Adiabatic model of $(d,p)$ reactions with expicitly energy-dependent non-local potentials.

In our two previous publications Tim13a ; Tim13b we have proposed a method for calculating $A(d,p)B$ cross sections when the interaction of the neutron and proton in deuteron with the target $A$ is energy-independent but non-local. This development has been motivated by the growing number of $(d,p)$ experiments performed at radioactive beams facilities with the aim of extracting spectroscopic information beyond the limits of $\beta$-stability, including cases important for astrophysical applications. The $(d,p)$ reaction is known PTJT to be dominated by those components of the total wave functions in which the separation between the neutron and proton in the incoming deuteron is less than the range of the $n-p$ interaction. Such components are often calculated in the adiabatic distorted wave approximation (ADWA) where the effective deuteron potential is derived from neutron and proton optical potentials taken at some fixed energy JS ; HJ . Since the introduction of ADWA the fixed neutron and proton energies used to calculate the adabatic deuteron potential were always taken to be precisely half of the deuteron incident energy JS ; JT ; Joh05 . Such a prescription seemed to be reasonable from the intuitive point of view that appeals to the low binding energy of deuteron.

The first indications for a need to go beyond the approximation of fixed nucleon energies came from the Faddeev study of transfer reactions in Refs. Del09a ; Del09b . The energy dependence of the nucleon optical potentials assumed there were either explicit Del09a or a result of the non-locality of energy-independent nucleon potential Del09b . Given that the Faddeev formalism is too complicated for the routine analysis of deuteron stripping reactions we have suggested Tim13a ; Tim13b a practical way to account for the non-locality of nucleon optical potentials in the ADWA developed in JS ; JT . It turned out Tim13b that for energy-independent non-local potentials of the Perey-Buck type PB a simple adiabatic deuteron potential can be constructed as a solution of a transcendental equation, similar to the one obtained for nucleon scattering in PB . Moreover, for $Z=N$ nuclei and isospin independent nucleon optical potentials this solution is related to the sum of nucleon potentials taken at an energy shifted with respect to the half the incident deuteron energy by about 40 MeV. We have shown that this large shift comes from the relative $n-p$ kinetic energy averaged over the short-range potential $V_{np}$ Tim13a ; Tim13b . The new effective deuteron potential is shallower than that traditionally used in the ADWA and this leads to a change in normalizaton of predicted $(d,p)$ cross sections with consequences for their interpretation in terms of nuclear structure quantities.

The simple prescription of Refs. Tim13a ; Tim13b to obtain the $d-A$ potential for $(d,p)$ reactions relies strongly on the assumption that the energy-dependence of nucleon optical potentials is a consequence of the non-locality of energy-independent potentials. However, the formal theory of optical potentials developed by Feshbach Fes58 shows that optical potentials are not only non-local but energy-dependent as well. The explicit energy dependence of Feshbach potentials comes from the coupling of channels with the target, $A$, in its ground state to channels in which $A$ is excited, i.e, the mechanism that gives rise to the imaginary part of the optical potential when the excited channels are open. The result is that global local energy-dependent potentials derived from analysis of experimental data may not be of the Perey-Buck type. We have checked this for optical potentials taken from frequently-used global systematics CH89 CH89 . We found that only their real parts satisfy the Perey-Buck form with the usual non-locality range of 0.85 fm while the imaginary parts are definitely not of the Perey-Buck type and do not follow any immediately obvious systematic rule. Therefore, it is unclear how to treat explicit energy dependence of the popular CH89 global potential in the ADWA. This may be relevant to other energy-dependent global optical potentials available in the literature and used in $(d,p)$ reaction calculations.

In this paper we suggest an approximate practical way of dealing with explicitly energy-dependent non-local optical potentials. For this purpose we first consider in Sec. II the connection between three-body model and underlying many-body problem that leads to the energy dependence of optical potentials. Then in Sec. III we give formal expressions for optical potential operators in two- and three-body systems. We derive in Sec. IV the link between the adiabatic $(d,p)$ models with energy-dependent and energy-independent optical potentials making important comments in Sec. V. We apply in Sec. VI the new model to calculation of the cross sections of the same $(d,p)$ reactions considered earlier by us in Refs. Tim13b . We discuss our model and the results obtained making conclusions in Sec. VII. Important derivations are given in the Appendix.

Explicit energy dependence of the nucleon optical potential arises as the result of internal structure of the target and the consequential possibility that the target can be excited. On the other hand, it is customary to model the $A(d,p)B$ reaction in terms of scattering state solutions of a three-body Schrödinger equation in which only the co-ordinates of the $n$ and $p$ appear explicitly. We can formalize this procedure by regarding the three-body wavefunction of the model as the projection, $\Psi^{(+)}_{\mbox{\boldmath$k$}_{d}}(n,p)\phi_{A}$, of the full $A+2$ many-body wavefunction corresponding to a deuteron incident on a nucleus $A$ in its ground state, $\phi_{A}$, onto the ground state of $A$. This projection has outgoing waves that describe elastic deuteron scattering and elastic deuteron break-up exactly and outgoing waves in any open stripping channel in which $B$ has a non-neglible component with $A$ in its ground state. The complete many-body wavefunction has components in which $A$ is not in its ground state. The existence of coupling to these components influences the projection $\Psi^{(+)}_{\mbox{\boldmath$k$}_{d}}(n,p)\phi_{A}$ but we will base our analysis on a model in which their explicit contribution to the $A(d,p)B$ transition matrix is neglected.

Within this framework an exact expression for $A(d,p)B$ transition matrix is

where $V_{np}$ is the $n-p$ interaction and $\Psi_{\mbox{\boldmath$k$}_{p},B}^{(-)}$ is a solution of the many-body Schrödinger equation corresponding to a proton with momentum $\mbox{\boldmath$k$}_{p}$ incident on $B$ but with $V_{np}$ deleted. This expression for the exact amplitude differs from the widely used alternative expression in which the transition operator is given by $V_{np}+\sum_{i\in A}V_{pi}-U_{pB}$ Satchler and the bra-vector contains the product of the wave function of nucleus $B$ and the distorted wave in the $p-B$ channel generated by the (arbitrary) optical potential $U_{pB}$. The advantage of Eq. (1) is that it contains the short-ranged transition operator $V_{np}$ only, which allows a simple approximation for the $\Psi^{(+)}_{\mbox{\boldmath$k$}_{d}}(n,p)$ to be developed. However, the final state wave function $\Psi_{\mbox{\boldmath$k$}_{p},B}^{(-)}$ in this expression becomes more complicated. To further discuss this issue we restrict ourselves to a three-body model, $n+p+A$, in which the internal degrees of freedom of $A$ are not treated explicitly. In this case $\Psi_{\mbox{\boldmath$k$}_{p},B}^{(-)}$ is a solution of a three-body problem, but it is a very special one in which (i) two of the bodies, $n$ and $p$, do not interact, and (ii) in practical applications to $A(d,p)B$ reactions $A$ is frequently much more massive than a nucleon. In fact, in the $A\rightarrow\infty$ limit excitation of $B$ is impossible in this model and the exact solution for $\Psi_{\mbox{\boldmath$k$}_{p},B}^{(-)}$ is the product of $\chi_{\mbox{\boldmath$k$}_{p}}^{(-)}(p)$, a proton scattering wave function distorted by the proton-target potential $V_{pA}$, and the wave function of the final nucleus $B$. The overlap of the latter with $A$ gives the neutron overlap function $I_{AB}$. Derivations of these results are given in RCJ09 where references to earlier work can be found. Corrections to this limit in the case of selected light nuclei $A$ have been evaluated in TJ99 using the adiabatic approximation to handle the excitation and break-up of $B$ by the recoil of $A$ and in Moro09 using the continuum-discretized coupled-channel method. It was shown in TJ99 how the recoil corrections modify the proton distorted wave, $\chi_{\mbox{\boldmath$k$}_{p}}^{(-)}(p)$, by a factor that goes to unity for large $A$ leaving a three-body wave function with a range limited in space by the neutron overlap function. We refer to TJ99 and Moro09 for further quantitative discussion of these results which show that recoil contributions to $\Psi_{\mbox{\boldmath$k$}_{p},B}^{(-)}$ can probably be ignored in the first instance except for the very lightest nuclei. The aim of the present paper is to clarify how a proper treatment of the wave function in the incident channel changes $(d,p)$ predictions when the wave function in the final channel is fixed. Although to date a complete solution to the problem of calculating $\Psi_{\mbox{\boldmath$k$}_{p},B}^{(-)}$ for finite $A$ has not yet been obtained the results of TJ99 and Moro09 give sufficient grounds for ignoring the finite $A$ complications in $\Psi_{\mbox{\boldmath$k$}_{p},B}^{(-)}$ for our purposes.

The emphasis will now be on the evaluation of $\Psi^{(+)}_{\mbox{\boldmath$k$}_{d}}(n,p)$, which by definition is a function of the coordinates of $n$ and $p$. We will make heavy use of the fact that the evaluation of the amplitude (1) requires knowledge of this function within the range of $V_{np}$ only and we will make approximations that are inspired by this observation.

Using projection operator techniques originally given by Feshbach Fes58 it is possible to derive explicit expressions for the effective interactions that appear in the three-body Hamiltonian that drives $\Psi^{(+)}_{\mbox{\boldmath$k$}_{d}}(n,p)$. This effective Hamiltonian is

where $T_{3}$ is the three-body kinetic energy operator for $n+p+A$ in the centre of mass system and the bra-ket notation here implies integration over the target nucleus co-ordinates to leave an operator in $n$ and $p$ co-ordinates only. The interactions $U^{0}_{N}(N)$ that appear in Eq. (2) depend only on the $n-A$ and $p-A$ coordinates, respectively, being diagonal in states of the target A, but otherwise arbitrary at this point. They have no influence on the ground state matrix elements of $U+U^{0}_{n}+U^{0}_{p}$ or on the three-body wave function. The importance of the introduction of the $U^{0}_{N}$’s in the present context is that their arbitrariness can be used to decouple the neutron and proton contributions to the complex many-body operator $U$, at least as far as the lowest multiple scattering contributions are concerned. How this is done will be discussed in Sec. IV.

The many-body operator $U$ in Eq. (2) accounts for all the effects of the target nucleus degrees of freedom. It satisfies the integral equation (see Appendix A)

The operator $Q_{A}$ projects onto excited states of the target and the energy denominator $e$ is given by

where $H_{A}$ and $E_{A}$ are the internal Hamiltonian and the ground state energy of the target $A$, respectively. $E_{3}$ is the three-body energy related to the incident centre of mass kinetic energy $E_{d}$ and deuteron binding energy $\epsilon_{0}$ by $E_{3}=E_{d}-\epsilon_{0}$.

A formal solution of Eq. (LABEL:U) is

$\langle\phi_{A}\mid U\mid\phi_{A}\rangle$ sums up all processes via excited target states and the deuteron ground and break-up states that are coupled to the target ground state by $\Delta v_{NA}$ and which begin and end on the target ground state. The expansion that appears on iteration of Eq. (LABEL:U) by repeated substitution for $U$ on the right-hand side makes this clear:

When averaged over the target ground state, and added to $U^{0}_{n}(n)+U^{0}_{p}(p)$, the first term on the right-hand-side of (6) will be recognised as providing an expression for the sum of the real nucleon optical potentials calculated in a folding model using the free-space interactions between $n$ and $p$ with the target nucleons. The second and higher order terms include the effect of target excitation. Because of the propagator $\frac{1}{e}$ they are complex and non-local in neutron and proton coordinates even when the nucleon-nucleon interactions $v_{Ni}$ are local. In addition there is an explicit dependence on the energy $E_{3}$ through the energy denominator $e$. The purpose of this $E_{3}$ dependence is to link the three-body energy of the system to the thresholds of all relevant channels correctly. For example, if $E_{3}$ is less than the energy of a coupled intermediate excited state then the corresponding contribution to $U$ will be real (Hermitean); otherewise, a complex (non-Hermitean) contribution will result. This is the physical reason for expecting the effective interaction to be explicitly energy dependent.

Our aim is to make a connection between $U^{0}_{n}(n)+U^{0}_{p}(p)+\langle\phi_{A}\mid U\mid\phi_{A}\rangle$ and the neutron and proton optical potentials and, in particular, to determine the energies at which phenomenological optical models should be taken to reproduce the main properties of $U^{0}_{n}(n)+U^{0}_{p}(p)+\langle\phi_{A}\mid U\mid\phi_{A}\rangle$ for application to evaluation of the $(d,p)$ transition amplitude (1). We shall see that this can only be done approximately.

It has been shown PTJT that for a range of incident deuteron energies of current experimental interest the $A(d,p)B$ amplitude (1) is dominated by the first Weinberg component of $\Psi^{(+)}_{\mbox{\boldmath$k$}_{d}}(n,p)$, i.e.,

where $\phi_{1}$ is defined in terms of the $n$-$p$ interaction $V_{np}$ by

in which $\phi_{0}$ is the deuteron ground state wavefunction. In (18), the integration is performed over the $n$-$p$ relative coordinate, $\mbox{\boldmath$r$}=\mbox{\boldmath$r$}_{n}-\mbox{\boldmath$r$}_{p}$, resulting in a function of the $n$-$p$ centre of mass coordinate, $\mbox{\boldmath$R$}=\frac{1}{2}(\mbox{\boldmath$r$}_{n}+\mbox{\boldmath$r$}_{p})$, only. The state $\phi_{1}$ restricts the $n$-$p$ relative coordinate to be less than the range of the $n$-$p$ interaction $V_{np}$.

In the ADWA approximation JT the first Weinberg component of $\Psi^{(+)}_{\mbox{\boldmath$k$}_{d}}(n,p)$ is determined by a distorted wave $\chi^{(+)}_{\mbox{\boldmath$k$}_{d}}(\mbox{\boldmath$R$})$ satisfying the equation

where $U_{\mathrm{eff}}$ is given by Eq. (16). We first consider the $U_{nA}$ term in the expression for $U_{\mathrm{eff}}$.

The dependence of the $U_{nA}$ on the proton coordinates is solely through the presence in $e$ of the operators $T_{p}+U^{0}_{p}$ and $V_{np}$. The values taken by $T_{p}+U^{0}_{p}+V_{np}$ influence the net neutron energy parameter in $e$ and hence determine the appropriate energy at which to choose the neutron optical potential. Our method here is to replace $T_{p}+U^{0}_{p}+V_{np}$ by an average value consistent with dominance of the first Weinberg component to the $(d,p)$ reaction. We deduce this value within the ADWA. Similarly, we will replace the $T_{n}+U^{0}_{n}+V_{np}$ operator in the $U_{pA}$ term for $U_{\rm eff}$ by an average value.

The ADWA potential in Eq. (20) has energy dependence that comes through the energy denominators in $\langle\phi_{1}\phi_{A}|\Delta v_{NA}Q_{A}(e-Q_{A}\Delta v_{NA}Q_{A})^{-1}Q_{A}\Delta v_{NA}|\phi_{0}\phi_{A}\rangle$ which contains the matrix elements $\langle\phi_{A}|P_{A}\Delta v_{NA}Q_{A}|\phi_{A}\rangle$ that are the functions of $\mbox{\boldmath$R$}\pm\mbox{\boldmath$r$}/2$. The range of variable $r$ is restricted by the range of $\phi_{1}\phi_{0}$ which is about 0.45 fm. Therefore, for most $R$ relevant for evaluating $\chi^{(+)}_{\mbox{\boldmath$k$}_{d}}(\mbox{\boldmath$R$})$ we have $r/2\ll R$. Therefore, to a first approximation the operators $Q_{A}\Delta v_{NA}P_{A}$ in the numerator can be replaced by their form at $\mbox{\boldmath$r$}=0$. Then the only matrix element involving integration over $r$ is $\langle\phi_{1}|(e-Q_{A}\Delta v_{NA}Q_{A})^{-1}|\phi_{0}\rangle$.

We show in Appendix C that within the ADWA we can consistently use the approximation

We consider the contribution from $N=n$ first. To evaluate the denominator in the right-hand-side of (21) we use definition (17) for $e$ and the result from Appendix D

where $T_{R}$ and $T_{r}$ are the kinetic energy operators associated with the coordinates $R$ and $r$ respectively and

We determine a reasonable value for the operator $T_{R}$ in Eq. (22) by recalling that it acts on the first Weinberg component $\chi^{(+)}_{\mbox{\boldmath$k$}_{d}}(\mbox{\boldmath$R$})$. Using Eq. (20) we get

We now recall that the potentials $U_{n}^{0}$ and $U_{p}^{0}$ are arbitrary provided that they commute with $Q_{A}$. We are therefore at liberty to choose

With this step Eq. (21) reduces to

where

The neutron contribution to the ADWA effective interaction (16) is given by

We see that the expression (LABEL:UADWeffn) is just the neutron contribution to the ADWA distorting potential calculated with a non-local neutron optical potential taken at energy ${\cal E}_{\mathrm{eff}}$. Similar conclusion can be made for proton contribution $\langle\phi_{1}\mid U_{\mathrm{eff}}^{p}(n,p)\mid\phi_{0}\rangle$ to the ADWA distorting potential. With the choice

we find in identical fashion that this contribution is obtained using the $p-A$ optical potential, including Coulomb terms, taken at the proton energy $E_{p}$ also equal to ${\cal E}_{\rm eff}=\frac{1}{2}(E_{d}+\langle T_{r}\rangle)$. We note that with the choices given by Eqs. (29) and (25) the neutron and proton contributions to the effective interaction of the three-body model in the single scattering approximation are independent of the $U_{N}^{0}$. The choice (29) for $U_{n}^{0}$ will influence higher order multiple scattering terms in Eq. (7). We will not consider these terms further here.

We have given arguments that suggest that for the purpose of calculating $(d,p)$ cross sections in the ADWA approximation when the nucleon optical potentials are explicitly energy dependent as well as non-local the neutron and proton kinetic energies used in the incident channel should be $\frac{1}{2}E_{d}+\frac{1}{2}\langle T_{r}\rangle$. This is acheved by averaging the proton kinetic energy in the incoming deuteron and adding the extra kinetic energy the neutron has because it must be close to proton in order to contribute to the $(d,p)$ amplitude. Thus the problem of calculating the ADWA amplitude for $A(d,p)B$ with energy-dependent nonlocal potentials is reduced to the problem of calculating this amplitude with energy-independent nonlocal potentials, the solution of which is found in Refs. Tim13a ; Tim13b . If a phenomenological non-local, explicitly energy dependent nucleon potential is available, the prescription of present paper together with that from Refs. Tim13a ; Tim13b can be used unambiguously.

The energy-independent non-local potential that replaces the original energy-dependent nonlocal potential is obtained from the latter by evaluating it at the energy shifted from the intuitively assumed $E_{d}/2$ value by one half of the average $n-p$ kinetic energy within the range of the $n-p$ potential, approximately equal to 57 MeV Tim13a . This value is close to, but not equal to, the energy shift $\Delta E\sim 40$ MeV, identified in Refs. Tim13a ; Tim13b as the shift to the $E_{d}/2$ value that provides the appropriate energy of energy-dependent equivalents of nonlocal potentials to be used in ADWA.

For all known phenomenological optical potentials the underlying energy-dependent non-local potentials are not known. It may be tempting to use the conclusion of the previous section for choosing in the ADWA calculations the phenomenological local potentials taken at energies ${\cal E}_{\rm eff}=\frac{1}{2}E_{d}+\frac{1}{2}\langle T_{r}\rangle$. This, however, would be a source of confusion and may lead to a wrong result. Indeed, if ${\cal U}(E,r,r^{\prime})$ is the underlying energy-dependent non-local potential for the phenomenological local energy-dependent equivalent $U_{phen}(E,r)$ then the ADWA needs to use the energy-independent non-local potentials ${\cal U}({\cal E}_{\rm eff},r,r^{\prime})$. This would give a different equivalent energy-dependent potential ${\tilde{U}}(E,r)$ which then should be taken at the energy $E_{d}/2+\Delta E$, as shown in Tim13a , Tim13b . The two potentials ${\tilde{U}}(E_{d}/2+\Delta E,r)$ and $U_{phen}({\cal E}_{\rm eff},r)$ are not the same. We will derive a link between them for a particular class of ${\cal U}(E,r,r^{\prime})$ in the next section. However, more generally, further knowledge of energy-dependent non-local potentials is needed.

It is interesting to compare our treatment of explicit energy dependent optical potentials with that of Ref. Del09a . In that work the Faddeev approach was used to solve the three-body problem. Energy dependence is treated in Del09a by taking the neutron (proton) optical potential at an energy equal to the variable neutron (proton) energy parameter used in evaluating the neutron (proton) $t$-matrix as it appears in the Faddeev equations. This seems physically plausible at first sight, but what kind of Schrödinger equation such an approach corresponds to is not clear. The Faddeev equations have been derived from the Schrödinger equation with energy-independent non-local pairwise potentials. We are not aware of a formal derivation of the Faddeev equations for explicitly energy-dependent pair potentials of the type that arise from many-body effects as discussed in Section II.

In our view, explicit energy dependence is an effect that can only be understood by explicitly recognising the internal degrees of freedom of the target as is done here and in Refs. Muk12 ; Joh05 . It should be added that the approximate prescription proposed here is intended for use only in the calculation of the incident channel distorting potential in a particular expression for the $A(d,p)B$ transition matrix and is not expected to be useful in any other context.

In this section we apply the formalism developed above to the ADWA calculation of the cross sections of the ${}^{16}{\rm O}(d,p)^{17}$O, ${}^{36}{\rm Ar}(d,p)^{37}$Ar and ${}^{40}{\rm Ca}(d,p)^{41}$Ca reactions at $E_{d}=15$, 9 and 11.8 MeV respectively. The cross sections of these reactions have been measured in o16 ; ar36 ; ca40 . These are the same reactions that have been considered in Ref. Tim13b using the energy-independent non-local nucleon optical potentials given by Giannini and Ricco (GR) systematics in Gia76 . The choice of targets made in Tim13b was to justify the use of the GR potential which is valid for $N=Z$ nuclei only. In the present work, we first use the Giannini-Ricco-Zucchiatti (GRZ) nonlocal potential GRZ which has an energy-dependent imaginary part. Then we make a suggestion of how to account for non-locality when only energy-dependent local phenomenological potentials are known but the underlying energy-dependent non-local potentials are not known. As an example we use the widely used CH89 systematics CH89 .