Transfer reactions and the dispersive optical-model
N.B.Nguyen, S.J.Waldecker, F.M.Nunes, R.J.Charity, W.H.Dickhoff
aa r X i v : . [ nu c l - t h ] O c t Transfer reactions and the dispersive optical-model
N. B. Nguyen , , S. J. Waldecker , F. M. Nunes , , R. J. Charity , and W. H. Dickhoff National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics, Washington University, St. Louis, Missouri 63130, USA and Department of Chemistry, Washington University, St. Louis, Missouri 63130, USA (Dated: October 3, 2018)The dispersive optical-model is applied to transfer reactions. A systematic study of ( d, p ) reac-tions on closed-shell nuclei using the finite-range adiabatic reaction model is performed at severalbeam energies and results are compared to data as well as to predictions using a standard globaloptical-potential. Overall, we find that the dispersive optical-model is able to describe the angulardistributions as well as or better than the global parameterization. In addition, it also constrains theoverlap function. Spectroscopic factors extracted using the dispersive optical-model are generallylower than those using standard parameters, exhibit a reduced dependence on beam energy, and aremore in line with results obtained from ( e, e ′ p ) measurements. I. INTRODUCTION
Rare isotopes have posed new challenges to our un-derstanding of nuclei as protons and neutrons organizedinto well-defined orbitals. Nucleon-nucleon (NN) cor-relations manifest themselves in different forms as onemoves away from the valley of stability. Nuclear reac-tions form one of the most important probes for the lim-its of stability and the role of the underlying fundamentalinteractions. Transfer reactions offer a plethora of op-portunities for studying shell structure because one canaccess ground as well as excited states (including res-onances), and by choosing appropriately the kinematicconditions, one can explore peripheral properties as wellas the nuclear surface. For this reason, they consti-tute an important part of the science program of manyrare isotopes facilities worldwide. Examples of recentstudies include
Sn( d, p ) Sn [1], C( d, p ) C [2], and , , Ar( p, d ) [3].The increased interest in using transfer reactions as atool to study these exotic nuclei has called for reaction-theory developments [4–8]. Whereas in the sixties thestandard approach to one-nucleon transfer was to relyon the first-order distorted-wave Born approximation(DWBA), it is now well understood that in ( d, p ) re-actions, breakup is important and first-order perturba-tion is insufficient to describe the process. A practicalmethod for including deuteron breakup was introducedby Johnson and Soper [9, 10] within a zero-range approx-imation, usually referred to as the adiabatic wave ap-proximation (ADWA). In addition to including deuteronbreakup to all orders, the ADWA depends exclusivelyon nucleon optical-potentials rather than the more am-biguous deuteron optical-potential as in DWBA. ADWAwas later extended by Tandy and Johnson [11] to in-clude finite-range effects (FR-ADWA). A recent system-atic study of ( d, p ) reactions within FR-ADWA [12] hasshown the importance of finite-range effects in consid-ering deuteron breakup in ( d, p ) reactions. The methodexplored in Refs. [11, 12] is based on the truncation of aWeinberg expansion. In Ref. [13] a systematic compar- ison between ( d, p ) angular distributions for FR-ADWAand those from the exact full Faddeev solution is per-formed. The results from FR-ADWA [12] are within 10%of the full solution at forward angle, for a wide range ofbeam-energies [13]. While the exact full solution is com-putationally intensive and expensive, FR-ADWA calcu-lations are clearly of practical use.The work in [13] determines the level of accuracy ofFR-ADWA concerning the solution of the three-body n + p + A dynamics in the reaction. For the (d,p) re-actions on Ca at 19 MeV and 56 MeV effects are 6%and 3% respectively [13]. With this level of accuracy itis now essential to have good control over the ingredientsto the problem, namely the effective interactions. Allapplications of FR-ADWA so far have relied on globaloptical potentials for the nucleon-target interactions and“standard single-particle parameters” for the effective in-teraction that determines the final neutron bound state.So far, these two types of input have been disconnected.Future extraction of nucleon properties in rare isotopeswill require presently unavailable knowledge of nucleon-target interactions in the continuum as well as bound-state information.An attractive feature of the recently implemented dis-persive optical-model (DOM) [14–16] is that it intrin-sically connects the scattering and bound states. Themethod, first introduced by Mahaux and Sartor [17], andalso known as the dynamic polarization potential (DPP),relies on the dispersion relation between the imaginaryand real parts of the nucleon self-energy. The work ofRef. [14] concentrated on the interaction between pro-tons and Ca isotopes and found larger correlations nearthe Fermi surface with increasing nucleon asymmetry.Neutrons were included in the fits reported in Ref. [15].That analysis raised questions concerning the standardparameterization of the imaginary part of the nucleonpotentials as a function of asymmetry ( N − Z ) /A , par-ticularly for neutrons. More recently, a non-local exten-sion of the DOM has been introduced that allows for abroader application to calculate observables below theFermi energy [18]. Insight from ab initio calculations ofthe nucleon self-energy are expected to provide furtherguidance on the details of non-locality [19].In the present work we will use only the local versionof the DOM. Its most recent implementation [16] yieldsan accurate representation of a wealth of data in severallocations of the chart of nuclides, including nuclei with Z = 20 ,
28 and N = 28 (FIT1), nuclei with N = 50,with Z = 50 (FIT2), and finally Z = 82 (FIT3). Whenseveral isotopes or isotones are included in the fit, theobserved nucleon asymmetry dependence of the poten-tials allows for an extrapolation to nuclei that are moreexotic [16]. The corresponding predictions can thus beprobed with future experiments that may allow furtherrefinements. This feature of data-driven extrapolationsto the respective drip lines and its capacity to link bothreaction and structure information make the DOM anexcellent framework to study rare isotopes.In DOM fits [14–16], nucleon elastic, total and reac-tion cross section data, as well as bound-state propertiesinferred from ( e, e ′ p ) measurements, are all included si-multaneously. The aim of the present work is to explorethe application of the DOM to transfer reactions. We in-clude a number of representative cases: Ca( d, p ) Ca, Ca( d, p ) Ca,
Sn( d, p ) Sn, and
Pb( d, p ) Pb.These reactions are studied at various beam energies forwhich data are available. We note that apart from the n − p interaction that binds the deuteron, all effectiveinteractions to the problem in the ADWA frameworkare provided by the DOM. We compare the results tothose obtained using a global optical-potential (CH89from Ref. [20]) and standard geometry for the boundstate.In Sec. II we summarize the important theoretical con-cepts pertaining to the ADWA and DOM. Details of theingredients of the reaction description are presented inSec. III A and the results of the transfer calculations arepresented in Sec. III B. A more detailed discussion onspectroscopic factors is presented in Sec. III C. Finally,we summarize and draw our conclusions in Sec. IV. II. THEORYA. Finite-range adiabatic-wave approximation
The adiabatic theory of Refs. [9, 11] for A ( d, p ) B startsfrom a three-body model of n + p + A . The deuteron scat-tering wavefunction in the incident channel is obtainedby solving the differential equation:[ E + iǫ − T r − T R − U nA − U pA − V np ]Ψ (+) ( r , R )= iǫφ d ( r ) exp( i K d · R ) , (1)with r = r p − r n ( R = ( r n + r p ) /
2) being the rela-tive coordinate (center-of-mass coordinate) of the n − p system. The neutron and proton coordinates, which aretaken at the center of mass of the target A , are givenby r n and r p , respectively. We take U nA ( r n ), U pA ( r p ), and V np ( r ) to be the neutron-target, proton-target, andneutron-proton interactions. In this work, we use theReid potential [21] for V np and DOM potentials for U pA and U nA (see Sect. II B).In this three-body approach, the solution of Eq. (1) isinserted into the exact transfer matrix element: T = h φ nA χ ( − ) pB | V np + ∆ rem | Ψ (+) i , (2)where φ nA is the bound state of the neutron-target sys-tem, and χ ( − ) pB is a proton scattering distorted wave inthe outgoing channel. The remnant operator is ∆ rem = U pA − U pB . Contributions from this term are often smallexcept for the lighter systems.Johnson and Tandy [11] noted from Eq. (2) that theexact three-body wavefunction Ψ (+) ( r , R ) is only neededwithin the range of the V np interaction, whenever theremnant contributions are negligible. For this reason,the Weinberg states, which form a complete square in-tegrable basis within the range of V np , offer an excellentrepresentation for the three-body scattering wavefunc-tion: Ψ (+) ( r , R ) = ∞ X i =1 φ i ( r ) χ i ( R ) . (3)Here φ i ( r ) denote Weinberg states, normalized by h φ i | V np | φ j i = − δ ij and satisfying:( T r + α i V np ) φ i ( r ) = − ǫ d φ i ( r ) , (4)where ǫ d is the deuteron binding energy and α i are theeigenvalues. Inserting Eq. (3) into Eq. (1) one arrives ata non-trivial coupled-channels scattering equation. How-ever, if only the first term in the Weinberg expansionof Eq.(3) is necessary to describe transfer, then Eq.(1)becomes a single-channel optical-model-type equation:( E + ǫ d + iǫ − T R − U JT ( R )) χ JT ( R ) = iǫN d exp( i K d · R ) , (5)where N d is the normalization coefficient defined as N d = −h φ | V np | φ d i . The potential U JT in Eq.(5) is calculatedby a folding procedure: U JT ( R ) = −h φ ( r ) | V np ( U nA + U pA ) | φ ( r ) i (6)where, φ ( r ) is the lowest eigenfunction of Eq. (4), pro-portional to the deuteron bound-state wavefunction φ d .The expansion of Eq.(3) truncated to the first term, isnow inserted into the transfer matrix element: T = h φ nA χ ( − ) pB | V np + ∆ rem | φ d [ χ JT ( R ) /N d ] i . (7)The validity of the truncation has been tested throughcomparisons with Faddeev solutions and the results sug-gest that for most cases of interest, the finite-rangeADWA works very well [13].One important advantage of ADWA over DWBA, isthat ADWA depends on nucleon optical-potentials ratherthan the deuteron optical-potential, which is far moreambiguous. ADWA also depends on the nucleon bind-ing potential, as does the DWBA. Since these nucleonoptical and binding potentials are connected by a dis-persion relation in the DOM approach, it is interestingto test how the DOM potentials perform in describingtransfer angular distributions for a wide range of ener-gies and targets. In the next subsection we give a briefsummary of the recent developments with DOM nucleonoptical-potentials. B. Dispersive optical-model (DOM)
The dispersive optical-model [17] develops a practicalrepresentation of the irreducible nucleon self-energy Σ ∗ from the Green’s-function approach to the many-bodyproblem [22]. The real part of the nucleon self-energyor optical-model potential can be decomposed into anenergy-independent non-local part and a dynamic con-tribution (energy-dependent), that can also be non-local, i.e. ,Re Σ ( r , r ′ ; E ) = Re Σ ( r , r ′ ; ε F ) + ∆ V ( r , r ′ ; E ) , (8)where ε F is the Fermi energy. The second term, referredto as the dispersive correction, can be obtained from theimaginary part through the subtracted dispersion rela-tion∆ V ( r , r ′ ; E ) = (9)+ 1 π P Z Im Σ ( r , r ′ ; E ′ ) (cid:18) E ′ − E − E ′ − ε F (cid:19) dE ′ , where P stands for the principal value and we note theconvention to employ the same sign for the imaginarypart of the self-energy above and below the Fermi en-ergy [17]. By definition in Eq. (8), the dispersive correc-tion is zero at the Fermi energy. The dispersive correctionvaries rapidly around E F and causes the valence single-particle levels to be focused towards the Fermi energy.Following a long tradition [23], the non-local energy-independent term ReΣ( r , r ′ ; ε F ) is approximated by alocal energy-dependent form, generally designated as theHartree-Fock (HF) potential V HF ( r, E ). While it doesdescribe mean-field properties, it is not strictly a HFcontribution, since it involves the complete self-energyat one particular energy. The energy derivative of V HF encodes a measure of the non-locality, which is related tothe momentum-dependent effective mass e m ( r, E ) m = 1 − d V HF ( r, E ) dE , (10)where m is the nucleon mass.The local approximation of the HF potential necessi-tates a scaled imaginary potential given by W = e m ( r, E ) m Im Σ (11) and a similarly scaled dispersive correction. For lack ofdetailed theoretical input, the imaginary part of the self-energy is also approximated by a local potential in keep-ing with a long-standing tradition of empirical opticalpotentials. We note that the dispersive correction is cor-respondingly local. The nucleon self-energy, as typicallyrepresented in the DOM, can then be written as U ( r, E ) = V HF ( r, E ) + ∆ V ( r, E ) + i W ( r, E ) . (12)The Fermi energy is defined as ε F = ε + F + ε − F ε + F = M A +1 − ( M A + m ) (14) ε − F = M A − ( M A − + m ) , (15)where ε + F and ε − F represent the binding energy for addingor removing a nucleon, or alternatively, the single-particleenergies of the valence particle and hole states.Since the current implementation of the DOM includesscattering data up to 200 MeV, a lowest-order relativisticcorrection is employed in solving the radial wave equa-tion [24] " d dρ + − e U ( ρ, E ) E tot − M − m − ℓ ( ℓ + 1) ρ ! u ( ρ ) = 0(16)with ρ = k r , where k = ME tot p T ( T + 2 m ), T is thelaboratory kinetic energy, E tot is the total energy in thecenter-of-mass frame, and M is the target mass. Thescaled potential is e U = γ U , γ = 2 ( E tot − M ) E tot − M + m . (17)We denote by ˜ ϕ nℓj ( r ) bound-state solutions to the ra-dial wave equation, where n refers to the correspondingstate in the A ± ϕ nℓj ( r ) = r e m ( r, ε nℓj ) m ˜ ϕ nℓj ( r ) , (18)where ε nℓj is the discrete energy solution to Eq. (16). Itshould be emphasized that these corrected wave functionscorrespond to overlap functions for adding or removingparticle from the target nucleus [18], e.g. p S nℓj ϕ nℓj ( r ) = h Ψ A +1 n | a † rℓj | Ψ A i , (19)where ϕ nℓj is normalized to one and S nℓj represents thespectroscopic factor (norm of the overlap function). Thesolutions ϕ therefore still need to be normalized by thespectroscopic factor which is obtained in the DOM from: S nℓj = Z ∞ ϕ nℓj ( r ) mm ( r, ε nℓj ) dr, (20)where the energy-dependent effective mass determinesthe reduction of strength from the mean-field value andis given by m ( r, E ) m = 1 − m e m ( r, E ) d ∆ V ( r, E ) dE . (21)The result for the spectroscopic factor in Eq. (20) wasderived in Ref. [17] and was shown to be an excellentquantitative approximation to the corresponding solutionof the Dyson equation that incorporates a non-local HFin Ref. [18]. We will also include results for the root-mean-square (rms) radius given by R rmsnℓj = sZ ∞ ϕ nℓj ( r ) r dr (22)in the DOM analysis.In the present work, we will extract the normalizationof the overlap functions from the analysis of the transferreactions by scaling the theoretical cross section to matchthe data. We will also compare these results with theDOM spectroscopic factors generated by Eq. (20). Forcomparison purposes, we will also consider the single-particle wave function produced with a Woods Saxonpotential with standard parameters (radius r = 1 . a = 0 .
65 fm) and the depth adjustedto reproduce the experimental binding energies. Thesewave functions will be referred to as ϕ W S . An importantadditional experimental constraint on the overlap func-tion at large distances (when available) is provided byan asymptotic normalization constant (ANC). The sin-gle particle ANC b nℓj is defined through the asymptoticbehavior of the single particle state ϕϕ nℓj ( r ) ⇒ b nℓj iκh ℓ ( iκr ) , (23)where κ = p µε nℓj , µ is the corresponding reduced massand ϕ nℓj ( r ) is mormalized to one. We note that thespherical Hankel function in Eq. (23) is only appropriatefor neutrons. III. RESULTSA. Details of the calculations
Separate DOM fits were produced for different partsof the chart of nuclides in Ref. [16]. For the Ca isotopes,the DOM fit included nucleon scattering and bound-statedata for nuclei with Z = 20 ,
28 and N = 28 (FIT1).Excluded were proton elastic scattering data on Cabelow 18 MeV, since for those cases the optical modeldoes poorly and there are fluctuations in the reactioncross section [25]. For the Sn isotopes, the DOM fit in-cludes proton elastic-scattering data on − Sn andneutron elastic-scattering data on
Sn,
Sn,
Sn,and
Sn (FIT2) [16]. Finally, for Z = 82, Pb data
TABLE I. Properties of overlap functions with a compari-son of the DOM overlap function, corrected for non-locality¯ ϕ with the Woods-Saxon single-particle wavefunction ϕ WS .These include the counting number n , the angular momenta( ℓj ) of the valence orbital, the separation energy S n , the rootmean square radius of the valence orbital R rms , and the mod-ulus of the single-particle ANC b nlj .Nucleus Overlap nℓj S n [MeV] R rms [ fm ] | b nlj | [fm − / ] Ca ϕ WS f / ϕ Ca ϕ WS p / ϕ Sn ϕ WS f / ϕ Pb ϕ WS g / ϕ are included (FIT3). The fitting procedure follows pre-vious works [14, 15] but the depth of the HF potentialwas adjusted to reproduce the valence particle levels forneutrons. Since transfer cross sections are strongly de-pendent on the binding energy of the final bound state,particular effort was made in all these fits, to reproducethe experimental binding energies. The DOM parame-terizations are subsequently used to calculate A ( d, p ) B within FR-ADWA. The DOM thus provides all opticalpotentials U nA , U pA , U pB as well as the neutron bound-state interaction V nA . The properties of the neutronstates considered in this work are summarized in Table I,including the rms radius and the single-particle asymp-totic normalization coefficient [4] [see Eq. (23)]. We use twofnr [26] to calculate the finite-range deuteron adia-batic potential and fresco [27] to calculate the transfercross sections.The DOM fit for Ca isotopes generated a radius param-eter of 1.18 fm while the corresponding standard Woods-Saxon potential used for comparison was fixed at 1.25fm. The non-locality correction almost completely can-cels this difference making the DOM overlap function¯ ϕ for Ca essentially identical to ϕ W S . When a nodeis present, the non-locality correction is even more pro-nounced making the rms radius of the 1 p / DOM wavefunction larger than its Woods-Saxon counterpart. We il-lustrate this in Fig.1 for the Ca ground state, where thesquare of the single-neutron overlap function obtainedwith the local DOM, corrected for non-locality, is com-pared to the Woods-Saxon counterpart with standard ge-ometry. The DOM result (FIT2) for the radius parame-ter for Sn nuclei is 1.24 fm and therefore a substantiallylarger rms radius is obtained for the DOM overlap func-tion compared to the standard Woods-Saxon one whenthe non-locality correction is applied for the 1 f / orbit(see Table I).As is usual in optical-model potentials, the imaginary | r ϕ (r) | [f m - ] WSDOM
FIG. 1. (Color online) Comparison of the square of the single-neutron overlap functions for Ca obtained with a Woods-Saxon potential ϕ WS (solid) and with the local DOM cor-rected for non-locality ¯ ϕ (dashed). potential in the DOM is separated into surface and vol-ume terms. The former is stronger at the lower energiesconsidered in this work and the latter becomes dominantonly at much higher energies. The volume term was as-sumed to have a small dependence on ± ( N − Z ) /A , wherethe positive and negative signs refer to protons and neu-trons. The magnitude of this term was determined fromthe fit to the lead data (FIT3 of Ref. [16]) and then waskept fixed when fitting the Sn and Ca data [16]. It is forthe study of n − Sn (derived from FIT2 in Ref. [16])that the asymmetry dependence becomes critical, as oneis extrapolating well outside the region of fitted data. Aspreviously seen for the Ca isotopes [15], and for FIT1in Ref. [16], the Sn DOM potentials contain an insignif-icant asymmetry dependence of the neutron imaginarysurface term, while that for protons exhibited a lineardependence on ( N − Z ) /A .A comparison between the DOM and CH89 opticalpotential reveals some similarities and some differences.Since the surface absorption for neutrons exhibits a dif-ferent (weaker) asymmetry dependence than assumed inRef. [20], it is not surprising that the Sn potentialsshow the largest differences, as this corresponds to asubstantial extrapolation from potentials constrained bydata (see Fig.2). In particular, the neutron (proton) sur-face absorption in the DOM is substantially less (more)than in the CH89 parameterization). The real parts ofthe CH89 potentials have the same radius parameter forboth protons and neutrons by decree, whereas in theDOM, due to the different surface absorption, the dis-persive correction makes the real proton potentials ex-tend farther than those for neutrons. We also note thatthe CH89 potentials are not dispersive, making a moredetailed comparison less productive. r [fm] −60−40−200 V [ M e V ] V n DOMV n CH89V p DOMV p CH89 (a) r [fm] −20−15−10−50 W [ M e V ] W n DOMW n CH89W p DOMW p CH89 (b)
FIG. 2. (Color online) Optical potentials for n-
Sn and p-
Sn (thin) at 4 . B. Transfer cross sections
We consider 9 reactions for which data exists: Ca( d, p ) Ca at E d = 20 and 56 MeV, Ca( d, p ) Caat E d = 2 , , . Sn( d, p ) Sn at E d = 9 .
46 MeV, and
Pb( d, p ) Pb at E d = 8 and20 MeV.FR-ADWA calculations for all these cases were per-formed for three interaction models: optical potentialsfrom CH89 [20] and the neutron overlap function as ϕ W S (CH89+WS); optical potentials from DOM but the neu-tron overlap function as ϕ W S (DOM+WS); and finallyboth the optical potentials and the neutron overlap func-tion from this local approximation of DOM corrected fornon-locality ¯ ϕ (DOM). In Figs. 3, 4, and 5 the angulardistributions for E d = 2 , . Ca( d, p ) Ca reaction.The normalization is performed at the peak of the data, i.e. at backward angles for reactions below the Coulombbarrier and at the first peak when the bombarding en-ergy is above it. For these cases the relevant angles were θ ≈ ◦ , θ ≈ ◦ and θ ≈ ◦ for E d = 2 , . θ [deg]0.00.20.40.60.8 d σ / d Ω [ m b / s r ad ] EXPCH89+WSDOM+WSDOM
FIG. 3. (Color online) Angular distributions are shown for thereaction Ca( d, p ) Ca at E d = 2 MeV. Theory predictionshave been normalized to the data at backward angles. θ [deg]010203040 d σ / d Ω [ m b / s r ad ] EXPCH89+WSDOM+WSDOM
FIG. 4. (Color online) Angular distributions are displayedfor the reaction Ca( d, p ) Ca at E d = 19 . θ [deg]02468 d σ / d Ω [ m b / s r ad ] EXPCH89+WSDOM+WSDOM
FIG. 5. (Color online) Angular distributions for the reaction Ca( d, p ) Ca at E d = 56 MeV are displayed. Theory pre-dictions have been normalized to the data at forward angles. θ [deg]051015 d σ / d Ω [ m b / s r ad ] EXPCH89+WSDOM+WSDOM
FIG. 6. (Color online) Angular distributions for the
Sn( d, p ) Sn reaction at a deuteron energy of E d = 9 . θ [deg]0246 d σ / d Ω [ m b / s r ad ] EXPCH89+WSDOM+WSDOM
FIG. 7. (Color online) Angular distributions for the
Pb( d, p ) Pb reaction are shown at a deuteron energy of E d = 20 MeV and normalized at the peak of the experimentaldata. MeV, respectively. Note that at 2 MeV, compound con-tributions to the transfer cross section can be important.However for this case, a close analysis of the experimen-tal angular distribution suggests such a contribution tobe negligible.The first thing to note is that DOM is able to describethe cross sections well. However, there is no significantdifference between the angular dependence predicted byDOM, DOM+WS and CH89+WS, even though there aredifferences in the shapes of the overlap functions as shownin Fig. 1. As a similar set of data was used to constrainthe CH89 optical potential [20], one can conclude thatthis set is sufficient to produce the correct angular dis-tribution for ( d, p ) reactions within FR-ADWA for thisnucleus. For the ( d, p ) reactions with Ca at the E d = 20and 56 MeV, DOM performs as well as CH89+WS andno significant difference is observed between the standardand the dispersive approach (not shown here).There were two cases for which we did see consider-able difference in the angular distributions between DOMand CH89, the Sn and
Pb. These distributions areshown in Figs. 6 and 7. It is clear that this is not dueto the modification in the shape of the overlap functions,but rather the optical potentials. The real part of theoptical potential in the DOM has a larger radius, whichshifts the diffraction pattern toward smaller angles. How-ever an increase in radius in the overlap function onlyaffects the magnitude at forward angles (shown by com-paring DOM+WS and DOM). While for
Sn DOM ap-pears to improve the angular distribution, for
Pb theangular distributions using DOM optical potentials moveaway from the data. We will address the reactions with
Pb in more detail later.
C. Spectroscopic factors
While the angular distributions predicted using DOMdo not differ considerably from those using CH89, thenormalization of the cross sections do. We determine anexperimental spectroscopic factor by taking the ratio of dσ/d
Ω(exp) over dσ/d
Ω(theory) for θ at the first peak ofthe distribution for all but sub-barrier energies. At suchenergies the normalization is determined at backward an-gles. We compare the results obtained in the various ap-proaches in Table II: the labels CH89+WS, DOM+WSand DOM are exactly as described in Sec. III B. Notethat the normalizations are calculated with an overlapfunction normalized to unity. However, the DOM pre-dicts overlap functions that are not normalized to unity,as correlations are already taken into account. The spec-troscopic factor coming directly from such an analysis isgiven in the last column of Table II and labeled DOM(th).One should keep in mind that for Ca, it is notoriouslydifficult to describe low-energy scattering. This fact ledto the exclusion of proton elastic scattering data at ener-
TABLE II. Spectroscopic factors obtained from the FR-ADWA analysis. The deuteron kinetic energy E d (lab. frame)is in MeV. Reference to the experimental data set used in theextraction is also givenNucleus E d data CH89+WS DOM+WS DOM DOM(th) Ca 20 [29] 0.96 0.85 0.86 0.7556 [30] 0.88 0.73 0.74 Ca 2 [31] 0.94 0.72 0.66 0.8013 [32] 0.82 0.67 0.6119.3 [33] 0.77 0.68 0.6256 [34] 1.1 0.70 0.62
Sn 9.46 [1] 1.1 1.0 0.72 0.80
Pb 8 [35] 1.7 1.5 1.2 0.7620 [36] 0.89 0.61 0.51 gies below 18 MeV from the DOM fit, as discussed above.Since in the ADWA the optical potentials are evaluatedat E d /
2, the DOM results for Ca( d, p ) at E d = 20MeV are not well constrained by elastic nucleon scatter-ing data. In all other cases, DOM potentials provide agood description of scattering observables.If one focuses first on the traditional CH89+WS ap-proach, one sees that a wide range of spectroscopic fac-tors can be obtained depending on the beam energy. Forexample with Ca, the spectroscopic factor ranges from S = 0 .
77 to 1.1. This unwanted energy dependence wasalready seen in the systematic study in [28]. Note thatspectroscopic factors extracted in [28] do not coincidewith those in our Table II due to additional approxima-tions in the reaction theory in [28], namely the zero-rangeADWA corrected within the local energy approximation(errors introduced in making those approximations havebeen quantified in [12]). The large energy dependence issignificantly reduced when DOM optical potentials areused, the exception being the the reactions on
Pb,which we will consider later.When comparing CH89+WS and DOM+WS, one seesthat the DOM optical potentials reduce the spectroscopicfactor by a non-negligible amount. Note that this is dueto the optical potentials alone, since the overlap functionsare the same in CH89+WS and DOM+WS. If one now re-places the single-particle wave function by that predictedfrom DOM, two things can happen: virtually no change(as in the case of Ca) or an important further reductionof the spectroscopic factor (as in all other cases). It is in-deed only for Ca that the wave functions are similar. Inall other cases, the different radius parameter obtainedin DOM fits, together with the non-locality correction,shift density from the interior and enhance the proba-bility in the surface region. A larger overlap function atthe surface, produces larger cross sections which then im-ply smaller spectroscopic factors. Values extracted usingDOM ingredients are much more in line with those from( e, e ′ p ) measurements [37] with the exception of Pb.Finally, we also show the spectroscopic factors pre-dicted directly by the DOM without reference to transferdata (last column). These are larger than those extractedfrom experiment for Ca and
Sn. This difference ismostly associated with the different choice in Ref. [16]for the domain where the imaginary part of the opticalpotentials vanish as compared to the choice in Ref. [15].This leads to a larger particle-hole gap in Ref. [16] ascompared to Ref. [15] for the same nuclei. Since spectro-scopic factors are particularly sensitive to the particle-hole gap, an increase of the order of 0.1 is obtainedin Ref. [16] somewhat similar to the difference betweencolumns DOM and DOM(th) in Table II for these nu-clei. If one adopts the perspective that for nuclei whereoptical potentials relevant for transfer reactions are well-constrained and the FR-ADWA provides an accurate de-scription, the present results suggest that future DOMfits should either reduce the particle-hole gap or make thecoupling at low energy stronger. Nevertheless, the con-
TABLE III. The square of the many-body asymptotic nor-malization coefficient C for various models (fm − ).Nucleus E d [MeV] CH89+WS DOM+WS DOM DOM(th) Ca 20 5.0 4.4 4.4 2.856 4.6 3.8 3.8 Ca 2 31.7 24.4 24.4 29.613 27.9 22.7 22.619.3 26.0 23.1 23.056 35.8 23.5 23.2
Sn 9.46 0.78 0.71 0.49 0.56
Pb 8 4.5 4.1 4.2 2.520 2.4 1.7 1.7 sistency of the extracted spectroscopic factors employingDOM ingredients is very encouraging (see also the dis-cussion for
Pb below).Most of the reactions studied in this work are notsensitive to details of the overlap function in the in-terior, but rather to the ANC [5]. For completeness,in Table III we present the square of the many-bodyANCs obtained directly from the spectroscopic factorspresented in Table II, multiplied by the square of thesingle-particle ANC ( C = Sb ). The ANC for Cawas determined to be C f / = 8 . ± .
42 fm − , usinga DWBA analysis of sub-Coulomb ( d, p ) data [5]. Theanalysis in Ref. [5] shows that with that large ANC,there is no consistency between the expected spectro-scopic factor for Ca and the Ca( d, p ) Ca data abovethe Coulomb barrier. In Ref. [5] it is also shown thatto obtain consistency between sub-Coulomb and aboveCoulomb energies one needs an ANC about half thatvalue. This work solves the discrepancy seen for Ca inRef. [5]. The ANC for Ca was determined in Ref. [38]from a DWBA analysis of sub-Coulomb ( d, p ) data tobe C p / = 32 . ± . − . This value is consistentwith the values we obtain using CH89+WS. Using theDOM instead, the ANC in Ref. [38] should be signifi-cantly reduced. An analysis of the Sn( d, p ) Sn reac-tion around the Coulomb barrier [39] using FR-ADWAand the CH89 optical potentials generates an ANC forthe ground state of C f / = 0 . ± .
07 fm − . Thisvalue would significantly be reduced if DOM were usedinstead. Finally, in Ref. [4] an ANC for Pb is extractedfrom heavy ion reactions C g / = 2 . ± .
16 fm − us-ing DWBA. This value is close to the DOM theoreticalprediction.The transfer results for Pb are problematic for tworeasons: i) there is a very large discrepancy betweenthe spectroscopic factors obtained at E d = 20 MeV and E d = 8 MeV, and ii) for the sub-Coulomb barrier energy,this value is much larger than unity. No issues arosein the DOM fitting for this nucleus and therefore theDOM fits can be considered reliable for the elastic andtotal cross-section data. Even though the transfer an- gular distributions are well described within FR-ADWA,we cannot exclude the possibility of target excitation. Inorder to test whether the cause for inconsistency is thereaction mechanism, we performed exploratory calcula-tions including the low-lying 3 − and 2 + states in Pband the strong octupole and quadrupole couplings be-tween these states and the ground state within a coupled-channel Born approximation (CCBA), using a global op-tical potential for the deuteron [40]. One should keep inmind that, within a CCBA approach, the extraction ofthe spectroscopic factor for a single orbital is obscuredby other components. We found the effects to be strongfor
Pb(d,p) at 20 MeV but rather weak at the lowerenergy. We also performed coupled reaction channel cal-culations, iterating the transfer coupling, but again foundthe effects to be weak at the lower energy while signifi-cant at higher energy. These results suggest that at leastfor the higher energy, the reaction mechanism goes be-yond the present implementation of ADWA. The unre-alistic SF obtained at the lower energy could indicatethe failure of ADWA. At present, for reactions involvingsuch large Coulomb fields, we cannot verify the validityof ADWA through the comparison with exact Faddeevcalculations. The present techniques used in solving theFaddeev equations are limited to targets with Z ≈ Pb( d, p ) at 20 MeVnecessitates the inclusion of target excitation, could thismechanism also play a role in our other test cases, even ifthe angular distributions are well described in our presentformulation? In Ref. [5] a study of target excitation in Ca( d, p ) Ca at low energy shows strong effects. How-ever we expect these to be small at 56 MeV. The Canucleus has a very weak transition to its first excitedstate, and therefore no significant effect of target excita-tion is expected. Target excitation was tested for the re-action on
Sn and found to be negligible. Nevertheless,the present results for
Pb do call for an extension ofthe FR-ADWA to include target excitation and deuteronbreakup in a consistent framework.
IV. CONCLUSIONS
We have tested the performance of the dispersiveoptical-model potentials and corresponding overlap func-tions in ( d, p ) reactions. We performed finite-range adia-batic calculations for a range of closed-shell nuclei cover-ing a wide range of beam energies. Within this descrip-tion, the DOM provides all necessary ingredients apartfrom the deuteron V np interaction, i.e. nucleon optical-potentials and the mean-field binding the neutron in thefinal state. We compare the results obtained with theDOM to those obtained using a global parameterizationof the nucleon optical-potential and a standard geometryfor the neutron single-particle overlap. We find that theDOM performs as well as the CH89 parameterization inthe description of the angular distributions. While spec-troscopic factors extracted within the standard approachcan be strongly dependent on the energy at which the( d, p ) data were obtained, this dependence is stronglysuppressed when DOM potentials are employed. The ex-ception is Pb, a case for which coupled-channel calcu-lations demonstrate the importance of target excitation.Overall, we find that the extracted spectroscopic factorsusing DOM are significantly reduced when compared tothe standard approach, bringing the values more in linewith those obtained from ( e, e ′ p ) measurements.Because DOM potentials can be extrapolated to rareisotopes and separate checks of their quality can be madeby performing elastic nucleon (proton) scattering experi-ments in inverse kinematics, the present framework of theDOM in which reaction and structure data are incorpo-rated on the same footing, provides an excellent platform to analyze transfer reactions involving rare isotopes. ACKNOWLEDGMENTS
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