Asymptotic normalization coefficients and spectroscopic factors from deuteron stripping reactions

We present the analysis of three deuteron stripping reactions, ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C},{}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}$, and ${}^{116}\mathrm{Sn}(d,p){}^{117}\mathrm{Sn}$ using the combined method [A. M. Mukhamedzhanov and F. M. Nunes, Phys. Rev. C 72, 017602 (2005)], in which each reaction is analyzed at low and significantly higher energies. At low energies all these reactions are peripheral and the experimental asymptotic normalization coefficients (ANCs) are determined with accuracy about $10\%$. At higher energies we determine the spectroscopic factors (SFs) by fixing the normalization of the peripheral parts of the reaction amplitudes governed by the ANCs found from the low-energy data. The combined method imposes a strict limitation on the variation of the geometrical parameters of the single-particle potential, which can be arbitrarily taken in the standard approach. By checking the compatibility of the ANCs and SFs using the combined method we reveal the flaw in the contemporary nuclear reaction theory in treating the nuclear interior, which is the most crucial part in the determination of the SFs.

Figure 1

Angular distributions of the DWBA, ADWA, and CDCC differential cross sections for the deuteron stripping ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}\left(2s_{1/2}\right)$ at $E_d=23.4$ MeV. The blue short and green dashed lines are the post-DWBA and -ADWA differential cross sections, correspondingly. The solid red line is the post-CDCC calculation. The theoretical differential cross sections are normalized to the experimental one at forward angles. The dots are the experimental data from Ref. [5].

Figure 2

$r_0$ dependence of the ANCs normalized to unity at $r_0=1$ in the DWBA, ADWA and CDCC for the deuteron stripping ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}\left(2s_{1/2}\right)$ at $E_d=23.4$ MeV. The blue square dots and the short dashed line is the ANC determined from the DWBA, the green open dots and the dashed line is the ANC obtained from the ADWA, and the solid red dots and the solid line is the ANC obtained from the CDCC. For simplicity, the subscripts denoting the quantum numbers in the ANC are omitted.

Figure 3

$r_0$ dependence of the SFs normalized to unity at $r_0=1$ in the DWBA, ADWA, and CDCC for the deuteron stripping ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}\left(2s_{1/2}\right)$ at $E_d=23.4$ MeV. The blue square dots and the short dashed line is the SF determined from the DWBA, the open green dots and the dashed line is the SF obtained from the ADWA, and the red solid dots and the solid line is the SF obtained from the CDCC. The magenta dotted line is the $r_0$ dependence of the normalized SF in the case of pure peripheral reaction. For simplicity, the subscripts denoting the quantum numbers in the SF are omitted.

Figure 4

Angular distributions of the DWBA, ADWA, and CDCC differential cross sections for the deuteron stripping ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}\left(2s_{1/2}\right)$ at $E_d=60$ MeV. The blue short and green dashed lines are the post-DWBA and -ADWA differential cross sections, correspondingly. The solid red line is the CDCC calculation. The theoretical differential cross sections are normalized to the experimental one at forward angles. The dots are the experimental data from Ref. [5].

Figure 5

$r_0$ dependence of the ANCs normalized to unity at $r_0=1$ in the DWBA, ADWA, and CDCC for the deuteron stripping ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}\left(2s_{1/2}\right)$ at $E_d=60$ MeV. The blue sqaue dots and short dashed line is the ANC determined from the DWBA, the green open dots and dashed line is the ANC obtained from the ADWA, and the solid red dots and the solid line is the ANC obtained from the CDCC. For simplicity, the subscripts denoting the quantum numbers in the ANC are omitted.

Figure 6

$r_0$ dependence of the SFs normalized to unity at $r_0=1$ in the DWBA, ADWA, and CDCC for the deuteron stripping ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}\left(2s_{1/2}\right)$ at $E_d=60$ MeV. The blue square dots and the short dashed line is the SF determined from the DWBA, the open green dots and dashed line is the SF obtained from the ADWA, and the red solid dots and the solid line is the SF obtained from the CDCC. The magenta dotted line is the $r_0$ dependence of the normalized SF in the case of pure peripheral reaction. For simplicity, the subscripts denoting the quantum numbers in the SF are omitted.

Figure 7

Dependence of the normalized CDCC differential cross sections $R_x$ on $r_{nA}$ for the deuteron stripping ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}\left(2s_{1/2}\right)$ at $E_d=23.4$ and 60 MeV. The green dashed and the magenta dotted lines are $R_X$ at $23.4$ and 60 MeV, correspondingly. To get these $R_x$, we calculated the post-CDCC differential cross section at the peak of the angular distribution. The radial integral over $r_{nA}$ is calculated for $r_{nA}^{\min }\le r_{nA}<\infty$. The calculated differential cross section is normalized to the full differential cross section calculated at the peak of the angular distribution. Similarly, the solid red and the blue short dashed lines are the post-CDCC $R_X$ calculated at $23.4$ and 60 MeV, correspondingly, in which the radial integral over $r_{nA}$ is calculated in the interval $0\le r_{nA}\le r_{nA}^{\max }$. Again, the calculated differential cross sections are normalized to the full differential cross sections at the corresponding energies. Hence, $r_{nA}$ on the abscissa is $r_{nA}^{\min }$ for the green dashed and magenta dotted lines and $r_{nA}^{\max }$ for the solid red and blue short dashed lines.

Figure 8

Comparison of the $r_0$ dependence of the ANCs in the CDCC for the deuteron stripping ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}\left(2s_{1/2}\right)$ at $E_d=23.4$ and 60 MeV. The open green dots is the ANC $r_0$ dependence from the $23.4$-MeV reaction and the solid red dots are the ANC $r_0$ dependence from the 60-MeV reaction. For simplicity, the subscripts denoting the quantum numbers in the ANC are omitted.

Figure 9

Angular distributions of the DWBA, ADWA, and CDCC differential cross sections for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(2p_{3/2}\right)$ at $E_d=10$ MeV. Notations are the same as in Fig. 1. Dots are the experimental data from Ref. [15]. All the calculated angular distributions are normalized to the experimental one at the first stripping peak.

Figure 10

$r_0$ dependence of the ANCs normalized to unity at $r_0=1$ in the DWBA, ADWA, and CDCC for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(2p_{3/2}\right)$ at $E_d=10$ MeV. Notations are the same as in Fig. 2. For simplicity, the subscripts denoting the quantum numbers in the ANC are omitted.

Figure 11

$r_0$ dependence of the SFs normalized to unity at $r_0=1$ in the DWBA, ADWA, and CDCC for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(2p_{3/2}\right)$ at $E_d=10$ MeV. Notations are the same as in Fig. 3. For simplicity, the subscripts denoting quantum numbers are omitted in the SF.

Figure 12

Angular distributions of the DWBA, ADWA, and CDCC differential cross sections for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(2p_{3/2}\right)$ at $E_d=56$ MeV. Notations are the same as in Fig. 1. Dots are the experimental data from Refs. [16, 17]. The calculated angular distributions are normalized to the experimental one at the forward peak.

Figure 13

Dependence of the normalized ADWA differential cross sections $R_x$ on $r_{nA}$ for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(1d_{3/2}\right)$ at $E_d=10$ and 56 MeV. The green dashed and magenta dotted lines are $R_X$ at 10 and 56 MeV, correspondingly. To get these $R_x$, we calculated the post-ADWA differential cross section at the peak of the angular distribution. The radial integral over $r_{nA}$ is calculated for $r_{nA}^{\min }\le r_{nA}<\infty$. The calculated differential cross section is normalized to the full differential cross section. Similarly, the solid red and blue short dashed lines are the post-ADWA $R_X$ calculated at 10 and 56 MeV, correspondingly, in which the radial integral over $r_{nA}$ is calculated in the interval $0\le r_{nA}\le r_{nA}^{\max }$. Again the calculated differential cross sections are normalized to the full differential cross sections at the corresponding energies. Hence, $r_{nA}$ on the abscissa is $r_{nA}^{\min }$ for the green dashed and magenta dotted lines and $r_{nA}^{\max }$ for the solid red and blue short dashed lines.

Figure 14

$r_0$ dependence of the ANCs normalized to unity at $r_0=1$ in the ADWA and CDCC for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(2p_{3/2}\right)$ at $E_d=56$ MeV. Notations are the same as in Fig. 2. For simplicity, the subscripts denoting quantum numbers are omitted in the ANC.

Figure 15

$r_0$ dependence of the normalized to unity at $r_0=1$ SFs in the ADWA and CDCC for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(2p_{3/2}\right)$ at $E_d=56$ MeV. Notations are the same as in Fig. 3. For simplicity, the subscripts denoting quantum numbers are omitted in the SF.

Figure 16

Comparison of $r_0$ dependence of the ANCs in the CDCC for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(2p_{3/2}\right)$ at $E_d=10$ and 56 MeV. The green dots is the ANC $r_0$ dependence from the 10-MeV reaction and the red dots from 56 MeV. For simplicity, in the ANC the subscripts denoting quantum numbers are omitted.

Figure 17

Comparison of the $r_0$ dependence of the ANCs in the ADWA for the deuteron stripping ${}^{58}\mathrm{Ni}(d,p){}^{59}\mathrm{Ni}\left(2p_{1/2}\right)$ at $E_d=10$ and 56 MeV. The green open dots is the ANC $r_0$ dependence from the 10-MeV reaction and the red solid dots is the ANC from 56 MeV. For simplicity, in the ANC the subscripts denoting quantum numbers are omitted.

Figure 18

The ADWA differential cross sections for the deuteron stripping ${}^{116}\mathrm{Sn}(d,p){}^{117}\mathrm{Sn}\left(3s_{1/2}\right)$ at $E_d=12.2$ MeV: green dashed line. Black dots are the experimental data from Ref. [18]. The calculated angular distribution is normalized to the experimental one at the first measured peak.

Figure 19

$r_0$ dependence of the ANC normalized to unity at $r_0=1$ in the ADWA for the deuteron stripping ${}^{116}\mathrm{Sn}(d,p){}^{117}\mathrm{Sn}\left(3s_{1/2}\right)$ at $E_d=12.2$ MeV: green open circles and dashed line. The solid dots and solid red line is the $r_0$ dependence of the normalized ANC obtained from $79.2$-MeV data. For simplicity, in the ANC the subscripts denoting quantum numbers are omitted.

Figure 20

$r_0$ dependence of the normalized to unity at $r_0=1$ SF in the ADWA for the deuteron stripping ${}^{116}\mathrm{Sn}(d,p){}^{117}\mathrm{Sn}\left(3s_{1/2}\right)$ at $E_d=12.2$ MeV is shown by the green open dots and dashed line. The solid dots and solid red line is the $r_0$ dependence of the normalized SF obtained from the $79.2$-MeV data. The magenta dotted line is the $r_0$ dependence of the normalized SF in the case of pure peripheral reaction. For simplicity, in the SF the subscripts denoting quantum numbers are omitted.

Figure 21

Angular distribution of the ADWA differential cross sections for the deuteron stripping ${}^{116}\mathrm{Sn}(d,p){}^{117}\mathrm{Sn}\left(3s_{1/2}\right)$ at $E_d=79.2$ MeV. ADWA calculations without any nonlocality effects are the magenta dotted line. ADWA calculations with nonlocality effects in the neutron bound-state potential are presented by the blue short dashed line. The green dashed line shows the ADWA calculations with the nonlocal corrections in the optical potentials. Finally, the solid red line is the ADWA angular distribution with all nonlocality effects included, which is normalized to the experimental one at the forward peak. The same normalization factor was applied to the three other curves. Black dots are the experimental data from Ref. [19].

Figure 22

Dependence of the normalized ADWA differential cross sections $R_x$ on $r_{nA}$ for the deuteron stripping ${}^{116}\mathrm{Sn}(d,p){}^{117}\mathrm{Sn}\left(3s_{1/2}\right)$ at $E_d=12.2$ and $79.2$ MeV. The green dashed and the magenta dotted lines are $R_x$ at $12.2$ and $79.2$ MeV, correspondingly. To get these $R_x$, we calculated the post-ADWA differential cross section at the peak of the angular distribution. The radial integral over $r_{nA}$ is calculated for $r_{nA}^{\min }\le r_{nA}<\infty$. The calculated differential cross section is normalized to the full differential cross section. Similarly, solid red and blue short dashed lines are the post-ADWA $R_x$ calculated at $12.2$ and $79.2$ MeV, correspondingly, in which the radial integral over $r_{nA}$ is calculated in the interval $0\le r_{nA}\le r_{nA}^{\max }$. Again the calculated differential cross sections are normalized to the full differential cross sections at the corresponding energies. Hence, $r_{nA}$ on the abscissa is $r_{nA}^{\min }$ for the green dashed and magenta dotted lines and $r_{nA}^{\max }$ for the solid red and blue short dashed lines.

Figure 23

Comparison of $r_0$ dependence of the ANCs in the ADWA for the deuteron stripping ${}^{116}\mathrm{Sn}(d,p){}^{117}\mathrm{Sn}\left(3s_{1/2}\right)$ at $E_d=12.2$ and $79.2$ MeV. Green open dots show the $r_0$ dependence of the ANC determined from the $12.2$-MeV reaction and red solid dots is the ANC from $79.2$ MeV. For simplicity, in the ANC the subscripts denoting quantum numbers are omitted.