SEARCHING FOR THE ORIGINS OF FLUORINE: A MEASUREMENT OF THE
19 F(t,3 He) REACTION
By
Amanda Margaret Prinke

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics
2012

ABSTRACT
SEARCHING FOR THE ORIGINS OF FLUORINE: A MEASUREMENT OF
THE 19 F(t,3 He) REACTION
By
Amanda Margaret Prinke
The astrophysical process responsible for fluorine nucleosynthesis has been a matter of
significant debate. The β-decay of 19 O∗ in core-collapse supernova has been previously suggested as a possible avenue for fluorine nucleosynthesis. In hot astrophysical environments,
it is possible to thermally populate low-lying excited states of 19 O, and the β-decay of these
excited states would enhance the overall β-decay rate into 19 F.
To examine this theory from an experimental nuclear physics perspective, the 19 F(t,3 He)
reaction differential cross section at 345 MeV was measured. This measurement was performed at the National Superconducting Cyclotron Laboratory using a secondary triton
beam impinged on a Teflon (CF2 ) target. The 3 He ejectiles were momentum-analyzed in the
S800 spectrograph. A Gamow-Teller strength distribution was extracted from the reaction
cross sections. The Gamow-Teller strength for the transition between the ground state of
19 F

and the 0.096 MeV excited state of 19 O was then used to find the 19 O∗ β-decay rate to

the 19 F ground state over a range of astrophysical temperatures.
The calculated β-decay rate for 19 O∗ (β − )19 F with the new experimental measurement
taken into account did show a modest enhancement for high stellar temperatures. However,
the magnitude of this enhancement was much lower than prior theory calculations had predicted. The β-decay rate enhancement is not large enough to make a significant impact on
future fluorine nucleosynthesis simulations.

TABLE OF CONTENTS

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

v

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

vii

Chapter 1 Astrophysics Motivation . . .
1.1 Introduction . . . . . . . . . . . . . . .
1.2 Nucleosynthesis . . . . . . . . . . . . .
1.3 Neutrino Nucleosynthesis of Fluorine .
1.4 Wolf-Rayet Star Production of Fluorine
1.5 Fluorine Found in AGB Stars . . . . .

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1
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14

Chapter 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Gamow-Teller Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Distorted Wave Born Approximation . . . . . . . . . . . . . . . . . . . . . .

22
22
26

Chapter 3 Experiment . . .
3.1 Triton Beam Production
3.2 Targets . . . . . . . . . .
3.3 S800 Spectrograph . . .

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34
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Chapter 4 Data Analysis . . . . . . . . . . . . . . . . .
4.1 Extracting Excitation Energy Spectra . . . . . . . . .
4.1.1 Particle Identification . . . . . . . . . . . . . .
4.1.2 Cathode Readout Drift Chamber Calibration .
4.1.3 Ejectile Parameters at the Secondary Target .
4.1.4 Missing Mass Calculation . . . . . . . . . . .
4.1.5 Excitation Energy Corrections . . . . . . . . .
4.2 Differential Cross Sections . . . . . . . . . . . . . . .
4.2.1 Relative Cross Sections . . . . . . . . . . . . .
4.2.2 Absolute Cross Sections . . . . . . . . . . . .
4.2.3 Isolating the 19 F Cross Section . . . . . . . .
4.3 Calculating Gamow-Teller Strengths . . . . . . . . .
4.3.1 Multipole Decomposition Analysis . . . . . . .
4.3.2 Extrapolating the Cross Section to Q = 0 . . .
4.3.3 Gamow-Teller Strength Distribution . . . . .

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45
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iii

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Chapter 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 β-decay Rate Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Discussion of Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Experimental Errors . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Theoretical Cross Section Errors . . . . . . . . . . . . . . . . . . . .
5.3.3 Gamow-Teller Strength Errors . . . . . . . . . . . . . . . . . . . . .
5.3.3.1 Statistical Error . . . . . . . . . . . . . . . . . . . . . . .
5.3.3.2 Systematic Error in the Multipole Decomposition Analysis
5.3.3.3 Systematic Error in the B[GT] Calculation . . . . . . . . .
5.3.3.4 Systematic Error from the Tensor Interaction . . . . . . .
5.3.3.5 Summary of Errors Related to Gamow-Teller Strengths . .
5.3.4 β-decay Rate Errors . . . . . . . . . . . . . . . . . . . . . . . . . .

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88
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101
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107
108
109

Chapter 6 Summary . . . . . . . .
6.1 Summary . . . . . . . . . . . .
6.2 Outlook . . . . . . . . . . . . .
6.2.1 Fluorine Nucleosynthesis
6.2.2 Nuclear Structure . . . .

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112
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Bibliography . . . . . . . . . . . . . . .

iv

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. . . . . . . . . . . . . . . . . . . 119

LIST OF TABLES

Table 1.1

Stellar classifications of AGB stars . . . . . . . . . . . . . . . . . . .

16

Table 2.1

Single-particle binding energies . . . . . . . . . . . . . . . . . . . . .

30

Table 2.2

Optical model potential parameters . . . . . . . . . . . . . . . . . .

32

Table 3.1

S800 spectrograph focal plane voltages . . . . . . . . . . . . . . . . .

44

Table 4.1

Measured ejectile parameters . . . . . . . . . . . . . . . . . . . . . .

46

Table 4.2

Calculated ejectile parameters . . . . . . . . . . . . . . . . . . . . .

47

Table 4.3

Particle identification plot corrections . . . . . . . . . . . . . . . . .

50

Table 4.4

Cathode readout drift chamber calibrations . . . . . . . . . . . . . .

53

Table 4.5

S800 spectrograph settings . . . . . . . . . . . . . . . . . . . . . . .

55

Table 4.6

Mass excesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Table 4.7

Excitation energy corrections . . . . . . . . . . . . . . . . . . . . . .

60

Table 4.8

Run-specific corrections . . . . . . . . . . . . . . . . . . . . . . . . .

61

Table 4.9

Monte-Carlo boundary equations . . . . . . . . . . . . . . . . . . . .

65

Table 4.10

Extrapolation to Q = 0 . . . . . . . . . . . . . . . . . . . . . . . . .

81

Table 4.11

Experimental Gamow-Teller strengths . . . . . . . . . . . . . . . . .

86

Table 4.12

Predicted Gamow-Teller strengths . . . . . . . . . . . . . . . . . . .

87

Table 5.1

19 F

energy levels relevant for decay rate calculations . . . . . . . . .

90

Table 5.2

Calculation of the β-decay rate for 19 O∗ (β − )19 F . . . . . . . . . . .

93

v

Table 5.3

Experimental 19 Og.s. (β − )19 F∗ decay rates . . . . . . . . . . . . . . .

Table 5.4

Errors from MDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

vi

96

LIST OF FIGURES

Figure 1.1

Core-collapse supernova . . . . . . . . . . . . . . . . . . . . . . . . .

6

Figure 1.2

19 O(β − )19 F

level scheme . . . . . . . . . . . . . . . . . . . . . . . .

9

Figure 1.3

Wolf-Rayet star . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Figure 1.4

Asymptotic giant branch star . . . . . . . . . . . . . . . . . . . . . .

15

Figure 1.5

Nuclear paths for fluorine nucleosynthesis in AGB stars . . . . . . .

20

Figure 2.1

Global unit cross section fit . . . . . . . . . . . . . . . . . . . . . . .

26

Figure 2.2

Kinematic factor of σGT . . . . . . . . . . . . . . . . . . . . . . . .
ˆ

27

Figure 3.1

Experiment layout . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Figure 3.2

A1900 fragment separator . . . . . . . . . . . . . . . . . . . . . . . .

37

Figure 3.3

S800 spectrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Figure 3.4

Cathode Readout Drift Chamber . . . . . . . . . . . . . . . . . . . .

43

Figure 4.1

Particle identification spectra . . . . . . . . . . . . . . . . . . . . . .

48

Figure 4.2

Mask schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Figure 4.3

Mask calibration data . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Figure 4.4

Focal plane shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Figure 4.5

Initial excitation energy spectra . . . . . . . . . . . . . . . . . . . .

62

Figure 4.6

Solid angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Figure 4.7

Monte-Carlo simulation boundaries . . . . . . . . . . . . . . . . . .

66

vii

Figure 4.8

Relative cross section . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Figure 4.9

Absolute scaling factor . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 4.10

Cross section before carbon subtraction . . . . . . . . . . . . . . . .

72

Figure 4.11

Angular distribution example . . . . . . . . . . . . . . . . . . . . . .

75

Figure 4.12

Differential cross sections . . . . . . . . . . . . . . . . . . . . . . . .

76

Figure 4.13

Differential cross sections, continued . . . . . . . . . . . . . . . . . .

77

Figure 4.14

Gamow-Teller strength for the first two exited states in 19 O . . . . .

83

Figure 4.15

Gamow-Teller strength distribution . . . . . . . . . . . . . . . . . .

85

Figure 5.1

Population of the first excited state of 19 O for stellar temperatures .

94

Figure 5.2

β-decay rate λtot

96

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

viii

Chapter 1
Astrophysics Motivation

1.1

Introduction

Experimental nuclear physics provides a unique opportunity to study the innermost workings of stars in the laboratory. The underlying physics for a nuclear reaction in a star has
much in common with nuclear reactions that are produced with a particle accelerator at a
research facility. One specific area where experimental nuclear physics has made a significant contribution to astrophysics is in the field of nucleosynthesis research. Nuclear reaction
rates measured in laboratories are incorporated into simulations that predict the elemental
abundances of stars.
This dissertation describes the measurement of a particular nuclear reaction, 19 F(t,3 He)
at 115 MeV/nucleon, to test a hypothesis on fluorine nucleosynthesis in core-collapse supernovae. The rest of this chapter discusses the motivation for investigating fluorine nucleosynthesis in general as well as the specific reason for studying the 19 F(t,3 He) in particular.
Theoretical principals that are important to charge-exchange reactions and relevant to the
experiment’s data analysis are explained in Chapter 2. Chapter 3 describes the experiment
that was undertaken for this work. The data analysis of this experiment is explicated in
Chapter 4; the results of the data analysis and their astrophysical implications are discussed
in Chapter 5. A summary of the dissertation is provided in Chapter 6.

1

1.2

Nucleosynthesis

The study of nucleosynthesis endeavors to answer one of the most fundamental questions
of science: where did we come from? We have a strong grasp on the origin of some elements
[1, 2, and references therein]. Hydrogen (both 1 H and 2 H) and most helium was produced
in the first moments of the universe during the Big Bang. A smaller fraction of helium is
generated by hydrogen-burning processes in stellar environments. Carbon is formed from
the triple-alpha process in helium-burning stars. Massive helium-burning stars also produce
16 O

and 18 O. The slow neutron capture process, referred to as the s-process [3], explains the

solar abundances of many isotopes in the mass range above iron up to bismuth, including
krypton, xenon, barium, and lanthanum.
More complex astrophysical processes account for still more elements and for specific
isotopes. The s-process is typically characterized by (n, γ) reactions at a time scale that
is much slower than the intervening β-decays (over several thousand years) and is thought
to occur primarily in asymptotic giant branch (AGB) stars (see Section 1.5). The rapid
neutron capture process, referred to as the r-process [3], may explain many of the isotopes
with 70 ≤ A ≤ 209 (where A is the nucleon number). The r-process is typically characterized
by neutron captures on a very short time scale compared to the relevant β-decays (less than
10 seconds) and is thought to occur in core-collapse supernovae (see Section 1.3) or in merging
neutron stars. Specifically, the r-process accounts for several neutron-rich stable nuclei that
are beyond the reach of the s-process, such as 122 Sn and 130 Te (the s-process is essentially
blocked for these nuclei because there are intervening unstable nucleus in their isotopic chains,
121 Sn and 129 Te respectively, which will β-decay before more s-process neutrons are captured,

meaning that 122 Sn and 130 Te are only accessible via rapid neutron captures). The r-process

2

may also account for naturally occurring thorium and uranium isotopes, which are too far
beyond the stable isotopes to be produced by the s-process. The neutrino nucleosynthesis
process occurs when supernovae release large amounts of neutrinos, and these neutrinos
interact with stellar material to produce low-abundance isotopes [4, 5]. The neutrino process
might account for the solar abundance of isotopes such as 138 La and 180 Ta, and it may
partially contribute to the solar abundance of 11 B [6].
Several other astrophysical processes are also being explored as explanations for observed
astrophysical abundance patterns. These include carbon, neon, oxygen, or silicon burning in
massive stars, as well as proton capture (p-process) and rapid proton capture (rp-process).
The origin of fluorine, however, has been a persistent mystery despite significant research
efforts.
Fluorine nucleosynthesis is a challenging area of research. Theoretical efforts to model
fluorine production in stellar environments have frequently been stymied by inadequate nuclear reaction rate data. Recently, there have been significant experimental improvements
in many of the nuclear reaction rates tied to fluorine production; several recent measurements will be discussed later in this chapter. Hopefully, these reaction rate improvements
(including the rate that is the subject of this dissertation) will pave the way for a more
definitive theoretical explanation for fluorine nucleosynthesis. However, the biggest obstacle
to creating a reliable model of fluorine production is arguably the “fragility” of the fluorine
nucleus, meaning that 19 F is easily destroyed in stellar environments via neutron capture
19 F(n, γ)20 F

, alpha capture

19 F(α, p)22 Ne

, or proton capture

19 F(p, α)16 O

. This

fragile nature leads to fluorine having a very low solar abundance relative to neighboring isotopes; it has the lowest abundance in the range of elements between carbon and calcium [7].
Efforts to explain fluorine production focus primarily on this problem and search for ways to
3

preserve significant quantities of fluorine from premature destruction. So far, no explanation
offered for fluorine production fully explains the measured astrophysical abundances.
Another major challenge in this field is the astrophysical observation of fluorine. The low
abundance of fluorine and its particular physical properties coalesce to make it very hard
to detect [8]. Most of our knowledge of galactic fluorine abundances comes from the stellar
absorption lines of HF molecules, which are found only in cool stars or in sunspots. Many
of the vibration-rotation absorption lines from HF overlap with lines from other common
molecules, like 12 C14 N [8, 9]. This makes the detection of a low-abundance element like
fluorine very difficult. It also leads to some model-dependence in fluorine abundance measurements, where different predictions of contaminants can heavily influence the final fluorine
abundance determination. Absorption lines from elemental fluorine have been observed in
hotter stars, but such measurements are difficult because these absorption lines fall on the
edge of the visible spectrum near the ultraviolet region [10, 11] or in the far ultraviolet region
of the spectrum [12, 13].
In fact, the one and only measure to date of our own Sun’s fluorine abundance was performed in 1969 by examining the spectral lines produced by the molecule HF in sunspots [14].
The molecule HF only forms in cool stars or in relatively cool locations like sunspots; most
fluorine in the Sun and similar stars is elemental. Hall and Noyes found a solar abundance1
of A(F) = 4.56 ± 0.33 in the Sun. The only other measure available for the solar system’s
fluorine abundance comes from abundance measurements performed on the CI group of car1 Stellar

abundances of elements are quantified using a common spectroscopic notation:
A(x) = log

N (x)
+ 12.00
N (H)

(1.2.1)

indicates how abundant an element x is compared to the hydrogen in the same star. The
scale is defined such that A(H) = 12.00.

4

bonaceous chondrite meteorites [15], which found a value of A(F) = 4.46 ± 0.06. At a glance,
the meteoric abundance seems like the best choice as a solar system standard for fluorine
abundance because of its lower uncertainty, but an earlier review of the meteoric data openly
questioned the quality of these measurements [16]. This suggests that the more conservative
sunspot measurement with a larger error may be the better option. Fortunately, the sunspot
and meteoric abundances are in agreement with each other, but the larger uncertainty of the
sunspot measurement can make it difficult to determine whether other stars are exhibiting
a significant overabundance of fluorine compared to our solar system.
Three astrophysical sites have been proposed for fluorine nucleosynthesis. All of these
sites were suggested because they offer an opportunity to preserve fragile 19 F from the
capture processes that would otherwise destroy it. Core-collapse supernovae might produce fluorine through an unusual neutrino nucleosynthesis process and eject it prior to its
destruction into the interstellar medium through a supernova explosion (see Section 1.3).
Wolf-Rayet stars might produce fluorine via helium-burning and then eject it into the interstellar medium in extreme solar winds, preserving it from destruction (see section 1.4).
Asymptotic giant branch (AGB) stars might produce fluorine via helium-burning and then
eject it into planetary nebulae (see Section 1.5).

1.3

Neutrino Nucleosynthesis of Fluorine

The first theory to attempt to tackle fluorine nucleosynthesis was proposed by Woosley
and Haxton [4]. They hypothesized that solar fluorine abundances could be produced in
the carbon and neon shells of core-collapse (Type II) supernovae due to interactions with
neutrinos (see Figure 1.1). Woosley and Haxton postulated that neutrinos produced in

5

Figure 1.1: This is an artist’s impression of the remnants of the core-collapse supernova SN
1987A. Image credit: ESO/L. Cal¸ada [17]. For interpretation of the references to color in
c
this and all other figures, the reader is referred to the electronic version of this dissertation.
the supernova go through inelastic collisions with the abundant 20 Ne found in the collapsing star. These inelastic collisions excite the 20 Ne nuclei, which can then emit a proton:
20 Ne(ν, ν

p)19 F. The cross section for inelastic neutrino collisions is very low: the average

cross section expected for a collision between a 20 Ne nucleus and a neutrino from a corecollapse supernova is less than one attobarn (10−18 barn) [18]. However, a core-collapse
supernova produces a large flux of neutrinos: roughly 1058 neutrinos are released in a corecollapse supernova, accounting for about 1053 ergs of released energy [6, 18]. This large flux
of neutrinos counteracts the low reaction cross section, making neutrino nucleosynthesis a
potentially important method of production for certain low-abundance nuclides such as 19 F,
138 La,

and 180 Ta. Early core-collapse supernova simulations [18, 19] gave promising results

for fluorine production via neutrino nucleosynthesis, and a study by Renda et al. [20] em-

6

ploying a galactic chemical evolution model indicated that solar fluorine abundances could
be explained fully by neutrino nucleosynthesis in core-collapse supernovae. Additionally,
several astrophysical observations of extrasolar fluorine abundances [21, 22] are consistent
with these early neutrino model predictions.
Despite this early success, more recent simulations and observations present a different picture. Heger et al. [6] found that revised core-collapse supernovae simulations predicted significantly less fluorine production than prior efforts. Advances in the field of neutrino physics, especially an enhanced understanding of neutrino oscillations, drove this revised study. Improved stellar models and nuclear reaction rates also impacted the newer
simulations. An experimental measurement of the Gamow-Teller strength distribution for
20 Ne(p, n)20 Na

[23] was included in this recent simulation (see Section 2.1 for an explanation

of Gamow-Teller strength). This experimental measurement had a particularly large impact
on the simulated fluorine production, because the measured Gamow-Teller strength distribution was found to be much lower than earlier shell-model calculations had predicted. The
full impact on predictions of core-collapse supernovae fluorine production was a reduction of
about 40% from the earlier estimates of Woosley et al. [6, 19]. An additional study by Austin
et al. on the sensitivity of neutrino nucleosynthesis to variations in certain poorly-known
reaction rates showed large fluctuations in fluorine production, suggesting that supernovae
account for 25% to 75% of the solar fluorine abundance. With the lower fluorine production
rates of Heger et al., one cannot attribute a solar fluorine abundance solely to core-collapse
supernovae. On the experimental front, Federman et al. [24] set out to measure ν-process
fluorine in stars that had formed in the remnants of known core-collapse supernovae, but they
did not see any evidence of enhanced fluorine abundances. The revised, significantly lower
theoretical predictions of fluorine abundance and the lack of direct experimental evidence for
7

neutrino nucleosynthesis of fluorine indicate that different methods of fluorine production
should be considered.
The reaction investigated in this dissertation, 19 F(t,3 He)19 O∗ , is most relevant to supernova environments. However, the method of fluorine production examined here differs
significantly from the traditional neutrino nucleosynthesis scenario described above. Instead,
this dissertation seeks to examine the possibility that fluorine is produced by neutron capture
on 18 O to form 19 O, followed by a β-decay to 19 F:

18 O(n, γ)19 O(β − )19 F.

(1.3.1)

The isotope 18 O is already produced primarily during helium-burning in the massive stars
that are the progenitors of core-collapse supernovae [5]. The neutrons needed for this process
could come from the supernova explosion itself. Core-collapse supernovae have the potential
to emit sufficient quantities of neutrons for r-process nucleosynthesis to occur [3, 5], so there
is potential for some 19 O to form and decay to 19 F during the supernova explosion.
Supernovae are particularly relevant to the 19 F(t,3 He)19 O∗ reaction because of the high
temperatures achieved during the explosion. At these high temperatures, a significant portion of 19 O will achieve equilibrium in the very low-lying first excited state at 0.096 MeV,
making it possible to β-decay from this excited state in 19 O∗ to 19 F. This is interesting because β-decay between the ground state of 19 O and the ground state of 19 F is forbidden (the
+

5
spin-parity assignment of the 19 O ground state is J π = 2 , while the spin-parity assignment
1+
of the 19 O ground state is J π = 2 [25]). However, the low-lying excited state in 19 O at
+

0.096 MeV has a spin and parity of J π = 3 , which means that β-decay from this 0.096
2
MeV excited state in 19 O to the ground state of 19 F is allowed (see Figure 1.2).

8

19

O Β

19

F Level Scheme
0.1 3 2
0.0 5 2
19

Excited state decays

O

Energy MeV

Ground state decays

32

1.6

52
12

0.2
0.0
19

F

Figure 1.2: This level scheme depicts the allowed β-decays from 19 O to 19 F. Each arrow
represents an allowed β-decay, while the thickness of the arrow roughly corresponds to the
expected branching ratio (thicker arrows signify more probable transitions). The spin and
parity for each level is indicated in parentheses. Information on these energy levels was taken
from Tilley et al. [25].

9

This opens the door for a significant increase in the β-decay rate for 19 O in a supernova
compared to the β-decay rate that has been measured in laboratory conditions. The first
excited state of 19 O decays to the ground state by γ-ray emission in roughly 1 ns, so it
isn’t viable to measure the excited state β-decay through a direct experiment. Chernyhk
et al. [26] performed a theoretical calculation to estimate the reaction rate increase that
occurs when the 0.096 MeV state in 19 O is taken into account. Their calculations suggested
that the 19 O(β − )19 F β-decay rate might increase by a factor of three at high stellar temperatures, which indicates that this reaction is an excellent candidate for further experimental
investigation. In Section 4.3.3 of this dissertation, the strength of the transition from the
0.096 MeV state of 19 O to the ground state of 19 F is extracted experimentally through study
of the 19 F(t,3 He)19 O∗ reaction.

1.4

Wolf-Rayet Star Production of Fluorine

A different astrophysical site proposed for fluorine nucleosynthesis is the Wolf-Rayet
star [27] (see Figure 1.3). A Wolf-Rayet star is a type of hot, giant star characterized by
powerful stellar winds. These stars typically have temperatures above 30,000 K and have 10
to 25 times the mass of the Sun (but they are occasionally much more massive) [28]. The
distinctive stellar winds of a Wolf-Rayet star are driven by photons transferring momentum
via absorption to material in the outer atmosphere of the star. This radiative pressure drives
very high winds that in turn remove large quantities of material from the star: winds of up to
6000 km have been measured, and calculations predict that the winds from Wolf-Rayet stars
s
eject material greater than the mass of the Earth in a year. This mass-loss mechanism makes
Wolf-Rayet stars ideal candidates for fluorine production; any fluorine that is produced by

10

Figure 1.3: This image shows a Wolf-Rayet star, WR124. It is surrounded by a nebula, M167, that was formed from material ejected by the Wolf-Rayet star’s powerful winds. Image
credit: ESO [17].
the star via helium-burning can be ejected into the interstellar medium by high stellar winds
before it is destroyed by α-capture.
In Wolf-Rayet stars, fluorine production occurs during early phase helium core burning [27].
In stars that are burning helium, fluorine is destroyed via the 19 F(α, p)22 Ne reaction, typically near the end of the helium-burning phase. However, the high stellar winds of a WolfRayet star preserve the fluorine by sweeping much of it out into the interstellar medium
before it can be destroyed in α-capture. In the early helium burn phase, fluorine production
is accomplished via the chains [27]:

14 N(α, γ)18 F(β + )18 O(p, α)15 N(α, γ)19 F,

14 N(α, γ)18 F(n, p)18 O(p, α)15 N(α, γ)19 F,

11

and

(1.4.1)

(1.4.2)

14 N(α, γ)18 F(n, α)15 N(α, γ)19 F

(1.4.3)

with neutrons and protons produced in the reactions 13 C(α, n)16 O and 14 N(n, p)14 C, respectively. The first chain, Equation 1.4.1, is the dominant fluorine-producing reaction in
Wolf-Rayet stars.
Simulations by Meynet and Arnould [27] supported the theory that Wolf-Rayet stars
could create a fluorine overabundance. In particular, Meynet and Arnould found that WolfRayet stars with masses ranging from 40 to 85 times the mass of the Sun and with metallicity
similar to the Sun would exhibit the highest fluorine yields. Lower metallicities lead to the
stars burning off the fluorine before stellar winds can carry it away. Higher metallicities
and higher masses lead to helium-burning cores that are too small to be uncovered by stellar
winds, so the fluorine cannot escape into the interstellar medium before it is destroyed. These
simulations indicated that Wolf-Rayet stars could potentially account for the solar abundance
of fluorine, and there were even some concerns expressed that the simulations could predict
too high of a solar fluorine abundance. The aforementioned galactic chemical evolution
model employed by Renda et al. [20] indicated that Wolf-Rayet stars were needed to match
experimentally observed overabundances of fluorine, along with neutrino nucleosynthesis and
AGB star production of fluorine.
Later, Palacios, Arnould, and Meynet revisited these Wolf-Rayet simulations of fluorine
abundance [29]. Updated models of Wolf-Rayet stars and updated nuclear reaction rates
triggered the re-examination. The updated stellar models indicated a significant reduction
in expected mass loss, meaning that more fluorine will be burned up in α-capture and less
will escape into the interstellar medium. Note that neutron densities in Wolf-Rayet stars are
considered negligible, so the reaction 19 F(α, p)22 Ne is considered the only significant source

12

of fluorine destruction. These mass loss changes lead to a decrease in fluorine production
in Wolf-Rayet stars by as much as a factor of 30. The authors note, however, that these
simulations are very sensitive to the rate of 19 F(α, p)22 Ne and to mass loss rates that may
be underestimated in this model. If these simulations prove correct, Wolf-Rayet stars are
not a significant source of galactic fluorine production. However, the rate of 19 F(α, p)22 Ne
was recently measured by Ugalde et al. [30] and they recommend a rate that is an order
of magnitude lower than the rate used in Palacios et al., which would lead back to higher
fluorine abundances in Wolf-Rayet stars.
Observationally, there is no direct measurement of fluorine in the stellar winds of a WolfRayet star. However, there are some measurements that indirectly support the Wolf-Rayet
production of fluorine. Observations of red giants in the Large Magellanic Cloud and in the
globular cluster ω Centauri by Cunha et al. [21] show a link between fluorine abundance
and metallicity. This link is predicted by the Wolf-Rayet simulations and by neutrino nucleosynthesis models of fluorine. Such a link between metallicity and fluorine abundance
is not predicted by the AGB star models of fluorine production that will be discussed in
greater detail later in this chapter. Later observations of red giants in the Galactic bulge,
also by Cunha et al. [22], supported Wolf-Rayet fluorine production over neutrino or AGB
production. These Galactic bulge red giants had a stronger link between metallicity and
fluorine abundance than could be explained through Heger et al.’s [6] neutrino nucleosynthesis calculations, and had no overabundance of the s-process elements that are expected
from AGB star fluorine production. A study of fluorine abundances in planetary nebulae
by Zhang et al. [11] found one planetary nebula with both a Wolf-Rayet central star and
a high fluorine overabundance. This particular Wolf-Rayet star planetary nebula deviated
significantly from the trend of the other planetary nebulae observed in Zhang et al.’s study,
13

in that it had a high abundance of fluorine and a high metallicity whereas the general trend
among the nebulae was a decrease in fluorine with higher metallicity. These observations
provide some qualitative support for Wolf-Rayet nucleosynthesis of fluorine, but lack a strong
quantitative argument.
The picture for fluorine production in Wolf-Rayet stars is thus anything but clear. There
is certainly room for Wolf-Rayet stars to play an important role in fluorine nucleosynthesis.
However, updated models for Wolf-Rayet fluorine production are needed in light of Ugalde
et al.’s measurement of the 19 F(α, p)22 Ne rate.
The measurements that are discussed in this dissertation are not expected to have an
impact on fluorine nucleosynthesis in Wolf-Rayet stars. The temperatures relevant to fluorine
production via helium-burning (T ≤ 0.2 GK) are too low for any significant amount of 19 O
to occupy an excited state (see Sections 1.3 and 5.1 for more details). Additionally, the low
neutron densities of Wolf-Rayet stars mean that they are the least likely of the three major
sites proposed for fluorine production to undergo significant rates of the alternative fluorine
production process, 18 O(n, γ)19 O(β − )19 F.

1.5

Fluorine Found in AGB Stars

Direct observational evidence of fluorine abundances greater than the solar abundance
was first recorded in asymptotic giant branch (AGB) stars by Jorissen et al. [8]. This
discovery spurred much of the activity in the field of fluorine nucleosynthesis in the last two
decades. AGB stars are cool, red giants that eject mass into the interstellar medium as they
burn (see Figure 1.4). Stars with a mass less than roughly 9 times the mass of the Sun will
go through an AGB phase as they near the end of their life. As an AGB star dies, the mass

14

Figure 1.4: This image shows the planetary nebula NGC 5189. Planetary nebulae like this
one are formed when an asymptotic giant branch (AGB) star loses its outer layers. Many
such nebulae are roughly spherical, giving rise to images that look like giant planets (hence
the name “planetary” nebula). This one has a more complex shape that resembles a spiral.
Image credit: ESO [17].
emitted forms a planetary nebula, and the core that remains behind becomes a white dwarf.
AGB stars have an inert, electron-degenerate core comprised of carbon and oxygen. This core
is surrounded by a helium-burning shell, which in turn is surrounded by a hydrogen-burning
shell, and the outermost layer of the star forms a convective envelope [28, 30]. Fluorine
nucleosynthesis occurs in the helium-burning shell of the AGB star. The region between the
hydrogen- and helium-burning shells is referred to as the intershell. The two burning regions
are largely isolated from each other by the intershell, but the complicated stellar dynamics
of AGB stars allow for material from the hydrogen shell to occasionally be mixed into
the helium-burning region, introducing protons that play an important role in the fluorine
nucleosynthesis process [8]. Some protons in the helium shell are also generated by the
15

Type Description
K
M
MS
S
SC
C
J

Orange stars slightly cooler than the Sun.
Cool, red stars.
Intermediate AGB star with more carbon than M stars and less than S
stars.
AGB star rich in s-process elements, with more carbon than M stars and
less than C stars.
AGB star with more carbon than S stars. Abundance of carbon and
oxygen is nearly equal.
Carbon-rich AGB stars.
Carbon AGB stars with high 13 C abundances.

Table 1.1: Brief descriptions of the stellar classifications that are relevant for a discussion of
asymptotic giant branch stars.
14 N(n, p)14 C

reaction. The neutrons needed for the preceding reaction and other neutron

captures in AGB stars are generated through the 13 C(α, n)16 O reaction [8] and potentially
through the 22 Ne(α, n)25 Mg reaction [30, 31].
The study by Jorissen et al. [8] covered red giant stars of several different spectral
types. Please refer to Table 1.1 for a brief definition of each spectral type. In this study,
it was found that K and M stars have fluorine abundances consistent with the Sun. Slight
overabundances were found in some MS and S stars, and high excesses of fluorine were
observed in barium stars, SC stars, most C stars, some J stars, and a few MS or S stars.
The excess fluorine abundances ranged as high as 100 times the solar fluorine abundance
and frequently fell within the range of 3 to 30 times the solar fluorine abundance. However,
Abia et al. [9, 32] recently released two re-evaluations of the fluorine abundances in several
stars from the Jorissen et al. study. Abia et al.’s 2009 paper [9] looked at three C stars
from the Jorissen et al. study and found much lower fluorine abundances that are consistent
with solar fluorine abundance. Abia et al. examined in detail the differences between their
measurement and Jorissen et al., and found that the most likely culprit for the discrepancy

16

is contaminants. Jorissen et al.’s fluorine abundance measurements relied heavily on two
specific spectral vibration-rotation absorption lines of the HF molecule that Jorissen et al.
state are mostly free of contamination, but the study by Abia et al. used a model that showed
significant contamination of these same lines in carbon-rich stars from 12 C14 N and 12 C12 C
molecules. Abia et al.’s abundance measurements relied instead on a different vibrationrotation line of HF that their model indicated had no contamination [9].
Abia et al. released a 2010 paper [32] that covered a wider variety of AGB stars, including
several J-type, SC-type, and C-type stars (see Table 1.1 for a description of these stellar
classifications). This study also showed systematic reductions in the fluorine overabundances
as compared to Jorissen et al.’s work. The revised fluorine abundances found by Abia
et al. for J and C stars [32] are mostly consistent with the solar fluorine abundance [14].
However, Abia et al. still see a significant fluorine overabundance in SC stars. Both Jorissen
et al.’s paper [8] and Abia et al.’s paper [32] note that the SC stars are unique in that
they have a nearly equal amount of carbon and oxygen. This leads to nearly all the carbon
and oxygen being bound up as CO molecules. In turn, very slight variations in carbon
and oxygen abundances in SC star models result in large variations in the final molecular
makeup predicted by the models, meaning that there is a far greater uncertainty associated
with these SC star measurements than with other AGB stars. This casts considerable doubt
as to whether the high fluorine abundances measured in SC stars are due to systematic errors
or genuine enhancements.
Several other extrasolar abundance measurements of fluorine have been performed, but
they also present conflicting information on how significant AGB stars are for fluorine production. In 2003, Cunha et al. [21] released a paper with fluorine abundance measurements
of red giants in the Large Magellanic Cloud and in the globular cluster ω Centauri. They
17

found that the abundance of fluorine in the ω Centauri stars was well below the solar abundance and many of the Large Magellanic Cloud star abundances were slightly below the solar
abundance. The ω Centauri stars are notable because they have high abundances of elements
produced in the s-process. Since AGB stars are thought to be the primary production site
for s-process elements, the high s-process abundance in ω Centauri stars implies that AGB
stars played a significant role in their chemical evolution. If this is the case, the low fluorine
abundances in ω Centauri stars suggests that AGB stars may not be strong contributors to
fluorine production. In another paper published in 2008, Cunha et al. [22] found fluorine
overabundances in red giants in the galactic bulge that had no s-process enrichment. Models
predict that, in AGB stars, fluorine production and s-process element production should be
correlated [33], so a lack of s-process enrichment suggests that the fluorine seen in the red
giants observed by Cunha et al. is not produced by AGB stars.
Conversely, measurements of fluorine abundances in planetary nebulae by Zhang and
Liu [11] show an enhancement of fluorine when compared with the solar fluorine abundance.
Planetary nebulae are composed of the material ejected into the interstellar medium by
AGB stars [1], so this measurement of fluorine overabundance provides evidence in favor
of AGB stars as a fluorine production site. Werner et al. [12] found very high fluorine
overabundances of up to 250 times the solar abundance in extremely hot, hydrogen-deficient
post-AGB stars, which also supports the hypothesis that AGB stars are a fluorine production
site. Pandey et al. [13] found high fluorine enhancements of about 100 times solar in a sample
of cool extreme helium stars (hydrogen-deficient supergiants), which may have previously
gone through an AGB star phase.
Jorissen et al. examined several different nuclear reaction paths that could produce 19 F

18

in an AGB star [8]. They concluded that the most likely path for fluorine production is:

14 N(α, γ)18 F(β + )18 O(p, α)15 N(α, γ)19 F,

(1.5.1)

where neutrons and protons are produced via the 13 C(α, n)16 O and 14 N(n, p)14 C reactions,
respectively (see Figure 1.5). This is the reaction path that is discussed in most literature on
fluorine production in AGB stars. However, Jorissen et al. also mention another potentially
viable path for fluorine nucleosynthesis:

14 N(α, γ)18 F(β + )18 O(n, γ)19 O(β − )19 F,

(1.5.2)

where neutrons are produced via the 13 C(α, n)16 O reaction, the 18 O(α, n)21 Ne reaction,
or the 22 Ne(α, n)25 Mg reaction [30, 31] (see Figure 1.5). At the time of publication of
Jorissen et al.’s paper, there were large uncertainties associated with the 18 O(n, γ)19 O rate.
Such a fluorine production path would also require a large initial 18 O content in the AGB
star. This path is constrained by the competing 19 F(n, γ)20 F rate, which destroys the 19 F.
Jorissen et al. expressed skepticism that the reactions 18 O(n, γ)19 O(β − )19 F could play a
significant role in AGB star fluorine nucleosynthesis because the rate of neutron capture on
18 O

is orders of magnitude lower than neutron capture on 19 F [8].

The specific measurements that are discussed in this dissertation are not expected to have
an impact on fluorine nucleosynthesis in AGB stars. The temperatures relevant to fluorine
production via helium-burning (T ≤ 0.2 GK) are too low for any significant amount of 19 O
to occupy an excited state (see Sections 1.3 and 5.1 for more details). However, the broader
process of 19 O(β − )19 F (see Figure 1.5) as a potential source of fluorine in AGB stars has not

19

16

F

17

F

18

F

19

F

20

F

15

O

16

O

17

O

18

O

19

O

14

N

15

N

16

N

17

N

18

N

16

F

17

F

18

F

19

F

20

F

15

O

16

O

17

O

18

O

19

O

14

N

15

N

16

N

17

N

18

N

Figure 1.5: Nuclear paths for fluorine nucleosynthesis in AGB stars. Grayed boxes indicate a
stable isotope in this slice of the chart of nuclides. The blue path in the top figure corresponds
to Jorissen et al.’s primary reaction path. The red path in the bottom figure is the alternative
path involving 19 O(β − )19 F discussed in the text.

20

received attention in modern AGB nucleosynthesis calculations. Recent experimental developments suggest that this alternate reaction path deserves renewed scrutiny: a new measurement of the 19 F(n, γ)20 F rate by Uberseder et al. [34] indicates a lower cross section than
prior estimates, a recalculation of the 22 Ne(α, n)25 Mg rate by Karakas et al. [31] reduces uncertainties on this reaction, and a measurement of 19 F(α, p)22 Ne rate by Ugalde et al. [30]
indicates an order of magnitude lower cross section than prior measurements. These small
shifts all favor higher abundances of fluorine and better conditions for β-decay from 19 O.
The 19 O(β − )19 F process should be included in future fluorine nucleosynthesis calculations
simply to determine its contribution (if any) to AGB star fluorine production.

21

Chapter 2
Theory
The primary goal of this research is to measure the Gamow-Teller strength distribution
for transitions from 19 F to 19 O via the 19 F(t,3 He)19 O∗ charge-exchange reaction. This
Gamow-Teller strength distribution will be used to calculate a weak decay rate for the
19 O∗ (β − )19 F

reaction for use in the astrophysics application discussed in Chapter 1. This

chapter explains the theoretical concepts that form the foundation of the 19 F(t,3 He)19 O∗
measurement. In addition to covering relevant theoretical arguments, this chapter will also
discuss the computer programs used to perform theoretical physics calculations needed in
the data analysis for Chapter 4.

2.1

Gamow-Teller Strength

When a nucleus undergoes β − -decay, it converts one neutron into a proton and emits an
electron and antineutrino in the process [2]:

19 O
8 11

→19 F10 + e− + νe .
¯
9

(2.1.1)

A β-decay involves a unit change in isospin (∆T = 1); this isospin change is represented
by the isospin operator τ . For the case of β − -decay, this is specifically an isospin-lowering
process represented by the operator τ− . There are two separate classes of β-decay transitions:
22

Gamow-Teller and Fermi. Gamow-Teller transitions occur when the ejected electron and
antineutrino have parallel spins. This means that the transition has a unit change in spin
(∆S = 1); such a spin transfer is represented by the spin operator σ, and the overall
Gamow-Teller transition is then represented by the operator στ . The corresponding β-decay
transitions where the ejected electron and antineutrino are antiparallel (∆S = 0) are called
Fermi transitions, which are represented simply by the operator τ .
Nuclear reactions that involve the exchange of a neutron for a proton, or vise versa, are
called change-exchange reactions. The simplest example is an (n, p) or (p, n) reaction, but
more complex probes such as (t,3 He) and (7 Li,7 Be) can also be used. These reactions are
also isovector transitions (∆T = 1) like β-decays. In cases where there is a change in spin
but no change in orbital angular momentum (∆S = 1, ∆L = 0), these strong-force mediated
reactions can also be represented by the στ operator and connect the same states as allowed
transitions in the β-decay mediated by the weak force. Under the right circumstances,
the cross section of a nuclear charge-exchange reaction is proportional to the strength of
the Gamow-Teller transition. This allows for an extraordinary situation where transition
strengths associated with the weak force can be studied directly in reactions mediated by
the strong force. Therefore, charge-exchange reactions like (t,3 He) can be used to study
Gamow-Teller β-decay transitions that are inaccessible to direct β-decay studies because of
Q-value limitations.
A significant amount of research has gone into establishing a firm relationship between
various charge-exchange reaction cross sections and Gamow-Teller strengths. Taddeucci
et al. [35] performed the first detailed investigation of the relationship between weak transition strengths and (p, n) cross sections at intermediate beam energies (120–200 MeV). In
Reference [35], they established a proportionality called a unit cross section (ˆGT ) between
σ
23

Gamow-Teller strengths and the (p, n) reaction cross section at zero momentum transfer in
the Eikonal approximation. This proportionality equation is written as:

dσ
(q = 0) = KN D |Jστ |2 (B[GT ]) = σGT (B[GT ]) ,
ˆ
dΩ

where

(2.1.2)

dσ
(q = 0) represents the reaction cross section at zero momentum transfer, B[GT]
dΩ

denotes Gamow-Teller strength, and KN D |Jστ |2 is the factorization of the unit cross section
(ˆGT ) proportionality coefficient under the Eikonal approximation. In that factorization, K
σ
represents a kinematic factor (see also Figure 2.2). N D is a distortion factor that is defined
by the ratio of distorted-wave to plane-wave cross sections; it accounts for the distortion of
the incoming plane wave of the projectile and outgoing plane wave of the ejectile by the
mean field of the target and residual nuclei, respectively. |Jστ |2 is the volume integral of
the central στ component of the effective interaction between the nucleons in the target and
projectile nuclei. Note that Gamow-Teller strength is defined here such that B[GT] = 3 for
the decay of the free neutron.
This proportionality between Gamow-Teller strength and charge-exchange cross section
holds only under certain conditions. As already mentioned, the charge-exchange cross section must be obtained for zero momentum transfer (q = 0), which requires that the cross
section be extracted for zero reaction Q-value (Q = 0) and at a scattering angle of zero
degrees (θcom = 0◦ ). A direct, single-step charge-exchange reaction mediated by pion exchange is needed with no multi-step contributions from knock-on interactions; this occurs
at intermediate beam energies above 100 MeV per nucleon [36, 37]. It has also been shown
that the proportionality of Equation 2.1.2 can break down if the transition strength is very
weak. This is likely caused by interference between the transition amplitudes mediated by

24

the στ and Tτ (tensor-τ ) components of the effective interaction, which is discussed in more
detail in Section 2.2. A more detailed discussion of the unit cross section is available in
References [35, 38, 39].
Further studies of charge-exchange reactions and Gamow-Teller strength proportionalities
have opened up additional experimental tools beyond the initial (p, n) probe. A similar
relationship has been established for probes such as (7 Li,7 Be) reactions [40, 41], (3 He, t)
reactions [38], and (t,3 He) reactions [42]. The significant similarities between (t,3 He) and
(3 He, t) reactions and the availability of high-quality data sets for both types of reactions
has permitted detailed examinations of the three factors (K, N D , and |Jστ |2 ) in the unit
cross section in Equation 2.1.2 [39].
To extract Gamow-Teller strengths from charge-exchange data, a value must be established for the unit cross section (ˆGT ). There are two methods to accomplish this. In
σ
the method that is most frequently used, σGT is determined through a direct calibration.
ˆ
A charge-exchange cross section is measured for a specific transition with a well-known
Gamow-Teller strength from direct β-decay studies. Alternatively, it is possible to interpolate the unit cross section from other measurements on targets with similar masses where
a unit cross section was established via the first method. This is useful when there is no
well-known Gamow-Teller β-decay measurement available (as is the case for the 19 F→19 O
transition). In Reference [38], a phenomenological relationship between the unit cross section and nuclear mass (A) was established for (3 He, t) reactions at 140 MeV/nucleon in the
region of 12 ≤ A ≤ 120 (see Figure 2.1):

σ=
ˆ

109
.
A0.65

25

(2.1.3)

Figure 2.1: This figure shows the nuclear mass dependence (A) of the measured unit cross
section (ˆGT ) for several different (3 He, t) and (t,3 He) reactions. The solid line is a pheσ
nomenological fit to the (3 He, t) data for masses of 12 ≤ A ≤ 120 that is discussed in detail
in Reference [38]. The deviation from this fit at low masses is an expected phenomenon,
dominated by the kinematic contribution (K) in Equation 2.1.2 (see also Figure 2.2). This
figure was taken from Reference [39].
This equation has also been found to work well for (t,3 He) reactions at 115 MeV/nucleon [39]
and was used to calculate a unit cross section for the present measurement of 19 F(t,3 He)19 O∗

2.2

Distorted Wave Born Approximation

The spectrum for a charge-exchange reaction measurement is not restricted to only
Gamow-Teller transitions. Several other transitions with different spin and angular momen-

26

Figure 2.2: This figure shows the nuclear mass dependence (A) of the kinematic factor (K)
in Equation 2.1.2 for the unit cross section (ˆGT ) for several different (3 He, t) and (t,3 He)
σ
Ei Ef kf
reactions. The solid line is defined as K =
, where Ei and Ef are the reduced
(π 2 c2 )ki
energies for the incoming and outgoing channel and ki and kf are the linear momentums of
the projectile and ejectile. Since this line trends toward zero for low masses, it causes masses
A < 12 to fall off the phenomonological fit described by Equation 2.1.3. The kinematic
factor and its impact on the unit cross section is discussed in more detail in Reference [39].
This figure was taken from Reference [39].

27

tum transfers contribute as well. The allowed Gamow-Teller transitions (∆S = 1, ∆L = 0)
must be separated from these other contributions during the data analysis. As an example,
transitions with a total angular momentum transfer of 1 (∆L + ∆S = ∆J = 1) can occur
as Gamow-Teller (∆S = 1, ∆L = 0) transitions or as quadrupole (∆S = 1, ∆L = 2)
transitions. The only way to achieve this separation of the Gamow-Teller transition is to
use a reaction model; the reaction model can be used to predict the angular distributions
for contributions to the spectra associated with different transfers of orbital angular momentum. The measured differential cross sections for the charge-exchange reaction are then
decomposed on the basis of these theoretical calculations (see Section 4.3.1) so that the
∆L = 0 component can be isolated. Further, once the ∆L = 0 component of the measured
differential cross sections are extracted, they must be extrapolated to zero momentum transfer (q = 0) so that the proportionality relationship in Equation 2.1.2 can be applied and
Gamow-Teller strengths can be extracted. This procedure also requires theoretical reaction
calculations, both to extrapolate from a finite reaction Q-value to Q = 0 and to quantify
the cross section at a scattering angle of zero (θcom = 0◦ ), which then fulfills the zero linear
momentum transfer condition.
As mentioned above, Tτ (tensor-τ ) terms can further complicate matters for weak GamowTeller transitions due to interference between the transition amplitudes mediated by the στ
and Tτ interactions. The angular distribution at forward scattering angles is not modified
by this interference, but the proportionality in Equation 2.1.2 becomes uncertain. The effect is strongest for weak excitations [42, 43], where the tensor contribution can become the
dominant term. The tensor contribution can be estimated through theory calculations.
The theoretical tool used for these reaction calculations was the three-part WSAW,
FOLD, and DWHI code package [44]. Angular distributions were calculated with this code
28

package through a first order distorted wave Born approximation (DWBA) [45, 46, 47, 48].
In general, a reaction cross section can be expressed in terms of transition amplitudes (Tf i )
from a DWBA calculation as:

dσ
=
dΩ

2 k
µ
f
|Tf i |2 ,
2
ki
2π

(2.2.1)

where µ is the reduced energy, ki is the incoming wave number, and kf is the outgoing wave
number [49]. The calculation of transition amplitudes requires several nuclear structure and
nuclear reaction inputs. The FOLD code is the core of this reaction model package; it computes form factors, which contain information about the interaction between the projectile
and target as well as nuclear structure information about relevant transitions. The WSAW
code generates radial wave functions that are used in FOLD’s calculation of form factors.
The DWHI code uses the form factors computed by FOLD and optical potentials to calculate
transition amplitudes and differential cross sections on the basis of the DWBA formalism.
The WSAW code is used to compute a set of single-particle radial wave functions (φ) for
a two-body particle system, such as the target-recoil system. A potential with a Coulomb,
Woods-Saxon, and spin-orbit term is input into the code for each single-particle state. Then
radial wave functions for each single-particle state are determined by allowing the depth of
the Woods-Saxon potential to vary while fitting the radial wave function to the Schr¨dinger
o
equation such that it reproduces a set of input single-particle binding energies. The other
parameters in the potential are fixed in the WSAW input file at the following values: diffuseness a0 = 0.65 fm, Coulomb radius rc = 1.25 fm, Woods-Saxon radius r0 = 1.25 fm,
spin-orbit potential depth coefficient VSO = 7.0 MeV. The single-particle binding energies
for the target-recoil system were calculated with the shell-model code OXBASH [50] us-

29

Model
18 O + p(0s

Eb (MeV)
1/2 )

18 O + p(0p

3/2 )

18 O + p(0p

1/2 )

18 O + p(0d
5/2 )
18 O + p(0d
3/2 )
18 O + p(1s
1/2 )
18 O + p(0f
7/2 )
18 O + p(0f
5/2 )
18 O + p(1p
3/2 )
18 O + p(1p
1/2 )
18 O + p(0g
9/2 )

-29.6
-17.9
-13.3
-5.9
-1.0
-3.1
-1.0
-1.0
-1.0
-1.0
-1.0

Model
18 O + n(0s

1/2 )
18 O + n(0p
3/2 )
18 O + n(0p
1/2 )
18 O + n(0d
5/2 )
18 O + n(0d
3/2 )
18 O + n(1s
1/2 )
18 O + n(0f
7/2 )
18 O + n(0f
5/2 )
18 O + n(1p
3/2 )
18 O + n(1p
1/2 )
18 O + n(0g
9/2 )

Eb (MeV)
-29.6
-17.7
-13.1
-6.1
-1.0
-4.4
-1.0
-1.0
-1.0
-1.0
-1.0

Table 2.1: This table lists the single-particle binding energies that were used as an input for
the WSAW radial wave form calculation. The model column indicates the core nucleus and
the added proton (left) or neutron (right) in the indicated single-particle level to form the
19 F or 19 O nucleus, respectively. These binding energies for the spsdpf model space were
calculated using the Skyrme SK20 interaction [51] with the shell-model code OXBASH [50].
When the calculated binding energies were greater than -1.0 MeV, a value of -1.0 MeV was
substituted to ensure that the generated wave functions were well-bound and the computations converged.
ing the Skyrme SK20 interaction [51]. The Skyrme SK20 interaction was selected for this
application because it is valid over a wide range nuclei and thus allows for comparisons
across a broad variety of charge-exchange reactions. The binding energies (Eb ) used for the
WSAW calculation on the 19 F–19 O system are listed in Table 2.1. The t–3 He system radial
wave functions were computed from Variational Monte Carlo results [52, 42] instead of with
WSAW.
The FOLD code produces form factors (F (r)) by double-folding the effective nucleonnucleon interaction (Vef f ) of Love and Franey [37, 53] at 140 MeV/nucleon over the one-body
transition densities of the projectile-ejectile and target-recoil systems. Form factors are a
function of the distance between the target and projectile (r). These form factors account
for the composite nature of all the nuclei involved in the 19 F(t,3 He)19 O∗ reaction when the
30

DWBA is performed. The one-body transition densities describe the overlap between initial
and final states in the projectile-ejectile and target-recoil systems. A one-body transition
density serves as a weighting factor for each one-particle, one-hole transition, determined by
the overlap between the initial and final nuclear states. One-body transition densities for
the 19 F–19 O system were calculated in the spsdpf -shell model space with the WBP effective
interaction [54] using the shell-model code OXBASH [50]. FOLD generates form factors from
the radial wave functions generated in WSAW (φ), one-body transition densities, kinematic
data, and spin-coupling information. Form factors are calculated with the equation [49]:

F (r) =

Φe Φr a† ac† c Φp Φt

φe φr Vef f φe φr ,

(2.2.2)

where the subscripts of t, p, e, and r indicate the wavefunctions of the target, projectile,
ejectile, and recoil particles respectively and the sum runs over all possible quantum numbers
for the system. The portion of the equation represented by Φe Φr a† ac† c Φp Φt

contains

the one-body transition densities as well as spin-coupling terms and creation and annihilation
operators.
The DWHI code uses the form factors found with FOLD calculations to compute transition amplitudes (Tf i ) and then angular distributions

dσ
dΩ

through a DWBA calculation.

Transition amplitudes for a specific transfer of spin (∆S), total angular momentum (∆J),
and orbital angular momentum (∆L) are calculated from the following equation [49]:

Tf i = χf (kf , r) F (r) χi (ki , r) ,

(2.2.3)

where χi (ki , r) and χf (kf , r) are the distorted waves for the initial and final states, respec-

31

Channel
19 F + t
19 O+3 He

Real
Depth Radius Diffuseness
(MeV) (fm)
(fm)

Imaginary
Depth Radius Diffuseness
(MeV) (fm)
(fm)

-18.8
-22.1

-36.3
-42.66

1.54
1.54

0.74
0.74

0.89
0.89

0.96
0.96

Table 2.2: This table lists the Woods-Saxon optical model potential parameters used
for the DWBA calculations. The parameters for the outgoing channel were measured in
Reference [55] for the incoming channel of the 16 O(3 He, t)16 F reaction at 140 MeV/nucleon.
The parameters for the incoming channel were derived from the parameters of the outgoing
channel as described in Reference [56].
tively. An optical potential is used in this calculation to account for the distortion effects
of the nuclear mean field. Since there are no optical potentials available for the specific
beam energy and reaction channels of this experiment, a suitable published optical potential
from a similar reaction was used instead. The best optical model available for this purpose
was found in Reference [55], in which Fujita et al. studied the 16 O(3 He, t)16 F reaction at
140 MeV/nucleon. As part of their analysis, Fujita et al. determined the optical potential
for the 16 O+3 He channel from elastic scattering data. The model chosen for this optical
potential consisted of real and imaginary Woods-Saxon terms, and any error due to the
choice of optical potential is expected to be negligible. The extracted parameters for the
16 O+3 He

channel were used in the current analysis for the 19 O+3 He outgoing reaction chan-

nel. For the incoming channel of 19 F + t, the real and imaginary Woods-Saxon potential
depths were scaled by a factor of 0.85, following the phenominological procedure described
in Reference [56]. The Woods-Saxon optical model potential parameters used for the DWHI
calculations are listed in Table 2.2. In addition to the optical model parameters, DWHI also
takes spin and kinematic information as inputs, including reaction Q-values. The calculated
transition amplitudes are then used to compute the desired differential cross section angular
distributions through Equation 2.2.1.

32

The differential cross section angular distributions produced by the WSAW/FOLD/DWHI
code package were used during the data analysis in a multipole decomposition analysis (see
Section 4.3.1). They were also used to extrapolate differential cross sections with finite linear
momentum transfer to zero linear momentum transfer (q = 0). This extrapolation meets the
necessary conditions for the determination of Gamow-Teller transition strengths using the
unit cross section proportionality described in Section 2.1.

33

Chapter 3
Experiment
The goal of this dissertation is to extract the Gamow-Teller strength of the 19 F(t,3 He)19 O∗
reaction (see Section 2.1 for a detailed explanation of Gamow-Teller strength). This chapter
will discuss the facility at which this experiment was performed and describe the experimental devices used to collect the data for this dissertation. A beam of tritons was impinged on a target that contained fluorine (CF2 ). The charge-exchange reaction of interest,
19 F(t,3 He)19 O∗ ,

was selected for by triggering on 3 He ejectiles with a high-resolution spec-

trograph. The spectrograph recorded position, angles, timing, and energy loss data that are
used in Chapter 4 to construct differential cross sections for the reaction over the excitation
energy (Ex ) range of 0 MeV ≤ Ex [19 O] ≤ 16 MeV, and those differential cross sections are
then used to calculate Gamow-Teller strengths.
This experiment is not the first measurement of the 19 F(t,3 He) reaction. Earlier work
by Pinder et al. [57] measured the cross section of the ground and first excited state of
19 O

produced by a (t,3 He) reaction with a 33 MeV beam of tritons. However, that data

set did not provide enough information to investigate the astrophysics application discussed
in Section 1.3. At the low beam energies used by Pinder et al., many two-step (indirect)
reactions occur; the proportionality relationship of Equation 2.1.2 discussed in Section 2.1
isn’t valid under such conditions.

34

Figure 3.1: This is a schematic of the devices used in experiment e09053 at the NSCL. This
figure shows the Coupled Cyclotron Facility (red), the A1900 fragment separator (orange),
the transfer line between theA1900 and the S800 spectrograph (green), the S800 spectrograph
analysis line (blue), and the S800 spectrograph (purple). For more details, refer to the text.
Image taken from Reference [48].

3.1

Triton Beam Production

This experiment was performed at the Coupled Cyclotron Facility at the National Superconducting Cyclotron Laboratory (NSCL). The experiment was designated with the number
e09053 and ran in February 2010. See Figure 3.1 for a schematic layout of the NSCL apparatuses used in this research.
The Coupled Cyclotron Facility [58, 59] consists of two superconducting cyclotrons (red
section of Figure 3.1): the K500 cyclotron and the K1200 cyclotron. In this experiment,
16 O3+

ions from an Electron Cyclotron Resonance ion source were injected into the smaller

of the two cyclotrons, the K500. The 16 O3+ ions underwent an initial acceleration to an
35

energy of 13 MeV/nucleon in the K500 cyclotron and were then injected through a coupling
line into the larger K1200 cyclotron. Inside the K1200 cyclotron, the ions hit a thin carbon
stripper foil that removed the remaining electrons, producing 16 O8+ . This fully-ionized
oxygen was accelerated in the K1200 cyclotron to an energy of 150 MeV/nucleon. It was
then extracted from the K1200 and impinged on a beryllium production target to create a
secondary radioactive beam through projectile fragmentation. A 3526 mg/cm2 production
target of 9 Be was used in this experiment to produce a 115 MeV/nucleon triton (3 H+ or t)
secondary beam. Prior research by Hitt et al. [60, 47] found that this method produces
the highest triton beam rate at an energy appropriate for charge-exchange experiments
(> 100 MeV/nucleon) at the NSCL.
A wide variety of fragments are produced when primary beam impinges on the beryllium
production target. To separate the desired secondary beam from all other fragments and
from the remaining primary beam, the secondary beam is sent through the A1900 fragment
separator [61, 62] (see Figure 3.2 for the layout of the A1900 fragment separator, which is
also pictured in the orange section of Figure 3.1). The A1900 separator consists of four
dipole magnets and eight quadrupole triplets and separates fragments by momentum. It
has high momentum and angular acceptance (∆p/p = 5% and ∆Ω = 8 msr, respectively).
It operates in an achromatic ion-optical mode, meaning that the final positions and angles
of the output beam particles are independent of momentum. The first half of the A1900
fragment separator selects fragments with a specific magnetic rigidity: Bρ = γmv/q [62],
where Bρ is magnetic rigidity, γ is the Lorentz factor, m is mass, v is velocity, and q is
charge. Further separation of fragments occurs when the beam passes through an energy
degrader, often referred to as a wedge [61]. Fragments that enter the wedge with a fixed
magnetic rigidity will emerge with magnetic rigidity that varies by atomic number due to
36

Figure 3.2: This figure shows a schematic of the A1900 fragment separator. This device
separates the desired secondary beam isotope from other fragments. See the text for a
detailed description of the separation process. Image taken from Reference [62].
differences in energy loss in the wedge. After the wedge, the A1900 fragment separator
disperses the fragments at its focal plane, and slits are used to remove the contaminants by
blocking particles with a momentum that varies too much from the desired isotope.
In this experiment, a 195 mg/cm2 aluminum wedge was used in the A1900 fragment
separator to remove the primary contaminants of the triton beam, 6 He and 9 Li. This method
of purifying the triton beam with a wedge was first tested in experiment 06032 by C. J. Guess
et al. [48]. Without this wedge, the triton beam had a purity of about 85%, and the 6 He
contamination created a background at the S800 spectrograph focal plane by breaking up
into 3 He + 3n at the reaction target. With the wedge inserted, all background from these
6 He

events was eliminated. The slits that purify the triton beam after the wedge were set to

restrict the momentum acceptance to ∆p/p = 0.5%. The final purified triton secondary beam
was directed through a transfer line (green section of Figure 3.1) toward the experimental
station.
There are two beam monitors of note near the cyclotrons and the A1900 that were used
37

in the data analysis for this experiment. These beam monitors did not intercept the triton
beam, but they produced signals that were proportional to the triton beam intensity. The
first probe, labeled Z001I-C, was a non-intercepting current probe located immediately after
the K1200 cyclotron that measured the primary beam intensity. The second probe, labeled
Z026R-C, was a copper bar located in the first dipole magnet of the A1900. It acted as a
Faraday cup to measure the current of the unreacted primary beam that passed through the
beryllium production target; the secondary triton beam was steered away from this probe by
the dipole magnet. These beam current measurements, along with other data, were used to
provide a relative scale for the secondary beam intensity on different reaction targets. This
information was needed because there was no direct measurement of the secondary beam
intensity available throughout the experiment. An estimate of the average absolute triton
beam intensity from the well-known 12 C(t,3 He) cross section gave a rate of roughly 7 × 106
particles per second. This rate was 30% lower than in experiment 06032 due to problems
with the triton beam tuning.

3.2

Targets

Four different reaction targets were used in the course of this experiment at the pivot point
of the S800 spectrograph. The reaction target of primary interest for this dissertation was an
11.2 mg/cm2 thick piece of CF2 , commonly known as Teflon. The goal of using this target
was to study the 19 F(t,3 He)19 O∗ reaction. CF2 was selected as the fluorine target material
because it has a high fluorine content and is easy to obtain. No notable contamination of
the CF2 target was detected. A separate CH2 reaction target with a thickness of 10 mg/cm2
was used to provide both calibration data and information on the 12 C background events

38

produced by the carbon in the CF2 target. The third target, a piece of aluminum coated
with ZnS (which fluoresces when hit by beam), was used as a viewer to verify that the beam
spot was properly aligned on the target at the pivot point. The fourth reaction target was
a foil of isotopically purified 56 Fe. Data taken on the 56 Fe target was analyzed in detail in
a separate study by Y. Shimbara et al. [63], and is only used in this dissertation for a few
calibrations.
In prior charge-exchange measurements at the NSCL, a large vacuum chamber with a
remote-controllable target ladder was used to implement easy transitions between different
reaction targets. The large vacuum chamber was not available for this experiment, so all
reaction target changes were done manually instead. Since manual target changes are timeconsuming, this means that there were limited opportunities to perform calibration checks
with the CH2 target and to inspect the beam spot at the S800 pivot point. However, enough
calibration runs were performed that this restriction did not add any significant error to the
data taken on the CF2 reaction target.

3.3

S800 Spectrograph

The experimental station used for this research was the S800 spectrograph [64, 65]. The
S800 spectrograph is a high resolution and high acceptance device that uses superconducting
magnets to measure the momentum of charged particles. It has two sections: the analysis
line (light blue area in Figure 3.1), and the high acceptance spectrograph with focal plane
detectors (purple area in Figure 3.1). The S800 spectrograph focal plane detectors provide
timing, position, and energy loss data for the reaction ejectile. This information is used to
reconstruct the events at the reaction target.

39

Figure 3.3: This figure shows the layout of the S800 spectrograph and its analysis line.
The triton beam entered the S800 analysis line at the object position and was momentumdispersed as it traveled through the analysis line so that it was fully dispersed at the reaction
target. When the desired (t,3 He) reaction occurs in the target, the 3 He ejectiles are bent
through the S800 spectrograph and then measured by detectors at the focal plane. Image
adapted from a figure in Reference [65].
The analysis line of the S800 spectrograph (see Figure 3.3) is composed of four dipole
magnets and five quadrupole triplet magnets. Two distinct operating modes are available
for the device. In focused mode, the analysis line is achromatic, which produces a chromatic
image at the focal plane. Focused mode provides high momentum acceptance of beam
particles at the expense of energy resolution in the measured ejectiles. In dispersion matching
mode, the entire system of analysis line and spectrograph is achromatic, meaning that the
momentum spread of the beam at the beginning of the analysis line is ideally cancelled out
at the focal plane. Thus the momentum variation that is measured at the focal plane is only
due to the momentum transfer that took place in the reaction target. This mode provides
better energy resolution for the ejectiles at the cost of lower momentum acceptance. The
reduced momentum acceptance in dispersion matching mode occurs because the beam is

40

momentum-dispersed on the reaction target at the S800 spectrograph pivot point, and the
S800 spectrograph dispersion is about 11 cm per 1% momentum spread. The layout of the
analysis line and the S800 spectrograph cannot fully accommodate a very large beam spot,
so in dispersion matching mode the momentum needs to be restricted to ∆p/p = 0.5% to
provide a manageable beam spot size (this corresponds to a beam spot roughly 5.5 cm tall)
whereas focused mode can support a momentum spread of up to ∆p/p = 3%. Dispersion
matching mode was used for this experiment because the high energy resolution was critical
to reconstructing the reaction on an event-by-event basis. The magnetic rigidity (Bρ) of the
analysis line was set to 4.8 Tm for this experiment.
The spectrograph itself (see Figure 3.3) consists of two quadrupole magnets and two
dipole magnets that focus the reaction ejectiles onto a set of detectors at the focal plane. The
magnetic rigidity set for the S800 spectrograph during this experiment was Bρ = 2.3293 Tm.
The focal plane detectors suite contains two Cathode Readout Drift Chambers (CRDCs), an
ion chamber, and a set of plastic scintillators [65]. At the time of this experiment, the focal
plane detectors were undergoing an upgrade. As a result, the ion chamber was offline and
the four plastic scintillators that are normally in the focal plane were replaced; both CRDCs
were fully functional. This did not adversely affect the experiment. The ion chamber is not
used for (t,3 He) experiments because the 3 He ejectile has a low proton number (Z = 2),
so it doesn’t lose enough energy in the ion chamber fill gas to produce a useful signal [47].
The four plastic scintillators were in the process of being replaced with a more sophisticated
hodoscope detector made of 32 crystal CsI(Na) scintillators [66] and a single new thin plastic
scintillator. The new thin plastic scintillator had been fully installed when this experiment
ran and provided sufficient timing and energy loss data to replace the old scintillators.
The two Cathode Readout Drift Chambers (CRDCs) measure the dispersive and non41

dispersive positions of particles [64]. They are separated by a distance of 1.073 meters along
the beam path so that the two sets of position measurements can be used to reconstruct the
flight path angle of particles. Each CRDC acts as single wire drift detector [64, 67]. The
drift chamber is filled with a gas mixture of 80% CF4 and 20% C4 H10 . This gas mixture
has low aging characteristics, high drift velocity, and low avalanche spread. A set of 225
cathode pads are arrayed along the dispersive direction and a single anode wire runs parallel
to them (see Figure 3.4). When an ejectile enters the CRDC, the gas is ionized, creating
electron - ion pairs. The electrons drift quickly along the electric field toward the anode
wire. As the electrons reach the anode wire, they induce a signal on both the anode and the
nearby cathode pads [67]. The amount of signal induced on each cathode pad will vary with
the distance from the ejectile dispersive position, so the CRDC determines the dispersive
position by performing a Gaussian fit to the current measured on each pad (see inset on
Figure 3.4). The non-dispersive position is determined by the drift time of electrons; the
detector measures the time difference between the data acquisition trigger (a signal from the
plastic scintillator) and the collection of electrons on the anode wire.
The CRDCs give a position resolution of approximately 0.5 mm in both directions, which
corresponds to an ideal scattering angle resolution of 2 mrad. In dispersion matching mode,
the ideal energy resolution of the S800 spectrograph for reconstructed events is 1 part in
10,000 [65]. In practice, experiments performed with the S800 spectrograph and radioactive
beams usually obtain a scattering angle resolution of 10 mrad and an energy resolution of up
to 1 part in 2,000. The difference between the ideal and actual resolution is due to factors
external to the S800 spectrograph, such as beam spot size and reaction target thickness. See
Sections 4.1.2, 4.1.3, and 4.1.4 for a detailed description of the extraction of reaction data
from CRDC measurements and a discussion of the resolutions obtained in this experiment.
42

Figure 3.4: This diagram illustrates how the CRDCs in the focal plane of the S800 spectrograph work. The CRDCs are used to measure the transverse positions of an event at two
different locations along the beam axis of the focal plane. This information can then be used
to reconstruct the dispersive and non-dispersive angles of the event’s trajectory. The red line
is an example of an event, and the thin black line shows the central axis of the S800 spectrograph focal plane. The bar-chart insert shows an example of an integrated image charge
as measured by the CRDCs. The integrated image charge is fitted with a Gaussian line to
obtain a specific dispersive position. The original CRDC image is taken from Reference [64]
and was adapted by G. W. Hitt in Reference [47].

43

Name
CRDC 1 Anode
CRDC 1 Drift
CRDC 2 Anode
CRDC 2 Drift
Grid
Upper Scintillator PMT
Lower Scintillator PMT

Voltage Applied (V)
1150
800
1150
800
25
1790
1760

Table 3.1: This table lists the voltages that were applied to the S800 spectrograph focal
plane detectors for this experiment.
Table 3.1 lists the voltages that were applied to the CRDCs during this experiment.
The plastic scintillator that was present in the S800 for this experiment was used to
measure relative energy loss and to provide a timing signal. This scintillator was a 1 mm thick
piece of C9 H10 plastic with photomultiplier tubes (PMT) attached to both ends. Ejectiles
that pass through the scintillator deposit energy proportional to their proton number squared
(Z 2 ) and dependent on their velocity. Some of that energy will excite electrons in the plastic.
As these electrons de-excite, they release photons that are detected in the PMTs. A timing
signal generated by the scintillator serves as a trigger for the data acquisition system and
as the start of a time-of-flight measurement. The cyclotron RF provides a corresponding
stop signal for the time-of-flight measurement. Energy loss and time-of-flight information
were used to make a particle identification (see Section 4.1.1 for more details) to ensure
measurement of the desired reaction channel and to exclude background events. Table 3.1
lists the voltages that were applied to the PMTs during this experiment.

44

Chapter 4
Data Analysis
This chapter will explain the data analysis process used for this dissertation. The data
collected with the S800 spectrograph (see Section 3.3 for details) from the 19 F(t,3 He)19 O∗
reaction at 115 MeV/nucleon was used to obtain excitation energy spectra for the 19 O∗ recoil
nucleus in Section 4.1. The excitation energy spectra were converted into differential cross
sections, up to 16 MeV in the excitation energy of the 19 O recoil nucleus, in Section 4.2. A
multipole decomposition analysis was performed to extract the ∆L = 0 contributions to the
differential cross sections as a function of excitation energy. In Section 4.3, the Gamow-Teller
strength distribution for the 19 F(t,3 He)19 O∗ reaction was extracted using the proportionality relationship between the ∆L = 0 component of the differential cross section at zero
momentum transfer and the Gamow-Teller strength (see Equation 4.3.1 and Section 2.1).
Finally, the Gamow-Teller strength associated with the excitation of the 0.096 MeV state in
19 O

was used to calculate the corresponding β-decay rate of 19 O∗ (β − )19 F in Section 5.1, be-

cause this specific low-lying state was important for the astrophysical application discussed
in Chapter 1.

4.1

Extracting Excitation Energy Spectra

Data extracted from the S800 spectrograph focal plane detectors were used to reconstruct
the excitation energy spectra and to determine the differential cross sections. The ejectile
45

Name
tof

de1u
de1d
de1
xg1
xg2
tac1
tac2

Description
time of flight of the ejectile: the difference between an initial timing
signal from a thin scintillator in the focal plane and a stop timing signal
from the radiofrequency of the K1200 cyclotron, in arbitrary units
energy loss signal from the upper photomultiplier tube on the thin scintillator, in arbitrary units
energy loss signal from the lower photomultiplier tube on the thin scintillator, in arbitrary units
derived energy loss in the thin scintillator, in arbitrary units:
de1 = (de1u)(de1d)
position in the dispersive plane recorded by CRDC 1, in terms of a pad
number
position in the dispersive plane recorded by CRDC 2, in terms of a pad
number
drift time recorded by CRDC 1, in nanoseconds; used to determine the
position in the non-dispersive plane
drift time recorded by CRDC 2, in nanoseconds; used to determine the
position in the non-dispersive plane

Table 4.1: A description of the ejectile parameters that were measured with the S800 spectrograph focal plane.
parameters measured by the S800 spectrograph at the focal plane are listed in Table 4.1.
The extracted energy loss and timing information was used to uniquely identify and separate the desired 3 He2+ ejectiles from other possible reaction products and background
events (described in Section 4.1.1). The position parameters that were measured at the S800
spectrograph focal plane were also calibrated (discussed in Section 4.1.2) and used to reconstruct the ejectile’s properties at the secondary target position (explained in Section 4.1.3).
Table 4.2 provides a summary of the reconstructed ejectile parameters at both the focal plane
and the secondary target position. These parameters are referred to and used in calculations
throughout this chapter. When a parameter is corrected or calibrated, a subscript “c” is
used to differentiate the corrected parameter from the original data.
The properties of the ejectile at the secondary target position were used to calculate
the excitation energy of the recoil nucleus through a missing mass calculation (described in
46

Name
afp
bfp
xfp
yfp
ata
bta
dta
yta

Description
angle in the dispersive plane at the focal plane, in radians
angle in the non-dispersive plane at the focal plane, in radians
position in the dispersive plane at the focal plane, in microns
position in the non-dispersive plane at the focal plane, in microns
angle in the dispersive plane at the target, in radians
angle in the non-dispersive plane at the target, in radians
fractional energy, closely related to the ejectile’s kinetic energy
position in the non-dispersive plane at the target, in meters

Table 4.2: A description of the ejectile parameters that were calculated from the measurements taken with the S800 spectrograph.
Section 4.1.4). Some corrections were applied to improve energy resolution (explained in
Section 4.1.5). The excitation energy spectra obtained through this method were used to
construct differential cross sections in Section 4.2.

4.1.1

Particle Identification

The S800 spectrograph detects particles with a magnetic rigidity within a narrow range.
In this experiment, a central magnetic rigidity setting of Bρ = 2.32930 Tm (±0.5% due to
momentum acceptance restrictions) was selected to correspond with the rigidity of the 3 He2+
ejectiles that are produced in the (t,3 He) reaction at a triton beam energy of 115 MeV/nucleon
in the excitation energy region of interest. The limited momentum acceptance of the S800
spectrograph prevented most ejectiles from alternative reaction channels in the secondary
target from entering the S800 spectrograph focal plane. However, there were still some
background events that needed to be separated from the desired (t,3 He) reaction channel
ejectiles. Many of these background events were due to unreacted tritons scattering off the
S800 spectrograph dipole magnets or ejectiles from triton reactions such as (t, d) or (t, p)
with the aforementioned magnets. A particle identification gate was necessary to select out

47

energy loss, de1c (arbitrary units)

energy loss, de1 (arbitrary units)

600

(A)

500

3
400

He

3

He

300
200

1000

(B)

3

He

800

3

He

600

400

200

100
0

0
-1000
-800
-600
-400
time of flight, tof (arbitrary units)

-1000
-800
-600
-400
time of flight, tofc (arbitrary units)

Figure 4.1: These figures show particle identification spectra. The z-axis is on a logarithmic
scale. Two cycles of the cyclotron RF are shown here, so two blobs in each figure correspond
to the desired 3 He2+ ejectiles. Figure A (left) shows the particle identification spectrum
before any corrections were applied. The white circles identify the 3 He2+ events. Figure B
(right) shows the improved separation of the 3 He2+ from the background after corrections
were applied. The white boxes illustrate the 3 He2+ gating used in the data analysis.
only the relevant 3 He2+ events from background events.
A particle identification spectrum was generated from data recorded at the thin scintillator in the S800 spectrograph focal plane. In this spectrum, the time of flight (tof ) is plotted
versus the energy loss (de1 ) as shown in Figure 4.1. This type of particle identification spectrum can be used to separate the ejectiles from different reaction channels in the secondary
target. The energy loss (de1 ) is proportional to the ejectile proton number squared (Z 2 ) as
determined with the Bethe-Bloch equation; energy loss allows for the selection of the desired
ejectile isotope (in this experiment, 3 He). The time of flight (tof ) is proportional to M ,
q
where M is ejectile mass and q is ejectile charge; the time of flight allows for the selection of
the desired reaction channel (in this experiment, (t,3 He) reactions at the secondary target
position are the desired reaction channel).

48

Events corresponding to the (t,3 He) reaction were easily determined in the particle identification spectrum. Data was taken on a target with a well-known excitation energy spectrum.
Then, each blob in the particle identification spectrum was gated on individually until the
expected excitation energy spectrum was found. In this case, a target of CH2 was selected
for this purpose because the excitation energy spectrum of the 12 C(t,3 He)12 B∗ reaction has
been studied previously and was easily identifiable. The blobs corresponding to 3 He ejectiles
are marked in Figure 4.1. In theory, if the other blobs were due to interactions in the reaction
target from an alternative reaction channel, that reaction channel could potentially be identified by closely examining the excitation energy spectrum and comparing it to known excited
states. However, the other blobs in the particle identification spectrum for this experiment
did not correspond to reactions in the secondary target. When these background blobs were
gated on, the measured trajectories exhibited unusual angular distributions, which strongly
implied that they were background events that did not originate at the secondary target, as
discussed above.
To clean up the particle identification spectrum so that the 3 He ejectiles were easily separated from background events, linear corrections were applied to the time of flight (tof )
and energy loss (de1 ). For a given ejectile isotope, the time of flight corrections removed the
dependence on momentum and path length, while the energy loss corrections removed the
dependence of energy loss on path length. The corrected parameters are referred to as tofc
and de1c . The specific corrections used in this experiment were determined phenomenologically and are listed in Table 4.3. The effect of these corrections is illustrated in Figure 4.1 B.
The gate for the 3 He ejectiles was chosen from this corrected particle identification spectrum
and used during further data analysis (the gate ranges used are also listed in Table 4.3).

49

Particle Identification corrections
tofc = tof + (1.95)(xg2 − xg1) + (0.169)(xg2) + (0.008)(xg1)
de1uc = de1u + (0.84)(xg2 − xg1)
de1dc = de1d − (1.07)(xg2 − xg1)
de1c = (de1uc )(de1dc )
Boundaries for the 3 He gate
de1 c range
tofc range
432 < de1c < 720
−830 < tofc < −751
432 < de1c < 720
−420 < tofc < −350
Table 4.3: This table lists the formulas used to make corrections to the time of flight (tof )
and energy loss (de1 ) to obtain a clean gate on 3 He in the particle identification spectrum.
The 3 He gate boundaries are also listed.

4.1.2

Cathode Readout Drift Chamber Calibration

The cathode readout drift chamber (CRDC) position measurements (xg1, xg2, tac1, tac2 )
were recorded in terms of pad numbers (in the dispersive plane) and electron drift time (in
the non-dispersive plane). To convert these parameters into physical distances, a mask calibration was performed. A tungsten plate with a distinct set of holes and slits (see Figure 4.2)
was inserted upstream of each CRDC sequentially. The distances between the holes and slits
in these masks were well-known. Ideally, the mask blocks all particles except those that pass
through the holes and slits, and thus the pattern of dots and lines observed in a masked
CRDC run can be used to determine the conversion between the position parameters and
actual distances with a linear parameterization.
However, the 3 He ejectile has a low proton number of Z = 2. The masks are not thick
enough to stop fast ejectiles with such a low Z, so the 3 He ejectiles passed through the
mask without stopping. This made it difficult to isolate the slit-hole pattern to calibrate
the CRDC positions. However, the 3 He ejectiles lost some energy as they passed through
the mask, causing the ejectiles to slow down slightly if they did not pass through a hole or

50

Figure 4.2: A schematic of the tungsten mask used for position calibrations of the CRDCs.
This figure was taken from [40].
slit. Then, when the 3 He ejectiles interacted with the thin scintillator in the focal plane, the
slower particles that had passed through the mask deposited more energy than the faster
particles that had passed through slits or holes. Thus, through careful gating on energy loss
(de1 ), ejectiles that had passed through the mask were separated from ejectiles that had
passed through a hole, so the slit-hole pattern was recovered and used for calibration.
The CH2 target was used during the mask calibration runs because it had the highest
rate of 3 He ejectiles, and thus produced the best statistics of any of the available target
options. However, the statistics for the mask calibration runs were relatively poor, so only
a few mask holes were visible in the calibration runs (the slits were still easily visible, see
Figure 4.3). To compensate for the difficulty of detecting the holes, information from a
prior S800 spectrograph experiment was used to establish an initial calibration. This prior
experiment, NSCL experiment number 08017, detailed in Reference [40], had higher statistics
for its mask calibration runs. This initial calibration from experiment 08017 made it possible
to identify a few holes and ensure that they were assigned correctly. The data from the mask

51

non-dispersive position, tac2c (ns)

Mask Spectrum for CRDC 2
20
10
0
-10

-200

-100
0
100
200
dispersive position, xg2c (mm)

Figure 4.3: Position measurements for the mask calibration of CRDC 2. The slits from the
mask were easily visible, but the holes were more difficult to distinguish.
runs in the present experiment were then used to refine and finalize the position calibration.
Since it is possible to fine-tune the mask calibrations by applying kinematical constraints
to the data sets, further mask calibration runs were unnecessary (see Section 4.1.5). The
calibration was a simple linear relationship with parameters of slope (M ) and intercept (B):

calibrated CRDC position = M (CRDC data) + B.

(4.1.1)

The values that were found for the parameters of the four calibrations are listed in
Table 4.4. The origin of the coordinates corresponds to the vertex of the L-shape seen in the
mask (see Figure 4.2), which is also the central trajectory through the S800 spectrograph
for particles with the appropriate magnetic rigidity (Bρ). The calibration in the dispersive
plane is stable over the duration of an experiment [47]. However, the slope in the nondispersive plane is dependent on the electron drift velocity in the CRDCs and is sensitive
52

Name
xg1c
xg2c
tac1c
tac2c

CRDC
CRDC 1
CRDC 2
CRDC 1
CRDC 2

Plane
Dispersive
Dispersive
Non-dispersive
Non-dispersive

M
2.54 mm/pad
2.54 mm/pad
-0.068 mm/ns
0.071 mm/ns

B
-278.88 mm
-278.99 mm
132.34 mm
-147.35 mm

Table 4.4: Parameters used to calibrate CRDC data to a physical position, in millimeters.
The values for M in the dispersive plane are fixed to the width of each pad at 2.54 mm.
to subtle changes in experimental conditions, so it tends to change slowly over the course
of an experiment. It was not practical to take multiple CRDC calibration runs during this
experiment, so these corrections were instead done on a run-by-run basis by examining the
data as discussed in Section 4.1.5.
By using these calibrated position parameters (xg1c , xg2c , tac1c , and tac2c ), angles and
positions at the focal plane (afp, bfp, xfp, and yfp) were calculated in the format needed for
the next step in the data analysis. The equations for the parameter transformations in the
dispersive and non-dispersive planes for the angles (afp and bfp, respectively) and positions
(xfp and yfp, respectively) are:

af p = arctan

xg2c − xg1c
,
1073

(4.1.2)

bf p = arctan

tac2c − tac1c
,
1073

(4.1.3)

xf p =

xg1c
, and
1000

(4.1.4)

tac1c
.
1000

(4.1.5)

yf p =

The factor of 1073 present in the calculations of afp and bfp comes from the distance between
the two CRDCs, which is 1.073 meters. The units chosen for these parameters were selected

53

for compatibility with the inverse transfer map matrix calculation detailed in Section 4.1.3.

4.1.3

Ejectile Parameters at the Secondary Target

Data on the position and the trajectory angle of the ejectile at the S800 spectrograph
focal plane can be converted into information on the ejectile’s properties as it leaves the
secondary target. Since the goal of this data analysis is to study the excitation energy
spectrum of the recoil particle, this conversion is a necessary step towards reconstructing
reaction cross sections. This parameter conversion was accomplished via an inverse transfer
map matrix. The inverse transfer map matrix transformed the positions (xfp, yfp) and angles
(afp, bfp) at the focal plane into the position in the non-dispersive plane (yta), angles in the
dispersive and non-dispersive planes (ata, bta), and fractional energy (dta) at the reaction
target position [65].
The inverse transfer map matrix was generated using a COSY INFINITY [68, 69] calculation. Inputs for this calculation include magnetic field maps of the S800 spectrograph as
well as experiment-specific magnetic rigidity settings and current settings. These experimentspecific settings were determined by the magnetic rigidity (Bρ) of the ejectile nucleus to be
measured, which dictated the current that was needed in the S800 spectrograph dipole and
quadrupole magnets. The values used for this experiment are listed in Table 4.5. Currently,
the COSY INFINITY program used to determine the inverse transfer map matrix for the
S800 spectrograph is accessible online at http://maps.nscl.msu.edu/~s800maps/ and uses
a fifth order calculation to generate the map matrix. The exact formula for the transform
can be found in Reference [65] and additional details on this calculation are available in
Reference [40].
The distance between the target and pivot point (d1 ) was nonzero mainly due to imper54

Parameter
I256QA
I258QB
I265DS
I269DS
Bρ
A
Z
d1
d2

Value
Description
-38.499 Amps current in a quadrupole magnet
35.342 Amps current in a quadrupole magnet
157.501 Amps current in a dipole magnet
157.519 Amps current in a dipole magnet
2.32930 Tm
magnetic rigidity of the S800 spectrograph
3
ejectile mass number
2
ejectile proton number
-0.08 m
distance between target and pivot point
0.0
focal plane shift

Table 4.5: These settings for the S800 spectrograph were taken from a run midway through
the experiment. This information was used in the COSY INFINITY calculation of the inverse
transfer map matrix.
fections in the COSY INFINITY calculation of the inverse transfer map matrix, though some
of the shift was likely due to minor uncertainty in the placement of the secondary target. It
was possible to experimentally determine this pivot point shift through examination of the
data. An unaccounted-for shift from the pivot point results in a noticeable rotation between
the angle in the non-dispersive plane (btac ) and the position in the non-dispersive plane
(ytac ). When the shift is corrected for, this rotation vanishes. The 56 Fe target had a thick
frame that illustrated this effect clearly (see Figure 4.4). A thick metal frame held the 56 Fe
target on both edges in the non-dispersive plane, and was thus expected to occupy a small,
fixed span of ytac on both sides of the target. Since the frame was much thicker than the
target, many background reactions occurred when the frame was hit by the beam, so events
in the frame were clearly distinguishable from 56 Fe target events in the spectra as shown
in Figure 4.4. When the pivot point shift was not corrected for in the COSY INFINITY
calculation (d1 = 0 cm), the two frame edges appeared diagonally in a btac vs. ytac plot.
When the shift from the focal plane was correctly accounted for (d1 = −8 cm), the frame position in the non-dispersive plane (ytac ) was independent of the angle in the non-dispersive
plane (btac ). The value of d1 found for the present experiment was similar to the pivot
55

non-dispersive angle, bta (rad)

non-dispersive angle, bta (rad)

0.1

(A)

0.05

0

-0.05

-0.1

0.1

(B)

0.05

0

-0.05

-0.1

-0.02 -0.01
0
0.01
0.02
non-dispersive position, yta (m)

-0.02 -0.01
0
0.01
0.02
non-dispersive position, yta (m)

Figure 4.4: These figures demonstrate how the pivot point shift (d1 ) was corrected. The
thick frame around the edges of the 56 Fe target created many background events (circled)
that were used to determine the pivot point shift. Figure A (left) shows the rotation between
ytac and btac that occurs when the pivot point shift was not accounted for. Figure B (right)
shows that, when this shift was corrected, the frame position in the non-dispersive plane was
independent of the angle in the non-dispersive plane.
point shift seen in a prior S800 spectrograph experiment and to prior simulation results by
G. W. Hitt [47].
The parameter for fractional energy (dta) is closely related to kinetic energy (Te ). Fractional energy (dta) is a measure of the ejectile’s energy relative to the energy of an ejectile
traveling along the central trajectory (E0 ) through the S800 spectrograph with the selected
magnetic rigidity. The central kinetic energy of a 3 He ejectile with Bρ = 2.32930 Tm was
calculated to be E0 = 327.995 MeV. The formula used to determine the ejectile’s kinetic
energy (Te ) from the fractional energy parameter (dta) is:

Te = (1 + dta)E0 .

56

(4.1.6)

A laboratory reference frame scattering angle was needed to construct differential cross
sections. This scattering angle (θlab ) was determined from the angle in the dispersive plane
(ata) and the angle in the non-dispersive plane (bta) at the target location through:

θlab = arctan

tan2 (ata) + tan2 (bta) .

(4.1.7)

The laboratory scattering angles (θlab ) were converted to the center-of-mass reference
frame (θcom ). At forward scattering angles, this transformation can be approximated with
a linear scaling factor (S) with negligible error, as in Equation 4.1.8. The values used for
the scaling factor were S = 1.281 for the 12 C target nuclei and S = 1.176 for the 19 F target
nuclei with the kinematics of this experiment.

θcom = Sθlab .

4.1.4

(4.1.8)

Missing Mass Calculation

Once the motion of the ejectile at the secondary target position was well-defined by ata,
bta, yta, and dta, a kinematic missing-mass calculation was used to recreate the excitation
energy (Ex ) of the recoil nucleus. The 4-momentum of the incoming projectile and the outgoing ejectile system (the target nucleus is immobile) were defined as follows.

57

Incoming projectile 4-momentum:

Tp =340 MeV,

(4.1.9)

pp1 c =0,

(4.1.10)

pp2 c =0, and

(4.1.11)

pp3 c = pp4 c = (Tp + Mp c2 )2 − (Mp c2 )2 ,

(4.1.12)

where ppi was the projectile momentum i-component, Tp was the projectile kinetic energy,
Mp was the projectile mass, and c was the speed of light. In this experiment, the exact
projectile kinetic energy (Tp ) was determined from calibration runs using a CH2 target. The
value of Tp was varied until the ground state excitation energy for the 12 C(t,3 He) reaction
corresponded to zero, which means that it was assumed that there was no systematic error
in the S800 spectrograph energy measurements. Additionally, setting pp1 and pp2 to zero
was an approximation; in practice, the incoming beam had distributions of momentum in
the transverse plane. This approximation decreased the experiment’s angular and energy
resolution.
Outgoing ejectile 4-momentum:

Te = (1 + dta)E0 ,

(4.1.13)

pe1 c = (pe4 c) sin (ata),

(4.1.14)

pe2 c = (pe4 c) sin (bta),

(4.1.15)

pe3 c = (pe4 c) cos (θlab ), and

(4.1.16)

pe4 c =

(Te + Me c2 )2 − (Me c2 )2 ,

58

(4.1.17)

where pei was the ejectile momentum i-component, Te was the ejectile kinetic energy, and
Me was the ejectile mass.
The missing energy (Emis ) was found from:

Emis = (Te + Me c2 ) − (Tp + Mp c2 + Mt c2 ),

(4.1.18)

where Mt was the mass of the target nucleus. The missing momentum (pmis ) was:

pmis =

p2 + p2 + (pe3 − pp3 )2 .
e1
e2

(4.1.19)

This gave the missing mass (Mmis ) and the recoil excitation energy (Ex ) as:

Mmis c2 =

2
Emis − (pmis c)2 and

(4.1.20)

Ex = Mmis c2 − Mr c2 ,

(4.1.21)

where Mr was the mass of the recoil in its ground state. All of the masses relevant for this
data analysis were taken from Reference [70] and are listed in the form of a mass excess1 in
Table 4.6.

4.1.5

Excitation Energy Corrections

The excitation energy of the recoil nucleus (Ex ) is independent of the scattering angle
(θlab ). In practice, small corrections to ata and bta are needed to achieve this. The kine1 The

mass of atom, M , is found with the formula M = A
where A is the atomic mass number and ME is the mass excess.

59

931.494 MeV/c2
1 amu

+ ME ,

Mass Excess
Nucleus
(MeV/c2 )
3H
3 He
1H

n

Nucleus
12 C

14.9498060
14.9312148
7.28897050
8.0713171

12 B
19 F
19 O

Mass Excess
(MeV/c2 )
0.0
13.3689
-1.48739
3.3349

Table 4.6: This table lists the mass excesses that were relevant for this data analysis, as
taken from Reference [70].
atac = ata(0.987) + 0.00042
btac = bta(1.03) − (1.005)yta + 0.00325
xg2c − xg1c
af pc = (0.95) arctan
1073
tac1c
yf pc = (0.95)
1000
Table 4.7: This table contains the kinematic corrections that were applied to the angle in
the dispersive plane (ata) and to the angle in the non-dispersive plane (bta) at the target
position. Similarly, a factor of 0.95 was applied earlier in the data analysis process to correct
the angle in the dispersive plane (afp) and position in the non-dispersive plane (yfp at the
S800 spectrograph focal plane).
matic line from p(t,3 He)n reactions in the CH2 calibration target were used to perform this
correction, because recoil energy is more strongly correlated to θlab in light nuclei than in
heavy nuclei. The first-order corrections that were required to ensure that the excitation
energy (Ex ) was independent of the scattering angle (θlab ) are listed in Table 4.7.
In addition, afp and yfp were scaled by a factor of 0.95 (see Table 4.7) to improve the
reconstruction of the excitation energy (Ex ). These corrections were likely related to minor
inaccuracies in the calculation of the inverse transfer map matrix that was discussed in
Section 4.1.3.
Some minor run-by-run corrections were required to obtain the optimal energy resolution.
These small shifts ensured that the excitation energy was independent of scattering angle
across the entire duration of the experiment, and that the scale for the excitation energy of
the recoil was consistent. These run-by-run corrections compensated for gradual changes in
60

Run
1012
1013
1014
1015
1016
1017
1018

exa shift ata shift yta shift
-0.016
0.8
0.0100
-0.018
1.7
0.0116
-0.015
-0.07
0.0113
-0.01
-0.3
0.0108
-0.005
0.4
0.0107
0.011
1.8
0.0105
0.014
-0.4
0.0107

Run
1019
1064
1067
1068
1069
1099

exa shift ata shift
-0.02
-0.17
0.017
0.12
-0.08
1.3
-0.08
3.6
-0.06
0.04
-0.079
-0.7

yta shift
0.0114
0.0050
0.0057
0.0052
0.0051
0.0018

Table 4.8: This table lists all run-specific corrections. These corrections compensated for
small changes that occurred during the experiment, such as small shifts in the beam alignment or changes in the gas of the CRDCs.
the CRDCs that caused variations in the calibration of the position in the non-dispersive
plane. Run-by-run corrections also compensated for minor changes over time in the triton
beam, such as shifts in the beam angular distribution. A run-by-run shift on yta was used to
center the distribution of the position in the non-dispersive plane on zero, which simplified
the acceptance corrections in Section 4.2.1.
The run-by-run corrections were linear and took the form of:

ytac = yta − yta shif t and

(4.1.22)

Exc = Ex − (atac )(ata shif t) − exa shif t,

(4.1.23)

where the parameters yta shift, ata shift, and exa shift are the measured corrections for each
run.
With these corrections in place, initial excitation energy spectra for recoil nuclei were
obtained. Figure 4.5 shows examples of the excitation energy spectra for the CF2 target,
with kinematics set for the 19 F(t,3 He)19 O∗ reaction (note that the excited states of 12 B due
to 12 C target nuclei were also visible in these spectra). Gamow-Teller transitions (∆L = 0)
are characterized by angular distributions that peak at forward angles. Thus, restricting
61

Counts

Counts

12

3

C(t, He)
ground state

(A)
600

70

12

(B)

3

C(t, He)
ground state

60

500
19

50

3

F(t, He)
0.096 MeV

400
300

40

19

F(t, 3He)
1.47 MeV

200

19

F(t, 3He)
0.096 MeV

30

19

F(t, 3He)
1.47 MeV

20

100

10

0

0

2

4

6

Excitation energy of

8
19

10

0

12

O (MeV)

0

2

4

6

Excitation energy of

8
19

10

12

O (MeV)

Figure 4.5: These figures show excitation energy spectra for (t,3 He) reactions on a CH2
target. Figure A shows the excitation energy spectrum over all scattering angles. Figure B
shows the corresponding spectrum for a scattering angle of 0–1 degrees in the center-ofmass frame, which gives the first indication of what the Gamow-Teller strength distribution
will be. Three strong Gamow-Teller states are visible in this figure; the first two are from
reactions on 19 F and the strongest state is from reactions with 12 C to the ground state of
12 B.
the center-of-mass frame scattering angle (θcom ) to a range of 0–1 degrees gave an initial
indication of where Gamow-Teller strength was located (see Figure 4.5 B). Much of the
higher-lying strength seen in Figure 4.5 A is dominated by dipole transitions (which peak at
higher scattering angles), and thus vanishes in Figure 4.5 B where Gamow-Teller strength is
dominant.

4.2

Differential Cross Sections

Differential cross sections are the next information needed to extract a Gamow-Teller
strength distribution for the 19 F(t,3 He)19 O∗ reaction. Since absolute beam intensities were
not measured in this experiment, relative differential cross sections were calculated in Section 4.2.1
62

from the excitation energy spectra found in Section 4.1. Data on the 12 C(t,3 He)12 B∗ reaction from a prior experiment were then used to scale relative differential cross sections to
absolute differential cross sections in Section 4.2.2. In Section 4.2.3, the carbon background
was subtracted from the absolute differential cross sections for the CF2 target data, which
produced the absolute differential cross sections for only the 19 F(t,3 He)19 O∗ reaction. This
final set of differential cross sections were then used in Section 4.3 to obtain a Gamow-Teller
strength distribution.

4.2.1

Relative Cross Sections

Relative differential cross sections were extracted from the measured excitation energy
spectra. First, corrections were made to account for the detector’s solid angle coverage and
acceptance limitations. Those corrections produced uncalibrated relative cross sections. The
uncalibrated relative cross sections were converted to absolute cross sections by establishing
a proportionality between the 19 F(t,3 He) reaction event rate and the 12 C(t,3 He) reaction
event rate while data was taken on the CF2 target.
To create differential cross sections, the excitation energy spectra were divided into angular bins with a scattering angle width of θlab = 10 mrad in the laboratory reference frame.
Eight bins covered an angular range of 0–80 mrad. The solid angle for an angular bin ranging
from θ1 to θ2 (see Figure 4.6) was calculated with the following equation:
2π

dΩ(θ1 ≤ θlab ≤ θ2 ) =

dφ
0

θ2
θ1

dθ sin (θ) = 2π cos (θ1 ) − 2π cos (θ2 ).

(4.2.1)

An acceptance correction was necessary to compensate for incomplete coverage of events
with large scattering angles in the S800 spectrograph. The S800 spectrograph coverage is
63

0.03 radians

0.02

0.01

0

Figure 4.6: This figure illustrates the solid angle coverage as a function of scattering angle
(θlab ). The red surface area covers a range of 0 to 10 mrad, the orange surface area covers
a range of 10 to 20 mrad, and the yellow surface area covers a range of 20 to 30 mrad.
The solid angle is proportional to these areas and to the number of events (in an isotropic
distribution).

64

Boundaries for ata
atac < (2.73244)dta + 0.1453
atac < (−0.2269)dta + 0.0626
atac > (400.6571)dta − 12.3447
atac > (−0.0115)dta − 0.0562
atac > (−0.9405)dta − 0.0817
atac < (21.8486)dta + 1.1976

Boundaries for bta
btac > (−33.4018)ytac − 0.33949
btac < (7.546039)ytac + 0.099082
btac < (−5.3610)ytac + 0.1124
btac < (−182.2980)ytac + 1.5828
btac > (6.1234)ytac − 0.0772
btac > (−0.6744)ytac − 0.0702

Table 4.9: This table lists the boundary equations used to constrain the angles ata and bta
in the Monte-Carlo acceptance correction simulation.
complete for ejectiles at scattering angles of θlab < 3.5◦ for the excitation energies studied in
this experiment. However, for scattering angles θlab ≥ 3.5◦ , some ejectiles were obstructed by
dipole magnets or fell outside of the range of the focal plane detectors. Instead of disregarding
angles with azimuthal coverage less than 2π, a Monte-Carlo simulation was performed to
model and correct for the partial loss of events. This procedure allowed for the extraction
of differential cross sections up to θlab = 5◦ [48].
The Monte-Carlo simulation produced an isotropic distribution of events over the angles,
positions, and energy ranges relevant to this data analysis. It compared the ideal distribution of events to a distribution of events with certain constraints on the angles in the
dispersive and non-dispersive planes to determine the probability, Pacceptance , of a valid
event successfully reaching the S800 spectrograph focal plane as a function of dta, yta, and
θlab . The angular constraints used for this simulation were established experimentally. These
constraints are represented by the white boundary lines in Figure 4.7 and the equations for
the constraints are available in Table 4.9. In the dispersive direction (ata), the angular
acceptance depends on the momentum of the particles (dta) as seen in Figure 4.7 A. In
the non-dispersive direction (bta), the angular acceptance depends on the non-dispersive
position of the particles (yta) as seen in Figure 4.7 B.
Once the solid angle (dΩ) and the acceptance correction (Pacceptance ) were established,
65

non-dispersive angle, bta (rad)

dispersive angle, ata (rad)

(A)
0.05

0

0.1 (B)

0.05

0

-0.05

-0.05
-0.1
-0.05 -0.025 0 0.025 0.05
fractional energy, dta (unitless)

-0.02 -0.01
0
0.01
0.02
non-dispersive position, yta (m)

Figure 4.7: These spectra show the boundaries used in the Monte-Carlo simulation that
determined the acceptance correction. These figures use data from the CF2 target.
the unscaled relative differential cross sections were determined from the excitation energy
spectra. Each count (N ) in a given excitation energy spectrum was divided by the solid
angle and acceptance correction:

N (θlab , dtac , ytac )
dσ
(arbitrary units) =
.
dΩ
dΩ(θlab ) · Pacceptance (θlab , dtac , ytac )

(4.2.2)

These differential cross sections, when compared to measured excitation energy spectra,
showed a significant enhancement at small scattering angles from the solid angle and a
moderate enhancement at large angles from the acceptance correction. See Figure 4.8 for
an example of the transformation from excitation energy spectra to differential cross section
for one particular transition.

66

Detected counts
100

Counts

80
60
40
20
0
0

1

2

3

4

5

ΘCOM degrees

Cross section arbitrary units

Relative cross section
60
50
40
30
20
10
0
0

1

2

3

4

5

ΘCOM degrees
Figure 4.8: These spectra illustrate the transformation from excitation energy spectra to
differential cross section for one specific excited state of 19 O. The top figure shows the
measured excitation energy spectrum for the 0.096 MeV excited state of 19 O. The bottom
figure shows the relative cross section for the same excited state after the solid angle and the
acceptance correction were applied. The solid angle enhances the contribution of the first
several angular bins, while the acceptance correction has the strongest effect on the three
data points with θcom > 3.5◦ .

67

4.2.2

Absolute Cross Sections

So far, the differential cross sections are in arbitrary units. They can be normalized
to an absolute scale by taking advantage of the carbon in both the CF2 and CH2 targets
and information from a prior experiment. In experiment 06032, performed at the NSCL
by C. J. Guess et al. [48, 71], an absolute cross section was measured for the reaction
12 C(t,3 He)12 B

g.s. .

This reaction was also observed in the current experiment with both the

CH2 and the CF2 targets. The data from the 12 B ground state in the current experiment
was scaled to the data from experiment 06032 to find absolute differential cross sections.
While it would have been ideal to compare the 12 B ground state data from experiment
06032 directly to the 12 B ground state data from the CF2 target in the current experiment
this was not possible due to background events from reactions on 19 F nuclei. Instead, the
12 B

ground state data from experiment 06032 was scaled to the CH2 data taken in the

current experiment with the scaling factor of abs . After that, an additional factor of prop
was used to scale the data taken on the CH2 target to the data taken on the CF2 target.
The 12 B ground state in the CH2 target data was well-separated from other 12 B states and
from events due to the hydrogen content of the target, so there were no notable background
events to contend with. Figure 4.9 shows a comparison of the absolute differential cross
section for 12 B ground state data from experiment 06032 with the distribution that was
extracted during this experiment. A scaling factor of abs = 0.048 ± 0.001 mb was found
from a linear fit between the two differential cross sections. This factor of abs provided
an absolute differential cross section for the data taken on the CH2 target in the current
experiment.
A relationship between the CH2 and CF2 data sets ( prop ) was established by taking

68

dσ/dΩ (mb/sr)

22.5

12

3

12

C(t, He) B(g.s.)

20
17.5

NSCL 06032
CH2 target, carbon

15
12.5

∆L=0+2

10
7.5

∆L=0

5
2.5 ∆L=2
0
0 1 2

3

4

5

6

7
θc.m. (deg)

Figure 4.9: This figure shows a comparison of the data from experiment 06032 [48] and
the CH2 target of the current experiment for the 12 B ground state after the scaling factor
abs was applied. The angular distribution for this experiment was in excellent agreement
with the results from experiment 06032. The ratio between the two results was used to
establish an absolute cross section for the data collected in the current experiment, because
the absolute cross section was carefully normalized in experiment 06032.
.

69

into account the differences in integrated beam intensities, data acquisition live times, and
target thicknesses. A direct measurement of the triton beam rate at the target position
was not available for this experiment, so three different indirect measurements of the beam
rate were used instead. Two of these beam rate measurements came from the primary beam
current probes that were described in Section 3.1, the Z001I-C probe and the Z026R-C probe.
These primary beam current measurements were expected to be directly proportional to the
secondary triton beam intensity. The third indirect measurement of the triton beam intensity
was made at the S800 spectrograph focal plane. As mentioned earlier, there are a significant
number of events seen in the S800 spectrograph particle identification spectra that are not
due to reactions in the experimental targets (see Figure 4.1). These background events
appeared to be unreacted triton particles that were scattered into the focal plane detectors,
and thus also proportional to the triton beam intensity. The background events were used
in aggregate to produce a beam rate ratio between the two different data sets for the CH2
and CF2 targets.
CH2
The beam rate ratios ( 1 =
) found were 0.35, 0.33, and 0.36 for the Z001I-C
CF2
probe, the Z026R-C probe, and the unreacted beam in the S800 spectrograph focal plane,
respectively. The beam was fairly stable over the entire experiment, with a run time of
3 hours for the CH2 target and 9.5 hours for the CF2 target. The average of these three
measurements was taken as the final value of the beam rate ratio, and a conservative error
of two standard deviations was used to reflect that these are indirect measurements, for a
final value of 1 = 0.35 ± 0.03.
The data sets taken with the two different reaction targets had notable differences in
overall detector count rates due to composition, which led to differences in data acquisition
live times. The CH2 target, which had a higher measured count rate due to the large reaction
70

cross section of triton on hydrogen, had an electronics dead time of 8.7%; the CF2 target
had a dead time of 3.7%. This gave a ratio of 2 = 0.9634 between the CH2 data set and the
CF2 data set related to data acquisition live time differences. The error in this dead time
measurement was neglected because it was much smaller than other sources of experimental
error.
Differences in target thickness and composition also needed to be taken into account to
establish a proportionality between the CH2 data set and the CF2 data set. The CH2 target
had a thickness of 10.0 mg/cm2 , while the CF2 target had a thickness of 11.2 mg/cm2 .
Fluctuations in target thicknesses were negligible sources of error because the beam spot
covered a large area of the target foils. The target composition differences were taken into
account by calculating the proportion of the molar mass of each target due only to carbon,
which was

12.0108
14.0266

for the CH2 target and

12.0108
50.0076

for the CF2 target. Combined,

this gave a target thickness scaling ratio between the CH2 data set and the CF2 data set
of 3 = 3.183. This scaling factor was only applicable to the differential cross section for
reactions on carbon in the targets. In order to scale the cross section due to reactions on
1
fluorine nuclei in the CF2 target, an additional factor of 2 was needed to account for there

being twice as many fluorine nuclei as carbon nuclei. Combining the three aforementioned
scaling factors gave prop = 1 2 3 = 1.06 ± 0.09 as an overall proportionality factor to
scale the carbon-target events in the CF2 data set from the CH2 data set. The error on this
proportionality factor was treated as a systematic error. It was not included in the statistical
error bars on differential cross section plots.

71

19

JΠ

C t,3 He

12

B ground state events

3

2
0.096 MeV

dΣ d

mb sr

12

19

F t, He O
at 115 MeV nucleon

15

10

3

JΠ

5

0

1

2
1.471 MeV

0

2

4

6

Excitation Energy of

8
19

10

12

O MeV

Figure 4.10: This figure shows the scaled absolute cross section for 19 O events from the CF2
target. Note that the 12 B events are not subtracted, and are incorrectly scaled by a factor
of two in this graph due to the difference in the number of carbon nuclei to fluorine nuclei
in the target. This spectrum covers the full angular range of the experiment. Prominent
features of the spectrum are marked, including the 12 B ground state and two low-lying 19 O
states.

72

4.2.3

Isolating the

19

F Cross Section

After the proportionality factors abs and prop were established to scale the data set
from the CF2 target to an absolute cross section for either target nucleus, final differential
cross sections for the 19 F(t,3 He)19 O∗ reaction were obtained through subtraction of events
due to the carbon nuclei in the CF2 target. This background subtraction was unnecessary
over a significant energy range due to the large difference in ground state Q-values for the
two reactions. The 19 F(t,3 He)19 Og.s. reaction has a Q-value of −4.802 MeV, while the
12 C(t,3 He)12 B

g.s.

reaction has a Q-value of −13.350 MeV. This placed the ground state

of 12 B at 8.5 MeV above the ground state of 19 O on an energy excitation spectrum (see
Figure 4.10 for an illustration). Since the experimental resolution was 480 keV, carbon
background subtraction was not needed before Ex 19 O = 8 MeV. No notable hydrogen
contamination was visible in the CF2 data set, but any events related to hydrogen contamination would have sat 4 MeV below the ground state of 19 O and so were not a significant
concern (the p(t,3 He)n reaction has a Q-value of −0.764 MeV).
For excitation energies of Ex 19 O < 8 MeV, the following formula was used to obtain
the absolute differential cross sections for 19 O∗ events:

dσ 19 ∗
O =
dΩ

N [CF2 ]
dΩ · Pacceptance

prop

1
2

abs .

(4.2.3)

For excitation energies of Ex 19 O ≥ 8 MeV, where carbon background subtraction was
necessary, this formula was used instead:

dσ 19 ∗
O =
dΩ

N [CF2 ]
dΩ · Pacceptance

prop

−

73

N [CH2 ]
dΩ ∗ Pacceptance

1
2

abs .

(4.2.4)

Differential cross sections as a function of scattering angle were created to calculate
Gamow-Teller transition strengths. Most of the states in 19 O couldn’t be clearly separated
from each other, so separate angular distributions were generated for different excitation
energy bin ranges. For excitation energies of 3.5 MeV ≤ Ex 19 O ≤ 16 MeV, data was
divided in 0.5 MeV wide bins, with the bin width roughly reflecting the experimental energy
resolution of 480 keV. The two low-lying states of 19 O beneath Ex 19 O = 3.5 MeV were
well-separated from other spectrum features, so they were treated differently. For the 0.096
MeV (first excited) state, an energy bin ranging from -0.52 MeV to 0.68 MeV was used; for
the 1.471 MeV (second excited) state, a bin range of 0.86 MeV to 2.1 MeV was used. For
angular distributions of differential cross sections, binning of the scattering angle was done
in increments of 10 mrad in the laboratory frame from 0 mrad to 80 mrad; 10 mrad bins
were chosen to reflect the angular resolution achieved in this experiment. Figure 4.11 shows
an example of an angular distribution for a differential cross section after carbon background
subtraction.

4.3

Calculating Gamow-Teller Strengths

Gamow-Teller transition strengths are extracted from charge-exchange differential cross
sections at zero momentum transfer through a well-established proportionality coefficient
that is referred to as the unit cross section, shown in Equation 4.3.1 (see also Section 2.1).
In this proportionality equation, σGT is the unit cross section, B[GT] is the Gamow-Teller
ˆ
dσ
strength, and
(q = 0) is the charge-exchange differential cross section at zero momentum
dΩ
transfer (q = 0).
dσ
(q = 0) = σGT (B[GT]) .
ˆ
dΩ
74

(4.3.1)

Cross section mb sr

1.4

0.096 MeV excited state of

1.2

19

O

1.0
0.8
0.6
0.4
0.2
0.0
0

1

2

3

4

5

ΘCOM degrees
Figure 4.11: This figure displays an angular distribution of the differential cross section for
the 0.096 MeV excited state of 19 O. This is an example of what the differential cross sections
look like after all scalings and corrections were performed and after carbon subtraction was
completed.
The differential cross sections extracted in Section 4.2 occurred at a nonzero momentum
transfer. Three factors must be taken into account to extract a zero momentum transfer
cross section

dσ
(q = 0)
dΩ

from the experimentally determined cross sections with finite

momentum transfer so that Equation 4.3.1 can be used to determine the Gamow-Teller
strength distribution of the 19 F(t,3 He)19 O∗ reaction. First, the measured (t,3 He) reactions
can occur at a variety of angular momentum transfers (∆L), but the Gamow-Teller states
only correspond to zero angular momentum transfer (∆L = 0). A multipole decomposition
analysis (MDA) was used to extract the portion of the differential cross section due to only
zero angular momentum transfer. Second, the extracted differential cross sections cover a
range of scattering angles, but the cross section at a scattering angle of zero (θcom = 0◦ ) is
required for the extraction of Gamow-Teller strengths, so this differential cross section value

75

F(t,3He) at 115 MeV/nucleon
for laboratory scattering
1
angles of 0 to 10 mrad

0.8
0.6
0.4
0.2
0

0

2

4

6

8

10

Excitation energy of
Cross section (mb/sr⋅100 keV)

Cross section (mb/sr⋅100 keV)

19

1.2

19

12 14

0.8
0.6
0.4
0.2

2

4

6

8

10

Excitation energy of

19

12 14

19

0.8
0.6
0.4
0.2

0

2

4

6

8

10

Excitation energy of

19

0

F(t,3He) at 115 MeV/nucleon
for laboratory scattering
1
angles of 10 to 20 mrad

O (MeV)

F(t,3He) at 115 MeV/nucleon
for laboratory scattering
1
angles of 20 to 30 mrad

0

1.2

0

16

Cross section (mb/sr⋅100 keV)

Cross section (mb/sr⋅100 keV)

1.2

16

O (MeV)

1.2

19

12 14

16

O (MeV)

19

F(t,3He) at 115 MeV/nucleon
for laboratory scattering
1
angles of 30 to 40 mrad

0.8
0.6
0.4
0.2
0

0

2

4

6

8

10

Excitation energy of

19

12 14

16

O (MeV)

Figure 4.12: These figures show the absolute differential cross sections for the 19 F(t,3 He)
reaction over an excitation energy range in 19 O of -0.5 MeV to 16 MeV. Each spectrum
covers a different scattering angle range, as listed in the figure. Strong Gamow-Teller states
are most visible in the first spectrum, which covers the most forward scattering angles of
0 mrad ≤ θlab ≤ 10 mrad. The figures continue onto the next page.

76

F(t,3He) at 115 MeV/nucleon
for laboratory scattering
1
angles of 40 to 50 mrad

0.8
0.6
0.4
0.2
0

0

2

4

6

8

10

Excitation energy of
Cross section (mb/sr⋅100 keV)

Cross section (mb/sr⋅100 keV)

19

1.2

19

12 14

0.8
0.6
0.4
0.2

2

4

6

8

10

Excitation energy of

19

12 14

19

0.8
0.6
0.4
0.2

0

2

4

6

8

10

Excitation energy of

19

0

F(t,3He) at 115 MeV/nucleon
for laboratory scattering
1
angles of 50 to 60 mrad

O (MeV)

F(t,3He) at 115 MeV/nucleon
for laboratory scattering
1
angles of 60 to 70 mrad

0

1.2

0

16

Cross section (mb/sr⋅100 keV)

Cross section (mb/sr⋅100 keV)

1.2

16

O (MeV)

1.2

19

12 14

16

O (MeV)

19

F(t,3He) at 115 MeV/nucleon
for laboratory scattering
1
angles of 70 to 80 mrad

0.8
0.6
0.4
0.2
0

0

2

4

6

8

10

Excitation energy of

19

12 14

16

O (MeV)

Figure 4.13: These figures are continued from the prior page. They show the absolute
differential cross sections for the 19 F(t,3 He) reaction over an excitation energy range in 19 O
of -0.5 MeV to 16 MeV. Each spectrum covers a different scattering angle range, as listed in
the figure. Strong Gamow-Teller states are most visible in the first spectrum, which covers
the most forward scattering angles of 0 mrad ≤ θlab ≤ 10 mrad.

77

must be extrapolated. The differential cross section angular distributions were extrapolated
to θcom = 0◦ . Third, the measured reactions took place at a nonzero Q-value (−4.802 MeV ≤
Q ≤ −20.802 MeV), so the extracted differential cross section angular distributions were
projected to Q = 0 to satisfy the zero momentum transfer condition. After these three
dσ
factors were taken into account to obtain dΩ (Q = 0, 0◦ ), as described in Section 4.3.1, the

final Gamow-Teller strength distribution was calculated in Section 4.3.3.

4.3.1

Multipole Decomposition Analysis

A multipole decomposition analysis (MDA) [40, 48, 71] was used to extract the ∆L = 0
component of the measured differential cross sections. As discussed in Section 2.2, theoretical
angular distributions with orbital angular momentum of 0 ≤ L ≤ 3 were calculated for
transitions in the 19 F(t,3 He)19 O∗ reaction at selected Q-values. The MDA was performed
by fitting a linear combination of the theoretical angular distributions that were generated
with a distorted wave Born approximation calculation (see Section 2.2) to the differential
cross sections obtained in Section 4.2.2 (see Figure 4.11).
Initial testing of the MDA fit procedure used fits with four variable parameters, one for
each possible value of ∆L:

dσ
dσ
dσ
dσ
dσ
=A
+B
+C
+D
,
dΩ exp
dΩ ∆L=0
dΩ ∆L=1
dΩ ∆L=2
dΩ ∆L=3

(4.3.2)

dσ
are the experimentally determined differential cross section data points; the
dΩ exp
dσ
letters A, B, C, and D are variable parameters in the fit, and
is one of the
dΩ ∆L=i
where

possible theoretical angular distributions with an angular momentum of ∆L = i, for i within
the range 0–3. Since there are only eight data points available, the fit was restricted to a
78

maximum of one component per ∆L to preserve some degrees of freedom (ν). The MDA
fit procedure was limited to components with ∆L ≤ 3 because excitations associated with
∆L > 3 are not expected to be highly populated for low momentum transfers and because
their angular distributions peak outside of the angular range of this experiment. However,
it was found that in these four-parameter fits, the best fit always had at least one parameter
scaled to zero, so the process was expanded to consider several alternative combinations
of angular momentum with only two or three varying parameters. Using several different
models for the MDA fitting procedure gave insight into the systematic error due to choice
of the model.
The best fit (chosen by the χ2 /ν fit parameter) from this MDA process was used as the
final set of angular distributions for each differential cross section energy bin. To estimate the
systematic error in the MDA procedure, the most extreme fits that still had a probability of
being a valid ≥ 5% were used. Here, “extreme fits” refers to the fits that gave the highest and
lowest possible values of the ∆L = 0 angular distribution at θcom = 0◦ , which corresponded
to the maximum and minimum possible values of Gamow-Teller strength. These maximum
and minimum possible Gamow-Teller strengths were then used as the upper and lower limits
of the systematic uncertainty.
Note that this MDA procedure was applied to differential cross sections for excitation
energy bins of width 0.5 MeV, ranging from 3.5 MeV to 16 MeV in the excitation energy
of 19 O. The systematic errors from the choice of the MDA fit model for these excitation
energy bins were found to be very large. The possibility of there being no Gamow-Teller
strength could not be excluded over this excitation energy range. Despite this, the results
of the MDA procedure for these excitation energy bins can be used to provide upper limits
on the Gamow-Teller strength distribution.
79

The two low-lying excited states of 19 O at 0.096 MeV and 1.471 MeV are well-separated
in the spectra from other transitions, and no transitions with odd values of orbital angular
momentum are expected for Ex < 3 MeV. The systematic error due to choice of the fit
model in these two particular MDA procedures was very small. However, an additional
source of systematic error affects the first and second excited states of 19 O. Interference
between the transition amplitudes due to the στ and Tτ (tensor-τ ) components of the
effective interaction creates a systematic error that cannot be accounted for with only the
MDA procedure. This systematic error was estimated with a comparison of theoretical
calculations with and without the Tτ interaction by following the procedure described in
Reference [42]. The systematic error from the tensor-τ term was found to be 3.4% (0.002)
for the first excited state of 19 O at 0.096 MeV and 2.0% (0.001) for the second excited state
of 19 O at 1.471 MeV. Since the systematic error from the MDA is so large for the other
extracted Gamow-Teller states, error from the tensor term was ignored as negligible in the
reported values for error in Table 4.11 for Gamow-Teller strengths with Ex ≥ 3.5 MeV.

4.3.2

Extrapolating the Cross Section to Q = 0

Once a best fit for the theoretical angular distributions was determined, the differential
cross section of the ∆L = 0 component at θcom = 0◦ was extracted:

dσ
(Qexp , 0◦ )experiment .
dΩ

The equation used to further extrapolate to Q = 0 [39] is:

dσ
(Q = 0, 0◦ ) =
dΩ

dσ
◦
dΩ (Q = 0, 0 )
dσ
◦
dΩ (Qexp , 0 ) theory

·

dσ
(Qexp , 0◦ )
dΩ

.

(4.3.3)

experiment

In this equation, the ratio between the theoretical angular distributions at Q = 0 and
Q = Qexp was used as the basis of the extrapolation. The values calculated for the ratio
80

Ex (19 O) (MeV)
0.096
1.471
3.5–4.0
4.0–4.5
4.5–5.0
5.0–5.5
5.5–6.0
6.0–6.5
6.5–7.0
7.0–7.5

Ratio
1.043 ‡
1.113 ‡
1.124
1.144
1.164
1.183
1.203
1.222
1.242
1.261

Ex (19 O) (MeV)
7.5–8.0
8.0–8.5
8.5–9.0
9.0–9.5
9.5-10.0
10.0–10.5
10.5–11.0
11.0–11.5
11.5–12.0
12.0-12.5

Ratio
1.281
1.300
1.320
1.340
1.359
1.378
1.398
1.417
1.437
1.456

Ex (19 O) (MeV)
12.5–13.0
13.0–13.5
13.5–14.0
14.0–14.5
14.5–15.0
15.0–15.5
15.5–16.0

Ratio
1.476
1.496
1.515
1.535
1.554
1.574
1.593

Table 4.10: This table lists the ratios that were calculated to extrapolate the differential
cross section at a finite reaction Q-value to Q = 0 in order to fulfill the condition of zero
momentum transfer in Equation 4.3.1 and calculate Gamow-Teller transition strengths, as
discussed in the text. The ‡ indicates that, for the first two excited states of 19 O, the precise
calculated ratio was used instead of a value interpolated from the fit.

of

dσ
◦
dΩ (Q = 0, 0 )
dσ
◦
dΩ (Qexp , 0 ) theory

for each Gamow-Teller transition were then used to construct a

linear fit so that values of this ratio could be obtained for an arbitrary energy bin. The
values used for this ratio in the calculation of the Gamow-Teller strength distribution are
listed in Table 4.10. For the first two excited states of 19 O, the precise calculated ratio was
used instead of a value interpolated from the fit.

4.3.3

Gamow-Teller Strength Distribution

With an extracted value for

dσ
(Q = 0, 0◦ ), the only variable needed in Equation 4.3.1 to
dΩ

find the Gamow-Teller strength is the unit cross section (ˆGT ). The unit cross section was
σ
calculated from a global fit of σGT as a function of nuclear mass (A) that was performed in
ˆ
Reference [38]:
σGT =
ˆ

109
,
A0.65

81

(4.3.4)

where A is the mass number of the target nuclei. This fit has an error of 5% [38]. This
equation gives σGT = 16.1 ± 0.8 mb/sr for the 19 F(t,3 He) reaction. The error in the unit
ˆ
cross section was treated as a systematic error.
Gamow-Teller strengths were then calculated with equation 4.3.1. The Gamow-Teller
strength distribution for the 19 F(t,3 He)19 O∗ reaction at 115 MeV (and associated errors) are
listed in Table 4.11. The transition from the ground state of 19 F to the first excited state of
19 O

at 0.096 MeV was found to have a Gamow-Teller strength of B[GT] = 0.068+0.011 . The
−0.009

error in this figure is discussed in greater detail in Section 5.3 and this result for Gamow-Teller
strength is further discussed in Section 5.1 in the context of its application to fluorine nucleosynthesis. The transition from the ground state of 19 F to the second excited state of 19 O at
1.471 MeV was found to have a Gamow-Teller strength of B[GT] = 0.057+0.011 . The results
−0.010
of the MDA fitting process for these two low-lying states are shown in Figures 4.14 A and B,
respectively. A Gamow-Teller strength distribution was extracted for higher lying strength
for 3.5 MeV ≤ Ex ≤ 16 MeV in 19 O as well. However, the systematic error on these
Gamow-Teller strengths is large enough that B[GT] = 0 cannot be excluded. Additionally, the statistical error for the Gamow-Teller strengths in the excitation energy bins with
8.0 MeV ≤ Ex ≤ 16 MeV in 19 O are significantly larger than those with Ex < 8.0 MeV
because of the carbon background subtraction process discussed in Section 4.2.3.
The Gamow-Teller strength distribution was also compared to theoretical calculations of
Gamow-Teller strength. These theoretical Gamow-Teller strength distributions were calculated using the shell-model code OXBASH [50] in the sd model space with the USDB [72]
interaction and in the spsdpf model space with the WBP [54] interaction. The calculated
Gamow-Teller strengths from OXBASH are listed in Table 4.12. Figures 4.15 A and B
show comparisons between these two different theoretical Gamow-Teller strength distribu82

1.4

0.096 MeV state of

mb sr

B GT

1.0

dΣ d

1.2

0.068

19

O

0.011
0.009

0.6

0.8

L

0

Sum

0.4
0.2

L

0.0
0

1

2

2

3

4

5

Θcom degrees
1.4

1.471 MeV state of

mb sr

0.6

B GT

1.0

dΣ d

1.2

0.057

19

O

0.011
0.010

0.8

0.4
0.2
0.0
0

Sum
L

0

L
1

2
2

3

4

5

Θcom degrees
Figure 4.14: These figures show the results of the multipole decomposition analysis for the
first two excited states of 19 O. The blue circles are data points with statistical error bars.
Figure A shows the decomposition of the 0.096 MeV first excited state of 19 and Figure B
shows the decomposition of the 1.471 MeV second excited state. The red lines are ∆L = 0
components of the angular distribution, used for determining the Gamow-Teller strength
of each state. The green lines are ∆L = 2 components of the angular distribution from
quadrupole transitions. The blue lines are sums of the separate components.
83

tions and the distribution extracted in this work. When comparing theoretical GamowTeller strengths to experimental Gamow-Teller strengths, it is expected that the theoretical
Gamow-Teller strengths will need to be quenched by a factor of (0.77)2 , as discussed in
References [42, 73, 74, 75]. This factor accounts for the degrees of freedom not considered
in the shell model calculations. When quenching is applied, the WBP interaction comes
closest to reproducing the experimentally determined Gamow-Teller strength of the first
excited state in 19 O at 0.096 MeV; the quenched strength of the WBP interaction is 1.1
standard deviations from experiment, while the quenched strength of the USDB interaction
is 2.1 standard deviations from the Gamow-Teller strength found experimentally. Both interactions do a poor job of reproducing the Gamow-Teller strength of the second excited
state in 19 O at 1.471 MeV; their quenched strengths are 4.1 (for the WBP interaction) and
4.4 standard deviations (for the USDB interaction) from the Gamow-Teller strength found
experimentally.

84

0.25 A

Comparison to USDB

B GT

0.20
0.15
0.10
0.05
0.00
0

5

10

Ex of
0.25 B

19

15

O MeV

Comparison to WBP

B GT

0.20
0.15
0.10
0.05
0.00
0

5

10

Ex of

19

15

O MeV

Figure 4.15: These figures show comparisons between theoretical and experimentally extracted Gamow-Teller strength distributions. Figure A compares the Gamow-Teller strengths
extracted in this experiment (blue points) with the strength distribution produced with the
USDB [72] interaction (yellow bars) in the sd model space. Figure B compares the GamowTeller strengths extracted in this experiment (blue points) with the strength distribution
produced with the WBP [54] interaction (green bars) in the spsdpf model space. Both
theoretical strength distributions were calculated with OXBASH [50]. In both cases, the
theoretical Gamow-Teller strengths are quenched by a factor of (0.77)2 as discussed in the
text.
85

Ex 19 O

B[GT]

B[GT]

Error (+)

(MeV)

B[GT]

Error (−)

0.096

0.068

0.011

0.009

1.471

0.057

0.011

0.010

3.5–4.0

0

0.018

0

4.0–4.5

0

0.013

0

4.5–5.0

0

0.038

0

5.0–5.5

0

0.052

0

5.5–6.0

0.022

0.036

0.028

6.0–6.5

0.011

0.032

0.017

6.5–7.0

0.049

0.030

0.058

7.0–7.5

0

0.027

0

7.5–8.0

0.037

0.032

0.045

8.0–8.5

0.111

0.091

0.132

8.5–9.0

0.146

0.076

0.172

9.0–9.5

0.046

0.054

0.059

9.5-10.0

0.068

0.036

0.080

10.0–10.5

0.085

0.047

0.096

10.5–11.0

0.041

0.043

0.053

11.0–11.5

0.011

0.055

0.025

11.5–12.0

0

0.070

0

12.0-12.5

0.096

0.079

0.111

12.5–13.0

0.037

0.111

0.058

13.0–13.5

0

0.151

0

13.5–14.0

0.099

0.092

0.118

14.0–14.5

0

0.156

0

14.5–15.0

0.007

0.034

0.026

15.0–15.5

0

0.169

0

15.5–16.0

0

0.149

0

Table 4.11: This table lists the calculated Gamow-Teller strengths for the 19 F(t,3 He)19 O∗
reaction at 115 MeV. Associated errors are also listed.

86

WBP

USDB

Jπ

E (MeV)

B[GT]

E (MeV)

B[GT]

3+
2

0.29

0.0979

0.12

0.0835

1+
2

1.47

0.0286

1.57

0.0234

3+
2

3.75

0.0017

3.80

0.0018

3+
2

5.53

0.0096

6.08

0.0155

1+
2

7.13

0.1445

7.86

0.2036

3+
2

8.22

0.0328

8.84

0.0282

3+
2

9.48

0.0025

10.36

0.0036

3+
2

10.49

0.0070

11.28

0.0003

1+
2

10.72

0.0114

11.24

0.0058

3+
2

12.71

0.0006

12.58

0.0000

1+
2

15.34

0.0018

16.06

0.0045

1+
2

16.02

0.0001

16.71

0.0001

Table 4.12: This table lists the predicted Gamow-Teller strengths for transitions in the
19 F(t,3 He)19 O∗ reaction for two different interactions, USDB [72] and WBP [54], along with
the predicted energy level for each transition. These calculations were both performed with
the shell-model code OXBASH [50] in the sd model space. These values are all unquenched;
when used in calculations, a quenching factor of (0.77)2 should be applied as described in
the text.

87

Chapter 5
Results

5.1

β-decay Rate Calculation

As discussed in Section 1.3, the Gamow-Teller strengths of the 19 F(t,3 He)19 O∗ reaction
have an application in fluorine nucleosynthesis. In hot stellar environments, it is possible to
thermally populate low-lying excited states of 19 O∗ . These excited states are not accessible
to a traditional β-decay experiment because they de-excite by γ-ray emission in nanoseconds,
whereas the β-decays of interest take place on a time scale of several seconds. As discussed
in Reference [26] and in Section 1.3, the lowest-lying excited state of 19 O at 0.096 MeV
potentially makes a significant impact on the total β-decay rate of stellar 19 O. This particular
excited state at 0.096 MeV is allowed to β-decay to the ground state of 19 F directly, whereas
the 19 O ground state to 19 F ground state is a second forbidden (∆π = 0, ∆L = 2) β-decay
by spin-parity considerations (see Figure 1.2 for the relevant level scheme).
In Section 4.3.3, the extracted Gamow-Teller strength of the transition from the ground
state of 19 F to the 0.096 MeV first excited state of 19 O was found to be B[GT] = 0.0676+0.011 .
−0.009
This Gamow-Teller strength can be used to determine the weak interaction rate for the
transition from the 0.096 MeV state in 19 O to the ground state of 19 F. To account for the
fact that this measured Gamow-Teller strength is for the 19 F →19 O (A → B) direction while
the weak decay proceeds in the opposite 19 O →19 F (B → A) direction, the Gamow-Teller

88

strength must be scaled by a factor of:

F =

(2JA + 1)
,
(2JB + 1)

(5.1.1)

1
where JA = 2 for the total angular momentum of the transition state in the fluorine nucleus,

JB = 3 for the total angular momentum of the transition state in the oxygen nucleus, for a
2
scaling factor of F = 0.5. It is possible to decay to other states in 19 F from the 0.096 MeV
state in 19 O, but those transitions were not accessible via the experiment performed in this
dissertation. There are five total energy levels in 19 F with an allowed spin-parity for β-decay
reactions and within the permitted Q-value range of such a β-decay; these 19 F energy levels
are listed in Table 5.1.
Theoretical shell-model calculations are used to determine the Gamow-Teller strengths for
decay transitions that go from the 0.096 MeV state in 19 O to an excited state i in 19 F (listed in
Table 5.1). These theoretical Gamow-Teller strengths allow for an estimate of the branching
ratios (Ii ) for each allowed transition. The shell-model calculations to obtain theoretical
Gamow-Teller strengths were performed in the sd -model space with the USDB interaction
[72] with the code OXBASH [50]. It is difficult to assign an error to the theoretical GamowTeller strengths. Fortunately, the main role of the theoretical Gamow-Teller strengths in this
calculation is only to determine a branching ratio for the possible β-decay transitions, and
that branching ratio is dominated (Iexp = 86%) by the measured transition. Several different
assumptions for the error on the theoretical Gamow-Teller strengths were tested ranging up
to 100% error, and it was found to have only a small influence on the error of the final
β-decay rate calculation. Ultimately, the theoretical Gamow-Teller strengths were assigned
an error of 36% because that corresponds to the variation found between the Gamow-Teller

89

19 F

Energy Level
(MeV)

Jπ

Q-value
(MeV)

B[GT]
USDB

Quenched
B[GT] USDB

0.000

1+
2

4.918

0.04177 ‡

0.0248 ‡

0.197

5+
2

4.721

0.00064

0.0004

1.554

3+
2

3.364

0.04538

0.0269

3.908

3+
2

1.010

0.39516

0.2343

4.550

5+
2

0.368

0.49186

0.2916

Table 5.1: This table lists the energy levels and associated spin-parity assignments of the
states in 19 F that are relevant for weak reaction rate calculations for the β-decay of the
3+
0.096 MeV excited state in 19 O (J π = 2 ). Q-values for β-decays from the 0.096 MeV state
in 19 O are listed in the third column. Gamow-Teller theoretical strengths (quenched by a
factor of (0.77)2 [73, 74, 75, 42], discussed in Section 4.3.3) are listed as well. The 19 F energy
levels and spin-parity assignments were taken from Reference [25]. The ‡ denotes that this
theoretical Gamow-Teller strength is not actually used in further calculations, because it is
replaced with the measured value B[GT] = 0.034+0.006 (with F = 0.5 as discussed in the
−0.004
text) for this transition from Section 4.3.3.

90

strength measured in this experiment and its corresponding (quenched; see Section 4.3.3)
theoretical value from the USDB interaction. The USDB interaction was chosen for this
purpose because it is generally regarded as a better model for sd-shell isotopes, even though
in this case it performed worse than the WBP interaction (see Section 4.3.3). In practice,
the choice of interaction in this calculation had no significant effect on the outcome.
Gamow-Teller strengths can be related to f t-values via the equation:

ft =

where

K
2
gv

,
2
B[GT] ga
gv

(5.1.2)

K
= 6143 ± 2 s and ga = −1.2694 ± 0.0028 [39]. Once an f t-value is calculated
gv
2
gv

for specific transition to a final state i, it can be related to the partial half-life (ti ) and the
partial decay rate λi via the equations [2, 76]:

ti = 10 log (f t)i −log (f )i and

λi =

ln (2)
,
ti

(5.1.3)

(5.1.4)

where log (f )i is a calculated value based on the Q-value of the β-decay reaction, the type of
β-decay (a β + reaction vs. a β − reaction, and whether the decay is allowed or forbidden),
and the proton number (Z) of the daughter nucleus. The log (f )i values used in these weak
reaction rate calculations were interpolated from tables in Reference [76], and any error in
them was considered to be negligible.
For the set λi of β-decay transitions from a single specific state a in 19 O to the set i of

91

allowed states in 19 F, the branching ratio to each state i (Ii ) is [76]:

Ii =

λi
,
λi

(5.1.5)

i

and the overall half-life (T1/2,a ) and weak decay rate (λa ) for all decays from the state a are
found from [2]:
T1/2,a = Ii ∗ ti for any state i and

(5.1.6)

λ
λa = i for any state i .
Ii

(5.1.7)

From these equations, the final β − decay rate from the 0.096 MeV first excited state
in 19 O was found to be λx = 0.038+0.011 s−1 (which corresponds to T1/2 = 18+5 s). See
−0.008
−4
Table 5.2 for the intermediate calculation values. The errors in this calculation were handled
through standard error propagation formula and are discussed in greater detail in Section 5.3,
especially Section 5.3.4. Errors in the log (f )i values were negligible.
The Boltzmann distribution gives the probability Pj for a nucleus in a stellar environment
with temperature T to be in an excited state j with spin Jj and excitation energy Ej as [2]:
−Ej
(2Jj + 1) e kT
Pj =
−Ej
(2J + 1) e kT

,

(5.1.8)

j

j

where k is the Boltzmann constant, which has a value of k = 8.6173 × 10−2 MeV/T9 [2].
The temperature notation T9 is used frequently in astrophysics in place of GK; 1.0 T9 is
equivalent to 109 kelvin. The fluctuation in the population of the ground and 0.096 MeV
first excited state of 19 O is pictured in Figure 5.1. As expected, at low temperatures around
92

Ef 19 F
(MeV)

Jπ

B[GT]

Error in
B[GT]

log(f t)

log(f )

λ
(s−1 )

I
(%)

0.000

1+
2

0.034 ‡

+0.006
−0.004

5.05

3.73

0.033

86.0

0.197

5+
2

0.0004

0.0001

7.00

3.65

0.00032

0.8

1.554

3+
2

0.027

0.007

5.15

2.99

0.0048

12.5

3.908

3+
2

0.23

0.06

4.21

0.77

0.00025

0.7

4.550

5+
2

0.29

0.08

4.12

-0.87

7.2E-6

0.02

Table 5.2: This table shows intermediate calculation stages used to determine a β − decay
rate (λx ) for the 0.096 MeV first excited state of 19 O. The first column lists the energy level
of a possible final state for this β-decay in 19 F. ‡: This particular Gamow-Teller strength
was extracted from the experiment performed for this dissertation. The other B[GT] values
in this column were determined from shell-model calculations as described in the text. The
log(f ) values were interpolated from tables in Reference [76].
T9 = 0.1, no significant amount of 19 O is in the first excited state. At T9 = 0.4, the excited
state starts to account for Px = 4% of the total 19 O population. At T9 = 1.6, the excited
state accounts for a more noticeable Px = 25% of the total 19 O population, and it levels off
near Px = 36% over the rest of the relevant stellar temperature range. For this calculation,
higher excited states of 19 O are ignored. Their impact was examined separately, and they
have only a minor impact on the excited state populations; the second excited state at
1.471 MeV only makes up 3% of the 19 O population at T9 = 10.0.
Looking at only the ground state (subscript g.s.) and first excited state (subscript x)
β-decay of 19 O, the total decay rate λtot as a function of temperature (T ) is [2]:

λtot =

Pj λj = Pg.s. (T )λg.s. + Px (T )λx
j

93

(5.1.9)

100
Population of the ground state of

Population

80

19

O

60
Population of the 0.096 MeV excited state of

40

19

O

20
0
0

2

4

6

8

10

T9 GK
Figure 5.1: This figure shows the population of the ground state and first excited state of 19 O
over a stellar temperature range. The red line indicates the percentage of 19 O in the ground
state at temperature T . The blue line indicates the percentage of 19 O in the first excited
state at 0.096 MeV for a given temperature. The first excited state is rapidly populated
around T9 = 1.6 and levels off near 36% for T9 > 2.

94

The β-decay rate for the ground state of 19 O is known from direct β-decay measurements
to be λg.s. = 0.02571 ± 0.00007 s−1 , which equates to T1/2 = 26.96 ± 0.07 s [25] (see also
Table 5.3).

5.2

Conclusions

As Figure 5.2 illustrates, the increase in the total β-decay rate (λtot ) over a range of
astrophysical temperatures is modest. At the top of the temperature range, this new rate
(λtot ) increases 18% beyond the ground state reaction rate (λg.s ). This small rate increase
is unlikely to have a significant influence on astrophysical simulations of fluorine production.
It is worthwhile to note that the 19 O(β − )19 F reaction is neglected entirely in many fluorine
nucleosynthesis simulations. Inclusion of this fluorine production mechanism has the potential to increase simulated fluorine yields, even though the rate enhancement examined in this
work was found to be relatively small.
Recall from Section 1.3 that a theoretical calculation of this same rate by Chernykh
et al. [26] yielded an increase of up to a factor of three. The large discrepancy between
the calculation in Reference [26] and those presented here results deserves scrutiny. The
measured Gamow-Teller strength of the 19 O[0.096 MeV] (β − )19 Fg.s. transition was found to
+0.006
be B[GT] = 0.034−0.004 . In Figure 4.15 and Section 4.3.3, the experimentally determined

Gamow-Teller strength was compared to OXBASH [50] calculations in the USDB [72] and
WBP [54] interaction, which gave quenched (unquenched) values of 0.0248 (0.0418) and
0.0289 (0.0490) respectively in the 19 O →19 F direction. This means that the experimentally determined Gamow-Teller strength is 2.0 standard deviations from the Gamow-Teller
strength predicted with the USDB [72] interaction and 1.1 standard deviations from the

95

Ei
(MeV)
0.000
0.197
1.554

Ii
log f t
B[GT]
(%)
(estimated)
≤ 4 ≥ 6.5
≤ 0.0012
45.4 5.384
0.0157
54.4 4.625
0.0904

Table 5.3: This table shows experimental β-decay information for 19 Og.s. (β − )19 F∗ . The
decay to the ground state of 19 F is forbidden; the branching ratios listed here were calculated
without considering forbidden states. These values were taken from Reference [25].

0.035

PtotΛtot

0.030
0.025

Λ s

1

Ground State

0.020
0.015
0.010

PgsΛgs
PxΛx

0.005
0.000
0

2

4

6

8

10

T9 GK
Figure 5.2: This graph shows the temperature dependence of the β-decay rate for
19 O(β − )19 F. The green dotted line indicates the β-decay rate for 19 O in the ground state,
which was known from prior experiments [25]. The solid black line shows λtot , the new rate
determined in this work for β-decays from either the ground or first excited state of 19 O. The
black dashed lines indicate the error in this new decay rate. The red and blue lines indicate
the contribution to the new rate from the ground and first excited state, respectively. The
black line is the sum of the red and blue lines. As the blue line indicates, the first excited
state doesn’t contribute much to the new decay rate at low temperatures because the excited
state isn’t heavily populated. At higher temperatures, the excited state contributes nearly
as much as the ground state to the total β-decay rate.

96

Gamow-Teller strength predicted with the WBP [54] interaction. Theoretical calculations
of B[GT] values in Chernykh et al.’s work used different interactions: CW, HBUMSD, and
HBUSD [75] with Gamow-Teller strengths of 0.0290, 0.0456, and 0.0333 respectively. These
Gamow-Teller strengths are also fairly close to the experimentally determined value. No
quenching factor was taken into account in Chernykh et al.’s work, which inflates the β-decay
rate (λtot ) by 70%, but does not account for the overall discrepancies between the decay
rates. Another slight difference between the decay rate in this work and Reference [26] is
that this work uses the experimentally determined value of λg.s. while Chernykh et al. use
a theoretical calculation of λg.s. , but the effect of this difference in approach is small.
Since the discrepancy between decay rates for this work and Chernykh et al.’s work does
not arise from a conflict between experimental measurement and theoretical predictions of
Gamow-Teller strength, it is difficult to determine exactly where it comes from. A close
examination of the calculations available in Reference [26] suggests that there was an error
in Chernykh et al.’s calculation of the Boltzmann distribution (Equation 5.1.8).
The new experimentally determined rate (λtot ) presented in this work is available for
future studies of fluorine nucleosynthesis that involve the 19 O(β − )19 F reaction. Additionally,
the Gamow-Teller strength distribution presented in 4.3.3 provides boundaries to guide future
studies of 19 Fg.s. to 19 O∗ transitions and to constrain shell-model calculations.

5.3

Discussion of Error Sources

There are many potential sources of error in a complex experiment like this one. Brief
notes have been included throughout this dissertation to address sources of error as they
appear. However, it is helpful to have a comprehensive review of how all known error

97

sources are handled. This section discusses all errors associated with the factors used in the
data analysis of Chapter 4 and in the calculation of the β-decay rate for the transition from
the 0.096 MeV first excited state in 19 O to the ground state of 19 F in Section 5.1.

5.3.1

Experimental Errors

There are many possible sources of experimental error. This section examines potential
errors in any part of the experiment or data analysis up to the conversion from experimental
count rates to differential cross sections (Chapter 3 and Section 4.1).
Ultimately, only the statistical error of the measurement was considered a significant
source of error in this portion of the data analysis. Other errors were disregarded as negligible because it was possible to verify the results of the data analysis for Section 4.1 by
looking at data taken on the CH2 calibration target. The cross section that was produced
for the 12 C(t,3 He)12 B∗ reaction was compared to a prior measurement of this same reaction
in experiment 06032 (see Section 4.2.2) and the results were in good agreement. This indicated that any uncertainties in Section 4.1 of the data analysis procedure must be relatively
minor. This validation process also indicated that there were no significant problems with
the experimental apparatus that was described in Chapter 3.
Resolutions were the only other significant factor in the experiment. The scattering angle
resolution of the S800 spectrograph was 10 mrad in the laboratory frame; this was dominated
by the large beam spot size and thickness of the target. The energy resolution of the S800
spectrograph was found to be 480 keV for this experiment by examining the width of the
peak for the 12 Cg.s. (t,3 He)12 Bg.s. transition. This energy resolution was lower than normal
for similar experiments due to problems with beam tuning.

98

5.3.2

Theoretical Cross Section Errors

Theoretical DWBA calculations of differential cross section angular distributions from the
WSAW/FOLD/DWHI code package were used in two different aspects of the data analysis.
They were used in the multipole decomposition analysis described in Section 4.3.1 to provide
line shapes for the MDA fit procedure. They were also used to perform an extrapolation
from the experimental cross section with a finite Q-value to a cross section with Q = 0 using
Equation 4.3.3 that was discussed in Section 4.3.2. This extrapolation was necessary to fulfill
the condition of zero linear momentum transfer (q = 0) required to calculate Gamow-Teller
strengths with Equation 4.3.1. The only difference between these two theory calculations
was the input Q-value, so the potential sources of error within the calculations themselves
were essentially identical. Thus, treatment of the error associated with these two different
calculated quantities varied only slightly.
When used in the MDA fit procedure, the magnitudes of angular distributions were
treated as fitted parameters. This means that errors of absolute magnitude in the DWBA
were not very relevant; variations in the line shapes of the angular distributions were much
more important. This insensitivity to absolute magnitudes means that errors in the DWBA
calculations were considered negligible in comparison to other sources of experimental error.
Error resulting from line-shape choices in the MDA procedure was much more critical, and
this is discussed in more detail in the next section (Section 5.3.3).
In the extrapolation to zero linear momentum transfer (q = 0) from Equation 4.3.3, the
two different calculated angular distribution values for Q = Qexp and Q = 0 at θ = 0◦
were divided by each other to form a ratio. Since only the ratio of the two values matters,
neither absolute magnitudes nor line shapes were relevant; as a result, this quantity was very

99

insensitive to errors in the DWBA calculation. In the final error calculation, any error in
this quantity was considered to be negligible.
The WSAW/FOLD/DWHI code package itself has been vetted by comparing its calculation results to numerous data sets. Any errors from uncertainties in the code are expected
to be negligible compared to other sources of experimental error. Similarly, the error associated with the various code inputs was treated as negligible, but these inputs are listed for
completeness.
WSAW inputs:
• Binding energies calculated in OXBASH
• A single-particle potential: Diffuseness, Coulomb radius, Woods-Saxon radius, and
Spin-orbit potential depth
• Quantum numbers: spin, total angular momentum, orbital angular momentum, and
charge
FOLD inputs:
• Kinetic energy of the beam particles
• Quantum numbers: total angular momentum, parity, isospin, isospin projection
• Effective nucleon-nucleon interaction (Love-Franey interaction)
• One-body transition densities from OXBASH calculations
DWHI inputs:
• Quantum numbers: total angular momentum, charge

100

• Kinetic energy of the beam particles
• Optical potential parameters
• Coulomb radii
• Q-value

5.3.3

Gamow-Teller Strength Errors

Gamow-Teller strength was calculated with Equation 4.3.1 as discussed in Section 4.3.
This required fitting the theoretical angular distributions to experimentally determined differential cross sections in a multipole decomposition analysis, as discussed in Section 4.3.1.
Four different factors contributed to the error of the Gamow-Teller strength measurements.
Statistical error arose from the number of events counted in the course of the experiment.
Systematic errors came from the multipole decomposition analysis (MDA) procedure that
was discussed in Section 4.3.1. More systematic errors (not related to the MDA) arose from
the conversion of experimental event counts into an absolute cross section. The tensor term
of the effective interaction also contributed a small systematic error.

5.3.3.1

Statistical Error

The statistical error was derived from the number of counts (N ) measured during the
experiment. These counts were separated into bins by scattering angle (θ) and excitation
energy in 19 O (Ex 19 O ). The relative statistical error for each of these bins was found with
√
√
N
the formula
, where N was the error of a bin with N counts in it. For the first excited
N
state at 0.096 MeV in 19 O, this error was 7.9%. For the second excited state at 1.471 MeV

101

in 19 O, this error was 13.6%. The statistical errors of the measurements for Ex 19 O > 3.5
MeV varied greatly and were generally large.
In Gamow-Teller strength calculations with Ex 19 O > 8.0 MeV, the carbon background
subtraction procedure that was described in Section 4.2.3 increased the statistical error. This
was handled through error propagation formulas derived from Equation 4.2.4. The higher
statistical error from carbon background subtraction contributed significantly to the larger
final errors in computed Gamow-Teller strengths in this region.
This statistical error was incorporated into the calculation of Gamow-Teller strength
through the multipole decomposition analysis fit. The fit parameters had a calculated error,
which came from the statistical error for each point in the fit (shown with error bars in
Figure 4.11 and Figure 4.14).

5.3.3.2

Systematic Error in the Multipole Decomposition Analysis

In order to account for the error in the MDA procedure due to line shape choices that
was mentioned in Section 5.3.2, many different line shapes from sets of angular distributions
were tested. The best-fitting set of line shapes was used to calculate Gamow-Teller strengths.
However, to account for the error in this procedure, all sets of angular distributions with
at least a 5% chance of being accurate according to a chi-squared goodness of fit test were
considered valid. From this expanded set of valid fits, the two fits that provided the highest
and lowest Gamow-Teller strengths were used to establish the systematic error bounds due
to the MDA procedure.
For the two low-lying states at 0.096 MeV and 1.471 MeV in the excitation energy of 19 O,
the error from the MDA process was negligible. At these energies, the level density is still
very low, so there are very few possible choices of line shapes for the multipole decomposition
102

analysis; additionally, the line shape options that are present do not vary much from each
other. The state at 0.096 MeV is well-separated from all but the ground state of 19 O.
Fortunately, this ground state contribution was expected to have only a ∆L = 2 component,
which did not strongly alter the ∆L = 0 component that determined the Gamow-Teller
strength. The state at 1.471 MeV is well-separated from all other states in 19 O.
At higher energies, the systematic error from the multipole decomposition analysis became much more important. This systematic error was large enough that no Gamow-Teller
strength could be conclusively detected for Ex [19 O] > 3.5 MeV. Individual states could not
be isolated in the data set at these energies due to relatively low energy resolution and to
lack of statistics. Additionally, many of the known states in 19 O with Ex [19 O] > 3.5 MeV
do not have firm total angular momentum or parity assignments. Some known states do not
match up well with shell-model level predictions from OXBASH calculations. All of these
factors increased the uncertainty associated with the MDA fits due to line shape choices.
The errors from the MDA procedure are listed for each measured Gamow-Teller strength in
Table 5.4. The contribution of the MDA error to the final error of a Gamow-Teller strength
measurement varied considerably, but averaged 80% and ranged from 30% to 100% for measurements with Ex [19 O] > 3.5 MeV. It was the dominant error factor for all but one of these
measurements (the one exception had extremely high statistical error).

5.3.3.3

Systematic Error in the B[GT] Calculation

Several potential systematic errors (not related to the MDA) arose from the conversion
of experimental event counts into an absolute cross section. Three of these errors were
large enough to be included in the error analysis calculation. The rest were considered to
be negligible. These systematic errors come from the portion of the data analysis that is
103

Ex (19 O)

B[GT]

MDA

Error (+)

(MeV)

MDA

Error (-)

0.096

0.068

N/A

N/A

1.471

0.057

N/A

N/A

3.5–4.0

0

0.018

0

4.0–4.5

0

0.013

0

4.5–5.0

0

0.038

0

5.0–5.5

0

0.052

0

5.5–6.0

0.022

0.029

0.022

6.0–6.5

0.011

0.026

0.011

6.5–7.0

0.049

0.020

0.049

7.0–7.5

0

0.027

0

7.5–8.0

0.037

0.024

0.037

8.0–8.5

0.111

0.070

0.111

8.5–9.0

0.146

0.051

0.146

9.0–9.5

0.046

0.042

0.011

9.5-10.0

0.068

0.025

0.068

10.0–10.5

0.085

0.036

0.085

10.5–11.0

0.041

0.031

0.041

11.0–11.5

0.011

0.042

0.011

11.5–12.0

0

0.070

0

12.0-12.5

0.096

0.065

0.096

12.5–13.0

0.037

0.090

0.037

13.0–13.5

0

0.151

0

13.5–14.0

0.099

0.072

0.099

14.0–14.5

0

0.156

0

14.5–15.0

0.007

0.009

0.002

15.0–15.5

0

0.169

0

15.5–16.0

0

0.149

0

Table 5.4: This table lists the calculated Gamow-Teller strengths for the 19 F(t,3 He)19 O∗
reaction at 115 MeV along with the errors due only to the multipole decomposition analysis
(MDA). These errors from the MDA procedure dominated the total errors for Gamow-Teller
strengths above 3.5 MeV in 19 O. On average, they accounted for 80% of the total error for
a measurement, ranging from a minimum of 30% to a maximum of 100%.

104

described in Section 4.2.
The solid angle angle calculation (dΩ) in Equation 4.2.1 of Section 4.2.1 had no significant error, as it was a straightforward geometrical calculation. There was an error associated
with the calculation of the acceptance correction (Pacceptance ) via a Monte Carlo simulation.
However, the error in this Monte Carlo simulation can be reduced by simulating a large
number of events, so 1 billion events were simulated. This reduced the expected errors for
the simulation to a point where they were negligible compared to other sources of error.
Any uncertainties in the simulation code itself were determined to be negligible compared
to statistical uncertainties by examining the calibration data taken on a CH2 target to validate the data analysis procedure, as was discussed in Section 5.3.1. The cross section that
was produced for the 12 C(t,3 He)12 B∗ reaction using the Monte Carlo acceptance correction was compared to a prior measurement of this same reaction in experiment 06032 (see
Section 4.2.2) and the results were in good agreement. This indicated that any uncertainties
in the data analysis procedure up to the relative cross section calculation must be minor.
The conversion from a relative cross section to an absolute cross section involved four
scaling factors ( abs , 1 , 2 , and 3 ) that were discussed in detail in Section 4.2.2. The
scaling factor of abs = 0.048 mb was determined by a fit between the 12 C(t,3 He)12 Bg.s.
reaction cross section that was measured in the current experiment and a prior measurement
during experiment 06032 (see Figure 4.9). The error in this factor (±0.001 mb, 2.2%) was
determined from the error in the fit. This error in abs was treated as a systematic error.
The scaling factor for beam rate ratios ( 1 = 0.35) was determined by taking the average
of the beam rate measurements from three different detectors (see Section 4.2.2); the error
in this term (±0.03, 8.8%) was determined by taking twice the standard deviation of this
average. This very conservative error was chosen (instead of a single standard deviation)
105

because these beam rate measurements were all indirect measurements and there was no
direct comparison to the actual beam rate available; it was treated as a systematic error.
The scaling factor for differences in data acquisition live time ( 2 = 0.9634) was considered to have negligible error; this decision was based past experience with the same data
acquisition system in similar experiments.
The scaling factor for target composition differences ( 3 = 3.183) was also considered to
have negligible error. The calculation of the scaling factor 3 involves two different sources of
potential error: the molar mass of the chemical compounds in the targets and the thickness
of the targets. The masses of all the isotopes present in the target were known to a high
precision, so this part of the error was negligible. The target thickness was not trivial to
measure accurately, but could be considered negligible for this experiment because the beam
spot was large. The large beam spot sampled the target thickness across a large area, so any
variations in thickness over a small area were deemed irrelevant.
To calculate a Gamow-Teller strength using Equation 4.3.1, the experimental cross sections must first be projected to Q = 0 to fulfill the requirement of zero linear momentum transfer (q = 0), as discussed in Section 4.3.2. This was accomplished with the ratio
dσ
◦
dΩ (Q = 0, 0 )
dσ
◦
dΩ (Qexp , 0 ) theory

from Equation 4.3.3. As discussed previously in Section 5.3.2, this

ratio was computed from theoretical angular distributions that were calculated with the
WSAW/FOLD/DWHI code package. Thus, any errors in this ratio were considered negligible compared to experimental errors.
In the proportionality relationship described by Equation 4.3.1, there was a significant
error associated with obtaining the unit cross section (ˆGT ) from the phenomenological
σ
global fit of Equation 4.3.4. The error associated with this unit cross section was 5%; the fit
and its error are discussed in Reference [38]. This error was treated as a systematic error.
106

5.3.3.4

Systematic Error from the Tensor Interaction

As mentioned briefly at the end of Section 4.3.1, the tensor interaction causes systematic
errors in Gamow-Teller strength measurements. Interference between the transition amplitudes due to the στ and Tτ (tensor-τ ) components of the effective interaction creates a
systematic error that cannot be accounted for with an MDA procedure. The error due to
the tensor component of the interaction is stronger for weak Gamow-Teller states because
of this interference dynamic (see Reference [42] for details). This systematic error was estimated by calculating theoretical angular distributions for a transition twice: once with the
full interaction (including the tensor interaction term), and once without the tensor interaction. The difference between these two results provided an estimate for the systematic
error. The procedure for examining error from the tensor interaction is outlined fully in
Reference [42], which discusses error from the tensor interaction in Gamow-Teller strength
measurements in greater detail. The error due to the tensor term was found to be +0.0026
for the first excited state at 0.096 MeV in 19 O (an error of 4%) and +0.0011 for the second
excited state at 1.471 MeV in 19 O (an error of 2%). In both cases, this error happened to
cause an underestimation of the Gamow-Teller strength, so it is applied only to the positive
error value; more generally, this error could be positive or negative.
Error from the tensor term in the effective interaction was ignored for the higher-lying
strength. Error from the MDA procedure dominated these measurements, as was discussed
earlier. Since it was not possible to definitively identify any Gamow-Teller strength above the
1.471 MeV state of 19 O in this data analysis, any additional error from the tensor interaction
was negligible compared to the MDA error contributions. If Gamow-Teller strengths could
be definitively established in a future experiment, then it would be necessary to examine the

107

tensor contributions to the errors.

5.3.3.5

Summary of Errors Related to Gamow-Teller Strengths

In summary, there were four types of error considered in the calculation of Gamow-Teller
strengths. These four error categories were: statistical error, systematic MDA error, systematic (non-MDA) errors, and tensor interaction systematic error. For the first two excited
states at 0.096 MeV and 1.471 MeV in 19 O, statistical errors were the dominant source of
error, followed by systematic (non-MDA) errors and then by tensor interaction systematic
error, with systematic MDA error considered negligible. In the higher-lying Gamow-Teller
strength measurements with Ex [19 O] > 3.5 MeV, the total error was dominated by systematic MDA error, followed by statistical error, followed by systematic (non-MDA) errors;
tensor interaction systematic errors were considered negligible.

108

The transition from the ground state of 19 F to the first excited state of 19 O at 0.096 MeV
was found to have a Gamow-Teller strength of B[GT] = 0.068+0.011 , which corresponded to
−0.009
relative errors of +16% . The transition from the ground state of 19 F to the second excited
−11%
state of 19 O at 1.471 MeV was found to have a Gamow-Teller strength of B[GT] = 0.057+0.011 ,
−0.010
which corresponded to relative errors of +19% .
−18%

5.3.4

β-decay Rate Errors

Calculation of the β-decay rate for the first excited state at 0.096 MeV in 19 O was a
multi-step process with several possible sources of error. Ultimately, only one factor added
significant error to the decay rate, beyond the error in the computed Gamow-Teller strength
that was discussed in detail in Section 5.3.3. This important factor was the choice of theoretical Gamow-Teller strengths and an associated error for the determination of branching
ratios (Equation 5.1.5). Other possible but negligible sources of error are discussed in this
section for completeness.
Equation 5.1.2 converted Gamow-Teller strengths (B[GT]) into f t-values. Equations 5.1.3
and 5.1.4 converted the f t-values into partial decay rates (λi ). Beyond the error of the experimentally determined Gamow-Teller strength, possible sources of error included the factors
ga
K
and
in Equation 5.1.2; these had relative errors of 0.2% and 0.03% respectively, which
2
gv
gv
were both negligible compared to the error in B[GT]. The log (f ) values determined from
Reference [76] had a relative error of 0.01%, which was also negligible.
Equation 5.1.5 determined branching ratios for the β-decay from the 0.096 MeV state
in 19 O to five possible states in 19 F. These branching ratio (Ii ) calculations used the experimentally determined Gamow-Teller strength to find the partial decay rate to the ground
state of 19 F and theoretical Gamow-Teller strengths to find the partial decay rates to four
109

excited states of 19 F (see Table 5.2). Then the final β-decay rate (λx ) for the first excited
state at 0.096 MeV in 19 O was calculated with Equation 5.1.7. To perform this calculation of
the branching ratios, theoretical Gamow-Teller strengths and associated errors were chosen
for transitions from the 0.096 MeV excited state in 19 O to four excited states in 19 F. The
theoretical Gamow-Teller strengths were calculated with the USDB interaction [72] using
OXBASH [50]. The final calculation of λx was not sensitive to the choice of interaction;
Gamow-Teller strengths computed using the WBP interaction [54] gave the same result.
The USDB interaction was chosen over the WBP interaction because it is generally a better
choice for sd-shell nuclei, and thus likely to be of interest to a wider audience, even though
it did not outperform the WBP interaction for the transition measured in this experiment.
A relative error of 36% was assigned to the theoretical Gamow-Teller strengths to reflect
the error in the USDB’s prediction of the Gamow-Teller strength for the experimentally
measured transition from the 0.096 MeV excited state of 19 O to the ground state of 19 F.
This was a somewhat arbitrary assignment of error, but it turns out that the calculation of
λx was not very sensitive to this error assignment, though it is still significant. The reason
for this can be seen in the branching ratios listed in Table 5.2. The measured transition
accounted for the large majority (86%) of the branching ratio. Only one other transition
(from the 0.096 MeV excited state of 19 O to the 1.554 MeV state of 19 F) had a significant
branching ratio (12.5%), and thus a significant effect on the calculation of λx .
The population of the first excited state at 0.096 MeV in 19 O in a stellar temperature
range was computed with Equation 5.1.8. Equation 5.1.9 determined the temperature dependence of the total β-decay rate for a stellar population of 19 O. The variables involved in
computing the Boltzmann distribution for Pj were the spins and energy levels of the ground
and first excited states, which were well-known, and the Boltzmann constant, which was also
110

well-known; thus, the Boltzmann distribution computation was not a significant source of
error. The β-decay rate for the ground state of 19 O (λg.s. ) was a potential source of error in
Equation 5.1.9, but it had a negligible error of 0.3% [25].
The final value of the β-decay rate for the 0.096 MeV excited state of 19 O was found
+0.011
to be λx = 0.038−0.008 s−1 , which corresponded to a relative error of +29% . This error
−21%

was dominated by the error in the Gamow-Teller strength measurement for the transition
from the 0.096 MeV first excited state of 19 O to the ground state of 19 F, but error from the
determination of the branching ratio with theoretical Gamow-Teller strengths also influenced
the final error significantly.

111

Chapter 6
Summary

6.1

Summary

This dissertation described the measurement of the charge-exchange reaction 19 F(t,3 He)
at 115 MeV/nucleon. The experiment was performed at the Coupled Cyclotron Facility
at the National Superconducting Cyclotron Laboratory (NSCL). A beam of tritons from
the Coupled Cyclotron Facility was impinged on a target that contained fluorine (CF2 ).
The 3 He ejectiles were momentum-analyzed in the S800 spectrograph. An overview of the
experimental apparatuses that were used in this measurement is available in Chapter 3 and
Figure 3.1. A more detailed explanation of the S800 spectrograph in particular was given in
Section 3.3.
This specific reaction was measured to test a hypothesis on fluorine nucleosynthesis in
core-collapse supernovae. As discussed in Chapter 1, there are several different theories regarding where and how the element fluorine is produced. There is currently no consensus
between these different theories of fluorine nucleosynthesis and available astrophysical observations. The β-decay of 19 O∗ in core-collapse supernova has been previously suggested [26]
as a possible avenue for fluorine nucleosynthesis. In hot astrophysical environments (such
as supernovae), it is possible to thermally populate low-lying excited states of 19 O, and the
β-decay of these excited states would enhance the overall β-decay rate into 19 F.

112

At the high temperatures of a core-collapse supernova, a significant portion of 19 O will
achieve equilibrium in the very low-lying first excited state at 0.096 MeV, making it possible
to β-decay from this excited state in 19 O∗ to 19 F. As was discussed in Section 1.3, this
is interesting because β-decay between the ground state of 19 O and the ground state of
19 F

is forbidden. However, β-decay from the low-lying excited state in 19 O at 0.096 MeV

excited state in 19 O to the ground state of 19 F is allowed (see Figure 1.2). This creates an
opportunity for a significant increase in the β-decay rate for 19 O in a supernova compared
to the β-decay rate that has been measured directly for the 19 O ground state in laboratory
conditions. Chernykh et al. [26] performed a theoretical calculation to estimate the reaction
rate increase that occurs when the 0.096 MeV state in 19 O is taken into account. Their
calculations suggested that the β-decay rate for 19 O(β − )19 F might increase by a factor of
three at high stellar temperatures.
To examine this theory, the 19 F(t,3 He)19 O∗ reaction at 345 MeV was measured. The
differential cross section of this charge-exchange reaction can be used to extract the GamowTeller strength distribution for (∆L = 0, ∆S = 1) transitions between 19 F and 19 O using
Equation 2.1.2, as discussed in Chapter 2. The procedure discussed in Section 2.1, when
applied to data from the 19 F(t,3 He)19 O∗ reaction, allowed for the extraction of the GamowTeller strength of the beta decay from the 0.096 MeV state in 19 O to the ground state of
19 F.

This particular Gamow-Teller strength was then used to calculate the same β − -decay

rate for 19 O(β − )19 F (λtot ) that was discussed in by Chernykh et al. in Reference [26].
The procedure for extracting differential cross sections from data taken with the S800
spectrograph was described in Sections 4.1 and 4.2. The differential cross sections were used
to compute Gamow-Teller strengths in Section 4.3. For the transition from the ground state
of 19 F to the 0.096 MeV state of 19 O, a Gamow-Teller strength of B[GT] = 0.068+0.011 was
−0.009
113

found. This corresponds to B[GT] = 0.034+0.006 for transitions in the 19 O →19 F direction
−0.004
(see also Figure 4.14 A). Gamow-Teller strengths for transitions from the ground state of
19 F

for excitation energies in 19 O of 0 MeV to 16 MeV are listed in Table 4.11 and plotted

in Figure 4.15.
The Gamow-Teller strength for the transition from the first excited state of 19 O at
0.096 MeV to the ground state of 19 F was used to calculate a weak decay rate in Section 5.1
to compare with the theoretical calculation of Chernykh et al. in Reference [26]. For the
transition from the 0.096 MeV excited state alone, this β − -decay rate was found to be
λx = 0.038+0.011 s−1 (which corresponds to T1/2 = 18+5 s). A total β − -decay rate over
−0.008
−4
stellar temperatures (T ) for 19 O in either the first excited or ground state was found with
Equation 5.1.9 and plotted in Figure 5.2.
As discussed in Section 5.2 and shown in Figure 5.2, the increase in the total β-decay rate
(λtot ) over a range of astrophysical temperatures is modest. At the top of the temperature
range, this new rate (λtot ) increases 18% above the ground state reaction rate (λg.s. ). This
small rate increase is unlikely to have a significant influence on astrophysical simulations
of fluorine production, but a full fluorine nucleosynthesis simulation would be needed to
confirm that. This result contradicts the theoretical results put forth by Chernykh et al. in
Reference [26]; the discrepancy might stem from a calculation error in Reference [26] in the
Boltzmann distribution. A detailed discussion of the error analysis for all values calculated
in this dissertation was provided in Section 5.3.
The new experimentally determined rate (λtot ) presented in this work is available for
future studies of fluorine nucleosynthesis that involve the 19 O(β − )19 F reaction. Additionally,
the Gamow-Teller strength distribution presented in Figure 4.15 provides boundaries to guide
future studies of 19 Fg.s. to 19 O∗ transitions and to constrain shell-model calculations.
114

6.2
6.2.1

Outlook
Fluorine Nucleosynthesis

As stated earlier, this new measurement of the β-decay rate of the first excited state of
19 O

to the ground state of 19 F is not expected to significantly influence fluorine production

in core-collapse supernovae. This cannot be definitively concluded until the new decay rate
is tested in astrophysical simulations, but it is clear that any increase from such decays will
be modest. However, as mentioned in Section 1.5, there has been little effort to simulate the
fluorine production from 19 O(β − )19 F at all in any astrophysical environment. This path of
fluorine production has not received much attention because, as Jorissen et al. observed in
Reference [8], production of 19 O via neutron capture on 18 O inevitably competes with the
much stronger rate of neutron capture on 19 F. This leads to the assumption that neutron
capture processes are a net loss for fluorine abundance.
While it is quite possible, even probable, that neutron capture processes are strictly a
net loss for fluorine abundance, that hypothesis should be examined in detailed astrophysical
simulations. Fluorine abundance predictions from simulations would provide stronger guidance as to whether the 19 O(β − )19 F process is a viable alternative for fluorine production.
Prior research into fluorine nucleosynthesis has shown that it is an unusual nucleosynthesis
process which depends closely on the stellar dynamics that govern the ejection of matter from
the star, so simple rate comparisons can be misleading. Simulated abundance predictions
would give clearer guidance as to whether this nucleosynthesis path merits further research.
Further progress in understanding fluorine nucleosynthesis will rely heavily upon astrophysical observations and astrophysical simulations. Asymptotic giant branch (AGB) stars
are currently the best understood sources of fluorine production. It is not yet clear if Wolf115

Rayet stars or core-collapse supernovae produce significant amounts of fluorine, so more
study is needed for both of these stellar environments. A study of galactic fluorine with a
chemical evolution model is needed to update the work of Renda et al. [20] with the progress
made in the field of fluorine nucleosynthesis in the past decade.
Another opportunity for improvement in fluorine nucleosynthesis measurements lies in
the study of our solar system’s fluorine abundance. Currently, the only solar measurement
of fluorine abundance available comes from a study of sunspot composition from a paper by
Hall et al. published in 1969 [14], and it has a fairly large uncertainty: A(F) = 4.56 ± 0.33
dex (see also Section 1.2). It should be possible to take advantage of improvements in the
understanding of solar physics from the past four decades to perform an improved version
of Hall et al.’s sunspot study of HF molecules. Improvements in technology might allow
for a modern study of solar fluorine abundances to be done with a UV-capable satellite
telescope that could view elemental fluorine. It might be possible to improve upon the solar
system fluorine abundance measured with carbonaceous chondrite meteorites [15], which has
a higher accuracy of 15% that still leaves room for improvement.

6.2.2

Nuclear Structure

The experiment described in this dissertation did not allow for a detailed study of the
nuclear structure of the 19 Fg.s. to 19 O∗ transitions, but it did suggest a potential future experiment to that end. The state predicted by shell-model calculations to have the strongest
Gamow-Teller strength (B[GT] ≈ 0.1) in the 19 F(t,3 He)19 O∗ reaction occurs at approximately 7.5 MeV in the excitation energy of 19 O (see Table 4.12). In the present experiment,
that state happens to fall very close to the peak for the 12 C(t,3 He)12 Bg.s. transition at
Ex 19 O = 8.5 MeV generated by the carbon nuclei in the CF2 target. If another experi116

ment were performed with a different target choice, it would likely be possible to measure the
Gamow-Teller strength and excitation energy of this strong state. Current results hint that
a strong Gamow-Teller state might exist near Ex 19 O = 8.5 MeV (see Figure 4.15), but
the high error precludes a firm measurement of B[GT]. An experiment with higher statistics
might also allow for a quantitative measurement of the Gamow-Teller strength distribution
for states with Ex 19 O > 3.5 MeV. Possible target options for such an experiment include
fluorspar (CaF2 ), a mineral, or hydrofluoric acid (HF+(H2 O)n ), a weak acid, among other
fluorine compounds.

117

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