Separable Representation of Phenomenological Optical Potentials
of Woods-Saxon Type
Background: 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. 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. While there exist several separable representations for the nucleon-nucleon interaction, the optical potential between a neutron (proton) and a nucleus is not readily available in separable form. The purpose of this paper is to introduce a separable representation for complex phenomenological optical potentials of Woods-Saxon type. Results: Starting from a global optical potential, a separable representation thereof is introduced based on the Ernst-Shakin-Thaler (EST) scheme. This scheme is generalized to non-hermitian potentials. Applications to nCa, nSn and nPb are investigated for energies from 0 to 50 MeV and the quality of the representation is examined. Conclusions: We find a good description of the on-shell t-matrix for all systems with rank up to 5. The required rank depends inversely on the angular momentum. The resulting separable interaction exhibits a different off-shell behavior compared to the original potential, reducing the high momentum contributions.
The TORUS Collaboration
A variety of applications of nuclear physics require the understanding of neutron capture on unstable nuclei. Due to the short lifetimes involved, direct measurement are currently not possible, and thus indirect methods using (d,p) reactions have been proposed (see e.g Refs. Escher et al. (2012); Cizewski et al. (2013); Kozub et al. (2012)). Single neutron transfer (d,p) reactions have also been the preferred tool to study shell evolution in nuclear structure (see e.g. Refs. Schmitt et al. (2012); Jones et al. (2011)). In all these cases, a reliable reaction theory is a critical ingredient.
Scattering and reaction processes involving deuterons either as projectile or as target are perhaps the most natural three-body problem in the realm of nuclear reactions. The binding energy of the deuteron is so small that its root-mean-square radius is significantly larger than the range of the force. That means, when a deuteron interacts with a compact nucleus, one may expect that it will behave like a three-body system consisting of proton , a neutron , and a nucleus . The obvious three-body reactions are elastic scattering, rearrangement and breakup processes. In order to describe those processes on the same footing, deuteron-nucleus scattering should be treated as a three-body problem with a three-body Hamiltonian governing the dynamics. This three-body Hamiltonian contains the well understood nucleon-nucleon (NN) interaction as well as the effective interactions between the nucleons and the target ( and ). It is common for these nucleons-nucleus interactions to take phenomenological optical potentials which fit a large body of elastic scattering data Varner et al. (1991); Weppner et al. (2009); Koning and Delaroche (2003); Becchetti and Greenlees (1969).
The application of momentum space Faddeev techniques to nuclear reactions has been pioneered in Ref. Deltuva and Fonseca (2009), and successfully applied to (d,p) reactions for light nuclei Deltuva (2009). However, when extending these calculations to heavier nuclei Nunes and Deltuva (2011); Upadhyay et al. (2012), it becomes apparent that techniques employed for incorporating the Coulomb interaction in Faddeev-type calculations of reactions with light nuclei can not readily be extended to the heaviest nuclei. Therefore, a new method for treating (d,p) reactions with the exact inclusion of the Coulomb force as well as target excitation was formulated in Ref. Mukhamedzhanov et al. (2012). This new approach relies on a separable representation of the interparticle forces.
Separable representations of the forces between constituents forming the subsystems in a Faddeev approach have a long tradition in few-body physics. There is a large body of work on separable representations of NN interactions (see e.g. Refs. Haidenbauer and Plessas (1983); Haidenbauer et al. (1986); Berthold et al. (1990); Schnizer and Plessas (1990); Entem et al. (2001)) or meson-nucleon interactions Ueda and Ikegami (1994); Gal and Garcilazo (2011). In the context of describing light nuclei like He Ghovanlou and Lehman (1974) and Li Eskandarian and Afnan (1992) in a three-body approach, separable interactions have been successfully used. A separable nucleon-C optical potential was proposed in Ref. Miyagawa and Koike (1989), consisting of a rank-1 Yamaguchi-type form factor fitted to the positive energies and a similar term describing the bound states in the nucleon-C configuration. However, we are not aware of any systematic work along this line for heavy nuclei, for which excellent phenomenological descriptions exist in terms of Woods-Saxon functions Varner et al. (1991); Weppner et al. (2009); Koning and Delaroche (2003); Becchetti and Greenlees (1969). For applications to (d,p) reactions there is a need for a procedure for deriving a separable representation that is sufficiently general for a variety of nucleon-nucleus optical potentials, as well as over a wide range of nuclei and energies, so that one can take advantage of the already existing extensive work on phenomenological optical potentials.
The separable representation of two-body interactions suggested by Ernst-Shakin-Thaler Ernst et al. (1973) (EST) seems well suited for achieving this goal. We note that this EST approach has been successfully employed to represent NN potentials Haidenbauer and Plessas (1983); Haidenbauer et al. (1986). However, the EST representation as derived in Ref. Ernst et al. (1973), though allowing energy dependence of the potentials Ernst et al. (1974); Pearce (1987), assumes that they are Hermitian. Therefore, the EST approach needs to be generalized in order to be applicable for optical potentials which are complex.
In Section II we present the generalization of the EST approach to non-Hermitian potentials which is needed so that the potential as well as the transition matrix fulfills the reciprocity theorem. First we explicitly show for a rank-1 separable potential the required redefinition of the separable ansatz of Ref. Ernst et al. (1973), and then generalize to separable potentials of arbitrary rank. Since all our calculations are carried out in momentum space, we sketch the explicit procedure to obtain separable transition matrices. This procedure follows closely the one laid out in Ref. Ernst et al. (1973), and we will refer to it as EST scheme.
In Section III.1 we present the results for the separable representations of optical potentials for Ca, Sn, and Pb based on the CH89 Varner et al. (1991) phenomenological optical potential. Note that the definition of the optical potential in coordinate space, as well as details on the Fourier transform and partial wave decomposition are presented in the Appendix. In Section III.2, we investigate the off-shell behavior of the separable representations and compare it to the original potential. Finally, we summarize our findings in Section IV.
Ii Formal Considerations
A scheme for constructing separable potentials from the solution of a Lippmann-Schwinger (LS) equation with arbitrary hermitian potentials was suggested by Ernst, Shakin, and Thaler Ernst et al. (1973). This separable potential is designed to represent the scattering matrix (or equivalently the scattering phase shifts) over a chosen energy range with the same quality as the original potential. Furthermore, this EST scheme provides a well-defined prescription for increasing the accuracy of the representation by increasing the rank of the separable potential.
The basic idea of the Ernst-Shakin-Thaler (EST) scheme for constructing a separable representation of the two-body transition amplitude is that one selects a fixed number of energy points in the energy interval in which the separable potential shall represent the scattering matrix. At these chosen points, the on-shell as well as half-shell t-matrices of the original potential are identical to those obtained with the separable potential. The corresponding half-shell t-matrices at these points then serve as form-factors of the separable representation. The number of these fixed points gives the rank of the separable potential.
The EST Ernst et al. (1973, 1974) scheme has been successfully applied to construct separable representations of several NN potentials Haidenbauer and Plessas (1983); Haidenbauer et al. (1986) defined below the pion production threshold, i.e. real potentials. Potentials intended to describe the scattering of neutrons or protons from nuclei are in general complex as a result of reactions channels not explicitly taken into account. In the following we show that the formulation given in Ref. Ernst et al. (1973) is not suitable for complex potentials, since it is not compatible with the reciprocity theorem. We illustrate how the requirement of time-reversal invariance leads to a modification of the definition of the separable potential.
ii.1 Separable Complex Potentials of Rank-1 and Time Reversal
For applications to the theory of nuclear reactions it is convenient to arrange that all potential operators satisfy
where is the time reversal operator appropriate to the system. This condition guarantees that the -matrix corresponding to is symmetric and that reaction amplitudes constructed from these potentials satisfy convenient reciprocity relations.
When is a central potential in the space of a spinless particle, can be chosen to be the antilinear complex conjugation operator , which in the coordinate space basis is defined by
and from which we deduce Note that for this particular we have .
We first consider a Hermitian interaction acting in a partial wave with angular momentum . It was shown in Ref. Ernst et al. (1973) that a rank-1 separable potential leading to a scattering wave function that is identical to that of potential (defined via a Hamiltonian ) at a specific energy (support point) is given as
Here is the regular radial scattering wave function, unique within an overall constant, for at energy , is the form factor, and the strength parameter.
For a Hermitian the radial function can be taken to be real in both coordinate and momentum space and the strength parameter is real. As a result the non-local potential of Eq. (3) is a Hermitian symmetric matrix in both momentum and coordinate space and satisfies .
If however is a complex potential, or more generally a non-hermitian operator the radial functions are no longer real and if is simply replaced by in Eq. (3) the resulting rank-1 non-local potential will neither be Hermitian nor satisfy .
To remedy this situation for a non-Hermitian potential we replace the definition of Eq. (3) by
where now the strength parameter is defined by .
Here is the unique regular radial wave function corresponding to and is the unique regular radial wavefunction corresponding to . By a suitable choice of arbitrary normalization constants we can arrange that is simply the complex conjugate of and hence .
If satisfies the definition of Eq. (4) gives a symmetric complex potential matrix that satisfies
where the round brackets mean that here acts only on the quantities within the brackets.
For a general energy and arbitrary potential we define an operator as the solution of
For the potential given by Eq. (4) we then obtain a rank-1 separable t-matrix in a given partial wave with matrix elements
Introducing , and , the partial wave t-matrix element can be written as
The scattering wavefunction satisfies . Using this we find , and hence
where angular momentum indices are omitted.
On the energy shell, i.e. for and , the separable t-matrix of Eq. (10) agrees with the t-matrix evaluated with the original potential , as it should. For any general the function of Eq. (8) is explicitly calculated as
Here the half-shell t-matrices are the momentum space solutions of a standard LS equation at the scattering energy .
Thus, the rank-1 separable potential as given in Eq. (4) leads to the desired rank-1 separable t-matrix, which fulfills the reciprocity theorem.
ii.2 Separable Complex Potentials of Arbitrary Rank
It remains to generalize the above formulation of a rank-1 separable complex potential to one of arbitrary rank. In analogy to the procedure followed in Ref. Ernst et al. (1973) we define a complex separable potential of arbitrary rank in a given partial wave as
Here is the unique regular radial wave function corresponding to the complex potential and asymptotic energy , and the unique regular radial wave function corresponding to . Note that may also be energy dependent. The matrix is defined and constrained by
The corresponding separable partial wave t-matrix must be of the form
where angular momentum indices are omitted for simplicity of notation. The coefficient matrix is constrained by
For we recover the expressions for the rank-1 potential of the previous section.
For the explicit calculation of the matrix , we define a matrix
so that the condition of Eq. (18) reads
from which follows
Using again that , and , the matrix elements are calculated in momentum space. With , the diagonal matrix elements are given as
The half-shell t-matrices are the momentum space solutions of a standard LS equation at the support points . At these energies , the on-shell t-matrix elements calculated from the separable potential agree with those calculated from the original complex potential .
The off-diagonal elements of the matrix are given by
The matrix elements , where determine the rank of the separable representation, are calculated at a fixed number of energy support points . The elements are then obtained by solving a system of linear equations . For this to be successful, the inverse of must exist. For calculating the form factors, the half-shell t-matrices at the support points are calculated by solving a LS equation with the momentum space Woods-Saxon potential derived in Appendix A. For calculating the matrix elements we use the cubic Hermite spline interpolation of Ref. Huber et al. (1997) to obtain the values at the required momenta. Since spline functions are by design linearly independent, the representation of the half-shell t-matrices in this basis guarantees that the matrix is always invertible.
In order to calculate the momentum space partial wave separable potential, one needs to evaluate Eq. (14),
together with the constraint of Eq. (15). The matrix elements
iii.1 On-Shell Behavior of the Optical Potential and its Separable Representation
To demonstrate the construction of a separable representation of a complex potential we apply the method to nCa, nSn and nPb, and use as a starting point the Chapel-Hill phenomenological global optical potential CH89 Varner et al. (1991). CH89 has been widely used in the literature in the last decades, and also in recent studies of (d,p) reactions Jones et al. (2011); Schmitt et al. (2012).
Using Woods-Saxon functions as parameterized forms, these phenomenological global optical potentials are most naturally given in coordinate space. For convenience, these global parameterizations assume local form factors of the interaction but must introduce an explicit energy dependence in the strengths.
In order to construct a separable momentum space representation of CH89 using the scheme outlined in the previous section, we first must derive a momentum space representation of the CH89 optical potential. The explicit evaluation of the Fourier transform of the CH89 potential is given in Appendix A, including the final expressions we have implemented, but there are a few essential points worth highlighting. The Fourier transform of the coordinate-space Woods-Saxon function into momentum space can be written as a series expansion. Fortunately, we found, by the explicit calculation of the leading terms of this expansion, that only the first two terms are necessary to obtain a converged result. These expressions were used as input to a momentum space LS equation, and the resulting phase shifts were compared to those computed with the coordinate space CH89 potential using FRESCO Thompson (1988). Agreement was found for three significant figures.
A separable t-matrix constructed within the scheme outlined in Section II from a t-matrix calculated as solution of a LS equation with a given potential is exact on-shell as well as half-shell at fixed support points . For any other energy it is then calculated using Eq. (16).
For studying the quality of the separable representation it is convenient to look at the partial wave S-matrix, given as
where represents the center-of-mass (c.m.) energy, . The dimensionless amplitude is given by
The on-shell momentum is defined via , and is the reduced mass of the system under consideration. The partial wave t-matrix, is either calculated directly from the CH89 potential or obtained via our general scheme to construct a separable representation thereof.
First we study in detail the scheme for constructing a separable representation of the n+Ca system for c.m. energies from 0 to 50 MeV. Our goal is to arrive at an excellent separable representation of the partial wave S-matrix starting from the local CH89 potential. For practical applications in e.g. three-body type calculations, it is desirable to achieve this with as low a rank as possible. As a representative case, we show the , partial wave S-matrix in Fig. 1. The S-matrix obtained from the solution of the LS equation with the original CH89 optical potential (solid line) shows a relatively mild variation with energy in the energy regime under consideration. If one is only interested in describing the very low energies, i.e. MeV, a rank-1 separable potential with a support point at 6 MeV is barely sufficient (dash-double-dotted line), while a rank-2 representation with support points at 6 and 12.5 MeV can already capture the range between 0 and 20 MeV relatively well. A rank-3 representation with support points at 5, 15, and 25 MeV captures the S-matrix up to roughly 35 MeV. However, for a high quality separable representation of at least 4 significant figures of the CH89 result a rank-4 representation with support points at 6, 15, 36, and 47 MeV is needed in this partial wave. The figure also shows that more support points are needed in the region where the S-matrix shows structure, and less points are necessary for the smooth region.
The next question for a practical implementation of the EST scheme is whether the optimum support points when including both central and spin-orbit interactions differ from the results when including the central interaction only. Usually the central part of an optical potential is larger than the spin-part. Thus one may expect it to be sufficient to find EST support points for the S-matrix computed using only the central part of the optical potential, and then use the same points for deriving the separable representation of the partial wave S-matrices. This is indeed the case, as is demonstrated for the partial wave S-matrix for the n+Ca system in Fig. 2. The dashed line shows the S-matrix calculated from the cental part of the CH89 optical potential, while the solid and dash-double-dotted lines represent the and the partial wave S-matrices. Our results show that it is indeed sufficient to determine the EST support points for the S-matrix computed from the cental part of the optical potential. Thus, once the support points are determined from a calculation including the central part, one only needs to replace the corresponding form factors, i.e. the half-shell t-matrices, at the support points with the ones containing the spin-orbit contribution to obtain the separable representation of the partial wave S-matrices with unchanged accuracy.
For the n+Ca system we find that for the lower partial waves a rank-4 separable representation is sufficient for energies up to = 50 MeV. Let us now consider what happens as we increase the angular momentum. Reaction calculations in the energy range MeV often require partial waves up to , but due to the centrifugal barrier with increasing angular momentum the t-matrix remains close to zero even at higher scattering energy. Therefore, one may expect, for the same accuracy, a lower rank separable representation to suffice for representing higher partial waves. This is indeed the case as demonstrated in Fig. 3. Here we show the , and , partial wave S-matrices and their corresponding separable representation. We find that for a rank-1 representation is sufficient up to = 50 MeV, whereas the partial wave S-matrix still requires a rank-2 representation. For the energy regime under consideration, we determined the angular momentum intervals in which separable representations of specific ranks represent the original partial wave S-matrices within 4 significant figures. Those angular momentum groups and their corresponding support points are listed in Table 1.
Support points need not be at positive energies, therefore a bound state may be included and this is exactly the situation in the so-called unitary pole approximation Lovelace (1964). Thinking about our original purpose (namely Faddeev description of (d,p) reactions), a good description of the final bound state will be necessary. To understand the effect of including a bound state in the separable representation, we consider the , partial wave S-matrix of the n+Ca system, which would correspond to the partial wave of the valence neutron in the ground state of Ca. When extrapolating the CH89 parameterization to , we set the imaginary part to zero and directly extrapolate the real part with no further fitting, just for the sake of illustrating the method here. With such an approach, we find two bound states with energies = -28.8 MeV and = -5.62 MeV. We now move the support point at 6 MeV to = -5.62 MeV, which would be the valence orbital in this system. The results are shown in Fig. 4. The solid line shows the exact calculation, the solid triangles its separable rank-4 representation. The so obtained rank-4 separable representation is given by the dashed line in Fig. 4. Though the bound state is relatively close to threshold, the deviation from the original representation and the exact S-matrix is quite large in the region between 1 and 7 MeV, indicating that the low energy support point is needed for capturing the structure of this partial wave S-matrix at low energies.
From this study we learn that though it is very easy to add bound states to the separable representation of a specific partial wave S-matrix, such a bound state, even if shallow, does not necessarily replace a low energy support point. For an application to (d,p) reactions, one should add a bound state corresponding to the final state populated through the reaction, thus increasing the rank by 1.
Up to now we studied the separable representations in detail for the n+Ca system. Next, we turn to a heavy nuclei and repeat the study for the n+Sn and n+Pb systems. In these cases, the partial wave S-matrices have more structure, and thus require more support points for an accurate representation. For different partial waves, we again find that we have groups of angular momenta for which a specific rank is required to represent the partial wave S-matrix to an accuracy of at least 4 significant figures. These groups and the energies of the corresponding support points are summarized in Table 1. First we note that a separable representation of rank-5 is sufficient for the low angular momentum states. Similar to the n+Ca system, the higher angular momentum states require successively less support points. However, in order for a rank-1 representation to be accurate in the energy regime under consideration, one must go up to .
Considering the number of support points needed for medium-mass (n+Ca) and heavy ( n+Sn and n+Pb) systems, we find it encouraging that a very good description of the heavy systems can be achieved by increasing the rank of the representation by only one, relative to the medium-mass case.
If we are interested in a ‘universal’ separable representation for partial waves t-matrices obtained from the CH89 optical potential, we can use the support points obtained for the n+Pb system, apply them to all other systems and obtain a high quality separable representation. This is shown in Fig. 5 for the partial wave S-matrix for the n+Ca, n+Sn and n+Pb systems. The points used here can also be used for the relevant higher angular momentum t-matrices.
iii.2 Off-Shell Behavior of the Optical Potential and its Separable Representation
After having established the scheme for finding separable representations of the t-matrices obtained from the CH89 phenomenological optical potential and checking its accuracy by comparing the S-matrix elements between 0 and 50 MeV, we now want to look at the resulting off-shell t-matrices. The partial wave off-shell t-matrices, , are calculated at a given energy using Eq. (16) sandwiched with arbitrary moment and .
In Fig. 6 we show the off-shell t-matrix, at = 36 MeV for the n+Ca system in the partial wave , . The left side panels ([a] and [c]) show the off-shell t-matrix as function of momenta and obtained as solution of the LS equation with the CH89 potential, while the right side panels ([b] and [d]) depicts their separable representation. The on-shell momentum at = 1.3 fm is indicated in each panel by the horizontal/vertical lines.
First, we point out that in both cases the t-matrix is symmetric around the line , which must be the case if the underlying potential fulfills the reciprocity theorem. This also shows that our generalization of the EST scheme to complex potentials is correct. Had we used the definition given in Ref. Ernst et al. (1973), the resulting off-shell t-matrix would not have this symmetry.
Second, we observe that overall the magnitude of the off-shell elements of the separable interaction is smaller than that of the t-matrix of the original potential. It was shown in Ref. Ernst et al. (1974) that, for a rank-1 separable potential, the off-shell t-matrix has the form , where is the on-shell momentum, the form factors are the half-shell t-matrices calculated at the support points and their on-shell value. This justifies why the magnitude of the off-shell elements of the separable t-matrix is smaller than the magnitude of those obtained from the solution of an LS equation with the original CH89 potential.
Furthermore, we note that the t-matrix obtained from the CH89 potential has significant non-vanishing values along the line even for momenta fm, which is typical for local potentials, while these off-shell matrix elements approach zero for the separable counterpart. The fact that the separable representation projects out high momentum components of the original t-matrix is reminiscent of renormalization group techniques Bogner et al. (2007), which in nuclear physics are typically applied to the NN force. We note that the off-shell dip around 1.5 fm is present in both cases, indicating that closer to the on-shell point, the EST scheme preserves the off-shell structure.
In Fig. 7 we show the off-shell t-matrix at = 21 MeV for the n+Pb system in the partial wave . The left panel again depicts the real and imaginary t-matrix obtained as solution of the LS equation with the CH89 potential, while the right side panels give their separable representation. The on-shell momentum = 1.0 fm is indicated by a horizontal/vertical lines. The scale of the exact solution is dominated by the strong dip close to the origin. This dip is also present in the separable representation, however almost a factor of four smaller in case of the real part and a factor of two for the imaginary part. The off-shell structure around 1.5 fm is again captured well by the separable representation. The exact solution also has non-zero contributions along the line , while the separable representation does not, however due to the scale this is not visible in the figure. For these heavier systems, the higher partial waves behave very similar to the ones in the n+Ca system. Therefore we do not show them separately.
Iv Summary and Conclusions
In this work we extended the well-known EST scheme Ernst et al. (1973) for creating separable representations of two-body transition matrix elements as well as potentials to the realm of complex potentials. Requiring that the separable transition matrix fulfill the reciprocity theorem, we identified a suitable rank-1 separable potential. In analogy to Ref. Ernst et al. (1973), we generalized this potential to arbitrary rank.
All calculations presented in this work are based on the Chapel Hill phenomenological optical potential CH89 Varner et al. (1991). Since the CH89 potential, as nearly all phenomenological optical potentials, is given in coordinate space using Woods-Saxon functions, we first give a semi-analytic Fourier transform of those Woods-Saxon functions in terms of a series expansion. In practice, it turns out that only two terms in the expansion are sufficient for achieving convergence. Note that our approach for deriving the momentum-space optical potential is general and can be applied to any optical potential of Woods-Saxon form. This momentum space CH89 potential is then used in the partial-wave LS integral equation to calculate half-shell t-matrices. These then serve as input to the the generalized scheme for creating separable representations for complex potentials.
The systematic studies in this paper include: n+Ca, n+Sn and n+Pb. We are able to provide, for all cases, a systematic classification of support points for partial-wave groups, so that the partial-wave S-matrices are reproduced to at least 4 significant figures compared to the original momentum space solution of the LS equation. We find the low partial waves of the n+Pb system require a rank-5 separable potential to be well represented in the energy regime between 0 and 50 MeV center-of-mass energy. The support points obtained for this case are well suited to represent all partial waves of the n+Pb as well as all lighter systems described by the CH89 optical potential.
We find that the rank required for achieving a good representation decreases with increasing angular momentum of the partial wave considered. We developed recommendations for both the rank and the locations of support points to be used when describing medium-mass and heavy systems generated from the CH89 potential. Our recommendations group together partial waves. We also demonstrated that it is sufficient to determine support points including only the central part of the optical potential; when the spin-orbit interaction is added and the form factors are accordingly modified, the same support points can be expected to yield a good representation.
We then investigated the off-shell behavior of the constructed separable representations, and found that overall, the high momentum components along the axis which are typical for local potentials are removed from the separable representation. Furthermore, the off-shell elements of the separable representation are smaller in magnitude, however follow the functional shape of the CH89 potential. Since off-shell matrix elements are not observables, only reaction calculations can show if the differences seen in the off-shell t-matrix have any consequences for e.g. three-body observables. Future work will address this question.
Appendix A Fourier Transform of Global Optical Potentials
a.1 Global Optical Potentials in Coordinate Space
Most global optical potentials are parameterized in terms of Wood-Saxon form factors and its derivatives. For convenience, here we provide the definitions of the form factor and standard parameterizations of phenomenological optical potentials. These definitions are also those for the CH89 potential Varner et al. (1991), which we use throughout this work. The short range, nuclear part is given by
The constants , , , and are the strength parameters, and and the diffuseness and the radius parameters given in Ref. Varner et al. (1991).
a.2 Fourier Transform of Woods-Saxon Functions
where is the momentum transfer defined as .
Introducing dimensionless variables , , and (with ), and inserting into Eq. (43) yields
Repeating the above step for all the expressions in Eqs. (39) yields expressions that are similar to Eq. (44). The are four distinct real integrals appearing in the Fourier transform expressions of Eqs. (39) and they are the real and imaginary parts of the two complex integrals and such that
The integrals and can be evaluated by contour integration. The closed integration loop is taken to be the boundary of the first quadrant (i.e from to along positive real axis, then a circular path from to , and finally from to along the positive imaginary axis). The contribution from the circular path integral is zero in this case. Both integrands have poles at with . The corresponding residues are given by
The total closed loop integral is obtained via the residue theorem, and the path from to is evaluated as series expansion of the integrand in terms of . This leads to
Similar expressions were derived in Ref. Rodriguez-Gallardo et al. (2008). The infinite sums can not be evaluated analytically and their convergence properties need to be studied numerically. In order to do so, let us define the finite sums , , , and as
The convergence of the above sums is studied as function of . The parameter crucial to the convergence is , the smaller , the faster the series will converge. Thus, the slowest convergence will be for the smallest values of for a given optical potential. For the CH89 optical potential Varner et al. (1991) the minimum value is slightly larger than 5.8 corresponding to mass number = 40. For all heavier nuclei is larger. The numerical values of the sums of Eqs. (62) are given in Table 2 as function of for the worst case of = 5.8. The Table demonstrates that for the CH89 global optical potential = 2 is sufficient to obtain convergence up to six decimal points.
The truncation discussed above focused on CH89. The conditions for convergence should be tested for other global optical potentials. We repeated the study for the Weppner-Penny Weppner et al. (2009) (WP) global optical potential, which was fitted to nuclei as light as C. Then, the required minimal value of is about 3, where we consider that is energy dependent. In Table 3, we present the convergence of the sums of Eqs. (62) for the fixed value = 3.0, when using the WP potential. We find that = 4 is sufficient to obtain convergence up to six decimal points. For the heavier nuclei, , the WP optical potential requires only = 2 similar to the CH89 optical potential.
a.3 The Partial Wave Optical Potential in Momentum Space
According to the considerations in the previous section the first two terms in the series expansion are sufficient to obtain momentum space expressions for the Fourier transforms of the Woods-Saxon functions of the CH89 potential which are sufficiently accurate as shown in Table 2. This leads to
for the Fourier transform of . Here stands for the magnitude of the momentum transfer as defined in the previous section.
For , the imaginary volume term, the same expression is obtained with , , and in the place of , , and .
For the surface term of Eq. (39) we obtain the following momentum space form
The Fourier transform of the spin-orbit term, , leads to