one-nucleon transfer reactions and the optical potentia lone-nucleon transfer reactions and the...

Varenna, June 2015

One-nucleon transfer reactions and the optical potential

Filomena Nunes Michigan State University

In collaboration with: Amy Lovell, Alaina Ross,

Jimmy Rotureau and Luke Titus Supported by: NSF, NNSA, DOE

•  Funded by DOE Office of Science Office of Nuclear Physics. T. Glasmacher, Project Director

•  Key Feature is 400kW beam power (5 x1013 238U/s)

•  Separation of isotopes in-flight •  Fast development time

for any isotope •  Suited for all elements

and short half-lives •  Fast, stopped, and

reaccelerated beams

Facility for Rare Isotope Beams, FRIB

Brad Sherrill

Facility for Rare Isotope Beams (FRIB)

FRIB timeline

•  FRIB project started 6/2009 •  Total project cost $730 million

–  $635.5M Department of Energy –  $94.5M Michigan State University and

State of Michigan •  Civil construction started 3/2014 •  Technical construction start 10/2014

–  Critical Decision CD-3b approved 8/2014 •  CD-4 (DOE project completion) 6/2022

–  Project managed for early completion 12/2020 •  NSCL will continue to operate as national user facility

until shortly before FRIB completion – funded by NSF –  Integration of NSCL facilities into FRIB within one year in ~2020

Brad Sherrill

Our nucleus factory

January 23, 2014 14:42 WSPC/INSTRUCTION FILE FRIB-theory˙16

FRIB Nuclear Theory 3

to 84Ni, thereby spanning the full range of neutrons in the pfg-shells. The key is forthe facility to deliver very high power heavy-ion primary beams,6 400 kW minimum,at energies of at least 200MeV/u to be used in the production of rare isotopes.7

These high-power beams will be generated by a superconducting linear acceleratorcoupled with a production area designed to operate at high current.8 The broadscientific program requires rare-isotope beams at energies ranging from stopped ionsin traps9 to ions at relativistic energies of hundreds of MeV/u. FRIB will have thesecapabilities by using full-energy beams following in-flight separation, stopped ionsthermalized using a variety of ion catcher schemes, and reaccelerated ions deliveredby the ReA superconducting linear post-accelerator.8 The reaccelerated beams willeventually reach energies of 12–20MeV/u. As a result, the full complement of directreactions (including high-` transfer and deep-inelastic reactions) will be accessiblefor experimentation. An advantage of the in-flight production and reaccelerationapproach is that isotopes of all elements will normally be available with very shortdevelopment times and high eciency (approaching 10 to 20%). The technique willalso provide beams of most isotopes, even those with short (tens of ms) half-lives.This o↵ers the possibility to perform experiments, for example, with beams of highlyrefractory elements along the N = 126 line of isotones below 208Pb, as required toimprove r-process nucleosynthesis models.

Produced at FRIBEstimated by theory

Fig. 1. Number of isotopes of elements up to Z = 92 estimated to be produced in sucientquantities at FRIB (green bars) to allow study of their structure and determine at least oneproperty other than simple observation. The number of isotopes estimated to exist (blue bars) istaken from the recent theoretical survey of Ref. 10. FRIB is predicted to produce nearly 80% ofall possible isotopes in this range.

The anticipated range of isotopes to be available at FRIB, as estimated usingthe LISE++ program,11 is shown in Fig. 1. It is compared to the predicted num-ber of possible isotopes from the average of the Density Functional Theory (DFT)predictions.10 With the 400 kW beams of FRIB, nearly 80% of all isotopes of el-ements up to uranium may become available for study. This includes many newisotopes estimated to lie along the drip lines (perhaps even up to element Z = 61as shown in the figure) and many nuclei with skins predicted to be greater than

Nearly 80% of all isotopes up to Uranium may be produced at FRIB

Big science questions

FRIB theory manifesto, Balantekin et al, MPLA 2014 (arXiv:1401.6435)

1) How did matter come into being and how does it evolve? 2) How does subatomic matter organize itself and what phenomena emerge? 3) Are the fundamental interactions that are basic to the structure of matter fully understood? and 4) How can the knowledge and technological progress provided by nuclear physics best be used to benefit society?

•  Reactions are one of the most diverse probes in nuclear physics

•  Reactions will continue to be an integral part of science programs addressing these big questions

•  Reliable reaction theory is an essential ingredient to extract most of the desired information

Examples of using (d,p) to study unstable nuclei

Schmitt et al, PRL 108, 192701 (2012), PRC 88, 064612 (2013)

ADWA

10Be(d,p)11Be @ 12-21 MeV

K. Jones et al, Nature 465 (2010) 454, PRC 84, 034601 (2011)

d(132Sn,133Sn)p@5 MeV/u

Halo nuclei

Neutron rich doubly magic nuclei

Reaction theory: from many body to few body

q  isolating the important degrees of freedom in a reaction q  connecting back to the many-body problem

Ø  effective nucleon-nucleus interactions

Solving the 3-body problem for (d,p) reactions

Ø  FAGS: Faddeev (exact treatment) Ø  ADWA: adiabatic wave approximation

(like CDCC but introducing the adiabatic approx. in Ex) Ø  CDCC: continuum discretized coupled channel

(elastic and breakup fully coupled)

Upadhyay, Deltuva and Nunes, PRC 85, 054621 (2012)

Reaction theory: from many body to few body

q  isolating the important degrees of freedom in a reaction q  connecting back to the many-body problem

Ø  effective nucleon-nucleus interactions (non-local and energy dependent)

Challenges with ab-initio optical potentials

•  Reactions are extremely sensitive to thresholds •  Reactions require basis that describe continuum correctly •  Microscopically derived optical potentials are non-local U(R,R’)

Rotureau, Hagen, Nunes, Danielewicz

Non-local potential?

•  Phenomenological optical potentials are usually made local U(R) •  Microscopically derived optical potentials are non-local U(R,R’)

•  Does non-locality make a difference in the reaction? •  Can we constrain non-locality with reactions?

Non-local Perey and Buck potential in (d,p)

Solve the single channel scattering problem with non-local optical potential Solve the single channel bound state problem with non-local mean field Construct the (p,d) amplitude within DWBA

Perey and Buck type non-locality

2

rection Factor (PCF).Recently, Timofeyuk and Johnson [26, 27] studied the

e↵ects of including an energy-independent non-local po-tential in (d, p) reactions within the Adiabatic DistortedWave Approximation (ADWA) [28]. Non-locality was in-cluded approximately through expansions to constructa local equivalent potential and solving the correspond-ing local Schrodinger’s equation. They found that aPerey-Buck type non-locality can be e↵ectively includedin (d, p) through a very significant energy shift in theevaluation of the local optical potentials to be used inconstructing the deuteron distorted waves. This can im-pact cross sections dramatically, and calls for further in-vestigations.

In this work, we determine the importance of non-locale↵ects in the various components of a nuclear reactionprocess, and assess the validity of the PCF by studying awide range of reactions, including neutron states boundto 16O, 40Ca, 48Ca, 126Sn, 132Sn, and 208Pb, and (p, p)and (p, d) on 17O, 41Ca, 49Ca, 127Sn, 133Sn, and 209Pbat 20 and 50 MeV.

The paper is organized in the following way. In Sec. IIwe briefly describe the necessary theory. Numerical de-tails can be found in Sec. III. The results are presentedin Sec. IV, starting with a discussion of local equivalentpotentials in Sec. IVA and of approximate local equiv-alent potentials in Sec. IVB. We consider the e↵ects ofnon-localities on scattering wave functions and ways tocorrect for non-localities in Sec. IVC. The e↵ects of non-localities on bound state wave functions are presented inSec. IVD. We then explore the e↵ects of non-localities ontransfer cross sections in Sec. IVE. We discuss the con-nection of this work with other relevant studies in Sec. V.Finally, in Sec. VI, conclusions are drawn.

II. THEORETICAL CONSIDERATIONS

Let us consider a nucleon scattering o↵ a compositenucleus. The e↵ective interaction between the nucleonand the nucleus is a non-local optical potential. In thiscase, the two-body Schrodinger equation takes the form

~22µ

r2 (r)+E (r) = Uo(r) (r)+

ZUNL(r, r0) (r0)dr0

(1)where µ is the reduced mass of the nucleon-nucleus sys-tem, E is the energy in the center of mass, Uo(r) is thelocal part of the potential, and (r) is the scatteringwave function. A particular form of the non-local poten-tial introduced by Frahn and Lemmer [21] is

UNL(r, r0) = UNLWS

r+ r0

2

exp

rr0

2

3/23, (2)

where, is the range of the non-locality, and typically

takes on a value of 0.85 fm. In this work, UNLWS is of a

Woods-Saxon form of the variable 12 |r+ r0|.

This type of potential was further investigated byPerey and Buck [4]. Making the approximation |r+r0| (r + r0) in UNL

WS allows for an analytic partial wave de-composition resulting in the partial wave equation

~22µ

d2

dr2 `(`+ 1)

r2

NL` (r) + E NL

` (r)

= Uo(r) NL` (r) +

Zg`(r, r

0) NL` (r0)dr0.(3)

Here, the kernel is explicitly given by:

g`(r, r0) =

2i`z

12

j`(iz) exp

r2 + r02

2

UNLWS

1

2(r + r0)

,

(4)where j` are spherical Bessel functions, and z = 2rr0/2.In our study, we assume the spin-orbit and Coulomb po-tentials are local, and therefore, Uo(r) = Vso(r)+Vcoul(r).For a non-local potential of the Perey-Buck type, the

depths of an approximate local equivalent potential canbe found from the relations [4]

V NLv = V Loc

v exp

µ2

2~2E Vc + V Loc

v

WNLd = WLoc

d exp

µ2

2~2E Vc + V Loc

v

. (5)

Here, Vv and Wd are the depths of the real volume andimaginary surface terms in the Woods-Saxon potential,respectively, and are positive constants. E is the centerof mass energy, and Vc is the Coulomb potential at theorigin for a solid uniformly charged sphere with radiusRc = rcA

1/3. Notice that even though the non-localpotential is energy-independent, the transformed localdepths are energy-dependent, which is a common featureof local global optical potentials.Through use of Eq.(5) and fits to neutron elastic scat-

tering data on 208Pb at low energies, the Perey-Bucknon-local potential was determined: the correspondingparameters are given in the first column of Table I. Theparameters in the Perey-Buck potential are both energyand mass-independent.For a given non-local potential, a local equivalent po-

tential can often be found. However, in the nuclear inte-rior, the wave function resulting from using a non-localpotential is reduced compared to the wave function re-sulting from using a local equivalent potential. This phe-nomenon is known as the Perey e↵ect [23]. Correctingfor the reduced amplitude is done via the PCF:

F (r) =

1 µ2

2~2ULE(r) Uo(r)

1/2

. (6)

2







~22µ

r2 (r)+E (r) = Uo(r) (r)+

ZUNL(r, r0) (r0)dr0


UNL(r, r0) = UNLWS

r+ r0

2

exp

rr0

2

3/23, (2)






~22µ

d2

dr2 `(`+ 1)

r2

NL` (r) + E NL

` (r)

= Uo(r) NL` (r) +

Zg`(r, r

0) NL` (r0)dr0.(3)


g`(r, r0) =

2i`z

12

j`(iz) exp

r2 + r02

2

UNLWS

1

2(r + r0)

,



V NLv = V Loc

v exp

µ2

2~2E Vc + V Loc

v

WNLd = WLoc

d exp

µ2

2~2E Vc + V Loc

v

. (5)





F (r) =

1 µ2

2~2ULE(r) Uo(r)

1/2

. (6)Perey correction factor: if the local momentum approximation is valid

3

Note that here, ULE(r) is the local equivalent potential.As we required the spin-orbit and Coulomb terms to beidentical in the local and non-local potentials, these termsin ULE exactly cancel Uo. Since F (r) ! 1 as r ! 1, thecorrection factor Eq.(6) only a↵ects the magnitude of thewave function within the range of the nuclear interaction.A derivation of Eq.(5) and Eq.(6) is given in AppendixA.

In the asymptotic limit, the wave function takes theform

asym` (r) =

i

2

H

` (, kr) S`jH+(, kr)

, (7)

where = Z1Z2e2µ/~2k is the Sommerfeld parameter, k

is the wave number, S`j is the scattering matrix element,and H and H+ are incoming and outgoing sphericalHankel functions, respectively. For neutrons, = 0.

In Sec. IVE, we use the Distorted Wave Born Ap-proximation (DWBA) to calculate the T-matrix for theB(p, d)A reaction, which, neglecting the remnant term,is written as

Tp,d = h ()dA d|Vnp| pBnAi , (8)

where ()dA is the deuteron scattering wave function, d

is the deuteron bound state, Vnp is the Reid soft corenp interaction [29], pB is the proton distorted wave,and nA is the neutron bound state wave function. (fordetails on the formalism, please check [30]).

Due to its simplicity, a common technique is to doa calculation with a suitable local equivalent potential,then introduce the non-locality by modifying the wavefunction with the PCF

PCF` (r) = F (r) Loc

` . (9)

This is precisely the approach we want to test in thisstudy.

III. NUMERICAL DETAILS

In this systematic study, we consider elastic scattering(p, p) on 17O, 41Ca, 49Ca, 127Sn, 133Sn, and 209Pb at20 and 50 MeV and the wave functions for a neutronbound to 16O, 40Ca, 48Ca, 126Sn, 132Sn, and 208Pb. Inboth cases, the full non-local equation is solved usingthe Perey-Buck potential, with the method described inAppendix B.

For the scattering process, a local equivalent potentialis determined by fitting the elastic scattering generatedfrom the non-local equation. This was done using thecode sfresco [31]. Using the local equivalent potential,the local scattering equation is solved to obtain Loc and,finally, the PCF is applied to the wave function Eq.(9).

The corrected wave function, PCF , is then compared tothe solution of the full non-local equation, NL.A similar procedure is followed for the bound states.

The full non-local equation is solved using a real Woods-Saxon form with a radius parameter of r = 1.25 fm,which is used to find the radius of the nucleus under con-sideration through the formula R = rA1/3. The di↵use-ness is set to a = 0.65 fm, and the non-locality parameteris fixed at = 0.85 fm. The depth is then adjusted to re-produce the physical binding energy of the system. Thelocal equation is solved with the local depth V Loc

ws nec-essary to reproduce the binding energy. We then applythe PCF to the resulting wave function, and renormal-ize to unity, to obtained the corrected bound state. Thecorrected wave function, PCF , is then compared to thesolution of the full non-local equation, NL.The bound and scattering states resulting from either

non-local or local potentials are then introduced into theDWBA T-matrix for (p, d), Eq.(8), for describing the pro-cess at 20 and 50 MeV. Angular distributions are calcu-lated using the code fresco [31]. Non-locality was onlyadded in the entrance channel, namely through the pro-ton distorted wave and the neutron bound state. The lo-cal global parameterization of Daehnick et al. was used toobtain dA in the exit channel. In principle, the deuteronoptical potential is also non-local due to breakup e↵ectsand the non-locality of the nucleon-nucleus optical poten-tial. The non-locality of the deuteron optical potentialwill be addressed in a future study. The scattering wavefunctions were solved by using a 0.05 fm radial step sizewith a matching radius of 40 fm. For the bound statesolutions, we used a radial step size of 0.02 fm. Thematching radius was half the radius of the nucleus underconsideration, and the maximum radius was 30 fm, ex-cept for a very low binding energy study, when a largervalue was necessary. The cross sections contain contri-butions of partial waves up to J = 30.

In the following subsection, we present the results andanalyze the e↵ect of non-locality and the approximatecorrection factor in detail.

IV. RESULTS

A. Local Equivalent Potentials

As described before, in order to study the correctionfactor, a local equivalent potential (LEP) needs to befound. A local potential is equivalent to a given non-localpotential if it produces the same S-matrix elements, thus,producing the same elastic scattering angular distribu-tion. A LEP is found by 2 minimization starting fromthe transformed local potential obtained by using Eq.(5).We required that the spin-orbit and Coulomb terms ofthe Perey-Buck non-local potential and the LEP be ex-actly the same, thus only the real volume and imaginarysurface terms were allowed to vary in the fit to find theLEP (a total of 6 parameters). For most cases we were

2







~22µ

r2 (r)+E (r) = Uo(r) (r)+

ZUNL(r, r0) (r0)dr0


UNL(r, r0) = UNLWS

r+ r0

2

exp

rr0

2

3/23, (2)






~22µ

d2

dr2 `(`+ 1)

r2

NL` (r) + E NL

` (r)

= Uo(r) NL` (r) +

Zg`(r, r

0) NL` (r0)dr0.(3)


g`(r, r0) =

2i`z

12

j`(iz) exp

r2 + r02

2

UNLWS

1

2(r + r0)

,



V NLv = V Loc

v exp

µ2

2~2E Vc + V Loc

v

WNLd = WLoc

d exp

µ2

2~2E Vc + V Loc

v

. (5)





F (r) =

1 µ2

2~2ULE(r) Uo(r)

1/2

. (6)

F. Perey and B. Buck, Nucl. Phys. 32, 353 (1962).

N. Austern, Phys. Rev. 137, 752 (1965)

Effect of non-locality on wavefunctions

Scattering wfn for PB: 49Ca+p at 50 MeV

Bound state wfn for PB: 48Ca+n 2p3/2

Non-local Perey and Buck potential: effect on (p,d)

6

non-locality in both the proton distorted wave and theneutron bound state, the dashed line corresponds to thedistribution obtained when only local equivalent interac-tions are used, and the crosses correspond to the crosssections obtained when the proton scattering state andthe neutron bound state are both corrected by the PCF.While the Perey correction improves upon the distribu-tion involving local interactions only, it is still unableto fully capture the complex e↵ect of non-locality. Theprominent changes at zero degrees was unique to thiscase, but the very significant changes between the uncor-rected local and the fully non-local around the main peakwas seen for most distributions studied.

We also show the separate e↵ect of including only non-locality in the proton scattering state (dotted) and theneutron bound state (dot-dashed). For this case, the non-locality in the proton distorted wave acts in a similar wayto the non-locality in the bound state, namely it increasesthe cross section at zero degrees and reduces the crosssection around 15.

0 5 10 15 20 25 30 35 40 45 50θc.m. (deg)

0.00.51.01.52.02.53.03.54.04.55.0

dθ/dΩ

[m

b/sr

]

Non-LocalCorrected LocalUncorrected LocalNon-Local Scattering StateNon-Local Bound State

FIG. 5: Angular distributions for 49Ca(p, d)48Ca at 50.0 MeV:inclusion of non-locality in both the proton distorted wave andthe neutron bound state (solid line), using LEP, then apply-ing the correction factor to both the scattering and boundstates (crosses), using the LEP without applying any correc-tions (dashed line); including non-locality only to the protondistorted wave (dotted line), and including non-locality in theneutron bound state only (dot-dashed line).

The reason for the large changes at small angles can beseen from an analysis of the scattering and bound wavefunctions of Figs. 2, 3, and 4. The existence of a node inthe bound state wave function influences the cross sectionin a complex manner. The radius that corresponds to thesurface for 49Ca occurs at a radius slightly larger thanthat where the bound state wave function is zero. Thebound wave function has a large slope in this region, sothe percent di↵erence between the non-local and localwave functions can be quite large in this region. For thiscase, the non-local bound wave function is smaller than

the local wave functions in this region, reducing the crosssection at the peak. On the other hand, the magnitudeof the bound wave function is larger for the non-localcase in the tail region, which enhances the cross sectionat forward angles.For the scattering wave functions, the largest di↵er-

ences were for partial waves that corresponded to thesurface. Also, the asymptotics of scattering partial waveswere di↵erent due to small di↵erences in the S-Matrix,mostly for surface partial waves. The larger the ampli-tude in the asymptotic region, the larger the cross sec-tion at forward angles. There is an interplay betweenthe real and imaginary parts of the scattering wave func-tion which influences the cross section at forward angles.In a very complex manner, the combination of all thesee↵ects produces the interesting behavior of the transfercross section at forward angles, and the changes in themagnitude of the cross section at the peak for this par-ticular reaction.In order to better understand this case, we artificially

modified the bound wave function. By changing the bind-ing energy we altered the Q-value of the reaction. Dif-ferent Q-values produced very di↵erent types of distri-butions, both in shape and in magnitude. Nevertheless,similar dramatic changes in the cross section due to non-locality were found. For very low binding energy, the nor-malization of the bound wave function was dominated bythe asymptotics, so the PCF did very little. The node inthe wave function altered the cross section in a very com-plex way. The PCF was not able to correct the boundwave function in the region around the node since thewave function and the PCF have a very large slope inthis region, so inadequacies of the PCF were amplified.Consider now the same target but lower energy. In

Fig.6 we present the transfer angular distribution for49Ca(p, d)48Ca at 20.0 MeV. Non-locality is seen to havea large e↵ect at small angles. Including non-locality inonly the bound state increases the cross section at for-ward angles, which is to be expected from Fig.4, whereit is seen that the magnitude of NL

nA is larger than PCFnA

and LocnA in the asymptotic region. Non-locality in only

the scattering state decreases the cross section, but onlyby a small amount. The net e↵ect of non-locality is anoverall increase in the cross section of 17.3% relative tothe cross section obtained with local interactions only.While the correction factor moves the transfer distribu-tion in the right direction, it falls short by 5.2%.Next we consider some heavier targets, 133Sn and

209Pb, and study (p, d) at 20 MeV. In both cases, theinclusion of non-locality in the scattering state decreasesthe cross section by a small amount. This is due to thelow energy of the proton, and the high charge of the tar-get; the details of the scattering wave function withinthe nuclear interior are not significant for the transfersince these details are suppressed by the Coulomb bar-rier. Non-locality in the bound state is very significant,and increases the cross section by a large amount in bothcases. In 133Sn, the correction factor does a fair job tak-

7

0 5 10 15 20 25 30 35 40 45 50θc.m. (deg)

05

10152025303540455055

dθ/dΩ

[mb/

sr]


FIG. 6: Angular distributions for 49Ca(p, d)48Ca at 20.0 MeV(descriptions of each line is given in the caption of Fig.5).

ing non-locality into account, but there is still a notice-able discrepancy between the full non-local and correctedlocal results. In 209Pb, there are discrepancies at forwardangles, but coincidentally the distributions resulting fromthe non-local potential and the local potential with thePCF agree quite well at the major peak of the distribu-tion. This agreement is accidental and comes from thenon-local e↵ect in the bound state canceling that in thescattering state.

0 5 10 15 20 25 30 35 40 45 50θc.m. (deg)

0

2

4

6

8

10

12

14

dθ/dΩ

[mb/

sr]


FIG. 7: Angular distributions for 133Sn(p, d)132Sn at 20.0MeV (descriptions of each line is given in the caption of Fig.5).

The percent di↵erences at the first peak of the transferdistributions for all the cases that were studied are sum-marized in Table II and III for the (p, d) reactions at 20and 50 MeV.

It is seen that for both energies and for nearly all cases,the inclusion of non-locality in the entrance channel can

0 10 20 30 40 50 60 70 80 90θc.m. (deg)

0.0

0.4

0.8

1.2

1.6

2.0

dθ/dΩ

[mb/

sr]


FIG. 8: Angular distributions for 209Pb(p, d)208Pb at 20.0MeV (descriptions of each line is given in the caption of Fig.5).

Corrected Non-Local

Elab

= 20 MeV Relative to Local Relative to Local17O(1d5/2)(p, d) 7.1% 18.8%17O(2s1/2)(p, d) 20.1% 26.5%

41Ca(p, d) 11.4% 21.9%49Ca(p, d) 10.4% 17.3%127Sn(p, d) 17.5% 17.3%133Sn(p, d) 18.2% 24.4%209Pb(p, d) 19.4% 20.8%

TABLE II: Percent di↵erence of the (p, d) transfer cross sec-tions at the first peak when using Eq.(6) (2nd column), or anon-local potential (3rd column), relative to the local calcu-lation with the LEP, for a number of reactions occurring at20 MeV.

have a very significant e↵ect on the transfer cross section,often times introducing di↵erences of 15 35%. Most ofthe time, adding non-locality increases the cross sectionat the first peak. In general, the correction factor movesthe distribution obtained with local interactions in thedirection of the distribution including the non-localinteractions. In the case of 127Sn(p, d) at 50 MeV, thecorrection factor overshoots at the first peak, but theoverall shape of the corrected distribution is in betteragreement with the exact result.

V. DISCUSSION

It should be noted that the PCF is only valid for non-local potentials of the Perey-Buck form. However, thereis no reason to expect that the full non-locality in theoptical potential will look anything like the Perey-Buckform. On physical grounds, the optical potential must

Titus and Nunes, PRC 89, 034609 (2014)

Non-local Perey and Buck potential: effect in (p,d)

8

Corrected Non-Local

Elab



TABLE III: Percent di↵erence of the (p, d) transfer cross sec-tions at the first peak when using Eq.(6) (2nd column), or anon-local potential (3rd column), relative to the local calcu-lation with the LEP, for a number of reactions occurring at50 MeV.

be energy dependent due to non-localities arising fromchannel couplings. While the specific form chosen forthe Perey-Buck potential is convenient for numerical cal-culations, a single Gaussian term mocking up all energy-independent non-local e↵ects is likely to be an oversim-plification.

In an earlier study, Rawitscher et al. [8] calculated theexchange non-locality in n16O scattering and examinedthe PCF. The wave functions obtained from their micro-scopically derived exchange non-locality were reasonablycorrected by the PCF. The exchange non-locality is basedon anti-symmetrized wave functions, which will naturallyreduce the amplitude of the wave function in the nuclearinterior due to the Pauli exclusion principle, similarly tothe PCF. Results in [8] show that the PCF is able toapproximately take into account the e↵ects of includingexchange. However, data suggests [33] that exchange isnot sucient and that channel coupling is also needed.

In another study by Rawitscher [17], the micro-scopic Feshbach optical potential from channel couplingis examined. The resulting potentials were strongly`dependent, had emissive (positive imaginary) parts,and the non-local part did not resemble a Gaussianshape. The PCF obtained from the Wronskian was alsostrongly angular momentum dependent, and was foundto be larger than unity in some cases. The channel cou-pling non-locality is therefore very di↵erent than the ex-change non-locality, and one should not expect it to becorrected for in the same way. In those studies [8, 17], theexchange and channel coupling non-localities were ana-lyzed separately. In [7], the e↵ect of channel coupling inlow energy scattering is studied, including a repulsive po-tential to account for the Pauli Principle. The resultingnon-local potentials were found to be very di↵erent fromthe Perey-Buck form.

Our results, together with [7, 8, 17], emphasize theneed for non-locality to be treated explicitly, contrary towhat has been preferred for more than 50 years. Sincewe have not yet found a good way to pin down non-locality phenomenologically, it would be extremely help-ful to have microscopically derived optical potentials to

guide further work. Microscopic nA optical potentialsbased on the nucleon-nucleon interaction are particularlyattractive because they immediately connect the intrinsicstructure of the target to the reaction.

VI. CONCLUSIONS

The long established Perey correction factor (PCF)was studied. To do so, the integro-di↵erential equationcontaining the Perey-Buck non-local potential was solvednumerically for single channel scattering and boundstates. A local equivalent potential was obtained byfitting the elastic distribution generated by the Perey-Buck potential to a local potential. Both the local andnon-local binding potentials reproduced the experimentalbinding energies. The scattering and bound state wavefunctions were used in a finite range DWBA calculationin order to calculate (p, d) transfer cross sections. ThePCF was applied to the wave functions generated withthe local equivalent potentials.For the (p, d) transfer reactions, we found that the ex-

plicit inclusion of non-locality to the entrance channelincreased the transfer distribution at the first peak by15 35%. The transfer distribution from using a non-local potential increased relative to the distribution fromthe local potential in most cases. In all cases, the PCFmoved the transfer distribution in the direction of thedistribution which included non-locality explicitly. How-ever, non-locality was never fully taken into account withthe PCF.

ACKNOWLEDGEMENT

We are grateful to Je↵ Tostevin for countless discus-sions and invaluable advice. We would also like to thankNicolas Michel and Ron Johnson for many useful sug-gestions. This work was supported by the National Sci-ence Foundation under Grant No. PHY-0800026 and theDepartment of Energy under Contracts No. DE-FG52-08NA28552 and No. DE-SC0004087.

APPENDIX A: DERIVING THE PEREYCORRECTION FACTOR

Here we provide details on the derivation of the PCF,Eq.(6). We also include the derivation of the transforma-tion formulas Eq.(5), as well as the correct radial versionof the transformation formulas which could be used totransform the non-local radius and di↵useness to theirlocal counterpart.We start from Eq.(1). Let us define a function F (r)

that connects the local wave function Loc(r), resultingfrom the potential ULE(r), with the wave function re-sulting from a non-local potential, NL(r)

NL(r) F (r) Loc(r). (A1)

7

0 5 10 15 20 25 30 35 40 45 50θc.m. (deg)

05

10152025303540455055

dθ/dΩ

[mb/

sr]


FIG. 6: Angular distributions for 49Ca(p, d)48Ca at 20.0 MeV(descriptions of each line is given in the caption of Fig.5).

ing non-locality into account, but there is still a notice-able discrepancy between the full non-local and correctedlocal results. In 209Pb, there are discrepancies at forwardangles, but coincidentally the distributions resulting fromthe non-local potential and the local potential with thePCF agree quite well at the major peak of the distribu-tion. This agreement is accidental and comes from thenon-local e↵ect in the bound state canceling that in thescattering state.

0 5 10 15 20 25 30 35 40 45 50θc.m. (deg)

0

2

4

6

8

10

12

14

dθ/dΩ

[mb/

sr]


FIG. 7: Angular distributions for 133Sn(p, d)132Sn at 20.0MeV (descriptions of each line is given in the caption of Fig.5).

The percent di↵erences at the first peak of the transferdistributions for all the cases that were studied are sum-marized in Table II and III for the (p, d) reactions at 20and 50 MeV.

It is seen that for both energies and for nearly all cases,the inclusion of non-locality in the entrance channel can

0 10 20 30 40 50 60 70 80 90θc.m. (deg)

0.0

0.4

0.8

1.2

1.6

2.0

dθ/dΩ

[mb/

sr]


FIG. 8: Angular distributions for 209Pb(p, d)208Pb at 20.0MeV (descriptions of each line is given in the caption of Fig.5).

Corrected Non-Local

Elab



TABLE II: Percent di↵erence of the (p, d) transfer cross sec-tions at the first peak when using Eq.(6) (2nd column), or anon-local potential (3rd column), relative to the local calcu-lation with the LEP, for a number of reactions occurring at20 MeV.

have a very significant e↵ect on the transfer cross section,often times introducing di↵erences of 15 35%. Most ofthe time, adding non-locality increases the cross sectionat the first peak. In general, the correction factor movesthe distribution obtained with local interactions in thedirection of the distribution including the non-localinteractions. In the case of 127Sn(p, d) at 50 MeV, thecorrection factor overshoots at the first peak, but theoverall shape of the corrected distribution is in betteragreement with the exact result.

V. DISCUSSION

It should be noted that the PCF is only valid for non-local potentials of the Perey-Buck form. However, thereis no reason to expect that the full non-locality in theoptical potential will look anything like the Perey-Buckform. On physical grounds, the optical potential must

Titus and Nunes, PRC 89, 034609 (2014)

Dispersive Optical Potential (DOM)

numerical values of all parameters together with a list of allemployed equations.Included in the present fit are the same elastic scattering

data and level information considered in Ref. [15]. Inaddition, we now include the charge density of 40Ca asgiven in Ref. [22] by a sum of Gaussians in the fit. Data fromthe (e, e0p) reaction at high missing energy and momentumobtained at Jefferson Lab for 12C [23], 27Al, 56Fe, and 197Au[24] were incorporated as well. We note that the spectralfunction of high-momentum protons per proton number isessentially identical for 27Al and 56Fe, thereby providing asensible benchmark for their presence in 40Ca. We merelyaimed for a reasonable representation of these cross sectionssince their interpretation requires further consideration ofrescattering contributions [25]. We did not include the resultsof the analysis of the (e, e0p) reaction from NIKHEF [26]because the extracted spectroscopic factors depend on theemployed local optical potentials. We plan to reanalyze thesedata with our nonlocal potentials in a future study.Motivated by the work of Refs. [18,19], we allow for

different nonlocalities above and below the Fermi energy,otherwise the symmetry around this energy is essentiallymaintained by the fit. The values of the nonlocality param-eters β appear reasonable and range from 0.64 fm above to0.81 fm below the Fermi energy for volume absorption.These parameters are critical in ensuring that particle numberis adequately described. We limit contributions to l ≤ 5below εF [19] obtaining 19.88 protons and 19.79 neutrons.We note the extended energy domain for volume absorptionbelow εF to accommodate the Jefferson Lab data. Surfaceabsorption requires nonlocalities of 0.94 fm above and2.07 fm below εF.The final fit to the experimental elastic scattering data is

shown in Fig. 1 while the fits to total and reaction crosssections are shown in Fig. 2. In all cases, the quality of thefit is the same as in Refs. [14] or [15]. This statement alsoholds for the analyzing powers.Having established our description at positive energies is

equivalent to our earlier work, but now consistent withtheoretical expectations associated with the nonlocal con-tent of the nucleon self-energy, we turn our attention to thenew results below the Fermi energy. In Fig. 3 we display thespectral strength given in Eq. (2) as a function of energy forthe first few levels in the independent-particle model. Thedownward arrows identify the experimental location of thelevels near the Fermi energy while for deeply bound levelsthey correspond to the peaks obtained from (p, 2p) [27]and (e, e0p) reactions [28]. The DOM strength distributionstrack the experimental results represented by their peaklocation and width. Neutron single-particle energies arelisted in Table I for levels near εF. The calculated levelsexhibit a deviation of about 1 MeV from the experimentalvalues similar to Ref. [15], except for the 1s1=2.For the quasihole proton states we find spectroscopic

factors of 0.78 for the 1s1=2 and 0.76 for the 0d3=2 level.

The location of the former deviates slightly from theexperimental peak as for neutrons which may requireadditional state dependence of the self-energy as expressedby poles nearby in energy [29]. The analysis of the (e, e0p)reaction in Ref. [30] clarified that the treatment of non-locality in the relativistic approach leads to differentdistorted proton waves as compared to conventional non-relativistic optical potentials, yielding about 10%–15%larger spectroscopic factors. Our current results are also

0 50 100 150

[mb/

sr]

Ω/dσd

510

1210

1910

2610

3310Ca40n+

< 10lab 0 < E < 20lab10 < E < 40lab20 < E < 100lab40 < E

> 100labE

0 50 100 150

Ca40p+

[deg]cmθ

FIG. 1 (color online). Calculated and experimental elastic-scattering angular distributions of the differential cross sectiondσ=dΩ. Panels shows results for nþ 40Ca and pþ 40Ca. Data foreach energy are offset for clarity with the lowest energy at thebottom and highest at the top of each frame. References to thedata are given in Ref. [15].

[MeV]LabE0 50 100 150 200

[mb]

σ

0

1000

2000

3000

4000

Ca40n+

totσreactσ

[mb]

σ

0

500

1000

1500

Ca40p+

reactσ

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.

PRL 112, 162503 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

25 APRIL 2014

162503-3

numerical values of all parameters together with a list of allemployed equations.Included in the present fit are the same elastic scattering

data and level information considered in Ref. [15]. Inaddition, we now include the charge density of 40Ca asgiven in Ref. [22] by a sum of Gaussians in the fit. Data fromthe (e, e0p) reaction at high missing energy and momentumobtained at Jefferson Lab for 12C [23], 27Al, 56Fe, and 197Au[24] were incorporated as well. We note that the spectralfunction of high-momentum protons per proton number isessentially identical for 27Al and 56Fe, thereby providing asensible benchmark for their presence in 40Ca. We merelyaimed for a reasonable representation of these cross sectionssince their interpretation requires further consideration ofrescattering contributions [25]. We did not include the resultsof the analysis of the (e, e0p) reaction from NIKHEF [26]because the extracted spectroscopic factors depend on theemployed local optical potentials. We plan to reanalyze thesedata with our nonlocal potentials in a future study.Motivated by the work of Refs. [18,19], we allow for

different nonlocalities above and below the Fermi energy,otherwise the symmetry around this energy is essentiallymaintained by the fit. The values of the nonlocality param-eters β appear reasonable and range from 0.64 fm above to0.81 fm below the Fermi energy for volume absorption.These parameters are critical in ensuring that particle numberis adequately described. We limit contributions to l ≤ 5below εF [19] obtaining 19.88 protons and 19.79 neutrons.We note the extended energy domain for volume absorptionbelow εF to accommodate the Jefferson Lab data. Surfaceabsorption requires nonlocalities of 0.94 fm above and2.07 fm below εF.The final fit to the experimental elastic scattering data is

shown in Fig. 1 while the fits to total and reaction crosssections are shown in Fig. 2. In all cases, the quality of thefit is the same as in Refs. [14] or [15]. This statement alsoholds for the analyzing powers.Having established our description at positive energies is

equivalent to our earlier work, but now consistent withtheoretical expectations associated with the nonlocal con-tent of the nucleon self-energy, we turn our attention to thenew results below the Fermi energy. In Fig. 3 we display thespectral strength given in Eq. (2) as a function of energy forthe first few levels in the independent-particle model. Thedownward arrows identify the experimental location of thelevels near the Fermi energy while for deeply bound levelsthey correspond to the peaks obtained from (p, 2p) [27]and (e, e0p) reactions [28]. The DOM strength distributionstrack the experimental results represented by their peaklocation and width. Neutron single-particle energies arelisted in Table I for levels near εF. The calculated levelsexhibit a deviation of about 1 MeV from the experimentalvalues similar to Ref. [15], except for the 1s1=2.For the quasihole proton states we find spectroscopic

factors of 0.78 for the 1s1=2 and 0.76 for the 0d3=2 level.

The location of the former deviates slightly from theexperimental peak as for neutrons which may requireadditional state dependence of the self-energy as expressedby poles nearby in energy [29]. The analysis of the (e, e0p)reaction in Ref. [30] clarified that the treatment of non-locality in the relativistic approach leads to differentdistorted proton waves as compared to conventional non-relativistic optical potentials, yielding about 10%–15%larger spectroscopic factors. Our current results are also

0 50 100 150

[mb/

sr]

Ω/dσd

510

1210

1910

2610

3310Ca40n+

< 10lab 0 < E < 20lab10 < E < 40lab20 < E < 100lab40 < E

> 100labE

0 50 100 150

Ca40p+

[deg]cmθ

FIG. 1 (color online). Calculated and experimental elastic-scattering angular distributions of the differential cross sectiondσ=dΩ. Panels shows results for nþ 40Ca and pþ 40Ca. Data foreach energy are offset for clarity with the lowest energy at thebottom and highest at the top of each frame. References to thedata are given in Ref. [15].

[MeV]LabE0 50 100 150 200

[mb]

σ

0

1000

2000

3000

4000

Ca40n+

totσreactσ

[mb]

σ

0

500

1000

1500

Ca40p+

reactσ

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.


25 APRIL 2014

162503-3

larger by about 10%–15% than the numbers extracted inRef. [26]. Introducing local DOM potentials in the analysisof transfer reactions has salutary effects for the extraction ofspectroscopic information of neutrons [31] and nonlocalpotentials should further improve such analyses.In Fig. 4 we compare the experimental charge density of

40Ca (thick line representing a 1% error) with the DOM fit.While some details could be further improved, it is clearthat an excellent description of the charge density ispossible in the DOM. The correct particle number isessential for this result, which in turn can only be achievedby including nonlocal absorptive potentials that are alsoconstrained by the high-momentum spectral functions.With a local absorption we are not capable to eithergenerate a particle number close to 20 or describe thecharge density accurately [8].We compare in Fig. 5 the results for the high-momentum

removal spectral strength with the Jefferson Lab data [24].We note that the high-energy data correspond to intrinsic

nucleon excitations and cannot be part of the presentanalysis. To further improve the description, one wouldhave to introduce an energy dependence of the radial formfactors for the potentials. Nevertheless we conclude that anadequate description is generated which corresponds to10.6% of the protons having momenta above 1.4 fm−1.Employing the energy sum rule [9] in the form given inRef. [32] yields a binding energy of 7.91 MeV per nucleonmuch closer to the experimental 8.55 MeV than the4.71 MeV found in Ref. [8]. The constrained presenceof the high-momentum nucleons is responsible for thischange [33]. The 7.91 MeV binding per nucleon obtainedhere represents the contribution to the ground-state energyfrom two-body interactions including a kinetic energy of22.64 MeV per nucleon and was not part of the fit. Thisempirical approach therefore leaves about 0.64 MeV pernucleon attraction for higher-body interactions about1 MeV less than the Green’s function Monte Carlo resultsof Ref. [34] for light nuclei.

-140 -120 -100 -80 -60 -40 -20 0

-310

-210

-110

1

1/2s

-140 -120 -100 -80 -60 -40 -20 0

1/2p

-140 -120 -100 -80 -60 -40 -20 0

]-1

S(E

) [M

eV

-310

-210

-110

1

3/2d

-140 -120 -100 -80 -60 -40 -20 0

3/2p

E [MeV]-140 -120 -100 -80 -60 -40 -20 0

-310

-210

-110

1

5/2d

E [MeV]-140 -120 -100 -80 -60 -40 -20 0

7/2f

FIG. 3 (color online). Spectral strength for protons in the ljorbits which are fully occupied in the independent-particle modelas well as the f7=2 strength associated with the first empty orbit inthis description. The arrows indicate the experimental location ofthe valence states as well as the peak energies for the distributionsof deeply bound ones.

r [fm]0 2 4 6 8

]-3

[e.fm

ρ

0

0.02

0.04

0.06

0.08

0.1

FIG. 4 (color online). Comparison of experimental chargedensity [22] (thick line) with the DOM fit (thin line).

[MeV]mE

0 100 200 300 400 500

]-1

sr

-4) [

MeV

m,p

mS

(E

-1410

-1310

-1210

-1110

-1010

= 250 [MeV/c]m

p= 330 [MeV/c]

mp

= 410 [MeV/c]mp= 490 [MeV/c]mp= 570 [MeV/c]mp= 650 [MeV/c]mp

FIG. 5 (color online). Spectral strength as a function of missingenergy for different missing momenta as indicated in the figure. Thedata are the average of the 27Al and 56Fe measurements from [24].

TABLE I. Quasihole energies in MeV for neutron orbits in 40Canear the Fermi energy compared with experiment.

Orbit DOM Experiment

1p1=2 −3.47 −4.201p3=2 −4.51 −5.860f7=2 −7.36 −8.360d3=2 −16.2 −15.61s1=2 −15.3 −18.3


25 APRIL 2014

162503-4Mahzoon et al, Phys Rev Lett 112, 162503 (2014)

Non-local DOM potential: effect on (p,d)

Ross, Titus, Nunes, Mahzoon, Dickhoff, Charity, submitted to PRC

40Ca(p,d)39Ca @ 50 MeV

Non-local DOM potential: effect on (p,d)


DOM

Perey and Buck

40Ca(p,d)39Ca @ 50 MeV

error quantification in reaction theory

•  finite range adiabatic methods are used to obtained spectroscopic factors

•  Faddeev calculations are used to determined error in reaction theory

[FN, Deltuva, Hong, PRC 2011]

error quantification in reaction theory

•  Determine best fit from minimizing Chi2 •  Assume Gaussian distribution around this minimum •  Draw 200 parameters sets from this distribution •  For each angle, remove 5 highest and lowest to define 95% confidence bands

[Lovell, FN, Sarich, Wild]

48Ca(d,d)48Ca at 23.2 48Ca(d,p)49Ca at 23.2 MeV

Summary and conclusions

•  Optical potentials are being developed from ab-initio coupled cluster theory – they are inherently non-local

•  Reaction codes are being upgraded to include non-locality •  Within DWBA, we showed non-local effects in (p,d) transfer

reactions are very important and need to be included explicitly •  Inclusion of non-locality in adiabatic theories is underway…

•  Optical potentials are uncertain – methods are being developed to go beyond hand-waving error estimates and to obtain an accurate quantification of the error on reaction observables

thankyou!

This work was supported by DOE Office of Science, NNSA and NSF

Alaina Ross

Luke Titus

Amy Lovell Jimmy Rotureau

Non-local DOM and PB potential: local equivalents


reducing the many body to a few body problem

q  isolating the important degrees of freedom in a reaction q  keeping track of all relevant channels q  connecting back to the many-body problem

q  effective nucleon-nucleus interactions (energy dependence/non-local)

q  many-body input

reaction methods: CDCC versus Faddeev formalism

Faddeev Formalism

Continuum Discretized Coupled Channel Formalism

reaction methods: comparing CDCC with Faddeev

Upadhyay, Deltuva and Nunes, PRC 85, 054621

FRIB theory center: Enhancing theory efforts nationally

FRIB theory center will: •  connect broadly across

fields •  bring focus to those

activities that are relevant •  identify and nurture the

best talent •  take advantage of high

performance computing •  be the natural home for

training in advanced nuclear theory

ORNL NS,NR,NA

LLNL NS,NR

MSU NS,NR,NA,NF

INT NS,NA

LANL NS,NA

ANL NS,NR

FRIB TC Director

TAMUC NR UTK

NS,NR

ISU NS

OSU NF

CMU NS,FS

UNC NS,FS

IU NA

LBNL NA,FS

UCSD NA

UWi NA,FS Other

Area

Other Area

A possible snapshot for the FRIB theory center (NF-nuclear forces; NS-nuclear structure; NR-nuclear reactions; NA-nuclear

Astrophysics; FS-fundamental symmetries)

UI NF,NS

WMU NS

FSU NS,NA

NCSU NS,NA

UND NS

OU NF,NR,NA UA

NF

WUSL NS

UMn NA

LSU NS

TAMU NR

one-nucleon transfer reactions and the optical potentia lone-nucleon transfer reactions and the...

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