Entrance channel effects on sub-barrier capture

Within the quantum diffusion approach, the capture process of a projectile nucleus by a target nucleus is studied at bombarding energies above and below the Coulomb barrier. The entrance channel effects on the partial and total capture cross sections and the mean angular momentum of the captured system are studied in the reactions leading to ${}^{156,160}\mathrm{Er}$, ${}^{170}\mathrm{Hf}$, ${}^{200}\mathrm{Pb}$, ${}^{216}\mathrm{Ra}$, and ${}^{220}\mathrm{Th}$ compound nuclei.

Figure 1

The nucleus-nucleus potentials calculated at $J=0$ (solid curve), 30 (dashed curve), 60 (dotted curve), and 90 (dash-dotted curve) for the ${}^{16}\mathrm{O}+{}^{204}\mathrm{Pb}$ reaction. The position $R_b$ of the Coulomb barrier, interaction radius $R_{\mathrm{int}}$, and external $r_{\mathrm{ex}}$ and internal $r_{\mathrm{in}}$ turning points for some value of bombarding energy $E_{\mathrm{c}.\mathrm{m}.}$ are indicated at $J=0$.

Figure 2

The calculated capture cross sections ${\sigma }_{\mathrm{cap}}$ and the mean angular momenta $\langle J\rangle$ of captured system vs $E_{\mathrm{c}.\mathrm{m}.}$ are compared with the experimental data for the ${}^{16}\mathrm{O}+{}^{152}\mathrm{Sm}$ reaction. The experimental cross sections are taken from Ref. [33] (closed squares) and the experimental values of $\langle J\rangle$ from Ref. [34] (closed squares). The calculations with and without taking into consideration the deformation effect are shown by solid and dashed lines, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$ and ${\beta }_2{(}^{152}\mathrm{Sm})=0.31$ [32]. The height $V_b$ of the Coulomb barrier for the spherical nuclei case is indicated by the arrow.

Figure 3

The calculated partial capture cross sections vs $J$ are compared with the experimental data from Ref. [34] (closed squares) for the ${}^{16}\mathrm{O}+{}^{152}\mathrm{Sm}$ reaction at indicated $E_{\mathrm{c}.\mathrm{m}.}$. The calculations with and without taking into consideration the deformation effect are shown by solid and dashed lines, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$ and ${\beta }_2{(}^{152}\mathrm{Sm})=0.31$ [32].

Figure 4

The same as in Fig. 2, but for the ${}^{16}\mathrm{O}+{}^{184}\mathrm{W}$ reaction. The experimental cross sections are taken from Refs. [35] (closed circles) and [36] (closed squares) and the experimental values of $\langle J\rangle$ from Ref. [36] (closed squares). The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$ and ${\beta }_2{(}^{184}\mathrm{W})=0.24$ [32].

Figure 5

The calculated partial capture cross sections vs $J$ are compared with the experimental data from Ref. [36] (closed squares) for the ${}^{16}\mathrm{O}+{}^{184}\mathrm{W}$ reaction at indicated $E_{\mathrm{c}.\mathrm{m}.}$. The calculations with and without taking into consideration the deformation effect are shown by solid and dashed lines, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$ and ${\beta }_2{(}^{184}\mathrm{W})=0.24$ [32].

Figure 6

The calculated capture cross sections ${\sigma }_{\mathrm{cap}}$ and the mean angular momenta $\langle J\rangle$ of captured system vs $E_{\mathrm{c}.\mathrm{m}.}$ are compared with the experimental data from Ref. [37] (closed circles) for the ${}^{19}\mathrm{F}+{}^{175}\mathrm{Lu}$ reaction. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{21}\mathrm{F})=0.56$ and ${\beta }_2{(}^{173}\mathrm{Lu})=0.33$. The height $V_b$ of the Coulomb barrier for the spherical nuclei case is indicated by the arrow.

Figure 7

The calculated partial capture cross sections vs $J$ for the ${}^{19}\mathrm{F}+{}^{175}\mathrm{Lu}$ reaction at indicated $E_{\mathrm{c}.\mathrm{m}.}$ are compared with the experimental data from Ref. [37] (closed squares) for evaporation residue cross sections. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{21}\mathrm{F})=0.56$ and ${\beta }_2{(}^{173}\mathrm{Lu})=0.33$.

Figure 8

The calculated capture cross sections ${\sigma }_{\mathrm{cap}}$ and the mean angular momenta $\langle J\rangle$ of captured system vs $E_{\mathrm{c}.\mathrm{m}.}$ are compared with the experimental data for the ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$ reaction. The experimental cross sections are taken from Refs. [38] (closed squares) and [39] (closed circles) and the experimental values of $\langle J\rangle$ from Refs. [38] (closed squares) and [40] (closed triangles). The calculations with and without taking into consideration the neutron transfer process are shown by solid and dashed lines, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{64}\mathrm{Ni})=0.09$, ${\beta }_2{(}^{66}\mathrm{Ni})=0.16$ [32], ${\beta }_2{(}^{98}\mathrm{Mo})=0.17$ [32], and ${\beta }_2{(}^{100}\mathrm{Mo})=0.23$ [32]. The height $V_b$ of the Coulomb barrier for the spherical nuclei case is indicated by the arrow.

Figure 9

The calculated partial capture cross sections vs $J$ are compared with the experimental data from Ref. [38] (closed squares) for the ${}^{64}\mathrm{Ni}+{}^{100}\mathrm{Mo}$ reaction at indicated $E_{\mathrm{c}.\mathrm{m}.}$. The calculations with and without taking into consideration the neutron transfer process are shown by solid and dashed lines, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{64}\mathrm{Ni})=0.09$, ${\beta }_2{(}^{66}\mathrm{Ni})=0.16$ [32], ${\beta }_2{(}^{98}\mathrm{Mo})=0.17$ [32], and ${\beta }_2{(}^{100}\mathrm{Mo})=0.23$ [32].

Figure 10

The calculated capture cross sections ${\sigma }_{\mathrm{cap}}$ and the mean angular momenta $\langle J\rangle$ of captured system vs $E_{\mathrm{c}.\mathrm{m}.}$ are compared with the experimental data in the evaporation residue channel for the ${}^{58}\mathrm{Ni}+{}^{60}\mathrm{Ni}$ reaction. The calculations with and without consideration of transfer are shown by solid and dashed lines, respectively. The experimental cross sections ${\sigma }_{\mathrm{cap}}$ are taken from Ref. [41] (closed squares) and the experimental values of $\langle J\rangle$ from Ref. [42] (closed squares). The following quadrupole deformation parameters are used: ${\beta }_2{(}^{58}\mathrm{Ni})=0.05$, ${\beta }_2{(}^{60}\mathrm{Ni})=0.21$ [32]. The height $V_b$ of the Coulomb barrier for the spherical nuclei case is indicated by the arrow.

Figure 11

The calculated partial capture cross sections vs $J$ are compared with the experimental data from Ref. [42] (closed squares) in evaporation residue channel for the ${}^{58}\mathrm{Ni}+{}^{60}\mathrm{Ni}$ reaction at indicated $E_{\mathrm{c}.\mathrm{m}.}$. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{58}\mathrm{Ni})=0.05$, ${\beta }_2{(}^{60}\mathrm{Ni})=0.21$ [32].

Figure 12

The calculated partial capture cross sections vs $J$ for the reactions ${}^{32}\mathrm{S}+{}^{94}\mathrm{Zr}$ (solid lines) and ${}^{32}\mathrm{S}+{}^{90}\mathrm{Zr}$ (dashed lines) at indicated $E_{\mathrm{c}.\mathrm{m}.}$. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{32}\mathrm{S})=0.31$ [32], ${\beta }_2{(}^{34}\mathrm{S})=0.25$ [32], ${\beta }_2{(}^{90}\mathrm{Zr})=0$, and ${\beta }_2{(}^{92}\mathrm{Zr})=0.1$ [32].

Figure 13

The calculated partial capture cross sections vs $J$ for the reactions ${}^{40}\mathrm{Ca}+{}^{94}\mathrm{Zr}$ (solid lines) and ${}^{40}\mathrm{Ca}+{}^{90}\mathrm{Zr}$ (dashed lines) at indicated $E_{\mathrm{c}.\mathrm{m}.}$. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{40}\mathrm{Ca})=0$, ${\beta }_2{(}^{42}\mathrm{Ca})=0.25$ [32], ${\beta }_2{(}^{90}\mathrm{Zr})=0$, and ${\beta }_2{(}^{92}\mathrm{Zr})=0.1$ [32].

Figure 14

The calculated capture cross sections ${\sigma }_{\mathrm{cap}}$ and the mean angular momenta $\langle J\rangle$ of captured system vs $E_{\mathrm{c}.\mathrm{m}.}-V_b$ are compared with the experimental data for the reactions ${}^{12}\mathrm{C}+{}^{144}\mathrm{Sm}$ and ${}^{64}\mathrm{Ni}+{}^{92}\mathrm{Zr}$ (compound nucleus ${}^{156}\mathrm{Er})$. The experimental cross sections for the ${}^{12}\mathrm{C}+{}^{144}\mathrm{Sm}$ reaction are taken from Refs. [44] (closed triangles), [45] (closed squares), and [46] (closed circles) and for the ${}^{64}\mathrm{Ni}+{}^{92}\mathrm{Zr}$ from Ref. [43] (open squares). The experimental values of $\langle J\rangle$ for the ${}^{64}\mathrm{Ni}+{}^{92}\mathrm{Zr}$ reaction are from Ref. [43] (open squares). The heights $V_b$ of the Coulomb barriers in the case of the spherical nuclei are 46.0 and 129.9 MeV, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{12}\mathrm{C})=-0.3$, ${\beta }_2{(}^{144}\mathrm{Sm})=0.05$, ${\beta }_2{(}^{64}\mathrm{Ni})=0.09$, and ${\beta }_2{(}^{92}\mathrm{Zr})=0.1$ [32].

Figure 15

The same as in Fig. 14, but for the reactions ${}^{16}\mathrm{O}+{}^{144}\mathrm{Nd}$, ${}^{37}\mathrm{Cl}+{}^{123}\mathrm{Sb}$, ${}^{64}\mathrm{Ni}+{}^{96}\mathrm{Zr}$, and ${}^{80}\mathrm{Se}+{}^{80}\mathrm{Se}$ (compound nucleus ${}^{160}\mathrm{Er})$. The experimental cross sections and mean angular momenta for the ${}^{16}\mathrm{O}+{}^{144}\mathrm{Nd}$ reaction are taken from Ref. [47] (squares), for the ${}^{37}\mathrm{Cl}+{}^{123}\mathrm{Sb}$ from Ref. [7] (triangles), for the ${}^{64}\mathrm{Ni}+{}^{96}\mathrm{Zr}$ from Ref. [43] (circles), and for the ${}^{80}\mathrm{Se}+{}^{80}\mathrm{Se}$ from Ref. [7] (rhombuses). The heights $V_b$ of the Coulomb barriers in the case of the spherical nuclei are 57.3, 101.2, 128.5, and 132.8 MeV, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$, ${\beta }_2{(}^{144}\mathrm{Nd})=0.12$ [32], ${\beta }_2{(}^{37}\mathrm{Cl})=0.05$, ${\beta }_2{(}^{123}\mathrm{Sb})=0.17$, ${\beta }_2{(}^{66}\mathrm{Ni})=0.16$ [32], ${\beta }_2{(}^{94}\mathrm{Zr})=0.09$ [32], and ${\beta }_2{(}^{80}\mathrm{Se})=0.23$ [32].

Figure 16

The same as in Fig. 14, but for the reactions ${}^{28}\mathrm{Si}+{}^{142}\mathrm{Ce}$, ${}^{32}\mathrm{S}+{}^{138}\mathrm{Ba}$, and ${}^{48}\mathrm{Ti}+{}^{122}\mathrm{Sn}$ (compound nucleus ${}^{170}\mathrm{Hf})$. The experimental cross sections for the reactions ${}^{28}\mathrm{Si}+{}^{142}\mathrm{Ce}$ (squares), ${}^{32}\mathrm{S}+{}^{138}\mathrm{Ba}$ (triangles), and ${}^{48}\mathrm{Ti}+{}^{122}\mathrm{Sn}$ (circles) are taken from Ref. [10]. The experimental values of $\langle J\rangle$ for the reactions ${}^{28}\mathrm{Si}+{}^{142}\mathrm{Ce}$ (squares), ${}^{32}\mathrm{S}+{}^{138}\mathrm{Ba}$ (triangles), and ${}^{48}\mathrm{Ti}+{}^{122}\mathrm{Sn}$ (circles) are taken from Ref. [48]. The heights $V_b$ of the Coulomb barriers in the case of the spherical nuclei are 96.0, 106.1, and 125.9 MeV, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{30}\mathrm{Si})=0.32$ [32], ${\beta }_2{(}^{140}\mathrm{Ce})=0.11$ [32], ${\beta }_2{(}^{34}\mathrm{S})=0.25$ [32], ${\beta }_2{(}^{136}\mathrm{Ba})=0.13$ [32], ${\beta }_2{(}^{50}\mathrm{Ti})=0.17$ [32], and ${\beta }_2{(}^{120}\mathrm{Sn})=0.11$ [32].

Figure 17

The same as in Fig. 14, but for the reactions ${}^{12}\mathrm{C}+{}^{204}\mathrm{Pb}$, ${}^{30}\mathrm{Si}+{}^{186}\mathrm{W}$, and ${}^{48}\mathrm{Ca}+{}^{168}\mathrm{Er}$ (compound nucleus ${}^{216}\mathrm{Ra})$. The experimental cross sections for the reactions ${}^{12}\mathrm{C}+{}^{204}\mathrm{Pb}$ (squares), ${}^{30}\mathrm{Si}+{}^{186}\mathrm{W}$ (triangles), and ${}^{48}\mathrm{Ca}+{}^{168}\mathrm{Er}$ (circles) are taken from Ref. [11]. The heights $V_b$ of the Coulomb barriers in the case of the spherical nuclei are 56.7, 115.7, and 151.2 MeV, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{12}\mathrm{C})=-0.3$, ${\beta }_2{(}^{204}\mathrm{Pb})=0$, ${\beta }_2{(}^{32}\mathrm{Si})=0.22$ [32], ${\beta }_2{(}^{184}\mathrm{W})=0.24$ [32], ${\beta }_2{(}^{48}\mathrm{Ca})=0$, and ${\beta }_2{(}^{168}\mathrm{Er})=0.34$ [32].

Figure 18

The same as in Fig. 14, but for the reactions ${}^{16}\mathrm{O}+{}^{204}\mathrm{Pb}$, ${}^{34}\mathrm{S}+{}^{186}\mathrm{W}$, ${}^{50}\mathrm{Ti}+{}^{170}\mathrm{Er}$, and ${}^{96}\mathrm{Zr}+{}^{124}\mathrm{Sn}$ (compound nucleus ${}^{220}\mathrm{Th})$. The experimental cross sections for the ${}^{16}\mathrm{O}+{}^{204}\mathrm{Pb}$ reaction are taken from Ref. [49] (squares) and for the ${}^{96}\mathrm{Zr}+{}^{124}\mathrm{Sn}$ from Ref. [50] (rhombuses). The heights $V_b$ of the Coulomb barriers in the case of the spherical nuclei are 73.6, 131.4, 165.5, and 199.2 MeV, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$, ${\beta }_2{(}^{204}\mathrm{Pb})=0$, ${\beta }_2{(}^{36}\mathrm{S})=0.17$ [32], ${\beta }_2{(}^{184}\mathrm{W})=0.24$ [32], ${\beta }_2{(}^{52}\mathrm{Ti})=0.27$, ${\beta }_2{(}^{168}\mathrm{Er})=0.34$ [32], ${\beta }_2{(}^{96}\mathrm{Zr})=0.08$ [32], and ${\beta }_2{(}^{124}\mathrm{Sn})=0.1$ [32].

Figure 19

The calculated partial capture cross sections versus $J$ are compared with the experimental data for the reactions ${}^{16}\mathrm{O}+{}^{144}\mathrm{Nd}$ at $E_{\mathrm{c}.\mathrm{m}.}-V_b=13$ MeV and ${}^{64}\mathrm{Ni}+{}^{96}\mathrm{Zr}$ at $E_{\mathrm{c}.\mathrm{m}.}-V_b=11$ MeV (compound nucleus ${}^{160}\mathrm{Er})$. The experimental partial evaporation residue cross sections for the ${}^{16}\mathrm{O}+{}^{144}\mathrm{Nd}$ reaction are taken from Ref. [47] (squares) and for the ${}^{64}\mathrm{Ni}+{}^{96}\mathrm{Zr}$ from Ref. [51] (circles). The heights $V_b$ of the Coulomb barriers in the case of the spherical nuclei are 57.3 MeV and 128.5 MeV, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$, ${\beta }_2{(}^{144}\mathrm{Nd})=0.12$ [32], ${\beta }_2{(}^{66}\mathrm{Ni})=0.16$ [32], and ${\beta }_2{(}^{94}\mathrm{Zr})=0.09$ [32].

Figure 20

The calculated capture cross sections ${\sigma }_{\mathrm{cap}}$ and the mean angular momenta $\langle J\rangle$ of captured system (lower part) versus $\eta$ for the reactions ${}^{16}\mathrm{O}+{}^{144}\mathrm{Nd}$ ($\eta =0.8$), ${}^{37}\mathrm{Cl}+{}^{123}\mathrm{Sb}$ ($\eta =0.538$), ${}^{64}\mathrm{Ni}+{}^{96}\mathrm{Zr}$ ($\eta =0.2$), and ${}^{80}\mathrm{Se}+{}^{80}\mathrm{Se}$ ($\eta =0$) (compound nucleus ${}^{160}\mathrm{Er})$ at indicated values of $E_{\mathrm{c}.\mathrm{m}.}-V_b$. The heights $V_b$ of the Coulomb barriers in the case of the spherical nuclei are 57.3 MeV, 101.2 MeV, 128.5 MeV, and 132.8 MeV, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$, ${\beta }_2{(}^{144}\mathrm{Nd})=0.12$ [32], ${\beta }_2{(}^{37}\mathrm{Cl})=0.05$, ${\beta }_2{(}^{123}\mathrm{Sb})=0.17$, ${\beta }_2{(}^{66}\mathrm{Ni})=0.16$ [32], ${\beta }_2{(}^{94}\mathrm{Zr})=0.09$ [32], and ${\beta }_2{(}^{80}\mathrm{Se})=0.23$ [32].

Figure 21

The same as in Fig. 20, but for the reactions ${}^{16}\mathrm{O}+{}^{204}\mathrm{Pb}$ ($\eta =0.855$), ${}^{34}\mathrm{S}+{}^{186}\mathrm{W}$ ($\eta =0.691$), ${}^{50}\mathrm{Ti}+{}^{170}\mathrm{Er}$ ($\eta =0.545$), and ${}^{96}\mathrm{Zr}+{}^{124}\mathrm{Sn}$ ($\eta =0.127$) (compound nucleus ${}^{220}\mathrm{Th})$. The heights $V_b$ of the Coulomb barriers in the case of the spherical nuclei are 73.6 MeV, 131.4 MeV, 165.5 MeV, and 199.2 MeV, respectively. The following quadrupole deformation parameters are used: ${\beta }_2{(}^{16}\mathrm{O})=0$, ${\beta }_2{(}^{204}\mathrm{Pb})=0$, ${\beta }_2{(}^{36}\mathrm{S})=0.17$ [32], ${\beta }_2{(}^{184}\mathrm{W})=0.24$ [32], ${\beta }_2{(}^{52}\mathrm{Ti})=0.27$, ${\beta }_2{(}^{168}\mathrm{Er})=0.34$ [32], ${\beta }_2{(}^{96}\mathrm{Zr})=0.08$ [32], and ${\beta }_2{(}^{124}\mathrm{Sn})=0.1$ [32].