$R$-matrix analysis of ${}^{16}$O compound nucleus reactions

$R$ -matrix analysis of $^{16}$ O compound nucleus reactions

R. J. deBoer, J. Görres, G. Imbriani, P. J. LeBlanc, E. Uberseder, and M. Wiescher

Phys. Rev. C 87, 015802 – Published 16 January 2013

Abstract

Background: Over the past 60 years, a large amount of experimental nuclear data have been obtained for reactions which probe the $^{16}$ O compound nucleus near the $α$ and proton separation energies, the energy regimes most important for nuclear astrophysics. Difficulties and inconsistencies in $R$ -matrix fits of the individual reactions prompt a more complete analysis.

Purpose: Determine the level of consistency between the wide variety of experimental data using a multiple entrance/exit channel $R$ -matrix framework. Using a consistent set of data from multiple reaction channels, attain an improved fitting for the $^{15}$ N ${(p, γ_{0})}^{16}$ O reaction data.

Methods: Reaction data for all available reaction channels were fit simultaneously using a multichannel $R$ -matrix code.

Results: Over the wide range of experimental data considered, a high level of consistency was found, resulting in a single consistent $R$ -matrix fit which described the broad level structure of $^{16}$ O below $E_{x} = 13.5$ MeV. The resulting fit was used to extract an improved determination of the low-energy $S$ factor for the reactions $^{15}$ N ${(p, γ)}^{16}$ O and $^{15}$ N ${(p, α)}^{12}$ C.

Conclusion: The feasibility and advantages of a complete multiple entrance/exit channel $R$ -matrix description for the broad level structure of $^{16}$ O has been achieved. A future publication will investigate the possible effects of the multiple-channel analysis on the reaction $^{12}$ C ${(α, γ)}^{16}$ O.

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Received 2 October 2012

DOI:https://doi.org/10.1103/PhysRevC.87.015802

Authors & Affiliations

R. J. deBoer^1,*, J. Görres¹, G. Imbriani^1,2, P. J. LeBlanc¹, E. Uberseder¹, and M. Wiescher¹

¹Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA
²Università degli Studi di Napoli “Federico II” and INFN, Napoli, Italy

^*rdeboer1@nd.edu

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Vol. 87, Iss. 1 — January 2013

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Figure 1
The diagram on the left shows the level structure of $^{16}$ O up to $E_{x} = 14$ MeV. Only levels and subthreshold states included in this analysis are shown (see Table of Appendix 8). On the right is an enlarged view of the energy region above the proton separation energy at $E_{x} = 12.13$ MeV. Entrance (exit) channels are displayed on the left (right). Particle separation energies are indicated by horizontal red dashed lines. Excitation energies are in units of MeV.Reuse & Permissions
Figure 2
The $^{15}$ N ${(p, γ_{0})}^{16}$ O data from Ref. [10]. The dotted-dashed, solid, and dashed lines correspond to Fits 6, 3, and 2 from Table , respectively.Reuse & Permissions
Figure 3
Fit to the angle-integrated $^{15}$ N ${(p, γ_{0})}^{16}$ O cross-section data of Refs. [5, 6, 10], and [26] labeled (a) through (d), respectively. Laboratory proton energy is given on the top horizontal axis and the excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 4
Fit to the total $^{15}$ N ${(p, γ)}^{16}$ O cross-section data of Refs. [35, 36] normalized to the $R$ -matrix fit. The inset shows the calculation at higher energies based on the sum of the contributions from the cascade transitions of Ref. [40]. Laboratory proton energy is given on the top horizontal axis and the excitation energy on the bottom horizontal axis of the plot.Reuse & Permissions
Figure 5
Fits to the $^{15}$ N ${(p, γ_{0})}^{16}$ O, $^{15}$ N ${(p, γ_{(6.050)})}^{16}$ O, $^{15}$ N ${(p, γ_{(6.130)})}^{16}$ O, and $^{15}$ N ${(p, γ_{(7.117)})}^{16}$ O data, labeled (a) through (d), respectively, reported in Ref. [40]. Laboratory proton energy is given on the top horizontal axis and the excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 6
Fits to the angle-integrated $^{15}$ N ${(p, α_{0})}^{12}$ C data of Refs. [26, 31], and [34] [labeled (a) through (c), respectively]. Laboratory proton energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 7
Fits to the $^{15}$ N ${(p, α_{0})}^{12}$ C differential cross-section data of Refs. [18] ( $θ_{lab} = 90^{\circ}$ ), [30] ( $θ_{lab} = 135^{\circ}$ ), and [21] ( $θ_{lab} = 159 . 5^{\circ}$ ) [labeled (a), (b), and (c), respectively]. Laboratory proton energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 8
Fits to the $^{15}$ N ${(p, α_{0})}^{12}$ C differential cross sections of Ref. [20] at $θ_{lab} = 25^{\circ}$ , 40 $^{\circ}$ , 60 $^{\circ}$ , 75 $^{\circ}$ , 90 $^{\circ}$ , 105 $^{\circ}$ , 120 $^{\circ}$ , and 140 $^{\circ}$ [labeled (a) through (h), respectively]. Laboratory proton energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 9
Fits to the $^{15}$ N ${(p, α_{0})}^{12}$ C angular distributions of Ref. [29] at $E_{p} = 0.838$ , 0.888, 0.938, 0.988, 1.038, 1.088, 1.1113, and 1.138 MeV [labeled (a) through (h), respectively].Reuse & Permissions
Figure 10
Fits to the $^{15}$ N ${(p, α_{0})}^{12}$ C angular distributions of Ref. [29] at $E_{p} = 1.163$ , 1.188, 1.213, 1.214, 1.238, and 1.263 MeV [labeled (a) through (f), respectively].Reuse & Permissions
Figure 11
Fits to the $^{15}$ N ${(p, α_{0})}^{12}$ C angular distributions of Ref. [31] at $E_{p} = 0.149$ , 0.247, 0.344, 0.450, 0.600, and 0.750 MeV [labeled (a) through (f), respectively].Reuse & Permissions
Figure 12
Fits to the $^{15}$ N ${(p, α_{0})}^{12}$ C yield-ratio data of Ref. [38] at $θ_{lab} = 105^{\circ}$ , 135 $^{\circ}$ , 150 $^{\circ}$ , and 165 $^{\circ}$ [labeled (a) through (d) respectively]. Laboratory proton energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 13
Fits to the $^{15}$ N ${(p, p)}^{15}$ N cross-section data from Ref. [19] (circles) at $θ_{lab} = 86^{\circ}$ , 122 $^{\circ}$ , and 159 $^{\circ}$ [labeled (a) through (c) respectively] and from Ref. [21] (pluses) at $θ_{lab} = 86^{\circ}$ and 159 $^{\circ}$ [labeled (a) and (c) respectively]. An uncertainty of 3 $%$ has been assumed for each data point. Laboratory proton energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 14
Fits to the $^{15}$ N ${(p, p)}^{15}$ N angular distributions of Ref. [21] at $E_{p} = 1.214$ and 1.643 MeV [labeled (a) and (b), respectively].Reuse & Permissions
Figure 15
Fits to the $^{15}$ N ${(p, p)}^{15}$ N angular distributions of Ref. [21] at $E_{p} = 0.650$ , 0.675, 0.700, 0.725, 0.750, and 0.775 MeV [labeled (a) and (f), respectively].Reuse & Permissions
Figure 16
Fits to the $^{15}$ N ${(p, p)}^{15}$ N angular distributions of Ref. [21] at $E_{p} = 0.80$ , 0.85, 0.95, 1.00, 1.05, and 1.10 MeV [labeled (a) and (f) respectively].Reuse & Permissions
Figure 17
Fits to the $^{15}$ N ${(p, p)}^{15}$ N angular distributions of Ref. [21] at $E_{p} = 1.15$ , 1.30, and 1.50 MeV [labeled (a) and (c) respectively].Reuse & Permissions
Figure 18
Fits to the $^{15}$ N ${(p, p)}^{15}$ N yield-ratio data of Ref. [38] at $θ_{lab} = 105^{\circ}$ , 135 $^{\circ}$ , 150 $^{\circ}$ , and 165 $^{\circ}$ [labeled (a) through (d), respectively]. Laboratory proton energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 19
Fits to the $^{15}$ N ${(p, α_{1})}^{12}$ C differential cross sections of Ref. [21] at $θ_{lab} = 86 . 2^{\circ}$ , 90 $^{\circ}$ , and 159.5 $^{\circ}$ [labeled (a), (b), and (c) respectively]. Laboratory proton energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 20
Fit to the $^{15}$ N ${(p, α_{1})}^{12}$ C angular distribution of Ref. [21] at $E_{p} = 1.214$ MeV.Reuse & Permissions
Figure 21
Fit to the $^{15}$ N ${(p, α_{1} γ_{1})}^{12}$ C angle-integrated cross-section data of Ref. [10]. Laboratory proton energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 22
Fits to the $^{12}$ C ${(α, p)}^{15}$ N differential cross sections of Ref. [52] at $θ_{lab} = 30^{\circ}$ , 40 $^{\circ}$ , 90 $^{\circ}$ , and 120 $^{\circ}$ [labeled (a) through (d), respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 23
Fits to the $^{12}$ C ${(α, α_{1})}^{12}$ C differential cross sections of Ref. [52] at $θ_{lab} = 40^{\circ}$ , 90 $^{\circ}$ , and 120 $^{\circ}$ [labeled (a) through (d), respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 24
Fits to the $^{12}$ C ${(α, α_{1})}^{12}$ C differential yields of Ref. [39] at $θ_{lab} = 24 . 0^{\circ}$ , 33.9 $^{\circ}$ , 38.9 $^{\circ}$ , 43.9 $^{\circ}$ , 63.9 $^{\circ}$ , 68.9 $^{\circ}$ , 74.0 $^{\circ}$ , and 79.0 $^{\circ}$ [labeled (a) through (h), respectively]. The $(α, α_{1})$ yields were divided by the corresponding $(a, a_{0})$ yields. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 25
Fits to the $^{12}$ C ${(α, α_{1})}^{12}$ C differential yields of Ref. [39] at $θ_{lab} = 80 . 8^{\circ}$ , 84.0 $^{\circ}$ , 85.8 $^{\circ}$ , 89.0 $^{\circ}$ , 90.8 $^{\circ}$ , 94.0 $^{\circ}$ , 95.8 $^{\circ}$ , and 99.0 $^{\circ}$ [labeled (a) through (h), respectively]. The $(α, α_{1})$ yields were divided by the corresponding $(a, a_{0})$ yields. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 26
Fits to the $^{12}$ C ${(α, p)}^{12}$ C differential yields of Ref. [39] at $θ_{lab} = 24 . 0^{\circ}$ , 33.9 $^{\circ}$ , 38.9 $^{\circ}$ , 43.9 $^{\circ}$ , 48.9 $^{\circ}$ , 54.0 $^{\circ}$ , 68.9 $^{\circ}$ , and 74.0 $^{\circ}$ [labeled (a) through (h), respectively]. The $(α, p)$ yields were divided by the corresponding $(a, a_{0})$ yields. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 27
Fits to the $^{12}$ C ${(α, p)}^{12}$ C differential yields of Ref. [39] at $θ_{lab} = 79 . 0^{\circ}$ , 80.8 $^{\circ}$ , 84.0 $^{\circ}$ , 85.8 $^{\circ}$ , 89.0 $^{\circ}$ , 90.8 $^{\circ}$ , 94.0 $^{\circ}$ , and 95.8 $^{\circ}$ [labeled (a) through (h), respectively]. The $(α, p)$ yields were divided by the corresponding $(a, a_{0})$ yields. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 28
Fits to the $^{12}$ C ${(α, p)}^{12}$ C differential yields of Ref. [39] at $θ_{lab} = 99 . 0^{\circ}$ , 100.8 $^{\circ}$ , 103.9 $^{\circ}$ , 105.8 $^{\circ}$ , 110.8 $^{\circ}$ , 115.8 $^{\circ}$ , 120.8 $^{\circ}$ , and 125.8 $^{\circ}$ [labeled (a) through (h), respectively]. The $(α, p)$ yields were divided by the corresponding $(a, a_{0})$ yields. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 29
Fits to the $^{12}$ C ${(α, p)}^{12}$ C differential yields of Ref. [39] at $θ_{lab} = 130 . 8^{\circ}$ , 140.8 $^{\circ}$ , 150.8 $^{\circ}$ , and 160.8 $^{\circ}$ [labeled (a) through (c), respectively]. The $(α, p)$ yields were divided by the corresponding $(a, a_{0})$ yields. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 30
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C differential cross-section data of Ref. [15] at $θ_{lab} = 24^{\circ}$ , 33.9 $^{\circ}$ , 38.9 $^{\circ}$ , 43.9 $^{\circ}$ , 48.9 $^{\circ}$ , 54.0 $^{\circ}$ , 58.9 $^{\circ}$ , and 63.9 $^{\circ}$ [labeled (a) through (h), respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 31
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C differential cross-section data of Ref. [15] at $θ_{lab} = 68 . 9^{\circ}$ , 74.0 $^{\circ}$ , 75.8 $^{\circ}$ , 79 $^{\circ}$ , 80.8 $^{\circ}$ , 84.0 $^{\circ}$ , 85.8 $^{\circ}$ , and 89.0 $^{\circ}$ [labeled (a) through (h), respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 32
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C differential cross-section data of Ref. [15] at $θ_{lab} = 90 . 8^{\circ}$ , 94.0 $^{\circ}$ , 95.8 $^{\circ}$ , 99.0 $^{\circ}$ , 100.8 $^{\circ}$ , 103.9 $^{\circ}$ , 105.8 $^{\circ}$ , and 110.8 $^{\circ}$ [labeled (a) through (h), respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 33
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C differential cross-section data of Ref. [15] at $θ_{lab} = 115 . 8^{\circ}$ , 120.8 $^{\circ}$ , 125.8 $^{\circ}$ , 130.8 $^{\circ}$ , 140.8 $^{\circ}$ , 150.8 $^{\circ}$ , 160.8 $^{\circ}$ , and 165.9 $^{\circ}$ [labeled (a) through (h) respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 34
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C differential cross-section data of Ref. [24] at $θ_{lab} = 71 . 55^{\circ}$ , 104.5 $^{\circ}$ , 106.6 $^{\circ}$ , 124.89 $^{\circ}$ , 136.06 $^{\circ}$ , and 167 $^{\circ}$ [labeled (a) through (f) respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 35
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C angular distribution data of Ref. [24] at $E_{α} = 6.600$ , 6.885, 6.965, 6.99, 7.064, and 7.093 MeV [labeled (a) through (f), respectively].Reuse & Permissions
Figure 36
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C angular distribution data of Ref. [24] at $E_{α} = 7.167$ , 7.400, 7.500, 7.700, 7.767, and 7.800 MeV [labeled (a) through (f) respectively].Reuse & Permissions
Figure 37
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C angular distribution data of Ref. [24] at $E_{α} = 7.845$ , 7.900, 7.960, 8.015, 8.050, and 8.113 MeV [labeled (a) through (f) respectively].Reuse & Permissions
Figure 38
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C angular distribution data of Ref. [24] at $E_{α} = 8.196$ , 8.247, 8.300, and 8.500 MeV [labeled (a) through (d) respectively].Reuse & Permissions
Figure 39
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C differential cross-section data of Ref. [33] at $θ_{lab} = 165^{\circ}$ ). Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of the plot.Reuse & Permissions
Figure 40
Fits to the phase shifts from Ref. [27] for $l = 0$ , 1, 2, 3, 4, and 5 [labeled (a) through (f) respectively] included in the $R$ -matrix analysis. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 41
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C yield data of Ref. [39] at $θ_{lab} = 24 . 0^{\circ}$ , 33.9 $^{\circ}$ , 38.9 $^{\circ}$ , 43.9 $^{\circ}$ , 48.9 $^{\circ}$ , 54.0 $^{\circ}$ , 63.9 $^{\circ}$ , and 68.9 $^{\circ}$ [labeled (a) through (h) respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 42
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C yield data of Ref. [39] at $θ_{lab} = 74 . 0^{\circ}$ , 75.8 $^{\circ}$ , 80.8 $^{\circ}$ , 84.0 $^{\circ}$ , 85.8 $^{\circ}$ , 89.0 $^{\circ}$ , 90.8 $^{\circ}$ , and 94.0 $^{\circ}$ [labeled (a) through (h), respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 43
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C yield data of Ref. [39] at $θ_{lab} = 95 . 8^{\circ}$ , 99.0 $^{\circ}$ , 100.8 $^{\circ}$ , 103.9 $^{\circ}$ , 105.8 $^{\circ}$ , 110.8 $^{\circ}$ , 115.8 $^{\circ}$ , and 120.8 $^{\circ}$ [labeled (a) through (h) respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 44
Fits to the $^{12}$ C ${(α, α_{0})}^{12}$ C yield data of Ref. [39] at $θ_{lab} = 125 . 8^{\circ}$ , 130.8 $^{\circ}$ , 140.8 $^{\circ}$ , 150.8 $^{\circ}$ , 160.8 $^{\circ}$ , and 165.9 $^{\circ}$ [labeled (a) through (f), respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 45
Fits to the angle-integrated cross section $^{12}$ C ${(α, γ)}^{16}$ O of Refs. [16, 37] for the ground state and 6.049-, 6.130-, 6.917-, and 7.117-MeV transitions and for the total capture cross section [labeled (a) through (f), respectively]. At energies above the proton separation energy, the cross section is determined by other available reaction-channel measurements. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 46
Fits to the $^{12}$ C ${(α, γ_{0})}^{16}$ O differential cross-section data at 45 $^{\circ}$ (Ref. [22]), 61 $^{\circ}$ (Ref. [25]), 90 $^{\circ}$ (Refs. [25, 28]), and 135 $^{\circ}$ (Ref. [25]) [labeled (a) through (d) respectively]. Laboratory $α$ -particle energy is given on the top horizontal axis and excitation energy on the bottom horizontal axis of each plot.Reuse & Permissions
Figure 47
Fits to the $^{12}$ C ${(α, γ_{0})}^{16}$ O angular distribution data of Refs. [22, 25] at $E_{α} = 7.06$ , 7.42, 7.70, 7.88, 8.00, and 8.10 MeV [labeled (a) through (f), respectively].Reuse & Permissions
Figure 48
Comparison of the $S$ factors for the different data sets and corresponding $R$ -matrix calculations in the energy region above $S_{p}$ . Part (a) shows the angle-integrated $^{12}$ C ${(α, γ_{0})}^{16}$ O data of Ref. [26], (b) the $^{12}$ C ${(α, α_{0})}^{12}$ C data of Refs. [24, 33, 48] at $θ_{lab} \approx$ 165 $^{\circ}$ , (c) the $^{15}$ N ${(p, γ_{0})}^{16}$ O data of Ref. [10], (d) the $^{15}$ N ${(p, α_{0})}^{12}$ C data of Refs. [26, 31], (e) the angle-integrated $^{15}$ N ${(p, α_{1} γ)}^{12}$ C data of Ref. [40], (f) the $^{15}$ N ${(p, p)}^{15}$ N data of Ref. [19] at $θ_{lab} = 158 . 7^{\circ}$ , (g) the $^{12}$ C ${(α, α_{1})}^{12}$ C data from Ref. [23] at $θ_{lab}$ $=$ 90 $^{\circ}$ , and (h) the $^{12}$ C ${(α, p)}^{15}$ N data of Ref. [23] at $θ_{lab}$ $=$ 90 $^{\circ}$ . The black dashed lines indicate the energies of the two 1 $^{-}$ levels at $E_{x} = 12.45$ and 13.09 MeV.Reuse & Permissions
Figure 49
Comparison of the different reaction components included in the fitting of the $^{15}$ N ${(p, γ_{0})}^{16}$ O data. Only the ground-state data from Ref. [10] are shown for clarity. The full fit is shown by the dashed black line, the resonance components by the solid red line for the 1 $^{-}$ levels and dot-dash-dashed blue line for the 2 $^{+}$ level, the ground-state external capture by the dot-dot-dashed black line, and the background pole by the dashed-dotted black line.Reuse & Permissions
Figure 50
An example $^{15}$ N ${(p, α_{0})}^{12}$ C angular distribution measurement at $E_{p} = 0.75$ MeV from Ref. [31]. The solid red line represents the current $R$ -matrix fit to the data including the two 1 $^{-}$ levels at $E_{x} = 12.45$ and 13.09 MeV and the 2 $^{+}$ level at $E_{x} = 12.97$ MeV. In contrast, the dashed blue line represents the angular distribution which would result if only the two 1 $^{-}$ levels were included in the calculation.Reuse & Permissions
Figure 51
Comparison of the different reaction components included in the fitting of the $^{15}$ N ${(p, α_{0})}^{12}$ C data. Only data from Refs. [26, 31] are shown for clarity. The full fit is shown by the dashed black line, the resonance components by the solid red for the 1 $^{-}$ levels, dot-dash-dashed blue line for the 2 $^{+}$ levels, and dot-dashed green line for the 3 $^{-}$ levels.Reuse & Permissions
Figure 52
Comparison of the different reaction components included in the fitting of the $^{15}$ N ${(p, α_{1})}^{12}$ C data. The data are from Ref. [40]. The full fit is shown by the dashed black line, the resonance components by solid red lines for the 1 $^{-}$ levels, the dot-dot-dashed maroon line for the 1 $^{+}$ level, the dot-dash-dashed blue line for the 2 $^{+}$ level, the dotted violet lines for the 2 $^{-}$ levels, and the dot-dashed green lines for the 3 $^{-}$ levels.Reuse & Permissions

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