SPECTROSCOPY OF 12 BE USING THE (7 LI, 7 BE) REACTION IN INVERSE
KINEMATICS
By
Rhiannon Meharchand

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics
2011

ABSTRACT
SPECTROSCOPY OF 12 BE USING THE (7 LI, 7 BE) REACTION IN
INVERSE KINEMATICS
By
Rhiannon Meharchand
The quenching of the N = 8 shell gap has been under investigation for more than 40 years,
yet there remains significant controversy over the extent of mixing between 0 ω and 2 ω
configurations in 12 Be. Recent neutron knockout and transfer measurements have helped
quantify 2 ω contributions to the ground and excited 0+ states. The current study measures
the 0 ω component of the 0+ wavefunctions for the first time.
The Gamow-Teller strength (B(GT)) of transitions from the ground state of 12 B to
the 0+ states in 12 Be has been measured using the (7 Li, 7 Be) charge-exchange reaction
in inverse kinematics. Since the ground state of 12 B is dominated by 0 ω configurations,
Gamow-Teller (∆L=0, ∆S=1 ) transitions to 0+ states in 12 Be selectively probe the 0 ω
component of the wavefunction.
The extracted B(GT) to the 0+ states in 12 Be were compared with shell model calculations, performed using the code OXBASH and employing a modified WBP interaction in
the spsdpf model space. To reproduce the observed B(GT) distribution, a 0 ω admixture of
25.48±5.49% is required in the ground state of 12 Be, and a 0 ω admixture of 59.83±4.78%
is required in the the 2.24 MeV 0+ state. These results are compared with the wavefunctions
extracted from neutron knockout and transfer measurements, and the theoretical predictions
of Fortune and Sherr, Barker, and Romero-Redondo et al.

To Natalia - dream big, work hard, and you can do ANYTHING.

iii

ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor, Remco Zegers. For providing encouragement and guidance over the past half-decade, and for being equal parts mentor and
supervisor. For never losing faith in me those first few years - had I had any other advisor, I’m not sure I would have made it through. For giving me ample opportunities to grow
personally and professionally, and for supporting my (many) extracurricular activities wholeheartedly. For suggesting projects and opportunities that kept me passionate about physics,
and for letting me see the world: Japan, Italy, and South Africa (thanks also to RCNP, the
Graduate School, ECT*, and AAPT for supplementary travel funds). Thanks also to the
National Science Foundation for financial support, and to B. Alex Brown, Alexandra Gade,
Norman Birge and Elizabeth Simmons, for serving on my guidance committee and sharing
your valuable time, expertise, and insight.
Thanking everyone who has helped me over the years is impossible, but I’m going to try.
Chronologically, it makes sense to start with those responsible for getting me to graduate
school in the first place. My parents, for their unwavering support and trust in my decisions
(even when they didn’t know what those decisions meant). My extended family, my earliest
role-models who taught me the value of education. The Caldwells, who took me into their
home and their hearts. The Meharchand-Quirt-Naidoo clan, a source of stability and support I am blessed to have in my life. My pre-graduate mentors and advisors: Coach Lock,
Kostas Kalogerakis, Philippe Collon, Sam Tabor, and Powell Barber.
Next, I’d like to thank those responsible for bringing me to MSU: the Physics and Astronomy department, especially Wolfgang Bauer, S.D. Mahanti, and Michael Thoennessen, for
iv

building a phenomenal graduate program; and the recruitment committee, especially Micha
Kilburn, and the non-official members who made MSU seem like the most fun place on earth
(even in the dead of winter), David (Floyd) Miller, Brian Martin, and Davitt Driscoll.
I’ve been fortunate to be surrounded by people who made my time here easy. Debbie
Barratt and the NSCL administrative assistants, who patiently answered all of my questions,
and took care of travel, reimbursement, and other administrative hurdles so that I could focus on my studies and research. The NSCL operators and staff, who worked around the clock
to get my thesis experiment up and running (twice). John Yurkon, who spent hours teaching me how to roll targets, and Jon Bonofiglio, the best maintenance group leader/tiramisu
maker/landlord ever. George Perdikakis and Dirk Weisshaar, who gave valuable technical
and career advice. The WaMPS 2009-2010 executive board, who put their time and heart
into the organization and made everything run smoothly.
When things weren’t so easy, it was the people at NSCL who made it all worth it. I have
to thank the graduate students and postdocs who moved on too soon: Wes Hitt, for setting a great example; Jeremy Armstrong, for being easily persuadable; Arnau Rios, for Olè,
London, and tomato bread; Carol Guess, for food, hiking, and all the advice; The Germans,
for being great sports and great hosts; the lunch crew, for the best hour of the day; Phil
Voss, for teaching me "my bad"; and Heather Crawford, an officemate for years but a friend
for life. Also the graduate students who make it hard to move on: Andrew Ratkiewicz, for
your loyal friendship and sympathetic ear; Shawna Valdez, for bringing your bright spirit to
the CEClub; Travis Baugher, for coffee, cherries, chocolate, and conversation; and too many
others to name. A special thanks to those young couples who make marriage seem merry:
Andrew and Kristen Klose, Karl and Kristen (soon to be) Smith, and Krista and Collin

v

Meierbachtol. I hope our friendships continue to grow as they have for the past two years.
Saving the best for last, there are two people who have been by my side through this
entire experience. My best friend from day one, Angelo Signoracci: resident structure expert,
personal tip calculator, and cured meat connoisseur. Thank you for being there every day my fondest memories of NSCL are the times I shared with you. And my ever patient pillar
of silent support, Tom Hutton: the man who makes me the luckiest girl in the world. I’m
not sure I will ever know all the things you did behind the scenes to make this achievement
possible but I know I could not have done it without you. Thank you.

vi

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Atomic Nucleus . . . . . . . . . . . . . . . . . . . . .
1.2 The Shell Model of Nuclear Structure . . . . . . . . . . . .
1.3 Charge-Exchange Reactions . . . . . . . . . . . . . . . . .
1.3.1 Proportionality Relationship . . . . . . . . . . . . .
1.3.2 History of Charge-Exchange Probes . . . . . . . . .
1.3.3 (7 Li, 7 Be) Charge-Exchange in Inverse Kinematics
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Motivation . . . . . . . . . . . . . . . . . .
2.1 Nuclear Structure Away from Stability
2.2 History of 12 Be . . . . . . . . . . . . .
2.3 12 B(7 Li, 7 Be)12 Be . . . . . . . . . . .

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3 Theory . . . . . . . . . . . . . . . . . . . .
3.1 Distorted Wave Born Approximation .
3.2 Theoretical Cross-Section Calculations
3.2.1 Transition Densities . . . . . . .
3.2.2 Form Factors . . . . . . . . . .
3.2.3 Cross-Sections . . . . . . . . . .
3.3 Extrapolation to q=0 . . . . . . . . . .
3.4 Tensor Contributions . . . . . . . . . .

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26
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4 Experiment . . . . . . . . . . . . . . .
4.1 Beam Preparation and Delivery . .
4.1.1 Ion Source . . . . . . . . . .
4.1.2 Coupled Cyclotrons . . . . .
4.1.3 A1900 Fragment Separator .
4.2 12 Be Production and Measurement
4.2.1 S800 Spectrograph . . . . .

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49
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65
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91

6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Multipole Decomposition Analysis . . . . . . . . . . . . . . .
6.1.1 Extraction of ∆L = 0 cross-section, Singles Data . . .
6.1.2 Extraction of ∆L = 0 cross-section, Coincidence Data
6.2 12 Be B(GT) Distribution . . . . . . . . . . . . . . . . . . . .
6.3 Comparison with Shell Model Predictions . . . . . . . . . . .
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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100
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113

4.2.2

4.2.1.1
4.2.1.2
4.2.1.3
Gamma
4.2.2.1

Dispersion Matching . . . .
Focal Plane Detectors . . .
Magnetic Field Corrections
Ray Detection . . . . . . . .
Barrel SeGA . . . . . . . .

5 Data Analysis . . . . . . . . . . . . . . . .
5.1 Calibrations . . . . . . . . . . . . . . .
5.1.1 S800 Calibrations . . . . . . . .
5.1.2 SeGA Calibrations . . . . . . .
5.1.2.1 Source Calibrations . .
5.1.2.2 In-Flight Calibrations
5.2 Particle Identification . . . . . . . . . .
5.3 Excitation Energy Reconstruction . . .
5.4 Cross-section Calculations . . . . . . .

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7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

viii

LIST OF TABLES

3.1

Angular momentum selection rules . . . . . . . . . . . . . . . . . . . . . . .

32

3.2

Optical potential parameters used to calculate distorted waves . . . . . . . .

33

3.3

Definitions for cases used in extrapolation to q=0 . . . . . . . . . . . . . . .

41

3.4

q=0 extrapolation scaling factors . . . . . . . . . . . . . . . . . . . . . . . .

42

3.5

Systematic error in ∆L=0 cross-section due to tensor interference . . . . . .

44

4.1

Properties of BicronT M BC-400 Scintillators . . . . . . . . . . . . . . . . . .

56

5.1

Mask calibration parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.2

Focal plane position and angle definitions . . . . . . . . . . . . . . . . . . . .

69

5.3

Gamma transitions used for energy and efficiency calibrations . . . . . . . .

70

5.4

PID background and charge-state removal gates . . . . . . . . . . . . . . . .

80

5.5

Energy and angular resolutions . . . . . . . . . . . . . . . . . . . . . . . . .

83

5.6

Angular distributions, singles data . . . . . . . . . . . . . . . . . . . . . . . .

98

5.7

Angular distributions, coincidence data . . . . . . . . . . . . . . . . . . . . .

98

6.1

DWBA multipole components included in MDA fits . . . . . . . . . . . . . . 102

6.2

MDA scaling factors, singles data . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3

∆L = 0 cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
ix

6.4

12 Be B(GT) distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.5

0 ω wavefunction percentages . . . . . . . . . . . . . . . . . . . . . . . . . . 114

x

LIST OF FIGURES

1.1

Differences in one-neutron separation energies . . . . . . . . . . . . . . . . .

3

1.2

Differences in one-proton separation energies . . . . . . . . . . . . . . . . . .

3

1.3

Energy of first 2+ state and B(E2) strength for even-even nuclei . . . . . . .

4

1.4

Neutron single-particle states . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.5

Transitions from the ground state of 7 Li to states in 7 Be . . . . . . . . . . .

14

2.1

Simplest ground-state configuration for 12 Be . . . . . . . . . . . . . . . . . .

17

2.2

Examples of 0 ω and 2 ω configurations in 12 Be . . . . . . . . . . . . . . .

19

2.3

Level scheme, 12 Be . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4

Calculated B(GT) distribution to 0+ states in 12 Be . . . . . . . . . . . . . .

25

3.1

DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(g.s., 0+ ), 7 Li(g.s.,
3/2− ) → 7 Be(429 keV, 1/2− ) transition . . . . . . . . . . . . . . . . . . . .

34

DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(2.11 MeV, 2+ ),
7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition . . . . . . . . . . . . . . .

35

DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(2.24 MeV, 0+ ),
7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition . . . . . . . . . . . . . . .

36

DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(2.68 MeV, 1− ),
7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition . . . . . . . . . . . . . . .

36

3.2

3.3

3.4

xi

DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(g.s., 0+ ), 7 Li(g.s.,
3/2− ) → 7 Be(g.s., 3/2− ) transition . . . . . . . . . . . . . . . . . . . . . . .

37

DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(2.11 MeV, 2+ ),
7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition . . . . . . . . . . . . . . . . . .

38

DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(2.24 MeV, 0+ ),
7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition . . . . . . . . . . . . . . . . . .

39

DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(2.68 MeV, 1− ),
7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition . . . . . . . . . . . . . . . . . .

40

Zero-degree ∆L=0 cross-section as a function of momentum transfer q . . . .

42

3.10 DWBA angular distributions for the 12 B(g.s., 1+ ) → 12 Be(g.s., 0+ ), 7 Li(g.s.,
3/2− ) → 7 Be(429 keV, 1/2− ) transition, without tensor . . . . . . . . . . .

44

4.1

Schematic of NSCL ion sources, cyclotrons and fragment separator . . . . . .

47

4.2

S800 Spectrograph including analysis line . . . . . . . . . . . . . . . . . . . .

50

4.3

Cartoon depiction of lateral dispersion-matching . . . . . . . . . . . . . . . .

52

4.4

Cathode-Readout Drift Counters (CRDCs) . . . . . . . . . . . . . . . . . . .

54

4.5

Barrel SeGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.1

CRDC mask pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.2

CRDC mask spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.3

SeGA absolute energy calibration . . . . . . . . . . . . . . . . . . . . . . . .

71

5.4

SeGA efficiency calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.5

Doppler-corrected gamma energy spectrum, gated on 11 Be . . . . . . . . . .

73

5.6

"Effective" vs "angle file" z-positions of SeGA segments . . . . . . . . . . . .

74

5.7

Difference between "effective" and "angle file" z-positions . . . . . . . . . . .

75

3.5

3.6

3.7

3.8

3.9

xii

5.8

Particle Identification (PID) plot . . . . . . . . . . . . . . . . . . . . . . . .

78

5.9

PID with master gate applied . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.10 Excitation energy spectrum, before and after angle corrections . . . . . . . .

84

5.11 "All ∆E" gate plotted as a function of 12 Be excitation energy . . . . . . . .

85

5.12 Illustration of background subtraction procedure . . . . . . . . . . . . . . . .

86

5.13 Final 12 Be excitation energy spectrum, coincidence and singles data . . . . .

87

5.14 Doppler-corrected gamma energy spectrum, gated on 12 Be . . . . . . . . . .

89

5.15 12 Be excitation energy spectrum, gamma ray gates applied . . . . . . . . . .

90

5.16 12 Be excitation energy, singles data, divided into angular bins . . . . . . . .

93

5.17 12 Be excitation energy, coincidence data, divided into angular bins

. . . . .

94

5.18 Illustration of peak-fitting procedure used to extract Nm . . . . . . . . . . .

97

5.19 Angular distributions, singles data . . . . . . . . . . . . . . . . . . . . . . . .

98

5.20 Angular distributions, coincidence data . . . . . . . . . . . . . . . . . . . . .

99

6.1

DWBA angular distributions for transitions to the ground state of 12 Be, for
the 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition, before and after resolution adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2

DWBA angular distributions for transitions to the ∼2 MeV state of 12 Be,
for the 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition, before and after
resolution adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3

DWBA angular distributions for transitions to the ground state of 12 Be, for
the 7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition, before and after resolution
adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4

DWBA angular distributions for transitions to the ∼2 MeV state of 12 Be, for
the 7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition, before and after resolution
adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
xiii

6.5

MDA fits, singles data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.6

MDA fits, coincidence data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.7

B(GT) ratio vs. 0 ω wavefunction percentage . . . . . . . . . . . . . . . . . 112

6.8

B(GT) ratio vs. 0 ω wavefunction ratio . . . . . . . . . . . . . . . . . . . . . 112

xiv

Chapter 1
Introduction

1.1

The Atomic Nucleus

The idea that everything in our universe can be described in terms of fundamental particles
was first developed by the Ancient Greeks and has persisted to the present day, evolving
into its current form through advances in technology and the contributions of visionary scientists. Atoms, thought indivisible by the earliest scholars, are now known to be made up
of negatively charged electrons orbiting a nucleus of positively charged protons and neutral
neutrons, where the nucleons in the nucleus are made up of quarks. The field of nuclear
physics is dedicated to understanding the structure and reactions of atomic nuclei, and in
the past century nuclear physicists have made significant strides in this direction. More than
3000 different isotopes have been discovered, and models explaining the behavior of the most
abundant, stable species have reached remarkable levels of accuracy. Yet there remains work
to be done - particularly with respect to rare isotopes, which only exist for short periods of
time before decaying into more stable species.
1

The two- and three-nucleon forces derived from stable isotope data have difficulty reproducing and predicting properties of rare isotopes, which indicates a lack of understanding
of the fundamental forces at play in nuclear systems. Fortunately, a revolution in rare isotope research is underway: scientists are studying more exotic isotopes than ever before,
producing data that can test and inform theoretical descriptions. The present study aims to
improve the understanding of the evolution of nuclear structure away from stability, using a
novel experimental technique to study the structure of the rare isotope Beryllium-12 (12 Be).

1.2

The Shell Model of Nuclear Structure

One of the foundations of chemistry is that the observed elemental properties can be welldescribed by a model in which the electrons of the atom are arranged in orbitals, or shells,
around the atomic nucleus. In nuclear physics an analogous model is used to describe
the structure and properties of isotopes - the Nuclear Shell Model. Developed in the midtwentieth century, the Nuclear Shell Model helped nuclear scientists explain the experimental
observations of the time - specifically, why it is more energetically expensive to remove a
nucleon from some nuclei than others. To illustrate this property, Figure 1.1 (1.2) plots
differences in in one-neutron (proton) separation energies ∆Sn (∆Sp ) as a function of neutron
number. Maxima in ∆S correspond to closed-shell configurations, occuring in nuclei with
2, 8, 20, 28, 50, 82 or 126 neutrons and/or protons. These are referred to as "magic" or
"semi-magic" nuclei. Other properties of "magic" nuclei are small ground-state quadrupole
moments, and for even-even nuclei, relatively high-lying first 2+ excited states and small
electric quadrupole transition strengths B(E2, 2+ → 0+ ) [1]. The last two properties are
shown in Figure 1.3.
2

∆ Sn (MeV)

6
4
2
0

0

50

100
Neutron Number

140

∆ Sp (MeV)

Figure 1.1: Differences in one-neutron separation energies as a function of neutron number.
Solid lines connect nuclei with constant proton number. The magic numbers are indicated
by dashed lines. Figure taken from Ref. [2].

8
6
4
2
0

0

50
Proton Number

100

Figure 1.2: Differences in one-proton separation energies as a function of proton number.
Solid lines connect nuclei with constant neutron number. The magic numbers are indicated
by dashed lines. Figure taken from Ref. [2].

The simplest form of the Nuclear Shell Model is the Independent Particle Model, in
which the behavior of one nucleon in the nucleus is modeled as a single particle inside an
average potential, created by the mean field of all the other nucleons. A typical potential
consists of the Coulomb potential, a central term, and a spin-orbit term
V = Vc (r) + Vo (r) + Vso (r) l · s
3

.

(1.1)

Energy of 21+ (MeV)

10

1
1

0.1

0

20

40

60

100
80
Neutron Number

20

40

60

100
80
Neutron Number

B(E2) (e2b2)

10

1

0.1

0.01

0.001
0

Figure 1.3: Energy of first 2+ state and B(E2) strength, for even-even nuclei, as a function
of neutron number. Arrows indicate magic numbers. Figure taken from Ref. [3].

4

The solution of the one-body Schrödinger equation using this potential has four good quantum numbers: the radial quantum number nr , indicating the number of nodes in the wavefunction; the orbital angular momentum quantum number l; the total angular momentum
quantum number j, corresponding to the addition of orbital angular momentum l and nucleon spin s; and the z-projection of the total angular momentum jz [2].
Using the notation of Ref. [2], and looking at each of the terms in Equation 1.1.
• The Coulomb term has the form of a Coulomb potential for a sphere of charge Ze and
radius Rc

Ze2
r
Ze2 3
r2
=
−
2
Rc 2 2Rc

for r ≥ Rc

Vc (r) =

(1.2)
for r ≤ Rc .

• The central term is most often one of two forms: a harmonic oscillator
1
Vo (r) = mω 2 r2 ,
2

(1.3)

which has the advantage of being completely analytically solvable, and for which the
energy eigenstates are characterized by the major quantum number N = 2nr + l; or a
Woods-Saxon potential
Vo (r) = Vo fo (r)
1
fo (r) =
,
1 + exp[(r − R0 )/a0 ]

(1.4)

which has the advantage of better representing the short-ranged nature of the nuclear
force, and removing the l-degeneracy present in the harmonic oscillator basis.
• The radial part of the spin-orbit term takes the form of a derivative of a Woods-Saxon
potential
Vso (r) =
5

Vso dfso (r)
,
r
dr

(1.5)

where fso has the same form as fo in Equation 1.4, but with diffuseness and radius
aso and Rso . The addition of the spin-orbit term removes all remaining degeneracies
[4,5], and produces large differences in energy between single-particle orbitals for magic
numbers of nucleons, shown in Figure 1.4.
In Equations 1.2 - 1.5, the radii are expressed as Ri = ri A1/3 and ri is typically taken to
be 1.25 fm.
The Independent Particle Model can predict many of the properties of nuclei with one
nucleon more or less than a magic number. However, when several valence nucleons occupy
orbits outside of a magic number "core" or when considering very light (no-core) nuclei, the
problem becomes significantly more complex due to strong forces between valence nucleons.
Modern shell model codes like OXBASH [6], NuShell [7], and NuShellX [8] account for valence
nucleon interactions by solving the Schrödinger equation
HΨ = EΨ

(1.6)

as a matrix equation, diagonalizing the many-body Hamiltonian H. The diagonalization
procedure is often performed in a reduced model space, since the dimensionality of H depends on the number of particles and orbits included in the calculation, which can be quite
large for medium and heavy nuclei. Off-diagonal elements result in mixing of many-body
configurations. While there is no exact expression for the nuclear Hamiltonian H, it can
be conveniently written as a combination of single-particle energies (SPEs) and two-body
matrix elements (TBMEs)
H = SP E + αβ|V |αβ ,

(1.7)

where α and β are single-particle states and V is an effective interaction obtained by fitting
SPEs and TBMEs such that experimental observables are reproduced.
6

0

Neutron Single-Particle Energies

single-particle energy (MeV)

[42] 112 N=5
-10
[30] 70 N=4

-20

[20] 40 N=3

[ 6] 112 2p
[14] 106 1f
[22] 92 0h
[ 2] 70 2s
[10] 68 1d
[18] 58 0g
[ 6] 40 1p

[12] 20 N=2

[14] 34 0f
[ 2] 20 1s
[10] 18 0d

-30
[ 6] 8 N=1

[ 6] 8 0p
-40

[ 2] 2 N=0
Harmonic
Oscillator
Potential

[ 2] 2 0s
Woods-Saxon
Potential

[ 2] 126 2p1
[ 6] 124 1f5
[ 4] 118 2p3
[10] 100 0h9
[ 8] 90 1f7
82 [12] 82 0h11
[ 2] 70 2s1
[ 4] 68 1d3
[ 6] 64 1d5
[ 8] 58 0g7
50
[10] 50 0g9
[ 2] 40 1p1
[ 4] 38 1p3
[ 6] 34 0f5
28
[ 8] 28 0f7
20
[ 2] 20 1s1
[ 4] 18 0d3
[ 6] 14 0d5
8
[ 2] 8 0p1
[ 4] 6 0p3
2
[ 2] 2 0s1
Woods-Saxon
Plus Spin-Orbit
Potential

-50
Figure 1.4: Neutron single-particle states in 208 Pb with three potential models, harmonic
oscillator (left), Woods-Saxon without spin-orbit (middle) and Woods-Saxon with spin-orbit
(right). The numbers in square brackets are the maximum number of neutrons in that each
level can contain, the major quantum number N = 2n + l, the Woods Saxon is labeled by
n, l and the Woods-Saxon with spin-orbit is labeled by n, l, 2j. Figure and caption taken
exactly from [2].

7

Despite its many successes, the Nuclear Shell Model has difficulties predicting the properties of unstable nuclei. The experiment and result presented here will provide information
about the N = 8 "semi-magic" nucleus 12 Be, shedding light on the evolution of shell structure in light, unstable systems.

1.3

Charge-Exchange Reactions

In a nuclear charge-exchange reaction, participating nuclei change proton number by one
(∆Z = ±1) but keep their mass number constant (∆A = 0). These are isovector transitions,
meaning there is a unit change of isospin (∆T = 1). Charge-exchange reactions can proceed
with or without spin transfer (∆S = 1 or 0), and with any transfer of angular momentum
(∆L). The charge-exchange processes discussed here involve direct conversion of a proton
into a neutron - or vice versa - via meson exchange (as opposed to indirect multi-nucleon
exchange processes).
Charge-exchange reactions have long been used to study the spin-isospin response of
nuclei (see e.g. Refs. [9, 10] and references therein). They are a particularly useful tool
for measuring Gamow-Teller strengths (B(GT)), where a Gamow-Teller transition is characterized by no transfer of orbital angular momentum and a unit transfer of spin (∆L = 0,
∆S = 1). The isovector spin-transfer operator for charge-exchange reactions is written as
±
OJM =

J
∆L Y
rk
∆L ⊗ σk M τ±k ,

(1.8)

k
where σ refers to the nucleon spin operator, Y∆L to the spherical harmonics, and the cross
symbol ⊗ denotes a Clebsh-Gordan product coupling to the total angular momentum J
and absorbing all M dependence of the spherical harmonics. τ± refers to the isospin rais8

ing/lowering operator, and the sum over k represents the sum over all nucleons in the nucleus.
When ∆L = 0, this operator is identical to the Gamow-Teller operator for β-decay
O(GT± ) =

σk τ±k .

(1.9)

k
In β-decay, the B(GT) is the square of the reduced matrix element involving the GamowTeller operator and connecting the initial i and final f states of the nucleus
B(GT± ) =

| < f ||O(GT± )||i > |2
,
(2Ji + 1)

(1.10)

where the matrix element has been reduced in orbital space following the convention of
Ref. [2]. The B(GT) can be directly related to the β-decay partial half-life (t), historically
written in terms of an f t-value
ft =

C
,
B(F ) + (gA /gV )2 B(GT )

(1.11)

which incorporates a phase-space factor f . In Equation 1.11, C is a combination of fundamental constants constrained by super-allowed β-decays [11], and the ratio of the axial and
vector coupling constants gA /gV is empirically determined by the decay of the neutron [12].
One major limitation of β-decay studies is that they are limited by the decay Q-value to a
small excitation energy window, and as a result they are unable to extract the full B(GT)
distribution. B(GT) distributions obtained in charge-exchange measurements are not subject
to this limitation.

1.3.1

Proportionality Relationship

Although β decay is governed by the weak force and charge-exchange by the strong force, the
operators for the two processes are similar, connecting the same final and initial states. A
9

phenomenological proportionality has been established relating the differential cross-section
measured in charge-exchange reactions (at the limit of vanishing momentum transfer) and
the Gamow-Teller strength:
dσ
= σ B(GT ) .
ˆ
dΩ q=0

(1.12)

This proportionality was established with (p,n) reaction data and is described in detail in
Ref. [13]. The unit cross-section σ can be parameterized using direct reaction theory into a
ˆ
kinematic term (K), a term describing distortion effects due to the mean field of the nuclei
involved (ND ), and a term related to the effective nucleon-nucleon interaction (Jστ )
σ = KND |Jστ |2 .
ˆ

(1.13)

The last term consists of a direct (D) and exchange (E) component
Jστ = JD + JE

(1.14)

where the exchange term is due to the antisymmetrization of target and projectile nucleons
and typically interferes destructively, reducing the total cross-section (compared to calculations where only the direct component is considered) [13].
In most cases it is not necessary to rely on reaction theory to calculate the terms K, ND ,
and Jστ . For (t,3 He) and (3 He, t) reactions, an empirical mass-dependent formula for the
unit cross-section σ has been established [14]. For cases in which there are several Gamowˆ
Teller transitions, the unit cross-section can be extracted locally and model-independently,
by calibrating Equation 1.12 with a known B(GT) (i.e. from a β-decay measurement). The
unit cross-section obtained from this calibration can be applied to all other Gamow-Teller
transitions, providing a direct link between the differential cross-sections measured in chargeexchange experiments and B(GT)s of interest for nuclear structure.
10

The validity of Equation 1.12 relies on several approximations, described in detail in
Ref [13]. These include:
• the Eikonal approximation, where collisions are assumed to occur at high beam energies
and scattered particles follow straight-line trajectories, which greatly simplifies the
distortion contribution to the unit cross-section; and
• the factorization of the cross section into K, ND , |Jστ |, and B(GT), meaning that the
kinematic, distortion, interaction, and structure terms are independent.
In practical terms, for Equation 1.12 to hold, the linear momentum transfer q needs to
be approximately equal to zero (validating the use of the Eikonal approximation), and the
incident beam energy needs to be ≥80 MeV/u (to largely eliminate contributions from multistep processes and to reduce the effect of distortion).
The proportionality between differential cross-section and B(GT) allows one to measure
B(GT) distributions without the Q-value limitations of β-decay studies. Despite the fact that
Equation 1.12 was derived for (p,n) charge-exchange probes, it appears to hold for composite
probes as well, and as a result over the past quarter-century many other charge-exchange
probes have been developed.

1.3.2

History of Charge-Exchange Probes

The first charge-exchange experiment took place in the late 1950s at the University of California Radiation Laboratory, where a low-energy proton beam was impinged upon deuterium
gas [15]. Since then, (p,n) reaction studies have been carried out at many different laboratories, resulting in several significant achievements including: a deeper understanding
of collective spin-isospin modes in nuclei [16–20]; development of a nucleon-nucleon inter11

action for use in Distorted Wave Born Approximation (DWBA) calculations [21–24]; and
the derivation of the linear relationship between (p,n) cross-sections at forward angles and
Gamow-Teller strengths [13, 25, 26].
As the usefulness of charge-exchange reactions became apparent, new probes were developed to excite other isospin modes and to achieve better energy resolution than (p,n)
reactions (which could only reach energy resolutions of approximately 300 keV FWHM). At
present, in the isospin-raising (∆Tz = 1) direction, probes include: (t,3 He), developed in
the 1970s [27]; (n,p) and (7 Li, 7 Be), developed in the 1980s [28–30]; and (d,2 He), developed in the 1990s [31–33]. In the isospin-lowering (∆Tz = −1) direction, the (3 He,t) probe
was developed in the mid-1980s [34–36]. More recently, heavy-ion charge-exchange probes
such as (12 C, 12 N) and (13 C,13 N) were developed, used primarily to investigate giant resonances [37–40].
Each charge-exchange probe has advantages and limitations. (p,n) and (n,p) probes benefit from a (relatively) simple nucleon-nucleus reaction mechanism, but yield significantly
worse resolution than (3 He,t) and (t,3 He) measurements. Composite probes can provide improved resolution (approximately a factor of 10), but the reaction mechanism is significantly
more complicated and less experimental data is available. Perhaps most importantly, all
probes mentioned here share one major limitation: until recently, charge-exchange studies
using these probes were limited to transitions from stable nuclei - these probes could not be
used to study rare isotopes.
The (7 Li, 7 Be) probe is the first ∆Tz = +1 charge-exchange probe to be developed for
use in inverse kinematics experiments at intermediate beam energies, expanding chargeexchange studies to rare isotopes. The first inverse kinematics charge-exchange experiment

12

utilizing a rare isotope beam took place at the National Superconducting Cyclotron Laboratory (NSCL) in 2002. In the experiment, a beam of 56 Ni was impinged upon a thick 7 Li
target and the exciation energy spectrum of 56 Co was reconstructed from Doppler-corrected
gamma rays. Unfortunately, this inaugural experiment was largely unsuccessful. In 2007, a
second attempt was made, this time aimed at extracting the B(GT) distribution of 34 Si. The
combination of a lighter projectile, much thinner target, and use of a high-resolution spectrometer to reconstruct the 34 Si excitation energy spectrum (rather than Doppler-corrected
gamma rays) yielded success, demonstrating the feasibility of the (7 Li, 7 Be) reaction for
inverse kinematics experiments [41].

1.3.3

(7 Li, 7 Be) Charge-Exchange in Inverse Kinematics

The (7 Li,7 Be) reaction has a long history of use in forward kinematics experiments [42–46].
As a result, at 7 Li energies of 65 MeV/u, the proportionality between B(GT) and crosssection at forward angles has been established at the ∼20% level [46]. This is expected to
improve at higher beam energies, due to a reduction of multi-step contributions and of the
tensor-τ component of the effective nucleon-nucleon interaction [23] (see Section 3.4).
The (7 Li, 7 Be) probe is spin-selective; that is, it is possible to isolate spin-transfer
(∆S = 1) events from all others. To illustrate how this works, Figure 1.5 shows the possible
transitions from the ground state of 7 Li to states in 7 Be. If a transition is made to the 7 Be
ground state, the reaction contains a mixture of spin-transfer and non-spin-transfer components (∆S = 1 and ∆S = 0). If the transition is made into the 7 Be excited state at 429
keV, the reaction is purely ∆S = 1 (for ∆L = 0 transitions). This means one can isolate or
"tag" the ∆S = 1 charge-exchange reaction channel, by measuring the reaction ejectile in
13

4.57 MeV
1587 keV

ΔS=1
B(GT) =1.07
g.s.

7

Li

3/2-

7/2α+3He

429 keV

1/2-

g.s.

γ

7
Be
ΔS=0,1
B(GT)=1.2

3/2-

Figure 1.5: Transitions from the ground state of 7 Be to states in 7 Be. The particle decay
threshold is indicated by the black dotted line, and the red dotted line refers to the deexcitation gamma-ray emitted from the 429 keV 7 Be excited state. For interpretation of the
references to color in this and all other figures, the reader is referred to the electronic version
of this dissertation.

coincidence with the 429 keV gamma ray emitted during the de-excitation of 7 Be. Events
registering a reaction ejectile and a 429 keV 7 Be de-excitation gamma ray are referred to
as "coincidence" events. Events that do not require a coincident 429 keV gamma ray are
referred to as "singles" events, and could involve transitions into the ground or excited states
of 7 Be.
In inverse kinematics experiments, coincidence data can be extremely useful for isolating
charge-exchange events from those produced via other reaction channels. It can also provide
better energy resolution than the singles data, because of an inherent 429 keV ambiguity in
the singles data excitation energy reconstruction (depending on whether the ground or 429
keV state in 7 Be is populated). That said, the limited detection efficiency for gamma rays
and the branching ratio to the 429 keV state make the overall count rate for coincidence
data lower than for singles data, and since rare isotope beam rates are already significantly
lower than those obtained in stable beam experiments, this additional reduction can be prohibitive.
14

Fortunately, in the current study, the Gamow-Teller transitions of interest in 12 Be are
located at low excitation energies (less than 3 MeV), making it possible to distinguish singles
events transitioning through the ground and 429 keV states in 7 Be from those transitioning
through the 4.57 MeV (and higher) excited states. Although the transition through the
ground state of 7 Be is associated with both ∆S = 1 and ∆S = 0, the magnitude of the
∆S = 0 operator is small compared to that of the ∆S = 1 operator at the beam energies
under consideration here [23]. Additionally, the transitions of interest for the present study
are from the 1+ ground state of 12 B to the 0+ states of 12 Be. Since the total change of
angular momentum ∆J = ∆L + ∆S = 1, the ∆L = 0 component of the cross-section for
these transitions is guaranteed to involve spin-transfer ∆S = 1, satisfying the requirements
for Gamow-Teller transitions. Consequently, for the study of the 12 B(7 Li, 7 Be)12 Be reaction, the singles data is as useful as the coincidence data, and much improved statistics are
achievable.

1.4

Outline

This chapter has introduced the atomic nucleus, the Nuclear Shell Model, and the basics
of charge-exchange reactions. It has provided an historical overview of charge-exchange
studies, and has introduced the general principles of (7 Li,7 Be) charge-exchange experiments
in inverse kinematics. Chapter 2 will motivate the study of 12 Be, and Chapter 3 will discuss
(7 Li, 7 Be) reaction theory in the Distorted Wave Born Approximation framework. Chapter
4 will detail the experiment itself, Chapter 5 will describe the data analysis procedure, and
Chapter 6 will present the result and compare to previous work. Chapter 7 will summarize
this dissertation and provide a brief outlook for possible future studies.
15

Chapter 2
Motivation
In the Independent Particle Model, 12 Be, with four protons and eight neutrons, is a "semimagic" nucleus, and the ground state configuration consists of neutrons completely filling
the 0s and 0p shells, as depicted in Figure 2.1. In reality, the situation is much more
complicated, and more than forty years of experimental and theoretical work have gone in
to understanding the structure of this light, unstable nucleus. This chapter will give a brief
overview of the reasons for configuration mixing in 12 Be. It will place the current study
in an historical context, and highlight the benefits of gaining a better understanding of the
Gamow-Teller strength distribution in 12 Be.

2.1

Nuclear Structure Away from Stability

One of the main goals of nuclear physics is to understand the evolution of nuclear structure
away from stability. In unstable systems, the ordering of the single particle orbitals shown
in Figure 1.4 changes, quenching the major shell gaps and creating new, smaller gaps at
different proton and neutron numbers. This rearrangement of single-particle orbitals can
16

0d3/2
1s1/2
0d5/2
0p1/2
0p3/2

0p1/2
0p3/2

0s1/2

0s1/2

protons

neutrons

Figure 2.1: Simplest ground-state configuration for 12 Be.
lead to complications when trying to predict nuclear structure, and as a result calculations
involving unstable systems often require large configuration spaces to reproduce experimentally observed properties1 .
Light unstable nuclei provide a fertile testing ground for nuclear structure models, as
exotic phenomena (neutron halos, borromean systems, parity inversion of low-lying states)
can be found in isotopes with only a few nucleons more or less than stable species. The
configuration spaces for light nuclei are relatively small, and calculations are less computationally taxing than those for medium or heavy isotopes. The configuration space of interest
for 12 Be includes the 0p, 1s, and 0d orbitals. The 0s and 1p0f orbitals were included in
all of the calculations presented here, but for the most part these orbitals can be ignored
for Gamow-Teller transitions, as excitations into the pf shell are impossible and transitions
involving the 0s nucleons are Pauli-blocked.
When contemplating the structure of 12 Be, it helps to consider the systematics of neighboring nuclei. Tensor and central components of the proton-neutron monopole interaction
1 While this single-particle description is illustrative and is often used to describe the
evolution of shell structure away from stability, the complete many-body problem is actually
much more complicated - for interested readers, a more robust description can be found in
Refs. [47, 48].
17

are stronger in 14 C and 16 O than in 12 Be, because more protons occupy the 0p3/2 orbital [49, 50]. This results in a higher energy 0p1/2 neutron orbit in 12 Be than in the more
stable N = 8 isotones [50, 51]. In the exotic Beryllium isotopes, the 1s1/2 and 0d5/2 orbits
are inverted, due to the smaller kinetic energy of the loosely-bound 1s1/2 orbital [52]. Together, these single-particle orbital shifts act to bring the 1s1/2 and 0p1/2 neutron orbitals
closer together, making configurations in which neutrons occupy 1s or 0d orbitals more energetically favorable for 12 Be than for stable isotopes in the region.
Using the notation of the harmonic oscillator basis, we can describe many-body configurations in terms of the number of nucleons occupying the single-particle orbitals. For 12 Be,
(0s)4 (0p)8 configurations are referred to as "0 ω" configuration. "2 ω" configurations are
those in which two nucleons are excited into the next harmonic oscillator shell, or one nucleon is excited up two oscillator shells. Examples of 2 ω configurations in 12 Be include
(0s)4 (0p)6 (1s0d)2 , (0s)3 (0p)8 (1s0d)1 , and (0s)4 (0p)7 (1p0f )1 . Due to the structure systematics discussed in the previous paragraph, (0s)4 (0p)6 (1s0d)2 is the most probable 2 ω
configuration, where the two nucleons in the 1s0d orbitals are neutrons.
The 0 ω configuration and most probable 2 ω configuration are shown in Figure 2.2.
Naïvely, one would expect 0 ω configurations to have a lower total energy than 2 ω configurations. However, because of pairing and alpha correlation effects [52], 0 ω and 2 ω
configurations in 12 Be are approximately degenerate in energy. As a result, the physical
ground and excited 0+ states in 12 Be consist of a mixture of these two configurations. If
the two configurations were exactly equal in energy, the ground and excited 0+ state would
consist of equal parts 0 ω and 2 ω configurations. The precise configuration mixing in 12 Be
is still under debate.

18

2ħω

0ħω
0d3/2
0d5/2
1s1/2
0p1/2

0p1/2
0p3/2

0p3/2

0s1/2

0s1/2
protons

0d3/2
0d5/2
1s1/2
0p1/2

0p1/2
0p3/2

0p3/2

0s1/2

neutrons

0s1/2
protons

neutrons

Figure 2.2: Example of 0 ω and 2 ω configurations in 12 Be.

2.2

History of

12

Be

12 Be was first discovered in 1965 in proton-irradiation experiments on 15 N [53]. Ten years
later, noting "considerable discrepancies" between calculated and experimental masses, excitation energies, and widths, F.C. Barker proposed a model for the T=2 states of A=12
nuclei in which only small components of the states belong to 0 ω configurations [54]. Based
on Talmi and Unna’s description of 11 Be as a 10 Be core plus a 1s1/2 neutron [51], Barker
predicted that low-lying 0+ states in 12 Be should be formed with the same 10 Be core and
a pair of neutrons in the 1s1/2 , 0p1/2 , or 0d5/2 orbit. Support for this hypothesis came
just two years later, when the β-decay half-life and excitation energy of the first 2+ state
were measured and found to be longer and lower than that predicted by a pure 0 ω calculation [55].
Over the next two decades, theorists strived to gain a more complete understanding of
12 Be, driven by experimental data. By the early 1980’s, three states in the 12 Be level scheme
had been confirmed: a 0+ ground-state, 2+ first excited state at 2.11 MeV, and another
state at 2.68 MeV [56] (later determined to have spin-parity 1− [57]). Interpreting 10 Be(t,p)
experimental data, H.T. Fortune et al. predicted that the low-lying states of 12 Be would
19

be dominated by 2 ω configurations, but emphasized the need for more serious shell-model
calculations with proper accounting for 0 ω+2 ω configuration mixing [58]. Other models
developed during this time described the observed 12 Be ground state properties in terms
two neutrons coupled to an excited 10 Be core, requesting (p,t) reaction measurements [59],
and attempted to understand the quenching of the N = 8 shell gap in terms of its effect on
β-decay half-lives and Gamow-Teller strength [60]. The latter work stressed "it would be
extremely interesting to measure the Gamow-Teller strength distribution in 12 Be".
Near the turn of the century, inverse kinematics experiments employing rare isotope
beams brought a flurry of experimental activity studying 12 Be. Inelastic proton scattering and inelastic 12 Be scattering measurements performed by Iwasaki et al. demonstrated
that the neutron-rich beryllium isotopes exhibited a tendency towards strong quadrupole
deformation [61] and identified unambiguiously the spin and parity of the 2.68 MeV 1−
state [57]. These observations supported the hypothesis that the N = 8 shell closure in
12 Be was significantly smaller than in stable N = 8 isotopes. The first direct evidence for
the breakdown of the N = 8 shell closure came from a (12 Be,11 Be+γ) neutron-knockout
measurement performed at NSCL [62], which suggested a 12 Be ground state configuration
that was 32% 0 ω and 68% 2 ω. This experiment also indicated a reduced role for 10 Be
core excitation than previous proposed [63]. A subsequent knockout experiment, designed
to measure the (0s)4 (0p)6 (0d)2 component of the wavefunction independently, verified the
NSCL results [64]. With the discovery of an isomeric 0+ state at 2.24 Mev [65], the spectrum
of 12 Be below the neutron decay threshold was completed. The 12 Be level scheme is shown
in Figure 2.3.
During this experimental renaissance, controversy over the extent of configuration mixing

20

3.69 MeV

Sn

2.68 MeV

11-

2.24 MeV
2.11 MeV

γ (100%)

02+ τ = 331 ns
21+

γ (21.5%)

e+e- (100%)

γ (100%)

g.s.

12

Be

01+

Figure 2.3: Level scheme, 12 Be.
in the 0+ states of 12 Be was escalating. Citing instabilities with the available shell-model
calculations, in 1999 R. Sherr and H.T. Fortune developed a "relatively simple" model for the
T = 2, 0+ states of A = 12 nuclei based on Coulomb shifts. Using a Woods-Saxon nuclear
potential and a uniform sphere Coulomb potential, Sherr and Fortune calculated the binding
energies of the T = 2, 0+ states for three configurations: a 0p1/2 , 1s1/2 , or 0d5/2 neutron
coupled to the appropriate A = 11, T = 3/2 state (1/2− , 1/2+ , 5/2+ ). Finding that the
(0s)4 (0p)8 and (0s)4 (0p)6 (0d)2 contributions to the 12 Be ground state were approximately
equal, they performed a one-parameter fit to the experimental binding energies to estimate
the (0s)4 (0p)6 (1s)2 fraction of the wavefunction (55%) [66]. The exact 0p : 1s : 0d ratio
in 12 Be was then determined by assuming the physical ground state must be a mixture of
the lowest energy 0 ω and 2 ω configurations, yielding a final ground state configuration of
32% 0 ω, 68% 2 ω. The wavefunction for the excited 0+ state was estimated by assuming
the two 0+ states sum to 100%: the 0+ wavefunction consists of 68% 0 ω and 32% 2 ω.
2
Fortune and Sherr claimed calculations using these wavefunctions reproduced experimental
cross-sections, excitation energies, and widths better than the original shell-model calcula21

tions performed by Barker in 1976 [67].
In 2009 Barker responded, arguing that Fortune and Sherr had interpreted the data incorrectly, leaving out weak states and making assumptions about spins and parities of states
in 12 C that were not yet definitively proven. Making minor modifications to his 20-year old
calculation, Barker claimed that a correct interpretation of the data in fact supported his
model, in which the ground state of 12 Be is still 32% 0 ω and 68% 2 ω, but the excited
0+ state is made up of 42% 0 ω and 58% 2 ω configurations [68]. Fortune and Sherr then
reitterated their beliefs that Barker’s interpretation was flawed and suggested that improved
experimental data could easily settle the issue [69]. During this period, several other models
for 12 Be were developed, including a 10 Be+n+n three-body calculation that compared well
with the discrete and continuum energy spectrum of 12 Be, and measured electric transition
strengths (which were not considered by Fortune and Sherr or Barker) [70].
As recently as 2010, 11 Be(d, p)12 Be transfer measurements explored the (0s)4 (0p)6 (1s)2
2 ω configurations of the 0+ levels in 12 Be [71]. The extracted 2 ω component of the ground
state is smaller than most models predict, and a recent paper by Fortune and Sherr has
questioned the result [72]. Clearly, the structure of the 0+ states in 12 Be remains an open
question, despite knockout, transfer, (t,p), and inelastic scattering experiments. A chargeexchange study of 12 Be could prove a valuable addition to the existing body of knowledge,
helping clear up some of the controversy.

2.3

12

B(7Li, 7Be)12Be

Much of the direct evidence for N = 8 shell quenching obtained to date has been based
on spectroscopic factors deduced from knockout and transfer experiments. Spectroscopic
22

factors can be defined as either an overlap of the many-body wave functions of the initial
and final states (theoretical spectroscopic factors), or as the ratio of the experimental crosssection to that of a particular reaction model (experimental spectroscopic factors). Knockout
and transfer measurements yield experimental spectroscopic factors, which are, by definition,
dependent upon the reaction model employed, and can have relatively large errors that should
be factored into any final conclusions. The spectroscopic factors from the neutron knockout
experiment performed at NSCL, for example, had errors of approximately 20% - and since
the ratio of these spectroscopic factors was used to deduce the ground state configuration
of 12 Be, the uncertainty in the wavefunction - not cited in Ref. [62] - would likely be large.
Since both knockout and transfer measurements probe the 2 ω component of the 0+ states
wavefunctions, an experimental technique should be developed that can selectively probe the
0 ω configurations of 12 Be. This measurement would provide a complement to the knockout
and transfer measurements, and may be able to confirm or exclude the previous results.
As described in Chapter 1, the Gamow-Teller transitions probed in charge-exchange
experiments are ∆L = 0, ∆S = 1 transitions. The 1+ ground state of 12 B is dominated
by 0 ω configurations. Gamow-Teller transitions into 12 Be, therefore, are: (1) restricted to
the 0+ and 2+ states in 12 Be, as ∆J = ∆L + ∆S = 1; and (2) capable of probing only the
0 ω component of the wavefunction. This selectivity makes these reactions an ideal tool for
exploring the 0 ω component of the ground and 0+ state wavefunction in 12 Be.
2
The sensitivity of the B(GT) distribution to configuration mixing in 12 Be can be seen in
Figure 2.4. Shown are predicted B(GT) values for transitions to 0+ states in 12 Be, calculated
using the shell-model code OXBASH [6] with the WBP interaction [73] in the spsdpf model
space. On the left is a calculation in which only 0 ω configurations are allowed. All 0+

23

strength is concentrated in the 12 Be ground state. Since the WBP interaction was not
originally derived for mixed n ω configurations (i.e. 0 ω mixed with 2 ω), it must be
altered slightly to account for the mixing of high-lying (n + 2) ω configurations with lowlying n ω states. In terms of energy, written explicitly, 2 ω admixtures push down 0 ω
states, but in addition 4 ω admixtures push down 2 ω states, and 6 ω admixtures push
down 4 ω states - one would need to go to 8-10 ω for a consistent calculation. Instead,
as described in Refs. [73–75], one can use the change in the ground state energy for 0 ω
and 0 ω+2 ω calculations (∆E) to estimate the effect of higher-lying admixtures on the
2 ω states, and lower the energy of all 2 ω configurations by this estimate. This procedure
has been performed for the calculations shown in the right panel of Figure 2.4 - all 2 ω
configurations have been shifted by ∆E = −3 MeV. Clearly, when 0 ω+2 ω mixing is
included in the calculation, the 0+ strength distribution is significantly altered, and the
strength formerly concentrated in the ground state is now fragmented amongst the two 0+
states. The total strength of each state, and also the ratio of Gamow-Teller strength of the
two 0+ states
R=

B(GT )(0+ , 2.24 MeV)
2
B(GT )(0+ , g.s.)
1

(2.1)

is very sensitive to the amount of 0 ω+2 ω configuration mixing in 12 Be. The aim of
the current work is to measure the 0+ B(GT) distribution in 12 Be, compare the results to
shell-model predictions to determine the appropriate shift for 2 ω configurations (∆E), and
compare the 0 ω component of the wavefunction in the ground and 2.24 MeV 0+ state from
these (shifted) calculations to those obtained in previous experimental and theoretical work.
The next three chapters will focus on the details of the theoretical cross-section calculations, experiment, and analysis - further discussion of the B(GT) distribution in 12 Be and

24

B(GT)

0.8

1

0ħω

0.8
B(GT)

1
0.6
0.4
0.2
0

(0+2)ħω
ΔE= -3 MeV

0.6
0.4
0.2
0

0 0.5 1 1.5 2 2.5 3 3.5
EX (MeV)

0 0.5 1 1.5 2 2.5 3 3.5
EX (MeV)

Figure 2.4: Calculated B(GT) distribution, for transition to 0+ states in 12 Be below the
neutron decay threshold (3.69 MeV). Values have been quenched according to Ref. [76]. 0 ω,
2 ω, and ∆E refer to the restrictions and shifts in the calculation - see text for details.
implications for the existing models will be continued in Chapter 6.

25

Chapter 3
Theory
The proportionality between differential cross-sections measured in charge-exchange experiments and Gamow-Teller strength (Equation 1.12) requires the isolation of transitions with
no transfer of orbital angular momentum (∆L = 0). To extract the ∆L = 0 component
from the measured differential cross-section, a Multipole Decomposition Analysis (MDA)
was performed using theoretical angular distributions. The MDA will be described in detail
in Chapter 6. This chapter will discuss the calculation of theoretical angular distributions
in the Distorted Wave Born Approximation (DWBA).

3.1

Distorted Wave Born Approximation

Nuclear reactions are modeled as one nucleus scattering off another - the projectile nucleus
is treated as an incoming wave, and the outgoing (scattered) wave is characterized by a
scattering amplitude f (θ, φ). This scattering amplitude is directly related to the differential
26

cross-section:
kf
dσ
∝
|f (θ, φ)|2 ,
dΩ
ki

(3.1)

where ki and kf refer to the momentum of the incoming and outgoing wave. The scattering
amplitude (written in terms of momentum vectors) can also be expressed in terms of a
transition amplitude between initial and final states Tf i
f (kf , ki ) = −

µ
T (k , k ) ,
2π f i f i

(3.2)

where µ is the reduced energy of the incoming projectile (1) and the target (2)
µ=

E1 E2
.
E1 + E2

(3.3)

These transition amplitudes are calculated in the Distorted Wave Born Approximation
(DWBA).
The starting point for calculating transition amplitudes in DWBA is the Schrödinger
equation, where the Hamiltonian contains standard kinetic and potential terms K and V
HΨ = (K + V )Ψ = EΨ .

(3.4)

For many cases, the potential V can be described as a combination of two terms: one term
(U1 ) that describes the distortion of incoming and outgoing waves due to the mean field of
the target nucleus, and another term (U2 ) that describes the interaction between individual
nucleons
V = U1 + U2 .

(3.5)

With a potential of this form, the transition amplitude Tf i can also be written as a combination of two terms [77]
Tf i = T 1 + T 2(1) ,
27

(3.6)

where T 1 describes all effects of the distorting potential U1 and T 2(1) describes all residual
effects after distortion has been accounted for. For this reason U2 is sometimes referred to
as the residual interaction. Since the initial and final states in a charge-exchange reaction
aren’t directly connected by U1 , the effect of the distorting potential can be factored into
the calculation by treating the incoming beam as already distorted by U1 . In terms of the
transition amplitude, the first term of Equation 3.6 drops out, leaving the residual term only
Tf i = T 2(1) .

(3.7)

Switching to bra-ket notation, this transition amplitude can be written in the prior form, in
terms of the initial (already distorted by U1 ) wave χ− |, the residual interaction U2 , and
the outgoing scattered wave |Ψ
Tf i = χ− |U2 |Ψ

(3.8)

Then, using Green’s function methods and the Lippmann-Schwinger equation for the exact
wavefunction solution Ψ (see e.g. [77, 78])
ˆ
Ψ = χ + G1 U2 Ψ

(3.9)

ˆ
G1 = [E − K − U1 ]−1 ,
one can expand the transition amplitude to obtain the Born Series
Tf i = χ− |U2 |χ + χ− |U2 G1 U2 |χ + ...

(3.10)

Since the residual interaction (U2 ) is weak relative to the distorting potential (U1 ), the
transition amplitude can be truncated to the first term in the series
Tf i = χ− |U2 |χ .

28

(3.11)

This approximation is known as (first order) DWBA.
The expression for the transition density given in Equation 3.11 is deceptively simple in
appearance. In practice, to calculate Tf i one needs both nuclear structure and reactions
input including: transition densities, which describe the overlap between the initial and final
states of both the projectile/ejectile and target/recoil systems; the effective nucleon-nucleon
interaction, which will ultimately be folded over the transition densities to account for the
composite nature of all particles involved, yielding form factors; and an optical model to
describe distortion effects and calculate distorted waves. Here, each of these components of
the transition amplitude is discussed separately, in the context of the WSAW/FOLD/DWHI
code package [79] used to obtain theoretical cross-sections for this study.

3.2

Theoretical Cross-Section Calculations

Theoretical cross-sections calculations were performed with the WSAW/FOLD/DWHI code
package [79], designed for charge-exchange studies with composite probes. First, one calculates transition densities for the projectile/ejectile and target/recoil systems using single particle radial wavefunctions generated in WSAW and one-body transition densities (OBTDs)
obtained from OXBASH [6]. Then, the program FOLD generates form factors, which are
combined with an optical potential in the code DWHI to calculate transition amplitudes and
separate angular distributions for each allowed value of total angular momentum transfer J.

3.2.1

Transition Densities

Transition densities describe the overlap between the initial and final states in the projectile/ejectile and target/recoil system. Essentially a way to connect single-particle wave29

functions and operators to a many-body problem, they have two major components: radial
wavefunctions for the one-particle and one-hole single-particle orbits, and one-body transition densities (OBTDs) that weigh the one-particle one-hole contributions.
The radial wavefunctions were generated in the program WSAW, which fits solutions
to the Schrödinger equation such that they match binding energies. The WSAW potential
contains Coulomb, Woods-Saxon and spin-orbit terms. The binding energies were calculated
using the Skx interaction [80] in the DENS subroutine of the OXBASH shell model code [6].
The OBTDs contain all information about couplings and spins and serve as a weighting factor for each one-particle, one-hole excitation, summed to construct the many-body
transition densities:
OBT D(f ikα kβ ) kα ||O(GT± )||kβ

f ||O(GT± )||i =

(3.12)

kα ,kβ
(here, ki refer to one-particle one-hole states and f , i, to many-body states). OBTDs used
in this analysis were calculated using the code OXBASH [6]. The CKII interaction [81] was
used in the p model space for the 7 Li → 7 Be OBTD calculation. The WBP [73] interaction
was used in the spsdpf model space for the 12 B → 12 Be OBTD calculation. The WBP
interaction was intended for "pure ω" calculations (i.e. no configuration mixing between
major oscillator shells), but the 0 ω and 2 ω states in 12 Be are known to be degenerate
in energy [62], and therefore matrix elements that mix n and (n + 2) ω configurations are
required. The interactions were changed to accurately calculate these mixed matrix elements,
by shifting all (n + 2) ω configurations by ∆E= -3 MeV, following the procedure described
in Chapter 2.

30

3.2.2

Form Factors

Using the transition densities described in the previous section, separate form factors are
calculated for each allowed transition, characterized by the relative transfer of total angular
momentum (∆Jr ), and the change in total angular momentum of the target and projectile
(∆Jp and ∆Jt ). For each combination of ∆Jr , ∆Jp and ∆Jt , the effective interaction Vef f is
double-folded over the transition densities for the projectile/ejectile and target/recoil systems.
This double-folding is necessary due to the composite probes involved - if the charge-exchange
reaction involved a single-nucleon probe, only a single-folding procedure would be necessary.
The effective interaction is a sum of two-body terms involving nucleons in the target and
projectile
vpt (s)(1 − Ppt ) .

Vef f =

(3.13)

p,t
Here, s refers to the separation between the target and projectile nucleon and Ppt is an
antisymmetrization operator. A short-range approximation is used for exchange terms, as
discussed in Ref. [23]. The nucleon-nucleon (NN) interaction vpt was derived via a phase shift
analysis of NN scattering data - the 100 MeV parameterization was used in the calculations
presented here [23,24]. This interaction was chosen for its factorization into spin- and isospintransfer terms, convenient for charge-exchange studies. The NN interaction contains central,
spin-orbit and tensor terms
vpt = v c (s) + v LS (s) + v T (s)
v c = vτ (s)(τp · τt + vστ (s)(σp · σt )(τp · τt )
v LS = vLSτ (s)(L · S)(τp · τt )
v T = vT τ (s)Spt (τp · τt )
(σp · s)(σt · s)
Spt =
− σp · σt ,
s2
31

(3.14)

Angular Momentum Selection Rules
∆Jt = Jti ⊕ Jtf
∆Jp = Jpi ⊕ Jpf
∆Jr = ∆Jp ⊕ ∆Jt
(−1)∆Lt = (πi πf )t
(−1)∆Lp = (πi πf )p
∆Lr = ∆Lp ⊕ ∆Lt
∆Jr = ∆Lt
Table 3.1: Angular momentum selection rules. Subscript r refers to relative transfers of
angular momentum. Subscript p refers to projectile/ejectile system, t to target/recoil system,
i to initial state and f to final state. Valid for central nucleon-nucleon interaction terms only;
tensor terms will be discussed in Section 3.4.

where only terms relevant to isovector transitions have been included. The central isoscalar
term (not included explicitly in Equation 3.14 but similar to v c , without τ components) is
responsible for mean field distortions and multistep processes.
The angular momentum selection rules for a each transition are given in Table 3.1, and
depend on the initial i and final f total angular momentum J and parity π for the projectile
p and target t. Form factors are calculated for each value of relative angular momentum
transfer ∆Jr . The last line of Table 3.1 is only valid for central terms of the effective nucleonnucleon interaction - spin-orbit terms do not play a significant role [82], and tensor terms
will be discussed in detail in Section 3.4. The program FOLD executed the double-folding
procedure and produced form factors formatted for input into DWHI.

3.2.3

Cross-Sections

The last step in the transition amplitude calculation involves accounting for the distortion
effects of the mean field. Usually, one would obtain an optical model by fitting elastic
scattering data measured under the same experimental conditions as the charge-exchange
32

V (MeV)

r(fm)

108

1.375

a0 (fm) Wv (MeV)
0.854
37.9

rw (fm) aw (fm)
1.671

0.758

Table 3.2: Optical potential parameters used in DWHI program to calculate distorted waves.
Taken from Ref. [83].

experiment (same beam, same target, and same incident energy). When this is not possible,
one can use a published optical potential derived from similar data. For the current study,
the most suitable optical potential available in the literature was a fit to elastic scattering
measurements of 7 Li on 12 C at 50 MeV/u [83](recall, the current study involves 7 Li and 12 B
at 80 MeV/u). The optical potential contained Coulomb, real and imaginary Woods-Saxon
terms
U (r) = Uc (r, rc ) − V f (r, r0 , a0 ) − iWv f (r, rw , aw )
1

f (r, rx , ax ) =
1 + exp

1/3
[(r−rx At )/ax ]

.

(3.15)

The Woods-Saxon optical potential parameters included in the calculation are given in Table
3.2. All three terms were used in the DWHI calculation to calculate distorted waves.
The distorted waves were combined with FOLD form factors to calculate transition amplitudes. The mass and kinematic factors of Equations 3.1 and 3.2 are also input into DWHI,
and angular distributions were calculated for each allowed value of total angular momentum
transfer. The WSAW/FOLD/DWHI angular distributions are shown in Figures 3.1 through
3.8. The DWBA cross-sections generally do a good job reproducing experimental crosssections obtained in charge-exchange experiments, but since approximations were used for
the optical potential and the energy shift ∆E, the absolute magnitudes of the components
are tuned during the MDA procedure described in Chapter 6.
All transitions originate in the ground states of 12 B and 7 Li. Figures 3.1 through 3.4 are
33

dσ/dΩ(mb/sr)

Total
All ΔJt=1

1

ΔJr=0, ΔJt=1
ΔJr=2, ΔJt=1

10-1
10-2
10-3

All ΔJt=2

0

2

4

6

8

θ

(deg)

cm

Figure 3.1: Angular distributions produced using the WSAW/FOLD/DWHI code package
for the 12 B(g.s., 1+ ) → 12 Be(g.s., 0+ ), 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition.
Black dots (all ∆Jt =1) are not visible because they are effectively underneath green stars
(total cross-section).
for transitions to the 429 keV excited state in 7 Be.

• Figure 3.1 shows the theoretical angular distributions for the transition to the ground
state of 12 Be. One of the simplest angular distributions calculated, it contains just
three components, one of which (∆Jr =2, ∆Jt =2, ∆Jp =1) is negligible. At zero degrees, the ∆Jr =0 term is nearly an order of magnitude larger than any other component.
• Figure 3.2 shows the theoretical angular distributions for the transition to the 2.11
MeV 2+ state of 12 Be. Due to the many possible combinations of ∆Jr , ∆Jp and ∆Jt ,
there are several more components contributing to the total cross-section, and it is
not sufficient at the largest angles to include only ∆Jr =0, ∆Jt =1 components for this
case. However, at zero degrees the ∆Jr =0 component is still dominant.
• Figure 3.3 shows the theoretical angular distributions for the transition to the 2.24
34

dσ/dΩ(mb/sr)

Total

All ΔJt=1, ΔJp=1
ΔJr=0, ΔJt=1, ΔJp=1
ΔJr=2, ΔJt=1, ΔJp=1
All ΔJt=1, ΔJp=2
All ΔJt=1, ΔJp=3
ΔJr=2, ΔJt=1, ΔJp=3
ΔJr=4, ΔJt=1, ΔJp=3
All ΔJt=2, ΔJp=1
All ΔJt=2, ΔJp=2
ΔJr=0, ΔJt=2, ΔJp=2
ΔJr=2, ΔJt=2, ΔJp=2
ΔJr=4, ΔJt=2, ΔJp=2
All ΔJt=2, ΔJp=3
ΔJr=2, ΔJt=2, ΔJp=3
ΔJr=4, ΔJt=2, ΔJp=3

10-1
10-2
10-3
10-4
10-5
10-6
0

2

4

6

8 θ

(deg)

cm

Figure 3.2: Angular distributions produced using the WSAW/FOLD/DWHI code package for
the 12 B(g.s., 1+ ) → 12 Be(2.11 MeV, 2+ ), 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition.
MeV 0+ state in 12 Be . As with the 12 Be ground-state calculation, there are three
components, ∆Jr =2 ∆Jt =2 ∆Jp =1 is negligible, and only ∆Jt =1 terms are needed
to adequately reproduce the total cross-section. ∆Jr =0 is again dominant at zero
degrees.
• Figure 3.4 shows the theoretical angular distribution for the transition to the 2.68 1−
state in 12 Be. The angular distributions are quite different from those shown in the
previous figures - they no longer peak at zero degrees, as expected for dipole states,
and the ∆Jr =1 and ∆Jr =3 components are now dominant. The total cross-section is
well-described using the ∆Jt =1 and ∆Jp =2 components only.
Figures 3.5 through 3.8 show transitions to the ground state of 7 Be.
• Figure 3.5 shows the theoretical angular distributions for the transition to the ground
state of 12 Be. There are more components than the coincidence case (transitions
35

dσ/dΩ(mb/sr)

Total

1

All ΔJt=1
ΔJr=0, ΔJt=1
ΔJr=2, ΔJt=1

10-1

10-2

All ΔJt=2

0

2

4

6

8

θ

(deg)

cm

dσ/dΩ(mb/sr)

Figure 3.3: Angular distributions produced using the WSAW/FOLD/DWHI code package for
the 12 B(g.s., 1+ ) → 12 Be(2.24 MeV, 0+ ), 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition.

1

Total
All ΔJt=1, ΔJp=2
ΔJr=1, ΔJt=1, ΔJp=2
ΔJr=3, ΔJt=1, ΔJp=2

10-1

All ΔJt=1, ΔJp=1

-2

10

All ΔJt=1, ΔJp=0
All ΔJt=2, ΔJp=2
ΔJr=1, ΔJt=2, ΔJp=2
ΔJr=3, ΔJt=2, ΔJp=2

-3

10

All ΔJt=2, ΔJp=1
ΔJr=1, ΔJt=2, ΔJp=1
ΔJr=3, ΔJt=2, ΔJp=1

10-4
10-5

0

2

4

6

8 θ

(deg)

cm

Figure 3.4: Angular distributions produced using the WSAW/FOLD/DWHI code package for
the 12 B(g.s., 1+ ) → 12 Be(2.68 MeV, 1− ), 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition.

36

dσ/dΩ(mb/sr)

Total

1

All ΔJt=1
ΔJr=0, ΔJt=1
ΔJr=2, ΔJt=1

10-1

10-2

All ΔJt=3
ΔJr=2, ΔJt=3
ΔJr=4, ΔJt=3

10-3

All ΔJt=2

10-4
10-5

0

2

4

6

8

θ

(deg)

cm

Figure 3.5: Angular distributions produced using the WSAW/FOLD/DWHI code package
for the 12 B(g.s., 1+ ) → 12 Be(g.s., 0+ ), 7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition.
through the 429 keV state in 7 Be, Figure 3.1), but the same trends are apparent:
∆Jr =2, ∆Jt =2 terms are negligible, and with the exception of the angular region
between 4 and 8 degrees, the total cross section can almost be reproduced by only
including ∆Jt =1 terms, and at zero degrees the ∆Jr =0 term is significantly larger
than any other component.
• Figure 3.6 shows the theoretical angular distributions for the transition to the 2.11
MeV 2+ state of 12 Be. This is the most complicated case calculated, and as with the
analogous case (Figure 3.2) it is not sufficient at large angles to include only ∆Jr =0,
∆Jt =1 components. However, at zero degrees the ∆Jr =0 component is still the largest
contribution to the total cross-section.
• Figure 3.7 shows the theoretical angular distributions for the transition to the 2.24
MeV 0+ state in 12 Be. As with the 12 Be ground state calculation, ∆Jr =2, ∆Jt =2 is
negligible, and with the exception of the angular region between 4 and 8 degrees, only
∆Jt =1 terms are needed to adequately reproduce the total cross-section. ∆Jr =0 is
37

dσ/dΩ(mb/sr)

Total
10-1
10-2
10-3
10-4
10-5
10-6
10-7
0

2

4

6

θ

8
(deg)

cm

All ΔJt=1, ΔJp=1
ΔJr=0, ΔJt=1, ΔJp=1
ΔJr=2, ΔJt=1, ΔJp=1
All ΔJt=1, ΔJp=2
All ΔJt=0, ΔJp=1
All ΔJt=3, ΔJp=1
ΔJr=2, ΔJt=3, ΔJp=1
ΔJr=4, ΔJt=3, ΔJp=1
All ΔJt=1, ΔJp=3
ΔJr=2, ΔJt=1, ΔJp=3
ΔJr=4, ΔJt=1, ΔJp=3
All ΔJt=3, ΔJp=2
ΔJr=2, ΔJt=3, ΔJp=2
ΔJr=4, ΔJt=3, ΔJp=2
All ΔJt=2, ΔJp=2
ΔJr=0, ΔJt=2, ΔJp=2
ΔJr=2, ΔJt=2, ΔJp=2
ΔJr=4, ΔJt=2, ΔJp=2
All ΔJt=3, ΔJp=3
ΔJr=0, ΔJt=3, ΔJp=3
ΔJr=2, ΔJt=3, ΔJp=3
ΔJr=4, ΔJt=3, ΔJp=3
ΔJr=6, ΔJt=3, ΔJp=3
All ΔJt=2, ΔJp=1
All ΔJt=2, ΔJp=3
ΔJr=2, ΔJt=2, ΔJp=3
ΔJr=4, ΔJt=2, ΔJp=3

Figure 3.6: Angular distributions produced using the WSAW/FOLD/DWHI code package
for the 12 B(g.s., 1+ ) → 12 Be(2.11 MeV, 2+ ), 7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition.

38

dσ/dΩ(mb/sr)

10

Total
All ΔJt=1
ΔJr=0, ΔJt=1
ΔJr=2, ΔJt=1

1
10-1

All ΔJt=3
ΔJr=2, ΔJt=3
ΔJr=4, ΔJt=3

10-2
10-3
10-4
10-5

All ΔJt=2
0

2

4

6

8

θ

(deg)

cm

Figure 3.7: Angular distributions produced using the WSAW/FOLD/DWHI code package
for the 12 B(g.s., 1+ ) → 12 Be(2.24 MeV, 0+ ), 7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition.
still dominant at zero degrees.
• Figure 3.8 shows the theoretical angular distribution for the transition to the 2.68 MeV
1− state in 12 Be. As with Figure 3.4, the angular distributions are quite different
from those shown in the previous three figures: the ∆Jr =0 component is no longer
the largest at zero degrees, but the total cross-section can still be estimated using the
∆Jt =1 and ∆Jp =2 components only.

3.3

Extrapolation to q=0

The proportionality relationship between differential cross-section and B(GT) (Equation
1.12) is only valid when there is no transfer of angular momentum (∆L = 0), at the limit of
zero momentum transfer (q = 0). To extract the ∆L = 0 component of the differential crosssection, a Multipole Decomposition Analysis (MDA) was performed using cross-sections
calculated in the DWBA formalism (Figures 3.1 through 3.8. The MDA procedure and
39

dσ/dΩ(mb/sr)

1

All ΔJt=1, ΔJp=2
ΔJr=1, ΔJt=1, ΔJp=2
ΔJr=3, ΔJt=1, ΔJp=2
All ΔJt=1, ΔJp=1
All ΔJt=1, ΔJp=0
All ΔJt=3, ΔJp=1
All ΔJt=0, ΔJp=1
All ΔJt=3, ΔJp=0
All ΔJt=2, ΔJp=2
ΔJr=1, ΔJt=2, ΔJp=2
ΔJr=3, ΔJt=2, ΔJp=2
All ΔJt=3, ΔJp=2
ΔJr=1, ΔJt=3, ΔJp=2
ΔJr=3, ΔJt=3, ΔJp=2
ΔJr=5, ΔJt=3, ΔJp=2
All ΔJt=2, ΔJp=1
ΔJr=1, ΔJt=2, ΔJp=1
ΔJr=3, ΔJt=2, ΔJp=1

Total

10-1
10-2
10-3
10-4
10-5
0

2

4

6

8

θ

(deg)

cm

Figure 3.8: Angular distributions produced using the WSAW/FOLD/DWHI code package
for the 12 B(g.s., 1+ ) → 12 Be(2.68 MeV, 1− ), 7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition.

extraction of ∆L = 0 cross-section will be discussed further in Chapter 6. However, as
the theoretical cross-sections shown here were used to obtain a scaling factor for the q →0
extrapolation, the extrapolation process is discussed here rather than in the context of the
MDA.
Experimentally, q = 0 corresponds to a scattering angle of zero degrees and zero reaction
Q-value - therefore, to extrapolate to q = 0, the total cross-section is measured at forward
angles, and the zero-degree ∆L = 0 cross-section is obtained from the MDA. Then, DWBA
cross-sections are used to obtain a scaling factor relating the measured zero-degree ∆L = 0
cross-section to that at zero Q-value:


dσ (Q = 0, θ = 0)
dσ
dσ

=  dΩ
×
(Q = Q, θ = 0)
dσ (Q = Q, θ = 0)
dΩ q=0
dΩ
measured
dΩ
DW BA
40

(3.16)

Case

7 Li → 7 Be Transition

12 B → 12 Be Transition

q (fm)−1

1
2
3
4

7 Li(g.s) → 7 Be(g.s.)
7 Li(g.s.) → 7 Be(429 keV)
7 Li(g.s.) → 7 Be(g.s.)
7 Li(g.s.) → 7 Be(429 keV)

12 B(g.s.) → 12 Be(g.s.)
12 B(g.s.) → 12 Be(g.s.)
12 B(g.s.) → 12 Be(2.24 MeV)
12 B(g.s.) → 12 Be(2.24 MeV)

0.153
0.159
0.180
0.186

Table 3.3: Definitions and momentum transfer q at zero degrees for four cases relevant to
q=0 extrapolation procedure (see text).

(all cross-sections refer to ∆L = 0 components only).
The dependence of the DWBA ∆L = 0 cross-section at zero degrees on momentum
transfer q is shown in Figure 3.9. To calculate scaling factors, four separate cases needed
to be considered, involving transitions to the ground and 429 keV states in 7 Be (related
to singles and coincidence data sets) and transitions to the ground and 2.24 MeV states
in 12 Be (the two 0+ states of interest). These cases are labeled in Table 3.3, and the
zero-degree momentum transfer associated with each case is provided for reference. All
theoretical cross-sections were adjusted for resolution effects (see Section 6.1) before scaling
factors were calculated. The coincidence data set uses the Case 2 scaling factor for transitions
to the ground state of 12 Be and Case 4 for transitions to the 2.24 MeV state; the singles
data uses a weighted average of Case 1 and 2 for transitions to the ground state of 12 Be
and a weighted average of Cases 3 and 4 for transitions to the 2.24 MeV state (to reflect the
population of both 7 Be states in the singles data set). The averaging procedure introduces
an error of approximately 1%. The scaling factors used for the coincidence and singles data
set are given in Table 3.4.

41

∆L=0 ∆dσ/dΩ (mb/sr)

2

Case 1

1.75

Case 2
Case 3
Case 4

1.5
1.25
1
0.75
0.5
0.25
0

0

0.05 0.1 0.15 0.2 0.25
q(fm-1)

Figure 3.9: Zero-degree ∆L=0 cross-section as a function of linear momentum transfer q.
Case labels are given in Table 3.3.

Coincidences

Singles

12 Be(g.s.)

1.428

1.419

12 Be(2.24 MeV)

1.620

1.602

Table 3.4: DWBA scaling factors used in Equation 3.16 to extrapolate the zero-degree,
∆L=0 cross-sections to zero Q-value (and therefore zero momentum transfer).

42

3.4

Tensor Contributions

For a given form factor, the contribution from the central-στ and tensor-τ terms of the
nucleon-nucleon interaction interfere. The last line of Table 3.1, which is valid for central
terms only, changes to
∆Jr = ∆Lt + K

(3.17)

when tensor terms are present, with K equal to two. This results in Gamow-Teller transitions
(∆Lt = 0) with relative angular momentum transfers ∆Jr = 2, which cannot be isolated
from other ∆Jr = 2 components in the data. If the tensor-τ term is relatively strong, such
as in the case of very weak transitions or low incident beam energies, the proportionality
between cross-section and B(GT) (Equation 1.12) can be significantly affected, and in some
cases can break down entirely [84].
To estimate the systematic uncertainty in the extraction of B(GT) due to tensor interference, calculations were performed with the tensor terms of the effective nucleon-nucleon
interaction removed. An example of a no-tensor calculation is shown in Figure 3.10, and
should be compared with Figure 3.1, which has the tensor term included. Comparing the
∆Jr =0 cross-section at zero degrees with (T) and without (NT) inclusion of the tensor allows one to quantify the effect of the interference terms and estimate the possible systematic
error in the ∆Lt =0 cross-section due to tensor contributions. As with the extrapolation to
q=0, a weighted average was used to estimate the singles data tensor errors.
The percent difference in the ∆L=0 cross-section at zero degrees with (T) and without
inclusion of the tensor (NT) is given by
%error = 1 −
43

T
× 100
NT

(3.18)

dσ/dΩ(mb/sr)

Total
All ΔJt=1
ΔJr=0, ΔJt=1
ΔJr=2, ΔJt=1

1

10-1
10-2
10-3

All ΔJt=2

0

2

4

8

6

θ

(deg)

cm

Figure 3.10: Angular distributions produced using the WSAW/FOLD/DWHI code package
for the 12 B(g.s., 1+ ) → 12 Be(g.s., 0+ ), 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition,
with the tensor part of the effective interaction removed from the calculation. Should be
compared with Figure 3.1.
Coincidences

Singles

12 Be(g.s.)

+9.22%

+5.25%

12 Be(2.24 MeV)

+15.09%

+10.16%

Table 3.5: Systematic error in ∆L=0 cross-section due to interference of tensor components.

These systematic errors are shown in Table 3.5. With the tensor error quantified, all futher
discussions regarding theoretical angular distributions will refer to the ∆Jr =0 and ∆L=0
interchangeably, with ∆L = ∆Lt implied.
This chapter has detailed the calculation of theoretical angular distributions in the
DWBA framework using the WSAW/FOLD/DWHI code package. These angular distributions will ultimately be used in a Multipole Decomposition Analysis (MDA) to extract the
∆L = 0 component of the measured differential cross-sections (Chapter 6).

44

Chapter 4
Experiment
Charged with investigating the B(GT) distribution in 12 Be, NSCL experiment 08017 - "Spectroscopy of 12 Be using the 12 B(7 Li,7 Be + γ) reaction in inverse kinematics" - ran in November 2009. This chapter details the production of 12 B and detection of 12 Be in experiment
08017.

4.1

Beam Preparation and Delivery

Rare isotope beams at NSCL are produced via fragmentation. In this process, a beam
consisting of one stable isotope species is accelerated to semi-relativistic speeds and impinged
upon a thick target. The resulting nuclear collisions produce a distribution of isotopes,
including some rare species. The fragmentation products then go through a purification
procedure, during which a beam consisting primarily of the isotope of interest is selected,
tuned to experimenter specifications, and transported to the experiment.
45

4.1.1

Ion Source

The first step of beam production is the removal of electrons from stable element atoms. At
NSCL, this is accomplished using Electron Cyclotron Resonance (ECR) ion sources. ECR
ion sources confine plasma of a stable element in a magnetic bottle long enough for the
particles to be ionized by collisions with moving electrons. The ionized particles are then
extracted with an intermediate charge state and injected into the coupled cyclotrons for
acceleration.
This experiment marked the first use of the Superconducting Source for Ions (SuSI)
in a non-commissioning experiment. SuSI was designed to improve the intensity of heavy
ion beams with medium charge states. It is a fully superconducting system with tunable
magnetic fields, low electric power consumption, minimized risk of demagnetization, and an
improved design with more room for attaching external devices [85]. For experiment 08017,
SuSI was used to ionize 18 O atoms to a charge state of 3+.

4.1.2

Coupled Cyclotrons

The Coupled Cyclotron Facility at NSCL [86] currently houses two particle accelerators1 :
the K500 and K1200 cyclotrons, where the numeric designation refers to the maximum
extraction energy for protons. In a cyclotron accelerator, an electric field is applied at a
given radio frequency to accelerate ions, and a magnetic field is applied to keep the ions
on an isochronous path. For ions heavier than protons, the maximum energy achievable
depends on the radial size of the cyclotrons, the strength of the magnetic field, and the
charge-to-mass (q/m) ratio of the particle - since the first two quantities are restricted by
1 A third accelerator, ReA3, is currently under construction and will ultimately be used
to reaccelerate stopped rare isotope beams to low energies (less than 6 MeV/nucleon).
46

Ion Sources

K500

20 Feet
10 Meters

K1200
A1900

Focal Plane
Image 3

Image 1
Intermediate Image

Figure 4.1: Schematic depicting NSCL ion sources, K500 and K1200 cyclotrons, and A1900
fragment separator. The production target is located in the blue box after the exit of the
K1200 cyclotron and the before the entrance of the A1900 fragment separator.

the design of the cyclotron, for optimum acceleration high charge states (few electrons per
atom) are preferable. To achieve these high charge states, the particles from the ion source
are first accelerated in the K500, then extracted at an intermediate energy and impinged
upon a thin carbon foil. Passage through this foil strips electrons from the beam particles,
resulting in a higher charge-state that is injected into the K1200 for further acceleration.
After acceleration, this "primary beam" is impinged upon a production target, and since
relatively little momentum in the direction transverse to the beam axis is transferred in the
fragmentation process [87], most of the isotopes produced continue forward into the A1900
Fragment Separator [88]. A schematic of the ion source, coupled cyclotrons, and fragment
separator can be found in Figure 4.1.
The 18 O primary beam used in experiment 08017 was fully ionized (charge state of 8+)
and reached a final energy of 120 MeV/u. A Beryllium production target with a thickness of
1904 mg/cm2 was used for fragment production.
47

4.1.3

A1900 Fragment Separator

The A1900 Fragment Separator consists of four dipole magnets and eight quadrupole triplets,
and functions as a two-stage selection system. In the first stage, immediately following the
production target, the first two dipole magnets are tuned to disperse particles according to
their magnetic rigidity Bρ:
Bρ = p/q .

(4.1)

Here momentum p = γm/q where m refers to the mass of the particle, v the velocity, q the
charge, and γ = 1/ 1 − (v/c)2 accounts for relativistic effects.
In the second selection stage, a wedge is placed in the path of the beam at the intermediate
image point of the A1900 (see Figure 4.1). Particles passing through the wedge deposit energy
according to the Bethe formula, shown here as given in Ref. [89]:
dE
4πe4 Z 2
=
n z
dx
m0 v 2 abs abs

ln

2m0 v 2
− ln 1 −
I

v2
c2

−

v2
c2

.

(4.2)

Here e is the electronic charge, m0 is the electron rest mass, Z and v refer to the atomic
number and velocity of the beam particle, and nabs , zabs , I refer to the number density,
atomic number, and average ionization potential of the absorber material, respectively. As
it depends on Z 2 , the energy lost by the beam introduces an elemental velocity shift. Therefore after passing through the wedge, as the beam particles traverse the remaining two dipole
magnets, they are further dispersed according to their new Bρ value. Following this second
dispersion, slits located in the A1900 focal plane are closed to isolate the rare isotope of
interest for further transmission.
This two-stage selection system is quite powerful, and in many cases the resulting "secondary beam" is nearly isotopically pure. For experiment 08017, a 405 mg/cm2 Aluminum
48

wedge was placed at the intermediate image point, and the 12 B secondary beam selected was
more than 99% pure, with small amounts of contamination from 9 Li and 11 Be. The beam
rate was approximately three million particles per second. The total momentum spread of
the beam was reduced to ±0.25 % using slits placed at the Image 1 position, to allow for
unobstructed transmission when dispersion-matching optical elements further downstream
(see Section 4.2.1.1).

4.2

12

Be Production and Measurement

The experimental end-station was located in the S3 vault of NSCL and consisted of the
S800 Spectrograph [90], a large-acceptance, high-resolution magnetic spectrometer used for
particle detection, and the Segmented Germanium Array (SeGA) [91], used to detect gamma
rays. Both detection systems were needed to isolate 12 Be particles from other sources of
background and to select the single-step, spin-transfer charge-exchange events discussed in
Chapter 1 from those produced via other reaction mechanisms. The particle identification
procedure and gamma analysis will be discussed in the following chapter - this section describes the technical details of particle and gamma ray detection.

4.2.1

S800 Spectrograph

4.2.1.1

Dispersion Matching

The 12 B secondary beam selected by the A1900 Fragment Separator was transported through
the analysis line of the S800 Spectrograph (Figure 4.2). The analysis line consists of four
dipole and five quadrupole triplet magnets, and has two modes of operation. To achieve
49

Focal Plane

Analysis Line

Spectrograph

Figure 4.2: Schematic depicting analysis line and spectrograph components of S800 Spectrograph.

the maximum momentum acceptance (±2%), one can operate in focused mode, where the
image at the focal plane of the spectrometer is chromatic, and the momentum resolution
is determined by folding the intrinsic momentum spread of the beam with the momentum
change induced by straggling in the target. This mode of operation provides no information
about the incoming momentum of each particle however, and unless the momentum of each
beam particle is tracked, the achievable energy resolution is limited to about 1 part in 1000.
Since the experimental method used for experiment 08017 relied solely upon the information from the spectrograph to identify 12 Be particles and reconstruct their energy on an
event-by-event basis, it was imperative that the maximum energy resolution possible was
achieved. Thus the second operation mode - dispersion matching mode - was employed, in
which the entire system (analysis line and spectrograph) is tuned achromatically. As shown
in Figure 4.3, in this mode the dispersion-matched spectrograph is set to compensate for the
momentum dispersion of the beam on the target.
The physical principles of dispersion-matched optics can be described relatively simply if
50

one considers that in first order, the size of the beamspot at the target position of the spectrograph (sta ) can be written in terms of the magnification (MAL ) and dispersion (DAL )
of the analysis line, the size of the beamspot at the object of the analysis line (s0 ) and the
dispersion of the beam itself (δ):
sta = MAL ∗ s0 + DAL ∗ δ .

(4.3)

Analogously, the size of the beamspot in the focal plane (sf p ) can be written in terms of
the spectrograph magnification and dispersion (MS and DS ):
sf p = MS ∗ sta + DS ∗ δ .

(4.4)

Substituting sta from Equation 4.3 into Equation 4.4, one can see the size of the beamspot
in the focal plane factorizes into two terms: one depending on the object beamspot size and
one depending on the dispersion of the beam:
sf p = s0 (MS ∗ MAL ) + δ(MS ∗ DAL + DS ) .

(4.5)

For perfect dispersion matching, the size of the beamspot in the focal plane should be
independent of the dispersion of the beam, requiring
DAL =

−DS
.
MS

(4.6)

This can be checked using time-of-flight and position measurements during beam tuning, or
by carefully adjusting the magnets to achieve the optimum resolution for a transition to a
particular final state. For the S800 Spectrograph DS = −9.536 cm/%, and MS = −0.892,
and the analysis line (in dispersion-matched mode) has a dispersion of DAL = −10.684
cm/%, so Equation 4.6 holds to better than 0.1%.
The maximum energy resolution achievable also depends on the dispersion of the spec51

Figure 4.3: Cartoon depiction of lateral dispersion-matching, taken from Ref. [92]. Dispersion
of rays with different momenta is compensated by the dispersion of the spectrometer.

trometer and the size of the beamspot at the object position:
2MS ∗ s0
∆E
=
RE =
E
DS

(4.7)

and so for an object beam spot size of 0.05 cm the maximum resolution achievable is 1/10,000.
Since the momentum spread of the beam on the target is large in dispersion-matched mode
(DAL = 10.684 cm/%), the momentum acceptance was limited to ±0.25% to ensure unobstructed transmission of the beam through the beamline. This results in an approximately
5.25 cm tall beamspot on the target.
The target used in experiment 08017 consisted of nat Li (92.5% 7 Li and 7.5% 6 Li) and was
placed at the pivot point of the S800. The final dimensions of the target were 3.81 cm (wide)
by 5.08 cm (tall) by 104 µm (thick).The target was rolled to 5.5 mg/cm2 (104 µm) from an
original thickness of 10.68 mg/cm2 (200 µm), and was kept in an oxygen-free environment
prior to rolling to avoid oxidation. The rolling procedure itself took place in an argon gas
environment, and the target was only exposed to air briefly while being placed at the pivot
52

point of the S800. Due to the short exposure time, oxygen contamination is not considered
a significant concern for the experiment. A visual inspection of the target following removal
from the beampipe supported this determination, as no discoloration due to oxidation was
present. Even if minute quantities of oxygen were present, the large difference in Q-values
for the 12 B(16 O, 16 F)12 Be reaction (Q=-27.125) and the 12 B(7 Li, 7 Be)12 Be reaction (Q=
-12.569) would result in all 16 O events lying far outside of the excitation energy region of
interest for the (7 Li, 7 Be) study (Ex (12 Be)= 0 - 3.169 MeV).

4.2.1.2

Focal Plane Detectors

Following interaction with the target, reaction residues were detected in the focal plane of
the S800 [93], equipped with a suite of detectors to measure position, energy loss, and timing
information.
The dispersive and non-dispersive positions of the particles were measured by two Cathode Readout Drift Counters (CRDCs) separated in the z (beam) direction by a distance
of 1073 mm, as pictured in Figure 4.4. Each CRDC has an active area of 30 cm (in the
non-dispersive/y direction) by 59 cm (in the dispersive/x direction) and an active depth of
1.5 cm. The detectors were filled with a mixture of 80% CF4 gas and 20% C4 H10 gas at
a pressure of 40.3 Torr. This gas mixture was chosen for its high drift velocity, slow aging
characteristics and reduced probability of charge-spreading due to photon creation.
The dispersive position was determined by measuring the induced charges on the segmented cathode lead, integrated on each cathode pad for a finite amount of time following
a trigger signal from the Data Aquisition System (DAQ). The center of gravity of the integrated charge distribution, shown in the inset of Figure 4.4, was taken to be the x-position of
the event. In the non-dispersive direction the position was inferred from the time difference
53

(X2, Y2)

Ev
e
Ce

nt

Integrated Image Charge

ntr

Tr
aj

ec
tor

y

al R

5000
4000
3000
2000
1000
0

ay

(X1, Y1)

Pad Number

Figure 4.4: Layout of CRDC detectors and cartoon of cathode position determination. Figure
taken originally from Ref. [93], modified by Wes Hitt in Ref. [94] .

54

between the DAQ trigger and the collection of secondary electrons on the anode wire of the
CRDCs. The dispersive and non-dispersive angles were calculated by combining the position
measurements from both CRDC detectors.
After passing through the CRDCs the particles entered an ionization chamber filled with
P-10 gas (90% argon and 10% methane) at 140 Torr. There, direct ionization of the filling
gas by the incident particles results in the creation of positive ion-electron pairs. The number
of ion pairs produced is strongly correlated with the amount of energy deposited in the gas
(energy deposition being governed by Equation 4.2) - on average, one ion pair is produced
for every 25-35 eV of energy deposited. The precise energy-ion pair relationship depends on
the nature of the incident radiation (velocity and charge) and the species of gas involved,
but only weakly - empirical observations have shown the energy needed to create an ion pair
is a remarkably constant parameter for many gases and radiation types [89]. Unfortunately,
due to the combination of low atomic number for the incident particles (Z(Be)=4) and low
density of absorber atoms in the fill gas (nabs in Equation 4.2), very little energy was deposited by the reaction residues in the ionization chamber, and the energy loss information
from the chamber was not used in the analysis of the experiment.
Upon leaving the ionization chamber the particles impinged upon a suite of plastic scintillator detectors, which provided energy loss, total energy and timing information. Unlike
the gas-filled ionization chamber, the solid plastic scintillators had a much higher number
density of absorber atoms, and the energy loss information provided by the scintillator suite
was crucial for particle identification. In experiment 08017, only two of the scintillators in
the focal plane were operational - a 3 mm thick detector attached to the back of the ionization chamber which provided the "start" signal for the DAQ, and a 5 mm thick detector

55

Light Output (% Anthracene)
Wavelength of Max Emission (nm)
Decay Constant (ns)
Attenuation Length (cm)
Refractive Index
H/C Ratio
Density
Softening or Flash Point (degrees C)

65
423
2.4
250
1.581
1.103
1.032
70

Table 4.1: Properties of BicronT M BC-400 Scintillators, as cited in Ref. [89]

further downstream. Both scintillators were made of commercially available BicronT M BC400 plastic (C9 H10 ), the properties of which are given in Table 4.1.
Charged particles passing through the scintillator deposit kinetic energy according to
Equation 4.2, most of which is dissipated via lattice vibrations or heat. A small fraction
however, is absorbed by the organic molecules in the scintillator material and excites the
electron configuration into an excited state. Transitions back to the ground state configuration result in the emission of scintillation light, which occurs quickly - on the order of
nanoseconds - and is therefore often referred to as prompt fluorescence. Each scintillator was
equipped with photomultiplier tubes on top and bottom to measure the prompt fluorescence
and output a voltage signal proportional to the energy deposited by the reaction residue.

4.2.1.3

Magnetic Field Corrections

Magnetic field aberrations for the S800 Spectrograph are largely due to the fringe fields of
the two focusing quadrupole and two large dipole magnets. These aberrations are calculated
and corrected for analytically, by modeling the fringe profiles with Enge functions of the
form
E(z) =

1
1 + expP (z)
56

.

(4.8)

These functions are fit to the measured magnetic field maps, with z being the direction of
a reference path through the center of all magnets and P (z) a fifth-order polynomial. The
order is chosen so that the error in the correction is comparable to the resolution of the
focal plane detectors of the S800 [90]. The fringe profiles are then input into the code COSY
INFINITY [95], which calculates the aberrations, inverts the polynomial matrix and applies
corrections event-by-event in the analysis software.
Another benefit of using this analytical approach is that the inverted matrix ("inverse
map") can be used to relate the positions and angles measured in the focal plane to the
position, energy, and angles at the target - a procedure known as "raytracing" which allows
for easy reconstruction of kinematically relevant parameters on an event-by-event basis. In
first order, the inverse map equation is:
(dta, yta, ata, bta) = S −1 (xf p, yf p, af p, bf p)

(dta|xf p) (dta|yf p) (dta|af p)


(yta|xf p) (yta|yf p) (yta|af p)

=

(ata|xf p) (ata|yf p) (ata|af p)


(bta|xf p) (bta|yf p) (bta|af p)


(dta|bf p) xf p




(yta|bf p) yf p






 af p
(ata|bf p) 



bf p
(bta|bf p)


(4.9)

where all "−f p" variables refer to the focal plane, and "−ta" variables refer to the target.
x (a) refers to the dispersive position (angle) and y (b) refers to the non-dispersive position
(angle). To deduce the energy at the target, one must assume the dispersive beam position at
the object of the analysis line is zero, since 1) the entire spectrograph is tuned achromatically;
and 2) the beam position in the dispersive plane at the object cannot be calculated. dta
therefore refers to the fractional energy at the target:
dta =

E − E0
,
E0
57

(4.10)

with E0 being the energy of a particle traveling along the central path of the spectrometer.
Clearly, the S800 Spectrograph is a powerful tool for particle detection and provides
valuable information about the 12 Be particles produced in 12 B(7 Li, 7 Be) reactions. To
ensure clean identification of charge-exchange events however, a secondary detection system
is needed - SeGA.

4.2.2

Gamma Ray Detection

In the 12 B(7 Li, 7 Be)12 Be reaction, there are two distinct cases for which gamma-ray detection is relevant. First is the decay of the 7 Be recoil, which results in the emission of a 429
keV gamma ray and can be used as a "tag" for spin-transfer only charge-exchange events,
as described in Chapter 1. Second is the in-flight decay of excited 12 Be particles, which
emit gamma rays between 2 and 3 MeV. Detecting the gamma rays from both sources (7 Be
and 12 Be) is important for the analysis and understanding of the data taken in experiment
08017, therefore this section will address not only the techical details of gamma ray detection
but also the motivation for choosing SeGA as the detection system.
There are three mechanisms by which gamma rays (photons) interact with material: photoelectric absorption, Compton scattering and pair production. In photoelectic absorption,
the photon interacts directly with an atom in the material, removing an electron from a
bound atomic shell. The products of this interaction are therefore the electron (a.k.a. "photoelectron"), and an ionized absorber atom with an electron "hole" where the photoelectron
was removed. This is the primary mode of interaction for gamma rays with relatively low
energy (on the order of hundreds of keV). In Compton scattering, an incident photon is
scattered from an electron in the material, yielding in a recoil electron with finite energy
58

and changing the energy and direction of the original photon. This is the primary mode of
interaction for intermediate-energy gamma rays (hundreds of keV to several MeV). Finally, in
pair production the incident photon has enough energy to produce an electron-positron pair
(at least double the mass of the electron, or 1.022 MeV), and the subsequent annihilation of
the positron results in the creation of two annihilation gamma rays with energies of 511 keV
each. Pair production is the primary interaction mechanism for high-energy photons (several MeV and higher), and must take place while in the Coulomb field of the atomic nucleus.
For all three processes, the probability of interaction increases with the atomic number of
the absorber material - for photoelectric absorption and Compton scattering simply because
there are more electrons present, and for pair production, because the Coulomb field of the
nucleus is extended. For the gamma rays emitted in experiment 08017, from both the target
recoil (7 Be) and in-flight (12 Be) sources, photoelectric absorption and Compton scattering
are the dominant interaction mechanisms.
Depending on the source of the gamma ray being detected, there were two technical
requirements for gamma ray detection in experiment 08017. For the photons emitted from
7 Be, it was important to achieve the maximum efficiency possible, to obtain maximum statistics for the coincidence data set (which requires both a 12 Be particle in the focal plane of
the S800 and a 429 keV photon in the gamma ray detector). The 12 Be particles decaying
in-flight however, required a detection system with good position resolution, capable of determining a precise angle for the emitted gamma ray. This angle is needed to correct the
measured gamma ray energy Eγ,lab for the Doppler-shift due to the motion of the particle
when it emits the photon:
Eγ =

Eγ,lab (1 − β cos θ)
.
1 − β2

59

(4.11)

Here, Eγ is the Doppler-corrected energy of the gamma ray, θ is the angle between the
emitted gamma ray and the emitting particle, and β = v/c, where v is the velocity of the
emitting particle and c is the speed of light.
When experiment 08017 took place, there were two gamma detection systems readily
available for use at NSCL. The Segmented Germanium Array (SeGA), which had been in
use for several years, and a new Caesium Iodide Array (CAESAR) that had yet to be used
in an NSCL experiment. When efficiency is a priority for gamma-ray detection, and optimal
resolution is not required, inorganic scintillators similar to the sodium-doped caesium iodide
(CsI(Na)) detectors in CAESAR are used. The density of the material and the relatively
high atomic number of iodine ensures that a large fraction of all interactions are photoelectric absorption, and therefore the full photon energy is absorbed and most of the events
lie under a clear "photopeak" located at the energy of the incident photon. However this
efficiency comes at a price - the resolution of CsI(Na) detectors is approximately 7% for a
1 MeV gamma transition, whereas high-purity germanium detectors such as those in SeGA
have intrinsic energy resolutions of just 0.2% at the same energy. Since both efficiency and
resolution were priorities for experiment 08017, the decision of which detector array to use
had to be carefully considered.
Fortunately, information was available from a previous (7 Li, 7 Be) charge-exchange experiment, which took place in February of 2007 and investigated the nucleus 34 Si [41]. In
that experiment, many excited states in 34 Si were populated, resulting in a complicated
gamma energy spectrum with several photopeaks and Compton edges (lower-energy shoulders indicating incomplete photon energy deposition) contributing to a large continuum at
low energies. Were it not for the excellent resolution of the SeGA detectors used in that

60

experiment, it would have been difficult to identify the 429 keV transition from the continuum and isolate the spin-transfer charge-exchange events. Furthermore, it would have been
nearly impossible to perform a Doppler correction and decompose the gamma energy spectrum, since the segmentation of SeGA greatly improves the angular resolution, as described
in further detail below. Unsure of whether resolution would play such a significant role in
the analysis of expeirment 08017, it was decided to err on the side of caution and employ
the germanium array again. To maximize efficiency however, the detectors were placed in a
closely-packed configuration near the target, known as Barrel SeGA.

4.2.2.1

Barrel SeGA

As previously mentioned, the SeGA array was chosen as the gamma detection system for experiment 08017 due to its superior energy resolution (as compared to inorganic scintillators
such as CsI(Na)). The excellent resolution of germanium detectors can be attributed largely
to the response of the material after interaction with a photon. Reminiscent of the ion pairs
produced in ionization chambers (described in Section 4.2.1.2), when incident gamma radiation interacts with germanium of semiconductor purity, the energy deposition leads to the
creation of electron-hole pairs, caused by excitation of electrons from the valence band to the
conduction band. These electron-hole pairs can be guided through the material and collected
using an electric field, and the number of pairs created corresponds to the energy deposited
in the material. Since semiconductor detectors have small ionization energies (around 3 eV),
more charge carriers are created per unit of energy deposited than for gas chambers or scintillating materials.
Additionally, semiconductors can be "doped" with small concentrations of impurities,
61

which have excess electrons (n-type semiconductors) or holes (p-type semiconductors) compared to the pure semiconductor material. Since the SeGA detectors are n-type semiconductors, we will only describe those here - a complete description of semiconductors and doping
is given in [89]. In an n-type superconductor, the addition of an impurity results in an excess of electrons, left after covalent bonds are formed between the impurity and pure atoms.
These electrons are very lightly bound to the lattice, and therefore can be easily knocked
into the conduction band. The added concentration of electrons in the conduction band
increases the rate of electron-hole recombination, altering the equilibrium between electrons
and holes and ultimately increasing the total number of charge carriers.
The increased number of charge carriers (as compared to gas chambers and scintillating
materials, as well as un-doped semiconductors) has two beneficial effects on the energy resolution [89]: the greater amount of charge per pulse raises the signal/noise ratio, the most
important contribution to the resolution at low energies; and the statistical fluctuation in the
number of charge carrier pairs per pulse decreases with more charge carriers produced, the
most important contribution to the resolution at medium-high energies. However, because
of the small band gap in germanium (0.7 eV), the detectors must be cooled to liquid nitrogen temperatures to reduce thermal excitations from the valence to the conduction band, a
source of noise that would spoil their excellent energy resolution.
Until now, all mention of resolution has referred to the intrinsic resolution of the germanium detectors. One of the main advantages of using SeGA however, is a significant
improvement in the Doppler-corrected energy resolution due to the detector design. Each
SeGA detector is a coaxial germanium crystal, with an external diameter of 7 cm and length
of 8 cm, electronically segmented into eight 1 cm wide disks along the cylinder axis and

62

four radial segments perpendicular to the cylinder axis [91]. Since the Doppler-corrected
energy resolution depends on three factors - the intrinsic energy resolution of the germanium
detector (∆Eintr ), the uncertainty in the source velocity due to the slowing down of the
projectile in the target (∆β), and the uncertainty in the emission angle of the gamma ray
due to the finite opening angle of the gamma-ray detector and ambiguity in the angle of the
scattered particle (∆θ) - the 32-fold segmentation of the SeGA crystals (and the resulting
reduction of ∆θ for each gamma ray event) significantly improves the Doppler-corrected energy resolution compared to non-segmented detectors.
For experiment 08017 SeGA was employed in Barrel configuration to optimize efficiency
while maintaining adequate energy resolution. Barrel SeGA consists of 15 Germanium crystals arranged in two concentric rings around the target, as shown in Figure 4.5. The forward/downstream ring has one less detector than the backward/upstream ring due to space
limitations at the target position of the S800 Spectrograph. The large opening angle coverage and closely-packed arrangement of the Barrel SeGA configuration results in a source
efficiency of approximately 11.75% for 429 keV gamma rays (see Section 5.1.2.1 for details
of efficiency measurement). However, despite the segmentation of the individual detectors,
due to the 5 cm tall beamspot incident on the target in experiment 08017 and the close
proximity of the detectors to the target, the angle ambiguities were still quite large, and the
Doppler reconstructed energy resolution was limited. Details of the Doppler reconstruction
for experiment 08017 can be found in Section 5.1.2.2.

63

Beam Line

Secondary Beam
Figure 4.5: Cartoon of Barrel SeGA configuration. Figure modified from Ref. [96].

64

Chapter 5

Data Analysis

This chapter outlines the transformation from raw experimental observables to calibrated
physical quantities, such as excitation energy, scattering angle and differential cross-sections.
The analysis was carried out with a combination of three programs: NewSpecTcl [97], an
object-oriented C++ framework used mainly for online analysis; tv [98], used solely for
fitting gamma peaks in calibration source spectra; and Physics Analysis Workstation (PAW)
version 2.14/04 [99], a Fortran based analysis program that executes elementary operations
on data using user-defined subroutines. The analysis procedure is divided into four stages:
calibrations, describing separately S800 and SeGA calibrations; particle identification, i.e.
isolation of 12 Be events; reconstruction of the 12 Be excitation energy spectrum; and crosssection calculations.
65

5.1

Calibrations

5.1.1

S800 Calibrations

As described in Chapter 4, two Cathode Readout Drift Counters (CRDCs) were used to
determine the positions and angles of reaction products in the focal plane of the S800. Since
these positions and angles were ultimately used to reconstruct the excitation energy and
scattering angle immediately following the reaction, it is important that the CRDC detectors
be carefully calibrated. Since the position measurement can shift over time, three CRDC
calibrations were performed during experiment 08017: before the first 12 Be production run,
during the middle of the experiment after a short facility breakdown, and at the end of the
12 Be production period. The CRDC calibration information allows one to reliably combine
position data from production runs taken at different times during the experiment.
To perform the CRDC calibration, a Tungsten plate with a well-defined pattern of slits
and holes (see Figure 5.1) was inserted remotely into the path of the beam upstream of each
CRDC. Particles passing through the plate were detected in the CRDC, leaving an imprint
of the plate design on the position spectrum of the detector. A sample imprint from a CRDC
calibration run is shown in Figure 5.2 - comparing with Figure 5.1, the hole and slit pattern
of the Tungsten plate is visible.
First-order polynomials were used to relate the channel number from the data stream
(units of pad, ns) with a physical position (units of mm)
x1,2 (mm) = m1,2 (mm/pad) × x1,2 (pad) + b1,2 (mm)

(5.1)

y1,2 (mm) = n1,2 (mm/ns) × y1,2 (ns) + c1,2 (mm) .

(5.2)

66

Figure 5.1: Diagram depicting hole and slit pattern of CRDC masks.

Figure 5.2: Sample spectrum from CRDC2 mask run taken during experiment 08017.

67

m1
2.54

b1
-279.628

m2
2.54

b2
-278.495

n1
-0.068

c1
138.821

n2
0.070

c2
-142.595

Table 5.1: Mask calibration parameter values obtained from first CRDC calibration run of
experiment 08017.

This was done separately for each CRDC. In both directions, the offsets b1,2 and c1,2 were
set such that the vertex of the "L" shape seen in Figure 5.1 was located at 0 mm. In the
x-direction, where the position is determined from a charge induced on the segmented cathode pads, the slope m1,2 was fixed by the geometry of the detector to 2.54 mm/pad. In
the y-direction, the slope n1,2 depended on the drift velocity of the electrons to the anode
wire. The drift velocity depends upon the gas composition and pressure, as well as other
parameters that can vary during the course of one experiment. Fortunately, in experiment
08017, n1,2 did not change much between each calibration run, indicating that the CRDC
environment was stable and a single mask calibration could be applied to all production
runs. Therefore, the first calibration was used as a reference and small corrections to x and
y were performed on a run-by-run basis. The mask calibration values adopted are given in
Table 5.1.
Run-by-run corrections of x and y were performed using the 12 B(4+) charge-state, produced when fully ionized 12 B beam particles capture electrons from target atoms. Compared
to nuclear charge-exchange, the atomic charge-exchange process does not involve significant
energy and momentum transfer, so the charge-state measured in the focal plane of the
spectrograph designates the beam axis to high accuracy and provides information about
properties such as the angular spread of the incoming beam. For each production run, the x
and y positions of the charge-state were shifted such that the average position for each run
68

Parameter
xf p
yf p
af p
bf p

Definition
xc
1
c
y1
c − xc )/d
(x1
2
c
c
(y1 − y2 )/d

Table 5.2: Definitions of focal plane position and angles used in Equation 4.9. c superscripts
refer to corrected and calibrated parameters, and d is the distance between the two CRDC
detectors in the focal plane (1073 mm).

matched that of the first production run after the initial mask calibration. This correction
resulted in significantly improved position resolution when production runs were combined.
After completing mask calibrations and run-by-run corrections, it was possible to define
the positions and angles in the focal plane (xf p, yf p, af p, bf p, in Equation 4.9) that will be
"raytraced" back to the target position. The definitions of these parameters are given in
Table 5.2.

5.1.2

SeGA Calibrations

5.1.2.1

Source Calibrations

To calibrate the absolute energy and efficiency of the SeGA detectors, measurements were
taken with gamma-ray sources 152 Eu and 226 Ra before and after the experiment. Each
source was placed at the target position, 1.75 cm upstream of the center of the detector
array, for one hour. The calibration spectra were analyzed using the program tv [98].
For each of the 15 SeGA detectors, the photopeak of the transitions listed in Table 5.3
were fit with a Gaussian function atop a linear background. The centroid from the fit was
taken as the channel number for that gamma-ray transition. The energy of the transition
69

Source γ transitions used in calibration (keV)
152 Eu
226 Ra

224.7 344.3 411.1 444.4 778.9 867.4 964.1 1112.1 1212.9 1299.1 1408.0
186.2 241.9 295.2 351.9 609.3 665.5 768.4 806.2 934.1 1120.3 1238.1
1280.9 1377.7 1509.2 1661.3 1729.6 1764.5 1847.2 2118.5 2204.1 2447.7

Table 5.3: Gamma transitions used for energy and efficiency calibrations.

was then assumed to be quadratically related to the channel x:
Eγ = a0 + a1 x + a2 x2 .

(5.3)

The calibration coefficients (a0 , a1 , and a2 ) were chosen to minimize the χ2 of the calculated
and known energies:
χ2 =

(Eγ − Eknown )2
,
σ2

(5.4)

where σ is the error associated with the peak fit. As shown in Figure 5.3, the 152 Eu and
226 Ra calibrations for data taken before and after the experiment are in good agreement.
Since the 152 Eu source has transitions in the immediate vicinity of the 429 keV 7 Be ‘tag’,
the calibration coefficients obtained using this source were applied to all subsequent gamma
analyses.
For the efficiency calibration, the detectors were split into two groups based on their
position relative to the target/source position - "downstream" detectors occupying the ring
closest to the spectrograph, and "upstream" detectors occupying the ring closest to the
analysis line. The efficiency of each group was calculated using the number of counts in the
photopeak Nγ , the activity of the source A, the duration of the calibration run t adjusted
for the DAQ deadtime, and the emission probability for the specific transition ( emit , taken
70

152

1200

152

γ Energy (keV)

1400

Eu before
Eu after
226
Ra before
226
Ra after

1000
800
600
400
200

200

400

600

800

1000
1200
1400
Channel (arb. units)

Figure 5.3: Channel-Energy relationship and fits for 152 Eu and 226 Ra calibrations, taken
before and after the experiment.

from [100]):
(%) =

Nγ
× 100 .
A × t × emit

(5.5)

The efficiency is plotted as a function of the gamma-ray energy for both sources in Figure
5.4. The absolute radioactivity of the 152 Eu source was known, as the activity was measured
to be 8.46 µCi on May 1, 1978 and 152 Eu has a well-established half-life of 13.537 ± 0.006
years. The absolute activity of the 226 Ra source was not known to such high accuracy, so
a common scaling factor was applied to the entire efficiency curve to ensure agreement with
the 152 Eu values in the energy region for which the two sources overlap. To estimate the
efficiency for the 429 keV 7 Be transition, the 152 Eu total efficiency curve was fit with the
power-law function
−0.685 ,
(Eγ ) = 749.01Eγ

(5.6)

shown as the solid line in Figure 5.4. Based on this fit, the calculated efficiency for the 429
keV 7 Be transition was 11.75% ± 0.05%.
71

Efficiency ( )
%

total 152 Eu
total 226 Ra
upstream 152 Eu
downstream 152 Eu
226
upstream
Ra
226 Ra
downstream
152 Eu Fit

20
17.5
15
12.5
10
7.5
5
2.5
0

0

250 500 750 1000 1250 1500 1750 2000 2250 2500
γ Energy (keV)

Figure 5.4: Efficiency curves for 152 Eu and 226 Ra calibration source measurements.
5.1.2.2

In-Flight Calibrations

The 12 Be particles exiting the target in experiment 08017 were traveling at approximately
38% of the speed of light. The gamma rays emitted from the moving particles were Doppler
shifted in the laboratory frame: for forward (backward) angles, the measured energies were
higher (lower) than in the projectile frame. Since the energy in the projectile frame characterizes the nuclear transition, the energies measured with SeGA must be Doppler-corrected
through Equation 4.11 to be physically meaningful.
Due to the large beamspot (∼5.25 cm) and close-packed configuration of the detectors,
the Doppler-corrected energy resolution was dominated by uncertainties in θ, the detection
angle of the gamma ray in the laboratory frame. To mediate this uncertainty, 1.75 hours of
data were taken with the spectrograph set to detect 11 Be particles. Using the well-known
320 keV transition in 11 Be, it was possible to tune the position of the 420 SeGA segments
and reduce angle ambiguities. As shown in Figure 5.5, even without tuning the segment
72

Counts

2500

Gated on

2000

11
Be

1500
1000
500
0

250 500 750 1000 1250 1500 1750 2000 2250 2500
γ Energy (keV)

Figure 5.5: Doppler-corrected gamma energy spectrum, gated on 11 Be and including all 15
SeGA detectors.
position the 320 keV transition can be clearly identified in the Doppler-corrected gamma
energy spectrum.
The cos θ in Equation 4.11 depends on all three cartesian coordinates. To simplify the
segment position calibration, the x- and y-positions of the segments were assumed to be in
exact agreement with those in the "angle file", which defines initial x, y, and z positions for
each segment in the Barrel SeGA configuration relative to a target placed at the exact center
of the array. Then, an "effective" z-position was calculated for each segment using Equation
4.11, by first calculating the segment angle that would be necessary to force the observed
peak in the 11 Be spectrum (Eγ,lab ) to exactly 320 keV (Eγ = 320 keV)
cos θ =

1
β

1−

320
(1 − β 2 )Eγ,lab

,

(5.7)

then calculating the z-position associated with this "effective" segment angle
Zγ,ef f.angle =

x2 + y 2
.
tan θ

(5.8)

A plot of the "effective" z-position Zγ,ef f.angle vs. the z-position provided by the original
angle file Zγ,anglef ile is shown in the top panel of Figure 5.6. If the segment positions in
73

Zγ eff. angle

15
10

100

5
0

10

-5
-10

Zγ eff. angle

-15

-10 -7.5

-5

-2.5

0

2.5

15

5
7.5 10
Zγ angle file

1

10
5
10

0
-5

-10
-15

-10 -7.5

-5

-2.5

0

2.5

5
7.5 10
Zγ angle file

1

Figure 5.6: "Effective" vs "angle file" z-positions of the 420 segments in the Barrel SeGA
array. (Top) Before any position corrections were applied; (Bottom) after position corrections
were applied.

the angle file were optimized, this plot would show a line with a slope of unity, with uniform
vertical width caused by the size of the beamspot and the finite angular coverage of each
segment. Comparing the top panel of Figure 5.6 (before position corrections) to the lower
panel (after position corrections were applied) it is clear that the correction procedure results
in effective segment positions that more closely resemble the optimization criteria.
To illustrate the corrections applied, spectra before (a,b) and after (c,d) corrections are
shown for a single detector in Figure 5.7. Plotted is the difference between the "effective"
and "angle-file" segment z-positions, henceforth referred to as Zdif f . A shift of Zdif f of
-0.1 cm corresponds to shifting the centroid of the (Doppler-corrected) 320 keV peak by 1
74

30
20
10
0 -5 0 5
Zdiff

25
20
15
10
5

(b)

10

50

8

(c)

40

6

30

4

20

2

Segment

30

Counts

40

(a)

Segment

Counts

50

10
0 -5 0 5
Zdiff

0 -5 0 5
0
Z diff

30

(d)

25
20
15
10
5
0

-5 0 5
Zdiff

9
8
7
6
5
4
3
2
1
0

Figure 5.7: Difference between "effective" and "angle file" z-positions, detector 9. (a) Entire
detector, prior to corrections; (b) as a function of segment number, prior to corrections; (c)
entire detector, after corrections were applied; (d) as a function of segment number, after
corrections were applied.

keV. The detector shown is an example of a worst-case scenario - one of three detectors (of
fifteen in the array) that required corrections of this magnitude.
Evident in Figure 5.7 (b) is a correlation between Zdif f and segment number. This is
likely due to the low-energy gamma ray used in the correction procedure. The interaction
length in Germanium for a 320 keV photon is approximately 2 cm [101]. The 320 keV gamma
rays emitted by 11 Be therefore tend to deposit their full energy in the SeGA segments
segments closest to the target, skewing the angles observed.
To summarize, small shifts to the effective z-positions of SeGA segments were made using
the 320 keV transitions in 11 Be as a reference. The magnitude and direction of the shifts
were determined by grouping segments together (to aggregate sufficient statistics), and then
comparing the z-position calculated from the Doppler-shift of the 320 keV line with the zposition assumed by the gamma angle file. The corrected segment positions reflect effective
angles for each segment, rather than physical positions. Using these effective angles, the final
Doppler-reconstructed energy resolution for the 320 keV transition in 11 Be was roughly 7%.
75

However, since the angle corrections applied depend on the interaction length of the gamma
rays in Germanium, and the interaction length is highly energy-dependent, the Dopplerreconstructed energy resolution for the ∼2 MeV gamma transitions in 12 Be was estimated
to be worse - approximately 10% - although making a precise estimate was difficult due to
extremely low statistics.

5.2

Particle Identification

The 12 B secondary beam used in experiment 08017 was more than 99% pure, containing
small amounts of 9 Li and 11 Be. The nat Li target consisted of 92.5% 7 Li and 7.5% 6 Li,
and was mounted on a circular plastic frame. Considering the large beamspot, it is probable that some of the beam particles interacted with the frame as well as the target. It
is likely therefore, that in addition to the 12 B(7 Li, 7 Be)12 Be reaction of interest, several
other reaction channels involving the beam and/or target contaminants, or reactions on the
target frame, were also populated during the experiment. Fortunately, by first tuning the
spectrograph to transport only particles with magnetic rigidities comparable to those of the
12 Be charge-exchange ejectiles to the focal plane, and then exploiting energy loss, timing,
position, and angle measurements, it was possible to isolate the 12 Be particles of interest
from other reaction products and contaminants.
The majority of the reaction products from non-charge-exchange channels did not make
it into the focal plane due to momentum selection in the S800 (Bρ =3.928 Tm, corresponding to a central-ray energy of 949.96 MeV for 12 Be). Based on the magnetic rigidity setting
of the analysis line, the 12 B beam energy was 959.96±4.8 MeV. The ground-state Q-value
for the 12 B(7 Li, 7 Be)12 Be reaction is -12.56 MeV, so the maximum possible energy for
76

12 Be(g.s.) particles produced in this reaction would be 952.20 MeV. The energy acceptance
of the spectrograph is 10%, so the range of 12 Be energies accepted were 895.47 MeV to
999.81 MeV, which would correspond to 12 Be excitation energies up to Ex =50 MeV. Since
the experiment was only able to resolve to states below the neutron decay threshold in 12 Be
(Ex =3.169 MeV), this energy range encompassed the entire region of interest for experiment
08017.
Recall that the plastic scintillators of the focal plane measure both timing information
(time-of-flight TOF, relative to the RF of the cyclotron) and energy-loss information (∆E).
The TOF allows for velocity determination, which combined with Equation 4.1 provides information about the mass-to-charge ratio of the detected species. ∆E is directly correlated
to the atomic number Z of the particle, as shown in Equation 4.2. Combined, these two
measurements provide all the information needed for unambiguous particle identification,
and the ∆E - TOF two-dimensional spectrum is therefore often referred to as a PID (Particle IDentification) plot.
Figure 5.8 shows the PID plot for experiment 08017. In the top panel is the raw data,
without any corrections applied. Four cyclotron RF cycles are visible, centered at -1500,
-1050, -600 and -150. Since each cycle contains the same reaction products, in the center panel these cycles were combined for ease of viewing and to aggregate statistics. In
the bottom panel, corrections have been applied to TOF and ∆E for optical and detectordependent effects. The TOF corrections account for particles of the same species that have
slightly different flight paths through the spectometer. The ∆E corrections account for the
β-dependencies of the Bethe formula (Equation 4.2).
It is worth noting that this PID plot is significantly less resolved than most others ob-

77

∆E (a.u.)

1000
900
800
700
600
500
400
300
200
100
0
-2000 -1750 -1500 -1250 -1000 -750 -500 -250 0
TOF (a.u.)

10
1

10 2
10
-700 -600 -500 -400 -300
TOF (a.u.)

1
-200
10 3

∆E (a.u.)

1000
900
800
700
600
500
400
300
200
100
0
-1000 -900 -800

10 2

10 3

∆E (a.u.)

1000
900
800
700
600
500
400
300
200
100
0
-1000 -900 -800

10 3

10 2
10
-700 -600 -500 -400 -300
TOF (a.u.)

1
-200

Figure 5.8: (Top) Raw (uncorrected) PID plot for experiment 08017. (Center) PID after
four RF cycles have been collapsed. (Bottom) PID after TOF and ∆E corrections have been
applied.

78

tained in NSCL experiments. Since this experiment involved relatively light particles (A=12,
whereas most NSCL experiments involving the S800 spectrograph measure A=30-70) the signals from the ionization chamber were not sufficient for ∆E determination, resulting in poor
PID resolution. Currently a high-pressure ionization chamber is installed in the S800 focal plane, which has capabilities for lower-mass ∆E measurements. Had this detector been
available for experiment 08017, a cleaner PID plot would likely have been possible.
By far, the strongest feature in the PID plot is the 12 B(4+) charge-state (TOF∼ -600,
∆E∼675). Although useful for calibration purposes, the presence of the charge-state in the
PID plot can obscure the charge-exchange events. Traditionally, once the PID plot has been
corrected, the charge-state is removed and the particle of interest becomes clearly visible.
Unfortunately, in experiment 08017 there was a significant contaminant of unknown origin
overlapping with12 Be in the PID plot. Utilizing the full focal plane detector suite and information from SeGA (to identify charge-exchange events) it was possible to create a master
gate that consisted primarily of 12 Be particles. The limits imposed in this master gate are
shown in Table 5.4 and the gate has been applied on the final PID plot shown in Figure
5.9. Examples of the separation possible using this gate and the removal of any remaining
background contamination are discussed in the following section.

5.3

Excitation Energy Reconstruction

From the reconstructed energies (dta), and angles (ata, bta) one can determine the excitation
energy of 12 Be and the scattering angle. The excitation energy was calculated via a missing
mass calculation. In the following derivation, the speed of light has been set to 1.
The missing mass is found using the missing energy and missing momentum after recon79

Purpose

Signal

Good Event Gate

Charge-state removal
Background removal

xfp
TOF

xfp < -0.033 AND xfp > -0.02
-620 < TOF < -570

Background removal

∆E1

Background removal

xfp

440 < ∆E1 < 640
xfp > -0.1

Background removal

CRDC1(2) anode

Background removal

∆E2

Background removal

All ∆E

(a1+a2) > 3500 OR
a1 < 4000 OR a2 < 4000 OR
a1 > 1500 OR a2 > 1500
700 < ∆E2 < 900
∆E1 (a1+a2)
IC
∆E2 × 500 × 4400 × 100 > 750

Table 5.4: Gates applied to remove charge-state and background contamination from PID.

10

∆E (a.u.)

1000
900
800
700
600
500
400
300
200
100
0
-1000 -900 -800

10

-700 -600

-500 -400

1
-300 -200
TOF (a.u.)

Figure 5.9: PID with master gate applied.

80

2

struction:
mmiss =

2
Emiss − p2
miss

(5.9)

and the excitation energy is determined by accounting for the mass of the recoil
Ex (12 Be) = mmiss − m(7 Be) .

(5.10)

For the coincidence case (a 12 Be particle detected in the S800 is measured in coincidence
with a 429 keV gamma ray detected in SeGA), the recoil excitation should also be subtracted
from the missing mass. However, since for the higher-statistics singles data (S800 events
only) it is impossible to disentangle excitations through the ground state of 7 Be from those
through the 429 keV state, the recoil excitation was not explicitly included in the missing
mass calculation. The end result is that for the coincidence data the 12 Be ground-state
calculated via the missing mass calculation will be shifted to 429 keV (rather than zero). For
the singles data, there is a 429 keV ambiguity resulting from the population of both states
below the neutron threshold in 7 Be. Considering each of the terms in Equation 5.9:

• The missing energy is the excess energy in the 12 B(7 Li, 7 Be)12 Be reaction:
Emiss = Ek (12 B) + m(12 B) + m(7 Li) − Ek (12 Be) − m(12 Be) ,

(5.11)

where Ek refers to kinetic energy. Ek (12 B) was estimated using the program LISE++
[102] and then calibrated such that the ground-state of 12 Be in the singles data set
was centered at 0 MeV. Ek (12 Be) was calculated on an event-by-event basis using the
energy of a central ray through the spectrograph E0 (calculated in LISE++ using the
Bρ setting of the S800) and the raytraced fractional energy dta:
Ek (12 Be) = E0 (1 + dta) .
81

(5.12)

All masses were taken from Ref. [103].
• The missing momentum is found by first calculating the magnitude of the beam and
ejectile momenta from their respective kinetic energies and masses
p=

E 2 + 2Ek m .
k

(5.13)

Then, defining the beam axis as the z-axis, the cartesian components of the incoming
beam momenta are given by
px (12 B) = 0
py (12 B) = 0

(5.14)

pz (12 B) = p(12 B) .
The outgoing 12 Be momenta are found using the raytraced dispersive and non-dispersive
angles
px (12 Be) = p(12 Be) × sin ata
px (12 Be) = p(12 Be) × sin bta
pz (12 Be) =

(5.15)

p2 (12 Be) − p2 (12 Be) − p2 (12 Be) ,
x
y

and then since the x- , y- and z- components of the incoming and outgoing momenta
are known, the components of the missing momenta can be found by subtraction:
pmiss = px,y,z (12 B) − px,y,z (12 Be) .
x,y,z

(5.16)

Calculating the scattering angle in the laboratory frame is simple by comparison:
Θlab =

ata2 + bta2 .

(5.17)

The excitation energy of 12 Be obtained from the missing-mass calculation is shown in the top
panel of Figure 5.10. Clearly visible at the low-energy range of the spectrum is the 12 B(4+)
82

Resolution (FWHM)
E
ata
bta

900 keV
12 mrad
9 mrad

Table 5.5: Intrinsic energy resolution, and dispersive and non-dispersive angular resolutions
in experiment 08017.

charge-state, included in the PID gate to provide an estimate for the intrinsic energy and
angular resolution of the measurement, given in Table 5.5. In addition to the charge state
there are clearly two peaks, likely the ground-state and a superposition of excited states in
12 Be. To improve the energy resolution, correlations between the excitation energy of 12 Be
and the dispersive, non-dispersive, and scattering angle (shown in the center panel of Figure
5.10) were removed. These correlations reflect imperfections in the raytracing procedure.
Also in the lower panel, the absolute beam energy was calibrated by shifting the entire spectrum such that the12 Be ground-state was centered at Ex = 0. The fact that this overall shift
places the charge-state at ∼ -13 MeV, corresponding to the ground-state Q-value for the
reaction plus minor modification for the energy lost in the target, is additional confirmation
that shifting Ex in this way is not unreasonable.
Looking closely at the region between -3 and -11 MeV, it is clear there is background
contamination present in the 12 Be spectrum. This is likely due to the inability of the master
gate discussed in Section 5.2 to remove all contamination in the 12 Be region of the PID. Figure 5.11 shows the "All ∆E" gate indicated in the last line of Table 5.4, the most effective of
the background gates applied. The "All ∆E" threshold is indicated by the purple line (good
events are above the threshold, background events are below). A close examination shows it
was impossible to remove all background contamination without significantly reducing the
83

Counts

1200
1000
800
600
400
200
0
-15

-5

0

5

10

15

20

25

30

Ex (12Be) (MeV)

Counts

Ex (12Be) (MeV)

35
30
25
20
15
10
5
0
-5
-10
-15
-0.05

-10

1400
1200
1000
800
600
400
200
0
-15

35

10

0 -0.05 0.05
0
0.05
ata (rad)
bta (rad)

-10

-5

0

5

10

15

0

0.05
Θ lab (rad)

1

20 25 30 35
Ex (12Be) (MeV)

Figure 5.10: (Top) Excitation energy spectrum as generated with missing mass calculation;
(middle) Dependence of excitation energy on dispersive, non-dispersive, and scattering angle.
The 12 B4+ charge-state has been removed to ease viewing of 12 Be events ; (bottom) Anglecorrected excitation energy spectrum.

84

“All ΔE” Gate (a.u.)

2000
1800
1600
1400
1200
1000
800
600
400
200
0
-15 -10

10

-5

0

5

10

15

20 25 30 35
Ex (12Be) (MeV)

1

Figure 5.11: "All ∆E" as described in Table 5.4 plotted as a function of 12 Be excitation
energy. A division between good events and background contamination is visible - the purple
line shows the "All ∆E" threshold indicated in Table 5.4.
number of 12 Be events as well.
The following procedure was used to remove the remaining (post-master gate) background
from the excitation energy spectrum: (1) the counts in the excitation energy spectrum were
divided into 10 angular bins of Θlab = 5 mrad; (2) the "All ∆E" gate was reversed and set
so that no signature of 12 Be events could be seen in the excitation energy spectrum; (3)
in each angular bin the "background-only" excitation energy spectrum was plotted and fit
with a fourth-order polynomial, and the fit function was scaled to match the background in
the -5 to -10 MeV region of the data; (4) this scaled background function was subtracted
from the data in each angular bin. The top panel of Figure 5.12 shows a comparison of the
raw data, background, and background-subtracted data for one angular bin (Θlab = 10-15
mrad). The total background-subtracted excitation energy spectrum including all 10 angular
bins (Θlab =0-50 mrad) is compared with the raw data in the bottom panel of Figure 5.12.
To this point, all excitation energy spectra shown have involved the singles data, meaning no coincidence was required with the 429 keV gamma from 7 Be. These events include
both spin-transfer (∆S = 1) and non-spin-transfer (∆S = 0) contributions. To make sure
85

Counts

100
80
60
40
20

Data
Scaled Fit
To Background
Data Background
Removed

6
4
2
0
-10

-8

-6

-4

-2

-10

-5

0

5

10

15

800
700
600
500
400
300
200
100
0
-15

20
25 30
35
12
Ex ( Be) (MeV)

-10

-5

0

5

10

15

20
25 30
35
Ex (12Be) (MeV)

Counts

0
-15

Figure 5.12: (Top) Excitation energy spectrum for 12 Be, Θlab = 10-15 mrad. Raw data
shown in black, scaled background function in red, and background-subtracted data shown
in blue. Inset shows the region of the excitation energy spectrum used to perform the
background scaling. (Bottom) Same, but for Θlab from 0 to 50 mrad, and not including the
scaled background fit.

86

Counts

60

Coincidences

40
20

Counts

800

Singles

600
400
200
0

-10 -5

0

5

10 15 20 25 30
Ex (12Be) (MeV)

Figure 5.13: 12 Be excitation energy spectrum.
Background-subtracted singles data.

(Top) Coincidence data; (Bottom)

the features of the singles spectrum are be properly interpreted, it is helpful to compare the
singles data with the coincidence data (events where 12 Be was detected in the focal plane
and the 429 keV gamma transition was detected in SeGA). These are ensured to be spintransfer charge-exchange events, and the excitation energy spectrum for the coincidence data
is shown in the top panel of Figure 5.13. The background-subtracted singles spectrum is
shown for direct comparison in the lower panel of Figure 5.13. Comparing the two spectra it
is clear that the two peaks in the singles data are due predominantly to 12 B(7 Li, 7 Be)12 Be
events. The high-energy shoulder visible in the singles data but not in the coincidence data
is mostly due to excitations in 7 Be and 12 Be above the particle decay threshold, although
minor contributions could be caused by 6 Be produced by charge-exchange on the 6 Li contamination in the target.
The peak centered at zero is the ground state of 12 Be. The peak centered around ∼2
MeV could have contributions from three states (labeled by Jπ , where x=1 for the state of
x
87

lowest energy for a specific Jπ , x=2 for the next-lowest, etc): 2+ (Ex = 2.11 MeV); 0+ (Ex =
1
2
2.24 MeV, τ = 331 ns); and 1− (Ex = 2.68 MeV). Since the 2+ and 1− states have relatively
1
1
1
short lifetimes, information from the Doppler-corrected gamma energy spectrum was used
to estimate the contribution of these states to the ∼2 MeV peak. Figure 5.14 shows the
Doppler-corrected gamma energy spectrum from 1.5 to 4 MeV, gated on different regions of
the 12 Be excitation energy spectrum, indicated in the figure. The majority of the counts
in the gamma spectrum correspond to the higher-lying shoulder region of the Ex spectrum,
and these are likely due to transitions through 7 Be and 12 Be states above the particle decay
threshold, or the small background contamination remaining after the master PID gate was
applied. Therefore to obtain an accurate estimate of how much the 2+ and 1− state con1
1
tribute to the ∼2 MeV peak, it is necessary to gate on the 2110 keV and 2680 keV regions
of the gamma energy spectrum and look back at their relative contributions to the 12 Be
excitation energy spectrum, rather than rely solely on the counts in the Doppler-corrected
gamma energy spectrum.
The gamma-gated excitation energy spectra are shown in the lower panels of Figure 5.15.
The ungated singles spectrum is shown in the top panel for comparison. To estimate the
contribution of the 2+ or 1− states to the ∼2 MeV peak, the number of counts in the
1
1
gamma-gated excitation energy spectra were corrected for the in-beam detection efficiencies for the 2110 keV (2+ ) and 2680 keV (1− ) transitions, 4.1% and 3.2% respectively. The
1
1
(efficiency-corrected) number of counts in the ∼2 MeV peak region was then compared to the
number of counts in the same region in the ungated singles data. As previously mentioned,
most of the counts in the gamma energy spectrum are not coming from direct population of
the 2+ and 1− states, but rather from states above the decay threshold in 7 Be and 12 Be,
1
1

88

Counts

20

-2.75 < Ex(12Be) (MeV) < 0.75
No Ex Gates

10

Counts

0
20

0.75 < Ex (12Be) (MeV) < 3.5
No Ex Gates

10

Counts

0
20

4.0 < Ex (12Be) (MeV)
No Ex Gates

10
0

1500

2000

2500

3000

3500 4000
γ Energy(keV)

Figure 5.14: Doppler-corrected gamma energy spectrum, gated on 12 Be. Black corresponds
to all counts (no Ex gates applied). Shown in red are gated spectra, where the Ex (12 Be)
gates are indicated in the figure.

89

800
600
400
200
0
6

Gated on 2.11 MeV γ

4
2

Counts

0
6

Gated on 2.68 Mev γ

4
2
0

-4

-2

0

2

12

4

Ex ( Be) (MeV)
Figure 5.15: 12 Be excitation energy spectrum. (Top) Raw Singles Data; (Center) Gated on
2.11 MeV region of gamma energy spectrum; (Bottom) Gated on 2.68 MeV region of gamma
energy spectrum.
or from the remaining background contamination in the PID gate applied. The counts in
the ground-state peak region of the gamma-gated excitation energy spectrum provide an
estimate for the non-direct contributions, as none of the counts in the 2110 keV or 2680 keV
gamma energy regions could come from a direct population of the ground state of 12 Be.
Subtracting the non-direct contribution results in a 10.21% (4.55%) contribution to the ∼2
MeV peak from the 2+ (1− ) states. The gamma gates used were relatively wide (400 keV),
1
1
to ensure all the counts from the 2110 keV and 2680 keV peaks would be included, so the
percentages listed here should be considered upper limits on the contributions from these
states.
Further support for the dominance of the 0+ state in the ∼2 MeV region comes from the
2
relatively narrow width of the peak - just 400 keV wider (FWHM) than the ground state
90

in the singles data. If the 12 Be particles were decaying in-flight, the ∼2 MeV peak would
be significantly broader than the ground state, due to the momentum "kick" experienced
when the decay photon is emitted. At the extremes (0 and 180 degree emission angles), this
emission of a 2.24 MeV gamma ray would boost the 12 Be particle energy by ±940 keV, much
more than the 400 keV broadening observed. This suggests most of the nuclei populated in
this excited state do not decay in flight before reaching the focal plane, and for the 0+ state,
2
the long lifetime (τ = 331 ns) means only ∼33% of nuclei excited to this state in the reaction
would decay in flight before detection. This information, combined with the gamma analysis described previously, provides strong evidence that the 0+ state is likely the dominant
component of the ∼2 MeV peak in the 12 Be excitation energy spectrum. Further discussion
of the relative 0+ , 2+ , and 1− contributions to the ∼2 MeV peak can be found in Chapter
2 1
1
6, in the context of the Multipole Decomposition Analysis (MDA).

5.4

Cross-section Calculations

The last step of the data analysis procedure involves calculating differential cross-sections
dσ/dΩ to each of the states seen in the 12 Be excitation energy spectrum, and generating
an angular distribution (dσ/dΩ vs. scattering angle Θcom ). To calculate the differential
cross-section, counts in the excitation energy spectra for the singles and coincidence data set
(shown in Figure 5.13) are divided into discrete angular bins - 10 bins of 5 mrad each for the
singles data, and 5 bins of 10 mrad each for the coincidence data. The number of bins and
bin size was chosen such that there were sufficient statistics in each bin to identify and fit
the ground state and ∼2 MeV peaks within reasonable error. The excitation energy spectra
for the singles and coincidence data in each angular bin are shown in Figures 5.16 and 5.17.
91

The differential cross section is given by
dσ
Nm
1
=
,
dΩlab
Ni Nt L dΩlab

(5.18)

where Nm refers to the integrated counts in the ground state and ∼2 MeV peaks (after
background subtraction, for the singles case), Ni refers to the number of incident 12 B ions,
Nt refers to the number of 7 Li atoms in the target, L is the average live-time of the data
acquisition system (0.9), and dΩlab is the opening angle subtended by the scattering angle.
The efficiency of the focal-plane detectors is assumed to be 100%. For the coincidence
data, the efficiency of detection for the 429 keV gamma ‘tag’ must also be included in the
denominator of Equation 5.18.
In the laboratory frame, dΩ is obtained from a simple geometric interpretation:
dΩ = 2π

Θf

sin Θ dΘ ,

(5.19)

Θi
where Θi and Θf are the boundaries of the angular bin. Nt is easily calculated from the
target thickness (5.5 mg/cm2 ), target composition (92.5% 7 Li), and mass of 7 Li (1.165 ×
10−20 mg):
Nt =

5.5mg/cm2 × 0.925
= 4.409 × 1020 cm−2 .
−20 mg
1.165 × 10

(5.20)

Ni was calculated on a run-by-run basis using a periodic rate calibration. Three times
during the experiment (at the beginning, after a short facility breakdown, and towards the
end of the 12 Be production runs), a plastic scintillator located at the object of the analysis
line was inserted into the path of the beam and the unreacted beam was directed into
the focal plane of the spectrograph. This allowed for simultaneous measurements of the
rate on the object scintillator (located at the entrance to the S800 analysis line) and the
secondary beam rate (measured in the focal plane of the S800), providing information about
92

Θ = 0 - 5 mrad
lab

Counts

40

20

50

0

Θ = 10 - 15 mrad
lab

Counts

100

50

Θ = 20 - 25 mrad
lab

Counts

100

Θ = 15 - 20 mrad
lab

100

0

Θ = 25 - 30 mrad
lab

100

50

50

0

Counts

0

50

0

Θ
lab

= 30 - 35 mrad

0

Θ = 35 - 40 mrad
lab

40

50

20
0

Θ = 40 - 45 mrad
lab

40
Counts

Θ = 5 - 10 mrad
lab

100

Θ = 45 - 50 mrad
lab

20

20

0

0

-5

0

5

10

0

-5

0
5
10
Ex (12Be) (MeV)

Figure 5.16: Excitation energy spectrum, singles data, divided into angular bins used in
93
cross-section calculation.

Θ = 0 - 10 mrad
lab

5
2.5
0

Θ = 10 - 20 mrad
lab

10
5

Counts

0
10

Θ = 20 -30 mrad
lab

5
0
4

Θ = 30 -40 mrad
lab

2
0
Θ = 40 - 50 mrad
lab

4
2
0

-4

-2

0

2

4

6 12 8
10
Ex ( Be) (MeV)

Figure 5.17: Excitation energy spectrum, coincidence data, divided into angular bins used
in cross-section calculation.

94

the transmission through the analysis line (∼33%) and a calibration between the object
scintillator rate and the secondary beam rate. Since the object scintillator could not be in
the path of the beam during the experiment, it was also necessary to correlate the object
scintillator rate to non-intercepting probes located near the exit of the K1200 cyclotron and
at the production target position. These probes were continuously monitored throughout
the course of the experiment, and the average primary beam current readings were used to
estimate the number of incident beam particles during each run. In total, summed over all
runs, Ni was determined to be 1.21×1012 .
The procedure used to obtain Nm is illustrated in Figure 5.18. Although the figure
shown is for one angular bin in the singles data set, the same procedure was followed for
both coincidence and singles data, in all angular bins. First, the Ex = -2.5 to 3.57 MeV region
was fit using two Gaussian functions representing the ground state and ∼2 MeV peaks. Then,
the area under the second (first) Gaussian function was subtracted from the histogram and
the counts in the first (second) peak were integrated. The limits of integration were set using
the Gaussian fit functions (g(x) must be > 0.01 to be included in integration region), with
the exception of the upper limit for the ∼2 MeV peak, set to 3.57 MeV for all angular bins
in the singles data.
To ensure the counts in the ∼2 MeV peak in the singles data set aren’t overestimated,
one must carefully consider (i) higher-lying excitations in 7 Be; and (ii) the 6 Li present in
the nat Li target. To determine whether or not higher-lying excitations in 7 Be could be
contributing to counts in the ∼ 2 MeV peak, it is helpful to calculate where 12 Be events
populated through these states would lie in the excitation energy spectrum. The first state
in 7 Be above the 1/2− state at 429 keV is located at 4.57 MeV, and has a width of 175

95

keV (FWHM). If the charge-exchange reaction producing 12 Be in its ground state went
through this state, the corresponding peak in the excitation energy spectrum would have
a width comparable to that of the coincidence data (∼860 keV, FWHM) plus the width of
the unbound state itself (175 keV) and would be centered at 4.57 MeV. Since the upper
limit of the ∼ 2 MeV peak was set to 3.570, approximately 1.25% of the counts from the
12 B(7 Li, 7 Be(4.57 MeV))12 Be reaction could make it into the integration region for the
∼2 MeV peak. However, the relative probability of populating the 4.57 MeV state in 7 Be
(compared to the states below the neutron decay threshold) is unknown, so it is impossible
to estimate exactly the number of counts that should be removed from the second peak,
and the 1.25% of 12 B(7 Li, 7 Be(4.57 MeV))12 Be events that make it into the ∼2 MeV peak
must be taken as a systematic error in the cross-section calculation. It is possible to get a
more definite estimate for the 6 Li contamination present in the ∼2 MeV peak, following the
same logical procedure. The Q-value for the 12 B(6 Li, 6 Be)12 Be reaction is -15.996 MeV,
placing the ground state 12 Be events produced in this reaction at Ex =3.427 MeV, within
the summing region for the ∼2 MeV peak. With 12.33 times more 7 Li than 6 Li in the
[104], 6 Li should be
1.45 times stronger than B(GT)6
target, and B(GT)7
Li→6 Be
Li→7 Be
populated ∼1/18 times as frequently as 7 Li. However, only 55% of the 6 Li peak width is
included in the summing region, and therefore 3.1% of the counts in the ∼2 MeV peak were
assumed to be due to 6 Li contamination and removed from the cross-section calculation.
The differential cross-sections obtained in this manner have to be transformed to the
center of mass frame to be compared with theoretical cross-section calculations (see: Chapter
6). This essentially boils down to a transformation of the opening angle from the laboratory

96

Counts

100
50

Counts

0
100
50
0
Counts

100
50
0
-5

-2.5

0

2. 5

5

7.5

10 12.5 15
Ex (12Be) (MeV)

Figure 5.18: Illustration of peak-fitting procedure used to extract Nm .
to center of mass frame, which for small angles is
dΩcom
1 − β2
.
=
dΩlab
(β − 1)2

(5.21)

The angular distributions for the ground and ∼2 MeV states in 12 Be for the singles data
set are shown in Figure 5.19 and Table 5.6. The coincidence data angular distributions are
shown in Figure 5.20 and Table 5.7.

97

dσ/d Ω(mb/sr)

Ground State

0.4
0.2

dσ/d Ω(mb/sr)

0

~2 MeV State

0.6
0.4
0.2
0

1

2

3

4

5

6

7

Θcom (deg)

Figure 5.19: Angular distributions, singles data.

Angular Bin (Θlab )
0 - 5 mrad
5 - 10 mrad
10 - 15 mrad
15 - 20 mrad
20 - 25 mrad
25 - 30 mrad
30 - 35 mrad
35 - 40 mrad
40 - 45 mrad
45 - 50 mrad

dσ (g.s.)
dΩ
0.442 ±
0.369 ±
0.272 ±
0.169 ±
0.132 ±
0.118 ±
0.078 ±
0.051 ±
0.036 ±
0.028 ±

(mb/sr)
0.041
0.023
0.015
0.012
0.009
0.008
0.006
0.005
0.003
0.003

dσ (∼2MeV) (mb/sr)
dΩ
0.613 ± 0.052
0.452 ± 0.025
0.339 ± 0.018
0.267 ± 0.013
0.225 ± 0.011
0.160 ± 0.009
0.119 ± 0.007
0.069 ± 0.005
0.050 ± 0.004
0.030 ± 0.003

Table 5.6: Angular distributions, singles data.

Angular Bin (Θlab )
0 - 10 mrad
10 - 20 mrad
20 - 30 mrad
30 - 40 mrad
40 - 50 mrad

dσ (g.s.) (mb/sr)
dΩ
0.129 ± 0.029
0.077 ± 0.013
0.060 ± 0.010
0.021 ± 0.010
0.010 ± 0.003

dσ (∼2MeV) (mb/sr)
dΩ
0.150 ± 0.037
0.089 ± 0.015
0.070 ± 0.011
0.020 ± 0.007
0.007 ± 0.002

Table 5.7: Angular distributions, coincidence data.

98

Ground State

0.12
0.08
0.04
0
0.2

dσ/d Ω (mb/sr)

dσ/d Ω(mb/sr)

0.16

~2 MeV State

0.15
0.1
0.05
0

1

2

3

4

5

6

7

Θcom (deg)

Figure 5.20: Angular distributions, coincidence data.

99

Chapter 6

Results

Angular distributions in hand, this chapter focuses on the extraction of the Gamow-Teller
strength distribution for 12 Be. First, the ∆L = 0 component of the cross-section was isolated via a Multipole Decomposition Analysis (MDA) using the theoretical cross-sections
described in Chapter 3. Then, the zero-degree ∆L = 0 cross-section was extrapolated to
zero momentum transfer using the DWBA scaling factors given in Section 3.3. The ∆L = 0,
q = 0 cross-section was input into Equation 1.12 to extract the B(GT) distribution for the
two 0+ states in 12 Be, with the unit-cross section σ derived from the 12 Be → 12 B β-decay
ˆ
half-life. Finally, the B(GT)(0+ , 2.24 MeV)/B(GT)(0+ ,g.s.) ratio obtained in the current
2
1
work was compared with shell-model predictions, and the 0 ω component of the wavefunction deduced is compared with previous measurements and models.

100

6.1

Multipole Decomposition Analysis

For the ground state and ∼2 MeV excited state in 12 Be, in both the singles and coincidence
data set, a Multipole Decomposition Analysis (MDA) was performed to extract the ∆L = 0
component of the cross-section at zero degrees. The angular distributions shown in Figures
5.19 and 5.20 were fit with a linear combination of theoretical (DWBA) cross-sections
dσ DW BA
dσ
dσ
dσ
dσ
=a
+b
+c
+d
+ ... (6.1)
dΩ total
dΩ ∆L=0
dΩ ∆L=1
dΩ ∆L=2
dΩ ∆L=3
where fitting parameters a, b, c, d, and so on were allowed to vary freely until the total
DWBA cross-section best fit the angular distribution from the data. "Best fit" was defined
by the lowest reduced χ2 value (χ2 /N , where N is the number of angular bins less the
number of components included in the fit). Recall, that the error due to the tensor force has
been quantified (Section 3.4), and therefore ∆Jr in the theoretical angular distributions will
be used synonymously with ∆L in the MDA discussion.
Looking back at the calculations of Chapter 3, it is clear that for both the coincidence
case (where there were five angular bins) and the singles data set (ten angular bins), many
more ∆L components were calculated than could be included in any reasonably constrained
fit. Since the angular distributions produced in DWBA calculations depend strongly on
L-transfer and weakly on nuclear structure input, the calculated cross-sections are generally
reliable with respect to the order of magnitude of each ∆L component. Therefore the
strongest ∆L components of each state below the neutron decay threshold in 12 Be (ground
state, 2+ , 0+ , and 1− ) were selected for use in the MDA fits - these components are shown in
1 2
1
Table 6.1. Different combinations of these multipole components were tested systematically
until either: (1) the lowest χ2 /N value was achieved; or (2) it was determined that the χ2 /N
101

Coincidences

Singles

Ground State

011, 211, 221
∼2 MeV State (2+ ) 211, 011, 212;
1
(0+ ) 011, 211, 221;
2
(1− ) 112, 312, 111
1

011, 211, 221, 231, 431
(2+ ) 211, 011, 212, 202, 231, 431;
1
(0+ ) 011, 211, 221, 231, 431;
2
− ) 112, 312, 111
(11

Table 6.1: Theoretical DWBA cross-sections included in MDA fits. Three-digit labels refer
to ∆Jr , ∆Jt , and ∆Jp , as defined in Chapter 3.

was independent of the component choice.
Strictly speaking, the singles data MDA should include multipole components from both
the 7 Li → 7 Be(g.s., 3/2− ) and 7 Li → 7 Be(429 keV, 1/2− ) DWBA calculations (Figures 3.1
through 3.8). However, according to total angular momentum selection rules (Table 3.1),
all of the 7 Be(429 keV, 1/2− ) multipole components are included in the 7 Be(g.s., 3/2− )
calculation, and since the absolute magnitude of each multipole component is tuned in the
MDA fitting procedure, it is reasonable to use only the 7 Li → 7 Be(g.s., 3/2− ) multipole
components (Figures 3.5 through 3.8) in the MDA of the singles data.
Prior to performing the MDA fits, the DWBA cross-sections were adjusted for two factors:
resolution effects, as the experimental angular resolution was comparable to the bin size;
and binning effects, as the fine binning used in the DWBA calculations would not be an
accurate reflection of the widely binned angular distribution of the data. To account for
the experimental angular resolution (12 and 9 mrad in the dispersive and non-dispersive
direction, respectively), the following steps were taken: (1) the DWBA cross-sections were
fit with an 11th-order polynomial; (2) the polynomial parameters were imported into a
FORTRAN code, which generated the angular distribution in the laboratory frame; (3) the
laboratory frame angular distribution was smeared with the experimental resolution; and (4)
102

0.5
0

0 2 4 6 8
θcm(degrees)

0.5

∆Jr=2
∆Jt=1
∆Jp=1

0.25
0

dσ/dΩ (mb/sr)

1

∆Jr=0
∆Jt=1
∆Jp=1

dσ/dΩ (mb/sr)

dσ/dΩ (mb/sr)

1.5

0 2 4 6 8
θcm(degrees)

0.01
0.008
0.006
0.004
0.002
0

∆Jr=2
∆Jt=2
∆Jp=1

0 2 4 6 8
θcm(degrees)

Figure 6.1: Original (black) and smeared (red) DWBA angular distributions for the 12 B(g.s.,
1+ ) → 12 Be(g.s., 0+ ), 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition. Only states
included in the MDA are shown.

the smeared distribution was converted back into the center-of-mass frame. The original and
post-smearing angular distributions are shown in black and red in Figures 6.1 through 6.4.
Note that the sharp features (peaks, valleys) of the original distributions become washed
out, and the smearing procedure has a significant effect on the ∆L(∆Jr ) = 0 component at
zero degrees.The effects of re-binning are not shown explicitly here but can be seen in the
MDA fits for the singles and coincidence data to follow.

6.1.1

Extraction of ∆L = 0 cross-section, Singles Data

The angular distributions of the ground and ∼2 MeV states in the singles data were best fit
with a combination of ∆L = 0, 2, and 4 components, and the final MDA fits are shown in
Figure 6.5. A relatively strong ∆L = 2 component was required to fit the angular region
between 3 and 5 degrees, and the ∆L = 4 component was needed to fill out the angular
distribution at the largest angles. At zero degrees the ∆L = 0 component is strongest. A
χ2 /N of 4.27 was achieved for the ground state, and for the ∼2 MeV state χ2 /N was 0.59.
The MDA scaling factors for the ∆L = 0 (a, Equation 6.1), ∆L = 2 (c, Equation 6.1) and
103

∆Jr=2
∆Jt=1
∆Jp=1

0.04
0.02
0

0.15
0.1

+

21
0

5
θcm (deg)
∆Jr=1
∆Jt=1
∆Jp=2

0.05 1 1
0
0
5
θcm (deg)

0

+

02
0

∆Jr=0
∆Jt=1
∆Jp=1
21+

0.05
0

0.004

5
θcm (deg)

0.01

0

dσ/dΩ (mb/sr)

5
θcm (deg)

0.1

0.004

dσ/dΩ (mb/sr)

0

dσ/dΩ (mb/sr)

0

∆Jr=2
∆Jt=1
∆Jp=1

0.2

5
θcm (deg)
11-

0.15
0.1

∆Jr=3
0.05 ∆Jt=1
0 ∆Jp=2
0
5
θcm (deg)

dσ/dΩ (mb/sr)

0.2

dσ/dΩ (mb/sr)

∆Jr=0
∆Jt=1
∆Jp=1
02+

dσ/dΩ (mb/sr)

dσ/dΩ (mb/sr)
dσ/dΩ (mb/sr)
dσ/dΩ (mb/sr)

0.4

02+

0.002 ∆Jr=2
∆Jt=2
0 ∆Jp=1
0
5
θcm (deg)
21+

0.002 ∆Jr=2
∆Jt=1
0 ∆Jp=2
0
5
θcm (deg)
∆Jr=1
∆Jt=1
∆Jp=1

0.01
0.005
0

110

5
θcm (deg)

Figure 6.2: Original (black) and smeared (red) DWBA angular distributions for transitions
to the ∼2 MeV state in 12 Be, for the 7 Li(g.s., 3/2− ) → 7 Be(429 keV, 1/2− ) transition.
Only states included in the MDA are shown.

104

0.02

0.2
0

0 2 4 6 8
θcm(degrees)

∆Jr=2
∆Jt=3
∆Jp=1

0.01
0

0.4

0 2 4 6 8
θcm(degrees)

dσ/dΩ (mb/sr)

0
0 2 4 6 8
θcm(degrees)

∆Jr=2
∆Jt=1
∆Jp=1

dσ/dΩ (mb/sr)

1

dσ/dΩ (mb/sr)

∆Jr=0
∆Jt=1
∆Jp=1

dσ/dΩ (mb/sr)

dσ/dΩ (mb/sr)

2

x10-4
0.4
0.2

∆Jr=2
∆Jt=2
∆Jp=1

0 2 4 6 8
θcm(degrees)

∆Jr=4

0.04 ∆Jt=3
0.02 ∆Jp=1
0

0 2 4 6 8
θcm(degrees)

Figure 6.3: Original (black) and smeared (red) DWBA angular distributions for the 12 B(g.s.,
1+ ) → 12 Be(g.s., 0+ ), 7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition. Only states included
in the MDA are shown.

Multipole Component

Scaling Factor, g.s.

Scaling Factor, ∼2 MeV state

∆L = 0
∆L = 2
∆L = 4

0.460 ± 0.037
0.309 ± 0.040
1.409 ± 0.189

1.529 ± 0.141
1.091 ± 0.092
3.205 ± 0.349

Table 6.2: Scaling factors for ∆L = 0, 2, and 4 components, as defined in Equation 6.1, for
calculations in which all 2 ω configurations were shifted by ∆E= -3 MeV.

∆L = 4 (not labeled, Equation 6.1) are given in Table 6.2.
For the ground state, fits were made including two, three, four and five multipole components. Including additional ∆L = 2 components merely fragments the total ∆L = 2 strength,
and has no significant impact on the ∆L = 0 cross-section at zero degrees. As the shape
of the ∆L = 2 components are similar (see Figure 6.3), exchanging the ∆Jr ∆Jt ∆Jp =211
component with the other ∆L = 2 combinations given in Table 6.1 only acts to increase the
magnitude of the ∆L = 2 coefficient in Equation 6.1 - as before, there is no significant change
in the ∆L = 0 cross-section at zero degrees. Since the choice of ∆L = 2 component was
somewhat arbitrary, the ∆Jr ∆Jt ∆Jp =211 component was selected, as it was the strongest
105

dσ/dΩ (mb/sr)

0.5
02 ∆Jr=0
∆Jt=1
0.25
∆Jp=1
0
5
0
θcm (deg)

0.2

dσ/dΩ (mb/sr) dσ/dΩ (mb/sr)

∆Jr=2
+ ∆Jt=3
02
∆Jp=1
0
0
5
θcm (deg)

0.02
0.01
0

+

0.005

∆Jr=0
+ ∆Jt=1
0.1
21
∆Jp=1
0
0
5
θcm (deg)

0.1
0

0.004
0.002
0

02+
0

∆Jr=2
∆Jt=1
∆Jp=1

5
θcm (deg)

∆Jr=4
∆Jt=3
∆Jp=1
0
5
θcm (deg)
02+

0.04
0.02
0

∆Jr=2
∆Jt=1
21+
∆Jp=1
0
5
θcm (deg)

∆Jr=2
∆Jr=2
∆Jt=1 0.002
∆Jt=0
21+ ∆Jp=2
21+ ∆Jp=1
0
0
5
0 0
5
θcm (deg)
θcm (deg)

dσ/dΩ (mb/sr)

∆Jr=2 0.004
∆Jr=4
21+
0.001
∆Jt=3 0.002
∆Jt=3
21+ ∆Jp=1
∆Jp=1
0
0
0
5
0
5
θcm (deg)
θcm (deg)
dσ/dΩ (mb/sr)

x10-4
∆Jr=2
0.2
∆Jt=2
+
0.1 02
∆Jp=1
0
0
5
θcm (deg)

∆Jr=3
∆Jt=1
0.1
11∆Jp=2
0
0
5
θcm (deg)

∆Jr=1
0.02
∆Jt=1
0.01
11∆Jp=3
0
0
5
θcm (deg)

∆Jr=1
∆Jt=1
11∆Jp=1
0
0
5
θcm (deg)

0.01

Figure 6.4: Original (black) and smeared (red) DWBA angular distributions for transitions
to the ∼2 MeV state in 12 Be, for the 7 Li(g.s., 3/2− ) → 7 Be(g.s., 3/2− ) transition. Only
states included in the MDA are shown.

106

0.4

dσ/dΩ (mb/sr)

dσ/dΩ (mb/sr)

0.5

0.8

∆Jr=0 ∆Jr=2 ∆Jr=4
∆Jt=1 ∆Jt=1 ∆Jt=3
∆Jp=1 ∆Jp=1 ∆Jp=1
χ2 = 4.27

0.3
0.2
0.1
00

2

4

6

0.6

θcm (deg)

χ2 = 0.59

0.4
0.2
0

8

∆Jr=0 ∆Jr=2 ∆Jr=4
∆Jt=1 ∆Jt=1 ∆Jt=3
∆Jp=1 ∆Jp=1 ∆Jp=1

0

2

4

6

8

θcm (deg)

Figure 6.5: MDA fits for the ground state (left panel) and ∼2 MeV state (right panel), singles
data.

of the three ∆L = 2 components calculated.
For the ∼2 MeV state, the situation is slightly more complicated, as hypothetically components from the 0+ , 2+ , and 1− states could all contribute to the angular distribution.
2 1
1
However, as seen on the right side of Figure 6.5, the ∆L = 0, 2, and 4 components from the
0+ state do an excellent job of reproducing the experimental angular distribution. No fit in
2
which a ∆L = 1 component was included converged, and fits including a ∆L = 3 component
converged but yielded ∆L = 0 cross-sections that agreed with the ∆L = 2 and 4 component
fits within error bars (difference in ∆L = 0 cross section was less than 1%). At forward
angles, a strong ∆L = 3 component is unlikely without a strong ∆L = 1 component, so
the ∆L = 2 and 4 components were chosen for the final MDA fit. This result supports the
conclusion made using the in-flight gamma energy spectrum, which placed an upper limit
of 5% on the contributions from the 1− state. The shape of the angular distribution for
1
multipole components with the same ∆L are very similar - so including other ∆L = 0, 2
or 4 components in the fit (including those from the 2+ state) has negligible effect on the
1
107

dσ
(mb/sr)
MDA
dΩ ∆L=0
Ground State
0.482 ± 0.039
+
2.24 MeV 02
0.496 ± 0.046

q=0
0.684 ± 0.055
0.796 ± 0.073

Table 6.3: ∆L = 0 cross-sections, from MDA and after extrapolation to q = 0.

∆L = 0 cross-section at zero degrees. This, combined with the 10% limit from the in-flight
gamma analysis, justifies the use of only the 0+ ∆L = 0, 2, and 4 components in the final
2
fit.
It is worth noting that the error in the ∆L = 0 cross-section at zero degrees is dominated
by the fit to the first two angular bins (θcm =0-1.6 degrees), and the second bin (θcm =0.80.16) carries more weight than the first. χ2 /N , however, depends on all ten bins, including
those at large angles. Therefore, the addition of higher multipoles like ∆L = 2, 3, 4 (which
peak at larger angles) to the MDA fit reduces the χ2 /N value and improves the overall fit,
but has very little effect on the ∆L=0 cross-section at zero degrees. This results in the
relatively small error bars for the ∆L = 0 component at zero degrees included in Table 6.3.
Since the angular distribution for the ∼2 MeV state can be adequately reproduced using
only components from the 0+ state in 12 Be, and the gamma analysis discussed in Chapter
2
5 limits the contributions of the 2+ and 1− states to 10 and 5%, respectively, in all further
discussions of the B(GT) distribution the ∼2 MeV state will be referred to as the 2.24 MeV
0+ state, and any contribution from the 2+ or 1− state will be neglected. The zero-degree
2
1
1
∆L = 0 cross-sections found using the MDA are given in Table 6.3. Also included in the
table are the ∆L = 0 cross-sections extrapolated to q = 0 using the scaling factors of section
3.3.
108

dσ/dΩ (mb/sr)

0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0

∆Jr=0 ∆Jr=2
∆Jt=1 ∆Jt=1
∆Jp=1 ∆Jp=1
χ2 = 1.61

4
2
θcm (deg)

6

8

∆Jr=0 ∆Jr=2
∆Jt=1 ∆Jt=1
∆Jp=1 ∆Jp=1
χ2 = 1.83

0

4
6
2
θcm (deg)

8

Figure 6.6: MDA fits for the ground state (left panel) and ∼2 MeV state (right panel),
coincidence data.

6.1.2

Extraction of ∆L = 0 cross-section, Coincidence Data

Limited to five angular bins in the coincidence data, at most four theoretical cross-section
components could be included in the MDA. However, due to the large statistical errors in
this data set, it was impossible to deduce which DWBA cross-sections should be included.
Based on the singles data MDA for the ground and excited state, ∆L = 0 and 2 components
were selected (see Figure 6.6), but since an independent MDA could not be performed the
∆L = 0 cross-sections at zero degrees for the coincidence data set were not ultimately used
to extract Gamow-Teller strengths.

6.2

12

Be B(GT) Distribution

The final step for extracting the B(GT) distribution in 12 Be is to calibrate the unit-cross
section σ using the proportionality relationship given in Chapter 1 [13]
ˆ
dσ
= σ B(GT ) .
ˆ
dΩ q=0
109

(6.2)

Ex (12 Be)
0 MeV (g.s)
2.24 MeV

Jπ
0+
1
+
02

B(GT)
0.184 ± 0.008
0.214 ± 0.051

Table 6.4: B(GT) distribution, 12 Be.

This relationship can be calibrated using the 12 Be(g.s.) → 12 B(g.s.) B(GT), calculated
from the β-decay ft-value,
ft =

C
,
B(F ) + (gA /gV )2 B(GT )

(6.3)

where log(f t)=3.834±0.017 [105], B(F )=0 for a 0+ → 1+ transition, C=6143±2 s [11] and
gA /gV =-1.2694±0.0028 [12]. After taking into account detailed balance
(2JBe + 1)
B(GT,12 B →12 Be) =
B(GT,12 Be →12 B)
(2JB + 1)

(6.4)

this results in a B(GT) of 0.184±0.007. Using this B(GT) and the ground state to ground
state ∆L = 0 cross-section at q = 0, one obtains a unit cross-section σ =3.722±0.334.
ˆ
Applying this unit-cross section to the 2.24 MeV 0+ state yields a B(GT) of 0.214±0.028.
2
The B(GT) distribution for 12 Be is given in Table 6.4. The ground state error consists
only of the error in the β-decay log(ft) value, propagated through the B(GT) calculation.
The error for the 2.24 MeV state includes: statistical and fitting errors coming from the
χ2 minimization errors on the ∆L = 0 component, propogated through the extrapolation
to q = 0 and conversion to B(GT) (8% for the ground state, 9% for the excited state);
and systematic errors from tensor contributions (11%, see section 3.4), q=0 extrapolation
assumptions (1%, see section 3.3), peak-fitting procedure (4%), and smearing of DWBA
angular distributions (1%).
110

6.3

Comparison with Shell Model Predictions

The Gamow-Teller strength for transitions to the 0+ states in 12 Be is very sensitive to the
mixing of 0 ω and 2 ω configurations. As discussed in Chapter 2, the ratio of B(GT) for
the 0+ states
B(GT )(0+ , 2.24M eV )
2
R=
B(GT )(0+ , g.s.)
1

(6.5)

can provide information about the 0 ω component of the ground and excited 0+ state
wavefunction. The B(GT) ratio obtained in the current study was 1.162±0.283. This ratio
was compared to that predicted by OXBASH shell model calculations, using the WBP
interaction in the spsdpf model space. As described in Section 2.3, all 2 ω configurations
were manually adjusted by ∆E to account for the effect of 4 ω and higher admixtures.
Thirty-one separate calculations were performed, with ∆E varying from -2.0 to -5.0 MeV
in increments of 0.1 MeV. Each calculation is one data point in the left (right) panel of
Figure 6.7, as a function of the percentage of the ground (excited) state wavefunction that is
made up of 0 ω configurations. The ratio found in the current study is indicated by the red
line, with the shaded red area representing the error. This ratio indicates the ground state
wavefunction is 25.48±5.49% 0 ω, and the excited state wavefunction is 59.83±4.78% 0 ω.
The B(GT) ratio is plotted as a function of the ratio of 0 ω configurations in the excited
state to 0 ω configurations in the ground state in Figure 6.8. As predicted by a twolevel mixing scheme, the influence of 2 ω configurations has an opposite effect on the 0 ω
percentage of the ground state and excited states. However, it is notable that the 0 ω
percentages of the 0+ wave functions do not sum to 100%.
111

Ground State

20 40 60 80
0ħω % of wavefunction

B(GT)(02+)/B(GT)(g.s.)

B(GT)(02+)/B(GT)(g.s.)

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

02+ State

20 40 60 80
0ħω % of wavefunction

B(GT)(02+)/B(GT)(g.s.)

Figure 6.7: Ratio of 0+ to 0+ B(GT) values, plotted as a function of the percentage of the
2
1
ground (left) and excited (right) state wavefunction that consist of 0 ω configurations. The
red line indicates the B(GT) ratio found in the current work; the red shaded area is the error
associated with this value.

2
1.5
1
0.5
0

0 1 2 3 4 5 6
0ћω% (02+)/0ћω% (g.s)

Figure 6.8: Ratio of 0+ to 0+ B(GT) values, plotted as a function of the ratio of 0 ω
2
1
wavefunction percentage between the two states. The red line indicates the B(GT) ratio
found in the current work; the red shaded area is the error associated with this value.

112

6.4

Discussion

The 0 ω wavefunction percentages obtained in the current work are compared with those
extracted from neutron knockout [62, 64] and (d,p) transfer [71] measurements in Table
6.5. Also included in Table 6.5 are the predictions of Barker [68], Fortune and Sherr [69],
and Romero-Redondo et. al [70]. The differences between these measurements and models
was discussed in detail in Chapter 2. The current study predicts 0 ω contributions to the
ground and 2.24 MeV 0+ states that are generally consistent with previous measurements and
2
models, with the exception of the recent (d, p) transfer study and the predictions of Barker,
both of which under-predict the 0 ω component of the 0+ state wavefunction. The transfer
2
result given is an upper limit for the 0 ω contribution - since the measurement was only
sensitive to (0s)(0p)6 (1s)2 component, the exact contribution from other 2 ω configurations
is unknown.
One major difference in the current measurement is that the 0 ω fraction of the 0+ and
1
0+ state’s wavefunctions do not sum to 100%. This suggests that models which deduce the
2
configuration of the 0+ state based on the assumption that the 0+ and 0+ state (or vice1
2
2
versa) should sum to 100% of the 0 ω component are may be relying on flawed assumptions.
Nevertheless, the current measurement clearly shows the ground state wavefunction of 12 Be
is predominantly made up of 2 ω configurations, and the 2.24 MeV 0+ state is predominantly
2
0 ω in nature, further evidence for the breakdown of the N = 8 shell closure and additional
fodder for the theoretical disputes of Fortune and Sherr, Barker, Romero-Redondo et al.,
and those to come.

113

12 Be(0+ , 2.24 MeV)
2
25.48±5.49
59.83±4.78
32
N/A
≤72
≤27
31
42
32
68
13-19
71-78
12 Be(g.s.)

Current Work
Knockout [62, 64]
Transfer [71]
Barker [68]
Fortune and Sherr [69]
Romero-Redondo et al [70]

Table 6.5: Percentages of ground and excited state wavefunctions that consist of pure 0 ω
configurations.

114

Chapter 7
Conclusion

7.1

Summary

The (7 Li, 7 Be) charge-exchange reaction has been employed in inverse kinematics to study
the Gamow-Teller strength distribution in 12 Be. The only (n,p)-type charge-exchange probe
currently available for use at intermediate energies with rare isotope beams, the (7 Li, 7 Be)
reaction is a powerful tool, providing detailed structure information in a largely modelindependent manner. It has been shown to be especially useful for examining nuclei near
neutron shell closures, with the first experiment providing further evidence that the N = 20
shell gap is not quenched for 34 Si, and the most recent work illuminating the extent of
quenching of the N = 8 shell gap in 12 Be.
The present work describes the second successful application of the (7 Li, 7 Be) reaction in inverse kinematics. Coupling a high-resolution magnetic spectrometer and a highly
segmented germanium array, it was possible to isolate 12 Be particles produced in chargeexchange processes from numerous other reaction channels, and to reconstruct their exci115

tation energy and scattering angle. Angular distributions were generated and decomposed
into multipole components using theoretical cross-sections produced in Distorted Wave Born
Approximation (DWBA) calculations. The ∆L = 0 cross-section at zero degrees was extrapolated to zero momentum transfer and used to extract Gamow-Teller strengths of transitions
from the ground state of 12 B to the 0+ (ground state) and 0+ (2.24 MeV) state of 12 Be. This
1
2
extraction was possible due to the existence of a well-established proportionality between the
∆L = 0 differential cross-sections at the limit of zero momentum transfer and Gamow-Teller
strength [13], calibrated model-independently using the 12 Be→12 B Gamow-Teller strength
from β-decay.
The extent of configuration mixing in the 0+ states of 12 Be has been an open question
for decades. The ratio of the Gamow-Teller strength of the 2.24 MeV 0+ state to the 0+
1
2
ground state was compared with shell model calculations, and was found to be very sensitive to the 0 ω component of the wavefunction. Exploiting this sensitivity allowed us to
pin-down the 0 ω component of the 12 Be ground state to 25.48±5.49%, and deduce for the
first time the 0 ω component of the 2.24 MeV 0+ state (59.83±4.78%). Both 0+ states
2
had smaller 0 ω configuration percentages than previous neutron knockout studies had indicated [62, 64] (although they agree with our result within 2σ error limits), and neither
the shell-model calculations of Barker [68], nor the Coulomb-shift model of Fortune and
Sherr [69], nor the three-body model of Romero-Redondo et. al [70] exactly reproduce our
observed wavefunctions, although the latter two are quite close.

116

7.2

Outlook

The future is bright for charge-exchange. In forward kinematics, NSCL continues to lead the
way in new probe development, exploring heavier composite probes such as (10 Be, 10 B+γ).
Similar to (7 Li, 7 Be), the (10 Be, 10 B) probe has a gamma transition (1.022 MeV, from the
0+ state at 1.74 MeV to the 1+ state at 0.718 MeV) that can be used to isolate non-spintransfer (∆S = 0) transitions. This non-spin-transfer selectivity makes the (10 Be, 10 B)
probe particularly useful for studying non-spin-transfer giant resonances such as the Isovector Giant Monopole Resonance (IVGMR). Such an experiment was recently proposed to
the NSCL Program Advisory Committee, to run with the new gamma ray tracking system
GRETINA.
In inverse kinematics, additional (7 Li, 7 Be) work has been proposed at NSCL and
RIKEN, to investigate areas of the nuclear chart where configuration mixing is not wellunderstood (such as near shell closures and the Island of Inversion). (p, n) programs are
now underway at NSCL, GSI, and RIKEN, in which low-energy recoil neutrons are used to
reconstruct excitation energy spectra. These experiments employ a liquid hydrogen target,
itself a significant technological achievement. Finally, in the future, a (d, 2 He) program in
inverse kinematics may be possible using an active target Time Projection Chamber, such
as the one currently being built at NSCL.
The (7 Li, 7 Be) reaction in inverse kinematics has proven useful for studying light and
medium-mass nuclei, and the current work has demonstrated that for certain cases, meaningful structure information can be extracted from singles data as well as data requiring a
charged particle+gamma coincidence. Despite this success, however, there is always room
for improvement, including: the use of a high-efficiency gamma ray detection system (such
117

as CAESAR) to allow for more coincidence statistics (as high-resolution was not necessary
to isolate the 429 keV 7 Be gamma from background); and the addition of a neutron detection array to kinematically reconstruct states beyond the neutron decay threshold in 12 Be.
The shell-model calculations performed could benefit from the use of a more realistic interaction than the modified WBP interaction described here, perhaps a new interaction for
the 0s0p1s0d shell, fit to data from stable and rare isotopes. If computational capacities
continue to expand, in the near future it may be possible to compare these results to ab
initio calculations as well.
For more than 40 years, nuclear scientists have been debating the extent of configuration
mixing in 12 Be, in an attempt to understand the evolution of nuclear structure away from
stability. This work brings us one step closer to that goal, quantifying the 0 ω component of
the 0+ states. This may be one small piece placed into a much larger puzzle, but in general,
the outlook is good for nuclear science - with new rare isotope facilities coming online and
novel experimental techniques being employed everyday, we are inching ever closer to the
ultimate goal: a fundamental understanding of the atomic nucleus.

118

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