arXiv:1011.2518v1 [nucl-ex] 10 Nov 2010
Constraining the S factor of15N(p,γ)16O at Astrophysical Energies
P. J. LeBlanc,1, ∗G. Imbriani,1,2J. G¨ orres,1M. Junker,3R. Azuma,1,4M. Beard,1D. Bemmerer,5A. Best,1C.
Broggini,6A. Caciolli,7P. Corvisiero,8H. Costantini,8M. Couder,1R. deBoer,1Z. Elekes,9S. Falahat,1,10A.
Formicola,3Zs. F¨ ul¨ op,9G. Gervino,11A. Guglielmetti,12C. Gustavino,3Gy. Gy¨ urky,9F. K¨ appeler,13A.
Kontos,1R. Kuntz,14H. Leiste,15A. Lemut,8Q. Li,1B. Limata,2M. Marta,5C. Mazzocchi,12R. Menegazzo,6
S. O’Brien,1A. Palumbo,1P. Prati,16V. Roca,2C. Rolfs,14C. Rossi Alvarez,6E. Somorjai,9E. Stech,1
O. Straniero,17F. Strieder,14W. Tan,1F. Terrasi,18H.P. Trautvetter,14E. Uberseder,1and M. Wiescher1
1University of Notre Dame, Department of Physics, Notre Dame, IN 46556, USA
2Universit` a degli Studi di Napoli “Frederico II”, and INFN, Napoli, Italy
3INFN, Laboratori Nazionali del Gran Sasso (LNGS), Assergi (AQ), Italy
4Department of Physics, University of Toronto, Toronto, Ontario M55 1A7, Canada
5Forschungszentrum Dresden-Rossendorf, Dresden, Germany
6INFN, Padova, Italy
7Universit´ a degli Studi di Padova and INFN, Padova, Italy
8Universit´ a degli Studi di Genova and INFN, Genova, Italy
9Institute of Nuclear Research (ATOMKI), Debrecen, Hungary
10Max-Planck-Institut f¨ ur Chemie, Mainz, Germany
11Universit´ a degli Studi di Torino and INFN, Torino, Italy
12Universit` a degli Studi di Milano and INFN, Sez. di Milano, Milan, Italy
13Forschungszentrum Karlsruhe, Institut f¨ ur Kernphysik, Karlsruhe, Germany
14Institute f¨ ur Experimentalphysik III, Ruhr-Universit¨ at Bochum, Bochum, Germany
15Forschungszentrum Karlsruhe, Institut f¨ ur Materialforschung I, Karlsruhe, Germany
16Universit` a degli Studi di Genova and INFN, Genova, Italy
17Osservatorio Astronomico di Collurania. Teramo, and INFN Napoli, Italy
18Seconda Universit` a di Napoli, Caserta and INFN, Napoli, Italy
(Dated: November 12, 2010)
The15N(p,γ)16O reaction represents a break out reaction linking the first and second cycle of the CNO
cycles redistributing the carbon and nitrogen abundances into the oxygen range. The reaction is dominated
by two broad resonances at Ep = 338 keV and 1028 keV and a Direct Capture contribution to the ground
state of16O. Interference effects between these contributions in both the low energy region (Ep < 338 keV)
and in between the two resonances (338 < Ep < 1028 keV) can dramatically effect the extrapolation to
energies of astrophysical interest. To facilitate a reliable extrapolation the15N(p,γ)16O reaction has been
remeasured covering the energy range from Ep=1800 keV down to 130 keV. The results have been analyzed
in the framework of a multi-level R-matrix theory and a S(0) value of 39.6 keV b has been found.
The energy production and nucleosynthesis in stars is characterized by nuclear reaction sequences which determine
the subsequent phases of stellar evolution. Energy production during the first hydrogen burning phase takes place
through the fusion of four protons into helium. This occurs either through the pp-chains or the CNO cycles. The pp-
chains dominate hydrogen burning in first generation stars with primordial abundance distributions and in low mass,
M≤1.5 M⊙, stars. The CNO cycles dominate the energy production in more massive, M≥1.5 M⊙, second or later
generation stars with an appreciable abundance of CNO isotopes. The CNO cycles are characterized by sequences of
radiative capture reactions and β-decay processes as shown in Figure .
At stellar temperatures the14N(p,γ)15O reaction is the slowest process in the cycle, defining the time scale and the
overall energy production rate [2–4]. This reaction is therefore of importance for the interpretation of CNO burning.
The proton capture by15N is relevant since it is a branch point linking the first CNO or CN cycle with the second
CNO, or NO cycle as shown in Figure  . This branching has always been a matter of debate since both reactions
are characterized by strong low energy resonances.
The reaction rate of15N(p,α)12C is determined by two broad low energy s-wave resonances at Ep = 338 keV
and 1028 keV, populating the Jπ= 1−states at 12.44 and 13.09 MeV, respectively, in the16O compound nucleus.
∗email at: email@example.com
FIG. 1: Diagram of the CNO bi cycle 
There have been a number of low energy measurements , , and  which provide the basis of the present rate
in the literature [9, 10]. Recently, three lower energy points were derived from an indirect “Trojan Horse Method
(THM)”  which are consistent with low energy data [6–8].
The competing15N(p,γ)16O reaction decays predominately to the ground state of16O and exhibits the same two
resonances but in addition is expected to have a strong non-resonant direct capture component . The presently
available low energy cross section data from , , and  differ substantially at lower energies. This poses
difficulties for a reliable extrapolation of the cross section towards the stellar energy range. An extrapolation was
performed  using a two level Breit-Wigner formalism taking into account the direct capture contribution. The
present reaction rate for15N(p,γ)16O  relies entirely on the predictions of .
To reduce the uncertainty in the strength of the direct capture term single-particle transfer reactions have been
performed  to determine the proton Asymptotic Normalization Coefficient (ANC) for the ground state of16O.
With this ANC value C2
resulted in smaller values for the low energy cross section than those obtained by . This result was confirmed by
an independent R-matrix analysis of the existing data . This conclusion was furthermore supported by a study
in which both the (p,α) and (p,γ) reactions were analyzed simultaneously with a multi-level, multi-channel R-matrix
Low energy data points were extracted from a re-analysis  of a study of14N(p,γ)15O performed at the LUNA
underground accelerator facility at the Gran Sasso laboratory. This experiment was performed using a windowless
differentially pumped gas target with natural nitrogen gas. For detecting the γ-ray signal, a large BGO scintillator
detector was used to observe the characteristic γ decay in summing mode . Since the natural abundance of15N
is low, the yield of the15N(p,γ) signal was weak and overshadowed by beam induced background yield from proton
capture reactions on target impurities and from the14N(p,γ) reaction. The measurements were therefore limited to
energies below 230 keV. The proposed cross section results are clearly lower than the values obtained by  but
also slightly lower than the results of  and . Given the strong background conditions, the results could not be
normalized to the known on-resonance yield at 338 keV and systematic errors in the data cannot be excluded.
Because of the inconsistencies in the existing data and the uncertainties of an R-matrix analysis based on these
existing data, we have performed a new study of the15N(p,γ)16O over a wide energy range using high resolution Ge
detectors. In the following section the experimental approach at the Notre Dame Nuclear Science Laboratory (NSL)
and the LUNA II facility at the Gran Sasso underground laboratory will be discussed. This will be followed by a
discussion of the experimental data. In the last section the stellar nuclear reaction rate based on the present data will
be calculated and compared with existing results.
p1/2= (192±26) fm−1, R-matrix fits to the15N(p,γ)16O data have been performed  which
A.Accelerators and Experimental Setup
The experiment was performed at two separate facilities. At the University of Notre Dame the 4 MV KN Van de
Graaff accelerator provided proton beams in an energy range of 700 to 1800 keV with beam intensities limited to
≤10 µA on target because of the high count rate in the Ge detector from the15N(p,α1γ)12C reaction. The energy
calibration of this machine was established to better than 1 keV using the well known27Al(p,γ)28Si resonance at
992 keV . The 1 MV JN Van de Graaff accelerator at Notre Dame was used in the range of 285 keV to 700 keV
with protons beams of 20 µA. The energy of the this machine was calibrated using the well known15N(p,α1γ)12C
resonance at 429 keV .
The LUNA II facility , located in the Gran Sasso National Laboratory, uses a high current 400 kV, Cockroft-
Walton type accelerator. The accelerator provided proton beam currents on target of up to 200 µA in the energy
range of 130 to 400 keV. In addition to the high beam output, the accelerator is extremely stable, and the voltage is
known with an accuracy of about 300 eV.
The experimental setup in both experiments was very similar. The targets were water cooled and mounted at 45◦
with respect to the beam direction. At Notre Dame the position of the beam on the target was defined by a set of
vertical and horizontal slits. The beam was swept horizontally and vertically across a target area of 1 cm2by steerers
in order to dissipate power over a large target area. At LUNA the ion beam optics provided a de-focused beam on
target and no beam sweeping was applied. To avoid the build-up of impurities on the target a Cu finger, cooled
to LN2 temperatures, was placed along the inside of the beamline extending as close to the target as possible. In
addition, a bias voltage of about -400 V was applied to the isolated cold finger to suppress the secondary electrons
ejected from the target due to proton bombardment.
The Ti15N targets were fabricated at the Forschungszentrum Karlsruhe by reactive sputtering of Ti in a Nitrogen
atmosphere enriched in15N to 99.95%. The stoichiometry was analyzed using Auger electron spectroscopy to confirm
the composition. This test was performed on two target spots, one which had been exposed to beam and one which
was not exposed. The results agreed within ≤2% with the nominal stoichiometry of 1:1. Isotopic abundances were
experimentally verified by comparing the yield of the14N(p,γ)15O, 278 keV resonance  from the enriched targets
with that obtained using a target produced with a natural nitrogen gas. The results of this measurement showed an
abundance of ≤ 2% of14N for the thin targets corresponding to a15N enrichment of ≥ 98% in agreement with the
quote of the supplier. For the thick target used at LUNA,14N and15N enrichment of 17.4 ± 2.0% and 82.6 ± 2.0%
were found, respectively, most likely caused by a contamination of the enriched gas during sputtering.
The thicknesses of all TiN targets were measured using the narrow15N(p,α1γ)12C resonance at 429 keV . The
target used at Notre Dame had a thickness of 7.2 ± 0.3 keV at Ep= 429 keV and the two targets used at LUNA had
thicknesses of 9.5 ± 0.4 keV and 24.8 ± 0.5 keV at 429 keV, respectively. The stability of the Notre Dame targets
was checked continuously during the course of the experiment using the15N(p,α1γ)12C resonance. The thin LUNA
target was monitored by rescanning the top of the 338 keV resonance in15N(p,γ)16O. The thick LUNA target could
also be monitored using the14N(p,γ)15O resonance at Ep= 278 keV because of the large14N content (17%) of this
target. Because of the relatively low power density delivered at Notre Dame, the target saw virtually no degradation
over the experiment with an accumulated charge of 5 C (see Figure 2) and no yield corrections were necessary. During
the LUNA experiment with significantly higher beam currents, the thickness of the thin target was reduced by 27%
after an accumulated charge of 17 C and that of the thick target by 30% after an accumulated charge of 65 C (see
At the NSL, the γ-rays were observed using a HPGe Clover detector, which consists of four HPGe crystals contained
in the same cryostat. This unique arrangement allows the separate detectors to be used in so called add-back mode .
At LUNA, a single crystal, 115% HPGe detector from Bochum, Germany was used for the detection of γ-rays. Several
sample spectra are given in Figure 4.
The primary advantage of the LUNA II facility is the low background environment created by the rock cover from
the Gran Sasso mountains. The rock shields from cosmic rays and therefore decreases the γ-induced background from
cosmic rays in the detector and has been shown  to suppress the Eγ > 3.5 MeV background count rate in a Ge
Proton Energy [keV]
Yield [arbitrary units]
Proton Energy [keV]
Yield [arbitrary units]
Target #1, Max Yield
Target #18, Max Yield
FIG. 2: Notre Dame target (left) has a measured width of approximately 7.2 keV at Ep = 430 keV. Squares, diamonds, and
stars correspond to 0.006 C, 3.4 C, and 4.8 C of accumulated charge on the target, respectively. LUNA thin (right, diamonds)
and thick (right, crosses) targets have measured widths of approximately 9.5 keV, and approximately 25 keV, respectively. The
difference in stoichiometry between the two targets can be seen by comparing the plateau heights from the same resonance
scan of each target (right).
FIG. 3: Targets scans of thick target (18) using the14N(p,γ)15O, Ep = 278 keV resonance. The target was stable until about
20 C, and went through significant deterioration. Squares represent the initial target scan, while down triangles represent the
scan after 65 C on target
110001150012000 12500 1300013500
Ep = 1300 keV
Ep = 700 keV
Ep = 130 keV
FIG. 4: Several sample spectra from the present15N(p,γ)16O experiment. The spectra were taken with Notre Dame’s KN
accelerator (Ep = 1300 keV, top), Notre Dame’s JN accelerator (Ep = 700 keV, middle), and the LUNA II accelerator (Ep =
130 keV, bottom). The15N(p,γ)16O ground state transition γ-ray peak is indicated in each spectrum with an arrow.
detector by three orders of magnitude. In each experiment the detectors were set up at an angle of 45◦with respect
to the beam direction, allowing the position of the detector to be set as close as possible to the reaction position. The
relative efficiency of the γ-ray detector systems was measured using radioactive sources along with well known capture
γ reactions. At Notre Dame the relative efficiencies for high energy γ-rays were established using the27Al(p,γ)28Si
resonances at 992 keV  and 1183 keV  and the23Na(p,γ)24Mg resonance at 1318 keV . The efficiency was
extended to a γ-energy of 12.79 MeV using the11B(p,γ)12C reaction at 675 keV and 1388 keV following the method of
reference . At LUNA the higher energy efficiencies were determined using the 278 keV resonance in14N(p,γ)15O
] and the 163 keV resonance in11B(p,γ)12C . The11B resonance has a small angular distribution , and the
correction for the relative intensity is less than 3%.
While summing is not of concern for the ground state transition of the15N(p,γ)16O reaction itself, summing plays a
significant role in the determination of the γ-efficiency. For this reason the efficiency measurements at both laboratories
were carried out at several different detector-target distances and the data were simultaneously fitted for all distances
following the procedure described by Imbriani et al.  (see Figure 5). For the 1 cm and 5 cm measurements, there
FIG. 5: Efficiency curves obtained at LUNA. Circles are measurements at a distance of 1 cm, squares at 3 cm, and diamonds
at 20 cm. Closed symbols represent data without summing corrections, open symbols include summing corrections.
is one point which has significant summing corrections. This point corresponds to the ground state transition in the
Ep= 278 keV14N(p,γ)15O reaction. While the ground state transition of this reaction has a small branching ratio,
each of the other cascades have significant probabilities, which lead to strong summing effects .
A.Cross Section Determination
The excitation function for the ground state transition of the reaction15N(p,γ)16O has been measured in the energy
range of 131 to 1800 keV. It consists of three distinct, overlapping sections, LUNA data from 131 keV to 400 keV, JN
data from 285 keV to 700 keV, and KN data from 700 to 1800 keV. The experimental yield Y (number of reactions
per projectile) at proton energy Epcorresponds to the cross section σ(E) integrated over the target thickness ∆:
Y (Ep) =
ǫ(E)dE = σ(Ep)
where ǫ is the stopping power , and f(E) represents the energy dependence of the cross section in the integration
σ(E) = σ(Ep)f(E), (2)
with σ(Ep) the cross section at Ep. This reduces to the well known thin target yield equation if the cross section is
constant over the target thickness, f(E)=1. However, at energies where the cross section varies significantly over the
target thickness,the yield has to be corrected to extract the cross section. This correction factor is given by:
FIG. 6: Cross section for the ground state transition of the15N(p,γ0)16O reaction versus center of mass energy.
and requires the knowledge of the energy dependence of the cross section in the energy interval ∆, f(E). This problem
can be solved by an iterative method. In a first step f(E)=1 is assumed resulting in an approximation of the cross
section which in turn can be used to calculate a new correction factor. This process is continued until no change
in the resulting cross section is obtained. This process quickly converges after 2 to 3 iterations. The resulting cross
section is shown in Figure 6.
An alternative method to determine the cross section was used in which the data were analyzed following the
approach described in . Here, a brief review of the method is given, which consists of the analysis of the shape of
The cross section, the stopping power of protons in TiN, the γ-ray efficiency, and the detector resolution all have
an influence on the γ-ray line shape observed in a spectrum. The observed energy of the γ-ray is related to the proton
energy by 
M + m+ Q − Ex− ∆ERec+ ∆EDop
The number of counts Niin channel i of the γ-spectrum, corresponding to the energy bin Eγito Eγi+ δEγ(δE =
dispersion in units of keV per channel) is given by the expression
for Epi≤ Ep(Epi= proton energy corresponding to channel i, Ep= incident proton energy), where σ(Epi) is the
capture cross section, ηfep(Eγi) is the γ-ray detection efficiency, ǫ(Epi) is the stopping power and bjis the branching
of the associated decay. The conversion from Eγito Epiincludes the Doppler and recoil effects. The result is folded
with the known detector resolution ∆Eγto obtain the experimental line-shape.
Finally, to infer the cross section in the energy window defined by the target thickness, the cross section was
written as the sum of two resonance terms and a constant, non-resonant term as described in . In these fits, the
free parameters were the non-resonant astrophysical S factor and the gamma-ray background parameters. The results
of both methods are in excellent agreement.
The absolute cross section for the15N(p,γ)16O reaction has been measured on top of the two broad resonances at
proton energies of 338 keV and 1028 keV using two independent well known reaction standards and two independent
methods. All measurements were performed at three distances (d = 1 cm, 5 cm, 20 cm) to check for systematic errors.
The exception to this protocol was the15N(p,γ)16O measurement at Ep= 338 keV, where the yield at 20 cm was too
low. In the first method the cross sections were determined relative to the thick target yield Y∞(27Al) of the well
known27Al(p,γ)28Si resonance at 992 keV:
M + m
with m (M) the mass of the projectile (target), λ the DeBroglie resonance wave length and ωγR the resonance
strength. Adopting the resonance strength of 1.93 ± 0.13 eV from [19, 29, 30] the cross sections values of σ338=6.7 ±
0.6 µb and σ1028=446 ± 40 µb are obtained. The errors include the statistical error (≤ 3 %), the error on the relative
efficiency (2 %), the error on the relative charge measurement (2%), the uncertainty on stopping power values (5%),
and the uncertainty of 7 % arising from the reference resonance strength of27Al.
In the second method the cross sections were determined relative to the well known 429 keV resonance in
15N(p,α1γ)12C . In this case the cross section was calculated relative to the integral over the yield curve of
the reference resonance, Aref. The resonance strength ωγ is related to Arefby:
with nt the number of target atoms per cm2which can be calculated from the measured target thickness and the
stopping power values. Using this relationship for Aref(Equation 7), along with Equation 6 and the thin target yield
approximation for the15N(p,γ)16O reaction, the following formula can be derived:
??M + m
with η the relative γ-ray efficiency. In this approach the result is independent of all target uncertainties. The
yield of the 4.4 MeV γ-rays from the15N(p,α1γ)12C was corrected for the well known angular distribution [31, 32].
Using the resonance strength of ωγR=21.2 ± 1.4 eV from , values of σ338=6.4 ± 0.6 µb and σ1028=436 ± 44 µb
are obtained. The errors include the statistical error (≤ 3 %), the error of the relative efficiency (2 %), the error of
the relative charge measurement (2%), the error of Aref owing to the numerical integration and angular distribution
correction (6%) and the uncertainty of 7 % of the reference value .
The weighted average of the results from the two methods gives final values of σ338 = (6.5 ± 0.3) µb (5%) and
σ1028= (438 ± 16) µb (4%) taking into account common error sources. The present results are compared to previous
values [12–14] in Table I and are in very good agreement. An exception is the cross section at 338 keV from Rolfs and
Rodney  which is about 50% higher than the other values. This discrepancy will be discussed in more detail in the
next section. Data from LUNA and the Notre Dame JN accelerators were normalized to the low energy resonance,
while data from the Notre Dame KN accelerator were normalized to the higher energy resonance. Good agreement
between the data sets was found at the overlapping energies around 700 keV.
In previous experiments small cascade transitions have been observed for the broad resonances at 338 keV and 1028
keV . For the 338 keV resonance a branching ratio of (1.2 ± 0.1) % for the transition to the 6050 keV level was
6.5 ± 0.3 9.6 ± 1.3
438 ± 16 420 ± 60
6.5 ± .3†
TABLE I: Summary of present resonance cross sections in comparison to previous results. *No uncertainty is given.
derived from given reproducibility of 5%.
found, in good agreement with the literature value of 1.2 ± 0.4 % . For the 1028 keV resonance, branches to the
6050 keV level (1.0 ± 0.3) % and to the 7118 keV level (2.8 ± 0.3) % were measured, and were also in good agreement
with literature value of (1.4 ± 0.4) % and (3.1 ± 0.8) %, respectively .
In previous analyses of the15N(p,γ0)16O reaction [12, 15, 16], a direct capture component was required to fit the
data. In each of these cases, only the li= 0 initial wave was included in the calculation. This component represents
s-wave capture, and therefore interferes with the resonant component of the cross section. The s-wave components
yield isotropic γ-ray distributions [12, 28]. However, an li= 2 direct capture component is allowed for E1 transitions,
and can contribute to the reaction, possibly introducing some anisotropy in the gamma decay. In looking for this
signature, the set-up was tested using the 1 MeV resonance, which is reported to be isotropic . Measurements were
made with the central Clover axis at several angles relative to the beam direction with a nominal distance of 5 cm. To
extract additional data points, the add-back feature was used to treat the left and right halves of the Clover system
as separate detectors, where Monte Carlo calculations using the code GEANT4  yielded an effective angular offset
of each half from the central detector axis of 15◦. Measurements with a calibrated137Cs source, and the isotropic 278
keV resonance of14N(p,γ)15O made corrections to the absorption effects possible. The results confirmed the isotropy
of the angular distribution of the ground state decay γ-rays from the 1028 keV resonance in15N(p,γ0)16O.
In the search for the lf= 2 component, angular distribution measurements of15N(p,γ0)16O were made at a proton
energy of 540 keV at detection angles of 0◦, 45◦, and 90◦relative to the beam axis. With two Clover segments, this
resulted in 6 data points. Two of these points were excluded due to large absorption effects from the shape of the
target chamber. The resulting angular distribution was fit with a function of the form W(θ) = aexp
where P2is the L = 2 Legendre polynomial, and aexp
= Q2a2, where Q2is the usual geometrical correction factor
due to the finite size of the detector . The Q2term can be directly determined using the16O(p,γ) DC → 495 keV
reaction . This reaction has a well known angular distribution of the form W(θ) = sin2(θ). Plotting the measured
W(θ) with respect to sin2(θ), the deviation of the slope from 1.0 gives directly the Q2value. Using the left and right
Clover halves to measure the16O(p,γ)17F DC→495 keV a value of Q2= 0.975±0.020 was found. This results in an
a2value for the15N(p,γ0)16O reaction at 540 keV of 0.08 ± 0.10, which is consistent with isotropy.
An R-matrix analysis was performed which mirrored the procedure of previous analyses [12, 15, 16]. This analysis
included the two broad 1−resonance levels and a direct capture contribution. The analysis was performed with the
multi-channel R-matrix code AZURE . Details of the theory and the nomenclature are given in  and references
therein. The best fit results can be seen in Figure 7 where the cross sections have been converted to the astrophysical
S-factor. This fit results in an S(0) value of 39.6 keV b.
Looking closer at the low energy region (see Figure 8) and comparing the results to previous experiments, the low
energy data of  are inconsistent with the present results. While this data set agrees very well at higher energies it
starts to deviate below a proton energy of 400 keV. There is no obvious reason for this inconsistency but it should be
noted that the low energy data of Rolfs and Rodney  carry a significant uncertainty in this region. The data of
Hebbard  are in reasonably good agreement above 230 keV. The data of Brochard  show good agreement except
for the three data points below the 338 keV resonance which show a large scatter.The recent Bemmerer data  are
systematically lower than the present data, and also seem to have a slightly different energy dependence.
The parameters for the best fit are given in Table II. With the exception of the first resonance, most parameters
are in fair agreement with previous results [15, 16]. The width of the first resonance is dominated by the alpha
width . This parameter is, therefore, well constrained by the present (p,γ) data. The resonance strength, however,
FIG. 7: Experimental S factor for the ground state transition of the15N(p,γ0)16O reaction shown together with the best fit
results of an R-matrix calculation using the code AZURE.
Jπ= l−lp = 0 lα = 1
Jπ= l−lp = 0 lα = 1
TABLE II: Best fit R-matrix parameters for the15N(p,γ)16O reaction (Ex in MeV, γ2
and aα = 6.500 fm were used in the fit. The boundary condition BC in the fit was chosen to be equal to the shift function
at the energy of the lowest Jπ=1−resonance. Because of this choice of BC, the Γγ1(in eV) can be calculated directly from
the reduced width amplitude while Γγ2is calculated from the Barker transformation of the reduced width amplitude . The
parameters shown in this table are mainly intended to enable the reader to reproduce the fit presented in this manuscript, and
should not be regarded as final. With respect to the physical interpretation of these fit parameters, a detailed discussion will
be presented in a forthcoming paper .
iin keV). Channel radii of ap = 5.030 fm
is determined by the product of the proton and γ width. Without including proton scattering data in the fit, these
parameters are not well constrained. However, it should be noted that tests showed that this ambiguity has no
influence on the extrapolation of the data. In these tests, the proton width (γp1) was fixed at different values while
allowing the other parameters to vary. For the higher energy resonance the proton and alpha width are comparable,
which provides more of a constraint on the parameters. In addition, the present result for the ground state ANC is
significantly larger than the value of  which cannot be attributed to the choice of radius (see below). While these
differences do not have an impact on the extrapolation of the S-factor to lower energies, a more thorough analysis is
warranted for the interpretation of the R-matrix parameters. This will be addressed in a forthcoming publication 
where the results of simultaneous multi-channel fits to all relevant reaction channels will be presented.
FIG. 8: Low energy region of present data set (filled circles) shown with AZURE R-matrix calculation along with previous
measurements of Rolfs (open squares), Hebbard (filled squares), Brochard (open triangles), and Bemmerer (open
diamonds) in the region of 0 - 400 keV
In the present parameter space the only non-s-wave contribution arises from the d-wave component of the direct
capture. Using the parameters for the best fit the γ-ray angular distribution was calculated at the minimum of the
cross section between the two resonance at 540 keV yielding a value of a2= 0.034. The result is consistent with the
experimental upper limit of 0.18 (see Section III C).
The sensitivity of the best fit was tested against several key parameters. In multi-parameter fitting, uncertainties
of specific parameters are determined in terms of confidence regions. For a nine parameter fit (Eλ,γifor both levels
plus ANC or θg.s.) a 70% confidence region for one of the parameters is defined by the range where χ2≤ χ2
10 . The degrees of freedom is 102 (113 data points, 9 parameters) resulting in a reduced χ2of 1.8.
Radii of rp= 5.03 fm, and rα= 6.5 fm, were taken from . The dependence of the fit on the proton radius was
tested, and the results are given in Figure 9(a). Any proton radius value between 4 and 6 fm is considered acceptable.
Variation of the radius over this range corresponds to only a 5% uncertainty in the S(0) extrapolation. This test also
showed that the ANC is not very sensitive to the choice of the radius. The best fit gives a ground state ANC of (23
± 3) fm−1/2(corresponding to a reduced width amplitude of θg.s.= 0.61). The error associated with the ANC was
determined by fixing the ANC at different values, and finding a new best fit. The results of this procedure can be
seen in Figure 9(b). Even though the results give a 13% uncertainty for the ANC, the variation in S(0) from this
procedure is only 4%.
To evaluate the uncertainty of the fit itself, fits were performed which were forced to result in different S(0) values
by inserting a fake data point at 1.5 keV with extremely small errors. The small error of this data point forces the
calculation to match this fictitious data point and thus vary the extrapolation. By varying the fictitious point and
observing the change in the χ2
tot, the experimental uncertainty of the extrapolation can be determined (see Figure 10).
This results in an uncertainty of ± 0.6 keV b. Including the 5% error of the absolute cross section gives a final S(0)
value of (39.6 ± 2.6) keV b. This value is compared in Table III with previous extrapolations [12, 13, 15–17]. The
present result is in good agreement with the original value of Hebbard , the Barker  analysis of the Hebbard
data (labelled HH in Table III, and the results of Mukhamedzhanov et al. . The extrapolation of the Rolfs and
Total χ2 Value
Proton Radius Value [fm]
S(0) [keV b]
Total χ2 Value
17 18 19 20 21 22 23 24 25 26 27 28 29 30
Fixed ANC [fm
S(0) [keV b]
FIG. 9: Dependence of AZURE fits on key parameters used in the fits. Parameters were varied noting that for a 9 parameter
fit, a 70% confidence region is defined for ≈ χ2= χ2
4.4 fm, 5.03 was chosen so as to be more comparable to Barker . ANC was found to be (23 ± 3) fm−1/2. Fixing the ANC
to the value found in  gives a χ2that is much higher than the current best fit value.
min+ 10 . Even though a lower χ2was found with a radius of about
Rodney data  (Rolfs and Rodney , Simpson  and Barker , labelled RR in Table III) are all significantly
higher because of the larger low energy cross sections used for their fit analysis.
Hebbard 1960 
Rolfs 1974 
Barker 2008 (RR) 
Barker 2008 (HH) 
S(0)γ (keV b)
64 ± 6
36.0 ± 6.0
39.6 ± 2.6
TABLE III: Summary of Previous Results for extrapolated S(0) values
Using the results from the AZURE extrapolations, the reaction rates can be numerically determined using the
formalism outlined in 
NA?σv? = 3.73 · 107µ−1
with µ the reduced mass, T the temperature in GK, and η the Sommerfeld parameter; the results are in cm3mole−1
s−1. The above function was numerically integrated for temperatures between T = 0.01 - 10 GK. The present data
covers an energy range of E = 0 to 1.8 MeV, which validates the integration up to a temperature of T = 1 GK. For
higher temperatures, the S-factor curve must be extended to higher energies beyond what the present experimental
data cover. We followed the procedure of the NACRE compilation  which handles this difficulty by equating all
higher energy S-factor values with the highest energy data point where the cross section varies only slowly with energy.
The results are given in Table IV. The present rate is at lower temperatures up to a factor two lower than previous
rates reflecting the change in the low energy S-factor.
FIG. 10: Variation of χ2with extrapolated S(0) values (for details, see text).
Following the example of the NACRE compilations, the above results were fit using the following parametrization
NA< σv > = a1109T−3
[1 + a4T + a5T2] + a6103T−3
exp(a7/T) + a8106T−3
where the best fit parameters can be found in Table V.
The result of this work clearly demonstrates that there are significant uncertainties in the low energy cross section
data of radiative capture processes of astrophysical relevance, despite many decades of low energy reaction studies.
These uncertainties affect directly our understanding and interpretation of solar and stellar hydrogen burning phe-
nomena. In this case the new results influence primarily the leakage rate from the CN to the ON cycle in stellar
burning via the15N(p,γ)16O radiative capture process, which is reduced by a factor of two compared to the previous
rate used traditionally in CNO nucleosynthesis simulations. In particular, the change in rate will modify the equilib-
rium abundance of16O, which is correlated with the leakage rate of15N(p,γ)16O from the CN cycle and the rate of
16O(p,γ)17F in the NO cycle. However a detailed study of the astrophysical impact of the present measurement goes
beyond the aim of the present work, but should benefit from recent studies of low energy reaction rates .
The reliability of stellar reaction rates depends critically on the quality of the experimental cross section data.
Direct measurements of the reaction cross sections at the Gamow range of stellar burning have been successful in
only a few cases of reactions between light nuclei. Considering the anticipated count rates for the CNO radiative
capture reactions we will continue to rely for most cases on the extrapolation of low energy measurements into the
Gamow range. The present analysis clearly demonstrates that this requires a two-fold approach, pursuing the direct
T9 [GK] NA?σv?Present NA?σv?NACRE NA?σv?CA88
0.070 3.747E-07 7.64E-076.14E-07
0.150 1.280E-032.46E-03 1.92E-03
0.200 1.908E-023.52E-02 3.09E-02
0.500 1.668E+01 2.58E+015.42E+01
2.000 4.735E+035.05E+03 6.61E+03
TABLE IV: Table of reaction rates for15N(p,γ0)16O. Present rates, along with the NACRE , and CA88  results are given.
a1 = 0.523
a2 = -15.240 a5 = -2.164 a8 = 3.048
a3 = 0.866a6 = 0.738 a9 = -9.884
a4 = 6.339 a7 = -2.913
TABLE V: Best fit parameters for the15N(p,γ0)16O reaction rate using the NACRE fitting formulations.
reaction measurements to the lowest possible energies in a background shielded environment but also expanding
the experimental range of the measurements to determine unambiguously the various reaction components of the
radiative capture process. The latter step is essential for minimizing the uncertainties in the R-matrix analysis of the
cross section and can be complemented by independent studies which explore independently the strength of specific
”hidden” reaction components such as the direct capture through ANC measurements and analysis.
The approach taken here for the study of the15N(p,γ)16O reaction has succeeded in combining both the efforts of
improving on the extent and quality of the low energy cross section data in underground accelerator experiments. At
the same time the study has improved on the detailed measurement of higher energy data providing a better constraint
on determining the external capture component and its impact on the low energy extrapolation of the reaction cross
section. The combination of these two complementary measurements successfully reduced the overall uncertainty in
the15N(p,γ)16O reaction rate.
We are extremely grateful for the help of the technical staff of both the Nuclear Science Laboratory at the University
of Notre Dame and that of the Gran Sasso facility. REA thanks the NSERC for partial financial support through
the DRAGON grant at TRIUMF. This work was funded in part by the National Science Foundation through grant
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