Separable representation of multichannel nucleon-nucleus optical potentials

Background: One important ingredient for many applications of nuclear physics to astrophysics, nuclear energy, and stockpile stewardship is cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not feasible, indirect methods, e.g., ($d,p$) reactions, should be used. Those ($d,p$) reactions may be viewed as three-body reactions and described with Faddeev techniques.

Purpose: Faddeev equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. Optical potentials representing the effective interactions in the neutron (proton) nucleus subsystem are usually non-Hermitian as well as energy dependent. Including excitations of the nucleus in the calculation requires a multichannel optical potential. The purpose of this paper is to introduce a separable, energy-dependent multichannel representation of complex, energy-dependent optical potentials that contain excitations of the nucleus and that fulfill reciprocity exactly.

Methods: Momentum space Lippmann-Schwinger integral equations are solved with standard techniques to obtain the form factors for the separable representation.

Results: Starting from energy-dependent multichannel optical potentials for neutron and proton scattering from ${}^{12}\mathrm{C}$, separable representations based on a generalization of the Ernst-Shakin-Thaler (EST) scheme are constructed which fulfill reciprocity exactly. Applications to $n+{}^{12}\mathrm{C}$ and $p+{}^{12}\mathrm{C}$ scattering are investigated for energies from 0 to 50 MeV.

Conclusions: We find that the energy-dependent separable representation of complex, energy-dependent phenomenological multichannel optical potentials describes scattering data with the same quality as the original potential.

Figure 1

The energy-dependent EST (eEST) separable representation of the $J^{\pi }=1/2^{+}$ multichannel $S$-matrix elements, $S_{\alpha {\alpha }_0}^J\left(E\right)$, for the $n+{}^{12}\mathrm{C}$ system as function of the laboratory energy. The solid (dashed) line represents the $S$-matrix elements obtained with a rank-2 eEST separable representation in the $0^{+}\otimes s_{1/2}\to 0^{+}\otimes s_{1/2}$ ($0^{+}\otimes s_{1/2}\to 2^{+}\otimes d_{3/2}$) channel. The support points are located at 6 and 40 MeV. The filled diamonds and squares represent the corresponding $S$-matrix elements directly evaluated with the Olsson DOMP [28].

Figure 2

The $S$-matrix elements of the $0^{+}\otimes s_{1/2}\to 0^{+}\otimes s_{1/2}$ channel for the $n+{}^{12}\mathrm{C}$ system as function of the laboratory energy. The solid line represent the rank-2 energy-dependent eEST separable representation, while the dashed line give the energy-independent EST separable representation. For both, the support points are at 6 and 40 MeV. The dash-dotted line shows the $S$-matrix elements obtained with the interpolated eEST scheme. The filled diamonds represent the calculation based directly on the Olsson DOMP [28].

Figure 3

The differential cross sections for scattering in the $n+{}^{12}\mathrm{C}$ system computed at different incident neutron energies with the eEST separable representation of the Olsson DOMP [28] (solid lines). The left-hand panel shows the differential cross section for elastic scattering, while the right-hand panel depicts the differential cross section for inelastic scattering to the $2^{+}$ state of ${}^{12}\mathrm{C}$. The filled diamonds represent the data taken from Ref. [28]. The cross sections are scaled up by multiples of 10. The results at 21.6 MeV are multiplied by 10, those at 20.9 are multiplied by 100, etc.

Figure 4

The real part of the $t$ matrix elements for $J^{\pi }=1/2^{+}$ in the $n+{}^{12}\mathrm{C}$ system at 20.9 MeV incident neutron energy. Panels (a), (b), (c), and (d) depict $T_{{\alpha }_0{\alpha }_0},T_{{\alpha }_1{\alpha }_0},T_{{\alpha }_0{\alpha }_1}$, and $T_{{\alpha }_1{\alpha }_1}$. The quantum numbers for the channels shown here are ${\alpha }_0=\{I=0,l=0,j_p=1/2\}$ and ${\alpha }_1=\{I=2,l=2,j_p=1.5\}$. The on-shell momentum is given by $k_0$ = 0.93 ${\mathrm{fm}}^{-1}$.

Figure 5

The real part of multichannel off-shell eEST separable $t$ matrix elements for the $n+{}^{12}\mathrm{C}$ system at 20.9 MeV incident neutron energy. Panels (a), (b), (c), and (d) depict $T_{{\alpha }_0{\alpha }_0},T_{{\alpha }_1{\alpha }_0},T_{{\alpha }_0{\alpha }_1}$, and $T_{{\alpha }_1{\alpha }_1}$. The quantum numbers for the states shown here are $J^{\pi }=1/2^{+}{\alpha }_0=\{I=0,l=0,j=0.5\}$, and ${\alpha }_1=\{I=2,l=2,j=1.5\}$. The on-shell momentum is given by $k_0$ = 0.93 ${\mathrm{fm}}^{-1}$.

Figure 6

Same as Fig. 8 but for the EST separable representation of the $t$ matrix.

Figure 7

The $J^{\pi }=1/2^{+}$ asymmetry for the $n+{}^{12}\mathrm{C}$ system evaluated at 20.9 MeV as function of the off-shell momenta $k^{\prime }$ and $k$. Panels (a)and (b) show the asymmetry for the energy-independent EST separable representations.

Figure 8

The differential cross sections for proton scattering off ${}^{12}\mathrm{C}$. Panels (a) and (c) depict the differential cross sections for elastic scattering at proton incident energies of 35.2 MeV (a) and 65 MeV (c). The corresponding cross sections for inelastic scattering to the $2^{+}$ state at 4.43 MeV are depicted in panels (b) and (d). The solid lines represent the calculations with the eEST separable representation, while calculations with the original optical potential by Meigooni [31] are given by the crosses. The support points for the separable representation at located at $E_{\mathrm{lab}}=$ 25, 45, and 65 MeV. The data represented by filled circles at 35 and 65 MeV are taken from Refs. [33] and [34] respectively.

Figure 9

The differential cross sections for proton scattering off ${}^{12}\mathrm{C}$. Panels (a) and (c) depict the differential cross sections for elastic scattering at proton incident energies of 35.2 MeV (a) and 65 MeV (c). The corresponding cross sections for inelastic scattering to the $2^{+}$ state at 4.43 MeV are depicted in panels (b) and (d). The solid lines represent the calculations with the eEST separable representation. The support points for the separable representation at located at $E_{\mathrm{lab}}=$ 25, 45, and 65 MeV. The filled triangles represent a calculation in which only the short range nuclear potential is deformed, while the short-ranged Coulomb potential spherical. The data at 35 and 65 MeV are taken from Refs. [33] and [34] respectively.

Figure 10

The real part of half-shell multichannel $t$ matrix elements $t_{\alpha 1}^{J^{\pi }}(k,k_1;E)$ for $J^{\pi }=1/2^{+}$ for proton scattering from ${}^{12}\mathrm{C}$ at incident proton energy 35.2 MeV. Panel (a) shows the half-shell $t$ matrix in the interval 0 ${\mathrm{fm}}^{-1}\le k\le 7 {\mathrm{fm}}^{-1}$, while panel (b) depicts the same half-shell $t$ matrix between $k=4$ and 10 ${\mathrm{fm}}^{-1}$. The channels 1 and 2 are represented as $1\equiv \{I=0,l=0,j=1/2\}$ and $2\equiv \{I=0,l=2,j=3/2\}$. The dashed line represents the $t$ matrix elements $t_{11}^{1/2^{+}}(k,k_1;E)$ calculated from the eEST separable representation of the Meigooni DOMP [31] for the on-shell momentum $k_1=1.2 {\mathrm{fm}}^{-1}$ as function of $k$. The solid line gives the $t$ matrix elements $t_{21}^{1/2^{+}}(k,k_1;E)$ obtained in the same fashion, but multiplied by a factor of 3 to enhance its features. For the calculations of the same channels, represented with the dotted and dash-dotted lines, the short-ranged Coulomb potential is omitted.

Figure 11

Same as Fig. 10. However, for the calculations represented with the dotted and dash-dotted lines, the short-ranged Coulomb potential is included but taken to be spherical. The $t$-matrix elements of the coupling channels are multiplied by a factor 3 to enhance their features.