Implications for (d,p) reaction theory from nonlocal dispersive optical model analysis of 40 Ca(d,p) 41 Ca
aa r X i v : . [ nu c l - t h ] O c t Implications for (d,p) reaction theory from nonlocal dispersive optical model analysisof Ca(d,p) Ca.
S. J. Waldecker and N. K. Timofeyuk Department of Chemistry and Physics, University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN 37403 Department of Physics, Faculty of Engineering and Physical Sciences,University of Surrey, Guildford, Surrey GU2 7XH, UK (Dated: February 17, 2018)The nonlocal dispersive optical model (NLDOM) nucleon potentials are used for the first timein the adiabatic analysis of a (d,p) reaction to generate distorted waves both in the entrance andexit channels. These potentials were designed and fitted by Mahzoon et al. [Phys. Rev. Lett.112, 162502 (2014)] to constrain relevant single-particle physics in a consistent way by imposingthe fundamental properties, such as nonlocality, energy-dependence and dispersive relations, thatfollow from the complex nature of nuclei. However, the NLDOM prediction for the Ca(d,p) Cacross sections at low energy, typical for some modern radioactive beam ISOL facilities, is about 70%higher than the experimental data despite being reduced by the NLDOM spectroscopic factor of0.73. This overestimation comes most likely either from insufficient absorption or due to constructiveinterference between ingoing and outgoing waves. This indicates strongly that additional physicsarising from many-body effects is missing in the widely used current versions of (d,p) reactiontheories.
PACS numbers: 25.45.Hi, 21.10.Jx, 27.40.+z
I. INTRODUCTION
One nucleon transfer reactions have been a tool fornuclear spectroscopic studies for half of a century. To-day, they are used in experiments with radioactive beamsand among them the (d,p) reactions are perhaps themost popular choice. Analysis of these reactions re-lies on (d,p) reaction theory, which is traditionally ei-ther the distorted-wave Born approximation (DWBA)[1] or adiabatic distorted wave approximation (ADWA)[2], the latter being a computationally inexpensive wayof taking deuteron breakup into account. The deuteronbreakup means that the (d,p) amplitude should containthe A + n + p degrees of freedom explicitly, which requiressolving the A + n + p three-body Schr¨odinger equation.Several methods exist to solve this equation exactly, theCDCC [3, 4] and the Faddeev approach [5] being the mostused. The usual assumption in these calculations is thatthe A + n + p Hamiltonian contains the p − A and n − A potentials (often taken at half the deuteron incident en-ergy) that describe nucleon elastic scattering. However,it has been shown in [6] that the n − A and p − A potentialsfor the A + n + p problem are very complicated objectswhich depend on the position and the energy of the thirdnucleon and are not equal to optical potentials taken athalf the deuteron energy. It was also shown in [6] thatin the case of (d,p) reactions the averaging over the firstWeinberg component (which is the same as making theadiabatic approximation) results in a simple prescriptionfor choosing the n − A and p − A potentials appropriatefor analysis of (d,p) reactions. This prescription is possi-ble due to the main contribution to the (d,p) amplitudecoming from small n − p separations.The prescription in [6] says that within the Feshbachformalism, the n − A and p − A potentials should be non- local energy-dependent potentials evaluated at half thedeuteron incident energy plus half of the n − p kineticenergy in deuteron averaged over the n − p potential,which is about 57 MeV. After evaluation of these poten-tials, they should be treated as energy-independent andnonlocal. A simple recipe to include such potentials intothe available (d,p) reaction scheme, based on the localenergy approximation, is given in [7].At the time when [6] was written, only one energy-dependent nonlocal potential had been known [8], de-rived from Watson multiple scattering theory. It has anenergy-independent real part and an energy-dependentimaginary potential. Soon after the publication of [6],a nonlocal version of the dispersive optical model (NL-DOM) became available for Ca [9]. The nonlocal struc-ture of NLDOM is more complicated than that from pre-vious nonlocal optical potentials in that it is described byseven different nonlocality parameters. Based on the nu-cleon self-energy from Green’s function many-body the-ory, the NLDOM potential contains both real and imag-inary dynamic terms that are connected through a dis-persion relation, which enforces causality and links thenegative and positive energy regions. This dispersion re-lation is important for constraining the NLDOM param-eters with both scattering and bound-state data while si-multaneously providing a good description of these data.The potential from Ref. [8] was also constrained withboth scattering and bound-state data but without incor-porating the dispersion relation.In this paper, we analyze the Ca(d,p) Ca reactionat 11.8 MeV using NLDOM to generate the distortingpotentials in both the entrance and exit channels. TheNLDOM potential has already been used in [10] to cal-culate the Ca(p,d) Ca cross sections but only withinthe DWBA (which means neglecting deuteron breakup)and no comparison to the experimental data was made.Our choice of the reaction is due to the availability of thep- Ca and n- Ca optical potentials needed to constructthe d- Ca potential. The choice of the deuteron energyis due to several radioactive beams facilities existing inthe world that use this low-energy range. Also, it is thislow-energy range where the dispersive relations cause themost prominent effects in the energy behavior of the op-tical potential. In addition, at these energies, spin-orbiteffects and finite range effects can be neglected and theprescription from [6] should be valid.In Sec. II, we review the NLDOM and show that, sim-ilar to the standard Perey-Buck case, a local-equivalentpotential exists for NLDOM and a generalization of thePerey factor can be introduced. In a similar fashion, weshow in Sec. III that the d- Ca distorting potential canbe constructed by extending the local scheme proposedin [7] to the case with several nonlocality parameters.We summarize the adiabatic approximation in lowest or-der and introduce first order corrections. In Sec. IV wecalculate the cross section of the Ca(d,p) Ca reactionat 11.8 MeV and show that, using the prescription from[6], the NLDOM strongly overestimates the experimentaldata. In Sec. V, we discuss the implications for the (d,p)reaction theory following from our analysis.
II. NONLOCAL DOM POTENTIAL ANDNUCLEON SCATTERING
The NLDOM potential from Ref. [9] models the ir-reducible nucleon self-energy Σ( r , r ′ ; E ) with real andimaginary parts that are both explicitly nonlocal. Thepotential contains eight terms, which were constrainedwith both bound-state and scattering data associatedwith Ca. It is written in the formΣ( r , r ′ ; E ) = U vol HF (˜ r ) H ( x ; β vol )+ U vol HF (˜ r ) H ( x ; β vol )+ U wbHF (˜ r ) H ( x ; β wb )+ U sur + dy (˜ r ; E ) H ( x ; β sur + )+ U sur − dy (˜ r ; E ) H ( x ; β sur − )+ U vol + dy (˜ r ; E ) H ( x ; β vol + )+ U vol − dy (˜ r ; E ) H ( x ; β vol − )+ U so ( r ; E ) (1)where ˜ r = | r + r ′ | / x = r ′ − r . Following Pereyand Buck [13], the nonlocality function H is assumed tobe of the form H ( x ; β ) = exp( − x /β ) / ( π / β ) , (2)where β is the nonlocality range.In Eq. (1) the U HF terms represent the static partof the self-energy and are purely real. For this reason,these terms are referred to as parts of the Hartree-Fock(HF) potential, but technically they do not form the true TABLE I. NLDOM nonlocality parameters.Parameter Nonlocality (fm) β vol β vol β wb β sur + β sur − β vol + β vol − HF potential, because in practice a subtracted dispersionrelationship is used [9]. The U dy terms represent the dy-namic part of the self-energy and are complex. The realpart of the dynamic self-energy is determined completelyfrom the dispersion integral of the imaginary part. Thedynamic self-energy consists of surface and volume terms,and these have different nonlocalities for energies abovethe Fermi energy E F (denoted with a ’+’ sign) and en-ergies below E F (denoted by a ’-’ sign). The inclusion ofseveral nonlocality parameters was based on the micro-scopic calculation in Ref. [11], which indicated differentdegrees of nonlocality in different energy regions. Table Ishows the value of each nonlocality parameter. Note thatsome of these parameters are about twice as large as thoseknown from traditional Perey-Buck potentials. The U so term is the spin orbit potential, which was assumed tobe local. It has a weak energy dependence that only be-comes important at high energies.Overlap functions can also be generated with the dis-persive optical model. For discrete states in the A + 1and A − Ca (d,p) Ca reactions, the calculatedbinding energy of the 0 f / neutron level in Ca mustmatch the experimental one of 8 .
36 MeV. However, in [9],such a constraint was not employed. For the purposes ofthis study, some of the parameters were refit in order toreproduce the experimental binding energy of the 0 f / neutron level. Only the parameters associated with the U HF terms were refit, and they were constrained by elas-tic scattering, charge density, and energy level data. Allother parameters were unchanged. The new parametersare shown in Table II and compared with those from theanalysis in [9]. The quality of the new fit is comparableto that obtained in [9].The nonlocal terms in Eq. (1) can be written moresuccinctly asΣ N ( r , r ′ ; E ) = X i U NA,i (˜ r ) H i ( x ) , (3)where the energy dependence of the dynamic terms isimplied, and N = p, n for proton and neutron potentials,respectively.In this paper we will construct an effective local modelfor the deuteron-target adiabatic potential using NL- TABLE II. Adjusted HF parameters used in the present fit.For a description of these parameters, refer to [9].Parameter New value Old value V HF [MeV] 106.15 100.06 r HF [fm] 1.14 1.10 a HF [fm] 0.58 0.68 β vol [fm] 0.84 0.66 β vol [fm] 1.55 1.56 x V wb [MeV] 12.5 15.0 ρ wb [fm] 2.06 1.57 β wb [fm] 1.04 1.10 DOM. Therefore, we first evaluate the effective local po-tential model for nucleon scattering. Following the pro-cedure in [13] for transforming a nonlocal potential to alocal equivalent, one finds that for a nonlocal potentialwith multiple nonlocalities of the Perey-Buck type thelocal-equivalent potential can be found by solving thetranscendental equation U loc ( r ) = X i U NA,i ( r ) exp (cid:20) − µ N β i h ( E − U loc ( r )) (cid:21) (4)where µ N is the reduced mass of the N + A system. Thisequation is obtained in the lowest order of the expansionof Σ N ( r , r ′ ; E ) over x and corrections to any order canbe built systematically using developments from [14]. Inparticular, the first order correction is∆ U N = ¯ h µ N "(cid:18) ∇ ff (cid:19) − ∇ ff , (5)where the function f ( r ) is the Perey factor explained be-low. For proton scattering, equation (4) must be cor-rected by reducing the centre-of-mass energy E in ther.h.s. of Eq. (4) by the local Coulomb interaction V coul ( r ).According to [14], the local spin-orbit term must alsobe corrected when transforming to a local-equivalent po-tential. For the present case of a potential with multiplenonlocality parameters, the new spin-orbit term, U leso , ofthe local-equivalent potential is U leso = U so / (1 − U ) (6)where U = − X i µ N β i h U NA,i ( r ) G i ( r, E ) ! (7)and G i ( r, E ) = exp (cid:20) − µ N β i h ( E − U loc ( r )) (cid:21) . (8)Figure 1 compares proton scattering differential crosssections (normalized by the Rutherford cross section)determined from solving the NLDOM scattering prob-lem exactly using the iterative procedure outlined in Ref. [15] and from solving the local-equivalent prob-lem. Both the Coulomb and spin-orbit corrections areincluded. The experimental data are from Refs. [16–19].Aside from small deviations at large angles, the resultsfrom using Eq. (4) are very similar to the exact solutions.The results from including ∆ U p are also shown. Overall,this correction improves the correspondence between theexact and approximate solutions of the nonlocal problemfor angles θ ≈ ◦ and above.For Perey-Buck potentials with one nonlocality param-eter the N − A wave function obtained from the phase-equivalent local model defined by a potential U ( r ) = U loc + ∆ U N differs in the nuclear interior from the exact N − A wave function by the Perey factor [14] f ( r ) = exp (cid:20) µ N β h U loc ( r ) (cid:21) . (9)Elastic scattering observables do not depend on thePerey factor. Transfer cross sections may depend onit if they are not peripheral. In the particular case of Ca(d,p) Ca, the internal part contributes up to 20%[20] for the energy being considered, and this contribu-tion is more important in the DWBA than in the ADWA.One can show it is also possible to derive the Perey factorfor optical potentials with multiple nonlocalities such asin NLDOM. In this case the Perey factor is f ( r ) = exp (cid:20) µ N β ( r )4¯ h U loc (cid:21) , (10)where β ( r ) = − U − loc ( r ) Z ∞ r dr P i β i U ′ NA,i ( r ) G i ( r , E )1 − U ( r , E ) . (11)This Perey factor has some effective r -dependent range β eff , which can be complex. The real and imaginaryparts are shown in Fig. 2 for the case of p - Ca elas-tic scattering at several proton energies. The imaginarypart is small and has a negligible effect on the (d,p)cross sections. We note that β eff decreases with energy.This decrease reflects the fact that β sur + is larger than β vol + . Since the volume imaginary potential dominatesat higher energies, the term in Eq. (4) with β i = β vol + be-comes more important with increasing energy. This termalso seems to dominate at large r , as Re β eff convergesto β vol + = 0 .
64 fm for all energies.The Perey factor for NLDOM is shown in Fig. 3, evalu-ated at the energy E p = 17 .
37 MeV, which is the center ofmass energy for the outgoing proton in the Ca(d,p) Careaction with E d = 11 . -2 -1 σ / σ r u t h nonlocallocallocal + ∆ U p -2 -1 θ c.m. (deg)10 -2 -1 σ / σ r u t h θ c.m. (deg)10 -3 -2 -1 E = 9.86 MeV E = 17.57 MeVE = 40 MeV E = 65 MeV(a) (b)(c) (d)
FIG. 1. (color online). Differential cross sections normalized by the Rutherford cross section for proton scattering on Ca at(a) 9.86 MeV, (b) 17.57 MeV, (c) 40 MeV and (d) 65 MeV, calculated using the fully nonlocal DOM potential (solid), using thelocal potential U loc from Eq. (4) with the Coulomb and spin-orbit potentials included (dot-dashed), and using this equivalentlocal potential but with the correction ∆ U p included as well (dashed). These are compared with the experimental data (dots). f ( r, E ) = p ˜ m ( r, E ) /m, (12)where ˜ m ( r, E ) /m is the so-called momentum-dependenteffective mass and is related to the LDOM Hartree-Fockpotential as ˜ m ( r, E ) m = 1 − d V HF ( r, E ) dE . (13)Figure 3 also shows the Perey factor calculated with thewidely used CH89 potential using Eq.( 9) and assuming β = 0 .
85 fm. The Perey factors from LDOM and CH89both have less effect in the surface region than the onecalculated with NLDOM.To calculate the Ca(d,p) Ca cross sections a choiceneeds to be made for the optical potential U p in the exitchannel. In principle, this potential is auxiliary, and itis believed that choosing U p that describes proton elas-tic scattering in the exit channel makes the remnantterm P i V pi − U p in the transfer amplitude to disap-pear [1]. Since NLDOM was not fit to p + Ca scattering data, one choice for the auxiliary p + Ca potential U p is to use NLDOM but evaluated with A = 41 instead of A = 40. An alternative, originally proposed in [24] andthen further explored in [25], stems from the argumentthat the remnant term can be removed from the tran-sition operator exactly, leading to a different model forthe exit state wave function. In this model, the three-body Hamiltonian associated with the exit channel con-tains the p − Ca optical potential, the n − Ca bound-state potential and no n − p interaction. In the limitof infinitely large core and in the zero-range approxima-tion, the corresponding three-body wave function con-tains the p − Ca distorted wave function calculated withthe p − Ca optical potential. Corrections due to recoilexcitation and breakup are considered in [25]. For lightnuclei the validity of the transfer amplitude with no rem-nant has also been confirmed by [3]. We analyzed the(d,p) reaction with both choices for U p , and the result-ing cross sections were found to differ at the peak byabout 1%. For the purposes of this study, both choicesgive practically the same result. Below, we choose to useNLDOM evaluated with A = 41 for U p . R e β e ff (f m ) r (fm) -0.2-0.100.1 I m β e ff (f m ) (a)(b) FIG. 2. (color online). (a) Real and (b) imaginary parts of β eff for E = 9.86 MeV (solid), 17.57 MeV (long dashed), 40MeV (long dot-dashed), 65 MeV (short dashed), 100 MeV(short dot-dashed), and 200 MeV (dot-dot-dashed). r (fm) | f | FIG. 3. (color online). Absolute value of the Perey factorevaluated at E cm = 17 .
37 MeV for NLDOM (solid), LDOM(dashed) and CH89 (dot-dashed).
III. THE DEUTERON-TARGET POTENTIALFOR ( d, p ) REACTIONS IN THE ADIABATICAPPROXIMATION
Following Johnson-Tandy [2] we retain only the firstWeinberg component of the A + n + p system in the( d, p ) transfer amplitude since this amplitude is sensi-tive only to those parts of this wave function in whichthe neutron n is close to the proton p . Recently, exactcontinuum-discretized coupled channel calculations con-firmed that this component indeed dominates [26]. The first Weinberg component is a product of the deuteronwave function φ ( r ) times the d − A relative motionwave function χ ( R ) which is the solution of the two-body Schr¨odinger equation with an adiabatic potentialconstructed from p − A and n − A optical potentials.The generalization of the deuteron adiabatic potentialfor the case of nonlocal, energy-independent N − A op-tical potentials of the Perey-Buck type is given in [7]. Ifnonlocal potentials (such as NLDOM) explicitly dependon energy then they should be evaluated at the energy E = E d / h T np i V np / h T np i V np / n − p kinetic energy in deuteron averaged over the short-ranged potential V np . The value of this term is about 57MeV [6], so for E d = 11 . n − p momentum, itbecomes clear that there is an additional kinetic energyin the N − A system which should be taken into accountwhen choosing the energy at which the N − A potentialshould be evaluated. A. Lowest order equivalent local model
The nonlocal Schr¨odinger equation for χ ( R ) from [7]can easily be generalized for the case of nonlocal opticalpotentials with multiple nonlocalities:( T R + U C ( R ) − E d ) χ ( R ) = − X N = n,p X i Z d s d x × φ ( x + α s ) U NA,i ( x − R ) H i ( s ) φ ( x ) χ ( α s R )(14)where R is the radius-vector between d and A , T R is thekinetic energy operator associated with R , α = A/ ( A +1), α = ( A + 2) / ( A + 1), φ is the deuteron ground statewave function and φ ( r ) = V np ( r ) φ ( r ) h φ | V np | φ i . (15)Solving the nonlocal problem (14) directly is certainlypossible, and has recently been done in Ref. [27]. How-ever, in this paper, we construct the local-equivalentmodel, as simplified local-equivalent models can provideuseful insight into the physics of a problem and makeavailable transfer reaction codes applicable to nonlocalproblems. The local-equivalent approximation of (14)can be obtained by expanding both U NA,i ( x − R ) and χ ( α s + R ) into Taylor series. In the lowest order ap-proximation, using U NA,i ( x − R ) ≈ U NA,i ( R ) we get( T R + U C ( R ) − E d ) χ ( R ) = − X i U dA,i ( R ) ˜ H (0) i ( T R ) χ ( R ) , (16) TABLE III. Coefficients β ( λ ) n (in fm) and moments M ( λ )0 (in fm λ ) for λ = 0 , β that are used in the NLDOM. n β = 0 .
64 fm β = 0 .
84 fm β = 0 .
94 fm β = 1 .
04 fm β = 1 .
55 fm β = 2 .
07 fm β (0)1 β (0)2 β (0)3 β (0)4 β (0)5 β (0)6 β (0)7 β (0)8 M (0)0 β (1)1 β (1)2 β (1)3 β (1)4 β (1)5 β (1)6 β (1)7 β (1)8 M (1)0 where U dA,i ( R ) = U nA,i ( R ) + U pA,i ( R ),˜ H (0) i ( T R ) = M (0)0 ,i ∞ X n =0 ( − ) n n ! (cid:18) µ d α h ( β (0) n,i ) T R (cid:19) n , (17)the coefficients β (0) n,i are defined by β (0) n,i = 1 √ " M (0)2 n,i (2 n + 1)!! M (0)0 ,i n , (18)and the moments M (0)2 n,i are defined by M (0)2 n,i = Z d s d x s n H i ( s ) φ ( x − α s ) φ ( x ) . (19)Eq. (16) is further simplified by introducing the local-energy approximation [1], T R ≈ T ( R ) = E d − U loc ( R ) − U C ( R ) , (20)where the local potential U loc ( R ) is defined as U loc ( R ) = X i U dA,i ( R ) ˜ H (0) i ( T ( R )) . (21)This approximation works very well for nucleon opticalpotentials with one nonlocality parameter [7]. We willshow that it remains good for NLDOM with its multiplenonlocalities, but we first present the results of calcula-tions of U loc ( R ) for E d = 11.8 MeV using the deuteron wave function from the Hult`en model, the same as in [7].It is pointed out in [7] that a realistic deuteron wave func-tion gives the moment M (0)0 which is very similar to thatobtained with the Hult´en wave function. In Table IIIwe show the calculated β (0) n,i and M (0)0 ,i terms. For verysmall nonlocalities, β (0) n,i should be approximately equalto β i / ∼ . − . ∼
10% smaller than the β i / n , allowing one to replace the β (0) n,i coefficients with a constant. We define this constantto be β d,i = β (0)1 ,i . (22)The summation in (17) then gives ˜ H (0) i ( T R ) an expo-nential form˜ H (0) i ( T R ) ≈ H (0) i ( γ i T R ) = M (0)0 ,i exp ( − γ i T R ) , (23)where γ i = µ d α β d,i h . (24)As β i increases beyond 1.0 fm, the approximation inEq. (22) becomes less valid and the coefficients β (0) n,i de-viate from β i / β = 2 .
07 fm,
R (fm) -80-60-40-200 U l o c ( R ) ( M e V ) E d = 11.8 MeV FIG. 4. (color online). Real and imaginary parts of the local-equivalent NLDOM potential calculated using the exponentialform (solid and dotted lines, respectively) and using the seriesform (thick short-dashed and long-dashed lines, respectively)with n max = 6. which is the largest nonlocality parameter used in NL-DOM. Thus, we solve Eqs. (17), (20), (21) by restrictingthe sum over n in (17) by some n max . We found that n max = 6 is sufficient to obtain a converged solution for U loc . However, we also found that using Eq. (22) leadsto practically the same result because the real and imagi-nary parts of the NLDOM terms associated with β = 2 . U loc potentials obtained from solution of (21) with (17) and(23) are shown in Fig. 4. Using Eq. (22) for all nonlocal-ity parameters, one obtains U loc from the transcendentalequation U loc = X i M (0)0 ,i U dA,i × exp " − µ d α β d,i h ( E d − U loc − U C ) . (25)The Coulomb potential U C is approximated by a con-stant, given by U C = − .
08 + 1 . ZA / , (26)which was used in Refs. [8, 29]. The difference betweenusing this approximation and a more realistic potential isonly about 1%, in terms of the peak cross section of theproton angular distribution for the Ca(d,p) Ca reac-tion at E d = 11 . B. Correction to the local-energy approximation inthe lowest order
It was shown in [7] that corrections to the lowest or-der local model beyond the local-energy approximationare small because they are the fourth-order effect of thenucleon nonlocality β , as the second-order terms canceleach other for Perey-Buck potentials with one nonlocality parameter. The NLDOM from [9] contains several non-localities, and second-order contributions may not can-cel. Moreover, some of these nonlocalities are large sothat the contributions beyond the local-energy approxi-mation are expected to be larger than those in [7]. Inthis section we study these corrections using results fromsections IV.C and A.4 of [7]. Including leading correctionterm, linear in the kinetic energy operator T R , and usingthe exponential form (23) for ˜ H (0) i ( T R ), the right-handside of Eq. (16) becomes X i U dA,i H (0) i ( γ i T R ) χ ( R ) ≈ X i U dA,i H (0) i ( γ i T ) × (cid:20) − γ i ( T R − T + ∆ i ) − ¯ h γ i µ d ∇ T · ∇ R (cid:21) χ ( R ) , (27)where the energy correction ∆ i , arising because T R and T do not commute, is given by∆ i = ¯ h γ i µ d (cid:18) T ′′ T ′ R − γ i T ′ (cid:19) . (28)The solution of Eq. (16) in this approximation is theproduct χ ( R ) = f ( R ) ϕ ( R ) , (29)where ϕ is the scattering wave of the local model( T R + U C − E d ) ϕ = − ( U loc + ∆ U ) ϕ. (30)The U loc term is discussed in the previous section, andthe correction term ∆ U is∆ U = T R f f + ¯ h µ d (cid:18) U T ′ − U (cid:19) − U ∆ − U (31)with U n ( R ) = X i U dA,i H (0) i ( γ i T ) γ ni , (32) U ∆ ( R ) = X i U dA,i H (0) i ( γ i T ) γ i ∆ i . (33)The function f is the analog of the Perey factor dis-cussed in Sec. II. It modifies the scattering wave function ϕ ( R ) in the nuclear interior and satisfies the first orderdifferential equation ∇ f f = − U ( R )1 − U ( R ) ∇ T , (34)with the boundary condition f ( R ) → R → ∞ . Thesolution of this equation is f ( R ) = exp (cid:18) Z ∞ R dR U ( R )1 − U ( R ) T ′ ( R ) (cid:19) . (35)Because of multiple nonlocalities, the analytical integra-tion in (35) cannot be done. So, it is difficult to see ifthe contributions to f from second-order terms on β d,i cancel. Most likely, they do not. R (fm) | f | FIG. 5. (color online). Perey factors f (solid), f (dot-dashed), and f (dashed) calculated with NLDOM for E d =11 . R e ∆ U ( M e V ) R (fm) -0.2-0.100.10.2 I m ∆ U ( M e V ) (a)(b) FIG. 6. (color online). (a) The real parts and (b) the imagi-nary parts of ∆ U (solid) and the first (dashed), second (dot-ted), and third (dot-dashed) terms in Eq. (31). The Perey factor f and the correction ∆ U to theequivalent local potential U loc are shown in Fig. 5 andFig. 6, respectively, for d + Ca at a deuteron incidentenergy of 11.8 MeV. The Perey factor increases the scat-tering wave in the nuclear interior by about 6%, whichis a couple of percent higher than the result in [7]. Thecorrection to U loc , however, remains small. Its real partis very close to the one obtained in [7] in the maximum,being about 150 keV, while the imaginary part is muchsmaller. Thus, for NLDOM the second order correctionsmost likely remain small and the local-energy approxi-mation remains good. C. First order corrections
The first order correction to the local-equivalentlowest-order model of Sec. III A is obtained by retainingtwo terms in the Taylor series expansion of the centralpotential U NA ( ± x − R ): U ( ± x − R ) ≈ U ( R ) ∓ x · ∇ U ( R ) . (36)In this case, using techniques of [7], we obtain the follow-ing:( T R + U C ( R ) − E d ) χ ( R ) = − X i U dA,i ( R ) H (0) i ( γ i T R ) χ ( R ) − X i ∇ [ U dA,i ( R )] H (1) i (˜ γ i T R ) ∇ χ ( R ) , (37)where H (1) i (˜ γ i T R ) = M (1)0 ,i M (0)0 ,i H (0) i (˜ γ i T R ) (38)and the moments M (1)2 n,i are defined as M (1)2 n,i = Z d s d x s n H i ( s ) φ ( x − α s ) φ ( x ) α s · x . (39)The new factor ˜ γ i arises from the fact that the moments M (1)2 n,i lead to new coefficients β (1) n,i that are also practi-cally independent of n (see Table III). Introducing a newconstant ˜ β d,i = β (1)1 ,i (40)the factor ˜ γ i can be written as˜ γ i = µ d α ˜ β d,i h . (41)The coefficients β (1) n,i are defined as β (1) n,i = 1 √ " M (1)2 n,i (2 n + 3)!! M (1)0 ,i n . (42)At this point we make the local-energy approximation(20) but we also include the correction to this approx-imation similar to that considered above. We expand H (0) i ( γ i T R ) in the first term of r.h.s of Eq. (37) as inEq. (27), but we use a simpler expansion for H (1) i (˜ γ i T R )in the second term, H (1) i (˜ γ i T R ) = H (1) i (˜ γ i T )(1 − ˜ γ i ( T R − T )) , (43)because H (1) i (˜ γ i T R ) is an order of magnitude smaller than H (0) i ( γ i T R ) so that the higher order terms on ˜ β d,i in termswith ∆ i and ∇ · ∇ R will be small. For a similar reasonwe keep only the leading correction to ( T R − T ) ∇ χ ( R ):( T R − T ) ∇ χ ( R ) ≈ ∇ T χ ( R ) . (44)With these approximations we can solve Eq. (37) byintroducing the same representation χ ( R ) = f ( R ) ϕ ( R )used both for proton scattering in Sec. II and for correc-tion to local-energy-approximation above. The scatteringwave ϕ is found from the local equation( T R + U C − E d ) ϕ = − ( U loc + ∆ U ) ϕ, (45)with the same U loc as before but corrected by∆ U = ∆ U + ∆ U , (46)in which the first term is the same as in Eq. (31), andthe second term is given by∆ U = T R f f + µ d ¯ h U (1 − U ) − U T ′ − U − U U T ′ (1 − U ) , (47)where U n ( R ) = X i ∇ [ U dA,i ( R )] H (1) i (˜ γ i T )˜ γ ni . (48)The Perey factor f is the solution of the first order dif-ferential equation ∇ ff = µ d ¯ h U − U − U T ′ − U (49)with the boundary condition f ( R ) → R → ∞ . Thesolution to this equation can be written as f ( R ) = f ( R ) f ( R ) (50)where f is given by Eq. (35) and f is given by f ( R ) = exp (cid:18) − µ d ¯ h Z ∞ R dR ′ U ( R ′ )1 − U ( R ′ ) (cid:19) . (51)The correction ∆ U and the four terms on the right-hand side of Eq. (47) are shown in Fig. 7. The Pereyfactors f and f are plotted in Fig. 5. The correction∆ U and the Perey factor f are comparable to the cor-responding quantities obtained in [7] for a Perey-Buckpotential with a single nonlocality. We can rewrite thePerey factor f in the form of Eq. (10). The correspondingeffective deuteron nonlocality, β d, eff is plotted in Fig. 8as a function of r for E d = 11 . β d, eff iscomplex, but, as in the case for protons, the imaginarypart is small and changes the cross section of the protonangular distribution by less than 0.5%. The real partlies between 0.50 and 0.60 fm, and these values are verysimilar to 0.56 fm, which is the value used for deuteronelastic scattering. R e ∆ U ( M e V ) R (fm) -0.4-0.200.20.4 I m ∆ U ( M e V ) (a)(b) FIG. 7. (color online). (a) The real parts and (b) the imagi-nary parts of ∆ U (solid) and the first (short-dashed), second(long-dashed), third (dot-dashed), and fourth (dotted) termsin Eq. (47). R (fm) -0.200.20.40.6 β d , e ff (f m ) FIG. 8. (color online). Real part (solid) and imaginary part(dashed) of effective deuteron nonlocality as a function of r for E d = 11 . IV. TRANSFER REACTION Ca(d,p) Ca at11.8 MEV
We calculated the proton angular distributions for the Ca(d,p) Ca reaction at E d = 11 . D , given by D = 15615 MeV fm , wasused. The distorted potentials both in the deuteron andproton channels, generated with NLDOM, were read intothe TWOFNR code [31]. There is no option in TWOFNRfor incorporating complex r -dependent effective nonlo-0 r (fm) I(r) (f m - / ) FIG. 9. (color online). Overlap functions calculated usingNLDOM (solid), LDOM (dashed), a Woods-Saxon potentialwith standard geometry (dot-dashed), and a Woods-Saxonpotential but corrected with a Perey factor with β = 0 .
85 fm(dot-dot-dashed). calities β eff ( r ). Therefore, in order to reduce the corre-sponding distorted waves in the nuclear interior, we mul-tiplied the NLDOM h Ca | Ca i overlap function (alsoread into the TWOFNR code) by the Perey factors ofthe proton and deuteron channels, given by Eqs. (10)and (50), respectively. This is legitimate in the zero-range approximation, where the integrand of the (d,p)reaction amplitude is a function of only one vector vari-able. In this case, the Perey factor for the proton channelhad to be calculated on a different grid.The overlap function I NLDOM ( r ), generated by NLDOMand read into TWOFNR, is compared in Fig. 9 to (i) theoverlap function I WS ( r ) obtained from a Woods-Saxonpotential with standard geometry ( r = 1 .
25 fm, a = 0 . I L DOM ( r ) generated withLDOM and (iii) the overlap function I NL WS ( r ), calculatedin a standard Woods-Saxon model employing a nonlocal-ity correction via the Perey factor with β = 0 .
85 fm. Allthese overlap functions are normalized to 0.73, which isthe spectroscopic factor calculated from NLDOM.We have found that I NLDOM ( r ) can be described very well(with about 1% accuracy) by a local two-body Woods-Saxon potential model that has the radius r = 1.252fm, diffuseness a = 0 .
718 fm and the spin-orbit strength V s.o. = 6 .
25 MeV. These parameters are very close to thestandard values of r = 1.25 fm and a =0.65 fm usedto generate I WS ( r ). However, relative to I WS ( r ), using I NLDOM ( r ) as the overlap function increases the transfercross section at the peak, σ peakd,p , by about 15% (for thereaction at E d = 11 . I NLDOM ( r )radius of 4.030 fm is slightly larger than that of I LDOM ( r ),which is 3.965 fm. Also, the single-particle ANC b ℓj for I NLDOM ( r ) is 2 .
48 fm − / , which is about 10% largerthan that of I L DOM ( r ). The spectroscopic factors forthese two overlaps are practically the same. As a result, θ c.m. (deg) d σ / d Ω ( m b / s r) Ca(d,p) Ca(7/2 - )E d = 11.8 MeV FIG. 10. (color online). Proton angular distributions for the Ca(d,p) Ca(7 / − ) reaction with E d = 11 . I NLDOM ( r ) produces a larger many-body ANC squared, C ℓj = S ℓj b ℓj , equal to 4 . − , whereas I L DOM ( r ) has C ℓj = 3 . − . Interestingly, the NLDOM value of C ℓj is very close to the prediction of C ℓj = 4 . − of the source term approach [32], which is based on theindependent-particle-model for Ca and Ca. This ap-proach accounts for correlations between nucleons viaan effective interaction potential of the removed nucleonwith nucleons in the core [33].The standard overlap I WS ( r ) is very close to I L DOM ( r )(see Fig. 9). The overlap I NL WS ( r ), which is sometimesused in (d,p) calculations, has a distinctly larger radiusand is not consistent with NLDOM. Below, in all our(d,p) calculations we use only the NLDOM overlap withits own normalization of 0.73, which allows for makingconclusions from comparison between theoretical and ex-perimental cross sections without any further renormal-ization.Proton angular distributions calculated using the NL-DOM potentials are presented in Fig. 10. The solid curvecorresponds to the lowest-order result. The long-dashedcurve shows that incorporating the first-order correctionsfor the proton channel, via Eqs. (5) and (10), reduces thelowest-order (d,p) peak cross sections σ peakd,p by 3 %. Fur-ther first-order corrections, coming from Eqs. (46) and(50) for the deuteron channel and shown by the short-dashed curve, reduces σ peakd,p by another 5 %.Finally, including the spin-orbit potential raises σ peakd,p and makes it comparable to the cross sections obtainedwith no corrections in the deuteron channel (the dot-dashed curve). The experimental data are from [34].The spread between all these calculations does not exceed112% and all of them considerably overestimate the exper-imental data, shown in Fig. 10 as well. This overestima-tion (by about 70% after normalizing overlap functionto 0.73) cannot come from the local approximations wehave made to solve the nonlocal problem. The first-ordercorrections of about 12% mean that the second-order cor-rections would most likely be around 1% or less. Thus,other reasons for this overestimation should be investi-gated.It was already noted in [7, 28] that the adiabatic modelwith nonlocal energy-independent potentials gives highercross sections, as compared to the standard adiabaticmodel, due to a weaker attraction in the deuteron chan-nel. The higher cross sections are confirmed in other(d,p) studies with such potentials [27]. Here, the overes-timation of the (d,p) cross sections using the NLDOM isstronger than in the case of energy-independent poten-tials. This can be seen in Fig. 11, which compares theNLDOM angular distribution with the angular distribu-tions from two nonlocal, energy-independent potentials,referred to as GR [29] and TPM [35]. Figure 11 alsoshows the angular distribution from another nonlocal,energy-dependent potential, referred to as GRZ [8] andused previously in [6]. This potential generates an an-gular distribution very similar to the one generated withNLDOM. The structure of the GRZ potential is not ascomplicated as the NLDOM potential, but it has a typicallow-energy behaviour of the imaginary part, vanishing at E → ∼
20% difference in σ peakd,p shown inFig. 11, we compare the entrance and exit channel poten-tials generated from the four nonlocal parameterizationsand present them in Fig. 12. The NLDOM, GRZ andGR generate real parts of similar depths and sizes both θ c.m. (deg) d σ / d Ω ( m b / s r) Ca(d,p) Ca(7/2 - )E d = 11.8 MeV FIG. 11. (color online). Proton angular distributions for the Ca(d,p) Ca(7 / − ) reaction with E d = 11 . in entrance and exit channels while the TPM produces areal part of a moderately larger radial extent. The imag-inary parts, however, show a marked difference, in boththe entrance and exit channels. The energy-independentparameterizations GR and TPM produce a much largerimaginary part in the surface region than the energy-dependent parameterizations NLDOM and GRZ. Thesmaller imaginary parts produce less absorption thus in-creasing σ peakd,p . This is even better seen in Fig. 13, whichshows the (d,p) angular distributions calculated with fourdifferent parameterizations for the exit proton channeland using NLDOM for the deuteron channel. In thisfigure, we have also added the calculations with LDOM[21] and the widely used local CH89 [36] parameteriza-tions in the proton channel while keeping the NLDOMin the deuteron channel. The LDOM and CH89 protonpotentials are shown in panels c) and d) of Fig. 12.Figure 13 shows that predictions with NLDOM,LDOM and GRZ form a different class from those ob-tained with GR, TPM and CH89. All the potentials fromthe first class have smaller imaginary parts and/or vol-ume integrals than the potentials from the second class.Thus, overestimation of the cross section calculated withNLDOM seems to be at least partly due to a weaker ab-sorption in the exit channel potential.The weaker absorption may not be the only reasonfor large cross sections obtained with NLDOM. It wasdiscussed in detail in [37] that a particular relation be-tween optical potentials in entrance and exit channelsresults in destructive interference between the ingoingand outgoing partial waves leading to l -localization ofradial (d,p) amplitudes in the adiabatic model. A simi-lar situation may occur here. Indeed, the standard adia-batic Johnson-Soper (JS) [38] calculations using both thelocal-equivalent NLDOM and LDOM potentials, taken at E = E d /
2, predict much lower cross sections (see Fig. 14)while the imaginary parts of the JS potentials, shown inFig. 15, are much smaller than those of local-equivalentdeuteron potentials obtained in this work (TJ). Thiscould be an indication of constructive interference be-tween the ingoing and outgoing partial waves generatedwith NLDOM potentials. A new procedure was proposedin Sec. VI.B of Ref. [6], explaining how phenomeno-logical local energy-dependent optical potentials can beused in (d,p) calculations if they represent local equiva-lents of nonlocal potentials. Using the LDOM potentialwithin this procedure and assuming a hidden nonlocalityof 0.85 fm gives very similar results to NLDOM both forthe real part of deuteron distorting potential (Fig. 15)and the (d,p) cross sections (Fig. 14) despite strongerimaginary part in the deuteron channel. Although theJS cross sections are close to experimental data, in lightof recent findings [6, 7, 27, 28], constructing the adia-batic potentials from nucleon optical potentials taken athalf the deuteron incident energy does not seem to bejustified anymore.2 R ea l P o t e n ti a l ( M e V ) I m a g i n a r y P o t e n ti a l ( M e V ) (a) (c)(b) (d) FIG. 12. (color online). Panels (a) and (b) show the real and imaginary parts of the entrance channel potentials for theNLDOM (solid), GRZ (dashed), GR (dot-dashed), and TPM (dotted) nonlocal parameterizations used in Fig. 11, while panelsc) and d) show these quantities for the exit channel. Panels (c) and (d) also show the potentials calculated using LDOM(dot-dot-dashed) and CH89 (dash-dash-dotted).
V. CONCLUSION
We presented the first adiabatic (d,p) calculations withthe NLDOM potential, which has been designed with theaim of forging the link between nuclear structure and nu-clear reactions in a consistent way. It has its roots inthe underlying self-consistent Green’s functions theoryand possesses the fundamental properties - nonlocality,energy-dependence and dispersion relations - that arisefrom the complex structure of the target. The NLDOMexplicitly takes into account a number of components ofnuclear many-body theory that many other optical mod-els do not.One could expect that using an advanced optical po-tential parametrization such as NLDOM would result inproperly fixed single-nucleon properties both below andabove the Fermi surface crucial for agreement betweenpredictions of (d,p) reaction theory and experimentaldata. However, we have shown that using the NLDOMto generate the distorting potentials entering the (d,p)amplitude strongly overestimates the (d,p) cross sectionsdespite the reduced strength of the NLDOM one-neutronoverlap function employed in the calculations. Moreover, the NLDOM predictions are very similar to those madewith a much simpler nonlocal potential GRZ derivedwithin Watson multiple scattering theory and Wolfen-stein’s parameterization of the nucleon-nucleon scatter-ing amplitude [8, 29]. The energy dependence is pre-sented in GRZ only in the imaginary part.Since we do not have strong reason to doubt the qualityof the NLDOM parameterization the main assumptionsof the (d,p) theory used in the present calculations shouldbe reviewed. We list them below: • The (d,p) amplitude contains a projection of thetotal many-body wave function into the three-bodychannel A + n + p only. Projections onto all excitedstates of A are neglected. • Only n − A and p − A potentials are used to calculatethe A + n + p projection. According to [6] there arealso multiple scattering terms playing the role of athree-body A + n + p force. These are neglected. • Averaged n − A and p − A potentials were obtainedusing the procedure from Ref. [6], which uses theadiabatic approximation. Corrections to this ap-3 θ c.m. (deg) d σ / d Ω ( m b / s r) Ca(d,p) Ca(7/2 - )E d = 11.8 MeV FIG. 13. (color online). Proton angular distributions for the Ca(d,p) Ca(7 / − ) reaction with E d = 11 . θ c.m. (deg) d σ / d Ω ( m b / s r) Ca(d,p) Ca(7/2 - )E d = 11.8 MeV FIG. 14. (color online). Proton angular distributions for the Ca(d,p) Ca(7 / − ) reaction at E d = 11 . E = E d / E = E d /
2, with both NLDOM(dashed) and LDOM (dot-dashed). proximation may change the energy value at whichthese potentials should be evaluated. • It was assumed that the (d,p) transition opera-tor contains the V np term only. Any other termspresent in this amplitude [1] are neglected. • It was shown in [25] that keeping V np only in the(d,p) transition operator modifies the proton chan-nel wave function. In our particular case, thiswould result in using p- Ca optical potential inthe p+ Ca channel. We have not seen any differ-ence in (d,p) cross sections when replacing Ca by Ca and this could mean that the averaging proce-dure, introduced in [6], when applied to the special R ea l P o t e n ti a l ( M e V ) R (fm) -20-15-10-50 I m a g i n a r y P o t e n ti a l ( M e V ) (a)(b) FIG. 15. (color online). (a) Real and (b) imaginary partsof the deuteron potential for the Ca(d,p) Ca(7 / − ) reac-tion at E d = 11 . E = E d / E = E d /
2, with both NLDOM (dashed) andLDOM (dot-dashed). A + n + p three-body model, that does not have V np and has different asymptotic conditions [25], mayresult in completely different requirements to theproton distorting potential in the exit channel. Us-ing proton optical potentials may not be justifiedanymore. • We used the adiabatic approximation to solve thethree-body Schr¨odinger equation.The deviation from the adiabatic approximation forsolving the Schr¨odinger equation has been studiedboth within the continuum-discretized coupled channelmethod [3] and using Faddeev equations [39]. Althoughthese corrections can be non-negligible, they cannot beresponsible for 70% overestimation of (d,p) cross sectionsobtained in this work. At E d / ∼
12 MeV these devi-ations were no more than 4%, while at a larger energyrange, 5 ≤ E ≤
56 MeV they could be up to 23%. Theunknown non-adiabatic corrections to optical n − A and p − A potentials entering the Schr¨odinger equation for the A + n + p model [6] can change both real and imaginaryparts of these effective potentials, which could affect the(d,p) cross sections. But given that the adiabatic approx-imation is a good first choice for the (d,p) reactions, most4likely, they will not explain the 70% difference betweenthe NLDOM predictions and experiment.The contributions from the remnant term in the (d,p)amplitude (all other terms that are not V np ) have beenstudied in an inert core model, where they were foundto be small [40]. We estimated the effect of the rem-nant term for the Ca(d,p) Ca reaction at E d = 11 . U p is not known.The strong overestimation of the Ca(d,p) Ca cross sections at 11.8 MeV implies that neglected parts of the(d,p) amplitude and/or its constituents are much moreimportant than was thought before. Given that thedeuteron energy, chosen for this work, is often used inmodern experiments with radioactive beams for spectro-scopic and astrophysical reasons, and that the dispersiveeffects are strong at this energy, further development ofdirect reaction theories is crucial to understand transferexperiments performed either recently or in the past andplanned for the future.
ACKNOWLEDGEMENT
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