Microscopic global optical potential for rare isotope reactions
MMicroscopic global optical potential for rare isotope reactions
T. R. Whitehead,
1, 2
Y. Lim,
3, 4, 5 and J. W. Holt
1, 2 Cyclotron Institute, Texas A&M University, College Station, Texas 77843, USA Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany
We construct from chiral effective field theory two- and three-body forces a microscopic globalnucleon-nucleus optical potential suitable for reactions involving radioactive isotopes. Within the im-proved local density approximation and without any adjustable parameters, we begin by computinglocal proton and neutron optical potentials for 1800 target nuclei in the mass range 12 < A < < E <
200 MeV. We then construct a global optical potentialparametrization that depends smoothly on the projectile energy as well as the target nucleus massnumber and isospin asymmetry. Elastic scattering observables calculated from the global opticalpotential are found to be in good agreement with available experimental data for a wide range of pro-jectile energies and target nuclei. Compared to traditional phenomenological optical potentials, wefind a strong energy dependence and shell structure features in the Woods-Saxon geometry param-eters. For target nuclei with small proton-neutron asymmetry, we find that the real and imaginaryoptical potential depths exhibit a clear linear dependence on the isospin-asymmetry and preservethe well-known Lane form up to high projectile energies. For nuclei with larger isospin asymmetries,we find evidence for a novel isoscalar term in the low-energy optical potential proportional to thesquare of the isospin asymmetry. These insights from microscopic many-body theory may be used toinform next-generation phenomenological optical potentials for proton- and neutron-rich isotopes.
Introduction - Nuclear physics is approaching an ex-citing new era in which rare isotope beam facilities, suchas FRIB, RIBF, FAIR, and Spiral2, will explore previ-ously inaccessible regions of the nuclear chart that areimportant for understanding the origin of the elements[1–4] and the properties of neutron stars [5, 6]. Rare iso-tope beam experiments will produce a flood of new datawhose interpretation and connection to nuclear structurewill be guided by theoretical modeling. Of particularimportance in the context of nuclear reaction studies isthe nuclear optical model [7–10], where the complicated(and in most cases intractable) problem of solving the N -body Schroedinger equation for nucleon-nucleus scatter-ing in terms of fundamental two- and three-body forcesis simplified by assuming the projectile nucleon interactswith an average single-particle potential generated by thetarget nucleus. Global phenomenological optical poten-tials [11–13] are the workhorse for theoretical modelingof nuclear reactions but are currently tuned to limitedexperimental data near nuclear stability. The worldwideradioactive ion beam program requires next-generationglobal optical potentials that can reach into unexploredregions of the nuclear chart, provide quantified uncer-tainty estimates for reaction observables [14, 15], andthat are informed by microscopic nuclear theory basedon high-precision nuclear forces [16–21].In the late 1960s the first phenomenological nucleon-nucleus optical potentials were limited to heavy isotopeswith mass numbers A >
40 and low scattering energiesof E (cid:46)
50 MeV. Phenomenological and semi-microscopicoptical potentials [10–13, 22–30] have improved dramat-ically since then, and today the most widely used op-tical potential of Koning and Delaroche [11] is suitable to describe scattering phenomena for stable nuclei with24 < A <
209 up to projectile energies of E (cid:39)
200 MeV.However, it remains an open question how reliable suchphenomenological optical potentials behave for reactionsinvolving exotic isotopes. This is particularly crucial forsimulating the late-time freeze-out phase of r-process nu-cleosynthesis, where photodissociation and radiative cap-ture processes are out of equilibrium and neutron-capturerates play an enhanced role in determining the final abun-dance pattern of r-process elements [31]. Neutron cap-ture cross sections on neutron-rich isotopes cannot bedirectly measured with today’s experimental techniques,but considerable efforts [32, 33] are being made towardmeasuring gamma strength functions and nuclear leveldensities that enter into the Hauser-Feshbach theory forradiative neutron capture in stellar plasmas [34]. Suchcalculations also require as input the neutron-nucleus op-tical potential, and especially its imaginary part at lowscattering energies [35].In the present work we construct the first microscopicglobal nucleon-nucleus optical potential based on an anal-ysis of 1800 isotopes in the framework of many-bodyperturbation theory with state-of-the-art nuclear interac-tions from chiral effective field theory. Compared to phe-nomenological [11, 13, 24, 26] or semi-microscopic opticalpotentials [12, 28, 29] that are directly fitted to nuclearreaction data, purely microscopic calculations may havegreater predictive power for reactions involving exoticisotopes. Constructing optical potentials via the nucleonself-energy in finite nuclei or nuclear matter from chiraleffective field theory (EFT) [18–21, 36–42] is a promisingroute of inquiry since chiral EFT features realistic nuclearinteractions based on the symmetries of QCD and a sys- a r X i v : . [ nu c l - t h ] O c t tematic expansion of the nuclear force [16, 17, 43, 44]that provides a method of quantifying theoretical un-certainties. In the present work we use a specific chi-ral nuclear interaction [45] with momentum-space cutoffΛ = 450 MeV in the calculation of the nucleon self-energythat is known to reproduce well the saturation energy anddensity of symmetric nuclear matter [45]. The low-energyconstants of the potential are fitted to nucleon-nucleonscattering phase shifts, deuteron properties, and the tri-ton binding energy and lifetime [45]. In future work weplan to implement next-to-next-to-next-to-leading order(N3LO) three-body forces [46–51] as well as a wider classof chiral potentials [44, 52] to better assess the theoreticaluncertainties. Formalism - We begin by calculating up to second or-der in many-body perturbation theory the nucleon self-energy Σ( k, E ( k )) for E >
0, which is equivalent [7] to theoptical potential for scattering states. The backgroundmedium is taken to be homogeneous nuclear matter withfixed density and isospin asymmetry in the thermody-namic limit. The calculation of the second-order dia-grams involves intermediate-state propagators whose en-ergies E ( k ) = k / (2 M ) + Σ( k, E ( k )) are computed selfconsistently with the on-shell self-energy. In general theresulting self-energy is complex and energy dependent.In order to construct a nucleon-nucleus optical poten-tial, we compute the nucleon self-energy over the range ofdensities and isospin asymmetries found in finite nuclei.Since the spin-orbit interaction vanishes in homogeneousnuclear matter, we employ the improved density matrixexpansion [53–56] at the Hartree-Fock level to calculatethe spin-orbit contribution to the nuclear energy densityfunctional. In this formulation the spin-orbit interactionis calculated at the Fermi energy and consequently doesnot have an explicit energy dependence. Density distri-butions for the target nuclei are calculated in mean fieldtheory with the Sk χ
450 Skyrme effective interaction [57]constrained by the chiral interaction used to calculate theoptical potential.The improved local density approximation (ILDA) isutilized to transition from a nuclear matter optical poten-tial to that of a finite nucleus by folding the density- andisospin-asymmetry-dependent self-energy with the targetnucleus density distribution U LDA ( E ; r ) = U NM ( E ; ρ ( r ) , δ ( r )) , (1)where ρ = ρ n + ρ p and δ = ( ρ n − ρ p ) /ρ . The ILDAis applied by integrating over the radial direction with aGaussian form factor to account for the nonzero range ofthe nuclear force [9, 30]: U ILDA ( E ; r ) = 1( t √ π ) (cid:90) U LDA ( E ; r (cid:48) ) e −| (cid:126)r − (cid:126)r (cid:48)| t d r (cid:48) , (2)where the range parameter t represents the characteristiclength scale of the interaction. The range parameter isderived in this work by calculating the root mean squareradii of the local chiral NN interactions presented in Ref.[58]. We use the average value of t C = 1 .
22 fm for the
FIG. 1. Isotopes used in constructing the Whitehead-Lim-Holt (WLH) global microscopic optical potential are shown inblue, and those used for parameterizing the Koning-Delarochephenomenological optical potential are shown in red. central terms of the optical potential and t SO = 0 .
98 fmfor the spin-orbit term. In Refs. [18, 42] the effect ofvarying these range parameters was shown to be small.Within the framework just outlined, we have developedin previous works [18, 36, 37, 42] proton and neutron op-tical potentials for stable calcium isotopes. In the presentwork we develop the first microscopic global optical po-tential, referred to as Whitehead-Lim-Holt (WLH), thatis built upon local optical potentials for 1800 target nu-clei with mass numbers 12 < A <
242 and projectileenergies 0 MeV < E <
200 MeV. This set includes allstable and long lived isotopes, light and medium-massbound isotopes out to the predicted neutron drip line ofiron [59], and heavier neutron-rich isotopes relevant tothe r-process [60]. The isotopes for which we have con-structed local optical potentials are shown as blue dots inFig. 1 in comparison with the stable isotopes that wereused in calibrating the Koning-Delaroche phenomenolog-ical optical potential (shown as red squares).We then fit the position- and energy-dependent opticalpotentials U ( r, E ) = U V ( r, E ) + iU W ( r, E ) + iU S ( r, E ) (3)+ U SO ( r, E ) (cid:126)(cid:96) · (cid:126)σ, to the commonly used Woods-Saxon form f ( r ; r i , a i ) = e ( r − A / ri ) /ai (for U V and U W ) and its derivative (for U S and U SO ). Functional forms for the A , E , and δ de-pendence of the Woods-Saxon geometry parameters andoverall strengths were chosen in order to minimize theleast squares fit while using as few parameters as possi-ble. We used the following functional forms to define the FIG. 2. The left and right top plots show the depth of the realvolume term at E = 0 MeV and E = 180 MeV respectively forneutron and proton potentials in red and blue as functions ofthe isospin asymmetry. The left and right bottom plots showthe imaginary volume and surface depths respectively at E =0 MeV for neutron and proton potentials in magenta and cyanas functions of the isospin asymmetry. The points representvalues for local potentials and the solid lines are values forthe WLH global potential, analogous Koning-Delaroche (KD)values are given by orange and green squares. global optical potential parametrization: U V = u V + u V E + u V E + u V E + u V E (4)+( u V + u V E + u V E + u V E ) δ + u V e u V E δ r V = r V + r V E + r V E + r V A − / a V = a V + a V E + a V E + a V E + a V E +( a V + a V δ ) δ U W = u W + u W E + u W E + ( u W + u W E ) δ (5) r W = r W + r W + r W Ar W + A + r W E + r W E a pW = a pW + a pW Ea pW − E + ( a pW + a pW E ) δa nW = a nW + a nW Ea nW − E + (cid:18) a nW + a nW E (cid:19) δ U pS = u pS + u pS E + ( u pS + u pS E ) δ (6) U nS = u nS + u nS E + u nS E + ( u nS + u nS E + u nS E ) δr S = r S + r S E + r S E + r S A − / a S = a S + a S E + a S E + a S A + a S A U SO = u SO + u SO A + u SO A + u SO A (7) r SO = r SO + r SO A − / a SO = a SO + a SO A + a SO A + a SO A . Values for the ∼
70 fit parameters in Eqs. 5 - 8 can befound in Table I of the supplementary material.
Results - The microscopic global optical potential con-structed in this work is found to have several differentfeatures compared to widely used phenomenological op-tical potentials [11, 13]. The most prominent differenceis found in the isovector terms, which are crucial for anaccurate description of neutron-rich isotopes. In Fig. 2we show the isospin asymmetry dependence of the realvolume U V , imaginary volume U W , and imaginary sur-face U S depths at E = 0 MeV. In the top left panel weshow the isospin asymmetry dependence of the WLH realvolume term which is similar to the results of the KD op-tical potential for low energy. However, as seen in the topright panel of Fig. 2 the two models differ at higher ener-gies as the WLH real volume term preserves the isovectorLane form, U = U + τ z U I δ , even after undergoing anisospin inversion at E = 115 MeV where the strength ofthe isovector term changes sign. As seen in the upper leftplot of Fig. 2 the WLH real depth exhibits an isoscalar δ dependence for low energies that is consistent with thenuclear matter calculations in Ref. [37]. This extensionof the standard Lane form is not accounted for in cur-rent phenomenological optical potentials, and is impor-tant for accurately modeling reactions with neutron-richisotopes. In further contrast to the KD optical poten-tial, the WLH potential has a clear isospin asymmetrydependence in both the imaginary volume and surfaceterms shown in the bottom left panel of Fig. 2. Theimaginary term must vanish at the Fermi energy, whichis E F = 0 for neutrons at the neutron drip line. As shownin the bottom right panel, the microscopic WLH imagi-nary surface term vanishes as δ increases toward valuescorresponding to the drip line, while the KD model doesnot. The isovector terms of the optical potential are cru-cial for describing isotopes off stability and in particularthe imaginary isovector term is of significant importanceto the neutron capture rates on exotic isotopes involvedin the astrophysical r-process [35].An additional distinction of the WLH optical poten-tial is energy dependent geometry parameters. In the lefttwo panels of Fig. 3 we show the energy dependence ofthe real and imaginary radius and diffuseness parametersfor Sn, which are indicative of all isotopes consideredin this work. By comparison, the assumed form of theKoning-Delaroche optical potential geometry parametersis not only energy independent but also identical for thereal and imaginary volume contributions. In general theenergy dependence of geometry parameters is as signifi-cant as the mass number dependence and more easily de-scribed in a closed form. One point of similarity, shownin the top middle panel, is that the WLH radius parame-ters exhibit the well-known [11, 13, 25] r ∝ A − / depen-dence. In the bottom middle panel we find that the massnumber dependence of the diffuseness parameters havecomplicated shell structure effects that would require toomany parameters to accurately describe. Instead we in-clude an explicit dependence on the isospin asymmetry, FIG. 3. The top (bottom) plots show the microscopic real and imaginary radius (diffuseness) parameters for proton and neutronprojectiles. The points represent values for local optical potentials and the solid lines are values for the WLH global potential,Koning-Delaroche (KD) values are given by a dashed line. The left panels are for a
Sn target plotted as a function of energy.The middle and right panels, at E = 100 MeV, are plotted as a function of mass number and isospin-asymmetry respectively.
20 40 60 80 100 (deg) d / d ( m b / s r ) A=40x 10 ( A A=42x 10 ( A A=44x 10 ( A A=46x 10 ( A A=48x 10 ( A WLH GlobalKD GlobalExperiment
20 40 60 80 100 (deg) A=50x 10 ( A A=52x 10 ( A A=54x 10 ( A A=56x 10 ( A A=58x 10 ( A A=60x 10 ( A WLH GlobalKD GlobalExperiment p+ A Ca, E=65MeV
FIG. 4. Proton elastic scattering cross sections at E =65 MeV for Ca - Ca. Results of the microscopic global op-tical potential constructed in this work are shown in blue andlabeled by WLH Global. Results of the Koning-Delarochephenomenological global optical potential are given by thegreen dashed line and labeled as KD Global, experimentaldata are shown as red dots. shown in the bottom right plot to have a positive correla-tion and roughly quadratic behavior. Finally, in the topright plot we show the isospin asymmetry dependence ofthe radius parameters to be very diffuse with bands atconstant values of r i that result from an interplay of shelleffects with the increasing r ∝ A − / dependence.In anticipation of future experiments involving radioac-tive nuclei, we show in Fig. 4 the proton differentialelastic scattering cross sections on calcium isotopes from A = 40 to A = 60. The results of the WLH global opticalpotential (blue lines) are close to those of the KD globaloptical potential (dashed green lines) and in good agree-ment with experimental data (red dots), except for the Ca data that are also in disagreement with other opticalpotentials [28]. The microscopic predictions of the WLHpotential become systematically smaller than the resultsof Koning-Delaroche as mass number increases. This isdue to the different behavior of the isovector terms inthe two optical potentials. In the supplemental materialwe show plots of neutron and proton elastic scatteringcross sections and analyzing powers for a selection of tar-get nuclei to give a more comprehensive demonstrationof the effectiveness of the WLH global optical potential.Experimental scattering cross sections are generally wellreproduced while analyzing powers stand to benefit fromthe inclusion of higher-order calculations of the spin-orbitterm.
Summary - We have constructed a microscopic globaloptical potential based on the many-body perturbationtheory calculation of the nucleon self-energy in nuclearmatter from chiral two- and three-nucleon interactions.The global optical potential is expressed as a functioncomposed of Woods-Saxon terms with parameters thatvary smoothly in E , A , and δ , which can be easily im-plemented into modern reaction theory codes. A pythonscript for generating input files needed to run TALYSwith the WLH global optical potential may be found at[61]. These dependences are represented by functionalforms parametrized to local proton and neutron opticalpotentials that span projectile energies 0 < E <
200 MeVand 1800 isotopes with mass numbers 12 < A < [1] A. B. Balantekin, J. Carlson, D. J. Dean, G. M.Fuller, R. J. Furnstahl, M. Hjorth-Jensen, R. V. F.Janssens, B.-A. Li, W. Nazarewicz, F. M. Nunes,W. E. Ormand, S. Reddy, and B. M. Sher-rill, Mod. Phys. Lett. A , 1430010 (2014),https://doi.org/10.1142/S0217732314300109.[2] C. J. Horowitz, A. Arcones, B. Cˆot´e, I. Dillmann,W. Nazarewicz, I. U. Roederer, H. Schatz, A. Apra-hamian, D. Atanasov, A. Bauswein, T. C. Beers, J. Bliss,M. Brodeur, J. A. Clark, A. Frebel, F. Foucart, C. J.Hansen, O. Just, A. Kankainen, G. C. McLaughlin, J. M.Kelly, S. N. Liddick, D. M. Lee, J. Lippuner, D. Martin,J. Mendoza-Temis, B. D. Metzger, M. R. Mumpower,G. Perdikakis, J. Pereira, B. W. O’Shea, R. Reifarth,A. M. Rogers, D. M. Siegel, A. Spyrou, R. Surman,X. Tang, T. Uesaka, and M. Wang, J. Phys. G: Nucl.Part. Phys. , 083001 (2019).[3] B. Cˆot´e, C. L. Fryer, K. Belczynski, O. Korobkin,M. Chru´sli´nska, N. Vassh, M. R. Mumpower, J. Lip-puner, T. M. Sprouse, R. Surman, and R. Wollaeger,Astrophys. J. , 99 (2018).[4] D. Kasen, B. Metzger, J. Barnes, E. Quataert, andE. Ramirez-Ruiz, Nature , 80 (2017).[5] F. J. Fattoyev, J. Piekarewicz, and C. J. Horowitz, Phys.Rev. Lett. , 172702 (2018).[6] B. A. Brown, Phys. Rev. Lett. , 122502 (2017).[7] J. S. Bell and E. J. Squires, Phys. Rev. Lett. , 96 (1959).[8] F. G. Perey, Phys. Rev. , 745 (1963).[9] J. P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rev.C , 80 (1977).[10] J. P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rep. , 83 (1976).[11] A. J. Koning and J. P. Delaroche, Nucl. Phys. A713 , 231(2003).[12] E. Bauge, J. P. Delaroche, and M. Girod, Phys. Rev. C , 024607 (2001).[13] R. L. Varner, W. J. Thompson, T. L. McAbee, E. J.Ludwig, and T. B. Clegg, Phys. Rep. , 57 (1991).[14] M. Catacora-Rios, G. B. King, A. E. Lovell, and F. M.Nunes, Phys. Rev. C , 064615 (2019).[15] G. B. King, A. E. Lovell, L. Neufcourt, and F. M. Nunes, Phys. Rev. Lett. , 232502 (2019).[16] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev.Mod. Phys. , 1773 (2009).[17] R. Machleidt and D. R. Entem, Phys. Rep. , 1 (2011).[18] T. R. Whitehead, Y. Lim, and J. W. Holt, Phys. Rev.C , 064613 (2020).[19] A. Idini, C. Barbieri, and P. Navr´atil, Phys. Rev. Lett. , 092501 (2019).[20] J. Rotureau, P. Danielewicz, G. Hagen, G. R. Jansen,and F. M. Nunes, Phys. Rev. C , 044625 (2018).[21] M. Vorabbi, P. Finelli, and C. Giusti, Phys. Rev. C ,064602 (2018).[22] F. D. Becchetti and G. W. Greenlees, Phys. Rev. ,1190 (1969).[23] D. Wilmore and P. Hodgson, Nucl. Phys. , 673 (1964).[24] S. P. Weppner, R. B. Penney, G. W. Diffendale, andG. Vittorini, Phys. Rev. C , 034608 (2009).[25] J. Rapaport, V. Kulkarni, and R. W. Finlay, Nucl. Phys.A , 15 (1979).[26] X.-H. Li and L.-W. Chen, Nucl. Phys. A , 62 (2012).[27] X. Li and C. Cai, Nucl. Phys. A , 43 (2008).[28] T. Furumoto, K. Tsubakihara, S. Ebata, and W. Hori-uchi, Phys. Rev. C , 034605 (2019).[29] R. Xu, Z. Ma, Y. Zhang, Y. Tian, E. N. E. van Dalen,and H. M¨uther, Phys. Rev. C , 034606 (2016).[30] E. Bauge, J. P. Delaroche, and M. Girod, Phys. Rev. C , 1118 (1998).[31] M. R. Mumpower, R. Surman, G. C. McLaughlin, andA. Aprahamian, Prog. Part. Nucl. Phys. , 86 (2016).[32] A. Spyrou, S. N. Liddick, A. C. Larsen, M. Guttormsen,K. Cooper, A. C. Dombos, D. J. Morrissey, F. Naqvi,G. Perdikakis, S. J. Quinn, T. Renstrøm, J. A. Rodriguez,A. Simon, C. S. Sumithrarachchi, and R. G. T. Zegers,Phys. Rev. Lett. , 232502 (2014).[33] S. N. Liddick, A. Spyrou, B. P. Crider, F. Naqvi,A. C. Larsen, M. Guttormsen, M. Mumpower, R. Sur-man, G. Perdikakis, D. L. Bleuel, A. Couture, L. Cre-spo Campo, A. C. Dombos, R. Lewis, S. Mosby, S. Nikas,C. J. Prokop, T. Renstrom, B. Rubio, S. Siem, and S. J.Quinn, Phys. Rev. Lett. , 242502 (2016).[34] T. RAUSCHER, International Journal of Modern Physics E , 1071 (2011),https://doi.org/10.1142/S021830131101840X.[35] S. Goriely and J. P. Delaroche, Physics Letters B ,178 (2007).[36] J. W. Holt, N. Kaiser, G. A. Miller, and W. Weise, Phys.Rev. C , 024614 (2013).[37] J. W. Holt, N. Kaiser, and G. A. Miller, Phys. Rev. C , 064603 (2016).[38] K. Egashira, K. Minomo, M. Toyokawa, T. Matsumoto,and M. Yahiro, Phys. Rev. C , 064611 (2014).[39] J. Rotureau, P. Danielewicz, G. Hagen, F. M. Nunes,and T. Papenbrock, Phys. Rev. C , 024315 (2017).[40] M. Toyokawa, M. Yahiro, T. Matsumoto, K. Minomo,K. Ogata, and M. Kohno, Phys. Rev. C , 024618(2015).[41] V. Durant, P. Capel, L. Huth, A. Balantekin, andA. Schwenk, Phys. Lett. B , 668 (2018).[42] T. R. Whitehead, Y. Lim, and J. W. Holt, Phys. Rev.C , 014601 (2019).[43] S. Weinberg, Physica A , 327 (1979).[44] P. Reinert, H. Krebs, and E. Epelbaum, Eur. Phys. J.A , 86 (2018).[45] L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt, L. E.Marcucci, and F. Sammarruca, Phys. Rev. C , 044321(2014).[46] I. Tews, T. Kr¨uger, K. Hebeler, and A. Schwenk, Phys.Rev. Lett. , 032504 (2013).[47] C. Drischler, A. Carbone, K. Hebeler, and A. Schwenk,Phys. Rev. C , 054307 (2016).[48] N. Kaiser and V. Niessner, Phys. Rev. C , 054002(2018).[49] N. Kaiser and B. Singh, Phys. Rev. C , 014002 (2019).[50] C. Drischler, K. Hebeler, and A. Schwenk, Phys. Rev.Lett. , 042501 (2019).[51] J. W. Holt, M. Kawaguchi, and N. Kaiser, “Imple-menting chiral three-body forces in terms of medium-dependent two-body forces,” (2020).[52] F. Sammarruca, L. E. Marcucci, L. Coraggio, J. W. Holt,N. Itaco, and R. Machleidt, arXiv:1807.06640 (2018).[53] S. K. Bogner, R. J. Furnstahl, and L. Platter, Eur. Phys.J. A , 219 (2009).[54] B. Gebremariam, T. Duguet, and S. K. Bogner, Phys.Rev. C , 014305 (2010).[55] B. Gebremariam, S. K. Bogner, and T. Duguet, Nucl.Phys. A , 17 (2011).[56] J. W. Holt, N. Kaiser, and W. Weise, Eur. Phys. J. A (2011).[57] Y. Lim and J. W. Holt, Phys. Rev. C , 065805 (2017).[58] A. Gezerlis, I. Tews, E. Epelbaum, M. Freunek, S. Gan-dolfi, K. Hebeler, A. Nogga, and A. Schwenk, Phys. Rev.C , 054323 (2014).[59] J. D. Holt, S. R. Stroberg, A. Schwenk, and J. Simonis,(2019), arXiv:1905.10475 [nucl-th].[60] M. R. Mumpower, R. Surman, G. C. McLaughlin, andA. Aprahamian, Prog. Part. Nucl. Phys. SUPPLEMENTARY MATERIAL
The WLH global optical potential is expressed in thefollowing form U ( r, E ) = U V ( r, E ) + iU W ( r, E ) + iU S ( r, E ) (8)+ U SO ( r, E ) (cid:126)(cid:96) · (cid:126)σ, where U V ( r, E ) = −U V f ( r ; r V , a V ) (9) U W ( r, E ) = −U W f ( r ; r W , a W ) U S ( r, E ) = 4 a S U S ddr f ( r ; r S , a S ) U SO ( r, E ) = U SO m π r ddr f ( r ; r SO , a SO ) , and f ( r ; r i , a i ) = 11 + e ( r − A / r i ) /a i . (10)The depth, radial size and diffuseness of each term aregiven by U i , r i , a i and A is the mass number. The pa-rameters U i , r i , a i are expressed as functions of A, δ, E in the following equations U V = u V + u V E + u V E + u V E + u V E (11)+( u V + u V E + u V E + u V E ) δ + u V e u V E δ r V = r V + r V E + r V E + r V A − / a V = a V + a V E + a V E + a V E + a V E +( a V + a V δ ) δ U W = u W + u W E + u W E + u W δ + u W δE (12) r W = r W + r W + r W Ar W + A + r W E + r W E a pW = a pW + a pW Ea pW − E + a pW δ + a pW δEa nW = a nW + a nW Ea nW − E + a nW δ + a nW δ E U pS = u pS + u pS E + u pS δ + u pS Eδ (13) U nS = u nS + u nS E + u nS E + u nS δ + u nS Eδ + u nS E δr S = r S + r S E + r S E + r S A − / a S = a S + a S E + a S E + a S A + a S A U SO = u SO + u SO A + u SO A + u SO A (14) r SO = r SO + r SO A − / a SO = a SO + a SO A + a SO A + a SO A . U V u pV = 54 . u pV = − . u pV = − . u pV = 1 . · − u pV = − . · − u pV = 20 . u pV = − . u nV = 53 . u nV = − . u nV = − . u nV = 1 . · − u nV = − . · − u nV = − . u nV = 0 . u pV = 0 . u pV = − . · − u pV = − . u pV = − . u nV = − . u nV = 3 . · − u nV = − . u nV = − . r V r pV = 1 . r pV = − . · − r pV = 1 . · − r pV = − . r nV = 1 . r nV = − . · − r nV = 1 . · − r nV = − . a V a pV = 0 . a pV = − . · − a pV = − . · − a pV = 5 . · − a pV = − . · − a pV = − . a pV = 0 . a nV = 0 . a nV = 0 . a nV = − . · − a nV = 2 . · − a nV = − . · − a nV = − . a nV = 0 . U W u pW = 3 . u pW = 0 . u pW = − . · − u pW = 11 . u pW = − . u nW = 2 . u nW = 0 . u nW = − . · − u nW = − . u nW = − . r W r pW = 0 . r pW = 48 . r pW = 0 . r pW = 54 . r pW = 0 . r pW = 9 . · − r nW = 0 . r nW = 86 . r nW = 0 . r nW = 84 . r nW = 0 . r nW = 2 . · − a W a pW = 0 . a pW = − . a pW = − . a pW = 0 . a pW = − . a nW = 0 . a nW = − . a nW = − . a nW = 0 . a nW = − . U S u pS = 0 . u pS = − . u pS = − . u pS = 0 . u nS = 1 . u nS = − . u nS = 1 . · − u nS = − . u nS = 0 . u nS = − . r S r pS = 1 . r pS = − . r pS = − . · − r pS = − . r nS = 1 . r nS = 0 . r nS = − . · − r nS = − . a S a pS = 0 . a pS = 0 . a pS = − . · − a pS = 0 . a pS = − . · − a nS = 0 . a pS = 0 . a nS = − . · − a nS = 5 . · − a nS = 3 . · − U SO u SO = 8 . u SO = 0 . u SO = − . · − u SO = 5 . · − r SO r SO = 1 . r SO = − . a SO a SO = 0 . a SO = 0 . a SO = − . · − a SO = 6 . · − TABLE I. WLH global optical potential fit parameters.
The fit parameters in the above equations are listedin Table I for neutron and proton projectiles. Theparametrization of the imaginary surface term is onlyvalid for neutron projectile energies of E (cid:46)
40 MeV andproton projectile energies of E (cid:46)
30 MeV. We find no mi-croscopic evidence of an imaginary surface peak beyondthese energies.We calculate scattering observables from the WLH mi-croscopic global optical potential and compare to resultsof the Koning-Delaroche phenomenological optical po-tential and experiment. In Figs. 5 and 6 we show neutronand proton elastic scattering for a wide range of isotopesand a variety of scattering energies to demonstrate theperformance of the WLH optical potential compared tophenomenology and experiment. Additionally we plotthe analyzing power for neutron and proton projectilesand a variety of target nuclei in Fig. 7. Neutron elastic scattering cross sections are on par with phenomenologyfor energies
E <
20 MeV past which the WLH resultsslightly overestimate for larger scattering angles. Thetop left panel of Fig. 5 shows neutron scattering with Nwhich is outside the mass range for the Koning-Delarochepotential and may indicate the extrapolative power ofphenomenological models. In Fig. 6 proton elastic scat-tering data are also well reproduced by the WLH cal-culations with tendency to slightly underestimate crosssections. In Fig. 7 analyzing powers calculated by theWLH potential generally reproduce the angular patternof experimental data, while often overestimating or un-derestimating the magnitude of peaks and troughs. Thismay be improved by inclusion of higher order perturba-tive terms and isovector contributions to the spin-orbitinteraction.
50 100 15010 d / d ( m b / s r ) n+ N, E=14MeV
WLH GlobalKD GlobalExperiment 50 100 15010 n+ O, E=15MeV
50 100 15010 n+ S, E=26MeV
50 100 15010 d / d ( m b / s r ) n+ Fe, E=26MeV
20 4010 n+ Zr, E=75MeV
50 100 15010 n+ Sb, E=14MeV
50 100 (deg) d / d ( m b / s r ) n+ Ba, E=3MeV
50 100 150 (deg) n+ W, E=6MeV
50 100 150 (deg) n+ Pb, E=8MeV
FIG. 5. Neutron elastic scattering cross sections for a selection of target isotopes at varied energies. Results of the microscopicglobal optical potential constructed in this work are shown in blue and labeled by WLH Global. Results of the Koning-Delarochephenomenological global optical potential are given by the green dashed line and labeled as KD global, experimental data areshown as red dots.
50 100 15010 d / d ( m b / s r ) p+ O, E=11MeV
WLH GlobalKD GlobalExperiment 25 50 75 10010 p+ Al, E=57MeV
50 100 15010 p+ Ti, E=14MeV
20 40 6010 d / d ( m b / s r ) p+ Ni, E=65MeV
50 100 15010 p+ Se, E=16MeV
10 20 30 40 5010 p+ Sn, E=104MeV
20 40 60 (deg) d / d ( m b / s r ) p+ W, E=65MeV
10 20 30 (deg) p+ Au, E=160MeV
25 50 75 100 (deg) p+ Pb, E=35MeV
FIG. 6. Proton elastic scattering cross sections for a selection of target isotopes at varied energies. Results of the microscopicglobal optical potential constructed in this work are shown in blue and labeled by WLH Global. Results of the Koning-Delarochephenomenological global optical potential are given by the green dashed line and labeled as KD global, experimental data areshown as red dots.
50 100 1501.00.50.00.51.0 A y p+ Si, E=21MeV
50 100 1501.00.50.00.51.0 p+ Ni, E=25MeV
50 100 1501.00.50.00.51.0 p+ Zr, E=20MeV
50 100 1501.00.50.00.51.0 A y p+ Pb, E=26MeV
25 50 75 100 1251.00.50.00.51.0 n+ Al, E=15MeV
WLH GlobalKD GlobalExperiment 50 100 1501.00.50.00.51.0 n+ Cu, E=10MeV
50 100 150 (deg) A y n+ Y, E=14MeV
50 100 150 (deg) n+ Sn, E=10MeV