11Be SPECTROSCOPY FROM THE FIRST INVERSE KINEMATICS TRANSFER
REACTION MEASUREMENT BY THE AT-TPC IN SOLARIS

By

Michael Zachary Serikow

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics—Doctor of Philosophy

2025

ABSTRACT

As evidenced by the field’s existence surpassing a 100 years, nuclei are varied and immensely

complex. New accelerator facilities are now able to produce rare isotopes that seldom appear in

nature and whose properties must be measured to lead to a better understanding of the nuclear

force. These facilities have naturally facilitated the need to switch to inverse kinematics. This,

coupled with the low intensity of rare isotope beams, has lead to the creation of a new type of

detector called an active target. These detectors are especially well-suited for rare isotope beams

due to their high luminosity, theoretical 4𝜋 solid angle coverage, and ability to determine the vertex

of a nuclear reaction.

One such active target is the Active Target-Time Projection Chamber (AT-TPC) housed at the

Facility for Rare Isotope Beams. This thesis reports the results from the first transfer reaction

measurement by the AT-TPC in inverse kinematics. Specifically, 11Be spectroscopy was performed

via the 10Be(𝑑, 𝑝) reaction. 11Be is an ideal candidate for such a commissioning measurement as the

structure of its low-lying states is well known, allowing for comparisons to the literature. Despite

this, the parity of its fourth excited state at 3.40 MeV has been debated. This thesis measured the

angular distributions of all states in 11Be up to and including this state. Their spectroscopic factors

were derived and compared to values in the literature and theoretical shell-model interactions.

Reasonable agreement was found between the average derived factors, literature, and WBP and

YSOX calculations. Although the coverage of the 3.40 MeV state’s angular distribution is too

limited for a definitive parity assignment, it and the spectroscopic analysis tentatively support a

positive parity. The implications of this assignment on the rotational structure, 0𝑑3/2 effective
single-particle energy, and spin-orbit splitting of the 0𝑑 orbitals in 11Be are explored.

Copyright by
MICHAEL ZACHARY SERIKOW
2025

ACKNOWLEDGMENTS

Beginning in my childhood, my parents invested much of their time into not only my academic

success but my future. My mom worked all day at home as a bookkeeper while simultaneously

keeping the house clean, cooking for the family, and helping me study. Even in the evening after

working since early in the morning, I remember her often quizzing me in preparation for exams.

She worked so hard but never complained because she loved her family. I love her and them, too.

The University of Notre Dame set the trajectory for my nuclear physics tenure. Tan Ahn was

pivotal in providing meaningful research experiences and a friendship that will last a lifetime. I am

continually surprised by the strength of my connections I made there. I hope I can return a fraction

of what has been given to me. It was also during this time that I studied under three different and

remarkable physicists: Alexandre Zagoskin, Harvey Brown, and Craig Woody. I am sorry that I

am not the best in keeping in touch. I will never forget you.

As a graduate student at Michigan State University, my advisor Daniel Bazin has given me

more opportunities, connections, and experiences than I could have ever imagined. Growing up

in northern Indiana among the cornfields never sowed the possibility of traveling the world and

encountering people so like myself in cultures so different.

Indiana is a special place, though.

During my last two years at Michigan State, I become friends with Gordon McCann. The impact of

his knowledge and guidance are not belied by his short period in my life, indicative of the magnitude

of his impact.

To my fiancée Amy Dao and her family: I love you all. You fueled my ability to perform. You

supported my well-being. These actions are eternal.

iv

TABLE OF CONTENTS

LIST OF TABLES .

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LIST OF FIGURES .

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vi

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
. . . . . . . . . . . . . . . . . . . .
1.1 Experimental nuclear physics with beams
. . . . . . . . . . . . . . . . . . . . . . .
1.2 Active targets for radioactive beams
1.3 Active-Target Time Projection Chamber
. . . . . . . . . . . . . . . . . . . . .
1.4 Coupling the AT-TPC with the SOLARIS solenoid . . . . . . . . . . . . . . .
1.5 Physics with the AT-TPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6

1
1
2
3
5
7
10Be + d experiment with the AT-TPC . . . . . . . . . . . . . . . . . . . . . . 15

CHAPTER 2

EXPERIMENTAL SETUP . . . . . . . . . . . . . . . . . . . . . . . . . 19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Ion chamber multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1 Setup of the AT-TPC .
2.2 Trigger .
2.3
2.4 Beam .

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CHAPTER 3

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ANALYSIS OF AT-TPC DATA . . . . . . . . . . . . . . . . . . . . . . 30
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Data structure .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Creating point clouds .
3.3 Clustering tracks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
.
3.4 Estimating clustered track kinematic parameters . . . . . . . . . . . . . . . . . 44
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Gain matching . .
3.6 Optimizing clustered track kinematic parameters . . . . . . . . . . . . . . . . . 51

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CHAPTER 4

AT-TPC SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 57
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Kinematics generator
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Detector simulation .
4.3 Efficiency factors .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
.
4.4 CM scattering angle error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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CHAPTER 5

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EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . 68
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
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5.1 Gates
. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
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5.2 Kinematics
5.3 Excitation spectra .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
.
5.4 Angular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Extraction of spectroscopic factors . . . . . . . . . . . . . . . . . . . . . . . . 76
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6 Discussion of results

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CHAPTER 6

CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

BIBLIOGRAPHY .

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v

LIST OF TABLES

Table 2.1 Experimental parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Table 3.1 Spyral point cloud phase parameters for the analysis of the experiment of this

thesis.

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Table 3.2 Spyral clustering phase parameters used for the analysis of the experiment of

this thesis. .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Table 3.3 Spyral estimation phase parameters used for the analysis of the experiment

of this thesis.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Table 3.4 Spyral interpolation phase parameters used for the analysis of the experiment
of this thesis. The proton parameters were used for the transfer reaction while
the deuteron parameters were used for the elastic and inelastic reactions. . . . . . 52

Table 4.1 Detector parameters used for the attpc_engine simulations of the experiment

of this thesis.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Table 4.2 Electronics parameters used for the attpc_engine simulations of the experi-

ment of this thesis.

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Table 5.1 OMPs determined from fitting the elastic scattering angular distribution at
different 10Be beam energies corresponding to the indicated locations in the
AT-TPC. There were no real nor imaginary spin-orbit potentials and 𝑟𝑐 was
fixed to 2.801 fm. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

. .

Table 5.2 Spectroscopic factors extracted from the measured angular distributions using
DWBA calculations for neutrons with transferred angular momentum ℓ to the
single-particle state 𝑛ℓ 𝑗. The listed OMPs were used in the incoming channel.
The reported spectroscopic factors used the beam energy at the midpoint of
the detector while their upper and lower limits used the beam energies at the
∗The average column only includes
window and micromegas, respectively.
the DA1p and An Cai OMPs. See the text for the details.

. . . . . . . . . . . . . 81

Table 5.3 Spectroscopic factors from other experiments and those theoretically calcu-
lated using shell-model interactions. The factors reported from [1] are an
average of those found. The author thanks Alex Brown for providing the WBP,
Cenxi Yuan for the YSOX, and Rebeka Lubna for the FSU calculations.

. . . . . 82

vi

LIST OF FIGURES

Figure 1.1 Comparison between a traditional passive target and an active target. Courtesy

of Daniel Bazin.

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Figure 1.2 Schematic of AT-TPC with some key components indicated. Reproduced

with permission from [2].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.3

Illustration of the readout plane composed of 10,240 pads. The inner pads
are smaller than the outer pads. Reproduced with permission from [2].

. . . . .

Figure 1.4

Illustration of GET electronics hierarchy connected to the sensor, or pad,
plane that the electrons induce signals on.

. . . . . . . . . . . . . . . . . . . .

Figure 1.5 Picture of the AT-TPC within the SOLARIS solenoid at FRIB. Courtesy of

Daniel Bazin.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

4

6

7

8

Figure 1.6 The difference between experimental two-proton and two-neutron separa-
tion energies and those predicted by the semi-empirical mass formula for
sequences of isotones and isotopes, respectively. Note the large spikes in the
difference at the magic numbers, where the model underpredicts the separa-
tion energies. Reproduced with permission from [3]. Copyright © 1988 by
John Wiley & Sons, Inc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 1.7 Geometry for the semi-classical stripping reaction considered. Note that
collisions are shown on both sides of the target nucleus as these result in a
diffraction pattern (a series of minima and maxima) in the angular distribu-
tion as seen in the experimental data shown in Fig. 1.8. Reproduced with
permission from [3]. Copyright © 1988 by John Wiley & Sons, Inc.

. . . . . . 18

Figure 1.8 Angular distributions from the 31P(𝑑, 𝑛)32S reaction that populate different
final states in 32S. They are compared to DWBA calculations with the indicated
transferred angular momentum values. The transferred angular momentum
is clearly correlated to the shape of the angular distribution and location of
the first peak. Reproduced with permission from [4]. . . . . . . . . . . . . . . . 18

Figure 2.1 Rack housing the CoBos, Mac Minis, MuTAnT module, power supplies for

the AsAd front-end boards, and NIMs for the trigger logic.

. . . . . . . . . . . 24

Figure 2.2 Rack housing the computer that controls the gas-handling system and the high
voltages applied to the MPGDs and the cathode. It also contains high-voltage
power supplies for the MPGDs, IC, and smart ZAP boards.

. . . . . . . . . . . 25

Figure 2.3 Rack housing the cathode high-voltage power supply and gas-handling sys-

tems for the AT-TPC and IC.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 26

vii

Figure 2.4 The IC is the cylinder atop the vacuum flange. Its electrical and gas leads can

be seen.

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Figure 2.5 Trigger logic for the experiment of this thesis. Courtesy of Daniel Bazin. . . .

. 28

Figure 2.6 Histogram of the energy loss of incoming unreacted beam particles seen
over the course of the entire experiment. The dotted vertical dotted lines
correspond to the gate used to select 10Be events. The Gaussian fits used to
subtract the 10B contamination from the 10Be counts in the gate are shown.
The small peak short of 2000 ADC units belongs to 15N. . . . . . . . . . . . . . 29

Figure 3.1 A pad’s trace taken from an event during the experiment shown in solid blue.
The baseline of the trace is removed using the outlined procedure to produce
the corrected trace shown in dotted red. . . . . . . . . . . . . . . . . . . . . . . 31

Figure 3.2 Point cloud of an event from the experiment. The spiral track was identified

as originating from a proton.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 3.3 Reconstructed beam-region mesh signal from an unreacted beam event. The
micromegas edge near 60 time buckets is from liberated electrons near the
MPGDs and the window edge near 400 time buckets is from liberated electrons
near the window. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

.

.

.

Figure 3.4 Drift velocity of experimental runs used in the analysis.

. . . . . . . . . . . . . 36

Figure 3.5 Time correction factor for pad 4632 as a function of its average signal ampli-
tude for various pulser voltages. The errors on the average signal amplitudes
are shown but are very small. Within error, the correction factor is well
described by a constant value. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.6 Track identified as originating from a proton shown without the electric field

correction (blue circles) and with it (red stars).

. . . . . . . . . . . . . . . . . . 39

Figure 3.7 Track identified as originating from a deuteron shown without the electric

field correction (blue circles) and with it (red stars).

. . . . . . . . . . . . . . . 40

Figure 3.8 HDBSCAN clustering of the point cloud shown in Fig. 3.2. Black points are

noise while blue and orange points belong to two distinct clusters. . . . . . . . . 42

Figure 3.9 Final clustering of the point cloud from Fig. 3.2 after the initial clustering
in Fig. 3.8 underwent merging and cleanup. All points belong to the same
singular cluster.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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viii

Figure 3.10 Geometry of the circle fit to the first arc of a splined track oriented with the
beam coming out of the page. The arrows on the circle show the direction
the track curves due to the magnetic field. The azimuthal angle shown in
blue is measured to the tangent line to the circle at the estimated vertex. The
angle shown in red is given by Eq. 3.18 and is measured to the radial line
connecting the circle’s center and the estimated vertex. . . . . . . . . . . . . . . 47

Figure 3.11 Example particle identification plot constructed by plotting energy vs. energy

loss.

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Figure 3.12 PID plot constructed from a subset of this experiment’s data. The bottom-
most band is made of protons while the band above it is made of deuterons.
The other bands correspond to heavier species. . . . . . . . . . . . . . . . . . . 50

Figure 3.13 PID plots for run 302 (a) and run 347 (b). Notice the clear difference in the
location of the energy loss hot spot of the two deuteron bands. The gain of
the MPGDs apparently changed over the course of the experiment.

. . . . . . . 55

Figure 3.14 L-BFGS-B track optimization to the cluster shown in Fig. 3.9. The cluster
consists of the blue points, while the optimized trajectory is composed of the
red points.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

. .

. .

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.

Figure 4.1 Efficiency per bin for the angular distributions of the observed states in 10Be

seen in this experiment. They were found using attpc_engine simulations. . . . 64

Figure 4.2 Efficiency per bin for the angular distributions of the observed states in 11Be

seen in this experiment. They were found using attpc_engine simulations. . . . 66

Figure 4.3 Spyral reconstructed 𝜃𝐶 𝑀 vs. attpc_engine simulated 𝜃𝐶 𝑀 of 10Be elasti-
cally scattering off a deuteron. The same attpc_engine simulated data to
determine the elastic scattering efficiency factors was used. Note that there
are clearly regions of the phase space where the analysis does worse, such as
simulated 𝜃𝐶 𝑀 near 20°. The width of the hot central band also changes as a
function of simulated 𝜃𝐶 𝑀, indicating that the resolution is not constant and
degrades at larger angles.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 67

Figure 5.1 Measured kinematics for the proton from the 10Be(𝑑, 𝑝) reaction in the lab-
oratory frame. The theoretical kinematic bands are overlaid for the ground,
1.78 MeV, and 3.40 MeV states. The bands for the 0.32 MeV and 2.65 MeV
states lie between those shown with significant overlap.

. . . . . . . . . . . . . 69

Figure 5.2 Measured kinematics for the deuteron from the 10Be(𝑑, 𝑑) reaction in the
laboratory frame. The theoretical kinematic bands are overlaid for the elastic
and first inelastic channels.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

ix

Figure 5.3

Figure 5.4

10Be excitation spectrum from 18-60°, inclusively, in the CM frame. The
individual fits for each observed state and the total fit to the spectrum are
shown, but their heavy overlap makes them hard to distinguish.
Indicated
energies are from the centroids of the fit.

. . . . . . . . . . . . . . . . . . . . . 71

11Be excitation spectrum from 19-31°, inclusively, in the CM frame. The
individual fits for each observed state, a term for states above 3.40 MeV,
and the total fit to the spectrum are shown. Indicated energies are from the
centroids of the fit. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Figure 5.5 Total fit to each angular bin in the CM frame for the 10Be(𝑑, 𝑑) reaction. The

fit includes the elastic and first inelastic states of 10Be. . . . . . . . . . . . . . . 73

Figure 5.6 Total fit to each angular bin in the CM frame for the 10Be(𝑑, 𝑝) reaction. The
fit includes the first five states of 11Be and a term for the 3.89 MeV and 3.96
MeV states.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

.

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Figure 5.7 Measured angular distributions for the first four states of 11Be compared to
DWBA calculations. The calculations used the beam energy at the center of
the AT-TPC and were scaled by their spectroscopic factors. The orange solid
lines use the DA1p OMP, the green dashed lines use the An Cai OMP, and
the red dotted lines use the OMP from the elastic fit.

. . . . . . . . . . . . . . . 75

Figure 5.8 Measured angular distribution for the 3.40 MeV state of 11Be compared to
DWBA calculations. The calculations used the beam energy at the center of
the AT-TPC and were scaled by their spectroscopic factors. The orange solid
lines use the DA1p OMP, the green dotted lines use the An Cai OMP, and
the red dotted lines use the OMP from the elastic fit. The two different bands
correspond to either negative (ℓ = 1) or positive (ℓ = 2) parity. Because the
points measured are few and not near a minima, the parity cannot be definitely
determined from the shape. However, the reduced chi-squared values of the
fits slightly favor a positive parity. . . . . . . . . . . . . . . . . . . . . . . . .

. 76

Figure 5.9 Measured angular distributions for the elastic and first inelastic states of 10Be
compared to those measured for the same reaction by Schmitt et al. [1] with
a 90 MeV 10Be beam.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

.

Figure 5.10 Histograms of the minimized objective function value for the target-like prod-
uct from the 10Be(𝑑, 𝑑) reaction (top) and 10Be(𝑑, 𝑝) reaction (bottom). The
red dotted line is where the gate was made for the sensitivity analysis. . . . . . . 78

Figure 5.11 Elastic angular distribution plotted alongside calculations using various OMPs,
including the DA1p, An Cai, and one fit to it. Each calculation took the beam
energy at the middle of the AT-TPC.

. . . . . . . . . . . . . . . . . . . . . . . 79

x

Figure 5.12 Vertex z-coordinate histograms of events with the indicated ranges of 11Be
excitation energies. The gate on the vertex z-coordinate described in Section
5.1 is applied and the errors shown are statistical. Overall, the distributions
are generally flat with the total counts divided evenly about the midpoint of
the AT-TPC at 0.5 m.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xi

CHAPTER 1

INTRODUCTION

1.1 Experimental nuclear physics with beams

For over a century, many nuclear physics experiments have been conducted using the same

simple setup: a beam of nuclei is made to collide with a target of nuclei. Although how these

beams and targets are produced, their stability, etc. have evolved over time, this core idea has not.

The reason for this is because the collision of two nuclei in a nuclear reaction is analogous to a

microscope, which is of course used to illuminate the structure of things too small for us to see

with our naked eyes. As such, nuclear physicists turned to nuclear reactions to study the structure

of nuclei. Specifically, a nuclear reaction reveals information about both its reactants and products.

Such information includes the collision probability of the two reacting nuclei, the nuclear products

they can produce, and the spatial distributions, energies, lifetimes, etc. of these products. Nuclear

theory can use these observables from the nuclei involved in the reaction to infer information about

their nuclear structure, such as the distributions and energies of their protons and neutrons.

Experimentally, nuclear reactions can be divided into two categories: normal and inverse

kinematics. Normal kinematics are the reaction type that have historically dominated laboratory

use. In normal kinematics, the target is the heavier nucleus and the beam is the lighter nucleus.

The main reason for their popularity was that nuclear physicists had been primarily interested in

the study of stable nuclei. Stable nuclei can be made into targets that do not radioactively decay.

Thus, one has the convenience of hitting this stable target with a beam made of a light nucleus that

is relatively easy to produce. Another benefit of using a light nucleus is that its nuclear reactions

are much easier to theoretically model as there are less degrees of freedom than reactions between

two heavy nuclei (this is still true in inverse kinematics). However, with the shift towards studying

rare radioactive nuclei, nuclear reactions cannot be done in normal kinematics. It is not possible to

build a target that would endure an experiment if it is made of a nucleus that has a lifetime many

orders of magnitude smaller than a single second.

To overcome the limitation of building a radioactive target, the radioactive nucleus can be

1

turned into the beam itself. When the beam nucleus is heavier than the target nucleus, the reaction is

referred to as occurring in inverse kinematics. The production of rare-isotope beams is significantly

more difficult than light stable beams and is only due to amazing advances in rare-isotope beam

science over the past few decades that is now spearheaded by the Facility for Rare Isotope Beams

at Michigan State University.

It is the age of rare-isotope beams. This is the context for my thesis work.

1.2 Active targets for radioactive beams

Radioactive beams provide unique challenges for the study of their constituent nuclei. First,

they often are low intensity due to the complexity of their production. This is of experimental

importance because it means that they must impinge upon thick targets if there is to be any hope of

observing a reasonable amount of nuclear reactions in a reasonable amount of time. The problem

with traditional passive targets is that by increasing their thickness to compensate for the low

intensity of the beam, a larger uncertainty in the energy of the reaction is introduced as there is no

way to know how deep in the target it occurred. Another problem with passive targets is that it is

extremely expensive to surround the target with enough detectors to cover a large solid angle while

simultaneously achieving a high angular resolution.

An elegant solution to the challenges posed by radioactive beams and passive targets is an active

target. An active target serves as both the target of the nuclear reaction and detection medium.

Gases are used for this purpose since an impinging beam can react with it as the target while the

gas can simultaneously be used as an ionization chamber for detecting the reactants and products

of the reaction. Active targets solve the problem of target thickness because the reaction vertex can

be accurately determined in the gas by reconstructing the particle tracks using their electron-ion

pairs created ionizing the gas. Active targets theoretically provide a 4𝜋 solid angle coverage as the

detecting gas completely encompasses the nuclei in the reaction. Fig. 1.1 compares a passive and

active target.

Active targets are phenomenal detectors for use with rare-isotope beams, but they are not without

their own drawbacks. Gas detectors are known to be slow since they must drift the electron-ion

2

Figure 1.1 Comparison between a traditional passive target and an active target. Courtesy of Daniel
Bazin.

pairs generated by the ionizing radiation to a measuring device at a boundary of the detector.

The time needed for electrons to drift these distances is long (electron drift velocities are on the

order of cm/𝜇s), thus the detector can only accept low beam intensities without experiencing a

large pile up of events. There often is the problem that the gas, while chosen to study a certain

nuclear reaction, is not very good for detection, i.e. it may not have a desirable amplification factor

when used with micro-pattern gaseous detectors (MPGDs), too slow a drift velocity, etc. There

is also the issue of energy loss. Because the reaction can occur anywhere in presumably a large

gas volume, the incoming beam will experience significant energy loss. Depending on the depth

in the detector, the reaction channel of interest may not be energetically allowed or lose detection

efficiency. Finally, as will be evident from the analysis of this thesis, data generated from active

targets is quite non-trivially related to the physical observables of interest.

1.3 Active-Target Time Projection Chamber

The Active-Target Time Projection Chamber (AT-TPC) was designed and developed at the

former National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University, now

known as the Facility for Rare Isotope Beams (FRIB). It is the marriage of an active target, as

described in Section 1.2, with a time projection chamber. As a time projection chamber, the AT-

TPC records the tracks of particles inside of it. It is the active target used in this work and now will

be described.

The AT-TPC is shown in Fig. 1.2. It is a large cylindrical active target with an approximately

3

Figure 1.2 Schematic of AT-TPC with some key components indicated. Reproduced with permission
from [2].

1 m long active volume whose boundary is denoted by green in the figure. Its radius is 29.2 cm.

The active volume houses the target gas. A beam of nuclei is shot into the active volume in the

direction indicated in the figure. The beam nucleus then has the length of the active volume to

undergo a nuclear reaction with a nucleus of the gas. If a reaction occurs, the products fly off into

the gas, ionizing it and creating electron-ion pairs along their paths. The AT-TPC only measures

the electrons, not the ions, which encode information related to both the location and energy loss

of the particle where they were created.

A uniform electric field is made between the cathode at one end of the active volume and the

anode at the other to drift the liberated electrons towards the readout plane underneath the MPGDs

(labeled in Fig. 1.2 as the “micromegas”). The uniformity of the field is ensured by a field cage

made of rings connected to each other in series with a 20 MΩ resistor between consecutive pairs.

In this way, the rings and resistors form a series of voltage dividers that create a uniform potential

gradient. The active volume is encapsulated by an outer cylinder and the space between the two is

filled with nitrogen gas. The outer cylinder prevents access to the high-voltage cathode while the

gas prevents sparking between the two. The nitrogen gas is at a lower pressure than the target gas to

4

ensure any leaks in the active volume result in gas flowing into the outer volume. This helps keep

the target gas pure. The gases are controlled by the intricate gas handling system of the AT-TPC

that continuously recycles the target gas to remove impurities.

The liberated electrons enter the MPGDs where the avalanche process occurs and orders of

magnitude more electrons are emitted that induce signals on the readout plane. It is vital that the

number of electrons is increased as the number of liberated electrons in the active volume is too

low to create a detectable signal on the readout plane. A pad that records a signal reveals the x-

and y-coordinates of the particle at the time those liberated electrons were created. Measuring how

long those electrons took to induce a signal on the pad, with knowledge of how fast electrons drift

in the gas, is used to calculate the z-position of the particle at that time. A variety of MPGDs in

different configurations can be used with the AT-TPC. This experiment used two multilayer thick

GEMs (M-THGEMs) stacked on top of a micromegas. The readout plane is shown in Fig. 1.3

and comprised of 10,240 equilateral triangular pads. As the figure illustrates, the inner hexagonal

region is made of smaller triangular pads with a height of 0.5 cm while the larger outer triangular

pads have a height of 1 cm. The inner hexagonal region covers where the beam enters the AT-TPC

and reacts with the gas, thus it contains the reaction vertex and the beginning of the products’ tracks.

A finer granularity is desirable in this region to better extract important kinematic variables, such

as the polar angle, from the tracks.

The AT-TPC uses the General Electronics for TPCs (GET) to readout the signals induced by the

electrons on the readout plane. GET is a hierarchical data collection system, as seen in Fig. 1.4,

where each pad on the readout plane is connected to a specific ASIC for GET (AGET) chip, ASIC

and ADC (AsAd) front-end board, and concentration board (CoBo) module. The specifics of GET

will not be elaborated on further in this work unless necessary, but Ref. [5] provides more details.

1.4 Coupling the AT-TPC with the SOLARIS solenoid

Section 1.3 describes the AT-TPC as a stand-alone detector. It certainly can be used this way to

study nuclear reactions. However, there are considerable advantages in coupling the AT-TPC to a

solenoid magnet as done for this thesis experiment. The AT-TPC was placed inside the SOLARIS

5

Figure 1.3 Illustration of the readout plane composed of 10,240 pads. The inner pads are smaller
than the outer pads. Reproduced with permission from [2].

solenoid in the ReA6 vault of the NSCL. SOLARIS is a former magnetic resonance imaging magnet

that can reach a maximum field strength of 4 T. Fig. 1.5 shows SOLARIS with the AT-TPC inside.

The benefits of a solenoid come from the addition of a magnetic field that curves the trajectories

of the charged particles in the AT-TPC. The trajectory lengths of the particles are effectively

increased because they can spiral longer distances before hitting the walls of the AT-TPC. A longer

trajectory means more information is gathered about the particle in the detector. The magnetic field

also allows the energy of a particle to be directly calculated from the magnetic rigidity measured

from its track. This is a significant advantage of the solenoid as without it the energy of a particle

would have to be calculated from its range in the gas, which cannot be done if the particle does

not stop in the AT-TPC active volume. Ensuring that particles deposit their entire energy in the

detector requires careful tuning of other experimental parameters, such as the gas pressure, that

may restrict the sensitivity of the AT-TPC to certain nuclear reactions and are altogether avoided by

6

Figure 1.4 Illustration of GET electronics hierarchy connected to the sensor, or pad, plane that the
electrons induce signals on.

the solenoid. The magnetic field curves the trajectories of the electrons ionized by the particles as

well. This has a focusing effect on the electrons as they drift through the detector and reduces their

transversal diffusion. This is because their radii of curvature are extremely small in the magnetic

fields typically used.

1.5 Physics with the AT-TPC

The AT-TPC is well-designed to study exotic radioactive nuclei. There is an assortment of

physical observables that can be extracted with the AT-TPC including differential and total cross

sections, excitation functions, excitation energy spectra, etc. This section will provide a cursory

overview of the physical measurements that were extracted from the experiment of this thesis using

the AT-TPC.

1.5.1 Cross section

Perhaps the most fundamental and important concept of nuclear reactions is the nuclear cross

section. They are important because they are a concrete observable for comparison with theoretical

models and can reveal a wealth of information including the spin-parities of nuclear states as

7

AsAdZAPSensor planeCoBoAGETAGETAGETAGETFigure 1.5 Picture of the AT-TPC within the SOLARIS solenoid at FRIB. Courtesy of Daniel Bazin.

discussed in Section 1.5.4. The cross section is the probability that a nuclear reaction will occur.

We can define the experimental cross section by imagining a beam of nuclei impinging on a target.

In this scenario, suppose that we have perfect detectors that always tell us when any reaction

between the beam and target occurs. Let 𝑅 represent the number of reactions per time per target

volume hit by the beam. Suppose we also know the beam intensity 𝐼 that has units of nuclei per

8

area per time. We could say that the probability that a reaction occurred is given by the simple

ratio 𝑅/𝐼 that has units of inverse length. This definition makes intuitive sense because we have

simply taken the ratio of the number of reactions to the number of possible reactions. The problem

with this definition is that the cross section is a property of the bulk target and not its constituent

nuclei. This is clear by considering that we could arbitrarily change the cross section by adding

or subtracting material to the target, hence altering 𝑅 while keeping 𝐼 the same. We thus need to

convert this macroscopic cross section to a microscopic cross section. We want the cross section to

be a property of the target nuclei. This can be accomplished by dividing out the number of nuclei

in the volume of the target seen by the beam, denoted by 𝑛 with units of nuclei per volume. The

reaction cross section is then given by

𝜎 =

𝑅
𝐼𝑛

(1.1)

and has units of area. The condition that 𝑅 includes all reactions can be restricted to only include

certain reaction channels.

A reaction product is scattered in some direction that can be described by the spherical coordinate

system angles 𝜃 and 𝜙 measured with respect to the reaction vertex. 𝑅 is thus a function of the

angles 𝜃 and 𝜙. To examine the dependence of the cross section on the solid angle the reaction

product is scattered into, we introduce the differential cross section 𝑑𝜎/𝑑Ω. Taking the derivative

of Eq. 1.1 with respect to the solid angle yields

𝑑𝜎
𝑑Ω

=

1
𝐼𝑛

𝑑𝑅(𝜃, 𝜙)
𝑑Ω

.

(1.2)

We typically do not measure the 𝜙 dependence of 𝑅 and can restrict the differential solid angle 𝑑Ω

to cones by integrating over all 𝜙. This results in

Plugging Eq. 1.3 into Eq. 1.2 yields

𝑑Ω =

∫ 2𝜋

0

sin 𝜃𝑑𝜃𝑑𝜙,

𝑑Ω = 2𝜋 sin 𝜃𝑑𝜃.

𝑑𝜎
𝑑Ω

1
2𝜋𝐼𝑛 sin 𝜃

𝑑𝑅(𝜃)
𝑑𝜃

.

=

9

(1.3)

(1.4)

An actual experiment cannot measure angles to infinite precision, so we will recast the differential

cross section as

𝑑𝜎
𝑑Ω

=

1
2𝜋𝐼𝑛 sin 𝜃

𝑅(𝜃, Δ𝜃)
Δ𝜃

(1.5)

where it is understood that 𝑅(𝜃, Δ𝜃) is the reaction rate measured by our detector at the nominal

angle 𝜃 that also includes counts from the angles around it in the range ±Δ𝜃/2. The differential

cross section in this form is commonly called the angular distribution.

The cross section also depends on the total kinetic energy of the colliding system. This

dependence can be made explicit by taking the derivative of Eq. 1.4 with respect to it. However,

because angular distributions are often measured at one beam energy, this dependence is not shown.

1.5.2 Spectroscopy

Nuclei are inherently quantum systems. They have excited states with quantized energies that

carry quantum numbers such as spin and parity. This is evident from the shell model, historically

the most successful nuclear structure theory. The shell model assumes the nucleons in a nucleus

experience a mean-field force they themselves create. Solving the Hamiltonian for a nucleon

in this mean field gives rise to discrete quantum levels for the nucleons to populate, with gaps

in their energies separating different shells (analogous to the shells electrons fill in atoms).

In

nuclear physics, a large gap between shells corresponds to a magic number, which are nuclei of

enhanced stability, and the shell model was the first to accurately reproduce those experimentally

known. Magic numbers are known from a variety of data. Fig. 1.6 shows the difference between

experimental two-nucleon separation energies and those predicted by the semi-empirical mass

model.

In the figure, the data shows structure at the magic numbers, where the experimental

separation energies are greater than predicted.

Nuclear reactions provide a tool for extracting observables that can be used to benchmark

nuclear structure models like the shell model. This includes measuring the spectrum of excited

state energies of a nucleus, which is called spectroscopy. The spectroscopic method used depends

on the nuclear reaction, measured nucleus, and detector. Here we discuss the missing-mass method

used by the AT-TPC for the two-body reaction 𝑎 + 𝑏 → 𝐴 + 𝐵. Let 𝑎 be the beam with momentum

10

only in one direction and 𝑏 be the stationary target. In the equations that follow, 𝐾 refer to kinetic

energies, 𝑚 to rest masses, and 𝑝 to momenta. Assume we are interested in the spectroscopy of

nucleus 𝐵 but only detect nucleus 𝐴. 1 The excitation energy 𝐸𝑒𝑥 of 𝐵 is

𝐸𝑒𝑥 = (𝑚∗

𝐵 − 𝑚𝐵)𝑐2

(1.6)

where 𝑚∗

𝐵 is the rest mass of 𝐵 in its excited state and 𝑐 is the speed of light. With the relativistic

energy-momentum relation Eq. 1.6 becomes

√︃

𝐸𝑒𝑥 =

𝐵 − ( 𝑝𝐵𝑐)2 − 𝑚𝐵𝑐2
𝐸 2

(1.7)

where 𝐸𝐵 is the total energy of 𝐵. Conservation of energy and momentum in the reaction requires

𝐸𝐵 = 𝐾𝑎 + 𝑚𝑎 + 𝑚𝑏 − 𝐾𝐴 − 𝑚 𝐴,

𝐵 = 𝑝2
𝑝2

𝑎 + 𝑝2

𝐴 − 2𝑝𝑎 𝑝 𝐴 cos 𝜃

(1.8)

(1.9)

where 𝜃 is the laboratory scattering angle of nucleus 𝐴. Notice that insertions of Eqs. 1.8 and 1.9

into Eq. 1.7 allow for the determination of 𝐸𝑒𝑥 without measuring 𝑚∗

𝐵, hence the term “missing
mass”. The missing-mass method is powerful because it can measure the excitation energy of

unbound states that prevent a direct measurement of their mass.

1.5.3 Spectroscopic factors

While the shell model equipped with a Wood-Saxon potential and spin-orbit interaction is quite

successful in describing stable nuclei near the magic numbers, it requires modifications for the vast

majority of nuclei. The nuclear force has residual interactions the mean-field potential does not

capture that play an important role in these nuclei. For example, such residual interactions include

the pairing and tensor forces. In this context, nuclear theorists have attempted to modify and/or

supplement the mean-field force to find potentials that better suit a nucleus or group of nuclei. To

characterize the effectiveness of a particular nuclear model, spectroscopic factors were introduced.

A spectroscopic factor 𝑆(+/−)

𝛼

is the probability that adding (+) a nucleon in the single-particle state

1This may sound contrived, but it is accurate for the AT-TPC. In inverse kinematics, the beam-like nucleus is
constrained to small forward scattering angles where the pad plane is insensitive as described in Section 2.1. The
target-like nucleus does not have this constraint and is much easier to detect.

11

𝛼 to the nucleus in state 𝜒 𝐴−1 or subtracting (−) a nucleon in the single-particle state 𝛼 from the

nucleus in state 𝜒 𝐴+1 yields the nucleus in state 𝜓 𝐴 with 𝐴 nucleons. Mathematically,

𝛼 = |⟨𝜓 𝐴|𝑎†
𝑆+

𝛼| 𝜒 𝐴−1⟩|2,

𝛼 = |⟨𝜓 𝐴|𝑎𝛼| 𝜒 𝐴+1⟩|2
𝑆−

(1.10)

where 𝑎†

𝛼 and 𝑎𝛼 are the creation and annihilation operators for the single-particle state 𝛼, respec-

tively. The spectroscopic factors can be related to the effective single-particle energy (ESPE) of 𝛼

via

𝜖𝛼 =

∑︁

𝑁

𝑆+
𝛼 (𝐸 − 𝐸𝑁 ) +

∑︁

𝑛

𝑆−
𝛼 (𝐸𝑛 − 𝐸)

(1.11)

where 𝐸, 𝐸𝑁 , and 𝐸𝑛 are the energies of the nuclei in states 𝜓 𝐴, 𝜒 𝐴−1, and 𝜒 𝐴+1, respectively [6].

The summations are over the set of all nuclei in either particle (𝜒 𝐴+1) or hole (𝜒 𝐴−1) states. ESPEs

are one of the most important quantities in nuclear structure for testing the validity of a model,

and Eq. 1.11 relates them to the excitation energies and spectroscopic factors of nuclei found from

spectroscopy experiments.

Spectroscopic factors are not experimentally observable but can be experimentally derived.

Imagine we have an ensemble of nuclei in state 𝜓. We now add a nucleon in the single-particle

state 𝛼 to each of them and record the final state of each resulting nucleus. From this information

we could determine the probability to produce the final state 𝜓 ⊗ 𝛼, which is a measure of the purity

of the single-particle state 𝛼, and compare it to some model’s spectroscopic factor. An analogous

process can be done for subtracting a nucleon from 𝛼 to produce 𝜓 ⊗ 𝛼−1. The experimental way

to add or subtract a nucleon to a nucleus is through pick-up or stripping reactions, respectively.

In a reaction, the probability to populate a certain state of a nucleus is given by its cross section.

We thus suspect that the cross section is related to the spectroscopic factor. However, the cross

section is not directly related to a pure spectroscopic nuclear matrix element and has kinematic

contributions from scattering [7]. Theoretically, if we have a good handle on the reaction theory,

we can divide out the kinematic dependence from our experimental cross section and be left with

the spectroscopic factor. For a pick-up reaction, the experimentally deduced spectroscopic factor

for the nucleus in state 𝜓 ⊗ 𝛼, 𝑆𝑒𝑥 𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡, is then related to the measured cross section, 𝜎𝑒𝑥 𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡,

12

through

𝑆𝑒𝑥 𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 =

𝜎𝑒𝑥 𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡
𝜎𝑡ℎ𝑒𝑜𝑟 𝑦

(1.12)

where 𝜎𝑡ℎ𝑒𝑜𝑟 𝑦 is the theoretical cross section for the production of the nucleus also in state 𝜓 ⊗ 𝛼.

The definition is the same for a stripping reaction except the nucleus is in the state 𝜓 ⊗ 𝛼−1.

Typically, 𝜎𝑡ℎ𝑒𝑜𝑟 𝑦 is calculated using a reaction model like the distorted-wave Born approximation

(DWBA). Eq. 1.12 makes apparent that 𝑆𝑒𝑥 𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 depends on the nuclear reaction model used to

calculate 𝜎𝑡ℎ𝑒𝑜𝑟 𝑦, again illustrating that spectroscopic factors are not physically observable. Despite

this, they have proven to be a powerful tool for examining the nuclear structure of single-particle

states when making comparisons with theory.

1.5.4 Angular momentum transfer in pick-up and stripping reactions

Pick-up and stripping reactions belong to a class of reactions called direct reactions and more

specifically transfer reactions when the kinetic energy of the system is close to the Fermi energy of

the nucleus to be studied. Direct reactions are characterized by the nuclear reaction taking place at

the surface of the nucleus over a very short period of time, on the order of 10−22 s. It turns out that

the angular distribution for a pick-up or stripping reaction reveals the quantized angular momentum

of the transferred nucleon. The transferred angular momentum is a key quantity in these types of

reactions as it can then be used to deduce the spin and parity of the nucleus that gained or lost a

nucleon in the reaction. The following is a widely used semi-classical calculation that illustrates

how the angular distribution is related to the transferred angular momentum and closely follows the

treatment given by Krane [3].

Consider a stripping reaction where an incident nucleus with momentum (cid:174)𝑝𝑎 hits a target nucleus

and transfers one nucleon to it. Denote the momentum of the transferred nucleon as (cid:174)𝑝 and the

momentum of the outgoing nucleus that lost the nucleon as (cid:174)𝑝𝑏. See Fig. 1.7 for the geometry of

the collision. The angular momentum of the transferred nucleon 𝐿 is given by the classical formula

where (cid:174)𝑅 is the radial vector from the center of the target nucleus to the collision point. Because

(cid:174)𝐿 = (cid:174)𝑅 × (cid:174)𝑝

13

this is a direct reaction, we will assume that the transfer of the nucleon occurs at the surface of the

nucleus, i.e. (cid:174)𝑅 and (cid:174)𝑝 are perpendicular. This leads to

| (cid:174)𝐿| = | (cid:174)𝑅|| (cid:174)𝑝|.

Quantizing the angular momentum yields

where 𝑙 is the angular momentum quantum number and ℏ is the reduced Planck constant. By

𝑙ℏ = | (cid:174)𝑅|| (cid:174)𝑝|

(1.13)

conservation of momentum

(cid:174)𝑝𝑎 = (cid:174)𝑝 + (cid:174)𝑝𝑏

| (cid:174)𝑝|2 = | (cid:174)𝑝𝑎 |2 + | (cid:174)𝑝𝑏 |2 − 2| (cid:174)𝑝𝑎 || (cid:174)𝑝𝑏 | cos 𝜃

(1.14)

where 𝜃 is the angle of the scattered nucleus that lost a nucleon to the target. We can insert Eq. 1.13

into the left-hand side of Eq. 1.14 to result in a relation between the scattering angle and transferred

angular momentum. We therefore expect that the angular distributions for pick-up and stripping

reactions should contain information on the transferred angular momentum. This is seen in Fig.

1.8 where the value of the transferred angular momentum greatly influences the location of the first

peak in each angular distribution and its overall shape. DWBA calculations are used to calculate

the angular distributions in the figure. These are completely quantum-based calculations unlike the

semi-classical example given and form the standard method for calculating the transferred angular

momentum in a pick-up or stripping reaction by comparison of the calculation to the experimental

cross section.

As previously mentioned, once the transferred angular momentum is determined, this informa-

tion can be used in calculations related to the spin and parity of the nucleus that a nucleon was

added to or subtracted from. Using the rules for the addition of angular momentum, we add the

spin of the initial nucleus 𝐽𝑖, the angular momentum of the transferred nucleon 𝑙, and the spin of

the transferred nucleon (1/2 for protons and neutrons) to yield

(cid:12)
(cid:12)
(cid:12)
(cid:12)

|𝐽𝑖 − 𝑙 | −

1
2

(cid:12)
(cid:12)
(cid:12)
(cid:12)

≤ 𝐽 𝑓 ≤ 𝐽𝑖 + 𝑙 +

1
2

14

(1.15)

where 𝐽 𝑓 is the final parity of the nucleus after a nucleon is added or subtracted. Considering

that the parity of the final nucleus can only be changed by the angular momentum added to it or

subtracted from it by the transferred nucleon (all nucleons have the same intrinsic parity, so this

quantity does not matter for the calculation) results in the relation

𝜋𝑖𝜋 𝑓 = (−1)𝑙

(1.16)

where 𝜋𝑖 is the parity of the initial nucleus and 𝜋 𝑓 is that of the final nucleus.

1.6 10Be + d experiment with the AT-TPC

The AT-TPC was used for the first time in the experiment of this thesis to measure a transfer

reaction in inverse kinematics in the ReA6 vault at the NSCL. The nuclear structure of the low-lying

11Be states was investigated via the 10Be(𝑑, 𝑝) reaction and the physics measurements described

in Section 1.5 performed.

11Be is an ideal nucleus for a transfer commissioning experiment

because the structure of its low-lying states is well understood with many experimentally derived

spectroscopic factors [1, 8, 9, 10]. The exception is its fourth excited state at 3.40 MeV whose

parity is debated. A 9Be(𝑡, 𝑝)11Be reaction performed by Liu et al. concluded that this state has

a negative parity [11]. The same assignment was given by Hirayama et al. using 11Li 𝛽-decay

measurements [12]. However, Coulomb breakup of 11Be on a carbon target done by Fukuda et

al. indicated that this state has a positive parity [13]. Recent ab initio calculations on Be isotopes

performed by Caprio et al. predict the 3.40 MeV state in 11Be to be either the second member of

the K𝑃 = 1/2+ rotational ground state band if the parity is positive or the K𝑃 = 3/2− band-head if

the parity is negative [14]. A positive parity is supported by the rotational analysis of Bohlen [15].

A definitive party assignment to the 3.40 MeV state would aid in determining the rotational band

it belongs to as well as the location of the 0𝑑3/2 ESPE in 11Be.

All 11Be states up to 3.40 MeV were populated in this experiment. The spectroscopic factors

were derived by comparing the experimentally measured angular distributions to theoretical DWBA

calculations. Their averages were compared to those in the literature and shell-model calculations

using the WBP [16], YSOX [17], and FSU [18] interactions. Generally, reasonable agreement was

found with the literature and the WBP and YSOX interactions. Despite a limited angular coverage

15

for the 3.40 MeV state, a tentative positive parity assignment was made and a lower limit for the

0𝑑3/2 ESPE calculated. The 0𝑑5/2 ESPE was also extracted using the 1.78 MeV state, enabling
exploration of the 0𝑑 spin-orbit splitting in 11Be and a comparison to that of its 𝑁 = 7 isotone 13C.

The success of this experiment demonstrates the ability of the AT-TPC to investigate nuclear

structure via transfer reactions and opens the door to future investigations of exotic areas of the

nuclear landscape using rare-isotope beams.

16

Figure 1.6 The difference between experimental two-proton and two-neutron separation energies
and those predicted by the semi-empirical mass formula for sequences of isotones and isotopes,
respectively. Note the large spikes in the difference at the magic numbers, where the model
underpredicts the separation energies. Reproduced with permission from [3]. Copyright © 1988
by John Wiley & Sons, Inc.

17

Figure 1.7 Geometry for the semi-classical stripping reaction considered. Note that collisions
are shown on both sides of the target nucleus as these result in a diffraction pattern (a series of
minima and maxima) in the angular distribution as seen in the experimental data shown in Fig. 1.8.
Reproduced with permission from [3]. Copyright © 1988 by John Wiley & Sons, Inc.

Figure 1.8 Angular distributions from the 31P(𝑑, 𝑛)32S reaction that populate different final states in
32S. They are compared to DWBA calculations with the indicated transferred angular momentum
values. The transferred angular momentum is clearly correlated to the shape of the angular
distribution and location of the first peak. Reproduced with permission from [4].

18

CHAPTER 2

EXPERIMENTAL SETUP

2.1 Setup of the AT-TPC

As described in Section 1.4, the AT-TPC was coupled to the SOLARIS solenoid.

It was

inserted and locked inside the solenoid such that the central axes of both were aligned. The AT-

TPC was oriented so that the readout plane was on the far end of the detector relative to where

the beam entered. The AT-TPC uses GET to record data, as mentioned in Section 1.3, and the

only components of this hierarchical system not directly attached to the AT-TPC are the CoBos

and back-end computers. Cables were used to connect the AsAd front-end boards, protruding from

behind the readout plane on the AT-TPC, to the CoBos kept on a nearby rack. The same rack had

power cables connected to the AsAd boards, which were connected to the pad plane via so-called

“ZAP” boards. The “simple” ZAP board is a printed circuit board (PCB) that connects each pad

on the readout plane to a grounded 100 MW resistor and capacitor. The purpose of this protection

circuit is to prevent large currents from damaging the expensive GET modules downstream. The

AT-TPC was also equipped with “smart” ZAP boards that provide extra functionality through an

onboard Arduino Nano. The Arduino Nanos are programmed to provide a biasing voltage to the

innermost pads of the pad plane that see a large amount of liberated electrons from the beam. The

centers of these beam pads lie within a circle of radius 2 cm from the center of the pad plane. The

biasing voltage effectively reduces the gain in this region by reducing the induced signals on the

pads. Reducing the gain in the beam region is a necessity for the way the trigger was constructed,

as described in Section 2.2.

The setup of the AT-TPC includes three electronics racks pictured in Figs. 2.1-2.3. One rack

houses the multiplicity, trigger and time (MuTAnT) module along with the CoBos, Mac Minis,

power supplies for the AsAd front-end boards, and nuclear instrumentation modules (NIMs) for the

trigger logic. The MuTAnT ensures the timestamps of each CoBo are synchronized and distributes

the trigger. This is necessary because each CoBo, which corresponds to a particular region of

the pad plane, has its own data acquisition system (DAQ); every CoBo starts and stops recording

19

data on its own and saves the raw data to its own connected Mac Mini. Each Mac Mini reads

a configuration file that controls the GET parameters for its electronic channels including their

sampling frequency, amplifier gain, and shaping time. These values were the same for each pad

across all CoBos and are given in Table 4.2. Each CoBo has 1,024 channels, so ten CoBos were

used during the experiment for the full 10,240 pads of the pad plane. An eleventh CoBo was

installed to record auxiliary signals such as those from the ion chamber (IC), which is described

in the next paragraph. Another rack houses a computer and the high-voltage power supplies for

the MPGDs, IC, and smart ZAP boards. The computer controls the gas-handling system and the

high voltages applied to the MPGDs and the cathode via two different programs designed by the

AT-TPC collaboration. The final rack contains the cathode high-voltage power supply and the gas

handling systems for the AT-TPC and IC.

The IC was installed upstream of the AT-TPC. Pictured in Fig. 2.4, it is a small cylindrical

volume that houses a gas that the beam nuclei ionize. The liberated electrons are drifted to the

anode via an electric field and the signal is read out. The energy loss of the beam nucleus that

traveled through it is proportional to the detected signal making the IC crucial for determining the

composition of the cocktail beam used in this experiment (see Section 2.4). It also provided the

time reference the AT-TPC needed for accurate position reconstruction of particle tracks.

For a summary of the experimental parameters see Table 2.1.

Parameter

Gas
Gas pressure (Torr)
Cathode voltage (kV)
Field cage ring voltage (V)
Micromegas voltage (V)
E-trans voltage (V)
M-THGEM voltage (V)
Smart ZAP bias voltage (V)
Gas
Gas pressure (Torr)
Voltage (V)

AT-TPC

IC

SOLARIS Magnetic field (T)

Table 2.1 Experimental parameters.

20

Value
Deuterium
600
60
100
420
100
850
130
CF4
≈ 200
50
3

2.2 Trigger

A detailed schematic of the trigger is presented in Fig. 2.5. At the highest level, the AT-TPC

uses the micromegas mesh signal to determine whether or not to record data. The liberated electrons

that drift towards the micromegas, which have already been multiplied by the M-THGEM above it,

induce a current on the micromegas mesh that is read out. This signal is sent through a preamplifier

and shaper before a single channel analyzer (SCA) determines if the amplitude of the signal passes

the set threshold to trigger data recording. As seen in Fig. 2.5, if the mesh signal generates a trigger,

the resulting signal from the SCA is then stretched to form an AND gate with the signal from the IC.

The IC signal is used in coincidence with the mesh signal to provide a time reference for when beam

particles enter the AT-TPC. This time reference is later used to reconstruct the absolute positions

of the particle. Like the mesh signal, the IC signal is passed through a preamplifier and shaper

before a SCA determines if the signal amplitude is above a set threshold to trigger. If both the IC

and mesh signals trigger, then the AND gate is triggered. Before this is described, let us consider

why the IC signal has a delay and the mesh signal is stretched.

The AND gate between the mesh and IC signals requires their signals to overlap to trigger. In

a slow gas detector like the AT-TPC, this condition will not happen without modifying the signals.

The mesh signal is generated by the liberated electrons from the ionizing particles that drift through

the AT-TPC at relatively slow speeds (on the order of cm/𝜇s). Without any modification, the IC

signal will always be faster than the mesh signal and no coincidence can be made between the two.

To remedy this, a modification must be made to the timing of each signal. The IC signal is delayed

by slightly more time than it would take an electron to drift the entirety of the AT-TPC. This is

done because the mesh signal can be created by electrons liberated anywhere inside the AT-TPC.

Delaying the IC signal ensures that the mesh signal has seen all the electrons liberated in the entire

AT-TPC volume belonging to an event. This delay still does not guarantee an overlap between

the IC and the mesh signals, though. It is possible to have liberated electrons slightly above the

MPGDs that would cause the mesh signal to trigger the SCA much earlier than the now delayed IC

signal. No coincidence would occur. To always ensure a coincidence between the two signals, the

21

mesh signal is stretched slightly longer than the delay on the IC signal.

The AND gate from the mesh and IC signals is proceeded by another AND gate. It checks if

the GET DAQ is busy writing data from a previous event to prevent interrupting it. If the DAQ

is not busy, an OR gate is encountered and the MuTAnT is triggered to record the data from the

current event. If a nuclear reaction occurs and is detected, the path hitherto described is the most

likely route its signals take to result in its data acquisition. We will now describe the uppermost

path of Fig. 2.5 in relation to unreacted beam events and how it may be taken for nuclear reactions.

An unreacted beam event occurs when the beam particle traverses the entirety of the AT-TPC

without undergoing a nuclear reaction. Recording unreacted beam events over the course of an

experiment is crucial for determining the intensity of the beam to measure cross sections. As

described in Section 3.2.3, they can also be used to calculate the electron drift velocity. The IC

signal of an unreacted beam event will trigger but the mesh signal will not. This is because the

electrons liberated by the beam induce signals on the central pads of the pad plane that have a

reduced gain from the smart ZAPs. Following the IC signal after the SCA, it is still delayed and

then sent to a downscaler. Because nuclear reactions have small cross sections, it is exceedingly

likely that the beam will not undergo a reaction in the AT-TPC. Recording all of these unreacted

beam events would overload the DAQ and be a waste of disk space as they all share the same

features. To avoid this, the downscaler recorded only one in every thousand presumably unreacted

beam events. “Presumably” is used as it is possible that a normal nuclear reaction may coincide

with the thousandth recorded event that is downscaled and recorded as an unreacted beam event.

The signal from the downscaler goes to an AND gate that checks if the DAQ is busy writing. If

not, a LEMO signal is generated that the current event is a beam event and the MuTAnT triggers,

recording its data. The LEMO signal is readout by the auxiliary eleventh CoBo mentioned in

Section 2.1 and uses a specially modified simple ZAP board equipped with LEMO connectors

connected to an AsAd front-end board.

22

2.3

Ion chamber multiplicity

The ion chamber may see multiple incoming beam nuclei within a single event. Unfortunately,

these events cannot be used. The AT-TPC lacked auxiliary detectors to identify the beam particle

that reacted when the multiplicity was greater than one. The time reference is also always with

respect to the beam particle that resulted in a trigger, as described in Section 2.2. Thus, for events

with ion chamber multiplicity greater than one, the z-coordinates of the reconstructed particle

trajectories may be wrong. The ion chamber multiplicity restriction was applied in the analysis,

not the physical trigger.

For this experiment, the characteristics of the beam were such that it was common for the ion

chamber to see multiple beam nuclei for a single event. Of the 6,131,566 detected events, only

≈ 34% of them had a multiplicity of one.

2.4 Beam

The 10Be beam for this experiment was provided by the ReA6 facility of the NSCL. It had an

energy of 9.6 MeV/u and a rate of approximately 1000-2000 pps. 10B and 15N contaminants were

present. Fig. 2.6 is a histogram of the energy loss of the unreacted beam particles through the IC

and clearly shows three nuclei in the beam.

The energy of the beam as it entered the AT-TPC was less than the 9.6 MeV/u provided by ReA6

as it lost energy traveling through the IC and the window of the AT-TPC. The pressure of the CF4

gas used in the IC along with the thicknesses of its two windows and the window of the AT-TPC

were not exactly known. This made theoretical calculations of the beam’s energy loss unreliable.

However, the beam energy was not a free parameter.

It was only needed for the calculation of

the excitation spectra described in Section 5.3. Because the only unknown quantity in calculating

the excitation spectra was the beam energy, it was adjusted until the peaks in the measured 11Be

spectrum approximately aligned with their known energies (see Fig. 5.4). This required the beam

energy right after the window of the AT-TPC to be approximately 9.3 MeV/u.

23

Figure 2.1 Rack housing the CoBos, Mac Minis, MuTAnT module, power supplies for the AsAd
front-end boards, and NIMs for the trigger logic.

24

Figure 2.2 Rack housing the computer that controls the gas-handling system and the high voltages
applied to the MPGDs and the cathode. It also contains high-voltage power supplies for the MPGDs,
IC, and smart ZAP boards.

25

Figure 2.3 Rack housing the cathode high-voltage power supply and gas-handling systems for the
AT-TPC and IC.

26

Figure 2.4 The IC is the cylinder atop the vacuum flange. Its electrical and gas leads can be seen.

27

Figure 2.5 Trigger logic for the experiment of this thesis. Courtesy of Daniel Bazin.

28

E20009 trigger logicIon ChamberPAShaperSingle Channel AnalyzerAT-TPCMeshPAShaperSingle Channel AnalyzerThresholdThresholdDownScaler/ 1000Delay100 µsStretcher120 µsLogicalANDLogicalORLogicalANDMutantNot busyLogicalANDMutantNot busyMutantTriggerBeamTriggerLEMODownscaled beam triggerAT-TPC mesh triggerThe purpose of this AND is to set the timing of the mesh trigger to the time of the IC signalIon ChamberLEMOMeshLEMO100 µs is the maximum drift time of the electrons Figure 2.6 Histogram of the energy loss of incoming unreacted beam particles seen over the course
of the entire experiment. The dotted vertical dotted lines correspond to the gate used to select 10Be
events. The Gaussian fits used to subtract the 10B contamination from the 10Be counts in the gate
are shown. The small peak short of 2000 ADC units belongs to 15N.

29

05001000150020002500Energy loss (ADC units)0100200300400500600700Counts / ADC unit10Be10BCHAPTER 3

ANALYSIS OF AT-TPC DATA

This chapter is dedicated to describing the arduous process of analyzing AT-TPC data. For the

experiment of this thesis, the data was analyzed using a modified version of the Python package

Spyral [19]. There is no attempt here at detailing the coding specifics of this package. Instead, the

main concepts of the analysis will be illustrated with references to the relevant portions of Spyral

where necessary. Due to this, the Spyral parameters used in each step of the analysis will be given

but not all of them may be discussed in the text. They are included for completeness.

3.1 Data structure

As mentioned in Section 1.3, the AT-TPC uses the GET system for data acquisition. For each

event of a run, a CoBo records a pad’s trace only if it had charge deposited above a threshold set

in the CoBo’s configuration file. A trace is composed of 512 time buckets. A time bucket in GET

does not represent an instantaneous measurement but rather the sum of measurements within the

time width of that bucket. A trace from this thesis’ experiment is shown in Fig. 3.1. GET is

highly flexible and the length of a time bucket can be changed via the sampling frequency. For this

experiment, the sampling frequency was set to 3.125 MHz so that each time bucket sampled 0.32

𝜇s. With 512 time buckets, the total time range covered by the GET system was 163.84 𝜇s, enough

to record the longest drift time of the electrons, which was about 100 𝜇s.

All events an AsAd records traces for in a run are output to a single binary file called a graw file.

The graw files from all AsAds, each representing a different group of pads on the pad plane, are

merged together into a single HDF5 file to form a complete record of each event in a run. This is

accomplished using a merger program. HDF5 is ideal for AT-TPC data because it is a hierarchical

data format that functions like a folder directory. The traces of all pads from each event are stored

together in one HDF5 file under what is called a group (the data itself is called a dataset in HDF5).

The merged HDF5 files have sizes on the order of tens of gigabytes as they contain tens of thousands

of events that each contain hundreds of pad traces that are each 512 time buckets long. The large

size of the HDF5 files makes them somewhat cumbersome to work with, but transforming the data

30

Figure 3.1 A pad’s trace taken from an event during the experiment shown in solid blue. The
baseline of the trace is removed using the outlined procedure to produce the corrected trace shown
in dotted red.

from traces to point clouds drastically reduces their size.

3.2 Creating point clouds

The first major step of analyzing AT-TPC data is converting an event from traces to a point

cloud. As a time projection chamber, the AT-TPC visualizes the trajectories of ionizing radiation

within it. The points along the trajectories are extracted from the traces of the pads that recorded

signals during an event. These are the trajectories that are analyzed. The Spyral parameters for

this phase are listed in Table 3.1.

3.2.1 Baseline correction

Before any peaks are extracted from a trace, the signal baseline is first removed. Correctly

removing the baseline is key for accurately calculating the total charge deposited on a pad. This is

needed to accurately determine the energy deposited by a particle over the pads that make up its

track. The best method found for calculating the baseline is applying a moving average to the raw

31

0100200300400500020040060080010001200Raw traceBaseline corrected traceTime bucketAmplitude (ADC units)Parameter
baseline_window_scale (time buckets)
peak_separation (time buckets)
peak_prominence (ADC units)
peak_max_width (time buckets)
peak_threshold (ADC units)
low_accept (time buckets)
high_accept (time buckets)

GET Ion chamber
20
5
20
100
30
N/A
N/A

100
5
30
20
300
60
411

Table 3.1 Spyral point cloud phase parameters for the analysis of the experiment of this thesis.

trace, as pointed out by Bradt [20]. The idea is to perform a discrete convolution of the raw trace

with a rectangle function over its 512 time buckets,

( 𝑓 ∗ 𝑔) (𝑡) =

511
∑︁

𝜏=0

𝑓 (𝜏)𝑔(𝑡 − 𝜏)

(3.1)

where ( 𝑓 ∗ 𝑔)(𝑡) is the discrete convolution yielding the noise in the raw trace at time 𝑡.

𝑓 is the

raw trace of the pad and 𝑔 is the rectangle function divided by its length 𝑎

𝑔(𝑡) =

1
𝑎





1,

|𝑡| ≤ 1
𝑎

0, elsewhere.

(3.2)

𝑎 is an integer controlling how many time buckets are taken for averaging and was set to 20. The

convolution theorem allows the discrete convolution to be calculated as

( 𝑓 ∗ 𝑔) (𝑡) = F −1{F { 𝑓 }F {𝑔}}

(3.3)

where F is the Fourier transform. This has the benefit of being extremely fast due to the speed

of the fast Fourier transform. It is also convenient that the Fourier transform of 𝑔 is given by the

normalized sinc function provided by many fast Fourier transform libraries.

One correction is needed for this procedure to work. The peaks in the raw trace must be masked

before applying the convolution to avoid them influencing the baseline correction. Bradt masked

any time bucket with a value larger than 1.5 standard deviations of the average value of the raw

trace with the trace’s average. The same procedure is used in Spyral. After masking the peaks

and calculating the discrete convolution, the result is subtracted from the raw trace. This yields the

baseline-corrected trace (see Fig. 3.1).

32

3.2.2 Pulse extraction

The baseline-corrected trace is used to extract information related to the electrons that induced

a signal on a pad. Peaks in a trace correspond to signals induced by the electrons. A peak contains

two valuable pieces of information: its location in time and amplitude. A peak finding algorithm

is able to provide both of these pieces of information, and the scipy [21] find_peaks function was

used. The location is related to how long the electrons drifted in the AT-TPC, and its amplitude

is proportional to the number of electrons detected at that time. The location cannot be measured

to finer accuracy than a time bucket because a time bucket is an integrated sampling and not an

instantaneous measurement. To account for this uncertainty, the peak location was randomly chosen

within its measured time bucket. This implies an error taken as half a time bucket’s width

𝜎(𝑡) =

1
2

.

(3.4)

The trajectory of the ionizing particle lives in 3D Euclidean space, so each point in the point

cloud of an event is described by three position coordinates. The coordinate system used for this

analysis has the origin at the center of the AT-TPC window with the positive z-direction towards

the pad plane. Two coordinates are immediately known from the pad the trace belongs to. The

pad plane of the AT-TPC is completely mapped out and each pad is assigned an x- and y-position

according to its centroid. The uncertainty in the xy-position can be characterized by the size of

the pads. The AT-TPC pad plane is composed of triangles of two different sizes (see Section 1.3).

The error is estimated by circumscribing each triangle with a rectangle. The error in the the x-

and y-directions is simply half the length of the respective side of the circumscribed rectangle.

Explicitly:

0.0025,

𝜎(𝑥) =




where the standard deviations are in meters.

0.0050, big pad

small pad

,

𝜎(𝑦) =





0.0029,

small pad

0.0058, big pad

(3.5)

The z-coordinate is calculated from the location in time of a peak. Combining the peak time

bucket 𝑡 with the drift velocity 𝑣𝑒 of the electrons (in units of meters per time bucket) allows the

33

z-coordinate 𝑧 (in units of meters) to be reconstructed via

𝑧 = 𝑣𝑒 (𝑤𝑒𝑑𝑔𝑒 − 𝑡)

(3.6)

where 𝑤𝑒𝑑𝑔𝑒 is the time bucket of the AT-TPC window in the GET electronics. Eq. 3.6 has 𝑧 = 0

at 𝑡 = 𝑤𝑒𝑑𝑔𝑒 as required by our coordinate system. Section 3.2.3 will explain how 𝑣𝑒 and 𝑤𝑒𝑑𝑔𝑒 are

calculated, but in theory the point cloud can now be fully constructed with each point consisting of

x- and y-coordinates given by the pad that detected the electrons and the z-coordinate by Eq. 3.6.

Fig. 3.2 shows a point cloud of an event from the experiment.

Figure 3.2 Point cloud of an event from the experiment. The spiral track was identified as originating
from a proton.

3.2.3 Electron drift velocity measurement

Construction of the point cloud requires knowledge of the electron drift velocity, as seen in Eq.

3.6. Ideally, the drift velocity would be experimentally measured continuously over the course of

34

2k4k6k8k10k12k14k16k18kCharge (ADC units)an experiment with a laser or some other device. This experiment did not have such a device, so

the drift velocity was calculated on a run-per-run basis using unreacted beam events.

Fig. 3.3 shows the reconstructed beam-region mesh signal of an unreacted beam event. The

Figure 3.3 Reconstructed beam-region mesh signal from an unreacted beam event. The micromegas
edge near 60 time buckets is from liberated electrons near the MPGDs and the window edge near
400 time buckets is from liberated electrons near the window.

leftmost edge corresponds to the micromegas edge 𝑚𝑒𝑑𝑔𝑒 and the rightmost edge to the window

edge 𝑤𝑒𝑑𝑔𝑒. The micromegas edge is created by the detection of liberated electrons immediately

above the stack of MPGDs. These electrons drift the shortest distance to the MPGDs and hence are

detected first. The window edge corresponds to the detection of electrons from near the window

of the AT-TPC and are detected last. 𝑚𝑒𝑑𝑔𝑒 and 𝑤𝑒𝑑𝑔𝑒 are the temporal locations of the physical

extremities of the AT-TPC. Subtracting them yields the time it took for electrons to drift the entirety

of the detector. The drift velocity (in units of meters per time bucket) is then

𝑣𝑒 =

𝑙
𝑤𝑒𝑑𝑔𝑒 − 𝑚𝑒𝑑𝑔𝑒

35

(3.7)

010020030040050024002600280030003200Time bucketAmplitude (ADC units)where 𝑙 is the length of the AT-TPC (in units of meters).

𝑚𝑒𝑑𝑔𝑒 and 𝑤𝑒𝑑𝑔𝑒 were found for multiple unreacted beam events in each run. The unreacted

beam events were gated to ensure that only one beam particle was present and that the beam particle

ionized the gas along its entire trajectory through the AT-TPC. These requirements helped ensure

a reconstructed beam-region signal like that of Fig. 3.3. The average of both edges for each run

were calculated. The errors on the average micromegas edge, 𝜎(𝑚), and window edge, 𝜎(𝑤), were

respectively found by the standard error of their mean

𝜎(𝑚/𝑤) =

𝑠
√
𝑛

(3.8)

where 𝑛 is the number of unreacted beam events and 𝑠 is their sample standard deviation. Fig. 3.4

shows the drift velocity, converted to units of cm/𝜇s, as a function of the run over the course of the

experiment.

Figure 3.4 Drift velocity of experimental runs used in the analysis.

36

1101200.910.920.930.940.95175200225250275300325350375Run numberDrift velocity (cm/s)3.2.4 GET time correction

Despite the electronics for each pad being theoretically the same, their response functions differ

due to known bugs in the CoBo firmware. Thus, if two signals are induced at exactly the same time

on two different pads, the signals are not guaranteed to peak at the same time bucket. This leads to

systematic offsets in the calculated z-positions of points determined from each pad that need to be

corrected.

To characterize the time correction of each pad, a pulse generator was used to pulse the

micromegas mesh. Each pulse uniformly induced a signal on each pad, thus any difference in the

time bucket of the signal’s peak between pads was due to their dissimilar responses. Multiple runs

were taken each with a different voltage ranging from 100-500 mV. The data from these pulser runs

had the same format as the real data and their point clouds were created using the same Spyral

analysis except that peak_max_width was set to 300 and peak_threshold was set to 100 in the GET

parameters (see Table 3.1). For each event in a pulser run, the time bucket of the earliest peak in

each pad’s trace was recorded and its average 𝑡 𝑝𝑎𝑑 calculated. The error on 𝑡 𝑝𝑎𝑑, 𝜎(𝑡 𝑝𝑎𝑑), for each

pad was found from the standard error of its mean (see Eq. 3.8). The average time bucket of the

earliest peak in that pulser run, 𝑡𝑎𝑣𝑔, was found from the inverse-variance weighted average of all

𝑡 𝑝𝑎𝑑. Its error, 𝜎(𝑡𝑎𝑣𝑔), is given by the standard error of the weighted mean

𝜎(𝑡𝑎𝑣𝑔) =

√︄

1
(cid:205)𝑖 𝜎𝑖 (𝑡 𝑝𝑎𝑑)−2

(3.9)

where the summation is taken over all pads. The time correction factor, 𝑐 𝑝𝑎𝑑, of each pad in a run

is then

𝑐 𝑝𝑎𝑑 = 𝑡𝑎𝑣𝑔 − 𝑡 𝑝𝑎𝑑

with an error, 𝜎(𝑐 𝑝𝑎𝑑), given from the standard error propagation formula as

𝜎(𝑐 𝑝𝑎𝑑) =

√︃

𝜎(𝑡 𝑝𝑎𝑑)2 + 𝜎(𝑡𝑎𝑣𝑔)2.

(3.10)

(3.11)

The time correction factor of each pad was found for each pulser voltage. Fig. 3.5 shows the

time correction factor for a pad as a function of its average signal amplitude from various pulser

37

Figure 3.5 Time correction factor for pad 4632 as a function of its average signal amplitude for
various pulser voltages. The errors on the average signal amplitudes are shown but are very small.
Within error, the correction factor is well described by a constant value.

voltages. It is not plotted as a function of the pulser voltage. At higher voltages the pads saturate

their analog circuits and ADCs, producing distortions in their electronic responses; saturation is

only evident in a pad’s signal amplitude. There is an error associated with the calculated average

signal amplitude of a pad for each voltage, but they were found to be smaller than 1% of the signal

amplitude on average and are negligible. Fig. 3.5 is well described by a linear fit with a very small

slope. The final time correction factor used for each pad was the the y-intercept of its linear fit.

3.2.5 Electric field correction

Due to a limitation of the maximum high voltage available on the MPGD high-voltage supply,

the electric field was non-uniform, altering the drift paths of the liberated electrons close to the outer

diameter of the field cage. Specifically, the voltage on the final ring of the field cage, noted in Table

2.1, was too low, which created a different voltage gradient between it and the top M-THGEM. The

net effect of this non-uniformity was that the reconstructed trajectories may not have accurately

represented the true paths of the ionizing particles in this region.

38

50010001500200025003000Signal amplitude (ADC units)0.160.180.200.220.24Time correction (time buckets)This non-uniformity of the electric field was corrected by performing a simple garfield++ [22]

calculation. The electric field was modeled with the rings of the field cage and its volume filled

with deuterium gas. The field cage was subject to the same magnetic field as in the experiment. The

inner volume of the field cage was meshed and stationary electrons were placed at each point. The

electrons were drifted through the gas by the electromagnetic fields and their intersections with the

AT-TPC pad plane recorded. This allowed for the creation of a mapping between initial and final

electron positions, which was used to correct the electric field non-uniformity. To avoid a large and

very fine mesh, this mapping was interpolated upon to estimate the correction for arbitrary points.

The electric field correction grows with increasing distance from the center of the AT-TPC and

was found to minimally impact the reconstructed proton tracks due to their low magnetic rigidity.

It heavily modified the deuteron tracks, though. Deuterons have twice the magnetic rigidity of

protons of the same velocity and can reach the fringes of the AT-TPC electric field with much less

energy. Figs. 3.6 and 3.7 show the effect of the correction on tracks identified as created by a

proton and deuteron, respectively.

Figure 3.6 Track identified as originating from a proton shown without the electric field correction
(blue circles) and with it (red stars).

39

−300−200−1000100200300−300−200−100010020030002004006008001000−300−200−100010020030002004006008001000−300−200−1000100200300X (mm)Z (mm)Z (mm)Y (mm)X (mm)Y (mm)XY ProjectionXZ ProjectionYZ ProjectionFigure 3.7 Track identified as originating from a deuteron shown without the electric field correction
(blue circles) and with it (red stars).

3.3 Clustering tracks

Reconstructing the point cloud for an event is not sufficient to analyze it. Multiple particle

trajectories can be recorded, as well as noise. Points must then be grouped together, or clustered,

into tracks based on the particle trajectory they belong to. This section details this process as done

by the cluster phase of Spyral. The parameters for this phase used in this analysis are shown in

Table 3.2.

Pre-clustering min_cloud_size

Parameter

Clustering

Merging

Cleaning

min_points
min_size_scale_factor
min_size_lower_cutoff
cluster_selection_epsilon
min_cluster_size_join
circle_overlap_ratio
outlier_scale_factor

Value
20
3
0.05
10
10
15
0.25
0.05

Table 3.2 Spyral clustering phase parameters used for the analysis of the experiment of this thesis.

40

−300−200−1000100200300−300−200−100010020030002004006008001000−300−200−100010020030002004006008001000−300−200−1000100200300X (mm)Z (mm)Z (mm)Y (mm)X (mm)Y (mm)XY ProjectionXZ ProjectionYZ Projection3.3.1 Pre-clustering

A point cloud must meet certain conditions and undergo some dimensional scaling before being

clustered. To begin, illegal points are removed from the cloud. These points are outside of the

active volume of the AT-TPC and can occur because the AT-TPC time recording window is longer

than the maximum drift time of the electrons. After removing the illegal points, a point cloud

must contain a minimum number of points, which, for this experiment, was chosen to be 20. This

is to prevent noisy events with small numbers of points and/or extremely short tracks from being

analyzed. These types of events typically arise when a beam particle scatters outside the beam

region (“leaky beam” events), or when a target-like recoil has just enough energy to emerge from

the beam region and stops not far from it (e.g. elastic scattering close to 90°).

The clustering algorithm used (described in Section 3.3.2) clusters points based on a distance

metric. However, there is a very large physical asymmetry in the AT-TPC: the length of the

cylindrical active volume is about twice its diameter! The z-coordinate can take on a much larger

range of values than the x- and y-coordinates. For spatial clustering, we want our data to be unified

to the same range to prevent biasing. This is accomplished by normalizing the z-coordinate of the

points in the cloud to the diameter of the AT-TPC.

3.3.2 HDBSCAN clustering

After pre-clustering, a point cloud is clustered using the Hierarchical Density-Based Spatial

Clustering of Applications with Noise (HDBSCAN) [23] algorithm. As the name suggests, HDB-

SCAN clusters the points in the cloud based on their densities. Instead of clustering based on the

Euclidean distance between points, it defines a new metric called the mutual reachability distance.

This metric is beneficial because it is chosen to push sparsely populated areas that more likely

correspond to noise away from more densely populated areas that more likely correspond to real

data [24]. After calculating the mutual reachability distance for all combinations of points in the

cloud, HDBSCAN finds its clusters that can be of varying local densities. This feature is needed

for the AT-TPC because the pad plane is not made of uniformly sized pads. The details of the

HDBSCAN algorithm can be found in Refs. [23, 24].

41

HDBSCAN in Spyral is controlled by four parameters listed in the “Clustering” section of

Table 3.2. minimum_points is related to how conservative the clustering is, with larger values

resulting in fewer clusters in only the densest areas [24]. minimum_size_scale_factor and min-

imum_size_lower_cutoff determine the minimum required number of points to form a cluster,

denoted as 𝑚𝑐𝑙𝑆𝑖𝑧𝑒. Because the number of points in an AT-TPC cloud, 𝑛𝑝𝑜𝑖𝑛𝑡𝑠, can wildly vary, this

minimum is determined on a per cloud basis as

𝑚𝑐𝑙𝑆𝑖𝑧𝑒 = max{minimum_size_scale_factor ∗ 𝑛𝑝𝑜𝑖𝑛𝑡𝑠, minimum_size_lower_cutoff}.

(3.12)

cluster_selection_epsilon sets a minimum distance between two clusters for them to be considered

unique.

If the two clusters are within this distance, they are merged. This parameter is useful

because it allows for 𝑚𝑐𝑙𝑆𝑖𝑧𝑒 to be small but prevents HDBSCAN from breaking up areas with high

densities of points into many small clusters [24]. Fig. 3.8 shows the clustering of a track produced

by HDBSCAN.

Figure 3.8 HDBSCAN clustering of the point cloud shown in Fig. 3.2. Black points are noise while
blue and orange points belong to two distinct clusters.

42

3.3.3 Merging clusters

A common occurrence with clustering AT-TPC point clouds is that a particle track will be

broken up into smaller clusters instead of being correctly identified as a single cluster. This can

occur because some tracks travel through the beam region that records highly suppressed signals.

HDBSCAN sometimes will also fragment a single track. The clustered fragments of a single

particle track must then be merged together into a single cluster.

The particle tracks in the AT-TPC curve due to the magnetic field of the solenoid. The

projections of the track fragments on the plane perpendicular to the beam (the sensor plane) should

then appear circular and all be concentric, sharing a common center point. Each fragment can be fit

with a circle and if the centers of two fragments are close, they are merged together. This method

works well but introduces an arbitrary parameter for how close two fragment centers must be that

depends on the geometry of the track, i.e. this method is scale dependent [25]. A scale independent

method is found by merging two clusters if the overlap area of their circle fits is greater than a set

percentage of the smaller circle’s area, which was chosen to be 0.25 for this analysis. However, to

avoid merging small clusters of noise with an actual track, a minimum number of points must be

present in a cluster. 15 points were required for a cluster to be able to merge in the analysis.

3.3.4 Cluster cleanup

The final operation applied to the clusters is the scikit-learn [26] LocalOutlierFactor function.

This function compares “the local density of a sample to the local densities of its neighbors” to

identify outliers in a cluster [26]. It is the last attempt at removing noise from a cluster. A scale

factor, outlier_scale_factor, is used for determining how many neighbors, 𝑛𝑛𝑒𝑖𝑔ℎ, are compared to

in the calculation due to the fact that the number of points in an AT-TPC cluster can wildly vary.

𝑛𝑛𝑒𝑖𝑔ℎ is calculated as

𝑛𝑛𝑒𝑖𝑔ℎ = max{outlier_scale_factor ∗ 𝑛𝑝𝑜𝑖𝑛𝑡𝑠, 2}

(3.13)

where 𝑛𝑝𝑜𝑖𝑛𝑡𝑠 is the number of points in the cluster. The identified outliers are removed from the

cluster. Fig. 3.9 shows the final clustering of the point cloud from Fig. 3.2 after the merging and

cleanup steps were performed on its initial HDBSCAN clustering shown in Fig. 3.8.

43

Figure 3.9 Final clustering of the point cloud from Fig. 3.2 after the initial clustering in Fig. 3.8
underwent merging and cleanup. All points belong to the same singular cluster.

3.4 Estimating clustered track kinematic parameters

After an event has been clustered, the kinematic parameters of its tracks are estimated. These

estimations will be feed to an optimizer afterwards for refinement, described in Section 3.6. This

section describes the process done by the Spyral estimation phase and the parameters used are

shown in Table 3.3.

Parameter
min_total_trajectory_points
smoothing_factor

Value
20
100.0

Table 3.3 Spyral estimation phase parameters used for the analysis of the experiment of this thesis.

As the reader goes through this section, there may be points in the following methodology

that they believe are not completely correct. For example, they might feel that an assumption is

made unjustly or conceive of an alternative method. Remember, what is done here is merely an

estimation. The hope is that no matter what method is used to guess the kinematic parameters of a

clustered track, they are close enough to their real values for the optimizer to converge to the lowest

44

value of the objective function.

3.4.1 Smoothing splines

As the clustered tracks currently stand, they are not ideal to work with. They are diffuse due to

the diffusion of charge and may have noise accidentally appended to them from poor clustering. The

net effect is that a clustered track depicts a blurry image of a particle’s trajectory. This is undesirable

for algorithms to estimate its kinematic parameters because they can be heavily influenced by non-

central and noisy points. To best estimate the actual path of the particle, its clustered track can

be smoothed using smoothing splines. The scipy [21] function make_smoothing_spline is used to

independently spline the x- and y-coordinates as a function of their z-coordinates. These splines

are then used to create a smoothed track by evaluating the splines at the z-coordinates of each point

in the original clustered track.

The splining can fail if the clustered track does not contain enough points or has multiple points

with the same z-coordinate. Such tracks are discarded. Importantly, the splined track is merely a

copy of the clustered track and does not replace it. The final optimization is done on the original

clustered track, which is what the AT-TPC actually records.

3.4.2 Vertex position

The first kinematic parameter to be extracted is a particle’s initial position. It is given by the

reaction vertex of its event. This thesis experiment could not measure the reaction vertex directly

because the signals of the beam region pads were highly suppressed. This means that the reaction

vertex had to be extrapolated from the characteristics of each track. The assumption is made that

the beam entered the AT-TPC perfectly straight and parallel to the z-axis (see Section 3.2.2 for an

explanation of the coordinate system used). This information can be used to guess whether the

particle’s splined track went forward or backward by calculating the distance of its two endpoints

to the z-axis; the endpoint with the smallest distance is taken as being closer to the vertex and thus

defines the track direction. The direction of the track is needed because it determines which points

in the cluster compose its first arc. The first arc of a splined track is defined as all points from its

endpoint closest to its vertex to its point farthest from the z-axis. The xy-projection of this first arc

45

is fit with a circle and the point on the fit closest to the z-axis gives the estimates for the x- and

y-coordinates of its vertex.

The z-coordinate of the vertex is estimated by applying a linear regression to a small segment

of the first arc. The regression includes either its first 10 points or half its total points, whichever

is larger, and is done in cylindrical coordinates. The distance 𝜌 of each point in the first arc from

the vertex is linearly fit as a function of the z-coordinate. The linear fit provides the slope 𝑚 and

intercept 𝑏 of the line defined by

𝜌 = 𝑚𝑧 + 𝑏.

The vertex has 𝜌 = 0 and its z-coordinate is given by

𝑧 =

−𝑏
𝑚

.

(3.14)

(3.15)

With the vertex fully estimated, its distance from the z-axis is calculated. If this distance is larger

than the beam region radius the splined track is discarded. Either the vertex algorithm failed or it

did not originate from a beam reaction.

A self-consistency check can be performed with the slope of the fit and the estimated direction

of the splined track. Recall that we have defined the coordinate system of the AT-TPC with the

window at 𝑧 = 0 m and the micromegas at 𝑧 = 1 m. This implies that 𝑚 > 0 corresponds to

forward-going tracks and 𝑚 < 0 to backward-going tracks. Failing this check sends the track

through the Spyral estimator again with the opposite direction.

3.4.3 Polar angle

The linear regression to find the z-coordinate of the vertex is also used to estimate the laboratory

polar angle 𝜃 of the splined track. 𝜃 is found from trigonometry as




In the highly unlikely situation that 𝑚 = 0, the splined track is discarded.

arctan (𝑚) + 𝜋, 𝑚 < 0.

arctan (𝑚),

𝑚 > 0

𝜃 =

(3.16)

46

3.4.4 Azimuthal angle

The azimuthal angle 𝜙 is estimated from the vertex and circle fit to the splined track’s first arc.

It can be found from the geometry shown in Fig. 3.10 and is given by the blue angle to the tangent

line of the circle fit at the estimated vertex. The dotted red radial line is perpendicular to the tangent

Figure 3.10 Geometry of the circle fit to the first arc of a splined track oriented with the beam
coming out of the page. The arrows on the circle show the direction the track curves due to the
magnetic field. The azimuthal angle shown in blue is measured to the tangent line to the circle at
the estimated vertex. The angle shown in red is given by Eq. 3.18 and is measured to the radial
line connecting the circle’s center and the estimated vertex.

line. With this knowledge, the angle in red, denoted as 𝛼, is related to the azimuthal angle by

𝛼 is found as

𝜙 =





𝛼 =





𝛼 < 3𝜋
𝛼 + 𝜋
2 ,
2
𝛼 − 3𝜋
2 , 𝛼 ≥ 3𝜋
2 .

𝛽,

𝛽 ≥ 0

𝛽 + 2𝜋, 𝛽 < 0

47

(3.17)

(3.18)

xCircle fit centeryEstimatedvertexwith

𝛽 = arctan2 (𝑦𝑣 − 𝑦𝑐, 𝑥𝑣 − 𝑥𝑐),

(3.19)

where (𝑥𝑐, 𝑦𝑐) and (𝑥𝑣, 𝑦𝑣) are the xy-coordinates of the circle fit center and the vertex, respectively.

3.4.5 Magnetic rigidity (kinetic energy)

Classically, a particle of mass 𝑚, charge 𝑞, velocity (cid:174)𝑣, and angle 𝜃 with respect to a magnetic

field of strength 𝐵 curves with a radius 𝜌. Its magnetic rigidity, 𝐵𝜌, is given by

𝐵𝜌 =

|(cid:174)𝑣|𝑚 sin 𝜃
|𝑞|

.

(3.20)

The Lorentz force used to derive Eq. 3.20 makes explicit that the particle curves in the plane

perpendicular to the magnetic field. This means that 𝜌 is measured in the plane parallel to the pad

plane for the AT-TPC. Although the AT-TPC does have an additional electric field, because it is

parallel to the magnetic field, Eq. 3.20 is still correct. For general orientations of magnetic and

electric fields Eq. 3.20 is not true.

Using the non-relativistic equation for kinetic energy 𝐸, we have

𝐵𝜌 =

√

2𝐸𝑚 sin 𝜃
|𝑞|

.

(3.21)

Since 𝐵𝜌 is proportional to the particle’s kinetic energy and the scattering angle gives the particle’s

initial direction of motion, these two quantities are equivalent to the particle’s initial velocity vector.

𝐵𝜌 is straightforward to estimate for a track. 𝐵 is an experimentally known parameter; only 𝜌 must

be extracted. 𝜌 is given by the radius of the circle fit to the first arc of the splined track.

3.4.6 Energy loss and particle identification (PID)

Measuring the particle’s initial position, polar angle, azimuthal angle, and magnetic rigidity

actually do not exhaust its kinematic variables. This is because its species is still unknown. This

information is encoded in its magnetic rigidity that depends on its mass and charge. However, neither

of these variables can be found without another relationship as the problem is underdetermined.

The well-known solution to determining the species of a particle whose kinetic energy is known

is to also measure its energy loss 𝑑𝐸/𝑑𝑥. This can be understood as follows. The energy loss of a

48

particle is given by the Bethe-Bloche formula, which for non-relativistic particles reduces to

𝑑𝐸
𝑑𝑥

∝

2

(cid:12)
(cid:12)
(cid:12)

𝑞
(cid:174)𝑣

(cid:12)
(cid:12)
(cid:12)

(3.22)

where 𝑞 is the charge and (cid:174)𝑣 the velocity of the particle [3]. Multiply this by the non-relativistic

kinetic energy 𝐸 to find

𝐸 ∗

𝑑𝐸
𝑑𝑥

∝ 𝑞2𝑚

(3.23)

where 𝑚 is the particle’s mass. As 𝑞 and 𝑚 are fixed for each nucleus, the right-hand side of Eq.

3.23 is constant. Plotting 𝐸 as a function of 𝑑𝐸/𝑑𝑥 gives hyperbolic curves as in Fig. 3.11.

Figure 3.11 Example particle identification plot constructed by plotting energy vs. energy loss.

The only hiccup with this approach is that the AT-TPC does not measure the kinetic energy of

a particle from its splined track, but its magnetic rigidity. Despite this, let us multiply 𝐵𝜌 by the

energy loss. This yields

𝐵𝜌 ∗

𝑑𝐸
𝑑𝑥

∝

(cid:12)
(cid:12)
(cid:12)

𝑞
(cid:174)𝑣

(cid:12)
(cid:12)
(cid:12)

𝑚 sin 𝜃.

(3.24)

Notice that the right-hand side would be constant if we removed the polar angle and velocity

dependencies. Taking the square root of Eq. 3.22 and plugging it in along with some rearranging

yields

𝐵𝜌
sin 𝜃

√︂ 𝑑𝐸
𝑑𝑥

∗

∝ 𝑚.

49

(3.25)

Plotting 𝐵𝜌/sin 𝜃 as a function of √︁𝑑𝐸/𝑑𝑥 will result in hyperbolic curves that correspond to

particles of the same mass. This is the particle identification plot used for this experiment (Fig.

3.12). Thus, to fully constrain the kinematics of a particle, its 𝑑𝐸/𝑑𝑥 must be estimated using its

Figure 3.12 PID plot constructed from a subset of this experiment’s data. The bottom-most band
is made of protons while the band above it is made of deuterons. The other bands correspond to
heavier species.

track.

The energy loss of a splined track is calculated by summing the deposited charge of each point

in its first arc as defined in Section 3.4.2. The only caveat is that points from large pads on the pad

plane are not included in the sum. The large pads have a different capacitance and collect more

charge due to their increased surface area, leading to a higher effective gain compared to the small

pads. The net effect is that large pads would not have the same weight in the sum as the small pads,

and this problem is avoided altogether by not including them. The length of the first arc up to its

50

020406080100dE/dx (arb. units)0.000.250.500.751.001.251.501.75B/sin (Tm)100101Countstransition to the large pads is estimated from a fine line integral using 1000 points generated from

the splined track’s smoothing splines. Its 𝑑𝐸/𝑑𝑥 is found by dividing the deposited charge along

the arc by this length.

3.5 Gain matching

While creating the PID plot for different runs of the experimental data, it became apparent that

the measured energy loss changed as seen in the PID plots of the two runs shown in Fig. 3.13.

This is due to the gain on the MPGDs changing, although it is not clear exactly why it did as their

experimental settings remained constant. One hypothesis is that impurities were introduced into

the deuterium gas through the gas-recycling system and outgassing. Regardless, this was corrected

by normalizing the energy loss of each run to one selected run, which was run 347 for this analysis.

The normalization factor for each run was determined using a scheme proposed by Joseph

Dopfer [27]. The deuteron band in the PID (Fig. 3.12) shows a hot spot for deuterons that

deposited the most charge near √︁𝑑𝐸/𝑑𝑥 = 80. Projecting the PID onto the √︁𝑑𝐸/𝑑𝑥 axis results in

a histogram with a peak in this region. The centroid of this peak can be found for each run and the

multiplicative factor to align it with that of the chosen run derived. The normalization factors were

found to range within 5 − 30% of the normalized value of 1.

3.6 Optimizing clustered track kinematic parameters

The estimated kinematic parameters from the estimation phase are fed into an optimizer for

refinement to find the combination of their values that produce a track closest to the one observed.

The optimizer produces solutions to the equations of motion of the particle in the AT-TPC and

determines the solution that matches a track the best via an objective function it attempts to

minimize. This is done by the interpolation phase of Spyral, whose parameters used in this

analysis are given in Table 3.4.

3.6.1 Equations of motion

Particle trajectories are generated by solving the equations of motion of the particle in the AT-

TPC. The relativistic equations of motion of a charged particle in the AT-TPC for this experiment

51

Parameter
ic_min_val (ADC units)
ic_max_val (ADC units)
n_time_steps
interp_ke_min (MeV)
interp_ke_max (MeV)
interp_ke_bins
interp_polar_min (°)
interp_polar_max (°)
interp_polar_bins

Value
Proton Deuteron

300
850
1300
0.01
40
800
0.1
179.9
500

300
850
1300
0.01
80
1600
0.1
89.9
250

Table 3.4 Spyral interpolation phase parameters used for the analysis of the experiment of this
thesis. The proton parameters were used for the transfer reaction while the deuteron parameters
were used for the elastic and inelastic reactions.

are given by the Langevin equation

𝑑 (cid:174)𝑝
𝑑𝑡

= 𝑞( (cid:174)𝐸 + (cid:174)𝑣 × (cid:174)𝐵) − (cid:174)𝜂(𝑡)

(3.26)

where (cid:174)𝐸 is the AT-TPC’s electric field, (cid:174)𝐵 the SOLARIS solenoid’s magnetic field, and 𝑞, (cid:174)𝑣, and

(cid:174)𝑝 are the particle’s charge, velocity, and three-momentum, respectively. (cid:174)𝜂(𝑡) is a time-dependent

damping force on the particle due to its energy loss through the target gas and was modeled using

the Python package pycatima [28] that interfaces with the well-known catima energy loss library.

Eq. 3.26 undergoes a large simplification as (cid:174)𝐸 and (cid:174)𝐵 each have only one component that is

anti-parallel to the incoming beam.

It is solved using a numerical ODE solver with the initial

kinematic parameter estimates found from the estimation phase in Section 3.4 and constrained to

the physical boundaries of the AT-TPC. The solver terminates when the linear distance the particle

would penetrate the target gas is 1 mm or less.

3.6.2

Interpolation mesh

Solving Eq. 3.26 for a particle in the AT-TPC is a well-posed problem, but how to do it

efficiently is not. Explicitly, there is a very large computational cost associated with optimizing

the solutions of an ODE. The optimizer will typically call the ODE solver many times while it is

searching for its minima, and numerical ODE solvers can be slow. The problem is exacerbated

by the fact that this experiment recorded terabytes of data. To analyze the data in any reasonable

52

amount of time required a different method for generating ODE solutions. Here we describe the

novel interpolation mesh method developed by Daniel Bazin.

To begin, we first recognize that two symmetries exist in subsets of solutions to Eq. 3.26.

Suppose a subset of all solutions is given by those that share the same species, kinetic energy,

scattering angle, and azimuthal angle in the AT-TPC. The trajectories of this subset must be the

same except that they begin at different vertices; they are translationally invariant. We now remove

the constraint that all trajectories have the same azimuthal angle. All trajectories of the resulting

subset are rotationally invariant in the azimuthal angle if they are first translated to the same vertex.

With these two observations, a drastic reduction in the space of solutions is achieved: solutions can

be generated at a fixed vertex with a fixed azimuthal angle and translated/rotated as needed. This

reduction in initial values needed to solve Eq. 3.26 strengthens the brute force option of simply

generating a set of many solutions that the optimizer will draw from to compare to the actual data.

The accuracy of this method heavily depends on the set size, and we would like it to be as large as

possible. We thus interpolate between members of this set to infer solutions that are not part of it,

effectively increasing its size.

For an input species and lists of kinetic energies and polar angles (the only two initial parameters

of importance), Spyral generates solutions to Eq. 3.26 for all their combinations evaluated at the

same time steps. These time steps are logarithmically spaced with more steps at the earlier times.

The x-, y-, and z-coordinates at each time step have their own interpolation meshes with the

particle’s kinetic energy and polar angle as their axes. Tracks are made by bilinearly interpolating

each position coordinate at each time step for the desired kinetic energy and polar angle.

3.6.3 Optimizer and objective function

A plethora of optimization algorithms exist, but, for the analysis of this thesis experiment, the

limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm with box constraints (L-BFGS-B)

[29] was chosen. L-BFGS-B was employed over a more traditional least-squares algorithm, like

Levenberg-Marquardt, because of its significantly greater computational speed.

The optimizer requires an objective function whose output value is minimized. For the analysis

53

of AT-TPC data, the objective function performs a comparison between the set of points 𝑇 of a

clustered track, with each point’s total positional error a member of set 𝜎(position), to the set of

points 𝐷 of a generated particle trajectory. Explicitly, the objective function is

𝑓 (𝑇, 𝐷, 𝜎(position)) =

(cid:32)

∑︁

𝑖

𝜎𝑖 (position)−1

(cid:33) −1

𝑑𝑚𝑖𝑛 (𝑇𝑗 , 𝐷)
𝜎𝑗 (position)

∑︁

𝑗

(3.27)

where 𝑇𝑗 is the 𝑗-th point of 𝑇 with total positional error 𝜎𝑗 (position) and 𝑑𝑚𝑖𝑛 (𝑇𝑗 , 𝐷) is the shortest

Euclidean distance between 𝑇𝑗 and some point in 𝐷.

The total positional error assigned to point 𝑗 is

𝜎𝑗 (position) =

√︃

𝜎(𝑥)2 + 𝜎(𝑦)2 + 𝜎𝑗 (𝑧)2.

(3.28)

The uncertainty in the x- and y-coordinates of the point are given by Eq. 3.5. The uncertainty in the

z-position is determined from propagating the errors in the point’s measured time bucket, window

edge time bucket, and micromegas edge time bucket through Eq. 3.6 with the drift velocity given

by Eq. 3.7. The error of a time bucket is given by Eq. 3.4 and that of the micromegas and window

edges by Eq. 3.8. The total error on the z-coordinate 𝜎𝑗 (𝑧) is then

𝜎𝑗 (𝑧) = 𝑙

√︄(cid:18)

1
𝑤𝑒𝑑𝑔𝑒 − 𝑚𝑒𝑑𝑔𝑒

(cid:19) 2

𝜎(𝑡)2 +

(cid:18)

𝑡 − 𝑚𝑒𝑑𝑔𝑒
(𝑤𝑒𝑑𝑔𝑒 − 𝑚𝑒𝑑𝑔𝑒)2
(cid:18)
𝑤𝑒𝑑𝑔𝑒 − 𝑡
(𝑤𝑒𝑑𝑔𝑒 − 𝑚𝑒𝑑𝑔𝑒)2

(cid:19) 2

𝜎(𝑤)2+

(3.29)

(cid:19) 2

𝜎(𝑚)2.

Fig. 3.14 shows an optimized track produced using the L-BFGS-B optimizer with tracks

generated from the interpolation mesh and the aforementioned objective function.

54

(a)

(b)

Figure 3.13 PID plots for run 302 (a) and run 347 (b). Notice the clear difference in the location
of the energy loss hot spot of the two deuteron bands. The gain of the MPGDs apparently changed
over the course of the experiment.

55

020406080100dE/dx (arb. units)0.000.250.500.751.001.251.501.75B/sin (Tm)100101Counts020406080100dE/dx (arb. units)0.000.250.500.751.001.251.501.75B/sin (Tm)100101CountsFigure 3.14 L-BFGS-B track optimization to the cluster shown in Fig. 3.9. The cluster consists of
the blue points, while the optimized trajectory is composed of the red points.

56

−0.15−0.1−0.0500.05−0.0500.050.10.150.20.480.50.520.540.560.580.60.62−0.15−0.1−0.0500.050.480.50.520.540.560.580.60.62−0.0500.050.10.150.2X (m)Z (m)Z (m)Y (m)X (m)Y (m)XY ProjectionXZ ProjectionYZ ProjectionCHAPTER 4

AT-TPC SIMULATIONS

The geometry of the AT-TPC imposes efficiency loses on the detection of specific trajectories

from certain reactions. For example, very forward or backward scattered particles will not escape

the beam region. The process of analyzing AT-TPC data is novel and complex, as described in

Chapter 3, and also introduces efficiency losses. To ensure that Spyral produces accurate results,

an AT-TPC simulation framework called attpc_engine [30] was developed by Gordon McCann

and the author in Python with the primary focus of extracting efficiency factors. attpc_engine is

divided into two halves: a kinematics generator and a detector simulator. This chapter discusses

the core operating principles of each while leaving out coding specifics.

4.1 Kinematics generator

attpc_engine is designed for two-body nuclear reactions with a stationary target where one

of the products can undergo a decay chain. Decays are modeled to happen instantaneously with

two nuclei in the exit channel, so three-body or more decays can only be modeled sequentially and

not democratically. attpc_engine randomly samples the kinematic phase space in the center-of-

momentum (CM) frame of the reaction and decay products.

The reaction vertex is first sampled before any kinematic parameters. The z-coordinate is

restricted to be within the length of the AT-TPC and is taken from a uniform distribution. The

kinetic energy of the beam nucleus is corrected by subtracting the energy lost traveling through the

gas to reach this z-coordinate using the Python package pycatima [28]. The x- and y-coordinates

are restricted to the beam emittance whose boundary is modeled as a circle. The xy-position is

found by normally sampling 𝑟 along the circle’s radius and uniformly sampling an angle Θ from

0 − 2𝜋. The x- and y-coordinates are then

𝑥 = 𝑟 cos Θ,

𝑦 = 𝑟 sin Θ.

(4.1)

(4.2)

The kinematic parameters are not all sampled for the same reaction or decay product. The

57

reaction and each decay only require one product be specified while the other is inferred from the

conservation of proton and neutron numbers. The input product of each step has the cosine of its

polar angle uniformly sampled to ensure the reaction is isotropic in 3D space. Its azimuthal angle

is uniformly sampled from 0 − 2𝜋 as well. The inferred product has its excitation energy sampled

from the input distribution, e.g. a Gaussian. These constitute all the sampled parameters for the

input reaction and decays.

Reaction and decay products cannot have any arbitrary combination of energies due to energy

conservation of the system. Because the excitation energy of products are sampled in attpc_engine,

it is possible that a reaction/decay is illegal by violating conservation of energy. For an 𝑛-body

reaction, the total CM energy 𝐸𝑐𝑚 must be greater than or equal to the sum of the product rest

energies. In attpc_engine we have the reaction 𝑎 + 𝑏 → 𝐴 + 𝐵 where 𝑎 is the projectile with

kinetic energy 𝐾𝑎 and 𝑏 is the stationary target. For this reaction

𝐸𝑐𝑚 ≥ (𝑚 𝐴 + 𝑚𝐵)𝑐2

(4.3)

where the 𝑚 denote rest masses and 𝑐 is the speed of light. 𝐸𝑐𝑚 can be calculated from the rest

masses of the reactants and the sampled kinetic energy of the beam as

𝐸𝑐𝑚 =

√︁

(𝑚𝑎 + 𝑚𝑏)2𝑐4 + 2𝑚𝑏𝑐2𝐾𝑎.

(4.4)

The energy condition of a decay requires the rest mass of the parent to be greater than or equal to

the sum of its daughters’ rest masses. Denoting the decay as 𝑐 → 𝐷 + 𝐸, the condition is

𝑚𝑐 ≥ 𝑚𝐷 + 𝑚𝐸 .

(4.5)

If the sampled excitation energies do not satisfy Eq. 4.3 and/or Eq. 4.5 for each appropriate step,

the reaction and decays are sampled again.

Assuming a valid reaction and decays were sampled, attpc_engine calculates the laboratory

four-momentum of each nucleus for all steps excluding the parent nuclei of a decay since they are

not part of the final products. The four-momentum of the target nucleus and beam nucleus are

trivial: the target is stationary and only has an energy component equal to is rest mass while the

58

beam nucleus has momentum only in the z-direction that is found from its input kinetic energy. If

𝐴 is the angle-sampled reaction product, it has a CM energy 𝐸 𝐴 given by

𝐸 𝐴 =

1
2𝐸𝑐𝑚

(𝑚2

𝐴𝑐4 − 𝑚2

𝐵𝑐4 + 𝐸 2

𝑐𝑚)

(4.6)

where 𝑚𝐵 includes the sampled excitation energy. The magnitude of 𝐴’s three-momentum |−→𝑝 𝐴| is

found using the energy-momentum relation and its components are

−→𝑝 𝐴 =

|−→𝑝 𝐴| sin 𝜃 cos 𝜙
|−→𝑝 𝐴| sin 𝜃 sin 𝜙
|−→𝑝 𝐴| cos 𝜃

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(4.7)

where 𝜃 and 𝜙 are its sampled polar and 𝜙 azimuthal angles, respectively. Together, eqs. 4.6 and

4.7 compose the four-momentum of 𝐴 in the CM frame. These equations also work for the angle-

sampled decay product 𝐷. The Python package vector [31] transforms both to the laboratory frame,

using the beam and target four-momenta for reaction products and the parent four-momentum for

decay products. vector then uses four-vector math to determine the laboratory four-momentum

of the other energy-sampled reaction/decay product. The kinematics generator writes the four-

momenta of the beam, target, and products along with the reaction vertex location to a dataset in

an output HDF5 file for each generated event.

4.2 Detector simulation

The detector simulation of attpc_engine takes an HDF5 file of generated events from the

kinematics generator and applies AT-TPC detector effects to them. The output is the simulated

point cloud of each generated event in the Spyral format for easy and guaranteed compatibility

with the Spyral analysis.

For a generated event, the trajectory of each nuclear product in the AT-TPC is determined by

solving the equations of motion given by Eq. 3.26 with initial conditions defined by the product’s

four-momentum. The track is constrained by the physical boundaries of the AT-TPC and has a

similar stopping condition to that described in Section 3.6.1. The energy lost at each point in the

trajectory must be calculated. The ODE solver returns the velocity (cid:174)𝑣 of the nucleus at a point 𝑛 in

59

its trajectory that is related to its kinetic energy 𝐸𝑛 via

where

𝐸𝑛 = 𝑚𝑐2(𝛾 − 1)

𝛾 =

1
√︁1 − (|(cid:174)𝑣|/𝑐)2

.

The energy lost at point 𝑛, Δ𝐸𝑛, is defined as

Δ𝐸𝑛 = 𝐸𝑛−1 − 𝐸𝑛.

(4.8)

(4.9)

(4.10)

The number of electrons made at point 𝑛 in the nucleus’ trajectory is found by sampling a

normal distribution with mean 𝜇 and standard deviation 𝜎 given by

𝜇 =

Δ𝐸𝑛
𝑤

,

𝜎 = √︁ 𝑓 𝜇

(4.11)

(4.12)

where 𝑤 is the W-value of the gas and 𝑓 is the Fano factor. The W-value is the average energy

needed to create an electron-ion pair in the gas, so 𝜇 is the expected number of electrons. The

formation of electrons by the nucleus losing energy is statistical and we should not expect exactly

𝜇 electrons to be created, hence the sampling of a normal distribution. It turns out the number of

electrons liberated deviates from purely statistical predictions [32]. The Fano factor attempts to

capture this behavior by modifying the statistically predicted standard deviation and depends on the

gas [32]. The number of electrons from the sampling is floored as only an integer amount can be

made. After sampling, the number of electrons made at each point is multiplied by the gain factor

of the MPGDs.

The electric field of the AT-TPC drifts the electrons made at point 𝑛 towards the pad plane. We

define the spatial coordinates of 𝑛, in meters, as (𝑥𝑛, 𝑦𝑛, 𝑧𝑛) and use the same coordinate system as

the analysis described in Section 3.2.2. Thus, 𝑧𝑛 is given relative to the window of the AT-TPC and

is converted to exact time buckets 𝑡 via

𝑡 =

𝑙 − 𝑧𝑛
𝑣𝑒

+ 𝑚𝑒𝑑𝑔𝑒

60

(4.13)

where 𝑙 is the length of the AT-TPC in meters, 𝑣𝑒 is the electron drift velocity in meters per time

bucket, and 𝑚𝑒𝑑𝑔𝑒 is the micromegas edge time bucket. attpc_engine determines the pad each

electron hits depending on the input ratio of the transverse diffusion to the charge mobility, denoted

as 𝑑𝑡, of the gas. If 𝑑𝑡 is zero, then all electrons made at point 𝑛 are projected onto the pad plane

with xy-coordinates given by (𝑥𝑛, 𝑦𝑛). If 𝑑𝑡 is not zero, the electrons made at point 𝑛 are smeared

onto the pad plane according to a discretized uniform 2D Gaussian distribution. Its mean is at

(𝑥𝑛, 𝑦𝑛) and its standard deviation is

𝜎𝑡 =

√︄2𝑑𝑡𝑣𝑒𝑡
| (cid:174)𝐸 |

(4.14)

where (cid:174)𝐸 is the electric field of the AT-TPC. Electrons that hit the beam pads are not recorded

because these pads were suppressed and not used in the experimental data. Regardless of the value

of 𝑑𝑡, the discrete time bucket of each electron recorded by GET is found by flooring Eq. 4.13.

Transporting the liberated electrons from all recorded nuclear products in the event to the pad

plane results in the raw trace for each pad. The raw trace of a pad consists of the number of

electrons that hit it in each of its 512 time buckets. Actual AT-TPC pad traces do not output

how many electrons a pad saw in each time bucket but rather their induced signal in ADC units.

Here an important choice has been made with attpc_engine. To faithfully create the point cloud

requires convolution of the raw trace with the GET response function and performing the point

cloud phase of Spyral on the result. This is not an attractive option. First, it requires explicitly

coupling attpc_engine to Spyral. Changes in Spyral might require changes in attpc_engine.

Second, the point cloud phase of Spyral is the slowest portion of the analysis pipeline. Hundreds

of thousands of events may need to be simulated to adequately explore the phase space of a reaction,

which is orders of magnitude more than the the number of events actually measured. For these

reasons, attpc_engine does not perform the full convolution and peak finding analysis of the point

cloud phase; instead, it estimates their results.

Every time bucket in a trace that sees electrons is made a point in the point cloud. Its z-position

is found from Eq. 3.6 using a randomly sampled time within its time bucket (see Section 3.2.2)

and its x- and y-coordinates are found from its pad’s centroid. The charge deposited at the point is

61

estimated from the amplitude and integral of the GET theoretical response function

𝑟 (𝑡) = 𝐴 exp

(cid:18)

−3

𝑡
𝜏

(cid:19) (cid:18) 𝑡
𝜏

(cid:19) 3

sin

(cid:19)

(cid:18) 𝑡
𝜏

(4.15)

where 𝐴 converts from electrons to ADC units and 𝜏 is the peaking time of the electronics [33]. 𝐴

is explicitly given by

𝐴 = 4095

𝑒𝑁𝑒
𝑔𝑎𝑚 𝑝

(4.16)

where 𝑒 is the electron charge, 𝑁𝑒 is the number of electrons in the time bucket, and 𝑔𝑎𝑚 𝑝 is the

GET amplifier gain in units of Coulombs. 𝜏, in units of time buckets, is found from

𝜏 = 𝑡𝑠ℎ𝑎 𝑝𝑖𝑛𝑔 · 𝑓𝑐𝑙𝑜𝑐𝑘 · 0.001

(4.17)

where 𝑡𝑠ℎ𝑎 𝑝𝑖𝑛𝑔 is the GET shaping time in ns and 𝑓𝑐𝑙𝑜𝑐𝑘 is the GET clock frequency in MHz. For

each point in an event’s cloud, attpc_engine writes the position, deposited charge, and any other

required Spyral attributes, like the size of a point’s pad, to a dataset in an HDF5 file. This is done

for each simulated event that has a point cloud consisting of at least one point.

4.3 Efficiency factors

attpc_engine was used to simultaneously correct efficiency losses due to the geometry of the

AT-TPC and the Spyral analysis. attpc_engine simulated 3 million 10Be(𝑑, 𝑑) events for both

the elastic and first inelastic states of 10Be and 1.5 million 10Be(𝑑, 𝑝) events for the first five states

of 11Be. The standard deviation of the normal distribution for sampling the xy-coordinates of the

reaction vertices was one-third of the beam region radius, which was 2 cm. Tables 4.1 and 4.2 list

the detector and electronics parameters used by attpc_engine, respectively.

The window and

micromegas edge time buckets chosen were the average of those across all the experimental runs.

For each event, the polar angle was simulated from 0 − 𝜋 in the CM frame. The excitation energy of

the Be species was sampled from a Gaussian distribution for particle-bound states and a relativistic

Breit-Wigner distribution for particle-unbound states.

The simulated point clouds were analyzed using Spyral with the same parameters as for the

experimental data given in Chapter 3. For each reaction channel, a histogram of the CM scattering

62

Parameter

length (m)
efield (V/m)
bfield (T)
mpgd_gain
diffusion (V)
fano_factor
w_value (eV)

Value
1
60,000
3
175,000
0
0.2
34

Table 4.1 Detector parameters used for the attpc_engine simulations of the experiment of this
thesis.

Parameter

clock_freq (MHz)
amp_gain (fC)
shaping time (ns)
micromegas_edge (time buckets)
windows_edge (time buckets)
adc_threshold (ADC units)

Value
3.125
900
1000
62
396
30

Table 4.2 Electronics parameters used for the attpc_engine simulations of the experiment of this
thesis.

angle of its Be isotopes was created from the Spyral analysis (see Section 5.4 for more details,

specifically on the calculation of 𝑅(𝜃𝐶 𝑀, Δ𝜃𝐶 𝑀) and the bounds of the excitation energy spectra).

Denote such a histogram by 𝑃 and its 𝑖-th bin by 𝑃(𝑖). Similarly, let 𝑆 represent the histogram

of simulated Be CM scattering angles and its 𝑖-th bin by 𝑆(𝑖). To be clear, 𝑆 is made from all

the attpc_engine simulated events produced from the kinematics generator, and 𝑃 are those that

then survived the detector effects of attpc_engine and the Spyral analysis. To compute the

efficiency factors, 𝑃 and 𝑆 both must have the same number of bins over the same angular range.

The efficiency factor of the 𝑖-th bin 𝑒(𝑖) is given by

𝑒(𝑖) =

𝑆(𝑖)
𝑃(𝑖)

.

(4.18)

For the efficiency factors to be accurate, 𝑃 must be made with the same gates used in the actual

analysis. These are detailed in Section 5.1. The error of each efficiency factor 𝜎(𝑒(𝑖)) results from

propagating the error of the counts in each histogram bin through Eq. 4.18. Thus,

𝜎(𝑒(𝑖)) = 𝑒(𝑖)

√︄

(cid:19) 2

(cid:18) 𝜎(𝑆(𝑖))
𝑆(𝑖)

+

(cid:18) 𝜎(𝑃(𝑖))
𝑃(𝑖)

(cid:19) 2

(4.19)

63

where 𝜎(𝑃(𝑖)) and 𝜎(𝑆(𝑖)) are the error in the number of counts in 𝑃(𝑖) and 𝑆(𝑖), respectively.

Figs. 4.1 and 4.2 show the relevant subsets of the inverse of the efficiency factors, or efficiencies,

for the states of 10Be and 11Be seen in this experiment, respectively. A dip in efficiency occurs for

Figure 4.1 Efficiency per bin for the angular distributions of the observed states in 10Be seen in this
experiment. They were found using attpc_engine simulations.

low CM scattering angles for all states, although at different places, due to the detected target-like

product becoming less energetic (this dip is off the graphs for the elastic and inelastic states of 10Be

as well as the ground and 0.32 MeV states of 11Be). Decreasing the target-like product’s energy

shrinks its phase space to leave the beam region and be detected. The angular threshold where

detection ceases depends on the reaction channel. For the 1.78 MeV state and above in 11Be, another

efficiency drop is seen for larger CM scattering angles. In these regions, the target-like product

scattered at or near 90° with a velocity perpendicular or nearly perpendicular to the magnetic

field, confining its motion to a plane. The pre-fitting process described in Section 3.4 fails for

such trajectories, namely the smoothing splines and vertex z-coordinate extraction from the linear

64

0.000.250.500.751.00Elastic1822263034384246505458CM (deg)0.000.250.500.751.003.37 MeVEfficiency (fraction)regression given by Eq. 3.14 as all the points in the trajectory have nearly the same z-coordinate.

Where the scattering angle of the target-like product approaches 90° depends solely on the reaction

channel.

4.4 CM scattering angle error

Accurate reconstruction of the CM scattering angle of the Be isotope, 𝜃𝐶 𝑀, is needed to create

its angular distribution (see Eq. 1.5). The attpc_engine calculations used for the efficiency factors

described in Section 4.3 were also used to calculate the error on the reconstruction of 𝜃𝐶 𝑀. Fig.

4.3 shows a 2D histogram of the reconstructed versus simulated 𝜃𝐶 𝑀 for the elastic scattering of

10Be off a deuteron. Despite a few regions of strange behavior, the reconstruction is generally good

but the width of the diagonal line is not constant; the error in 𝜃𝐶 𝑀 clearly evolves and degrades at

larger angles.

Using the language of Section 4.3, the error on 𝜃𝐶 𝑀 was calculated for the 𝑖-th bin of its

respective histogram 𝑃 and is denoted as 𝜎(𝜃𝐶 𝑀 (𝑖)). Recall that 𝑃 is the histogram of all the

analysis reconstructed 𝜃𝐶 𝑀. If 𝐶 (𝑖) is the set of the simulated 𝜃𝐶 𝑀 for all the reconstructed 𝜃𝐶 𝑀 in

𝑃(𝑖), then

𝜎(𝜃𝐶 𝑀 (𝑖)) = IQR(𝐶 (𝑖))/2

(4.20)

where IQR stands for interquartile range. Graphically, 𝜎(𝜃𝐶 𝑀 (𝑖)) is half the IQR of all the elements

in the horizontal band corresponding to the 𝑖-th bin of 𝑃 in Fig. 4.3. The IQR is a robust measure

of dispersion in a data set and is better than the standard deviation in this case because of large

outliers seen in Fig. 4.3 away from the diagonal.

65

Figure 4.2 Efficiency per bin for the angular distributions of the observed states in 11Be seen in this
experiment. They were found using attpc_engine simulations.

66

0.000.250.500.751.00G.S.0.000.250.500.751.000.32 MeV0.000.250.500.751.001.78 MeV0.000.250.500.751.002.65 MeV10131619222528313437CM (deg)0.000.250.500.751.003.40 MeVEfficiency (fraction)Figure 4.3 Spyral reconstructed 𝜃𝐶 𝑀 vs. attpc_engine simulated 𝜃𝐶 𝑀 of 10Be elastically scattering
off a deuteron. The same attpc_engine simulated data to determine the elastic scattering efficiency
factors was used. Note that there are clearly regions of the phase space where the analysis does
worse, such as simulated 𝜃𝐶 𝑀 near 20°. The width of the hot central band also changes as a function
of simulated 𝜃𝐶 𝑀, indicating that the resolution is not constant and degrades at larger angles.

67

020406080100120140160180Simulated CM (deg)020406080100120140160180Reconstructed CM (deg)100101102103CountsCHAPTER 5

EXPERIMENTAL RESULTS

This chapter derives and presents the results from the 10Be(𝑑, 𝑝) reaction measured by the AT-TPC.

It also discusses results from the 10Be(𝑑, 𝑑) reaction that was simultaneously recorded. Elastic

cross sections are well known and provide a benchmark to ensure the accuracy of the analysis.

They, along with inelastic cross sections, can also provide input for theoretical calculations used to

extract spectroscopic factors.

5.1 Gates

Only one gate was applied to the raw data. The vertex z-position was constrained to 0.004 m ≤

𝑧 ≤ 0.958 m. This range was chosen to exclude nuclear reactions from the two extremities of the

AT-TPC on the window and MPGDs.

Another natural gate to consider is on the minimized objective function value returned by the

optimizer for each clustered track, which is a figure of merit for the goodness of the minimization.

In the case of the Spyral analysis, this is the error-weighted average distance between the points

in the clustered track and the generated trajectory. However, the implementation of such a gate is

not straightforward. The first problem is determining the limits of a gate on the returned objective

function value. We have observed that seemingly small changes in its limits can result in large

differences in certain bins of the angular distributions for certain reaction channels. Another

problem is propagating this gate to the simulated data used for the determination of the efficiency

factors and error on 𝜃𝐶 𝑀. As expected, the distribution of minimized objective function values

from simulated data is different from that of real data. The values are also typically much smaller.

A transformation of this gate from the real to the simulated data would need to be devised, and it

is not obvious how to do so.

Still, the minimized objective function value does provide relevant information 1. If changing

the limits of its gate produces drastic differences for some bins in a reaction channel’s angular

distribution, this implies that the optimizer has difficulties in those regions for that channel and

1We will sidestep the issue of transforming the gate from the real to simulated data by assuming they are the same.

68

their points should be assigned a larger systematic error. This idea was applied to the angular

distributions in Section 5.4.

5.2 Kinematics

Figs. 5.1 and 5.2 are kinematic plots of the kinetic energy of the target-like product vs.

its

laboratory scattering angle for the 10Be(𝑑, 𝑝) and 10Be(𝑑, 𝑑) reactions, respectively. The scattering

Figure 5.1 Measured kinematics for the proton from the 10Be(𝑑, 𝑝) reaction in the laboratory frame.
The theoretical kinematic bands are overlaid for the ground, 1.78 MeV, and 3.40 MeV states. The
bands for the 0.32 MeV and 2.65 MeV states lie between those shown with significant overlap.

angle of the target-like product was found from its optimized trajectory while its kinetic energy 𝐾

was calculated from

𝐾 =

√︁( 𝑝𝑐)2 + (𝑚𝑐2)2 − 𝑚𝑐2

(5.1)

where 𝑚 is its rest mass, 𝑝 its momentum found using Eq. 3.20, and 𝑐 the speed of light. The

theoretical kinematic lines for some of the states are shown in the figures and are derived from the

relativistic conservation of energy and momentum for a two-body reaction. Notice that they are not

kinematic lines but bands. This is due to the variety of beam energies present in the AT-TPC that

come from the energy lost by the beam particle as it travels through the gas to its reaction vertex.

69

708090100110120130140150160 (deg)0246810Kinetic energy (MeV)G.S.1.78 MeV3.40 MeV100101CountsFigure 5.2 Measured kinematics for the deuteron from the 10Be(𝑑, 𝑑) reaction in the laboratory
frame. The theoretical kinematic bands are overlaid for the elastic and first inelastic channels.

5.3 Excitation spectra

The 10Be and 11Be excitation spectra were found using Eqs. 1.7-1.9 with the extracted kinematic

parameters and are shown in Figs. 5.3 and 5.4, respectively. pycatima corrected the beam energy

at the AT-TPC window by the energy lost to reach the vertex position. States are observed in 10Be

up to 7.3 MeV 2 and in 11Be up to 3.4 MeV (see Section 5.4 for a discussion about the higher-lying

states). The full width at half maximum (FWHM) of these states evolves as a function of 𝜃𝐶 𝑀, as

evident from the kinematic plots shown in Figs. 5.1 and 5.2 (see also the discussion in Section 4.4).

The evolution of the FWHM is a manifestation of the complex interplay between the geometric

shapes of the tracks and how their closest calculated trajectory is optimized by the analysis. The

dynamic range of the AT-TPC is incredibly large, recording nuclei from multiple reaction channels

with energies ranging from MeV to tens of MeV. Unsurprisingly, and unfortunately, not all regions

in this phase space have tracks as equally optimized. FWHM is not a useful metric for the AT-TPC.

2The 10Be spectrum was limited to 5.25 MeV because only the elastic and first inelastic states would be fit to
determine optical model parameters for reaction calculations as discussed in Section 5.5. However, higher-lying states
are observed and their kinematic lines can be seen in Fig. 5.2.

70

0102030405060708090 (deg)051015202530Kinetic energy (MeV)G.S.3.37 MeV100101102CountsFigure 5.3 10Be excitation spectrum from 18-60°, inclusively, in the CM frame. The individual fits
for each observed state and the total fit to the spectrum are shown, but their heavy overlap makes
them hard to distinguish. Indicated energies are from the centroids of the fit.

5.4 Angular distributions

Angular distributions were made using Eq. 1.5. The number of counts per angular bin,

𝑅(𝜃𝐶 𝑀, Δ𝜃𝐶 𝑀), were found by fitting the bin’s excitation spectrum and integrating the counts

contained within the state of interest. Figs. 5.5 and 5.6 show the fit to each angular bin for the

10Be(𝑑, 𝑑) and 10Be(𝑑, 𝑝) reactions, respectively.

The distributions were limited to the bins

shown in the figures as they contained the majority of the recorded statistics. The excitation energy

was restricted to −2.0 MeV ≤ 𝐸𝑥 ≤ 5.25 MeV for 10Be and −1.0 MeV ≤ 𝐸𝑥 ≤ 4.3 MeV for

11Be. Particle-bound states were fit with Gaussians while particle-unbound states were fit with

Voigts. The particle-bound states have a shape dominated by the Gaussian-like response of the

detector/analysis while the particle-unbound states have a shape made from the convolution of their

intrinsic Breit-Wigner shape with this Gaussian-like response. The excitation spectra for 10Be and

11Be in Figs. 5.3 and 5.4, respectively, show the total fits with the aforementioned profiles for

each state. For the 11Be spectrum, a Voigt term was added above the 3.40 MeV state to capture

71

0.04 MeV 0+3.56 MeV 2+Figure 5.4 11Be excitation spectrum from 19-31°, inclusively, in the CM frame. The individual
fits for each observed state, a term for states above 3.40 MeV, and the total fit to the spectrum are
shown. Indicated energies are from the centroids of the fit.

possible counts from the 3.89 MeV and 3.96 MeV states. However, because these states cannot be

separated by the resolution of the AT-TPC and lie near the edge of the experimental acceptance, no

conclusions are drawn from them.

For convenience, we will move units around in the denominator of Eq. 1.5 so that 𝐼 is the

number of 10Be beam particles per time and 𝑛 is the areal density of the target nuclei seen by the

beam. 𝐼 was found using unreacted beam events. The counts in the 10Be peak of Fig. 2.6 had an

IC multiplicity of one and were summed from 300-850 Δ𝐸. Note that both these same IC gates

were applied to the data during the optimization phase of the analysis described in Section 3.6 (the

IC amplitude gate was explicitly set in Table 3.4 while the IC multiplicity gate was implicitly set to

one in the Spyral analysis). The areal density of the gas 𝑛 was trivially determined from the ideal

gas law and the length of the AT-TPC active volume. It is apparent from Fig. 2.6 that the 10Be

and 10B peaks were not completely separated. To estimate the amount of 10B that leaked into the

10Be gate, these two peaks were fit with Gaussians and the one corresponding to 10B was integrated

72

-0.07 MeV 1/2+ 0.27 MeV 1/2-1.78 MeV 5/2+2.71 MeV 3/2-3.36 MeV 3/2(+/-)Figure 5.5 Total fit to each angular bin in the CM frame for the 10Be(𝑑, 𝑑) reaction. The fit includes
the elastic and first inelastic states of 10Be.

over the gated region. The leakage was less than 1.3% of the total summed 10Be counts and was

subtracted.

Figs. 5.7, 5.8, and 5.9 show the measured angular distributions for the indicated states of 10Be

and 11Be populated in this experiment with their efficiency factors applied (see Section 4.3).

The horizontal error bar of each bin results from the uncertainty in the reconstruction of 𝜃𝐶 𝑀 (see

Section 4.4). The vertical error results from propagating various uncertainties through Eq. 1.5

73

202402000400018.0°-20.0°202402000400020.0°-22.0°202401000200022.0°-24.0°20240800160024.0°-26.0°20240600120026.0°-28.0°2024040080028.0°-30.0°2024020040030.0°-32.0°2024015030032.0°-34.0°202406012034.0°-36.0°20240408036.0°-38.0°20240408038.0°-40.0°20240408040.0°-42.0°20240408042.0°-44.0°20240408044.0°-46.0°20240408046.0°-48.0°20240408048.0°-50.0°20240306050.0°-52.0°20240306052.0°-54.0°20240255054.0°-56.0°20240204056.0°-58.0°20240204058.0°-60.0°CM (deg)Counts / 112 keVFigure 5.6 Total fit to each angular bin in the CM frame for the 10Be(𝑑, 𝑝) reaction. The fit includes
the first five states of 11Be and a term for the 3.89 MeV and 3.96 MeV states.

using the error propagation formula. For each bin, these include the statistical uncertainty of its

counts along with the error from its efficiency factor given by Eq. 4.19. The error on the number of

beam particles 𝐼 was less than 1% and 𝐼 was so large that its contribution to the total vertical error

was negligible, thus it was safely ignored. The areal density of the target nuclei 𝑛 was calculated

using the ideal gas law and depended on the pressure and temperature of the AT-TPC active volume.

The pressure was kept constant via continuous feedback from the vacuum system and assigned no

74

10123402040608010012014016010.0°-13.0°10123405010015020025030035040013.0°-16.0°101234010020030040050016.0°-19.0°101234010020030040050019.0°-22.0°101234010020030040022.0°-25.0°10123405010015020025030035040025.0°-28.0°10123405010015020025030028.0°-31.0°10123405010015020025031.0°-34.0°10123402040608010012014016034.0°-37.0°CM (deg)Counts / 133 keVFigure 5.7 Measured angular distributions for the first four states of 11Be compared to DWBA
calculations. The calculations used the beam energy at the center of the AT-TPC and were scaled
by their spectroscopic factors. The orange solid lines use the DA1p OMP, the green dashed lines
use the An Cai OMP, and the red dotted lines use the OMP from the elastic fit.

error. The temperature was assumed to be room temperature at 293.15 K but was not measured, so

no error was assigned.

The vertical error bars also take into account the sensitivity of the bin to a gate on the minimized

objective function value 𝑑𝑎𝑣𝑔, although the final distributions included no such gate. This was done

to find regions of the distributions where the optimizer performs poorly (see Section 5.1). Fig.

75

0816G.S.010200.32 MeV040801.78 MeV01020304050CM (deg)08162.65 MeVd/d (mb / sr)Figure 5.8 Measured angular distribution for the 3.40 MeV state of 11Be compared to DWBA
calculations. The calculations used the beam energy at the center of the AT-TPC and were scaled
by their spectroscopic factors. The orange solid lines use the DA1p OMP, the green dotted lines use
the An Cai OMP, and the red dotted lines use the OMP from the elastic fit. The two different bands
correspond to either negative (ℓ = 1) or positive (ℓ = 2) parity. Because the points measured are
few and not near a minima, the parity cannot be definitely determined from the shape. However,
the reduced chi-squared values of the fits slightly favor a positive parity.

5.10 shows the histograms of 𝑑𝑎𝑣𝑔 for the target-like products from the 10Be(𝑑, 𝑑) and 10Be(𝑑, 𝑝)

reactions. The gate was set such that 𝑑𝑎𝑣𝑔 ≤ 10−4 m. The resulting gated angular distributions

were subtracted from the ungated distributions. Their differences were added in quadrature to the

other aforementioned errors.

The number of bins in each distribution is not the same. This is primarily due to the efficiency

of the detector not being constant for each reaction channel as illustrated in Figs. 4.1 and 4.2.

5.5 Extraction of spectroscopic factors

Spectroscopic factors were derived for the states of 11Be from their measured angular distribu-

tions and theoretical DWBA calculations with Eq. 1.12. The DWBA calculations were performed

with the finite-range code ptolemy [34] using the AV18 deuteron wave function [35]. These cal-

culations require the excited states of 11Be to be bound, which only its ground and 0.32 MeV states

76

ℓ=2ℓ=1Figure 5.9 Measured angular distributions for the elastic and first inelastic states of 10Be compared
to those measured for the same reaction by Schmitt et al. [1] with a 90 MeV 10Be beam.

are. The unbound states were given a -200 keV binding energy to extend these DWBA calculations

by a simple, but by no means completely correct, method. The Koning-Delaroche [36] global

optical model potential (OMP) was used for the outgoing channel. The DA1p [37] and An Cai

[38] OMPs were used for the incoming channel. Because the 10Be(𝑑, 𝑑) reaction was measured

simultaneously, the elastic and inelastic cross sections can also be used to extract an OMP. ptolemy

cannot do a simultaneous fit to both states, so Table 5.1 presents the optimized parameters fit to

only the elastic channel. These fits used the DA1p parameters as initial values. Fig. 5.11 compares

the measured elastic angular distribution to calculations, performed with ptolemy, using the three

OMPs for the incoming channel.

77

101102103ElasticPresentSchmitt et al.1923273135394347515559CM (deg)04812163.37 MeVd/d (mb / sr)Figure 5.10 Histograms of the minimized objective function value for the target-like product from
the 10Be(𝑑, 𝑑) reaction (top) and 10Be(𝑑, 𝑝) reaction (bottom). The red dotted line is where the
gate was made for the sensitivity analysis.

Beam energy
(MeV/u)
9.3 (window)
8.7 (midpoint)
8.1 (micromegas)

𝑉𝑣
(MeV)
98.977
101.380
95.550

𝑎𝑣
𝑟𝑣
(fm)
(fm)
2.107 0.863
2.299
0.703
2.135 0.789

𝑊𝑣
(MeV)
2.734
7.807
9.303

𝑟𝑤
(fm)
3.612
4.094
3.600

𝑎𝑤
(fm)
1.442
0.686
1.026

𝑊𝑠
(MeV)
3.659
3.304
3.233

𝑎𝑠
𝑟𝑠
(fm)
(fm)
4.428 0.417
3.792 0.914
3.305 0.958

Table 5.1 OMPs determined from fitting the elastic scattering angular distribution at different 10Be
beam energies corresponding to the indicated locations in the AT-TPC. There were no real nor
imaginary spin-orbit potentials and 𝑟𝑐 was fixed to 2.801 fm.

The beam energy for the DWBA calculations is complicated by the fact that the beam particle

lost energy traversing the large AT-TPC active volume to reach its reaction vertex. The 10Be beam

energy varied from 9.3 MeV/u at the entrance window to 8.1 MeV/u at the micromegas. This energy

difference affects the amplitude and shape of the calculated angular distributions, and the question

becomes what energy should be used. This is directly related to the fact that the AT-TPC actually

probes both the angular and energy dependencies of the cross section simultaneously. The best

way to handle the explicit energy dependence would be to the bin cross section in both angle and

78

10110210310410Be0.000.250.500.751.001.251.501.752.00Average distance (m)1e410010110210311BeCounts / 4e7 mFigure 5.11 Elastic angular distribution plotted alongside calculations using various OMPs, includ-
ing the DA1p, An Cai, and one fit to it. Each calculation took the beam energy at the middle of the
AT-TPC.

energy (or, equivalently, vertex z-position). This was not possible for this experiment because the

statistics were too low. Therefore, the angular distributions reported by this thesis include events

with beam energies from the entire range of the detector and are energy integrated. Thus, no beam

energy will be entirely correct for the extraction of spectroscopic factors.

With this important limitation understood, we make the compromise of taking the beam energy

at the center of the AT-TPC, 0.5 m deep, to capture the “average” behavior. If the cross section

changes slowly as a function of energy, we should expect that roughly half the events will be within

the first and last halves of the AT-TPC. To first order, this assumption is justified from the vertex

z-coordinate histograms created by scanning various 11Be excitation energy ranges shown in Fig.

5.12. To provide bounds on the spectroscopic factors, DWBA calculations were also performed

with the beam energies at the beginning and end of the detector. Thus, three DWBA calculations

for each state were fit to its measured angular distribution with a scaling factor (the spectroscopic

factor) that was optimized by a least-squares algorithm. Table 5.2 presents the spectroscopic factors

for the states of 11Be whose angular distributions were measured by this thesis experiment. The

79

202530354045505560CM (deg)101100101102103d/d (mb / sr)DA1pAn CaiFitExperimentFigure 5.12 Vertex z-coordinate histograms of events with the indicated ranges of 11Be excitation
energies. The gate on the vertex z-coordinate described in Section 5.1 is applied and the errors
shown are statistical. Overall, the distributions are generally flat with the total counts divided evenly
about the midpoint of the AT-TPC at 0.5 m.

DWBA calculations using the beam energy at the midpoint of the AT-TPC, with their spectroscopic

factors applied, are plotted on the angular distributions in Figs. 5.7 and 5.8.

The upper and lower bounds on the spectroscopic factors in Table 5.2 are generally the same for

the DA1p and An Cai OMPs. This results from the smooth evolution of these OMPs as a function

80

0.00.20.40.60.81.0010203040-1.00 MeV to -0.65 MeV0.00.20.40.60.81.00204060-0.65 MeV to -0.29 MeV0.00.20.40.60.81.0050100-0.29 MeV to 0.06 MeV0.00.20.40.60.81.001002000.06 MeV to 0.41 MeV0.00.20.40.60.81.00501000.41 MeV to 0.77 MeV0.00.20.40.60.81.002550751000.77 MeV to 1.12 MeV0.00.20.40.60.81.00501001.12 MeV to 1.47 MeV0.00.20.40.60.81.001002003004001.47 MeV to 1.83 MeV0.00.20.40.60.81.001002003004001.83 MeV to 2.18 MeV0.00.20.40.60.81.00501001502002.18 MeV to 2.53 MeV0.00.20.40.60.81.00501001502.53 MeV to 2.89 MeV0.00.20.40.60.81.00501001502.89 MeV to 3.24 MeV0.00.20.40.60.81.00501001503.24 MeV to 3.59 MeV0.00.20.40.60.81.00501003.59 MeV to 3.95 MeV0.00.20.40.60.81.00501003.95 MeV to 4.30 MeVVertex z-coordinate (m)Counts / 0.05 mE𝑥 (MeV)

ℓ

𝑛ℓ 𝑗

DA1p

An Cai

Elastic fit Average∗

0

0.32

1.78

2.65

3.40(−)

3.40(+)

0

1𝑠1/2
1 0𝑝1/2
0𝑑5/2
2
1 0𝑝3/2
0𝑝3/2
1
0𝑑3/2

2

0.85+0.12
−0.16
0.82+0.10
−0.10
0.87+0.08
−0.08
0.32+0.05
−0.05
0.51+0.08
−0.08
0.23+0.02
−0.02

0.99+0.14
−0.19
0.64+0.09
−0.09
0.68+0.09
−0.08
0.25+0.04
−0.04
0.39+0.07
−0.07
0.18+0.02
−0.02

0.98−0.30
−0.09
0.77+0.27
−0.07
0.84+0.16
−0.04
0.28+0.15
−0.03
0.44+0.27
−0.04
0.22+0.05
−0.01

0.92+0.13
−0.17
0.73+0.10
−0.09
0.78+0.08
−0.08
0.28+0.05
−0.05
0.45+0.08
−0.08
0.20+0.02
−0.02

Table 5.2 Spectroscopic factors extracted from the measured angular distributions using DWBA
calculations for neutrons with transferred angular momentum ℓ to the single-particle state 𝑛ℓ 𝑗. The
listed OMPs were used in the incoming channel. The reported spectroscopic factors used the beam
energy at the midpoint of the detector while their upper and lower limits used the beam energies at
∗The average column only includes the DA1p and An
the window and micromegas, respectively.
Cai OMPs. See the text for the details.

of the incident kinetic energy. However, a large asymmetry is observed for those using the elastic

fit OMP. The distributions are rapidly evolving as a function of the energy, perhaps indicative that

the fits do not describe the data well. As such, the averages listed in the table only include the DA1p

and An Cai OMPs.

5.6 Discussion of results

The average derived 11Be spectroscopic factors from Table 5.2 can be compared to those in

Table 5.3, which lists the factors found from other experiments and predicted from shell model

calculations. The lower range for the ground state factor agrees with the values in the literature

and all three shell-model interactions. Within the given range, the factor for the 0.32 MeV state

has good agreement with the literature, excluding the large value reported by Ref. [9], and each

interaction. The 1.78 MeV state’s factor is larger than the literature and FSU values, which are

consistent with each other, but close to the WBP and YSOX values. For the 2.65 MeV state, the

literature value and each interaction are consistent while the factor deduced from this experiment

is larger than all of them.

The spectroscopic factor for the 3.40 MeV state is complicated. Not enough points near a

minima were measured by the experiment to determine the parity from the shape of the distribution.

However, the reported reduced chi-squared values do slightly favor a positive parity when fitting the

81

E𝑥 (MeV)

𝐽 𝜋

𝑆𝑒𝑥 𝑝

0

0.32

1.78

2.65

3.40

1/2+

1/2−

5/2+

3/2−

3/2−

3/2+

0.72 ± 0.04 [1], 0.77 [9], 0.73 ± 0.06 [10]
0.62 ± 0.04 [1], 0.96 [9], 0.63 ± 0.15 [10]
0.58(8) [8], 0.50 [9]

≈ 0.12 [8]

≈ 0.05 [8]
0.10(1) [8]

𝑆𝑡ℎ
WBP YSOX FSU

0.74

0.67

0.71

0.14

0.77

0.72

0.70

0.11

0.79

0.67

0.56

0.14

0.02 < 0.01

0.02

0.11

0.16

0.02

Table 5.3 Spectroscopic factors from other experiments and those theoretically calculated using
shell-model interactions. The factors reported from [1] are an average of those found. The author
thanks Alex Brown for providing the WBP, Cenxi Yuan for the YSOX, and Rebeka Lubna for the
FSU calculations.

data with the DWBA calculations using the midpoint beam energy (see Fig. 5.8). Spectroscopic

factors were then derived for this state for each parity. Two observations are made about the

averages. First, the negative parity results in an order of magnitude larger factor when compared

to previous measurements and theory. Second, the factor for the positive parity is near both the

literature value and those predicted by the WBP and YSOX interactions. Both of these results,

while not as direct as a measurement of the shape of the angular distribution, are consistent with a

positive parity assignment to the 3.40 MeV state.

Overall, the average experimentally derived spectroscopic factors reasonably agree with the

values in the literature, excluding the 1.78 MeV state, and with the WBP and YSOX interactions.

Using these factors, the ESPE of the 0𝑑3/2 orbital in 11Be can be roughly estimated. Since only

one 3/2+ state is definitively observed up to 4 MeV, only a lower limit can be provided for the

ESPE. The assumption is made that the unobserved spectroscopic strength is in the 3.89 MeV state.

Performing the spectroscopic weighted average of the energies of the 3.40 MeV and 3.89 MeV

states yields 3.79−0.01

+0.01 MeV (the superscript used the spectroscopic factor at the window while the
subscript used the factor at the micromegas). This same procedure can be done for the ESPE of

the 0𝑑5/2 orbital producing 2.25−0.17

+0.17 MeV. Assuming that these ESPE lower limits are not too far
from their actual values, their difference yields an estimate of the strength of the spin-orbit splitting

82

between the 0𝑑 orbitals in 11Be, which is 1.54+0.16
−0.16

MeV.

This splitting can be compared to the 11Be isotone 13C. Most of the 13C 0𝑑5/2 strength resides

in its 3.85 MeV and 6.86 MeV states [39]. Considering only them, the spectroscopic factors of

Darden et al. yield a 0𝑑5/2 ESPE of 4.51 MeV [39]. Darden et al. also measured all the 0𝑑3/2

strength in the 8.40 MeV state for a 0𝑑3/2 ESPE of 8.40 MeV. The spin-orbit splitting of the 0𝑑
orbitals in 13C is found to be 3.89 MeV. Although this experiment could only measure ESPE lower

limits for the 0𝑑 orbitals, if we assume that these limits are near their true values, then the splitting

in 11Be appears much smaller. This might be evidence for a reduction of the 0𝑑 splitting in the

𝑁 = 7 isotones. A similar phenomenon has been reported for the 0𝑝 orbitals in the 𝑁 = 17, 19, 21

isotones [40].

83

CHAPTER 6

CONCLUSION

This thesis experiment successfully measured the 10Be(𝑑, 𝑝) reaction with the AT-TPC. Importantly,

it marks the detector’s first measurement of a transfer reaction in inverse kinematics. Angular

distributions for the first five states of 11Be were measured along with those from elastic and

inelastic scattering of 10Be on deuterons. This is a highlight of AT-TPC transfer reaction studies:

the incoming elastic and inelastic channels are recorded for free to constrain the OMP of the

incoming channel for reaction calculations.

Incidentally, this 10Be(𝑑, 𝑑) data also led to the

identification of a new 1− resonance at 7.27 MeV in 10Be [41]. It can be seen in Fig. 5.2 as the hot

band near 40° under 5 MeV.

Comparisons of the measured 11Be angular distributions with theoretical DWBA calculations

revealed nuclear structure details such as spectroscopic factors, which were compared to those in the

literature and calculations from various theoretical shell-model interactions. The derived average

factors generally agree with the literature and WBP and YSOX interactions. Despite the inability

to definitively assign a parity to the 3.40 MeV state in 11Be, evidence was produced in support

of a positive parity. This would place the 3.40 MeV state as the second member of a K𝑃 = 1/2+

rotational ground state band, as predicted by ab initio calculations from Caprio et al. [14], and a

lower limit of 3.79−0.01

+0.01 MeV on the 0𝑑3/2 ESPE. The simultaneous measurement of the majority of
the 0𝑑5/2 strength allowed the spin-orbit splitting of the 0𝑑 orbitals in 11Be to be estimated, which
was found to be less than its 𝑁 = 7 isotone 13C. This might be evidence for the evolution of the

spin-orbit splitting of the 0𝑑 orbitals in these isotones.

The success of this experiment has paved the way for a new era of AT-TPC transfer experiments.

Some of these have already been completed, such as 15C+𝑑 and 15C+𝑝 measurements at Argonne

National Laboratory, and some are yet to come. However, it also provides a means for future

improvements. Ultimately, the angular distribution of the 3.40 MeV state is too limited. A lack of

statistics cut off larger CM angles where the cross section is small. This cutoff was worsened by

the inability of the Spyral analysis to measure protons scattered near 90°. Small CM angles also

84

faced restrictions. The beam energy of this experiment was too low to allow for the protons at the

smaller CM angles to have sufficient kinetic energy to exit the beam region of the AT-TPC and be

detected. A higher beam energy would also have allowed for the population of higher-lying states

for a better determination of the ESPEs of the 0𝑑 orbitals.

Except for analyzing protons nearly perpendicular to the beam, the other two problems have

straightforward solutions. They would be fixed by increasing the requested beam time with a beam

of higher energy. Unfortunately, beam time is a valuable commodity in high demand. That said,

there may be an avenue to increase the effective statistics of the AT-TPC given the same beam time.

The AT-TPC was limited to events with only one incoming beam particle (see Section 2.3). This

resulted in a significant loss of 66% of all the recorded experimental data. Even reducing this loss

by a factor of two would have nearly doubled the amount of data used in this thesis experiment’s

analysis. The implementation of auxiliary detectors might be able to achieve this goal. Currently,

the use of a 0° silicon detector on the beam axis for this purpose is being investigated. An increase

in the statistics would also have allowed for a more accurate extraction of spectroscopic factors.

The cross sections could then be binned as a function of energy instead of having their energy

dependence integrated out (see Section 5.5).

11Be is a nucleus that exhibits some of the most interesting known phenomena in nuclei,

including a neutron halo ground state and level inversion. This nucleus is deserving of a more

advanced theoretical analysis than what has been provided in this thesis to capture these features.

Such an analysis should go beyond DWBA calculations to include deuteron breakup and continuum

effects, both of which can be modeled in adiabatic distorted wave approximation calculations. The

continuum effects can also be introduced in coupled channels Born approximation calculations that

fit both the measured elastic and inelastic distributions to determine the OMP for the incoming

channel, which ptolemy was unable to do. With the angular distributions measured by this thesis,

these advanced calculations can be done and are presently being explored by colleagues. The results

will be published in a future paper, along with more in-depth comparisons to ab initio no-core shell

model calculations.

85

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