Bayesian Estimation of the SFactor and Thermonuclear Reaction Rate for16Op17F Christian Iliadis and Vimal Palanivelrajan Department of Physics Astronomy University of North Carolina at Chapel Hill NC 27599-3255 USA

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Bayesian Estimation of the SFactor and Thermonuclear Reaction Rate for 16O(p,γ)17F

Christian Iliadis and Vimal Palanivelrajan

Department of Physics & Astronomy, University of North Carolina at Chapel Hill, NC 27599-3255, USA

Triangle Universities Nuclear Laboratory (TUNL), Durham, North Carolina 27708, USA

Rafael S. de Souza

Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory,

Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China

(Dated: October 27, 2022)

The 16O(p,γ)17F reaction is the slowest hydrogen-burning process in the CNO mass region. Its

thermonuclear rate sensitively impacts predictions of oxygen isotopic ratios in a number of astrophys-

ical sites, including AGB stars. The reaction has been measured several times at low bombarding

energies using a variety of techniques. The most recent evaluated experimental rates have a reported

uncertainty of about 7.5% below 1 GK. However, the previous rate estimate represents a best guess

only and was not based on rigorous statistical methods. We apply a Bayesian model to ﬁt all reli-

able 16O(p,γ)17F cross section data, and take into account independent contributions of statistical

and systematic uncertainties. The nuclear reaction model employed is a single-particle potential

model involving a Woods-Saxon potential for generating the radial bound state wave function. The

model has three physical parameters, the radius and diﬀuseness of the Woods-Saxon potential, and

the asymptotic normalization coeﬃcients (ANCs) of the ﬁnal bound state in 17F. We ﬁnd that

performing the Bayesian Sfactor ﬁt using ANCs as scaling parameters has a distinct advantage

over adopting spectroscopic factors instead. Based on these results, we present the ﬁrst statistically

rigorous estimation of experimental 16 O(p,γ)17 F reaction rates, with uncertainties (±4.2%) of about

half the previously reported values.

I. INTRODUCTION

The 16O(p,γ)17F reaction (Q= 600.27 ±0.25 keV [1])

is the slowest process among all proton-induced reactions

in the CNO mass region [2, 3]. The lowest-lying reso-

nance is located at relatively high laboratory energy of

≈2.7 MeV [4]. Below this energy, the 16O(p,γ)17F reac-

tion is a prime example of the nonresonant direct radia-

tive capture process, which assumes that the proton is

captured via a single-step process into a ﬁnal-state orbit

outside a closed 16O core [5, 6]. This reaction has been

measured many times at low bombarding energies using

a variety of techniques, including the activation method,

in-beam detection of prompt γrays, and experiments in

inverse kinematics. A comprehensive analysis of the most

reliable data has been presented in Refs. [7, 8]. The re-

ported thermonuclear reaction rate uncertainty in Ref. [7]

is about 7.5% at temperatures below 1 GK. Knowledge

of the rate at a few-percent uncertainty level is desirable

because the 16O(p,γ)17F reaction inﬂuences sensitively

the 17O/16O abundance ratio and, to a lesser degree, the

18O/16O ratio. This information directly impacts the in-

terpretation and paternity of oxygen isotopic ratios mea-

sured in presolar stardust grains [7, 9–11].

Previous evaluations of the 16O(p,γ)17F rate [2, 7, 12]

were performed with methods that were conventionally

employed at the time. The data from diﬀerent experi-

ments were ﬁtted independently because it was not clear,

within the χ2method used, how to perform a common

ﬁt across several diﬀerent data sets. Also, it was neither

clear how to treat independent contributions from statis-

tical and systematic uncertainties, nor how to combine

total cross section data in the analysis with data on in-

dividual transitions. For these reasons, a number of ad

hoc assumptions were made by Ref. [2, 7, 12] that were

not rigorous in a statistical sense.

The advent of thermonuclear rates based on hierar-

chical Bayesian models has improved this situation sig-

niﬁcantly. The method was ﬁrst presented in Ref. [13]

and subsequently applied to reactions of interest to

Big Bang Nucleosynthesis (BBN). In the simplest cases,

the Bayesian models employed either polynomial ﬁtting

functions or predictions from microscopic nuclear re-

action models [14–16]. The method was extended in

Refs. [17, 18] to implement a one-level R-matrix approx-

imation into the Bayesian ﬁtting. Here, we report on the

implementation of a single-particle potential model into

the Bayesian framework.

As will be seen below, the hierarchical Bayesian model

solves a number of problems that plagued previous work

(see, e.g., Ref. [7]): (i) it allows for the straightforward

implementation of the total cross section data of Ref. [19]

and the single datum of Ref. [20] for the ﬁrst-excited-state

transition (both disregarded previously); (ii) it facilitates

a combined ﬁt of all data (results from diﬀerent experi-

ments were previously ﬁtted independently); (iii) it fully

accounts for the independent contributions of statistical

and systematic uncertainties (which were previously com-

bined into a single uncertainty); (iv) it makes no ad hoc

assumptions on how to combine ﬁts from diﬀerent data

sets or γ-ray transitions.

A primary goal of this work is to include the parame-

ters of the single-particle potential model in the random

sampling, i.e., we will be exploring the sensitivity of the

arXiv:2210.14354v1 [nucl-th] 25 Oct 2022

Sfactor to these quantities. It will be demonstrated that

the uncertainty of the ﬁtted Sfactor is relatively large

when the Bayesian analysis is performed using spectro-

scopic factors, but is signiﬁcantly reduced when asymp-

totic normalization coeﬃcients (ANCs) are employed in-

stead.

The data selection is discussed in Sec. II. The nuclear

reaction formalism is given in Sec. III and the hierarchi-

cal Bayesian model is presented in Sec. IV. Fits of the

Sfactor are found in Sec. V and thermonuclear rates

calculated in Sec. VI. Section VII provides a concluding

summary.

II. DATA SELECTION

Consistent with previous Bayesian reaction rate esti-

mates [13–18], we will consider only data for which statis-

tical and systematic uncertainties can be estimated sepa-

rately. This is the case for four data sets: Hester, Pixley

and Lamb [19]; Chow, Griﬃths and Hall [21]; Becker et

al. [20]; and Morlock et al. [22]. The ﬁrst work [19] only

measured the total cross-section, while the third [20] re-

ported only the cross section for the transition into the

ﬁrst-excited state (Ex= 495.33 ±0.10 keV [4]) at a sin-

gle bombarding energy. The data of Refs. [19, 20] were

not taken into account in the previous 16O(p,γ)17F rate

evaluation [7], because, at the time, it was neither clear

how to ﬁt the total cross section [19] together with those

for the individual transitions, nor how to reliably include

a data set consisting of a single data point only [20]. As

will be seen in Sec. IV, all of these data can be rigorously

included in a hierarchical Bayesian model. We discuss

below the four data sets individually.

Hester, Pixley and Lamb [19] measured the total cross-

section of the 16O(p,γ)17F reaction at six center-of-mass

energies between 132 and 160 keV. These represent the

lowest-energy data points among all the data sets taken

into account in the present work. The reported statistical

uncertainties range from 14% to 40% for the highest and

lowest energy, respectively. The cross sections have been

corrected using modern stopping powers, as discussed in

[7], and we adopt these corrected values in the present

work. From their quoted uncertainties in the measured

beam current (6%), counter eﬃciency (7%), and stopping

power (10%), we estimate a total systematic uncertainty

of 14%.

Chow, Griﬃths and Hall [21] measured the cross sec-

tion for the transition to the ground state at four center-

of-mass energies between 1288 and 2404 keV, and for

the transition to the ﬁrst-excited state at seven energies

between 795 and 2404 keV. The statistical uncertainties

range from 3% to 12%. The systematic eﬀects in their

measurement were discussed by Ref. [7], including γ-ray

eﬃciency (3%), escape peak detection (1%), angular dis-

tribution correction (1%), and eﬀective bombarding en-

ergy (3%). Consequently, we adopt a value of 5% for the

combined systematic uncertainty.

Becker et al. [20] measured the 16O(p,γ)17F reaction

in inverse kinematics at a single center-of-mass energy

of 853 keV. Although not mentioned explicitly, their re-

ported cross section refers to the transition to the ﬁrst-

excited state only. The statistical uncertainty amounts

to 13%. The main sources of systematic uncertainty arise

from the strength of their adopted standard resonances in

19F(p,α2)16O and the γ-ray eﬃciency in their extended

gas target. We estimate an overall systematic uncertainty

of 5% for their reported cross section.

Finally, Morlock et al. [22] reported 16O(p,γ)17F cross

sections below a center-of-mass energy of 3.5 MeV. The

lowest energy measured was 365 keV for the ground state

transition and 222 keV for the ﬁrst-excited state one.

These data are presented in Fig. 3 of Ref. [22], which

displays statistical uncertainties only. Subsequently, the

Morlock et al. data [22] have been corrected by Ref. [7]

for coincidence summing, and these corrected data have

been adopted for the present analysis. More detailed

information regarding the cross section uncertainties of

these data is given in the caption of Fig. 2.37 in Ref. [23],

which states “...the statistical errors are with few excep-

tions between 1.5% and 3%. One has to add 10% sys-

tematic uncertainty (scattering measurement, energy de-

terminations, error propagation)...” Consequently, in the

present work, we adopt a systematic uncertainty of 10%.

The lowest-lying 16O(p,γ)17F resonance is located near

a center-of-mass energy of 2.5 MeV, and, therefore, we

only took their data points below an energy of 2.4 MeV

into account. The data at higher energies are irrelevant

for stellar burning.

We note that the four measurements discussed above

provide independent estimates of the 16O(p,γ)17F cross

section. In particular, only the work of Ref. [19] relied

on stopping power corrections, while the other measure-

ments of the direct capture cross section [20–22] were per-

formed relative to the Rutherford scattering yield, thus

obliviating the eﬀects of target stoichiometry or stopping

powers.

The cross sections, σ, discussed above were con-

verted to astrophysical S-factors, deﬁned by S(E)≡

σ(E)Ee2πη, with ηdenoting the Sommerfeld parameter.

The experimental S-factors were then analyzed with our

Bayesian model.

III. NUCLEAR REACTION MODEL

The 16O(p,γ)17F reaction cross section below 2.4 MeV

center-of-mass energy is considered as a standard case

for the direct radiative capture (DC) model since the

seminal works of Christy and Duck [24] and Rolfs [5].

Subsequent analyses using the direct capture model for

16O(p,γ)17F can be found in Refs. [6, 7, 25, 26], and

references therein. The study of Ref. [7] demonstrated

that the potential model and the R matrix model provide

nearly identical data ﬁts at low energies. We will adopt

in the present work a single-particle potential model, as

discussed below.

The potential model assumes a single-step process,

where the proton is directly captured, without the for-

mation of a compound nucleus, into a ﬁnal bound state

with the emission of a photon. The dominant E1 contri-

bution to the theoretical (p,γ) cross section (in µb) for

capture from an initial scattering state with orbital an-

gular momentum `ito a ﬁnal bound state with orbital

angular momentum `fand the principal quantum num-

ber n(i.e., the number of wave-function nodes), is given

by [5]

σDC

sp (E1, n, `i, `f) = 0.0716µ3

2Zp

Mp−Zt

Mt2E3

2×

(2Jf+ 1)(2`i+ 1)

(2jp+ 1)(2jt+ 1)(2`f+ 1)(`i010|`f0)2R2

n`i1`f(1)

Rn`i1`f=Z∞

us(r)OE1(r)ub(r)dr (2)

where µis the reduced mass, Zt,Zpand Mt,Mpare the

charges and masses (in amu), respectively, of target and

projectile; jp,jt,Jfare the spins of projectile, target and

ﬁnal state, respectively; Eand Eγare the center-of-mass

energy and the emitted γ-ray energy, respectively; OE1is

the radial part of the E1 multipole operator; and usand

ubare the radial wave functions of the initial scattering

state and ﬁnal bound state, respectively, where ub(r= 0)

= 0 and R∞

0u2

bdr = 1. We disregarded any M1 and E2

contributions, which amount to less than 0.1% compared

to the dominant E1 Sfactor [5, 27].

Bound-state wave functions were generated by using

a potential consisting of a Woods-Saxon term and a

Coulomb term, given by

V(r) = −V0

1 + e(r−R)/a +VC(r) (3)

where R=r0A1/3

tand aare the Woods-Saxon potential

radius and diﬀuseness, respectively; Atis the mass num-

ber of the target nucleus; VCcorresponds to a uniformly

charged sphere of radius R. The well depth, V0, was cho-

sen to reproduce the binding energy of the ﬁnal state.

Several works (see summary in Table I of Ref. [6]) have

employed bound-state square-well potentials instead of

Woods-Saxon potentials. However, Refs. [28, 29] found

that the adoption of square-well potentials over-predicts

the calculated single-particle direct capture cross sections

by up to a factor of 3.

Scattering-state wave functions were computed using a

hard-sphere nuclear potential, which gives similar results

as a zero-energy nuclear potential at the low energies ex-

plored here [6]. The insensitivity of the 16O(p,γ)17F cross

section to the choice of scattering potential at low ener-

gies has also been reported in Ref. [30]. The hard-sphere

nuclear potential radius was set equal to the Woods-

Saxon radius, R, of the bound state potential. The radial

integration in Eq. (2) was extended to 500 fm, because

the integrand has a maximum located far beyond the

nuclear radius at the lowest center-of-mass energies ex-

plored here. For the same reason, we used the exact

expression for the radial part of the E1 operator, OE1,

instead of its long-wavelength approximation.

For a zero-spin target nucleus, such as 16O, the direct

capture to a speciﬁc ﬁnal state proceeds via a unique

orbital angular momentum, `f, but may involve several

values of `i. In this case, the direct capture cross section

is given by an incoherent sum,

σDC =C2S`fX

σDC

sp (E1, n, `i, `f) (4)

with S`fand Cdenoting the spectroscopic factor and

isospin Clebsch-Gordan coeﬃcient, respectively. For 16O

+p, we have C2= 1. To avoid confusion with other

symbols, we will not mention further the isospin Clebsch-

Gordan coeﬃcient.

For the 16O(p,γ)17Freaction data considered in the

present work, the population of the 17F ground state,

DC →0 keV (Jf= 5/2+), proceeds predominantly via

E1 radiation and orbital angular momenta of `i= 1, 3

and `f= 2, while the transition to the ﬁrst excited state,

DC →495 keV (Jf= 1/2+), proceeds via E1 radiation

and angular momenta of `i= 1 and `f= 0. We assume

that the proton is transferred into the 1d5/2and 2s1/2

shell-model orbitals for the transition to the ground and

ﬁrst-excited state, respectively. In the following, we will

label the spectroscopic factor (or later the ANC) for a

given ﬁnal level by the orbital angular momentum of the

bound state alone, and suppress other quantum numbers

(such as total spin). Hence, Sgs ≡S`f=2 and Sfes ≡

S`f=0 for the transition to the ground and ﬁrst-excited

state, respectively.

The calculated single-particle cross section, σDC

sp , will

depend strongly on the adopted choice of the Woods-

Saxon potential radius parameter, r0, and diﬀuseness, a.

This means that the spectroscopic factor, S`f, which is

derived from a ﬁt to experimental data, will also be sensi-

tive to these very same parameters. We will demonstrate

below the diﬃculty arising when the data are ﬁtted in

terms of the spectroscopic factor using Eq. (4).

Previous works have demonstrated [5, 22] that, for the

16O(p,γ)17F reaction at low energies, the integrand in

Eq. (2) peaks far outside the nuclear radius. For such a

peripheral reaction, the single-particle radial bound state

wave function is asymptotically given by [30]

ub,`f(r)→b`fW−η,`f+1/2(2κr) (5)

where b`fis the single-particle asymptotic normaliza-

tion coeﬃcient (spANC) and Wis the Whittaker func-

tion [31]; κis the bound state wave number, with κ2=

2µEb/~2, where µis the reduced mass and Eb=Q−Ex

the binding energy of the ﬁnal state; η=eZpZtµ/(κ~2)

is the bound state Coulomb parameter.

In a microscopic nuclear model, the capture cross sec-

tion can be described in terms of the overlap, IB, of

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BayesianEstimationoftheSFactorandThermonuclearReactionRatefor16O(p,)17FChristianIliadisandVimalPalanivelrajanDepartmentofPhysics&Astronomy,UniversityofNorthCarolinaatChapelHill,NC27599-3255,USATriangleUniversitiesNuclearLaboratory(TUNL),Durham,NorthCarolina27708,USARafaelS.deSouzaKeyLaboratoryforRes...

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Bayesian Estimation of the SFactor and Thermonuclear Reaction Rate for16Op17F Christian Iliadis and Vimal Palanivelrajan Department of Physics Astronomy University of North Carolina at Chapel Hill NC 27599-3255 USA

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