Separable Representation of Multichannel Nucleon-Nucleus Optical Potentials
One important ingredient for many applications of nuclear physics to astrophysics, nuclear energy, and stockpile stewardship are 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.
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.
Momentum space Lippmann-Schwinger integral equations are solved with standard techniques to obtain the form factors for the separable representation.
Starting from energy-dependent multichannel optical potentials for neutron and proton scattering from C, separable representations based on a generalization of the Ernst-Shakin-Thaler (EST) scheme are constructed which fulfill reciprocity exactly. Applications to nC and pC scattering are investigated for energies from 0 to 50 MeV.
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.
Nuclear reactions are an important probe to learn about the structure of unstable nuclei. Due to the short lifetimes involved, direct measurements are usually not possible. Therefore indirect measurements using () reactions have been proposed (see e.g. Refs. Escher et al. (2012); Cizewski et al. (2013); Kozub et al. (2012)). Deuteron induced reactions are particularly attractive from an experimental perspective, since deuterated targets are readily available. From a theoretical perspective they are equally attractive because the scattering problem can be reduced to an effective three-body problem Nunes and Deltuva (2011). Traditionally deuteron-induced single-neutron transfer () reactions have been used to study the shell structure in stable nuclei. Nowadays experimental techniques are available to apply the same approaches to exotic beams (see e.g. Schmitt et al. (2012)). Deuteron induced or reactions in inverse kinematics are also useful to extract neutron or proton capture rates on unstable nuclei of astrophysical relevance. Kankainen et al. (2016). Given the many ongoing experimental programs worldwide using these reactions, a reliable reaction theory for reactions is critical.
One of the most challenging aspects of solving the three-body problem for nuclear reactions is the repulsive Coulomb interaction. While for very light nuclei, exact calculations of (d,p) reactions based on momentum-space Faddeev equations in the Alt-Grassberger-Sandhas (AGS) Alt et al. (1967) formulation can be carried out Deltuva and Fonseca (2009) by using a screening and renormalization procedure Deltuva et al. (2005a, b), this technique leads to increasing technical difficulties when moving to computing (d,p) reactions with heavier nuclei Nunes and Upadhyay (2012). Therefore, a new formulation of the Faddeev-AGS equations, which does not rely on a screening procedure, was presented in Ref. Mukhamedzhanov et al. (2012). Here the Faddeev-AGS equations are cast in a momentum-space Coulomb-distorted partial-wave representation instead of the plane-wave basis. Thus all operators, specifically the interactions in the two-body subsystems, must be evaluated in the Coulomb basis, which is a nontrivial task. The formulation of Ref. Mukhamedzhanov et al. (2012) requires the interactions in the subsystems to be of separable form. The same reference suggests an extension of the Faddeev-AGS equations to take excitations of the nucleus into account. Faddeev-AGS calculations taking into account rotational excitations were carried out for (d,p) reactions with Be and Mg in Refs.Deltuva (2013, 2016), showing that the inclusion of excited states in those nuclei could improve the description of experimental data.
To include excitations of the nucleus in the formulation of Ref. Mukhamedzhanov et al. (2012), separable representations of the effective neutron and proton interactions with the nucleus which include those excitations need to be constructed. We follow here the work already carried out for single-channel optical potentials that are complex Hlophe et al. (2013) as well as energy dependent Hlophe and Elster (2016). These separable representations have roots in the work by Ernst, Shakin, and Thaler Ernst et al. (1973) (EST), who derived a scheme for constructing separable representations using scattering wave functions at given energies as basis for their expansion. In momentum space, the scattering wave functions are related to half-shell matrices. Thus, the EST choice guarantees that, at those fixed energies, the separable expansion is not only on-shell but also half-shell exact. The generalization to complex, energy-dependent optical potentials of Refs. Hlophe et al. (2013); Hlophe and Elster (2016) fulfills the same conditions and in addition guarantees that reciprocity is fulfilled.
Taking into account excitations of the nucleus leads to multichannel optical potentials. Although an extension of the EST scheme to multichannel potentials was already carried out in Ref. Pieper (1974), that work was limited to Hermitian potentials. In this manuscript we extend the work of Refs. Hlophe et al. (2013); Hlophe and Elster (2016) to obtain a separable representation of complex, energy-dependent multichannel optical potentials. First we derive the formulation for multichannel neutron-nucleus optical potentials in Sec. II and apply the formulation to neutron elastic and inelastic scattering off C. In Sec. III we extend this formulation to charged particle scattering and show as application the elastic and inelastic scattering of protons from C. Our findings are summarized in Sec. IV. The manuscript is accompanied by three appendices explaining analytic and numerical details.
Ii Energy-Dependent Multichannel Neutron-Nucleus Optical Potentials
ii.1 Formal Considerations
ii.1.1 Coupled-Channel Formalism
For setting up the multichannel problem, let us consider a nucleus characterized by the spin-parity . The nuclear wavefunction is represented by , where is the projection of along the -axis. A neutron interacting with the nucleus through a potential has an angular momentum , with being the relative orbital angular momentum and its spin. The total, conserved angular momentum of the system takes values , and its projection along the z-axis is given by . The states of conserved angular momentum are thus given by
The functions are the spherical harmonics, are the corresponding spinors, and Clebsch-Gordon (C.G.) coefficients. The quantities , , and collectively characterize a particular configuration of the system but are not individually conserved. These configurations make up the angular momentum channels and will be denoted using the Greek letters etc. Since the potential couples different angular momentum channels, we need to employ a coupled-channel formulation to describe a scattering process. The multichannel wavefunction is characterized by the total angular momentum , the parity , as well as the incident angular momentum channel , and obeys a coupled set of Lippmann-Schwinger (LS) equations,
Here represents a projection of the wavefunction onto the -channel. The free propagator in a given channel has the form with being the nonrelativistic kinetic energy in the incident channel and the reduces mass. Here designates the nuclear excitation energy in channel . Using the states of Eq. (3) in the operator LS equation, , one obtains a set of coupled matrix equations
Here and are elements of the multichannel potential and matrix respectively. The explicit LS equation for the half-shell matrix in momentum space takes the form
where the propagator is given by . Here is the center of mass (c.m.) energy and the reduced mass in channel . Further details on the scattering of neutrons from deformed nuclei are given in Appendix B.
ii.1.2 Separable Representation of complex, energy-dependent Multichannel Potentials
For single-channel separable representations, the work of Ref. Hlophe et al. (2013) already showed that the EST scheme presented in Ref. Ernst et al. (1973) leads to the violation of reciprocity when applied to complex potentials, which can be resolved if both, incoming and outgoing scattering states are used in the separable expansion. However, the separable potentials from Ref. Hlophe et al. (2013) still do not completely fulfill reciprocity for energy-dependent potentials. A further generalization of the EST scheme for potentials that are both complex and energy-dependent had to be developed Hlophe and Elster (2016), which involves introducing an energy-dependent separable potential
Here is an outgoing single-channel scattering wavefunction corresponding to at the energy , and is an incoming single-channel scattering wavefunction corresponding to at the energy . The energies are referred to as the EST support points. As Eq. (6) shows, the energy dependency is introduced into the coupling matrix elements , which obey the constraint
where the matrix elements between in- and out-going scattering states at energies and are given as
Defining the matrix elements
leads to a more compact form for Eq. (9),
where represents the transpose of . This constraint ensures that the eigenstates of and coincide at the EST support points. The coupling matrix is symmetric so that , a condition necessary for the fulfillment of reciprocity. Since is explicitly energy dependent, we coin this representation as energy-dependent EST (eEST) scheme. For single-channel, energy-dependent optical potentials the on-shell matrix elements obtained with the eEST scheme and its energy-independent approximation (EST) agree quite well Hlophe and Elster (2016).
To generalize the eEST scheme to multichannel potentials, we proceed analogously to Ref. Pieper (1974) and replace the single-channel scattering wavefunctions with their multichannel counterparts, leading to a multichannel separable potential
The indices and stand for the EST support points. Using the definition of the multichannel half-shell matrix Gloeckle (1983),
Eq. (14) can be recast as
To determine the constraint on , we first generalize the matrices and defined by Eqs. (10) and (11) to multichannel potentials. This is accomplished by replacing the single-channel scattering states by the multichannel wavefunctions so that
The dependence of matrix elements and is omitted for simplicity. One one hand, Eq. (20) shows that the matrix depends only on the support energies and . On other hand, we see from Eq. (19) that depends on the projectile energy as well as the support energies. The constraint on the separable potential is obtained by substituting the multichannel matrices and into Eq. (13) leading to
The coupling matrix elements fulfill
The expression for the matrix is analogous to the one obtained in Ref. Hlophe and Elster (2016) for the single-channel case except for the extra channel indices. The numerical implementation of Eqs. (25)-(30) is discussed in Appendix B.
The eEST scheme simplifies considerably for energy-independent potentials. From Eqs. (19) we see that the dependence of the separable potential on the scattering energy arises from the energy dependence of . Consequently, the separable potential is independent of the energy if is energy-independent. To derive the multichannel EST separable representation for energy-independent potentials we set
in Eq. (20). This leads to
which implies that the matrices and are identical. As a consequence, the matrix in Eq. (30) reduces to an identity matrix. The energy-independent separable potential takes the form
with the constraint
The matrix elements are given as
Since the separable potential does not depend on the scattering energy , we refer to this scheme as the energy-independent EST separable representation. Although it is derived for energy-independent potentials, it can be applied to energy-dependent ones as well since only the multichannel half-shell matrices are required as input. The consequences of applying the energy-independent EST separable representation scheme to energy-dependent potentials were investigated in Ref. Hlophe and Elster (2016) for single-channel optical potentials. It was determined that the off-shell matrix was not symmetric and thus violated the reciprocity theorem. Here a similar study will be carried out for multichannel neutron optical potentials.
ii.2 Elastic and Inelastic Scattering of Neutrons from C
To illustrate the implementation of the multichannel eEST separable representation scheme, we consider the scattering of neutrons from the nucleus C. The C nucleus possesses selected excited states, with the first and second levels having and and being located at 4.43 and 14.08 MeV above the ground state. The collective rotational model Thompson and Nunes (2009) is assumed to the coupling between the ground state and these excited states. We consider here elastic scattering and inelastic scattering to the rotational state. To test the multichannel eEST separable representation we use the deformed optical potential model (DOMP) derived by Olsson et al. Olsson et al. (1989) and fitted to elastic and inelastic scattering data between 16 and 22 MeV laboratory kinetic energy. DOMPs are generally constructed by introducing angular dependence to the optical model. Some details are provided in Appendix C.
ii.2.1 -matrix Elements and Differential Cross Section
First we want to consider -matrix elements in well defined channels to study how well the energy-dependent multichannel eEST scheme can represent them and what rank is required to do so. In Fig. 1 the S-matrix elements, , are shown for the and channels for neutron scattering from C as function of the laboratory energy. The diagonal channel is represented by the solid line and the coupling to the by the dashed line. The corresponding calculations using directly the Olsson 89 DOMP are represented by the solid diamonds and squares. The agreement between the energy-dependent separable representation and the original calculation is excellent. It should be noted that in this case already a rank-2 representation with support points at 6 and 40 MeV is sufficient to achieve this high quality agreement between 0 and 50 MeV laboratory energy.
The energy-independent EST scheme is a simplification of the eEST scheme leading to an energy-independent separable representation. To illustrate the difference between the two schemes, Fig. 2 shows the -matrix elements in the diagonal channel computed in the eEST scheme (solid line) and the energy-independent EST scheme (dashed line) using the same support points (rank). As reference the calculation with the original Olsson 89 potential is given by the filled diamonds. The figure clearly shows that the representation with the EST scheme is of lesser quality than the eEST scheme. This finding is consistent with observations for representations of single-channel optical potentials Hlophe et al. (2013). This suggests that the EST scheme might be improved by increasing the rank of the representation. However, in the multichannel case there is an additional complication, since in the energy-independent scheme coupling matrix elements are not symmetric, i.e. , as will be illustrated later.
The numerical effort needed to evaluate the matrix elements increases rapidly with the number of coupled channels. To simplify the implementation of the eEST scheme, one can avoid the evaluation of these matrix elements for every energy one wants to compute. Instead can be computed at fixed energies and an interpolation scheme used to determine its value elsewhere. Following Ref. Hlophe and Elster (2016), we choose the fixed energies to coincide with the EST support points. Such a choice has the advantage that the potential matrix elements are needed only at the support points. The results obtained with the interpolated eEST scheme are shown by the dash-dotted lines in Fig. 2. We observe that the results agree remarkably well with those obtained with the exact eEST scheme. To show the overall quality of the eEST separable representation in all partial wave matrix elements, we compute cross sections for elastic and inelastic scattering. In Fig. 3 the differential cross sections for elastic and inelastic scattering for the C system are shown at various incident neutron energies. The left hand panel shows the differential cross section for elastic scattering, and the right hand panel the differential cross section for inelastic scattering to the state of C. The support points are at 6 and 40 MeV. The separable representation describes both differential cross sections very well. In addition, it is in good agreement with the coupled-channel calculations shown in Fig. 1 of Ref. Olsson et al. (1989). The dashed lines indicate cross sections computed with the spherical Olsson 89 Olsson et al. (1989) OMP.
ii.2.2 Off-shell matrices
The reciprocity theorem requires that the off-shell matrix be invariant under simultaneous interchange of channel indices and momenta. It is thus imperative that we investigate the properties of the off-shell matrix elements computed with the separable representation schemes. First, we calculate the multichannel off-shell matrix with the original Olsson 89 DOMP and show the real parts of the matrix for the C system at 20.9 MeV incident neutron energy in Fig. 4. The matrix elements correspond to the quantum numbers and . We observe that the matrices exhibit high momentum components in all channels, which is characteristic of local potentials. In the coupled channels they are invariant under the simultaneous interchange of channel indices and momenta, as required by the reciprocity theorem.
Next, we explore the off shell properties of matrix obtained with the eEST separable representation scheme. In Fig. 5 the real part of the multichannel eEST separable matrix is shown for the C system at 20.9 MeV incident neutron energy for the same channels. First we observe that the separable representation does not contain high-momentum components in either channel. We also see that matrix elements obtained with the eEST separable representation are invariant under a simultaneous interchange of channel indices and momenta, as required by the reciprocity theorem. To illustrate that the energy-independent EST separable representation is deficient in that respect, Fig. 6 shows the off-shell matrix for the same channels. It is quite obvious that the channel-coupling matrices, panels (b) and (c), are not invariant under a simultaneous interchange of channel indices and momenta. However, even the diagonal channels, panels (a) and (d) are not symmetric under the exchange of and . This violation was already found for single channel energy-independent EST calculations Hlophe and Elster (2016). In multi-channel calculations this violation of symmetry seems to be enhanced.
To determine the extent to which reciprocity is violated (as in Ref. Hlophe and Elster (2016)), we define an asymmetry relation as
In Fig. 7 this asymmetry is shown for the C system computed at 20.9 MeV as function of the off-shell momenta and for the energy-independent EST representation. Panel (b) clearly shows that even at the on-shell point ( 0.93 fm) the asymmetry in the channel is non-zero. This behavior can not be repaired by increasing the rank of the separable representation. The same calculation for the eEST representation will give exactly zero for all values of and in all channels. The panels would be white and are therefore not shown. From this we conclude that for a separable representation of energy-dependent multi-channel complex optical potential, the eEST scheme must be applied if reciprocity should be fulfilled.
Iii Separable representation of energy-dependent Multichannel Proton-Nucleus Optical Potentials
iii.1 Formal Considerations
The interaction of protons with nuclei comprises of the strong nuclear force and the Coulomb potential. The nuclear interaction is given by the complex, energy-dependent optical potential and usually has the same form as the neutron optical potential. The Coulomb interaction consists of a long-ranged point-Coulomb potential and a short-ranged piece. The latter arises from the charge distribution of the nucleus and is commonly approximated by a uniformly charged sphere. The point-Coulomb potential is long-ranged and affects the asymptotic behavior of the scattering wavefunctions. Consequently, the proton-nucleus scattering problem can not be treated with the same techniques employed for neutron scattering in Section II. According to the Goldberger-Gell-Mann relation Rodberg and Thaler (1967), the scattering amplitude for scattering separates into two parts. The first part is the Rutherford amplitude corresponding to the point Coulomb potential. The second part is the nuclear amplitude corresponding to the short-range potential , consisting of the nuclear interaction and the short-range Coulomb potential. While the Rutherford amplitude is known analytically, the nuclear amplitude must be evaluated numerically. The nuclear amplitude is evaluated in a basis of Coulomb wavefunctions according to Eq. (95). When working in momentum space, this is a very challenging task since the Coulomb wavefunctions are singular. We employ the techniques presented in Ref. Elster et al. (1993), where the authors showed that, in the Coulomb basis, the Coulomb-distorted nuclear matrix fulfills a LS-type equation of the same form as in the basis of plane waves. The momentum-space matrix elements of the potential in the Coulomb basis are obtained via Fourier transform from coordinate space. For multichannel potentials, the nuclear matrix fulfills the coupled set of LS-type equations shown in Eq. (112).
To derive a separable representation for the short-ranged proton-nucleus potential , we modify the multichannel eEST scheme of Section II.1.2. The multichannel scattering wavefunctions are replaced by the Coulomb-distorted multichannel scattering wavefunctions . This leads to the separable potential
Here are the scattering Coulomb wavefunctions corresponding to the channel momentum . The Coulomb-distorted nuclear transition matrix fulfills the LS equation
and the energy-independent matrix
The coupling matrix elements then fulfill
Here represents the transpose of . The separable representation of the matrix is then given as
Here the indices represent angular momentum channels. It is noteworthy that the expressions for the Coulomb distorted separable multichannel nuclear matrix elements have the same form as the ones obtained for neutron-nucleus systems in Section II.1.2. The only difference is that all quantities are evaluated in the Coulomb-basis. Moreover, the behavior of the off-shell nuclear matrices under a transposition does not depend on the chosen basis, and thus the multichannel eEST separable representation for proton-nucleus systems fulfills the reciprocity in the same fashion as the one for neutron-nucleus systems.
iii.2 Elastic and Inelastic Scattering of Protons from C
To illustrate the implementation of the multichannel separable expansion presented in Section III.1, we consider the scattering of protons from C including both the the ground state and excited state. A rigid rotor model is adopted to describe the structure of the C nucleus as in Section II.2. The proton-C interaction is given by a deformed OMP plus the point-Coulomb potential. The proton OMP includes a short-range contribution arising from the nuclear charge distribution. Deformations are introduced using multipole expansions as outlined in Appendix C. In this work, we employ the Meigooni 85 Meigooni et al. (1985) DOMP which is presented in Appendix C.2. A uniformly charged sphere is assumed for the nuclear charge distribution so that the short-ranged Coulomb potential has the form
The values of the Coulomb radius are adopted from Ref. Weppner et al. (2009). Here is the atomic number and the electromagnetic coupling constant.
First, the differential cross sections are evaluated using the Meigooni 85 DOMP and compared to the ones obtained using its separable representation. In Fig. 8 the differential cross section for elastic proton scattering from C as function of the center of mass (c.m.) angle is shown for proton incident energies at 35.2 MeV (panel (a)) and 65 MeV (panel (c)). The solid lines show the eEST separable representation while the crosses represent calculations based on the original Meigooni 85 DOMP. The support points for the separable representation are chosen to be at 25, 45, and 65 MeV. The differential cross sections for inelastic scattering to the state are shown in panels (b) and (d) for the same energies. We observe that the cross sections computed with the eEST separable representation of rank-3 agree very well with the ones computed directly from the Meigooni 85 DOMP.
When incorporating the rotational excitation of C in the calculations presented in Fig. 8, we deformed the nuclear part of the optical potential as well as the short-ranged Coulomb potential, since the charge distribution should undergo the same deformation as the nuclear short-ranged potential. In order to investigate if deforming the short-ranged Coulomb potential affects the cross sections, we carry out the same calculations as before, but keeping the short-range Coulomb potential spherical. The result of this calculation is shown in Fig. 9 by the filled upward triangles. It is interesting to note that the differential cross sections for inelastic scattering do not show any effect of this simplification. This may be due to the still relatively small charge (Z=6) of C where a deviation from a spherical charge has a small effect. In the framework of (d,p) reactions on Mg it was shown that for Z=12 the deformation of the charge only leads to a very small effect in the transfer cross section Deltuva (2016), thus our finding is consistent. However, since the strength of the short-ranged Coulomb force depends on nuclear charge, its deformation would have a larger effect on the cross sections for heavier nuclei.
Differential cross sections are summed up over all partial waves. The fact that the eEST scheme represents the cross sections directly obtained from the Meigooni 85 DOMP very well implies that overall the partial waves must be well represented. To study further details we now concentrate on the state and study the half matrices calculated at the incident proton energy 35.2 MeV in more detail. Here can take the values and . The real parts of the matrices in those channels are depicted in Fig. 10. Panel (a) shows the half-shell matrix elements for the channels ‘11’ and ‘21’ in the interval fm fm for the full calculation and a calculation in which the short-range Coulomb potential is omitted. The matrix elements in the ‘21’ channel are multiplied with a factor 3 to be roughly of the same size as the matrix elements in the ‘11’ channel. Since the fall-off behavior of the matrix elements for large momenta may be important for reaction calculations, we depict in panel (b) the matrix elements in the momentum interval fm fm. Here we see that the short-range Coulomb potential mainly influences the diagonal ‘11’ channel. At 6 fm the matrix calculated with the nuclear potential only is essentially zero, while the short-ranged Coulomb potential still gives a small contribution. In the ‘21’ channel both matrices fall off to zero, indicating that the deformation of the short-ranged Coulomb potential has little effect for a nucleus as light as C. This may however be different when considering heavier nuclei and will have to be investigated further.
Finally, we separately investigate the effects of the deformation of the short-ranged Coulomb potential on the half-shell multichannel matrix elements. To do so, the coupled-channel calculation is carried out with a spherical short-ranged Coulomb potential and compared with the full calculation. This comparison is shown in Fig. 11. We plot the matrices in the same momentum ranges as in Fig. 10. In the curves in panel (a) we can not discern between the two calculations. Only when considering a much smaller scale in panel (b) the calculations in the ‘21’ channel are slightly different. However, the differences are so small that using a spherical short-range Coulomb potential can be considered a good approximation for C, as already seen in the cross section at high momentum transfer in Fig. 9.
Iv Summary and Conclusion
In this work we introduce separable representations of complex, energy-dependent multichannel optical potentials for neutron as well as proton scattering from nuclei. To fulfill reciprocity exactly, the separable representation must be energy-dependent. This is achieved by having energy-dependent coupling constants.
The first part of the manuscript concentrates on the separable expansion of neutron-nucleus Deformed Optical Model Potentials (DOMPs). This is achieved by generalizing the energy-dependent EST (eEST) scheme of Ref. Hlophe and Elster (2016) to multichannel potentials. To illustrate the implementation of the multichannel eEST scheme, we considered neutron scattering from C. In order to describe the structure of the C nucleus, the rigid rotor model is assumed, and the Olsson 89 Olsson et al. (1989) DOMP was adopted to describe the effective neutron-C interaction. The multichannel eEST scheme was then used to construct a separable representation for the Olsson 89 DOMP. In this case a rank-2 separable expansion was sufficient to describe elastic and inelastic scattering cross sections between 0 and 50 MeV incident neutron energies.
To demonstrate the necessity of using an energy-dependent multichannel separable representation, we also constructed a separable expansion based on the energy-independent multichannel EST scheme, which needed to be of rank-3 to match the representation of -matrix elements between 0 and 50 MeV with the same quality as the eEST scheme. An examination of the off-shell matrix elements showed that only the multichannel eEST separable representation fulfills reciprocity, i.e. the matrix is invariant upon a simultaneous interchange of momenta and channel indices. In contrast, the energy-independent multichannel expansion yields an asymmetric off-shell matrix, as was already observed in single channel expansions in Ref. Hlophe and Elster (2016).
The cost of implementing the eEST scheme for multichannel potentials increases with the number of channels, since the matrix elements must be calculated at each desired energy . However, it turns out that it is sufficient to compute this matrix element at fixed energies and interpolate on the energy to obtain its value at arbitrary energies. As already observed in Ref. Hlophe and Elster (2016) for single-channel potentials, the results obtained with the interpolated eEST scheme agree very well with the ones obtained without the interpolation.
The second part of the manuscript focuses on the separable representation of proton-nucleus DOMPs. The proton-nucleus potential consists of a nuclear piece as well as the Coulomb interaction. The Coulomb interaction further separates into a short-ranged part, usually represented as a charged sphere, and a the long-ranged point-Coulomb force. The point-Coulomb potential is incorporated by working in the Coulomb basis in accordance with the Gell-Mann-Goldberger Rodberg and Thaler (1967) relation. In order to employ the eEST scheme in the same fashion as for neutron-nucleus scattering, we need to solve a Lippmann-Schwinger type equation to obtain half-shell matrices in the Coulomb basis. While the Coulomb propagator is quite simple in this basis, the evaluation of potential matrix elements is more involved. We followed Ref. Elster et al. (1993) to evaluate the potential matrix elements.
To demonstrate an implementation of an eEST representation of a proton-nucleus optical potential, we considered proton scattering off C and used the Meigooni 85 Meigooni et al. (1985) DOMP as a starting point. Differential cross sections for elastic and inelastic scattering were computed using the eEST scheme, showing that a rank-3 separable expansion is sufficient to represent the Meigooni 85 Meigooni et al. (1985) DOMP. Note that the only difference between the eEST schemes for neutron and proton optical potentials is the basis employed for the separable expansion. This implies that the discussions on reciprocity given in Section II.2 for neutron-nucleus potentials apply to proton-nucleus systems as well.
In the Meigooni 85 DOMP the short-ranged Coulomb potential is deformed in the same fashion as the short-range nuclear potential. To study the effects of deforming the short-ranged Coulomb potential, the coupled-channel calculations were repeated with the Coulomb deformation parameter set to zero. It was observed that the differential cross sections are not significantly altered by using a spherical short-ranged Coulomb potential. The effect on the half-shell matrix elements for elastic scattering is also negligible. There is a minimal change to the half-shell matrix elements for inelastic scattering. However, those changes are so small that when considering a nucleus as light as C a deformation of the charge distribution may be safely neglected. This insight is consistent with the finding in Ref. Deltuva (2016). Most likely, for heavier nuclei this will not be the case and a deformation of the short-ranged Coulomb potential will be mandatory.
Acknowledgements.This work was performed in part under the auspices of the U. S. Department of Energy under contract No. DE-FG02-93ER40756 with Ohio University. The authors thank F.M. Nunes for thoughtful comments and careful reading of the manuscript.
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Appendix A Nucleon Scattering from a Deformed Nucleus
Let us consider a neutron scattering from a nucleus possessing a rotational energy spectrum characterized by the spin-parity . The spectrum corresponds to collective rotational states with wavefunctions . These are given by Bohr and Mottelson (1998)
where are the Wigner rotation matrices. Here are the angles specifying the orientation of the nucleus. The interaction between the neutron and the nucleus leads to couplings between states of different spin-parity. Here we treat couplings to selected rotational states explicitly, while the imaginary part of the optical potential accounts for couplings to all other channels besides those that are included. To compute scattering observables for such systems, the coupled-channels formalism A. Bohr (1953) must be adopted.
a.1.1 States of conserved angular momentum
To construct states of conserved angular momentum we adopt the so-called -coupling scheme. The neutron spin is coupled to the relative orbital angular momentum to yield . The corresponding projections along the -axis and fulfill the condition . The angular momentum is in turn coupled to the nuclear spin to yield the total angular momentum . The angular momentum and its projection along the -axis are conserved. To obtain the states of conserved angular momentum , we first define
where the functions are the spherical harmonics and is the corresponding spinor. The quantity is the Clebsch-Gordon (C.G.) coefficient. The inverse relation is given by
The sates are constructed by coupling Eq. (66) to the rotational state so that