SPECTROSCOPY OF NEUTRON UNBOUND STATES IN NEUTRON RICH
CARBON
By
Shea Mosby

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 NEUTRON UNBOUND STATES IN NEUTRON
RICH CARBON
By
Shea Mosby
Neutron unbound states in 19,21 C have been populated by nucleon removal from a
beam of 22 N at 68 MeV/u, and the invariant mass method was used to reconstruct
decay energies of the reaction products. Charged fragments were bent from the
beam axis by a dipole magnet and their properties measured by a suite of charged
particle detectors sufficient to identify individual reaction products and reconstruct
momentum vectors at the reaction target. Neutrons were detected at forward angles
by the Modular Neutron Array (MoNA), which measured their trajectory and time
of flight.
A resonance at 76 ± 14 keV was observed in the decay spectrum of 19 C, corresponding to an excited state at 383 ± 15 keV. Comparison to shell model calculations

results in the assignment of this state to the 5/2+ state in 19 C, and the resulting level
1
scheme indicates a breakdown in existing shell model descriptions of this nucleus. In
the case of 21 C, an s-wave line shape was used to describe the experimental decay
energy spectrum, resulting in a scattering length limit of |as | < 1.5 fm. This result
suggests that 22 C is bound by less than 100 keV.

DEDICATION

To Michelle

iii

ACKNOWLEDGMENT
I would like to thank Michael Thoennessen for serving as my advisor during my time
as a graduate student. Thank you for making the effort to direct my work while
allowing me the freedom to follow my own trains of thought and make many of my
own mistakes. I would also like to thank my committee members (Artemis Spyrou,
Hironori Iwasaki, Scott Bogner, James Linnemann, and Phillip Duxbury) for their
time spent reviewing my work. In particular, Artemis was a valuable source of advice
and motivation during my time at the NSCL.
The NSCL operations staff was invaluable during the execution of my experiment,
providing a high quality beam well above the predicted rates which ultimately made
my results tractable. I am indebted to the MoNA Collaboration for the construction
and maintenance of MoNA, without which my thesis work would never have existed.
My fellow MoNA graduate students (Greg Christian, Jenna Smith, Jesse Snyder,
and Michael Strongman) provided invaluable support during the execution of my
experiment, as well as uncounted hours of input and discussions. Thanks is also due
to the other members of the NSCL graduate student body - too many students to
mention by name - who assisted in the work presented here, whether it be a clever
programming trick or an encouraging word on an otherwise discouraging day.
Warren Rogers deserves particular thanks for his role in starting me on the path
that led to this point. As an academic advisor, role model, and friend he awoke my
interest in Physics as more than a hobby and has served as an immense source of
information, advice, and inspiration.
Words fail to express the debt I owe my wife, Michelle, for her contributions to
this work. Thank you for the countless hours you’ve spent supporting me on this
endeavor while working toward your own doctorate - I never could have done this
without you. Your influence permeates all aspects of this work, whether it was logic
checking a thought I had, encouraging me to try “just one more” idea out on a
iv

particularly difficult analysis challenge, or suffering through and providing emotional
support during the difficult times. Finally, I’d like to acknowledge the rest of my
family for their support during this whole process. Without you all I wouldn’t be the
person I am today.

v

TABLE OF CONTENTS

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

viii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Theory and Motivation . . . . . . . . .
2.1 Shell Model Calculations . . . . . . . . . .
2.2 Shell Evolution Near N = 14 . . . . . . . .
2.2.1 N = 14 shell gap . . . . . . . . . .
2.2.2 The N=15 Probe . . . . . . . . . .
2.2.3 22 C as a Two-neutron Halo . . . .
2.3 Decay Energy Lineshapes . . . . . . . . .
2.3.1 Breit-Wigner Resonance . . . . . .
2.3.2 S-wave Dynamics . . . . . . . . . .
2.3.3 Nonresonant Contributions . . . . .
2.4 Estimators: Maximum Likelihood Method

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3 Method . . . . . . . . . . . . . . .
3.1 Invariant Mass Reconstruction . . . . .
3.2 Beam Production . . . . . . . . . . . .
3.3 Experimental Setup . . . . . . . . . . .
3.3.1 Beamline Detectors . . . . . . .
3.3.2 Charged Fragment Detection . .
3.3.3 MoNA . . . . . . . . . . . . . .
3.3.4 Electronics and DAQ . . . . . .

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34
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4 Analysis . . . . . . . . . . . . . . . . . . . . .
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Calibrations . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Timing Scintillators . . . . . . . . . . . . . .
4.2.1.1 RF Timing Shifts . . . . . . . . . .
4.2.1.2 Virtual Target Scintillator Timing
4.2.2 Tracking Detectors . . . . . . . . . . . . . .
4.2.3 dE, E Scintillators . . . . . . . . . . . . . .
4.2.4 Ionization Chamber . . . . . . . . . . . . . .
4.2.5 MoNA . . . . . . . . . . . . . . . . . . . . .
4.3 Particle Identification . . . . . . . . . . . . . . . . .
4.4 Reconstruction . . . . . . . . . . . . . . . . . . . .
4.4.1 Fragment Momentum Four-Vector . . . . . .
4.4.2 Neutron Momentum Four-Vector . . . . . .

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4.5

Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 19 C Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 21 C Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 110
Bibliography

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vii

LIST OF FIGURES

1.1

1.2

2.1

2.2

2.3

2.4

2.5

2.6

Schematic rendering of the development of a shell model, with example
energy level shifts for 24 O. . . . . . . . . . . . . . . . . . . . . . . . .

2

Chart of nuclides with “magic numbers” illustrated and this work’s
beam and reaction products of interest indicated. . . . . . . . . . . .

4

Collapse of the N=14 shell gap demonstrated in two ways by [12]. Top,
a comparison of the measured 2+ energies as a function of neutron
number shows that at N=14 in oxygen (a) an increase occurs which is
not matched in the carbon isotopic chain (b). Bottom, ESPEs calculated in the WBP interaction demonstrate the large gap between the
ν0d5/2 and ν1s1/2 shells for N=14 gap in oxygen (c). In carbon, this
gap is significantly diminished (d). . . . . . . . . . . . . . . . . . . .

10

Comparison of 18,20 C experimental level schemes to WBT, WBT*
interactions from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Resulting γ-ray spectrum for bound excited states in 19 C from [16]
and [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Resulting decay energy spectrum for unbound excited states in 19 C
from [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Experimental level scheme for 19 C based on results from [12, 16, 17].
For interpretation of the references to color in this and all other figures,
the reader is referred to the electronic version of this dissertation. . .

16

Evolution of the ν1s1/2 − ν0d5/2 shell gap for N=15 oxygen, nitrogen,
and carbon isotones. The dashed line shows a linear extrapolation of
the gap to carbon following the prescription of [20], and experimental
data are taken from [13, 18, 19]. . . . . . . . . . . . . . . . . . . . . .

18

viii

2.7

Matter radii for carbon isotopic chain, indicating a large increase for
22 C [26]. Inset show the matter density and demonstrates a large
matter tail thought to be a neutron halo. . . . . . . . . . . . . . . . .

19

Correlations between the two-neutron separation energy and rms neutron halo radius in 22 C for given positions of a virtual state in 21 C
from [27]. Each curve is calculated for a specific energy of the 21 C
virtual state, and range from 0 to 100 keV in steps of 10 keV. . . . .

20

Level scheme indicating the predicted locations of 21,22 C in energy
relative to a 20 C core [27]. . . . . . . . . . . . . . . . . . . . . . . . .

21

2.10 Coordinate system and associated parameters for s-wave calculations.

26

2.11 Diagrammatic representation of how nonresonant background arises in
the decay spectrum for 18 C in coincidence with neutrons. Highly excited states in 20,21 C are populated by nucleon removal from the 22 N
beam, and these evaporate neutrons to other high-lying states until
19 C∗ is reached (thick arrows). 19 C∗ decays via emission of a neutron
with large decay energy (thin arrow), which is unlikely to be observed
because of the geometric efficiency of the setup. Evaporation between
continuum states results in neutron emission with lower relative energy,
which is likely to be observed and results in background. . . . . . . .

30

3.1

Neutron evaporation schematic. . . . . . . . . . . . . . . . . . . . . .

35

3.2

Beam production at the CCF. . . . . . . . . . . . . . . . . . . . . . .

37

3.3

Experimental setup diagram. . . . . . . . . . . . . . . . . . . . . . . .

39

3.4

Schematic of CRDC operation from [43]. The incident charged particle
ionizes the gas in the active volume, and the freed electrons move
toward the anode wire under the drift field. A Frisch grid reduces
position dependent charge collection effects, and the charge collection
on the anode induces charge on the aluminum pads which determines
the dispersive position of the interaction. The drift time of the electrons
determines non-dispersive position. . . . . . . . . . . . . . . . . . . .

41

3.5

Trigger logic diagram. . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.1

Timing drifts as a function of event number for all beam line timing
signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.8

2.9

ix

4.2

The downscaled RF timing signal showing up as three peaks in an uncorrected time spectrum (a). In (b), the 43.18 ns shift corresponding
the cyclotron frequency does not match the middle RF peak, while adjusting the shift parameters slightly (see text) results in good agreement. 51

4.3

Resolution comparison between the target chamber and thin scintillators for a no-target run. The beam particle peaks in the spectrum
utilizing the target chamber scintillator (right solid) is a factor of 2.5
wider than the corresponding peaks in the time spectrum utilizing the
thin scintillator (left) despite the A1900 to thin flight path being longer.
This widening is entirely due to resolution effects. The virtual target
scintillator (right dashed) improves the resolution by ∼20%. . . . . .

52

4.4

Notational reference for virtual target scintillator calculations . . . .

54

4.5

Velocity relationships calculated by LISE for all beam-line elements
necessary for virtual target scintillator calculations. All trends are
linear, and the CRDC trends lie very near each other because the
energy loss is minimal through those detectors. . . . . . . . . . . . . .

57

Sample mask hole position spectrum for CRDC2. The positions of
the holes are known which enables the mapping of the raw detector
information onto physical units. . . . . . . . . . . . . . . . . . . . . .

58

Drift time of ions in all CRDCs as a function of event number. The
discontinuity near event 2·106 resulted from a technical issue changing
the drift velocity of ions in the focal plane CRDCs. . . . . . . . . . .

59

TCRDC2 charge collection issue (left) and the correlation between
TCRDC positions used to replace the physical device’s x position information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Thin (top left) and thick (top right) response as a function of x position
and the corresponding correction curves. The bottom panels show
the correction taking effect for both scintillators, which results in a
flattened spectrum with position correlations removed. . . . . . . . .

63

4.10 Sample ionization chamber pad signal vs. Y position for all pads. . .

66

4.11 Position and time dependence of pad 1. . . . . . . . . . . . . . . . . .

67

4.6

4.7

4.8

4.9

x

4.12 Resolution comparison of uncorrected ion chamber and physical target
scintillator (a) to corrected ion chamber and virtual target scintillator
(b). Element separation is only possible in (b), and the improved
timing resolution allows clean separation of 22 N missing the target. .

68

4.13 γ-ray timing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.14 Beam profile and beam ID spectra. . . . . . . . . . . . . . . . . . . .

72

4.15 Element separation by two different methods - ∆E vs. Time of Flight
and ∆E vs. E. Both methods are combined to reduce contamination
from unreacted beam. . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.16 Particle tracks for different carbon isotopes through the sweeper magnet (a), and a traditional particle ID attempt from plotting dispersive
angle vs. time of flight (b). Correlations between angle, position, and
time of flight preclude such a treatment (see text). . . . . . . . . . . .

77

4.17 Example Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.18 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.19 Projection of correlations into 2D. . . . . . . . . . . . . . . . . . . . .

81

4.20 ”Emittance” parameter vs. time of flight. . . . . . . . . . . . . . . . .

82

4.21 Final corrected time of flight with grey regions indicating particle gates. 83
4.22 Consistency check for magnetic field map transformations showing reasonable agreement between the forward and inverse tracking processes.

86

4.23 Comparison of simulation to data for 22 N reaction fragments. . . . .

90

4.24 Comparison of simulation to data for 20 C reaction fragments. . . . .

92

4.25 Comparison of simulation to data for 18 C reaction fragments. . . . .

93

Comparison of level schemes for 19 C using old and new mass evaluations. Note that only the neutron unbound 5/2+ is affected by the
2
change in relative mass. The new level scheme is better described
by WBP*, consistent with observations for other neutron rich carbon
isotopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5.1

xi

5.2

Best fit to experimental data (solid line) along with the relative contributions of a 76±14 keV resonance (dashed) constituting 54% of
the spectrum and nonresonant background from high lying continuum
states (dotted) modelled as a Maxwellian distribution with Θ=2.44 MeV. 96

5.3

Level scheme of 19 C compared to theory relative to 18 C+n with both
the Elekes state (solid red) and our state (dashed) included. . . . . .

97

Experimental spectrum from [17] with the estimated strength of the
5/2+ shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1

99

5.4

5.5

Our experimental spectrum with a best fit including the 5/2+ state
2
from [17] with relative population determined by cross section estimates from [63]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.6

Calculated neutron stripping cross sections of the 5/2+ relative to the
1
total [63]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.7

Reconstructed decay energy spectrum for 21 C (a), where squares represent data points and triangles represent an s-wave curve of as = -0.05
fm with the same binning as the data. Panel (b) demonstrates the sensitivity of the setup to a low-lying virtual state by showing simulated
decay energy line shapes for s-waves with differing scattering lengths.
The inset shows the geometric efficiency of the setup as a function of
decay energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.8

Experimental level schemes of 20,21 C and theoretical level schemes of
21,22 C, with 20 C data from [65, 12] and N3 LOW predictions from [66].
The lack of a resonance structure in the 21 C decay energy spectrum
indicates that the virtual state’s strength is broadly spread as indicated
by the grey region. This is consistent with WBP* and N3 LOW calculations which predict 21 C to be unbound by 1.7 MeV and 1.6 MeV,
respectively. The calculations show a 1.2 MeV spread in the predicted
22 C binding energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.9

Correlations between the two-neutron separation energy and rms neutron halo radius in 22 C for given positions of a virtual state in 21 C
from [27]. The shaded grey area indicates the region of interest to
find the 22 C S2n and rn . Constraints on rn are extracted from the
measured reaction cross section of 22 C. . . . . . . . . . . . . . . . . . 107
xii

5.10 Evolution of the ν1s1/2 −ν0d5/2 shell gap for N=15 Oxygen, Nitrogen,
and Carbon isotones. The dashed line shows a linear extrapolation of
the gap to Carbon following the prescription of [20], while the dotted
and dashdotted lines show the results of WBP* and WBP calculations
respectively. Experimental data taken from [18, 13, 19]. . . . . . . . . 108

xiii

Chapter 1
Introduction
The study of nuclear structure asks a simple question with a complex answer: how do
combinations of protons and neutrons behave when they are assembled into an atomic
nucleus? The nature of this problem precludes a simple answer - a full description
of the atomic nucleus requires solving an n-body problem (where n is an arbitrary
number of nucleons), which alone is a significant computational challenge. Furthermore, the binding of protons and neutrons together is largely mediated by the strong
force, a force which is not well-understood at present and therefore a complete and
robust form is not available for such calculations. Work on both of these challenges
is ongoing, and progress is being made for the lightest nuclei where the calculation
space is still tractable given current computational constraints [1]. Moving beyond the
lightest of systems requires appealing to some other, more approximate model. One
very well tested model is the nuclear shell model, which treats nuclei as collections of
protons and neutrons filling discretized energy levels within some sort of mean field
in a description which is qualitatively similar to the description of electrons orbiting
the nucleus in atomic physics.

1

N=3
1p1/2
1p3/2
0f5/2

(2) [40]
(4) [38]
(6) [34]

0f7/2

1p

(8) [28]

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

(4) [20]
(2) [16]
(6) [14]

0f
N=2

1s
0d

0p

0p1/2
0p3/2

0s

0s1/2

(2) [2]

Harmonic
Oscillator

Woods-Saxon

(2) [16]
(6) [14]

(2) [8]
(4) [6]

N=0

(4) [20]

1s1/2
0d5/2

N=1

0d3/2

Woods-Saxon Plus
Spin-Orbit

Z=8 Neutron Rich
Example

Figure 1.1: Schematic rendering of the development of a shell model, with example energy level shifts for 24 O.
2

This model is shown schematically in Figure 1.1. The motivation to construct
such a model began in the 1930’s, when measurements began demonstrating that certain “magic” numbers of protons and neutrons resulted in increased stability. Initial
skepticism that the nucleus could be described with a mean-field approach fell away
under the weight of experimental evidence over the next decade, with M. GoeppertMayer noting the existence of increased stability at nucleon numbers 8, 20, 50, 82
and 126 in 1948 [2]. Simple potentials only reproduced the lightest magic numbers of
8 and 20, but in 1949 two different groups independently demonstrated that adding
a spin-orbit term to a harmonic oscillator potential could reproduce all of the magic
numbers [3, 4]. The swapping of a harmonic oscillator for a Woods-Saxon potential first developed 5 years later [5] further improved the model, which is capable of
describing nuclei near stability very well across the entire nuclear chart. Figure 1.1
demonstrates that the shell model describes these magic numbers (indicated by bold
face lettering) as corresponding to shell closures, followed by a large energy gap to
the next level.
Near the limits of existence, this theoretical description begins to break down.
In particular, the level structure changes as one moves to more asymmetric nuclear
matter (extreme A/Z ratios). One such example is neutron-rich oxygen, as demonstrated in Figure 1.1. In particular, the neutron sd shell splits apart, resulting in a
new large shell gap at N=16 for 24 O. In recent years, significant effort has been put
forth from both theoretical and experimental camps to understand and constrain this
changing structure for very asymmetric nuclei. Support from the experimental side
is necessary for continued progress in this area, and the present work explores the
structure of 19,21 C as indicated in Figure ??.

3

16

10
14

Ne 17 Ne 18 Ne 19 Ne 20 Ne 21 Ne 22 Ne 23 Ne 24 Ne 25 Ne 26 Ne 27 Ne 28 Ne 29 Ne 30 Ne 31 Ne 32 Ne 33 Ne 34 Ne

F

15

F

16

F

17

F

18

F

19

F

20

F

21

F

22

F

23

F

24

F

25

F

26

F

27

F

28

F

29

F

30

F

31

12

O

13

O

14

O

15

O

16

O

17

O

18

O

19

O

20

O

21

O

22

O

23

O

24

O

25

O

26

O

N

11

N

12

N

13

N

14

N

15

N

16

N

17

N

18

N

19

N

20

N

21

N

22

N

23

N

24

N

25

N

C

11

C

12

C

13

C

14

C

15

C

16

C

17

C

18

C

19

C

20

C

21

C

22

C

B

10

B

11

B

12

B

13

B

14

B

15

B

16

B

17

B

18

B

19

B

A

S Stable

A

S Beam

A

S Unstable

A

S Fragment

A

S Unbound

8
10
8

6
6

Z

5

4
3

Li

1

H

Be

4
3

2

B

C

9

C

10

7

B

8

B

9

6

Be

Li

5

He

4

2

H

1

7

Be

Li

6

He

5

3

H

8

Be

Li

7

He

6

4

H

9

Be

Li

8

He

7

5

H

10

Be

11

Li

10

He

9

Li

9

He

8

6

H

7

Be

12

Be

Li

11

He

10

Li

13

Be

12

14

Be

15

Be

16

Be

Li

He

H

n
2

F

4

6

8

10

12

14

16

18

20

22

24

N
Figure 1.2: Chart of nuclides with “magic numbers” illustrated and this work’s beam and reaction products of interest
indicated.
4

Chapter 2
Theory and Motivation
2.1

Shell Model Calculations

As mentioned in Chapter 1, a successful method of describing the atomic nucleus is
the nuclear shell model, in which the nucleons comprising the nucleus are treated as
independent particles in a mean field. The choice of mean field for the present work’s
calculations is that of a Woods Saxons potential:

V (r) =

V0
1 + e(r−R)/a

(2.1)

where V0 is the potential depth of order -50 MeV, R is the nuclear radius of order
r0 A1/3 where r0 ≈ 1.2 − 1.4 fm, and a is the diffuseness and generally is between 0.6
and 0.8 fm. A spin-orbit potential is also included to better reproduce data:
1
VSO = − V (r) · s
r

(2.2)

as shown in Figure 1.1, this potential breaks the degeneracy of given -value states and
performs well in describing the major shell structures near stability. More correlations
must be included to reproduce observables away from closed shells as their structure
5

is not so simple. These typically take the form of two-body interactions and enter
calculations by re-expressing the Hamiltonian as

H = H0 + Hres

(2.3)

Where H0 is the original mean-field Hamiltonian and Hres includes the modifications
to the interaction
1
Hres =
2

A

A
Vij +

i,j=1

U (ri )

(2.4)

i=1

where Vij is the two-body interaction term and U (ri ) is a central potential term.
In principle the many-body system of equations can be solved with the above equations, but this is not a tractable problem for all but the lightest systems with current
computational facilities. Therefore, some truncation of the model space is generally
imposed, and an active area for the calculation defined where the two-body interactions are calculated. In this space, the total wavefunction Ψk for a system with
quantum numbers k = (n, , j) is given by a linear combination of its basis states

Ψk =

α

akα |Ψα

(2.5)

where α represents the basis state index and |Ψα represent the basis states themselves. Then, the Schr¨dinger equation may be written as
o

H|Ψk
(H0 + Hres )

α

aαk |Ψα

6

= Ek |Ψk
= Ek

α

(2.6)
aαk |Ψα

(2.7)

and the inner product with Ψβ | =

α,β

β aβk Ψβ | taken:

aαk aβk Ψβ |H|Ψα = Ek

aαk aβk δαβ

(2.8)

α,β

This represents a system of equations which can be solved to find the eigenvalues Ek
as well as matrix elements

Hαβ = Ψβ |H|Ψα = Ψβ |H0 |Ψα + Ψβ |Hres |Ψα

(2.9)

Here, the two elements on the right side of 2.9 are known as the bare (spherical) single
particle energies (SPE)
(0)
(0)
Ψβ |H0 |Ψα = E δαβ = Eα
αβ

(2.10)

and two-body matrix elements (TBME) Ψβ |Hres |Ψα . With this information in
hand, the wavefunctions and other spectroscopic information can be calculated. The
interaction used to generate the TBME can in principle be obtained from fundamental
nucleon-nucleon interactions, and some progress is being made in this area (e.g. [6]).
The interactions used for shell model calculations in this work are data-driven, which
is to say that they are the result of fitting the interaction parameters over a large
set of measured binding energies and low-lying states for nuclei in the vicinity of the
calculation space. In particular, the WBP and WBT interactions used in this work
have both been developed in this way, with special care taken because the interactions
must describe psd cross-shell nuclei [7]. The interaction must cross over the proton
p and neutron sd shells to properly describe the nuclei in this region. The WBP
and WBT interactions differ only in the specific routine used to fit the interaction
parameters.
In the following discussions, the contribution of the tensor force to Hres will be
7

particularly relevant. This force arises from spin-isospin coupling between nucleons,
and takes the form
VT = (τ1 · τ2 )([s1 s2 ](2) · Y (2) )f (r)

(2.11)

where τ is isospin, s is spin, and [ ](2) represents the coupling of two operators to rank
2. This force is a significant driver in changes of nuclear structure away from stability
[8, 9, 10]. The tensor force can be either attractive or repulsive between protons and
neutrons depending on their orbital and total angular momenta, and modifies the
binding energy of different orbitals accordingly. If the proton and neutron orbitals
are denoted by j± and j± respectively (j± =
for spin-flip partners j± and j

2.2

± 1/2), then the force is attractive

and repulsive for j± and j± .

Shell Evolution Near N = 14

The formation and disappearance of nuclear shell gaps in exotic nuclei is currently
a topic of great interest, as it affects numerous phenomena in nuclear physics and
provides a tool to constrain theoretical descriptions of the nucleus. In light neutronrich nuclei, it is now known that those nuclei with holes in the π0p1/2 and π0p3/2
subshells experience a smaller neutron-neutron interaction than theory predicts. This
effect has been observed in Z=5-7 and N=10-15 [11, 12, 13]. In particular, the USD
portion of the WBT interaction [7] has been shown to need empirical reductions of
12.5 - 25% of its TBME to describe the bound excited states of these nuclei. Similar
reductions in TBME have been needed to reproduce magnetic moments measured in
17 B and 17 N [14].
8

2.2.1

N = 14 shell gap

N=14 has been shown to be a closed subshell for the oxygen isotopes [15]. The
resulting large energy gap in 22 O is thought to occur because of attractive monopole
matrix elements Vnn d
d5/2 5/2 increasing the binding of the ν0d5/2 relative to the
ν1s1/2 . Figure 2.1 demonstrates the evolution of the shell gap as proton number
is reduced by comparing the even-even Z=8 isotopic chain with the Z=6 one in two
ways (taken from [12]). Panels (a) and (b) show the evolution of the 2+ excited
state energies in Z=8 and Z=6 taken from experimental data. Both follow a similar
trend for N=8,10,12 but significantly differ for N=14, where Z=8 shows a significant
increase in 2+ energy while Z=6 does not. The energy of the 2+ can be related to
the energy needed to excite a neutron across the ν0d5/2 − ν1s1/2 gap, which is why
it grows for the closed subshell existing in Z=8. The lack of a significant rise in this
energy for Z=6 suggests the gap is greatly diminished.
Figure 2.1(c) and (d) show this reduction from a slightly different perspective,
and compare the effective single particle energies (ESPE) for the two isotopic chains
calculated by shell model. It is readily observed that at N=14, there is a large
energy gap between the ν0d5/2 and ν1s1/2 orbitals for Z=8 which is greatly reduced
in Z=6. The driver for this vanishing gap is thought to be the tensor force, which
serves to attractively couple the π0p1/2 and ν0d5/2 orbitals while leaving the ν1s1/2
unchanged. In Z=6 the attractive couplings are gone, which causes the ν0d5/2 to be
less bound relative to the ν1s1/2 .

9

Figure 2.1: Collapse of the N=14 shell gap demonstrated in two ways by [12]. Top,
a comparison of the measured 2+ energies as a function of neutron number shows
that at N=14 in oxygen (a) an increase occurs which is not matched in the carbon
isotopic chain (b). Bottom, ESPEs calculated in the WBP interaction demonstrate
the large gap between the ν0d5/2 and ν1s1/2 shells for N=14 gap in oxygen (c). In
carbon, this gap is significantly diminished (d).

10

Figure 2.2: Comparison of 18,20 C experimental level schemes to WBT, WBT* interactions from [12].
11

While this serves to describe the overall structure of Z=6 relative to Z=8, it does
not capture all changes. This can be seen in Figure 2.2, which compares experimental
level schemes for 18,20 C with theoretical predictions [12]. The first comparison to be
made is between experiment and the shell model interaction labelled WBT, which is
a standard interaction to use when describing nuclei in this psd space. It is readily
observed that the experimental level scheme is compressed relative to the theoretical
one. This effect is observed systematically for Z=5-7 and N=10-15, and suggests a
breakdown in the description of psd cross-shell nuclei within this theoretical framework. Empirical reductions of the neutron sd TBME associated with this interaction
are capable of reproducing the level density in this region, with reductions varying
between 12.5% for the Z=7 isotopic chain to 25% for Z=5,6. It has been suggested
that the root cause of this systematic breakdown is related to the phenomenological
nature of the WBT [11]. In particular, the very neutron rich psd cross shell nuclei are
more loosely bound and have larger matter radii than those nuclei which were fit over
to develop the interaction. The reduction in TBME for this interaction suggests that
the interaction binds these nuclei too much. Therefore it is interesting to continue
characterizing the nuclei in this region in order to more fully understand how the
theoretical description of these nuclei breaks down.
19 C is a bound nucleus and lies adjacent to the N=14 nucleus 20 C on the nuclear
chart, making it an interesting case to search for excited states with its proximity
to the N=14 subshell closure and location within the psd cross-shell space described
above. This nucleus has been the subject of prior study, of which three experiments
have obtained excited state spectroscopic information by utilizing two different reaction mechanisms and experimental techniques. The first two experiments searched
for bound excited states through γ-ray spectroscopy, with one [12] utilizing fragmentation and another [16] using proton inelastic scattering (p,p’) on a liquid hydrogen
target. The resulting γ-ray spectra are shown in Figure 2.3, with the former reaction
12

producing a γ line at 201 ± 15 keV and the latter reaction populating two lines at
72 ± 4 keV and 197 ± 6 keV.
Fast neutron spectroscopy has also been used in conjunction with (p,p’) reactions
on a liquid hydrogen target to look for neutron unbound excited states in 19 C [17].
The resulting decay spectrum is shown in Figure 2.4 and shows a clear resonance
near 1 MeV. All together, the three experiments claim to have observed three different excited states in 19 C, and the resulting level scheme compared to theoretical
calculations in Figure 2.5. The 200 keV γ-ray line has been assigned to a 3/2+ , the
72 keV to a 5/2+ lying above the 3/2+ , and the unbound resonance to the second
5/2+ . There is, however, a question of why the fragmentation γ-ray spectroscopy
2
experiment did not observe the 72 keV line that the (p,p’) experiment did (shown in
the figure as the red line corresponding to the 5/2+ state). No explanation is given
1
in either publication to justify this difference, and as shall be seen in Section 5.1 cross
section estimations for the two different reaction mechanisms cast further doubt on
the 72 keV result.
As shown in Figure 2.5, a low lying triplet is predicted by shell model, followed
by a large energy gap to the next state. Further, within the uncertainty of the
theoretical calculations, it is possible that at least one state in the low-lying triplet
be neutron unbound. Therefore, a new neutron spectroscopy experiment could remove
the ambiguity in the observation of the 5/2+ state, and attempt to reproduce the
1
location of the 5/2+ unbound state with a different reaction mechanism. Observation
2
of a near-threshold resonance would demonstrate that the 5/2+ state is in fact neutron
1
unbound, leaving 19 C with two bound states.

2.2.2

The N=15 Probe

The evolution of a shell gap for neutron number N can be extracted from the single
particle or single hole levels in the N-1 or N+1 isotones, so the N=14 gap can be
13

(a) (p,p’)

(b) Fragmentation
Figure 2.3: Resulting γ-ray spectrum for bound excited states in 19 C from [16] and
[12].
14

Figure 2.4: Resulting decay energy spectrum for unbound excited states in 19 C from
[17].

15

Exp.
Energy (MeV)

2

WBP

2+
5/2+

5/2+

1
3/2+

0+

5/2+
3/2+
1/2+

0
18

19

C+n

5/2+
1/2+

C

Figure 2.5: Experimental level scheme for 19 C based on results from [12, 16, 17]. For
interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this dissertation.

16

probed by exploring the structure of N=15 isotones. The ν1s1/2 − ν0d5/2 energy
gap has been determined for 23 O and 22 N [18, 19], and it is useful to extend this
measurement to 21 C with its empty πp1/2 subshell. Prior to this work no spectroscopic information about 21 C was known. Figure 2.6 shows the measured energy gap
for Z=8,7, where the level schemes are generated from measured states in [18, 13, 19].
A linear extrapolation to Z=6 as first used by [20] is shown as the dashed line, and
predicts that the gap will vanish, suggesting degeneracy of the ν1s1/2 and ν0d5/2
orbitals. Within the uncertainty of the extrapolation it is possible for the levels to
invert. The driver for this vanishing gap is again thought to be the tensor force,
which serves to attractively couple the π0p1/2 and ν0d5/2 orbitals while leaving the
ν1s1/2 unchanged. For Z=8, the π0p1/2 is full with its two protons, and Z=7 is one
proton short and the ν0d5/2 is correspondingly less bound relative to the ν1s1/2 .
Therefore, the removal of a second proton would be expected to cause the ν0d5/2
to be less bound by approximately the same amount as moving from Z=8 to Z=7.
Measuring the ν1s1/2 − ν0d5/2 shell gap in 21 C would directly test this.
Because Z=6 corresponds to a closed πp3/2 subshell and N=15 naively corresponds to a full νd5/2 shell plus one extra neutron, the energy difference between
the 1/2+ (ν1s1/2 particle state) and 5/2+ (ν0d5/2 hole state) in 21 C directly determines this energy gap. Therefore, it is interesting to populate these two states in
21 C to experimentally determine the N=14 ν1s
1/2 − ν0d5/2 energy gap for carbon.

2.2.3

22

C as a Two-neutron Halo

Since the discovery of the large matter radius of 11 Li [21], neutron halos have been
a topic of intense study near the neutron drip line. The halo structure results from
valence nucleon(s) being very loosely bound which, combined with the short range of
the nuclear force, allows them to have a large probability of being found at distances
much greater than the normal nuclear radius [22]. The separation energy for the
17

Energy (MeV)

3

N=15

2

0d5/2

2.79

1.93

1.02

0

2–

0.19
0

1

3–

1
0–

–

N

C

1s1/2
O

Figure 2.6: Evolution of the ν1s1/2 −ν0d5/2 shell gap for N=15 oxygen, nitrogen, and
carbon isotones. The dashed line shows a linear extrapolation of the gap to carbon
following the prescription of [20], and experimental data are taken from [13, 18, 19].

18

Figure 2.7: Matter radii for carbon isotopic chain, indicating a large increase for 22 C
[26]. Inset show the matter density and demonstrates a large matter tail thought to
be a neutron halo.

valence nucleon(s) is a critical parameter in understanding the halo structure [23, 24,
25].
Recently a large matter radius has been extracted from the measured reaction cross
section of 22 C via finite-range Glauber analysis under an optical-limit approximation,
suggesting that this nucleus exhibits a two-neutron halo [26]. Figure 2.7 shows matter
radii for the carbon isotopic chain as a function of neutron number including the
latest measurement of 22 C from [26]. For N < 16, a fairly regular and compact form
is observed but a large discontinuity exists at N = 16, which corresponds to 22 C.
19

Figure 2.8: Correlations between the two-neutron separation energy and rms neutron
halo radius in 22 C for given positions of a virtual state in 21 C from [27]. Each curve
is calculated for a specific energy of the 21 C virtual state, and range from 0 to 100
keV in steps of 10 keV.

Inset, the figure shows the extracted proton (solid line) and neutron (dotted) density
line shapes. The neutron density extends out to a great distance from the core of the
nucleus and suggests a halo with two neutrons in the ν1s1/2 orbital.
The two-neutron separation energy (S2n ) has not yet been measured, but it has
been shown that constraints on the S2n of 22 C can be placed by measuring the energy
of the virtual state in 21 C. Recently, Yamashita et al. derived a relationship between
the 22 C two-neutron separation energy and the energy of the virtual state in 21 C
20

Energy (MeV)

2
2+
1
1/2+

+

0

0

0+

-1
20

C+2n

21

C+n

22

C

Figure 2.9: Level scheme indicating the predicted locations of 21,22 C in energy relative to a 20 C core [27].

within the renormalized zero-range three-body model of [27]. Figure 2.8 demonstrates
this relationship as lines of constant decay energy for the virtual state in 21 C through
a phase space of two-neutron separation energy S2n and 22 C halo radius rn . This
sort of correlation has been observed in other halo systems [28]. The lines shown
correspond to 21 C virtual state energies ranging from 0 to 100 keV in steps of 10 keV.
The hashed region indicates constraints on this parameter space from [26].
Prior to this work no spectroscopic information about 21 C was known, but Yamashita et al. suggest that the virtual state should be unbound by less than 100
keV [27]. Their reasoning is shown in Figure 2.8. The 100 keV virtual state curve
crosses into the experimentally constrained region at approximately S2n = 100 keV,
which limits the range of predicted virtual state energies in 21 C to E ≤ 100 keV for
this calculation space. Their predictions are illustrated schematically in Figure 2.9,
which shows the predicted ground states of 21,22 C relative to 20 C as a level scheme.
Therefore it is interesting to measure the location of the virtual state in 21 C to test
21

this model and provide constraints on the S2n for 22 C.

2.3

Decay Energy Lineshapes

In order to extract information from the measured decay energy spectra, the observed
line shapes must be modelled in such a way that the input parameters can be related
to physics observables. In the present work, there were three possible contributions
to the line shapes: narrow resonance structures resulting from the breakup of
2 states, broader virtual resonances arising from the decay of

=

= 0 states, and

nonresonant background contributions from the decay of high-lying continuum states.

2.3.1

Breit-Wigner Resonance

It is possible to model the decay of a neutron unbound state as the scattering of
a neutron from the fragment nucleus with the energy and angle as variables. This
description of the breakup can be constructed mathematically within the R-matrix
framework [29], and an expression for the cross section as a function of energy σ(E)
can be derived. While the full derivation of the most general case is beyond the scope
of this work, the relevant portions for this work are highlighted below (full details
available in [29], and the notation here is kept consistent wherever possible).
The scattering of the neutron from the charged fragment can be modelled as the
neutron scattering elastically from some central potential. The radial Schr¨dinger
o
equation for this situation then reads:
( + 1) 2M
d
−
−
(V − E) uλ (r) = 0
2
2
dr
r2

(2.12)

This equation has solutions in the form of incoming and outgoing waves I and O
which are Coulomb Hankel functions [30]. The R-matrix formalism is constructed
22

by exploiting the properties of the wavefunction at the core’s “surface,” defined as
the minimum radial distance between the neutron and core at which the nuclear
interaction is unimportant:
1/3
1/3
a = r0 (Ac + An )

(2.13)

Here, r0 is a parameter describing the nuclear radius and is taken to be 1.4 fm, and
Ac and An are the mass numbers for the core and neutron respectively. A Green’s
theorem relation can be used to construct an expression for the relationship between
the wavefunction anywhere in the interior of the core and its derivative at the surface
for any given energy E:
uE (r) = G(r, a)(aduE /dr)a
where

2
G(r, a) =

2M a

λ

uλ (r)uλ (a)
Eλ − E

(2.14)

(2.15)

and λ is the index for eigenfunctions for the interior wavefunction which will combine
with a shift function to define the locations of observed resonances. The R-matrix
then can be defined:
R = G(a, a) =
λ

2
γλ /(Eλ − E)

(2.16)

where
γλ = ( 2 /2M a)1/2 uλ (a)

(2.17)

This can be related to a collision matrix U
∗
I 1−L R
U =
= exp(2iδ )
O 1−L R

23

(2.18)

where L is the logarithmic derivative of O :

L =

out
ρu
uout

= S + iP

(2.19)

r=a

where S is known as a shift factor and P as a penetration factor. In the one-level
approximation where it is assumed that the spacing between resonances is much larger
than their width, the phase shift then is:
δ = tan−1

1
2 Γλ
Eλ + ∆λ − E

−φ

(2.20)

where φ is the hard-sphere scattering phase shift, and

Γλ
∆λ

2
= 2γλ P
2
= −γλ S

(2.21)
(2.22)

Γλ determines how fast the phase changes as E passes through a resonance and
therefore defines the width, while ∆λ determines how far the resonance energy is
shifted from its eigenvalue Eλ . With the collision function U thus completely defined, the cross section can be expressed:

σ

=
=

σ(θ)dΩ =

π
k2

(2 + 1)|1 − U |2

Γλ Γ
π
λ
gj
2 (E + ∆ − E)2 + 1 Γ2
k
λ
λ
4 λ

(2.23)
(2.24)

Because only the relative intensity rather than the absolute cross section is needed
for fitting purposes, we collect all constants which serve as scaling factors and write

σ =A

Γ
(E + ∆ − E)2 + 1 Γ2
4
24

(2.25)

where
Γ = 2γ 2 P

(2.26)

∆ = −γ 2 S

(2.27)

and

which is the Breit-Wigner distribution with an energy dependent width. For this
expression, we make the one-level approximation stronger by restricting γ to only
include a single eigenfunction of the interior wavefunction. When the resonances are
widely spaced this approximation is valid.

2.3.2

S-wave Dynamics

The Breit-Wigner line shape derived above does not describe

= 0 transitions well,

as the lack of an angular momentum barrier causes them to have a broad energy
distribution - particularly for high lying states. Therefore, some other mechanism
for the description of these transitions must be used if we are to account for their
presence in the decay spectra. It is possible to treat the neutron evaporation as the
breakup of the projectile by the neutron being knocked out from the projectile by
the neutron-target interaction, and then re-interacting from its final state with the
core. This model is developed by Blanchon et al. in [31], and the highlights of that
work are presented here. The coordinate system used for the following calculations
is displayed in Figure 2.10.
First order time-dependent perturbation theory defines a transition amplitude for
inelastic-like excitations of the form we wish to describe,
1
Af i =
i

∞
−∞

dt ψf (r, t) V2 (r − R(t)) ψi (r, t)

(2.28)

where ψi and ψf are the wavefunctions for the initial and final states respectively,
25

Neutron

r

v

Core

Target
bc

R(t)

Figure 2.10: Coordinate system and associated parameters for s-wave calculations.

and ψf can be either a bound or continuum state. The potential V2 (r − R(t)) is the
interaction which causes the transition and moves past the core/neutron system with
velocity v in the z-direction with impact parameter bc , which is taken to be in the
− i it
, and introducing variables
x-direction with y = 0. Factorizing ψi (r, t) = φi (r)e
z − vt = z and t = (z − z )/v, the transition amplitude is re-expressed as
1
Af i =
i v

∞
t∞

dxdydzdz φ∗ (x, y, z)φi (x, y, z)eiq(z−z ) V2 (x − bc , y, z )
f

(2.29)

where
q=

f− i
v

(2.30)

Choosing V2 (r) to be a delta function V2 (r) = v2 δ(x)δ(y)δ(z) with strength v2 , the
26

integrals for x and y are can be calculated:
v
Af i = 2
i v

∞

dzφ∗ (bc , 0, z)φi (bc , 0, z)eiqz
f

(2.31)

−∞

The δ-function approximation holds best for peripheral reactions (see [31] for more
−2m i / as the decay length of the initial state

detail). Further defining γ =
and k =

2m f / as the final momentum in the continuum, and now imposing the

restriction of i = f = 0 results in the following expressions for the wavefunctions:
C e−γr
φi (bc , 0, z) = − i
4π r
Cf k (−)
(+)
i (h0 (kr) − Sh0 (kr))
φf (bc , 0, z) =
4π 2

(2.32)
(2.33)

where C∗ are normalization coefficients and S is the S-matrix corresponding to the
final state interaction of the neutron with the core. With this, the expression for the
transition amplitude reads:
v Ci Cf
Af i = − 2
I(k, q)
v 8π
where
I(k, q) =

∞
−∞

dz

e−(γ−ik)r − S ∗ e−(γ+ik)r
cos(qz)
r2

(2.34)

(2.35)

The probability spectrum to excite a final state then can be expressed:
2
dPin
2 v2 2 m
1
¯
=
C
|1 − Smi ,mf |2 |Imi ,mf |2
2v2 i k 2 i + 1
df
π
mi ,mf

(2.36)

where
¯
S = Se2iv = e2i(δ+v)

27

(2.37)

Finally, the expression for the cross section reads
dσ−1n
= C 2S
df

dbc

dPin (bc )
Pct (bc )
df

(2.38)

where
Pct (bc ) = |Sct |2 = e− ln 2exp[(Rs −bc )/a]

(2.39)

1/3
1/3
and the strong absorption parameter Rs ≈ 1.4(Ap + At ) fm is the distance of
closest approach for a trajectory which is 50% absorbed from the elastic channel and
a = 0.6 fm is the diffuseness. For large impact parameters, the approximation r =
b2 + z 2 ≈ bc + z 2 /2bc can be made, which allows for a simple analytic expression
c
for the cross section:




dσ
1
 k cos δ + γ sin δ 
∼
d
k(γ 2 + k 2 )
γ 2 + k2

(2.40)

In principle the phase shift δ comes from the calculated S-matrix, but an approximation of δ = as k can be used as a parameterization. As discussed in [31] this
approximation slightly over predicts the cross section for higher decay energies, but
as discussed in [32] still agrees very well with a full calculation. We therefore have
an expression for the decay energy as a function of the variables as , γ, and k, where
as is the scattering length, γ is the decay length and is related to the beam’s binding
energy as discussed above, and k is related to the decay energy. For any given decay
energy calculation, as and γ are fixed, which results in a decay energy line shape
which can be used for fitting experimental data.

2.3.3

Nonresonant Contributions

In the case of 19 C, there is the possibility of background which is uncorrelated with
the population of a particular state. This background arises from the population
28

of high-lying continuum states in 20,21 C where the level density is large. These
states can evaporate neutrons until the 18 C core is reached, and if the neutron(s)
resulting from this decay scheme are detected, they will contribute to some sort of
background distribution. Figure 2.11 diagrammatically shows this effect. The thick
lines correspond to chains of high-lying continuum states evaporating neutrons to
other continuum states by the emission of a fairly low energy neutron. These neutrons
are more likely to be detected because their relative energy is small and therefore are
more likely to make it into the geometric acceptance of the experimental setup. The
thin line represents the final decay mode to the ground state of 18 C from a high-lying
continuum state. This mode is unlikely to be observed because of its high relative
energy kick, which will very likely cause its trajectory to lie outside of the setup’s
acceptances.
Therefore, the background neutrons are unlikely to be directly correlated with the
18 C, and result from neutron evaporations through regions of high level density. This
lends itself to a statistical model approach, and can be modelled as a Maxwellian distribution [33]. The general approach to this sort of model is summarized below, with
full details being covered in [34]. The notation below will conform to [34] wherever
possible.
The probability of system C to decay through a specific reaction channel β can
be written
GC (β) =

2
kβ σC (β)
2
γ kγ σC (γ)

(2.41)

where the sum of over all possible channels γ, k = λ−1 is the wave number, and
σC is the cross section for C to decay. Then, the kinetic energy for channel β with
residual nucleus Y is expressed:
∗
β = nY − Eβ
29

(2.42)

22

18

C

19

20

C

C

N

21

C

Figure 2.11: Diagrammatic representation of how nonresonant background arises in
the decay spectrum for 18 C in coincidence with neutrons. Highly excited states in
20,21 C are populated by nucleon removal from the 22 N beam, and these evaporate
neutrons to other high-lying states until 19 C∗ is reached (thick arrows). 19 C∗ decays
via emission of a neutron with large decay energy (thin arrow), which is unlikely to
be observed because of the geometric efficiency of the setup. Evaporation between
continuum states results in neutron emission with lower relative energy, which is likely
to be observed and results in background.

30

where
nY = + Qβ

(2.43)

and

is the energy of the incoming beam and Qβ is the Q-value of the reaction,
∗
and Eβ is the excitation of the residual nucleus. For a high density of states, the
distribution of emitted neutrons between

and + d is

Gn ( )d =

GC (β)

(2.44)

< β < +d
Because we are interested in a line shape rather than an absolute cross section, we
consider only the numerator of 2.41 and write the relative intensity I as

In ( )d ∼ σC (β)wY ( nY − )d
where

(2.45)

2
replaces kβ and w is related to the level density. The shape of I can be

estimated by taking the Taylor expansion of logarithm of w about nY :
G ( nY − ) = log w( nY − ) = G ( nY ) −

dG
+ ...
dE E=
nY

(2.46)

this expansion results in the following expression for I

d
In ( )d ∼ σC ( )exp −
Θ( nY )

(2.47)

where σC is a slowly varying function which can be absorbed into the constant, and
1
dG
=
Θ(E)
dE

(2.48)

Therefore, Θ has units of energy and can be interpreted as a nuclear temperature
of a thermal neutron source. The dominant feature of this expression is the e− /Θ
31

exponential term. The function used in this work’s background modelling is slightly
different [35]:
/Θ3 e− /Θ

f ( ) = const

(2.49)

with parameter Θ a parameter which is fit to the data. The differences in line shape
between 2.47 and 2.49 are irrelevant once experimental resolutions are accounted for.

2.4

Estimators: Maximum Likelihood Method

Physics observables were extracted in this work by comparing simulated datasets to
the experimental data. Some statistical estimator method was needed to provide
tests to compare the simulated and experimental datasets, and the method chosen
for this work is the maximum likelihood method. The simulated dataset served to
generate a decay energy probability density functional (p.d.f.) f (θ), where θ is the
set of parameters being fit over. The likelihood can then be defined as in [36]:
N
L(X|θ) =
i=1

f (Xi |θ)

(2.50)

where X = X1 , X2 , ..., XN is the set of N observations and f (Xi |θ) is the value of the
p.d.f. evaluated at each data point. While L(X|θ) and the log-likelihood ln L(X|θ)
both provide good test statistics, ln L(X|θ) was used as a test statistic for this work:
N
ln(L(X|θ)) =
i=1

ln(f (Xi |θ))

(2.51)

L(X|θ) was maximized by generating a likelihood surface over the relevant decay
parameters by populating grid points in the decay parameter phase space with individual simulation runs from which the maximum likelihood could be extracted. The
32

nσ confidence intervals defined as the regions where

ln(L(X|θ))max − ln(L(X|θ)) ≤

n2
2

(2.52)

In order to reduce systematic errors due to modelling deficiencies, the simulated
datasets were large and the simulation verified for simple cases before modelling the
reactions of interest (more detail in Section 4.5).

33

Chapter 3
Method

3.1

Invariant Mass Reconstruction

Neutron unbound states decay on very short timescales (order 10−21 s), which prevents the direct observation of these nuclei. One way the study of such nuclei can
proceed is by the observation of decay products and reconstruction of the properties of the original state. The technique used to perform this reconstruction is that
of invariant mass spectroscopy, which is derived from relativistic mechanics. The
relativistic four-momentum P is defined:

P = (E, p)

(3.1)

Where E is the total energy and p is the three-momentum vector. For state i decaying
to products f and n (schematically shown in Figure 3.1), conservation of momentum
is expressed as
Pi = Pf + Pn
34

(3.2)

and the invariant mass M can be defined in terms of the squares of 3.2
M 2 = (Pf + Pn )2 = (Ef + En )2 − |pf + pn |2
= m2 + m2 − 2(Ef En − pf · pn )
n
f

(3.3)
(3.4)

The decay energy can be found by subtracting off the rest masses of the fragment
and neutron

Edecay = M − mf − mn =

m2 + m2 − 2(Ef En − pf · pn ) − mf − mn (3.5)
n
f

In order to reconstruct the decay energy, the full four-momentum of the fragment and
neutron must be measured. Therefore, the objectives of the experimental setup are
twofold: to identify particular reaction products and to measure the decay residue
four-momenta.

pn
Lab Frame
CoM Frame

Θ
pf

Figure 3.1: Neutron evaporation schematic.

35

3.2

Beam Production

Population of states in 19,21 C was performed by single proton removal from a beam
of 22 N, a nucleus which is β-unstable with a half-life of 20 ± 2 ms [37]. The NSCL
makes use of the fast fragmentation method [38] to produce beams with such short
half-lives. The application of this method at the Coupled Cyclotron Facility (CCF)
[39] is schematically shown in Figure 3.2.
The primary 48 Ca beam was injected from the ion source into the K500 cyclotron,
where it was accelerated to 12.5 MeV/A. From there, it was sent through a thin foil
which fully stripped it of electrons and injected to the K1200 cyclotron, where 48 Ca
was accelerated to the nominal primary beam energy of 140 MeV/u. The beam was
then extracted from the K1200 and transported to the object position of the A1900
fragment separator [40], where it impinged on a 2068 mg/cm2 9 Be production target.
The A1900 separates reaction fragments by their momentum and m/q ratios by
setting the four dipole magnets of the separator to the expected magnetic rigidity
of the final 22 N secondary beam. In this experiment, the final dipole was set to a
rigidity of 3.856 Tm. A 1057 mg/cm2 Al wedge degrader was placed after the second
dipole of the A1900 to selectively disperse other reaction products relative to the
desired 22 N and improve the final separation. Momentum slits are inserted at the
dispersive image, and serve to tune the acceptance of the device. These slits were
at 2.5% momentum acceptance for this experiment. The beam was delivered at an
energy of 68 MeV/u and particle rate of 37/s with a purity of 32%. The primary
contamination was from light ion production in the Al wedge, while the strongest
heavy ion contaminants were 26 F at 6% and 20 C at 2.8%. All of these could be
separated on an event by event basis offline.
36

Figure 3.2: Beam production at the CCF.

37

3.3

Experimental Setup

A schematic of the experimental setup is shown in Figure 3.3. After exiting the
A1900, the secondary beam passed through two position-sensitive cathode readout
drift chambers (CRDCs). Downstream of the CRDCs, a quadrupole triplet magnet
focused the beam onto the 481 mg/cm2 9 Be reaction target. A 0.254 mm plastic scintillator was placed immediately upstream of the reaction target to determine beam,
charged fragment, and neutron time of flight. Neutron-unbound isotopes produced in
the reaction target immediately decayed into charged fragments and one or more neutrons. A large-gap superconducting dipole magnet [41] bent the charged fragments
away from the beam axis, and the neutrons were detected at forward angles by the
Modular Neutron Array (MoNA) [42].
Downstream of the magnet, two more CRDCs separated by 1.82 m provided fragment trajectory information. An ionization chamber and thin plastic scintillator
provided energy loss information for element separation, while a thick plastic scintillator provided a total kinetic energy measurement. Additionally, the thin scintillator
provided time of flight of the fragments for isotope separation.

38

Quadrupole
Triplet

Beamline
CRDCs
3m

Timing Scintillator
Reaction Target

CRDCs
Sweeper
Magnet

MoNA
Ion
Chamber

Thin Scintillator
Thick Scintillator

Figure 3.3: Experimental setup diagram.

39

3.3.1

Beamline Detectors

The incoming beam’s trajectory is measured by a pair of CRDCs separated by 2.18 m
whose operation is schematically shown in Figure 3.4. The active volume of each
detector is filled with a mixture of 80% CF4 and 20% C4 H10 at a pressure of 100 Torr,
and incident charged particles ionize this gas. The freed electrons drift under a -150 V
drift field toward an anode wire held at 1100 V which collects the charge. Drift time to
the anode wire determines position in the non-dispersive plane. The charge collection
on the anode in turn induces charge on a series of 64 aluminum pads placed near
the anode wire with 2.54 mm pad pitch. A Frisch grid reduces position dependent
response effects.
The distribution of charge on each pad can be used to determine the particle’s
dispersive position in pad space. The charge collected as a function of pad number
can be fitted with a Gaussian function whose centroid corresponds to the position
of the strike. There is a linear correlation between pad and physical position space,
and the slope for that transformation is set by the pad pitch of 2.54 mm. The offset
for the dispersive position transformation as well as nondispersive slope and offset
are determined by calibration runs taken with a tungsten mask with holes in known
locations placed in front of the detector.
A quadrupole triplet magnet was located downstream of the beamline CRDCs
to focus the beam onto the 481 mg/cm2 reaction target. The magnetic field of the
triplet is mapped so a forward transformation can be applied to the CRDC trajectory
information to construct target emittance parameters.
Three timing signals provide information about the beam time of flight. First,
the cyclotron RF signals provide timing information which can be correlated with the
primary beam particles impinging on the production target. A pair of scintillators at
the A1900 and target chamber positions provide secondary beam time of flight over
40

Figure 3.4: Schematic of CRDC operation from [43]. The incident charged particle
ionizes the gas in the active volume, and the freed electrons move toward the anode
wire under the drift field. A Frisch grid reduces position dependent charge collection
effects, and the charge collection on the anode induces charge on the aluminum pads
which determines the dispersive position of the interaction. The drift time of the
electrons determines non-dispersive position.

41

a flight path of 11.52 m. Both scintillators are made of the plastic organic scintillator
BC-404 [44] and are coupled to one PMT. The A1900 scintillator is 1008 µm thick, and
the target chamber scintillator is 254 µm thick. When charged particles pass through
an organic scintillator, energy loss through the material causes transitions to excited
states of the organic molecule’s electron structure, and the prompt fluorescence (τ =
1.8 ns) resulting from their subsequent de-excitation is converted to an electronic
signal by the PMT.

3.3.2

Charged Fragment Detection

The large-gap superconducting dipole Sweeper magnet [41] has a bending radius of
1 m, a bend angle of 43◦ , and a maximum rigidity of 4 Tm. Its 14 cm vertical gap
allows for neutron transmission to MoNA, with the consequence of some nonuniformities in the magnetic field which necessitated the careful mapping of its magnetic
field properties. The magnetic field was set to a rigidity of 3.7581 Tm for maximum
transmission of the 20 C reaction fragments during this experiment.
Another pair of CRDCs are located downstream of the sweeper magnet. The basic
operation of CRDCs is explained in Section 3.3.1, but the specifications for these
CRDCs is somewhat different. These detectors have an active area of 30 × 30 cm2
with 128 aluminum pads of pad pitch 2.54 mm for dispersive position measurement,
are separated by 1.82 m, and are operated at a pressure of 50 Torr of the same gas
mixture as the beamline CRDCs. Additionally, the drift field was set to 500 V and
the anode wire held at 1050 V.
Immediately downstream of the CRDCs is an ionization chamber which is used
to measure energy loss for element separation (see Section 4.3). This detector has an
active volume filled with P-10 gas (90% Ar, 10% CH4 ) and is operated at a pressure
of 300 torr. Like the CRDCs, a drift field is applied to ionization generated by
incident charged particles, and the ionization is recorded by charge collection pads
42

after passing through a Frisch grid to reduce charge collection position dependence.
In this case the drift field is 800 V, the anode is held at 50 V, and 16 collection pads
with pad pitch 3.08 cm are arranged from front to back along the device. The active
area is 40 × 40 cm with an interaction length of 53 cm [43]. The charge collected by
each pad is summed to create the complete energy loss measurement.
Two more organic plastic scintillators are located immediately downstream of the
ionization chamber - one 4.5 mm scintillator to measure the time of flight of reaction
products for mass ID (see Section 4.3), and one 150 mm scintillator to stop the beam
and provide a total residual energy measurement. These scintillators have a large
active area (40 × 40 cm) and therefore have four PMTs each to collect scintillation
light. Their operation is otherwise the same as described for the beam line scintillators
in Section 3.3.1 as the scintillation material used is BC-404.

3.3.3

MoNA

MoNA consists of 144 10 × 10 × 200 cm3 BC-408 [44] plastic scintillator bars with
light guides and PMTs attached to each end. The modules were arranged in walls
that were 16 modules tall and centered on the beam axis. Walls of 2 by 16 modules
each were positioned with their front faces at 5.90 m, 6.93 m, and 7.95 m from the
reaction target as shown in Figure 3.3. A block of three walls was placed at 8.65 m.
Neutrons are detected indirectly by the same method described for charged particles in Section 3.3.1. The uncharged neutrons elastically scatter from protons (hydrogen nuclei) in the hydrocarbon making up the plastic, causing the charged proton
to recoil and deposit energy in the scintillation material. De-exciting molecular levels
produce photons which travel to each end of a bar and are collected by a PMT and
turned into an electric signal which can be recorded. Each PMT has two output
signals. The anode signal is sent to a constant fraction discriminator (CFD) to determine the time of the pulse, which is in turn fed into a time to digital converter
43

(TDC) which measures the time of the signal by using the target scintillator as a
common stop. Neutron time of flight was measured from the mean time of the two
PMT signals of the detector module that detected a hit, while position across the
bar was measured by the time difference between the signals. The dynode signal is
sent to a charge to digital converter (QDC) and measures the energy deposition of
neutron strikes.

3.3.4

Electronics and DAQ

Detailed descriptions of the electronics and data acquisition (DAQ) for the MoNA/Sweeper
set-up can be found in [45, 46] so a brief overview will be included in this work. The
Sweeper and MoNA electronics and DAQ are in principle independent and can be
operated separately, but are coupled for the purpose of neutron coincidence experiments. Figure 3.5 schematically highlights the important aspects of the detector
subsystems from a trigger logic / DAQ perspective.
The trigger logic was implemented by two logical levels of field programmable gate
arrays (FPGAs). “Level 1” decides whether MoNA observed a neutron; this is determined by the logical OR of all MoNA CFD signals. “Level 2” makes the final decisions
on whether to process the event. The target scintillator signal served as a common
stop for the MoNA TDCs and allowed the neutron time of flight measurement. The
master trigger was formed by the output of the CFD attached to the thin scintillator’s
upper left PMT, which indicated that a charged fragment entered the sweeper focal
plane chamber. As soon as Level 2 received a trigger, it sent out a “veto” signal to
stop all other electronics processing until a decision is made if the data was worth
recording. Because the beam rate was low for this experiment, the only requirement
for data to be recorded was the presence of the master trigger signal, so data was
recorded to disk for every charged fragment event regardless of whether a neutron
was in coincidence. A “Computer Go” signal was sent out to all electronics which
44

MoNA Electronics Rack
TDC
MoNA

CFD
Level 1
Level 2

DAQ
MoNA Stop
Target

Thin

CFD

Trigger

CFD
TDC

Sweeper Electronics Rack
Figure 3.5: Trigger logic diagram.

45

Veto
Computer Go

allowed them to communicate with the DAQ software to record their information.

46

Chapter 4
Analysis
4.1

Overview

Online data analysis during the experiment was performed using SpecTcl [47], while
offline analysis was performed post-experiment using a combination of ROOT [48],
SciPy [49], and IPython [50]. Figures and final plots were created in PyX [51].
Calibration data were taken before or during the experiment for different detector
systems and calibration parameters extracted in offline analysis. After the calibrations
were complete, particle identification could be performed and the neutron-fragment
decay reconstructed. Once reconstruction was complete, physics information was
extracted by using a Monte Carlo simulation package to model the entire experimental
response to differing physics input.

4.2
4.2.1

Calibrations
Timing Scintillators

All beam timing signals were calibrated using knowledge of the electronics modules
processing each signal and beam properties. As discussed in Section 3.3, the timing
47

signals were processed by TDCs, which had a linear response and therefore were
calibrated with slopes and offsets. The slopes were directly determined from the
TDC range (400 ns) and number of channels (4096), leading to a slope of 0.0977
ns/channel. The raw timing signals included a 20 ns jitter resulting from FPGA
processing, as the FPGA operated on a 50 MHz clock. To rectify this, all beam
timing signals were converted to nanosecond scale by multiplying by the relevant
slope (0.977 ns/ch for all detectors in this setup), and the signal from the upper left
PMT of the thin scintillator was then subtracted from all other PMT signals.
As the set-up only measured relative times, some choice had to be made for T
= 0 in order to define the offsets; for this analysis, T = 0 was chosen to be beam
crossing through the target. To determine the timing offsets for each detector, a
run with the target removed and beam tuned to travel through the center of all
charged particle detectors was used. The magnetic rigidities of all magnets in the
beam line then defined beam velocity at each point in the beam line and therefore
the time of beam particles hitting each timing detector relative to the target. Unlike
the other timing detectors, the thin scintillator had four PMTs contributing to its
timing signal. Therefore, the individual PMT timing signals were calibrated for each
event, and signals from those PMTs which fired were averaged to provide the final
thin timing signal.
All beam line timing signals drifted over the course of the experiment and had to
be shifted on an event by event basis to remove the drift effects. Figure 4.1 displays the
centroids of the raw timing signals for each of the major timing detectors as a function
of event number throughout the course of the experiment. The thin scintillator is not
shown because it was the triggering detector and therefore showed no apparent drift.
The drift structures observed were removed with a shift calculated event-by-event by
interpolating over a running average created by averaging each detector’s time signals
in bins 1·105 events wide.
48

Time (TDC Channels)

250
Target Chamber - Thin Left Up
A1900 - Thin Left Up
RF - Thin Left Up

225
200
175
150
0

5 · 105

1 · 106

1.5 · 106
Event Number

2 · 106

2.5 · 106

Figure 4.1: Timing drifts as a function of event number for all beam line timing signals.

49

4.2.1.1

RF Timing Shifts

The RF timing signal is downscaled by a factor of two, so it is possible to see multiple
RF timing peaks in the raw RF time spectrum. As shown in Figure 4.2a, there were
three RF peaks in the timing spectrum for this experiment, corresponding to two
actual timing peaks. The middle peak corresponds to one actual RF signal peak, while
the left and right correspond to another peak being split because the physical timing
peak is located on the edge of the electronic time gate for accepting signals in the TDC.
Lying on the edge of the timing window causes some sensitivity to the gate generation
and signal propagation, so a simple shift of the cyclotron frequency (1./23.16 MHz =
43.18 ns) imperfectly matched and small additional shifts were needed to properly line
up the peaks and match the line shape of the central peak. Figure 4.2b demonstrates
the difference between the two techniques. The shifts corresponding to the cyclotron
frequency (dotted line) do not line up with or match the line shape of the middle
peak (solid), but agreement is achieved with an additional shift of 0.5 ns to the left
peak and 0.7 ns to the right peak (dashed line).

50

3 · 105

3 · 105

2 · 105

2 · 105

1 · 105

Middle Peak
43.18 ns Shift
Modified Shift

1 · 105

0

−475

−450

−425 −400
RF Time (ns)

0

−375

(a) RF Time before shifts

−425

−420
−415
RF Time (ns)

(b) RF Timing shifts

Figure 4.2: The downscaled RF timing signal showing up as three peaks in an uncorrected time spectrum (a). In (b), the
43.18 ns shift corresponding the cyclotron frequency does not match the middle RF peak, while adjusting the shift parameters
slightly (see text) results in good agreement.

51

4.2.1.2

Virtual Target Scintillator Timing

(a) A1900 to Thin
750
500
250
0

600
400

110
120
130
140
150
Time of Flight from A1900 to Thin (ns)
(b) A1900 to Target
Physical Tgt. Scint
Virtual Tgt. Scint

200
0

80
90
100
110
Time of Flight from A1900 to Target (ns)

Figure 4.3: Resolution comparison between the target chamber and thin scintillators
for a no-target run. The beam particle peaks in the spectrum utilizing the target
chamber scintillator (right solid) is a factor of 2.5 wider than the corresponding peaks
in the time spectrum utilizing the thin scintillator (left) despite the A1900 to thin
flight path being longer. This widening is entirely due to resolution effects. The
virtual target scintillator (right dashed) improves the resolution by ∼20%.
The target chamber timing scintillator had poor timing resolution during the
experiment which critically degraded the quality of isotopic separation. Figure 4.3
demonstrates the degraded resolution by showing two time of flight spectra for a
run where the target was removed, the momentum slits of the A1900 were reduced to
52

0.5%, and the beam was sent through the middle of all charged particle detectors. The
left spectrum shows the resulting time of flight from the A1900 scintillator to the thin
scintillator, while the right spectrum displays time of flight from the A1900 scintillator
to the target chamber scintillator. Despite the A1900 to thin scintillator path length
being longer, the A1900 to target chamber timing spectrum has significantly wider
peaks corresponding to the various incoming beam particles - a factor of roughly 2.5.
This widening is entirely due to the timing resolution of the target chamber scintillator
during this experiment. Therefore, a virtual timing signal was constructed event by
event from the timing signals of the RF and A1900 using knowledge of the beam
properties and LISE++ calculations to account for the various locations where the
beam lost energy. Figure 4.4 will serve as a notational reference for the following
discussion, and shows what beam line elements come into play with the calculations
that follow.

53

Index:

0

Name:
Prod. Tgt.
Velocity (cm/ns): 12.05
Time (ns):
-417.1

1

2

3

4

5

6

7

8

Wedge
11.00
-272.0

A1900 Scint.
10.84
-106.9

TCRDC1
10.84
-50.5

TCRDC2
10.83
-30.4

Target Scint.
10.79
-0.6

CRDC1
10.75
16.0

CRDC2
10.71
32.6

Thin Scint.
0
41.2

Figure 4.4: Notational reference for virtual target scintillator calculations

54

In order to properly calculate the timing signal of this virtual target scintillator,
the relationship between velocities of the beam going into and exiting from each
beam line element must be understood. Energy loss calculations were performed using
LISE++ near the beam velocity for the achromatic wedge, the A1900 scintillator, and
beamline CRDCs. Figure 4.5 demonstrates this relationship for every point where
the beam lost energy before the target chamber. In all cases the relationship is linear,
which simplifies the problem of calculating a beam velocity and propagating it to
the target. For the discussion below, the following notation will be used: Vij will
represent the velocity between points i and j, Tij will be the time of flight for beam
particles between those points, Dij will represent the distance between the beam
line elements, mi will represent the slope for the linear relationship between velocity
before and after beam line element i, and bi will represent the offset.

Constructing the virtual target chamber scintillator begins with the time of flight
between the RF signal (assigned to correspond to the location of the primary beam
production target) and A1900 scintillator. There is one issue however: the achromatic
wedge mentioned in Section 3.2 is between the primary target and A1900 scintillator, and the unmeasured energy loss there can change the predicted beam velocity
by approximately 8% and therefore introduce systematic errors in the calculation.
Therefore, let us consider the problem where we have points 0, 1, and 2 (see Figure 4.4) and wish to calculate the velocity between 1 and 2. If we can calculate this
quantity, the aforementioned velocity relationships can be used to propagate beam
velocity and time of flight anywhere in the beam line. We are given T02 and V12 =
m1 V01 + b1 . Then,
D12
D01
+
V01
m1 V01 + b1
T02 V01 (m1 V01 + b1 ) = D01 (m1 V01 + b1 ) + D12 V01
T02 = T01 + T12 =

55

Rearranging,
2
T02 m1 V01 + (T01 b1 − D12 − D01 m1 )V01 − D01 b1 = 0
Therefore we must simply solve the quadratic formula for the correct root and exploit
the linear velocity relationships:

V01 =

b2 − 4ac
2a

−b ±

a = T02 m1
b = T01 b1 − D12 − D01 m1
c = −D01 b1
V12 = m1 V01 + b1
The positive root has the correct behavior in the limit ∆V = 0 (no energy loss, or
wedge thickness = 0) so that one is used. With this in hand, we have the velocity of
the beam between the wedge and the A1900 scintillators. Because we know that, the
correlations between velocities everywhere the beam loses energy, and the distances
between all the beam elements we can predict what the time of flight between the
A1900 and target chamber scintillators was without having to use the signal from the
physical target chamber scintillator.
In Figure 4.3, the dashed line corresponds to the virtual target scintillator being used for time of flight calculations. The improvement in timing resolution was
∼20% over the physical scintillator, which made isotope separation possible. Velocity
correlations must be calculated for each beam particle as each beam component has
different energy loss characteristics, so only the 22 N beam component is shown.
56

Post-Element Velocity (cm/ns)

13
Wedge
A1900
TCRDC1
TCRDC2

12.5
12
11.5
11
10.5
10
10

11
12
Pre-Element Velocity (cm/ns)

13

Figure 4.5: Velocity relationships calculated by LISE for all beam-line elements necessary for virtual target scintillator calculations. All trends are linear, and the CRDC
trends lie very near each other because the energy loss is minimal through those
detectors.

4.2.2

Tracking Detectors

Linear calibration parameters were used to calibrate all CRDCs, both in the beam
line and focal plane chamber. The slope in the x direction was fixed by the pad pitch
of 2.54 mm. The x offset, y slope, and y offset for a given detector were determined
by sweeping beam across the face of the detector with a mask placed in front of it
with holes drilled at known positions. An example calibration spectrum is shown
in Figure 4.6. Well defined holes and slits are observed which can then be used to
calculate the relevant calibration parameters.
The x position in each CRDC was determined by the distribution of charge on its
57

2000
Drift Time (Channels)

1800
1600
1400
1200
1000
800
600
0

20

40
60
80
100
Charge Distribution Centroid (Channels)

120

Figure 4.6: Sample mask hole position spectrum for CRDC2. The positions of the
holes are known which enables the mapping of the raw detector information onto
physical units.
pads. The pad data were read out every 25 ns, and the sum for each pad was then
determined after a pedestal subtraction. Beam swept across the detector was used to
gain match the pads relative to each other. A Gaussian fit to the charge distribution
for each event was used to determine the x position of that particle.
A technical issue with the gas handling system for the focal plane CRDCs resulted
in a discontinuity near event 2·106 demonstrated in Figure 4.7, which shows the centroids for the drift times of each CRDC as a function of event number throughout
the experiment. The events near the discontinuity were discarded, as the rapidly
changing detector properties resulted in unacceptable resolution losses when attemping to correct the detectors in that region. The drift time signal which generates the
y position information for each CRDC was multiplicatively corrected on an event by
event basis by interpolation over a histogram describing the time drift.

58

Time (TDC Channels)

1500
TCRDC1
TCRDC2
CRDC1
CRDC2

1000

500

0
0

5 · 105

1 · 106

1.5 · 106
Event Number

2 · 106

2.5 · 106

Figure 4.7: Drift time of ions in all CRDCs as a function of event number. The discontinuity near event 2·106 resulted from
a technical issue changing the drift velocity of ions in the focal plane CRDCs.

59

The beam line tracking CRDCs suffered a less severe change in the drift time signal
demonstrated in Figure 4.7, which was corrected with a multiplicative scaling factor
in the same fashion as the focal plane CRDCs. Additionally, one of the TCRDCs
had a technical issue with its charge collection such that the x position spectrum was
distorted. The issue and its resolution are shown in Figure 4.8. On the left, the x
position spectrum for a no-target run shows sharply peaked structures in place of a
smooth distribution of beam particles. The shaded regions show where the detector
was functioning reliably, albeit at a very low efficiency.
The event by event position information from that TCRDC was not salvageable
at a useful efficiency, but removing the events in the spikes allowed a correlation
between the x position spectra of both tcrdcs to be observed as shown in the right
hand spectrum of Figure 4.8. This correlation was used to calculate the incoming
beam trajectory in the x plane without using the affected TCRDC on an event by
event basis. The trajectory of the beam particles could still be propagated forward to
the target chamber for use with the reconstruction process later in the analysis. The
y position information from the detector was not affected by this charge collection
issue and could be utilized without further treatment.

4.2.3

dE, E Scintillators

The eight PMTs of the ∆E and total E scintillators were gain matched and corrected
for position dependence due to light attenuation in the plastic. Each PMT was
corrected for drift event by event. A no-target run with the beam centered in the focal
plane was then used to gain match all PMTs with the beam travelling straight through
the middle of the both scintillators. To produce the complete energy signal for each
scintillator, the top and bottom PMTs were separately averaged and then the root
mean square of the top and bottom pairs taken. X and y correction parameters for
this signal were then obtained by sweeping beam across the face of the detectors and
60

400
300
200
100

TCRDC2 X position (mm)

0
−40

−20
0
20
TCRDC2 X Position (mm)

40
X2 = 1.270X1
30
20
10
0
−10
−20
−30
−40
−40 −30 −20 −10 0 10 20 30
TCRDC1 X Position (mm)

40

40

Figure 4.8: TCRDC2 charge collection issue (left) and the correlation between
TCRDC positions used to replace the physical device’s x position information.

61

using the CRDC information to obtain positions on each scintillator. The dominant
position dependence was in the x direction, and this dependence is shown for both
detectors in Figure 4.9. In both cases, the total energy signal for each scintillator
was plotted as a function of x position at the detector, and a fit function generated
to describe the position dependence. This function was then used to generate a
multiplicative correction event by event, the results of which are also demonstrated
in Figure 4.9.

62

250
200
150
100
50

1200
Uncorrected EThick (arb units)

Uncorrected EThin (arb units)

300

0
−200−150−100 −50 0
50 100 150 200
Thin X Position (mm)

200
100
0
−200−150−100 −50 0
50 100 150 200
Thin X Position (mm)

900
800
700

1500
Corrected EThick (arb units)

Corrected EThin (arb units)

300

1000

600
−200−150−100 −50 0
50 100 150 200
Thick X Position (mm)

500
400

1100

1400
1300
1200
1100
1000
900
−200−150−100 −50 0
50 100 150 200
Thick X Position (mm)

Figure 4.9: Thin (top left) and thick (top right) response as a function of x position and the corresponding correction curves.
The bottom panels show the correction taking effect for both scintillators, which results in a flattened spectrum with position
correlations removed.
63

4.2.4

Ionization Chamber

As discussed in Section 3.3.2, the ionization chamber has 16 charge collection pads
whose signals are summed to obtain a ∆E measurement for element separation of
reaction products. Therefore, the energy loss calibration of the ionization chamber
proceeded by:
15
∆E =

qi fi (y, t)

i=0
fi (y, t) = S/di (y, t)

(4.1)
(4.2)

Where qi is the raw charge collection for an individual pad, fi (y, t) is a gain matching
function, S is a scaling factor for setting the overall scale of the calibrated energy loss
spectrum, and di (y, t) is function characterizing the pad response as a function of
position. The function fi (y, t) required special treatment because of the particular
behaviour of this device during the experiment.
The primary cause of the more challenging calibration of this detector is demonstrated in Figure 4.10, which shows each charge collection pad’s signal as a function
of y position in the ion chamber. Pads 7 and 9-12 either collected no or very little
charge, and therefore were not used in the calculation of the total ∆E measurement.
All other pads show a severe position dependence which contributed to the poor resolution in Figure 4.12. In particular, the individual pad responses were sufficiently
different to require independent treatment, so each pad was characterized separately
and the pad signals combined after gain matching.
Finally, Figure 4.11 shows the response of one pad as a function of y position
over the first 20 hours of the experiment. It is clear that the overall structure of the
detector response does not change, but smaller substructures develop and evolve as
a function of time. Because of the sharp rising and falling regions of the response
64

function, it was important to capture the time evolution of each pad’s response with
as fine a time binning as possible. Therefore, the data was divided into sections
corresponding to timescales of roughly 45 minutes, each overlapping by half that
value. Each pad’s response was captured as a function of position for each section,
and the sections chained together to form a surface characterizing the pad response
as a function of position and time. Bilinear interpolation across this surface was then
used to generate a multiplicative correction parameter, which could be combined
with a global scaling parameter to create an event-by-event fi (y, t) to gain match
each pad. Pads 0, 8, and 13 were beyond this technique’s ability to recover and their
inclusion in the total ion chamber signal spectrum degraded it, so fi (y, t) was set to
0 for those pads. Therefore, pads 1-6 and 14-15 were used to generate the final ∆E
signal from the ion chamber. The results of this correction method are summarized
in Figure 4.12, where the right panel demonstrates reasonable Z separation while the
left does not.

65

Pad 0

Pad 1

Pad 2

Pad 3

Pad 4

Pad 5

Pad 6

Pad 7

Pad 8

Pad 9

Pad 10

Pad 11

Pad 12

Pad 13

Pad 14

Pad 15

Uncorrected Ion Chamber Pad Signal

500
0
500
0
500
0
500
0

−100

0

100

−100 0
100
−100 0
100
Ion Chamber Y Position (mm)

−100

0

Figure 4.10: Sample ionization chamber pad signal vs. Y position for all pads.
66

100

400
20

350

Ti
e

m

10
)

rs
(h

5
0

50

300
−50
−25
)
0
(mm
ion
25
osit
YP
ber
ham
nC
Io

Figure 4.11: Position and time dependence of pad 1.

67

Signal

15

Pad 1 Unco
rrected

450

600
Corrected ∆EIC (arb units)

Uncorrected ∆EIC (arb units)

600
500
400
300
200
100
0
38

40
42
44
46
48
Uncorrected Time of Flight (ns)

500
400

(a) Before Corrections

miss

22 N

300
200
100
0
38

50

22 N

40
42
44
46
48
Corrected Time of Flight (ns)

50

(b) After Corrections

Figure 4.12: Resolution comparison of uncorrected ion chamber and physical target scintillator (a) to corrected ion chamber
and virtual target scintillator (b). Element separation is only possible in (b), and the improved timing resolution allows clean
separation of 22 N missing the target.

68

Counts/ns

200

neutron peak
γ peak

100

0

0

50
100
Time of Flight (ns)

150

Figure 4.13: γ-ray timing.

4.2.5

MoNA

MoNA provided information about the neutron time of flight, trajectory, and deposited energy. Each PMT on each bar output an energy and time signal. Cosmic
muons were used to gain match and perform a linear calibration of the energy signals
for each PMT as muons deposit roughly 20.5 MeVee (MeV electron equivalent) in 10
cm of BC-408. The time signal from each PMT is ultimately sent into a TDC. The
response of all MoNA TDCs was linear, so their signals were calibrated using slopes
and offsets. The slopes were determined by pulsing the electronics with a timing
calibrator, and offsets from cosmic muons travelling through multiple bars.
Time of flight was measured by the mean time of both PMT signals from a struck
bar, and required a timing offset for MoNA relative to the target. A thick collimator
was placed in the target chamber and γ-rays from the target striking the front face
of MoNA were used to determine this offset. Figure 4.13 shows an example time of
flight spectrum for the collimator run, where two peaks are noticeable. The sharply
defined early peak results from prompt γ-rays and is used to determine the offset,
while the slower, broad peak results from neutrons.
69

Finally, to ensure the veracity of the γ timing, CFD walk in the MoNA electronics
had to be accounted for. It has been observed in several experiments that the mean
timing signal from MoNA is systematically shifted for low energy deposition events.
This effect is demonstrated in Figure 4.13, which shows the time of flight for a γ timing
run as a function of energy deposition [52]. There is a clear correlation between time
and energy deposition for the events which deposit a small amount of energy. This
effect has been characterized and shown to be a consistent property of the MoNA
electronics, so while this experiment did not accrue sufficient statistics in a γ timing
run to independently characterize the walk, correction parameters could be taken
from other experiments and used with confidence. The walk shift function took the
form:


−2.56625e−0.62272/q + 1.6531 for q < 1.8



tcorr = t −

 2.861


− 1.761
q

(4.3)

for q > 1.8

Here, t is the uncorrected time of flight, q is the energy deposited in MoNA in MeVee,
and tcorr is the walk corrected time of flight. This completed the characterization
and calibration of MoNA. With all calibrations performed, the momentum of neutrons
arriving at MoNA could be fully described for use in the reconstruction process.
The y and z position of neutron hits in MoNA were determined by which bar
was struck. The time difference between signals at each end of a bar was used to
determine x-position, and was calibrated using cosmic muons.

4.3

Particle Identification

In order to reconstruct the reactions of interest for the 18,20 C + n systems, all
fragments entering the sweeper had to be separated in Z and A. The incoming beam
70

was not pure, so the first step was the selection of events correlated to the 22 N
incoming beam. Figure 4.14a demonstrates the constituents of the beam for this
experiment by showing ∆E versus time of flight from the A1900 scintillator to the
thin scintillator for a no-target run with beam running through the center of all
charged particle detectors. The 22 N is labelled and corresponds to the single most
intense particle spot. This component comprised 32% of the total beam, and arrived
at the target with a rate of 37/s. The remaining fraction of the beam was dominated
by light ions created in the achromatic wedge at the central point of the A1900 which
populate the lower left section of the figure. The nearest heavy ion contaminant to
the beam of interest was 20 C, which comprised roughly 2.8% of the total beam and
could be separated in a time of flight spectrum. Beam selection for the production
runs is shown in Figure 4.14b, where ∆E in the target chamber scintillator is plotted
versus the time of flight from the A1900 scintillator to the target chamber scintillator.
The primary separation is in time of flight, and ∆E is used to reduce contamination
from lighter ions. Once again the 22 N is labelled and corresponds to the single
brightest particle spot in the spectrum. The sharp cut structure on the high end
of ∆E corresponds to the signal coming from the PMT on the target scintillator
saturating its QDC for some events, which did not affect the timing or the separation
for lighter ions.

71

800
2000

600

∆Epot (QDC channels)

∆EIC (arb units)

700
22 N

500
400
300
200

1500
1000
22 N

500

100
0
100

110
120
130
140
150
Beam ToF A1900 to Thin Scint (ns)

0
70

160

(a) Beam Profile.

80
90
100
110
Beam Time of Flight (ns)
(b) Beam ID Spectrum.

Figure 4.14: Beam profile and beam ID spectra.

72

120

The next step is Z separation of the reaction fragments. A simplification of the
2
Bethe-Bloch formula results in the relation ∆E∝ Z2 , which suggests that Z separav
tion is best achieved through an energy loss measurement with its Z 2 dependence.
Energy loss is correlated with the square of the particle’s velocity, which further
suggests that two dimensional separation of energy loss versus a parameter correlated to velocity would be optimal. The experimental setup has two independent
measurements correlated with velocity - a time of flight and a total kinetic energy
measurement in the thick scintillator. Both of these are plotted against the energy
loss in the ionization chamber as shown in Figure 4.15. In both cases the different
elements appear as slanted regions through the parameter space, indicative of the
correlations mentioned above. The 22 N secondary beam appears as the most intense
region for both spectra, and results from unreacted secondary beam reaching the focal plane detectors. This can be used as a reference, and defines the Z=6 reaction
fragments of interest to be the locus of points just below the beam spot as indicated
on both spectra. Gates on both spectra were applied to reduce contamination levels,
and a neutron coincidence gate used to further reduce the amount of unreacted beam
entering the reaction gate.

73

600
500

400
300

∆EIC (arb units)

∆EIC (arb units)

500
Z=7
Z=6

200

Z=7

300
200

Z=6

100

100
0
38

400

40
42
44
46
48
Corrected Time of Flight (ns)

0
600 650 700 750 800 850 900 950 1000
Deposited Energy (arb units)

50

(a) Z ID from ∆E vs. Time of Flight.

(b) Z ID from ∆E vs. E.

Figure 4.15: Element separation by two different methods - ∆E vs. Time of Flight and ∆E vs. E. Both methods are combined
to reduce contamination from unreacted beam.

74

Isotope separation for the reaction fragments was achieved using the time of flight
information provided by the virtual target scintillator combined with the post-target
trajectory information. Figure 4.16a illustrates simulated particle tracks corresponding to three different isotopes of the same element travelling through the sweeper
magnet. The different isotopes have different magnetic rigidity, which causes them
to bend differently in a dipole field and therefore travel different trajectories, which
in turn makes separation possible. Complications arise from spreads angle and momentum distribution, which smear out the individual isotopic signatures through the
magnet. The definition of magnetic rigidity serves to illustrate this:

Bρ =

mv
q

t = A

(4.4)
Du
Bρq

(4.5)

The time of flight t for each isotope is related to its track length D and magnetic
rigidity Bρ, and these are affected by the emittance of the incoming beam, straggling
through the target, reaction dynamics and subsequent neutron evaporation, and the
nonuniform nature of the sweeper’s magnetic field. Therefore, within each isotope
D and Bρ have a width which smears out the time of flight. These effects can
be correlated with the reaction fragment’s emittance in the focal plane, and in a
traditional spectrometer such as the S800 this appears as correlations with the focal
plane angle. The right panel of Figure 4.16 demonstrates the application of this
method to our setup, showing a plot of x (dispersive) angle versus time of flight.
No hint of separation can be observed, which can be attributed to the properties
of the sweeper magnet. Position and angle measured at the focal plane are highly
correlated; consequently, a naive 2D separation is impossible without reducing the
correlation space by at least one parameter. A systematic mechanism to visualize
and account for these strong correlations has been developed (see [52] for the original
75

work).

76

More Rigid
Less Rigid
(a) Particle tracks.

(b) Traditional PID.

Figure 4.16: Particle tracks for different carbon isotopes through the sweeper magnet (a), and a traditional particle ID attempt
from plotting dispersive angle vs. time of flight (b). Correlations between angle, position, and time of flight preclude such a
treatment (see text).

77

The strongest correlations that need to be accounted for are those between time,
measured position, and measured angle in the focal plane. An example visualization
of the correlations is shown in Figure ??, where bands can be seen in a 3D scatter
plot of those parameters [53]. In this experiment, the low beam rate and previously
discussed technical reasons result in Figure 4.18, in which the correlation curves can
still be seen albeit more faintly due to the smaller number of counts in the spectrum.
Reducing this separation to two parameters proceeds by determining the correlations
between the angle and position parameters, which is accomplished by first projecting
the 3D scatter plot into the angle/position plane.
This is accomplished by dividing the angle/position plane into a number of small
regions, and calculating the average time of particles entering those regions. This
calculation results in the plot shown in Figure 4.19, where the angle/position plane
make up the y and x axes and the average time is represented by color. The correlation between the angle and position in the focal plane is seen as lines of constant color
running through the image. A curve can be drawn which then represents this correlation, and a reduced parameter created which accounts for the correlations between
angle and position:
e = θx − f (x)

(4.6)

This emittance parameter e can be plotted against time of flight to obtain the first
separation. Figure 4.20a shows a high statistics example to illustrate the technique,
while 4.20b shows how it looked for this experiment. Bands can be seen forming along
a diagonal line through this parameter space. To improve the separation, these 2D
correlations can be further reduced to a single parameter (corrected time of flight)
and that parameter plotted against all other charged particle parameters to remove
correlations. To make this reduction, a simple linear projection of the form

tcor = t − e/s
78

(4.7)

Figure 4.17: Example Correlations

79

Figure 4.18: Correlations

is performed, where s is the slope the bands follow in Figure 4.20. With this in place
and other correlations with nondispersive position and angle removed, the corrected
time of flight can be used for isotope identification. Figure 4.21a demonstrates a
high statistics example result from this technique, while 4.21b demonstrates particle
separation for the Z=6 reaction fragments of this experiment. The separation is more
limited in this experiment than the example, but there are clear indications of three
isotopes in the spectrum. These have been identified as 18-20 C, with the unresolved
momentum tail of 17 C seen as the tail on the left side of 4.21b. Because of the
somewhat limited separation, conservative isotope gates were placed on the corrected
time of flight and are represented by the grey regions. Both 18 C and 20 C fragments
were resolved at the cost of some 20% of their counts and a cross contamination 2%.
19 C was contaminated from both 18,20 C fragments and therefore its contamination
level was 16%. As a result, detailed analysis was performed on 18,20 C fragments but
80

Figure 4.19: Projection of correlations into 2D.
not 19 C.

81

(a) Example Reduction.

(b) Reduction.

Figure 4.20: ”Emittance” parameter vs. time of flight.

82

80

Counts / 0.4 ns

70
18 C

60

19 C

50

20 C

40
30
20
10
0

(a) Example PID.

36

38
40
42
44
46
Corrected Time of Flight (ns)
(b) PID.

Figure 4.21: Final corrected time of flight with grey regions indicating particle gates.

83

48

4.4

Reconstruction

As discussed in Section 3.1, calculating the decay energy of neutron unbound states
by the invariant mass method requires the measurement of the neutron and charged
fragment momentum four-vectors. Constructing these vectors proceeds by an indirect method utilizing knowledge of the magnetic field of the sweeper for the charged
fragment, and by ray-tracing for the neutron.

4.4.1

Fragment Momentum Four-Vector

The post-magnet trajectory of the fragment is directly measured by the experimental
setup, and the quantities needed for reconstructing a decay energy are those of the
fragment at the target (Section 3.1). Therefore, the post-magnet trajectory information must be tracked back to the target by some means. Because these two frames are
related by ion optics through a magnetic field, this is a tractable problem if the magnetic field is known. The magnetic field of the sweeper was measured by seven Hall
probes evenly spaced and mounted vertically on a movable cart which was stepped
through the track of the sweeper at regular intervals (more details given in [45]).
The ion-optics code COSY INFINITY [54] generated a matrix M which related
the post-magnet trajectory information to a target momentum 4-vector:


xcrdc1













crdc1
θx
y crdc1





xtarget







 target 

 θx







 = M  y target 






 target 
crdc1 
θy

 θy




target
L
E

(4.8)

To define a momentum 4-vector, the target quantities needed from Equation 4.8
target
target
are E target , θx
, and θy
and as mentioned above, none of these is directly
84

crdc1 , y crdc1 , and θcrdc1 , while
measured. Post-sweeper tracking provides xcrdc1 , θx
y
the forward tracking from the beam line CRDCs provides xtarget and y target (the
angle information is destroyed by the reaction in the target). Direct matrix transformations to rearrange for the desired quantities in terms of the measured ones is
impossible because the track length L is unknown. An approach commonly used by
COSY to overcome this obstacle is to assume xtarget = 0, However, this assumption
is only valid for systems where the beam spot is small, and breaks down for the larger
beam distributions encountered in this setup (order of 2 cm).
It is however possible to perform a partial inversion of M to obtain the needed
transformation matrix. In particular, matrix operations were performed to exchange
pairs of terms on each side of Equation 4.8 (see [55] for details). A condition for the
successful swapping of terms is that they have a high degree of correlation with each
other. The resulting expression is:


target
θx




 target 


 y




 target 
 = Mp 
 θ


 y






L






target
E


xcrdc1











crdc1 
θy


target
x
crdc1
θx
y crdc1

(4.9)

where now quantities measured by the setup are on the right side and can be related
to quantities we are interested in on the left though a transformation matrix which
can be calculated with COSY.

85

40

30

30

Inverse Tracked θy (mrad)

Inverse Tracked θx (mrad)

40
20
10
0
−10
−20
−30
−40
−40 −30 −20 −10 0
10 20 30
Forward Tracked θx (mrad)

20
10
0
−10
−20
−30
−40
−40 −30 −20 −10 0
10 20 30
Forward Tracked θy (mrad)

40

40

Figure 4.22: Consistency check for magnetic field map transformations showing reasonable agreement between the forward
and inverse tracking processes.

86

Performing this reconstruction process required transformations between and alignment of three different coordinate systems (pre-quadrupole, target chamber, and postsweeper), so it was necessary to perform consistency checks in a case where the ion
optics should be straightforward. Therefore, data was taken with the production target removed, and the beam placed at a defined point in the focal plane. Example
comparison spectra are shown in Figure 4.22 for the beam travelling near the center
of the magnet and focal plane. The lines show the perfect correlation line between
the forward and inverse tracked parameters. The agreement of the data is reasonable,
with the systematic deviations seen being insufficient to affect the reconstruction of
the decay energy. The magnetic field is best understood near the center of the sweeper
which results in the best agreement. As a result, the sweeper is generally tuned such
that the primary reaction fragments of interest are focused near the center.

4.4.2

Neutron Momentum Four-Vector

The neutron four-momentum must be determined to fully reconstruct the mass of
the initial unbound state. MoNA is time and position sensitive, so it is possible to
construct this vector by measuring the neutron flight path and time of flight (ToF).
The full position vector (x,y,z) must be determined for each neutron, where the
target is taken to be (0,0,0). The y and z components were determined by the
measured location of the struck bar. Neutron strikes were assumed to occur at the
center of each bar in the y and z directions, leading to a 5 cm uncertainty for those
components of the position measurement. The x component was determined by the
calibrated time difference between the two PMTs on the struck detector module,
resulting in a FWHM resolution of 7.5 cm [46].
The neutron ToF was determined from the calibrated average of the timing signals
from the PMTs on each end of a bar. The start time for each TDC came from
the PMTs on the struck bar, while the stop time resulted from the target chamber
87

scintillator. The ToF was calibrated using cosmic ray muons and γ rays from the
target as described in Section 4.2.5. The calculation of the neutron kinetic energy
then proceeded by first determining the velocity:

v=

R
T oF

(4.10)

Where R is the length of the position vector of the neutron strike. Then from relativistic mechanics:

β = v/c
γ =

1
1 − β2

KEneut = Mn (γ − 1)

(4.11)
(4.12)
(4.13)

Where Mn is the rest mass of the neutron. With the trajectory and kinetic energy of
the neutron thus determined, the four-vector can be fully defined and combined with
the fragment four-vector to determine the decay energy.

4.5

Simulation

The measured decay energy spectrum is heavily influenced by resolution and acceptance effects resulting from the characteristics of the experimental setup. This
smearing was accounted for numerically with a Monte Carlo simulation which included resolution and acceptance effects for each detector in the setup. Reaction dynamics and decay properties were used as inputs to create the neutron and charged
fragment within this framework, and those particles were propagated to their respective detectors. Particles which missed a detector used in the analysis were flagged,
and the response functions for each detector encountered were folded into the gen88

eration of a simulated dataset which could be analyzed using the same software the
experimental data used. Therefore the two datasets could be directly compared and
free parameters adjusted until spectra matched between the datasets.
The simulation begins with 22 N beam particles generated at the position of
TCRDC1 with some energy and emittance distribution. The parameters associated
with these degrees of freedom are fixed by comparing the simulated datasets to runs
taken with the target removed and beam sent down the center of all detectors. The
response of TCRDC1 (actual particle position convoluted with the x and y resolutions
of the detector) is recorded, and the particle propagated forward through TCRDC2
to the reaction target. The reaction in the target is treated by a Goldhaber model
[56] with a friction term [57]. The neutron decay is then modelled by one of the
evaporation mechanisms discussed in Section 2.3. The neutron and fragment final
products are treated separately from this point forward.
The neutron is first propagated to the window at the downstream end of the
sweeper magnet, where the geometric acceptances of the window flag events which
would have struck the steel encasing the window. From there the neutron is propagated to the front face of MoNA, where the acceptance of MoNA flags neutrons which
miss it. The discretization in the y and z directions and position resolution along the
length of the struck MoNA bar (x direction) are all accounted for in the final event
properties.
The charged particle is propagated from the reaction target to CRDC1 by a transformation matrix generated by COSY as discussed in Section 4.4.1. Here the detector’s resolutions and acceptances are accounted for, and the particle allowed to freely
drift to CRDC2 and finally the thin scintillator, where their respective responses are
recorded.
Because of the technical issue with the TCRDC2 x position spectrum, the incoming beam angular distribution could not be constrained directly. Instead, the
89

500
Counts

Counts

500
250

Counts

Counts

0
−100
0
100
CRDC1 X (mm)
400
200
0

250
0
−100
0
100
CRDC1 Y (mm)
500
250
0

−50
0
50
CRDC1 θx (mrad)

−50
0
50
CRDC1 θy (mrad)

Figure 4.23: Comparison of simulation to data for 22 N reaction fragments.

90

position distribution for TCRDC1 was matched, and the angle and energy distributions matched by comparing to the focal plane CRDC spectra for a no-target,
centered beam run. Figure 4.23 shows the comparison of simulation to data for this
run. The agreement in x is quite good while the agreement in y degrades because of
shortcomings in the magnetic field map’s treatment of the y direction.
Reaction model parameters can then be verified by comparing charged fragment
focal plane and inverse tracked spectra to data. Because of the relative mass between
the neutron and fragment, the fragment focal plane emittance spectra are insensitive
to the neutron evaporation. These checks are performed for each reaction fragment
being analyzed, and the resulting spectra are shown for 20,18 C in Figures 4.24 and
4.25 respectively. The statistics are much lower than the unreacted beam run, but
the agreement is still reasonable. Once again the best agreement is in x because of
the properties of the magnetic field map. It should be mentioned that the comparison
to reaction fragments probes a much wider region of the simulation’s validity, as the
individual emittance distributions are wider because of the reaction and the 18 C
fragments populated the edge of the fragment acceptances which probes the fringe of
the magnetic field map.

91

30
Counts

Counts

40
20

20
10
0

−50
0
50
CRDC1 θx (mrad)

20
Counts

Counts

20
0

20
10

10
0
−100
0
100
CRDC1 Y (mm)

Counts

Counts

0
−100
0
100
CRDC1 X (mm)

20

−50
0
50
CRDC1 θy (mrad)

10

0
0
−50
0
50
−50
0
50
Inverse Tracked θx (mrad) Inverse Tracked θy (mrad)
Figure 4.24: Comparison of simulation to data for 20 C reaction fragments.

92

0

Counts

Counts

25

50
25

50
25
0
−100
0
100
CRDC1 Y (mm)
75

−100 0
100
CRDC1 X (mm)
Counts

Counts

50

0
−100
0
100
CRDC1 θx (mrad)

50
25
0

−50
0
50
CRDC1 θy (mrad)

20
10

Counts

Counts

30
20

0

0
−100 0
100
−100
0
100
Inverse Tracked θx (mrad) Inverse Tracked θy (mrad)
Figure 4.25: Comparison of simulation to data for 18 C reaction fragments.

93

Chapter 5
Discussion
As discussed in Section 2.2, 19 C was studied with the intent of determining whether
its 5/2+ excited state was bound with respect to neutron decay and resolve questions
1
raised by prior work. Nothing is know about 21 C beyond its instability, and populating states in this nucleus could provide constraints on the halo nucleus 22 C as well
as probe the ν1s1/2 − ν0d5/2 shell gap for N=15 isotones.

5.1

19

C Results

In the intervening time between when prior measurements which extracted spectroscopic information for 19 C were published and this work, a new mass evaluation’s
results became available [58]. The new neutron separation energy of 19 C was 308±5
keV, compared to the old value of 580±9 keV [59]. As discussed in Section 3.1, the
excitation energy of a neutron unbound state of a bound nucleus depends on the neutron separation energy E ∗ = Edecay + Sn and as a result the position of the 5/2+
2
state in 19 C was shifted down by roughly 300 keV with the new evaluation. This
effect is shown in the level scheme of Figure 5.1 which compares the level schemes
generated from the two mass evaluations to theoretical calculations. The new mass
94

Exp.

Energy (MeV)

2

Old AME

Theory

New AME

WBP

WBP*
5/2+

5/2+

5/2+

5/2+
1
3/2+
0

5/2+
3/2+
1/2+

5/2+
3/2+
1/2+
19

5/2

+

1/2+

3/2+
5/2+
1/2+

C

Figure 5.1: Comparison of level schemes for 19 C using old and new mass evaluations.
Note that only the neutron unbound 5/2+ is affected by the change in relative mass.
2
The new level scheme is better described by WBP*, consistent with observations for
other neutron rich carbon isotopes.

causes the 19 C level scheme to be described better by the WBP* interaction, which
puts it in agreement with the systematic compression of level schemes observed for
the even-even neutron rich carbon isotopes. The position of the bound states was unaffected as excitation energy proceeds directly from the γ-ray energy associated with
their de-excitation. The evaluation of the following results utilizes the new evaluated
masses.
19 C was produced by 1p2n removal from the 22 N beam. While part of the cross
section proceeded via a direct reaction, other reaction channels were also available.
In particular, it was possible to populate high lying continuum states in 20,21 C
which would decay by neutron emission to a variety of states in 19 C, schematically
represented by Figure 2.11. Therefore, it was expected that in addition to any possible
resonance structures, there would be an additional background contribution of the
95

Normalized Counts

0.6
Data
Resonant
Nonresonant
Sum

0.4

0.2

0

0

0.5

1

1.5
2
Energy (MeV)

2.5

3

Figure 5.2: Best fit to experimental data (solid line) along with the relative contributions of a 76±14 keV resonance (dashed) constituting 54% of the spectrum and
nonresonant background from high lying continuum states (dotted) modelled as a
Maxwellian distribution with Θ=2.44 MeV.
form discussed in Section 2.3.3. Figure 5.2 demonstrates the best fit to data (solid
line) utilizing the techniques set forth in Section 2.4. This fit had two components:
a Breit-Wigner resonance (2.3.1) at 76±14 keV (dashed line) comprising 54% of the
spectrum, and a non-resonant contribution in the form of a Maxwellian distribution
(2.3.3) with a temperature of 2.44 MeV (dotted line). The cross section to populate
the resonance was estimated to be 0.8±0.2 mb. The fit was not sensitive to the width
of the distribution. As mentioned in 2.3.1, the energy dependent width of the Breit
Wigner line shape for l=2 has only a weak dependence on the input width, and this
combined with the resolution effects of the setup to make any constraint impossible.
Therefore, the fits were performed using the expected single particle width of 10 keV.
Assignment of this state was guided by shell model calculations. As shown in
the level scheme of Figure 5.3 and discussed in 2.2, a low lying triplet of states is
expected, and within the uncertainties of these calculations it is possible that they
96

Energy (MeV)

2

Exp.

+

2

WBP*

5/2+

5/2+

5/2+
3/2+
1/2+

3/2+
5/2+

1

+

0
0

18

19

C+n

1/2+

C

Figure 5.3: Level scheme of 19 C compared to theory relative to 18 C+n with both the
Elekes state (solid red) and our state (dashed) included.
be neutron unbound. Given the agreement between [12, 16] regarding the placement
and assignment of the 3/2+ and the determination of the ground state to be spin
and parity 1/2+ [60], the best explanation for a near-threshold resonance would be
the 5/2+ at 383±15 keV (shown as the dashed line in Figure 5.3). This immediately
1
begs two questions. The first is why this work’s spectrum differs from that of [17] as
displayed in Figure 5.4, while the second is how to handle the prior assignment of the
5/2+ to a bound state at 269±7 keV [16].
1
Resolving the questions regarding our results compared to the other neutron spectroscopy work is a matter of interpreting the effects of reaction mechanisms. In particular, the two experiments used different reaction mechanisms to populate excited
states in 19 C - [17] using (p,p’) and this work using fragmentation. The selectivity
of these reaction mechanisms is different. In particular, for (p,p’) it is possible to
compare the relative population of different transitions with the same ∆l in a fashion
97

described by [61] by writing the cross section as:
σpp (θ) ≈ |Mpp |2 A(θ)

(5.1)

M
= bh Mp + bh Mn
p
n
pp

(5.2)

where

and M∗ are transition matrix elements which can be calculated from shell model and
b∗ are interaction strengths which can be taken from literature. In particular, from
[61], bp ≈ 0.25, bn ≈ 0.75. Therefore, the relative population strength can be defined:
σ2 /σ1 = |M
|2 /|M
|2
pp 2
pp 1

(5.3)

Because the transitions of interest have the same ∆l (both states being 5/2+ and
populated from the 1/2+ ground state), the angular form factor A(θ) for their cross
sections should be approximately the same and therefore will cancel. Calculating
the relative cross section for the 5/2+ states therefore is a matter of obtaining the
∗
relevant matrix elements from shell model. These calculations were performed with
the WBP and WBT interactions, and the resulting relative population estimation was
σ2 /σ1 ≥ 13. The resulting contribution of the 5/2+ state to the RIKEN spectrum
1
is displayed as the red bars on Figure 5.4. Within their estimated background it is
possible that this state would go unnoticed.
To determine why this work did not observe the 5/2+ , a 1p-2n stripping cross
2
section estimate following the Eikonal method of [62] expanded to account for three
particle stripping was performed for two cases [63]. First, a lower limit for the
removal of a π1p1/2 , ν2s1/2 , ν1d5/2 cluster from 22 N was estimated to be 0.32 mb,
and should dominantly feed the 5/2+ .This result is consistent with our estimated
1
cross section of 0.8±0.2 mb. The cross section to remove a π1p1/2 , 2ν1d5/2 cluster
98

Figure 5.4: Experimental spectrum from [17] with the estimated strength of the 5/2+
1
shown in red.

99

Normalized Counts

0.6
Data
Without RIKEN
RIKEN Res.
With RIKEN

0.4

0.2

0

0

0.5

1

1.5
2
Energy (MeV)

2.5

3

Figure 5.5: Our experimental spectrum with a best fit including the 5/2+ state from
2
[17] with relative population determined by cross section estimates from [63].
from the 22 N beam was calculated to be 0.69 mb, a process which could populate
either the ground state or the 5/2+ in 19 C. Using WBP calculations, we estimated
2
the overlap to the ground state to be twice that to the 5/2+ , which results in a
2
+ . Folding this state into our simulated
cross section of 0.23 mb to populate the 5/22
data set with the relative population constrained to σ2 /σ1 = 0.23 mb/0.32 mb =
0.71 results in the solid red line in Figure 5.5. This shows a comparison between
best fit results with and without the inclusion of the 5/2+ , and which demonstrates
2
that the experimental setup’s efficiency and resolution for that state would not be
sufficient to resolve it from nonresonant background given the limited statistics of the
measurement.
This leaves the question of prior assignment of the 5/2+ to a bound state. A
1
recent cross section measurement for neutron stripping from a 20 C beam producing
18 C requires the 5/2+ to be neutron unbound to reproduce the magnitude of the
1
cross section [63]. The table for their 20 C stripping results is shown in Figure 5.6,
100

Reaction
(20 C(0+ ),19 C(Jπ ))
S1n (20 C)=2.90 MeV

Ex (MeV)

Jπ

0.190

5/2+
Inclusive

σ−1n(e) mb σexp (mb)
111.17
192.2

126(10)

Rs
0.66(5)

Figure 5.6: Calculated neutron stripping cross sections of the 5/2+ relative to the
1
total [63].
and the predicted contribution of the 5/2+ shown relative to the total predicted
1
cross section. In particular, it is worth noting that the 5/2+ constitutes roughly 60%
1
the total calculated cross section. Without that contribution their description of the
stripping process breaks down, and would show a stark aberration from the cross
section systematics observed in that work.
Furthermore, Elekes et al. quote a cross section of 4.2 ± 0.5 mb to populate the

5/2+ , which is a factor of 2 lower than that of the 5/2+ (8.6±0.4 mb) [17]. This
2
1

relative population ratio of 2 is a factor of 6 lower than the ratio of 13 calculated
predicted by shell model calculations.
The level scheme in Figure 5.3 demonstrates that the WBP interaction does not
capture the known level density well, and the WBP*, while performing better still
incorrectly predicts the ordering of the excited states. The performance of the WBP
interaction can be understood as the phenomenological description of the sd shell
neutrons breaking down for such asymmetric nuclear matter. In particular, the reduction in TBME for WBP* suggests that the neutrons are less bound because of
the relative absence of protons compared to the nuclei which were fit over to produce
the neutron sd shell interaction utilized by WBP. It is possible that extending the
fit range of the WBP to include more recent neutron rich data would improve its
description of these level schemes.
However, the large degree of reduction and incorrect ordering suggests there might
be physics beyond the interaction mechanisms going into the WBP interaction. In
particular, [6] performed calculations on the Z=8 isotopic chain and was able to
101

reproduce the binding of the neutron rich isotopes and correctly predict the location of
the oxygen drip line by including three body forces in an interaction model developed
from fundamental N N forces. It is possible that calculations in a more realistic
interaction framework such as this would better describe the neutron rich carbon
isotopes.

5.2

21

C Results

Figure 5.7(a) shows the experimentally determined decay energy for 21 C. The data
(black squares) are distributed over a broad energy range and do not exhibit any
obvious resonance. A sharp resonance was not expected because the = 2 states were
not expected to be populated in the proton removal reaction from the 22 N ground
state, which is accepted to possess a J π of 0− [13, 64]. The calculated spectroscopic
factors for the = 2 5/2+ and 3/2+ excited states in 21 C are 0 and 0.05, respectively.
In contrast, the spectroscopic factor for populating the

= 0 1/2+ state is 0.75.

Therefore, the decay energy spectrum was fit assuming a pure s-wave decay, with
the line shape for the decay determined as in [31]. The scattering length was allowed
to freely vary in the fitting process from 0 to −100 fm with all other parameters
constrained to fixed values. Because of the low number of counts in the spectrum,
the fitting process used an unbinned likelihood technique. The fitting process favored
the limit of |as | < 1.5 fm, with the best fit being as = −0.05 fm (shown in Figure 1(a)
as grey triangles). Folding geometrical acceptances, detector resolutions, and live time
together we estimate the production cross section for this state to be 1.1 ± 0.23 mb.
Figure 5.7(b) demonstrates the sensitivity of the present set-up to any potential low
energy virtual state. It compares the line shape for an s-wave at the limit of as =
−1.5 fm (dot-dashed line) with the line shapes for s-wave decays with as = −15 fm
(dashed line) and as = −50 fm (solid line). It is clear that any low lying s-wave state
102

(a) 21 C

0.3

20

data
s-wave fit

C+n

0.1

0
3

Acceptance

Normalized Counts

0.2

2

0.5
0
0

1

2

(b) s-wave lineshapes
as = −50 fm
as = −15 fm
as = −1.5 fm

1

0

0

0.5

1

1.5
2
Energy (MeV)

2.5

3

Figure 5.7: Reconstructed decay energy spectrum for 21 C (a), where squares represent
data points and triangles represent an s-wave curve of as = -0.05 fm with the same
binning as the data. Panel (b) demonstrates the sensitivity of the setup to a low-lying
virtual state by showing simulated decay energy line shapes for s-waves with differing
scattering lengths. The inset shows the geometric efficiency of the setup as a function
of decay energy.

103

would have been apparent in the decay energy spectrum.

104

Energy (MeV)

Exp.

WBP*

2

N3 LOW

1/2+

+

2
1

WBP* N3 LOW

1/2+
0+

0+

0

0+
-1

20

C+2n

21

C+n

22

C

Figure 5.8: Experimental level schemes of 20,21 C and theoretical level schemes of 21,22 C, with 20 C data from [65, 12] and
N3 LOW predictions from [66]. The lack of a resonance structure in the 21 C decay energy spectrum indicates that the virtual
state’s strength is broadly spread as indicated by the grey region. This is consistent with WBP* and N3 LOW calculations
which predict 21 C to be unbound by 1.7 MeV and 1.6 MeV, respectively. The calculations show a 1.2 MeV spread in the
predicted 22 C binding energy.
105

Figure 5.8 compares the present experimental results with two different shell model
calculations. First, we performed calculations with NuShellX@MSU [67] in a truncated s−p−sd−pf model space with a modified WBP [7] interaction labeled WBP*.
In this interaction the neutron sd TBME were reduced to 0.75 of their original value in
order to reproduce a number of observables in neutron rich carbon isotopes [12]. The
second model is based on an effective Hamiltonian from the chiral N3 LOW nucleonnucleon potential [66]. As shown in the figure, the models agree in their description
of 21 C, but have a 1.2 MeV spread in predicted binding for 22 C. This suggests a
breakdown in the shell model description of 22 C.
As discussed in Section 2.2.3, Yamashita et al. derived a relationship between the
22 C two-neutron separation energy and the energy of the virtual state in 21 C within
the renormalized zero-range three-body model of [27]. Figure 5.9 demonstrates this
relationship as lines of constant decay energy for the virtual state in 21 C through a
phase space of two-neutron separation energy S2n and 22 C halo radius rn . This sort of
correlation has been observed in other halo systems [23]. The lines shown correspond
to 21 C virtual state energies ranging from 0 to 100 keV in steps of 10 keV.
The shaded grey region indicates where 22 C is likely to reside in this parameter
space given current experimental constraints. The limits on rn were calculated from
the matter radius of 22 C (extracted from the reaction cross section via finite-range
Glauber analysis under an optical-limit approximation) assuming a 20 C core [26], and
the upper boundary is a consequence of Sn < 0 for 21 C. The original work restricted
calculations to S2n ≥ 100 keV. As shown in Figure 5.9, the 100 keV virtual state curve
crosses into the experimentally constrained region at approximately S2n = 100 keV,
which limits the range of predicted virtual state energies in 21 C to E ≤ 100 keV for
this calculation space. A 100 keV virtual state energy corresponds to as = −15 fm,
which is not supported by our data.
If the calculations were extended to S2n < 100 keV, it might be possible to
106

Figure 5.9: Correlations between the two-neutron separation energy and rms neutron
halo radius in 22 C for given positions of a virtual state in 21 C from [27]. The shaded
grey area indicates the region of interest to find the 22 C S2n and rn . Constraints on
rn are extracted from the measured reaction cross section of 22 C.

107

Energy (MeV)

3

Extrapolation
WBP
WBP*

1s1/2

2.79

2

3−

1.93

1

2−

1−
0−

0.19
0

0
C

N

0d5/2
O

Figure 5.10: Evolution of the ν1s1/2 − ν0d5/2 shell gap for N=15 Oxygen, Nitrogen,
and Carbon isotones. The dashed line shows a linear extrapolation of the gap to
Carbon following the prescription of [20], while the dotted and dashdotted lines show
the results of WBP* and WBP calculations respectively. Experimental data taken
from [18, 13, 19].

calculate correlations in the S2n / rn space which are consistent with our limit on
the scattering length. Given the crossing point for the 100 keV virtual state energy
curve at S2n = 100 keV, such a curve calculated for a virtual state with as = −1.5 fm
100 keV. Therefore, we conclude that the limit of |as | < 1.5 fm
placed on the virtual state energy in 21 C constrains the 22 C S2n to be much less than
100 keV, which further reinforces the current evidence that 22 C is a loosely bound

would require S2n

halo system.
It would be interesting to measure the ν1s1/2 −ν0d5/2 shell gap for N=15 isotones.

In 21 C, this amounts to measuring the energy gap between the 1/2+ (one particle in
108

the νs1/2 ) and 5/2+ (one hole in the νd5/2 ). While this experiment did not populate
the 5/2+ , it is possible to speculate on where the 5/2+ might appear in a decay
energy spectrum for an experiment that would populate that state. As discussed in
[18], one method of predicting the evolution of the shell gap is a linear extrapolation
first put forth in [20]. The dashed line in Figure 5.10 demonstrates the result of this
method, whereby one observes that the gap disappears. Within the uncertainty of the
extrapolation, the levels could in fact invert in a fashion similar to the level inversion
observed in 15 C [20]. By this prediction, an experiment populating the 5/2+ would
observe the resonance above 800 keV. The dash-dotted and dotted lines in Figure 5.10
show the WBP and WBP* shell model predictions of 0.6 MeV and 1.1 MeV for the
energy gap respectively. There is some precedent for shell gaps to bend over near
the neutron dripline for N=7 [68] which lends credence to the shell model prediction.
By this prediction, the 5/2+ state should be above 1.6 MeV in decay energy. Given
the spread in possible locations for this state, it would be interesting to populate
the 5/2+ . Shell model calculations predict the ground state wavefunction of 20 C to
posses a mixed configuration which may allow 20 C(d,p)21 C to populate both states.

109

Chapter 6
Summary and Conclusions
In this work, the excitation energy of a neutron unbound excited state of 19 C was
measured, and a limit placed on the scattering length of a virtual state in 21 C. These
states were populated by nucleon removal reactions from a 22 N beam produced by
fast fragmentation at the Coupled Cyclotron Facility at the NSCL. Charged fragments resulting from these reactions were bent from the beam axis by a large-gap
dipole magnet and identified by a suite of charged particle detectors. Neutrons were
detected at forward angles by MoNA, and the decay energy of the neutron evaporation reconstructed using the invariant mass method. A Monte Carlo simulation was
used to account for resolution and acceptance effects in the modelling of the data, and
the decay energy spectra were fit using combinations of several different line shapes
depending on the physics case.
In the case of 19 C, the fitting took the form of a Breit-Wigner resonance at
76 ± 14 keV comprising 54% the total spectrum, and a Maxwellian distribution for
the remaining non-resonant background. Utilizing the latest evaluated mass tables,
this state was assigned to be an excited state of 19 C at 383 ± 15 keV.
Shell model calculations were performed to aid in the interpretation of this result.
Calculations were performed in a truncated s−p−sd−pf model space using both the
110

WBP and a modified WBP* interaction, where WBP* had empirical modifications
performed to reproduce observables for neutron rich carbon isotopes. Comparison to
these calculations results in the assignment of the state in 19 C to the 5/2+ state. The
1
resulting experimental level scheme suggests physics beyond the WBP* interaction,
and it is speculated that more realistic interactions developed from fundamental 2and 3-body forces may be able to reproduce the level schemes.
21 C was fit using a pure s-wave line shape, and the fit favored the limit of |a | <
s
1.5 fm. This result was compared to predictions from a renormalized zero-range threebody model which calculated correlations between the energy of the 21 C virtual state,
the matter radius of the halo nucleus 22 C, and the two-neutron separation energy for
22 C. This comparison suggests that 22 C is bound by much less than 100 keV, which
reinforces the current evidence that 22 C is a loosely bound halo system.
This work has put a constraint on one of the two levels in 21 C necessary to
measure the N=14 shell gap in carbon. Preliminary calculations suggest that a future
experiment utilizing 20 C(d,p)21 C would likely populate both the 1/2+ and the 5/2+
and be able to determine this energy gap.

111

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112

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