# Weight Pruning and Uncertainty in Radio Galaxy Classification

Devina Mohan
Department of Physics & Astronomy
University of Manchester, UK
<EMAIL_ADDRESS>
Anna Scaife
Department of Physics & Astronomy
University of Manchester, UK
<EMAIL_ADDRESS>
The Alan Turing Institute, 96 Euston Rd, London, UK<EMAIL_ADDRESS>
###### Abstract

In this work we use variational inference to quantify the degree of epistemic
uncertainty in model predictions of radio galaxy classification and show that
the level of model posterior variance for individual test samples is
correlated with human uncertainty when labelling radio galaxies. We explore
the model performance and uncertainty calibration for a variety of different
weight priors and suggest that a sparse prior produces more well-calibrated
uncertainty estimates. Using the posterior distributions for individual
weights, we show that signal-to-noise ratio (SNR) ranking allows pruning of
the fully-connected layers to the level of 30% without significant loss of
performance, and that this pruning increases the predictive uncertainty in the
model. Finally we show that, like other work in this field, we experience a
cold posterior effect. We examine whether adapting the cost function in our
model to accommodate model misspecification can compensate for this effect,
but find that it does not make a significant difference. We also examine the
effect of principled data augmentation and find that it improves upon the
baseline but does not compensate for the observed effect fully. We interpret
this as the cold posterior effect being due to the overly effective curation
of our training sample leading to likelihood misspecification, and raise this
as a potential issue for Bayesian deep learning approaches to radio galaxy
classification in future.

##  1 Introduction

A new generation of radio astronomy facilities around the world such as the
Low-Frequency Array [LOFAR; 1], the Murchison Widefield Array [MWA; 2], the
MeerKAT telescope [3], and the Australian SKA Pathfinder (ASKAP) telescope [4]
are generating increasingly larger and larger data rates. In order to extract
scientific impact from these facilities on reasonable timescales, a natural
solution has been to automate the data processing as far as possible and this
has lead to the increased adoption of machine learning methodologies.

In particular for new sky surveys, automated classification algorithms are
being developed to replace the _by eye_ approaches that were possible
historically. In radio astronomy specifically, studies looking at
morphological classification using convolutional neural networks (CNNs) and
deep learning have become increasingly common, in particular with respect to
the classification of radio galaxies.

The Fanaroff-Riley (FR) classification of radio galaxies was introduced over
four decades ago [5], and has since been widely adopted and applied to many
catalogues since then. The morphological divide seen in this classification
scheme has historically been explained primarily as a consequence of differing
jet dynamics. Fanaroff-Riley type I (FRI) radio galaxies have jets that are
disrupted at shorter distances from the central super-massive black hole host
and are therefore centrally brightened, whilst Fanaroff-Riley type II (FRII)
radio galaxies have jets that remain relativistic to large distances,
resulting in bright termination shocks. These observed structural differences
may be due to the intrinsic power in the jets, but will also be influenced by
local environmental densities [6, 7].

Intrinsic and environmental effects are difficult to disentangle using radio
luminosity alone as systematic differences in particle content, environmental
effects and radiative losses make radio luminosity an unreliable proxy for jet
power [8]. Hence the use of morphology for inferring the environmental impact
on radio galaxy populations is therefore important for gaining a better
physical understanding of the FR dichotomy, and of the full morphological
diversity of the population.

From a deep-learning perspective, the ground work for morphological
classification in this field was done by [9]. This was followed by other works
involving the use of deep learning in source classification [e.g. 10, 11, 12].
More recently, [13] showed that an attention-gated CNN could perform
classification of radio galaxies with equivalent performance to other
applications in the literature, but using $\sim$50% fewer learnable
parameters, [14] showed that using group-equivariant convolutional layers that
preserved the rotational and reflectional isometries of the Euclidean group
resulted in improved overall model performance and stability of model
confidence for radio galaxies at different orientations, and [15] generated
synthetic populations of radio galaxies using structured variational
inference.

With the exception of [14], there has been little work done on understanding
the degree of confidence with which CNN models predict the class of individual
radio galaxies; however, for radio astronomy, where modern astrophysical
analysis is driven by population analyses, quantifying the confidence with
which each object is assigned to a particular classification is crucial for
understanding the propagation of uncertainties within that analysis.

This work In this work we use variational inference (VI) to quantify the
degree of epistemic uncertainty in model predictions of radio galaxy
classification. This differs from the approach of [14] who used dropout as a
Bayesian approximation [16] to estimate model confidence as a function of
image orientation, which is just one specific aspect of model performance and
not directly comparable to this work. We compare the variance of our posterior
predictions to qualifications present in our test data that indicate the level
of human confidence in assigning a classification label and show that model
uncertainty is correlated with human uncertainty.

An added advantage of Bayesian CNNs is their ability to identify the
significance of individual weights within the model. In this work we show that
weight signal-to-noise ratio (SNR) allows pruning of our fully-connected
layers to the level of 30% without significant loss of model performance. We
also find that pruning based on a Fisher analysis is able to thin our model
more effectively, to the level of 60%, but that both pruning methods increase
the uncertainty calibration error of the model in different ways.

Finally we show that, like other work in this field, we experience a _cold
posterior_ effect [see e.g. 17]. As suggested by [18], we examine whether a
PAC Bayes approach adapting the cost function in our model to accommodate
model misspecification can compensate for this effect, but find that it does
not make a significant difference. Other works in this area have suggested
that unprincipled data augmentation could be a contributing factor to the cold
posterior effect [e.g. 19, 20]. We examine the effect of principled data
augmentation and find that the cold posterior effect observed in our work
_reduces_ slightly with data augmentation. Consequently we suggest that the
cold posterior effect in this case is likely to be due to the manner in which
the training sample is curated, rather than model misspecification.

##  2 Uncertainty in radio galaxy classification

For this work we use the MiraBest radio galaxy dataset111The data used in this
work is provided under a Creative Commons license at
https://zenodo.org/record/4288837 [21], which is based on the catalogue of
[22], who used a parent galaxy sample taken from [23] that cross-matched the
Sloan Digital Sky Survey [SDSS; 24] data release 7 [DR7; 25] with the Northern
VLA Sky Survey [NVSS; 26] and the Faint Images of the Radio Sky at Twenty
centimetres [FIRST; 27]. The parameters of the dataset itself and the image
preprocessing are described in [14]. For this work we extract the objects
labelled as FRI and FRII galaxies. This creates a binary dataset and we do not
employ sub-classifications. Each object within the MiraBest dataset was
assigned a confidence qualification: _Confident_ or _Uncertain_ by its
original human classifiers. Training is performed on the _Confident_ subset.

We use an expanded LeNet-5 architecture incorporating two additional
convolutional layers with 26 and 32 channels, respectively, and implement the
variational inference approach described in [28] using the Adam optimiser with
an initial learning rate of $5.10^{-5}$. Hyper-parameters were tuned using a
grid search. We use a Gaussian variational approximation to the true posterior
and we consider four different prior distributions over the model parameters:
a simple Gaussian, a two component Gaussian mixture model (GMM) prior, a
Laplace prior and a Laplace Mixture Model (LMM) prior. We find that model
performance is optimised using a Laplace prior. Using a GMM prior allows us to
achieve comparable accuracy to the model trained with a Laplace prior but it
has worse uncertainty calibration error and experiences a more significant
cold posterior effect, see Figure 1. In the following analysis we report
results for the Laplace prior.

We train the model in batches of 50 images and implement an early stopping
criterion based on validation accuracy. We note that while other works have
used deeper models for classification of radio galaxies, no significant
difference in performance is seen for binary FR classification. We also note
that, unlike previous works, we do not use any data augmentation in our
training. Models were trained on an Intel i5 CPU for 500 epochs. Each training
run took approximately $1$ hour.

Figure 1: The “cold posterior” effect: for the MiraBest classification problem
presented here we can improve the generalization performance significantly by
cooling the posterior with a temperature $T\ll 1$, deviating from the Bayes
posterior. Data are shown for the BBB models with no data augmentation and the
original ELBO cost function trained with a Laplace prior (solid blue line),
and trained with a GMM prior (orange dashed line).

The cold posterior effect It has been observed in the literature that in order
to obtain good predictive performance from Bayesian neural networks, it is
necessary to down-weight or _lower the temperature_ of the Bayesian posterior.
Several explanations have been proposed to explain this effect, including
model or prior misspecification [17], and data augmentation or curation issues
[29]. We also observe this effect in the classification of radio galaxies in
this work, see Figure 2.

Figure 2: The “cold posterior” effect: Data are shown for the BBB model
trained with a Laplace prior with no data augmentation and the original ELBO
cost function (solid blue line), the BBB model with no data augmentation and
the Masegosa posterior cost function (red dashed line), the BBB model with
data augmentation and the original ELBO function (orange dot-dash line), and
the BBB model with data augmentation and the Masegosa posterior cost function
(green dotted line).

Assuming that all models are likely to be misspecified at some level, [18]
suggested that the use of more generalised posteriors that treat the true
Bayes posterior as a solution of a loose PAC-Bayes generalization bound on the
predictive cross-entropy might provide a more desirable approximation target
than the Bayes posterior itself. We retrain our models using the PAC-Bayes
cost function defined in [18], but find that this does not make a significant
difference to the observed cold posterior effect.

If we include data augmentation using random rotations from 0 to 360 degrees,
the cold posterior effect observed in our work _reduces_ slightly, see Figure
2. We suggest that this is because we have augmented the MiraBest dataset
using principled methods that correspond to an informed prior for how radio
galaxies are oriented, as radio galaxy class is assumed to be equivariant to
orientation and chirality [see e.g. 30, 14]. Since the Masegosa posterior is a
more complete test of model misspecification than the data augmentation used
here is of likelihood misspecification, we suggest that a key element for
exploring this problem in future may be the availability of radio astronomy
training sets that do not only present average or consensus target labels, but
instead include all individual labels from human classifiers. For the
following results we use a posterior temperature of $T=10^{-2}$.

Model Confidence As well as the MiraBest _Confident_ test set, we use 49
samples from the MiraBest _Uncertain_ subset to test the trained model’s
ability to correctly represent epistemic uncertainty. These samples can be
considered to be drawn from the same data generating distribution as the
MiraBest _Confident_ samples, but have a lower degree of belief in their
classification. Using Monte Carlo samples obtained from the trained predictive
posterior distribution, we obtain 200 Softmax probabilities for each test
sample. Following [16], we find that the predictive uncertainty (epistemic +
aleatoric), as quantified by the entropy, and the model uncertainty
(epistemic), as quantified by the mutual information of these samples, show
that on average test data samples labelled as _Uncertain_ have higher
epistemic uncertainty than those labelled as _Confident_ , see Figure 3. To
quantify aleatoric uncertainty for in-distribution data samples for classical
NNs [31] demonstrate that the entropy of a single pass can be used. Here we
extend that definition to our Bayesian NN and take the average entropy for a
single input using MC samples to capture the aleatoric uncertainty associated
with each data point.

We also examine the uncertainty metrics for the MiraBest _Hybrid_ subsample,
which contains galaxies that have characteristics of both classes. We find
that all uncertainty measures for this sample are even higher than for
_Uncertain_ FRI and FRII type objects, see Figure 3. We also note that
Confidently classified _Hybrid_ samples have higher predictive uncertainty
than the Uncertain _Hybrid_ samples. This could be because the Uncertain
samples are more like the FRI/FRII galaxies that the model has seen during
training, i.e. their classification as a _Hybrid_ was considered Uncertain by
a human classifier because the morphology was biased towards one of the
standard FRI or FRII classifications. In which case their predictive
uncertainty might be expected to be lower since the model was trained to
predict those morphologies.

Figure 3: Distribution of predictive uncertainty (entropy; left), epistemic
uncertainty (mutual information; centre) and aleatoric uncertainty (average
entropy; right) for the _Confident_ (MBFR_Conf) and _Uncertain_ (MBFR_Uncert)
test samples. Data are also shown for the _MiraBest_ Hybrid test sample.

Uncertainty calibration We quantify the calibration of our posterior
uncertainties using the class-wise expected Uncertainty Calibration Error
(cUCE; [32]). We find that predictive entropy and average entropy for the
unpruned model are better calibrated than the mutual information, see Table 1.

##  3 Weight pruning based on posterior variance

For a typical variational posterior such as the Gaussian distribution, the
number of parameters in a Bayesian model will double compared its standard
counterpart because both the mean and standard deviation values need to be
learned. Consequently, to reduce the computational cost and memory overhead
during deployment, network pruning approaches which remove uninformative
parameters are often used. Several authors have also considered pruning to
improve the generalisation performance of the network [e.g. 33].

Following [28], we calculate the signal-to-noise ratio (SNR) of individual
weights as $|\mu|/\sigma$, where the trained posterior distribution on a
weight is $\mathcal{N}(\mu,\sigma^{2})$. By ranking all of the model weights
in order of SNR, we are able to prune a given percentage of the lowest SNR-
valued weights. We then calculate the error on the test set using the average
of 100 forward passes.

The choice of prior directly influences the shape of the SNR distribution and
the proportion of weights that can be pruned without affecting model
performance on the test set. For the Laplace prior, we find that up to 30% of
the weights in the fully-connected layers can be pruned without a significant
drop in performance. However, following the work of [34], we also explore a
pruning approach that uses the Fisher Information Matrix (FIM) of the weights.
As also observed by [34], pruning the weights based on the Fisher information
alone does not allow for a large number of parameters to be pruned effectively
because many values in the FIM diagonal are close to zero; however, combining
Fisher-based pruning and magnitude-based pruning allows for a larger number of
weights to be pruned - up to 60% in this case.

We find that both the SNR and Fisher pruned models have higher predictive
uncertainty than the unpruned model. This is also reflected in their
distributions of aleatoric uncertainty. The epistemic uncertainty narrows
slightly for both the pruning methods. Pruning also affects the uncertainty
calibration error, see Table 1. We find that while cUCE of mutual information
does not change significantly, the predictive entropy and average entropy
values have a higher calibration error, the effect being worse for SNR
pruning.

Table 1: Percentage classwise Uncertainty Calibration Error (cUCE) on MiraBest Confident test set for our BBB-CNN model trained with a Laplace prior. Results are shown for the unpruned model and the model pruned to its threshold limit for SNR and FIM-based pruning. The percentage cUCE is shown separately for the predictive entropy (PE), mutual information (MI) and average entropy (AE) as calculated on the MiraBest Confident test set. |  | % cUCE
---|---|---
Prior | Pruning | PE | MI | AE
Laplace | Unpruned | 9.69 | 16.37 | 10.84
| SNR | 14.35 | 16.82 | 13.93
| FIM | 13.43 | 15.29 | 11.25

##  4 Conclusions

In this work we find that whilst Bayesian neural networks using variational
inference are useful for returning the degree of model confidence in an
individual classification, and that on average this seems to agree with more
general degrees of confidence assigned by human classifiers, there is still
work that needs to be done on the development of these models. Most
specifically, the effect of overly curated training data samples requires
additional investigation and we suggest that a key element for exploring this
problem in future may be the availability of radio astronomy training sets
that do not only present average or consensus target labels, but instead
include all individual labels from human classifiers.

Code for this work is available at:
https://github.com/devinamhn/RadioGalaxies-BBB.

## Acknowledgments and Disclosure of Funding

AMS gratefully acknowledges support from an Alan Turing Institute AI
Fellowship EP/V030302/1.

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figuret

11institutetext:  E. Butler (11email<EMAIL_ADDRESS>22institutetext: W.
Bertsche 33institutetext: M. Charlton 44institutetext: A.J. Humphries
55institutetext: N. Madsen 66institutetext: D.P. van der Werf 77institutetext:
D. Wilding 88institutetext: Department of Physics, Swansea University, Swansea
SA2 8PP, United Kingdom 99institutetext: G.B. Andresen 1010institutetext: P.D.
Bowe 1111institutetext: J.S. Hangst 1212institutetext: Department of Physics
and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark 1313institutetext:
M.D. Ashkezari 1414institutetext: M.E. Hayden 1515institutetext: Department of
Physics, Simon Fraser University, Burnaby BC, V5A 1S6, Canada
1616institutetext: M. Baquero-Ruiz 1717institutetext: C.C. Bray
1818institutetext: S. Chapman 1919institutetext: J. Fajans 2020institutetext:
A. Povilus 2121institutetext: C. So 2222institutetext: J.S. Wurtele
2323institutetext: Department of Physics, University of California, Berkeley,
CA 94720-7300, USA 2424institutetext: C.L. Cesar 2525institutetext: R. Lambo
2626institutetext: Instituto de Física, Universidade Federal do Rio de
Janeiro, Rio de Janeiro 21941-972, Brazil 2727institutetext: T. Friesen
2828institutetext: M.C. Fujiwara 2929institutetext: R. Hydomako
3030institutetext: R.I. Thompson 3131institutetext: Department of Physics and
Astronomy, University of Calgary, Calgary AB, T2N 1N4, Canada
3232institutetext: M.C. Fujiwara 3333institutetext: D.R. Gill
3434institutetext: L. Kurchaninov 3535institutetext: K. Olchanski
3636institutetext: A. Olin 3737institutetext: J.W. Storey 3838institutetext:
TRIUMF, 4004 Wesbrook Mall, Vancouver BC, V6T 2A3, Canada 3939institutetext:
W.N. Hardy 4040institutetext: Department of Physics and Astronomy, University
of British Columbia, Vancouver BC, V6T 1Z4, Canada 4141institutetext: R.S.
Hayano 4242institutetext: Department of Physics, University of Tokyo, Tokyo
113-0033, Japan 4343institutetext: S. Jonsell 4444institutetext: Fysikum,
Stockholm University, SE-10609, Stockholm, Sweden 4545institutetext: S. Menary
4646institutetext: Department of Physics and Astronomy, York University,
Toronto, ON, M3J 1P3, Canada 4747institutetext: P. Nolan 4848institutetext: P.
Pusa 4949institutetext: Department of Physics, University of Liverpool,
Liverpool L69 7ZE, United Kingdom 5050institutetext: F. Robicheaux
5151institutetext: Department of Physics, Auburn University, Auburn, AL
36849-5311, USA 5252institutetext: E. Sarid 5353institutetext: Department of
Physics, NRCN-Nuclear Research Center Negev, Beer Sheva, IL-84190, Israel
5454institutetext: D.M. Silveira 5555institutetext: Y. Yamazaki
5656institutetext: Atomic Physics Laboratory, RIKEN Advanced Science
Institute, Wako, Saitama 351-0198, Japan
Graduate School of Arts and Sciences, University of Tokyo, Tokyo 153-8902,
Japan

# Towards Antihydrogen Trapping and Spectroscopy at ALPHA

E. Butler  G.B. Andresen  M.D. Ashkezari  M. Baquero-Ruiz  W. Bertsche  P.D.
Bowe  C.C. Bray  C.L. Cesar  S. Chapman  M. Charlton  J. Fajans  T. Friesen
M.C. Fujiwara  D.R. Gill  J.S. Hangst  W.N. Hardy  R.S. Hayano  M.E. Hayden
A.J. Humphries  R. Hydomako  S. Jonsell  L. Kurchaninov  R. Lambo  N. Madsen
S. Menary  P. Nolan  K. Olchanski  A. Olin  A. Povilus  P. Pusa  F. Robicheaux
E. Sarid  D.M. Silveira  C. So  J.W. Storey  R.I. Thompson  D.P. van der Werf
D. Wilding  J.S. Wurtele  Y. Yamazaki
ALPHA Collaboration

(Received: date / Accepted: date)

###### Abstract

Spectroscopy of antihydrogen has the potential to yield high-precision tests
of the CPT theorem and shed light on the matter-antimatter imbalance in the
Universe. The ALPHA antihydrogen trap at CERN’s Antiproton Decelerator aims to
prepare a sample of antihydrogen atoms confined in an octupole-based Ioffe
trap and to measure the frequency of several atomic transitions. We describe
our techniques to directly measure the antiproton temperature and a new
technique to cool them to below 10 K. We also show how our unique position-
sensitive annihilation detector provides us with a highly sensitive method of
identifying antiproton annihilations and effectively rejecting the cosmic-ray
background.

###### Keywords:

antihydrogen antimatter CPT Penning trap Atom trap

††journal: Hyperfine Interactions

##  1 Introduction

Antihydrogen, the bound state of an antiproton and a positron, is the simplest
pure antimatter atomic system. The first cold (non-relativistic) antihydrogen
atoms were synthesised by the ATHENA experiment in 2002 by combining
antiprotons and positrons under cryogenic conditions in a Penning trap
Amoretti  _et al._ (2002). The neutral antihydrogen atoms formed were not
confined by the electric and magnetic fields used to hold the antiprotons and
positrons as non-neutral plasmas, but escaped to strike the matter of the
surrounding apparatus and annihilate. Detection of the coincident antiproton
and positron annihilation signals was used to identify antihydrogen in these
experiments. However, before performing high-precision spectroscopy, it is
highly desirable, perhaps even necessary, to confine the antihydrogen in an
atomic trap.

##  2 Atom Trap

Atoms with a permanent magnetic dipole moment $\vec{\mu}$ can be trapped by
exploiting the interaction of the dipole moment with an inhomogeneous magnetic
field. A three-dimensional maximum of magnetic field is not compatible with
Maxwell’s equations, but a minimum is. Thus, only atoms with $\mu$ aligned
antiparallel to the magnetic field (so-called ‘low-field seekers’) can be
trapped.

ALPHA creates a magnetic minimum using a variation of the Ioffe-Pritchard
configuration Pritchard (1983), replacing the transverse quadrupole magnet
with an octupole Bertsche  _et al._ (2006). The octupole and the ‘mirror
coils’ that complete the trap are superconducting and are cooled to 4 K by
immersing them in liquid helium. The depth of the magnetic minimum produced is
approximately 0.8 T, equivalent to a trap depth of $0.6~{}\mathrm{K}\times
k_{\mathrm{B}}$ for ground state antihydrogen.

ALPHA’s scheme to detect trapped antihydrogen is to quickly release trapped
atoms from the atomic trap and detect their annihilation as they strike the
apparatus. Having the antihydrogen atoms escape over a short time minimises
the background from cosmic rays that can mimic antihydrogen annihilations (see
section 5), so the magnet systems have been designed to remove the stored
energy in as short a time as possible. The current has been measured to decay
with a time constant of 9 ms for the octupole and 8.5 ms for the mirror coils.

The atom trap is superimposed on a Penning trap, which is used to confine the
charged particles used in antihydrogen production. The Penning trap electrodes
are also cooled by a liquid helium reservoir and reach a temperature of
approximately 7 K. In the absence of external heating sources, the stored non-
neutral plasmas should come into thermal equilibrium at this temperature.

Introduction of the multipolar transverse magnetic field modifies the
confinement properties of the Penning trap. In the most extreme case, this
manifests as a ‘critical radius’ Fajans, Madsen, and Robicheaux (2008),
outside which particles can be lost from the trap simply because the magnetic
field lines along which the particles move intersect the electrode walls. Even
if particles are not lost, the transverse field results in a higher rate of
plasma diffusion Gilson and Fajans (2003). As the plasma diffuses and expands,
electrostatic potential energy is converted to thermal energy, resulting in a
temperature higher than would be otherwise expected.

ALPHA chose to use an octupole instead of the prototypical quadrupole in its
Ioffe trap to reduce the transverse fields close to the axis of the Penning
trap, where the non-neutral plasmas are stored. Though this choice can
significantly ameliorate the undesirable effects, it does not eliminate them
entirely. Other sources of heating, notably the coupling of the particles to
electronic noise Beck, Fajans, and Malmberg (1996), will also increase the
temperature. This highlights the importance of direct, absolute measurements
of the particle temperature to accurately determine the experimental
conditions.

##  3 Cooling and temperature measurements of antiprotons

The temperature of a plasma can be determined by measuring the distribution of
particles in the tail of a Boltzmann distribution - a technique common-place
in non-neutral plasma physics Eggleston  _et al._ (1992). This measurement has
the advantage of yielding the absolute temperature of the particles without
recourse to supporting measurements (for example, of the density
distribution), unlike measurements of the frequencies of the normal plasma
modes Amoretti  _et al._ (2003), which can only give a relative temperature
change. The plasmas typical in ALPHA have densities in the range $10^{6}$ to
$10^{8}~{}\mathrm{cm^{-3}}$, with collision rates high enough to ensure that
the plasma comes to equilibrium in a few seconds. In equilibrium, the energy
of the particles conforms to a Boltzmann distribution.

To measure the temperature, the particles are released from a confining well
by slowly (compared to the axial oscillation frequency) reducing the voltage
on one side of the well. As the well depth is reduced, particles escape
according to their energy; the first (highest-energy) particles to be released
will be drawn from the tail of a Boltzmann distribution. As the dump
progresses, the loss of particles causes redistribution of energy and, at
later times, the measured distribution deviates from the expected Boltzmann
distribution. The escaping particles can be detected using a micro-channel
plate as a charge amplifier, or for antimatter particles, by detecting their
annihilation. The temperature is determined by fitting an exponential curve to
the number of particles released as a function of energy, such as in the
example measurement shown in Fig. 1.

513,265)554,265) 1547,265)1506,265) 1513,334)533,334) 1547,334)1527,334)
513,374)533,374) 1547,374)1527,374) 513,402)533,402) 1547,402)1527,402)
513,425)533,425) 1547,425)1527,425) 513,443)533,443) 1547,443)1527,443)
513,458)533,458) 1547,458)1527,458) 513,471)533,471) 1547,471)1527,471)
513,483)533,483) 1547,483)1527,483) 513,493)554,493) 1547,493)1506,493)
10513,562)533,562) 1547,562)1527,562) 513,602)533,602) 1547,602)1527,602)
513,631)533,631) 1547,631)1527,631) 513,653)533,653) 1547,653)1527,653)
513,671)533,671) 1547,671)1527,671) 513,686)533,686) 1547,686)1527,686)
513,699)533,699) 1547,699)1527,699) 513,711)533,711) 1547,711)1527,711)
513,722)554,722) 1547,722)1506,722) 100513,790)533,790) 1547,790)1527,790)
513,830)533,830) 1547,830)1527,830) 513,859)533,859) 1547,859)1527,859)
513,881)533,881) 1547,881)1527,881) 513,899)533,899) 1547,899)1527,899)
513,914)533,914) 1547,914)1527,914) 513,928)533,928) 1547,928)1527,928)
513,939)533,939) 1547,939)1527,939) 513,950)554,950) 1547,950)1506,950)
1000513,1018)533,1018) 1547,1018)1527,1018) 513,1059)533,1059)
1547,1059)1527,1059) 513,1087)533,1087) 1547,1087)1527,1087)
513,1109)533,1109) 1547,1109)1527,1109) 513,1127)533,1127)
1547,1127)1527,1127) 513,1143)533,1143) 1547,1143)1527,1143)
513,1156)533,1156) 1547,1156)1527,1156) 513,1168)533,1168)
1547,1168)1527,1168) 513,1178)554,1178) 1547,1178)1506,1178)
10000513,265)513,306) 513,1178)513,1137) 0720,265)720,306) 720,1178)720,1137)
0.1927,265)927,306) 927,1178)927,1137) 0.21133,265)1133,306)
1133,1178)1133,1137) 0.31340,265)1340,306) 1340,1178)1340,1137)
0.41547,265)1547,306) 1547,1178)1547,1137)
0.5513,1178)513,265)(1547,265)(1547,1178)(513,1178)

Integrated annihilation counts

Well depth
[V]513,1073)518,1073)(518,1070)(528,1070)(528,1060)(539,1060)(539,1050)(549,1050)(549,1036)(559,1036)(559,1024)(570,1024)(570,1011)(580,1011)(580,999)(590,999)(590,985)(601,985)(601,967)(611,967)(611,956)(621,956)(621,941)(632,941)(632,926)(642,926)(642,907)(652,907)(652,890)(663,890)(663,870)(673,870)(673,851)(683,851)(683,831)(694,831)(694,820)(704,820)(704,800)(714,800)(714,787)(725,787)(725,766)(735,766)(735,745)(745,745)(745,722)(756,722)(756,701)(766,701)(766,680)
766,680)776,680)(776,659)(787,659)(787,633)(797,633)(797,615)(807,615)(807,605)(818,605)(818,602)(828,602)(828,580)(838,580)(838,567)(849,567)(849,527)(859,527)(859,527)(869,527)(869,493)(880,493)(880,493)(890,493)(890,493)(900,493)(900,483)(911,483)(911,458)(921,458)(921,458)(931,458)(931,443)(942,443)(942,443)(952,443)(952,443)(962,443)(962,443)(973,443)(973,443)(983,443)(983,425)(993,425)(993,425)(1004,425)(1004,425)(1014,425)(1014,425)(1024,425)(1024,425)
1024,425)1035,425)(1035,425)(1045,425)(1045,425)(1055,425)(1055,425)(1066,425)(1066,425)(1076,425)(1076,425)(1086,425)(1086,425)(1097,425)(1097,425)(1107,425)(1107,402)(1117,402)(1117,402)(1128,402)(1128,402)(1138,402)(1138,374)(1148,374)(1148,374)(1159,374)(1159,374)(1169,374)(1169,374)(1179,374)(1179,374)(1190,374)(1190,374)(1200,374)(1200,374)(1210,374)(1210,374)(1221,374)(1221,374)(1231,374)(1231,374)(1241,374)(1241,374)(1252,374)(1252,374)(1262,374)(1262,374)(1272,374)(1272,374)(1283,374)(1283,374)
1283,374)1293,374)(1293,374)(1303,374)(1303,374)(1314,374)(1314,374)(1324,374)(1324,374)(1334,374)(1334,374)(1345,374)(1345,374)(1355,374)(1355,374)(1365,374)(1365,374)(1376,374)(1376,374)(1386,374)(1386,374)(1396,374)(1396,374)(1407,374)(1407,374)(1417,374)(1417,374)(1427,374)(1427,374)(1438,374)(1438,374)(1448,374)(1448,374)(1458,374)(1458,374)(1469,374)(1469,374)(1479,374)(1479,374)(1489,374)(1489,374)(1500,374)(1500,374)(1510,374)(1510,374)(1520,374)(1520,374)(1531,374)(1531,374)(1541,374)(1541,374)
1541,374)1547,374)(1547,374)
680,848)680,848)(691,829)(701,810)(711,791)(722,773)(732,754)(743,735)(753,717)(764,698)(774,679)(785,660)(795,642)(805,623)(816,604)(826,585)(837,567)(847,548)(858,529)(868,511)(879,492)
513,1178)513,265)(1547,265)(1547,1178)(513,1178)  Figure 1: An example
temperature measurement of approximately 45,000 antiprotons, after separation
from the cooling electrons and with the inhomogeneous trapping fields
energised. The straight line shows an exponential fit to determine the
temperature, which in this case, is $\left(310~{}\pm~{}20\right)~{}\mathrm{K}$

The actual process of manipulating the trap potentials can change the
temperature of the particles as the measurement takes place. Particle-in-cell
(PIC) simulations of the measurement process have predicted that the
temperature obtained from the fit is around 15% higher than the initial
temperature for a typical antiproton cloud. For the denser electron and
positron plasmas, the measured temperature can be as much as factor of two
higher than the initial temperature. We can apply the corrections determined
from these simulations to the measured temperature to find the true
temperature. This temperature diagnostic has been applied to all three
particle species used in ALPHA - antiprotons, positrons and electrons. The
lowest temperatures measured for electron or positron plasmas at
experimentally relevant densities $\left(10^{6}~{}\mathrm{cm^{-3}}\text{or
more}\right)$ is of the order of 40 K.

Electrons are used to collisionally cool the antiprotons, which, due to their
larger mass, do not effectively self-cool via synchrotron radiation. Before
mixing the antiprotons with positrons to produce antihydrogen, the electrons
must be removed. If the electrons were allowed to remain, they could
potentially deplete the positron plasma by forming positronium, destroy
antihydrogen atoms through charge exchange, or destabilise the positron plasma
by partially neutralising it.

Electron removal is accomplished through the application of electric field
pulses. These pulses remove the confining potential on one side of the well
holding the antiproton/electron two-component plasma, typically for 100-300
ns. The electrons, moving faster than the antiprotons, escape the well. The
well is restored before the antiprotons can escape, so they remain trapped.
However, the process does not avoid disturbing the antiprotons. The electron
removal process has been the focus of a significant portion of experimental
effort at ALPHA, and the coldest antiproton temperatures obtained have been
around 200-300 K.

##  4 Evaporative Cooling

Antiprotons at a few hundred Kelvin will have a very small probability of
forming low-energy, trappable, antihydrogen atoms. To further cool the
antiprotons, ALPHA has implemented a technique of forced evaporative cooling.
Evaporative cooling is a common-place technique in neutral particle trapping,
and has been instrumental in the production of Bose-Einstein condensates Davis
_et al._ (1995). However, evaporative cooling has found limited application to
charged particles.

Before evaporative cooling, a cloud of antiprotons, containing 45,000
particles, with a radius of 0.6 mm, density $7.6\times
10^{6}~{}\mathrm{cm^{-3}}$, and initial temperature of
$\left(1040~{}\pm~{}45\right)~{}\mathrm{K}$ was prepared in a 1.5 V deep
potential well. The collision rate between antiprotons was of order 200
$\mathrm{s}^{-1}$, high enough to ensure that the temperatures in the parallel
and perpendicular degrees of freedom had equilibrated before evaporative
cooling commenced.

To perform evaporative cooling, the confining potential on one side of the
well is slowly (with respect to the equilibration rate) lowered. Particles
with kinetic energy higher than the instantaneous well depth escape the trap,
carrying with them energy in excess of the mean thermal energy. The
distribution then evolves towards a Boltzmann distribution with lower
temperature, and the process continues.

Starting with $45,000$ antiprotons at 1040 K, we have obtained temperatures as
low as (9 $\pm$ 4) K with $\left(6\pm 1\right)\%$ of the particles stored in a
10 mV deep well. Measurements of the temperature, number of particles and
transverse size of the clouds were made at a number of points between the most
extreme well depths. The temperatures and number of particles remaining at
each measurement point are shown in Fig. 2.

431,265)451,265) 1336,265)1316,265) 431,320)451,320) 1336,320)1316,320)
431,359)451,359) 1336,359)1316,359) 431,390)451,390) 1336,390)1316,390)
431,414)451,414) 1336,414)1316,414) 431,435)451,435) 1336,435)1316,435)
431,453)451,453) 1336,453)1316,453) 431,469)451,469) 1336,469)1316,469)
431,484)472,484) 1336,484)1295,484) 10431,578)451,578) 1336,578)1316,578)
431,633)451,633) 1336,633)1316,633) 431,672)451,672) 1336,672)1316,672)
431,703)451,703) 1336,703)1316,703) 431,727)451,727) 1336,727)1316,727)
431,748)451,748) 1336,748)1316,748) 431,766)451,766) 1336,766)1316,766)
431,782)451,782) 1336,782)1316,782) 431,797)472,797) 1336,797)1295,797)
100431,891)451,891) 1336,891)1316,891) 431,946)451,946) 1336,946)1316,946)
431,985)451,985) 1336,985)1316,985) 431,1015)451,1015) 1336,1015)1316,1015)
431,1040)451,1040) 1336,1040)1316,1040) 431,1061)451,1061)
1336,1061)1316,1061) 431,1079)451,1079) 1336,1079)1316,1079)
431,1095)451,1095) 1336,1095)1316,1095) 431,1110)472,1110)
1336,1110)1295,1110) 1000431,1204)451,1204) 1336,1204)1316,1204)
431,1259)451,1259) 1336,1259)1316,1259) 431,265)431,285) 431,1259)431,1239)
451,265)451,285) 451,1259)451,1239) 469,265)469,285) 469,1259)469,1239)
484,265)484,306) 484,1259)484,1218) 10588,265)588,285) 588,1259)588,1239)
648,265)648,285) 648,1259)648,1239) 691,265)691,285) 691,1259)691,1239)
725,265)725,285) 725,1259)725,1239) 752,265)752,285) 752,1259)752,1239)
775,265)775,285) 775,1259)775,1239) 795,265)795,285) 795,1259)795,1239)
812,265)812,285) 812,1259)812,1239) 828,265)828,306) 828,1259)828,1218)
100932,265)932,285) 932,1259)932,1239) 992,265)992,285) 992,1259)992,1239)
1035,265)1035,285) 1035,1259)1035,1239) 1068,265)1068,285)
1068,1259)1068,1239) 1096,265)1096,285) 1096,1259)1096,1239)
1119,265)1119,285) 1119,1259)1119,1239) 1139,265)1139,285)
1139,1259)1139,1239) 1156,265)1156,285) 1156,1259)1156,1239)
1172,265)1172,306) 1172,1259)1172,1218) 10001275,265)1275,285)
1275,1259)1275,1239) 1336,265)1336,285) 1336,1259)1336,1239)
431,1259)431,265)(1336,265)(1336,1259)(431,1259)

Temperature [K]

On-axis well depth [mV]1047,951)1047,962) 1027,951)1067,951)
1027,962)1067,962) 840,694)840,740) 820,694)860,694) 820,740)860,740)
772,646)772,680) 752,646)792,646) 752,680)792,680) 674,553)674,630)
654,553)694,553) 654,630)694,630) 570,549)570,585) 550,549)590,549)
550,585)590,585) 490,367)490,520) 470,367)510,367) 470,520)510,520)
1231,1106)1231,1118) 1211,1106)1251,1106) 1211,1118)1251,1118)
431,1259)431,265)(1336,265)(1336,1259)(431,1259)  (a)

390,265)410,265) 1336,265)1316,265) 390,347)410,347) 1336,347)1316,347)
390,410)410,410) 1336,410)1316,410) 390,461)410,461) 1336,461)1316,461)
390,505)410,505) 1336,505)1316,505) 390,543)410,543) 1336,543)1316,543)
390,576)410,576) 1336,576)1316,576) 390,606)431,606) 1336,606)1295,606)
0.1390,803)410,803) 1336,803)1316,803) 390,918)410,918) 1336,918)1316,918)
390,999)410,999) 1336,999)1316,999) 390,1063)410,1063) 1336,1063)1316,1063)
390,1114)410,1114) 1336,1114)1316,1114) 390,1158)410,1158)
1336,1158)1316,1158) 390,1196)410,1196) 1336,1196)1316,1196)
390,1229)410,1229) 1336,1229)1316,1229) 390,1259)431,1259)
1336,1259)1295,1259) 1390,265)390,285) 390,1259)390,1239) 411,265)411,285)
411,1259)411,1239) 429,265)429,285) 429,1259)429,1239) 446,265)446,306)
446,1259)446,1218) 10554,265)554,285) 554,1259)554,1239) 617,265)617,285)
617,1259)617,1239) 662,265)662,285) 662,1259)662,1239) 697,265)697,285)
697,1259)697,1239) 725,265)725,285) 725,1259)725,1239) 749,265)749,285)
749,1259)749,1239) 770,265)770,285) 770,1259)770,1239) 789,265)789,285)
789,1259)789,1239) 805,265)805,306) 805,1259)805,1218) 100913,265)913,285)
913,1259)913,1239) 977,265)977,285) 977,1259)977,1239) 1021,265)1021,285)
1021,1259)1021,1239) 1056,265)1056,285) 1056,1259)1056,1239)
1085,265)1085,285) 1085,1259)1085,1239) 1109,265)1109,285)
1109,1259)1109,1239) 1130,265)1130,285) 1130,1259)1130,1239)
1148,265)1148,285) 1148,1259)1148,1239) 1165,265)1165,306)
1165,1259)1165,1218) 10001273,265)1273,285) 1273,1259)1273,1239)
1336,265)1336,285) 1336,1259)1336,1239)
390,1259)390,265)(1336,265)(1336,1259)(390,1259)

Fraction of particles remaining

On-axis well depth [mV]1226,1240)1226,1249) 1206,1240)1246,1240)
1206,1249)1246,1249) 817,1035)817,1066) 797,1035)837,1035) 797,1066)837,1066)
536,615)536,650) 516,615)556,615) 516,650)556,650) 747,983)747,1015)
727,983)767,983) 727,1015)767,1015) 644,827)644,857) 624,827)664,827)
624,857)664,857) 1034,1171)1034,1198) 1014,1171)1054,1171)
1014,1198)1054,1198) 452,439)452,482) 432,439)472,439) 432,482)472,482)
390,1259)390,265)(1336,265)(1336,1259)(390,1259)  (b)

Figure 2: The temperature (a) and the fraction of the initial number of
particles (b) after evaporative cooling to a series of well depths. The
minimum temperature is (9 $\pm$ 4) K

The evaporation process can be described using simple rate equations for the
number of particles $N$ and the temperature $T$;

$\frac{\mathrm{d}N}{\mathrm{d}t}=-\frac{N}{\tau_{ev}},$ (1a) |  $\frac{\mathrm{d}T}{\mathrm{d}t}=-\alpha\frac{T}{\tau_{ev}}.$ (1b)
---|---

Here, $\tau_{ev}$ is the characteristic evaporation timescale and $\alpha$ is
the excess energy carried away by an evaporating particle, in multiples of
$k_{\mathrm{B}}T$. At a given time, the distribution of energies can be
thought of as a truncated Boltzmann distribution, characterised by a
temperature $T$, and the well depth $U$. $\tau_{ev}$ is linked to the mean
time between collisions, $\tau_{col}$ as Currell and Fussmann (2005)

$\frac{\tau_{ev}}{\tau_{col}}=\frac{\sqrt{2}}{3}\eta e^{\eta},$ (2)

where $\eta=U/{k_{\mathrm{B}}T}$ is the rescaled well depth. We note the
strong dependence of $\tau_{ev}$ on $\eta$, indicating that this is the
primary factor determining the temperature in a given well. We find values of
$\eta$ between 10 and 20 over the range of our measurements. The value of
$\alpha$ can be calculated using the treatment in reference Ketterle and van
Druten (1995). We have numerically modelled evaporative cooling in our
experiment using these equations and have found very good agreement between
our measurements and the model Andresen  _et al._ (2010a).

Measurements of the transverse density profile were made by ejecting the
particles onto an MCP/phosphor/CCD imaging device Andresen  _et al._ (2009).
It was seen that, as evaporation progressed, the cloud radius increased
dramatically - see Fig. 3. We interpret this effect to be due to escape of the
evaporating particles principally from the radial centre of the cloud, and the
conservation of the total canonical angular momentum during the subsequent
redistribution process. Inside the cloud, the space charge reduces the depth
of the confining well. This effect is accentuated closer to the trap axis,
with the result that the well depth close to the axis can be significantly
lower than further away. The evaporation rate is exponentially suppressed at
higher well depths (eqn. 2), so evaporation is confined to a small region
close to the axis, causing the on-axis density to become depleted. This is a
non-equilibrium configuration, and the particles will redistribute to replace
the lost density. In doing so, some particles will move inwards, and to
conserve the canonical angular momentum, some particles must also move to
higher radii O’Neil (1980). Assuming that all loss occurs at $r=0$, the mean
squared radius of the particles, $\left<r^{2}\right>$, will obey the
relationship

$N_{0}\left<r_{0}^{2}\right>=N\left<r^{2}\right>,\vspace{-0.5cm}$ (3)

where N is the number of particles, and the zero subscript indicates the
initial conditions.

As seen in Fig. 3, this model agrees very well with the measurements. This
radial expansion can be problematic when attempting to prepare low kinetic
energy antiprotons to produce trappable antihydrogen atoms, as the energy
associated with the magnetron motion grows with the distance from the axis,
and the electrostatic potential energy released as the radius expands can
reheat the particles. The effect can be countered somewhat by taking a longer
time to cool the particles, resulting in a higher efficiency and, thus, a
smaller expansion, but we find that the efficiency depends very weakly on the
cooling time.

390,265)431,265) 1546,265)1505,265) 0390,431)431,431) 1546,431)1505,431)
0.5390,596)431,596) 1546,596)1505,596) 1390,762)431,762) 1546,762)1505,762)
1.5390,928)431,928) 1546,928)1505,928) 2390,1093)431,1093)
1546,1093)1505,1093) 2.5390,1259)431,1259) 1546,1259)1505,1259)
3390,265)390,306) 390,1259)390,1218) 0621,265)621,306) 621,1259)621,1218)
0.2852,265)852,306) 852,1259)852,1218) 0.41084,265)1084,306)
1084,1259)1084,1218) 0.61315,265)1315,306) 1315,1259)1315,1218)
0.81546,265)1546,306) 1546,1259)1546,1218)
1390,1259)390,265)(1546,265)(1546,1259)(390,1259)

Plasma Size [mm]

Fraction of Antiprotons LostMeasured Size1046,1177)1251,1177)
1046,1197)1046,1157) 1251,1197)1251,1157) 1477,1027)1477,1093)
1457,1027)1497,1027) 1457,1093)1497,1093) 1419,729)1419,795)
1399,729)1439,729) 1399,795)1439,795) 1280,629)1280,696) 1260,629)1300,629)
1260,696)1300,696) 1084,563)1084,629) 1064,563)1104,563) 1064,629)1104,629)
991,497)991,563) 971,497)1011,497) 971,563)1011,563) 656,431)656,497)
636,431)676,431) 636,497)676,497) 390,431)390,497) 370,431)410,431)
370,497)410,497) Predicted Size1046,1094)1251,1094)
390,464)390,464)(402,465)(413,466)(425,467)(437,468)(448,469)(460,470)(472,471)(483,472)(495,474)(507,475)(518,476)(530,477)(542,478)(553,480)(565,481)(577,482)(589,483)(600,485)(612,486)(624,488)(635,489)(647,490)(659,492)(670,493)(682,495)(694,497)(705,498)(717,500)(729,501)(740,503)(752,505)(764,507)(775,508)(787,510)(799,512)(810,514)(822,516)(834,518)(845,520)(857,523)(869,525)(880,527)(892,529)(904,532)(915,534)(927,537)(939,539)(950,542)(962,545)
962,545)974,548)(986,551)(997,554)(1009,557)(1021,560)(1032,563)(1044,567)(1056,570)(1067,574)(1079,578)(1091,582)(1102,586)(1114,590)(1126,595)(1137,599)(1149,604)(1161,609)(1172,615)(1184,620)(1196,626)(1207,632)(1219,639)(1231,646)(1242,653)(1254,661)(1266,669)(1277,677)(1289,687)(1301,697)(1312,707)(1324,719)(1336,731)(1347,745)(1359,760)(1371,776)(1383,794)(1394,814)(1406,836)(1418,861)(1429,891)(1441,924)(1453,964)(1464,1013)(1476,1073)(1488,1150)(1499,1254)(1500,1259)
390,1259)390,265)(1546,265)(1546,1259)(390,1259)  Figure 3: The measured size
of the antiproton cloud using a MCP/phosphor/CCD device as a function of the
number of particles lost. This is compared to the size predicted from eqn 3

Colder antiprotons are of great utility in the effort to produce cold
antihydrogen atoms. Antihydrogen production techniques can be broadly
categorised as ‘static’ - in which a cloud of antiprotons is held stationary
and positrons, perhaps in the form of positronium atoms are introduced
Charlton (1990), or ‘dynamic’ - where antiprotons are passed through a
positron plasma Gabrielse  _et al._ (1988). In the first case, the advantages
of cold antiprotons are obvious, as the lower kinetic energy translates
directly into lower-energy antihydrogen atoms. In the second case, the colder
temperature allows the manipulations used to ‘inject’ the antiprotons into the
positrons to produce much more precisely defined antiproton energies.
Indirectly, this will also permit these schemes to produce more trappable
antihydrogen.

##  5 Annihilation vertex detector

Among the most powerful diagnostic tools available to experiments working with
antimatter are detectors capable of detecting matter-antimatter annihilations.
Antiproton annihilations produce an average of three charged pions, which can
be detected by scintillating material placed around the trap. The passage of a
pion through the scintillator produces photons, which trigger a cascade in a
photo-multiplier tube to produce a voltage pulse. Individual voltage pulses
can be counted to determine the number of annihilations.

A further technique uses a position-sensitive detector to reconstruct the
trajectories of the pions and find the point where the antiproton annihilated
(usually called the ‘vertex’). The ALPHA annihilation vertex detector
comprises sixty double-sided silicon wafers, arranged in three layers in a
cylindrical fashion around the antihydrogen production and trapping region.
Each wafer is divided into 256 strips, oriented in orthogonal directions on
the p- and n- sides. Charged particles passing through the silicon result in
charge deposits, and the intersection of perpendicular strips with charge
above a defined threshold marks the location a particle passed through the
silicon.

Each module is controlled by a circuit that produces a digital signal when a
charge is detected on the silicon. If a coincidence of modules is satisfied in
a 400 ns time window, the charge profile is ‘read-out’ and digitised for
further analysis. Each readout and associated trigger and timing information
comprises an ‘event’. The pion trajectories are reconstructed by fitting
helices to sets of three hits, one from each layer of the detector. The point
that minimises the distance to the helices is then identified as the
annihilation vertex. An example of an annihilation event is shown in Fig.
4(a).

Figure 4: (a) an example reconstruction of an antihydrogen annihilation and
(b) a cosmic ray event. The diamond indicates the position of the vertex
identified by the reconstruction algorithm, the polygonal structure shows the
locations of the silicon wafers, the dots are the positions of the detected
hits, and the inner circle shows the radius of the Penning trap electrodes.
Also shown are annihilation density distributions associated with antihydrogen
production (c, e) and deliberately induced antiproton loss (d, f). (c) and (d)
are projected along the cylindrical axis, with the inner radius of the
electrodes marked with a white circle, while (e) and (f) show the azimuthal
angle $\phi$ against the axial position $z$

Examination of the spatial distributions of annihilations can yield much
insight into the physical processes at work. ATHENA established that
antihydrogen production resulted in a characteristic ‘ring’ structure - an
azimuthally smooth distribution concentrated at the radius of the trap
electrodes Fujiwara  _et al._ (2004), shown in 4(c) and (e). In contrast, the
loss of bare antiprotons occurred in spatially well-defined locations, called
‘hot-spots’, examples of which are shown in 4(d) and (f). This was interpreted
to be due to microscopic imperfections in the trap elements. These produce
electric fields that break the symmetry of the trap and give rise to preferred
locations for charged particle loss. When antihydrogen is produced in a
multipole field, antiprotons generated by ionisation of weakly-bound
antihydrogen also contribute small asymmetries Andresen  _et al._ (2010b).
These features are present in Fig. 4(c) and (e).

The vertex detector is also sensitive to charged particles in cosmic rays.
When passing through the detector, they are typically identified as a pair of
almost co-linear tracks (Fig. 4(b)), and can be misidentified as an
annihilation. Cosmic-ray events when searching for the release of trapped
antihydrogen thus present a background.

To develop a method to reject cosmic ray events, while retaining
annihilations, we compared samples of the events using three parameters, shown
in Fig. 5. Cosmic rays have predominantly two tracks, while antiproton
annihilations typically have more. 95% of cosmic events have two or fewer
identified tracks, while 58% of antiproton annihilations have at least three.
A significant number of antiproton annihilations can have only two tracks, so
it is not desirable to reject all these events as background.

Figure 5: Comparison of the distributions of event parameters for antiproton
annihilations (solid line) and cosmic rays (dashed line). Shown are (a) the
number of identified charged particle tracks, (b) the radial coordinate of the
vertex, and the squared residual from a linear fit to the identified positions
for the events with (c) two tracks and (d) more than two tracks. The shaded
regions indicate the range of parameters that are rejected to minimise the
p-value as discussed in the text

We determine if the tracks form a straight line by fitting a line to the hits
from each pair of tracks, and calculating the squared residual value. As seen
in Fig. 5(c) and (d), cosmic events have much lower squared residual values
than annihilations. This is to be expected, since particles from cosmic rays
have high momentum and pass through the apparatus and the magnetic field
essentially undeflected, while the particles produced in an annihilation will,
in general, move in all directions. In addition, annihilations occur on the
inner wall of the Penning trap, at a radius of $\sim$2.2 cm, and as shown in
Fig. 5(b), reconstructed annihilation vertices are concentrated here, whereas
cosmic rays pass through at a random radius.

By varying the ranges of parameters for which events are accepted, we could
optimise the annihilation detection strategy. The point where the ‘p-value’ –
the probability that an observed signal is due to statistical fluctuations in
the background Amsler  _et al._ (2008) – was minimised requiring the vertex to
lie within 4 cm of the trap axis, and the squared residual value to be at
least 2 $\mathrm{cm}^{2}$ or 0.05 $\mathrm{cm}^{2}$ for events with two tracks
and more than two tracks, respectively.

These thresholds reject more than 99% of the cosmic background, reducing the
absolute rate of background events to 22 mHz, while still retaining the
ability of identify $\sim 40\%$ of antiproton annihilations. While this method
effectively removes cosmic rays as a source of concern, other background
processes, including mirror-trapped antiprotons must also be considered when
searching for trapped antihydrogen. Our cosmic-ray rejection method has been
applied to data taken from the 2009 ALPHA antihydrogen trapping run, and a
full discussion of the results obtained will be made in a forthcoming
publication.

##  6 Conclusions and outlook

In this paper we have described two of the most recent techniques developed by
the ALPHA collaboration in our search for trapped antihydrogen. Evaporative
cooling of antiprotons has the potential to greatly increase the number of
low-energy, trappable atoms produced in our experiment. The use of our unique
annihilation vertex imaging detector to discriminate with high power between
annihilations and cosmic rays will be a vital tool to identify the first
trapped antihydrogen atoms. We have integrated both of these techniques into
our experiment and are hopeful of soon being able to report detection of
trapped antihydrogen.

###### Acknowledgements.

This work was supported by CNPq, FINEP/RENAFAE (Brazil), ISF (Israel), MEXT
(Japan), FNU (Denmark), VR (Sweden), NSERC, NRC/TRIUMF, AIF (Canada), DOE, NSF
(USA), EPSRC and the Leverhulme Trust (UK). We are also grateful to the AD
team for the delivery of a high-quality antiproton beam, and to CERN for its
technical support.

### Note added in proof

: Since the preparation of this article, trapping of antihydrogen atoms has
been achieved by the ALPHA collaboration Andresen  _et al._ (2010c)

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