Forging the link between nuclear reactions and nuclear structure
M. H. Mahzoon, R. J. Charity, W. H. Dickhoff, H. Dussan, S. J. Waldecker
aa r X i v : . [ nu c l - t h ] D ec Forging the link between nuclear reactions and nuclear structure
M. H. Mahzoon , R. J. Charity , W. H. Dickhoff , H. Dussan , S. J. Waldecker Department of Physics and Chemistry , Washington University, St. Louis, Missouri 63130 andDepartment of Physics , University of Tennessee, Chattanooga, TN 37403 (Dated: September 18, 2018)A comprehensive description of all single-particle properties associated with the nucleus Ca isgenerated by employing a nonlocal dispersive optical potential capable of simultaneously reproduc-ing all relevant data above and below the Fermi energy. The introduction of nonlocality in theabsorptive potentials yields equivalent elastic differential cross sections as compared to local ver-sions but changes the absorption profile as a function of angular momentum suggesting importantconsequences for the analysis of nuclear reactions. Below the Fermi energy, nonlocality is essentialto allow for an accurate representation of particle number and the nuclear charge density. Spectralproperties implied by ( e, e ′ p ) and ( p, p ) reactions are correctly incorporated, including the energydistribution of about 10% high-momentum nucleons, as experimentally determined by data fromJefferson Lab. These high-momentum nucleons provide a substantial contribution to the energyof the ground state, indicating a residual attractive contribution from higher-body interactions for Ca of about 0.64 MeV/ A . PACS numbers: 21.10.Pc,24.10.Ht,11.55.Fv
The properties of a nucleon that is strongly influencedby the presence of other nucleons have traditionally beenstudied in separate energy domains. Positive energy nu-cleons are described by fitted optical potentials mostlyin local form [1, 2]. Bound nucleons have been analyzedin static potentials that lead to an independent-particlemodel modified by the interaction between valence nu-cleons as in traditional shell-model calculations [3, 4].The link between nuclear reactions and nuclear structureis provided by considering these potentials as represent-ing different energy domains of one underlying nucleonself-energy. This idea was implemented in the dispersiveoptical model (DOM) by Mahaux and Sartor [5]. By em-ploying dispersion relations, the method provides a crit-ical link between the physics above and below the Fermienergy with both sides being influenced by the absorptivepotentials on the other side.The DOM provides an ideal strategy to predict proper-ties for exotic nuclei by utilizing extrapolations of thesepotentials towards the respective drip lines [6, 7]. Themain stumbling block so far has been the need to uti-lize the approximate expressions for the properties of nu-cleons below the Fermi energy that were developed byMahaux and Sartor [5] to correct for the normalization-distorting energy dependence of the Hartree-Fock (HF)potential. By restoring the proper treatment of nonlocal-ity in the HF contribution, it was possible to overcomethis problem [8] although the local treatment of the ab-sorptive potentials yielded a poor description of the nu-clear charge density and particle number.In the present work we have for the first time treatedthe nonlocality of these potentials for the nucleus Cawith the aim to include all available data below the Fermienergy that can be linked to the nucleon single-particlepropagator [9] while maintaining a correct description ofthe elastic-scattering data. The result is a DOM poten- tial that can be interpreted as the nucleon self-energyconstrained by all available experimental data up to 200MeV. Such a self-energy allows for a consistent treat-ment of nuclear reactions that depend on distorted wavesgenerated by optical potentials as well as overlap func-tions and their normalization for the addition and re-moval of nucleons to discrete final states. The re-analysisof such reactions may further improve the consistency ofthe extracted structure information. Extending this ver-sion of the DOM to N = Z will allow for predictionsof properties that require the simultaneous knowledge ofboth reaction and structure information since at presentfew weakly-interacting probes are available for exotic nu-clei [10].The self-energy Σ ℓj provides the critical ingredient tosolve the Dyson equation for the nucleon propagator G ℓj .Employing an angular momentum basis, it reads G ℓj ( r, r ′ ; E ) = G (0) ℓj ( r, r ′ ; E ) + Z d ˜ r ˜ r Z d ˜ r ′ ˜ r ′ (1) × G (0) ℓj ( r, ˜ r ; E )Σ ℓj (˜ r, ˜ r ′ ; E ) G ℓj (˜ r ′ , r ′ ; E ) . The noninteracting propagators G (0) ℓj only contain kineticenergy contributions. The solution of this equation gen-erates S ℓj ( r ; E ) = Im G ℓj ( r, r ; E ) /π , the hole spectraldensity, for negative continuum energies. The spectralstrength at E , for a given ℓj , is given by S ℓj ( E ) = Z ∞ dr r S ℓj ( r ; E ) . (2)For discrete energies one solves the eigenvalue equationfor the overlap functions ψ nℓj ( r ) = (cid:10) Ψ A − n (cid:12)(cid:12) a rℓj (cid:12)(cid:12) Ψ A (cid:11) , forthe removal of a nucleon at r with discrete quantum num-bers ℓ and j [8]. The removal energy corresponds to ε − n = E A − E A − n with the normalization for such a solu-tion α qh given by S nℓj = (1 − ∂ Σ ℓj ( α qh , α qh ; E ) /∂E | ε − n ) − . We note that from the solution of the Dyson equationbelow the Fermi energy, one can generate the one-bodydensity matrix by integrating the non-diagonal imaginarypart of the propagator up to the Fermi energy and there-fore access the expectation values of one-body operatorsin the ground state including particle number, kineticenergy and charge density [9]. The latter is obtainedby folding the point density with the nucleon form fac-tors [11]. For positive energies, it was already realizedlong ago that the reducible self-energy provides the scat-tering amplitude for elastic nucleon scattering [12].The self-energy fulfills the dispersion relation which re-lates the physics of bound nucleons to those that propa-gate at positive energy [9]. It contains a static correlatedHF term and dynamic parts representing the couplingin the A ± ε + F = E A +10 − E A ) and removal( ε − F = E A − E A − ), respectively. The latter featureis particular to a finite system and allows for discretequasi particle and hole solutions of the Dyson equationwhere the imaginary part of the self-energy vanishes.It is convenient to introduce the average Fermi energy ε F = (cid:2) ε + F − ε − F (cid:3) and employ the subtracted form of thedispersion relation calculated at this energy [5, 8]Re Σ ℓj ( r, r ′ ; E ) = Σ ℓj ( r, r ′ ; ε F ) (3) −P Z ∞ ε + F dE ′ π Im Σ ℓj ( r, r ′ ; E ′ ) (cid:20) E − E ′ − ε F − E ′ (cid:21) + P Z ε − F −∞ dE ′ π Im Σ ℓj ( r, r ′ ; E ′ ) (cid:20) E − E ′ − ε F − E ′ (cid:21) , where P represents the principal value. The beauty ofthis representation was recognized by Mahaux and Sar-tor [5, 13] since it allows for a link with empirical infor-mation both for the real part of the nonlocal self-energyat the Fermi energy (probed by a multitude of HF cal-culations) as well as through empirical knowledge of theimaginary part of the optical potential also constrainedby experimental data. Consequently Eq. (3) yields a dy-namic contribution to the real part linking both energydomains around the Fermi energy. Empirical informationnear ε F is emphasized by Eq. (3) because of the E ′− -weighting in the integrands. The real self-energy at theFermi energy will be denoted in the following by Σ HF .We now provide a more detailed description of thechanges that are necessary in the conventional applica-tion of the DOM in order for the resulting potential toyield a realistic description of the single-particle prop-erties below the Fermi energy. In particular we referto previous papers for a description of ingredients thathave not changed from the purely local treatment of theDOM [14, 15]. The nonlocal treatment of the HF poten-tial was discussed in Ref. [8]. The present form readsΣ HF ( r , r ′ ) = − V volHF ( r , r ′ ) + V wbHF ( r , r ′ ) , (4) where the volume term is given by V volHF ( r , r ′ ) = V HF f (cid:0) ˜ r, r HF , a HF (cid:1) (5) × [ x H ( s ; β vol ) + (1 − x ) H ( s ; β vol )] , allowing for two different nonlocalities with differentweight (0 ≤ x ≤ r = ( r + r ′ ) / s = r − r ′ . A wine bottle ( wb ) shape producing Gaus-sian is introduced replacing the surface term of Ref. [15] V wbHF ( r , r ′ ) = V wb exp (cid:0) − ˜ r /ρ wb (cid:1) H ( s ; β wb ) . (6)This Gaussian centered at the origin helps to representoverlap functions generated by simple potentials that re-produce corresponding Monte Carlo results [16]. Non-locality is represented by a Gaussian form H ( s ; β ) = exp (cid:0) − s /β (cid:1) / ( π / β ) (7)first suggested in Ref. [17]. As usual we employ Woods-Saxon form factors f ( r, r i , a i ) = [1 + exp (cid:16) r − r i A / a i (cid:17) ] − .Equation (4) is supplemented by the Coulomb and localspin-orbit interaction as in Ref. [15].The introduction of nonlocality in the imaginary partof the self-energy is well-founded theoretically bothfor long-range correlations [18] as well as short-rangeones [19]. Its implied ℓ -dependence is essential in re-producing the correct particle number for protons andneutrons. The nonlocal part of this imaginary compo-nent has the formIm Σ( r , r ′ , E ) = − W vol ( E ) f (cid:0) ˜ r ; r vol ; a vol (cid:1) H ( s ; β vol )+4 a sur W sur ( E ) H ( s ; β sur ) dd ˜ r f (˜ r, r sur , a sur ) . (8)We also include a local spin-orbit contribution as inRef. [15]. The energy dependence of the volume absorp-tion has the form used in Ref. [15] whereas for surfaceabsorption we employed the form of Ref. [14]. The solu-tion of the Dyson equation below the Fermi energy wasintroduced in Ref. [8]. The scattering wave functionsare generated with the iterative procedure outlined inRef. [20] leading to a modest increase in computer timeas compared to the use of purely local potentials. Neu-tron and proton potentials are kept identical in the fitexcept for the Coulomb potential for protons. The nu-merical values of all parameters together with a completelist of all employed equations is available in Ref. [21].Included in the present fit are the same elastic scat-tering data and level information considered in Ref. [15].In addition, we now include the charge density of Caas given in Ref. [22] by a sum of Gaussians in the fit.Data from the ( e, e ′ p ) reaction at high missing energyand momentum obtained at Jefferson Lab for C [23], Al, Fe, and
Au [24] were incorporated as well. Wenote that the spectral function of high-momentum pro-tons per proton number is essentially identical for Aland Fe thereby providing a sensible benchmark for their [ m b / s r ] Ω / d σ d Ca n+ < 10 lab lab
10 < E < 40 lab
20 < E < 100 lab
40 < E > 100 lab E Ca p+ [deg] cm θ FIG. 1. (Color online) Calculated and experimental elastic-scattering angular distributions of the differential cross sec-tion dσ/d
Ω. Panels shows results for n + Ca and p + Ca.Data for each energy are offset for clarity with the lowestenergy at the bottom and highest at the top of each frame.References to the data are given in Ref. [15]. presence in Ca. We merely aimed for a reasonable rep-resentation of these cross sections since their interpreta-tion requires further consideration of rescattering contri-butions [25]. We did not include the results of the analy-sis of the ( e, e ′ p ) reaction from NIKHEF [26] because theextracted spectroscopic factors depend on the employedlocal optical potentials. We plan to reanalyze these datawith our nonlocal potentials in a future study.Motivated by the work of Refs. [18, 19], we allow fordifferent nonlocalities above and below the Fermi energy,otherwise the symmetry around this energy is essentiallymaintained by the fit. The values of the nonlocality pa-rameters β appear reasonable and range from 0.64 fmabove to 0.81 fm below the Fermi energy for volume ab-sorption. These parameters are critical in ensuring thatparticle number is adequately described. We limit con-tributions to ℓ ≤ ε F [19] obtaining 19.88 protonsand 19.79 neutrons. We note the extended energy do-main for volume absorption below ε F to accommodatethe Jefferson Lab data. Surface absorption requires non-localities of 0.94 fm above and 2.07 fm below ε F [21].The final fit to the experimental elastic scattering datais shown in Fig. 1 while the fits to total and reaction crosssections are shown in Fig. 2. In all cases, the quality ofthe fit is the same as in Refs. [14] or [15]. This statementalso holds for the analyzing powers given in Ref. [21].Having established our description at positive ener-gies is equivalent to our earlier work, but now consistentwith theoretical expectations associated with the nonlo- [MeV] Lab E [ m b ] σ Ca n+ tot σ react σ [ m b ] σ Ca p+ react σ FIG. 2. (Color online) Total reaction cross sections are dis-played as a function of proton energy while both total andreaction cross sections are shown for neutrons. cal content of the nucleon self-energy, we turn our atten-tion to the new results below the Fermi energy. In Fig. 3we display the spectral strength given in Eq. (2) as a func-tion of energy for the first few levels in the independent-particle model. The downward arrows identify the exper-imental location of the levels near the Fermi energy while -140 -120 -100 -80 -60 -40 -20 0 -3 -2 -1
101 1/2 s -140 -120 -100 -80 -60 -40 -20 01/2 p -140 -120 -100 -80 -60 -40 -20 0 ] - S ( E ) [ M e V -3 -2 -1
101 3/2 d -140 -120 -100 -80 -60 -40 -20 03/2 p E [MeV] -140 -120 -100 -80 -60 -40 -20 0 -3 -2 -1
101 5/2 d E [MeV] -140 -120 -100 -80 -60 -40 -20 07/2 f FIG. 3. (Color online) Spectral strength for protons in the ℓj -orbits which are fully occupied in the independent-particlemodel as well as the f / strength associated with the firstempty orbit in this description. The arrows indicate the ex-perimental location of the valence states as well as the peakenergies for the distributions of deeply bound ones. r [fm] ] - [ e . f m ρ FIG. 4. (Color online) Comparison of experimental chargedensity [22] (thick line) with the DOM fit (thin line). for deeply bound levels they correspond to the peaks ob-tained from ( p, p ) [27] and ( e, e ′ p ) reactions [28]. TheDOM strength distributions track the experimental re-sults represented by their peak location and width.For the quasi-hole states we find spectroscopic factorsof 0.78 for both the 1 s / and 0.76 for the 0 d / level.The location of the former deviates slightly from the ex-perimental peak which may require additional state de-pendence of the self-energy as expressed by poles nearbyin energy [29]. The analysis of the ( e, e ′ p ) reaction inRef. [30] clarified that the treatment of nonlocality in therelativistic approach leads to different distorted protonwaves as compared to conventional non-relativistic opti-cal potentials, yielding about 10-15% larger spectroscopicfactors. Our current results are also larger by about 10-15% than the numbers extracted in Ref. [26]. Introducinglocal DOM potentials in the analysis of transfer reactionshas been shown to have salutary effects for the extractionof spectroscopic information of neutrons [31] and nonlo-cal potentials should further improve such analyses.In Fig. 4 we compare the experimental charge den-sity of Ca (thick line representing a 1% error) withthe DOM fit. While some details could be further im-proved, it is clear that an excellent description of thecharge density is possible in the DOM. The correct par-ticle number is essential for this result which in turn canonly be achieved by including nonlocal absorptive poten-tials that are also constrained by the high-momentumspectral functions. With a local absorption we are notcapable to either generate a particle number close to 20or describe the charge density accurately [8].We compare in Fig. 5 the results for the high-momentum removal spectral strength with the JeffersonLab data [24]. We note that the high-energy data corre-spond to intrinsic nucleon excitations and cannot be part [MeV] m E ] - s r - ) [ M e V m , p m S ( E -14 -13 -12 -11 -10
10 = 250 [MeV/c] m p = 330 [MeV/c] m p = 410 [MeV/c] m p = 490 [MeV/c] m p = 570 [MeV/c] m p = 650 [MeV/c] m p FIG. 5. (Color online) Spectral strength as a function miss-ing energy for different missing momenta as indicated in thefigure. The data are the average of the Al and Fe mea-surements from [24]. of the present analysis. To further improve the descrip-tion, one would have to introduce an energy dependenceof the radial form factors for the potentials. Neverthe-less we conclude that an adequate description is gener-ated which corresponds to 10.6% of the protons havingmomenta above 1.4 fm − . Employing the energy sumrule [9] in the form given in Ref. [32], yields a bindingenergy of 7.91 MeV/ A much closer to the experimental8.55 MeV/ A than found in Ref. [8]. The constrained pres-ence of the high-momentum nucleons is responsible forthis change [33]. The 7.91 MeV/ A binding obtained hererepresents the contribution to the ground-state energyfrom two-body interactions including a kinetic energy of22.64 MeV/ A and was not part of the fit. This empiricalapproach therefore leaves about 0.64 MeV/ A attractionfor higher-body interactions about 1 MeV/ A less thanthe Green’s function Monte Carlo results of Ref. [34] forlight nuclei.In conclusion we have demonstrated that the nucleonself-energy for Ca requires a nonlocal form and can thenwith reasonable assumptions represent all relevant single-particle properties of this nucleus.This work was supported by the U.S. Department ofEnergy, Division of Nuclear Physics under grant DE-FG02-87ER-40316 and the U.S. National Science Foun-dation under grants PHY-0968941 and PHY-1304242. [1] R. L. Varner, W. J. Thompson, T. L. McAbee, E. J.Ludwig, and T. B. Clegg, Phys. Rep. , 57 (1991)[2] A. J. Koning and J. P. Delaroche, Nucl. Phys. A ,231 (2003)[3] B. A. Brown, Prog. Part. Nucl. Phys. , 517 (2001)[4] E. Caurier, G. Martnez-Pinedo, F. Nowacki, A. Poves,and A. P. Zuker, Rev. Mod. Phys. , 427 (2005) [5] C. Mahaux and R. Sartor, Adv. Nucl. Phys. , 1 (1991)[6] R. J. Charity, L. G. Sobotka, and W. H. Dickhoff,Phys. Rev. Lett. , 162503 (2006)[7] R. J. Charity, W. H. Dickhoff, L. G. Sobotka, and S. J.Waldecker, Eur. Phys. J. A49 , in press (2013)[8] W. H. Dickhoff, D. Van Neck, S. J. Waldecker, R. J.Charity, and L. G. Sobotka, Phys. Rev. C , 054306(2010)[9] W. H. Dickhoff and D. Van Neck, Many-Body Theory Ex-posed!, 2nd edition (World Scientific, New Jersey, 2008)[10] W. H. Dickhoff, J. Phys. G: Nucl. Part. Phys. ,064007 (2010)[11] B. A. Brown, S. E. Massen, and P. E. Hodgson,J. Phys. G: Nucl. Phys. , 1655 (1979)[12] J. S. Bell and E. J. Squires, Phys. Rev. Lett. , 96 (1959)[13] C. Mahaux and R. Sartor, Phys. Rev. Lett. , 3015(1986)[14] R. J. Charity, J. M. Mueller, L. G. Sobotka, and W. H.Dickhoff, Phys. Rev. C , 044314 (2007)[15] J. M. Mueller, R. J. Charity, R. Shane, L. G. Sobotka,S. J. Waldecker, W. H. Dickhoff, A. S. Crowell, J. H.Esterline, B. Fallin, C. R. Howell, C. Westerfeldt,M. Youngs, B. J. Crowe, and R. S. Pedroni, Phys. Rev.C , 064605 (2011)[16] I. Brida, S. C. Pieper, and R. B. Wiringa, Phys. Rev. C , 024319 (2011)[17] F. Perey and B. Buck, Nucl. Phys. , 353 (1962)[18] S. J. Waldecker, C. Barbieri, and W. H. Dickhoff,Phys. Rev. C , 034616 (2011)[19] H. Dussan, S. J. Waldecker, W. H. Dickhoff, H. M¨uther,and A. Polls, Phys. Rev. C , 044319 (2011)[20] N. Michel, Eur. Phys. J. A42 , 523 (2009)[21] M. H. Mahzoon, R. J. Charity, W. H. Dickhoff,H. Dussan, and S. J. Waldecker, “Non-local dis-persive optical model ingredients for Ca,” (2013),arXiv:nucl-th/1312.4886 [22] H. de Vries, C. W. de Jager, and C. de Vries,At. Data Nucl. Data Tables , 495 (1987)[23] D. Rohe, C. S. Armstrong, R. Asaturyan, O. K. Baker,S. Bueltmann, C. Carasco, D. Day, R. Ent, H. C.Fenker, K. Garrow, A. Gasparian, P. Gueye, M. Hauger,A. Honegger, J. Jourdan, C. E. Keppel, G. Kubon,R. Lindgren, A. Lung, D. J. Mack, J. H. Mitchell,H. Mkrtchyan, D. Mocelj, K. Normand, T. Petitjean,O. Rondon, E. Segbefia, I. Sick, S. Stepanyan, L. Tang,F. Tiefenbacher, W. F. Vulcan, G. Warren, S. A.Wood, L. Yuan, M. Zeier, H. Zhu, and B. Zihlmann,Phys. Rev. Lett. , 182501 (2004)[24] D. Rohe, Habilitationsschrift (University of Basel, 2004)[25] C. Barbieri, D. Rohe, I. Sick, and L. Lapik´as, Phys. Lett.B , 49 (2006)[26] G. J. Kramer, H. P. Blok, J. F. J. van den Brand, H. J.Bulten, R. Ent, E. Jans, J. B. J. M. Lanen, L. Lapik´as,H. Nann, E. N. M. Quint, G. van der Steenhoven, P. K. A.de Witt Huberts, and G. J. Wagner, Phys. Lett. B ,199 (1989)[27] G. Jacob and T. A. J. Maris, Rev. Mod. Phys. , 6(1973)[28] S. Frullani and J. Mougey, Adv. Nucl. Phys. , 1 (1984)[29] W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. , 377 (2004)[30] J. M. Udias, P. Sarriguren, E. Moya de Guerra, E. Gar-rido, and J. A. Caballero, Phys. Rev. C , 3246 (1995)[31] N. B. Nguyen, S. J. Waldecker, F. M. Nunes, R. J. Char-ity, and W. H. Dickhoff, Phys. Rev. C , 044611 (2011)[32] A. E. L. Dieperink and T. de Forest Jr., Phys. Rev. C , 543 (1974)[33] H. M¨uther, A. Polls, and W. H. Dickhoff, Phys. Rev. C , 3040 (1995)[34] S. C. Pieper and R. B. Wiringa, Annu. Rev.Nucl. Part.Sci.51