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Particle Decay Branching Ratios for States o f Astrophysical Importance in 19Ne
A Dissertation Presented to the Faculty o f the Graduate School
o fYale University
in Candidacy for the Degree o f Doctor o f Philosophy
. B yDale William Visser
Dissertation Director: Peter D. Parker
DECEMBER 2003
© 2002, Dale William Visser. This work, originally published as "Particle Decay Branching Ratios for States o f Astrophysical Importance in I9Ne" is distributed under the terms o f the Public Library o f Science Open Access License version 1.0b, a copy o f which can be found at http://www.publiclibraryofscience.org. This work is archived at arXiv.org (http://arxiv.org).
ABSTRACT
Particle Decay Branching Ratios for States o f Astrophysical Importance in 19Ne
by Dale William Visser 2003
Reactions through resonances in 19Ne are thought to be important participants in explosive
stellar environments. In x-ray bursters, 15O(0C,Y) has been suggested as a possible pathway out
o f the hotCNO “ bottleneck” to processes that generate more energy. At the lower
temperatures and densities in novae, in particular oxygen-neon novae, 18F(p,7) and 18F(p,0t)
play a role that is important in y-ray astronomy. The amount o f 18F remaining in a nova’s
expanding envelope o f ashes depends sensitively on the rates o f these two reactions.
It is difficult to directly measure astrophysical reaction rates, even at the relatively high energies
o f novae (~500 keV) and x-ray bursters (~l-2 MeV). For the reactions considered here, the
participant nuclei are radioactive, creating more challenges. At the Wright Nuclear Structure
Laboratory, indirect methods are employed for measuring astrophysical reaction rates. Nuclei
are populated through alternate reactions, and the properties o f their states are measured. For
this study, 19Ne was populated via the 19F(3He,t) reaction with a 25 MeV 3He beam incident on
80 Jig/cm2 o f CaF2 deposited on 10 |ig/cm2 o f carbon. A large area segmented silicon array,
the Yale LampShade Array (YLSA), was assembled for the purpose o f measuring proton- and
Ct-decays from astrophysically relevant states in 19Ne. The reaction rates are linearly dependent
on the branching ratios to these channels. YLSA detected the decay particles at backward lab
angles in coincidence with tritons at 0=0°. Angular distributions were fit to the coincidence
data, and integrated in order to determine branching ratios to the proton and alpha channels.
These results were compared to similar efforts to measure branching ratios using different
reactions and techniques, and also to a previous (3He,t) effort that used different instruments.
In addition to standard statistical methods, a Bayesian technique was employed for averaging
the data due to the physical constraints on allowed values.
A C K N O W L E D G E M E N T S
I would like to thank many people, without whom this work would have been much more
difficult. At the top o f the list, o f course, is Peter Parker. I cannot imagine having a more
supportive or effective mentor. Beyond his extensive knowledge o f the field, Peter’s great
strength as an advisor is his ability to accommodate the different working styles o f those he
supervises. I believe this is something that cannot be taught.
I am very grateful to the succession o f post-doctoral researchers who have worked in our
research group for all o f their help and guidance. Ken Swartz eased my transition into the life
o f a graduate researcher. The focal plane detector works today largely because o f his efforts.
By creating Jam, he gave me the chance to feed the code-hacker monkey that lives on my back
and do work important for our research at the same time. After he left to work in industry, he
allowed me to appreciate what our west coast has to offer by hosting a couple o f my vacations.
William Bradfield-Handler arrived at Yale with a freshly minted Ph.D. from the U.K He
proceeded to kick-start the “ coincidence project” and organized all the major hardware
purchasing and design. He also made certain that we all had fun while we were doing our
work. It is because o f him that I learned how to get properly drunk and party all night (well,
most o f it, anyway) in Edinburgh. Jac Caggiano was Will’s successor, and was no less valuable
in making sure that my experiment was pulled o ff successfully. He has definitely taught me
patience in the face o f electronics debugging, and also the importance o f using appropriate
tools for data analysis. I’ll miss his ability to produce an appropriate quote from “The
Simpsons” for any situation.
O f course, I am indebted to all the graduate students who have worked with me. As Peter has
often stated, we all teach each other as much as any faculty member. For that, and for their
hard work on experiments (especially the night shifts), I am grateful to Alan Chen, Rachel
Lewis and Anuj Parikh. I don’t know what’s going to happen to the social life in the lab when
Rachel leaves.
The support staff o f WNSL are invaluable. Thomas Barker, Richard Wagner and Craig Miller
all created portions o f the physical and electronic infrastructure o f YLSA. Mary Anne Schulz,
Karen Defelice and their assistants have made sure that work-related travel, purchasing and
other day-to-day concerns are taken care o f smoothly and professionally.
Dr. James Perlotto at the university health plan has been a very important person in my life
here. When I arrived, I was in very poor health, with asthma that was out o f control. Dr.
Perlotto’s aggressive coordination o f my health care is the reason I’m as healthy as I am today.
He literally helped me survive graduate school.
I am very lucky to have a family that has supported me all the way through this process. My
Connecticut cousins, W illiam Kraus, Lynne Yeannakis, James Kraus, Julie Kraus, Sylvia
Sanders and Alice Hobolth (“ Grammy”) have provided me with a family and support network
in New Haven. Uncountable moral and financial support for my education has come from my
parents, Howard and Georgia Visser, as well as from my uncle, Richard Hilliker. My sister,
Heather, has been my good friend and my closest confidant throughout graduate school,
despite our 800-mile separation. Finally, I am profoundly grateful to my grandfather, Fred
Froman, whose generosity towards his grandchildren in his estate allowed me to pursue my
education without ever having to worry about where the tuition money was going to come
from.
For my grandfather, Fred Ftvman
T A B L E O F C O N T E N T S
Acknowledgements...................................................................................................ivTable o f Contents...................................................................................................... 5List o f Figures............................................................................................................7List o f Tables............................................................................................................10Chapter 1: Introduction to Explosive Nucleosynthesis......................................... 111.1. Stellar nucleosynthesis...................................................................................... 111.2. White dwarves and novae.................................................................................141.3. Neutron stars and x-ray bursters......................................................................16Chapter 2: Present Status o f 19Ne’s Role in Nuclear Astrophysics....................... 182.1. Nuclear burning through resonances............................................................... 182.2. Production o f 18F in Novae..............................................................................21
2.2.1. Sensitivity to resonances in 19Ne............................................................. 212.2.2. Proton resonances in 19Ne....................................................................... 22
2.3. Breakout from HotCNO via 150(a,7).............................................................. 252.4. Summary............................................................................................................27Chapter 3: Experimental Setup for 19F(3He,t)19Ne(x,)x2......................................... 293.1. The Experiment................................................................................................293.2. Accelerator........................................................................................................ 293.3. Enge Split-Pole Spectrograph.......................................................................... 303.4. Focal Plane Detector.........................................................................................323.5. Yale Lamp Shade Array.................................................................................... 353.6. Protection o f YLSA.......................................................................................... 40
3.6.1. Beam Scatter............................................................................................ 403.6.2. Electrons..................................................................................................41
3.7. Data Acquisition............................................................................................... 413.7.1. Signal Processing.....................................................................................413.7.2. Conversion and Data Processing........................................................... 42
3.8. The four data sets............................................................................................. 43Chapter 4: Results and Analysis...............................................................................454.1. YLSA Efficiency...............................................................................................454.2. Particle ID .........................................................................................................454.3. Kinematic Selection o f Decays........................................................................ 474.4. Extraction o f Branching Ratios....................................................................... 50
4.4.1. Angular Distribution Theory...................................................................504.4.2. Angular distribution fitting..................................................................... 52
4.5. Branching ratios above the 01-threshold (Ex=3.529 M eV)............................. 534.5.1. Ex=4.379 MeV.........................................................................................534.5.2. Ex=4.549 MeV.........................................................................................544.5.3. Ex = 4.600 MeV and 4.635 MeV............................................................ 564.5.4. Ex=4.712MeV.........................................................................................574.5.5. Ex=5.092 MeV.........................................................................................58
4.6. Branching ratios above the proton threshold (Ex=6.411 MeV).....................59
5
4.6.1. Ex=6.742MeV.........................................................................................594.6.2. Ex=6.861 MeV........................................................................................ 614.6.3. Ex=7.070 MeV........................................................................................ 624.6.4. Ex=7.500 MeV........................................................................................ 63
Chapter 5: Discussion and Conclusions.................................................................665.1. Comparison to prior results............................................................................. 66
5.1.1. The standard approach............................................................................665.1.2. An alternative Bayesian approach.......................................................... 675.1.3. Application o f the Bayesian approach....................................................705.1.4. Discussion................................................................................................72
5.2. Future Directions..............................................................................................78Appendix A: Monte Carlo Algorithm for Determining Efficiency.......................80Appendix B: Mathematica fitting example: W(0) for To/T o f Ex=4.600 M eV 82Bibliography............................................................................................................. 85
6
LIST O F FIGURES
Number Page
Figure 1: Flow diagram for the CNO cycles. Purple boxes represent the valley o f P-stability. CNO-I is the dominant cycle, while the other numbered CNO cycles are additional side cycles through which processing may occur. Hot CNO is a process that produces energy at a greater rate at higher temperatures away from the valley o f P-stability. The rate o f energy generation in hot CNO reaches its maximum at approximately 100 MKat which point it is limited by the waiting points [Wiescher, Gorres et al. 1999]...........12
Figure 2: 19Ne level diagram. The spins and parities, where known, are given in parentheses after the level excitations. The orbital angular momenta for 15O g s+ ( X are given below the proton threshold. Above the proton threshold, the orbital angular momenta are for% + P ................. v.......................................................................................................... 23
Figure 3: Layout o f the Wright Nuclear Structure Lab................................................................ 30Figure 4: Enge split-pole spectrograph at zero degrees with the scattering chamber and YLSA
schematically shown......................................................................................................... 31Figure 5: Side view o f focal plane detector...................................................................................33Figure 6 : Top view o f focal plane detector. The arrows schematically illustrate a the possible
trajectories for projectiles from a certain reaction channel, if the front wire notpositioned at the focal plane. §4.2 discusses how corrections may be made................. 34
Figure 7: A segment o f YLSA. The length dimensions are given in cm. There are actually 1-mm gaps between strips................................................................................................... 36
Figure 8: CAD design o f YLSA frame with detectors mounted................................................. 37Figure 9: Protective measures in the scattering chamber to prevent charging o f exposed
insulator on YLSA by electrons. The tuning collimators rotated out o f the way after tuning the beam. +500 volts were applied to the target ladder, shield and mountwhenever beam was present Note the orientation o f the segments in the blowup 38
Figure 10: A Monte Carlo simulation o f the solid angle o f YLSA. 105 events were generated isotropically in the lab frame, and events which intersected YLSA were counted as hits.There is significant overlap in 0 between adjacent strips................................................39
Figure 11: Schematic view o f electronics used in the experiment Amplifiers are shown as triangles, with pre-amps designated by Tre’, shaping amps designated by a Gaussianpulse and sum amps designated by “+ ’ or ‘Sum’............................................................. 44
Figure 12: Particle identification in the focal plane detector showing triton gates. The top panel shows energy detected in the scintillator vs. position along the focal plane, gated on AE vs. Bp. The vertical axis o f the bottom panel shows energy loss in the isobutane gas. The AE vs. Bp plot has been gated on E vs. Bp. The horizontal bands in thebottom pane are due to hardware gates in the trigger electronics..................................46
Figure 13: “Theta” parameter vs. focal plane position before and after correcting for aberrations in the spectrograph focusing. “Theta” is essentially a difference betweenposition measured at the rear wire and position measured at the front wire.................47
Figure 14: Kinematic selection o f the decay channels out o f 19Ne. The 2D plot is o f events gated on the time peak in Figure 15. Candidate events for Ot-decay to the ground state
7
o f lsO are outlined in blue in the lower panel and plotted in blue in the upper panel, while candidates for p-decay to the ground state o f 18F are indicated similarly with green. The top pane shows all tritons in black, giving a spectrum o f excited states in 19Ne, starting with the strongest peak on the right being the ground state and increasing in excitation with decreasing channel number. One 13N state, i.e. l3C(3He,t)13N*(p)l2C, is the strongest contaminant and is circled in black in the lowerpanel. (Data from 1st July run.)........................................................................................49
Figure 15: Time spectrum, summed from all channels in YLSA. The black spectrum is plotted for all l9Ne events, and the blue spectrum is plotted for all kinematically selectedcandidate events for 18F . (Data from 1st July run.)........................................................ 50
Figure 16: The Ex=4.549 MeV confidence regions for the fit parameters. The best fit coordinates are marked by the cross. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively. Most o f the area inside thecontours lies inside the physical region for p1/2...............................................................55
Figure 17: The Ex=4.600 MeV confidence regions for the fit parameters. The best fit coordinates are marked by the cross. The inner and outer contours are the boundariesfor the 68.27% and 90% confidence levels, respectively................................................ 56
Figure 18: The Ex=4.712 MeV confidence regions for the fit parameters. The best-fit coordinates are marked by the cross. The inner and outer contours are the boundariesfor the 68.27% and 90% confidence levels, respectively................................................57
Figure 19: The Ex= 5.092 MeV confidence regions for the fit parameters. The best-fit coordinates are marked by the cross. The inner and outer contours are the boundariesfor the 68.27% and 90% confidence levels, respectively................................................58
Figure 20: Angular distributions o f decay Ot's below the proton threshold. The blue curves arefits to equation (4.2)......................................................................................................... 59
Figure 21: The Ex=6.742 MeV confidence regions for the fit parameters. Circles mark the best- fit coordinates. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively, (a) The left panel shows the initial fit, which resulted in a non-physical value for p1/2. The physically allowed values are below and to the left o f the lines, (b) The right panel shows the confidence regions obtained by notallowing p1/2>0.5.............................................................................................................. 60
Figure 22: The Ex=6.861 MeV confidence regions for the fit parameters. The best fit coordinates are marked by the cross. The inner and outer contours are the boundariesfor the 68.27% and 90% confidence levels, respectively................................................61
Figure 23: The Ex=7.070 MeV confidence regions for the fit parameters. The best fit coordinates are marked by the smallest ellipse. The larger contours are the boundanesfor the 68.27% and 90% confidence regions.................................................................. 62
Figure 24: The Ex=7.500 MeV confidence regions for the parameters o f the fit to the proton decay data. Each plot is a projection taken at the best fit value for the missing 3rd parameter. The best fit coordinates are marked by the cross. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively. 64
Figure 25: The Ex=7.500 MeV confidence regions for the parameters o f the fit to the alpha decay data. Each plot is a projection taken at the best fit value for the missing 3rd parameter. The best fit coordinates are marked by the small central ellipses. The inner and outer large contours mark the boundaries for the 68.27% and 90% confidence levels, respectively.............................................................................................................64
8
Figure 26: Measured decay angular distributions above the proton threshold. Red curves are fits for a decays, and violet curves are fits for proton decays. All curves are fits assuming a certain spin and parity, except for the curves for Ex=7500 keV, which aresimple fits to Legendre polynomials.................................................................................65
Figure 27: Bayesian posterior probability distributions calculated from the data in Table 8 using formula (5.10). The distributions are normalized so that each sums to unity. If one accepts a Bayesian interpretation as valid, i.e., that measurement results make a statement about the likelihood o f the actual value o f a measured quantity, then the results o f the procedure described in these sections give these probability distributionsfor the values o f the branching ratios.............................................................................. 71
Figure 28: The 15O(0C,Y) reaction rate calculated from the averages reported in Table 9. The 90% upper limit o f Davids et al. for the Ex=4.033 MeV state is plotted [Davids, van den Berg et al. 2003]. The Fy used in the calculation come from the same paper 74
Figure 29: Effect o f the new measurements on the 15O(0C,Y) reaction rate uncertainty. The top and bottom dashed curves represent the uncertainty in the reaction rate as calculated before incorporating the present measurements. The middle solid curve is the ratio o f the reaction rate shown in Figure 28 with the older rate. The top and bottom curvesrepresent only the uncertainties in the reaction rates due to the error bars for r j T ............ 74
Figure 30: The calculated 18F(p,"$ reaction See the discussion for the details on the propertiesused for the calculation..................................................................................................... 75
Figure 31: Effect o f the new measurement on the 18F(p,7) reaction rate uncertainty. The top and bottom dashed curves represent the uncertainty in the reaction rate as calculated before incorporating the present measurements. The middle solid curve is the ratio of the reaction rate shown in Figure 30 with the older rate. The top and bottom curvesrepresent only the uncertainties in the reaction rates due to the error ban for IfT . 76
Figure 32: The calculated ,8F(p,0t) reaction See the discussion for the details on the propertiesused for the calculation..................................................................................................... 76
Figure 33: Effect o f the new measurement on the l8F(p,Y) reaction rate uncertainty. The top and bottom dashed curves represent the uncertainty in the reaction rate as calculated before incorporating the present measurements. The middle solid curve is the ratio of the reaction rate shown in Figure 31 with the older rate. The top and bottom curves represent only the uncertainties in the reaction rates due to the error bars for FJT. 77
9
LIST O F TABLES
Number Page
Table 1: Radioisotopes thought to be capable o f generating astronomically observable y-rays innova explosions................................................................................................................ 15
Table 2: Previously determined parameters o f 18F+p resonances. Except where noted, allbranching ratios are from measurements by Utku et al. [Utku, Ross et al. 1998]..........24
Table 3: Properties o f 18F+p resonances inferred from mirror assignments with states in 19Ne. Assuming 6 2a C N e ) = 6 2a C F ) , Tr C*Ne) = Tr {n F ) and 0 * =0.1(0.01) for even(odd) paritystates. [Utku, Ross et al. 1998].........................................................................................25
Table 4: Previous experimentally determined CX-branching ratios for states in the Gamowrange..................................................................................................................................28
Table 5: Summary o f experimental runs. See §3.8 for an explanation o f the 4th column 43Table 6 : Results o f fitting to the decay angular distribution data for Ex=7500 keV. The
asymmetric error bars on fit parameters were made symmetric by setting them to the larger o f the upper and lower error bars in order to calculate the uncertainty in thebranching ratios................................................................................................................ 63
Table 7: Summary o f branching ratio measurement results. The error bars give the limits o f 68.27% (la) confidence intervals. Above the proton threshold (Ex= 6.411 MeV) the quoted resonance energies are proton resonance energies. The 659 keV proton resonance branching ratio comes from a simultaneous fit o f proton and CX-decayssubject to r«+rp=r......................................................................................................... 65
Table 8: Measured values for branching ratios in 19Ne. States where only limits have been measured are ignored. The values in the last column are the presently measured values listed in Table 7. Table 2 and Table 4 give more detailed information about the sources o f the results in the other columns. Results that have error bars implying greater than5% probability lies outside the physical region are highlighted in red............................67
Table 9: Results o f averaging the data in Table 8.The 3rd column gives the result o f the standard least-squares method o f averaging, with the Y statistic in the next column. The tail probability for the f distribution is given in the 5th column. The value inferred using the Bayesian method described in this chapter is also given along with the probability that all the given measurements have underestimated error bars................72
10
Chapter 1
Chapter 1: Introduction to Explosive Nucleosynthesis
1.1. Stellar nucleosynthesis
“...a star is a gravitationally controlled thermonuclear reactor in the sky.”
-Claus Rolfs & William Rodny[Rolfs and Rodny 1988]
Many astrophysical phenomena require detailed knowledge o f nuclear reactions for adequate
explanation. For example, it is well known that the sun, the source of nearly all the energy
which fuels our society, is itself fueled by the fusing o f hydrogen into helium. Observations of
solar neutrinos at Kamioka, Japan [Fukuda, Fukuda et al. 2001] and in Sudbury, Ontario,
Canada [Ahmad, Allen et al. 2001] have confirmed this fact beautifully.
Balancing gravitational pressure with radiative pressure produced by the heating o f their
interiors by nuclear reactions stabilizes stars like our sun. Put another way, the stellar matter
(mosdy hydrogen) is squeezed to such high temperatures (average energy) and densities that
nuclei are able to overcome their electrostatic repulsion and fuse with one another. The most
typical pathway for this is called the pp-I chain:
p { p Z v ) d ( p , y y H e C H e , 2 p Y H eNet result: 4 'H -> *He + 2e+ + 2v + 26.73 MeV
In our sun, two other sequences, the pp-II and pp-III chains, are responsible for another 14%
of the fusions. The amounts o f energy lost to the neutrinos, which escape without interacting,
differ slighdy among the chains.
11
► CNOI ► CNO II ► £ N Q 4 I ^ ► CNO IV ► Hot CNO
O
NC
6 7 8 9 10 11 12«
Figure 1: Flow diagram for the CNO cycles. Purple boxes represent the valley of P-stability. CNO-I is the dominant cycle, while the other numbered CNO cycles are additional side cycles through which processing may occur. Hot CNO is a process that produces energy at a greater rate at higher temperatures away from the valley o f P-stability. The rate o f energy generation in hot CNO reaches its maximum at approximately 100 MK at which point it is limited by the waiting points [Wiescher, Gorres et al. 1999].
A second important fusion process is burning through the C N O cycles. This process catalyzes the fusion using existing stable carbon and nitrogen in the star. Mosdy, it bums through the
CNO-I cycle:
n C ( p , y ) n N ( e +v ) n C ( p , y ) u N ( p , Y) XiO (e +v ) l5N ( p , a ) 12C. (1.2)
At low temperatures and densities (T < 50 M K , p<100 g/cm3), the rate of this cycle is limited
by the 14N(p,^ reaction [Wiescher, Gorres et al. 1999]. For approximately one cycle in 1000,
15N captures a proton instead cycling back to 12C in the CNO-II cycle:
14 N ( p , Y )'50 ( e +v ) 15 N ( p , y ) 160 ( p , / ) 17 F ( e +v ) l l O ( p , a ) 14iV. (1.3)
1 2
There is some additional leakage out o f CNO-II to the nudeus 18F via the CNO-III cyde:
X10{p , y) ,8F (e V ) lsO(p, a ) 'sN. (1.4)
For about one cyde in 150, the lsO may capture a proton instead. If the resulting 19F cydes
back to lciO via (p,CC), this is called the CNO-IV cyde. Alternatively, the 19F may capture a
proton into 20Ne upon which protons may be converted into alphas by the faster Ne-Na cycle.
The ratio o f the 19F(p/$ rate to the 19F(p,a) rate is small and very uncertain [Wiescher, Gorres
et al. 1999].
At high enough temperatures (e.g. 150 MK at densities o f 100 g em'3), the CNO cycles are
turned into the “ hotCNO” cycle. In the hotCNO cyde, proton capture proceeds to the drip
line [Wiescher, Gorres et al. 1999]:
/ : nC(p, y)13N{p, y) 140(eV ) "N(p, y) l50(e+v)'5N(p, a) 12C
II : i60(p, y^Fip, y) nNe(e'v) »F(j>, a) i50
The hotCNO cyde is the primary energy generation process in novae. Note that the rate at
which this cycle operates is limited by the P+-lifetimes o f the 140 (1=1.70 minutes) and lsO
(X=2.94 minutes) nudei. These lifetimes, unlike the capture reaction rates, are independent of
temperature. Any point in a nuclear burning process where P-decay is the dominant path is
called a “waiting point” . The waiting point nudei 140 , lsO and 18F shown in (1.2), (1.3), (1.4)
and (1.5) may have important roles in explosive nucleosynthesis. We will focus on the latteritwo: ,sO and ,8F . (
One proposed alternative path to the A>18 region is the| alpha-capture reaction on lsO. One
o f the purposes o f the measurement described in this document is to constrain the possible
values for this reaction rate. The rate o f 18F(p,a) is one factor determining the temperatures
and densities at which the hotCNO cycle may operate. In novae, proton-induced reactions on
18F play an astronomically observable role described in more detail bdow [Wiescher, Gorres et
al. 1999].
13
1.2. White dwarves and novae
‘White dwarf matter is so highly compressed that a bowling ballfilled with the stuff would tip the scales at over a thousand tons—about the weight of a small ocean liner. —Laurence A.Marschall [MarschaU 1988]
In normal stars, the relationship between pressure, density and temperature is well-described
by the ideal gas law[Rolfs and Rodney 1988]. This relationship holds as long as nuclear fuel
remains to provide heat and energy. Once the fuel runs out, however, the star no longer is
able to support itself, and its material then collapses.
Chandrasekhar showed that if the remaining mass is less than 1.4 solar masses, the object may
be supported from further collapse by electrons behaving as a Fermi gas [Chandrasekhar
1939]. The electrons in a star are the most numerous particles, especially in a white dwarf.
White dwarves are primarily ashes from the later stages o f nuclear burning, such as carbon,
oxygen, neon and magnesium.
As fermions, electrons are governed by the Pauli exclusion principle, which says that no two
identical fermions may inhabit the same quantum state. In a white dwarf, the electrons have
filled up all available quantum states up to the Fermi energy. A solar-mass white dwarf has a
radius o f approximately 8000 km (only slighdy larger than the earth), making these objects very
dense (~106 g'cm 3) [Rolfs and Rodney 1988].
When it is in a close binary system with an actively burning star, the dwarfs strong
gravitational field will gradually pull material onto its surface from the less dense star. The
falling matter also heats up the surface with the kinetic energy it gains. A normal star would
expand as it heats up, lowering the density. The degeneracy (i.e., T /T f« l ) o f the white dwarf
[Pathria 1996] prevents this, however. Thermonuclear runaway may occur if the temperature
dependence o f nuclear energy generation exceeds the temperature dependence o f energy loss
through cooling [Wiescher, Gorres et al. 1999].
de-->dT
f e c o o l
dT(1.6)
14
Periodically, the heating will bring the surface material to a sufficient temperature for explosive
nuclear burning to occur, that is, condition (1.6) is satisfied. (If the mass accretion is fast
enough, such as with a red giant partner, the explosion may be violent enough to completely
disperse the dwarf. These events are known as Type la supemovae [Rolfs and Rodney 1988].)
The burning layers generate energy at such a large rate that they expand rapidly, i.e., the surface
layers are not degenerate like the bulk o f the white dwarf. The burning continues as long the
expansion hasn’t reduced the temperature and density too far. The peak o f visible light
observed by astronomers actually occurs during this expansion and cooling phase, because the
luminosity o f a radiating object is proportional to its surface area. O f particular interest in this
study is the gamma-ray flux potentially observable from such objects.
Table 1: Radioisotopes thought to be capable of generatingastronomically observable y-rays in nova explosions.
Radioisotope v-rav energy tkeV) Lifetime18F 511 158 min
“ Na 1275 3.75 yr26AI 1809 1.04 Myr
Table 1 shows the proton-rich radioactive nuclei that may contribute significandy to
observable y-rays from novae [Hemanz, Jose et al. 1999]. Any left over nuclei on the proton
rich side o f the valley o f (i-stability may be sources o f 511-keV y-radiation, but 18F and “ Na are
thought to dominate, the 511-keV flux. The waiting point nucleus, 13N, has a lifetime so short
that it has mostly decayed before the expanding matter becomes transparent to the y-rays. The
“ Na is present only in oxygen-neon novae and its long lifetime relative to 18F makes its 511-
keV emission weaker; however it’s daughter, “ Ne, emits a unique 1275-keV y-ray which may
eventually be observable [Clayton and Hoyle 1974]. 18F, which decays direcdy to the ground
state o f 180 , has a 2.5-hour lifetime, allowing it to be strongly observable if it is produced in
sufficient quantities. The effect o f 18F+p resonances in 19Ne on 18F production in novae is one
o f the subjects o f this thesis.
15
1.3. Neutron stars and x-ray bursters
“If our bodies were packed as solidly as the atomic nuclei they contain, our masses wouldincrease by ten trillion times, and each of us would have more mass than Mount Everest. ”—D. Goldsmith <&N. Cohen [Goldsmith and Cohen 1991]
If the remnant o f stellar collapse has a mass greater that the Chandrasekhar limit o f 1.4
the degeneracy pressure from the electrons is not sufficient to prevent further gravitational
collapse. Collapse resumes until a density o f approximately 1015 g 'C m ’ 3 is reached, at which
point the electrons and nuclei are packed so closely and collide so frequently that the electrons
fuse with protons and “neutronize” the matter. For stellar masses in the range 1.4
Msolar<M.<1.8 Msola the remnant at roughly nuclear density is held from further collapse by
neutron degeneracy pressure. If the mass exceeds 1.8 MMlar, however, this neutron degeneracy
is insufficient to prevent the star from collapsing into a black hole. [Rolfs and Rodney 1988]
In the collapse o f a pre-supernova star (with r ~ 1.5T06 km and a rotational period o f ~ 30
days) to a neutron star (with r = 150 km), the conservation o f angular momentum leaves the
remnant with a period o f only = 30 msec. (If the neutron star is accreting material from a
companion, it may be further spun up by the additional angular momentum gained with the in
falling material.) These rapidly spinning neutron stars are observed as pulsars in the visible
and/or radio wavelengths. A famous example is the pulsar at the center o f the Crab Nebula
(SN1054), which pulses at 30 Hz in both radio and visible light.
The large amount o f energy imparted to the in-falling material by the strong gravitational field
also causes x-ray emission. Many point sources o f x-rays in the spiral arms o f our galaxy are
thought to be neutron stars. Neutron stars in close binary systems may occasionally be sites for
thermonuclear explosions similar in principle to novae. These explosions are thought to be
what we observe as “x-ray bursters” . Astronomers observe sudden (= 1 sec) increases by an
order o f magnitude or more in x-ray intensity, which take seconds or minutes to decay back
down to normal levels.
The gravitational field o f a neutron star is thought to be too strong for x-ray bursters to enrich
the interstellar medium with their ashes. Astrophysicists are, however, interested in being able
to accurately model the observable properties o f x-ray bursters, such as the observed x-ray flux
16
as a function o f time. The risetime and peak luminosity o f the light curve are dependent on
the energy generation in the x-ray burster, which is dependent in large part on nuclear
reactions. Energy generation is limited to approximately 1014 erg-g s"1 in the CNO cycles by
waiting point nuclei, independent o f temperature. In order for the observed luminosities to be
reproduced, greater amounts o f energy are needed. One possible path to greater energy
generation is “breakout” through 15O(0C,j) [Wiescher, Gorres et al. 1999].
The purpose, then, o f the measurement made in this thesis, is to better understand the reaction
rates o f 150((X,'$, 18F(p,j) and ,8F(p,OC). The next chapter will introduce the mathematical
formalism for calculating the dependencies o f these reaction rates on the properties o f isolated,
narrow resonances in 19Ne. Then it will review the recent literature to give the latest
information on the known or calculated resonance properties. The rest o f the chapters will be
devoted to describing the experiments which we have recently earned out and their results,
and to interpreting those results.
17
Chapter 2
Chapter 2: Present Status o f 19Ne’s Role in Nuclear Astrophysics
2.1. Nuclear burning through resonances
The general expression for a reaction rate per particle pair in a stellar environment is [Rolfs
and Rodney 1988]:
(<xv) = <t>(y)vo(v)dv (2.1)
where v is the relative velocity o f the particle pair, <7 (v) is the velocity dependent cross-section
for the reaction, and 0 (v) is the velocity distribution o f the particles.
Consider a nuclear reaction, I+a— F+b, where I and F are the initial and final “heavy
nucleus” , while a and b are the incoming and outgoing particles, which here may be a ’s, p’s or
Ys. The total reaction rate is:
r = NINa(<Tv) (2 .2)
where the abundances o f the two types o f particles are denoted N[ and Na.
The lifetime o f I in the burning environment due to the given reaction is:
N,(cn)
At high enough temperatures, both the reaction and its inverse may occur, I+ak->F+b. The
convention here is that the forward reaction, I+a—*F+b, is the exothermic reaction. The net
energy generation, usually expressed in erg-g '-s’ 1, is then:
18
^net ^ la ^Fb ~ ( ria rFb ) Qf P (2.4)
In equation (2.4), Q is the energy liberated in the forward reaction in ergs, and p is the
density in g-cm’3.
For the temperatures and densities o f interest here, we will assume that isolated and narrow
resonances dominate the reaction rate (which is not always true, see §5.1.4). Formally, this
means T □ ER, where Er is the resonance energy and T is its width. For a Maxwell-
Boltzmann velocity distribution, the narrow-resonance reaction rate per pair is expressed,
using equation (2 .1):
(<7v) = ' 8 V/2n p )
1'E r exp
(kT)m K A kT)(E)dE (2.5)
In equation (2.5), T is the temperature, p = m]maJ(m] +ma) is the reduced mass o f the
interacting nuclei, and <JBW is the Breit-Wigner expression for the energy dependent cross
section through a resonance. Substituting the Breit-Wigner expression into (2.5), we obtain:
(2.6)
where:
( 0 -2J + 1
r.r.y — a b1 r
(2.7)
The factor, / is an electron screening factor and reflects the enhancement o f the rate by the
Coulomb potential. It can be calculated using atomic physics considerations [Salpeter 1954].
All the nuclear properties are contained in the resonance energy (Eres) and the resonance
strength ( OJy). The resonance strength contains two factors, the first o f which is just statistical,
dependent on the spins of the incoming particles and on the spin o f the resonance. (The19
term takes care o f the case o f identical particles.) The second factor summarizes the nuclear
structure information in terms o f the partial widths, r a and , and the total width o f the
resonance, T . If a and b are the only open channels, then T = Ta + TA; furthermore, if
r a Q Ta , then 07/ ~ 0)Ta and the uncertainty in Fa dominates the uncertainty in resonance
strength.
As described by Iliadis [Uiadis 1997], a single-particle partial width may be parameterized as
follows:
r„ = 2 p.u a 2. “ c c
c'setsp(2.8)
The subscript c refers to the single-particle c hannel (the single particle having a mass number
Ac) in or out o f a nucleus with mass number A. Technically, the OC-channel is not single-
particle, but a similar formula applies for that case. Pc contains the Coulomb and centrifugal
penetrability; flc is the reduced mass, and ac = a0 [A 3 + AlJ3 j is the channel radius, where
some value around 1.2 ffn is chosen for ao- C is the isospin Clebsch-Gordan coefficient for
coupling the nuclear state to the decay channel S is the single-particle spectroscopic factor,
which may be roughly understood as the overlap integral between the actual state
wavefunction and a coupled-channel wavefunction describing the initial nucleus plus c
[Brussard and Glaudemans 1977]. The final factor 0fp is the dimensionless single-particle
reduced width and is calculated by evaluating the single-particle radial wave function o f the /
orbit at ac. This wave function is normalized to unity inside the channel radius. Q2sp may vary
between zero and one, one being the Wigner limit [Iliadis 1997].
It is immediately apparent from (2.6) that the reaction rate depends most strongly on the
resonance energy. There have been many experiments that have precisely located important
resonances in ,9Ne. Next in importance is the linear dependence on the resonance strength.20
The experiment described in this thesis measures proton- or alpha-decays from excited states.
The measured decays are normalized to the total detected population o f the state, so that we
actually measure the branching ratios. It is possible to rewrite the expression for y
This new form shows proportionality to 3 factors. The first two factors are the dimensionless
branching ratios.
In addition to measuring the branching ratios, the experiment is sensitive to the angular
distribution o f the decays. In certain cases, this information may be used to determine the spin
and parity o f the states in 19Ne.
2.2. Production of 18F in Novae
Recalling the discussion in chapter 1, 18F production in novae is an interesting problem
because the beta-decay o f 18F is thought to be the prime source o f the 511-keV line (from
annihilation in the positronium singlet state) and sub-511-keV continuum y-ray flux [Hemanz,
Jose et al. 1999]. The continuum is due to Compton scattering o f the line emission and also to
the 3-photon decays o f triplet-state positronium. Using the resonance strength information
from Utku et al. [Utku, Ross et al. 1998], Hemanz et al. find that current satellites should be
able to detect this radiation out to 0.8 kpc and that the satellite INTEGRAL should be
sensitive out to 3 kpc [Hemanz, Jose et al. 1999]. So far no detection o f this radiation from
novae has been made.
2.2.1. Sensitivity to resonances in 19Ne
Using the latest reaction rate compilations [Angulo, Amould et al. 1999] and the most up-to-
date information on 18F+p resonances [Utku, Ross et al. 1998; Bardayan, Blackmon et al. 1999]
available at the time, Coc et al. [Coc, Hemanz et al. 2000] varied reaction rates by their error
bars to determine the sensitivity o f 18F production on the various reaction rates. For a 1.25 M0
ONe nova, they found that 18F+p reactions give an uncertainty o f a factor o f about 300 in the
21
production o f 18F. For resonances with unmeasured strengths, most notably the broad 38 keV
resonance, the uncertainty is estimated by varying Tp between zero and the Wigner limit1.
This corresponded to a factor o f 15 in the 18F(p/$ rate and a factor o f 30 in the 18F(p,a) rate.
While the properties o f 18F are not studied in this experiment, it should be noted that Coc et al.
find an additional factor o f 10 uncertainty in 18F production comes from the unknown proton
widths o f the 66-keV and 179.5-keV 170 + p resonances.
Iliadis et al. [Iliadis, Champagne et al. 2002] confirmed these results in a survey o f the
sensitivities o f isotope production factors to varying reaction rates. They used several
published nova models to provide temperature-density-profiles for their reaction network
calculations. They find a factor o f as much as 100 sensitivity for 18F production due to the
current ranges for 18F(p,a), 17O(p,0l) and nO(p,Y) reaction rates in CO novae. In ONe novae,
the sensitivities to vO(p,f) and 18F(p,^ are as much as 500. Their sensitivity to 170(p,a) is only
as much as 110 in nova models with 1.15 MQ< MVVD <1.25 M0. They also found a factor o f as
much as 600 sensitivity in 18F production to the reaction nF(p,Y) in the subset o f models with
peak temperatures greater than 250 MK and M ^^l.25 M0. nF(p,7) produces 18Ne, which may
then decay to 18F with a lifetime o f 2.41 seconds.
In summary, for understanding the production o f 18F in novae, the three most important
nuclear reaction rates to measure are 18F(p,^, 17F(p,T) and 18F(p,a) in approximately that order.
The branching ratios measured in this experiment help to more precisely determine the
reaction rates for 18F+p.
2.2.2. Proton resonances in 19Ne
Table 2 s11mmari7.es the direct experimental knowledge o f proton resonance properties in 19Ne.
Additional inferences may be made about the properties o f these states by appealing to isospin
symmetry, which claims that a nucleus and its isospin mirror should have identical properties.
19Ne’s isospin mirror, obtained by swapping the proton and neutron numbers, is 19F. Utku et
al. [Utku, Ross et al. 1998] performed a set o f isospin mirror reaction studies in order to make
1 See the discussion below equations (2.8) for a definition of Wigner limit.
22
isospin mirror assignments between levels in the two nuclei: lcO(6Li,t) / 160 ( 6Li,3He) and
20Ne(d,t) / 20Ne(d,3He). Their results, along with the properties that have been derived from
them, are displayed in Table 2 and Table 3. These results have been supported by an additional
study o f the isospin mirror reactions 12C(10B,t)19Ne / 12C(10B,3He)’9F at WNSL [Lewis,
Caggiano et al. 2002].
><DxHI
8000-1
7500-
7000-
6500-
6000-
5500-
5000-
4500-
4000-
3500- a -threshold 3529
19,Ne
Figure 2: 19Ne level diagram. The spins and parities, where known, are given in parentheses after the level excitations. The orbital angular momenta for ,5Ogj+0t are given below the proton threshold Above the proton threshold, the orbital angular momenta are for ,8Fg.s.+p.
23
Table 2: Previously determined parameters o f 18F+p resonances. Except where noted, all branching ratios are from measurements by Utku et al. [Utku, Ross et al. 1998]
Ere,1(keV)
ii* [MeV Ere, [keV] (keV)
ryr ryr r(keV)
ojy(p,a)(keV)
6.41926(9) 6.437 26(9)2 216(19)39(6) 6.450287(6) 6.698 <0.9-10-3330(6) 6.742 (3/2,1 /2)- 331(7)2 1.04(8)
324(7)3 3.5(16)-103332(17)4 1.48(46)-103
450(6) 6.861 450(7)2 <0.025 0.96(8) <0.8-103659(7) 7.070 3/2+ 656(9)2 0.37(4) 0.64(6)
664.7(16)5 0.39(2)5 39.0(16) 6.2(3)657.5(18)3 34.2(25) 4.7(2)
653(7)6 32(20) 2.7(7)638(15)7 0.4-0.6 5.6(6)
762(5) 7.173 799(20)2827(6) 7.238 842(10)2
915(15)21089(9) 7.500 0.84(4) 0.16(2) 16(16)1120(11) 7.531 1120(15)? 2 0.67(8) 0.33(6) 31(16)1197(11) 7.608 1205(16)2 0.97(4) 0.04(2) 45(16)1233(12) 7.644 0.37(6) 0.64(4) 43(16)1408(11) 7.819 0.19(9) 0.81(11) 22(16)
1 [Utku, Ross et al. 1998]
2 [Tilley, Weller et al. 1995]
3 [Graulich, Binon et al. 1997]
4 [Bardayan, Batchelder et al. 2002] J* = 3/2- assumed. See §4.6.1 for more information.
5 [Bardayan, Blackmon et al. 2001]
6 [Rehm, Paul et al. 1996]
1 [Coszach, Cogneau et al. 1995]
24
Table 3: Properties o f 18F+p resonances inferred from mirror assignments with states in 19Ne. Assuming 0 * ( l9iVie) = 6 £ (19F ) ,
r rCNe) = r rC9F) and = 0 .1 (0 .0 1 ) for even(odd) parity
states. [Utku, Ross et al. 1998]
E_ro ExfF ) J* r r n rtt Ty ojy(p,a) ajy(p,Y)
(keV) (keV) (eV) (cV) (eV) (eV) (eV) (eV)
8 6497 3 /2+ <1 103 3.50 1034 <1 103 0.85 2.3 10'34 2.0 10‘37
26 6429 1/ 2 ' * 6.6 lO'20 * 1.1 lO’25 2.2 -lO'20 1.1 lO'25
38 6528" 3/2+ 4.3 103 2.5 10n 4.3 103 1.2 1.7 1011 5.0 1015
287 6838** 5/2+ 1.2 103 0.27 1.2 103 0.33 0.27 7.5-lO'5
330 6787 3/2 ' 2.7 103 6 .0 ' 2.7 103* 5.5 3.5 8.1 1 0 3
450 6927 7/2- 3.1 103 1.6 10’2 3.1 103 2.4 2.1 10'2 1.7 lO'5
659 7100 3 /2+ * 2.4 10'1 * * * 2.4 lO'1
762 7166" 11/ 2 ' 12 9.4 10 s 12 0.17 1.9 10-4 2.7 -lO'6
827 7262 3 /2+ < 104 <4 103 < 6 103
2.3. Breakout £rom HotCNO via
Sections 1.1 and 1.3 described how the waiting point nuclei in the hotCNO processes can be a
botdeneck for energy generation, and how in order for the expected rates o f energy generation
to occur, there must be some kind o f breakout to higher-mass burning processes. 15O(0C,Y has
been discussed as a possible path for breakout [Wallace and Woosley 1981; Wiescher, Gorres
et al. 1999]. The a-threshold in 19Ne lies at 3.5294 MeV, well below the p-threshold at 6.4112
* Measured directly, see Table 2.
" Very tentative assignment
1 [Bardayan, Batchelder et al. 2002] recendy reported a direct measurement of 18F(p,a) giving Tp = 22+0.1 eV.
* Inferred from branching ratio in Table 2.
25
MeV and the n-threshold at 11.639 MeV. Therefore, for states o f importance in iS0(CL,f), i.e.,
states with 4 MeV < Ex < 5.5 MeV, the only partial widths to consider are Ty and Ta.
Table 4 shows the first 10 states above the a-threshold. Recalling the earlier discussion of
reaction rate dependence on resonance energy, these are the only states one needs to consider
in order to calculate the 150(a,7) reaction rate. The important resonance energies should lie in
the Gamow window from E0— A to E0 + A , defined [e.g., Rolfs and Rodney 1988] by the
peak in probability resulting from the product o f the Maxwell-Boltzmann distribution o f
kinetic energies and the Coulomb tunneling probability:
E0 = 1.22(Z,2Z 22flT( ) ' /3 keV
A = Q.lA9{z\Z\pLTl) ‘ /6 keV
where / / = AiA2j^Al +A2) , the Ai and Z( refer to the mass and charge numbers o f the two
reacting nuclei, and T6 is the temperature in MK. For possible peak temperatures o f 0.2-0.35
GK in novae [Davids, van den Berg et al. 2003], the Gamow window ranges from 300-700
keV, implying that only the first three resonances in Table 4 may play a significant role. The
high L=4 centrifugal barrier for the 611- and 668-keV resonances precludes them from playing
a significant role compared to the L=1 504-keV resonance. Davids et al. predict from their
measurement o f r j T and shell model predictions o f Fy using Coulomb excitation data
[Hackman, Austin et al. 2000] that the 504-keV resonance dominates the total reaction rate up
to temperatures o f 0.5 GK. It is not, however, predicted to outpace the P+ decay rate (T = 176
sec) in novae, even if one uses the upper limit estimated by Rehm et al. [Rehm, Wuosmaa et al.
2002; Rehm 2003].
On neutron stars, where x-ray burster peak temperatures are estimated to range up to 1-2 GK
[Davids, van den Berg et al. 2003], the Gamow window range is 775-2450 keV, and the
resonances above 668 keV in Table 4 may play a role, as well as other higher-lying resonances
where the strengths are not yet measured. Davids et al. predict that 15O(0C,'$ plays a role in
breakout in this case.
26
(2.10)
2.4. Summary
It has been my goal in this chapter to show how in both novae and x-ray bursters, there are
unresolved questions about both the fueling o f the explosions and the observable ashes that
are left over, and that answering these questions depends, in part, on understanding the
properties o f resonances an 19Ne through which p- and CL- burning nuclear reaction may occur.
The following chapters describe the 19F(3He,t) coincidence experiment for measuring the
branching ratios to the p0 and CXq channels.
27
Table 4: Previous experimentally determined a-branching ratios for states in the Gamow range.
r a / r measurements
F res
(keV)
Ex
(MeV)r L 19F(3He,t)
1
d(,8Ne,p)2
p(21Ne,t)3
20Ne(3Hc,OC)4
503.7(25) 4.033 3 / 2 + 1 <0.01 <4.3 10"4 <7.6 10"4
611(4) 4.140 (9/2)- (4) <0.01
667.9(25) 4.197 (7/2)- (4) <0.01
849.9 (23) - 3 9 7/2+ 3 0 .044(32) <3.0 10'3 0.016(5)
1020(4) 4.549 (1 / 2,3/2)- (0,2) 0.07(3) 0.16(4)
1071(4) 4.600 (5/2+) (3) 0.25(4) 0.32(3) 0.32(4)
1106(4) 4.635 13/2+ 7
1183(10) 4.712 (5/2-) (2) 0.82(15) 0.85(4)
1254(20) 4.783
1563(6) 5.092 5/2+ 3 0.90(9) 1.8(9) 0.90(6) 0.80(10)
1 [Magnus 1988; Magnus, Smith et al. 1990]
2 [Laird, Ostrowski et al. 2001] The confidence level on the upper limits is not given.
3 [Davids, van den Berg et al. 2003] The confidence level on the upper limits is 90%.
4 [Rehm, Wuosmaa et al. 2002; Rehm 2003] preliminary. The upper limit is a conservative 90% upper limit chosen here usingfigure la from [Narsky 2000] and based on [Rehm 2003] where it was stated there were zero coincidence counts observed out of 3300 events (100% efficiency assumed).
28
Chapter 3
Chapter 3: Experimental Setup for 19F(3He,t)19Ne(x1)x2
3.1. The Experiment
On the basis o f what we have seen in the previous chapter, the astrophysical reaction rates for
18F(p,a), 18F(p,Y) and 150(CX/y) need to be known with greater certainty. As shown in section
2 .1, we can improve our knowledge o f the reaction rate by measuring particle branching ratios
in 19Ne. Particle branching ratios in 19Ne have previously been measured using the reaction
19F(3He,t) [Magnus, Smith et al. 1990; Utku, Ross et al. 1998]. In the current study, these
measurements were repeated, using an array o f large solid angle ion-implanted silicon strip
detectors to measure decay protons and alphas from the residual 19Ne nucleus, i.e., to detect
19Ne —>l8F+p and 19Ne —>15O + 0C. By measuring the number o f these decays more efficiently, it
was possible to determine the proton and alpha partial widths o f these states more accurately.
3.2. Accelerator
The Yale ESTU (Extended Jtretched TransUranium) Tandem Van de Graaf accelerator was
used to accelerate the beam o f 3He to 25 MeV. The negative ions (3He j were produced by a
duo-plasmatron ion source [e.g., Rolfs and Rodney 1988]. These were transmitted to the high
voltage terminal in the ESTU, where a voltage o f approximately 8.33 MV was maintained. A
thin carbon foil at the terminal strips off all three electrons from the negative 3He ions to give
them a charge o f + 2e, after which they are further accelerated while exiting the accelerator.
An analyzing magnet at the high-energy end (see Figure 3) o f the ESTU determines the purity
and energy o f the beam. Charged particles in a region o f constant magnetic field move in a
circular path determined by the following equation:
p = 10.18— 2 2 2 7 — (3.1)QB
29
Control Room
H 1 1L u< n|
L J
— I Ion ESTUSource Tandem
Figure 3: Layout o f the Wright Nuclear Structure Lab.
J
AnalyzingMagnet
In equation (3.1), p is the radius o f the path in cm, m is the mass o f the particle in amu, Tis its
kinetic energy in MeV, Q is its charge number, and B is the magnetic field in Tesla. The
analyzing magnet has a central bending radius o f 179 cm, and a typical aperture o f acceptance
5 mm wide. Regulating the Van de Graaf voltage from the aperture slit currents, which are
produced by the small tails o f the beam distribution, gives a AE/E o f about 2 10 for our 3He
beam. The accuracy o f the calibration is also about 1CT4, using the precisely known T=3/2
12C+p elastic scattering resonance at Ep=14.231 MeV [Overley, Parker et al. 1969].
3.3. Enge Split-Pole Spectrograph
The Enge spectrograph at Yale is an instrument for precise measurement o f the momenta of
charged particles. It can be used to measure the population o f excited states o f nuclei in beam-
induced nuclear reactions. It does this by focusing particles from the target position to a
30
position along a focal plane (which is a function of momentum, charge and B-field). It was
designed to maximize acceptance without sacrificing resolving power [Enge 1979]. When
making use of ray-tracing in the detector (see Figure 13), the spectrograph accepts up to
A0=16O mrad and A<|)=80 mrad or a total solid angle of 12.8 msr at the entrance.
Figure 4: Enge split-pole spectrograph at zero degrees with the scattering chamber and YLSA schematically shown.
As its name indicates, the spectrograph consists of two pole pieces (see Figure 4) enclosed by a
single coil [Spencer and Enge 1967; Enge 1979]. The edges of the pole pieces are shaped to
provide 2nd order focusing of particles from the same reaction channel, but of slightly differing
momentum and direction from the target [Enge 1979]. The dispersion, D = d x / d ( A p / p 0 ) ,
which determines the amount of displacement at the focal plane per fractional change in
momentum, has been made constant across the entire focal plane by a corrective insert after
the second pole piece. Significant 3rd order horizontal 0 dependence remains, but we are able31
to correct this aberration in offline data analysis using trajectory information from a second
position sensitive wire in the focal plane detector (see below). The magnifications of the beam
spot in the horizontal and vertical directions, respectively, are 0.39 and 2.9. The spectrograph is
sensitive to all charged particles of appropriate momenta passing through the entrance
aperture with 51.1 cm</K92.0 cm. The maximum attainable B-field is 16.3 kG. The field used
for this experiment was approximately 12.9 kG.
The spectrograph was set at 0=0° in order to simplify the geometry and interpretation of the
coincident decay data. The beam axis is the axis of symmetry for our silicon detector array, the
Yale Lampshade Array (YLSA, see §3.5). With the spectrograph at 0°, the kinematics
simplifies, since the average velocity of the decaying 19Ne is on the beam axis. The full
acceptance of the spectrograph was used in order to maximize the experiment’s yield.
3.4. Focal Plane Detector
The focal plane detector is an isobutane gas-filled ionization chamber (see Figure 5, Figure 6).
It is designed to separate and identify different species of light ions, as well as precisely
measuring their position along the focal plane. A cathode plate covers the bottom of the entire
active region of the detector, and is given a negative bias. A set of anodes at ground are along
the top, as well as the two position-sensitive wires. A set of equally spaced thin wires is at the
entrance, along with an identical set is at the downstream exit, connected by copper strips
along the sides. These wires and strips are biased in steps from the cathode voltage up to
ground in order to reduce the fringe fields, maximizing the useful area of the detector. In
addition, two grids, between the incident ions and the top of the detector, are biased to
provide optimal field gradient for the collection of electrons at the position-sensitive wires.
32
AE-2Front Wire Rear Wire
r L ^ L U r L
EntranceWindow
Grids
Field Shaping WiresExit Window
\
Cathode
//
PlasticScintillator
Figure 5: Side view of focal plane detector.
A AE signal, representing the energy lost by incident ions in the gas, is read off of the cathode.
It is produced when the ionization electrons pass into the region between the above-
mentioned grids. A AE signal is also available from the anode section between the two
position-sensitive wire assemblies. This anode produces signals when the ionization electrons
pass out of the region between the grids. At the downstream end of the entire detector there is
placed a plastic scintillator (6.35 mm thick BC-404), which provides a residual energy (E)
signal. The light produced in the plastic by a stopping ion propagates down to photomultiplier
tubes placed at each end. The scintillator’s time resolution is roughly 10 ns, whereas the gas
detector’s time resolution is hundreds of nanoseconds. Therefore, for this experiment the
scintillator also played a crucial role in defining the fast trigger and timing START for all the
channels of electronics devoted to measuring the times of coincident events in YLSA.
33
Figure 6: Top view of focal plane detector. The arrows schematically illustrate a the possible trajectories for projectiles from a certain reaction channel, if the front wire not positioned at the focal plane.§4.2 discusses how corrections may be made.
The position along each of the two wires is determined by a delay-line method. The wires are
0.001” diameter gold-plated tungsten and have a large positive bias on them, typically 1500 V.
They capture an avalanche of electrons when a passing particle ionizes the isobutane. Parallel
to the wire, a set of conducting pads (see [Chen 1999] for further description) picks up an
image of this charge. The current pulse picked up on the pads travels through a delay line
(Allen Avionics LC050Z100B tapped lumped constant delay-line chips, 50 nsec/chip, 1
chip/inch or 1.97 |Xsec/m: the FW has 26 chips and the RW has 28 chips). The time
differences between the signals at each end of the delay lines determines the positions. The
upstream position-sensitive wire is placed at the focal plane of the spectrograph. The
downstream wire is 14 cm further downstream (10 cm displacement between the parallel
wires), and is used mainly for the trajectory corrections described in §3.3.
The position of the focal plane is determined by the kinematic shift parameter, k , for the
reaction channel of interest The position of the focal plane has a simple linear relationship
34
with k, and has been calibrated using an alpha source (£=0) and several different reactions. It
is defined by [Enge 1979]:
z = z „ - X \ k \
1 d p a (ln (p )) P-2)p d d d 0
In (3.2), z is the position of the focal plane; A is a constant; p (6 ) is the momentum of the
particle detected in the focal plane detector; and 6 is the lab angle in radians. In general, k
varies for different states over the sensitive region of the detector. The detector can be moved
back and forth with a motorized assembly, where a variable resistor reads the position. The
position is set so that the most interesting states are in the best focus. The procedure for
correcting for the higher order aberrations in the focusing is described in §4.2.
3.5. Yale Lamp Shade Array
The Yale Lamp Shade Array (YLSA) was designed and constructed for this experiment It
contains five silicon strip detectors of the type used in the Louvain-Edinburgh Detector Array
(see Figure 7) [Davinson, Bradfield-Smith et al. 2000]. (From this point forward, “segment”
will refer to one of the five detectors.) Each detector has 16 annular strips (5mm wide from
r=40 mm to r=130 mm in a typical flat array) corresponding to a total of 80 energy signals and
80 associated time signals for the full array. The detectors are approximately 500 ]im thick,
making them capable of stopping ~8 MeV protons or ~32 MeV alphas. They are
manufactured by Micron Semiconductor Ltd. (UK). Members of our research group have
some prior experience with this type of detector, having measured resonance strengths in
18Ne[Bardayan 1999; Bardayan, Blackmon et al. 1999; Blackmon, Bardayan et al. 2001],
19Ne[Bardayan, Blackmon et al. 2000] and ^MgfBradfield-Smith 1999; Bradfield-Smith,
Davinson et al. 1999; Bradfield-Smith, Davinson et al. 1999] in radioactive ion beam
experiments at the Holifield Radioactive Ion Beam Facility (HRIBF) and Louvain La Neuve
(LLN).
35
The detectors are arranged in a “lampshade” configuration, Le., they are folded forward so as
to form the sides of a pyramid with a pentagonal base (see Figure 8). This particular
configuration has the advantage of a relatively large solid angle in a confined vertical and
horizontal space, and was chosen because of the limited space in the scattering chamber for
the spectrograph. A disadvantage is that the annular strips are no longer simple intervals in 0,
which makes the data analysis more difficult In practice, it was possible to understand the
efficiency of the array using Monte Carlo simulations (see Figure 10 and §4.1 for details). In
order to improve the statistics in the final data analysis, the data was grouped into four bins
containing four strips each. This had the further advantage of reducing the significance of the
overlapping angles. Monte Carlo simulations determined for each resonance the mean 0CM for
each of these bins.
Figure 7: A segment of YLSA. The length dimensions are given in cm. There are actually 1-mm gaps between strips.
Bias is applied to the detectors on the large “backplane” contact, which is the side opposite the
p-n junctions for the individual strips. It was found during initial experiments with YLSA in a
“high” beam current environment (several pnA or more) that 8-electrons from the target
36
charge up the non-conductive surfaces of YLSA and impair detector function [Bradfield-
Smith, Lewis et al. 2002]. The strip contacts are normally on the side facing the target. In our
experiment, however, in order to protect them from 5-electrons, we chose to face them away
from the target, so that the particles enter the back of the detector (i.e., the side opposite the p-
n junctions). This is not a significant problem, but it does mean that we had to fully deplete the
detectors with a bias voltage in order to get the full energy signals. It was determined with an
alpha source that 70 volts was an optimal bias. Changing the bias from 70 V down to 60 V
results in a slight worsening of resolutions and very litde change in peak position. We biased
the detectors with 100 V during the experiment, in order that any unexpected rise in leakage
current would still allow the detectors to remain fully depleted.
Figure 8: CAD design o f YLSA frame with detectors mounted.
Energy calibration o f YLSA is achieved using a 228Th CC-source which has several well-known
peaks. Software was developed to automatically find these peaks in as many of the 80 channels
as possible, which are then each fitted by a Gaussian shape. The positions of the strongest
peaks are linearly fitted to expected energy deposited in YLSA, as calculated assuming a
37
nominal aluminum dead layer thickness of 0.2 [im [Davinson, Bradfield-Smith et al. 2000]. For
a time calibration, a pulser is used to create several self-triggered time peaks in the time spectra.
Those peaks are fit in a similar fashion to yield the time calibration.
Figure 9: Protective measures in the scattering chamber to prevent charging of exposed insulator on YLSA by electrons. The tuning collimators rotated out o f the way after tuning the beam. +500 volts were applied to the target ladder, shield and mount whenever beam was present. Note the orientation of the segments in the blowup.
38
Lab e (°)
Figure 10: A Monte Carlo simulation of the solid angle of YLSA. 10s events were generated isotropically in the lab frame, and events which intersected YLSA were counted as hits. There is significant overlap in 0 between adjacent strips.
YLSA is placed at backward angles in the scattering chamber with its active region extending
from 0=131° to 9=166°, subtending approximately 14% of lab solid angle (see Figure 10).
This is done in order to avoid the large forward fluxes of protons, deuterons and alphas from
stripping and pickup reactions. If the array were placed in the forward hemisphere ($ ^ A= 14°-
49°) without any protective absorbers, the flux of scattered beam and reaction products would
soon damage the detector so that it couldn’t be used. On 12C, cross sections to (3He,p), (3Fle,d)
and (3He,a) were estimated from reported measurements near Ebeam=25 MeV at the forward
angles to be 10 mb/sr, 900 mb/sr and 500 mb/sr respectively [Flahn 1967; Matsuda,
Nakanishi et al. 1968]'. On Ca, the same quantities were estimated as 0.3, 300 and 200 mb/sr
1 [Hahn 1967] Ebeam-24.8 MeV on 12C; (Matsuda, Nakanishi et al. 1968] Ebom-25.3 MeV on 12C
39
[Schlegel, Schmitt et al. 1970; Dzhamalov and Dolinskii 1972; Apell, Gemeinhardt et al.
1975]1’2 . These cross-sections were estimated here to be 50 times measured cross-sections to
the ground states, based on the pattern in those references when cross-sections to excited
states were measured, too . At the experimental beam current of 20 pnA, this would add up to
a rate of 3400 particles cm'2 -sec'1 in YLSA. It takes the implantation of approximately 2.0-109
heavy ions cm'2 to kill a silicon junction detector [Deamaley 1963]. However, serious
deterioration (resulting in worse energy resolution) shows up at one-tenth that dosage. The
above rate would reach this point within a day. With protective absorbers that eliminate the
lower energy (majority) fluxes, the energy loss and straggling of the interesting protons and
alphas is so severe as to prohibit measuring any useful branching ratios. The fitted DWBA
angular distributions typically dropped by 2-4 orders of magnitude at the backward angles,
giving a much longer useful life for the detectors. For this reason, we performed the
experiment with the array at backward angles. We were also careful to expose these detectors
to beam conditions only when there was a good chance of getting useful data.
3.6. Protection of YLSA
3.6.1. Beam Scatter
A schematic of the scattering chamber configuration may be seen in Figure 9. The beam was
ultimately incident on a target at the center of the chamber consisting of 80 |lg/cm2 CaF2 on
10 |ig/cm2 carbon. The beam was tuned a diameter of 2 mm. There is a protective sheet of
aluminum placed between the detectors and the spectrograph entrance, to protect YLSA from
backscattering from the downstream collimator during tuning.
There is also a tuning collimator at the chamber entrance upstream of the array. We opted for
this in order to avoid the detectors having a view of a target position collimator during tuning.
Scattering off of a metal collimator at the target position would be fatal to the functioning of
the detectors. In order to avoid damage to YLSA due to beam scattered from the upstream
' [Schlegel, Schmitt et al. 1970] £ ^ = 1 8 MeV on 40.42.«.44,«ca
2 [Dzhamalov and Dolinskii 1972] Ebeam=24.3 MeV on 40Ca
J [Apell, Gemeinhardt et al. 1975] £ ^ = 2 5 MeV on ^" 'C a
40
collimator, the mount completely shields YLSA from such scattered beam. We have tested
that using the two collimators (up- and downstream) still gives an acceptable tune (~2 mm
beam spot) at the target position. An arm fixed to a gear holds the collimators, so that they
may be rotated out of the way after tuning.
3.6.2. Electrons
Commissioning runs were carried out using a 160 beam. In our initial tests, the leakage current
in YLSA was found to grow whenever beam was on target As described elsewhere, the main
cause of this is a high flux of electrons incident on YLSA [Bradfield-Smith, Lewis et al. 2001].
One way to reduce the flux is to use a fully stripped beam. Initial commissioning runs with
YLSA revealed charging up of the exposed insulating surfaces on the detectors by electrons
ejected from the target. To eliminate this, rare-earth magnets were mounted to the aluminum
sheet to either side of the target The sides of the detectors with the aluminum annular strip
contacts face away from the target This protects the exposed SiOz between the strip contacts.
The target ladder and detector mount are biased using a power supply at +500 volts.
(Subsequent experiments now use +600 V from dry cells, which reduces electronic noise.)
Biasing the target ladder suppresses the ejection of many of the electrons, and lowers the
energy of others enough so that they will spiral around the magnetic field lines between the
rare-earth magnets. Bias is also applied to the detector mount and attached foil sheets placed
parallel to and behind the segments. This provides in a strong attractive electric field away
from the vulnerable surfaces. The electrons are harmlessly pulled into the foil, and there is no
more charging of the detectors. The combination of all these steps has enabled us to operate
with no significant increase in detector leakage current during a several-day run.
3.7. Data Acquisition
3.7.1. Signal Processing
All signals from the focal plane detector are handled via standard electronics, with all tasks
after any pre-amplification stage handled by NIM-standard modules in the data collection area
(see Figure 11). YLSA’s signals are handled by more integrated circuitry. The signals from
strips in YLSA are brought through 34-pin vacuum-feedthrough connectors on the top of the
41
scattering chamber. Motherboards1 on the outside carry the signals through as many as 34
separate charge pre-amplifiers2 (32 strips, and 2 backplane “sum” signals). The biases for
YLSA are also supplied through these motherboards to the backplanes of up to 2 segments
each.
The pre-amplified signals are carried through approximately 15 meters of shielded “twist-n-
flat” 34-line ribbon cable to the amplifiers in the data collection area. Shaping amplifiers in a
special crate3 process the signals. The amplifiers are organized with 8 channels (on replaceable
cards) to a module. Each amplifier channel includes a discriminator as well, which provides the
time signal for each separate strip. The 80 time signals are ECL-logic and are carried through
300-nsec ECL delay lines housed in NIM modules4. After the delay circuits, the time signals
are brought into a TDC (time-to-digital converter) that takes its START from the fast OR of
the scintillator’s photomultiplier tubes. The 80 energy signals and the signals from the focal
plane detector are brought into the ADC’s, which are gated by the same scintillator event The
ADC’s and TDC’s are VME-6U modules5 with 32 channels each.
In addition, a scaler module in the VME crate5 is fed logic signals to count the trigger rates,
beam current, and a clock.
3.7.2. Conversion and Data Processing
A computer running the Vx-Works operating system takes the data from the ADC and TDC
buffers and stores it in an 8-kbyte memory buffer. When this buffer fills, the computer ships
the events over a private Ethernet to a workstation, then clears the buffer for more data. The
workstation runs a Java-based acquisition and display package called Jam [Swartz, Visser et al.
2001], which stores the event data to a hard drive. At the same time, it uses a sort routine to
sort the data into 1-D and 2-D spectra. These spectra may contain 1-D and 2-D gates drawn
by the experimenter. These gates are used for the particle identification in the focal plane
1 Designed and produced by RJS Corporation in Knoxville, TN (Model RJS-13040Y).
2 Purchased from the Instrumentation Division at Brookhaven National Laboratory in Upton, NY.
3 Crate and shaping amplifiers (RAL109) purchased from Rutherford Appleton Laboratory in Oxfordshire, UK.
4 The ECL delay modules were designed and built in the WNSL electronics shop by Tom Baker and Richard Wagner.
5 ADC’s (VX785), TDC’s (VX775), and scaler module (V260) produced by CAEN in Viareggio, Italy.
42
detector, and to select out the true proton decay and alpha decay events by tuning and
kinematic constraints. Additionally, Jam interacts with the VME computer to display
information about the scalers. At the end of each run, a file is saved which stores all the
produced spectra, gates, and scaler values.
3.8. The four data sets
The data that were used for the final analysis presented here were divided into four parts, each
of which was analyzed separately. For the first data, we used ADC thresholds on the YLSA
channels equivalent to 200 keV in deposited energy. The second set of data was collected
immediately after the first, but with the thresholds lowered to 100 keV. The third set of data
was taken with the same 100 keV thresholds as the second set, but is considered separately
because the B-field in the spectrograph was (inescapably) slighdy different For the fourth set
of data, we removed the ADC which processed the focal plane detector signals from the
control bus that the other ADC’s and TDC’s share. This should have allowed us to shorten
the conversion window in the YLSA ADC’s to 2 (Is from the 8 |Is which the focal plane
signals require. In this way, we hoped to reduce the pileup in the YLSA when two particles
interact with the detector within the conversion window. It was discovered, when analyzing
the data, that our ADC’s and TDC’s randomly dropped a fixed percentage of coincidence
events (see Table 1). Section 4.4.2 explains how these numbers were determined.
Table 5: Summary o f experimental runs. See §3.8 for an explanationof the 4th column.
Data Set Total Hours Iavg (pnA) Coincidence
Efficiency (%)
1st June 15.35 19.7 86.712.7
2nd June 27.76 19.2 88.812.6
1st July 26.33 19.4 88.812.6
2nd July 51.32 20.0 58.111.8
43
PMT Anode L PMT Anode RCathodeA node
FW-L FW-R RW-L RW-R FWPH RWPH
PMT Dynode L PMT Dynode R
Delay
Delay
p > ~
? > ■TimingAmp
250 nsec delayYLSA
GateGen.
"In
•Veto
GateGen.
Figure 11: Schematic view o f electronics used in the experiment. Amplifiers are shown as triangles, with pre-amps designated by ‘Pre’, shaping amps designated by a Gaussian pulse and sum amps designated by “+ ’ or ‘Sum’.
44
Chapter 4
Chapter 4: Results and Analysis
4.1. YLSA Efficiency
The efficiency of the detector array varies for different states and decay channels in 19Ne, due
to differing kinematics for the decays. These efficiencies were determined with a Monte Carlo
code that simulates the kinematics of 19Ne production and decay. See Appendix A for details
on the algorithm used.
4.2. Particle ID
The (3He,t) reaction is particularly clean in terms of identifying particles with the focal plane
detector. With beam energy of 25 MeV, the reaction Q-values are such that only deuterons and
tritons are incident on the focal plane of the spectrometer. All other particles are much less
magnetically rigid and get bent off the focal plane.
Tritons are selected using two-dimensional plots of the position (momentum, or Bp), energy
(E) detected in the scintillator, and energy loss (AE) in the gas (see Figure 12). Two-
dimensional plots using an additional AE from the anode section of the detector provided
useful secondary information. A trajectory condition is also applied, by using a gate on front-
wire position (FW) vs. rear-wire position (RW) and gates front-Y (FY) vs. rear-Y and FY vs.
FW.
Corrections are applied to the FW spectrum using RW information. After defining
THETA=RW-FW-offset, we plot FW vs. THETA, and fit a 4th order polynomial
FW=P4(THETA) for various strong triton peaks across the spectrum. This relationship as a
function of position is then interpolated to re-sort the data and give a corrected FW spectrum
(see Figure 13).
45
AE
(Cat
hode
) En
er9V
(Sci
ntill
ator
)
Bp (FW Position)
Figure 12: Particle identification in the focal plane detector showing triton gates. The top panel shows energy detected in the scintillator vs. position along the focal plane, gated on AE vs. Bp. The vertical axis o f the bottom panel shows energy loss in the isobutane gas. The AE vs. Bp plot has been gated on E vs. Bp. The horizontal bands in the bottom pane are due to hardware gates in the trigger electronics.
46
(0*<b
sz
Position (FW )
Figure 13: “Theta” parameter vs. focal plane position before and after correcting for aberrations in the spectrograph focusing. “Theta” is essentially a difference between position measured at the rear wire and position measured at the front wire.
4.3. Kinematic Selection of Decays
Decay particles are identified by the energy deposited in Y L S A for a coincidence event The resonance’s excess energy above the particle threshold is released as kinetic energy shared
between the decay particle and the residual nucleus. In the case of wN e — ^O^+CX, the lsO
gets about 4/19 of the total energy, and the CL gets the other 15/19. Likewise for
47
Corrected
,9Ne —»,SF +p, the proton gets approximately 18/19 of the total energy and the 18F the
remaining 1/19.
We detect the proton or (X, and not the heavy residual nucleus, which is moving so slowly that
it stops in the target Also, when the decay particle is traveling backwards into YLSA, the
residual nucleus travels forward. In YLSA, the peaks due to particular decays are spread out
due to the finite angular acceptance (kinematic broadening) and energy straggling in the target
and detector dead layer. The lower panel of Figure 14 shows how protons to the 18F ground
state and alphas to the lsO ground state were selected on the basis of their kinematics. This
plot shows the sum of data from all the segments and strips o f YLSA; the energy deposited in
YLSA is further spread out due to a slight angular dependence in the kinetic energy in the
decay particles. In this plot, the horizontal axis is equivalent to the momentum of the triton,
and the vertical axis is the energy of the decay particles, so one can see that the separation into
diagonal bands expresses the energy conservation condition
Ttn,on + Tdecay ~ Tbeam + Q ~ ( T N e + T final )(4.1)
where Q is the Q-value for the reaction (-Ex [19Ne]-3.2571 MeV), Tbeam is the beam energy,
and 7)ve is the lab energy of the 19Ne and T finai is the lab energy of the final daughter nucleus.
The left hand side of (4.1) is roughly constant, since 7//e and Tfinai is are small relative to the
other terms.
Once a gate has been set on the decay channel, a gate may be drawn on the summed time
spectrum (see Figure 15). Gates are also drawn outside the time peak in order to analyze
background “accidental” coincidences.
48
20 * M
eV
Dep
osite
d C
ount
s
10
10“
10
10
10120
100
80
60
40
20
0
Figure 14: Kinematic selection of the decay channels out o f 19Ne. The 2D plot is o f events gated on the time peak in Figure 15. Candidate events for CX-decay to the ground state o f lsO are outlined in blue in the lower panel and plotted in blue in the upper panel, while candidates for p-decay to the ground state o f 18F are indicated similarly with green. The top pane shows all tritons in black, giving a spectrum of excited states in l9Ne, starting with the strongest peak on the right being the ground state and increasing in excitation with decreasing channel number. One 13N state, Le. I3C(3He,t)13N '(p)12C, is the strongest contaminant and is circled in black in the lower panel. (Data from 1st July run.)
1500Position (Channels)
49
50 100 150 200Time (=2.5 nsec/ch)
Figure 15: Time spectrum, summed from all channels in YLSA. The black spectrum is plotted for all 19Ne events, and the blue spectrum is plotted for all kinematically selected candidate events for 18Fgs.(Data from 1st July run.)
4.4. Extraction of Branching Ratios
4.4.1. Angular Distribution Theoty
When the tritons which tag a particular state in 19Ne are selected at roughly 0=0°, the analytic
forms of the possible angular distributions are greatly simplified, since only the m=0 substate
of the orbital angular momentum may be populated at 0°. The simplest example to consider
here for deriving possible angular distributions is the breakup of a 19Ne excited state into
150 &s +0C, because the OC and I50 have spins of 0+ and 1/2* respectively . It takes the form
[Satchler 1980]:50
where P m = P -m is the population of the M-substate of the resonance. The fits are restricted to
M < 3 / 2 because the 3He beam, 19F target and 3H ejectile all are spin 1/2 and the selection of
m=0 for the orbital angular momentum mentioned above. The angular distributions of the
decay (X’s are fit with (4.2) using r a/r (the overall scale) and the P m as free parameters, with /
(the orbital angular momentum of the decay channel) set to the minimum possible value for
coupling to the Jn of the 19Ne state.
In the case of 19N e —> I8F s + p , the V2+ spin of the proton adds some more complexity.
Now there are two possibilities for the channel spin, s, the total spin of the two decay
products. Since the ,8F has a spin of V , the two possibilities for s are l /2 + and 3 /2+. The
angular distribution is calculated as follows [Ferguson 1965; Noe, Balamuth et al. 1974]:
f f r W = T , A P > ( c ° s 6 ) ( « )*=0,2-..
where / \ i s the fc’th order Legendre polynomial, and the coefficients are given by:
J J k \ ( l I k M l J s '
0 0 0 \ J I k \(4.4)
and
f { J k ls M ) = [ - ) 2J+5~M ( 2 J +1) [ 2 k +1) (21 + 1) (4.5)
The last factor of equation (4.4) is the Wigner 6J symbol for coupling 3 angular momenta,
while the preceding two factors are Wigner 3J symbols for coupling 2 angular momenta. J is
the total spin of the decaying 19Ne state, p u is the probability of populating the M-substate of
the resonance, and p s is the probability of populating a particular channel spin, s. The
minimum value of angular momentum, /, which couples the outgoing particles to the 19Ne
51
state is chosen. Because states are isolated and assumed to have definite parity, only even
values of k have A k ^ 0 , giving W (6) symmetric about 90° [Noe, Balamuth et al. 1974],
When the spin and parity of the 19Ne state are known, expressions (4.3) through (4.5) are fit to
the angular distribution with p m and p s as free parameters. When nothing is known about the
spin or parity of the state, a fit to expression (4.3) is performed, using the A t as free
parameters.
4.4.2. Angular distribution fitting
Events that pass through the kinematic and time gates described in §4.3 were used to produce
focal plane spectra corresponding to events where a decay was detected in one of four angular
bins in YLSA (strips 0-3, 4-7, 8-11, and 12-15). Background spectra were also produced for
each of the angular bins, using the gates drawn on the flat background outside the time peak
(see Figure 15). Sections of full triton spectrum were then fit using gf3 l . All peak parameters
other than intensity and a constant background term were fixed to the results from the full
spectrum, and then fits were performed on the coincidence and background spectra.
Using the Monte Carlo simulations described earlier to define the mean 6 c m and efficiency of
each of the four bins, a spreadsheet was used to calculate dn/dQ. as a function of bin,
normalized such that when fit with an angular distribution, integration over 9 direcdy yields
the branching ratio. This was done for YLSA simultaneously for each of four datasets. The
Ex=5.351 MeV state in 19Ne is known to be a x/z state which decays =100% by an a to the
ground state of lsO. The decay for this state is predicted to be isotropic using the machinery of
equation (4.2):
Substituting in the Clebsch-Gordan coefficients and the Tim's gives:
1 The program gf3 is part o f RadWare, a software package developed by David Radford for interactive graphical analysis of y- ray coincidence data. See h ttp :// radware.phv.oml.gov/ for more information.
52
= — (cos2 # + s in 2 0 + s in 2 0 + c o s 2 0 ) = — 8 / r ' ’ 4k
(4.7)
This distribution and all other distributions derived from equation (4.2) sum to unity when
integrated over all angles. The only free parameter in equation (4.7) is the overall scale (not
shown), which is equal to Fo/r with our choice of units. For Ex= 5.351 MeV, r a/r is extracted
this way; the resulting r a/r is less than one and is a function of the hardware triggering used
for the experiment. Because of this, all angular distributions were renormalized using the same
factor that would be necessary to correct our measured Ex=5.351 MeV branching ratio to
100%. The general agreement of the branching ratios extracted using this procedure with
previously measured values is considered evidence that this procedure is valid and insensitive
to excitation energy in 19Ne. The loss of efficiency versus trigger type is listed in Table 5 with
details given in §3.8.
For each resonance and decay channel, the steps taken in the previous paragraph result in 4
data points, each with a 0CM and a d n /d Q . with associated error bar. M athem atica 4.1 ’
worksheets were used to fit to equation (4.2) or (4.3), as appropriate. In the case of Ex=7.500
MeV, nothing was known about the spin or parity of the state, so equation (4.3) was simply
used with the An as the free parameters.
In the following two sections where the analysis of specific resonances is detailed, any
discussion comparing the resulting branching ratios to other branching ratio measurements is
delayed until the next chapter.
4.5. Branching ratios above the a-threshold (Ex=3.529 MeV)
A detailed example of the fitting with Mathematica is given in Appendix B.
4.5.1. Ex=4.379 MeV
The lowest state for which it was possible to unambiguously extract a branching ratio from my
data was the 1020-keV a-resonance (Ex[19Ne]=4.549 MeV, see the next subsection and Figure
1 See http:/ / www.wolfram.com/products/mathematica/ for details.
53
2). For the resonance below this, £^=668 keV (Ex=4.379 MeV), the signal was very close to
the energy threshold for the true coincidence events. Applying isotropic assumption to the
data would yield 0.010(12) for the branching ratio. In addition, the efficiency was highly
uncertain, due to energy threshold effects in YLSA. Since the energy calibration of YLSA was
performed using 5-9 MeV a ’s, the extrapolation to low energies was uncertain. The actual
energy threshold is somewhere between 250 and 500 keV in YLSA. This range is exactly what
Monte Carlo simulations predict for deposited CC-energies from decay of the 4.379 MeV state.
Therefore, a further 50%(±25%) reduction in efficiency is estimated, resulting in an estimate
for Ta/r of 0.018(25). A 68.27% confidence interval is inferred to be 0.003<ra/r<0.043
[Feldman and Cousins 1998]. This is consistent with past measurements.
4.5.2. Ex=4.549 MeV
The state at Ex=4.549 MeV (Em=1020 keV) has an angular distribution in (3He,a)
experiments that indicates 1=1 in a DWBA analysis. Because 20Ne and “He have 0+ ground
states and 3He has a l /2 + ground state, the scattering data mean J* must be 1/2' or 3/2\
Measurement of the isospin-mirror capture reaction, 15N(0t,y), show the two corresponding
resonances at EX[19F]=4.550 and 4.556 MeV to have ] n= 5 / 2 + and 3/2' respectively [Magnus,
Smith et al. 1987]. Since both the 4.549 MeV state in ,9Ne and the 4.556 MeV state in 19F y-
decay mainly to the 1/2 member of the ground state triplet, they have been assumed to be
isospin mirror states [Davidson and Roush 1973]. Similarly, the same study showed that both
the 4.600 MeV state in 19Ne and the 4.681 MeV state in 19F decay primarily to the 5/2+
member of the ground state triplet.
Because of the y-branching information, a J7t= 3/2’ (/=2) assumption is adopted here, and fits
were performed to the angular distribution of Ct-decays seen for the Ex=4.549 MeV state. The
minimum ^2/v=0.43 (V=2), and the corresponding fit curve is shown in Figure 20. The
contours enclosing the 68.27% and 90% confidence regions (A%2=2.30 and 4.61, respectively)
are shown in Figure 16. From Figure 16, r a/T= 0.04 'JJ . The other parameter, p i / 2 , is totally
unconstrained by the fit at a 68.27% confidence leveL The error bars are asymmetric due to
the strong anti-correlation between F«/r and p i / 2 , which is itself due to a combination of low
54
statistics and the lack of a measurement of the flux near 0=90° to constrain the shape of the fit
curve (see Figure 20).
r itA a '1
Figure 16: The E,=4.549 MeV confidence regions for the fit parameters. The best fit coordinates are marked by the cross. The mner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively. Most o f the area inside the contours lies inside the physical region for p i/2
55
r«/r
Figure 17: The E*=4.600 MeV confidence regions for the fit parameters. The best fit coordinates are marked by the cross. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively.
4.5.3. Ex = 4.600 MeV and 4.635 MeV
The J3l=5/2+ assignment for the 1071 keV resonance (Ex=4.600 MeV) is corroborated by its
measured y-decays; the primary y-decays are to states known to haveJn= 5/2+ (90±5% of Ys)
and 3/2+ (10±5% of Ys) [Davidson and Roush 1973]. The y 2 surface for the fit is shown in
Figure 17. The best fit was obtained for ra/r=0.186^0^ and p i / 2 = 0 . 1 with %2/v=1.9.
The best-fit curve is shown in Figure 20. The value determined here is significandy lower than
previously measured values (see Table 4). This, along with the fact that this peak is not
resolved from the 4.635 MeV state, means that the measurement should possibly not be
trusted for this state.
The 4.635 MeV state (Eres=1106 keV) has been well established as a J,t=13/2+ (/=7) state,
which rules it out as a contributor to the 150((X,y) reaction rate due to its high centrifugal
harrier. Our data shows To/r=0.036(20) if one assumes isotropy; this is much higher than one56
would expect due to the suppression from the centrifugal barrier. This state is not resolved
completely from the 4.600 MeV state, so it cannot be ruled out that contamination resulted in
the decay counts. In fact, both resonances’ branching ratios differ from expectations in the
direction one would expect if they are contaminating each other due to being unresolved.
Therefore, no claims are made here about measured branching ratios of the Ex=4.600 MeV or
Ex=4.635 MeV states.
4.5.4. Ex=4.712 MeV
Based on excitation energy alone, the Ex=4.712 MeV state in t9Ne (Ercs=1183 keV) is
tentatively identified with the f f —5 / 2 ' Ex—4.682 MeV state in 19F [Davidson and Roush 1973].
The X2 surface for the fit is shown in Figure 18. The best fit was obtained for ra/r=0.68^ J2
and p i / 2 — 0.33^ with X2/v=0.37. The best-fit curve is shown in Figure 20.
ra/r
Figure 18: The Ex=4.712 MeV confidence regions for the fit parameters. The best-fit coordinates are marked by the cross. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively.
57
4.5.5. Ex=5.092 MeV
The Jn= 5/2+ character of the Ex=5.092 MeV state in 19Ne (Ercs=1563 keV) is well established
[Fortune, Bishop et aL 1979; Tilley, Weller et al. 1995]. The y j surface for the fit is shown in
Figure 19. The best fit was obtained for r a/r=0.76^J j and p m ~ 0 .3 (T ^ . The final fit
yielded a %2/v statistic of 1.32 (see Figure 20).
r / r(X
Figure 19: The £*=5.092 MeV confidence regions for the fit parameters. The best-fit coordinates are marked by the cross. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively.
58
0c* O e«<#)
Figure 20: Angular distributions o f decay Ct's below the proton threshold. The blue curves are fits to equation (4.2).
4.6. Branching ratios above the proton threshold (Ex=6.411 MeV)
The angular distributions discussed in this section are shown in Figure 26 along with their
best-fit curves. The lowest state for which proton branching ratios could be measured was the
Ex=7.070 MeV state, due to kinematics and the energy thresholds in YLSA. The two states
lower in excitation therefore were only analyzed for r a/r. All quoted resonance energies in this
section refer to 18F+p resonances.
4.6.1. Ex=6.742 MeV
The 6.742 MeV state (£^=330 keV) is thought to have JIt= 1/2' or 3/2' due to the /= 1
character of its 2ONe(3He,0c) angular distribution [Garrett, Middleton et aL 1970]. A comparison
of fits to the (X-decay of this state using these assignments is shown in Figure 26. The j"=3/2‘
fit clearly does the better job, having a X2/ v statistic of 1.96 versus 13.3 for the JIt= l /2“ fit
59
Besides the total area, the only free parameter is p i / 2 , with a best fit value of 0.58^ (see
Figure 21a). Since this lies outside the allowable range of 0-0.5, the best fit is found for
Pl /2—0-5, and a new confidence region is found (see Figure 21b). The final result fit was
ra/r = 0.888 030 withp i / 2 > 0.44 at a 68.27% confidence leveL
07 08 09 1 0.875 0.9 0 9Sra/r ra/rFigure 21: The Ej=6.742 MeV confidence regions for the fit parameters. Circles mark the best-fit coordinates. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively, (a) The left panel shows the initial fit, which resulted in a non-physical value for p i/2. The physically allowed values are below and to the left o f the lines, (b) The right panel shows the confidence regions obtained by not allowing p i/2>0.5.
Comparison of the isospin-mirror reactions 20Ne(d,t) and 20Ne(d,3He), as well as 160 (6Li,3He)
and 16O^T.i,t) support identifying this state with the Jn=3/2 state at Ex=6.787 MeV in 19F
[Utku, Ross et al. 1998]. Studies of 12C('°B,t) and 12C(10B,3He) at WNSL support this
assignment as well [Lewis, Caggiano et al. 2002], The fit to data shown here, when viewed in
light of the other studies, may be viewed as a confirmation of the J7t=3/2' character of the
Ex=6.742 MeV state. A direct measurement of the 18F(p,<X) resonance strength for this state
was performed recently [Bardayan, Batchelder et aL 2002], which assumed }"=3/2' and ra=2.7 keV (from the known r o for 19F and the assumption of identical single-particle reduced
60
widths). They reported rp=2.22(69) eV, which we may interpret as meaning r a/r=0.9992(5)
(statistical error only), which is marginally consistent with the present measurement and
completely consistent with the earlier measurement [Utku, Ross et al. 1998].
4.6.2. E„=6.861 MeV
The Ex=6.861 MeV state in 19Ne (Ercs=450 keV) is believed from the study mentioned earlier
[Utku, Ross et aL 1998] to be the isospin analog of the Jn=7/2' Ex= 6.927 MeV state in 19F.
Assuming this, a fit to equation (4.2) was made, and the confidence region for the fit
parameters is shown in Figure 22. The best fit value was r a/r= 0.925^ q33 corresponding to
Pi /2 ~ 0-476^q3° . The goodness-of-fit statistic was %2/v=0.38.
r j r
Figure 22: The E,=6.861 MeV confidence regions for the fit parameters. The best fit coordinates are marked by the cross. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respecnvely.
61
4.6.3. Ex=7.070 MeV
The Ex=7.070 MeV state in 19Ne is very well studied, most notably in the high precision direct
measurement of 18F(p,p) and 18F(p,d) at Oak Ridge National Laboratory [Bardayan, Blackmon
et al 2001]. It is clear from this and other direct studies [Coszach, Cogneau et al. 1995; Rehm,
Paul et al. 1996; Bardayan, Blackmon et al. 2000; Graulich, Cherubini et aL 2000] that JIt=3/2+,
making this an 1=0 resonance for 18F(p,x) reactions. Likewise, a-decay of the state has an l —\
angular distribution. For the proton decays with /=0, only a spin channel of 3/2+ and k=0
yield n on -vanishing coefficients in equation (4.3), so the angular distribution formula is
isotropic. A simultaneous fit to the OC-decay and proton decay data was performed assuming
rp+ra=l. This is justified because Ty is estimated using known widths in the isospin mirror
19F to be around 1 eV, orders of magnitude below the width of 39 keV [Utku, Ross et al.
1998]. Fitting both curves simultaneously in this way gives a much better constraint on
possible shapes for the r a/r angular distribution, and lowers overall branching ratio
uncertainties. The fit (see Figure 23) had %2/v=1.85 (v=6) and yielded P//2—0.258+0.025 and
Tp/T = 0.3712 019 (or r a/ r = 0.6292^020 one prefers), in agreement with the Oak Ridge result
ofrp/r=0.39±0.02.
rp/r
Figure 23. The £*=7.070 MeV confidence regions for the fit parameters. The best fit coordinates are marked by the smallest ellipse. The larger contours are the boundaries for the 68.27% and 90% confidence regions.
62
4.6.4. Ex=7.500 MeV
The E1=7.500 MeV state in 19Ne (£^=1089 keV) was the highest excitation for which a
calibrated energy was possible. An isospin mirror for this state has not yet been identified in
19F, nor has any measurement restrained its spin or parity. Looking at the possibilities in 19F,
there are four states in the region with no identified analog: 7.364 MeV (1/2+), 7.540 MeV
(5/2+ or 3/2), 7.56 (7/2+) and 7.59 (5/2+). Due to this uncertainty, the CL and proton decay
angular distribution data were fit to (4.3) with the Au as the free parameters. The results are
listed in Table 6 and the confidence regions for the fit parameters are plotted in Figure 24 and
Figure 25. In section 4.5.3, it was decided that only resolved peaks were trustworthy for
branching ratio determinations. This peak was unresolved in our experiment from a higher
lying state at Ex=7.531 MeV [Utku, Ross et aL 1998], so the result is discarded in Table 7 and
Table 8.
Table 6: Results o f fitting to the decay angular distribution data for E*=7500 keV. The asymmetric error bars on fit parameters were made symmetric by setting them to the larger o f the upper and lower error bars in order to calculate the uncertainty in the branching ratios.
Ao a 2 a 4 X2 (v=l) Branching
Ratio
ra/r 0.012±0.004 0.021 ±0.010 -0 .0 1 7 3 “ ? 0.165 0.18±0.06
ryr 0.057±0.007 0-0653™ -0.01 ±0.04 0.074 0.86±0.12
63
A ) A
Figure 24: The £<=7.500 MeV confidence regions for the parameters of the fit to the proton decay data. Each plot is a projection taken at the best fit value for the missing 3rd parameter. The best fit coordinates are marked by the cross. The inner and outer contours are the boundaries for the 68.27% and 90% confidence levels, respectively.
A A A
Figure 25: The £<=7.500 MeV confidence regions for the parameters o f the fit to the alpha decay data. Each plot is a projection taken at the best fit value for the missing 3rd parameter. The best fit coordinates are marked by the small central ellipses. The inner and outer large contours mark the boundaries for the 68.27% and 90% confidence levels, respectively.
64
drVdfl
/ c
ounts
In
state
[sr ’
) dn
/dO
/ cou
nts
In sta
te (s
r')
Table 7: Summary of branching ratio measurement results. The error bars give the limits o f 68.27% (lo) confidence intervals. Above the proton threshold (E,=6.411 MeV) the quoted resonance energies are proton resonance energies. The 659 keV proton resonance branching ratio comes from a simultaneous fit o f proton and a- decays subject to r a-tTp= r .
E x [M e V ] E J k e V ] r j r4.379 850 0 - 0 1 8 ^
4.549 1020 0 .0 4 ^
4.712 1183 0.68^;25.092 1563 0 .7 6 ^6.742 330 A OOO+0-091 U. OOO ^Q
6.861 450 0.925
7.070 659 0.629^020
•„n
Figure 26: Measured decay angular distributions above the proton threshold. Red curves are fits for a decays, and violet curves are fits for proton decays. All curves are fits assuming a certain spm and panty, except for the curves for E,=7500 keV, which are simple fits to Legendre polynomials.
65
Chapter 5
5.1. Comparison to prior results
5.1.1. The standard approach
The results from the previous chapter are shown along with other reported measurements in
Table 8. The first five columns of Table 9 show weighted averages of the measurements, with
asymmetric error bars treated as described in the introduction to the Review of Particle Physics
[Particle Data Group 2002]. Disagreement of the measurements is defined here as the
following condition:
P(jp >c2)<0.10where
«?
is assumed to be distributed as a £ distribution with N - l degrees of freedom, and the H j± (J j
are the individual measurement values. The implicit assumption is that the reported results are
describing normal distributions with mean H i and standard deviation (Tj. All reported
measurements in Table 8 are in agreement according to relation (5.1).
With the branching ratio measurements, there are definite physical boundaries on the allowed
values they may take. They must lie in the interval between zero and one, inclusive. Any
measured value (or average) that doesn’t meet this condition is highlighted in red in Table 8.
Here, this is arbitrarily defined as <1.645 or — CT <1.645 , Le., no more than
5% of the tail o f a Gaussian of mean H, and variance <72 may reside in the non-physical
region.
Chapter 5: Discussion and Conclusions
66
(5.1)
Table 8: Measured values for branching ratios in ,9Ne. States where only limits have been measured are ignored. The values in the last column are the presently measured values listed in Table 7. Table 2 and Table 4 give more detailed information about the sources of the results in the other columns. Results that have error bars implying greater than 5% probability lies outside the physical region are highlighted in red.
Ej (MeV) Channel Magnus, et al. Laird et al. Rehm et al. Davids et al. Utku et al. Bardayan et al. This Thesis
4.379 a 0.044±0.032 0.016+0.0050.0187
4.549 a 0.07±0.03 0.1640.040.047
4.712 a 0.8240.15 0.8540.040.687
5.092 a 0.9040.09 1.8=0.9 0.8040.10 0.9040.060.767
6.742 a 1.0440.080.8887
6.861 a 0.9640.080.9257
7.070 0.6440.06
7.070 p 0.3740.04 0.3940.020.3717
5.L2. An alternative Bayesian approach
A method is needed for consistendy averaging discrepant data or for interpreting results that
are too close to physical boundaries for the usual interpretation. One method that will be used
here is rooted in Bayesian statistics, and was suggested in Unsolved Problems in Astrophysics [Press
1997]. The results of experiments have already been defined as H ftO i in Equation (5.2),
denoted as H , because they may be considered a set of hypotheses for the true value. They will
also be collectively referred to as the data, D .
The probability of a hypothesized true value, H o, may be written as
P ( H 0 \ D ) = ^ P ( H , p v \ D ) (5.3)
where p is the “community-wide probability of doing a correct observation” [Press 1997], v is
a vector of zeros and ones denoting which of the experiments are right (1) and which are
wrong (0). The notation, P (A \B ), means the probability of A given that B is true, and P (A B )
67
means the probability of A and B being true. Equation (5.3) makes use of the fact that each
complete hypothesis, H ^ p v , is disjoint from the others and that all such hypotheses form an
exhaustive set o f possibilities. The method to be described is called Bayesian because it uses
Bayes’s theorem to convert data and explicit prior assumptions into posterior probability
distributions. Bayes’s theorem may be stated as
To simplify computation, the denominator may be ignored, turning the relationship into a
proportionality and requiring distributions to sum to unity. Using Bayes’s theorem and the rule
for calculating P (A B ), which is P (A B )= P (A )P (B \A ), we may arrive at
P ( / / J D ) - 5 ) / > ( D | / / oP5 )P (/f0)/>(P | / / 0)/> (v |H 0p). (5.5)P,V
For the second term, the prior distribution for H o, we assume it’s equally likely for H o to take
any value between zero and one:
/>(H0) = e ( t f „ ( l - H 0)) (5.6)
where 0 , the Fleaviside unit step function, constrains the branching ratios to physical values.
The quantity p describes the accuracy of the experimenters’ assigned error bars, and it is
assumed that its probability distribution is independent of H o, so P (p \ H o ) is replaced by P (p )-
We assume complete ignorance of p , and so assign P (p )~ U (0 ,l) , the uniform distribution
between zero and 1, or P (p )= H o t values between zero and one, and zero outside that interval.
Using the same arguments, v is also assumed independent of H o, so the last term of (5.5) may
be replaced by P (v \ p ) , which is just
l u r K ' - p ) (57>v , = l v ,= 0
68
for a given particular choice of v . P ( D \ H 0p v ) is the probability of our data given v , which
experiments are right and which are wrong, and also given particular values for p and Ho- It is
modeled as [Press 1997]:
P ( D \ H , p v ) = ( » . ) f l A, ( « . )V = 1 v( = 0 (5.8)
where N(jU , (Y ) is a Gaussian with mean p. and variance (? . S is a “large but finite number
characterizing the standard deviation o f ‘wrong’ measurements (e.g., plausible range in which a
wrong measurement could have survived the refereeing process)” [Press 1997]. S is given a
more precise definition when this method is applied in the next section.
Pulling formulas (5.5) through (5.8) together, we get a calculable expression for the posterior
distribution of H q:
P ( H , \ D ) ~ e ( H , ( 1 - f l , ))/<$>£ n a - r i M * . ).v,=l
(5.9)
It can be simplified for easier calculation by performing the sum over v in advance:
P ( H „ ID ) ~ e(ff, ( ! - « „ ) ) j d p n * ( H , ) + ( \ - p ) P s, (Zf„) (5.10)
Normalization is obtained by demanding an integral over H o to sum to unity. The method also
allows for calculating the probability that a particular measurement is good, given the weight of
evidence of the other measurements. The procedure is similar. Start with the relation:
P ( v \ D ) = ^ P { H t p v \ D ) (5.11)u 0 ,p
which is the probability of a particular v , given the data. The same arguments that led to the
proportionality relation (5.10) give the following:
69
11 ( Y ^P ( y \ D ) ~ \ \ d H J p \ [ p P a ( H , ) r K 1- # ) ^ . ^ . ) (5'12>
oo Vv,=1 J \ v'=° j
where the step function in (5.6) is used to define the limits of integration. The probability of a
particular measurement, i , being good is obtained by calculating and summing (5.12) for all v
with /—l. Dividing this by the sum over all v will provide the normalization. This last sum
may be simplified by replacing the integrand in (5.12) with the integrand in (5.10), which turns
it into the expression for that sum. By calculating (5.12) for the case of v =0, none of the
experiments being correct, we obtain (after n ormalization) a confidence level for the average.
That confidence level is the probability that none of the experiments is correct. Ideally this
would be a small number below 5%.
Finally, one may obtain the posterior distribution for p very easily from the expression:
P ( p \ D ) = J , P ( H ap v \ D ) o . jd H , r K , ( « . ) + ( l - p ) n , ( H . ) C m )H0,v o V ' /
easily derived by the same procedure that yielded (5.10) and (5.12).
5.13. Application of the Bayesian approach
Before applying the approach given in the previous section, a choice for S in equation (5.8) is
needed. The method is fairly robust, giving similar results for any reasonable choice. It was
decided that “plausibly surviving the refereeing process” would be defined as follows: S is
assigned as the maximum reported error bar in the data or as twice the standard deviation of
the H i, whichever was greater. The standard deviation was calculated without considering the
error bars, i.e., it was just the scatter in the values. For the Ex=5.092 MeV state, the
d(l8Ne,19Ne)p measurement of Ba= 1.8(9) [Laird, Cherubini et aL 2002] was left out of this
analysis, because of its much lower precision relative to the other measurements.
The best estimate, the mean of the posterior distribution, is easily found by integration. For
quotable “1 -sigma” error bars, the points on the posterior distributions for H o were found that
gave tail probabilities of 0.1587 (Le., an interval containing the same 68.26% probability as
70
Proba
bility
Dens
itybetween ±10 on a Gaussian). In the case of Ex—6.742 MeV, (l — j j " <70 " < 1.645, so a
68.26% confidence lower limit was suggested instead.
Branching Ratio
Figure 27: Bayesian posterior probability distributions calculated from the data in Table 8 using formula (5 10). The distributions are normalized so that each sums to unity. If one accepts a Bayesian interpretation as valid, Le., that measurement results make a statement about the likelihood of the actual value of a measured quantity, then the results o f the procedure described in these sections give these probability distributions for the values of the branching ratios.
71
Table 9: Results o f averaging the data in Table 8.The 3rd column gives the result of the standard least-squares method of averaging, with the X2 statistic in the next column. The tail probability for the j } distribution is given in the 5th column. The value inferred using the Bayesian method described in this chapter is also given along with the probability that all the given measurements have underestimated error bars.
E* (MeV) Channel Average X*/V P-value Bayes Avg. Bayes P
4.379 a 0.01710.004 0.38 0.68 0.03410.020 0.30
4.549 a 0 096+o °32u.U7U_0O26 1.95 0.14 0.1010.04 0.19
4.712 a 0.8310.05 0.91 0.40 O 00 o Lt>
© © 0.19
5.092 a 0.8510.04 0.96 0.24 0.8410.05 0.13
6.742 a ft Q7+0.08 'J y '-0.04 1.68 0.40 >0.89 0.27
6.861 a 0 930+o°31V/.7 ju_oo32 0.16 0.92 0.9310.04 0.27
7.070 P 0.37910.013 0.26 0.77 0.37810.016 0.18
5.1.4. Discussion
Most of these resulting averages seem very reasonable. In general, it may be observed that the
Bayesian averages are not very different from the standard averages. They are more
conservative because the skepticism about the measurement error bars introduced in the
averaging procedure. All subsequent discussion will refer to the results from the standard
method o f averaging.
The error bars on the present measurement represent a slight improvement over the
previously reported 19F(3He,t) coincidence measurements [Magnus, Smith et al. 1990; Utku,
Ross et al. 1998]. Recendy reported measurements through different reaction channels have
already reported better precision, though [Bardayan, Blackmon et al. 2001; Rehm, Wuosmaa et
al. 2002; Davids, van den Berg et al. 2003; Rehm 2003]. The precision of the present
measurement could have been much improved by the addition of charged particle detectors
near 0^=90°, because of the additional efficiency as well as better constraints on the angular
distribution fits.
72
The measurement result reported here for the Ex=6.742 MeV state, as mentioned in §4.6.1,
should be considered in light of the recent direct measurement of Tp at Oak Ridge National
Laboratory [Bardayan, Batchelder et aL 2002]. It is wise in this case to abandon statistical
arguments and defer to the result of the direct measurement, which with any reasonable values
Ta and Ty inferred from the mirror for this state yields Fp/r ~ r y r ~ 1 0 3 [Utku, Ross et al.
1998]. Even if the direct measurement erred by an order of magnitude, r a/r must be within
10'2 of unity.
The r a/r ratio inferred for the Ex=6.861 MeV state is inconsistent with properties inferred
from the mirror nucleus, 19F. Reasonable values for r«, r p and r y inferred from the mirror
give r p/r~10'5, r y r - 10° and r a/r~ l [Utku, Ross et al. 1998]. There is also a previously
measured upper limit of r p/r<0.025 [Utku, Ross et aL 1998]. The average value for the
measurements o f r a/r is over 2G below the expected value, with most of the discrepancy
resulting from the present measurement
The resonant 15O(0C,' rate has been calculated for the average values given in Table 9 and
using the Ty for the states as given in Davids et al. [Davids, van den Berg et aL 2003]. The
result has been plotted in Figure 28. The 90% reported upper limit for the Ex=4.033 MeV state
has been used rather than the 99.73% upper limit which was not given explicidy in their paper.
This upper limit (Ta/r < 4.3X1O 4) has been interpreted as Ta/r = (2.15±2.15)xl0^* for the
calculation. Figure 29 shows the ratio of the total reaction rate in Figure 28 and a reaction rate
calculated by taking an average of all results but the present results. The present measurements
have not modified the reaction rate significandy.
73
Temperature [K]
Figure 28: The lsO(a,y) reaction rate calculated from the averages reported in Table 9. The 90% upper limit o f Davids et al. for the E*=4.033 MeV state is plotted [Davids, van den Berg et al. 2003] The T7 used in the calculation come from the same paper.
Temperature [K]
Figure 29: Effect o f the new measurements on the lsO(0C,7) reaction rate uncertainty. The top and bottom dashed curves represent the uncertainty in the reaction rate as calculated before incorporating the present measurements. The middle solid curve is the ratio o f the reaction rate shown in Figure 28 with the older rate. The top and bottom curves represent onty the uncertainties in the reaction rates due to the error bars for r j Y .
74
2 x 1 0 8 4 x l 0 8 6 x l 0 8 8 x 1 0 s l x l O 9
Temperature [K]
Figure 30: The calculated l8F(p,Y) reaction See the discussion for the details on the properties used for the calculation.
The resonant and non-resonant 18F(p,Y) rate has been calculated for the new average value of
r p/ r for the Ex=7.070 MeV state given in Table 9. The rates are shown in Figure 30. The
properties used for the Ex=6.698 MeV state were Tp= 3.8+3.8x1 O'8 MeV, Fy=0.33 eV and
r a—1.2 keV [Utku, Ross et aL 1998; Coc, Hemanz et al. 2000]. Fp=2.22±0.69 eV, Fy=5.5 eV
and Ta=2.7 keV were used for the Ex=6.742 MeV state [Utku, Ross et aL 1998; Bardayan,
Batchelder et al. 2002]. The Ex=6.861 MeV state properties are taken from Table 3 [Utku,
Ross et al. 1998]. The direct capture reaction rate calculated by Utku et aL is also plotted [Utku,
Ross et aL 1998]. A comparison between the new reaction rate and one calculated from an
average that does not include the present measurements is shown in Figure 31.
75
Log10 (
NA<
av>
[cm3 m
ol*,
s'1]
) (NA
<ov>)/
(NA<
av>)
i
Temperature [K]
Figure 31: Effect o f the new measurement on the 18F(p,Y) reaction rate uncertainty. The top and bottom dashed curves represent the uncertainty in the reaction rate as calculated before incorporating the present measurements. The middle solid curve is the ratio o f the reaction rate shown in Figure 30 with the older rate. The top and bottom curves represent only the uncertainties in the reaction rates due to the error ban for r / T .
2x10° 4x108 6x108 8x108 lx lO 9
Temperature [K]
Figure 32: The calculated l8F(p,a) reaction See the discussion for the details on the properties used for the calculation.
76
Figure 32 shows a calculated reaction rate versus temperature for 18F(p,Ct). It was calculated
using the same resonance properties as were used to calculate the 18F(p,y) rate. The tail
corrections calculated by Utku et al. for the two broad states have also been included [Utku,
Ross et aL 1998]. The relative errors of the temperature-dependent reaction rate both before
and after the present measurement are shown in Figure 33. Considering a typical 1.35 M
ONeMg white dwarf as a nova progenitor, the most important temperature range for
determining production of 18F is about 0.2-0.5 GK [Bardayan, Batchelder et al. 2002]. The
current measurement has made little difference in the knowledge of the reaction rate at these
temperatures.
Temperature [K]
Figure 33: Effect of the new measurement on the 18F(p,y) reaction rate uncertainty. The top and bottom dashed curves represent the uncertainty in the reaction rate as calculated before incorporating the present measurements. The middle solid curve is the ratio o f the reaction rate shown in Figure 31 with the older rate. The top and bottom curves represent onfy the uncertainties in the reaction rates due to the error ban for r / V .
77
5.2. Future Directions
Other experiments to explore branching ratios in nuclei of astrophysical interest are planned at
WNSL. For example, “Al+p resonances in “Si are of astrophysical interest and could
potentially have their proton branching ratios measured by YLSA. If “Al^/y) outpaces the (F
decay of 25Al in novae, then less astronomically observable “Al (via the 1.8 MeV y-ray line
following its ground state |F decay) will be produced. It would be valuable to know the
branching ratios of the states just above the proton threshold at 5.518 MeV, especially the
three known states between 5.678 and 5.945 MeV [Bardayan, Blackmon et al. 2002; Caggiano,
Bradfield-Smith et al. 2002], Reactions such as 24Mg(’°B,8Li) are being explored as ways to
populate 26Si in a kinematically favorable way for coincidence detection in YLSA. Alternatively,
an amount of “Al equal to ~0.15 |lCi of activity is available and will be deposited to give ~10
Jig/cm2 of “Al on a thin carbon foil. This will then be used to perform an “Al(3He,t)“Si(x)
coincidence experiment.
s Mg is similar in importance to 19Ne in that it provides a potential breakout path from the
hotCNO cycles via 18Ne((X,p). Measuring Ba and Bp for states above the 8.14 MeV OC-threshold
would greatly reduce uncertainties in this reaction rate. It also is important in understanding
the production of 2Na in novae, which depends sensitively on the 21Na(p,yj rate and may
produce an astronomically observable 1.275 MeV y-ray line. This reaction rate would be
understood much better by knowing proton branching ratios for states from the 5.50 MeV
threshold up to the recendy discovered Ex=6.051 MeV state [Caggiano, Bradfield-Smith et aL
2002]. A measurement of branching ratios in ^Mg was already attempted with YLSA using the
12C(160 ,6He) reaction that had previously allowed the discovery of many new states above the
(X-threshold [Chen 1999; Chen, Lewis et al. 2001]. That reaction, however, proved to be too
forward focused kinematically for effidendy detecting decay particles with YLSA. This might
be explored again when additional detectors augment YLSA as described below. Alternate
paths into ^Mg, such as two-neutron pickup reactions on 24Mg might be explored as
possibilities [Bateman, Abe et aL 2001; Parikh, Caggiano et aL 2002] for branching ratio
78
measurements. However, direct resonance strength measurements have just been performed at
TRIUMF1 by the DRAGON collaboration [Bishop, Azuma et aL 2003].
YLSA will be augmented with a pair of 16x16 position-sensitive single-sided silicon strip
detectors (PSSSD) placed at or near 0bb=9O°. It was never feasible to place YLSA in the
forward hemisphere, but the PSSSD’s will add some detection efficiency at more forward
angles. They will add sensitivity to the angular distributions near 0CM=9O°, with the 256 pixels
providing well defined bins in 0CM and useful constraints on the angular fits. The PSSSD’s
should also provide excellent energy resolution.
It has been shown that large area silicon detectors, usually used with low intensity radioactive
beams, are also useful when used in relatively high-intensity stable beam experiments. YLSA
and similar arrays should prove useful instruments for exploring charged particle branching
ratios for the next several years at WNSL.
Recendy, the 19F(3He,t) experiment has been repeated with four of the detectors “stacked” so
that the two rear detectors could veto any energetic particles that punch through the two front
detectors. Analysis of this data by others is still in progress, so it is as yet unknown whether
this is useful information to have.
1 Canada's National Laboratory for Particle and Nuclear Physics. See http://www.triumf.ca/
79
A p p e n d i x A : M o n t e C a r l o A l g o r i t h m f o r D e t e r m i n i n gE f / i c i e n c j
1. Define beam energy and species, target composition and thickness, and the reaction
product to be detected in the spectrograph. Also define decay particle, resonance
populated and excitation energy in final nucleus. Finally, define at what angle the
spectrograph is sitting and what acceptance its aperture is opened up to.
2. Randomly pick a distance through the target at which the reaction takes place.
Calculate the beam energy loss up to this point
3. Randomly pick a direction in entering the spectrograph aperture for the projectile (the
3H), isotropically in the lab frame.
4. Perform the relativistic kinematics calculations that populate your state of interest with
the projectile going in your selected direction. Also calculate the trajectory of the
residual excited nucleus.
5. Isotropically and randomly pick a direction in the CM frame of the residual nucleus for
the decay product to fly out. Boost this momentum vector to the lab frame, and
propagate it out of the target calculating energy loss.
6. Determine if the trajectory of the decay product will intersect with YLSA. Determine
it’s energy loss in the backplane dead layer (assumed to be 0.2 |Xm).
7. If the decay product intercepts a strip in the detector, and leaves any energy in the
strip, consider it a hit
8. Keep track, by segment number and strip number, of total simulated events where the
projectile makes it into the spectrograph, and the decay product deposits energy in a
strip.
80
A p p e n d i x B: M a t b e m a t i c a f i t t i n g e x a m p l e : W(6) for /’a/7-' o fE x - 4 . 6 0 0 M e V
W : =Function] (J , 1, p ),
^ j Extract [p, (M* J + 1)] . ^ Abe[ClebsciGordan[{l/2, m), (1 , M -m ), (J, M>) . a in m i [ 1. H-m, theta, 0]] A2|]
5/2+ state in 19Ne, no M=5/2 allowed due to assumption it's suppression.dist = Collect [Simplify [W[5/2, 3, {0, 1/2-pi, pi, pi, 1/2-pi, 0}], Os theta], pi]3 (19 - 4 Cos [2 theta] - 15 Cos [4 theta]) 3 pi (22 + 56 Cos [2 theta] + 50 Cos [4 theta])
2 5 6 t t + 2 5 6 t t
sum := Integrate [2 Pi dist Sin [theta], (theta, 0, Pi}]noonDist = area* A* dist /. Solve [A* sum = 1, A] [[1]]
; 3 ( 1 9 - 4 C os [2 t h e t a ] - 15 Cos (4 t h e t a ] ) 3 p i (22 + 56 C os [2 t h e t a ] + 50 C os [4 t h e t a ] ) ia r e a -------------------------------------------------------------------- + -----------------------------------------------------------------------------
' 256 t t 256 t t 1distSum = Integrate [2 Pi normDi st Sin [theta], (theta, 0, Pi}]areaSetDirectoxy [ "D: \Nel9\J\ily20~ I\analysis\alpiha0 " ]D: \N e l9 \J u ly 2 0 ~ l\a n a ly s is \a lp h a 0 dataTable = Inport [ "4600 .csv" ]{{165.12, 0.0147836, 0 .00391632). {158.963, 0.0153095. 0.00243057), {150.882, 0 0235768, 0 00384083), {142.373. 0.0161131, 0.00288579))
angles = Transpose [dataTable] [ [1] ] * Pi /180{2 .88189 , 2 .77443 , 2 .63339, 2.48488} anglesDeg = Transpose [dataTable] [ [1] ]{165 .12 , 158 .963 , 150 .882, 142.373} dndW = Transpose [dataTable] [ [2] ]{0 .0147836, 0 .0153095, 0 .0235768, 0.0161131} dataToFit = Transpose [ {angles, dndW} ]{ { 2 .8 8 1 8 9 , 0 .0 1 4 7 8 3 6 ) , { 2 .7 7 4 4 3 , 0 .0 1 5 3 0 9 5 ] , { 2 .6 3 3 3 9 , 0 .0 2 3 5 7 6 8 ] , { 2 .4 8 4 8 8 , 0 .0161131} }
errors = Transpose[dataTable] [ [3] ]{0.00391632, 0.00243057, 0 .00384083, 0.00288579}weights = errarsA-l{255 .342 , 411 .427 , 260 .361, 346.525}Remove [ NanlinearRegress]« Statistics' NonldnearFit'resultsTable = HcolioeazSegresstdataTCFit, mrncdnt, (theta), {(area, 0, 1), (p i, 0, 1 / 2 } ) , Fteigtats -* weights]
82
{ B e s tF i tP a r a m e t e r s -» ( a r e a -
P a ra m e te rC IT a b le -> a r e a P i
0 .1 6 7 2 2 3 , p l -
E s t i n a t e0 .1 6 7 2 2 30 .1 7 4 6 6 7
0 .1 7 4 6 6 7 } ,
A sy m p to tic SE0 .02 9 2 3 7 40 .0 8 0 7 0 3 2
(0 .0 4 1 4 2 4 5 , 0.293021), ( - 0 .1 7 2 5 7 1 , 0 .5 2 1 9 0 5 )
E s t im a te d V a r ia n c e -> 0 .0 0 5 6 0 8 5 6 , ANOVATable-M odelE r r o rU n c o r r e c te d T o ta l C o r r e c t e d T o ta l
DF2243
SumOfSq0 .3 7 5 7 1 50 .0 1 1 2 1 7 10 .3 8 6 9 3 20 .0 1 3 9 4 8 1
M eanSq0 .1 8 7 8 5 80 .0 0 5 6 0 8 5 6 ,
A s y n p t o t i c C o r r e l a t i o n M a t r i x -I -0 .7 1 5 3 1 6
-0 .7 1 5 3 1 6 v1. J F i tC u r v a tu r e T a b le • Max I n t r i n s i c
Max P a r a m e te r - E f f e c t s9 5 . % C o n f id e n c e R e g io n
C u rv a tu r e 2 .1 0 0 4 8 x 10-15 1 0 .9 7 9 0 .2 2 9 4 1 6
citable = ParameterCITable /. resultsTableEstimate Asymptotic SE Cl
area 0.167223 0.0292374 {0.0414245, 0.293021}pi 0.174667 0.0807032 {-0.172571, 0.521905}rules = Statistics' Cannon' RegiessianCcmnarf BestFitParameters /. resultsTable [ [1] ] {area-> 0.167223, pi-> 0.174667} finalDist = normDist /. rules0.000623774 (22.8427 + 5.78134 Cos[2 theta] - 6.26666 Cos[4 theta] ) Remove [ ErrorListPlot]« Graphics'Graphics' plotData := Transpose [{angles, dndW, errors}]g l ;= ErrorListPlot[plotUata, Fraas-* True, GrldLines -»flatcnnf.ic, Amstobftl -»{pe (radians]", "dn/dQ (counfcs/sr]"}] g2 := Plot [finalDist, (theta. Pi / 2, Pi), True, GridLines -» totamtic, JUoealabel -» ("8 [radians]", "dn/dQ (count s/sr]")]Shcw[gl, g2, PlotRange-* {0, 0.03}]
dn / dS5 [ counts / sr ]
- Graphics -= Sum[( ((finalDist/. theta-. angles [ [1]])-dm3H[[i]]) / errors! [i] ]) A2, {i, 1, Iength[angles]} ]
3.1922283
dof = Length[angles] - 22chiSq/ dof1.59611
Since ‘distSum’ is completely independent of'pi', the only term in the error of the branching ratio is the error in 'area'.
84
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