# Lifetime measurements of excited states in 15O

B. Frentz  A. Aprahamian  A.M. Clark  C. Dulal  J.D. Enright  R.J. deBoer  J.
Görres  S. L. Henderson  K.B. Howard  R. Kelmar  K. Lee  L. Morales  S. Moylan
Z. Raman  W. Tan  L. E. Weghorn  M. Wiescher  Department of Physics,
University of Notre Dame, Notre Dame, Indiana 46556 USA  The Joint Institute
for Nuclear Astrophysics, University of Notre Dame, Notre Dame, Indiana 46556,
USA

###### Abstract

The CNO cycle is the main energy source in stars more massive than our sun, it
defines the energy production and the cycle time that lead to the lifetime of
massive stars, and it is an important tool for the determination of the age of
globular clusters. One of the largest uncertainties in the CNO chain of
reactions comes from the uncertainty in the 14N$(p,\gamma)^{15}$O reaction
rate. This uncertainty arises predominantly from the uncertainty in the
lifetime of the sub-threshold state in 15O at $E_{x}$ = 6792 keV. Previous
measurements of this state’s lifetime are significantly discrepant. Here, we
report on a new lifetime measurement of this state, as well as the excited
states in 15O at $E_{x}$ = 5181 keV and $E_{x}$ = 6172 keV, via the
14N$(p,\gamma)^{15}$O reaction at proton energies of $E_{p}=1020$ keV and
$E_{p}=1570$ keV. The lifetimes have been determined with the Doppler-Shift
Attenuation Method (DSAM) with three separate, nitrogen-implanted targets with
Mo, Ta, and W backing. We obtained lifetimes from the weighted average of the
three measurements, allowing us to account for systematic differences between
the backing materials. For the 6792 keV state, we obtained a $\tau=0.6\pm 0.4$
fs. To provide cross-validation of our method, we measured the known lifetimes
of the states at 5181 keV and 6172 keV to be $\tau=7.5\pm 3.0$ and
$\tau=0.7\pm 0.5$ fs, respectively, which are in good agreement with previous
measurements.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

The measurements of solar neutrinos McDonald (2004) have confirmed the
prediction of the $pp$-chains as the dominant energy source of our sun. Only
about 1% of the solar energy in the sun comes from the CN cycle,

$\displaystyle\ce{{}^{12}C}\left(p,\gamma\right)\ce{{}^{13}N}$
$\displaystyle\ce{{}^{13}N}\left(\beta^{+}\nu\right)\ce{{}^{13}C}$
$\displaystyle\ce{{}^{13}C}\left(p,\gamma\right)\ce{{}^{14}N}$
$\displaystyle\ce{{}^{14}N}\left(p,\gamma\right)\ce{{}^{15}O}$
$\displaystyle\ce{{}^{15}O}\left(\beta^{+}\nu\right)\ce{{}^{15}N}$
$\displaystyle\ce{{}^{15}N}\left(p,\alpha\right)\ce{{}^{12}C},$

but it quickly becomes the dominate source of energy production in stars with
masses $M\gtrsim M_{\odot}$. One of the major uncertainties in the description
of the sun, in the framework of the standard solar model Bahcall and
Pinsonneault (2004); Bahcall et al. (2006), is the metallicity of the solar
core, which is determined by its carbon, nitrogen, and oxygen content Asplund
et al. (2006). The expected element abundances, based on the spectroscopic
analysis of the solar atmosphere, disagree with the solar profiles of sound
speed and density as well as the depth of the convective zone and the helium
abundance obtained by helioseismology measurements Bahcall et al. (2005). It
has been pointed out that a direct study of the CN neutrinos, coming from the
$\beta$ decay of 13N and 15O, can provide an independent measure of the solar
metallicity Haxton and Serenelli (2008). However, the CN neutrino flux not
only depends on the CN abundance in the solar interior, but also on the
associated CN reaction rates, such as 12C($p,\gamma$)13N and
14N($p,\gamma$)15O respectively.

The BOREXINO collaboration has succeeded in the first measurement of the CNO
neutrinos associated primarily with the 15O $\beta$ decay Agostini et al.
(2020). The abundance of 15O depends critically on the 14N$(p,\gamma)^{15}$O
reaction rate, which is the slowest in the CN cycle, and therefore determines
the 15O equilibrium abundance. Extensive experimental efforts have been
undertaken to determine a reliable reaction rate at the solar energy range.

The first comprehensive study of the 14N$(p,\gamma)^{15}$O reaction was
performed by Schröder et al. (1987), covering the proton energy range from
$E_{p}$ = 0.2 to 3.6 MeV. The total $S$-factor at zero energy, $S$(0), was
determined to be 3.20 $\pm$ 0.54 keV b, where the transitions to the ground
state and to the $E_{x}$ = 6.79 MeV excited state in 15O were dominant,
contributing $S_{g.s.}(0)=1.55\pm 0.34$ keV b and $S_{6.79}(0)=1.41\pm 0.02$
keV b, respectively. Fig. 1 depicts the level structure of the 15O compound
nucleus. However, an $R$-matrix analysis by Angulo and Descouvemont (2001)
drastically changed the extrapolated $S$-factor for both transitions to
$S_{g.s.}(0)=0.08^{+0.13}_{-0.06}$ keV b and $S_{6.79}(0)=1.63\pm 0.17$ keV b.
Overall, this reduced the total $S$-factor by a factor of 1.7. A number of
more recent measurements at the LUNA underground accelerator Imbriani et al.
(2005); Lemut et al. (2006); Bemmerer et al. (2006); Marta et al. (2008)
expanded the reaction data to lower energies, suggesting an even lower
$S$-factor, while an independent study by Runkle et al. (2005) indicated a
higher value for the ground state transition. An independent $R$-matrix
analysis of the various reaction channels over a wide range of energies by
Azuma et al. (2010) and Li et al. (2016) failed to confirm the LUNA
collaboration measurements and suggest a higher low energy $S$-factor, in
particular for the transition to the ground state. This discrepancy translates
into a substantial uncertainty in the prediction of the reaction rate and the
associated neutrino flux from the decay of 15O. The very low energy
contributions in the 14N$(p,\gamma)^{15}$O reaction rate come from the tail
contribution of the subthreshold state at $E_{x}$ = 6.79 MeV in 15O, which is
determined by its radiative width, $\Gamma_{\gamma}$.

Figure 1: Level scheme of the 15O compound nucleus. The beam (laboratory
frame) and resonance (center of mass frame) energies and corresponding
excitation energies of the important states are given. In this work, the
lifetimes of the states at 6792 keV, 6172 keV, and 5181 keV are reported.
These states all decay with 100% branching to the ground state.

The 6.79 MeV excited state decays with 100% probability to the ground state by
$\gamma$-ray emission. Therefore, the total width of the state results solely
from the radiative width, $\Gamma=\Gamma_{\gamma}$. The width can be
determined by measuring the level lifetime, $\tau$, where $\Gamma=\hbar/\tau$.
The first precision measurement of this lifetime was performed by Bertone et
al. (2001), using the Doppler Shift Attenuation Method (DSAM) in forward
kinematics with targets of N implanted into Ta. They populated the state
through resonant capture of protons to the $E_{p}$ = 278 keV resonance of the
14N$(p,\gamma)^{15}$O reaction and observed the decaying $\gamma$-rays at
three different angles. The measured lifetime for the 6.79 MeV state was
determined to be $\tau=1.60^{+0.75}_{-0.72}$ fs, corresponding to a width of
$\Gamma=0.41^{+0.34}_{-0.13}$ eV, where the reported uncertainties correspond
to a 90% confidence limit. The authors also noted that their reported value
would have increased by 100%, to $\tau=3.2\pm 1.5$ fs, if they used the
density of the TaN compound (as their target was implanted TaN) instead of
pure Ta in their calculations.

Yamada et al. (2004) performed another measurement of the radiative width of
this level at RIKEN in 2004, Coulomb excitation. This measurement of 15O
nuclei was performed by fragmentation of an 16O beam incident on a 9Be target
and then inelastically scattered off of a thick lead target. The de-excitation
$\gamma$-rays were detected with an array of NaI (Tl doped) scintillation
detectors. In order to disentangle the 6.79 MeV peak of interest from the 6.86
MeV peak in their spectrum, the authors compared the experimentally obtained
spectrum with the results of a Monte Carlo simulation Yamada et al. (2004).
Due to the poor energy resolution of the detector system, the authors
ultimately had difficulty separating the peaks and were unable to observe
isolated $\gamma$-rays from the 6.79 MeV state. The authors only reported an
upper bound on the width, $\Gamma=0.95^{+0.60}_{-0.95}$ eV.

Schürmann et al. (2008) published a new measurement of the lifetime of the
same state, using the DSAM technique and an improved experimental setup. This
measurement used a single high-purity germanium (HPGe) detector on a rotating
track and measured the angular distribution of the depopulating $\gamma$-ray
at several angles ranging from $40-116\degree$, allowing for a check of
asymmetries around the $90\degree$ point and reducing systematic uncertainties
from the use of different detectors. Similar to the measurement of Bertone et
al. (2001), the authors also used implanted TaN targets and populated the
state of interest via the resonance at $E_{p}=278$ keV in the
14N$\left(p,\gamma\right)^{15}$O reaction. They obtained an upper limit of
$\tau<0.77$ fs for the lifetime with 68.3% confidence. In contrast with the
result of Bertone et al. (2001), however, the authors note only a 2% change in
the reported lifetime by changing the target density to TaN in their analysis.

Galinski et al. (2014) performed another DSAM measurement at the TRIUMF
facility. The major difference between past measurements was that the reaction
was performed in inverse kinematics, emulating a measurement from over 50
years ago Gill et al. (1968). The 15O in this experiment was produced via the
3He$(^{16}$O$,\alpha)^{15}$O reaction with a 3He-implanted gold-foil. The de-
excitation $\gamma$-rays were detected with a clover HPGe detector and
filtered in coincidence with the reaction $\alpha$-particles. The authors
performed a Bayesian line-shape analysis of the spectra and, using the maximum
likelihood method, recommended an upper limit of $\tau<1.8$ fs with a 68.3%
confidence limit. The authors, however, acknowledge that their result was
statistics limited.

The only other recent measurement of this lifetime comes from a doctoral
dissertation by Michelagnoli (2013). Similar to that of Galinski et al.
(2014), this measurement was done in inverse kinematics, with the
$d(^{14}$N$,^{15}$O$)n$ reaction. The $\gamma$-rays from this measurement were
detected with the state-of-the-art AGATA Demonstrator array using $\gamma$-ray
tracking for precision angular data. A line shape analysis of the data was
performed by comparing with the results of a Monte Carlo simulation, also
obtaining an upper-bound on the lifetime, $\tau<1.0$ fs with a 68.3%
confidence limit.

In aggregate, these results are unsatisfactory due to their disagreement. The
DSAM results in forward kinematics had neglected the systematic uncertainties
arising from the stopping power of the targets in their ultimate lifetime
calculation. The forward kinematics reactions Bertone et al. (2001); Schürmann
et al. (2008) both used the same type of nitrogen-implanted tantalum-foils for
the experiment (saturated to a composition of Ta2N3), but calculated the
lifetimes assuming target densities and stopping powers equal to those of pure
tantalum, calculated in SRIM Ziegler et al. (2010). However, the authors note
that accurate densities would have changed the inferred lifetimes of their
works by 100% Bertone et al. (2001) and 2% Schürmann et al. (2008),
respectively.

This work reports on a new measurement of the the lifetime of the 6.79 MeV
excited state in 15O by populating it with the same 14N$(p,\gamma)^{15}$O
reaction in forward kinematics at proton energies ranging from $E_{p}=1.0-1.6$
MeV using the DSAM technique. Due to the discrepancies between the Bertone et
al. (2001) and Schürmann et al. (2008) measurements, discrepancies which
occurred despite the use of the same target type, reaction, and method, we
have used three different types of nitrogen implanted targets with backings of
Mo, Ta, and W to reduce systematic uncertainties. The targets were produced
with the 5 MV Sta. Ana accelerator at the University of Notre Dame Nuclear
Science Laboratory. We report measurements of the lifetimes of this state as
well as the 5.18 MeV and 6.17 MeV states in 15O, which are in the fs range and
thus used to validate the results for the state at 6.79 MeV.

In this work, the experimental details are first discussed in Sec. II,
covering the target implantation and lifetime measurements. In Sec. III, a
Monte Carlo simulation is described followed by the calculation of the
lifetime via the observed Doppler shifts. In order to demonstrate the effect
of the lifetime measurement and its uncertainty on the 14N$(p,\gamma)^{15}$O
reaction, preliminary $R$-matrix calculations are described in Sec. IV.
Summary and conclusions are presented in Sec. V.

##  II Experiment

###  II.1 Implanted targets

Isotopically pure targets were made by implantation using the 5 MV Sta. Ana
accelerator at the University of Notre Dame’s Nuclear Science Laboratory
(NSL). 350 keV 14N1+ ions were implanted into Ta, W, and Mo backings of 0.5 mm
thickness. A similar implantation technique was used in another study Seuthe
et al. (1987) that showed a deeper distribution of nitrogen in the target. The
Ta was chosen to compare with the earlier measurements of Bertone et al.
(2001); Schürmann et al. (2008), which used Ta2N3, while the W and Mo were
used to identify and highlight any systematic differences arising from the
choice of backing material. This also provided a method of disentangling the
effects of the high beam currents used in the measurement. Before the
implantation, the target backings were cut and cleaned using ethanol, acetone,
and with an oxygen plasma. Throughout the implantation, the beam was rastered
over the backing surface in order to produce a uniform implantation.
Additionally, in order to reduce the carbon buildup on the target face during
the implantation, a liquid-nitrogen cooled copper cold trap was employed. Due
to the beam intensities used, the target backings were water cooled.

The nitrogen content in the targets was assessed by scanning the 1057 keV
resonance in the 14N$(p,\gamma)^{15}$O reaction and comparing the yield with
that of a target previously used by Li et al. (2016). The Ta, Mo, and W
targets were found to have nitrogen contents of 21.3 $\pm$ 1.3, 13.1 $\pm$
0.7, and 13.4 $\pm$ 0.5 $\times 10^{17}$ atoms / cm2. Earlier targets were
produced to saturation of the nitrogen in interstitial lattice sites,
resulting in targets of 60% nitrogen. These targets, however, were produced to
36, 26, and 22 percent, respectively for the Ta, Mo, and W targets. As the
backing materials saturate with nitrogen to different compounds, a lower
nitrogen content was used to allow for improved consistency between the
different targets.

###  II.2 Lifetime Measurement Method

The excited 5.18 MeV, 6.17 MeV, and 6.79 MeV excited states in 15O were
populated with the 14N$(p,\gamma)^{15}$O reaction, where protons of $E_{p}$ =
1020 keV and $E_{p}$ = 1570 keV were delivered to the targets. At both
energies, the reaction proceeds primarily through the direct-capture mechanism
and these energies were chosen to provide the best signal-to-background
discrimination for the states of interest. The energy spread of the beam was
approximately 1 keV and the current on target throughout the experiment ranged
from 20 - 50 $\mu$A, dependent on accelerator stability. Targets were mounted
on a $45\degree$ target holder, relative to the beam axis, and due to the high
beam currents, the backings were constantly cooled with recirculating de-
ionized water. A copper cold finger, biased to $-400$ V and cooled with a
liquid nitrogen reservoir, was utilized to limit carbon build-up and suppress
secondary electrons throughout the measurements. To keep the beam focused and
centered through the runs, some of which lasted several hours, pairs of
upstream slits were utilized. These prevented the beam from drifting to the
edges of the implanted area on the targets where the nitrogen content was
lower.

The de-exciting $\gamma$ rays were observed with one of the GEORGINA
detectors, a coaxial n-type HPGe detector, of 109% relative efficiency. 1.5 mm
lead shielding was placed in front of the crystal’s face to attenuate the low-
energy x rays entering the detector, reducing the count rate from the low
energy background. The lifetimes were measured by the DSAM method. A single
detector was used to measure the depopulating $\gamma$-rays at different
angles of
$0\degree,45\degree,60\degree,75\degree,90\degree,111\degree,135\degree$, and
$-90\degree$ relative to the beam direction. The detector was placed 25.4 cm
away from the target on a rotating table with angular precision such that the
detector’s center could be placed within $\pm$ 0.5$\degree$ of the intended
angle. Fig. 2 shows a schematic of the experimental setup. Data at
$-90\degree$ were taken and compared to those at $90\degree$ to prove that the
beam was centered on target, since $\gamma$ rays detected at these
corresponding angles are unshifted. These two spectra are included in Fig. 3.
Additionally, the data at $90\degree$ were used to calibrate the spectra in
the detector.

Figure 2: Schematic of the experimental setup. It is important to note that
the same HPGe detector was set on a rotating table and set to each of the
angles, not seven different detectors.  Figure 3: Measured energy spectrum of
the 6.79 MeV transition at $\pm$ 90$\degree$. At these two angles, the energy
of the $\gamma$ ray is not Doppler shifted. Therefore, by comparing the
position of the peaks at these angles, we confirm accurate position of the
detector relative to the target and centering of the beam on target.

In order to prevent any peak shifting from smearing the peaks at a given
angle, the data were saved hourly to track the stability of the detector. This
allowed the monitoring of the detector for gain shifts throughout the
experiment. Additionally, a CeBr detector was fixed at a backward angle of
$-135\degree$ in order to provide a consistency check throughout the
experiment and monitor for any drifting. By monitoring background lines, we
found no signs of instability of the detector or electronics. The data from
both detectors were collected with a Mesytec MDPP-16 module, which amplified
and digitized the data.

##  III Analysis and results

###  III.1 Monte Carlo simulation

For this analysis, a DSAM program was written to simulate the expected Doppler
shift. The approach utilized Monte Carlo sampling to simulate the radioactive
decay of a nucleus with a given lifetime according to the normal exponential
decay law:

$P(t)=e^{-t/\tau}.$ (1)

The generator function for exponential decay, takes the random number, $r$,
within the range [0, 1], and propagates it to the simulated decay time through

$t=\dfrac{1}{\lambda}\ln\left(\dfrac{1}{1-r}\right),$ (2)

where $\lambda=1/\tau$ is the width of the state. To demonstrate this, Fig. 4
provides an example of the simulated decay probability as a function of time
for radioactive nuclei following the decay law of Eq. (1).

Figure 4: An example of the Monte Carlo method for a simulated decay curve.
For 100,000 decays with $\lambda$ = 50 (chosen simply for demonstration), the
normalized decay curve agrees with the exponential decay law.

These simulated lifetimes (with the target backings etc.) were used in
conjunction with the output from the SRIM stopping power software Ziegler et
al. (2010). Targets were generated in the software and then used to simulate
ion recoil tracks within the target after the reaction. The initial energy of
the recoil was randomly assigned to a reasonable range based on the energy
loss in the target and all recoils were assumed to have an initial velocity
parallel to the incident beam axis. For 10,000 different ions, the entire
travel path of the recoil through the target was simulated and saved. The
energy of the recoiling nuclei, in the case of this experiment, was not high
enough to warrant relativistic corrections, like the method of Galinski et al.
(2014), so the SRIM output was sufficient.

Based on the SRIM simulation, further information about each individual ion’s
travel was calculated. Utilizing the position of each interaction, the
particle’s trajectory was calculated between each position. The recorded
energy at each interaction was used to calculate the ion’s speed in the step.
Finally, these data allow for the time of the interaction to be calculated (in
femtoseconds). This provided a complete picture of any single recoiling ion’s
motion in the target. Additionally, in Ziegler et al. (2010), the SRIM
software quotes stopping power uncertainties of 5%. The code took this
uncertainty into account and propagated it through to an uncertainty in the
initial energy of the recoil ion.

Next, the program used Eq. (2) to randomly simulate a decay time to the
decaying nucleus (for a specified lifetime) and randomly assigned this decay
time to one of the SRIM tracks. Then, the code interpolated between
interaction points to find the position and velocity of the decaying nucleus
at the generated instant of decay. With this information, the Doppler shifted
de-exciting $\gamma$ energy, $E_{\gamma}$ was calculated by

$E_{\gamma}=E_{\gamma}^{o}\left(1+R\beta F(\tau)\cos(\theta)\right),$ (3)

where $E_{\gamma}^{o}$ was the unshifted $\gamma$ ray energy, $R$ represents a
correction to account for the detector size and resolution (obtained from
experimental spectra), $\beta=v/c$ was the recoil’s velocity relative to the
speed of light, $F(\tau)$ was the attenuation factor (encapsulating the
relationship between the lifetime of the decaying state and the medium through
which the nucleus moves), and $\theta$ was the angle at which the $\gamma$ was
observed. This process was repeated 50,000 times for each combination of
detection angle, lifetime, and target backing material in order to build up a
complete Monte Carlo simulation of the experiment.

An example of a complete simulation for the 6.79 MeV state de-excitation at
all angles, $\tau=0.5$ fs, and the Mo target backing is shown in Fig. 5,
demonstrating the Doppler shift of the peak energy with detection angle and
demonstrating correct behavior with an energy boost at forward angles and a
reduction in energy at backward angles. Using the relationship between the
centroid of the peak and angle, as generated by these simulations, the
attenuation factor, $F(\tau)$, was determined as a function of the nuclear
lifetime, $\tau$, for each individual target. This relationship is plotted in
Fig. 6. This was ultimately used to determine the experimental lifetime by
comparing the measured attenuation factors to those of the simulated curves to
extract the corresponding lifetime and uncertainty.

Figure 5: Simulated energy deposition histogram for the 6.79 MeV state de-
excitation, at all detection angles, $\tau=0.5$ fs, and the Mo target backing.
This figure clearly illustrates the simulated Doppler shifting of the
$\gamma$-rays’ energy with angle, the relationship upon which the entire DSAM
was predicated.  Figure 6: The simulated attenuation factors, $F(\tau)$, for
the various experimental backings, as a function of the state’s lifetime,
$\tau$. This was used to calculate the measured lifetime from the experimental
attenuation factors. The simulations were also carried out to lifetimes well
beyond what was expected from literature values to ensure that the
relationship was robust around the lifetimes of interest.

A significant difference between this procedure and other, previous methods
for calculating the lifetimes was that they only used numerical calculations.
This method is only a discrete calculation at values chosen across the
relevant lifetime landscape. Therefore, these attenuation factors require an
external fit to provide the final relationship. To minimize any artificial
bias from the choice of simulated lifetimes, they were chosen to range beyond
expected values to ensure that the relationship between the lifetime and
attenuation factor is robust in the relevant lifetime range.

###  III.2 Doppler shifts and lifetimes

In the measurement at each of the seven angles
($0\degree,45\degree,60\degree,75\degree,90\degree,111\degree$ and
$135\degree$), the centroid energy of each of the three secondary transition
peaks at 5.18 MeV, 6.17 MeV, and 6.79 MeV were determined. Example spectra of
the peak Doppler shifting is shown in Fig. 7. From this, the relationship
between the cosine of the measurement angle, $\cos(\theta)$, and the observed
energies, $E_{\gamma}$, was determined. With uncertainties in both the angle
and the centroid energy, the data were fit with orthogonal distance
regression, instead of the traditional least-squares method, because it takes
into account the uncertainties in both dependent and independent variables,
leading to higher fidelity fits. The relationship between shifted $\gamma$-ray
energy and $\cos(\theta)$ is shown in Fig. 8. These figures are representative
of the other two targets as well.

Figure 7: Doppler shifting spectra for the 6.79 MeV secondary transition with
detection angle for the Ta target, with all angles
($0\degree,45\degree,60\degree,75\degree,90\degree,111\degree$ and
$135\degree$) plotted together.  Figure 8: Doppler shifted $\gamma$-ray energy
plotted against $\cos(\theta)$ to determine $F(\tau)$, the attenuation factor.
The red line represents the orthogonal distance regression best fit and the
band represents a two-sigma uncertainty of the fit, incorporating the
uncertainty in both the slope and intercept.

The experimentally determined attenuation factors and corresponding lifetimes
are shown in Table 1. Additionally, the weighted average of the lifetimes has
also been calculated. This analysis leads to lifetimes for each of the 5.18
MeV, 6.17 MeV, and 6.79 MeV excited states in 15O. Table 1 results show no
systematic differences with the lifetimes extracted from the various target
materials. This allowed the exclusion of backing materials as sources of
uncertainties in comparison to earlier measurements.

Table 1: Experimental $F(\tau)$ and corresponding lifetime values obtained in the present work along with a weighted average of the lifetime for the various targets. Note that since the attenuation factor is dependent on the backing material, only the average of the lifetimes may be used.  | Mo | Ta | W | $\tau_{\text{average}}$
---|---|---|---|---
$F(\tau)_{5.18}$ | $9.040\pm 0.013$ | $0.911\pm 0.016$ | $9.120\pm 0.015$ |
$\tau_{5.18}$ (fs) | $7.1^{+4.8}_{-2.3}$ | $7.1\pm 5.5$ | $8.0\pm 6.7$ | $7.5\pm 3.0$
$F(\tau)_{6.17}$ | $0.992\pm 0.014$ | $0.976\pm 0.017$ | $0.988\pm 0.016$ |
$\tau_{6.17}$ (fs) | $0.4^{+0.7}_{-0.4}$ | $1.4\pm 1.0$ | $0.6^{+0.9}_{-0.6}$ | $0.7\pm 0.5$
$F(\tau)_{6.79}$ | $0.995\pm 0.019$ | $0.983\pm 0.019$ | $0.978\pm 0.015$ |
$\tau_{6.79}$ (fs) | $0.2^{+0.7}_{-0.2}$ | $0.7^{+0.9}_{-0.7}$ | $0.9\pm 0.6$ | $0.6\pm 0.4$

The experimentally determined lifetimes are shown in comparison with previous
measurements in Table 2. The mean lifetime of the 5.18 MeV state agrees well
with previous measurements, but has a much larger uncertainty. This large
uncertainty arises in part from the double escape peak at $E_{\gamma}=5150$
keV (from the $E_{\gamma}=6172$ keV full energy peak) that complicates the
background determination (and therefore the peak centroids for the Doppler
Shift). For the lifetime of the 6.17 MeV and 6.79 MeV states, we find good
agreement with the results reported in Schürmann et al. (2008) and Galinski et
al. (2014). For each of the states, our recommended lifetimes are taken to be
the weighted average of the measurements taken from each of the targets. These
results are depicted in Figs. 9, 10, and 11.

Table 2: Experimentally determined lifetime values (in fs) obtained in this work (as the weighted average of the different targets) and their comparison with literature values. $E_{x}$ (keV) | Present | Ref. Bertone et al. (2001) | Ref. Schürmann et al. (2008) | Ref. Galinski et al. (2014)
---|---|---|---|---
5181 | $7.5\pm 3.0$ | 9.67${}^{+1.34}_{-1.24}$ | 8.40$\pm$1.00 | -
6172 | $0.7\pm 0.5$ | 2.10${}^{+1.33}_{-1.32}$ | $<0.77$ | $<2.5$
6793 | $0.6\pm 0.4$ | 1.60${}^{+0.75}_{-0.72}$ | $<0.77$ | $<1.8$
Figure 9: Lifetimes of the 5.18 MeV state in 15O obtained in this work
compared with previous measurements. The measurement labelled TUNL corresponds
to that of Bertone et al. (2001), while Bochum to Schürmann et al. (2008).
Figure 10: Lifetimes of the 6.17 MeV state in 15O obtained in this work
compared with previous measurements. The measurement labelled TUNL corresponds
to that of Bertone et al. (2001), Bochum to Schürmann et al. (2008), and
TRIUMF to Galinski et al. (2014).  Figure 11: Lifetimes of the 6.79 MeV state
in 15O obtained in this work compared with previous measurements. The
measurement labelled TUNL corresponds to that of Bertone et al. (2001), Bochum
to Schürmann et al. (2008), TRIUMF to Galinski et al. (2014) and Michelagnoli
to Michelagnoli (2013), which has not been peer reviewed.

##  IV $R$-Matrix Fit

To examine the impact of these new lifetime measurements, the data were
incorporated into a set of limited $R$-Matrix fits using the AZURE2 code Azuma
et al. (2010). These represent fits to a selection of data to study the
influence of this lifetime measurement in particular; a full $R$-Matrix fit
incorporating these results and new low-energy cross-section results will be
presented in a forthcoming paper. In the fits presented here, shown in Figs.
12 and 13, most of the information about the levels was taken from Ajzenberg-
Selove (1991) or Daigle et al. (2016) where updated. The width of the 6.79 MeV
excited state in 15O, corresponding to values within the range of our lifetime
measurement, was fixed at a selection of different values. This collection of
fits, therefore, serves primarily as an illustration of the ways in which
these new lifetime measurements impact the low energy extrapolations of the
cross section.

A channel radius of 5.5 fm was adopted for this work, which matches the
analyses done by Adelberger and García (2011), Li et al. (2016), and Wagner et
al. (2018). Information about the levels and their parameters as used in
AZURE2 are contained in Table 3.

The cross-section data utilized in the fitting routine were from measurements
at LUNA Formicola et al. (2004); Imbriani et al. (2005); Marta et al. (2008,
2011), TUNL Runkle et al. (2005), Bochum Schröder et al. (1987), and the
University of Notre Dame Li et al. (2016). All of these data sets were left
without scaling during the fits. The Bochum data from Schröder et al. (1987)
were corrected as detailed in SFII Adelberger and García (2011). Additionally,
the data from Li et al. (2016) are a differential cross section taken at
45$\degree$ and are treated as such in the fits. This dataset was scaled by a
factor of 4$\pi$ in the plotting only to compare to the angle integrated data.

Table 3: Levels used in the R-matrix fits. Bold values indicate parameters which were allowed to vary during the fit. The signs on the partial widths and ANCs indicates the relative interferences. The dividing line demarcates the proton separation energy at $E_{x}$ = 7.2968(5) MeV Ajzenberg-Selove (1991). Levels where all parameters are fixed are not shown in this table for brevity but were included in the fits. a) Indicates the partial width of the 6.79 MeV state, measured in this experiment to between $\Gamma$ = 0.66 - 3.29 eV. For each individual fit, this width was fixed. However, between each fit, this width was varied to different values within our range to explore how the uncertainty in this measurement affects the low energy extrapolation. These different fits are shown in Fig. 12 and are otherwise identical.  $E_{x}$ (Ref. Ajzenberg-Selove (1991)) |  $E_{x}$ (fit) | $J^{\pi}$ | Channel | l | s | ANC (fm-1/2) / Partial Width (eV)
---|---|---|---|---|---|---
0.0 | 0.0 | 1/2- |  14N+p | 1 | 1/2 | 0.23
|  |  |  14N+p | 1 | 3/2 | 7.4
6.7931(17) | 6.7931 | 3/2+ |  14N+p | 0 | 3/2 | 4.75
|  |  |  15O+$\gamma_{0.00}$ | E1 | 1/2 | 2.50${}^{\text{a}}$
8.2840(5) | 8.2848 | 3/2+ |  14N+p | 2 | 1/2 | -92.2
|  |  |  14N+p | 0 | 3/2 | 4.013$\times$103
|  |  |  14N+p | 2 | 3/2 | -509
|  |  |  15O+$\gamma_{0.00}$ | E1 | 1/2 | 0.244
8.9821(17) | 8.98 | 5/2- |  14N+p | 1 | 3/2 | -5.872$\times$103
|  |  |  15O+$\gamma_{0.00}$ | E2 | 1/2 | -0.303
|  |  |  15O+$\gamma_{6.79}$ | E1 | 3/2 | -0.001
9.484(8) | 9.488 | 3/2+ |  14N+p | 2 | 1/2 | 77.69$\times$103
|  |  |  14N+p | 0 | 3/2 | 126.685$\times$103
|  |  |  14N+p | 2 | 3/2 | -7.822$\times$103
|  |  |  15O+$\gamma_{0.00}$ | E1 | 1/2 | 6.92
9.488(3) | 9.4905 | 5/2- |  14N+p | 3 | 1/2 | 0.979$\times$103
|  |  |  14N+p | 1 | 3/2 | -6.576$\times$103
|  |  |  14N+p | 3 | 3/2 | -0.985$\times$103
|  |  |  15O+$\gamma_{0.00}$ | E2 | 1/2 | -0.307
|  |  |  15O+$\gamma_{6.79}$ | E1 | 3/2 | -0.0123
9.609(2) | 9.6075 | 3/2- |  14N+p | 1 | 3/2 | -13.821$\times$103
|  |  |  15O+$\gamma_{0.00}$ | M1 | 1/2 | 1.24
|  |  |  15O+$\gamma_{6.79}$ | E1 | 3/2 | -0.044
| 15 | 3/2+ |  14N+p | 0 | 3/2 | 4.722$\times$106
|  |  |  15O+$\gamma_{0.00}$ | E1 | 1/2 | 327.3
Figure 12: $R$-Matrix fits exploring the uncertainty of our lifetime
measurements to the low energy extrapolation. The width of the 6.79 MeV
excited state in 15O is fixed during each fit and changed in each subsequent
iteration to another value within our uncertainty range. This clearly shows
that even though our lifetime result provides the most stringent limitation on
the lifetime of this state, it still has an outsized effect on the low energy
behavior of this reaction. The Schröder data are from Schröder et al. (1987),
while the LUNA data represents the measurements Formicola et al. (2004);
Imbriani et al. (2005); Marta et al. (2008, 2011), the Runkle data are from
Runkle et al. (2005), and the Li data are from Li et al. (2016). Of these, the
Li data are differential and were treated as such in the fitting but scaled up
by 4$\pi$ for plotting purposes. Figure 13: $R$-Matrix fits comparing our best
fit to those performed in previous works. Our fit used a lifetime value for
the 6.79 MeV excited state in 15O within our measured range range,
$\Gamma=2.75$ where the fits to the data provide good agreement with data
above and below the 278 keV resonance. This plot is limited to the low energy
region. The Schröder data are from Schröder et al. (1987), while the LUNA data
represents the LUNA measurements Formicola et al. (2004); Imbriani et al.
(2005); Marta et al. (2008, 2011), the Runkle data are from Runkle et al.
(2005), and the Li data are from Li et al. (2016). Of these, the Li data are
differential and were treated as such in the fitting but scaled up by 4$\pi$
for plotting purposes. The fits from previous works come from Refs. Runkle et
al. (2005); Azuma et al. (2010); Adelberger and García (2011); Li et al.
(2016).

In examining the capture to the ground state in 15O, the $R$-matrix fits show
the effect of our lifetime measurement. Specifically, in Fig. 12, we present
fits showing the whole range of lifetimes for the 6.79 MeV state of
$\tau=0.6\pm 0.4$. This shows that despite this measurement providing the most
stringent limit on the lifetime, this range still translates to dramatic
changes in the low energy behavior of the $S$-factor. Our fits, however, agree
well with the capture data. One of the best fits using our lifetimes is shown
alongside fits from Refs. Runkle et al. (2005); Azuma et al. (2010);
Adelberger and García (2011); Li et al. (2016) in Fig. 13; the impact of
higher energy data on the fit in the low energy range as presented in Li et
al. (2016) will be addressed in a forthcoming paper that presents a broad
range of new low energy cross section measurements.

##  V Summary and conclusions

The 14N$(p,\gamma)^{15}$O reaction was used to populate excited states at 5.18
MeV, 6.17 MeV, and 6.79 MeV in 15O. The nitrogen targets were made by
implantation on backings of Mo, Ta, and W. The Doppler shift of the
$\gamma$-rays emitted by the decaying recoils were measured using the
different targets and at seven different angles. A Monte Carlo simulation was
applied to reproduce the experimental shifts and extract the lifetimes from
the measured attenuation factors. By using multiple implanted targets of
different backings, we were able to take a weighted average of our
measurements to reduce the overall systematic uncertainty. Additionally, the
Monte Carlo approach allowed us to recreate the depth profile of implanted
targets with a high degree of accuracy, making the subsequent analysis based
on the target composition more robust. The simulation also propagates
uncertainties throughout every step, allowing it to reflect the experimental
conditions more accurately. This is an improvement over previous measurements
and their treatment of their targets.

The results show no evidence of systematic variations with previous
measurements arising from the choice of backing materials. This work shows a
larger uncertainty for the lifetime of the 5.18 MeV state but agrees within
the uncertainties of the previous measurements. For the other transitions at
6.17 MeV and 6.79 MeV, the present measurement agrees well with the values
reported by Bertone et al. (2001); Schürmann et al. (2008); Galinski et al.
(2014). Our work represents the only measurement of a finite lifetime for the
6.79 MeV state and provides even more stringent constraints on the level
lifetimes. The discrepancies in previous measurements were resolved in this
measurement with three different backings. These results, alongside recent
measurements of the low-energy 14N$(p,\gamma)^{15}$O reaction capture cross-
section will be used in a forthcoming paper to carry out a full $R$-Matrix fit
over a very broad energy range to extrapolate the $S$-factor to astrophysical
energies. This complete fit of and extrapolation from the data will allow for
a more confident determination of this reaction’s parameters and, therefore, a
more complete understanding of this crucial reaction in the CNO cycle.

###### Acknowledgements.

Fig. 1 has been created using the SciDraw scientific figure preparation system
[M. A. Caprio, Comput. Phys. Commun. 171, 107 (2005), http://scidraw.nd.edu].
This research utilized resources from the Notre Dame Center for Research
Computing and was funded by the National Science Foundation through Grant No.
PHY-2011890 (NSL), the Joint Institute for Nuclear Astrophysics - Center for
the Evolution of the Elements Grant No. PHY-1430152, and the U.S. Department
of Energy’s (DOE) National Nuclear Security Administration (NNSA, Grant # DE-
NA0003888).

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