Nonlocal nucleon-nucleus interactions in (d,p) reactions: Role of the deuteron D-state

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by  G. W. Bailey

Nonlocal nucleon-nucleus interactions in (d,p) reactions: Role of the deuteron D-state
G. W. Bailey, N. K. Timofeyuk, and J. A. Tostevin
Department of Physics, Faculty of Engineering and Physical Sciences,
University of Surrey Guildford, Surrey GU2 7XH, United Kingdom
(Dated: January 23, 2017)
Theoretical models of the(d,p) reaction are exploited for both nuclearastrophysics and spectro-
scopic studies in nuclear physics. Usually, these reaction models use local optical model potentials
todescribethenucleon-anddeuteron-targetinteractions. Withinsuchaframeworktheimportance
of the deuteron D-state in low-energy reactions is normally associated with spin observables and
tensor polarization effects - with very minimal influence on differential cross sections. In contrast,
recent work that includes the inherent nonlocality of the nucleon optical model potentials in the
Johnson-Tandy adiabatic-model description of the (d,p) transition amplitude, which accounts for
deuteron break-up effects, shows sensitivity of the reaction to the large n-p relative momentum
contentof thedeuteron wavefunction. Thedominance of thedeuteron D-statecomponent at such
high momenta leads to significant sensitivity of calculated (d,p) cross sections and deduced spec-
troscopic factors to the choice of deuteron wave function [Phys. Rev. Lett. 117, 162502 (2016)].
Wepresent detailsof theJohnson-Tandyadiabatic modelof the(d,p) transferreaction generalized
7
toincludethedeuteronD-stateinthepresenceofnonlocal nucleon-targetinteractions. Wepresent
1
0 exact calculations in this model and compare these to approximate (leading-order) solutions. The
2 latter, approximate solutions can be interpreted in terms of local optical potentials, but evaluated
at a shifted value of the energy in the nucleon-target system. This energy shift is increased when
n
including the D-state contribution. We also study the expected dependence of the D-state effects
a
ontheseparationenergyandorbitalangularmomentumofthetransferrednucleon. Theirinfluence
J
on the spectroscopic information extracted from (d,p) reactions is quantified for a particular case
0
of astrophysical significance.
2

]
h I. INTRODUCTION channel. These deuteron breakupeffects are treatedeffi-
t cientlyintheADWAmethod[2],thatdevelopsthed A
- −
l A primary interest in (d,p) reactions, especially those effectivepotentialfromthoseofthe n Aandp Asys-
c − −
tems. Until very recently, these n A and p A optical
u studied at modern radioactive beam facilities, is their − −
potentials used to describe the n+p+A entrance chan-
n ability to reveal single-particle spectra of rare isotopes
[ and to determine the angular momentum content and nelwere assumedto be local. In the ADWA, nonlocality
spectroscopic strengthof single-particle states near their in the proton(exit) channel can be included in the same
1
manner as in the DWBA. This proton channel nonlocal-
v Fermi-surfaces. This information is crucial also in nu-
ity has been treated exactly in recent calculations [6, 7].
3 clear astrophysics applications. These angular momenta
5 and associatedspectroscopicstrengths are deduced from However,constructingthed Aeffectivepotentialswhen
−
8 comparisonsofthemeasuredcrosssectionswiththeoret- including nonlocal nucleon-target(N A) potentials re-
−
5 quiredadditionalformaldevelopments,aspresentedonly
ical predictions. Differences between theoretical models
0 relatively recently [8–11]. In addition, earlier work that
of the reaction thus impact the interpretation of experi-
1. mentaldataandstudiesofthesensitivitiesofcalculations included N A nonlocalities using Faddeev framework
−
0 to model assumptions are vital. three-body calculations showed an improved description
7 of (d,p) reactioncross sections on a range of closed-shell
Necessary inputs to direct reaction models of the
1 targets [12].
A(d,p)B transfer process, such as the distorted-
:
v waves Born approximation (DWBA) [1] and adiabatic It was shown in Refs. [8, 9] that including nonlo-
i
X distorted-wavesapproximation(ADWA)[2]methods,are cal p A and n A potentials in the adiabatic model
− −
complex effective interactions (optical potentials) be- of the A(d,p)B reaction generates a nonlocal adiabatic
r
a tween the reactants in the entrance (d A) and exit d A potential. This model could be further reduced,
− −
(p B) channels. Feshbach theory clarifies that these to a local one, in a similar way to that originally intro-
−
interactions should be both complex and nonlocal, aris- duced by Perey and Buck [4]. This revised local adi-
ingfromthemany-bodynatureofthenucleiAandBand abatic potential Uloc is different to that which is usu-
dA
the effects ofinelastic channelcouplings upon the elastic allyconstructedintheADWAmethod,thatusesenergy-
channel wave functions [3]. Within the DWBA, taking dependentphenomenologicallocalnucleonopticalpoten-
accountofnonlocalitiesofthePerey-Bucktype[4]results tials and then evaluates these at half the energy of the
in a multiplication of the entrance and exit channel dis- incident deuteron, E . Instead, Refs. [8, 9] show, for
d
tortedwavesby a Perey factor [5]. However,the DWBA Z = N targets, that Uloc should be constructed from
dA
neglectsimportantcontributionsduetotransferfromthe nucleon optical potentials evaluated at a shifted energy
deuteronbreakupcontinuumthatrequireaconsideration E = E /2+∆E. The required shift, ∆E, is related to
d
of the nucleon-target degrees of freedom in the entrance the value of T , a measure of the n p relative ki-
h npiV −
2
netic energy T in the deuteron ground-state φ inside model d A potential. In Sec. III we then present the
np 0
−
the range of the n p interaction, V , that binds the formalism for the nonlocal deuteron adiabatic potential.
np
−
deuteron. Specifically, this value is In Sec. IV we compare the present results, made in a
lowest-order approximation, to those obtained in the lo-
φ V T φ
0 np np 0
hTnpiV = h φ| V φ| i ≡hφ1|Tnp|φ0i, (1) cal model proposed in Refs. [8, 9]. In Sec. V we present
h 0| np| 0i the present exact calculations for several targets, focus-
where we have defined ing on how D-state effects evolve with the separation
φ =V φ / φ V φ . (2) energyandorbitalangularmomentumofthe transferred
1 np 0 0 np 0
| i | i h | | i neutron. Implicationsforextractedspectroscopicfactors
Given the short-range nature of the nucleon-nucleon
are discussed for a specific reaction of astrophysical in-
(NN)interactionandφ ,amajorcontributionto T
1 h npiV terest, on a 26Al target. Conclusions are drawn in Sec.
arises from high n p relative momenta.
− VI. Other relevantdetails arepresentedin anAppendix.
InRef. [9],valuesof T and∆E wereobtainedas-
h npiV
sumingtheS-statewavefunctionofthepurelyattractive,
phenomenologicalcentralNNinteractionofHulth´en[13],
II. D-STATE IN ADWA WITH LOCAL
whereas realistic deuteron wave functions have a mod-
NUCLEON OPTICAL POTENTIALS
est D-state component with probability P 4 7%.
D
≈ −
Though modest, this D-state component can dominate
Before presenting our nonlocal potential plus D-state
the wave function at high n p momenta with impor-
tant implications for calculati−ons of Uloc and (d,p) cross (d,p) model we comment on D-state contributions to
dA standardlocalADWAcalculations. TheADWAincludes
sections. The intrinsic nonlocality of optical potentials
deuteronbreakupeffectsthroughathree-body(A+n+p)
thus presents a distinct and novel source of D-state and
descriptionofthedeuteronchannel. The(d,p)transition
n pmomentum sensitivity ofcrosssections forsuchre-
− amplitude in the three-body model is
actions. ThisisincontrastwithpreviousD-statestudies,
e.g. [14, 15], that focused on the effects of the D-state T =√C2S χ(−)φ V Ψ(+) , (3)
component of the reaction vertex V φ in the DWBA (d,p) h p n| np| d i
np 0
| i
amplitude. The conclusions there, for low energy reac- where Ψ(+)(R,r) is the deuteron channel three-body
tions,arethatDWBAcrosssectionsandvectoranalysing d
wavefunction,Risthevectorseparationofthedeuteron
powers are insensitive to the deuteron D-state, the pri- and the target, r the neutron-proton separation and
marysensitivitybeing onthe tensorpolarizationobserv-
V is the neutron-proton interaction in the deuteron.
ables. np
Following the convention used in Ref. [9], we define
In this paper we develop exact adiabatic model (d,p)
r = r r and R = (r +r )/2 where r and r
reaction calculations that use energy-independent non- n − p − n p n p
are the positions of the incident neutron and proton rel-
local nucleon optical potentials and that include the
ative to the mass A target. Thus, with this convention,
deuteron D-state. We derive formalexpressionsand cal- r = r/2 R and r = r/2 R. In the final state,
culate the nonlocal deuteron channel potential UdA and thne proton−channel dipstort−ed wa−ve χ(−) is a function of
the corresponding d A distorted waves. The effects p
− R , the position of the outgoing proton relative to the
on (d,p) cross sections are discussed and compared with p
productnucleusB(=A+n). φ isthenormalisedbound-
thoseobtainedfromtheearlier,approximatelowest-order n
local model. Our key findings, applied to a 26Al target, statewavefunctionofthetransferredneutroninthefinal
state (more generally, the neutron overlapfunction) and
were presented in Ref. [16] and focused on the sensitiv-
C2S is its spectroscopic factor.
ity of T and the corresponding (d,p) cross sections
h npiV The ADWA makesuseofa Weinberg statesexpansion
to high n p momenta, which is different between NN
models. It−was shown that, in some cases, cross sections of Ψ(d+) [2], valid for r values within the range of Vnp, as
canchangesignificantlywithdifferentchoicesofdeuteron required to evaluate T(d,p). It was shown [17] that with
wavefunction, and that these changescorrelatewith the this basis, the transition amplitude converges rapidly
D-state component. Here, we present full details of the and only the first Weinberg state needs to be retained.
model calculations and extend the model’s application With this approximation Ψ(+)(R,r) χ(+)(R)φ (r)
to include 40Ca and 28Si targets. For 28Si we explore and T in the ADWA is d → d 0
(d,p)
a range of neutron separation energies and different or-
bitalangularmomentumtransfers. We restrictourselves T =√C2S χ(−)φ V χ(+)φ , (4)
(d,p) h p n| np| d 0i
to low-energy(d,p)reactions,relevantto ISOLfacilities,
where spin-orbit terms of the nucleon optical potentials where χ(+)(R) describes the center-of-massdistortionof
d
and finite-range effects of the transition interaction can the incident np-pair in the presence of deuteronbreakup
beneglected,allowingaclearerevaluationoftheD-state effects. When the nucleon-target potentials U (N =
NA
effects. n,p) are local, χ(+)(R) is calculated from the adiabatic
InSec. IIwereviewtheroleoftheD-stateonthed A d
− distorting potential
distortedwaveandcrosssectionswithinthestandard(lo-
cal)ADWAandintheDWBAwiththeWatanabefolding U = φ U ( r/2 R)+U (r/2 R)φ . (5)
dA 1 pA nA 0
h | − − − | i
3
Because of the short range of φ , of Eq. (2), the main 0
1
contributionsinEq. (5)comefromvaluesr 0. Letting
r 0 connects the Weinberg states techni≈que and the -20 Imaginary
→ V)
earlierJohnson-Soperadiabaticformalismwhere,assum- e
M
ainngd,aiznerwoh-riacnhgleimVintp,,tUhdeJASa(dRi)ab=atUicnApo(Rte)n+tiaUlpiAs(sRee)n[1t8o] ntial ( -40 a) 40Ca + d
e
be independent of details of the assumed deuteron wave ot -60
p
function. Asisnowshown,fullcalculationsofthecentral al
terms of UdA of Eq. (5), with different realistic deuteron Centr -80 Real AAddiiaabbaattiicc,, HAVul1th8en
wavefunctionsandlocalnucleon-targetinteractions,give Watanabe, Hulthen
Watanabe, AV18
verysimilarresultsandshowessentiallyno sensitivityto -100
the D-state.
0 2 4 6 8
Below we show U for the deuteron wave function of R (fm)
dA
the Argonne V (AV18) NN interaction [19], with both
18
S- and D-state components. In the presence of the D-
state, we write the deuteron ground state wave function 6
φMd, with angular momentum projection M , as 40 41
0 d 5 Ca(d,p) Ca
u (r)
φMd(r)= (l λ s σ J M ) ld Yλd(rˆ)χσd,
0 ldXλdσd d d d d| d d r ld sd mb/sr)4 AAWddaiiaatabbnaaattiibcce,, ,HA HVuul1tlh8theenn
(6) Ω (3 Watanabe, AV18
d
with ldλd,sdσd the orbital and spin angular momenta σd/2
and their projections coupled to J (= 1), χ is the np
d
spinor and the uld(r) are the deuteron S- and D-state 1 b)
radial wave functions. The vertex function V φMd has
np 0
an identical form but with the uld(r) replaced by short- 00 30 60 90 120 150 180
ranged radial vertex functions vld(r). θc.m. (degrees)
ThecalculatedadiabaticpotentialU forthed 40Ca
dA
−
systematEd =10MeV,usingtheChapelHill89(CH89) FIG. 1. (a) Adiabatic and Watanabe potentials for d−40Ca
phenomenological local optical potential for the UNA atEd =10MeV,and(b)40Ca(d,p)41Cadifferentialcrosssec-
[20], is shown in Fig. 1a. For comparison, the UdA cal- tionsforEd =10MeV,usingtheChapelHill89phenomeno-
culated using the S-state Hulth´en wave function is also logicallocalopticalpotentialfortwoNNpotentialmodels: (i)
shown. The two potentials are very similar, as are the thecentralHulth´enpotentialwithanS-statedeuteron(dots)
corresponding40Ca(d,p)41Cacrosssections,presentedin and (ii) the realistic AV18 potential and deuteron with both
theS- and D-states (solid lines).
Fig. 1b. All of the NN potential models used in Ref.
[16] lead to this same conclusion. We add that there
is also negligible D-state sensitivity in the central terms
(−)
InthefollowingwecalculateT whenbothχ and
of the deuteron-target interaction and the transfer reac- (d,p) p
tion cross sections in the no-breakup limit of the d A χ(+) are generated by nonlocal potential models - the
− d
scattering - the Watanabe folding model [21] - when the latterincluding realisticdeuteronwavefunctions andD-
distorting potential is state effects through the adiabatic model entrance chan-
UWat = φ U ( r/2 R)+U (r/2 R)φ . (7) nel effective interaction.
dA h 0| pA − − nA − | 0i
The calculated Watanabe potentials are also shown in
Fig. 1a for both the Hulth´en and AV18 cases. The D-
III. THE NONLOCAL SCATTERING PROBLEM
state and NN-model insensitivity of the corresponding
(DWBA) cross sections is shown in Fig. 1b.
Inthissectionwedescribethecomputationsofthedis-
In the cross section calculations above, and through-
out this paper, we use the zero-range approximation, torted waves χp(−) and χ(d+) for the ADWA calculations
D(r) = r V φ D δ(r) to the transition interac- in the nonlocal model case. They satisfy the inhomoge-
np 0 0
tionwhenhc|alcu|latiing≈T . ThevolumeintegralsD are neous Schr¨odinger-like equation, with α=p or d,
(d,p) 0
determined for each NN model, and are given in Table
(T +U (R) E)χ (R)=
I for the models used here. The zero-range approxima- α c − α
tion to T(d,p) is very accurate for reactions of low energy dR′ (R,R′) χ (R′), (9)
deuteronbeams,wherefinite-rangecorrectionsaresmall, −Z Uα α
e.g. [22]. Hence, we compute the transition amplitudes
whereE,T andU arethecenterofmassenergy,kinetic
α c
T =D √C2S χ(−)φ χ(+) . (8) energyoperatorandCoulombinteractionand (R,R′)
(d,p) 0 h p n| d i Uα
4
is the nonlocalnuclear potential in channel α. Through- where µ = R R′/R R′ and P is the Legendre poly-
p· p p p L
out this work we adopt an energy-independent Perey- nomialoforderL. Withtheneglectofspin-orbitinterac-
Buckparameterizationofthenonlocalnucleon-targetpo- tions,asassumedhere,theJ =L 1/2channeldistorted
±
tentials. Equation(9)is solvediteratively,followingpar- waves are of course identical.
tial wave decomposition, from a trial complex and local In all of the nonlocal ADWA calculations presented,
starting potential U (R), namely the exactsolutions of Eq. (12) are readinto the transfer
α
reactions code twofnr [23]. Comparisonsbetween such
(T +U (R)+U (R) E)χ(i+1)(R)= exactsolutions and those from a phase-equivalentmodel
α α c − α can be found in Ref. [6].
dR′ (R,R′) U (R)δ(R R′) χ(i)(R′)
−Z Uα − α − α
(cid:2) (cid:3)
B. Deuteron Channel
with χ(0) the solutionofthe homogeneousequationwith
α
the appropriate scattering boundary conditions. Our The nonlocal deuteron channel potential (R,R′)
treatment of the proton and deuteron channel nonlocal UdA
is constructed using nonlocal nucleon-target optical po-
potentials is described in detail below.
tentials with the Perey-Buck form of Eq. (10). The for-
mal expression for the Johnson-Tandy adiabatic model
potential (R,R′) in terms of the nucleon potentials
dA
A. Proton Channel (R ,UR′ )isgivenbyEq. (12)ofRef. [9]-afolding-
UNA N N
type integralwhere the argumentsofthe nucleonoptical
The nonlocal interaction in the proton channel de- potentialsarereexpressedintermsofthe deuteronchan-
scribes the motion of the outgoing proton with respect nelvariablesR, R′ andr. Here,wetakethetargetmass
the resultantnucleus B(=A+n). This is of Perey-Buck A to be infinitely large in the more general expression
type, i.e. of Ref.[9]. The partial wave form of dA(R,R′) is more
U
complicatedwhentheD-stateispresentandtherequired
R +R′ expansion is most easily achieved by use of the variables
(R ,R′)=H(R R′ )U | p p| , (10) R and S =R R′. We obtain
UpA p p | p− p| pA(cid:18) 2 (cid:19) −
MdMd′(R,S)=8H(2S) dxφ∗Md(x 2S)
with H a normalized Gaussian (in 3 dimensions) with a UdA Z 1 −
x x
range β, U ( R)+U ( R) φMd′(x), (15)
×h nA 2 − pA 2 − i 0
H(x)=(√πβ)−3exp( x2/β2). (11)
′
− where Md,Md are the projections of the intrinsic angu-
lar momentum J (=1) of the deuteron referred to the
ThepotentialformfactorsU arecomplexwithconven- d
NA incident beam direction.
tionalWoods-Saxonrealpartsandsurface-peakedderiva-
We multipole expand the vertex function φ and the
tive Woods-Saxon imaginary parts. This nonlocal inter- 1
nonlocalnucleon-targetpotentialformfactorsintermsof
action is used directly in the source term of the inho-
the vectors x, S and R, and separate their radial and
mogeneous equation, Eq. (9). Thus, the proton channel
angularcomponents. After summation overangularmo-
partial wave functions χJ, with J = L+s , s = 1/2,
L p p mentumprojections, istheoperatorindeuteronspin
satisfy UdA
space
(cid:16)Tp(L)+Uc(Rp)−Ep(cid:17)χJL(Rp)= UdA(R,S)=4π vla1l2(R,S) (−1)a−αTa−α
R ∞dR′ R′ (p)(R ,R′)χJ(R′) (12) lX1l2a Xα
− p 0 p pUL p p L p
R Y (Rˆ) Y (Sˆ) , (16)
×(cid:20) l1 ⊗ l2 (cid:21)
with aα
T(L) = ~2 d2 L(L+1) (13) wthheersepaTckeqoifsstphieniJrredwuitchibmleatternixsoerleompeernattsor of rank k in
α −2m (cid:20)dR2 − R2 (cid:21) d
α α α
J M J M′ =kˆ(J M′kq J M ).
and m is the reduced mass in channel α. The potential h d d|Tkq| d di d d | d d
α
kernel is The functions va contain all information on the
l1l2
deuteron wave function and nonlocal potential form fac-
UL(p)(Rp,Rp′)=2πZ−11Rdµ+PLR(µ′)H(|Rp−R′p|) ttRohresar,neddxeptRaain′l,ssitoohfnewnohfdicvehlar1ilav2reeasnptdrheeYselr2ne(tqSeˆud)i,rinendoawn(rAawdpitipahelnrvedasirpxi.aecbFtluetsro)-
U | p p| , (14) kernel of ,
pA dA
× (cid:18) 2 (cid:19) U
5
1
(2l +1)! 2
ULJ′L′′(R,R′)= 2π aˆ2Jˆdˆl1ˆl2Lˆ2Lˆ′(cid:20)(2τ2)!(2η)!(cid:21) (τ0L0|L′0)(η0L0|L′′0)(l10L′0|L′0)W(L′Ll2η;τL′′)
aXl1l2 XLL′
τ+η=l2
1 va R, R R′
×W(L′′L′al1;l2L′)W(L′aJJd;L′′Jd)RτR′ηZ−1 l1l2(cid:0)R (cid:12)(cid:12)R′−l2 (cid:12)(cid:12)(cid:1)PL(µ) dµ. (17)
−
(cid:12) (cid:12)
(cid:12) (cid:12)
Here, in standard notations, xˆ = √2x+1, W is the where m is the deuteron channel reduced mass, β is
d d
Racah coefficient and µ = R R′/RR′. These nonlocal theeffectivedeuteronnonlocalityrange[9]andM isthe
0
kernels, J (R,R′), enter t·he Schr¨odinger-like equa- zeroth-order moment of the nonlocality factor
tion for tUhLe′Ld′′euteron distorted waves χJ (R) for total
L′L
angular momentum J. Explicitly, M = ds dxH(s)φ (x s)φ (x). (22)
0 1 0
Z Z −
(Td(L′)+Uc(R)−Ed)χJL′L(R) In Eq. (21) and below, Ud(R)=UnA(R)+UpA(R).
∞ Further, it was shown, for Z = N nuclei, that Uloc
=−R Z dR′R′ ULJ′L′′(R,R′)χJL′′L(R′) (18) can be constructed from phenomenologicallocal nucledoAn
XL′′ 0 optical potentials that describe elastic scattering at an
where Td(L′) is given by Eq. (13). eEnde/r2g,yofshhiaftlfedthbeyin∆ciEdenfrtodmeuttheerounseunaellrygya.sTsuhmroeudgvhaMlue0,,
areTohfeccoeunrtsreadlitaegromnsaloifnUthdAe,orcboirtraelspanognudlianrgmtoomaen=tum0, twheisfisnhdiftth∆atEEiqs.d(e2t1er)mainndeditsbyshhiTftnepdi-VenoefrgEyq.so(l1u)t.ioHnerree-,
quantum number and mainvalidwhenadeuteronD-stateispresent. However,
the values of T , M and ∆E are strongly affected
ULJ′L′′(R,R′)=2πδL′L′′ lXLL′ LˆL2ˆL′ˆ2′ˆl (cid:20)((22τl)+!(21η))!!(cid:21)12 TNbyaNbtlhmeeoIpdsrehelossewwnscihtethhnoaapftitDVhth-esetDa∆t0-seEtcaovteamlcupoeomsnefponortn,tethnyetp.nicFeaoulrtlyreoxsnpa,amfnrpoalmen,
τ+η=l
′ ′ ′ ′ ′ ′ interval from 38–75 MeV, significantly larger than the
(τ0 0 0)(η0 0L0)(l0 0L0)W( lη;τL)
× L |L L | L | LL ∆E of 32 MeV from the S-wave Hulth´en NN model.
1 v0 R, R R′
RτR′η ll − PL(µ) dµ. (19)
× Z−1 (cid:0)R(cid:12)(cid:12) R′ l (cid:12)(cid:12)(cid:1) NNModel PD D0 h TnpiV ∆E
− 3
(cid:12) (cid:12) % MeV fm2 MeV MeV
The smaller potent(cid:12)ial terms(cid:12) with a = 2, of tensor char-
Hulth´en 0 −126.15 106.6 31.7
acter, arise only when the deuteron D-state is included.
Reid soft core 6.46 −125.19 245.8 75.3
ArgonneV18 5.76 −126.11 218.0 66.8
CD-Bonn 4.85 −126.22 112.5 37.6
IV. LOWEST-ORDER APPROXIMATION TO
THE NONLOCAL MODEL
TABLE I. D-state percentages PD, volume integrals D0 of
the transfer vertex D(r), and short-ranged n-p kinetic en-
theBepfootreentciaallcucalalctuinlagtetdheisneaUndAappexroaxctimly,atwioen,incvaellsetdigtahtee eTrhgeynheuTtnrpoinVefnoerrgdyiffsehriefntst∆NEN-amroedceallciuntlaetreadctfioonrsthuesedd+h40eCrea.
lowest-order (LO) limit. Here, due to the short range of systematEd =11.8MeVandarecomputedusingthelowest-
φ in the folding integral, Eq. (15), we replace order methodology discussed here. The proton energy shifts
1 are larger by the Coulomb energy, ≈6.8 MeV for 40Ca + p
U (x/2 R) U (R) (20) scattering. DetailscanbefoundinsectionsIV.AandIV.Bof
NA NA
| − | → Ref. [9].
as was also used in [8, 9], there assuming an S-wave
deuteron. This limit calculates the leading-order contri- Below we will present, as typical, calculations for the
butions to the potential and the (d,p) cross section and phenomenological AV18 NN potential. The other NN
provides insight into the nature of the derived entrance potentials studied, see Table I and also Table I of Ref.
channel interaction. It was shown previously [9], in this [16],givebothlargerandsmaller∆E. Wenotealsothat,
limit and within the local-energy approximation (LEA), if we include only the S-wave part of the AV18 wave
that the adiabatic potential is local and that this local function, then the values of both M and ∆E are very
0
potential, Uloc, solves the transcendental equation similar to those of the Hulth´en wave function. Thus,
dA
the primary difference between the AV18 and Hulth´en
Uloc =M U (R)exp mdβd2(E Uloc U ) ,(21) model results arises from the D-state component of the
dA 0 d (cid:20)− 2~2 d− dA − c (cid:21) wave function.
6
0 We have solved the Schr¨odinger equation for the dis-
Imaginary torted waves χL(R) in this limit and then constructed
V)-20 the trivially-equivalent local potentials (TELPs)
e
M
ential (-40 40Ca + d UTELP(R)= RR dR′R′χUL((RR),R′)χL(R′) (24)
ot L
p AV18, full
al AV18, LO
ntr-60 AV18, TJ which can be compared with the lowest-order adiabatic
Ce Hulthen, full
Hulthen, LO potentials of the approximation used by Timofeyuk and
-80 Real Hulthen, TJ Johnson (TJ) in Ref. [9]. This comparison, for the
d 40Ca system at E = 11.8 MeV, is shown in Fig. 2.
d
0 2 4 6 8 −
In these and all subsequent calculations we use the non-
R (MeV)
local nucleon-nucleus potential parameterization of Gi-
annini and Ricco [24], that we denote GR76. We find
that the calculated TELP are essentially independent of
L and differ from Uloc by no more than 1% and 2% for
8 dA
40Ca(d,p)41Ca theAV18andHulth´enpotentials,respectively. Thiscon-
firms that, in leading order, the adiabatic potentials can
sr)6 AAVV1188,, fLuOll also be obtained from local nucleon potentials by apply-
mb/ AV18,TJ-LO ing an appropriate energy shift. Since this shift is larger
Ω (4 HHuulltthheenn,, fLuOll when the deuteron D-state is included, being 67 MeV
σ/d Hulthen, TJ-LO for AV18, compared to 32 MeV for the Hulth´en case,
d Hulthen, TJ-NLO the adiabatic optical potentials should be shallower - as
2 confirmed by the direct calculations shown in Fig. 2a.
The cross sections for the 40Ca(d,p)41Ca reaction using
(LO), shown in Fig. 2b, are also very close to those ob-
00 30 60 90 120 150 180 UdA
θ (degrees) tained with Uloc of the TJ approach, given by Eq. (21).
c.m. dA
Including the D-state is seen to increase the computed
FIG. 2. (a) Trivially-equivalent local adiabatic potentials for (d,p) cross sections.
d−40Ca,and(b)the40Ca(d,p)41Cadifferentialcrosssections We note that, by making only the LO approximation
obtainedfromfull(solidlines)andlowest-order(dashedlines) of Eq. (20) to the full expression for (of Eq. (15))
dA
calculations at Ed = 11.8 MeV. Calculations assume the S- we take into account all corrections beUyond the LEA in
wave Hulth´en potential (blue) and the more realistic AV18 the TJ derivation of Uloc. However, as was shown in
NN potential with both S- and D-states (black). These are dA
[9], the first-order correction to the LEA involves the
compared to the lowest-order TJ results for the same NN
fourth power of the (small) nonlocality range parameter
potentials. For the Hulth´en potential, the TJ cross sections
β, suggesting this and higher order corrections will be
in next-to-leadingorderarealso shown bythedashed-dotted
(magenta) curve. negligible. This expectation is indeed confirmed by our
comparisons of Uloc with the exact LO results in Fig. 2.
dA
In the LO limit, the nonlocal potential of Eq. (16)
simplifies to
V. RESULTS FROM FULL NONLOCAL
Ud(LAO)(R,S)=√4πUd(R) νa(S)(−1)a−αTa−αYaα(Sˆ), CALCULATIONS
Xaα
We now present the results of calculations that
where
compute the exact solutions of the nonlocal integro-
Jˆ differential problem, as given by Eqs. (17) and (18).
ν (S)=8H(2S) d ˆl′W(al J s ;l′J )
a aˆ d d d d d d We notethatsuchsolutionscouldalsobe approachedby
Xldl′d systematic development of the higher-order corrections
∞ to the LO model of the previous section. For example,
×Z dxxυ˜l(′dlda)(x,2S)ul′d(x). (23) for the pure S-state deuteron case, the next-to-leading
0
order (NLO) corrections were discussed above and in
The central terms (a = 0) of this LO nonlocal potential Ref. [8] and, for the Hulth´en wave function and the
arediagonalinLwithULJL′(R,R′)=ULJ(R,R′)δLL′,and 40Ca(d,p)41Ca reaction we have shown that these dif-
fer from the exact calculations by 3%, see Fig. 2b.
1 ≈
J(R,R′)=2πU (R) ν R R′ P (µ) dµ. However, no such systematic development has yet been
UL d Z−1 0(cid:0)(cid:12) − (cid:12)(cid:1) L carried out when the D-state is present.
(cid:12) (cid:12)
7
60
0.8 L = 0 b/sr)
µ
0.4 (sor 40 AV18, full
n
AV18, full a) +te AV18, LO
0 AV18, LO entral
Ωc 20
0.8 L = 0 d
σ/
d
0.4 Hulthen, full - ntral 0
2R)| Hulthen, LO b) Ωce
χ(L 0 σ/d
| 000...4880 LL == 44 AAVV1188,, fLuOll c) dFccasoerdIonrGisa-tt2s.rbe0aars40lmte.ictcsteirDp(omsnoioffstsle,eid(3nrfa0etolniria=nclEeet)0dh.b)ae=Tta6twh0ni1ende1ecθ.cnwl8cou.mridMtt.r hhe(edes9seeVp0ttgh4oh,r0eneecCedafsDauiln)c(l-ldgu1s,l2t(alpaa0et)tae4e=dd1iCgnw0eagn,i1t2eod5h)rr0iadffonteeneorrdlney(lnLotttec1Oihnaa8e)-ll0
potential results are shown by thedashed line.
0.4
Hulthen, full
d)
Hulthen, LO wave S-matrices SJ . The latter have maximum val-
00 5 10 15 20 ues of order 10−2 iLn′Lthe cases studied here. These ten-
R (fm) sor force effects are included fully in the exact distorted
wave functions input to the transfer reaction calcula-
FIG. 3. The Ed = 11.8 MeV, d−40Ca distorted waves in tions. The importance of these terms, for low energy
L = 0 (a,b) and L = 4 (c,d) partial waves, calculated using (d,p) reaction cross sections was assessed by comparing
theAV18(a,c)andHulth´en(b,d)NNmodelpotentials. The
calculationsthat include and neglectthe a=2 contribu-
exactnonlocal calculations areshown bythesolid lineswhile
tions to the adiabatic potential. This difference, for the
theapproximate,lowest-order(LO)calculationsareshownas
40Ca(d,p)41Ca cross sections, in Fig. 4, is less then 50
dashed lines.
µb/sr andrepresents a change in the calculateddifferen-
tial cross sections of 0.6% or less. Thus, the tensor force
effects in the nonlocal , arising from the deuteron
dA
A. Adiabatic distorted waves and tensor force U
D-state, are insignificant for cross-section-based nuclear
effects
spectroscopy and astrophysicalstudies.
The magnitude of the beyond-LO effects in the exact
calculationswhenthe deuteronD-stateispresentareas- B. D-state effect on cross sections: transferred
sessed in Fig. 3. The deuteron channel partial waves for orbital angular momentum and separation energy
L = 0 and L = 4 are shown for both the Hulth´en and dependence
AV18 NN potentials. We note that the wave functions
ofthe fullcalculationsaresmallerthantheLOresultsin Allcalculationsabovewereforthe 40Ca(d,p)41Ca(g.s)
the nuclear interior,as could be described by a Pereyef- reaction, carried out in zero-range approximation. The
fect. Theoscillationsofthewavefunctionsobtainedwith single-particle bound state wave function of the trans-
AV18 are also more pronounced, indicative of a reduced ferred1f neutronwasobtainedinthestandardpoten-
7/2
deuteron channel absorption. The TELPs deduced from tial model description, the Woods-Saxon binding poten-
these wave functions are presented in Fig. 2a showing tial having a radius parameter r = 1.25 fm, diffuseness
0
a complicated dependence of their depths and surface a = 0.65 fm and spin-orbit depth V = 6 MeV. Anal-
0 SO
diffuseness on both the NN interaction model and the ogous calculations for the 26Al(d,p)27Al reaction, popu-
presence of higher-order terms in the presence of the D- latingseveralfinalstates,werepresentedinRef. [16]. As
state. noted there, the magnitude of the cross section changes
These wave functions and TELPs from the exact cal- when using realistic deuteron wave functions, with a D-
culations include the small contributions from the a=2 state, depends on the details for the final state of the
(rank-2 spin-tensor) terms of , that generate off- transferred neutron. We now discuss a more systematic
dA
U
diagonal contributions to the nonlocal potential kernels study of this observed sensitivity.
J (R,R′) and the associated deuteron channel partial As an example we use the 28Si(d,p)29Si reaction at
UL′L
8
103 C. Uncertainty of spectroscopic information
extracted from (d,p) reactions: 26Al(d,p)27Al(7806
28 29
Si(d,p) Si (g.s.) keV) case.
102
Jπ = 1/2+ AV18 To illustrate how including the D-state in the non-
b/sr) 101 Hulthen local adiabatic model can affect the spectroscopic fac-
m tors extracted from (d,p) reactions, we present calcula-
Ω ( tions of the 26Al(d,p)27Al∗(7806 keV) reaction. The re-
σ/d 100 action populates the mirror of an astrophysically impor-
d
tant state in 27Si, relevant to the destruction of 26Al in
10-1 Wolf-Rayetand Asymptotic Giant Branchstars [25, 26].
The differentialcrosssectionfor this transitionis an(in-
coherent) combination of l = 0 and l = 2 single-particle
10-2 transfers; however, only the l = 0 part is important for
0 30 60 90 120 150 180
θ (degrees) characterizing the low-energy 26Al + p resonance in the
c.m. 27Simirror. SpectroscopicfactorsofS =9.3(19) 10−3
l=0
FIG. 5. The 28Si(d,p)29Si(g.s.) reaction differential cross and Sl=2=6.8(14) 10−2 were deduced in Ref. [25×] from
×
sections, at Ed =10 MeV,obtainedusingtheUdA calculated new high-precision data using an analysis that used the
withtheS-stateHulth´enandS+D-stateAV18deuteronwave ADWA and local global nucleon optical potentials [27].
functions. These calculated cross sections (black curves) and the
experimental data are shown in Fig. 7. The figure also
shows the cross sections obtained from the present non-
E = 10 MeV, 29Si having a rich excitation spectrum local adiabatic model for two choices of NN potential,
d
withlow-lyingstatesfromseveralsingle-particleorbitals. the pure S-state Hulth´en and the more realistic AV18
We perform calculations using the GR76 nonlocal nu- potential. All of the calculated l = 0 and l = 2 transfer
cleonopticalpotentialsandthebindingpotentialgeome- contributions have been scaled using the spectroscopic
trystatedabove. Fig. 5showsthepredicted29Siground- factors of Ref. [25] above. The nonlocal model calcu-
state (Jπ =1/2+) differential cross section calculated in lations use the same geometries for the neutron bound
the S-state Hulth´en and AV18 deuteron cases. The D- states potentials as in Ref. [25], with radius parameters
stateisseentoleadnotonlytoanenhancedcrosssection r =1.159fmand1.263fmforthel =0andl=2states,
0
intheforwardpeakbutalsotoamodifiedangulardistri-
butionatlargerangles. So,changesarecomplexandnot
simply a scalingandcomparisonswith data may depend
60
sensitively on the available range of measured angles.
To explore this cross section D-state sensitivity fur-
ther, plus its dependence on the neutron separation en- 40 l = 2
ergy,wehaveperformedaseriesofcalculationswithboth %) l = 0
the Hulth´en and AV18 wave functions. As above, these s (
e
are for the 28Si(d,p)29Si reaction at 10 MeV. Here we ng 20
a
have varied the assumed neutron separation energy be- h
c l = 1
tween 1 to 21 MeV for four assumed transitions of dif- al
n 0
ferent orbital angular momentum, namely: 2s12 (l = 0), ctio
2p23 (l =1), 1d23 (l =2) and 1f27 (l =3). The fractional Fra-20 l = 3
changes (as %) in the differential cross sections (at their
first peak) of calculations with the Hulth´en and AV18
deuteron wave functions are shown in Fig. 6. These ra- -40
0 4 8 12 16 20
tios depend on the neutron separation energy showing
S (MeV)
that inclusion of the D-state can result in cross section n
changes of up to 30%. The cross section changes for an-
FIG. 6. Fractional changes (as %) in the differential cross
other(fixed)center-of-massanglearealsopresented(see
sections (at theirfirst peak)of calculations with theHulth´en
caption to Fig. 6) showing that the cross section shapes
and AV18 deuteron wave functions (closed circles and solid
may also change and that the peak value ratios may not lines). Resultsareforthe28Si(d,p)29Sireaction. Thechanges
represent an simple overall scaling. In all cases the de- are shown as functions of the assumed separation energy of
pendence on the neutron separationenergy is significant the transferred neutron with orbital angular momenta l =
and changes can reach values of around 50%. Such dif- 0,1,2,3. Thechangesinthesamecrosssections,butatafixed
ferences would certainly affect the interpretation of the center-of-mass angle, are also shown (open circles connected
experimental data in terms of a deduced spectroscopic bydashedlines). Thefixedanglesusedwere0,16,31and44
strength. degrees for l=0,1,2,3, respectively.
9
TABLE II. Deduced spectroscopic factors for the 26Al(d,p)27Al(7806 keV) reaction from the nonlocal adiabatic potential
analyses using different NN potential models. The nonlocal model spectroscopic factors were deduced so as to reproduce the
local adiabatic model calculations of Ref. [25], the solid black line in Fig. 7.
Work of Ref. [25] Present work
Hulth´en AV18 CD-Bonn RSC
Sl=0 9.3(19)×10−3 8.6(1)×10−3 8.2(2)×10−3 9.0(3)×10−3 8.2(2)×10−3
Sl=2 6.8(14)×10−2 5.8(2)×10−2 3.3(2)×10−2 4.5(4)×10−2 2.8(2)×10−2
Sl=0/Sl=2 0.14 0.15(1) 0.25(2) 0.20(2) 0.29(2)
respectively, diffuseness a =0.7 fm and a spin-orbit po-
0
tential depth V =6 MeV.
SO 1 26Al(d,p)27Al*(7.806)
The summed l =0 and l =2 partial cross sections are
larger than the experimental data and the earlier local-
Ref. [25]
model calculations and hence the deduced spectroscopic 0.8
AV18
factorsfromthe nonlocalmodelaresmaller. Therevised sr) Hulthen
spectroscopicfactorsfromthe presentnonlocalanalyses, b/
m0.6
fitted so as to reproduce the angular distribution from Ω (
the local analysis (the black solid curve) are shown in d
Table II. The errors shown for the nonlocal calculations σ/0.4
d
are those associated with this fit of the different theo-
retical calculations and do not include the uncertainties 0.2
associated with the fitting to the data points, shown for
the local-model analysis. The spectroscopic factors ob-
0
tained from analyses using the CD-Bonn [28] and Reid 0 2 4 6 8 10 12 14
θ (degrees)
Soft Core (RSC) [29] NN potentials are also tabulated. c.m.
It was shown in Ref. [16] that these CDB and RSC
crosssectionsessentiallyprovideupperandlowerbounds FIG.7. Calculatedl=0(dashedlines)andl=2(dot-dashed
to those calculated with the other NN potentials stud- lines)differentialcrosssectionsandtheirsums(solidlines)for
the 26Al(d,p)27Al (7806 keV) reaction at 12 MeV. The red
ied there. One notes that the revised S are reduced
l=0 andorangecurvesresult fromnonlocalpotentialanalysesus-
by up to 12% from those of the local potential analy-
ing the Hulth´en and AV18 NN wave function models. These
sis [25]. Moreover, while the S obtained with the S-
l=2 calculationshavebeenscaledbythespectroscopicfactorsde-
state Hulth´en potentialare similar,the reductioncanbe
ducedfromthedatausingthelocal potentialanalysisofRef.
greater depending on the NN potential choice. Overall, [25], shown by theblack lines.
we find this choice introduces an 60% uncertainty in
≈
the deduced S , which is significantly larger than the
l=2
quoted experimental uncertainties. While important for
considerations of the structure of 27Al, this uncertainty adiabatic model wave functions. To clarify the D-state
doesnotaffectthemainconclusionofRef. [25]-thatthe dependence we have also performed lowest-order calcu-
26Al(p,γ)27Sicaptureand26Aldestructionmechanismin lations in which the nonlocal nucleon optical formfac-
novae is dominated by the l=0 channel. tors UNA are evaluated at the n p center of mass posi-
−
tion. This approximation is shown to generate the lead-
ing modifications to the (d,p) cross sections and to pro-
vide insight into the physical picture. Namely, it clar-
VI. CONCLUSIONS
ifies that the deuteron channel adiabatic potential can
also be generated from local nucleon optical potentials
We have extended the nonlocal adiabatic model of if these are evaluated at energies that are shifted with
A(d,p)B reactions to include the deuteron D-state. respect to the usually-used value, E /2. Inclusion of
d
Whereas adiabatic model deuteron channel potentials the deuteron D-state, through the use of realistic NN
generatedfromlocalnucleonopticalpotentialsareinsen- forces and deuteron wave functions, is shown to increase
sitive to the deuteron D-state, the nonlocality of the nu- this energy shift leading to shallower and less absorp-
cleon optical potentials emphasizes the high-momentum tive deuteron channel distorting potentials compared to
partsofthedeuteronwavefunctionsinwhichtheD-state those calculated using a purely S-state deuteron wave
componentplaysanimportantrole. Asaresult,the(d,p) function. Crosssections calculated using this leading or-
crosssections calculatedin the nonlocaladiabatic model der approximation differ from the exact calculations by
are significantly affected by the D-state component. 12% and 10% for deuteron wave function models with
We have presented exact calculations of the nonlocal and without a D-state, respectively.
10
The degree to which the significant D-state effects Appendix: Multipole expansions
upon the central terms of the deuteron channel interac-
tion affect the (d,p) cross section magnitudes and an- The multipole expansion of the nonlocal deuteron
gular distributions depend on the transferred angular channel potential in Eq. (16), expressed as a func-
dA
momentum and the neutron separation energy of the fi- tion of the variabUles R and S, includes the function
nal state. Our calculations show that they can be as va (R,S) of the radial variables. This is given by the
large as 50% for some cases and, when two values of fol1lllo2wing expression
lj are allowed by the selection rules, they add ambigu-
ity to the interpretation of experimental data. For ex- va (R,S)=8H(2S)ˆl Jˆ (l 0k0l′0)
ample, when the 26Al(d,p)27Al reaction populating the l1l2 1 dlXdl′dk 1 | d
astrophysically-relevant27Al(7806keV)stateisanalyzed
ˆl′kˆW(al′l k;l l )W(al J s ;l′J )
inour nonlocalmodel, the deduced spectroscopicfactors × d d 2 d 1 d d d d d
∞
faonrabnoatlhyslis=w0ithaonudtlth=e 2deturtaenrsofnerDs a-srteatreedruedceudc.esWthheislee ×Z dxxυ˜k(lld2)(x,2S)U˜l1(x/2,R)ul′d(x). (A.1)
0
spectroscopic factors by no more than 16%, inclusion of Inspection shows that the angular momentum couplings
the D-state results in a dramatic reduction of the ex- in the second Racah coefficient, with Jd = sd = 1 and
′
tractedl =2 spectroscopic strength,by up to a factorof ld,ld = 0,2, restrict the spin tensor terms in UdA of Eq.
two. The uncertainty associated with different S + D- (16)to a=0 (central)anda=2(rank-2tensor)compo-
state deuteron ground state wave function models is of nents.
order 60%. By contrast,the effects of spin-tensor poten- In Eq. (A.1), υ˜(ld) arises from the multipole expan-
k1k2
tial terms induced by the deuteron D-state, even when sionoftheradialcomponents,vld,ofthedeuteronvertex
thenucleonopticalpotentialsarecentral,areincludedin function,Vnpφ0,asdetailedinandfollowingEq. (6). Ex-
the calculated deuteron channel distorted waves and are plicitly,
shown to have negligible effects on the calculated (d,p)
1
cross sections. υ˜(ld) (x,2S)= (−1)k2 eˆ2 (2ld+1)! 2
k1k2 2 φ V φ (cid:20)(2c)!(2d)!(cid:21)
h 0| np| 0ie,cX+d=ld
(c0e0k 0)(d0e0k 0)W(k el d;ck )
1 2 1 d 2
× | |
1 v (x 2S )
Apart from the 26Al(d,p)27Al reaction case, we have xc(2S)d ld | − | P (µ)dµ (A.2)
presented D-state results only for the AV18 wave func- × Z−1 x 2S ld+1 e
| − |
tion. The other NN models studied predict smaller D-
with µ=x S/xS.
stateeffects[16],sothepresentresultscanbeconsidered ·
as a reasonable upper limit. Calculations for 26Al using Finally, the U˜l in Eq. (A.1) are the multipoles of the
sumthe protonandneutronnonlocalpotentialformfac-
the CD-Bonn potential give the smallest (d,p) cross sec-
tors, i.e. U˜ (x/2,R)=U˜n(x/2,R)+U˜p(x/2,R), where
tionsofthemodelsstudiedandcouldsimilarlybeconsid- l l l
ered as a lower limit, but nevertheless show the impor-
1
tance of the D-state contribution to the spectroscopic U˜N(x/2,R)= dµU x/2 R P(µ) (A.3)
factors obtained. l Z−1 NA(cid:0)(cid:12) − (cid:12)(cid:1) l
(cid:12) (cid:12)
with µ=x R/xR.
·
Finally,thepresentstudyhasusedenergy-independent ACKNOWLEDGEMENT
nonlocal nucleon potentials. Explicit energy-dependence
of the nonlocal nucleon optical potentials, discussed in We are grateful to Professor R.C. Johnson for many
Ref. [30], may significantly modify model predictions - useful discussions. This work was supported by the
aswasshowninRef. [31]forthe caseofapurelyS-state UnitedKingdomScienceandTechnologyFacilitiesCoun-
deuteron. cil (STFC) under Grant No. ST/L005743/1.
[1] N. Austern, Direct Nuclear Reaction Theories, Wiley, [5] F. Perey, Direct interactions and nuclear reaction mech-
NewYork, 1970. anisms (Gordon and Breach, N.Y.1963), p.125
[2] R.C. Johnson and P.C. Tandy, Nucl. Phys. A235, 56 [6] L.J.Titus, F.M.Nunes, Phys.Rev.C 89, 034609 (2014).
(1974). [7] A. Ross, L.J. Titus, F.M. Nunes, M.H. Mahzoon, W.H.
[3] H.Feshbach, Ann.Rev. Nucl.Sci. 8, 49 (1958). Dickhoff, R.J. Charity, Phys.Rev.92, 044607 (2015).
[4] F. Perey and B. Buck,Nucl. Phys. 32, 353 (1962). [8] N.K.TimofeyukandR.C.Johnson,Phys.Rev.Lett.110,
112501 (2013).