p}}$/shot, every 100 s) from the AD,
decelerated to 120 keV by a radio frequency quadrupole decelerator [49] and
further to $\sim$10 keV with degrader foils. The slow extraction cycle
($\sim$140 s) started with antiproton trapping in the MUSASHI (Monoenergetic
Ultra-Slow Antiproton Source for High-precision Investigation) trap, followed
by electron cooling for 40 s without radial compression. The antiprotons were
then released by lowering the trapping potential over 30–40 s, as described in
ref. [50]. A continuous $\mathrm{\bar{p}}$ beam of 150 eV energy, lasting for
about 20 s, was extracted and transported through the positron-antiproton
mixing trap to the annihilation target, as shown in fig. 1. The magnetic field
of the double cusp, used for production of spin-polarized beam for the
antihydrogen experiment, was set to the nominal value [51, 52].

Figure 1: Schematic overview of the experimental setup. The x-axis points to
the reader.

The antiprotons were annihilated on three different targets: 2
${\upmu\mathrm{m}}$ thick diamond-like-carbon (DLC) and molybdenum foils, and
a gold foil of 1 ${\upmu\mathrm{m}}$ thickness. The stopping range of 1.15 keV
antiprotons, according to ref. [53], is $\sim$46 nm for carbon and $\sim$57 nm
for gold. The beam was centered and focused on the target with an Einzel lens
(see fig. 1), where both the entrance and exit electrodes have additional
diagonal cuts. Consisting of five electrodes, this lens allows for two-
dimensional steering perpendicularly to the beam axis.

The on-target beam size was determined by first measuring the antiproton beam
on the bismuth germanium oxide (BGO) crystal detector used for the
antihydrogen experiment [54], and by using the results of $\sim$30 mm FWHM in
x and y direction [55] as an input to simulations. As the foils then replaced
the BGO for the current measurements, the beam size on the target foils was
estimated from the measured size of the beam, adding, in the Geant4
simulations, the 1 kV electric potential applied on the foil for further
focusing. The outcome of this combined work is shown in Fig.
LABEL:fig:XY_pbar_beam_trig_timea and LABEL:fig:XY_pbar_beam_trig_timeb, where
the x-y beam profile and its horizontal (x) and vertical (y) projections are
shown, respectively. A beam size of 10.5 mm in x and y (FWHM) was obtained,
with 92.4$\%$ of the extracted antiprotons annihilating on the 2$\times$2
$\mathrm{cm^{2}}$ target. These results were also confirmed with SIMION [56]
simulations, starting from the same initial conditions and including the
voltage bias on the foil.

Figure 3: Schematic overview of the hodoscope, the annihilation foil and the
Timepix3 detector with the corresponding dimensions.

###  2.1 Detection and identification of charged pions and heavy fragments

The charged annihilation products were simultaneously detected by two
detectors, a cylindrical hodoscope surrounding the target and a pixel detector
placed 1 cm behind it (Timepix3 quad), as shown in Fig. 3. Pions and heavier
fragments (p, d, t, He, Li, etc.) in the forward direction were impinging on
the 2$\times$2 array of Timepix3, with a total area of 2.8$\times$2.8
$\mathrm{cm^{2}}$. The ASICs (application-specific integrated circuit),
containing 256$\times$256 pixel channels of 55$\times$55
${{\upmu\mathrm{m}}^{2}}$ each, were coupled to a 500 ${\upmu\mathrm{m}}$
thick silicon sensor for this application. Providing a simultaneous
information of the time-of-arrival (ToA) and the time-over-threshold (ToT) for
each pixel, with a nominal time resolution of 1.56 ns, the detector is able to
detect mixed radiation fields and has a long history of a variety of
applications [57, 58, 59, 60, 61]. Its wide dynamic spectral range allows for
detection of all charged particles depositing energy above the threshold. In
this experiment, the detection threshold was set to
$\sim$$\mathrm{1,000~{}e^{-}}$ of collected charge per pixel, corresponding to
a minimum required energy of $\sim$3.6 keV for X rays and low-energy gamma
rays, tens of keV for electrons, hundreds of keV for protons and $\sim$MeV for
heavy ions [62]. The higher threshold for heavier particles originates from
the aluminum electrode, approximately 500 nm in thickness, that is deposited
on top of the silicon sensor for biasing purposes. For details about the ASIC
and the readout system see ref. [63, 64, 65].

Omnidirectional charged pions were detected with the hodoscope encircling the
foil (Fig. 3), consisting of two layers of scintillating plastic bars
(material EJ-200) parallel to the beam direction, and two layers of
scintillating fibres (material Saint Gobain BCF-12) perpendicular to the beam
and encircled by each bar layer. The widths of the bars in the inner and outer
layer are 20 mm and 35 mm, respectively, with a thickness of 5 mm. Both bar
layers have an octagonal x-y cross section and are composed of eight panels,
each composed of four scintillating bars. Every bar is read out on both ends
by silicon photomultipliers (SiPMs, KETEK 3350TS) of 3$\times$3
$\mathrm{{mm}^{2}}$ active surface. The bar hodoscope is capable of measuring
the time-of-flight (ToF) for particles crossing both layers [27]. The fibres,
forming two layers of concentric cylinders are also read out with SiPMs (KETEK
PM3350-EB), bonded with optical cement on one end of each fibre bundle. The
fibre part of the hodoscope provides an increased position resolution in
z-direction. A detailed description is given in ref. [66].

The two detectors were time synchronised, and the data acquisition coincided
with the start of the slow extraction cycle. Every time a hit was detected in
both inner and outer bar layers of the hodoscope, a trigger was issued and
this event was time-stamped in the Timepix3 data stream, where every hit in
this detector was also recorded. The time distribution of these events
(triggers) for one extraction cycle is shown in Fig. LABEL:fig:triggers_hitsa,
revealing the 20 s duration of the antiproton beam extracted from MUSASHI
within the 140 s long cycle. About 1,000 $\mathrm{\bar{p}}$ annihilation
events per slow extraction cycle were detected, resulting in a total of
$\sim$100,000 annihilation events for each target foil. The triggers outside
the slow extraction window are produced by cosmic rays or, sporadically, from
secondaries of other origin in the AD experimental hall crossing the
hodoscope. The two small peaks in the Timepix3 distribution are from two
cosmic particles that traversed both the hodoscope and the Timepix3 detectors.
The data analysis was performed only within the slow extraction time window.

The absence of degraders to moderate the antiprotons prevents pion
contamination from upstream beam annihilations, observed in previous
measurements [45, 46, 67], restricting the background contamination to
essentially cosmic rays.

The hits in the Timepix3 were assigned to a particular event (trigger)
according to their ToA. The interval between two successive triggers ranged
from 10 ms to a few 10 ms, as illustrated in Fig. LABEL:fig:triggers_hitsb.
The observed cutoff at 10 ms stems from the hodoscope data acquisition limit
of 100 Hz. The nominal cosmic background was measured to be $\sim$1.3 Hz. When
combined with the duration of one extraction (20 s) it gives a total of 26
cosmics events, which is only 2.6% of the average number of annihilation
events recorded per run. For the Timepix3 the cosmic ray background is fully
negligible.

####  2.1.1 Minimum ionizing particles in the hodoscope

Minimum ionizing particles (MIPs) were detected with the hodoscope, covering
$\sim$80% of the full solid angle. The majority of these particles originate
from charged pions generated in the annihilation process. However, a notable
fraction of the identified tracks are generated by other particles such as
electrons, positrons, and energetic protons (see Sec. 3). The $e^{+}e^{-}$
pairs arise from the conversion of the two 67 MeV $\gamma$-rays from the
$\pi^{0}$ decay, or from Dalitz pair decay $\pi^{0}\rightarrow\gamma
e^{+}e^{-}$. Some of the high-energy protons, with hundreds of MeV of energy
originating from the final state interactions can also cross all four layers
of the hodoscope and produce tracks.

A three–dimensional (3D) tracking algorithm was employed to ascertain the
number of MIPs for individual annihilation events. The algorithm groups
corresponding hits in the fiber and bar detector into subsets and identifies
track candidates by combining these hits. Subsequently, a linear fit is
applied to all candidates within the track collection, and those candidates
that best conform to a straight line are chosen. Further procedural
intricacies are outlined in detail in ref. [66]. Additionally, the determined
tracks are used to reconstruct the annihilation vertex. The resolution
$\sigma_{z}$ in the z-direction for the vertex reconstruction is $\sim$50 mm.

####  2.1.2 Charged pions and heavier fragments in Timepix3

In the pixel detector, charged particles were detected in $\sim$25% of the
full solid angle. Based on the ToT proportionality to the amount of charge
collected in each electrode, a per pixel energy calibration was carried out
using test pulses. The deposited energy was extracted from the known value of
the test pulse capacitor, assuming an energy of 3.6 eV for the creation of one
electron-hole pair in silicon. The calibration results were validated with
data from a $\mathrm{{}^{241}Am}$ radioactive source and cosmic rays. In Fig.
LABEL:fig:Am241_energya, the lower end of the $\mathrm{{}^{241}Am}$ spectrum,
converted to energy units, is presented with a Gaussian fit to the
characteristic 59.5 keV peak, determined to be 59.3$\pm$2.6 keV.

The energy deposit and morphology of the pixelated track a charged particle
generates in the Timepix3 (also referred to as pixel cluster) are influenced
by factors such as the particle type, its kinetic energy, and the incidence
angle. The graph in Fig. LABEL:fig:Am241_energyb illustrates the $dE/dx$
distribution for cosmic rays, both in measurements and simulations,
exclusively for clusters with high ($>$0.8) eccentricity, a measure of the
linearity of the shape of a particle track. These elongated tracks correspond
to cosmic rays that cross the Timepix3 at large angles with respect to its
normal. The distribution is modeled with a Landau-Gaussian fit and compared to
Geant4 simulations. The most probable value of $\sim$1.4
$\mathrm{{MeVg^{-1}cm^{2}}}$ agrees with the typical value characteristic for
MIPs in silicon [68].

Due to the geometry of the set-up (see Fig. 3), the annihilation prongs hit
the pixel detector only at acute angles with respect to its normal. The
maximum angle is $\sim$55$\degree$ for prongs emerging from the foil’s center,
and up to $\sim$67$\degree$ for prongs originating from annihilations close to
the edges of the target. Most of the particles impinge on the Timepix3 at
angles smaller than these, which results in the formation of mostly blob-like
clusters and relatively short tracks, with a maximum length of less than 1.5
mm. Combined with the variety of particles produced in the annihilation and
their diverse ranges of energies and incident angles, this allowed only a
broad classification of heavily ionizing particles (HIPs) as opposed to MIPs.
Ongoing efforts from a number of users of Timepix ASICs are aimed at
developing tools for more precise particle identification in mixed radiation
fields [62].

In this study, the differentiation of MIPs in the Timepix3 was achieved by
first implementing cuts based on the energy deposit in a cluster. These cuts
were derived from the analysis of cosmic data. The energy thresholds for the
highest pixel energy ($E_{1}$) within the cluster and the mean pixel energy
($E_{mean}$) were extracted from the $E_{mean}$ versus $E_{1}$ distribution
depicted in Fig. LABEL:fig:cosmics_cutsa, by calculating its cumulative
distribution function (CDF), shown in Fig. LABEL:fig:cosmics_cutsb. The
threshold was set to account for $>$99$\%$ of the cosmic rays data sample. The
remaining clusters underwent further evaluation based on their morphology and
were categorized as small blobs, large blobs, or heavy tracks before being
identified as HIPs. The efficiency of removing MIPs from the annihilation
clusters using only energy cuts from cosmics was estimated, from simulations,
to be $\sim$97$\%$. However, it was also revealed that some high-energy
protons can be mistakenly identified as MIPs after applying these cuts, as
shown in Fig. LABEL:fig:Etot_MIPs_tag. The comparison plots show the deposited
energy by charged pions (solid line) and by MIPs tagged after applying the
energy cuts obtained from simulated cosmic rays (dashed lines) for two Geant4
models ((a) FTF and (b) CHIPS), for all three targets. Additionally, the
difference between the two energy depositions is presented at the bottom. The
excess counts are attributed to the aforementioned protons, the quantity of
which depends on the specific simulation model. The percentage of these
protons incorrectly tagged as MIPs relative to the total number of detected
HIPs varies between 2% for Mo in CHIPS to 15% for C and Au in FLUKA.

In data, this issue was addressed by introducing an additional cut on the
number of halo pixels (explained in Sec. 3), extracted from the cosmic data in
a similar way as the energy cuts. This way, blob-like and heavy tracks
produced by protons, were distinguished among the clusters initially tagged as
MIPs with the energy cuts. The efficiency of the cut on the halo pixels could
not be assessed in simulations, since the halo is an ASIC-specific effect
present only in the measured data. However, the extracted HIPs after the
implementation of this cut in data amount to 10% of the total number of HIPs
in C and Mo, and to 12% in gold. These values are well within the limits of
the model’s predictions for protons incorrectly tagged as MIPs. In Fig.
LABEL:fig:hitmaps_tpx3a, the hitmap shows clusters from 280 annihilation
events that were classified as HIPs, while fig. LABEL:fig:hitmaps_tpx3b shows
those discarded as MIPs.

##  3 Monte Carlo simulations

The Monte Carlo simulations were performed using three of the physics lists
available in Geant4, as well as FLUKA (fluka 4-2.1). The
$\mathrm{QGSP\\_BERT\\_CHIPS}$ model was simulated in Geant4.9.6.p04,
$\mathrm{FTFP\\_BERT\\_EMY}$ in Geant4.10.05.p01, and the most recent one,
$\mathrm{FTFP\\_INCLXX\\_EMZ}$ in Geant4.11.2.1. The four models adopt
different approaches when describing antiproton-nucleus annihilation at rest.

In CHIPS, any excited hadronic system, such as an incident particle and target
nucleons or nuclear matter, is considered to be a quasmon - a bubble
containing massless quarks (quark-parton plasma). The fragmentation of a
quasmon into hadrons occurs through a quark fusion mechanism [33]. This is how
the mesons are produced when the antiproton annihilates on a peripheral
nucleon. The final state interactions that follow between these mesons and the
residual nucleus are treated in a similar way, by generating quasmons whenever
some of these mesons are absorbed by the nucleus. The CHIPS model, as opposed
to INC, does not develop cascades, and the final state particles are created
through hadronization of the quasmons inside the same nucleus [69].

The underlying idea of Fritiof is based on the hadronic string approach which
is used in Monte Carlo event generators like Pythia [70]. Here it is combined
with the Glauber theory for nucleus-nucleus interactions, which is used to
calculate cross sections for processes that involve the nucleus, such as the
antiproton-nucleus annihilation. The initially created string between the
antiproton and the target nucleus breaks up into several clusters, which
subsequently decay into final-state particles. Even though it can simulate the
production of intermediate resonances during the annihilation process, which
adds to the understanding of the underlying nuclear structure, the model does
not account for the FSI between the primarily produced mesons and the residual
nucleus, which is important for predicting the multiplicity distributions of
the final-state particles.

In FLUKA, $\mathrm{\bar{p}}$A annihilation at rest starts with the
antinucleon-nucleon process, modeled through the production and decay of two
or more intermediate states whose branching ratios are adjusted to reproduce
multiplicities for pions, kaons and resonances from experiments. The
subsequent interaction of these particles with the remaining nucleus, as well
as all nuclear effects are treated by the custom preequilibrium cascade
(PEANUT) model.

The latest update in the Geant4 simulation toolkit is the recent extension of
the Intranuclear Cascade de Liège (INCL) model [71, 72] for
$\mathrm{\bar{p}}$A annihilation at rest [39]. The frequencies and types of
the primary annihilation products are determined from experimental data. The
subsequent intranuclear cascade is based on binary collisions, particle decay
during flight and interactions at the nuclear surface. The classical approach
in INCL can result in the generation of states that are not physically
possible, such as those prohibited by the Pauli principle. This is resolved by
applying Pauli-blocking tests to the generated collisions.

The simulations presented here include a complete geometry of the set-up with
the Timepix3 and hodoscope detectors down to their smallest components, as
well as vacuum pipes, various support structures etc. To ensure a direct
comparison, identical analysis tools were applied to both data and
simulations. Achieving this required extracting the same type of signal from
the simulations–specifically, converting the step-based energy deposits
characteristic of Geant4 and FLUKA into hits in the hodoscope bars and fibers,
and into pixels and clusters in the Timepix3 quad.

For the hodoscope a straightforward threshold on the deposited energy was
implemented to define a hit (0.7 MeV for the bars and of 0.55 MeV for fibres),
determined from the deposited energy by simulated cosmic rays. Both Geant4 and
FLUKA simulations consistently indicate that most of the tracks in the
hodoscope are from charged pions. However, a significant portion is ascribed
to other high–energy particles crossing the four layers of the hodoscope. In
FTFP, CHIPS and INCL their contribution remains constant at $\sim$35%,
$\sim$50% and $\sim$40% respectively, irrespective of the target. In contrast,
for FLUKA the values range between $\sim$25% and $\sim$35%, increasing with
the mass of the nucleus. In CHIPS and FTFP models $\sim$95% of the non-pion
tracks are generated by electrons or positrons, each with tens of MeV of
energy, resulting from the conversion of $\gamma$-rays emitted from the decay
of $\pi^{0}$ from $\mathrm{\bar{p}}$ annihilation. In FLUKA however, the
surplus of 3D reconstructed tracks is primarily attributed to high-energy
protons. In INCL most of the non-pion tracks originate from electrons or
positron, but proton contribution is significant with $\sim$40%. Fig.
LABEL:fig:MIPs_distros_gold_sims shows the distribution of the number of 3D
tracks per annihilation event for $\mathrm{\bar{p}}$-Au, across the four
models. Additionally, the graph includes a breakdown of the particles fully
traversing the hodoscope, depositing in each layer energy above the defined
threshold. It is evident that the distribution of charged pions alone does not
reproduce the number of 3D reconstructed tracks, underscoring the contribution
of other particles crossing the detector.

The Timepix3 simulated data were digitised using the $\mathrm{AllPix^{2}}$
modular simulation framework, applying the same bias voltage and charge
threshold as during the actual measurements [73, 74]. The clusters generated
from MIPs in the cosmic rays simulations reproduced the morphology and the
total cluster energy of the measured ones, as observed in Fig.
LABEL:fig:cosmics_alphas_digia. On the other hand, the digitisation could not
consistently reproduce the measured signal from HIPs, due to features of the
Timepix3 ASIC, such as the halo, plasma and the volcano effect [58, 75, 76,
77, 78]. These effects arise from substantial energy deposits, and simulating
them accurately is challenging because HIPs do not always trigger each of
them. The halo (or "skirt"), indicating induced low energy signal in the
neighboring pixels is consistently present. However, when heavier or more
energetic ions deposit a significant amount of charge over short distances,
two additional phenomena may occur: the plasma effect, which extends the
charge collection time and consequently increases the track width, or the
volcano effect, where pixels with energy deposits exceeding $\sim$500 keV
register significantly lower values. The measured cluster energy is then
notably lower than the actual deposited energy in the silicon sensor. In
addition, the stochastic nature, in particular of the volcano effect, makes
their qualitative and quantitative replication in the digitisation unfeasible.
Therefore, these effects had to be individually addressed in the data
analysis.

The halo pixels, which are easily identifiable due to the very low energy
deposits, were excluded implementing an energy cutoff of 5 keV, based on the
pixel energy distribution derived from $\mathrm{{}^{241}Am}$ data. Fig.
LABEL:fig:cosmics_alphas_digib shows a good agreement of the energy deposition
by $\alpha$ particles from a $\mathrm{{}^{241}Am}$ source between data (after
halo removal) and Geant4 simulation, both before and after digitisation,
despite the significantly larger clusters observed in data due to the presence
of halo pixels. The small discrepancy is likely attributed to the
uncertainties in measuring the distance between the source and the detector
during the measurements.

Even though the volcano effect manifests in individual pixels, its occurrence
in data was examined in relation to the total cluster energy, to reveal the
total deposited energy for which this effect has a minimal impact on the
measured energy. Such correlation allows to establish an upper limit on the
cluster energy that ensures a reliable comparison to simulations. The analysis
is presented in Fig. 5, where the fraction of clusters with volcano effect is
plotted as a function of the deposited energy. Only the clusters produced by
HIPs from annihilation in carbon, molybdenum and gold were considered. The
figure indicates that up to 3 MeV, the volcano effect has a negligible impact,
affecting $<$5%, while above 5 MeV it is present in $>$10% of the clusters. An
upper limit of 5 MeV was chosen for further comparison of the energy deposited
by HIPs to simulations.

Figure 5: Fraction of clusters containing volcano pixels, from annihilation
measurements with the three different targets, as a function of total energy
deposited within the cluster.

##  4 Results and discussion

###  4.1 Multiplicities

The charged pion multiplicity in antiproton-nucleus annihilation at rest is
influenced by the final state interactions of the primary pions with the
remaining nucleus. The number of pions that interact with the nucleus is
predominantly determined by simple geometric factors, such as the proximity of
the annihilation point to the nuclear surface and the size (mass) of the
nucleus. The number of surviving, and detectable pions is related to the
primary pions through:

$\displaystyle\langle N_{\pi^{\pm}}\rangle=\langle
N_{\pi^{\pm}}^{init}\rangle\Bigl{(}1-\frac{\Omega}{4\pi}\Bigl{)}$ (1)

where $\langle{N_{\pi^{\pm}}^{init}}\rangle$ = 3.1 is the charged pion
multiplicity produced in the initial $\mathrm{\bar{p}}$-nucleon annihilation
[9], and $P=(1-\frac{\Omega}{4\pi})$ is the average survival probability for
the emitted charged pions [11]. ${\Omega}$ is the solid angle under which the
remaining nucleus is seen from the annihilation point, which is close to the
nuclear surface.

Due to the limited geometrical acceptance and efficiency of the hodoscope,
only a fraction of the surviving charged pions are measured in this
experiment. The distributions of MIP particles presented through the number of
reconstructed 3D tracks in the hodoscope are shown in Fig.
LABEL:fig:MIPs_distros. No model reproduces accurately the distributions
obtained from the measurements for all three targets, with FLUKA, CHIPS and
INCL constantly demonstrating more accurate predictions when compared to data,
outperforming FTFP. To ensure an authentic comparison with data, the analysis
of the simulated data encompassed all events, irrespective of whether
antiprotons annihilated in the foil or in its mechanical support ($\sim$7% of
the annihilations), as previously mentioned in Sec. 1. For illustration,
excluding antiprotons annihilating on the support structure from the analysis
in the Monte Carlo data leads to a marginal change of about 0.2% of the
average MIP multiplicity.

A comparison between measurements and simulations of the average MIPs
multiplicities detected with the hodoscope are given in Table 1, along with
the statistical errors. Fig. 8 illustrates these multiplicities as a function
of the target atomic mass. The predicted dependence on A is slowly decreasing
in FLUKA but is constant for FTFP and CHIPS. INCL constantly underestimates
the MIP multiplicities while showing the strongest decrease with A among the
models. Previous experimental studies for annihilation of stopped antiprotons
indicate a rapid decrease in charged pion multiplicity from A=2 to A$<$80,
following an exponential law, after which it remains nearly constant for
higher mass numbers [15]. In our results, the contribution from particles
other than charged pions to the 3D tracks leads rather to a slowly decreasing
dependence on A, in line with the FLUKA prediction. However, the disagreement
between data and predictions increases with A. For FLUKA, it ranges from
$<$1$\%$ for carbon, $4\%$ for molybdenum to 8$\%$ for gold. The CHIPS model
predictions are almost as precise as FLUKA for carbon and molybdenum, but are
off by $12\%$ for gold. The results obtained with the FTFP model show a
discrepancy between about 7$\%$ and 20$\%$ with our experimental data. The new
INCL model underestimates the MIP multiplicity between 7% and 10%.

Figure 8: Average number of 3D tracks from MIPs in antiproton-nucleus annihilation, detected in the hodoscope, versus atomic mass for carbon, molybdenum and gold targets. Measurements and different Monte Carlo simulation models are compared. | C | Mo | Au
---|---|---|---
Data |  2.775(7) |  2.642(3) |  2.491(4)
FLUKA |  2.780(6) |  2.739(6) |  2.686(6)
FTFP |  2.981(6) |  2.973(6) |  2.987(6)
CHIPS |  2.793(6) |  2.732(6) |  2.804(6)
INCL |  2.580(6) |  2.356(6) |  2.306(6)
Table 1: Average multiplicities of MIPs for antiproton annihilation in carbon,
molybdenum and gold, measured with the hodoscope detector.

The heavily ionizing particles, including protons and heavier nuclear
fragments, were detected with the Timepix3 and tagged with the procedure
described in Sec. 2.1.2. The low energy antiprotons in this experiment were
stopped in the first 100 nm of the target, allowing, for example, protons of
minimum $\sim$200 keV ($\sim$500 keV) energy to emerge from the 2
${\upmu\mathrm{m}}$ (1 ${\upmu\mathrm{m}}$) thick carbon (gold) foil [79]. The
minimum kinetic energy for a $\mathrm{{}^{4}He}$ ion to escape the same
targets is 800 keV (2 MeV), on the other hand, a $\mathrm{{}^{12}C}$ ion would
need a minimum energy of 5 MeV to break free from the carbon target [80].

In Fig. LABEL:fig:HIPs_distros the multiplicity distributions of HIPs are
compared between data and Monte Carlo simulations for the three target foils.
Each model is most successful in describing the HIP multiplicity for carbon,
the lightest of the three nuclei in this work. FLUKA stands out as the model
that provides the best description of the HIPs multiplicity across all three
targets, followed by CHIPS. In contrast, FTFP exhibits generally poor
agreement with the measured data, and this disparity intensifies with
increasing atomic mass.

| C | Mo | Au
---|---|---|---
Data |  0.521(3) |  0.730(5) |  0.598(4)
FLUKA |  0.575(3) |  0.88(1) |  0.701(4)
FTFP |  0.412(3) |  0.120(2) |  0.047(1)
CHIPS |  0.679(3) |  1.290(7) |  0.268(3)
INCL |  0.383(8) |  0.399(8) |  0.225(3)
Table 2: Average multiplicities of HIPs for antiproton annihilation in carbon,
molybdenum and gold, measured with the Timepix3 quad detector.

For quantitative comparison, Table 2 presents the average multiplicities for
HIPs in the three target foils. The prediction by FLUKA for
$\mathrm{\bar{p}}$-C annihilation is 10% lower than the measured value, while
CHIPS and INCL overestimate and underestimate it by 30%, respectively. FLUKA
and CHIPS overestimate the average number of detected HIPs from
$\mathrm{\bar{p}}$-Mo annihilation by 20% and 77% respectively, while INCL
underestimates it by 45%. For $\mathrm{\bar{p}}$-Au, FLUKA agrees within 17%,
whereas CHIPS and INCL underestimate the data by more than 50%. For
annihilation in carbon, FTFP disagrees with data by $\sim$20%, while for
molybdenum and gold it underestimates the HIP multiplicity by a factor of 6
and 12, respectively.

Fig. LABEL:fig:sims_HIPs_detail provides a breakdown of the various HIPs
generated in $\mathrm{\bar{p}}$ annihilation and detected by the Timepix3,
focusing on FLUKA and CHIPS, the two models that most accurately describe the
average multiplicity. Protons are dominant nuclear fragments in both models.
In FLUKA, about 50% of HIPs for $\mathrm{\bar{p}}$-C and $\mathrm{\bar{p}}$-Au
annihilation are protons, increasing to around 60% for $\mathrm{\bar{p}}$-Mo.
In CHIPS, protons account for nearly 80% of all HIPs in $\mathrm{\bar{p}}$-C
annihilation and exceed 95% for Mo and Au. While both models feature fragments
with Z=2 as the second most represented heavy prongs, they differ by larger
factors for Z$>$2\. In FLUKA, Z$>$2 fragments constitute between 2% and 6% of
the total HIPs, while in CHIPS they are similar for $\mathrm{\bar{p}}$-C
annihilation ($\sim$2.5%), but drop to $<$0.1% for molybdenum and gold.

###  4.2 Energy deposit from heavy prongs

The energy deposit from a HIP inside the silicon sensor of the Timepix3 quad
detector depends on the type of particle, its kinetic energy as well as the
angle under which the particle hits the detector (Sec. 2.1). The comparison
between data and Monte Carlo models encompasses all HIPs without distinction,
and is shown in Fig. LABEL:fig:HIPs_Etot. The bottom part of the plots shows
the difference between data and models in terms of the standard deviation
units. As detailed in Sec. 3, the volcano effect, an artifact of the Timepix3
ASIC, constrains the comparison up to 5 MeV.

In $\mathrm{\bar{p}}$A annihilation at rest, protons are emitted through two
different processes: evaporation and direct emission. Evaporated protons
typically possess energies below $\sim$30 MeV, peaking at $\sim$15 MeV
according to ref. [16]. This energy corresponds to an average deposited energy
of $\sim$3 MeV when protons cross the Timepix3 detector perpendicularly, and
to $\sim$6.7 MeV when impinging on the detector at the maximum possible angle
(67$\degree$) from its normal.

In our data, energy distributions for HIPs show a broad peak around 1 MeV,
which, for illustration, is the average deposited energy for protons with
approximately $\sim$60 MeV ($>$100 MeV) traveling perpendicularly (at
67$\degree$ from its normal) through the detector. Protons, deuterons and
tritons with energies up to about 10 MeV, 15 MeV, and 22 MeV respectively,
deposit more than 5 MeV in the 500 ${\upmu\mathrm{m}}$ thick silicon sensor
[81, 45], and are thus excluded form this comparison. $\mathrm{{}^{3}He}$ and
$\mathrm{{}^{4}He}$ with energies below 30 MeV are fully stopped inside the
Timepix3, releasing all of their energy.

Despite the quantitative underestimation of HIP production by FTFP, the energy
deposits predicted by this model show the smallest deviation from data for
annihilations across all three nuclei. FLUKA exhibits a satisfactory agreement
with measurements, with a consistent shift of the maximum of the distribution
towards lower energies. This behavior is also characteristic of INCL, whose
distributions closely follow those of FLUKA. CHIPS emerges as the model with
the least accurate predictions for the deposited energy from HIPs,
particularly as the atomic mass of the nucleus increases.

##  5 Conclusions

In this study we measured antiproton-nucleus annihilation at rest in thin
targets, with thicknesses of 1 ${\upmu\mathrm{m}}$ or 2 ${\upmu\mathrm{m}}$,
using a slowly extracted antiproton beam of 150 eV.

To validate Monte Carlo simulation models in the widely-used packages Geant4
and FLUKA, we measured charged particles emitted from the annihilation of
antiprotons in carbon, molybdenum, and gold targets. The comparison showed
that overall, FLUKA demonstrated the best performance in describing the
production of MIPs and HIPs, two quantities tightly bound to the final state
interactions between initially produced mesons and residual nuclei. FLUKA’s
predictions agree with experimental data within about 20% relative precision,
which was confirmed also in ref. [44]. Among the Geant4 models, CHIPS and INCL
show similar performance, outperforming FTFP in reproducing fragment
multiplicities. However, CHIPS has been discontinued in the latest versions of
Geant4 and is no longer maintained. The primary drawback of FTFP is the
underestimation of HIP production by large factors, ranging up to 12. Despite
the low rate of produced HIPs, its predictions of the deposited energy of
heavy prongs are the most accurate among the examined models.

Annihilation data at rest using new technologies are needed to test the
annihilation mechanisms [82]. Future measurements aiming to study the final
state interactions will require a more detailed identification of the types of
outgoing heavily ionizing particles, for which limited data currently exists.
Measuring annihilations in nuclei containing few nucleons, such as Be or C
with this approach can efficiently identify particular FSIs of the
annihilation mesons and nucleons. This will be considered in our next project,
involving a detailed, systematic study of $\mathrm{\bar{p}}$A annihilation at
rest, covering approximately 15 different nuclei. It will be conducted at the
AD/ELENA [83] facility at CERN, using the newly developed, dedicated beamline
for slow extracted antiprotons at the ASACUSA experiment. The total
multiplicity in nearly 4$\pi$ solid angle for charged prongs of various types
will be measured, along with their kinetic energy and angular distribution.
The project will provide an in-depth investigation of the FSIs, which evolve
with the atomic number, enhancing our understanding of antiproton-nucleus
interactions at low energies. Moreover, the acquired dataset will be used for
detailed validation of the current, but also future models describing
antiproton-nucleus annihilation at rest.

## Acknowledgments

This work is supported by JSPS KAKENHI Fostering Joint International Research
No. B 19KK0075, No. A 20KK0305, Grant-in-Aid for Scientific Research No. B
20H01930, No. A 20H00150; the Austrian Science Fund (FWF) Grant Nos. P 32468,
W1252-N27, and P 34438; Special Research Projects for Basic Science of RIKEN;
Università di Brescia and Istituto Nazionale di Fisica Nucleare; the European
Union’s Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie Grant Agreement No. 721559; the Research Grants Program of
the Royal Society and the Foundation for Dutch Scientific Research Institutes.
The authors express their gratitude to the Medipix Collaboration at CERN for
providing the Timepix3 ASICs for the measurements, and to Alberto Ribon from
the Geant4 group at CERN for the fruitful discussions and advises. This
research was funded in whole or in part by the Austrian Science Fund (FWF) [P
34438]. For open access purposes, the author has applied a CC BY public
copyright license to any author accepted manuscript version arising from this
submission.

## References

  * [1] H.S. Plendl, H. Daniel, T. von Egidy, T. Haninger, F.S. Hartmann, P. Hofmann, Y.S. Kim, H. Machner, G. Riepe, J. Jastrzȩbski, A. Grabowska, W. Kurcewicz, P. Lubinski, A. Stolarz, A.S. Botvina, Y.S. Golubeva, A.S. Iljinov, V.G. Nedorezov, A.S. Sudov, and K. Ziock, _Antiproton-nucleus annihilation at rest_ , Physica Scripta 48 (1993), pp. 160–163, Available at https://doi.org/10.1088/0031-8949/48/2/006.
  * [2] P. Lubiński, A. Grochulska, T. von Egidy, K. Gulda, F.J. Hartmann, J. Jastrzȩbski, W. Kurcewicz, L. Pienkowski, A. Stolarz, and A. Trzcińska, _Gold fragmentation induced by stopped antiprotons_ , Phys. Rev. C - Nuclear Physics 66 (2002), p. 14.
  * [3] D. Polster, D. Hilscher, H. Rossner, T. von Egidy, F.J. Hartmann, J. Hoffmann, W. Schmid, I.A. Pshenichnov, A.S. Iljinov, Y.S. Golubeva, H. Machner, H.S. Plendl, A. Grochulska, J. Jastrzȩbski, W. Kurcewicz, P. Lubiński, J. Eades, and S. Neumaier, _Light particle emission induced by stopped antiprotons in nuclei: Energy dissipation and neutron-to-proton ratio_ , Phys. Rev. C 51 (1995), pp. 1167–1180, Available at https://link.aps.org/doi/10.1103/PhysRevC.51.1167.
  * [4] M.R. Clover, R.M. DeVries, N.J. DiGiacomo, and Y. Yariv, _Low energy antiproton-nucleus interactions_ , Phys. Rev. C 26 (1982), pp. 2138–2151, Available at https://link.aps.org/doi/10.1103/PhysRevC.26.2138.
  * [5] A. Iljinov, V. Nazaruk, and S. Chigrinov, _Nuclear absorption of stopped antiprotons: Multipion-nucleus interactions_ , Nuclear Physics A 382 (1982), pp. 378 – 400, Available at https://doi.org/10.1016/0375-9474(82)90352-9.
  * [6] J. Bondorf, A. Botvina, A. Iljinov, I. Mishustin, and K. Sneppen, _Statistical multifragmentation of nuclei_ , Physics Reports 257 (1995), pp. 133–221, Available at https://www.sciencedirect.com/science/article/pii/037015739400097M.
  * [7] Z.Q. Feng and H. Lenske, _Particle production in antiproton-induced nuclear reactions_ , Phys. Rev. C 89 (2014), p. 044617, Available at https://link.aps.org/doi/10.1103/PhysRevC.89.044617.
  * [8] C. Amsler, _Proton-antiproton annihilation and meson spectroscopy with the Crystal Barrel_ , Rev. Mod. Phys. 70 (1998), pp. 1293–1339, Available at https://link.aps.org/doi/10.1103/RevModPhys.70.1293.
  * [9] E. Klempt, C. Batty, and J.M. Richard, _The antinucleon-nucleon interaction at low energy; annihilation dynamics_ , Physics Reports 413 (2005), pp. 197 – 317, Available at http://www.sciencedirect.com/science/article/pii/S0370157305001055.
  * [10] G. Bendiscioli, V. Filippini, A. Rotondi, P. Salvini, A. Venaglioni, S. Bossolasco, M. Bussa, L. Busso, L. Fava, L. Ferrero, A. Maggiora, D. Panzieri, G. Piragino, R. Piragino, F. Tosello, Y. Batusov, I. Falomkin, G. Pontecorvo, A. Rozhdestvensky, M. Sapozhnikov, V. Tretyak, C. Guaraldo, E. Rizzini, A. Haatuft, A. Halsteinslid, K. Myklebost, J. Olsen, F. Breivik, T. Jacobsen, S. Sørensen, F. Balestra, and A. Zenoni, _Pionic annihilation of antiprotons stopped on $\mathrm{{}^{3}{H}e}$_, Nuclear Physics A 518 (1990), pp. 683–708, Available at https://www.sciencedirect.com/science/article/pii/037594749090185O.
  * [11] J. Cugnon, S. Wycech, J. Jastrzȩbski, and P. Lubiński, _Geometrical effects in antiproton annihilation on nuclei_ , Phys. Rev. C - Nuclear Physics 63 (2001), p. 4.
  * [12] D. Mancusi, S. Lo Meo, N. Colonna, A. Boudard, M.A. Cortés-Giraldo, J. Cugnon, J.C. David, S. Leray, J. Lerendegui-Marco, C. Massimi, and V. Vlachoudis, _On the role of secondary pions in spallation targets_ , The European Physical Journal A 53 (2017), p. 80, Available at https://doi.org/10.1140/epja/i2017-12263-0.
  * [13] A.M. Rozhdestvensky and M.G. Sapozhnikov, _Selected topics of antiproton-nucleus annihilation_ , Surveys in High Energy Physics 6 (1992), pp. 115–140, Available at https://doi.org/10.1080/01422419208241237.
  * [14] P. Montagna, G. Bendiscioli, V. Filippini, A. Fontana, A. Rotondi, P. Salvini, A. Panzarasa, M. Bargiotti, A. Bertin, M. Bruschi, M. Capponi, S. De Castro, L. Fabbri, P. Faccioli, D. Galli, B. Giacobbe, U. Marconi, M. Piccinini, N. Semprini Cesari, R. Spighi, S. Vecchi, V. Vagnoni, M. Villa, A. Vitale, A. Zoccoli, A. Bianconi, G. Bonomi, M. Bussa, A. Donzella, L. Venturelli, A. Zenoni, C. Cicalò, A. De Falco, A. Masoni, G. Puddu, S. Serci, G. Usai, O. Gorchakov, A. Rozhdestvensky, V. Tretyak, P. Gianotti, C. Guaraldo, A. Lanaro, V. Lucherini, C. Petrascu, V. Ableev, C. Cavion, G. Vedovato, F. Balestra, L. Busso, P. Cerello, O. Denisov, L. Ferrero, R. Garfagnini, A. Grasso, A. Maggiora, D. Panzieri, M. Astrua, E. Botta, T. Bressani, D. Calvo, A. Feliciello, A. Filippi, S. Marcello, M. Agnello, F. Iazzi, and N. Mirfakhrai, _Single and multinucleon antiproton–4he annihilation at rest_ , Nuclear Physics A 700 (2002), pp. 159–192, Available at https://www.sciencedirect.com/science/article/pii/S0375947401013021.
  * [15] G. Bendiscioli and D. Kharzeev, _Antinucleon-nucleon and antinucleon-nucleus interaction. a review of experimental data_ , La Rivista Del Nuovo Cimento Series 3 17 (1994), pp. 1–142, Available at http://dx.doi.org/10.1007/BF02724447.
  * [16] P. Hofmann, F. Hartmann, H. Daniel, T. von Egidy, W. Kanert, W. Markiel, H. Plendl, H. Machner, G. Riepe, D. Protić, K. Ziock, R. Marshall, and J. Reidy, _Charged-particle spectra from antiproton annihilation at rest in A = 12–238 nuclei_ , Nuclear Physics A 512 (1990), pp. 669 – 683.
  * [17] I.N. Markiel, H. Daniel, T. von Egidy, F. Hartmann, P. Hofmann, W. Kanert, H. Plendl, K. Ziock, R. Marshall, H. Machner, G. Riepe, and J. Reidy, _Emission of helium ions after antiproton annihilation_ , Nuclear Physics 485 (1988), pp. 445–460.
  * [18] E. Moser, H. Daniel, T. von Egidy, F. Hartmann, W. Kanert, G. Schmidt, M. Nicholas, and J. Reidy, _Yields of residual nuclei after antiproton annihilation in 95Mo and 98Mo_, Physics Letters B 179 (1986), pp. 25 – 29.
  * [19] T. von Egidy, H. Daniel, F.J. Hartmann, W. Kanert, E.F. Moser, Y.S. Golubeva, A.S. Iljinov, and J.J. Reidy, _Yield of residual nuclei after antiproton annihilation in Ba_ , Zeitschrift für Physik A Atomic Nuclei 335 (1990), pp. 451–457, Available at https://doi.org/10.1007/BF01290193.
  * [20] E.F. Moser, H. Daniel, T. von Egidy, F.J. Hartmann, W. Kanert, G. Schmidt, E.S. Golubeva, A.S. Ilinov, M. Nicholas, and J.J. Reidy, _Residual Nuclei After Anti-proton Annihilation in Mo and Ho_ , Z. Phys. A333 (1989), pp. 89–105.
  * [21] P. Lubiński, J. Jastrzȩbski, A. Grochulska, A. Stolarz, A. Trzcińska, W. Kurcewicz, F.J. Hartmann, W. Schmid, T. von Egidy, J. Skalski, R. Smolańczuk, S. Wycech, D. Hilscher, D. Polster, and H. Rossner, _Neutron halo in heavy nuclei from antiproton absorption_ , Phys. Rev. Lett. 73 (1994), pp. 3199–3202, Available at https://link.aps.org/doi/10.1103/PhysRevLett.73.3199.
  * [22] J. Jastrzȩbski, H. Daniel, T. von Egidy, A. Grabowska, Y. Kim, W. Kurcewicz, P. Lubiński, G. Riepe, W. Schmid, A. Stolarz, and S. Wycech, _Signature of a neutron halo in 232Th from antiproton absorption_, Nuclear Physics A 558 (1993), pp. 405 – 414.
  * [23] R. Schmidt, F.J. Hartmann, B. Ketzer, T. von Egidy, T. Czosnyka, J. Jastrzȩbski, M. Kisieliński, P. Lubiński, P. Napiorkowski, L. Pieńkowski, A. Trzcińska, B. Kłos, R. Smolańczuk, S. Wycech, W. Pöschl, K. Gulda, W. Kurcewicz, and E. Widmann, _Composition of the nuclear periphery from antiproton absorption using short-lived residual nuclei_ , Phys. Rev. C 60 (1999), p. 054309, Available at https://link.aps.org/doi/10.1103/PhysRevC.60.054309.
  * [24] A. Trzcińska, J. Jastrzȩbski, P. Lubiński, F.J. Hartmann, R. Schmidt, T. von Egidy, and B. Kłos, _Neutron density distributions deduced from antiprotonic atoms_ , Phys. Rev. Lett. 87 (2001), p. 082501, Available at https://link.aps.org/doi/10.1103/PhysRevLett.87.082501.
  * [25] J. Cugnon, P. Deneye, and J. Vandermeulen, _Multipion dynamics following antiproton annihilation on nuclei_ , Nuclear Physics 500 (1989), pp. 701–730.
  * [26] J. Cugnon and J. Vandermeulen, _Antiproton-nucleus interaction_ , Ann. Phys. Fr. 14 (1989), pp. 49–87, Available at http://dx.doi.org/10.1051/anphys:0198900140104900.
  * [27] C. Sauerzopf, _et al._ , _Annihilation detector for an in-beam spectroscopy apparatus to measure the ground state hyperfine splitting of antihydrogen_ , Nucl. Instrum. Meth. A 845 (2017), pp. 579–582, Available at https://doi.org/10.1016/j.nima.2016.06.023.
  * [28] Y. Nagata, _et al._ , _Monte-Carlo based performance assessment of ASACUSA’s antihydrogen detector_ , Nucl. Instrum. Meth. A 910 (2018), pp. 90–95, Available at https://doi.org/10.1016/j.nima.2018.09.013.
  * [29] J. Storey, _et al._ , _Particle tracking at 4K: The Fast Annihilation Cryogenic Tracking (FACT) detector for the AEgIS antimatter gravity experiment_ , Nucl. Instrum. Meth. A 732 (2013), pp. 437–441, Available at https://doi.org/10.1016/j.nima.2013.05.130.
  * [30] G. Andresen, M. Ashkezari, W. Bertsche, P. Bowe, E. Butler, C. Cesar, S. Chapman, M. Charlton, A. Deller, S. Eriksson, J. Fajans, T. Friesen, M. Fujiwara, D. Gill, A. Gutierrez, J. Hangst, W. Hardy, M. Hayden, R. Hayano, A. Humphries, R. Hydomako, S. Jonsell, L. Jørgensen, L. Kurchaninov, N. Madsen, S. Menary, P. Nolan, K. Olchanski, A. Olin, A. Povilus, P. Pusa, E. Sarid, S.S. el Nasr, D. Silveira, C. So, J. Storey, R. Thompson, D. van der Werf, and Y. Yamazaki, _Antihydrogen annihilation reconstruction with the alpha silicon detector_ , Nucl. Instrum. Meth. A 684 (2012), pp. 73 – 81.
  * [31] A. Capra, _et al._ , _Design of a Radial TPC for Antihydrogen Gravity Measurement with ALPHA-g_ , JPS Conf. Proc. 18 (2017), p. 011015, Available at https://doi.org/10.7566/JPSCP.18.011015.
  * [32] T. Aumann, W. Bartmann, O. Boine-Frankenheim, A. Bouvard, A. Broche, F. Butin, D. Calvet, J. Carbonell, P. Chiggiato, H. De Gersem, R. De Oliveira, T. Dobers, F. Ehm, J.F. Somoza, J. Fischer, M. Fraser, E. Friedrich, A. Frotscher, M. Gomez-Ramos, J.L. Grenard, A. Hobl, G. Hupin, A. Husson, P. Indelicato, K. Johnston, C. Klink, Y. Kubota, R. Lazauskas, S. Malbrunot-Ettenauer, N. Marsic, W.F. O Müller, S. Naimi, N. Nakatsuka, R. Necca, D. Neidherr, G. Neyens, A. Obertelli, Y. Ono, S. Pasinelli, N. Paul, E.C. Pollacco, D. Rossi, H. Scheit, M. Schlaich, A. Schmidt, L. Schweikhard, R. Seki, S. Sels, E. Siesling, T. Uesaka, M. Vilén, M. Wada, F. Wienholtz, S. Wycech, and S. Zacarias, _PUMA, antiproton unstable matter annihilation_ , The European Physical Journal A 58 (2022), p. 88.
  * [33] P.V. Degtyarenko, M.V. Kossov, and H.P. Wellisch, _Chiral invariant phase space event generator. I. nucleon antinucleon annihilation at rest_ , Eur. Phys. J. A 8 (2000), pp. 217–222, Available at https://doi.org/10.1007/s100500070108.
  * [34] P.V. Degtyarenko, M.V. Kossov, and H.P. Wellisch, _Chiral invariant phase space event generator. II. Nuclear pion capture at rest and photonuclear reactions below the ${\Delta}$(3,3) resonance._, Eur. Phys. J. A 9 (2000), pp. 411–420, Available at https://doi.org/10.1007/s100500070025.
  * [35] P.V. Degtyarenko, M.V. Kossov, and H.P. Wellisch, _Chiral invariant phase space event generator. III. Modeling of real and virtual photon interactions with nuclei below pion production threshold_ , Eur. Phys. J. A 9 (2000), pp. 421–424.
  * [36] B. Andersson, G. Gustafson, and B. Nilsson-Almqvist, _A model for low-pt hadronic reactions with generalizations to hadron-nucleus and nucleus-nucleus collisions_ , Nuclear Physics B 281 (1987), pp. 289 – 309, Available at https://doi.org/10.1016/0550-3213(87)90257-4.
  * [37] B. Nilsson-Almqvist and E. Stenlund, _Interactions between hadrons and nuclei: The Lund Monte Carlo - FRITIOF version 1.6 -_ , Computer Physics Communications 43 (1987), pp. 387 – 397, Available at https://doi.org/10.1016/0010-4655(87)90056-7.
  * [38] A. Galoyan and V. Uzhinsky, _Simulations of light antinucleus–nucleus interactions_ , Hyperfine Interactions 215 (2013), pp. 69–76, Available at https://doi.org/10.1007/s10751-013-0780-2.
  * [39] D. Zharenov, _Interaction of antiprotons with nuclear matter within the incl model_ , Ph.D. diss., Université Paris-Saclay, 2023, Available at https://theses.hal.science/tel-04511526.
  * [40] G. Battistoni, T. Boehlen, F. Cerutti, P.W. Chin, L.S. Esposito, A. Fassò, A. Ferrari, A. Lechner, A. Empl, A. Mairani, A. Mereghetti, P.G. Ortega, J. Ranft, S. Roesler, P.R. Sala, V. Vlachoudis, and G. Smirnov, _Overview of the FLUKA code_ , Annals of Nuclear Energy 82 (2015), pp. 10–18, Available at https://www.sciencedirect.com/science/article/pii/S0306454914005878, joint International Conference on Supercomputing in Nuclear Applications and Monte Carlo 2013, SNA + MC 2013. Pluri- and Trans-disciplinarity, Towards New Modeling and Numerical Simulation Paradigms.
  * [41] C. Ahdida, D. Bozzato, D. Calzolari, F. Cerutti, N. Charitonidis, A. Cimmino, A. Coronetti, G.L. D’Alessandro, A. Donadon Servelle, L.S. Esposito, R. Froeschl, R. García Alía, A. Gerbershagen, S. Gilardoni, D. Horváth, G. Hugo, A. Infantino, V. Kouskoura, A. Lechner, B. Lefebvre, G. Lerner, M. Magistris, A. Manousos, G. Moryc, F. Ogallar Ruiz, F. Pozzi, D. Prelipcean, S. Roesler, R. Rossi, M. Sabaté Gilarte, F. Salvat Pujol, P. Schoofs, V. Stránský, C. Theis, A. Tsinganis, R. Versaci, V. Vlachoudis, A. Waets, and M. Widorski, _New Capabilities of the FLUKA Multi-Purpose Code_ , Front. Phys. 9 (2022), p. 788253, Available at https://cds.cern.ch/record/2806210.
  * [42] https://fluka.cern.
  * [43] C. Amsler, _Nucleon-antinucleon annihilation at LEAR_ (2019).
  * [44] S. Aghion, C. Amsler, A. Ariga, T. Ariga, G. Bonomi, P. Bräunig, R. Brusa, L. Cabaret, M. Caccia, R. Caravita, F. Castelli, G. Cerchiari, D. Comparat, G. Consolati, A. Demetrio, L.D. Noto, M. Doser, A. Ereditato, C. Evans, R. Ferragut, J. Fesel, A. Fontana, S. Gerber, M. Giammarchi, A. Gligorova, F. Guatieri, S. Haider, A. Hinterberger, H. Holmestad, T. Huse, J. Kawada, A. Kellerbauer, M. Kimura, D. Krasnický, V. Lagomarsino, P. Lansonneur, P. Lebrun, C. Malbrunot, S. Mariazzi, V. Matveev, Z. Mazzotta, S. Müller, G. Nebbia, P. Nedelec, M. Oberthaler, N. Pacifico, D. Pagano, L. Penasa, V. Petracek, C. Pistillo, F. Prelz, M. Prevedelli, L. Ravelli, B. Rienaecker, O. RØhne, A. Rotondi, M. Sacerdoti, H. Sandaker, R. Santoro, P. Scampoli, M. Simon, L. Smestad, F. Sorrentino, G. Testera, I. Tietje, S. Vamosi, M. Vladymyrov, E. Widmann, P. Yzombard, C. Zimmer, J. Zmeskal, and N. Zurlo, _Measurement of antiproton annihilation on Cu, Ag and Au with emulsion films_ , Journal of Instrumentation 12 (2017), p. P04021, Available at https://doi.org/10.1088\%2F1748-0221\%2F12\%2F04\%2Fp04021.
  * [45] S. Aghion, O. Ahlén, A.S. Belov, G. Bonomi, P. Bräunig, J. Bremer, R.S. Brusa, G. Burghart, L. Cabaret, M. Caccia, C. Canali, R. Caravita, F. Castelli, G. Cerchiari, S. Cialdi, D. Comparat, G. Consolati, J.H. Derking, S.D. Domizio, L.D. Noto, M. Doser, A. Dudarev, R. Ferragut, A. Fontana, P. Genova, M. Giammarchi, A. Gligorova, S.N. Gninenko, S. Haider, J. Harasimowicz, T. Huse, E. Jordan, L.V. JØrgensen, T. Kaltenbacher, A. Kellerbauer, A. Knecht, D. Krasnický, V. Lagomarsino, A. Magnani, S. Mariazzi, V.A. Matveev, F. Moia, G. Nebbia, P. Nédélec, N. Pacifico, V. Petráček, F. Prelz, M. Prevedelli, C. Regenfus, C. Riccardi, O. RØhne, A. Rotondi, H. Sandaker, A. Sosa, M.A.S. Vasquez, M. Špaček, G. Testera, C.P. Welsch, and S. Zavatarelli, _Detection of low energy antiproton annihilations in a segmented silicon detector_ , Journal of Instrumentation 9 (2014), p. P06020, Available at https://doi.org/10.1088\%2F1748-0221\%2F9\%2F06\%2Fp06020.
  * [46] S. Aghion, O. Ahlén, C. Amsler, A. Ariga, T. Ariga, A.S. Belov, G. Bonomi, P. Bräunig, J. Bremer, R.S. Brusa, L. Cabaret, C. Canali, R. Caravita, F. Castelli, G. Cerchiari, S. Cialdi, D. Comparat, G. Consolati, J.H. Derking, S.D. Domizio, L.D. Noto, M. Doser, A. Dudarev, A. Ereditato, R. Ferragut, A. Fontana, P. Genova, M. Giammarchi, A. Gligorova, S.N. Gninenko, S. Haider, J. Harasimovicz, S.D. Hogan, T. Huse, E. Jordan, L.V. Jørgensen, T. Kaltenbacher, J. Kawada, A. Kellerbauer, M. Kimura, A. Knecht, D. Krasnický, V. Lagomarsino, A. Magnani, S. Mariazzi, V.A. Matveev, F. Moia, G. Nebbia, P. Nédélec, M.K. Oberthaler, N. Pacifico, V. Petráček, C. Pistillo, F. Prelz, M. Prevedelli, C. Regenfus, C. Riccardi, O. Røhne, A. Rotondi, H. Sandaker, P. Scampoli, A. Sosa, J. Storey, M.A.S. Vasquez, M. Špaček, G. Testera, D. Trezzi, R. Vaccarone, C.P. Welsch, and S. Zavatarelli, _Prospects for measuring the gravitational free-fall of antihydrogen with emulsion detectors_ , Journal of Instrumentation 8 (2013), p. P08013, Available at https://doi.org/10.1088\%2F1748-0221\%2F8\%2F08\%2Fp08013.
  * [47] https://asacusa.web.cern.ch.
  * [48] E. Widmann, C. Amsler, S. Arguedas Cuendis, H. Breuker, M. Diermaier, P. Dupré, C. Evans, M. Fleck, A. Gligorova, H. Higaki, Y. Kanai, B. Kolbinger, N. Kuroda, M. Leali, A.M.M. Leite, V. Mäckel, C. Malbrunot, V. Mascagna, O. Massiczek, Y. Matsuda, D.J. Murtagh, Y. Nagata, A. Nanda, D. Phan, C. Sauerzopf, M.C. Simon, M. Tajima, H. Spitzer, M. Strube, S. Ulmer, L. Venturelli, M. Wiesinger, Y. Yamazaki, and J. Zmeskal, _Hyperfine spectroscopy of hydrogen and antihydrogen in asacusa_ , Hyperfine Interactions 240 (2018), p. 5, Available at https://doi.org/10.1007/s10751-018-1536-9.
  * [49] A.M. Lombardi, W. Pirkl, and Y.V. Bylinsky, _First operating experience with the CERN decelerating RFQ for anti-protons_ , Conf. Proc. C 0106181 (2001), pp. 585–587.
  * [50] N. Kuroda, H.A. Torii, Y. Nagata, M. Shibata, Y. Enomoto, H. Imao, Y. Kanai, M. Hori, H. Saitoh, H. Higaki, A. Mohri, K. Fujii, C.H. Kim, Y. Matsuda, K. Michishio, Y. Nagashima, M. Ohtsuka, K. Tanaka, and Y. Yamazaki, _Development of a monoenergetic ultraslow antiproton beam source for high-precision investigation_ , Phys. Rev. ST Accel. Beams 15 (2012), p. 024702, Available at https://link.aps.org/doi/10.1103/PhysRevSTAB.15.024702.
  * [51] Y. Nagata and Y. Yamazaki, _A novel property of anti-helmholz coils for in-coil syntheses of antihydrogen atoms: formation of a focused spin-polarized beam_ , New Journal of Physics 16 (2014), p. 083026, Available at https://dx.doi.org/10.1088/1367-2630/16/8/083026.
  * [52] N. Kuroda, _et al._ , _Antihydrogen synthesis in a double-cusp trap_ , JPS Conf. Proc. 18 (2017), p. 011009.
  * [53] S.P. Møller, A. Csete, T. Ichioka, H. Knudsen, U.I. Uggerhøj, and H.H. Andersen, _Antiproton stopping at low energies: Confirmation of velocity-proportional stopping power_ , Phys. Rev. Lett. 88 (2002), p. 193201, Available at https://link.aps.org/doi/10.1103/PhysRevLett.88.193201.
  * [54] Y. Nagata, N. Kuroda, M. Ohtsuka, M. Leali, E. Lodi-Rizzini, V. Mascagna, M. Tajima, H. Torii, N. Zurlo, Y. Matsuda, L. Venturelli, and Y. Yamazaki, _Direct detection of antihydrogen atoms using a BGO crystal_ , Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 840 (2016), pp. 153–159, Available at https://www.sciencedirect.com/science/article/pii/S0168900216310440.
  * [55] E. Widmann, Tech. Rep. CERN-SPSC-2018-003. SPSC-SR-226, CERN, Geneva, 2018, Available at http://cds.cern.ch/record/2300138.
  * [56] D. Manura and D. Dahl, _SIMION (R) 8.1 User Manual (Scientific Instrument Services, Inc. Ringoes, NJ 08551_. Available at http://simion.com/,2008).
  * [57] R. Ballabriga, M. Campbell, and X. Llopart, _Asic developments for radiation imaging applications: The medipix and timepix family_ , Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 878 (2018), pp. 10–23, Available at https://www.sciencedirect.com/science/article/pii/S0168900217307714.
  * [58] T. Campbell-Ricketts, M. Kroupa, and L. Pinsky, _Spectroscopy of high-energy ions with timepix3_ , Journal of Instrumentation 11 (2016), pp. P11007–P11007, Available at https://doi.org/10.10882F1748-02212F112F112Fp11007.
  * [59] B. Bergmann, M. Pichotka, S. Pospisil, J. Vycpalek, P. Burian, P. Broulim, and J. Jakubek, _3d track reconstruction capability of a silicon hybrid active pixel detector_ , The European Physical Journal C 77 (2017), p. 421, Available at https://doi.org/10.1140/epjc/s10052-017-4993-4.
  * [60] B. Bergmann, T. Billoud, C. Leroy, and S. Pospisil, _Characterization of the radiation field in the atlas experiment with timepix detectors_ , IEEE Transactions on Nuclear Science 66 (2019), pp. 1861–1869.
  * [61] S. Gohl, B. Bergmann, C. Granja, A. Owens, M. Pichotka, S. Polansky, and S. Pospisil, _Measurement of particle directions in low earth orbit with a Timepix_ , Journal of Instrumentation 11 (2016).
  * [62] C. Granja, J. Jakubek, S. Polansky, V. Zach, P. Krist, D. Chvatil, J. Stursa, M. Sommer, O. Ploc, S. Kodaira, and M. Martisikova, _Resolving power of pixel detector Timepix for wide-range electron, proton and ion detection_ , Nucl. Instrum. Meth. A 908 (2018), pp. 60 – 71.
  * [63] T. Poikela, J. Plosila, T. Westerlund, M. Campbell, M.D. Gaspari, X. Llopart, V. Gromov, R. Kluit, M. van Beuzekom, F. Zappon, V. Zivkovic, C. Brezina, K. Desch, Y. Fu, and A. Kruth, _Timepix3: a 65k channel hybrid pixel readout chip with simultaneous ToA/ToT and sparse readout_ , Journal of Instrumentation 9 (2014), p. C05013, Available at https://doi.org/10.1088/1748-0221/9/05/C05013.
  * [64] https://medipix.web.cern.ch/collaboration/medipix3-collaboration. [Online].
  * [65] J. Visser, M. Beuzekom, H. Boterenbrood, B.d. Heijden, J. Muñoz, S. Kulis, B. Munneke, and F. Schreuder, _SPIDR: a read-out system for Medipix3 & Timepix3_, JINST 10 (2015), p. C12028, Available at https://doi.org/10.1088/1748-0221/10/12/C12028.
  * [66] B. Kolbinger, _Machine Learning for Antihydrogen Detection in ASACUSA_ , Ph.D. diss., University of Vienna, 2018, Available at http://cds.cern.ch/record/2670395.
  * [67] S. Aghion, C. Amsler, M. Antonello, A. Belov, G. Bonomi, R. Brusa, M. Caccia, A. Camper, R. Caravita, F. Castelli, G. Cerchiari, D. Comparat, G. Consolati, A. Demetrio, L.D. Noto, M. Doser, C. Evans, M. Fanì, R. Ferragut, J. Fesel, A. Fontana, S. Gerber, M. Giammarchi, A. Gligorova, F. Guatieri, P. Hackstock, S. Haider, A. Hinterberger, H. Holmestad, A. Kellerbauer, O. Khalidova, D. Krasnický, V. Lagomarsino, P. Lansonneur, P. Lebrun, C. Malbrunot, S. Mariazzi, J. Marton, V. Matveev, S. Müller, G. Nebbia, P. Nedelec, M. Oberthaler, N. Pacifico, D. Pagano, L. Penasa, V. Petracek, F. Prelz, M. Prevedelli, B. Rienaecker, J. Robert, O. RØhne, A. Rotondi, H. Sandaker, R. Santoro, L. Smestad, F. Sorrentino, G. Testera, I.C. Tietje, E. Widmann, P. Yzombard, C. Zimmer, J. Zmeskal, and N. Zurlo, _Antiproton tagging and vertex fitting in a Timepix3 detector_ , Journal of Instrumentation 13 (2018), pp. P06004–P06004, Available at https://doi.org/10.1088/1748-0221/13/06/P06004.
  * [68] B. Bergmann, T. Billoud, P. Burian, C. Leroy, P. Mánek, L. Meduna, S. Pospíšil, and M. Suk, _Particle tracking and radiation field characterization with Timepix3 in ATLAS_ , Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 978 (2020).
  * [69] M. Kossov, _Simulation of antiproton-nuclear annihilation at rest_ , IEEE Transactions on Nuclear Science 52 (2005), pp. 2832–2835.
  * [70] H.U. Bengtsson and T. Sjöstrand, _The lund monte carlo for hadronic processes — PYTHIA version 4.8_ , Computer Physics Communications 46 (1987), pp. 43–82, Available at https://www.sciencedirect.com/science/article/pii/0010465587900361.
  * [71] A. Boudard, J. Cugnon, J.C. David, S. Leray, and D. Mancusi, _New potentialities of the liège intranuclear cascade model for reactions induced by nucleons and light charged particles_ , Phys. Rev. C 87 (2013), p. 014606, Available at https://link.aps.org/doi/10.1103/PhysRevC.87.014606.
  * [72] A. Boudard, J. Cugnon, S. Leray, and C. Volant, _Intranuclear cascade model for a comprehensive description of spallation reaction data_ , Phys. Rev. C 66 (2002), p. 044615, Available at https://link.aps.org/doi/10.1103/PhysRevC.66.044615.
  * [73] S. Spannagel, K. Wolters, D. Hynds, N.A. Tehrani, M. Benoit, D. Dannheim, N. Gauvin, A. Nürnberg, P. Schütze, and M. Vicente, _Allpix2: A modular simulation framework for silicon detectors_ , Nucl. Instrum. Meth. A 901 (2018), pp. 164 – 172.
  * [74] G. Costantini, _Simulations and reconstruction of antiproton-matter annihilation events in antimatter experiments_ , Ph.D. diss., University of Brescia, 2023, Available at https://iris.unibs.it/retrieve/ff311c0a-f8e4-479c-adb1-72c5ae6a0b57/PhD_thesis_Costantini.pdf.
  * [75] P. Tove and W. Seibt, _Plasma effects in semiconductor detectors_ , Nuclear Instruments and Methods 51 (1967), pp. 261 – 269.
  * [76] P. Mánek, B. Bergmann, P. Burian, L. Meduna, S. Pospíšil, and M. Suk, _Randomized computer vision approaches for pattern recognition in timepix and timepix3 detectors_ (2019).
  * [77] S. George, _Dosimetric applications of hybrid pixel detectors_ , Ph.D. diss., University of Wollongong, 2015.
  * [78] H. Holmestad, _Data analysis, simulations, and reconstruction of antiproton annihilations in a silicon pixel detector_ , Ph.D. diss., University of Oslo, 2018, Available at http://cds.cern.ch/record/2670395.
  * [79] M.J. Berger, M. Inokuti, H.H. Andersen, H. Bichsel, D. Powers, S.M. Seltzer, D. Thwaites, and D.E. Watt, _ICRU Report 49: Stopping Powers and Ranges for Protons and Alpha Particles_ , Journal of the International Commission on Radiation Units and Measurements os25 (1993), pp. NP–NP, Available at https://journals.sagepub.com/toc/crub/os-25/2.
  * [80] _Stopping power of matter for ions_ , https://www-nds.iaea.org/stopping (2024). [Online].
  * [81] J. Han and G. Guo, _Characteristics of energy deposition from 1-1000 MeV proton and neutron induced nuclear reactions in silicon_ , AIP Advances 7 (2017), p. 115220.
  * [82] J.M. Richard, _Antiproton physics_ , Frontiers in Physics 8 (2020).
  * [83] S. Maury, W. Oelert, W. Bartmann, P. Belochitskii, H. Breuker, F. Butin, C. Carli, T. Eriksson, S. Pasinelli, and G. Tranquille, _ELENA: The extra low energy anti-proton facility at CERN_ , Hyperfine Interactions 229 (2014), pp. 105–115, Available at https://doi.org/10.1007/s10751-014-1067-y.


# Mode Bifurcation and Fold Points of Complex Dispersion Curves for the
Metamaterial Goubau Line

P. L. Overfelt  Klaus Halterman<EMAIL_ADDRESS>Simin Feng  D. R.
Bowling  Research and Intelligence Department, Physics Division, Naval Air
Warfare Center, China Lake, California 93555, USA

###### Abstract

In this paper the complex dispersion curves of the four lowest-order
transverse magnetic modes of a dielectric Goubau line ($\epsilon>0,\mu>0$) are
compared with those of a dispersive metamaterial Goubau line. The vastly
different dispersion curve structure for the metamaterial Goubau line is
characterized by unusual features such as mode bifurcation, complex fold
points, both proper and improper complex modes, and merging of complex and
real modes.

The Goubau line (G-Line) has been known and studied since Sommerfeld and later
Goubau considered applications using non-radiating surface waves on
transmission lines sommer ; gball . While Sommerfeld analyzed a long
cylindrical metallic wire as the transmission line of interest, Goubau
realized that adding a dielectric outer sheath to the wire reduced the radial
extent of the electromagnetic (EM) field and thus the dimensions of the
associated excitation device. Interestingly, while the Sommerfeld wave can
exist only on a conductor of finite conductivity, the Goubau wave can exist
even when the inner conductor is assumed to have perfect conductivity. The
G-Line has been investigated since the first part of the twentieth century,
and its guided modes are well known sommer ; gball ; strat ; wald ; semen . As
with all open waveguide structures, the G-Line spectrum consists of a finite
discrete set of guided modes with purely real longitudinal propagation
constants and an infinite continuum of radiation modes. Also present on open
lossless structures are leaky waves marc ; marc2  characterized by discrete
complex longitudinal propagation constant solutions to the dispersion
equation, but which are improper solutions of Maxwell’s equations in that
these solutions decay longitudinally but do not obey the transverse radiation
condition and thus may only be used in restricted regions of space. Improper
waves are not considered part of an open waveguide spectrum and are often
referred to as “nonmodal” or “nonspectral” marc . Despite this fact, leaky
waves have found great usefulness in certain applications, particularly those
related to leaky wave antennas gold . Some authors have referred to the
complex solutions of the circular dielectric rod and the standard G-Line
dispersion equations as leaky modes and have considered them on a more or less
equal footing with the guided modes kim ; kim2 . The leaky waves of even the
standard G-Line are still not well characterized (only the transverse magnetic
(TM) solutions have been considered in detail kim ) but on that structure, all
complex leaky wave solutions of the characteristic equation have EM field
components that diverge as the radial coordinate increases to infinity and are
thus improper.

Figure 1: Schematic of the Goubau line geometry. The metal core of radius $a$
is a perfect conductor.

In this Letter the G-Line geometry (see Fig. 1) is used with a negative index
of refraction dispersive metamaterial (NIM) elef ; shad  replacing the usual
dielectric layer. Under these circumstances the metamaterial G-Line spectrum
consists of guided modes, radiation modes, improper complex waves, and proper
complex modes. In the following we consider only the symmetric transverse
magnetic (${\rm TM}_{0n}$) solutions of the metamaterial G-Line (MM G-line).
The transverse electric (${\rm TE}_{0n}$) and hybrid modes of the MM G-line
will be considered subsequently.

Using the geometry in Fig. 1, the characteristic equation for the G-Line with
perfectly conducting inner conductor and assuming a $\theta,z,t$ dependence of
$e^{-i(m\theta+\gamma z-\omega t)}$, can be written in the general form,

$\displaystyle\overline{\gamma}^{2}m^{2}\Bigl{(}\frac{1}{s^{2}}-\frac{1}{w^{2}}\Bigr{)}^{2}{\cal
Z}{\cal Y}+{\cal S}{\cal T}=0,$ (1)

where $\overline{\gamma}=\overline{\beta}+i\overline{\alpha}$ is the
normalized complex longitudinal propagation constant. The longitudinal phase,
$\overline{\beta}$, and attenuation $\overline{\alpha}$ are both normalized by
$k_{0}\equiv\omega/c$, the free space wavenumber. Considering the $m=0$ case,
Eq. (1) reduces to ${\cal T}=0$ for TM modes, where,

$\displaystyle{\cal T}=\frac{\epsilon_{2}{\cal W}}{w}-\epsilon_{3}{\cal
Z}g=0,$ (2)

${\cal S}={\mu_{2}{\cal X}}/{w}-\mu_{3}{\cal Y}g$, and $g\equiv-
H_{1}^{(2)}(s)/(sH_{0}^{(2)}(s))$. We also define, ${\cal
Z}=J_{0}(w)Y_{0}(v)-J_{0}(v)Y_{0}(w)$, ${\cal
Y}=-J_{0}(w)Y_{1}(v)+J_{1}(v)Y_{0}(w)$, ${\cal
W}=-J_{1}(w)Y_{0}(v)+J_{0}(v)Y_{1}(w)$, and ${\cal
X}=J_{1}(w)Y_{1}(v)-J_{1}(v)Y_{1}(w)$, with the $J$’s, $Y$’s and $H$’s the
usual Bessel functions of the first and second kind, and the Hankel functions
of the second kind, respectively. The dimensionless wavenumbers are,
$v=k_{2}a$, $w=k_{2}b$, and $s=k_{3}b$. The transverse propagation wave
numbers are given by
$k_{j}=k_{0}\sqrt{\epsilon_{j}\mu_{j}-\overline{\gamma}^{2}}$, for $j=2,3$.
For these ${\rm TM}_{0n}$ modes, the EM fields in the $j$th region are
generally written as, ${\bm{H}}_{j}=H_{j,\theta}\hat{\bm{\theta}}e^{-i(\gamma
z-\omega t)}$, and
${\bm{E}}_{j}=(E_{j,\rho}\hat{\bm{\rho}}+E_{j,z}\hat{\bm{z}})e^{-i(\gamma
z-\omega t)}$, for propagation in the positive $z$ direction. Maxwell’s
equations and the cylindrical symmetry result in the following transverse
fields, $H_{j,\theta}=-i\epsilon_{j}k_{0}/\kappa_{j}^{2}\partial
E_{j,z}/\partial\rho$, and $E_{j,\rho}=-i\gamma/\kappa_{j}^{2}\partial
E_{j,z}/\partial\rho$. The forms of (1) - (2) above are most suitable for
emphasizing material parameter changes in both $\epsilon$ and $\mu$ of Regions
2 and 3, although they have been derived previously in alternate forms semen .

Figure 2: Dispersion for standard G-line

We solve Eq. (2) numerically for its complex zeros as a function of frequency
by employing a recently developed complex root finding algorithm. Fig. 2 shows
the four lowest-order ${\rm TM}_{0n}$ ($n=0,1,2,3$) modes as functions of
frequency (in GHz) for the standard G-Line. Part (a) shows the normalized
longitudinal phase vs. frequency while Part (b) shows normalized longitudinal
attenuation vs. frequency. In Fig. 2 the parameters $\epsilon_{2}=5$,
$\mu_{2}=1$, $a=5$ mm, $b=10$ mm were used to compare our results with those
in Ref. kim, . In particular, note the lowest-order ${\rm TM}_{00}$ mode
(black curve) that has no cutoff frequency and $\alpha=0$. Except for this
mode, our results are consistent with Ref. kim, . We have assumed that the
outside region is air and thus $\epsilon_{3}=\mu_{3}=1$. At a given frequency,
when the imaginary part of a root is zero, that root represents a guided or
bound solution. When the imaginary part is nonzero, that root represents a
complex leaky wave solution of (2). Upon calculating the EM field components
associated with certain leaky wave solutions in Fig. 2, all discrete complex
solutions of the standard G-line (with Region 2 index of refraction greater
than one) are improper marc ; kim . Because the outside region is air, cutoff
occurs when $\overline{\gamma}=1$ or $s=0$. When
$1<\overline{\beta}<\sqrt{5}$, and $\overline{\alpha}=0$, the well-known
guided modes result. When $\overline{\beta}<1$, $\overline{\alpha}\neq 0$, the
waves are leaky. Also note that an infinite number of higher order leaky waves
can occur at lower frequencies These leaky waves have high frequency cutoffs
but no low frequency cutoffs. Generally these waves have
$|\overline{\alpha}|\gg 1$ and thus are not so far found to be particularly
useful. In Fig. 2, the real and imaginary parts of the longitudinal
propagation constant are correlated using color. Thus the lowest-order mode
(black) has zero attenuation and is purely guided [Fig. 2(a)], the second
lowest-order mode (magenta) is leaky at the lower frequencies with its
attenuation magnitude increasing as the frequency decreases (see Fig. 2(a) and
(b)). For this mode as the frequency increases up to cutoff at about 14.9 GHz,
the associated attenuation decreases in magnitude until this mode becomes a
guided mode once cutoff is passed (magenta $\rightarrow$ blue). The turquoise
blue and red curves show the next higher-order modes which for much of the
frequency range are leaky with $|\alpha|\gg 1$.

If the dielectric layer of Region 2 is replaced by a negative index of
refraction MM, the TM mode structure is vastly different. The MM is
characterized by the following dispersion formulas,
$\mu_{2}(f)=\mu_{l}+Ff^{2}/(f_{0}^{2}-f^{2}-i\Gamma f)$, and,
$\epsilon_{2}(f)=\epsilon_{h}+f_{p}^{2}/(f_{r}^{2}-f^{2}-i\Gamma f)$. We take
the filling factor, $F$, to be $0.25$, and the damping factor, $\Gamma$, to be
zero. The low and high frequency limits are $\mu_{l}=-1$ and
$\epsilon_{h}=-5$, respectively. The resonance frequencies are, $f_{0}=0.75$
GHz, $f_{r}=0.5$ GHz, and $f_{p}=0.25$ GHz.

Figure 3: Dispersion for the MM G-line. The curves in (c) and (d) give a
magnified view of $\overline{\beta}$ and $\overline{\alpha}$, respectively,
near the bifurcation point (Point 4).

Figure 3 contains the four lowest-order TM modes assuming a metamaterial
G-Line with both $\epsilon_{2}$ and $\mu_{2}$ negative over the frequency
range of interest. Fig. 3(a) shows the normalized longitudinal phase vs.
frequency, part (b) shows the normalized longitudinal attenuation vs.
frequency, where $|\overline{\alpha}|$ is small, and also shows the normalized
longitudinal attenuation on a larger scale. In Fig. 3, $\epsilon_{2}\approx-5$
and $\mu_{2}\approx-1$. The radii, $a$ and $b$, are the same as for the
standard G-line in Fig. 2. Referring to the various mode “paths” or “branches”
in Fig. 3, note that the green curve now has a leaky part (below about 3 GHz)
and the portion of the green curve with $\overline{\beta}>1$ now has
$\overline{\alpha}\neq 0$. As the frequency decreases, the attenuation peaks
in magnitude between 4 and 5 GHz (green curve) and then starts to decrease.
Thus the green path is a wave with a complex $\overline{\gamma}$. This path is
also improper (until it merges with the black curves at much higher
frequency). Thus the fundamental guided mode of the standard G-line is
replaced by a complex mode that is improper. The mode represented by the
magenta path is also complex with $\overline{\alpha}$ nonzero until point 4 is
reached. However this path is proper. At point 4 ($f\approx 10.025$ GHz),
there is a mode bifurcation [see Fig. 3(c) and (d)], as has been observed in
catastrophe and bifurcation theories. Catastrophe theory attempts to study how
the qualitative nature of solutions of equations depends on the parameters
appearing in those equations hanson ; poston . These techniques are invaluable
in analyzing the qualitative changes in system dynamical characteristics by
small perturbations. From this standpoint, the dispersion relationship, ${\cal
T}=0$, is cast as a nonlinear equation in $\overline{\beta}$,
$\overline{\alpha}$, and $f$ with parameters $\epsilon_{1}$, $\epsilon_{2}$,
$\mu_{1}$, $\mu_{2}$, $a$, and $b$. Using catastrophe theory, areas of unusual
qualitative interest such as the fold point at Point 4 can be determined.
Using numerical techniques alone to solve the dispersion relation over a
frequency range is difficult due to its “dense” set of zeros at each
frequency. Point 4 is a complex fold point (a point where, because no material
losses in the line have been assumed as yet, $\overline{\gamma}$,
$-\overline{\gamma}$, $\overline{\gamma}^{*}$, and $-\overline{\gamma}^{*}$
all meet at the same frequency) yak ; hanson ; poch ; poston . As the
frequency increases past the fold point, $\overline{\alpha}\rightarrow 0$
along both upper and lower black paths. The black paths near points 2 and 3
are therefore both guided modes immediately after bifurcation.

Thus, in Fig. 3 there are an improper complex set of roots, a proper complex
set of roots, and a mode bifurcation at a fold point in (a) near $f\approx 10$
GHz past which there are two guided modes (i.e., $\overline{\alpha}=0$ on both
black paths after 10 GHz). By considering the two guided paths (2 and 3) where
$\overline{\alpha}=0$, we find that the black upper path always has a slope
opposite in sign to the lower path until the first cutoff frequency, $f\approx
14.9$ GHz, and thus is backward traveling in this frequency region. At cutoff
the lower path near point 3 has zero slope and after cutoff, the slope becomes
positive. Also after cutoff, this path becomes complex with
$|\overline{\alpha}|\ll 1$, as shown by the blue curve in Fig. 3(b). The red
and turquoise blue curves are quite similar to those in Fig. 2 over the given
frequency range but they, too, exhibit the bifurcation and fold point
phenomena at higher frequencies None of the above situations occur for the
standard G-line.

Figure 4: Normalized power (see text) as a function of frequency for a few of
the dispersion curves in Fig. 3, and using the same color scheme. In (a), the
radial power flux increases, consistent with the attenuation growth for the
mode. In (b), the longitudinal power flux in the MM region declines rapidly
with frequency. The modes in (c) and (d) are strictly guided, with the EM
energy flowing in the MM mainly for those modes in the upper branch of the
dispersion curve of Fig. 3.

Certain effects similar to those in Fig. 3 have been seen in both grounded
metamaterial slab geometries shu ; lovat  and in complicated grounded
dielectric-loaded, open geometries yak ; bac ; lamp ; maj ; tamir . In the
case of the MM G-line, we have a transition from a complex proper mode
(magneta curve in Fig. 3(a)) to a pair of guided modes at higher frequencies.
We see in both the normalized phase and attenuation the characteristic
intersection of a parabola and a straight line [Fig. 3(c) and (d)]. The phase
has a straight line to parabola transition characteristic of a fold point in
catastrophe theory yak ; hanson ; poch ; poston . Note that the attenuation
for the MM G-line in Fig. 3(b) has changed sign for the red, magenta, and
turquoise blue curves when compared to the standard G-line. while the
attenuation shows a parabola to straight line transition.

The guided and leaky modes of the G-line and MM G-line can be described as
slow modes and fast modes, respectively. The guided modes are slow in that
they occur in the spectral region
$\sqrt{\epsilon_{3}\mu_{3}}<\overline{\beta}<\sqrt{\epsilon_{2}\mu_{2}}$.
Since $\overline{\beta}>1$, $c\beta/\omega>1$ but $\omega/(c\beta)<1$, thus
their phase velocity is less than the speed of light. In contrast the leaky
waves are fast waves. Note that with reference to both Fig. 2 and Fig. 3 there
is a point at the low frequency end of each graph where (although we cut the
graphs off at $\sqrt{\epsilon_{2}\mu_{2}}$) there are complex solutions of the
dispersion relation that have $\overline{\beta}>\sqrt{\epsilon_{2}\mu_{2}}$,
and $\overline{\alpha}\neq 0$. For the standard G-line this region is a
“forbidden” region. For the MM G-line of Fig. 3,
$\overline{\beta}>\sqrt{\epsilon_{2}\mu_{2}}$ is no longer considered
forbidden, and “superslow modes” can arise shad .

The total power through the various regions can be described by the normalized
power in the radial and longitudinal directions,
$\overline{P}_{j,\sigma}=P_{j,\sigma}/P_{\rm tot}$, for $\sigma=\rho,z$, and
$j=2$ or $3$. The total power, $P_{\rm tot}$ is the sum of power flux, defined
as, $P_{\rm tot}\equiv\sum_{j,\sigma}|P_{j,\sigma}|$. The power flux through
the surface of a cylinder of radius $R$ and length $L$ encompassing a segment
of the structure is determined from the spatially integrated time-averaged
Poynting vector ${\bm{S}}_{j}$, with components,
${S}_{j,z}(\rho,z)=(c/8\pi)e^{2\alpha z}{\rm Re}[E_{j,\rho}H_{j,\theta}^{*}]$,
or ${S}_{3,\rho}(\rho,z)=-(c/8\pi)e^{2\alpha z}{\rm
Re}[E_{3,z}H_{3,\theta}^{*}]$. Thus, the longitudinal power transmitted
through the two circular regions is, $P_{j,z}|_{z=L}=2\pi e^{2\alpha
L}\int_{0}^{R}\rho d\rho S_{j,z}(\rho,0)$, and
$P_{j,z}|_{z=0}=-2\pi\int_{0}^{R}\rho d\rho S_{j,z}(\rho,0)$. For the surface
normal to $\hat{\bm{\rho}}$, integration gives, $P_{3,\rho}=2\pi R\,e^{\alpha
L}(1/\alpha)\sinh(\alpha L)S_{3,\rho}(R,0)$. We take $R/b=10$. In Fig. 4, we
illustrate the normalized power as a function of frequency for several of the
modes shown in Fig. 3. In panel (a), power flows radially, with maxima at
frequencies correlated to when $|\alpha|$ peaks for that mode (Fig. 3(b)).
Although the power flow is predominately in the air region, is is evident that
the counter-directed longitudinal power flux in the MM region declines rapidly
towards zero with increased frequency [panel (b)]. This indicates that power
emission from the waveguide takes place at a slight angle, $\vartheta$ from
the $z$ axis over a relatively small bandwidth. As for the proper mode
spectra, $|\overline{P}_{j,z}|=1/2$ for both regions, but counter-directed. At
the bifurcation point ($f\approx 10$ GHz), the leaky wave transforms into a
strictly guided mode at higher frequencies, where the two counter propagating
regions are clearly identified by their sign [Fig. 4(c) and (d)]. The lower
branch (see Fig. 3) in the guided mode dispersion clearly maps to power flow
that is predominately in the air region. This is in contrast to modes
occupying the upper branch in Fig. 3, where again in Fig. 4(c) and (d) we see
the power distribution shift from both regions equally to reside completely in
the MM at higher frequencies. This also verifies that the MM G-line has the
remarkable property that certain guided modes can propagate below the first
waveguide cutoff and can be of the backward as well as of the forward wave
type hrab ; hrab2 ; lub .

In conclusion, dispersion curve behavior for a dielectric versus MM G-Line has
been shown to be vastly different. The MM G-line dispersion is characterized
by unusual mode bifurcations, complex fold points a la catastrophe theory, and
both proper and improper complex modes.

The authors acknowledge G. A. Lindsay, Z. Sechrist, and G. Ostrom for valuable
discussions and the support from ONR, as well as NAVAIR’s ILIR program from
ONR. This work is also supported in part by a grant of HPC resources as part
of the DOD HPCMP.

## References

  * (1) A. Sommerfeld, Ann. Physik 15, 673 (1904); G. Goubau, J. Appl. Phys. 21, 1119 (1950).
  * (2) G. Goubau, IRE Trans. Antennas Propag. 7, S140 (1959).
  * (3) J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  * (4) R. A. Waldron, Theory of Guided Electromagnetic Waves (Van Nostrand Reinhold, London, 1969).
  * (5) N. A. Semenov, Radio Eng. Electr. Phys. 9, 989 (1964); J. G. Fikioris and J. A. Roumeliotis, IEEE Trans. Microwave Theory Tech. 27, 570 (1979); K. Sakina and J. Chiba, IEICE Trans. Electron.76, 657 (1993).
  * (6) N. Marcuvitz, IRE Trans. Antennas Propag. 4,192 (1956); T. Tamir and A. A. Oliner, Proc. IEE 110, 310 (1963).
  * (7) D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, Orlando, FL, 1974); A. W. Snyder, Appl. Phys. 4, 273 (1974).
  * (8) L. O. Goldstone and A. A. Oliner, IRE Trans. Antennas Propag. XX, 307 (1958 or 1959); R. E. Collin and F. J. Zucker, Antenna Theory, Part 2 (McGraw-Hill Book Company, New York, 1969); R. C. Johnson, Antenna Engineering Handbook (McGraw-Hill Book Company, New York, 1993) Third Edition, Chapter 10; R. E. Collin, Field Theory of Guided Waves, 2nd ed. (New York, IEEE Press, 1991); A. Alu, et al., IEEE Trans. Antennas Propag. 55, 1698 (2007).
  * (9) K. Y. Kim, Microwave Opt. Tech. Lett. 50, 523 (2008);
  * (10) K. Y. Kim, et al., Electr. Lett. 39, 61 (2003);K. Y. Kim, Guided and Leaky Modes of Circular Open Electromagnetic Waveguides, PhD Thesis (2004).
  * (11) G. V Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials (IEEE Press, Hoboken, NJ, 2005); N. Engheta and R. W. Ziolkowski, Metamaterials (IEEE Press, Hoboken, NJ, 2006).
  * (12) I. V. Shadrivov, et al., Phys. Rev. E67,057602 (2003); K. Halterman, et al., Phys. Rev. A76, 013834 (2007).
  * (13) A. B. Yakovlev and G. W. Hanson, IEEE Trans. Microwave Theory Tech. 45, 87 (1997).
  * (14) G. W. Hanson and A. B. Yakovlev, Radio Science 33, 803 (1998).
  * (15) I. E. Pochanina, et al., Radiofizica 32, 1000 (1989).
  * (16) T. Poston and I. Stewart, Catastrophe Theory and Its Applications (London, Pitman, 1978); R. Gilmore, Catastrophe Theory for Scientists and Engineers (New York, Wiley, 1981).
  * (17) W. Shu and J. M. Song, PIER 65, 103 (2006).
  * (18) G. Lovat, et al., Metamaterials 34 (2008), doi:10.1016/ j.metmat.2008.09.002.
  * (19) P. Baccarelli, et al., IEEE Trans. Microwave Theory Tech. 53, 32 (2005).
  * (20) P. Lampariello, et al., IEEE Trans. Microwave Theory Tech. 38, 1831 (1990).
  * (21) S. Majumder, et al., IEEE Trans. Microwave Theory Tech. 45, 2296 (1997).
  * (22) T. Tamir and F. Y. Kou, IEEE J. Quantum Electr. 22, 544 (1986).
  * (23) S. Hrabar and G. Jankovic, IEEE MELECON, 280, Malaga, Spain, May 16-19 (2006).
  * (24) S. Hrabar, et al., IEEE Trans. Antennas Propag. 53,110 (2005).
  * (25) G. Lubkowski, et al., IET Microwave Antennas Propag. 1, 165 (2007).