THE SEARCH FOR CLUSTER STRUCTURE IN
WITH THE PROTOTYPE AT-TPC
By
Adam Louis Fritsch

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy
2014

14

C

ABSTRACT
THE SEARCH FOR CLUSTER STRUCTURE IN
WITH THE PROTOTYPE AT-TPC

14

C

By
Adam Louis Fritsch
Certain light nuclei are known to have inherent cluster structuring, and the nature of
the triple-α structure of carbon isotopes is a subject of active discussion in nuclear physics.
Clustering in neutron-rich nuclei is of particular interest as such work could shed light on
how neutrons affect α clustering, making
ter structure in

14

14

C a logical candidate for such a study. Clus-

C was investigated at the University of Notre Dame with the prototype

Active Target-Time Projection Chamber (AT-TPC). A 38.2 MeV secondary beam of

10

Be

was incident on an active target volume containing He:CO2 90:10 gas at 1 atm, in which
trajectories of beam particles and reaction products were measured using the tracking capabilities of the prototype AT-TPC. Angular correlations of

10

Be and α particles were used

to reconstruct kinematics of scattering. Excitation functions and angular distributions were
measured in both elastic and inelastic channels, which unraveled a number of resonances in
14

C. Spin-parity assignments have been made for the elastic resonances by R-matrix analysis.

Evidence of a positive-parity rotational band has been indicated by the 14 C resonances. The
proposed level scheme and the strong resonance strength observed in the inelastic channel
are in line with the prediction by the antisymmetrized molecular dynamics (AMD) method
indicating linear-chain structure in

14

C. The results also demonstrate unique and powerful

potentials of active-target technology in radioactive-beam experiments and are an important
step toward the construction of the full-scale AT-TPC for the ReA3 facility at NSCL.

ACKNOWLEDGMENTS

There are many people I would like to thank, starting with my advisor, Wolfi Mittig.
Throughout my time as a graduate student, he has been a source of constant support,
guidance, and encouragement. It’s been a real pleasure working for him. I also thank
Daisuke Suzuki, who has also been extremely helpful and gracious, and his work in helping
setup and run my thesis experiment and his advice and guidance during my analysis have
been phenomenal. Saul Beceiro-Novo also deserves much thanks, as he helped me greatly in
my understanding of the physics of my work over the last year. Among the numerous other
scientists who assisted me in making this dissertation become a reality, I want to specifically
thank Tan Ahn, Daniel Bazin, Jim Kolata, and John Yurkon for their great help.
In addition to my group members, many of my fellow graduate students have been great
to spend time with in and out of school. Jack Winkelbauer, Zach Meisel, Mike Scott, Andrew
Klose, Shawna Valdez, and many others have all been amazing friends and colleagues during
my time here at NSCL, and I’m forever thankful for their help and time.
From my first days at NSCL as a college freshman and continuing to today, I’ve had a fantastic experience interacting with many people, including Jim Brown, Michael Thoennessen,
and Artemis Spyrou, who were instrumental in introducing me to the field of nuclear physics.
Remco Zegers, Scott Pratt, and Vladimir Zelevinsky have also been very helpful with their
advice regarding graduate school and my future career plans. I’d also like to thank Carl
Bromberg and Mark Olson for their guidance during my time as a TA, and Rique Campa,
Danny Caballero, and Norman Birge for helping me grow as a teacher.
My family deserves the most thanks. Their support has been unconditional, starting
from my early years and continuing still. Thank you.
iii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Chapter 1 Introduction . . . . . . . . .
1.1 Nuclear Reactions . . . . . . . . . . .
1.2 Elastic Collisions . . . . . . . . . . .
1.3 Inelastic Collisions . . . . . . . . . .
1.4 Cross Sections . . . . . . . . . . . . .
1.5 Excitation Functions and Resonances
1.6 Optical Potentials . . . . . . . . . . .
1.7 Relation to Current Work . . . . . .

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1
1
2
4
7
11
16
18

Chapter 2 Motivation . . . . . . . . . . . . . . . . . . . . . .
2.1 Molecular Clustering of Carbon Isotopes . . . . . . . . . .
2.2 Development of Prototype AT-TPC and AT-TPC Project
2.2.1 Context of the AT-TPC Project . . . . . . . . . . .
2.2.2 Time Projection Chambers . . . . . . . . . . . . . .
2.2.3 Micromegas . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Active Target Mode . . . . . . . . . . . . . . . . .

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19
19
22
22
23
25
28

Chapter 3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Antisymmetrized Molecular Dynamics . . . . . . . . . . . . . . . . . . . . .
3.2 Description of β-γ Constraint AMD + GCM . . . . . . . . . . . . . . . . . .
3.2.1 AMD Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Parity and Angular Momentum Projections . . . . . . . . . . . . . .
3.2.3 Constraint Variation on Quadrupole Deformation Parameters, β and γ
3.2.4 Generator Coordinate Method . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Energy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 β-γ Plane Structures . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Comparison to 12 C and Discussion of Excess Neutrons . . . . . . . .
3.4 Relation to Current Work . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30
30
31
31
33
34
36
37
38
38
41
44
45
47

Chapter 4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Beam Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 MC-SNICS Ion Source . . . . . . . . . . . . . . . . . . . . . . . . . .

48
48
49

iv

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52
54
55
57
59
63
68
70
77
78
85
87
89
91

Chapter 5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Ray Tracing and Angular Determinations . . . . . . . . . . . . . . . . . . . .
5.2 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Beam Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Reaction Energy Calibration . . . . . . . . . . . . . . . . . . . . . . .
5.4 Cross Section and Excitation Function Calculations . . . . . . . . . . . . . .
5.4.1 Energy and Angle Frame Conversion . . . . . . . . . . . . . . . . . .
5.4.2 Acceptance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Elastic and Inelastic Cross Sections and Excitation Functions . . . .
5.5 Theoretical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 S˜ao Paulo Optical Model Calculation . . . . . . . . . . . . . . . . . .
5.5.2 R-Matrix Calculations . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Resonance Spin and Parity Assignments . . . . . . . . . . . . . . . . . . . .
5.6.1 Method of Making Assignments . . . . . . . . . . . . . . . . . . . . .
5.6.2 3.0 MeV, 2+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3 4.7 MeV, 3− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.4 5.6 MeV, 5− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.5 6.5 MeV, 7− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.6 7.0 MeV, 4+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.7 7.9 MeV, 5− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.8 8.7 MeV, 5− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.9 Experimental Results Compared to Theory: Positive-Parity Rotational
Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.10 Experimental Results Compared to Theory: Negative-Parity Rotational Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.11 Comparison to Literature . . . . . . . . . . . . . . . . . . . . . . . .

95
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101
106
109
112
113
113
115
119
124
125
129
139
139
142
142
143
143
143
144
144

4.2

4.3

4.4

4.1.2 FN Tandem Van de Graaff Accelerator
4.1.3 TwinSol . . . . . . . . . . . . . . . . .
Design of the Prototype AT-TPC . . . . . . .
4.2.1 Exterior Chamber . . . . . . . . . . . .
4.2.2 Field Cage . . . . . . . . . . . . . . . .
4.2.3 Cathode . . . . . . . . . . . . . . . . .
4.2.4 Micromegas . . . . . . . . . . . . . . .
4.2.5 Readout Electronics . . . . . . . . . .
Experimental Setup . . . . . . . . . . . . . . .
4.3.1 MCP and Beam Ladder . . . . . . . .
4.3.2 Position Sensitive Si Detector . . . . .
4.3.3 Prototype AT-TPC . . . . . . . . . . .
4.3.4 Gases . . . . . . . . . . . . . . . . . .
Trigger and Data Collection . . . . . . . . . .

v

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144
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152

Chapter 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
BIBLIOGRAPHY

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vi

. . . . . . . . . . . . . . . . . 158

LIST OF TABLES

Table 4.1

Table of trigger pad locations for each quadrant. . . . . . . . . . . .

92

Table 4.2

Table of high gain pads by quadrant. . . . . . . . . . . . . . . . . .

92

Table 5.1

Optical model parameter set [78] used with DSigmaIV. The real and
imaginary terms use the same radius and diffuseness, given in columns
two and three, respectively. Column six shows the Coulomb radius
rc , and the last column displays the channel radius Rc , which was
adopted as a different value than in Ref. [78]. . . . . . . . . . . . . . 136

Table 5.2

Table of elastic scattering resonances, as obtained from the R-matrix
calculation fit to the data. All of the spin-parity assignments are without ambiguity. The spectroscopic factors SF all are assigned a 50%
0
| is the transuncertainty as they are not precisely calculated. |Ucc
mission factor for each resonance, as determined by the absorption
chosen in the calculation. Γres is the width of the resonance, as defined in Eq. 5.27. Γconv is the width after convoluting the calculation
with the energy resolution of around 150 keV. . . . . . . . . . . . . 142

Table 5.3

Comparison of resonances from this work with literature. Reaction
mechanism for each column is given under the reference. The last
column show states which have been observed to α-decay. Table
adapted from Ref. [85]. . . . . . . . . . . . . . . . . . . . . . . . . . 153

vii

LIST OF FIGURES

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 2.1

Figure 2.2

Allowed kinematical angles for 10 Be and 4 He nuclei after the elastic
collision of a 10 Be projectile nucleus incident on a 4 He target nucleus.
This curve is independent of the incident energy of 10 Be. The angles
are given in the lab frame. . . . . . . . . . . . . . . . . . . . . . . .

3

Allowed kinematical angles for 10 Be projectile nucleus incident on a
4
He target nucleus. The black curve is for elastic scattering, which
is energy independent. The other three curves are for different 10 Be
incident energies of inelastic scattering to the first excited state in
10
Be at 3.368 MeV. The red dashed curve is at 35 MeV, the green
dotted at 25 MeV, and the blue dash-dotted at 15 MeV incident
energy, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Elastic and inelastic scattering of 240 MeV α particles incident on
12
C. Comparisons of the data, in colored circles, to various reaction
calculations, in solid and dashed lines, are shown. Figure adapted
from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Excitation functions of α+12 C elastic scattering at various angles.
Resonances in the compound nucleus 16 O are seen. The points are
experimental data. The solid lines are a theoretical fit. Figure from
Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

The density distribution of the excess neutrons in 14 C on the xy plane
before the angular momentum projection, based on the molecularorbit model. The spheres show the positions of the α clusters. The
distance between the centers of the α clusters is set to 3 fm, forming
an equilateral-triangle shape. The density is integrated over the z
axis. Figure adapted from Ref. [22]. . . . . . . . . . . . . . . . . . .

20

Density distribution of protons in positive-parity states of 14 C of intrinsic wave functions at (β cos γ, β sin γ) = (0.93, 0.04) on the β − γ
plane, showing a linear-chain structure with a 10 Be + α correlation.
The size of the box is 10 × 10 fm2 . Figure adapted from Ref. [13]. .

viii

21

Figure 2.3

Figure 2.4

Figure 2.5

Schematic diagram of the ionization process for the MAYA detector,
a type of TPC. The electrons drift down against the electric field
to the wire plane, where they are detected. The charged particle
trajectories can then be reconstructed. Figure from Ref. [34]. . . . .

24

Schematic of Micromegas amplification technique. Figure adapted
from Daniel Bazin. . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Photograph of the Micromegas. The inset shows a magnified view of
the central region. The central pad has a diameter of 5 mm, while
the other coaxial strips have a pitch of 2 mm. The pads are separated
by a 0.25 mm gap. Figure from Ref. [27]. . . . . . . . . . . . . . . .

27

Figure 2.6

llustration of tracking. The beam particle (indicated by A and red)
collides with a gas nucleus and produces reaction products (B, green
and C, blue). The ionization electrons (dashed lines) are transported
by the electric field (E), making a projected image on the surface of
the Micromegas. The energy deposit profile of the beam particle is
measured by the central pad, while those of particles B and C are
measured by the coaxial strips. The track is segmented by 2 mm
pitch strips in the R direction, thus providing a resolution of 2 mm or
less, while the azimuth φ information is absent. Figure from Ref. [27]. 28

Figure 3.1

Energy surfaces of 14 C on the β − γ plane. The top panel shows the
energy for the positive-parity states before the total angular momentum projection and the bottom panel shows that for the 0+ states
after the total-angular-momentum projection. Figure from Ref. [13].

39

Energy surfaces of 14 C on the β − γ plane. The top panel shows the
energy for the negative-parity states before the total angular momentum projection and the bottom panel shows that for the 3− states
after the total-angular-momentum projection. Figure from Ref. [13].

40

Figure 3.2

Figure 3.3

Density distributions of the intrinsic wave functions for the positiveparity states of 14 C. The proton density ρ˜p , neutron density ρ˜n , and
difference between the neutron and proton densities ρ˜n − ρ˜p are illustrated in the left, middle, and right columns, respectively. The density distributions of the intrinsic wave functions at (a) (β cos γ, β sin γ) =
(0.00, 0.00), (b) (β cos γ, β sin γ) = (0.23, 0.04), (c) (β cos γ, β sin γ) =
(0.45, 0.17), (d) (β cos γ, β sin γ) = (0.25, 0.35), (e) (β cos γ, β sin γ) =
(0.78, 0.22), and (f) (β cos γ, β sin γ) = (0.93, 0.04) on the β − γ plane
are shown. The size of the box is 10 × 10 fm2 . Figure from Ref. [13]. 42

ix

Figure 3.4

Density distributions of the intrinsic wave functions for the negativeparity states of 14 C. The proton density ρ˜p , neutron density ρ˜n , and
difference between the neutron and proton densities ρ˜n − ρ˜p are illustrated in the left, middle, and right columns, respectively. The density distributions of the intrinsic wave functions at (a) (β cos γ, β sin γ) =
(0.08, 0.04), (b) (β cos γ, β sin γ) = (0.20, 0.09), (c) (β cos γ, β sin γ) =
(0.25, 0.35), (d) (β cos γ, β sin γ) = (0.60, 0.17), and (e) (β cos γ, β sin γ) =
(0.93, 0.04) on the β − γ plane are shown. The size of the box is
10 × 10 fm2 . Figure from Ref. [13]. . . . . . . . . . . . . . . . . . . . 43

Figure 3.5

Energy levels for the positive-parity states of 14 C and the density
distributions of the intrinsic wave functions of bands. The dotted lines
on the left and right show the experimental and theoretical 10 Be +
α threshold energies, respectively. The proton density ρ˜p , neutron
density ρ˜n , and difference between the neutron and proton densities
ρ˜n - ρ˜p are illustrated. Figure from Ref. [43]. . . . . . . . . . . . . .

45

Energy levels of the negative-parity states in 14 C. The five columns
on the left are the experimental data and the five columns on the
right are the calculated results. The dotted lines on the left and right
show the experimental and theoretical 10 Be + α threshold energies,
respectively. Figure from Ref. [13]. . . . . . . . . . . . . . . . . . . .

46

Part of the beam-line layout at the University of Notre Dame’s Nuclear Science Laboratory. During this work, the prototype AT-TPC
was located in the vault to the left of, or beyond, TwinSol. Figure
from Ref. [55]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

A schematic drawing of the MC-SNICS Source at the Nuclear Science
Laboratory. Figure courtesy Brad Mulder. . . . . . . . . . . . . . .

50

A schematic drawing of the FN Tandem Van de Graaff accelerator at
the Nuclear Science Laboratory. Figure courtesy Brad Mulder. . . .

51

A schematic drawing of the FN Tandem Van de Graaff accelerator at
the Nuclear Science Laboratory with included detail on the stripper
foil mechanism. Figure courtesy Brad Mulder. . . . . . . . . . . . .

52

A schematic drawing of the beam line including the FN Tandem Van
de Graaff accelerator and analyzing magnet at the Nuclear Science
Laboratory. Figure courtesy Brad Mulder. . . . . . . . . . . . . . .

53

Figure 3.6

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

x

Figure 4.6

Figure 4.7

A schematic drawing of TwinSol at the Nuclear Science Laboratory.
Figure from Ref. [59]. . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Schematic drawings of the two modes of operation of TwinSol. The
top image shows cross-over mode, while the bottom is of parallel
mode. Figure from Ref. [55]. . . . . . . . . . . . . . . . . . . . . . .

56

Figure 4.8

Cross-sectional view of the prototype AT-TPC. Figure from Ref. [27]. 57

Figure 4.9

Image of the exterior chamber of the device. It’s 70 cm in length and
has a 40 cm diameter. . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Image of the downstream ISO 400 flange. Readout electronics for the
Micromegas are mounted on the vertical feedthroughs on the center
of the flange. On the left of the flange, there are the both a gas inlet
and outlet, as well as a KF40 flange with four SHV feedthroughs. . .

59

Schematic view of the field cage with equipotential rings. For presentation purposes, only one pair of inner (purple and smaller) and
outer (red and larger) rings are shown. The equipotential rings are
secured to the main frame by (a) the aluminum pin stands and (b) the
supports of G10. The pin stands are arranged in a zigzag pattern to
increase the inter-stand distance and to reduce the risk of sparking.
Resistors are installed along the dashed lines between neighboring
stands to step down the potential by voltage division. The stands
bore through the G10 tube to put the inner and outer rings at the
same potential. The joints between the stands and the tube are hermetically sealed by epoxy resin. Figure from Ref. [27]. . . . . . . . .

62

Electric field simulations by the Garfield code [61]. The cylindrical
detector volume is approximated by an infinite box with the same
cross-sectional geometry. Equipotential surfaces are plotted by the
solid lines as a function of the radius with respect to the symmetry
axis (Z). The cathode plate is located at Z = 50 cm. Along the
horizontal radius axis, the inner rings are located near 13 cm, the
outer rings near 15 cm, and the exterior chamber wall near 20 cm.
Figure from Ref. [27]. . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Simulated electron drift velocities for P10 (dotted line) and He:CO2
90:10 mixtures (solid line) as a function of electric field. The simulation was made using the Magboltz code [62]. Figure from Ref. [27]. .

66

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

xi

Figure 4.14

Schematic diagram of the readout circuit for the Micromegas. For
simplicity, only one pad is shown for each quadrant (Q1.a, Q2.b, Q3.c
and Q4.d). The pad signal Q#.X (# = Quadrant ID, X = Pad ID)
can be routed to the T2K electronics, the quadrant output ‘Q#’ and
the central output ‘Central,’ which can be selected by the jumpers
A. Signals can be split and sent to multiple outputs by installing
resistive jumpers. The potential of individual pads can be selected to
be ground, or a different choice of high voltages HV1 or HV2 by the
jumpers B. Figure from Ref. [27]. . . . . . . . . . . . . . . . . . . .

71

Diagram of the writing and readout of the Sampling Capacitor Array
(SCA). The SCA is continuously filled in a circular manner. When
a trigger signal is received, all 511 cells are read out. Figure from P.
Baron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Figure 4.16

A photo of the ZAP board before mounting. . . . . . . . . . . . . .

74

Figure 4.17

A photo of the shielded ZAP board and FEC as mounted on the
prototype AT-TPC. The unshielded FEM is perpendicularly mounted
on the back of the FEC. . . . . . . . . . . . . . . . . . . . . . . . .

75

Electron amplification gain of the Micromegas as a function of micromesh voltage. The data were measured in He:CO2 90:10 mixtures
at 0.25 atm (blue triangles), 0.5 atm (green boxes), and 1 atm (orange
circles). Figure from Ref. [27]. . . . . . . . . . . . . . . . . . . . . .

76

Schematic drawing of the experimental setup of the prototype ATTPC. The inset is a magnified view of the segmented anode pad plane
of the Micromegas detector near the beam axis. Figure adapted from
Ref. [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Image of the ISO160 tee with the retracting beam ladder. The MCP
is mounted on the top flange and is install inside of the tee. The
black beam steerer can be seen located a few inches upstream of the
tee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

A side view of the copper plates and resistor chain on the MCP
housing. The MCP and foil are both located on the green PCB on
the top of the chain. . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Figure 4.15

Figure 4.18

Figure 4.19

Figure 4.20

Figure 4.21

xii

Figure 4.22

Figure 4.23

Figure 4.24

Figure 4.25

Figure 4.26

Figure 4.27

Figure 4.28

Figure 4.29

Figure 4.30

Figure 4.31

Image of the MCP mounted on its housing and flange. Inside of the
red rectangle are two of the small permanent magnets mounted on
one of the soft iron walls. The orange rectangle shows the location of
the back of the MCP anode, while the black rectangle below identifies
the double-sided-aluminized polypropylene foil. . . . . . . . . . . . .

80

The MCP installed in the ISO160 tee, slightly downstream from the
beam ladder. Behind the tee, iron and steel plates are mounted to
minimize the effects of stray fields from the beam steerer on electron
traveling from the foil to the MCP. The foil can be seen inside of the
magenta square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

The beam ladder installed in the sleeve of the ISO160 tee. On it are
mounted, from left, a stacked pair of Si detectors, a 228 Th 3 µCi α
source, and an iris collimator. . . . . . . . . . . . . . . . . . . . . .

83

The PSSD mounted past the ISO160 tee, at the eventual location of
the beam window of the prototype AT-TPC. . . . . . . . . . . . . .

85

Spectrum of E − ∆E, showing separation of isotopes in the beam.
The relative intensities of the isotopes somewhat change as beam
conditions change. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

The alignment plate as mounted on the back of the prototype ATTPC. Clear plastic wrap is taped over the flange under the plate
to keep dust and dirt out of the interior of the device. The bolts
that secure the TPC to the table and allow for height adjustment
are highlighted inside the red square. The bolts used to horizontally
adjust the TPC on the table are shown inside of the cyan oval. More
such bolts are found on the opposite side of the table. . . . . . . . .

88

A schematic of various distances between the parts of the setup during
the experimental run. . . . . . . . . . . . . . . . . . . . . . . . . . .

89

An image of the gas handlers used for flowing He:CO2 90:10 (top) and
nitrogen (bottom) as the active target and as buffer gas, respectively.

90

Schematic of the first 17 pads of a quadrant of the Micromegas, with
trigger pads in yellow. Pad 1 is common to all quadrants. One of
the four quadrants uses trigger pads 6, 7, and 12 as it has a bad
connection to pad 15. . . . . . . . . . . . . . . . . . . . . . . . . . .

93

The trigger scheme used in this work. . . . . . . . . . . . . . . . . .

94

xiii

Figure 5.1

Figure 5.2

Figure 5.3

Reconstructed trajectories of reaction products in the prototype ATTPC. The path of the incoming 10 Be nucleus is seen, as are the paths
of the reaction products 10 Be and 4 He. The ray traces of the latter
two nuclei allow for their angles to be calculated. . . . . . . . . . . .

96

A ray trace through pad signals in time for a triggered quadrant. The
relative time of each pad signal is denoted by a black diamond. The
linear fit is shown in red. . . . . . . . . . . . . . . . . . . . . . . . .

96

Illustrations showing the ray trace algorithm. For each panel, the
red dot represents the signal from pad i, from which lines are drawn
to each of the other pad signals. The top two panels show that, for
a perfect fit, different pads i will results in lines to the other pads
that have equivalent slopes. In the bottom panels, however, there is
a spurious signal. If drawing from that spurious signal (bottom right
panel), the slopes of the lines will all be very different. If drawing
from one of the good signals (bottom left panel), only the line to
the spurious signal has a slope that is different from the other lines.
The algorithm takes advantage of this difference in the slopes of lines
betweens pads to exclude a spurious signal like the one seen in the
bottom panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Figure 5.4

Angular data of all events where both reaction products are successfully tracked through the detector volume. The x-axis denotes the
angle associated with the reaction product of lower charge Z, while
the y-axis is of the higher Z reaction product. . . . . . . . . . . . . 100

Figure 5.5

Plot of the energy deposit profile of a beam particle, in green. The
horizontal blue line at 3,000 represents the amplitude threshold. The
red region represents the ∆E sampling for beam particle determination. The shape of the energy deposit profile is described by a Bragg
curve. The value of ∆E is different for different nuclei. A histogram
of ∆E values is shown in Fig. 5.6. . . . . . . . . . . . . . . . . . . . 103

Figure 5.6

Histogram of beam species, as gated by ∆E after beam stabilization.
The identification of the various species labeled in this plot is done by
comparison to the kinematics of reactions with target 4 He particles
in the angular correlations matrix. . . . . . . . . . . . . . . . . . . . 104

Figure 5.7

Angular data of events gated on the 4 He peak in Fig. 5.6. The kinematics curve for elastic scattering overlays the data in black. The
4
He + 4 He scattering data are useful for confirming calibration of the
angles of the two reaction products. . . . . . . . . . . . . . . . . . . 106

xiv

Figure 5.8

Angular data of events gated on the 7 Li peak in Fig. 5.6. The kinematics curve for elastic scattering overlays the data in black. . . . . 107

Figure 5.9

Angular data of events gated on the 10 Be peak in Fig. 5.6. Kinematics
curves overlay the data: elastic scattering is in black, and the first
excited state (3.368 MeV in 10 Be) inelastic scattering is in red. . . . 108

Figure 5.10

Two dimensional histogram of beam particles. While the 10 B particles
stop in the chamber, the 10 Be particles punch through. . . . . . . . 110

Figure 5.11

Projection of Fig. 5.10 onto the x-axis. The penetration depth of 10 B
is 360 ± 5 mm and is 480 ± 5 mm for 10 Be, though this latter value
reflects the total active target volume length and not the stopping
distance of 10 Be. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Figure 5.12

Four plots of possible reaction scenarios. All figures depict particle
trajectories in red, a blue active target volume, a dark blue beam column, and an orange Micromegas plane at the bottom of the volume.
Fig. 5.12(a) shows a binary reaction that would successfully trigger
the electronics and thus would be recorded. Fig. 5.12(b) would not
trigger as one of the reaction products has too small of an angle to
leave the beam column. In Fig. 5.12(c), the energy of the reaction is
low so that the reaction products do not travel over trigger pads, resulting in the event not being recorded. The reaction in Fig. 5.12(d)
takes place so close to the Micromegas that the products quickly leave
the chamber and are thus not detected. The acceptance calculation
described in this section accounts for these various scenarios for the
normalization of the elastic and inelastic reaction cross sections. . . 116

Figure 5.13

Acceptance calculation for elastic scattering of 10 Be on 4 He. The color
scale represents the probability that a reaction at a given center-ofmass energy and angle would be recorded. . . . . . . . . . . . . . . . 117

Figure 5.14

Acceptance calculation for inelastic scattering of 10 Be on 4 He. The
color scale represents the probability that a reaction at a given centerof-mass energy and angle would be recorded if it were to take place
in the detector. The energy cutoff seen in the plot is due to the
3.368 MeV excitation energy of the first excited state in 10 Be. . . . . 118

Figure 5.15

Normalized experimental elastic cross section as a function of energy
and angle, given in mb/sr. . . . . . . . . . . . . . . . . . . . . . . . 121

xv

Figure 5.16

Normalized experimental inelastic cross section as a function of energy and angle, given in mb/sr. . . . . . . . . . . . . . . . . . . . . . 122

Figure 5.17

Elastic excitation function from 100◦ to 105◦ , a projection onto the
y-axis from Fig. 5.15. An x-axis projection produces an angular cross
section plot. The uncertainties are calculated from Eq. 5.15. . . . . . 123

Figure 5.18

Comparison of the effect of different imaginary depths on the transmission probability |S| as a function of angular momentum L and
center-of-mass energy for the SP calculation. . . . . . . . . . . . . . 127

Figure 5.19

Elastic SP cross section calculation as a function of energy and angle,
given in mb/sr. This output is used as a reference for the R-matrix
calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 5.20

Level scheme of 14 C with various exit channels. Neutron separation
occurs at 8.18 MeV, while 10 Be + α starts at 12.01 MeV. The energy
domain studied in this work ranges from about 14 MeV to 22 MeV,
over which the 13 C + n cross section is low and flat. Thus, neutron
emission can be modeled as background and absorbed with the optimal model used in DSigmaIV. Figure from [73]. . . . . . . . . . . . . 130

Figure 5.21

Elastic cross section calculation with DSIV using the Li and Hird
parameter set, given in mb/sr. The results are similar to the SP
calculation shown in Fig. 5.19. . . . . . . . . . . . . . . . . . . . . . 131

Figure 5.22

Comparison of various optical model parameter sets, given in mb/sr:
(a) S˜ao Paulo from Fig. 5.19, (b) final choice from Fig. 5.21, (c)
10
Be + α, 6 MeV CM [74], (d) 10 B + α, 16.07 MeV CM [75], (e)
9
Be + α, 6.06 MeV CM set one [76], (f) 9 Be + α, 6.06 MeV CM set
two [76]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 5.23

DSIV elastic cross section calculation, given in mb/sr, reproducing
the elastic data in Fig. 5.15. . . . . . . . . . . . . . . . . . . . . . . 138

Figure 5.24

Side-by-side comparison of the data, given in mb/sr, from Fig. 5.15
(left) with the calculation from Fig. 5.23 (right). . . . . . . . . . . . 139

xvi

Figure 5.25

Angular cross sections for four elastic resonances seen in the data.
The data are given in black, the squares of the Legendre polynomials
are in red, and the DSigmaIV calculations are in blue. There is very
good agreement in the oscillations of the angular distribution of the
data with the red and blue curves. Of particular note in this figure,
the good agreement of the data with the red PL2 curves show that the
spin and parity assignments are not dependent on the optical model
parameterization in DSigmaIV and are thus unambiguous. . . . . . . 140

Figure 5.26

0
| as a function of angular
Plot of transmission probability |S| = |Ucc
momentum L and center-of-mass energy for the final elastic R-matrix
calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 5.27

Comparison of resonances in this work (left column labeled “Experiment”) with predictions for a rotational band with linear-chain structure [13] (right column labeled “Theory”), given in MeV above the
10
Be + α threshold. The theory predictions are in good agreement
with the experimental results. . . . . . . . . . . . . . . . . . . . . . 145

Figure 5.28

Overlaps between linear-chainlike states and artificial 10 Be + α states.
Tables from Ref. [80]. . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Figure 5.29

Experimental excitation functions, given in 4 × mb/sr, from this
work: elastic in red, inelastic in black. Note the large cross section
in inelastic at 7.2 MeV, which coincides with the 7.2 MeV elastic
resonance (which is found to be a 7.0 MeV 4+ resonance from the Rmatrix calculation). This excitation function shows a strong branching to the inelastic for this 4+ resonance, as predicted in Ref. [80] for
the linear-chain rotational band. . . . . . . . . . . . . . . . . . . . . 146

Figure 5.30

Preliminary inelastic R-matrix excitation function from 85◦ to 95◦ .
The inelastic resonances in the calculation are 7.2 MeV 4+ and 7.8 MeV
5− . Further calculations will be made for the inelastic data. . . . . . 147

Figure 5.31

Comparison of negative-parity resonances of this work (left column)
and from Ref. [25], observed via a (7 Li,d) reaction. The rotational
band from Ref. [25] is predicted to be a prolate structure in 14 C. Poor
agreement is seen for 3− and 5− , though it is better for 7− . . . . . . 149

Figure 5.32

Band gradient comparison of 7− resonance from this work with proposed prolate and oblate rotational bands in 14 C from Ref. [25]. Close
agreement is seen with the prolate band. . . . . . . . . . . . . . . . 150

xvii

Chapter 1
Introduction
Atomic nuclei are of fundamental importance to our understanding of the universe. Researching nuclear physics tells us about the type of matter that comprises the visible universe and
how that matter is created. Knowledge of matter provides limitless potential for the advancement of science and for real-world application, with medicine, energy, and defense being a few
examples. One of the most effective ways to study the nucleus is through nuclear reactions.
In this chapter, some of the basic tools used in nuclear reactions studies are described.

1.1

Nuclear Reactions

By colliding nuclei with sufficient energy, information such as the internal structure of an
individual nucleus as well as the interactions between nuclei can be discovered. There are a
variety ways to investigate nuclear reactions, but all of them share some common characteristics.
For a given reaction, there is a projectile nucleus and a target nucleus. The projectile is
accelerated to some energy, anywhere from hundreds of keV up to even GeV, and is directed
towards the target. In the event that the projectile strikes the target nucleus, a reaction
takes place. This reaction could be a simple elastic collision, where the projectile and target
simply bounce off each other and both nuclei remain in the same internal states as before
the collision. Other possibilities include transfer reactions, where one of the nuclei loses a
1

proton or neutron to the other nucleus, and fusion reactions, in which the two nuclei combine
to form one larger nucleus. There are many other types of reactions, but only elastic and
inelastic collisions will be covered in further detail in this dissertation.

1.2

Elastic Collisions

As mentioned above, reactions where the projectile nucleus and incident nucleus do not
change from their initial states are referred to as elastic collisions. Such a collision can be
written as

A + B → A + B,

(1.1)

that is, the incident projectile nucleus A enters and exits the reaction in the same nuclear
state, as does the target nucleus B. A common convention is to write Eq. 1.1 as

B(A, A)B,

(1.2)

where again A represents the incident nucleus and B the target nucleus.
The individual kinetic energies of the two nuclei can change after the reaction, and often
do, but the internal states of the nuclei do not. For example, if A is incident on B at 15 MeV,
the elastic collision results in many possible exit angles and kinetic energies. The following
non-relativistic conservation laws of mass, energy, and momentum determine the allowed
exit angles and kinetic energies:
2

mA + mB = mA + mB

(1.3)

EA = EA + EB

(1.4)

pA = pA + pB

(1.5)

where the unprimed notation denotes the states of the particle before the reaction and the
primed after. The target B has no initial energy or momentum in the lab frame; if calculating

90

70
60
50
40
30

10

Be Scattering Angle [deg]

80

20
10
0
0

10

20
4

30
40
50
60
70
He Scattering Angle [deg]

80

90

Figure 1.1: Allowed kinematical angles for 10 Be and 4 He nuclei after the elastic collision of
a 10 Be projectile nucleus incident on a 4 He target nucleus. This curve is independent of the
incident energy of 10 Be. The angles are given in the lab frame.

3

in the center-of-mass frame, these values must of course be included. Angles are calculated
from the momentum vectors.
An important feature of elastic scattering is that for a given elastic scattering system A
and B, the angular kinematics are independent of incident energy and only depend on the
masses of A and B. An example plot of allowed exit angles where A is a

10

Be nucleus and

B is a 4 He nucleus is shown in Fig. 1.1. From the figure, one sees that kinematically the
scattered beam particle

10

Be can never have an exit angle larger than about 23◦ in the lab

frame, owing to its larger mass in comparison to the target nucleus 4 He.

1.3

Inelastic Collisions

If during the reaction, either or both of A and B undergo a change in internal state without
changing mass or charge, the reaction is called an inelastic collision. In this case, the changed
nucleus or nuclei enter an excited state where the nucleus transitions to a higher energy state
from its initial ground state, the lowest energy state. The spin and parity of the nucleus also
may change. To denote an excited state, an asterisk (*) is put after it. Thus, if A transitions
into an excited state during the reaction, one writes

B(A, A∗ )B.

The conservation laws are then

4

(1.6)

90

70
60
50
40
30

10

Be Scattering Angle [deg]

80

20
10
0
0

10

20
4

30
40
50
60
70
He Scattering Angle [deg]

80

90

Figure 1.2: Allowed kinematical angles for 10 Be projectile nucleus incident on a 4 He target
nucleus. The black curve is for elastic scattering, which is energy independent. The other
three curves are for different 10 Be incident energies of inelastic scattering to the first excited
state in 10 Be at 3.368 MeV. The red dashed curve is at 35 MeV, the green dotted at 25 MeV,
and the blue dash-dotted at 15 MeV incident energy, respectively.

5

(1.7)

mA + mB = mA∗ + mB
EA = EA∗ + EB + Q

(1.8)

p A = p A∗ + p B

(1.9)

where Q represents the Q-value of the reaction, given by

Q = (mA + mB − mA∗ − mB )c2

(1.10)

= (mA − mA∗ )c2 .

(1.11)

Since the excited state A* is at a higher energy than the initial state A, the Q-value is
negative. The Q-value can be interpreted in this case as the energy threshold needed to be
overcome by the incident nucleus A in order to inelastically scatter off of the target B; if EA
is less than the Q-value, the inelastic reaction is not allowed.
Therefore, unlike elastic collisions, inelastic collisions kinematics depend on the incident
energy of nucleus A. Fig. 1.2 compares the allowed kinematical angles of the elastic collision
and those of an inelastic collision of the type found in Eq. 1.6 at three different beam energies.
As in the elastic example, here A represents a

10

Be nucleus and B 4 He. Notice that as the

energy of the reaction decreases, the angular domain in which the particles

10

Be* and 4 He

may scatter becomes smaller.
The two outcomes of elastic and inelastic scattering for the A+B system refer to possible
exit channels of the reaction. There is one exit channel for elastic scattering as written in

6

Eq. 1.2. In inelastic collisions, however, there is one exit channel per possible inelastic
reaction. For example,

B(A,A∗ )B

(1.12)

B(A,A)B ∗

(1.13)

B(A,A∗ )B ∗

(1.14)

represent three separate exit channels. If there is more than one possible excited state in A
or B, and there often are, then there are even more exit channels that can be populated.
Approaching nuclear reactions calculations by analyzing each exit channel separately allows
for the comparison of the relative influence of each channel on the reaction.

1.4

Cross Sections

Elastic and inelastic collisions can both provide information about the system being studied. A common method used to quantify the reaction rates of various exit channels is by
computing the cross section σ for each channel. Cross section is a measure of the probability
of a nuclear reaction occurring; thus, comparing cross sections of various exit channels gives
the relative probability of the reaction taking each exit channel. When considering cross
section as a function of angle, it is referred to as a differential cross section. The integrated
cross section is found by taking the integral of the differential cross section over all angles.
The sum of the integrated cross sections for all possible exit channels is then called the total
cross section.

7

The differential cross section dσ(θ, φ)/dΩ can be written as

σ(θ, φ) ≡

N
dσ(θ, φ)
=
,
dΩ
Ninc nt sin θdθdφ

(1.15)

where N is the number of reactions in a given exit channel for a given solid angle (θ, φ), Ninc
is the number of incident nuclei, and nt is the number of target nuclei per unit area. The
notation dσ(θ, φ)/dΩ is often used to clearly illustrate that the differential cross section has
units of area per solid angle. The unit often used when reporting a differential cross section
is millibarn per steradian [mb/sr], where 1 mb = 10−27 cm2 . It’s also common to report
the elastic cross section as a ratio to the Rutherford, or point-Coulomb, cross section [1]:

σ(θ)Ruth =

η2
4k 2 sin4 (θ/2)

(1.16)

where η is defined as

Zp Zt e2
2η
=
E
k

(1.17)

for charges Zp and Zt of the projectile and target, elementary charge e, reaction energy E,
and wave number k. The variables θ, E, and k are all in center-of-mass.
A sample plot of elastic and inelastic differential cross sections is given in Fig 1.3. A
more detailed description of how differential cross sections are calculated experimentally will
be given in the following chapters.
8

D.T. Khoa et al. / Physics Letters B 695 (201

Fig. 1. Elastic and inelastic α + 12 C scattering data at E α = 240 MeV [12] measured
+
Figure
1.3: 2Elastic
and inelastic scattering of 240 MeV α particles incident on 12 C. Comfor the
1 state at 4.44 MeV in comparison with the OM, DWBA and CC results
parisons
of the data, in colored circles, to various reaction calculations, in solid and dashed
given by the complex double-folded OP and inelastic FF.

lines, are shown. Figure adapted from Ref. [2].

+
+
−
the ground state 0+
1 to the 21 , 02 and 31 states agree reasonably with the experimental values. As illustrated in Figs. 1 and 2
9
below, the AMD nuclear transition densities also give good description of the corresponding inelastic (α , α ′ )12 C cross sections.
+

Fig. 2. Inel
9.64 MeV [
plex double
the 3−
1 and

observed
spectrum
ergy is a
width of

Similarly, the integrated cross section is

σ=

dσ(θ, φ)
dΩ.
dΩ

(1.18)

In addition to energy, the properties angular momentum, spin, and parity are used to
describe the internal state of the nucleus. The convention used to represent a nucleus’ given
state is J π for

J = L + S,

(1.19)

where total angular momentum is represented by J, angular momentum by L, spin by S,
and parity by π = (−1)L . The angular momentum terms are added as vectors in Eq. 1.19.
By employing partial wave scattering theory [1], the single channel elastic cross section
can be separated into a sum of many partial waves, one for each relative angular momentum
L value:

dσ
1
=
dΩ el
k

2

∞
iδL

(2L + 1)PL (cos θ)e

sin(δL )

(1.20)

L=0

for wave number k, Legendre polynomials PL (cos θ), and partial wave phase shift δL [1]. Note
that the relative angular momentum L in Eq. 1.20 is not the same as the intrinsic angular
momentum L in Eq. 1.19.

10

At small angles, the Coulomb potential dominates and causes the cross section to tend
to infinity. Because of this, Eq. 1.20 is often divided by the point-Coulomb cross section
from Eq. 1.16, allowing for the non-Coulomb processes present in elastic scattering to be
quantified.
Integrating Eq. 1.20 over all angles yields [1]

σel =

4π
k2

∞

(2L + 1) sin2 δL .

(1.21)

L=0

Quantifying cross sections as a function of exit angle is a useful method for assigning
angular momentum states L to various exit channels in a reaction. To make determinations
about the energy of excited states that are populated in a reaction, the dependence of cross
section on reaction energy must be studied.

1.5

Excitation Functions and Resonances

In addition to measuring cross sections at various angles, it is often useful to examine them
as a function of reaction energy. Such cross sections

σ(E) ≡

dσ(E)
dE

(1.22)

are then called excitation functions [1]. If the excitation function is not fully integrated over
all angles, it is then expressed as

11

dσ 2 (E, θ, φ)
.
dΩdE

(1.23)

Example plots of experimental excitation functions are given in Fig. 1.4.
ELAS TI C SCA T TERI NG QF

pARTI CLES

375

e=I7I 2'Q. M.

2.0—
l.5—
I.

O-

0.0
K

O. I5-O

I

7.0

60

5.0

0.0

e

I

Iy79

GM

tel

Irn

ro

K
K

aIS
oI-

P.IOI

0.05 00
0.00
OIO—
~

0.05—

7.0

e

~

~

Ã

o 0.00

I

6.0

5.0

0.20
O.

I

ly0. 8

Q.M.

~ ~

7.0

e= l2&.2'Q. M.

I5-

O. IO~ ~

0.05—
~y ~

0.00

~ ~ ~

'~

1/0 ~ \ Is

O.I0

.0

Q. M.

0.056.0

5.0

4. 0

ALPHA

p yo,

7.0

(MEV LAB)

ENERGY

D'gerential
section of carbon for alpha p
elastic scattenng
1
are a theoretical 6t.
~
1iness a
ntal data. The so l'd
The points are expenmenta
&,

12 about
ANALYSISangles. Resonances
III. DATAatAND
4 kev
s read o
of 6 to
resulted in functions
a sprea
Figure 1.4:analyzer
Excitation
of 11.
α+
C elastic scattering
various
of 171.2'
extreme back ang lee o
value.
the centrala va
ue. At the 16
oints of Fig. 1 show the center-of-mass cross
in the compound
nucleus
O are
The points
are experimental
data.
The solid lines
kev
ev for energies
. seen.
aced at intervals
of 85
points were spaced
a]pha-particle
ora
sections aas a function of j aboratory
2'
6
6.
60
to
Mev,
energy. For corn p actness th e 171. cross sections are
are a theoretical fit. Figure from Ref. [3].
scaleb afactoro f tenrealative to the others.
l'df or
e een a pie
except in regio
points were takenn less frequently
q
where structure
angle data.

was expecte d because

of th e back-

loss in the gas. No correction as
ic abundance
isotoopic
for the one percent natura 1 &so

12

of

C"

When looking at an excitation function, the structure of the data can reveal a resonance.
While it is not a bound state, the resonance forms a discontinuity in the excitation function
near the energy Er . Resonances can be described in part by their energy widths Γ, which
are related to the lifetime τ over which the resonance occurs. This relation is constrained
by the Heisenberg Uncertainty Principle [4]

∆E∆t ∼

for the reduced Planck constant

(1.24)

= 6.58 × 10−16 eV·s. Thus, the longer lived the resonance

is, the narrower its peak will be:

Γ ∼ /τ.

(1.25)

Experimentally, Γ is defined as the full width at half maximum (FWHM) of the resonance
peak found at Er in the excitation function. The contribution of the resonance to the
integrated elastic cross section is [5]

σelres =

Γ2 /4
4π
(2L
+
1)
.
k2
(E − Er )2 + Γ2 /4

(1.26)

Such a description is called a Breit-Wigner resonance. Eq. 1.26 is valid in its given form only
if the background phase shift δbg (E) is near zero and thus negligible; in other cases, there

13

may be interference effects between the resonance and the background potential scattering
that must be considered [1].
If there is also an inelastic exit channel for the same resonance as seen in the elastic
channel, Eq. 1.26 becomes [4]

res
=
σinel

4π
Γc Γc /4
.
(2L
+
1)
k2
(E − Er )2 + Γ2 /4

(1.27)

Here, Γc represents the partial width from the elastic channel, Γc represents the partial
width from the inelastic channel, and Γ = Γc + Γc is the total width of the resonance. Also,
the total cross section would become the sum of the two integrated cross sections [1]:

res
res
σtot
= σelres + σinel
.

(1.28)

If necessary, any other possible contributions from other exit channels must be included
res
of
when calculating the total width Γ of the resonance and thus the total cross section σtot

the reaction.
Furthermore, the partial width of the resonance of a given exit channel c for angular
momentum L can be expressed as [5]

Γc (L, E) = 2Pc (L, E)γc2 (L)

14

(1.29)

for Pc (L) as the penetrability and γc2 (L) as the reduced width of the channel. The penetrability describes the barrier penetration of the two nuclei during the reaction. The reduced
width is defined, for a single-particle state, as [5]

γc2 (L) =

2 2
uL (R)

(1.30)

2µR

where µ is the reduced mass of the system and uL (R) is the energy eigenstate of the radial
Schr¨odinger equation [1]

uL (R) =

L(L + 1) 2µ
+ 2 (V (R) − E) uL (R)
R2

(1.31)

at the interaction radius

1/3

1/3

R = 1.2 A1 + A2

fm

(1.32)

for mass numbers A1 and A2 of the two nuclei [5]. Mass number is the number of nucleons
in a given nucleus and is dimensionless. The value of the partial width goes to zero at the
energy threshold of the resonance.
An important note is that the preceding expressions for cross section all assume that
the projectile and target both have a spin of zero. If the two nuclei have non-zero spin,
these spins must be taken into account in the cross section calculations by adding the spin
weighting function [4]
15

2J + 1
(2Sp + 1)(2St + 2)

g=

(1.33)

where J is the total angular momentum and Sp and St are the entrance channel spins of
the projective and target, respectively. Notice that when Sp = St = 0, Eq. 1.33 simplifies to
2L + 1, as has been used previously. The projectile and target investigated in this work are
both spin-zero particles, and thus have cross sections that are well described by the equations
found in this chapter where g = 2L + 1.

1.6

Optical Potentials

When investigating nuclear reactions, it’s useful to pick a model describing the reaction that
has an effective way to separate the elastic collision from other reaction mechanisms. One
such way to do so is by using an optical potential.
Optical potentials describe the interaction between colliding nuclei with a real and imaginary term, where the imaginary term functions to absorb the flux to any non-elastic reaction
channels, which is analogous to the absorption of light in a semi-transparent medium; hence
the name, optical potential [6].
The complex potential has the form

V (R) + iW (R).

The real part is given by
16

(1.34)

−Vr

V (R) =

1 + exp

R−Rr
ar

,

(1.35)

for the central depth of the potential Vr , diffuseness ar , and nuclear radius Rr for a distance
R between the nucleon and nucleus [1]. The free parameters Vr , ar , and Rr have typical
values of

Vr ∼ 40 − 50 MeV

(1.36)

ar ∼ 0.6 fm

(1.37)

Rr ∼ 1.2A1/3 fm

(1.38)

where A is the mass number of the nucleus. Similarly, the imaginary term is [1]

−Vi

W (R) =

1 + exp

R−Ri
ai

(1.39)

for

Vi ∼ 10 − 20 MeV

(1.40)

ai ∼ 0.6 fm

(1.41)

Ri ∼ 1.2A1/3 fm.

(1.42)

17

The shape of these potentials is called the Woods-Saxon, or Fermi, shape [1]. The
values of the parameters are found by fitting the full complex potential to data and are
thus phenomenological. Generally, optical models must account for spin, but as discussed
previously, the spin of both of the projectile and target in the entrance channel in this work
are zero, and thus is ignored here.
Eq. 1.34 is inserted as the potential in the radial Schr¨odinger equation from Eq. 1.31,
allowing cross sections to be calculated over a continuous energy domain.
By utilizing an optical potential, elastic scattering can be well described and its contribution to the total cross section understood. Other exit channels can then be investigated
in proper context to the elastic channel.

1.7

Relation to Current Work

The concepts and equations discussed in this chapter are of great importance to the following work in this dissertation. They will be referred to often, and understanding them
is essential to follow the arguments made going forward. For more detailed discussions of
nuclear reactions, cross sections, and resonances, as well as more general topics in nuclear
physics, see Refs. [1, 4, 5].

18

Chapter 2
Motivation
The experiment and analysis discussed in this work has two main motivations: one of physics,
and one of technical development. The physics of interest is the study of the clustering of 14 C
via resonant scattering of

10

Be incident on 4 He. The device used to measure the resonances

was the prototype Active Target-Time Projection Chamber. The following sections with
delve into detail on both of these motivations.

2.1

Molecular Clustering of Carbon Isotopes

Cluster structure in nuclei has long been predicted; α clustering in particular has been
discussed since at least the mid-1930s [7]. It is still at the forefront of nuclear physics,
as shown by the impressive number of recent publications [8, 9, 10, 11, 12]. Clustering is
often described as molecular in nature, as the clustering of nuclei into a composite nucleus
is analogous to atoms forming a molecule [2, 13, 14, 15, 16]. For example, linear-chain
configurations have been proposed in nuclei over the years [17, 18].
One such nucleus that may be described via molecular clustering is 12 C. How the three α
particles cluster to form 12 C is important to understanding the properties of the energy levels
of

12

C. Of particular interest is the Hoyle state in

is the second excited state in

12

12

C [2, 19, 20, 21]. The 0+ Hoyle state

C and is found at 7.6542 MeV. This energy is only 287 keV

above the three-α breakup threshold and can only be formed when a resonant state in 8 Be
19

triangle, the excess neutron is located also between two
’s, and its density spreads farther outside the 3 triangle.
Consequently, the neutron density distribution comes up
with an inverse triangular shape, which can be contrasted
with a possible naive expectation that the neutron density

trated near the center of the 3
to the description of yrast states
distances. Figure 2 exhibits the
ergies for this pz 2 configuratio
equilateral-triangular shape is a
while their distance is varied
shown for comparison for 12 C,
are present.

FIG. 1 (color). The density distribution of the excess neutron
FIG. 2. The 0 and 3 K 3 en
14 C (solid lines) as a function
on
the
xy
plane
before
the
angular
momentum
projection
The
andplane
14
Figure 2.1: The density distribution of the excess neutrons in C on the xy
before
equilateral triangular configuration
spheres show the positions of the
clusters. The distance
the angular momentum
projection,
based
on
the
molecular-orbit
model.
The
spheres
show
14 C with the p 2 c
lines represent
between the centers of the clusters is set to 3 fm. The density
z
the positions of theisαintegrated
clusters.over
Thethedistance
between
the
centers
of
the
α
clusters
is
set
to dashed lines repr
neutrons, and the
z axis.

3 fm, forming an equilateral-triangle shape. The density is integrated over the z axis. Figure
adapted from Ref. [22].
142501-2
pairs with a free α [1]. A narrow resonance of 8.3 eV breadth, the Hoyle state is essential
in explaining the observed nuclear abundances of the universe; without it, not only would
there be less or no

12

C, but all of the nucleosynthesis processes that build on

12

C would be

impacted. In fact, it’s from studying abundances of heavier isotopes that prompted Hoyle
to suggest an excited state in

12

C near the three-α breakup threshold in the first place [19].

A complimentary nucleus to study of relevance to

12

C is

14

C. By analyzing a three-α

system with two additional neutrons, the role that the neutrons play in the clustering of 14 C
can shed light on how all C isotopes are structured [13].
An important feature of excited three-α-cluster states is the configuration of the nucleus.
Various theoretical works have been made to describe these configurations [13, 22, 23, 24, 25].
Equilateral-triangle shapes in

14

C have been predicted [22] based on the molecular-orbit

20

the
hat

rgy
of
9).
ges
and
ely
ace
the
dly
ity
on,
ate

(f)

Figure 2.2: Density distribution of protons in positive-parity states of 14 C of intrinsic wave
functions at (β cos γ, β sin γ) = (0.93, 0.04) on the β − γ plane, showing a linear-chain structure with a 10 Be + α correlation. The size of the box is 10 × 10 fm2 . Figure adapted from
Ref. [13].

FIG. 3. (Color online) Density
wave functions for the positive-pa
density ρ˜p , neutron density ρ˜n , and
and proton densities ρ˜n − ρ˜p are
and right columns, respectively. T
intrinsic wave functions at (a)
(b) (β cos γ , β sin γ ) = (0.23, 0.0
(0.45, 0.17),
(d)
(β cos γ ,
(β cos γ , β sin γ ) = (0.78, 0.22),
(0.93, 0.04) on the β-γ plane are
10 × 10 fm2 .

model (Fig. 2.1), as have linear-chain structures [13] based on antisymmetrized molecular
dynamics (Fig. 2.2), among other possible configurations.

In Ref. [13], linear-chain structure is predicted not only for one state, but for an entire
rotational band, with a strong overlap with

10

Be + α. Up to now, no experimental evidence

has been found for the existence of linear-chain structure in

14

C.

Chapter 3 describes the theory presented in Ref. [13] that predicts linear-chain structure,
and includes more details on the properties of the rotational band.

21

2.2

Development of Prototype AT-TPC and AT-TPC
Project

In addition to physics motivation, the technical development of the Active Target-Time
Projection Chamber, or AT-TPC, project is also important. The AT-TPC is a newly-built
device used to study nuclear reactions. The development of the project has included the
designing and building of a test chamber [26] and prototype AT-TPC [9, 27].
For this work, the prototype AT-TPC was used, and it has provided many important
details about the performance of the device and how to develop the full-scale AT-TPC.
Other prior tests and experiments [9, 26, 27] have been carried out to investigate the device’s
performance and scientific promise. The large angular coverage and active target nature of
the prototype AT-TPC is well suited to investigate α clustering in

14

C, as various resonance

energies can be probed while discriminating between elastic and inelastic scattering of

10

Be

on α particles.
In this section, a basic description of how the device functions will be given, as well as its
impact on the development of the full-scale AT-TPC. The details of the exact experimental
setup used in the collection of data in this work will be described in later chapters.

2.2.1

Context of the AT-TPC Project

The current and next generation of accelerators are able to provide radioactive isotope beams
at a large variety of energies and intensity, allowing for more types of nuclear reactions to
be studied [28, 29, 30, 31]. The development of the AT-TPC project has been undertaken to

22

provide higher luminosity, efficiency, and resolution in nuclear reactions experiments through
the use of active target and time projection chamber capabilities [27].
The full-scale device, installed at NSCL, provides (1) a large active volume defined by
an electron drift region of 100 cm along the beam axis and a circular readout plane of
60 cm in diameter, (2) a high electric field (1 kV/cm) for fast electron drift to minimize
the detector dead time, (3) high granularity of the readout plane (10,240 pads), (4) efficient
waveform digitization of signals by the General Electronics for TPC (GET) currently under
development in collaboration with IRFU/SPhN CEA-Saclay, CENBG, GANIL, NSCL, and
RIKEN laboratories [32], (5) magnetic rigidity analysis using a superconducting solenoid
magnet, and (6) the application of a recently-developed electron amplification technology,
Micromegas [26, 33], to an active target [27].
This work was done using the prototype AT-TPC, which uses 253 readout pads in a
cylindrical volume 50 cm long and 28 cm in diameter. It is a larger and more efficient device
than other existing active target TPCs [34, 35, 36].
The following subsections will describe how the prototype device achieves these features.

2.2.2

Time Projection Chambers

One of the defining features of the prototype AT-TPC is that it is a time projection chamber,
or TPC [34, 37, 38, 39, 40]. A TPC functions as a gas-filled ionization drift chamber that
allows for the reconstruction of the trajectory of a charged particle [39]. The TPC is filled
with a neutral gas, which is ionized by charged particles traversing the gas volume. By
setting a fixed electric field across the chamber, the freed electrons then drift to one end of
the TPC, where an electron amplifier is located. The segmentation of the electron amplifier
gives the position for the two in-plane spatial dimensions. The third spatial dimension is
23

simulated (upper panel) and experimental (lower panel) p( He, He)p
reaction at 3.9 A MeV, within 1060 mbar of C4H10. The reaction takes
place at 100 mm from the entrance window, with a recoil angle for the
proton of 60 . The area of each dot is proportional to the charge collected
in the corresponding pad.

Fig. 2. Upper view of hexagonal cathode pads. The arrow shows the
direction of incoming beam particles.

Fig. 3. Schematic diagram of the ionization process and the MAYA detection principle.

Figure 2.3: Schematic diagram of the ionization process for the MAYA detector, a type of
TPC. The electrons drift down against the electric field to the wire plane, where they are
detected. The charged particle trajectories can then be reconstructed. Figure from Ref. [34].
obtained by measuring the electron drift time. The time can be converted to a distance by
multiplying this time by the drift velocity of the electrons in the electric field. Fig. 2.3 gives
a schematic of the TPC principle as seen in MAYA, a TPC used at GANIL [34].
As the charged particles travel through the gas in the TPC, they lose energy. This
energy loss is dependent in part on the length of the particle tracks. By measuring the
length elements dx of a track that a charged particle has traveled in the TPC, the particle’s
energy loss dE can be calculated. The rate of energy loss per unit length is given by the
Bethe formula [41]

−

4πe4 z 2
dE
=
NB
dx
m0 v 2

24

(2.1)

where

B ≡ Z ln

v2
2m0 v 2
− ln 1 − 2
I
c

−

v2
c2

(2.2)

for velocity v and charge ze of the charged particle, number density N and atomic number
Z of the tracking gas in the TPC, electron mass m0 , electric charge e, speed of light c, and
average ionization potential I of the gas [41].
The trajectories and energy losses of charged particle tracks can be measured in TPCs.
How the electrons freed by the ionization of the gas by the charged particles are detected
will now be discussed.

2.2.3

Micromegas

The number of primary electrons released in the ionization of the gas by charged particles
is too low to be measured; thus, an amplification of these electrons is needed before the
electronics can process the signal. For different TPC devices, various types of electron
multipliers can be used. Some examples include a Frisch grid with proportional wires [34]
and GEM detectors [42]. For the prototype AT-TPC, the detector plane is constituted by
Micromegas [33]. The name is derived from the term MICRO-MEsh-GAseous Structure,
and it is a type of gas-electron amplification device [33].
Micromegas consists of a micromesh held over an anode readout plane [27]. For the
Micromegas used in this work, the micromesh is held at a uniform distance of 128 µm above
the anode readout plane. The mesh itself is made of 18 µm thick stainless steel wire woven

25

Figure 2.4: Schematic of Micromegas amplification technique. Figure adapted from Daniel
Bazin.
at a pitch of 63 µm. Insulated pillars are placed 2 mm apart on the readout plane in a grid
configuration to hold the micromesh at a uniform distance above the readout plane [27].
The electric field between the micromesh and the anode plane is much higher than that
found in the rest of the detector; the field in the Micromegas is on the order of 10 to
100 kV/cm while it’s only on the order of 1 kV/cm in the volume where the charged particles
ionize the gas in the TPC. This greatly increased field allows for amplification of the electrons,
which produces a much larger signal that then can be read out [27]. Typical amplification
factors can reach up to 10,000, depending on the gas, the gas pressure, and the electric field.
The anode readout plane is segmented into pads. Each pad has its own dedicated readout
electronics channel. Gain can be varied pad by pad by applying a unique voltage to each pad.
This gain variation capability of the Micromegas was essential for the triggering of reaction

26

44

D. Suzuki et al. / Nuclear Instruments and Methods in Physics Research A

electrons. The par
from the center to
series of coaxial str
give the energy dep
strip as well as z,
reaction products.
the radial direction
250 mm
the range and thus
A trade-off is the ab
design is best suit
center-of-mass sca
scattered particle
angle of the recoilin
necessary. A compl
for binary reaction
state since these
angles on the react
quadrants.
The anode pads
two with 100 ins a
board. These conn
endplate. An adapt
field cage to the M
usually placed abo
Fig. 5. Photograph of the Micromegas. The inset shows a magnified view of the
of the field cage. I
region. The central pad has a diameter of 5 mm, while the coaxial strips
Figure 2.5:central
Photograph
of the Micromegas. The inset shows a magnified view of the central
have a pitch of 2 mm. The pads are separated by a 0.25 mm gap.
mounte
region. The central pad has a diameter of 5 mm, while the other coaxial strips have astand-offs
pitch
screws are inserted
of 2 mm. The pads are separated by a 0.25 mm gap. Figure from Ref. [27].
with the threads of
an SHV feedthroug

events investigated in this work. With the prototype in this work, there are 252 concentric

2.6. aReadout circui
pads separated into quadrants; the full-scale device uses 10,240 much smaller pads over

larger area. The 2 mm width of the pads used with the prototype allows for radial resolution
Fig. 7 shows th
the signal readout.
of also about 1 mm, achieving good precision in determining the range and thus kinetic
the micromesh si
readout electronic
energy of charged particles in the device [27]. The radial structure of the pads is conducive
signals from all of
analog memory us
to measuring two-body reactions. A picture of the Micromegas used in this work is given
analog-to-digital c
in Fig. 2.5. An illustration of the TPC technique for measuring a two-body reactionmesh
with signal is eith
generate a trigger
the Micromegas is shown in Fig. 2.6. More details on the specific Micromegas used in this

27
Fig. 6. Illustration of tracking. The beam particle (indicated by A and red) collides

2.6.1. Micromesh c
The micromesh
supply. A resistor
going through the
micromesh is rea

Fig. 5. Photograph of the Micromegas. The inset shows a magnified view of the
central region. The central pad has a diameter of 5 mm, while the coaxial strips
have a pitch of 2 mm. The pads are separated by a 0.25 mm gap.

endplate. An adapt
field cage to the M
usually placed abou
of the field cage. I
stand-offs mounted
screws are inserted
with the threads of
an SHV feedthroug

2.6. Readout circui

Fig. 7 shows th
the signal readout.
the micromesh si
readout electronic
signals from all of
analog memory us
analog-to-digital c
mesh signal is eith
generate a trigger

2.6.1. Micromesh c
The micromesh
supply. A resistor
going through the
Fig. 6. Illustration of tracking. The beam particle (indicated by A and red) collides
micromesh is rea
with a gas nucleus and produces reaction products (B, green and C, blue). The
Figure 2.6:ionization
llustration
of tracking. The beam particle (indicated by A and red) collides
preamplifier, mode
electrons (dashed lines) are transported by the electric field (E), making
with a gas anucleus
and
produces
reaction
products
(B, green
and deposit
C, blue).
The
of 20 nF is used. An
projected
image
on the surface
of the
Micromegas.
The energy
profile
of ionization
electrons (dashed
lines)
are
transported
by
the
electric
field
(E),
making
a
projected
image
on from the mic
the beam particle is measured by the central pad, while those of particles B and C
10 nF
by the coaxial
Thedeposit
track is segmented
pitch
strips in
the surface are
of measured
the Micromegas.
Thestrips.
energy
profile of by
the2 mm
beam
particle
is measured
to reduce cross-talk
the R
direction,
providing
a resolution
mmmeasured
or less, while
by the central
pad,
whilethus
those
of particles
B andofC2are
by the
theazimuth
coaxialfstrips.anode
The pads [22]. T
information is absent. (For interpretation of the references to color in this figure
track is segmented by 2 mm pitch strips in the R direction, thus providing a resolution
of coupled to
tively
caption, the reader is referred to the web version of this article.)
2 mm or less, while the azimuth φ information is absent. Figure from Ref. [27].
over the anode pl
avalanche creates
pads
consist
of a 5 mm
central
pad and
mm in
pitch
work, as well
as the
Micromegas
circuitdiameter
and readout
electronics,
are2given
Sectionspulses
4.2.4 in all the oth
when multiple pad
coaxial strips, ach spaced by a gap of 0.25 mm. The coaxial strips
amplitude of the cr
and 4.2.5. are divided into four quadrants and the five inner-most strips
inter-pad capacitan
into eight octants as shown in the inset of Fig. 5. Accordingly
individual quadrants have 63 coaxial strips, totaling 253 pads
the mesh and grou
over the entire plane. An illustration of tracking is shown in Fig. 6.
cing this effect is to
2.2.4 Active Target Mode
The beam particle travels perpendicular to the anode plane. The
to ground. Normall
central pad and its neighboring coaxial strips record the energy
by the mesh-to-ano
Another defining
feature
of
the
prototype
AT-TPC
is
that
it
acts
as
an
active
target,
that
is, Micromega
deposit profile as a function of time, and the position along the
present
beam axis (z) is deduced from the drift time of ionization
model of parallel p

the gas medium used for TPC tracking also serves as the reaction target for beam nuclei [27].
When using an active target, the pressure at which the gas is held determines the density of
gas particles and thus the thickness of the target. If the study of the entire energy domain of
28

a given beam energy is desired, the gas pressure should be sufficiently high to allow for the
stopping of the beam in the device. The energy loss profile of the beam particles is described
by the Bethe formula, defined previously in Eq. 2.1.
Since the gas is used as both a tracking medium and target, the choice of gas varies
experiment to experiment. In this work to study

10

Be scattered by 4 He, a He:CO2 90:10

mixture at around 1 atm of pressure is used. The

10

Be beam particles are incident on the

gas at about 4 MeV/u. As they travel through the active target and lose energy, the

10

Be

nuclei will, with some probability, scatter off of target 4 He nuclei. The position of the vertex
of the reaction along the beam axis can be calculated with the TPC capabilities of the
device. The vertex is converted to the energy of the reaction via Eq. 2.1. Since the

10

Be

nuclei lose nearly all of their kinetic energy by the time they traverse the full length of the
detector, reactions in the prototype AT-TPC occur across almost the entire energy domain
of the beam. The number of reactions that occur at a given vertex, and thus reaction energy,
gives the cross section dσ(E)/dΩ of the reaction. The active target capability of the device
results in nearly the entire energy spectrum of the entering 10 Be beam particles being probed,
allowing for

10

Be + 4 He reactions at many different energies to be studied.

Scattering of

10

short ranges of the

Be on the quench gas nuclei
12

C and

16

12

C and

16

O was not observed due to the

O nuclei in the gas. Thus, such scattering events were not a

source of background scattering in the data.
As described above, the prototype AT-TPC was used in active target mode for this work.
If so desired, for other types of experiments a solid foil target can be placed in the detector.
In this case, the prototype device would act as a TPC. The prototype has not yet been
used in this fixed target mode. The full-scale AT-TPC in particular is designed with such a
capability.
29

Chapter 3
Theory
This chapter describes the theoretical method of β-γ constraint Antisymmetrized Molecular
Dynamics (AMD) and the Generator Coordinate Method (GCM), or the β-γ constraint
AMD + GCM, presented by Suhara and Kanada-En’yo in their papers Refs. [13, 43]. Their
discussion of the structures of ground and excited states in

14

C motivated this work. For

more details on how the results of Ref. [13] motivated this work, see Section 2.1.

3.1

Antisymmetrized Molecular Dynamics

First mentioned in Ref. [15] and described in Ref. [44], Antisymmetrized Molecular Dynamics
(AMD) is a theoretical framework used to model molecule-like structure of nuclear states,
among other approaches [22, 45, 46]. In this section, the basic formulation of AMD as used
in Ref. [13] is presented.
The origins of AMD are in other microscopic simulation methods, such as Quantum
Molecular Dynamics (QMD) [47], an N -body theory that can describe the mean field effect and the dynamics of fragment formation. However, since QMD represents nucleons as
distinguishable Gaussian wave packets in phase space, ground states of nuclei are described
semi-classically by imitating the Thomas-Fermi model of phase space distribution. Various
improvements have been made to QMD to improve its treatment of the fermionic nature of
nucleons, including introducing Pauli potentials [48, 49]. The introduction of an antisym30

metrized version of molecular dynamics, called Fermionic Molecular Dynamics (FMD) [14],
created a method that treats the fermionic nature of nucleons exactly. In FMD, a Slater
determinant of Gaussian wave packets of nucleons is used to build the wave function of
a many-body system, and the time-dependent variational principle is used to describe the
time development of the wave packet parameters [50]. Modified versions of FMD, such as
found in Ref. [15], where the time variations of the width parameters and the spin wave
functions are neglected have been shown to make the FMD method free of assumptions such
as axial symmetry and even clustering structure. The incorporation of two-nucleon collision
processes has been successfully implemented into this modified FMD, which is now referred
to as Antisymmetrized Molecular Dynamics (AMD) [44].

3.2
3.2.1

Description of β-γ Constraint AMD + GCM
AMD Wave Functions

The AMD method describes an A-nucleon system’s basis wave function |Φ as a Slater
determinant of single-particle wave functions |ϕi as [13]

1
|Φ = √ det {|ϕ1 , ..., |ϕA }.
A!

(3.1)

Each single-particle wave function |ϕi is described by its spatial |φi , spin |χi , and
isospin |τi components as

31

|ϕi = |φi |χi |τi .

(3.2)

√
The spatial term is described as a Gaussian wave packet centered at Zi / ν of the form

r|φi =

2ν
π

3/4

Zi
exp − ν r − √
ν

2

1
+ Zi2
2

(3.3)

for width parameter ν common to all single-particle Gaussian wave functions [13]. The
parameter ξi determines spin orientation as

|χi = ξi↑ |↑ + ξi↓ |↓

(3.4)

and isospin is fixed to be either up (proton) or down (neutron) [13]:

|τi = |p or |n .

(3.5)

For a basis wave function |Φ , the complex variational parameters

{X} ≡ {Z, ξ} = {Z1 , ξ1 , Z2 , ξ2 , ..., ZA , ξA }

32

(3.6)

are determined by energy optimization [13].

3.2.2

Parity and Angular Momentum Projections

Projections of the AMD wave function |Φ onto the parity and angular momentum eigenstates are done with the parity projection operator Pˆ ± and angular momentum projection
J
operator PˆM
K , respectively. The operators are defined as [13]

1 ± Pˆ
Pˆ ± ≡
2

(3.7)

for parity operator Pˆ and

2J + 1
J
PˆM
K ≡
8π 2

J∗
ˆ
dΩ DM
K (Ω)R(Ω)

(3.8)

J∗
for Euler angles Ω = (α, βγ), Wigner’s D function DM
K (Ω), and rotation operator

ˆ
ˆ
ˆ
ˆ
R(Ω)
= e−iαJz e−iβ Jy e−iγ Jz .

(3.9)

Note that the operator given in Eq. 3.8 is written for the case of axial symmetry. Variation
is performed for the parity projected wave function

33

|Φ± ≡ P ± |Φ ,

(3.10)

after which |Φ± is projected onto the total angular momentum eigenstates. Thus, the
parity projection is made before the variation, while the total angular momentum projection
is made after [13].

3.2.3

Constraint Variation on Quadrupole Deformation Parameters, β and γ

When the energy variation is performed, constraints are placed on the quadrupole deformation parameters, β and γ. These constraints are made so that the cluster and shell-model
structures in ground and excited states of

14

C can be described with the AMD model and

have their deformation characterized [13]. The deformation parameters are defined as

β cos γ ≡

√

5π 2 zˆ2 − xˆ2 − yˆ2
,
3
R2
5π xˆ2 − yˆ2
,
3
R2

β sin γ ≡
R2 ≡

5
3

xˆ2 + yˆ2 + zˆ2

,

(3.11)
(3.12)
(3.13)

ˆ for an intrinsic wave function |Φ is represented
where the expectation value of an operator O
ˆ . The inertia principal axes xˆ, yˆ, and zˆ are chosen such that
as O

34

yˆ2 ≤ xˆ2 ≤ zˆ2 ,

(3.14)

xˆyˆ = yˆzˆ = zˆxˆ = 0,

(3.15)

xˆyˆ
yˆzˆ
zˆxˆ
= 2 = 2 =0
2
R
R
R

(3.16)

with the constraint

necessary to satisfy Eq. 3.15 [13].
A constraint potential

Vconst ≡ η1 (β cos γ − β0 cos γ0 )2 + (β sin γ − β0 sin γ0 )2
+ η2

xˆyˆ
R2

2

+

yˆzˆ
R2

2

+

zˆxˆ
R2

2

(3.17)

is introduced to the total system energy in the energy variation to obtain the energy minimum
state under the constraint condition in Eq. 3.16, where η1 and η2 are of sufficiently large
±
values. Thus, the constrained energy Econst
is defined as

±
Econst
≡

ˆ ±
Φ± |H|Φ
+ Vconst ,
Φ± |Φ±

35

(3.18)

ˆ is the Hamiltonian, to determine {X}. Once the variation is made on the conwhere H
straints, the optimized wave function |Φ± (β0 , γ0 ) are obtained for each set of parameters
(β, γ) = (β0 , γ0 ) [13].

3.2.4

Generator Coordinate Method

To calculate energy levels, the parity and total angular momentum projected AMD wave
J
±
±
functions PˆM
K |Φ (β, γ) are superposed. The final wave function for the Jn state is therefore

a linear combination of the basis wave functions:

|ΦJ±
=
n

K

i

J
±
fn (βi , γi , K)PˆM
K |Φ (βi , γi ) ,

(3.19)

where the coefficients fn (βi , γi , K) are determined with the Hill-Wheeler equation,

δ

ˆ ΦJ± − En ΦJ± ΦJ±
ΦJ± H

= 0.

(3.20)

Therefore, the wave function |ΦJ±
is the superposition of multiconfigurations described
n
by parity and total angular momentum projected AMD wave functions. In the limit of
sufficient basis wave functions on the β-γ plane, |ΦJ±
corresponds to the Generator Con
ordinate Method (GCM) with the two-dimensional generator coordinates of the quadruple
deformation parameters, β and γ [13].

36

3.2.5

Hamiltonian

ˆ mentioned in the previous section is composed of a kinetic term as well
The Hamiltonian H
as three effective two-body interactions:

ˆ =
H
i

tˆi − TˆG +

Vˆijspin-orbit +

Vˆijcentral +
i<j

i<j

VˆijCoulomb ,

(3.21)

i<j

for central force Vˆijcentral , spin-orbit force Vˆijspin-orbit , and Coulomb force VˆijCoulomb . The central
force is described as

2

Vˆijcentral =
k=1

vk exp −

rˆij
ak

2

(W + B Pˆσ − H Pˆτ − M Pˆσ Pˆτ ),

(3.22)

the Volkov No. 2 interaction [51], for v1 = −60.65 MeV, v2 = 61.64 MeV, a1 = 1.80 fm, and
a2 = 1.01 fm. The spin-orbit term is given by the G3RS interaction [52] as

2

Vˆijspin-orbit =
k=1

uk exp −

rˆij
bk

2

ˆ · Sˆ
Pˆ (3 O), L

(3.23)

a two-range Gaussian with projection operator

1 + Pˆσ 1 + Pˆτ
Pˆ (3 O) =
,
2
2

37

(3.24)

for b1 = 0.600 fm, b2 = 0.477 fm, spin exchange operator Pˆσ , and isosopin exchange operator
Pˆτ [13].
The interaction parameters used in the calculations presented in Ref. [13] are the same
as those used in Ref. [23], and they are the same as those adopted in many other studies,
including 9 Be [53], 10 Be [54], and C isotopes [22, 24], save for a small change in the spin-orbit
force strength. They are adjusted to reproduce the α + α phase shift, the binding energy of
the deuteron, and the α + n phase shift. The width parameter of single-particle Gaussian
wave packets in Eq. 3.3 is set to a value of ν = 0.235 fm−2 , which is determined from a 9 Be
ground state variational calculation in Ref. [53].

3.3

Calculations

With the AMD method described in Sec. 3.2, calculations were made for

14

C [13]. Positive-

and negative-parity state calculations are presented separately.

3.3.1

Energy Surfaces

Variational calculations were made with the β − γ constraint of the β − γ plane triangle
lattice at 196 mesh points. From these calculations, energy surfaces as functions of β and γ
are obtained. Positive-parity surfaces are shown in Fig. 3.1, while negative-parity surfaces
are found in Fig. 3.2. In both figures, the top panels show states before the total angular
momentum projections, while the bottom panels show states after the projection [13].
In Fig. 3.1, the top panel is called the positive-parity energy surface, and the bottom
panel is called the 0+ energy surface. The positive-parity energy surface’s minimum point
is located at (β cos γ, β sin γ) = (0.00, 0.00), indicating a spherical shape. The 0+ energy

38

TES IN 14 C

PHYSICAL REVIEW C 82, 044301 (2010)

ers

kinetic term and
spin-orbit
Vˆij

j

(16)

the central force,
spectively. As the
action [25],

Pˆσ Pˆτ ),

(17)

eV, a1 = 1.80 fm,
we used the spinhich is a two-range
onto the triplet odd

3

ˆ
O) Lˆ · S,

FIG. 1. Energy surfaces of 14 C on the β-γ plane. The top
14
shows
the ofenergy
positive-parity
before
the the energy
Figure 3.1: panel
Energy
surfaces
C onforthethe
β−
γ plane. The states
top panel
shows
for the positive-parity
states
before
the
total
angular
momentum
projection
and
total-angular-momentum projection and the bottom panel shows thatthe bottom
(18) showsforthat
panel
the 0+after
states
the total-angular-momentum
projection. Figure
the after
total-angular-momentum
projection.
the for
0+ states
from Ref. [13].

A. Energy surfaces

(19)

fm. Here, Pˆσ
change operators,

eters as those in
arameter M = 0.6
er B = 0.125, and

We performed variational calculations with the β-γ constraint at 196 mesh points of the triangle lattice on the
β-γ plane. Energy surfaces as functions of β and γ are
obtained. The calculated energy surfaces on the β-γ plane
for positive-parity states and negative-parity states are shown
in Figs. 1 and 2, respectively. 39
In Fig. 1, the top panel shows the energy of the positiveparity states before the total-angular-momentum projection,
+

TADAHIRO SUHARA AND YOSHIKO KANADA-EN’YO

ρ˜p

(a)

(b)

(c)

(d)

(e)
FIG. 2. Energy surfaces of 14 C on the β-γ plane. The top
14
Figure 3.2: panel
Energyshows
surfaces
of
C
on
the
β
−
γ
plane.
The
top
panel
shows
the
energy
for
the energy for the negative-parity states before the
the negative-parity
states before theprojection
total angular
momentum
projection
total-angular-momentum
and the
bottom panel
showsand
thatthe bottom
−
panel showsforthat
for
the
3
states
after
the
total-angular-momentum
projection.
Figure
the 3− states after the total-angular-momentum projection.
from Ref. [13].

respectively. The minimum point of the negative-parity energy
surface is at (β cos γ , β sin γ ) = (0.08, 0.04), and that of
the 3− energy surface is at (β cos γ , β sin γ ) = (0.20, 0.09).
Again, the deformation of the energy minimum state changes
from an almost spherical shape to a triaxial one before and
after the total-angular momentum projection. In the largely
deformed prolate region, the behavior of the energy surface
for the negative-parity states40is different from that of the
positive-parity states. Along the γ = 0◦ line, the energy rapidly
increases and there is no flat region in the negative-parity

(f)

FIG. 3. (Color o
wave functions for
density ρ˜p , neutron
and proton densitie
and right columns,
intrinsic wave func

surface’s minimum point is located at (β cos γ, β sin γ) = (0.23, 0.04), resulting in a deformation of the energy minimum state from spherical before the projection to triaxial after
the projection due to higher correlations beyond the mean field that are incorporated by
the projection. Similarly for negative-parity states in Fig. 3.2, the minimum point before
projection is (β cos γ, β sin γ) = (0.08, 0.04) and (β cos γ, β sin γ) = (0.20, 0.09) after, again
changing from nearly spherical to triaxial in shape. While there is a flat region in the 0+
energy surface near (β cos γ, β sin γ) = (0.90, 0.04), there is no flat region in the 3− energy
surface. This flat region in the 0+ energy surface is important since a rotational band with
large prolate deformation is constructed by wave functions in this region after the GCM
calculation is made, and it is this prolate state that is theorized to be in the form of a
linear-chain. No such linear-chain structures are predicted for negative-parity states [13].

3.3.2

β-γ Plane Structures

The intrinsic structures of the

14

C states are obtained by the β − γ constraint AMD. The

density distributions of protons and neutrons are analyzed, as is the difference in density
between neutrons and protons so that excess neutron behavior can be observed. The densities
are defined as

ρ˜(x, z) ≡

for

41

dy ρ(r)

(3.25)

PHYSICAL REVIEW C 82, 044301 (2010)

NADA-EN’YO

β-γ plane. The top
ity states before the
ttom panel shows that
tum projection.

ρ˜p

ρ˜n

ρ˜n − ρ˜p

[1/fm2 ]

(a)

(b)

(c)

(d)

(e)

gative-parity energy
(f)
0.04), and that of
n γ ) = (0.20, 0.09).
mum state changes
FIG. 3. (Color online) Density distributions of the intrinsic
xial one before and
C.the
The
proton
wave functions
for the
positive-parity
of 14for
Figure 3.3: Density
distributions
of the
intrinsic wavestates
functions
positive-parity
states
tion. In the largely
14
density density
ρ˜n , and ρ˜difference
betweenbetween
the neutron
ρ˜p , neutron
of C. The density
proton density
ρ˜p , neutron
the neutron and
n , and difference
the energy surface
proton
ρ˜n − ρ˜p densities
are illustrated
middle, in
andthe
right
ρ˜p the
areleft,
illustrated
left,columns,
middle,respectively.
and proton
ρ˜n − in
t from that of
the densities
The density and
distributions
of
the
intrinsic
wave
functions
at
(a)
(β
cos
γ,
β
sin
= (0.00, 0.00),
right columns, respectively. The density distributions ofγ)the
e, the energy rapidly
(b) (β cos γ,intrinsic
β sin γ) =wave
(0.23, functions
0.04), (c) (β
(d) (β
cos γ, β sin γ) =
at cos
(a)γ, β(βsin
cosγ)γ=
, β(0.45,
sin γ 0.17),
) = (0.00,
0.00),
the negative-parity
(0.25, 0.35), (b)
(e) (β(β
coscos
γ, βγ ,sin
γ) =
0.22),
and (f)(c)
(β cos(β
γ,cos
β sin
β sin
γ )(0.78,
= (0.23,
0.04),
γ ,γ)β =
sin(0.93,
γ ) =0.04) on the
mentum projection,
2
β − γ plane (0.45,
are shown.
The
size
of
the
box
is
10
×
10
fm
.
Figure
from
Ref.
[13].
0.17),
(d)
(β cos γ , β sin γ ) = (0.25, 0.35),
(e)
y deformed prolate

ane

(β cos γ , β sin γ ) = (0.78, 0.22), and (f) (β cos γ , β sin γ ) =
(0.93, 0.04) on the β-γ plane are shown. The size of the box is
42
10 × 10 fm2 .

uctures obtained by

functions for positive-parity states are illustrated in Fig. 3.

ATES IN 14 C

he origin; therefore,
between the shellThis feature is also
ed intrinsic spin of
the ideal value 4/3
and larger than the

bution for a triaxial
m the proton density
e squared intrinsic
res develop. Excess
ributing along the
eles-triangle which
orbitals make this
a K π = 0+ band
his triaxial state are

PHYSICAL REVIEW C 82, 044301 (2010)

ρ˜p

ρ˜n

ρ˜n − ρ˜p

[1/fm2 ]

(a)

(b)

(c)

ee α-cluster cores
s of three α clusters
(d)
parameters, β and
for typical density
e deformed states,
riangular structure,
(e)
ear-chain structure
ate [Fig. 3(d)], the
e state [Fig. 3(f)],
n values of squared
FIG. 4. (Color online) Density distributions of the intrinsic
an 0.10, which
is 3.4: Density
14 the negative-parity states
Figure
distributions
of the
intrinsic wave states
functions
C. The proton
wave functions
for the
negative-parity
of for
14 It
e (0s)4 α cores.
of C. The density
proton density
ρ
˜
,
neutron
density
ρ
˜
,
and
difference
between
the neutron and
p
n
the neutron
ρ˜p , neutron density ρ˜n , and difference between
tribute around
two
proton densities
ρ˜n − ρ˜p densities
are illustrated
middle, and
right
columns,
ρ˜p the
areleft,
illustrated
in the
left,
middle,respectively.
and proton
ρ˜n − in
ate [Fig. 3(e)]Theand
density distributions of the intrinsic wave functions at (a) (β cos γ, β sin γ) = (0.08, 0.04),
and right columns, respectively. The density distributions of the
(b) (β cos γ, β sin γ) = (0.20, 0.09), (c) (β cos γ, β sin γ) = (0.25, 0.35), (d) (β cos γ, β sin γ) =
e + α correlations.
intrinsic wave functions at (a) (β cos γ , β sin γ ) = (0.08, 0.04),
(0.60,
and (e) (β cos γ, β sin γ) = (0.93, 0.04) on the β − γ plane are shown. The size
uster structure,
the0.17), (b)
β sin γfrom
) = (0.20,
0.09), (c) (β cos γ , β sin γ ) =
10 ×(β
10 cos
fm2γ. ,Figure
Ref. [13].
he deformedof10the
Bebox is(0.25,
0.35), (d) (β cos γ , β sin γ ) = (0.60, 0.17), and (e)
mention here that
(β cos γ , β sin γ ) = (0.93, 0.04) on the β-γ plane are shown.
he base of the flat
The size of the box is 10 × 10 fm2 .
rmation. After the
ig. 3(f)] constructs
43
intermediate between the shell-model structure and the cluster
of negative-parity
structure.
ve-parity states are
Figures 4(c), 4(d), and 4(e) show intrinsic density distribu-

ρ(r) ≡ Φ(β, γ)|

i

δ(r − rˆi )|Φ(β, γ) .

(3.26)

The positive- and negative-parity density distributions are shown in Fig. 3.3 and Fig. 3.4,
respectively [13].
Of particular interest to the work presented in this dissertation is Fig. 3.3(f), where a
linear-chain structure is shown. Such a structure is not seen in Fig. 3.4, indicating that
the prolate deformed negative-parity states are different from the prolate deformed positiveparity states, though both show three α-cluster cores with

3.3.3

10

Be + α correlation [13].

Energy Levels

Energy spectra are calculated by superposing the β − γ constraint AMD wave functions at
196 mesh points on the β − γ plane by using the GCM. Positive-parity states are shown with
rotational band assignments in Fig. 3.5 [43], with negative-parity states given in Fig. 3.6
[13].
The GCM calculation results in multiple rotational bands. Consider the rotational band
+
+
formed by the 0+
5 , 26 , and 46 states in Fig. 3.5, labeled as band (3). The main component

of the band comes from the AMD wave function located at (β cos γ, β sin γ) = (0.93, 0.04),
the linear-chain structure, with 64% overlap in the 0+
5 state. Due to the large deformation
of the linear-chain structure of three α clusters, the band is predicted to have strong E2
4
4
+
+
+
2
2
transitions with B(E2, 2+
6 → 05 ) = 67.2 e fm and B(E2, 46 → 26 ) = 88.3 e fm . Since

the ratio of these B(E2) values is 1.31, which is consistent with the rigid rotor model value

44

Many-body correlations from dilute to dense nuclear systems (MBC 2011)
IOP Publishing
Journal of Physics: Conference Series 321 (2011) 012047
doi:10.1088/1742-6596/321/1/012047

Figure 1.Figure
Energy
the for
positive-parity
states
of 14
3.5: levels
Energyfor
levels
the positive-parity
states
ofC14and
C andthe
thedensity
density distributions
distributions of the
intrinsic wave
of bands.
The
dotted The
linesdotted
on the
left
right
show
experimental
of thefunctions
intrinsic wave
functions
of bands.
lines
onand
the left
and
rightthe
show
the
10
10
and theoretical
Be+α
respectively.
Therespectively.
proton density
ρ˜p , neutron
experimental
andthreshold
theoretical energies,
Be + α threshold
energies,
The proton
density density
ρ˜p , neutronbetween
density ρ˜the
difference
between
thedensities
neutron and
proton
densities
ρ˜n - ρ˜p are
n , and
ρ˜n , and difference
neutron
and
proton
ρ˜n −
ρ˜p are
illustrated.
illustrated. Figure from Ref. [43].

groups byofanalyzing
the intrinsic
structures
E2 transition
It is found
10/7, this band
is considered
to be the and
rotational
band of the strengths.
linear-chain structure
withthat the
ground band (1) has a shell-model-like structure, while three α clusters develop well in most
the 10 Be
+ α correlation
[13]. and third column is the K π = 0+ and K π = 2+ side band
of the excited
states.
The second
(2), which has a triaxial structure. In these states, three α clusters have the isosceles-triangle
configuration and excess neutrons occupy the sd-like orbitals. The fourth column is the K π = 0+
12
to α
C
and Discussion
of Excessstructure.
Neutrons
band (3), 3.3.4
in whichComparison
developed three
clusters
have a linear-chainlike
It is interesting
that the excess neutrons distribute around two of the three α clusters. This indicates the 10 Be
12 the linear-chain states. For the excited states of 14 C, there
correlation
are many experimental
In in
C, no similar linear-chain rotational band is predicted; only the 0+
3 state has a linear10
studies that indicate the appearance of Be+α cluster states. Our results suggest that excited
14
10 Be+α
chain
configuration
in thisare
model
while anfor
entire
rotational
states with
correlation
the [13],
candidates
these
states.band exists in C. The
existence of triaxial bands in

14

C is essential to the appearance of the positive-parity linear-

-73

chain band since they prevent the
14 linear-chain band from bending due to orthogonality. The

C

Energy [MeV]

states that prevent bending in the 2+ and 4+ states in
fact, the excess neutrons found in

14

-78

14

C, however, do not exist in

Intrinsic(-6.9 MeV)
Positive(-6.9 MeV)
0+
45
0

1

C. In

C are essential for constructing the triaxial bands that

prevent bending [13].

-83

12

2
3
d2 - d1 [fm]

4

5

CLUSTER STRUCTURES OF EXCITED STATES IN 14 C

Energy [MeV]

-85
-90
-95
-100

1-- 24
10
Be(2+)+α 1- 2(2-)
1 2
1- 10
+
2Be(0 )+α
4- 2
(3-) + 3
1
4-(5-, 6 )
2
2
(3 )
(1- -) (5--) (2- ,3- )
4
1
(3- ) 1 ,2
3
- 2
2-1
3
0
1
-

3-33-

4-- 4- 5
4

-7

3
3

-

4-

3- 4- 5
3

3

Energy [MeV]

-80

-

-105
-110

Experiment

-8

Theory

FIG. 6. Energy levels of the negative-parity
states in 14 C. The five
14
Figure 3.6: Energy levels of the negative-parity states in C. The five columns on the left
columns on the left are the experimental data and the five columns
are the experimental data and the five columns on the right are the calculated results. The
onlines
theonright
areand
theright
calculated
The and
dotted
lines 10
onBethe
dotted
the left
show the results.
experimental
theoretical
+ α left
threshold
10
energies,
respectively.
Figure
from
Ref.
[13].
and right show the experimental and theoretical Be + α threshold
energies, respectively.

interpreted as the intermediate between the cluster structure
and the shell-model structure with the almost p3/2 -subshell
closed-proton configuration and the 1p-1h excited neutron
−
configuration. The calculated strength, B(E2, 3−
1 → 11 ) =
1.3 e2 fm4 is reasonable compared with the experimental
−
4
2
value, B(E2, 3−
1 → 11 ) = 3.046± 1.2 e fm , though the level
ordering is somehow in disagreement with the experimental
one. As seen in the E2 transitions in Table III, E2 transitions
10

-8

FIG. 7. (
parity states
strengths in
of Jiπ states
B(E2) 20
50.0 e2 fm4 <
green bold s
line shows th

Calculations from Ref. [24] imply that the linear-chain state is perhaps unstable in

14

C.

Their work shows that the linear-chain is unstable against the bending mode due to the
lack of a local minimum in the energy curve. The work presented in Ref. [13], however,
takes into account the orthogonality to lower states in their GCM calculations. Further,
the work in Ref. [24] assumes the three α clusters are separated by equal distances with
the excess neutrons occupying an orbital moving around the 3α core. The β-γ Constraint
AMD + GCM used in Ref. [13] makes no assumptions on the clustering of α particles. The
various 3α + 2n structures they obtain are a result of their energy variation method, with a
preference for a

10

Be + α configuration.

The 10 Be + α structure of 14 C was first proposed in Ref. [25], and a parity doublet linearchain band is predicted. The work of Ref. [13] only predicts a positive-parity band, as the
10

Be core can easily rotate in the negative-parity

10

Be + α states, which results in strong

mixing of the linear-chain structure with bending

10

Be + α configurations. Also, the band

predicted in Ref. [25] is below the 10 Be + α threshold, while it is above threshold in Ref. [13].

3.4

Relation to Current Work

By the 10 Be+α entrance channel chosen in the present work, structures of the type 10 Be+α in
14

C clearly appear. These resonances in

14

C are investigated in this work and are compared

to the model predictions of Ref. [13].

47

Chapter 4
Experiment
The experimental data described in this work were collected at the University of Notre
Dame’s Nuclear Science Laboratory over a two-week span in October, 2011. The device and
electronics were set up and installed from October 11 to October 22, 2011. Beam tuning
and electronics setup were completed from October 22 to October 25. Finally, physics data
were collected from October 25 to October 31.
This chapter will discuss details of the beam facility at Notre Dame, the experimental
setup and detailed design of the prototype AT-TPC, and the data collection process.

4.1

Beam Facility

The Nuclear Science Laboratory is part of the University of Notre Dame’s Institute for
Structure and Nuclear Astrophysics, or ISNAP. The first accelerator for use in nuclear physics
research was built in the 1930’s on the campus, and the university has continued to produce
active research in the field ever since [56].
With the use of the Multi-Cathode Source of Negatives Ions by Cesium Sputtering (MCSNICS) ion source at ISNAP [57], a primary beam of

11

B was extracted from a

11

B + Ag

ion source. Of the two accelerators available at ISNAP, the FN Tandem Van de Graaff
accelerator [58] was used to accelerate the 11 B primary beam. The FN delivered the primary
beam to TwinSol, a low-energy radioactive nuclear beam (RNB) apparatus [55], where it
48

between the magnets. Here, an aperture and/or
energy-loss absorber (similar to that used in

forward-pe
utilized an
beam on a
production
intense (>0
the type of
utilize Be,
one-nucleo
9
Be (7Li, 6
reasonably
(105/s). Ho
requires us
8
B)n with a

4. Recent r

4.1. Low-e

Fig. 1. Beam-line layout at the UND FN tandem accelerator
Figure 4.1:lab
Partshowing
of the beam-line
at the
of Notre
Dame’s
Nuclear Science
present layout
location
of University
the TwinSol
RNB
apparatus
Laboratory. During this work, the prototype AT-TPC was located in the vault to the left A numb
and the new extended beam line to the low-background n=g
of, or beyond, TwinSol. Figure from Ref. [55].
8
cave.
Li and 6H
impinged upon 13 C foils, from which a secondary beam of 10 Be was extracted. The secondary
10

Be beam was then delivered to the prototype AT-TPC, where reactions between the

10

Be

beam particles and target α particles were measured. A schematic showing the beam-line
from starting from SNICS and ending at the end of TwinSol is shown in Fig. 4.1.
The three apparatuses mentioned above, in addition to the prototype AT-TPC, are described below.

4.1.1

MC-SNICS Ion Source

For the experiment conducted in this work, a primary beam of

11

B was extracted from

the MC-SNICS Source at the University of Notre Dame’s Nuclear Science Laboratory. An

49

Figure 4.2: A schematic drawing of the MC-SNICS Source at the Nuclear Science Laboratory.
Figure courtesy Brad Mulder.
example of a sputter ion source, MC-SNICS, or Multi-Cathode Source of Negatives Ions
by Cesium Sputtering [57], is the primary ion source for the FN Tandem Van de Graaff
accelerator. Fig. 4.2 shows a schematic of MC-SNICS.
Multiple identical ion source cathodes, in this case

11

B + Ag cathodes, are loaded onto

a cathode wheel. This wheel, which comes in both 20- and 40-cathode options, is of great
convenience to the operation of MC-SNICS; the cathode wheel can be rotated remotely under
vacuum as it allows for multiple cathodes to be available for use without opening the device,
greatly lessening the amount of time needed to switch from a spent cathode to a new one
[57].
Once a cathode is in place and the device is under vacuum, a cesium reservoir is heated to
90 ◦ C [57]. A vapor is formed, and gaseous cesium atoms condense on both a heated spherical

50

Figure 4.3: A schematic drawing of the FN Tandem Van de Graaff accelerator at the Nuclear
Science Laboratory. Figure courtesy Brad Mulder.
ionizer and the cathode surface. Those on the ionizer are quickly boiled off, stripping them
of an electron. These positively charged ions are then attracted to the negatively-charged
cathode and focused onto the cathode with an immersion lens, where they impact the source
material and cause the material to be sputtered, including the desired

11

B. Some of the

11

B

atoms gain an electron as they pass through the cesium that coats the cathode, allowing the
now-negatively charged

11

B to be extracted out of the source and focused and accelerated

towards the FN Tandem Van de Graaff accelerator [57].

51

Figure 4.4: A schematic drawing of the FN Tandem Van de Graaff accelerator at the Nuclear
Science Laboratory with included detail on the stripper foil mechanism. Figure courtesy Brad
Mulder.

4.1.2
As the

FN Tandem Van de Graaff Accelerator
11

B ions exit the MC-SNICS ion source, they are transported to the FN Tandem

Van de Graaff accelerator. The FN Tandem is 40 ft long and 12 ft in diameter and can be
operated at voltages of up to 10 MV [58]. It is shown in Fig. 4.3.
The 11 B ions enter the FN Tandem in a 1− charge state. At the center of the FN Tandem
is a terminal, a metal electrode held up to a high voltage of +10 MV. This terminal attracts
the negatively charged

11

B ions [58]. As the ions approach the terminal, they travel down

the FN column. This column focuses the charged ions, provides a vacuum for the ions to
travel in, and acts as mechanical support to hold the terminal in place in the center of the
FN Tandem by highly compressing it from either side. The part of the column where the
52

Figure 4.5: A schematic drawing of the beam line including the FN Tandem Van de Graaff
accelerator and analyzing magnet at the Nuclear Science Laboratory. Figure courtesy Brad
Mulder.
1−

11

B ions travel is called the low energy, or LE column. The other half is called the high

energy, or HE column [58].
Upon entering the terminal shell, the 1−

11

B ions traverse a stripper foil, as seen in

Fig. 4.4. This foil, made of 3 µg/cm2 carbon, is used to remove electrons from the

11

B ions,

thus making it a positively-charged ion [58]. Multiple charge states may be produced; in
this work, fully-stripped

11

B was desired. Thus, with a charge state of 5+ , an energy of up

to 60 MeV is achievable.
The positive

11

B ions, which are now completely free of electrons, accelerate away from

the terminal through the HE column. As various charge states may have been produced in
the FN Tandem, and other contaminants from MC-SNICS may also be present, the nuclei

53

are sent from the FN Tandem to an analyzing magnet to be sorted (Fig. 4.5). The magnitude
of the field in the analyzing magnet is tuned to only accept nuclei of the desired magnetic
rigidity; other beam particles are stopped [58]. The beam slits at the exit of the analyzing
magnet provide a signal to stabilize the high voltage at the terminal of the accelerator. The
chosen nuclei are then sent further down the beam-line to TwinSol.

4.1.3

TwinSol

In front of the TwinSol device, the beam impinges on the production target at a beam current
of about 1 µA. For this work, 13 C foils were used as the production target. The enriched 13 C
foils were supplied by the Arizona Carbon Foil Co., Inc. and assembled to target frames at
the NSCL using the fishing technique on water. A stack of four 100 µg/cm2
chosen as no single 400 µg/cm2

13

13

C foils was

C foil was commercially available. The beam of 10 Be nuclei

formed at the production target, referred to as the secondary beam, is then sent through
TwinSol.
TwinSol is a twin superconducting solenoid magnet low-energy radioactive nuclear beam
(RNB) apparatus [55]. The magnet is a 6 T superconducting solenoid comprised of two
“lenses,” the twin magnets in the device, as shown in (Fig. 4.6), which focus the justproduced secondary beam and deliver it to an experimental detector [55].
There are two configurations in which TwinSol is operated: parallel mode and crossover mode (Fig. 4.7). In cross-over mode, the first magnet focuses the secondary beam
between it and the other magnet where either an aperture and/or an energy loss absorber
is used to further purify the secondary beam [55]. In parallel mode, the beam is not focused
between the magnets. Additionally, an absorber may be placed between the magnets to stop
unwanted neutrons from passing down the beam-line through TwinSol. In both modes, the
54

Figure 4.6: A schematic drawing of TwinSol at the Nuclear Science Laboratory. Figure from
Ref. [59].
secondary beam is focused after the second magnet in TwinSol to the entrance window of the
experimental detector [55], which in this case is the prototype AT-TPC. The direct beam is
blocked by a beam stopper or slits. Between TwinSol and the prototype AT-TPC entrance
window is a beam steerer, which aids in the focusing of the beam onto the window. In the
experiment described in this work, TwinSol was operated in parallel mode.

4.2

Design of the Prototype AT-TPC

Before discussing how the prototype AT-TPC was installed and used at the Nuclear Science
Laboratory for this work, the design of the device will be described in detail. The device’s
most basic components are the exterior chamber, the field cage, the corona ring, the Micromegas, and the readout electronics. A schematic of the device is given in Fig. 4.8. It
was built to test the essential components of the AT-TPC at a smaller scale (1:2) for lower
cost. Much of the discussion in this section is based on Ref. [27], which additionally provides
information on the commissioning of the prototype AT-TPC.

55

truments and Methods in Physics Research A 505 (2003) 377–380

ome recent
details of
ation may
/twinsol/

sts of two
solenoid
Fig. 2. Schematic diagrams of the cross-over and parallel
esign with
4.7:
Schematic
drawingsoperation
of the two[2].
modes of operation of TwinSol. The top image
modes
of TwinSol
mize LHeFigure
shows cross-over mode, while the bottom is of parallel mode. Figure from Ref. [55].
his permits
a helium The exterior
chamber of the detector is 70 cm long and 40 cm in diameter, and contains
dipole-based
RNB systems) may be used to better
e magnetsa cylindrical
define
andmeasuring
purify 50the
field cage
cm secondary
in length and 28RNB.
cm in diameter, which is centered
de. Details
ef. [3]. Theon the beam axis. The field cage is filled with a gas, which acts as an active target as well
medium. A 3.7of
µmlow-energy
thick para-aramid
foil is used as the beam window into
ansport ofas a tracking
3. Production
RNBs
of experi-the device to separate the vacuum of the beam line (∼ 10−5 torr) from the gas that fills the
aforemenUnlike high-energy RNBs (i.e. E/A>20 MeV
detector (∼ 1 atm = 760 torr). This foil thickness can also be expressed as 0.536 mg/cm2
on-optical
per nucleon) which can be produced relatively
density
of 1.45
g/cm3 . Incident beam
particles pass through
this window
ode and agiven para-aramid’s
effectively
using
non-selective
fragmentation
reacFig. 2) theand entertions,
primary-beam
fragmentation
is not usually
the active
target gas in the field
cage [27].
B produceffective at low energies [2]. Instead, a selective
The beam particles can collide with target gas nuclei, resulting in a nuclear reaction. Both
2
few mm )
one- or two-nucleon transfer reaction with a large,
nuclei and the reaction cross-section
products, being charged
particles,
ionize thebe
gas. Due to
ure and/orthe beamforward-peaked
must
generally
t used inthe applied
utilized
andthedone
in reverse
kinematics
(heavy
electric field,
freed electrons
drift to the
Micromegas, which
amplifies the
beam on a light target) where feasible [2]. Also, the
electron signal and allows for the reconstruction of the trajectories of the charged particles
production target must be robust to withstand the
intense (>0.5 mA) primary beam. This again limits
56
the type of production reaction
to those which can
utilize Be, C or similar targets. As examples, the
9

7

8

ctron amplification
target.
ersion of the future
recently been conding the Prototype
aspects envisioned
pt. The cutting edge
ove involve several
the present article
volume of 50 cm in
as quality and field
of the high cathode
e a field gradient of
cromegas in helium
waveform digitizers
ng-term stability of
imental conditions.
s to deliver a highe for physics meacharacteristics, the
) The active volume
ve target TPCs and
(2) The Prototype
m bins in the radial
viding precise meaenergy of reaction

voltage, typically several hundred volts, which creates a high field
gradient of 10–50 kV/cm over the gap, amplifying the primary charge
through electron avalanches. The anode plane is segmented into
multiple pads to read out and locate avalanche electrons. The charge
distribution over the anode plane gives a projected image of the

Beam duct

Beam entrance Corona ring

Cathode

Field cage

Micromegas
Endplate

Cathode voltage feedthrough
Fig. 1. Cross-sectional view of the Prototype AT-TPC.

Figure 4.8: Cross-sectional view of the prototype AT-TPC. Figure from Ref. [27].
[27]. For a more general discussion of both the active target and TPC concepts, see the
previous descriptions in Section 2.2.
The following subsections will go into detail on each of the prototype AT-TPC’s basic
components.

4.2.1

Exterior Chamber

A stainless steel design, the outer chamber of the detector is cylindrical in shape and is 70 cm
long with a 40 cm diameter, as shown in Fig. 4.9. ISO 400 flanges are mounted on each end of
the chamber. The chamber houses the cathode, field cage, and Micromegas, described later
in this chapter. Inside of the chamber, a bellows is used to attach the upstream flange to
the beam duct and cathode high-voltage feedthrough. The difference in length between the
57

Figure 4.9: Image of the exterior chamber of the device. It’s 70 cm in length and has a 40 cm
diameter.
chamber and its internal components makes the bellows a practical choice for this situation
[27].
On the other end of the chamber, the downstream flange serves both as an endplate for
mounting the Micromegas and as a piece to secure the field cage inside the chamber, which
is done in such a way to allow for the Micromegas to be removed without dismounting the
field cage. A feedthrough on the endplate allows for anode signals from the Micromegas
to be relayed directly to exterior electronics, minimizing the addition of noise to the signal
between the anode and the preamplifiers. A gas inlet and a gas outlet are mounted on the
endplate for flowing the target gas through the field cage, as well as a KF40 flange with
four SHV feedthroughs. These SHV feedthroughs provide connections to a thermocouple
used to measure the gas temperature, supply bias and grounding and monitor the voltage
divider circuit of the field cage, and supply the bias voltage and readout the signal from the
micromesh of the Micromegas [27]. An image of the downstream flange is given in Fig. 4.10.

58

4.2.2

Field Cage

Inside of the exterior chamber is the cylindrical field cage, which is 50 cm long and 28 cm in
diameter. The purpose of the field cage is twofold: it provides both (1) a gas-tight volume in
which the active target gas can be filled and thus where nuclear reactions can take place and

Figure 4.10: Image of the downstream ISO 400 flange. Readout electronics for the Micromegas are mounted on the vertical feedthroughs on the center of the flange. On the left
of the flange, there are the both a gas inlet and outlet, as well as a KF40 flange with four
SHV feedthroughs.

59

(2) an interior and exterior surface on which to mount equipotential rings that step down
the voltage of the cathode across of the entire active target volume, allowing for a uniform
electric field that guides ionization electrons to the Micromegas. The direction of the field
is anti-parallel to the direction of the beam [27].
While the target for this work is 4 He, CO2 is added as a quenching gas to stabilize the
operation in practice and to increase the dielectric strength of the target gas [27]. At 1 atm,
pure helium gas has a dialectic strength of about 5 kV/cm, much lower than that of the
nitrogen insulation gas (34 kV/cm) and CO2 (28 kV/cm) [60]. The ratio of He to CO2 is 9
to 1, resulting in a net dielectric strength of 7.3 kV/cm, as the strengths add linearly [60].
In addition to the active target gas that fills the field cage, a second gas between the field
cage and exterior chamber is also present. This gas serves as insulation to avoid discharge
from the cathode to the exterior chamber. Thus, a gas with a high dielectric strength, such
as nitrogen, is used as the insulating gas [27].
The field cage is made of grade G10 non-conducting glass reinforced epoxy laminate,
both for its high mechanical strength and low outgassing characteristics. While not flame
resistant like FR4, G10 is not doped with bromine, the flame resistant material found in
FR4. As bromine can be be a strong source of electron absorption when diffused into the
the gas volume, G10 was used [27].
An aluminum flange is glued to the downstream end of the field cage, securing the field
cage to the endplate of the exterior chamber. On the upstream flange is glued a cathode
plate. Both the aluminum flange and cathode plate are hermetically sealed with an epoxy
resin glue, ensuring that the target gas inside of the field cage and the insulating gas outside
of it do not mix [27].

60

The design of the field cage does not allow for a large pressure differential between the
inside and outside of the field cage. A large differential could result in the breaking of the
seals on the end of the field cage or substantial damage to the mechanical integrity of the field
cage itself. To protect the field cage in such a scenario, a thin polypropylene foil of thickness
70 µg/cm2 is located between the two gas volumes. A high pressure differential would then
break the foil, allowing the pressures the quickly equilibrate before any damage to the field
cage could occur. The foil is located outside of the exterior chamber in a piece of vacuum
tubing that bypasses the inner and outer gas volumes. This location makes replacement
of the foil straightforward and simple. The foil is held in place in a KF25 flange, and its
differential pressure limit was measured to be about 0.1 atm [27].
As mentioned above, equipotential rings are mounted on the field cage, as shown in
Fig. 4.11. The rings serve as electrodes, connected via a series of resistors, that step the
voltage provided from the cathode at the front of the field cage down to zero at the opposite
end of the field cage where the Micromegas are located. The interior and exterior surfaces
of the cage each have 50 rings mounted on them, for a total of 100. The electrodes all are
made of aluminum rod with a cross-sectional diameter of 0.5 cm. The interior rings have
a diameter of 26.0 cm while the exterior rings are 29.5 cm across. The spacing between
neighboring rings is about 1 cm [27].
All rings are supported by both aluminum pin stands (Fig. 4.11a) and two additional G10
supports (Fig. 4.11b). The stands and supports are grooved into a V-shape, into which the
rings can snap into place. Once in place, the rings are self-supporting. Interior and exterior
rings are located at the same position along the field cage axis, and secured into place at the
opposite ends of the aluminum pin stand. The stands are secured to the field cage with an
epoxy resign glue and are thus hermetically sealed [27].
61

step down the electric potential from the cathode plate, which can
be biased up to À 50 kV, toward the Micromegas. This double-array
structure was adopted to reduce the non-uniformity of the electric
field. The required non-uniformity was less than 1% to ensure that
the resulting uncertainty of the position along the beam axis is
comparable or smaller than the spatial bin size of 2 mm in the radial

2.3. Field cage
The main frame of the field cage is made of a cylindrical tube
with a length of 50 cm and a diameter of 28 cm. A glass reinforced
epoxy laminate was chosen as the material because of its high

~2 cm

Fig. 2. Schematic view of the field cage with equipotential rings. For presentation purpose, only one pair of inner (purple and smaller) and outer (red and larger) rings are
shown. The equipotential rings are secured to the main frame by (a) the aluminum pin stands and (b) the supports of G10. The pin stands are arranged in a zigzag pattern
to increase the inter-stand distance and to reduce the risk of sparking. Resistors are installed along the dashed lines between neighboring stands to step down the potential
by voltage division. The stands bore through the G10 tube to put the inner and outer rings at the same potential. The joints between the stands and the tube are
hermetically sealed by epoxy resin. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Figure 4.11: Schematic view of the field cage with equipotential rings. For presentation
purposes, only one pair of inner (purple and smaller) and outer (red and larger) rings are
shown. The equipotential rings are secured to the main frame by (a) the aluminum pin stands
and (b) the supports of G10. The pin stands are arranged in a zigzag pattern to increase
the inter-stand distance and to reduce the risk of sparking. Resistors are installed along the
dashed lines between neighboring stands to step down the potential by voltage division. The
stands bore through the G10 tube to put the inner and outer rings at the same potential.
The joints between the stands and the tube are hermetically sealed by epoxy resin. Figure
from Ref. [27].
In order for the position uncertainty of the electrons that drift down the field parallel
to the beam axis to be at most 2 mm radially, or the spatial bin size of the Micromegas
anode strips described below in Section 4.2.4, the field non-uniformity must be below 1%.
Mounting the rings in the double-array structure described above helps to reduce to field
non-uniformity to the required 1% [27].
Starting from the cathode, the electric potential is stepped down the rings via voltage
division with a series of resistors. Each ring is connected to its neighbors by a 20 MΩ

62

resistor. The first ring is also connected to the cathode by a 20 MΩ resistor. The resistors
are connected across the pin stands on the outside of the field cage. They are represented by
dashed lines in Fig. 4.11a. Each pair of interior and exterior rings for a given pin stand are
thus held at the same potential. Note that the pin stands are arranged in a zigzag pattern
to lower the sparking risk between stands. The last downstream ring is connected to the
exterior chamber with a terminating resistor. The resistance of the terminator is chosen
such that the electric field remains uniform all the way to the micromesh of the Micromegas,
which is held at a negative potential [27]. For this work, the cathode was held at -40 kV and
the micromesh at -245 V, and a terminator with a resistance of 25.5 MΩ was used.
Garfield code [61] was used to simulate the electric field inside of the device [27]. The
code is limited to simulations of two-dimensional geometries, and therefore an infinite box
with a cross-section equivalent to that of the exterior chamber was used as an approximation.
Further, the rings and cathode plate were approximated in the simulation as an assembly
of infinitely long wires. Fig. 4.12 shows the equipotential surfaces produced by the code.
These surfaces reveal a very uniform field inside of the interior rings, located near a radius
of 13 cm in the figure. The uniformity is achieved by the screening of electric field lines from
the grounded exterior chamber at about 20 cm in radius by the outer and inner rings [27].

4.2.3

Cathode

An important feature of the prototype AT-TPC is its 1 kV/cm electric field. Such a high
field both limits the dead time of the detector, results in fast responses from it, and limits
the longitudinal and lateral diffusion straggling. To generate this field, a negative potential
of up to -50 kV is held on the cathode of the prototype AT-TPC and then stepped down the

63

in a zigzag pattern to maintain a larger distance. The voltage
divider circuit formed along the ring array is terminated by a
resistor between the most downstream ring and the vacuum
chamber. The resistance of the terminator is chosen in a way that

drift velocity of a fe
atmospheric pressu
ode voltage of À 1
100 cm and À 50 kV
insulation of such
diameter of the vac
70 cm by the bore
the detector assemb
therefore paramoun
TPC. Special care w
Prototype AT-TPC.

6

Drift velocity [cm/µs]

5
4
3
2
1
0

0

0.2

Fig. 3. Electric field simulations by the Garfield code [24]. The cylindrical detector
volume
is approximated
by anbyinfinite
box withcode
the [61].
same The
cross-sectional
Figure 4.12:
Electric
field simulations
the Garfield
cylindrical detector
Fig. 4. Simulated electr
geometry. Equipotential surfaces are plotted by the solid lines as a function of
volume is approximated by an infinite box with the same cross-sectional geometry. Equipomixtures (solid line) as a
the radius with respect to the symmetry axis (Z). The cathode plate is located at Z
tential surfaces
are
plotted
by
the
solid
lines
as
a
function
of
the
radius
with
respect
to the
the
Magboltz code [31].
¼ 50 cm.

symmetry axis (Z). The cathode plate is located at Z = 50 cm. Along the horizontal radius
axis, the inner rings are located near 13 cm, the outer rings near 15 cm, and the exterior
chamber wall near 20 cm. Figure from Ref. [27].

equipotential rings [27], as described above. As mentioned previously, the cathode plate is
glued on the upstream flange of the field cage.
Even though the insulating gas between the field cage and exterior chamber has a dielectric strength that is much greater than the voltage on the cathode, sharp edges or other
pointed structures of conducting material can greatly increase the field gradient, causing a
large non-uniformity in the field. Since the cathode plate itself has sharp edges, the design

64

must limit its contribution to field non-uniformity. Thus, a corona ring is used, shown in
Fig. 4.8. Built out of aluminum by Ross Engineering, Inc., the corona ring has a diameter
of 30 cm and is toroidal in shape, hence its name. Part of the toroid is cut away such that
the ring encloses the edges of the cathode plate, providing better shielding of the edges. The
cross-sectional diameter of the ring is 3.8 cm; a diameter this large ensures that the electric
field on the corona ring surface is below the discharge threshold of the insulation gas. The
ring’s maximum voltage is rated at 130 kV in air [27].
A cathode voltage feedthrough mounted on the upstream flange of the exterior chamber
allows for the voltage to be fed to the cathode from a power supply, as seen in Fig. 4.8. The
feedthrough is custom-made and based on the feedthrough used for the beam deflector of
the NSCL’s K500 cyclotron. It has a maximum 80 kV rating. The feedthrough consists of a
hollow cylinder made of machined Macor, an insulating ceramic material. The radius of the
surface of the cylinder varies in thickness from 3 to 6 mm in an alternating wavy pattern.
The cylinder guides the power cable of the voltage supply to a conducting tip located at
the end of the cylinder. Via a conducting spring, this tip is connected to a 3.8 cm wide
aluminum sphere that is in contact with the corona ring, which is held at the same potential
as the cathode plate [27].
In order to allow beam particles to enter the field cage, a beam duct is located between
the upstream flange of the exterior chamber and the cathode plate. The duct is made of
alumina ceramic tubing and is 93 mm long with a 64 mm inner diameter and is built from a
CeramTec vacuum isolator. It has a 60 kV maximum voltage rating. The entrance window is
a hole 7.6 mm in diameter which is centered on an aluminum plate of thickness 12.7 mm that
is located at the interface of the duct and cathode plate. The duct itself if hermetically sealed
from all other volumes of the device [27]. Over the entrance window, epoxy resin is used
65

the detector assembly. Efficient insulation of the cathode voltage is
therefore paramount to maximize the active area of the future ATTPC. Special care was given to the isolation of the cathode of the
Prototype AT-TPC. The four components described in Sections

6

P10
He:CO2 90:10

Drift velocity [cm/µs]

5
4
3
2
1
0

The cylindrical detector
same cross-sectional
lines as a function of
ode plate is located
at Z
Figure

0

0.2

0.4

0.6
0.8
1
Electric field [kV/cm/atm]

1.2

1.4

Fig. 4. Simulated electron drift velocities for P10 (dotted line) and He:CO2 90:10
mixtures (solid line) as a function of electric field. The simulation was made using
4.13:
Simulated electron drift velocities for P10 (dotted line) and He:CO2
the Magboltz code [31].

90:10
mixtures (solid line) as a function of electric field. The simulation was made using the
Magboltz code [62]. Figure from Ref. [27].
to glue a thin foil of para-aramid, or Mictron, produced by Toray Industries, Inc. Mictron
has a density of 1.45 g/cm3 . The specific foil used as the beam window has a thickness of
0.536 mg/cm2 , or 3.7 µm.
To use the prototype AT-TPC for a wide variety of nuclear reactions studies, many
different types of gases must be available to use as active target gases. In order to achieve
this versatility in gas choice, the high electric field capability of the prototype AT-TPC is
vital due to the sometimes radically different electron drift velocities of various gases. Lower
drift velocities cause a longer detector dead time, limiting the beam particle rate that can
be delivered to the device.
Magboltz [62], a code used to simulate the transport of electrons in gas mixtures, is used
to compare the drift velocities of electrons in both He:CO2 90:10 and Ar:CH4 , or P10, gas
66

mixtures. The comparison is shown in Fig. 4.13. In the figure, one sees a sharp peak at low
electric fields in the P10 simulation, while for He:CO2 90:10 a smooth rise in drift velocity as
a function of electric field is seen. Near 0.2 kV/cm/atm, the drift velocity in P10 is about an
order of magnitude higher than that in He:CO2 90:10. This electric field value is the typical
operating voltage of many large scale TPCs [63]. At this value, for a given drift length of
100 cm, a drift velocity of 0.5 cm/µs results in a 200 µs maximum dead time. Such a dead
time, coupled with a beam particle rate of 1 kHz, results in a pile up ratio of 20% [27].
In order to avoid such a large dead time scenario, an electric field that is higher than the
typical value of 0.2 kV/cm must be used. By utilizing a field of 1 kV/cm at atmospheric
pressure, the electron drift velocity in He:CO2 90:10 is near 2.5 cm/µs. To achieve this field
across the 50 cm length of the field cage, a voltage of -50 kV is applied to the cathode [27].
As described in the preceding sections, insulating gas and the corona ring ensure that
there is minimal risk for sparking. Tests were performed to rate the maximum voltage on
the cathode. The active target gas was He:CO2 90:10 with an insulating nitrogen gas, both
held at atmospheric pressure. The tests were made with a 880PL-10 mA-A unit, a portable
DC insulation tester used to supply power to the cathode. The tester was manufactured by
Hipotronics. The voltage was increased up to -60 kV before any internal sparking occurred.
This -60 kV value corresponds to the maximum voltage rating of the beam duct, and is 20%
larger than the nominal -50 kV operating voltage of the cathode. Thus, the safety margin
over the maximum operating voltage of the cathode is 20% [27].
The nitrogen insulation gas has a dielectric strength of 34 kV/cm/atm [60]. At 1 atm
and with an approximately 6 cm separation between the field cage wall and exterior chamber
wall, the insulation gas has a strength of about 200 kV, well above the 50 kV rating of the
cathode [27].
67

4.2.4

Micromegas

A basic description of the operating principles of the Micromegas electron amplification
device is found in Section 2.2.3. This section will describe more details about the specific
Micromegas used in this work.
The Micromegas is mounted on the downstream end of the field cage. The two basic
components of the device are the anode readout plane and the micromesh. Made of 18 µm
diameter SDC 45/18 stainless steel wire manufactured by Bopp, the wire is woven with a
63 µm pitch into a mesh plane. Insulated pillars 0.3 mm in diameter are arranged in a 2 mm
by 2 mm grid pattern on the anode readout plane to support the mesh at a uniform 128 µm
distance above the plane [27]. A field of about 50 kV/cm is held over this distance, which
serves as the amplification gap for the electrons.
The anode readout plane is made of printed circuit board (PCB) and was designed
and fabricated at the NSCL. The electrodes for the anode consist of copper that has been
electroplated with 3 µm of nickel and 1 µm of gold via immersion. This electroplating
was done to prevent oxidation damage of the surface of the electrodes and to provide a
smooth surface [27]. The assembly of the anode PCB and the micromesh was performed at
SEDI/IRFU CEA-Saclay using bulk technology [64].
The anode plane’s layout was designed to optimize resolution for kinematical reconstruction of binary events. The anode plane is circular and centered on the beam axis. The active
area of the plane is 250 mm in diameter. This area is divided into individual readout channels, referred to as pads. The segmentation of the pads is radially symmetric from the center
to optimize radial resolution. The central pad is circular with a 5 mm diameter. All other
pads are coaxial in shape and 2 mm wide, and are spaced with a 0.25 mm gap. The coaxial

68

pads are separated into four quadrants, with the five inner-most pads further separated into
eight octants for improved resolution near the beam axis. Each quadrant is comprised of 63
pads, resulting in 253 pads in total [27].
The pads are routed to three surface mount connectors that are soldered on the back
of the PCB. Two of the connectors have 100 inputs, and the third has 60, enough in total
for all 253 pads. The three connectors couple to the feedthrough on the endplate where the
Micromegas is mounted. If so desired, an adapter can be inserted between the endplate and
the Micromegas to adjust the placement of the Micromegas relative to the field cage. For
this work, the Micromegas was located about 5 mm downstream from the last field cage ring
electrode. Six threaded stand-offs are mounted on the back of the PCB for mounting to the
endplate. The endplate has six holes drilled into it that are aligned with the stand-offs. The
Micromegas are the held to the endplate with six self-sealing screws that are inserted from
the backside of the endplate. The endplate’s KF40 flange contains an SHV feedthrough that
provides a voltage to the micromesh [27].
As first shown in Section 2.2.3, Fig. 2.5 is a photograph of the Micromegas used in this
work, while Fig. 2.6 illustrates the tracking detection technique of the Micromegas.
The beam particles enter the field cage perpendicular to the Micromegas. As a

10

Be

beam particle traverses the active target, the central pad and some of the inner coaxial
pads record the energy deposit profile of the particle in time, from which the position of the
particle along the beam axis (z) can be calculated given the 24.0 mm/µs drift velocity of the
electrons in the active target gas. After an elastic scattering reaction, the outgoing 10 Be and
4

He particles travel over multiple coaxial pads in opposite quadrants as they leave the beam

axis. For each particle, the strips over which it travels give the energy deposit profile as a
function of radius r from the center in addition to z, from which the polar angle and total
69

range of the particle can be calculated. The azimuthal angle cannot be calculated with this
design of Micromegas, but for binary reactions, the two products have opposite azimuthal
angles, ensuring that they are recorded in opposite quadrants and thus have separable energy
deposit profiles [27].

4.2.5

Readout Electronics

The schematic of the electric circuit used for the signal readout of the Micromegas is given
in Fig. 4.14. The circuit handles both the anode pad signals and the micromesh signal.
The micromesh signal is used as a trigger for reading out the pad signals. The pad signals
are digitized and the waveforms of each of the 253 pads (252 separated into quadrants
and one central pad) are individually recorded. The electronics includes a 511 channel
sampling capacitor array for analog memory, an analog-to-digital converter (ADC), and
zero-suppresion [27].
A DC power supply holds a negative bias voltage on the micromesh, and a 100 MΩ
resistor is included to limit discharge currents that may occur in the Micromegas. A 2001A
Canberra charge-sensitive preamplifier is used, with a 20 nF decoupling capacitor, to amplify
the signal of the collected charge on the micromesh [27].
One contribution to the signals in the Micromegas is cross-talk due to the capacitive
coupling of the pads. The pads are coupled because the micromesh spans the entire anode
plane. Thus, when one pad creates a signal due to an electron avalanche between it and the
mesh, reverse polarity pulses are induced on the other pads. The induced pulses can distort
the signals on the nearby pads if they also have avalanche signals simultaneously with other
pads. The cross-talk pulse amplitude is proportional to the ratio of the capacitance Cpad-mesh
and the total capacitance Cmesh-ground between the mesh and ground. This amplitude is
70

D. Suzuki et al. / Nuclear Instruments and Methods in Physics Research A 691 (2012) 39–54

45

Fig. 7. Schematic diagram of the readout circuit for the Micromegas. For simplicity, only one pad is shown for each quadrant (Q1.a, Q2.b, Q3.c and Q4.d). The pad signal
Q#.X (# ¼Quadrant ID, X¼Pad ID) can be routed to the T2K electronics, the quadrant output ’Q#’ and the central output ‘Central’, which can be selected by the jumpers A.
Signals can be split and sent to multiple outputs by installing resistive jumpers. The potential of individual pads can be selected to be ground, or a different choice of high
voltages HV1 or HV2 by the jumpers B.

Figure 4.14: Schematic diagram of the readout circuit for the Micromegas. For simplicity,
only one pad is shown for each quadrant (Q1.a, Q2.b, Q3.c and Q4.d). The pad signal Q#.X
feedthrough
of the endplate interfaces
the readout system
10
inserted in parallel
toX
C = Padincreases
C
and
(# decreases
=nFQuadrant
ID,
ID) can
be routed
toThe
the
T2K electronics,
the quadrant
output
and the Micromegas plugged in to the opposite end. The input of
the C
=C
ratio, thus reducing the effect
‘Q#’
and
the cross-talk.
centralTheoutput
‘Central,’
which the
can
selected
the jumpers
A.endplate
Signals
FECbe
is connected
to theby
feedthrough
of the readout
of the
capacitive
capacitance
value is determined
via an adapter board, referred to as ZAP. The FEC and the ZAP
a compromise between reduction of the cross-talk and the risk
canby
splitby and
sent
tostored
multiple
outputs
by installing
resistive
potential
boards are sandwiched
by ajumpers.
pair of copper The
plates to
shield these of
ofbe
damage
discharge
of the
energy, which
increases
boards
from
electromagnetic
interference
and
tightly
with
C
.
individual pads can be selected to be ground, or a different choice of high voltages connect
HV1 or
the ground of the FEC to that of the enclosure of the Prototype
HV2 by the jumpers B. Figure from Ref. [27]. AT-TPC.
mesh2anode

pad2pad

mesh2ground

mesh2ground

mesh2ground

The ZAP board was developed at the NSCL. The role of this
board is twofold. The first is to have the ability to put a unique
2.6.2. Anode circuit
bias on individual pads. The gain of a pad is governed by the
The signals of the anode pads are read out by the electronics
potential difference between the given pad and the micromesh.
[28] developed for TPCs of the Tokai-to-Kamioka (T2K) experiThus, we can have different gains for different pad positions by
ment for neutrino oscillation measurements [29]. The T2K elecincreasing or decreasing the pad potential individually [22]. In
tronics consists of the Front End Card (FEC) for pulse shaping and
nuclear reactions, there can be a large difference between the
digitization,
and the Front-end Mezzanine Card (FEM) for data
mesh-ground
mesh-anode
proton number (Z) of the beam and the target. In direct reactions
transfer. The FEC contains four ASICs (application-specific intein inverse kinematics, for instance, the beam nucleus is usually
grated circuits) of 72 inputs each. Each channel has a preamplifier,
much heavier than the target nucleus. The higher Z beam particle
a shaping amplifier and an analog memory to record waveforms
mesh-anode
loses a larger amount of energy than the recoiling target ion over
of input signals. The memory is based on the Switching Capacitor
a given distance. If a uniform gain is set over the entire anode
Array (SCA) consisting of 511 capacitor cells. Each capacitor cell
plane, a very large dynamic range is required for the readout
stores a charge proportional to the amplitude of the input signal
electronics. Furthermore the high charge density induced by the
within a time window programmable from 0.02 to 1 ms. The
beam increases the possibility of discharges. Biasing individual
ensemble of the capacitor cells thus stores waveforms over a time
pads allows one to lower the gain in the beam region while
range of 10–511 ms. The SCA serves as a circular buffer in which
mesh-ground
retaining a higher gain in the outer region, which allows simultathe sampling
resumes from the first buffer cell after one sampling
neous recording of both beam and recoil particles with no
cycle ends with the 511th cell. Once an external trigger is
saturation of the electronics and less risk of discharges. Each
received by the FEM, the SCA stops the sampling and sequentially
pad has three possibilities of bias voltages, selectable via a shunt
sends the charges stored in the buffer cells to a 12-bit analog-tojumper labeled (B) in Fig. 7. The second role of the ZAP board is to
digital converter on the FEC.

minimized by increasing the detector capacitance relative to ground. The largest contributor
to C

value of C

is typically the capacitance C

between the mesh and anode plane. The

, when modeled in vacuum with a parallel plate configuration, is about

4 nF. Therefore, an additional 10 nF capacitor is added between the mesh and ground to
increase C

and thus decrease cross-talk [26]. The value of the capacitor was limited

to 10 nF to reduce the risk of damage by discharge of the energy stored in the capacitor and
to preserve a good quality signal of the mesh preamplifier [27].

71

The readout electronics [65] used were originally developed for TPCs used in the Tokai-toKamioka (T2K) experiment for neutrino oscillation measurements [66]. The Front End Card
(FEC) handles pulse shaping and digitization, and data transfer is done with the Front-End
Mezzanine card (FEM). Four Application-Specific Integrated Circuits (ASICs), each with 72
inputs, are located on the FEC. A preamplifier, shaping amplifier, and analog memory for
recording input signals and waveforms is included for each pad signal channel. A Sampling
Capacitor Array (SCA) built of 511 capacitor cells is used for the analog memory storage.
The charge stored in each cell is proportional to the input amplitude. The time window over
which the signal is stored in the SCA is programmable to between 0.02 and 1 µs, resulting
in a total time range of 10 to 511 µs . Acting as a circular buffer, the SCA samples the
input signal over its 511 cells. When the FEM receives an external trigger, the writing in
the SCA is stopped. After a read order, the stored charges are sent in sequence to the FEC,
where they are converted to a digital number by a 12-bit ADC [27]. A diagram depicting
the actions of the SCA is given in Fig. 4.15.
An adapter board, called the ZAP board [27], is used to connect the FEC to the
feedthrough from the Micromegas pads located on the back of the endplate of the prototype AT-TPC. The ZAP board and FEC are shielded form electromagnetic interference
by two copper plates. The plates also ensure that the FEC is grounded to the exterior of
the prototype AT-TPC. A photo of the ZAP board is shown in Fig. 4.16.
The FEM is mounted on the back of the FEC and is unshielded. The fully mounted
electronics setup used in this work is shown in Fig. 4.17.
Developed at NSCL, the ZAP board serves two purposes: (1) it allows for each pad to
have a specific gain and (2) it relays signals from the pads to the FEC for generating a

72

Stop
(trigger)

ci+1 511
ci
0

Write phase

AFTER-like readout

Readout
phase
Write phase:
Tdrift ≤ 512 / Fsampling
Read phase:
trigger [27]. In this work inSCAcells
particular, the gain
= variation
512 feature of the ZAP was necessary
Figure 4.15: Diagram of the writing and readout of the Sampling Capacitor Array (SCA).
The SCA is continuously filled in a circular manner. When a trigger signal is received, all
511 cells are read out. Figure from P. Baron.

to trigger binary events.
Gain variation among pads is achieved in the ZAP board by applying specific biases to
individual pads. The gain is a function of the potential difference between the pad and the
micromesh [26]. On the ZAP board, a shunt jumper for each pad is used to select one of
three possible bias voltages. These jumpers are labeled Jumper (B) in Fig. 4.14. A data plot
showing the relationship between gain and the applied voltage on the Micromesh is given in
Fig. 4.18.
The utility of gain variation comes from the difference in the energy deposit profile of
nuclei with different atomic number Z. As described by the Bethe formula in Eq. 2.1, the
energy loss of charged particles in a gas is proportional to Z 2 . Considering the nuclei studied
in this work, a

10

Be particle will deposit four times the energy per unit length as 4 He at
73

P.B

Figure 4.16: A photo of the ZAP board before mounting.
the same velocity. There are two scenarios for keeping a uniform gain across all pads in this
case, and both are limited by the finite dynamic range of the readout electronics. One is that
the energy deposit profiles of

10

Be particles are successfully recorded. However, this would

result in the 4 He signal being in the noise, and thus not recordable. On the other hand, if
the uniform gain is increased to allow for the 4 He energy deposit profiles to be measured,
the pads over which the 10 Be particles traverse would become highly saturated, resulting not
only in an unreliable measure of deposited energy on a given pad, but also in large cross-talk
between neighboring pads and channels, rendering the

10

Be pad signals essentially useless.

The solution to this gain dilemma is to vary the gain on specific pads. The majority of
the pads are kept at a gain suitable for recording the energy deposit profile of

10

Be. Every

fifth pad from the center, however, has an increased gain in order to record the energy
deposit profile of 4 He particles, allowing for binary events to be triggered. A more detailed
description of the triggering scheme is found in Section 4.3 later in this chapter.
The second purpose of the ZAP board is to route signals from the pads to the electronics.
There are three options for signal routing [27]:

74

Figure 4.17: A photo of the shielded ZAP board and FEC as mounted on the prototype
AT-TPC. The unshielded FEM is perpendicularly mounted on the back of the FEC.
1. Output each individual signal to the T2K electronics.

2. Output Q1, Q2, Q3, and Q4, each corresponding to one quadrant. The pad signals are
therefore separated only by which quadrant they are found in. These quadrant output
signals can then be used as a condition for triggering on product nuclei as they travel
through the TPC away from the beam axis.

75

uclear Instruments and Methods in Physics Research A 691 (2012) 39–54

10000

nics for generating

abeled (A) in Fig. 7.
divide and route a

Gain

drant. The pads are
ants. This path is
radius of 2 cm or
l can be used as a
m the beam axis.
all the pads, thus
ds independent of
m from the center,

1000

0.25 atm

0.5 atm

1 atm

100

10

1
150

200

250
300
350
400
Micromesh voltage [-V]

450

500

Fig. 8. Electron amplification gain of the Micromegas as a function of micromesh
voltage.
The data
were measured
in He:CO
mixtures atas
0.25
0.5 micromesh
2 90:10
4.18:
Electron
amplification
gain
of the
Micromegas
a (triangles),
function of
(boxes), and 1 atm (circles).

nd transferredFigure
to a
uter are interfaced
voltage. The data were measured in He:CO2 90:10 mixtures at 0.25 atm (blue triangles),
d on the ML507
0.5 atm (green boxes), and 1 atm (orange circles). Figure from Ref. [27].
h a duplex optical
e latter. The optical
respectively. At all three pressures, some of the alpha particles
3. Output ‘Central.’ This option sums the signals of all pads set to ‘Central’ and is only
-end components
deposit all the kinetic energy on the active area of the Microthe DCC and the available
megas,
corresponding
amplitude
of the
signalsis thus used
to pads
within 2 cmtoofthe
the maximum
center of the
Micromegas.
‘Central’
The DCC controls
observed. The primary charge (Q1) created by the energy deposidata to reduce the to record
tion of
MeVdeposit
is estimated
fC. Here we assume a mean
the5.5
energy
profile to
of be
the29
beam.
ionization energy (W-value) of 30 eV, which is the typical value
ctly coupled to the
for helium mixtures with a small ratio of a quencher [30]. The
The
three
are selected
the use of a shunt jumper, labeled as Jumper
as used for offline
gainoutput
(Gmes )options
was deduced
as Gby
mes ¼ Q 2 =Q 1 . The results shown in
acquisition rate of
Fig. 8 clearly demonstrate that the gain has an exponential
(A)
in
Fig.
4.14. If the jumper
used is resistive,
the signal
can be divided
andatthus routed
ch is not suited for
dependence
as a function
of micromesh
voltage.
The data
e FEM-DCC system,
1 atm are consistent with the results of our previous study [22], in
to multiple output options simultaneously [27]. The exact options chosen for this work are
in Section 4.1, was
which we tested a Micromegas detector with the same anode–
micromesh
gap size as the present Micromegas, but with a much
described in
Section 4.3.
shorter drift distance (3 cm) compared to the present setup
(50then
cm).digitizes
This indicates
there the
is read
no significant
absorption
of to a comThe FEC
the data during
phase and transfers
the data
241
Am
electrons over the distance of 50 cm in the field cage of the
puter for collection
through
the FEM. The FEM outputs the data to the Data Concentrator
Prototype
AT-TPC.
Note that the maximum gain in Fig. 8 was limited by the
Card (DCC)
via an optical
fiber.
fiber decouples
theThe
shared
of theofFEC, FEM,
dynamic
range of
the The
preamplifier
output.
real ground
upper limit
as characterized at
the gain was governed by discharge, which induced a continuous
and
AT-TPC
from
DCC and
to isolate
analog electronics
a standard 241
Amprototype
current
in both
thethe
cathode
and the
the computer
micromesh.
Once athe
discharge
cular plastic disk of
took place, the current remained until the power supplies were
from
potential
sources of noise. The DDC is based on the ML507 Evaluation Platform of
as centered on the
turned off. We presume that the discharges are triggered by
ve diameter of the
ultraviolet rays emitted from electron avalanches in the Micro76
d by a 50 mm thick
megas. When these ultraviolet rays hit metallic parts, such as the
holes in order to
cathode or voltage divider electrodes, they produce electrons
counts per second.
through the photo-electric effect. The photo-electrons are trans-

Xilinx, Inc., and interfaces the FEM with the computer via an ethernet cable [27]. The
collection of data in this work is further described in Section 4.4.
The parameter settings used for the T2K electronics in this work include a 0.5 µs peaking
time, a 40 µs time window, and a gain of 360 fC over all 4,000 digitalized channels.

4.3

Experimental Setup

During the experiment, the prototype AT-TPC was installed in the vault beyond TwinSol,
as shown in Fig. 4.1. A schematic of the prototype AT-TPC including the Micromegas used
in this work is found in Fig. 4.19.

Figure 4.19: Schematic drawing of the experimental setup of the prototype AT-TPC. The
inset is a magnified view of the segmented anode pad plane of the Micromegas detector near
the beam axis. Figure adapted from Ref. [9].

77

4.3.1

MCP and Beam Ladder

As mentioned in Section 4.1.3, a beam steerer is located after TwinSol. A high vacuum
ISO160 tee, shown in Fig. 4.20 was placed after the beam steerer. This tee houses a MicroChannel Plate (MCP) detector and a beam ladder, on which various items may be mounted.
An MCP is a fast amplification device. It consists of a bundle of glass tubes with
diameters and spacing on the order of a few microns. The upper and lower sides are plated
with Ni. The plate is typically held at a voltage of 500 to 1000 V. Each channel acts like a
miniature photomultiplier tube (PM tube [41]). When a particle, such as an electron, enters
one of the channels of the MCP, a cascade of electrons is formed, amplifying the signal. The
MCP used in this work is made of Ni-Cr and was built by tectra GmbH. Its model number
is MCP-33-D-R-A. The design employs two identical MCPs. The channels in each plate are
angled at 12◦ ± 1◦ , and the plates are stacked in a Chevron style such that the channels
of one plate are rotated 180◦ with respect to the channels of the other plate, forming a ‘V’

Figure 4.20: Image of the ISO160 tee with the retracting beam ladder. The MCP is mounted
on the top flange and is install inside of the tee. The black beam steerer can be seen located
a few inches upstream of the tee.

78

Figure 4.21: A side view of the copper plates and resistor chain on the MCP housing. The
MCP and foil are both located on the green PCB on the top of the chain.
shape. This stacking method is chosen to prevent positive ions from traveling back through
the plates. The channels have a diameter of 12 ± 0.5 µm and a pitch of 14.5 ± 0.5 µm. The
plates are 33 mm in diameter with a 25 mm active diameter. When biased to 2 kV, the pair

Due to the geometry of the flange, the last few ma
plates of the housing had to be moved about a cent
As part of the setup of the Protoype AT-TPC, the MCP is used as a beam diagnostic
locations. I do not expect this movement to greatly a
tool. Its role is to measure the incoming beam rate into the AT-TPC and to output a trigger
furthest from the target, but one can check the fie
signal for the incoming beam particles.
efficiency seems low during testing. The field sho
The beam is measured by the MCP without being intercepted. A double-sided-aluminized
possible.
polypropylene foil of thickness 70 µg/cm is placed in the path of the beam and held at a
• Any materials not located in the experimental area
internal cables and spare resistors, are stored in boxes
clean room in the 79detector lab. All necessary mate
conducted at NSCL should be located in these boxes a
•

has a gain of about 5 × 106 .

2

Figure 4.22: Image of the MCP mounted on its housing and flange. Inside of the red rectangle
are two of the small permanent magnets mounted on one of the soft iron walls. The orange
rectangle shows the location of the back of the MCP anode, while the black rectangle below
identifies the double-sided-aluminized polypropylene foil.
negative potential of −2400 V. As the beam particles pass through the foil, they lose negligible energy in producing secondary electrons off of the thin foil before continuing onward into
the prototype AT-TPC. A 0.5 kV/cm electric field is established parallel to the beam axis,
and the electrons freed from the foil are accelerated in this field. Superimposed is a perpen80

Figure 4.23: The MCP installed in the ISO160 tee, slightly downstream from the beam
ladder. Behind the tee, iron and steel plates are mounted to minimize the effects of stray
fields from the beam steerer on electron traveling from the foil to the MCP. The foil can be
seen inside of the magenta square.
dicular 65 G magnetic field. This E × B field transports the electrons a lateral distance of
4.5 cm to the MCP. The upper side of the MCP is at −2300 V, and the electrons therefore
have an energy of 100 eV when hitting the MCP. This energy corresponds to the maximum
detection efficiency of the MCP for electrons. The electric field is created by stepping down
2.4 kV over twelve copper plates connected in series with a chain of 10 MΩ resistors, seen
in Fig. 4.21. The plates are shaped such that there is an open path along the beam axis
and for also an open path for the electrons to be transported from the foil to the MCP. The
magnetic field is made by placing twenty small permanent magnets on the soft iron outer
walls of the housing, creating a uniform magnetic field between the two walls. While the
81

electric and magnetic fields are of proper strength to accelerate and bend the electrons to
the MCP, they have a negligible effect on the nearly 40 MeV beam particles. The assembled
housing as used in this work is shown in Fig. 4.22, and Fig. 4.23 shows it installed in the
ISO160 tee.
Once the electrons reach the MCP, they are amplified and the signal is sent to a threestep cascade fast amplifier. From the amplifier, the signal is sent through a leading edge
discriminator and then a gate and delay generator, where a 10 µs window is set. This
window, in principle, acts as a trigger window for reactions in the prototype AT-TPC. For
this experiment, however, the MCP signal could not be used as trigger. The beam rate
through the MCP was about 10 kpps (kilo-particles per second), while the beam rate inside
of the TPC, as measured by the mesh signal (described later) was only 1 kpps, an order of
magnitude less. This difference in rate is due to the large emittance of the secondary beam;
most beam particles that pass the MCP were not focused in the small 7 mm diameter beam
window of the prototype AT-TPC.
While the MCP did not give the rate of the beam delivered to the prototype AT-TPC,
the signal was used to determine how much beam was produced from TwinSol. In other
experiments where the beam had smaller emittance, the MCP signal was used as trigger.
In addition to the MCP, the ISO160 tee housed the beam ladder, a retractable arm
on which various items can be mounted. The ladder was manufactured by Kurt J. Lesker
Company, Part# KLSDDA, Screw Drive KLSD Series (Bellows Sealed), 6” travel, CF1.33”.
To accommodate the ladder, a KF40 sleeve was welded onto the ISO160 tee, perpendicular
to the beam axis and parallel to the floor. During this experiment, a pair of Si detectors, a
228

Th 3 µCi α source, and an iris collimator were mounted on the ladder, as seen in Fig. 4.24.

82

Figure 4.24: The beam ladder installed in the sleeve of the ISO160 tee. On it are mounted,
from left, a stacked pair of Si detectors, a 228 Th 3 µCi α source, and an iris collimator.
The Si detectors are stacked on top of each other, one on each side of the beam ladder.
The detector further upstream acts as a ∆E detector, with the one downstream acting as an
E detector. Both detectors have an active area of 450 mm2 and are Ortec models. The ∆E
detector is model 43-017J and is 50 µm thick, while the E detector is model 31-493A with
a 1000 µm thickness. Together, the pair is used for beam diagnostics along with another
detector, described in detail in Section 4.3.2. A detailed description of Si detectors can be
found in Ref. [41].
A 3 µCi 228 Th α source was installed on the beam ladder for testing the detection efficiency
of the MCP. In the testing, the α particles travel through the foil mounted on the MCP
housing and are detected by a Si detector mounted on the back of the housing. The number
of events registered by the MCP is compared with that of the Si detector, from which an
83

efficiency is calculated. The Si detector used was model TB-24-450-300-5. The tests yielded
a detection efficiency of between 80% and 85% for the MCP. Stray fields from the beam
steerer can alter the paths of electrons to the MCP, lowering the device’s efficiency. Iron
and steel plates placed between the ISO160 tee and beam steerer minimize the stray fields,
alleviating this issue. Some of the plates can be seen clamped behind the tee in Fig. 4.23.
The last item on the beam ladder was an iris collimator, or iris diaphragm. Model #53911 from Edmund Optics, the diaphragm is made of stainless steel and has an outer diameter
of 37.00 mm. Its maximum aperture is 25.00 mm in diameter, and its minimum aperture is
0.90 mm in diameter. The purpose of the iris is to use its adjustable aperture to collimate
the beam before the MCP and prototype AT-TPC if needed. For this experiment, the iris
did not prove necessary.
The three items on the ladder were positioned on the ladder as follows, with respect to
the distance settings of the ladder: 136.84 mm to the center of the Si detectors, 99.3 mm to
the center of the α source, and 56.97 mm to the center of the iris. Thus, when the ladder was
retracted to 136.84 mm, the Si detectors were centered on the beam axis, and similarly for
the α source and iris. When the ladder was set to 17.00 mm, there were 35 mm of clearance
around the beam axis, which is larger than the 33 mm foil on the MCP housing. Therefore,
when the beam is delivered to the prototype AT-TPC, the ladder was set anywhere between
0 and 17 mm to clear the beam passage.
The MCP and beam ladder were both aligned on the beam axis with the use of a telescope
permanently installed in the vault where the experiment was held. The crosshairs of the
telescope are centered on the beam axis, and adjustment knobs on the telescope are available
to measure how far off axis an object is both vertically and horizontally. For example, by
viewing the foil on the MCP housing through the telescope, its relation to the beam axis
84

can be seen and measured. The MCP mounted is then adjusted until it is on axis. This
measurement and adjustment method is done iteratively until the foil is aligned to a precision
of about 1 mm from the axis center. The same method is done with the beam ladder. The
alignment of the prototype AT-TPC is similar and will be discussed in more detail later.

4.3.2

Position Sensitive Si Detector

Once the MCP and beam ladder were installed in the ISO160 tee, a Position Sensitive Si
Detector (PSSD) was temporarily placed after the tee for measuring the energy of beam
particles. The PSSD was placed at the location where the entrance window of the prototype
AT-TPC would later be located, about 36.8 cm downstream from the centering ring on the
downstream flange of the ISO160 tee. Thus, the PSSD provides a measure of the energy of
beam particles as they enter the prototype AT-TPC. Fig. 4.25 shows the installation of the
PSSD as used in this work.

Figure 4.25: The PSSD mounted past the ISO160 tee, at the eventual location of the beam
window of the prototype AT-TPC.

85

ΔE (arb. units)

10B

10Be

7Li
4He

E (arb. units)
Figure 4.26: Spectrum of E − ∆E, showing separation of isotopes in the beam. The relative
intensities of the isotopes somewhat change as beam conditions change.
The PSSD is 230 µm thick and was custom-built by the UC-Berkeley detector group. It
is stacked with a thinner Si detector used as a ∆E detector. The ∆E detector is an Ortec
24-008A model, and is 50 µm thick.
The position sensitivity of the PSSD was calibrated with the α source on the beam ladder.
An asymmetric mask was placed in front of the PSSD. The resulting image on the PSSD
provided the position determination of the device. Once calibrated, the mask was removed
and the α source retracted. A 10 mm-diameter circular collimator was then placed in front
of the PSSD, centered on the beam axis. Beam was then delivered to the PSSD to determine
its energy. The E − ∆E detectors provided separation of different isotopes in the beam, as
shown in Fig. 4.26. From the measurement of E − ∆E on the beam ladder, the energy of
10

Be was found to be between 38.765 and 38.799 MeV, while the E − ∆E with the PSSD was
86

39.543 MeV, resulting in an energy determination of 39.0+0.50
−0.25 MeV for

10

Be beam particles

incident on the beam window of the device. This value was found to be consistent with the
energy measured by the E − ∆E Si detectors mounted on the beam ladder, as well.
Due to energy loss in the beam window, the energy of the beam delivered inside of the
prototype AT-TPC is 38.2 ± 0.5 MeV after the beam window. Confirmation of this beam
energy via analysis of the length of particle tracks in the active target volume is described
in detail in Section 5.3.1.

4.3.3

Prototype AT-TPC

Once the energy calibration measurement was finished, the PSSD was removed from the
beam line, and the prototype AT-TPC was put in place. The TPC was mounted on an
adjustable table that was bolted to the floor for stability. A bellows was used to join the
ISO160 tee with the front flange of the prototype AT-TPC. The pitch of the TPC was leveled
by adjusting the vertical height of the four bolts that connect the device to the table. Once
leveled, the TPC was then aligned on the beam axis. An image of the TPC during the
alignment process is shown in Fig. 4.27. Alignment was done in an iterative process with
the beam telescope, similar to the alignment procedures for both the beam ladder and MCP,
described in Section 4.3.1. The TPC alignment was more difficult, however, as both the
entrance window at the front of the device and the center of the Micromegas located on the
back flange must be aligned.
During the alignment procedure, it was necessary to remove the back flange of the TPC
so that the beam window was visible. To keep dirt and debris from entering the chamber,
plastic wrap was placed over the open back flange. The beam window was then centered on
the beam axis. Once the beam window was centered, an alignment plate was installed on the
87

Figure 4.27: The alignment plate as mounted on the back of the prototype AT-TPC. Clear
plastic wrap is taped over the flange under the plate to keep dust and dirt out of the interior
of the device. The bolts that secure the TPC to the table and allow for height adjustment
are highlighted inside the red square. The bolts used to horizontally adjust the TPC on the
table are shown inside of the cyan oval. More such bolts are found on the opposite side of
the table.
back flange of the TPC. The center of the plate corresponds to the center of the Micromegas.
The plate was then centered on axis. The plate would then be removed to check that the
alignment of the entrance window was not lost. If necessary, this process would be repeated
until both the beam window and rear alignment plate were centered on the beam axis to
within a couple of mm. The final distances between various components of the TPC and
ISO160 tee as mounted are shown in Fig. 4.28.
Once aligned, the back flange on which the Micromegas were mounted was installed on
the prototype AT-TPC. The ZAP, FEC, and FEM were then installed on the back flange
and connected to the DDC and computer.

88

~47 mm
10Be

experiment
117 mm

~327 mm

267 mm
224 mm

Replaced with a bellows
595 mm

Figure 4.28: A schematic of various distances between the parts of the setup during the
experimental run.

4.3.4

Gases

In addition to mounting the Micromegas electronics, the He:CO2 90:10 active target gas
and nitrogen insulation gas must be also be setup and installed. Bottles of each gas were
connected to gas handlers, seen in Fig. 4.29, that controlled the pressure and flow of the
gases in the detector. The bottles of He:CO2 90:10 were pre-mixed. During the experiment,
the pressure of He:CO2 90:10 inside the field cage was 695.5 ± 0.1 torr, while the pressure
of the nitrogen surrounding the field cage was 754 ± 4 torr. These pressures, however, are
uncalibrated. From Weather Underground’s historical data, the sea level pressure during the
experiment was 30.05 inHg, or 760.3 torr. Accounting for the elevation of 211 m of Notre
Dame’s campus, the local atmospheric pressure becomes 743 torr. The ambient pressure

89

Figure 4.29: An image of the gas handlers used for flowing He:CO2 90:10 (top) and nitrogen
(bottom) as the active target and as buffer gas, respectively.

90

measured with the gauge for the inner volume at the beginning of the experiment was
699.8 torr. During the data collection it read 695.5 ± 0.1 torr. This difference of 4.3 torr
is then scaled by (1 + (743 - 699.8)/743) to give 4.5 torr below atmosphere, or about
738.5 ± 0.1 torr. This pressure and the recorded 88◦ F temperature inside the chamber give
a target gas density of 0.3113 kg/m3 .
The target gas was circulated to minimize the possibility of adverse effects on the Micromegas signal due to contaminant buildup in the gas. In contrast, the nitrogen gas wasn’t
circulated.
As described in Section 4.2.2, the two gases are connected with a bypass valve that breaks
when an elevated pressure differential is achieved. The purpose of this valve is to protect
the field cage in the scenario where the two gases achieve a pressure differential of about
0.1 atm, or about 76 torr.

4.4

Trigger and Data Collection

As the beam nuclei enter the chamber, many types of scattering and reactions can occur.
In order to select the events of interest, a trigger must be implemented. The design of the
trigger was constrained by the limited tolerance in trigger rate of the T2K electronics, which
is practically around 10 Hz to ensure high live time. For this experiment, the trigger relied
on signals from (1) the micromesh of the Micromegas to measure the beam and (2) the pads
of the Micromegas to measure reactions.
The signal from the micromesh is not segmented, and it is used in the trigger as a method
of determining when a beam particle enters the TPC. A discriminator after the micromesh
signal amplifier delivers a 4 µs gate.

91

Table 4.1: Table of trigger pad locations for each quadrant.

Quadrant
Q1
Q2
Q3
Q4

Trigger Pads [Distance from center]
6
6
6
6

[9.85
[9.85
[9.85
[9.85

mm]
mm]
mm]
mm]

7
7
7
7

[9.85
[9.85
[9.85
[9.85

mm]
mm]
mm]
mm]

15
15
12
15

[24.55
[24.55
[18.25
[24.55

mm]
mm]
mm]
mm]

Table 4.2: Table of high gain pads by quadrant.

Quadrant
Q1
Q2
Q3
Q4

High Gain Pads
15,
15,
12,
15,

20,
20,
20,
20,

25,
25,
25,
25,

30,
30,
30,
30,

35,
35,
35,
35,

40,
40,
40,
40,

45,
45,
45,
45,

50,
50,
50,
50,

55,
55,
55,
55,

60
60
60
60

The trigger scheme for the pad signals takes advantage of the binary nature of the reaction
and thus the quadrant design of the Micromegas. Since α + 10 Be elastic scattering is a binary
event, the quadrant the α particle traverses will always be directly opposite the quadrant
that the

10

Be traverses. The trigger logic therefore requires coincident signals from opposite

quadrants for an event to be identified as binary. The quadrant signals are from pads near
the center of the Micromegas, as described in Section 4.2.5 and seen in Fig. 4.14.
Table 4.1 lists the trigger pads used for each quadrant, and Fig 4.30 shows the layout of
the trigger pads in a schematic. For Q3, pad 12 is used instead of 15 due to a bad connection
of pad 15. Pad 12 was chosen as it was the nearest eligible trigger pad on the ZAP. In order
to trigger on the α particles, the gain on the outer-most trigger pads for each quadrant was
increased. The higher gain was achieved by biasing the pads to +100 V. The micromesh was
held at −245 V.

92

Figure 4.30: Schematic of the first 17 pads of a quadrant of the Micromegas, with trigger
pads in yellow. Pad 1 is common to all quadrants. One of the four quadrants uses trigger
pads 6, 7, and 12 as it has a bad connection to pad 15.

93

In addition to the outer-most trigger pads, other pads were also held at a high gain.
They are found in Table 4.2. Each high gain pad is spaced 10 mm from its nearest neighbor,
except for the first high gain pad in Q3. These pads cover nearly the entire active area of
the Micromegas, making the tracking of α particles possible.
If there is a coincidence between Q1 and Q3 or Q2 and Q4 during the gate of the mesh
trigger, an Event Trigger is sent to the T2K electronics. This stops the write in the SCA,
and is followed by a read of the SCA for all pads. The trigger scheme is show in Fig. 4.31.
Typical trigger rates were around 5 to 10 Hz, with live times near 85%.
The analysis performed on the recorded events is discussed in the following chapter.
Q1 Trigger

Preamp

Amp

Disc.

Logic

Q2 Trigger

Preamp

Amp

Disc.

Q1 + Q3

Q3 Trigger

Preamp

Amp

Disc.

or

Q4 Trigger

Preamp

Amp

Disc.

Q2 + Q4
Logic
AND

Mesh Trigger

Amp

Disc.

Gate
Generator

Figure 4.31: The trigger scheme used in this work.

94

Event
Trigger

Chapter 5
Analysis
Binary events that are accepted by the trigger are recorded. Software is then used to analyze
the events and compute cross sections for the various exit channels of α + 10 Be. This chapter
includes discussions on particle identification, energy calibration, angular determination, and
cross section calculations.

5.1

Ray Tracing and Angular Determinations

Electrons ionized by charged particles in the field cage drift towards the Micromegas at a
velocity of 2.4 cm/µs. By keeping track of the relative time of when the electrons reach the
Micromegas and which pads receive an amplified charge, the three-dimensional trajectories
of the nuclei are reconstructed. An example of

10

Be scattering on 4 He is given in Fig. 5.1.

The discrete reconstruction of the 4 He trajectory in Fig. 5.1 demonstrates the importance of
adopting gain variation on individual pads for the reaction trigger. The reaction angles are
then calculated.
To calculate the angles, the trajectories are fit with a line, called a ray trace. The ray
trace extends from the furthest radial pad with signal back to the center of the beam axis,
as seen in Fig. 5.2. These traces are then converted into angles from beam axis with basic
trigonometry.

95

Scattering Tracks in Prototype AT-TPC
10
10Be
10Be

θ(10Be)
θ(α)

0

α

-5

Radius [cm]

5

-10
40

Time [µs]

0

Figure 5.1: Reconstructed trajectories of reaction products in the prototype AT-TPC. The
path of the incoming 10 Be nucleus is seen, as are the paths of the reaction products 10 Be and
4
He. The ray traces of the latter two nuclei allow for their angles to be calculated.

Figure 5.2: A ray trace through pad signals in time for a triggered quadrant. The relative
time of each pad signal is denoted by a black diamond. The linear fit is shown in red.

96

Certain conditions are placed on the ray trace. First, pads within the beam column are
excluded from the trace to ensure that signals from the beam or other the reaction product
do not affect the trace. For example, the two innermost pads in Fig. 5.2 are excluded for
being in the beam column.
Also excluded are signals from pads whose nearest neighbors do not have signal. Such a
signal is likely spurious and does not result from the trajectory of the nucleus being traced
in the quadrant. For example, the pad signal near 125 mm in Fig. 5.2 is excluded for this
reason.
Other spurious signals may be found among pads with real signals, however. In this case,
an algorithm is used to sort the signals. From each pad i with signal, a line is drawn to every
other pad j with signal, and the slopes of each of these lines are calculated. This calculation
is performed for all N pads with signal from i = 1 to i = N . A standard deviation is
calculated for each pad i, defined as

σ(i) =
j

(mj,i − mj,i )2
N

(5.1)

where mj,i is the slope of the line to each pad j from a given pad i and mj,i is the average
of all such slopes mj,i for a given pad i.
The values of σ(i) for each pad i in a trace are then compared. The smallest such value
is denoted by σ< (i). By comparing each σ(i) to σ< (i), the deviation in time of pad i from
the other pads in the trace is determined. If the deviation is too large, the pad is thrown out
of the ray trace. The process is then repeated until all signals which deviate too far from

97

40

Time [µs]

Time [µs]

40

20

0

0

60
Pad Position [mm]

0

120

20

0

0

60
Pad Position [mm]

120

60
Pad Position [mm]

120

40

Time [µs]

Time [µs]

40

0

20

60
Pad Position [mm]

20

0

120

0

Figure 5.3: Illustrations showing the ray trace algorithm. For each panel, the red dot
represents the signal from pad i, from which lines are drawn to each of the other pad signals.
The top two panels show that, for a perfect fit, different pads i will results in lines to the
other pads that have equivalent slopes. In the bottom panels, however, there is a spurious
signal. If drawing from that spurious signal (bottom right panel), the slopes of the lines will
all be very different. If drawing from one of the good signals (bottom left panel), only the
line to the spurious signal has a slope that is different from the other lines. The algorithm
takes advantage of this difference in the slopes of lines betweens pads to exclude a spurious
signal like the one seen in the bottom panels.

98

the ray trace are excluded, at which point a final ray trace is made. Fig. 5.3 illustrates the
algorithm. The maximum deviation was empirically determined to be

σ(i) < σ< (i)

N
.
2

(5.2)

The threshold set in Eq. 5.2 was chosen to be slightly above the level of standard statistical
fluctuations in the signals. In Fig. 5.2, the signal near 70 mm and 35 µs is excluded from
the ray trace due to this slope algorithm.
The ray trace is independent of pad signal amplitude and area, that is, it does not depend
on the amount of charge deposited on a pad as long as it is above noise. In principle, the
amount of deposited charge is proportional to the energy of the nucleus being measured.
Thus, measuring the charge allows for the energies of the products to be calculated, which
would be a useful tool for identifying particles based on the Bethe formula, Eq. 2.1. In this
work, however, the use of high gain pads for triggering makes the calculation of reaction
product energies unfeasible. When a

10

Be nucleus deposits charge on a high gain pad,

the pad becomes highly saturated. Also, pads near the saturated high gain pad pick up
some of the signal via crosstalk, further complicating charge calculations. This results in
an unreliable calculation of reaction product energies. Therefore, in this work the reaction
product energies are not used in analysis.
If ray traces are successfully obtained in opposite quadrants of a triggered event, the
angles of the two traces are calculated, as in Fig. 5.1. This procedure is repeated for all
triggered events. Then, all calculated angle pairs are put into the matrix seen in Fig. 5.4.

99

High Z Reaction Product Scattering Angle [deg]

90

100

80

90
80

70

70

60

60
50
50
40
40
30

30

20

20

10
0
0

10
10
20
30
40
50
60
70
80
Low Z Reaction Product Scattering Angle [deg]

90

0

Figure 5.4: Angular data of all events where both reaction products are successfully tracked
through the detector volume. The x-axis denotes the angle associated with the reaction
product of lower charge Z, while the y-axis is of the higher Z reaction product.

100

The structures seen in Fig. 5.4 are due to different reactions. By comparing these structures with basic kinematics for various reactions, the reactions for each structure can be
identified.
While the reaction product energies are not used in the analysis presented in this work,
the deposited charge information in the quadrants is still of use. If the two reaction products
of an event are not separable by their charge Z, the structures seen in Fig. 5.4 would be
symmetric with respect to the two axes, making kinematic comparisons more difficult. Even
with the saturation issues on high gain pads, the difference in deposited charge between two
nuclei with different charge Z is measurable. Thus, the matrix in Fig. 5.4 can be filled by
considering the relative charge of the two reaction products. When adding an entry to this
matrix, the total charge deposited in the two quadrants where the ray traces are made are
compared. The reaction product in the quadrant with a larger total charge across all of its
pads is labeled as the High Z Reaction Product, and the other reaction product is labeled
Low Z Reaction Product.

5.2

Particle Identification

Separating reaction products by their relative charges is only a first step in particle identification. The next task is identifying and separating beam nuclei.
Beam particles of interest in this work are

10

Be nuclei. As with most secondary beam

facilities, the purity of the beam produced at the Nuclear Science Laboratory at Notre Dame
is not perfect. As such, separating the desired beam particles from contaminants is important
for distinguishing various nuclear reactions. The isotopes present in the beam in this work
include

11

B,

10

B,

10

Be, 7 Li, and α particles.

101

Electrons ionized by beam particles in the field cage drift at a velocity of 2.4 cm/µs. The
beam column, the region defined as where the beam signal is measured by the Micromegas,
extends from the central pad to pad 15 of each quadrant, resulting in a 4.91 cm diameter.
The energy deposit profile read by the Micromegas is recorded over a 40 µs time window,
which is about double the full drift time of an electron in the 50 cm cage. The energy deposit
profile of a given beam particle is sampled between 1.6 µs and 3.2 µs after the profile is first
triggered on the micromesh. This sampling is called the ∆E of the beam, and is shown in
Fig. 5.5. For nuclei with different charge and mass, the value of ∆E will be different. Fig. 5.6
shows the separation of beam species by ∆E. While various structures are clearly visible
in Fig. 5.6, the identification of these beam nuclei relies on analyzing the kinematics of the
structures seen in Fig. 5.4.
A beam stabilization routine was implemented on the ∆E spectrum. This was done to
account for possible changes in electron absorption as the active target gas degrades over
time in the field cage. The

10

Be peak was used as reference for stabilization. An initial

center-of-gravity and boundary limits are set on the peak. If a data point filled into the
histogram in Fig. 5.6 is located within the boundaries of the target peak, a new center-ofgravity c is calculated based on the value of ∆E and the previous center-of-gravity c< by
the expression

c = c< +

∆E − c<
,
N

(5.3)

where N = 1000 is a constant weight. The ratio of the new and old center-of-gravity values is
called the gain g = c/c< . The boundaries of the target peak are then scaled by this gain. If,

102

3

30 ×10

Deposited Charge [arb. units]

25

20

15

10

5

0

0

5

10

15
20
25
Drift Time [µs]

30

35

40

Figure 5.5: Plot of the energy deposit profile of a beam particle, in green. The horizontal blue
line at 3,000 represents the amplitude threshold. The red region represents the ∆E sampling
for beam particle determination. The shape of the energy deposit profile is described by a
Bragg curve. The value of ∆E is different for different nuclei. A histogram of ∆E values is
shown in Fig. 5.6.

103

8000

4

He
10

7000

Be

6000
5000
4000
9

3000

Be

2000
10

B

7

1000

Li

3

0

50

100

150
200
250
∆ E [arb. units]

300

350

×10
400

Figure 5.6: Histogram of beam species, as gated by ∆E after beam stabilization. The identification of the various species labeled in this plot is done by comparison to the kinematics
of reactions with target 4 He particles in the angular correlations matrix.

104

however, the data point is outside of the boundaries, its ∆E is simply scaled by the current
gain value. This process is then repeated for all events that fill the histogram, and it corrects
for the slow drift in pulse height due to gas degradation.
For example, consider the first peak in Fig. 5.6 from 0 to 50,000. When plotting angular
correlations with a gate on this peak, the data seen in Fig. 5.7 are obtained. Overlaying
kinematics curves on the matrix reveals the two particles in the reaction. In this case, both
particles are 4 He, meaning that the first peak Fig. 5.6 is a 4 He beam contaminant.
The identification of a 4 He beam contaminant is useful in verifying the calibration of the
calculation of angles of the two reaction products. For 4 He + 4 He elastic scattering, the sum
of the two angles must be 90◦ . The data in Fig. 5.7 nicely follow this sum rule, confirming
the suitability of the ray tracing method described in Sec. 5.1.
This comparison of kinematics is done for all of the peaks seen in Fig. 5.6. Further
examples are given in Fig. 5.8 and Fig. 5.9, the latter of which reveals both elastic and
inelastic scattering of

10

Be on 4 He.

The separation of elastic and inelastic scattering in Fig. 5.9 is particularly important for
this work, as it allows for separate analysis of the two exit channels. The black curve in
Fig. 5.9 represents elastic scattering, and the red curve shows inelastic scattering to the first
excited state in

10

Be, which is a 2+ state at 3.368 MeV. There are higher lying inelastic

states as well. The data for these states are seen as a grouping near 15◦ in
4

10

Be and 25◦ in

He. Currently, they haven’t been further investigated as separating these unbound states

has proven difficult due to their natural widths and low statistics. Thus, for convenience,
when further instances of inelastic data are mentioned, they refer only to the first inelastic
state, as shown in the red kinematical curve.

105

He Scattering Angle [deg]

100

80

90

70

80

60

70

50

60

40

50

30

40

20

30

10

20

4

90

0
0

10

20

30
40
50
60
70
4
He Scattering Angle [deg]

80

90

10

Figure 5.7: Angular data of events gated on the 4 He peak in Fig. 5.6. The kinematics curve
for elastic scattering overlays the data in black. The 4 He + 4 He scattering data are useful
for confirming calibration of the angles of the two reaction products.
Analysis of ∆E of the beam and reaction kinematics has resulted in the separation of elastic and inelastic 10 Be scattering on 4 He from other triggered events. Next, the determination
of the reaction energy will be discussed.

5.3

Energy Calibration

To determine the energy at which a reaction occurs, the location of the reaction along the
beam axis must be determined. This location, called the reaction vertex, represents the
distance the

10

Be beam particle has traveled through the active target before interacting

106

90

7

80

6

5
60
50

4

40

3

30
2

7

Li Scattering Angle [deg]

70

20
1

10
0
0

10

20

30
40
50
60
70
4
He Scattering Angle [deg]

80

90

0

Figure 5.8: Angular data of events gated on the 7 Li peak in Fig. 5.6. The kinematics curve
for elastic scattering overlays the data in black.

107

Be Scattering Angle [deg]

100

80

90

70

80

60

70

50

60

40

50

30

40

20

30

10

20

10

90

0
0

10

20

30
40
50
60
70
4
He Scattering Angle [deg]

80

90

10

Figure 5.9: Angular data of events gated on the 10 Be peak in Fig. 5.6. Kinematics curves
overlay the data: elastic scattering is in black, and the first excited state (3.368 MeV in
10
Be) inelastic scattering is in red.

108

with a target 4 He nucleus. Thus, the reaction vertex can be converted into an energy by
comparing it to energy loss calculations of

10

Be in He:CO2 90:10 of density 0.3113 kg/m3 .

First, however, the energy of the incoming beam particle must be determined. Both of these
calibrations are described below.

5.3.1

Beam Energy Calibration

The beam energy was measured at the location of the beam window into the active target
volume with a Position Sensitive Silicon Detector (PSSD), described in Sec. 4.3.2. The

10

Be

beam energy, as measured by the PSSD, was found to be 39.0+0.50
−0.25 MeV. This is the energy
incident on the beam window, and after accounting for the energy loss in the window, the
energy becomes 38.2 ± 0.5 MeV. The energy loss in the window was computed using SRIM
(the Stopping and Range of Ions in Matter), which is software that calculates the transport
of ions in matter based on experimental databases [67, 68].
The value of this beam energy as measured by the PSSD was checked by measuring the
penetration depth of the various beam particles in the active target and comparing the depths
to SRIM and LISE++ calculations. LISE++ is software [69, 70] that, among other features,
is used to calculate penetration depth of ions in matter, similar to SRIM. Both softwares are
commonly used for this purpose. Thus, comparing calculations with each software provides
an independent beam calculation from that of the PSSD.
Data were taken without the binary reaction trigger, and instead just with a micromesh
trigger on the beam. There are two main beam components seen in the spectrum:
10

10

Be and

B. They were identified by their ∆E values seen in Fig. 5.10. Lighter beam particles, such

as 4 He and 7 Li contaminants seen in Fig. 4.26, did not trigger the micromesh as they have
a lower charge than

10

Be, which is the nucleus the trigger level was set to measure. A 4+
109

3

300 ×10

50

250
40

11

B
10

∆ E [a.u.]

200

B

30
150
20
100
10

Be

50

0
0

100

200
300
400
500
Beam Penetration Depth [mm]

Figure 5.10: Two dimensional histogram of beam particles. While the
the chamber, the 10 Be particles punch through.
charge state is assumed for

10

10

600

10

0

B particles stop in

Be and B isotopes. The data in Fig. 5.10 are then projected

onto the x-axis for determination of penetration depth in the active target volume, as seen
in Fig. 5.11. While present in both figures, the
of

10

Be as analysis of the low statistics

strong presence of

10

11

11

B isotopes were not used for calibration

B contaminant was not necessary considering the

B.

In Fig. 5.11, the penetration depth of

10

B beam particles is 360 ± 5 mm, while it is

480 ± 5 mm for 10 Be particles. However, this 480 mm depth is the full length of the chamber,
which suggests that the

10

Be beam does not stop in the active target gas. To calculate its

energy, a calculation must first be made on

10

B. With a penetration depth of 360 mm, the

110

10

1000

B

10

Counts

800

Be

600

400

200
11

B

0
0

100

200
300
400
500
Beam Penetration Depth [mm]

600

Figure 5.11: Projection of Fig. 5.10 onto the x-axis. The penetration depth of 10 B is 360 ±
5 mm and is 480 ± 5 mm for 10 Be, though this latter value reflects the total active target
volume length and not the stopping distance of 10 Be.

111

10

B beam energy after the window was calculated using SRIM to be 40.150 MeV. Making a

calculation of the energy before the beam window gives 41.335 MeV. Since TwinSol selects
nuclei by their charge-to-mass ratio, or Bρ value,

10

Be has the same energy as

10

B when

incident on the window since it has the same mass and 4+ charge state. The energy loss of
10

Be is less in the window due to its lower charge, and results in an energy of 40.554 MeV

after the foil, which would have a penetration depth of 537.9 mm if the chamber were long
enough to contain it.
Using LISE++, the

10

B energy was found to be 39.399 MeV after the window, and

40.539 MeV before it. The same energy in

10

Be results in an energy of 39.79 MeV after the

window with a total range of 541.9 mm.
The particle track length approach to calculating beam energy is not as precise as the
E − ∆E method of the PSSD, and since the values of 39.79 MeV and 40.55 MeV from
LISE++ and SRIM, respectively, are quite similar to the value of 38.2 ± 0.5 MeV from the
PSSD, the PSSD value is adopted as the calibrated beam energy of

10

Be. This implies an

estimated energy uncertainty in the center-of-mass frame of 0.5 × 4/14 = 0.14 MeV.

5.3.2

Reaction Energy Calibration

With the energy of the

10

Be beam after the window calibrated, the reaction energy of data

can be calculated. A SRIM calculation is made to convert the reaction vertex to the beam
energy as a function of penetration depth. The penetration depth is defined as the difference
in distance between the entrance of the beam particle into the active target and the reaction
vertex. The particle entrance is determined by the first signal over threshold of the incoming
beam particle, as shown in Fig. 5.5. Then, calculating the reaction vertex gives, via SRIM,
the reaction energy.
112

The vertex of a given reaction is computed by taking the beam axis intercept of the
reaction product 4 He ray trace. An average of the intercepts of both 10 Be and 4 He ray traces
for a given event was considered, but the shorter paths of the

10

Be particles result in a less

precise ray trace than 4 He. Thus, only the 4 He ray trace intercept is used to compute the
vertex. Uncertainty in the calculation is 20 mm with the preferred method, while using an
average results in an uncertainty a few mm larger.

5.4

Cross Section and Excitation Function Calculations

With energies and angles calculated and calibrated in the lab frame, they must be converted
into the center-of-mass frame in order to compute cross sections and excitation functions.
The acceptance of the detector, beam rate, and detector live time must also be taken into
account.

5.4.1

Energy and Angle Frame Conversion

Converting energy to center-of-mass is quite simple. The conversion depends only on the
masses of the system and the beam energy in the lab frame, defined as

ECM =

2
m4 He
ELab = ELab .
m4 He + m10 Be
7

(5.4)

Converting angles from lab to center-of-mass is more complicated, however. The expression below is derived from energy and momentum conservation, and it is fully relativistic.
The center-of-mass angle is defined as

113

tan θCM =

for





α,

if α ≥ 0




π − α, otherwise

P3 sin θ3

α=

γ P3 cos θ3 − β

(5.5)

(5.6)

m23 + P32

where

2
ELab
+ 2m1 ELab
m1 + m2 + ELab
1
m1 + m2 + ELab
γ=
=
2m2 ELab − (m1 + m2 )2
1 − β2

β=

P3 =

sin θ4

2
ELab
+ 2m1 ELab
sin(θ3 + θ4 )

(5.7)
(5.8)
(5.9)

where ELab is the lab frame beam energy and the masses m1 , m2 , m3 , and m4 refer to the
the incident

10

Be beam particle, the at-rest 4 He target particle, the 4 He recoil particle, and

the 10 Be ejectile, respectively. Similarly, the lab angles are labeled as θ3 for the recoil and θ4
for the ejectile. All expressions use units of

= c = 1.

For elastic scattering, the center-of-mass angle does not depend on reaction energy, and
can be calculated from the 4 He recoil angle θ3 alone, provided the target nucleus is at rest in
the lab frame. In this case, the center-of-mass angle in the non-relativistic approximation is

114

θCM = 2

π
− θ3 .
2

(5.10)

Thus, Eq. 5.10 was used for elastic data, and Eq. 5.5 was used for the inelastic data.

5.4.2

Acceptance Calculation

Reactions that take place in the active target do not all have an equal likelihood of being
measured. For example, the paths of the reaction products may simply have angles that
are so far forward that the products do not escape the beam column as they travel through
the gas, resulting in a lack of trigger. Also, if a reaction occurs at a very low energy, the
two reaction products may simply have kinetic energies that are so low that the particles’
ranges are too short to traverse trigger pads on the Micromegas. To account for these
situations, pictured in Fig. 5.12, acceptance calculations of elastic and inelastic scattering in
the chamber were made. These acceptance calculations are then used to normalize the cross
section calculations.
The acceptance calculations, one each for elastic and inelastic scattering of

10

Be on 4 He,

are Monte Carlo simulations. In a calculation, the detector volume and gas are defined
as found in the experimental setup, and N = 1, 000, 000 reactions are simulated inside of
it at random energies and angles. The energy losses of the particles in the simulation are
obtained from SRIM calculations. A trigger is set in the simulation to mimic that of the
actual detector. The trigger used in the simulation requires both reaction products to reach
the first trigger pad in their respective quadrants. The probability of simulated reactions
in a given energy bin and angular bin to pass the simulation’s trigger condition gives the
115

a

b

c

d

Figure 5.12: Four plots of possible reaction scenarios. All figures depict particle trajectories
in red, a blue active target volume, a dark blue beam column, and an orange Micromegas
plane at the bottom of the volume. Fig. 5.12(a) shows a binary reaction that would successfully trigger the electronics and thus would be recorded. Fig. 5.12(b) would not trigger
as one of the reaction products has too small of an angle to leave the beam column. In
Fig. 5.12(c), the energy of the reaction is low so that the reaction products do not travel
over trigger pads, resulting in the event not being recorded. The reaction in Fig. 5.12(d)
takes place so close to the Micromegas that the products quickly leave the chamber and are
thus not detected. The acceptance calculation described in this section accounts for these
various scenarios for the normalization of the elastic and inelastic reaction cross sections.

116

1

10

0.9

9
ECM [MeV] in 0.1 MeV steps

0.8

8
0.7

7
0.6

6
0.5

5

0.4

4

0.3

3

0.2

2

0.1

40

60

80
100
o
θCM [deg] in 5 steps

120

140

Figure 5.13: Acceptance calculation for elastic scattering of 10 Be on 4 He. The color scale
represents the probability that a reaction at a given center-of-mass energy and angle would
be recorded.
detector acceptance for reactions at such energies and angles. The acceptances for elastic
and inelastic scattering are plotted for center-of-mass energies and angles in Fig. 5.13 and
Fig. 5.14, respectively.
The simple trigger used in the acceptance calculations does not fully replicate the trigger
system of the experiment. Thus, the calculations are not fully reliable in regions where
acceptance is low, such as angles far from 90◦ and for low energies. An absolute uncertainty
of 10% is assumed for the calculations.

117

1

10

0.9

9
ECM [MeV] in 0.1 MeV steps

0.8

8
0.7

7

0.6

6

0.5
0.4

5

0.3

4
0.2

3
2

0.1

40

60

80
100
o
θCM [deg] in 5 steps

120

140

0

Figure 5.14: Acceptance calculation for inelastic scattering of 10 Be on 4 He. The color scale
represents the probability that a reaction at a given center-of-mass energy and angle would
be recorded if it were to take place in the detector. The energy cutoff seen in the plot is due
to the 3.368 MeV excitation energy of the first excited state in 10 Be.

118

5.4.3

Elastic and Inelastic Cross Sections and Excitation Functions

With the calculated detector acceptance and the center-of-mass energies and angles calibrated, elastic and inelastic cross sections and excitation functions can be computed.
The experimental cross section, adapted from Eq. 1.15, is calculated separately for elastic
and inelastic scattering as

σ(θ) =
DAQ

N
,
det Ninc nt 2π sin θ∆θ

(5.11)

where N is the number of reactions at angle θ within an angular bin of width ∆θ, Ninc is the
number of incident nuclei, nt is the number of target nuclei per unit area,
acquisition efficiency, and

det

DAQ

is the data

is the detector acceptance from the Monte Carlo simulation

described above in Sec. 5.4.2. These values are defined as follows:

Ninc = f∆E

10

Be NTPC

= 0.571 · 558, 872, 962
= 319, 116, 461

where f∆E

10

Be is the fraction of beam particles identified as

(5.12)

10

Be by ∆E in Fig. 5.6 and

NTPC is the number of total beam particles that triggered the micromesh in the prototype
AT-TPC;

119

nt = ρdx = ρ(dx/dE)∆E

(5.13)

where ρ = 2.15 × 1019 1/cm3 is the target gas density, dx/dE is the energy loss function of
10

Be in He:CO2 90:10 as calculated in SRIM, and ∆E is the energy bin;

DAQ

TimeLive
Time100
Time100 - DeadTime
=
Time100
33, 577, 009
=
38, 416, 240
=

= 0.874

(5.14)

where Time100 is the total amount of time the data acquisition (DAQ) electronics is running,
DeadTime is the time during which the DAQ is processing data and not accepting new trigger
events, and TimeLive is the amount of time the data acquisition electronics is accepting new
trigger events.
By filling a histogram of energy and angular bins with values calculated from Eq. 5.11,
the cross sections of the elastic and inelastic channels are obtained. The elastic scattering
data are shown in Fig. 5.15; the inelastic are given in Fig. 5.16.
In both cross section plots, resonances are present. To analyze these resonances, projection onto each axis are made. Projections onto the x-axis yield differential cross sections for
a given energy bin, while y-axis projections produce excitation functions for a given angular
bin. These projections are useful for comparing the data to theoretical calculations, which

120

Absolute Elastic Cross Sections
10

60

9
ECM [MeV] in 0.10 MeV steps

50

8
40

7
6

30

5

20

4
10

3
2

40

60

80
100
120
o
θCM [deg] in 5.0 steps

140

0

Figure 5.15: Normalized experimental elastic cross section as a function of energy and angle,
given in mb/sr.

121

Absolute Inelastic Cross Sections
10

60

9
ECM [MeV] in 0.10 MeV steps

50

8
40

7
6

30

5
20

4
10

3
2

40

60

80
100
120
o
θCM [deg] in 5.0 steps

140

0

Figure 5.16: Normalized experimental inelastic cross section as a function of energy and
angle, given in mb/sr.

122

Excitation Function 100 to 105 Deg CM
60

dσ/dΩ [mb/sr]

50

40

30

20

10

0
2

3

4
5
6
7
8
ECM [MeV] in 0.10 MeV steps

9

10

Figure 5.17: Elastic excitation function from 100◦ to 105◦ , a projection onto the y-axis from
Fig. 5.15. An x-axis projection produces an angular cross section plot. The uncertainties
are calculated from Eq. 5.15.
will be discussed in the following section. An example excitation function from elastic data
is shown in Fig. 5.17.
The projections include an uncertainty in the cross section values. This uncertainty is
defined, for each energy bin i and angular bin j, as

∆σi,j (E, θ) = σi,j (E, θ)

(∆

det )

2

+

123

Ni,j
Ni,j

2

+

σs.d. (Ninc )
Ninc

2

(5.15)

for an acceptance uncertainty of ∆
bin Ni,j , and the number of incident

= 0.1, number of counts per energy and angular

det
10

Be beam particles Ninc . The uncertainty in Ninc is

given by the standard deviation of various beam measurements taken over the course of
the experiment. For each beam measurement, the percentage of

10

Be beam particles was

calculated based on the ∆E spectrum. The standard deviation of these percentages of all
the beam measurements is called σs.d. (Ninc ), and it is included in Eq. 5.15 with a value of
σs.d. (Ninc )/Ninc = 0.0283/0.571 = 0.0496.

5.5

Theoretical Calculations

Theoretical calculations to compare to the data are done in the following manner:
1. A non-resonant elastic scattering calculation is made based on the S˜ao Paulo (SP)
potential, an optical model potential.
2. An elastic R-matrix calculation is made without any resonances and matched to the
output of the SP calculation.
3. Resonances are added to the R-matrix elastic calculation and compared to the elastic
data.
4. A similar R-matrix calculation is then made to compare to the inelastic data.
5. Spins and parities are assigned to the elastic and inelastic resonances, and their widths
Γres and spectroscopic factors SF are computed.
6. The relevance of the resonance assignments to

14

C cluster predictions are discussed.

The following subsections will describe these points in detail.
124

5.5.1

S˜
ao Paulo Optical Model Calculation

Before making theoretical calculations of the resonances seen in the data, a reliable description of Coulomb and nuclear elastic scattering is desired. An optical model (see Sec. 1.6)
calculation is made based on the S˜ao Paulo (SP) potential [71]. The code used to make this
calculation was provided by Luiz Chamon of the Departamento de F´ısica Nuclear, Instituto
de F´ısica da Universidade de S˜ao Paulo.
Within the SP model, the nuclear interaction between two nuclei is described with a
density-dependent folding potential and energy-dependent exponential term as

VN (R, E) = VF (R)e−4V

2 /c2

,

(5.16)

where c is the speed of light. The velocity relates to the energy by

V 2 (R, E) =

2
[E − VC (R) − VN (R, E)]
µ

(5.17)

for the reduced mass µ of the system and Coulomb potential VC . By iteratively solving
Eq. 5.16 and Eq. 5.17, the SP potential is obtained. The folding potential is defined as

VF (R) =

ρ1 (r1 )ρ2 (r2 )V0 δ(R − r1 + r2 )dr1 dr2 ,

125

(5.18)

where V0 = −456 MeV · fm3 , R is the relative displacement vector of the two nuclei, and
r1 and r2 are the radii of the two nuclei [71]. The nuclear density is modeled as a Fermi
distribution [72]

ρ0

ρ(r) =

1 + exp

r−R0
a

.

(5.19)

Matter and charge densities are both computed, and they have average diffuseness values
of a = 0.56 and 0.53 fm, respectively. These densities are calculated without any adjustable
parameters, meaning that the resulting cross sections are predictions and not fits to data,
an important characteristic of the SP potential [71].
For this work, the SP potential is used for the real part of the optical model. The value
of the imaginary term is set as a scaled value of the real depth. Multiple scaling factors
around the value of W = 0.8VN suggested in Ref. [71] were considered, with the suggested
value ultimately adopted. A comparison of these values and their effect on the transmission
probabilities |S| for each L is shown in Fig. 5.18.
Fig. 5.19 is a plot of the elastic output from the SP calculation. At low energies and
forward angles, the cross section due to Coulomb scattering is higher than found in the elastic
data in Fig. 5.15. This discrepancy is likely due to the acceptance calculation’s simplified
trigger condition. Further work may be done to address this issue.
This elastic SP calculation does not include the option of including resonances. Thus, a
separate R-matrix code is used. The SP calculation does, however, serve as a starting point
reference for the potential scattering of the R-matrix calculations done in this work.

126

1

L=2
0.8
L=1
L=0

|S|

0.6

0.4

0.2

+ NI = 0.5
x NI = 0.8
o NI = 1.1

0
2

4

6
EnergyCM [MeV]

8

10

12

Figure 5.18: Comparison of the effect of different imaginary depths on the transmission
probability |S| as a function of angular momentum L and center-of-mass energy for the SP
calculation.

127

Sao Paulo Elastic Cross Section [mb/sr]
60

10

ECM [MeV] in 0.5 MeV steps

9

50

8
40

7
6

30

5
20

4
10

3
2
40

60

80
100
o
θCM [deg] in 1 steps

120

140

Figure 5.19: Elastic SP cross section calculation as a function of energy and angle, given in
mb/sr. This output is used as a reference for the R-matrix calculation.

128

5.5.2

R-Matrix Calculations

In order to understand the resonances present in the data, interference effects among the
resonances and potential scattering must be modeled. R-matrix theory can be used to
calculate the cross sections of the resonances in the Breit-Wigner form shown in Eq. 1.27 by
modeling the interaction during which the resonance is formed [5].
In R-matrix analysis, the cross sections for resonances are calculated through the use of a
unitary and symmetric collision matrix U. Each matrix element of U defines the interaction
between incoming channel c and outgoing channel c as

Ucc =

0 U0
Ucc
cc

δcc +

i Γc (E)Γc (E)
Er − E − 2i Γ(E)

(5.20)

for potential scattering elements U0 , channel orthogonality condition δcc = c|c , partial
widths Γc and Γc , resonance energy Er , and total width Γ(E) = Γc (E) + Γc (E) for energy
E [77]. The differential elastic cross section is then

dσel.
1
=
dΩ
(2I1 + 1)(2I2 + 1) M

2

fC (Ω)δM1 M1 δM2 M2 +
1 M2

(cM1 M2 )
fcM M
(Ω)
1 2

M1 M2

for Coulomb scattering amplitude fC and elastic scattering amplitudes

129

(5.21)

Figure 5.20: Level scheme of 14 C with various exit channels. Neutron separation occurs at
8.18 MeV, while 10 Be + α starts at 12.01 MeV. The energy domain studied in this work
ranges from about 14 MeV to 22 MeV, over which the 13 C + n cross section is low and flat.
Thus, neutron emission can be modeled as background and absorbed with the optimal model
used in DSigmaIV. Figure from [73].
130

DSIGMA_IV Convoluted Elastic Cross Section [mb/sr]

10

60

9

ECM [MeV] in 10 keV steps

50

8
40

7
6

30

5
20

4
10

3
2

40

60

80
100
θCM [deg] in 5o steps

120

140

0

Figure 5.21: Elastic cross section calculation with DSIV using the Li and Hird parameter
set, given in mb/sr. The results are similar to the SP calculation shown in Fig. 5.19.

131

Sao Paulo Elastic Cross Section [mb/sr]

DSIGMA_IV Convoluted Elastic Cross Section [mb/sr]

10

50

50

ECM [MeV] in 10 keV steps

8
40

7
6

30

5
20

4

8
40

7
6

30

5
20

4
10

10

3

3

2
40

60

a

80
100
θCM [deg] in 1o steps

120

2

140

40

60

80
100
θCM [deg] in 5o steps

b

DSIGMA_IV Elastic Cross Section [mb/sr]

120

140

9

9

50

ECM [MeV] in 0.1 MeV steps

40

7
30

6

60

10

8

5
20

4

50

8
40

7
30

6
5

20

4

10

10

3
2

3

40

60

c

80
100
θCM [deg] in 5o steps

120

2

140

DSIGMA_IV Elastic Cross Section [mb/sr]

60

80
100
θCM [deg] in 5o steps

120

140

DSIGMA_IV Elastic Cross Section [mb/sr]
60

9

9

50

8
40

7
30

6

60

10

ECM [MeV] in 0.1 MeV steps

ECM [MeV] in 0.1 MeV steps

40

d

10

5
20

4

50

8
40

7
30

6
5

20

4

10

10

3
2

e

0

DSIGMA_IV Elastic Cross Section [mb/sr]
60

10

ECM [MeV] in 0.1 MeV steps

60

9

9
ECM [MeV] in 0.5 MeV steps

10

60

3

40

60

80
100
θCM [deg] in 5o steps

120

2

140

f

40

60

80
100
θCM [deg] in 5o steps

120

140

Figure 5.22: Comparison of various optical model parameter sets, given in mb/sr: (a) S˜ao
Paulo from Fig. 5.19, (b) final choice from Fig. 5.21, (c) 10 Be+α, 6 MeV CM [74], (d) 10 B+α,
16.07 MeV CM [75], (e) 9 Be + α, 6.06 MeV CM set one [76], (f) 9 Be + α, 6.06 MeV CM set
two [76].

132

(cM1 M2 )
fcM M
(Ω)
1 2

=i

√

π
k

(2 + 1)1/2 ei(σ
Jπ

I

+σ )

I

(I1 I2 M1 M2 |IM )

× (I M 0|JM )(I1 I2 M1 M2 |I M )(I M M − M |JM )
× (δc c δI I δ

− UcJπI

,cI

)Y M −M (Ω).

(5.22)

The differential inelastic cross section is [77]

π
dσc→c
1
= 2
dΩ
k (2I1 + 1)(2I2 + 1)

BL (E)PL (cos θ)

(5.23)

L

for

BL (E) =

1
4π

(−1)I−I ei(σ
Jπ I L J π I

+σ −σL −σL )

L

× Z( JLJ , Iλ)Z( JL J , I λ)UcJπI

,cI

(E)UcJ IπL∗ ,cIL (E),

(5.24)

where Z are the 6j-symbols, also called the Racah coefficients. The integrated inelastic cross
section has the form

σc→c =

π
1
2
k (2I1 + 1)(2I2 + 1)

UcJπI

(2J + 1)
Jπ

133

I

I

2
,cI

(E) ,

(5.25)

explicitly showing the dependence of the cross section on the modulus squared of the collision
matrix.
The partial width Γc from Eq. 1.29 now includes the spectroscopic factor SF as a scaling
factor in the form

Γc (L, E) = 2Pc (L, E)γc2 (L)SF.

(5.26)

Then, the calculated partial width Γres for a resonance is simply the partial width Γc times
the transmission factor:

0
Γres = Γc |Ucc
|.

(5.27)

Previously when discussing the SP calculation, the transmission factor was denoted as
0
| for this R-matrix calculation.
|S|. Now, it is explicitly given as |Ucc

Elastic R-matrix calculations in this work were made with code provided by Wang Hongwei of the Shanghai Institute of Applied Physics (SINAP) at the Chinese Academy of Sciences, based on the Lane and Thomas [5] description of R-matrix theory. Reference calculations were made by my advisor Wolfgang Mittig of the National Superconducting Cyclotron
Laboratory and Daisuke Suzuki of Institut de Physique Nucl´eaire Orsay with their own
codes, based on the Lane and Thomas description and Descouvement and Baye description
[77], respectively. Preliminary inelastic calculations were made with the code of Suzuki.

134

For a clear and complete description of R-matrix theory, see both the Lane and Thomas
description [5] and the Descouvement and Baye description [77].
As input for Wang’s R-matrix code, called DSigmaIV, resonances are entered for a given
energy, spin, parity, and spectroscopic factor. A Coulomb radius parameter must also be
given, and a choice between hard sphere scattering and an optical model parameterization is
made. Ideally, the SP output would be used as input for DSigmaIV, but the SP output is not
of the Woods-Saxon form required by DSigmaIV. Thus, various optical model parameter sets
were considered and used to compute the non-resonant scattering in DSigmaIV. By design,
the non-resonant scattering must represent exit channels other than 10 Be+α, such as 13 C+n
by the absorption part of the optical model. The neutron separation energy is 8.18 MeV
above the 14 C ground state, and resonances in 13 C+n continue throughout the energy domain
probed in

10

Be + α. However, in the energy region of interest for this work in

cross section in

13

10

Be + α, the

C + n is low and flat (Fig. 5.20), allowing for it to be modeled by the

imaginary part of the optical model.
No reliable optical model parameter set exists for elastic scattering of 10 Be on 4 He, so sets
describing similar reactions were considered, including 9 Be + 4 He and

12

C + 4 He at various

energies.
An optical model parameter set from Li and Hird [78] best reproduced the SP output,
and was thus adopted for this work. The Li and Hird set describes 9 Be + 4 He at Eα =
48 MeV. The parameters are given in Table 5.1, and are found in Ref. [78] in Table I(a).
The DSigmaIV calculation with the Li and Hird parameter set without any resonances is
shown in Fig. 5.21; comparisons to other optical model parameter sets that were considered
are shown in Fig. 5.22.

135

Table 5.1: Optical model parameter set [78] used with DSigmaIV. The real and imaginary
terms use the same radius and diffuseness, given in columns two and three, respectively.
Column six shows the Coulomb radius rc , and the last column displays the channel radius
Rc , which was adopted as a different value than in Ref. [78].
V [MeV] r [fm] a [fm] Wvol [MeV] Wsur [MeV] rc [fm] Rc [fm]
70.9

1.550

0.660

23.87

0

1.25

4.68

Table 5.1 also includes the channel radius Rc used in the DSigmaIV calculation. The
value of Rc was varied to best fit the DSigmaIV output to the SP calculation. Variations
between 3 and 5 fm were considered, as this range is common in literature [8, 78, 79]. A value
of Rc = 4.68 fm was found to produce the best fit to SP. This corresponds to a Coulomb
1/3

radius of rc = 1.25 fm in the expression Rc = rc A1

1/3

+ A2
1/3

model calculation sometimes use the expression Rc = rc A1

= 3.74 rc . Other optical

= 2.15 rc for defining channel

radius.
With the optical model parameter set chosen, DSigmaIV was used to compute cross
sections, shown in Fig. 5.23, comparable to that of the elastic data seen in Fig. 5.15. A sideby-side comparison of the two plots are given in Fig. 5.24. The DSigmaIV output shown
in Fig. 5.23 is convoluted to the energy resolution of the data. The convolution assumes a
Gaussian shape for the resonances, and incorporates both the uncertainty in beam energy
and the uncertainty in the location of the reaction vertex. The convoluted cross section
σconv (Ei , θ) for the ith energy bin is defined in terms of the cross section σDSIV (E, θ) from
the DSIV calculation as

σconv (Ei , θ) =
j

σDSIV (Ej , θ) · w(i, j) · N (i)
136

(5.28)

where N (i) is a normalization factor and the Gaussian weight function is

w(i, j) = e

−(Ei −Ej )2
2∆E 2

(5.29)

for

∆E =

∆R
α
E

2

+ ∆Eb2 .

(5.30)

In Eq. 5.30, ∆Eb = 143 keV is the uncertainty in beam energy in center-of-mass, ∆R =
20 mm represents the uncertainty in the vertex position, and α is a constant relating the
first term to the Betha formula. Values of ∆E are in the range of 200 to 350 keV.
Including energy convolution allows for the R-matrix calculation to be more reliably
compared to the data. Since the angular resolution of the data is better than the 5◦ binning
used in DSigmaIV, angular convolution is not necessary.
Since both

10

Be and 4 He are 0+ , the shapes of the resonances in the excitation functions

are of the form of the square of the Legendre polynomial PL for a given L value if potential
scattering is neglected. Thus, in a first step the experimental angular distribution was
compared to PL2 to provide a first guess of the angular momentum of the resonance. These
initial comparisons ended up being qualitatively very similar to the output of DSigmaIV,
which includes potential scattering. Thus, assigning spin and parity for each resonance was
quite straightforward and not reliant on the optical model parameterization. Four examples

137

DSIGMA_IV Convoluted Elastic Cross Section [mb/sr]

10

60

9

ECM [MeV] in 10 keV steps

50

8
40

7
6

30

5
20

4
10

3
2

40

60

80
100
θCM [deg] in 5o steps

120

140

0

Figure 5.23: DSIV elastic cross section calculation, given in mb/sr, reproducing the elastic
data in Fig. 5.15.

138

Absolute Elastic Cross Sections

DSIGMA_IV Convoluted Elastic Cross Section [mb/sr]

10

10

60

9

60

9

8
40

7
6

30

5

20

4

50

ECM [MeV] in 10 keV steps

ECM [MeV] in 0.10 MeV steps

50

8
40

7
6

30

5
20

4
10

3
2

10

3

40

60

80
100
120
θCM [deg] in 5.0o steps

140

2

0

40

60

80
100
θCM [deg] in 5o steps

120

140

0

Figure 5.24: Side-by-side comparison of the data, given in mb/sr, from Fig. 5.15 (left) with
the calculation from Fig. 5.23 (right).
are given in Fig. 5.25. Calculating the partial widths and spectroscopic factors and their
associated uncertainties is described in the following section.

5.6
5.6.1

Resonance Spin and Parity Assignments
Method of Making Assignments

Since the experimental resonances follow quite perfectly the square of the Legendre polynomials, spin and parity assignments are easily made and unambiguous. Modeling the width
and cross sections of the data by varying SF in DSigmaIV is the next step. Due to interference of the potential scattering with the resonances, perfect agreement cannot be expected
between the calculation and experimental data. Many iterations of parameters of energy and
spectroscopic factor were attempted, and the results in Table 5.2 and Fig. 5.23 represent the
best current fit to data. The error are estimated from the variations of these parameters,
139

Elastic Differential Cross Section 3.30 to 3.40 MeV CM

60
60

3.0 MeV 2+

40
40

40
40

30
30
20
20
10
10

00

4.7 MeV 3-

dσ/dΩ [mb/sr]

50
50

dσ/dΩ [mb/sr]

50
50

dσ/dΩ [mb/sr]

dσ/dΩ [mb/sr]

60
60

Elastic Differential Cross Section 5.10 to 5.20 MeV CM

30
30
20
20
10
10

40
40

60
80
100
120
60
80
100
120
θCM [deg] in 5.0o steps
θCM [deg] in 5o steps

00

140
140

40
40

Elastic Differential Cross Section 7.10 to 7.20 MeV CM

60

7.0 MeV 4+

40
40

40
40

30
30
20
20

8.7 MeV 5-

dσ/dΩ [mb/sr]

50
50

dσ/dΩ [mb/sr]

50
50

30
30
20
20
10
10

10
10

00

140
140

Elastic Differential Cross Section 8.60 to 8.70 MeV CM

60
60

dσ/dΩ [mb/sr]

dσ/dΩ [mb/sr]

60

60
80
100
120
60
80
100
120
θCM [deg] in 5.0o steps
θCM [deg] in 5o steps

40
40

60
80
100
120
60
80
100
120
θCM [deg] in 5.0o steps
o
θCM [deg] in 5 steps

00

140
140

40
40

60
80
100
120
60
80
100
120
θCM [deg] in 5.0o steps
o
θCM [deg] in 5 steps

140
140

Figure 5.25: Angular cross sections for four elastic resonances seen in the data. The data
are given in black, the squares of the Legendre polynomials are in red, and the DSigmaIV
calculations are in blue. There is very good agreement in the oscillations of the angular
distribution of the data with the red and blue curves. Of particular note in this figure,
the good agreement of the data with the red PL2 curves show that the spin and parity
assignments are not dependent on the optical model parameterization in DSigmaIV and are
thus unambiguous.

140

1

L=6
0.8

0.6
|S|

L=0

0.4

0.2

0
2

4

6
EnergyCM [MeV]

8

10

0
Figure 5.26: Plot of transmission probability |S| = |Ucc
| as a function of angular momentum
L and center-of-mass energy for the final elastic R-matrix calculation.

141

Table 5.2: Table of elastic scattering resonances, as obtained from the R-matrix calculation
fit to the data. All of the spin-parity assignments are without ambiguity. The spectroscopic
0
| is
factors SF all are assigned a 50% uncertainty as they are not precisely calculated. |Ucc
the transmission factor for each resonance, as determined by the absorption chosen in the
calculation. Γres is the width of the resonance, as defined in Eq. 5.27. Γconv is the width after
convoluting the calculation with the energy resolution of around 150 keV.

E [MeV]

Jπ

SF

3.0 ± 0.15
4.7 ± 0.15
5.6 ± 0.15
6.5 ± 0.15
7.0 ± 0.15
7.9 ± 0.15
8.7 ± 0.15

2+
3−
5−
7−
4+
5−
5−

0.3 ± 0.15
0.2 ± 0.1
0.4 ± 0.2
1.6 ± 0.8
0.2 ± 0.1
0.1 ± 0.05
0.4 ± 0.2

0
|Ucc
|

Γres [keV]

Γconv [MeV]

0.50
0.40
0.85
0.95
0.35
0.50
0.40

292
273
100
22
344
104
566

1.3 ± 0.5
1.1 ± 0.2
0.6 ± 0.2
0.6 ± 0.2
0.7 ± 0.3
0.5 ± 0.3
0.9 ± 0.3

within the limits of their agreement with the data. A plot of transmission factors is given in
Fig. 5.26.
A short discussion is given for each resonance appearing in Table 5.2, after which evidence
of rotational band structure is discussed.

5.6.2

3.0 MeV, 2+

This resonance is the only 2+ resonance identified in the analysis. The apparent partial
width Γconv is difficult to precisely measure due to the low energy of the resonance near
the strong potential scattering background, which means that Γres for this resonance is less
precise than for the others. Interference effects result in the apparent energy to be 3.3 MeV.

5.6.3

4.7 MeV, 3−

The apparent partial width Γconv is quite sharp for this resonance, resulting in confidence in
the value of Γres . The forward-angle peak matches the data quite well, while there is slight
142

disagreement at back angles. The data have an increase in cross section near 135◦ that is
not present in the calculation, though the calculation does reproduce the small increase near
115◦ . Interference effects result in the apparent energy to be 4.9 MeV.

5.6.4

5.6 MeV, 5−

The calculation reproduces well each of the two backward-angle peaks in the data, both in
width and cross section, though the peak near 75◦ is not clearly apparent.

5.6.5

6.5 MeV, 7−

This is the highest L resonance assignment in this work. The peaks near 50◦ and especially
80◦ are well reproduced by a 7− assignment, which other spin-parity choices could not
accomplish. With a narrow partial width of 22 keV, this resonance is very sensitive to
changes in SF, energy, and interference.

5.6.6

7.0 MeV, 4+

While the spin-parity assignment of this resonance is clear due to the location of the three
peaks of the resonance in the data and calculation, the calculated width and SF are less
precise. Interference puts the energy at 7.2 MeV. A substantial cross section in the inelastic
data is found at this energy, implying a strong branching ratio to the 2+ state in
predicted for a linear-chain state [80].

143

10

Be, as

5.6.7

7.9 MeV, 5−

The peak near 50◦ and 80◦ are reproduced in the calculation, as is a small peak near 140◦ .
The resonance appears at 8.0 MeV due to interference. There is also a resonance at this
energy in the inelastic data, implying a branching ratio to the 2+ state in

5.6.8

10

Be.

8.7 MeV, 5−

While the peaks near 50◦ and 80◦ are reproduced in the calculation, the calculation predicts
larger cross sections at backward angles than is seen in the data. The resonance in the
data at this energy may perhaps be two 5− resonances, but they are not separable given the
resolution of the data. Increasing the width of this 8.7 MeV calculation strongly increases
the 140◦ peak, in disagreement with the data, so a relatively narrow width compared to the
data is used. Interference puts the resonance at 8.8 MeV.

5.6.9

Experimental Results Compared to Theory: Positive-Parity
Rotational Band

As mentioned previously in Chapter 3, Ref. [13] predicts a positive-parity rotational band
for linear-chain structures in 14 C with 10 Be + α correlations. The resonances in this band are
0+ , 2+ , and 4+ , as first shown in Fig. 3.5. A plot of this band in comparison to the 3.0 MeV
2+ and 7.0 MeV 4+ resonances from the DSigmaIV calculation is given in Fig. 5.27.
In Fig. 5.27, there is very good agreement between the experimental and theoretical 4+
resonances. The 2+ state differs by less than 1 MeV. The lack of a 0+ resonance in the
data does not rule out the assignment of these resonances to the linear-chain band, for two
reasons: (1) with the substantial potential scattering background in the data at low energies,
144

January 28, 2010

22:54

WSPC - Proceedings Trim Size: 9in x 6in

hnp09

10
Experiment

Theory

8

3

+

4

4

6
-80
10

Be(2+)+α

4 -85
Energy [MeV]

Energy [MeV]

+

2

-90
-95

0
-100
-105

(1++)
(1 )
(6+,5-)

10
++
+ Be(0 )+α
(4 )
+

2

4+

(4 )

2+
0+

+

4+ 2++
+ 0
Be+ +2α
2 0+

10 1

2+
4++ 0+
2+ 4+
5++ 0 2+
2+
2++
4++
0
03
0+
+
2

2+

0+
+

0
Figure 5.27: Comparison
column labeled “Experiment”)
-110 of resonances
Experimentin this work (left
Theory
with predictions for a rotational band with linear-chain structure [13] (right column labeled
“Theory”), given in MeV above
the 10 Be + α threshold. The theory predictions are in good
14
Fig. 1. Energy
levels
of C. Four
columns on the left are the experimental data and five
agreement
with the
experimental
results.

columns on the right are the calculated results. The broken lines in the experimental-data
and calculated-results regions show the experimental and theoretical 10 Be+α threshold
energies, respectively. The details are described in the text.
Table 1.
J π of 14 C
0+
2+

Overlaps between linear-chainlike states and artificial
(J ′ , M ′ ) of
(0, 0)
(2, 0)
(0, 0)
(2, 2)
(2, 1)
(2, 0)

10 Be

Overlap
28.9%
7.7%
28.2%
0.3%
6.7%
10.1%

J π of 14 C
4+

10 Be+α

(J ′ , M ′ ) of
(0, 0)
(2, 2)
(2, 1)
(2, 0)

10 Be

states.
Overlap
15.7%
1.2%
14.0%
13.1%

10

10
To5.28:
discuss
components,
calculated
overlaps
theoretical
Figure
OverlapsBe+α
between
linear-chainlikewe
states
and artificial
Be +of
α states.
Tables
J±
10
from
Ref.|Φ
[80]. ⟩ with Be+α cluster wave functions defined as
states

n

′

J±
J + 10
|10 Be J ′± , M ′ + α⟩ ≡ PM
M ′ PM ′ 0 | Be⟩ ⊗ |α⟩ ,

(2)

which consists of a subprojected 10 Be
145 and an α cluster. The distance be10
tween Be and an α cluster is set to 5 fm. The calculated overlaps for
linear-chainlike states are listed in Table 1. In other states, 10 Be+α com-

Inelastic Excitation Function 80 to 90 Deg CM
240
220
200
180
160
140
120
100
80
60
40
20
0

2

3

4
5
6
7
8
ECM [MeV] in 0.10 MeV steps

9

10

Figure 5.29: Experimental excitation functions, given in 4 × mb/sr, from this work: elastic
in red, inelastic in black. Note the large cross section in inelastic at 7.2 MeV, which coincides
with the 7.2 MeV elastic resonance (which is found to be a 7.0 MeV 4+ resonance from the
R-matrix calculation). This excitation function shows a strong branching to the inelastic for
this 4+ resonance, as predicted in Ref. [80] for the linear-chain rotational band.

146

o

Inelastic R-matrix Calculation, 85 to 95o
30

Cross Section [mb/sr]

25

20

15

10

5

0
5

5.5

6

6.5
7
7.5
8
8.5
ECM [MeV] in 10 keV steps

9

9.5

10

Figure 5.30: Preliminary inelastic R-matrix excitation function from 85◦ to 95◦ . The inelastic
resonances in the calculation are 7.2 MeV 4+ and 7.8 MeV 5− . Further calculations will be
made for the inelastic data.

147

the 0+ resonance could be below the energy detection threshold, and (2) there is a strong
absorption for L = 0 resonances. Thus this resonance will be much more difficult to see
above background scattering.
The theory predicts overlap factors of 0.282 for the elastic 2+ resonance and 0.157 for
the elastic 4+ resonance. From the elastic R-matrix calculation, values for spectroscopic
factors of 0.3 ± 0.15 for the elastic 2+ resonance and 0.2 ± 0.1 for the elastic 4+ resonance
were obtained, in very good agreement with theory. The predicted linear-chain band also
has a large inelastic component [80], as shown in the tables of Fig. 5.28. Theory predicts an
overlap factor of 0.283 for this inelastic channel. In the data, there is strong branching to
the inelastic channel for the 4+ resonance, as seen in Fig. 5.29 and the R-matrix calculation
in Fig. 5.30. The current inelastic R-matrix calculations are preliminary. Even without a
detailed analysis it is clear that the large cross section in the inelastic channel is compatible
only with a large spectroscopic factor.
The good agreement between the predicted band and resonances shown in Fig. 5.27,
along with the strong branching to the inelastic for the 4+ , make these resonances excellent
candidates for a positive-parity linear-chain rotational band in

5.6.10

14

C.

Experimental Results Compared to Theory: Negative-Parity
Rotational Band

With five negative-parity resonances in Table 5.2, various types of rotational band structure
have been considered. As of yet, no final determination of band structure has been made.
A few preliminary proposals are described below.

148

10
This Work

von Oertzen

Energy [MeV]

8
753-

6
4

(7-)
(5-)

2
(3-)

10

0

Be + α

Figure 5.31: Comparison of negative-parity resonances of this work (left column) and from
Ref. [25], observed via a (7 Li,d) reaction. The rotational band from Ref. [25] is predicted to
be a prolate structure in 14 C. Poor agreement is seen for 3− and 5− , though it is better for
7− .

149

Be + α Threshold [MeV]

10

Energy above

20
718

(7-)
(5-)

16

5-

14
-

(3 )
12

4This work
Prolate
Oblate

-

10

3

8
0

10

20

30
40
J(J+1)

50

60

Figure 5.32: Band gradient comparison of 7− resonance from this work with proposed prolate
and oblate rotational bands in 14 C from Ref. [25]. Close agreement is seen with the prolate
band.
• One possibility is that the 3− , first 5− , and 7− resonances constitute a rotational band.
These resonances are within 2 MeV of each other, resulting in a small band gradient of
around 60 keV compared to the positive-parity band discussed above. Band gradient
analysis allows for the comparison of different bands by moment of inertia, and is often
shown by plotting energy as a function of J(J + 1). Other works [81, 82, 83, 84] do
suggest sub-100 keV band gradients for α clustering, so further investigation of this

150

possible band structure is planned.

• A second consideration is prolate structure. Ref. [25] proposes a 3− , 5− , and 7− prolate
band in

14

C above the

10

Be + α threshold. A comparison of this band with the data is

shown in Fig. 5.31. While the resonances do not generally agree in energy level spacing,
the 7− states are quite close in energy. The proposed band’s 3− and 5− resonances are
on the edge of the energy domain probed in this work, and are thus likely not to be
seen in the data. One explanation for the discrepancy is that the 7− seen in the data
belongs to this proposed prolate band, and is not part of a band with the 3− and 5−
resonances. Fig. 5.32 compares this 7− resonance with predicted prolate and oblate
band gradients from Ref. [25], showing much better agreement with the prolate band.

• Another view is that the 3− resonance forms a band with one of the two higher 5− resonances, as this would result in a more natural band gradient. Determining which 5−
resonance to pair with it may prove difficult; more work must be done on this possibility.

• One possible explanation is that the data do not show clear band structure of the
negative-parity resonances. Ref. [13] predicts that while there is a positive-parity linearchain rotational band, the negative-parity linear-chain band fractions into multiple
negative-parity resonances. This interpretation is fully compatible with the data. In
this case, it is natural to see multiple negative-parity states that have substantial α
decay widths, but do not seem to constitute a distinctive rotational band.

151

5.6.11

Comparison to Literature

In Table 5.3, the resonances from this work in Table 5.2 are listed with observed resonances
in 14 C from other reactions and works. The energies are given in MeV in 14 C, resulting in an
increase of 12.01 MeV compared to Table 5.2. While some of the resonances from this work
may be found at similar resonance energies, six have unique spin-parity assignment compared
to literature while the 19.91(15) MeV resonance (7.9 MeV in Table 5.2) is assigned 5− , as
is the 19.76 MeV resonance from Ref. [85]. The selectivity of different reactions is not the
same, and therefore exact correspondence is not expected. The results of this work provide
the most reliable spin-parity assignments due to (1) the simple and well-understood reaction
mechanism of resonant scattering and (2) clearly identified diffractive peak angles.

152

Table 5.3: Comparison of resonances from this work with literature. Reaction mechanism
for each column is given under the reference. The last column show states which have been
observed to α-decay. Table adapted from Ref. [85].
This Work
10
Be + α

10

Jπ

E

E

[85]
Be + α
Jπ

[86, 87]
Various
E

Jπ

11.666
4−
11.73
5−
11.9(3) (1− )
12.20
1−
12.61
2−
12.863
12.963 (3− )
(13.50)
13.7
2−

[25]
2n Transfer

13

[88]
C(p, n)

9

[25]
Be(7 Li, d)

Ref.

E

Jπ

E

Jπ

E

Jπ

[87]
[87]
[86]
[86]
[86]
[87]
[87]
[87]
[86]

11.66
11.73

4−
4+

11.7

4−

11.66
11.73

4−
4+

12.96(4)

3−

13.7(1)
14.0(1)

[89, 90, 91]
α-decay
E

Jπ

14.3(1)

[91]

14.8(1)

[89, 90, 91]

15.55(10)

[89, 90]

15.9

[90]

16.43(10)

[89, 90, 91]

17.3(1)

[91]

(18.1)
18.5(1)
18.6(1)

[91]
[89, 90]
[91]

(19.07(1))
(19.3(1))
19.83(1)

[89, 90, 91]
[91]
[89, 90, 91]

(20.3(1))
20.6(1)

[91]
[89, 91]

12.58 (3− )
12.86
12.96

12.96
13.56

14.03

14.54
(1− )
4+
(1+ )

[86]
[86]
[86]

15.20
4−
(15.37)
15.44
15.56
3−
15.8
(1− )
15.91
4+
(16.02)

[87]
[87]
[87]
[86]
[86]
[86]
[87]

16.43
(16.57)
16.715 (1+ )
17.30
4−
(17.5) (1+ )

[87]
[87]
[87]
[87]
[87]

14.63
14.717
14.91
15.01(15)

14.67(5)

6+

14.87(4)

5−

14.87

5−

14.67

6+

14.87

5−

15.18

(5− )

15.40

(2+ )

+

2

15.11
15.35
15.51
15.91

15.75(8)

16.22

16.71(15)

3−
−

17.50 (3 )
17.61(15)

(6+ )

18.6(1)

17.91 (2+ )
18.03 (7− )
(4− ) 18.39
18.60

16.72(5) (6+ )
17.3

4−

5−
17.98 (4+ )

18.51(15)

17.5(1)

16.43 6+
16.53 (2+ )
16.72 (1+ )
17.30 4−
17.52 (2+ )

16.4(1)

17.95
18.10

[87]
[87]

7−

18.53
18.8
18.86

19.01(15)

4+

19.91(15)

5−

20.71(15)

5−

4−

5−
19.14

19.76

5−

20.66

6+

19.8(1)
20.4

[87]

22.1

[87]

24.3
24.5

[87]
[87]

21.4(2)
22.5(3)
23.5

153

2+

19.73
20.02
20.75
21.00
21.41
23.2

(2+ )

21.43(1) [89, 90, 91]
22.45(3)
[89, 91]
(23.15(1))
[89, 91]
24.0(3)
[89]

Chapter 6
Conclusion

6.1

Summary

By studying elastic and inelastic scattering of 38.2 MeV
gas with the prototype AT-TPC, various resonances in

10

14

Be nuclei on a 4 He active target

C were observed. The beam was

delivered to the prototype AT-TPC by TwinSol at the University of Notre Dame. The
Micromegas detector in the prototype AT-TPC employed individual pad biasing, and the
geometry of the segmentation of the pads was optimized for triggering binary reactions. A
2+ elastic scattering resonance at 3.0 MeV and a 4+ elastic scattering resonance at 7.0 MeV
are strong candidates for a predicted [13] linear-chain positive-parity rotational band in
14

C. Preliminary analysis performed on the inelastic data shows strong branching of the

7.0 MeV 4+ resonance, also in agreement with the linear-chain prediction. These resonances
provide the first strong experimental evidence of a linear-chain structure in 14 C. This evidence
provides important information and insight into the more general discussion of α clustering in
light nuclei. This result also strongly supports the reliability of the β −γ constraint AMD for
the clustering of carbon isotopes. Another important conclusion from the study by Suhara
and Kanada-En’yo [13] is that the 0+
3 state of

12

C has a linear-chain structure, which is not

true in the molecular-orbit model [22]. Our results thus imply that this conjecture of 12 C by
the AMD framework is very probable and merits further experimental investigation. This

154

possible state may be a one-dimensional gas-like state, a novel nuclear structure proposed
by another work of Suhara [11], and this work points to a new horizon for cluster studies.
Negative-parity elastic resonances have also been investigated for possible band structure.
Preliminary analysis shows that a 6.5 MeV 7− elastic resonance may be part of a suggested
[25] prolate rotational band. The resonance may instead form a rotational band with the
observed 4.7 MeV 3− and 5.6 MeV 5− elastic resonances. There are also two additional 5−
elastic resonances observed at 7.9 and 8.7 MeV, with the former possibly having branching
to the inelastic. The general features of the negative-parity states are compatible with the
mixing of many configurations of rotational-like band structure.
This work has demonstrated the effectiveness of the principles of the AT-TPC. A He:CO2
90:10 active target gas successfully allowed for the measurement of resonances in

14

C across

an energy domain of nearly 40 MeV in the lab frame, about 10 MeV in the center-ofmass frame. The TPC capabilities provided large angular coverage for the measurements,
with probed center-of-mass angles ranging from 35 to 145◦ . This work employed the first
demonstrated use of the Micromegas amplification device’s gain variation for specific pads,
which was essential for triggering the reactions in the prototype AT-TPC. Along with other
experimental work performed with the prototype AT-TPC [9, 26, 27], this work has provided
important information for the design and construction of the now-built full-scale AT-TPC.

6.2

Outlook

The prototype AT-TPC will be of continued use as a complimentary device to the full-scale
AT-TPC. Its smaller size makes it portable for work at other labs, providing the opportunity
for studies with a large variety of beams.

155

While spin-parity assignments have been made for the seven of the observed elastic
resonances in

14

C, the values of their partial widths and spectroscopic factors could be

determined with better precision if needed. Further analysis will be done for the inelastic
data, including performing detailed R-matrix calculations similar to those made for the
elastic data. In addition to the strong evidence of linear-chain structure in a predicted
positive-parity rotational band, other band structures will be further investigated among the
observed elastic and inelastic resonances.
Many other light systems show specific quasi-molecular structures. A proposed experiment for 8 He + α scattering has been accepted for beam time at TRIUMF-ISAC, and will
be performed in the near future.

156

BIBLIOGRAPHY

157

BIBLIOGRAPHY
[1] I. J. Thompson and F. N. Nunes. Nuclear Reactions for Astrophysics. Cambridge
University Press, 2009.
[2] D. T. Khoa, D. C. Cuong, and Y. Kanada-En’yo. Hindrance of the excitation of the
12
Hoyle state and the ghost of the 2+
C. Phys. Lett. B, 695:469, 2011.
2 state in
[3] J. W. Bittner and R. D. Moffat. Elastic Scattering of Alpha Particles by Carbon. Phys.
Rev., 96:374, 1954.
[4] J. M. Blatt and V. F. Weisskopf. Theoretical Nuclear Physics. Dover, 1991. Originally
published by Springer-Verlag in 1979.
[5] A. M. Lane and R. G. Thomas. R-Matrix Theory of Nuclear Reactions. Rev. Mod.
Phys., 30:257, 1958.
[6] S. S. M. Wong. Introductory Nuclear Physics. Wiley-Interscience, 2nd edition, 1998.
[7] J. A. Wheeler. Molecular Viewpoints in Nuclear Structure. Phys. Rev., 52:1083, 1937.
[8] H. Yamaguchi, D. Kahl, Y. Wakabayashi, S. Kubono, T. Hashimoto, S. Hayakawa,
T. Kawabata, N. Iwasa, T. Teranishi, Y. K. Kwon, D. N. Binh, L. H. Khiem, and N. N.
Duy. α-resonance structure in 11 C studied via resonant scattering of 7 Be + α and with
the 7 Be(α, p) reaction. Phys. Rev. C, 87:034303, 2013.
[9] D. Suzuki, A. Shore, W. Mittig, J. J. Kolata, D. Bazin, M. Ford, T. Ahn, F. D. Becchetti,
S. Beceiro Novo, D. Ben Ali, B. Bucher, J. Browne, X. Fang, M. Febbraro, A. Fritsch,
E. Galyaev, A. M. Howard, N. Keeley, W. G. Lynch, M. Ojaruega, A. L. Roberts, and
X. D. Tang. Resonant α scattering of 6 He: Limits of clustering in 10 Be. Phys. Rev. C,
87:054301, 2013.
[10] H. Feldmeier and T. Neff. Clusters, Halos, and S-Factors in Fermionic Molecular Dynamics. EPJ Web of Conferences, 63:02021, 2013.
[11] T. Suhara, Y. Funaki, B. Zhou, H. Horiuchi, and A. Tohsaki. One-Dimensional α
Condensation of α-Linear-Chain States in 12 C and 16 O. Phys. Rev. Lett., 112:062501,
2014.
158

[12] W. Broniowski and E. R. Arriola. Signatures of α Clustering in Light Nuclei from
Relativistic Nuclear Collisions. Phys. Rev. Lett., 112:112501, 2014.
[13] T. Suhara and Y. Kanada-En’yo. Cluster structures of excited states in 14 C. Phys. Rev.
C, 82:044301, 2010.
[14] H. Feldmeier. Fermionic Molecular Dynamics. Nucl. Phys. A, 515:147, 1990.
[15] H. Horiuchi. Microscopic Study of Clustering Phenomena in Nuclei. Nucl. Phys. A,
522:257, 1991.
[16] P. Descouvemont and D. Baye. Microscopic theory of the 8 Be(α, γ)12 C reaction in a
three-cluster model. Phys. Rev. C, 36:54, 1986.
[17] H. Morinaga. Interpretation of Some of the Excited States of 4n Self-Conjugate Nuclei.
Phys. Rev., 101:254, 1956.
[18] A. H. Wuosmaa, R. R. Betts, B.B. Back, M. Freer, B. G. Glagola, Th. Happ, D. J.
Henderson, and P. Wilt. Evidence for Alpha-Particle Chain Configurations in 24 Mg.
Phys. Rev. Lett., 68:1295, 1992.
[19] F. Hoyle. On Nuclear Reactions Occurring in Very Hot Stars. 1. The Synthesis of
Elements from Carbon to Nickel. Astrophys. J. Suppl. Ser., 1:121, 1954.
[20] C. W. Cook, W. A. Fowler, C. C. Lauritsen, and T. Lauritsen. B12 , C12 , and the Red
Giants. Phys. Rev., 107:508, 1957.
[21] M. Freer, H. Fujita, Z. Buthelezi, J. Carter, R. W. Fearick, S. V. F¨ortsch, R. Neveling,
S. M. Perez, P. Papka, F. D. Smit, J. A. Swartz, and I. Usman. 2+ excitation of the
12
C Hoyle state. Phys. Rev. C, 80:041303, 2009.
[22] N. Itagaki, T. Otsuka, K. Ikeda, and S. Okabe. Equilateral-Triangular Shape in
Phys. Rev. Lett., 92:142501, 2004.

14

C.

[23] T. Suhara and Y. Kanada-En’yo. Quadrupole Deformation β and γ Constraint in a
Framework of Antisymmetrized Molecular Dynamics. Prog. Theor. Phys., 123:303,
2010.
[24] N. Itagaki, S. Okabe, K. Ikeda, and I. Tanihata. Molecular-orbital structure in neutronrich C isotopes. Phys. Rev. C, 64:014301, 2001.

159

[25] W. von Oertzen, H. Bohlen, M. Milin, Tz. Kokalova, S. Thummerer, A. Tumino,
R. Kalpakchieva, T. N. Massey, Y. Eisermann, G. Graw, T. Faestermann, R. Hertenberger, and H.-F. Wirth. Search for cluster structure of excited states in 14 C. Eur.
Phys. J. A, 21:193, 2004.
[26] D. Suzuki, D. Bazin, W. Mittig, W. G. Lynch, C. Hewko, A. Roux, D. Ben Ali,
J. Browne, E. Galyaev, M. Ford, A. Fritsch, J. Gilbert, F. Montes, A. Shore, G. Westfall, and J. Yurkon. Test of a micromegas detector with helium-based gas mixtures for
active target time projection chambers utilizing radioactive isotope beams. Nucl. Instr.
and Meth. Phys. Res. A, 660:64, 2011.
[27] D. Suzuki, M. Ford, D. Bazin, W. Mittig, W. G. Lynch, T. Ahn, S. Aune, E. Galyaev,
A. Fritsch, J. Gilbert, F. Montes, A. Shore, J. Yurkon, J. J. Kolata, J. Browne,
A. Howard, A. L. Roberts, and X. D. Tang. Prototype AT-TPC: Toward a new generation active target time projection chamber for radioactive beam experiments. Nucl.
Instr. and Meth. Phys. Res. A, 691:39, 2012.
[28] H. Sakurai. Physics opportunities with the RI Beam Factory at RIKEN. Eur. Phys. J.
Special Topics, 150:249, 2007.
[29] R. York. Realization of FRIB at Michigan State University. APS April Meeting 2010,
2010.
[30] D. Leitner, J. Popielarski, A. Lapierre, F. Montes, G. Perdikakis, S. Schwarz,
W. Wittmer, and X. Wu. Status of the ReAccelerator Facility ReA for Rare Isotopes.
2011 Fall Meeting of the APS Division of Nuclear Physics, 2011.
[31] S. Gales. SPIRAL2 at GANIL: A world leading ISOL facility at the dawn of the next
decade. Progress in Particle and Nuclear Physics, 59:22, 2007.
[32] E. Pollacco, S. Anvar, H. Baba, P. Baron, D. Bazin, C. Belkhiria, B. Blank, J. Chavas,
P. Chomaz, E. Delagnes, F. Druillole, P. Hellmuth, C. Huss, E. Galyaev, B. Lynch,
W. Mittig, T. Murakami, L. Nalpas, J. Pedroza, R. Raabe, J. Pibernat, B. Raine,
A. Rebii, A. Taketani anf F. Saillant, D. Suzuki, N. Usher, and G. Wittwer. GET: A
Generic Electronic System for TPCs for nuclear physics experiments. Physics Procedia,
37:1799, 2012.
[33] Y. Giomataris, Ph. Rebourgeard, J. P. Robert, and G. Charpak. MICROMEGAS: a
high-granularity position-sensitive gaseous detector for high particle-flux environments.
Nucl. Instr. and Meth. Phys. Res. A, 376:29, 1996.

160

[34] C. E. Demonchy, M. Caama˜
no, H. Wang, W. Mittig, P. Roussel-Chomaz, H. Savajols,
M. Chartier, D. Cortina-Gil, A. Fomichev, G. Fr´emont, P. Gangnant, A. Gillibert,
L. Giot, M. S. Golovkov, B. Jurado, J. F. Libin, A. Obertelli, E. Pollaco, A. Rodin,
Ch. Spitaels, S. Stepantsov, G. Ter-Akopian, and R. Wolski. MAYA: An active-target
detector for binary reactions with exotic beams. Nucl. Instr. and Meth. Phys. Res. A,
583:341, 2007.
[35] Y. Mizoi, T. Fukuda, Y. Matsuyama, T. Miyachi, J. Nakano, N. Fukuda, M. Hirai,
H. Kobinata, Y. X. Watanabe, H. Sakurai, Y. Watanabe, and A. Yoshida. Multiplesampling and tracking proportional chamber for nuclear reactions with low-energy radioactive isotope beams. Nucl. Instr. and Meth. Phys. Res. A, 431:112, 2010.
[36] K. Yamaguchi, H. Ishiyama, M. H. Tanaka, Y. X. Watanabe, H. Miyatake, Y. Hirayama,
N. Imai, H. Makii, Y. Fuchi, S. C. Jeong, T. Nomura, Y. Mizoi, S. K. Das, T. Fukuda,
T. Hashimoto, and I. Arai. Development of the GEM-MSTPC for measurements of
low-energy nuclear reactions. Nucl. Instr. and Meth. Phys. Res. A, 623:135, 2010.
[37] D. R. Nygren. The Time-Projection Chamber - A new 4π detector for charged particles.
Proceedings of the 1974 PEP Summer Study, PEP-144, page 58, 1974.
[38] D. R. Nygren. The Time-Projection Chamber. Proceedings of the 1975 PEP Summer
Study, PEP-198, page 126, 1975.
[39] D. Fancher, H. J. Hilke, S. Loken, P. Martin, J. N. Marx, D. R. Nygren, P. Robrish,
G. Shapiro, M. Urban, and W. Wenzel. Performance of a Time-Projection Chamber.
Nucl. Instr. and Meth., 161:383, 1979.
[40] S. Benso and L. Rossi. A Time Projection Chamber for Precise Vertex Determination.
Nucl. Instr. and Meth., 187:355, 1981.
[41] G. F. Knoll. Radiation Detection and Measurement. Wiley, 4th edition, 2010.
[42] F. Sauli. GEM: A new concept for electron amplification in gas detectors. Nucl. Instr.
and Meth. Phys. Res. A, 386:531, 1997.
[43] T. Suhara and Y. Kanada-En’yo. Cluster structures of excited states in
of Physics: Conference Series, 321:012047, 2012.

14

C. Journal

[44] A. Ono, H. Horiuchi, T. Maruyama, and A. Ohnishi. Antisymmetrized Version of
Molecular Dynamics with Two-Nucleon Collisions and Its Application to Heavy Ion
Reactions. Prog. Theor. Phys., 87:1185, 1992.
161

[45] K. Varga, Y. Suzuki, and R. G. Lovas. Microscopic multicluster description of neutronhalo nuclei with a stochastic variational method. Nucl. Phys. A, 571:447, 1994.
[46] Y. Ogawa, K. Arai, Y. Suzuki, and K. Varga. Microscopic four-cluster description of
10
Be and 10 C with the stochastic variational method. Nucl. Phys. A, 673:122, 2000.
[47] J. Aichelin and H. St¨ocker. Quantum Molecular Dynamics - A Novel Approach to
N -Body Correlations in Heavy Ion Collisions. Phys. Lett. B, 176:14, 1986.
[48] A. Ohnishi, T. Maruyama, and H. Horiuchi. Near- and Far-Side Contributions to Ejectile
Polarization. Prog. Theor. Phys., 87:417, 1992.
[49] C. Hartnack, L. Zhuxia, L. Neise, G. Peilert, A. Rosenhauer, H. Sorge, J. Aichelin,
H. St¨ocker, and W. Greiner. Quantum Molecular Dynamics: A Microscopic Model from
UNILAC to CERN Energies. Nucl. Phys. A, 495:303, 1989.
[50] P. Kramer and M. Saraceno. Geometry of the Time-Dependent Variational Principle in
Quantum Mechanics. Lecture Notes in Physics, 140:41, 1981.
[51] A. B. Volkov. Equilibrium Deformation Calculations of the Ground State Energies of
the 1p Shell Nuclei. Nucl. Phys., 74:33, 1965.
[52] N. Yamaguchi, T. Kasahara, S. Nagata, and Y. Akaishi. Effective Interaction with
Three-Body Effects. Prog. Theor. Phys., 62:1018, 1979.
[53] S. Okabe and Y. Abe. The Structure of 9 Be by a Molecular Model. II. Prog. Theor.
Phys., 61:1049, 1979.
[54] N. Itagaki and S. Okabe. Molecular Orbital Structures in 10 Be. Phys. Rev. C, 61:044306,
2000.
[55] F.D. Becchetti, M.Y. Lee, T.W. O’Donnell, D.A. Roberts, J.J. Kolata, L.O. Lamm,
G. Rogachev, V. Guimar˜aes, P.A. DeYoung, and S. Vincent. The TwinSol low-enegy
radioactive nuclear beam apparatus: status and recent results. Nucl. Instr. and Meth.
Phys. Res. A, 505:377, 2003.
[56] University of Notre Dame. http://library.nd.edu/chemistry/about/history/physics.shtml.
[57] ISNAP: MC-Snics Sputter Ion Source. http://isnap.nd.edu/html/research SNICS.html.

162

[58] ISNAP:
FN
Tandem
Van
http://isnap.nd.edu/html/research FN.html.

de

Graaff

Accelerator.

[59] ISNAP: Radioactive Nuclear Beams (RNB). https://isnap.nd.edu/html/research radio.html.
[60] W. M. Haynes, editor. CRC Handbook of Chemistry & Physics. CRC Press, 92nd
edition, 1991.
[61] R. Veenhof. Garfield, recent developments. Nucl. Instr. and Meth. Phys. Res. A,
419:726, 1998.
[62] S. F. Biagi.
Magbolts – transport
http://magboltz.web.cern.ch/magboltz/.

of

electrons

in

gas

mixtures.

[63] M. Anderson, F. Bieser, R. Bossingham, D. Cebra, E.L. Hjort, S.R. Klein, S. Kleinfelder,
C.Q. Vu, and H. Wieman. A readout system for the STAR time projection chamber.
Nucl. Instr. and Meth. Phys. Res. A, 499:679, 2003.
[64] I. Giomataris, R. De Oliveria, S. Andriamonje, S. Aune, G. Charpak, P. Colas,
G. Fanourakis, E. Ferrer, A. Giganon, Ph. Rebourgeard, and P. Salin. Micromegas
in a Bulk. Nucl. Instr. and Meth. Phys. Res. A, 560:405, 2006.
[65] P. Baron, D. Calvet, E. Delagnes, X. de la Broise, A. Delbart, F. Druillole, E. Mazzucato,
E. Monmarthe, F. Pierre, and M. Zito. AFTER, an ASIC for the Readout of the large
T2K Time Projection Chambers. IEEE Transactions on Nuclear Science, 55(3), 2008.
[66] Y. Kudenko. The near neutrino detector for the T2K experiment. Nucl. Instr. and
Meth. Phys. Res. A, 598:289, 2009.
[67] SRIM: The Stopping and Range of Ions in Matter. http://www.srim.org.
[68] J. F. Ziegler, M. D. Ziegler, and J. P. Biersack. SRIM - The stopping and range of ions
in matter. Nucl. Instr. and Meth. Phys. Res. B, 268:1818, 2010.
[69] LISE++:
Exotic
Beam
http://lise.nscl.msu.edu/lise.html.

Production

with

Fragment

Separators.

[70] O. B. Tarasov and D. Bazin. LISE++: Radioactive beam production with in-flight
separators. Nucl. Instr. and Meth. Phys. Res. B, 4657:266, 2008.
[71] L. C. Chamon. The S˜ao Paulo Potential. Nucl. Phys. A, 787:198, 2007.
163

[72] L. C. Chamon, B. V. Carlson, L. R. Gasques, D. Pereira, C. De Conti, M. A. G. Alvarez,
M. S. Hussein, M. A. Cˆandido Ribeiro, Jr. E. S. Rossi, and C. P. Silva. Toward a global
description of the nucleus-nucleus interaction. Phys. Rev. C, 66:014610, 2002.
[73] TUNL Nuclear Data Project. http://www.tunl.duke.edu/nucldata/.
[74] V. Avrigeanu and P. E. Hodgson. Global optical potentials for emitted alpha particles.
Phys. Rev. C, 49:2136, 1994.
[75] P. David, J. Debrus, H. Mommsen, and A. Riccato. Elastic Scattering of α-Particles on
10
B for Eα = 5 − 30 MeV. Nucl. Phys. A, 182:234, 1972.
[76] F. P. Brady, J. A. Jungerman, and J. C. Young. Elastic Alpha-Particle Scattering at
8.75, 9.35, and 10.15 MeV. Nucl. Phys. A, 98:241, 1967.
[77] P. Descouvemont and D. Baye. The R-matrix theory. Rep. Prog. Phys., 73:036301,
2010.
[78] T. Y. Li and B. Hird. Analysis of the C12 (p, α)B9 Reaction at 44.5 MeV. Phys. Rev.,
174:1130, 1968.
[79] M. Freer, N. L. Achouri, C. Angulo, N. I. Ashwood, D. W. Bardayan, S. Brown, W. N.
Catford, K. A. Chipps, N. Curtis, P. Demaret, C. Harlin, B. Laurent, J. D. Malcolm,
M. Milin, T. Munoz-Britton, N. A. Orr, S. D. Pain, D. Price, R. Raabe, N. Soi´c, J. S.
Thomas, C. Wheldon, G. Wilson, and V. A. Ziman. Resonances in 11 C observed in the
4
He(7 Be, α)7 Be and 4 He(7 Be, p)10 B reactions. Phys. Rev. C, 85:014304, 2012.
[80] T. Suhara and Y. Kanada-En’yo. 14 Be + α Correlation in 14 C. Proceedings of Workshop
on Hadron and Nuclear Physics, HNP09, page 366, 2010.
[81] P. Chevallier, F. Scheibling, G. Goldring, I. Plesser, and M. W. Sachs. Breakup of O16
into Be8 + Be8 . Phys. Rev., 160:827, 1967.
[82] Y. Suzuki, H. Horiuchi, and K. Ikeda. Study of α Chain States through Their Decay
Widths. Prog. Theor. Phys., 47:1517, 1972.
[83] M. Freer, M. P. Nicoli, S. M. Singer, C. A. Bremner, S. P. G. Chappell, W. D. M. Rae,
I. Boztosun, B. R. Fulton, D. L. Watson, B. J. Greenhalgh, G. K. Dillon, R. L. Cowin,
and D. C. Weisser. 8 Be + 8 Be decay of excited states in 16 O. Phys. Rev. C, 70:064311,
2004.

164

[84] M. Freer, N. M. Clarke, N. Curtis, B. R. Fulton, S. J. Hall, M. J. Leddy, J. S. Pople,
G. Tungate, R. P. Ward, P. M. Simmons, W. D. M. Rae, S. P. G. Chappell, S. P. Fox,
C. D. Jones, D. L. Watson, G. J. Gyapong, S. M. Singer, W. N Catford, and P. H.
Regan. 8 Be and α decay of 16 O. Phys. Rev. C, 51:1682, 1995.
[85] J. Malcolm. Study of Excited States in 14 C via the 4 He(10 Be, α)10 Be Resonance Scattering Reaction. PhD dissertation, The University of Birmingham, 2012.
[86] D. A. Resler, H. D. Knox, P. E. Koehler, and R. O. Lane. Structure of 14 C via elastic
and inelastic neutron scattering from 13 C: Measurement and R-matrix analysis. Phys.
Rev. C, 39:766, 1989.
[87] F. Ajzenberg-Selove. Energy Levels of Light Nuclei A = 13 − 15. Nucl. Phys. A, 523:1,
1991.
[88] E. Korkmaz, S. E. Vigdor, W. W. Jacobs, T. G. Throwe, L. C. Bland, M. C. Green,
P. L. Jolivette, and J. D. Brown. 12,13 C(p, π ± ) reactions at Tp = 200 MeV. Phys. Rev.
C, 40:813, 1989.
[89] N. Soi´c, M. Freer, L. Donadille, N. M. Clarke, P. J. Leask, W. N Catford, K. L. Jones,
D. Mahboub, B. R. Fulton, B. J. Greenhalgh, D. L. Watson, and D. C. Weisser. 4 He
decay of excited states in 14 C. Phys. Rev. C, 68:014321, 2003.
[90] D. L. Price, M. Freer, S. Ahmed, N. I. Ashwood, N. M. Clarke, N. Curtis, P. McEwan,
C. J. Metelko, B. Novatski, S. Sakuta, N. Soi´c, D. Stepanov, and V. Ziman. Alpha-decay
of excited states in 13 C and 14 C. Nucl. Phys. A, 765:263, 2006.
[91] D. L. Price, M. Freer, N. I. Ashwood, N. M. Clarke, N. Curtis, L. Giot, V. Lima,
P. McEwan, B. Novatski, N. A. Orr, S. Sakuta, J. A. Scarpaci, D. Stepanov, and
V. Ziman. α decay of excited states in 14 C. Phys. Rev. C, 75:014305, 2007.

165