R.A. Broglia

Milano University and

The Niels Bohr Institute, Copenhagen

F. Barranco

Sevilla University

Par$cle-vibra$oncouplinginthecon$nuumandthespectrumof11Be

G. Potel

Michigan State University

Figure 1. Relative energy of the 1d5/2 (green), 2s1/2 (blue) and 1p1/2 (red) shells for the

N = 7 isotones in the region of 10Li.

Transfer reactions are essential tools to probe selected components of the nuclear wave functions. Thanks to that they have been used in the past and are still crucial for nuclear spectroscopy purposes. With the advent of radioactive beam facilities, the new opportunity to explore nuclear phenomena far from the stability valley has driven a renewed interest to transfer reactions, to be studied in inverse kinematics. On one hand there is a specific attention to understand how the transfer mechanism is influenced by the reduction of the binding energy of the projectile [3,4]. On the other hand, the measured spectra could reveal unexpected features due to nucleon correlations which are beyond the mean field description of nuclear structure [5]. Examples of such phenomena have been recently found in light neutron rich nuclei such as the Li isotopes.

2. Past attempts to study the d(9Li,p)10Li reaction Recently 10Li has been the subject of different theoretical and challenging experimental studies [6-8], including two attempts to explore the resonant energy spectrum by the d(9Li,p)10Li transfer reaction at 2.35 AMeV incident energy at REX-ISOLDE and at 20 AMeV at NSCL-MSU, respectively.

In the experiment at 2.35 AMeV of Ref. [6], the energy spectrum of 10Li was measured up to about 1 MeV excitation energy in an angular range between 100° and 140° in the centre of mass. Due to the low beam intensity of about 5 × 104 pps, a relatively thick target of deuterated polyethylene CD2 (660 μg/cm2 thick) was used in order to maximize the yield, which slightly exceeded 100 events in total. The recoiling protons were detected by large area telescopes of position sensitive silicon detectors located between 18° and 80° in the laboratory. At backward laboratory angles, the energy of the protons was too low to be detected, thus excluding in the collected data the crucial region corresponding to forward angles in the centre of mass. Despite the low statistics and the not optimal angular range explored, an excitation energy resolution of about 300 keV (FWHM) was obtained and the authors could draw some conclusions about the role of the s1/2 and p1/2 neutron orbitals in 10Li around the neutron emission threshold.

In the NSCL-MSU experiment [7], a similar experimental technique was used but at higher incident energy (20 AMeV). The beam intensity was even lower than in the REX-ISOLDE experiment (~ 7 × 103 pps) and the emittance considerably worse. The CD2 target was about 2 mg/cm2 thick. The recoiling protons were measured at backward angles by a series of large area silicon detectors in

2

Theparityinversionbetween½+and½-

11Be within NCSMC: Discrimination among chiral nuclear forces

24

Feb 17 2016 Angelo Calci

11Be with NCSMC

2

exp.

n+10Be(0+)

Robert Roth - TU Darmstadt - February 2015

9Be: NCSM vs. NCSMC

! NCSMC shows much better Nmax convergence

! NCSM tries to capture continuum effects via large Nmax

! drastic difference for the 1/2+ state right at threshold

10

6 8 10 12 exp. 12 10 8 6

-2-1

01

2

34

5

6

7

8

.

Eth

r.[M

eV]

(a)

NCSM NCSMC

7 9 11 exp. 11 9 7Nmax Nmax

0123456789

10

.

Eth

r.[M

eV]

(b)

NCSM NCSMC

6 8 10 12 exp. 12 10 8 6

-2-1

01

2

34

5

6

7

8

.

Eth

r.[M

eV]

(a)

NCSM NCSMC

7 9 11 exp. 11 9 7Nmax Nmax

0123456789

10

.

Eth

r.[M

eV]

(b)

NCSM NCSMC

NCSM NCSMC NCSM NCSMCnegative parity positive parity

Nmax Nmax Nmax Nmax

NN+3

N ful

lα

= 0

.062

5 fm

4 , ħΩ

= 2

0 M

eV,

E3m

ax =

14

Langhammer, Navrátil, Quaglioni, Hupin, Calci, Roth; Phys. Rev. C 91, 021301(R) (2015)

E thr

. [MeV

]3Nind 3N(400)

n+10Be(2+)

n+10Be(2+)

N2LOsatexp. 3N(350)

exp.3Nind 3N(400) N2LOsatexp. 3N(350)

7

-1

0

1

2

3

4

5

6

.

Eχ

[MeV

]

8

1/2+1/2-

5/2+

3/2-

3/2-

5/2-

3/2+

preliminary

A. Calci, P. Navratil, G. Hupin, S. Quaglioni, R. Roth et al., in preparation

Parity inversion

9/2+

Typicalsphericalmean-fieldresultswithSkyrmeforces(Sagawa,Brown,EsbensenPLB309(93)1)

ParityinversioninN=7isotonesisnotreproducedbysphericalmeanfieldcalcula$ons

Apossibleexplana$onofparityinversion:dynamicalcouplingbetweenthecoreandthelooselyboundneutron

Thecore:sphericalordeformed?Importantroleoffluctua$onsexpectedNeedforadynamicaldescrip$on

G.F.EsbensenandH.Sagawa,Phys.RevC51(1995)1274F.M.NunesandI.Thompson,Nucl.Phys.A703(2002)593Blanchonetal.,PhysRev.C82(2010)034313Myoetal,PRC86(2012)024318

S.For$eretal.Phys.Le_.B461(1999)22J.S.Winfieldetal.,Nucl.Phys.A683(2001)48

T.Aumannetal.PRL84(2000)35

p(11Be,10Be)d9Be(11Be,10Be+γ)X

Theadmixtureofd5/2x2+configura$oninthe1/2+g.s.of11Beisabout15%

J.S.Winfieldetal.,Nucl.Phys.A683(2001)48

Acarefulanalysisoftransferreac$onsisneededtoes$matephononadmixturesinthewavefunc$ons

62 J.S. Winfield et al. / Nuclear Physics A 683 (2001) 48–78

Fig. 7. Theoretical angular distributions calculated under the DWBA obtained with single-particleSE form factors for states in 10Be. The points are the experimental angular distributions.

calculation. The largest-angle points were not used in the extraction of spectroscopicfactors in Ref. [21], neither are they so used in the present paper.

5. Analysis of angular distributions

5.1. Optical-model potentials

Different combinations of optical potentials for the entrance and exit channels have beentried in the calculations presented below, in order to test the sensitivity of the extractedspectroscopic factors to the input parameters. All the optical potentials used in the presentanalysis have the standard Woods–Saxon or Woods–Saxon derivative form.For the entrance channel, three principal optical potentials have been used. The most-

recent global nucleon–nucleus optical parameterisation is the “CH89” one of Varner etal. [44]. This has dependences on energy, mass and isospin, adjusted for a range of stablenuclei from masses A = 40 to 209. However, data from recent proton elastic scattering

J.S. Winfield et al. / Nuclear Physics A 683 (2001) 48–78 49

Keywords: NUCLEAR REACTIONS; 1H(11Be, 10Be), E/A = 35.3 MeV, 1H(15N, 14N) E/A = 38.9 MeV;Measured σ (θ); Deduced spectroscopic factors; Vibrational coupling model; Radioactive beam

1. Introduction

The nucleus 11Be is of especial interest for several reasons. As is well known, theground-state spin parity is 1/2+ in contradiction to the simple shell model and sphericalHartree–Fock prediction of 1/2−. This “parity inversion” is correctly predicted by, forexample, recent psd-shell calculations of Brown [1]. The 2s1/2 intruder orbital is loweredby the noncentral part of the particle–hole interaction [2]. Moreover, 11Be is often regardedas the classic one-neutron halo nucleus: the small single-neutron separation energy of505 keV together with an assumed s-wave nature of the valence neutron leads to a veryextended spatial distribution [3,4].Several calculations of the 11Be ground-state structure have been performed. The

theoretical approaches include: the shell model [5,6], the variational shell model [7], theGenerator Coordinate model [8], and coupling of the neutron with a vibrational [9–11] orrotational core [12,13]. Most of these models correctly reproduce the parity inversion andhigh-energy reaction data, but make very different predictions about the degree of couplingof an s1/2 neutron to the 10Be 0+ ground-state core relative to a d5/2 neutron coupled toa 2+ excited core (the first excited state of 10Be at 3.368 MeV).A direct test of the models for the structure of 11Begs may be made by measuring the

relative cross sections of one-neutron pick-up reactions feeding the 0+ and 2+ statesof 10Be. Transfer cross sections depend on the overlap between the wave functionsof the initial and final states through the radial neutron form factors ulj (r). Standarddistorted wave Born approximation (DWBA) analyses assume that these form factors areproportional to single-particle wave functions U

splj (r), so that one may calculate cross

sections independently of any prior assumption about the structure of initial and final states,apart from an overall normalisation factor. The latter is the spectroscopic factor, which isdefined as the product of the overlap integral

!u2lj (r)r

2 dr and a factor (n+ 1) [14], wheren in the present case is the neutron occupation number of the 2s1d shell in 10Be. If oneexpresses the wave function of the 1/2+ 11Be ground state as the sum of the single-particleand core excited components

""11Begs#= α

""10Be$0+%

⊗ 2s#+β

""10Be$2+%

⊗ 1d#, (1)

the spectroscopic factors S(0+) and S(2+) for transfer to the ground and first excitedstate of 10Be should be directly related to α2 and β2, respectively, assuming negligiblepopulation of the 2s1d orbitals by 10Be core neutrons. 2 Table 1 gives spectroscopic factorsdeduced from the various models cited above. These spectroscopic factors vary widely. Forexample, the standard Shell Model [5,6] predicts S(0+) = 0.74 and S(2+) = 0.19, while

2 Strictly speaking, α and β should be equal to the fractional parentage coefficients, the squares of which addup to unity. The relation between these and spectroscopic factors is given in Appendix A and Ref. [14].

Goodagreementwith2+crosssec$onsisobtainedinDWBAwithβ2=0.17consideringthecouplingeffectsonthetransferfromfactor;usingβasasimplespectroscopicfactoronefindsβ2=0.28

Dynamicaldescrip$onbasedonpar$cle-rotormodel(closedcore)

174 F.M. Nunes et aLI Nuclear Physics A 596 (1996) 171-186

where/3 is the deformation of the core, and we include a standard spin-orbit term,

( h ) z _~ d [ ( r - R s o ) ] - ' Vso(r) = - ~ (2 / - s ) drr 1 +exp (6)

\ aso / J

The potential used assumes that the nuclear interaction is the same for different states of the core (static rotor approximation). We expand Vw,~(r, 0, ~b) in spherical harmonics permitting again the separation of the valence neutron variables from the core variables when calculating the potential matrix elements. As another approximation one could have also used derivative expressions, but for large deformation the shapes of the form factors are not as diffuse as those obtained through the spherical harmonic expansion.

In our calculations we assume that the lsl/2 and the lp3/2 shells are full. We examine the lowest energy states for each spin and parity, and ignore any with a large 1sl/2 and lp3/2 overlap. We understand that shell model calculations [13] predict that the lp3/2 subshell is only partially filled and our simple treatment of the Panli principle is not fully correct. Nevertheless, we find the effective interactions in the restricted space where the 1p3/2 is full, because we believe that the degree of complexity involved in the correct treatment of the Panli principle would prevent a good insight to the effect of core excitation within a two body model.

2.2. The Sturrnian basis, bound states and scattering states

In order to solve the coupled channels equation (4) we use a Sturmian basis for convenience. This method offers very good convergence not only for the eigenvalues but also for the wavefunctions, and it can easily be applied to problems where the number of channels is large, when direct integration has numerical difficulties. It has been successfully applied to several nuclear physics problems in the past [14,15]. The Sturmian eigenfunctions are generated by a Schrtktinger-like equation (Ref. [ 15], p. 301),

h 2 0 2 h 2 l ( l + 1) 2l, Z Or 2 + 2/zr 2 + O~nlji~turm(r) + ei Snlji(r) = EoSntji(r) n = O, 1 . . . .

(7)

with exponentially decaying boundary conditions at large radii for to (E0 - El) < 0. The Sturmians Sntji(r) have (n - 1) nodes and are orthogonal with respect to the

potential as weight factor. Here, for a given E0 and ei, the eigenvalues are the strengths antji of the potential for all r. One is free to choose the shape of the potential as long as it is always the same sign for all r. For values of r where the potential is non-zero, the Sturmians Sntji(r) form a complete set. A useful feature of this basis is that they can be chosen to match the known asymptotic form of the eigenfunctions of the hamiltonian.

We next expand the radial wavefunction in Sturmians

ald'JM~ Z ~-~Cij iSnzj i (r) [ [Yl(? ) @ )(i/2(Xspin)]J @dol~,(Xcore)] TM , (8) lji n

178

~/2" 7.2~.

3/2- 2.49

EM. Nunes et al./ Nuclear Physics A 596 (1996) 171-186

1"--15 k e y 3/2- 3.46 3/2+

F=125 keV 1/2,3/2.5/2 + 2.91 3/2-

F=200 keV 1/2.3/2.5/2" 2.19

3.2 1"=250 keV

2.7 1"--80 keV

F=100 keV (3/2,5/2~ + 1.28 ~/2+ 1.11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5/2_* _ _ -0~09_ _ _

1/2- -0 .18 . . . . . . . . . . . . . . . . 1/2- -0 .18 . . . . . . . . . 1/2- -0 .18

1/2 ÷ - 0 . 5 . . . . . . . . . 1/2 + - 0 . 5

Shell Model Experiment Core-Particle (Oxbash)

Fig. 3. Calculated energy levels for nBe versus a shell model calculation [ 13] and the measured values [22]. The calculations presented in this figure take into account core excitation. The dotted line separates bound states from scattering states and represents the threshold of nBe to neutron emission. Each state is denoted by its spin, parity and energy in MeV. Note that the vertical scale is approximate.

The continuum spectrum is calculated at low energies, where the number of channels is still reasonably small. The resonant energies we present here were obtained from the energy dependence of the phase shifts calculated through direct integration of the radial equations. The results for resonant energies and widths are shown in Fig. 3 and we get reasonable agreement with experiment [22] . Shell model calculations performed with OXBASH [ 13,27] using the WBN interaction [20] produce similar results for the negative parity but fail to invert the parity of the ground state.

We would like to underline that, although one could have obtained a better picture for the positive parity states (as in Ref. [ 12] ), our aim is to find an effective interaction describing all states as well as possible, which can be used in three body calculations. An interaction describing positive parity states alone is insufficient.

The B(E1) distribution from the ground state to the continuum is also calculated (Fig. 4) and compared with recent data from Coulomb dissociation [28] . The results from our theoretical calculations (solid curve) are close to the data if one takes into account the 20% uncertainty due to neutron detection efficiency and the simple model used in Ref. [28] to exclude nuclear contributions from the total cross section. We believe that more accurate data is needed for a detailed comparison. Using for the ground state a Yukawa wave function with the correct tail normalisation, and using a plane wave for the continuum, allows an analytic expression for the B(EI ) distribution (dotted curve). This last curve is increased by "~ 20% when the ground state couplings are included but is decreased by ~ 20% by final state interactions. Both ground state and continuum couplings are therefore important for the final result.

F.M. Nunes et al./Nuclear Physics A 596 (1996) 171-186 177

Table 1 Calculated r.m.s, radius of the g.s. and B(E1 ) for UBe. The first column assumes no deformation for the core (pure) [23], the second column has the results obtained using the deformable core model here developed (deformed), the third column has the results for a variational shell model calculation (VSM) [5], and the last column contains the measured values

IIBe Pure [23] Deformed VSM [5] Experiment

r.m.s, matter radius of the g.s. (fm) 3.45 2.93 2.6 2.73 4- 0.05 [24] 3.1 i 0 . 3 8 [25]

B ( E I , 1 / 2 - ~ 1/2 + ) (e 2 fm 2) 0.480 0.150 << 0.1 0.115 4- 0.01 [4]

for both the mean square radius [24] and the B(E1 ) [4]. In this table we also present the results of the VSM [5] calculations which underestimate both the radius and the B(E1) .

The binding energies within this model depend strongly on the deformation parameter of the core (see Fig. 2) and it is mainly through this dependence that the parity inversion is obtained. The ground state of ~ Be is known to be a positive parity intruder state [ 5] not predicted by the spherical mean field calculations [8] . Note that the ~o is stronger that the traditionally used values to help impose the state inversion. It is important to note that this inversion is not obtained if a standard V~o is imposed. Such a strong ~o was not needed in Ref. [ 12] because only the positive parity states were studied. The r.m.s, radius is not sensitive to fl but the B(E1 ) shows a very strong dependence.

The curves presented in Fig. 2 are very much like those from a Nilsson plot (Ref. [26] , p. 221). The main difference is that, JIBe being a halo nucleus, these states are very close to threshold. The sl/2 and Pl/2 have also become closer due to the choice of the parameters, in particular to the rather strong spin orbit force. The shape of the even parity curve for negative fl differs from the usual behaviour of the 2S1/2 c u r v e seen in Ref. [26] due to the effect of the threshold.

2 . 0 , , r , . . . . , •

1.5 " ~ . 1/2+ - - - 1 / 2 - ,%

1 . 0 \ X

\ 0 . 5 \ / s

-0.5 \

- 1 . 0 \ / \

- 1 . 5 \ . " /

- 2 . 0 ~ ' ~ J

- 2 . 5 , , , , J . . . . J , , ~ , i . . . . - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

f~

Fig. 2. Calculated energies for the ground state of i I Be ( 1/2+), and the first excited state ( 1/2- ) as functions of the deformation parameter for the core (/3).

F.Nunesetal.Nucl.Phys.A609(1996)43

Rota$onalcouplingtodeformedcore

I.Hamamoto,andS.Shimoura,J.Phys.G34(2007)2715

RAPID COMMUNICATIONS

NUCLEAR ROTATION IN THE CONTINUUM PHYSICAL REVIEW C 93, 011305(R) (2016)

1/2 3/2 5/2 7/2 9/2 11/2 13/2J

0

5

10

Energy(MeV)

11Ber = +i

r = -i

E4+d

E2+d

E0+d

FIG. 1. Calculated yrast bands of 11Be with J ! 13/2 withsignature r = −i (squares) and r = +i (triangles) compared to theexperimental ground-state band of 10Be core (horizontal dashedlines). Some excited (yrare) states of 11Be are marked by stars.

with jr " 4. (Experimentally the 2+1 state in 10Be is particle

bound, and the 4+1 level has a fairly small width of 121 keV.)

The parameters of the deformed pseudopotential have beenfit to the 1/2+ and 5/2+ members of the yrast band of11Be [36]. We checked that these states collapse to the sameband-head energy in the adiabatic limit (I → ∞), i.e., they aremembers of the same rotational band. It is worth noting thatthe location of higher-lying band members is very uncertain,although candidates for the 3/2+

1 , 7/2+1 , and 9/2+

1 stateshave been suggested [37,38]. The optimized parameters ofthe WS pseudopotential are radius R0 = 2.08 fm; diffusenessa = 1.1 fm; strength V0 = −59.36 MeV; spin-orbit strengthVso = 15.09 MeV; and quadrupole deformation β2 = 0.53. Wechecked that the general conclusions of our study are inde-pendent of the precise values of these parameters, as long asthe one-neutron threshold is reproduced. For instance, similarresults were obtained by using a = 0.77 fm, R0 = 2.55 fm,and β2 = 0.52.

Results. The calculated lowest states of 11Be are shownin Fig. 1. The large splitting between the favorite-signatureband [r = exp(−iπJ ) = −i] and unfavored band (r = +i)is consistent with the results of a microscopic multiclustermodel [30,39] and large-scale shell model [19,40]. It isinteresting to note that Q(J,jr ) < 0 for |J − jr | = 1/2 for theyrast states in 11Be. For instance, the 3/2+

1 and 5/2+1 levels of

11Be are predicted to lie below the yrast 2+ state of 10Be. Thismeans that the ℓ = 0 neutron emission channel is blocked forboth r = −i and r = +i bands.

The rotational structure of the ground-state band of 11Be isrevealed by looking at the weights of individual K componentsof valence neutron density. It turns out that the K = 1/2component is dominant in most cases; in particular for thefavored band. For the 7/2+, 11/2+, and 15/2+ states, theK = 5/2 and 3/2 components dominate. In most cases, anappreciable degree of K mixing is predicted. This suggeststhat a “K = 1/2” label often attached to this band should betaken with a grain of salt.

To get further insight into the structure of the yrast lineof 11Be, the density ρJK (r) of the valence neutron in the coreframe is plotted in Fig. 2 for selected states and K components.In the adiabatic limit, we checked that all band members

0.0 0.2 0.4 0.6 0.8 1.0x(fm)

0.0

0.2

0.4

0.6

0.8

1.0

z(fm

)

-5

0

51/2+(a) (b)

(c) (d)

(e) (f)

I→ ∞

1/2+

K = 1/2

-5

0

53/2+

K = 1/2

5/2+

K = 1/2 0.0

3.0

6.0

8.9

-5 0 5

-5

0

57/2+

K = 5/2-5 0 5

15/2+

K = 3/2 0.0

0.6

1.2

FIG. 2. Partial densities ρJK (r) (in 10−2 fm−3) with K = 1/2 forthe three first ground-state band members of 11Be in panels (b)–(d),and for selected states of the unfavored r = i band (in 10−3 fm−3)in panels (e) J = 7/2, K = 5/2 and (f) J = 15/2,K = 3/2. Theadiabatic limit (I → ∞) is shown in panel (a).

have indeed the same K = 1/2 intrinsic density shown inFig. 2(a), whereas all K > 1/2 components vanish. For thefavored band, for which the K = 1/2 component dominates,the densities ρJK=1/2 for J = 9/2, 13/2, and 17/2 are similarto that for the 5/2+ state shown in Fig. 2(d). This is not the casefor the 7/2+, 11/2+, and 15/2+ band members, which showan appreciable K mixing. Indeed, as illustrated in Figs. 2(e)and 2(f), the dominant neutron distributions for the 7/2+ and15/2+ states are very different. Figure 2 also shows that thecalculated valence neutron states in 11Be are all fairly welllocalized within the range of the neutron-core potential. Wechecked that this holds for the yrare states as well.

Figure 3(a) shows the (real part) of the norms nℓj =!jr

nℓjjrfrom various partial waves (ℓj ) to the yrast band

of 11Be. It is seen that the alignment pattern of the valenceneutron is governed by a transition from the s1/2 wave, whichdominates at low spins, to d5/2, which governs the rotation athigher angular momenta. In the high-spin region, J " 7/2,the yrast line of 11Be can be associated with the weakcoupling of neutron’s angular momentum, j = 5/2, to theangular momentum jr of the core resulting in the full rotationalalignment of j with j r . Indeed, as seen in Fig. 1, the computedenergies EJ of the r = −i band appear close to the energiesof 10Be with jr = J − 5/2. The higher partial waves, such asg9/2, appear for J > 15/2. The structure of yrare states shownin Fig. 3(c) is dominated by d5/2 at low spins, and by s1/2 athigher angular momenta.

To estimate one-neutron decay rates, we compute decaywidths. An accurate way to compute the total decay width isto use the so-called current expression [26,27,41], that gives

011305-3

K.Fossezetal,PRC93011305(2016)

Nilssonmodel

Rota$onalbandsInthecon$nuum

TheNFTHamiltonian:

H=Hc+Hp+Hint

1

The NFT Hamiltonian:

H = Hc +Hp +Hint. (1)

Hc =X

µ

h![+.µµ + 1/2] (2)

Hp = h2/2m d2/dr2 + V (r) + Vls(r) (3)

Hint. =X

µ

rdV/drYµ[+.µ + (1)µµ] (4)

The channels:

H a = E a (5)

a = [ xa + ( C

b · )ja + ...]AGS (6)

xa = (Rx

a(r)/r)jama (7)

( Cb · )ja = (RC

b (r)/r)(jb · )ja (8)

where AGS is the ground state of the nucleus of even mass number A (we assume that it

can be described by a single Slater determinant), while denotes the wave function of an

excited state (phonon), calculated using the Tamm-Danco↵ approximation. Let us assume

the simple case of two channels, a and b :

[ h2

2m

@2

@r2+ Va(r) + 0h!]Rx

a(r) + a,b(rdV/dr)RCb (r) = ERx

a(r) (9)

a,b(rdV/dr)Rxa(r) + [ h2

2m

@2

@r2+ Vb(r) + 1h!]RC

b (r) = ERCb (r) (10)

where

Va(r) = V (r) + Vls(r) + Vcent(r) (11)

and

a,b = hjama

X

µ

Yµ[+.µ + (1)µµ][jb · ]jamai (12)

We can project these equations on a complete single-particle basis (labeled by quantum

Collec$ve(𝝘𝝺𝝻+createsaphonon)

Single-par$cle

Linearinterac$on

Def.parameter

1

The NFT Hamiltonian:

H = Hc +Hp +Hint. (1)

Hc =X

µ

h![+.µµ + 1/2] (2)

Hp = h2/2m d2/dr2 + V (r) + Vls(r) (3)

Hint. =X

µ

rdV/drYµ[+.µ + (1)µµ] (4)

The channels:

H a = E a (5)

a = [ xa + ( C

b · )ja + ...]AGS (6)

xa = (Rx

a(r)/r)jama (7)

( Cb · )ja = (RC

b (r)/r)(jb · )ja (8)

where AGS is the ground state of the nucleus of even mass number A (we assume that it

can be described by a single Slater determinant), while denotes the wave function of an

excited state (phonon), calculated using the Tamm-Danco↵ approximation. Let us assume

the simple case of two channels, a and b :

[ h2

2m

@2

@r2+ Va(r) + 0h!]Rx

a(r) + a,b(rdV/dr)RCb (r) = ERx

a(r) (9)

a,b(rdV/dr)Rxa(r) + [ h2

2m

@2

@r2+ Vb(r) + 1h!]RC

b (r) = ERCb (r) (10)

where

Va(r) = V (r) + Vls(r) + Vcent(r) (11)

and

a,b = hjama

X

µ

Yµ[+.µ + (1)µµ][jb · ]jamai (12)

We can project these equations on a complete single-particle basis (labeled by quantum

1

The NFT Hamiltonian:

H = Hc +Hp +Hint. (1)

Hc =X

µ

h![+.µµ + 1/2] (2)

Hp = h2/2m d2/dr2 + V (r) + Vls(r) (3)

Hint. =X

µ

rdV/drYµ[+.µ + (1)µµ] (4)

The channels:

H a = E a (5)

a = [ xa + ( C

b · )ja + ...]AGS (6)

xa = (Rx

a(r)/r)jama (7)

( Cb · )ja = (RC

b (r)/r)(jb · )ja (8)

where AGS is the ground state of the nucleus of even mass number A (we assume that it

can be described by a single Slater determinant), while denotes the wave function of an

excited state (phonon), calculated using the Tamm-Danco↵ approximation. Let us assume

the simple case of two channels, a and b :

[ h2

2m

@2

@r2+ Va(r) + 0h!]Rx

a(r) + a,b(rdV/dr)RCb (r) = ERx

a(r) (9)

a,b(rdV/dr)Rxa(r) + [ h2

2m

@2

@r2+ Vb(r) + 1h!]RC

b (r) = ERCb (r) (10)

where

Va(r) = V (r) + Vls(r) + Vcent(r) (11)

and

a,b = hjama

X

µ

Yµ[+.µ + (1)µµ][jb · ]jamai (12)

We can project these equations on a complete single-particle basis (labeled by quantum

Odddressedneutron

Single-par$clepart

Phononadmixture

p1/2p3/2

2+Eshim=+2.5MeV

Pauli blocking of core ground state correlations

Self-energy

Eshim=-2.5MeV

2+

s1/2

s1/2

d5/2

3MeV 0MeV

½-

½+

Level inversion

+

11Be

p1/2

N.VinhMau,Nucl.Phys.A592(1995)33

Thespin-orbitstrengthisdoubledforp-states

First-orderanddiagonalapproxima$on

Fermionic degrees of freedom:

•  s1/2, p1/2, d5/2 Wood-Saxon levels in a box, with a unique Woods-Saxon potential

Bosonic degrees of freedom:

•  2+ and 3- QRPA solutions; residual interaction: multipole-multipole separable with the coupling constant tuned to reproduce E(2+)=3.36 MeV and β2

nuc ∼ 0.65, β2Coul ∼ 1.15 in 10Be

Main ingredients of our calculation 11Be

included the coupling to the 2+ and 3− phonons; neglectingthe 3− states only slightly affects the results.The interweaving of single-particle motion and of collec-

tive degrees of freedom arising from the particle-vibrationHamiltonian has been treated within the framework of theNFT making use of Bloch-Horowitz perturbation theory[12,15]. The eigenvalues of the dressed single-particle stateswere thus obtained by diagonalizing (energy dependent) ma-trices of the order of 102!102 (cf. Fig. 1) whose elementsconnect a basis of unperturbed states containing both boundand continuum solutions of Eq. (3), corresponding to thequantum numbers s1/2 ,p1/2, and d5/2.The process depicted in Fig. 1(b) describes the emission

and reabsorption of a phonon by a valence neutron havingangular momentum and parity j", and leads to an increase ofthe binding energy of the neutron. The process of Fig. 1(c)accounts instead for the indistinguishibility of the 10Be coreand of the valence neutrons, that is, for the Pauli principle.Due to the particle-vibration coupling, there will be pro-cesses contributing to the ground state energy of 10Be, inwhich a neutron, initially in a hole state below the Fermienergy, virtually emits a phonon and jumps into an unoccu-pied state j" above the Fermi energy, finally returning to itsinitial state by reabsorbing the phonon. This process be-comes forbidden when a valence neutron is added in the statej" to form 11Be. Therefore the correlation energy associatedto the core fluctuation must be subtracted in calculating thebinding energy of the valence neutron in 11Be, and this isprecisely the contribution of Fig. 1(c) [9].The self-energy process (b) represented in Fig. 1 involves

mainly the positive parity 1 /2+ and 5/2+ states lying close tothe Fermi energy, lowering their energy through the couplingto the 2+ core excitation. On the other hand, the contributions

associated to the graph (c) of Fig. 1, concern mostly the 1/2−state, and lead to a reduction of its binding energy, becauseits partial occupation in 11Be and in 12Be blocks the groundstate correlations of the 10Be core. In fact, the blocking oftransitions from the hole state 1p3/2 to the particle state 1p1/2reduces the energy of the 1/2− state by about 3 MeV, andturns out to be essential to explain the parity inversion ob-served in 11Be. The final results for the single-particle ener-gies are listed in Table I, in comparison with the experimen-tal findings. The calculated matter radius is 3.2 fm, to becompared with the experimental value of 2.9±0.05 [16]. Theresulting wave function for the 1/2+ ground state has a 15 %admixture with the !d5/2!2+;1 /2+" configuration. This valuecompares well with the experimental information which hasrecently become available through the measurement of the!11Be,10Be+#" knockout reaction [17] and of the populationof the 2+ state of 10Be in the p!11Be,10Be"d one-neutrontransfer reaction [19] (in the latter case a “best estimate” of16 % was obtained for the admixture). From our wave func-tion we have also obtained the spectroscopic factors S#1/2+$and S#1/2−$ associated with the reaction 10Be!d ,p"11Be[20,21], populating both the 1/2+ and 1/2− levels in 11Be;the calculation is discussed below in connection with a simi-lar calculation carried out for the reaction 12Be!9Be,9Be+n+#"11Be [7]. The value of S#1/2+$ can also be inferred fromthe population of the 0+ state in the p!11Be,10Be"d one-neutron transfer reaction [19]. The extraction of the spectro-scopic factors depends on the theoretical model assumed forthe reaction analysis so that the experimental values listed inTable I are subject to a rather large uncertainty. In particular,the analysis of Ref. [22] leads, in the case of 11Be, to muchsmaller values !S#1/2+$=0.36–0.44".

TABLE I. Comparison of experimental binding energy and spectroscopic factors with those resultingfrom our calculation and from an independent particle model. For 12Be, we also show the components of theresulting ground state wave function. The experimental spin-parity assignment of the state at 1.28 MeV in11Be is only tentative [18,21].

TheoryExpt. Particle vibration Mean field

Es1/2 −0.504 MeV −0.48 MeV %0.14 MeVEp1/2 −0.18 MeV −0.27 MeV −3.12 MeV

411Be7 Ed5/2 1.28 MeV %0 MeV %2.4 MeV

S#1/2+$ 0.65–0.80 [19] 0.87 10.73±0.06 [20]0.77 [21]

S#1/2−$ 0.63±0.15 [20] 0.96 10.96 [21] 1

S#5/2+$ 0.72 1

S2n −3.673 MeV −3.58 MeV −6.24 MeV

412Be8 s2 ,p2 ,d2 23% ,29% ,48% 0% ,100% ,0%

S#1/2+$ 0.42±0.10 [7] 0.31 0S#1/2−$ 0.37±0.10 [7] 0.57 2

PARITY INVERSION AND BREAKDOWN OF SHELL… PHYSICAL REVIEW C 69, 041302(R) (2004)

RAPID COMMUNICATIONS

041302-3

G.Gorietal.,Phys.Rev.C69(1994)041302

1

The NFT Hamiltonian:

H = Hc +Hp +Hint. (1)

Hc =X

µ

h![+.µµ + 1/2] (2)

Hp = h2/2m d2/dr2 + V (r) + Vls(r) (3)

Hint. =X

µ

rdV/drYµ[+.µ + (1)µµ] (4)

The channels:

H a = E a (5)

a = [ xa + ( C

b · )ja + ...]AGS (6)

xa = (Rx

a(r)/r)jama (7)

( Cb · )ja = (RC

b (r)/r)(jb · )ja (8)

where AGS is the ground state of the nucleus of even mass number A (we assume that it

can be described by a single Slater determinant), while denotes the wave function of an

excited state (phonon), calculated using the Tamm-Danco↵ approximation. Let us assume

the simple case of two channels, a and b :

[ h2

2m

@2

@r2+ Va(r) + 0h!]R

xa(r) + a,b(rdV/dr)RC

b (r) = ERxa(r) (9)

a,b(rdV/dr)Rxa(r) + [ h2

2m

@2

@r2+ Vb(r) + 1h!]R

Cb (r) = ERC

b (r) (10)

where

Va(r) = V (r) + Vls(r) + Vcent(r) (11)

and

a,b = hjama

X

µ

Yµ[+.µ + (1)µµ][jb · ]jamai (12)

We can project these equations on a complete single-particle basis (labeled by quantum

1

The NFT Hamiltonian:

H = Hc +Hp +Hint. (1)

Hc =X

µ

h![+.µµ + 1/2] (2)

Hp = h2/2m d2/dr2 + V (r) + Vls(r) (3)

Hint. =X

µ

rdV/drYµ[+.µ + (1)µµ] (4)

The channels:

H a = E a (5)

a = [ xa + ( C

b · )ja + ...]AGS (6)

xa = (Rx

a(r)/r)jama (7)

( Cb · )ja = (RC

b (r)/r)(jb · )ja (8)

where AGS is the ground state of the nucleus of even mass number A (we assume that it

can be described by a single Slater determinant), while denotes the wave function of an

excited state (phonon), calculated using the Tamm-Danco↵ approximation. Let us assume

the simple case of two channels, a and b :

[ h2

2m

@2

@r2+ Va(r) + 0h!]Rx

a(r) + a,b(rdV/dr)RCb (r) = ERx

a(r) (9)

a,b(rdV/dr)Rxa(r) + [ h2

2m

@2

@r2+ Vb(r) + 1h!]RC

b (r) = ERCb (r) (10)

where

Va(r) = V (r) + Vls(r) + Vcent(r) (11)

and

a,b = hjama

X

µ

Yµ[+.µ + (1)µµ][jb · ]jamai (12)

We can project these equations on a complete single-particle basis (labeled by quantum

Thecoupledequa$ons(nogroundstatecorrela$ons):

Thecoupling:

OnecanexpandoveraHFbasisinabox(onlyunoccupiedlevels,eai>eF):

2

numbers ai, i = 1...N , with energies eai), obtained by solving the Hartree-Fock equation

in a box. This basis is characterised by an e↵ective mass mk (Fock exchange potential) .

We expand the solutions on wave functions lying above Fermi energy, (eai and ebi > eF ):

Rxa(r) =

X

i

xaiRHFai

(r) ; RCb (r) =

X

i

CbiRHFbi

(r) (13)

obtaining

0

BBBBBB@

ea1 0 ha1,b1 ha1,b2

0 ea2 ha2,b1 ha2,b2

ha1,b1 ha2,b1 eb1 + h! 0

ha1,b2 ha2,b2 0 eb2 + h!

1

CCCCCCA

0

BBBBBB@

xa1

xa2

Cb1

Cb2

1

CCCCCCA= Ea(n)

0

BBBBBB@

xa1

xa2

Cb1

Cb2

1

CCCCCCA(14)

where

hai,bj = a,b

ZRHF

ai (r)(rdV/dr)RHFbj (r)dr (15)

This condition in fact makes the basis incomplete, and thus the found solution does not

strictly verifies the di↵erential equation. In some way the Pauli exclusion condition can be

thought as a repulsive interaction which inhibits to some extent the impinging particle to

penetrate the nuclear volume, a repulsion which in fact alters the waves.

People solving the di↵erential equation directly in rspace impose the orthogonality

condition < Rxa|RHF

i >= 0, < RCa |RHF

i >= 0 for any of the i-occupied states, which is

exactly equivalent to our condition eai and ebi > eF in the RHF expansion.

Generalised CCC

This generalisation is required in order to take into account ground state correlations using

RPA rather than TDA, also in order to conserve the EWSR. In this case one introduces a

radial wave function RD expanded over the occupied states:

RDc (r) =

X

i

DciRWSci (r); eci < eF (16)

and obtains

2

numbers ai, i = 1...N , with energies eai), obtained by solving the Hartree-Fock equation

in a box. This basis is characterised by an e↵ective mass mk (Fock exchange potential) .

We expand the solutions on wave functions lying above Fermi energy, (eai and ebi > eF ):

Rxa(r) =

X

i

xaiRHFai

(r) ; RCb (r) =

X

i

CbiRHFbi

(r) (13)

obtaining

0

BBBBBB@

ea1 0 ha1,b1 ha1,b2

0 ea2 ha2,b1 ha2,b2

ha1,b1 ha2,b1 eb1 + h! 0

ha1,b2 ha2,b2 0 eb2 + h!

1

CCCCCCA

0

BBBBBB@

xa1

xa2

Cb1

Cb2

1

CCCCCCA= Ea(n)

0

BBBBBB@

xa1

xa2

Cb1

Cb2

1

CCCCCCA(14)

where

hai,bj = a,b

ZRHF

ai (r)(rdV/dr)RHFbj (r)dr (15)

This condition in fact makes the basis incomplete, and thus the found solution does not

strictly verifies the di↵erential equation. In some way the Pauli exclusion condition can be

thought as a repulsive interaction which inhibits to some extent the impinging particle to

penetrate the nuclear volume, a repulsion which in fact alters the waves.

People solving the di↵erential equation directly in rspace impose the orthogonality

condition < Rxa|RHF

i >= 0, < RCa |RHF

i >= 0 for any of the i-occupied states, which is

exactly equivalent to our condition eai and ebi > eF in the RHF expansion.

Generalised CCC

This generalisation is required in order to take into account ground state correlations using

RPA rather than TDA, also in order to conserve the EWSR. In this case one introduces a

radial wave function RD expanded over the occupied states:

RDc (r) =

X

i

DciRWSci (r); eci < eF (16)

and obtains

2

numbers ai, i = 1...N , with energies eai), obtained by solving the Hartree-Fock equation

in a box. This basis is characterised by an e↵ective mass mk (Fock exchange potential) .

We expand the solutions on wave functions lying above Fermi energy, (eai and ebi > eF ):

Rxa(r) =

X

i

xaiRHFai

(r) ; RCb (r) =

X

i

CbiRHFbi

(r) (13)

obtaining

0

BBBBBB@

ea1 0 ha1,b1 ha1,b2

0 ea2 ha2,b1 ha2,b2

ha1,b1 ha2,b1 eb1 + h! 0

ha1,b2 ha2,b2 0 eb2 + h!

1

CCCCCCA

0

BBBBBB@

xa1

xa2

Cb1

Cb2

1

CCCCCCA= Ea(n)

0

BBBBBB@

xa1

xa2

Cb1

Cb2

1

CCCCCCA(14)

where

hai,bj = a,b

ZRHF

ai(r)(rdV/dr)RHF

bj(r)dr (15)

This condition in fact makes the basis incomplete, and thus the found solution does not

strictly verifies the di↵erential equation. In some way the Pauli exclusion condition can be

thought as a repulsive interaction which inhibits to some extent the impinging particle to

penetrate the nuclear volume, a repulsion which in fact alters the waves.

People solving the di↵erential equation directly in rspace impose the orthogonality

condition < Rxa|RHF

i >= 0, < RCa |RHF

i >= 0 for any of the i-occupied states, which is

exactly equivalent to our condition eai and ebi > eF in the RHF expansion.

Generalised CCC

This generalisation is required in order to take into account ground state correlations using

RPA rather than TDA, also in order to conserve the EWSR. In this case one introduces a

radial wave function RD expanded over the occupied states:

RDc (r) =

X

i

DciRWSci (r); eci < eF (16)

and obtains

Theini$al2s1/2stateisnotbound.Amerthefirstdiagonaliza$on,the½+statebecomesbound,1/2+=0.95s1/2+0.3d5/2x2+whilethe5/2+statecontainsanadmixtureofthe2s1/2x2+configura$on.Byitera$ng,inthecalcula$onofthedressed5/2+statewetakeintoaccountthe2-phononcontribu$onandthefactthatthe1/2+stateislocalized.

Inthepreviouscalcula$on,many-phononcontribu$onshavebeenneglected

Empiricalrenormaliza$on:insteadofitera$ng,takeempiricalstatesasintermediatestates,andchecktheconsistencyofthefinalsolu$on

Es$mateofthequadrupolecouplingstrengthbetweenaneutronandthe10Becore

B(E2)=10.4±1.2e2fm4 𝛿p=1.80±0.25fm

The particle-vibration coupling strength is proportional to

nRdUdr = n

dUdr

.

The deformation length n = R for a neutron or a proton is estimated

by

n 4

3eR

0.7Mp + 0.3Mn

0.7Z + 0.3Np

4

3eR

0.3Mp + 0.7Mn

0.3Z + 0.7N

From the experimental E2 transition strength,

em =

4

3eR

Mp

Z= 2.92 fm

From inelastic proton scattering, p = 1.80 fm

Then n = 2.35 fm.

Using R = 1.2(10)1/3 = 2.58 fm, n = 0.9

1

Parameterop$miza$on

Weperformthemany-bodycalcula$onforagivensetofparameters:-Depth,diffusenessoftheneutron-corepoten$alandstrengthofspin-orbitcoupling,-Effec$vemassatthecenterofthenucleus,mk(r=0)-Strengthofthequadrupoleandoctupolecoupling

andthenvarytheparameterstoobtainthebestagreementwiththeexperimental1/2+,1/2-and5/2+statesin11Be

Renormaliza$onprocessesac$ngons-,p-andd-wavesCouplingtopairvibra$onsincluded

Poten$alparametersarereferredastandardparameteriza$onoftheneutron-corepoten$al(Bohr-Mo_elson)

4 The charge radius

.The charge radius in Be isotopes has been precisely measured. In particular, R

c

(10 Be)= 2.361 fm and R

c

(11 Be) = 2.466 fm,In the independent particle limit, adding a neutron to 11Be in an orbital of radius r

n

3 fm, should lead to a very small modification of the charge radius of 10Be due to the recoilof the center of mass:

< r

2>11Be

=< r

2>10 Be+ (r

n

/11)2 2.3612 + 0.07fm2 (13)

The coupling to phonons brings an extra correction which is proportional to the admixture↵ of the 2+ phonon in the ground state. One obtains

< r

2>11Be

=< r

2>10Be

+(3/11)2 + ↵

2< r

2>10Be

(14)

= 2.3612 + 0.07 + 0.2 0.72 2.3612 = 2.3612 + 0.07 + 0.546 = 6.19 = (2.49)2fm2, (15)

where we have used the value 2 = 0.7 for the deformation of the 10Be, and which is in goodagreement wth the experimental value. Actually, the charge radius is probably the best testof the admixture factor.

5 Calculations using a restricted number of single-particle

levels

.Using the program SIMAN, we can study in detail the renormalization e↵ects using a

basis of just five levels :1p3/2 (occupied in 10Be) and 2s1/2, 1p1/2, 1d5/2 and 1d3/2 (empty).The deep 1s1/2 level is not included.We use a reference Saxon-Woods potential, with V0 = 43.9 MeV ; R= 2.693 fm, a = 0.7

fm, which produces the following single-particle energies in a box of R = 20 fm (in MeV):

1p3/2= - 8.05; 2s1/2= 0.21; 1p1/2= - 3.25; 1d5/2= 1.54; d3/2

= 1.70,

where the positive energies of course depend on the box radius.The values of the single-particle energies are similar to those obtained according to Bohr-

Mottelson’s parametrisation of the Saxon-Woods potential, which is slightly more attractive:

V0 = 51 +33(N Z)

A

MeV ; a = 0.67 fm; ; R = 1.27A1/3, (16)

corresponding to V0 = 44.4 MeV ; R= 2.74 fm for Z = 4 and A = 10, are

1p3/2 = - 8.95 ; 2s1/2 = -0.17 ; 1p1/2 = - 3.91 ; 1d5/2 = 1.44 ; d3/2

= 1.70.

The phase shifts obtained in the two potentials are shown in Fig. 1. It is seen that thed5/2 phase shift has a resonant character; 5/2 = /2 for E 3.5 MeV. The old value of 5MeV was produced using a=0.8 fm and V

ls

= 0.9.

8

(𝛃2)0=0.67

Op$malparameters(notunique!)

m

k

(0)/m 2/(2)0 3/(3)0 V

WS

/(VWS

)0 V

ls

/(Vls

)0 a R

WS

R

n

2Rn

V

WS

0.7 1.3 0.8 1.60 0.8 0.81 2.10 2.69 126.6

Table 3: Optimal values of the deformation parameters 2, 3, and of the depth, radius,di↵useness and spin-orbit strength of the mean field potential V ,V

ls

,a,R, determined for agiven value of the e↵ective mass m

k

(0) at the centre of the nucleus. For each value of mk

(0),we compare the results obtained using fragmented or non fragmented (no frag.) intermediatestates. The values of V and V

ls

are given as the ratio to the parameters of the referencepotential introduced in Section 5 ((V

WS

)0 = 43.9 MeV), and the deformation parameteras the ratio to the experimental value (nuc

2,p )0 = 0.67, or to the self-consistent value (3)0 =0.35.

m

k

(0)/m E(p3/2) E(p1/2) E(1/2) S(1/2) E(s1/2) E(1/2+) S(1/2+) E

res

(d5/2) E(5/2+)0.7 -6.39 -3.01 -0.21 0.81 0.14 -0.56 0.83 - 1.27

Table 4: Unperturbed energies E(p3/2), E(p1/2) and E(s1/2), renormalized energies E(1/2)and E(1/2+) (calculated in a box of radius R

box

= 30 fm), resonant single-particle energiesE

res

(d5/2), renormalised resonant energies E(5/2+) (all in MeV), admixtures with the lowest2+ phonon A(1/2) and A(1/2+), obtained with the optimal values reported in Table 3.

m

k

(0)/m (2/(2)0) (3/(3)0) (VWS

/(VWS

)0) (Vls

/(Vls

)0) (aWS

) (RWS

)0.5 1. 0.3 1.85 0.74 0.81 2.150.6 1.0-1.2 0.3-0.8 1.6-1.8 0.48-1.44 0.63-0.90 1.90-2.250.7 1.1.-1.2 0.3-0.8 1.4-1.7 0.56-1.36 0.66-0.90 1.89-2.250.8 1.2 0.4-0.8 1.3-1.4 0.70-1.12 0.75-0.90 2.10-2.25

Table 5: Variation of the parameters which leads to a value of 2 smaller than 0.05 (cf. text)for di↵erent values of the e↵ective mass..

13

Structure and reactions of

11Be studied with Nuclear Field Theory: many-body basis for single-neutron halo 8

0 5r (fm)

-70

-60

-50

-40

-30

-20

-10

0

V (M

eV)

VHF(SGII)VWS (mk(0) = 0.7)

0 5r (fm)

0.7

0.75

0.8

0.85

0.9

0.95

1

mk/m

VHF (SGII)VWS(mk(0)=0.7)

Figure 2: The central potential (left panel) and the effective mass calculated with aSaxon-Woods potential parameterized as reported in Table ?? with m

k

(0) = 0.7 arecompared to the corresponding quantities obtained in the HF approximation with theSGII Skyrme interaction.

2 4 6 8 10E (MeV)

0

1

2

δ

1 2 3 4 5 6 7 8 9 100

1

2

Figure 3: Single-particle d5/2 phase shifts calculated with the various potentials listedin Table ??, and associated with different values of m

k

(0).

Thecentralpoten$al

m

k

(0)/m 2/(2)0 3/(3)0 V

WS

/(VWS

)0 V

ls

/(Vls

)0 a R

WS

R

n

2Rn

V

WS

0.7 1.3 0.8 1.60 0.8 0.81 2.10 2.69 126.6

Table 3: Optimal values of the deformation parameters 2, 3, and of the depth, radius,di↵useness and spin-orbit strength of the mean field potential V ,V

ls

,a,R, determined for agiven value of the e↵ective mass m

k

(0) at the centre of the nucleus. For each value of mk

(0),we compare the results obtained using fragmented or non fragmented (no frag.) intermediatestates. The values of V and V

ls

are given as the ratio to the parameters of the referencepotential introduced in Section 5 ((V

WS

)0 = 43.9 MeV), and the deformation parameteras the ratio to the experimental value (nuc

2,p )0 = 0.67, or to the self-consistent value (3)0 =0.35.

m

k

(0)/m E(p3/2) E(p1/2) E(1/2) S(1/2) E(s1/2) E(1/2+) S(1/2+) E

res

(d5/2) E(5/2+)0.7 -6.39 -3.01 -0.21 0.81 0.14 -0.56 0.83 - 1.27

Table 4: Unperturbed energies E(p3/2), E(p1/2) and E(s1/2), renormalized energies E(1/2)and E(1/2+) (calculated in a box of radius R

box

= 30 fm), resonant single-particle energiesE

res

(d5/2), renormalised resonant energies E(5/2+) (all in MeV), admixtures with the lowest2+ phonon A(1/2) and A(1/2+), obtained with the optimal values reported in Table 3.

m

k

(0)/m (2/(2)0) (3/(3)0) (VWS

/(VWS

)0) (Vls

/(Vls

)0) (aWS

) (RWS

)0.5 1. 0.3 1.85 0.74 0.81 2.150.6 1.0-1.2 0.3-0.8 1.6-1.8 0.48-1.44 0.63-0.90 1.90-2.250.7 1.1.-1.2 0.3-0.8 1.4-1.7 0.56-1.36 0.66-0.90 1.89-2.250.8 1.2 0.4-0.8 1.3-1.4 0.70-1.12 0.75-0.90 2.10-2.25

Table 5: Variation of the parameters which leads to a value of 2 smaller than 0.05 (cf. text)for di↵erent values of the e↵ective mass..

13

Resultscomparedtotheunperturbedstar$ngpoint

Rbox=30fm

0 5 10 15 20 25 30r (fm)

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

ψ

Initial 1d5/2 w.f.Final 5/2+ w.f.

0 5 10 15 20 25 30r (fm)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

ψ

Initial 2s1/2 w.f.Final 1/2+ w.f.

0 5 10 15 20 25 30r (fm)

0

0.1

0.2

0.3

0.4

0.5

ψ

Initial 1p1/2 w.f.Final 1/2+ w.f.

Ini$alandfinalsingle-par$clewavefunc$onsRbox=30fm

B(E1)

M(E1) X

1/2-

cm

λ

1/2-

1/2+

1/2-1/2-

d5/2+

1/2+

bkM(E1) X M(E1) XM(E1) X

M(E1) X

1/2+

bkbk

1/2-

1/2-

d5/2+

p3/2-

B(E1)

M(E1) X

1/2-

cm

λ

1/2-

1/2+

1/2-1/2-

d5/2+

1/2+

bkM(E1) X M(E1) XM(E1) X

M(E1) X

1/2+

bkbk

1/2-

1/2-

d5/2+

p3/2-

p3/2

2+

s1/2

ME1

d5/2

p1/2

p3/2

2+

s1/2

ME1

d5/2

p1/2

∑k , l

∑m

xm h(s1m ;d5k ,2 l )

(es1−(ed5k+ℏωl))

∑k

xn h(.2 l ; p3 , p1n)

(e s1−(ed5k−ep3+ e p1))×(d5 /2k , p−1

3/2∣ME1∣gs)×.

⟨5/2,(3/2 ;1/2)2 ;1/2∣(5/2,3/2)1,1/2 ;1/2⟩

∑k , l

∑k

xn h(2l ,( p3 , p1n) ;0gs)

(es1−(e s1+ℏωl−e p3+ e p1))

∑m

xmh(s1m ,2 l ; d5k )

(e s1−(ed5k−e p3+ e p1))×(d5 /2k , p−1

3/2∣ME1∣gs)×.

⟨1/2,(2 ;2)0 ;1/2∣(1/2,2)5/2,2 ;1 /2⟩×.

⟨5 /2,(3/2 ;1/2)2 ;1/2∣(5 /2,3/2)1,1/2 ;1/2⟩

p3/2

2+

s1/2

ME1

d5/2

p1/2

p3/2

2+

s1/2

ME1

d5/2

p1/2

∑k , l

∑m

xm h(s1m ;d5k ,2 l )

(es1−(ed5k+ℏωl))

∑k

xn h(.2 l ; p3 , p1n)

(e s1−(ed5k−ep3+ e p1))×(d5 /2k , p−1

3/2∣ME1∣gs)×.

⟨5/2,(3/2 ;1/2)2 ;1/2∣(5/2,3/2)1,1/2 ;1/2⟩

∑k , l

∑k

xn h(2l ,( p3 , p1n) ;0gs)

(es1−(e s1+ℏωl−e p3+ e p1))

∑m

xmh(s1m ,2 l ; d5k )

(e s1−(ed5k−e p3+ e p1))×(d5 /2k , p−1

3/2∣ME1∣gs)×.

⟨1/2,(2 ;2)0 ;1/2∣(1/2,2)5/2,2 ;1 /2⟩×.

⟨5 /2,(3/2 ;1/2)2 ;1/2∣(5 /2,3/2)1,1/2 ;1/2⟩

2 ++

Figure 9: Main processes contributing to the dipole transition between the first excited stateand the ground state of 11

Be.

20

1.95efm -0.26efm -0.19efmB(E1)=0.17e2fm2

B(E1)=0.10e2fm2

B(E1)(exp.)=0.102±0.002e2fm2

Strengthofthedipoletransi$onbetween½+and½-states

Thisresultissensi$vetothedetailsofthemeanfieldpoten$al

2 4 6 8 10E (MeV)

0

1

2

δ

1 2 3 4 5 6 7 8 9 100

1

2

0 0.5 1 1.5 2 2.5 3E (MeV)

0

1

2

3

∼ δ5/

2+ Eres = 1.23 MeV

tan δ = Γ /2(Eres -E)

Γ = 160 keV

d5/2phaseshiminthebarepoten$al

5/2+phaseshiminthebarepoten$al

0 5 10 15 20θcm

1

5

dσ/dΩ

(mb/

sr)

2+

11Be(p,d), E/A=35.3 MeV

J.S.Winfieldetal.,Nucl.Phys.A683(2001)48

11Be(p,d)10Be(2+)atE/A=35.3MeV

gs. 2+

Testofthecollec$vecomponentofthemany-bodywavefunc$on(butweshouldcalculatetheop$calpoten$al!)

0 2 4 6 8 10 12 14r(fm)

0.2

0.4

0.6many-bodys.p.

Halo Nucleus 11Be: A Spectroscopic Study via Neutron Transfer

K. T. Schmitt,1,2 K. L. Jones,1 A. Bey,1 S. H. Ahn,1 D.W. Bardayan,2 J. C. Blackmon,3 S.M. Brown,4 K.Y. Chae,5,2,1

K. A. Chipps,6 J. A. Cizewski,7 K. I. Hahn,8 J. J. Kolata,9 R. L. Kozub,10 J. F. Liang,2 C. Matei,2,* M. Matos,3 D. Matyas,11

B. Moazen,1 C. Nesaraja,2 F.M. Nunes,12 P. D. O’Malley,7 S. D. Pain,2 W.A. Peters,7 S. T. Pittman,1 A. Roberts,9

D. Shapira,2 J. F. Shriner, Jr.,10 M. S. Smith,2 I. Spassova,7 D.W. Stracener,2 A.N. Villano,13 and G. L. Wilson4

1Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA2Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

3Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA4Department of Physics, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom

5Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea6Physics Department, Colorado School of Mines, Golden, Colorado 80401, USA

7Department of Physics and Astronomy, Rutgers University, New Brunswick, New Jersey 08903, USA8Department of Science Education, Ewha Womans University, Seoul 120-750, Korea9Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA

10Department of Physics, Tennessee Technological University, Cookeville, Tennessee 38505, USA11Department of Physics and Astronomy, Denison University, Granville, Ohio 43023, USA

12National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University,East Lansing, Michigan 48824, USA

13Physics Department, University of Michigan, Ann Arbor, Michigan 48109, USA(Received 18 January 2012; published 8 May 2012)

The best examples of halo nuclei, exotic systems with a diffuse nuclear cloud surrounding a tightly

bound core, are found in the light, neutron-rich region, where the halo neutrons experience only weak

binding and a weak, or no, potential barrier. Modern direct-reaction measurement techniques provide

powerful probes of the structure of exotic nuclei. Despite more than four decades of these studies on the

benchmark one-neutron halo nucleus 11Be, the spectroscopic factors for the two bound states remain

poorly constrained. In the present work, the 10Beðd;pÞ reaction has been used in inverse kinematics at four

beam energies to study the structure of 11Be. The spectroscopic factors extracted using the adiabatic

model were found to be consistent across the four measurements and were largely insensitive to the optical

potential used. The extracted spectroscopic factor for a neutron in an n‘j ¼ 2s1=2 state coupled to the

ground state of 10Be is 0.71(5). For the first excited state at 0.32 MeV, a spectroscopic factor of 0.62(4) is

found for the halo neutron in a 1p1=2 state.

DOI: 10.1103/PhysRevLett.108.192701 PACS numbers: 25.60.Je, 25.45.Hi, 25.60.Bx

Nuclear halos are a phenomenon associated with certainweakly bound nuclei, in which a tail of dilute nuclearmatter is distributed around a tightly bound core [1–3].This effect is possible only for bound states with no strongCoulomb or centrifugal barrier, and which lie close to aparticle-emission threshold. Although excited-state halosexist, the number of well-studied halo states is predomi-nantly limited to a handful of light, weakly bound nucleiwhich exhibit the phenomenon in their ground state.

The neutron-rich nucleus 11Be is a brilliant example ofthis phenomenon, with halo structures in both of its boundstates, and light enough to be modeled with an ab initioapproach. It is well documented that the 1=2þ ground stateand the 1=2% first excited state in 11Be are inverted withrespect to level ordering predicted from a naıve shell model.There has been considerable theoretical effort put towardreproducing this level inversion in a systematic manner,while maintaining the standard ordering in the nearbynuclide 13C, where the 1=2þ state lies more than 3 MeVabove the 1=2% ground state. A variational-shell-model

approach [4] and models which vary the single-particleenergies via vibrational [5] and rotational [6] core cou-plings reproduce this level inversion in a systematic man-ner. Common to the success of these models is the inclusionof core excitation. Ab initio no-core shell model calcula-tions [7] have been unable to reproduce this level inversion,although a significant drop in the energy of the 1=2þ state in11Be is reported with increasing model space. In all of thesemodels, the wave functions for the 11Be halo states show aconsiderable overlap with a valence neutron coupled to anexcited 10Beð2þÞ core, in addition to the naıve n &10Beð0þgsÞ component. Despite decades of study, the extentof this mixing is not well understood, with both structurecalculations and the interpretation of experimental resultsranging from a few percent to over 50% core-excitedcomponent.Experimentally, it is possible to gain quantitative insight

into the mixed configuration of a state by studying reac-tions which provide observables that are sensitive todifferent components of the nuclear wave function. By

PRL 108, 192701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending11 MAY 2012

0031-9007=12=108(19)=192701(5) 192701-1 ! 2012 American Physical Society

potential [34]. In the current work, the elastic-scatteringdata were analyzed using three deuteron optical potentials:Fitz [36], Sa [34], and P-P [35]. The potentials producequite different angular distributions, and no calculation isable to adequately describe the elastic-scattering data at allfour energies. This indicates that another mechanism, pos-sibly breakup, is playing an important role which thedeuteron optical potentials fail to describe in the dþ10Be reaction. Furthermore, as the Auton data were nor-malized to one of these calculations, the spectroscopicfactors extracted from those data are subject to a significantsystematic uncertainty. When the Auton deuteron elastic-scattering data taken at Ebeam ¼ 15 MeV are scaled to thepresent data [shown as stars in panel (b) of Fig. 4],

excellent agreement is reached. An alternate analysis ofthese data using a continuum discretized coupled channelapproach has been completed and will be presented in afuture publication.In summary, elastic-scattering and neutron-transfer

measurements, performed with a primary 10Be beam inci-dent on deuterated plastic targets at Ed ¼ 12 to 21.4 MeV,have been used to study the structure of 11Be halo states ina systematic manner. The poor reliability of deuteronoptical potentials for these reactions is reflected in thescatter of the spectroscopic factor S extracted from thetransfer data and its dependence on the choice of opticalpotential. Including deuteron breakup through ADWA-FRreduces both of these problems and results in a morereliable extraction of S. The average S extracted fromour data using the ADWA-FR formalism are 0.71(5) forthe ground state and 0.62(4) for the first excited state. Thequoted uncertainties are solely from experimental consid-erations. The improved energy independence of the ex-tracted S using the present data and analysis is in starkcontrast to extractions from previous transfer data.This work was supported by the U.S. Department of

Energy under Contract Nos. DE-FG02-96ER40955 (TTU),DE-AC05-00OR22725 (ORNL), DE-FG02-96ER40990(TTU), DE-FG03-93ER40789 (Colorado School ofMines), DE-FG02-96ER40983 and DE-SC0001174 (UT),DE-AC02-06CH11357 (MSU), and DE-SC0004087(MSU); by the National Science Foundation underContract Nos. PHY0354870, PHY0757678 (Rutgers),PHY-1068571 (MSU), and PHY0969456 (Notre Dame);by the NRF of the MEST (Korea) under ContractNo. 2010-0027136; and by the UK Science andTechnology Facilities Council under Contract No. PP/F000715/1. This research was sponsored in part by theNational Nuclear Security Administration under theStewardship Science Academic Alliance program throughDOE Cooperative Agreement No. DE-FG52-08NA28552(Rutgers, ORAU, MSU).

FIG. 3 (color online). Spectroscopic factors for 11Be, extracted from (d;p) data with DWBA [panels (a) and (b)] and ADWA-FR[panel (c)] formalisms. Panels (a) and (b) illustrate the effect of choosing different potentials in the entrance and exit channels,respectively. The error bars are from experimental uncertainties only. See text for further details.

2468

1012141618

2468

1012141618

(a) Auton Schmitt Auton (Scaled)FitzSatchlerP-P

(b)

02468

1012141618

0 30 60 90

(c)

30 60 90

(d)

dσ/d

Ω/d

σ R/d

Ω

CM angle (degrees)

FIG. 4 (color online). Elastic-scattering differential cross-sections for equivalent deuteron energies of 12.0 (a), 15.0 (b),18.0 (c), and 21.4 (d) MeV, including the data of Auton [8] andthe present study, and optical model calculations using theparameters of Fitz [36], Satchler [34], and Perey and Perey [35].

PRL 108, 192701 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending11 MAY 2012

192701-4

K. T. SCHMITT et al. PHYSICAL REVIEW C 88, 064612 (2013)

TABLE VI. First excited-state spectroscopic factors extracted foreach set of optical model parameters in the exit channel. Uncertaintiesinclude only experimental contributions.

Ed (MeV) CH89 K-D

12 0.68 ± 0.08 0.61 ± 0.0715 0.68 ± 0.07 0.64 ± 0.0718 0.52 ± 0.06 0.48 ± 0.0521.4 0.71 ± 0.07 0.63 ± 0.06Average 0.65 ± 0.05 0.59 ± 0.05

as well as most of the breakup and Coulomb dissociationmeasurements [5–7]. The value of S for the ground stateextracted by Lima et al. [8], despite its relatively large errorbar, is lower than the value presented here. The calculationby Vinh Mau [21] using vibrational couplings significantlyoverestimates the value of S for the ground state, whereasthat of Nunes, Thompson, and Johnson [23] using rotationalcouplings gives a value in agreement with the current value.

There are fewer measurements related to the first excitedstate, and here the current value agrees with that from Autonet al. [10]. This agreement was not necessarily expectedbecause the normalization procedure in the work of Ref. [10]was subject to significant systematic uncertainties. The currentvalue of S for the first excited state is lower than that extractedfrom the transfer measurement of Zwieglinski [13] and the twotheoretical values used here for comparison [21,23].

0

0.5

1

Spe

ctro

scop

ic F

acto

r

CH89 K-D

Ground state

CH89 K-D0

0.5

1

0.32 MeV state

CH89 K-D

FIG. 9. (Color online) Spectroscopic factors extracted from thecurrent data using the optical potentials of Varner (CH89) [42] andKoning and Delaroche (K-D) [43]. The points are for spectroscopicfactors extracted from the data at equivalent deuteron energies of12.0 MeV (circles), 15.0 MeV (triangles), 18.0 MeV (invertedtriangles), and 21.4 MeV (squares). The boxes are centered on theaverage and show the uncertainty on the average value.

0

0.5

1S

pect

rosc

opic

Fac

tor

Ground state

Expt Theory0

0.5

1

0.32 MeV state

Expt Theory

FIG. 10. (Color online) Spectroscopic factors extracted from thecurrent data (solid stars) compared to previous measurements (black)and theory (red) for the ground and first excited states. Thecomparison points are the same works as those included in Table I,from Ref. [10] (triangles), Ref. [13] (asterisks), Ref. [5] (square),Ref. [6] (circle), Ref. [7] (open cross), Ref. [8] (open diamond),Ref. [21] (red squares), and Ref. [23] (red triangles).

Transfer to the d5/2 resonance at 1.78 MeV was calculated[54] in a similar manner to transfer to the bound statesdescribed above, using the FR-ADWA framework and theCH89 optical potential. The n-10Be potential depths, bothcentral and spin orbit, were adjusted to reproduce the resonanceenergy. This resulted in a real width of the resonance of0.192 MeV, almost twice the measured value of 0.1(0.01) MeV[17,55]. The final state was described in the calculation as anenergy bin with the width set to the same size as that usedin the model for the resonance, such that all the strength ofthe resonance could be captured. In principle, the calculationshould be insensitive to the size of this fictitious energy bin.To test this, the calculation was repeated with the bin set to be50% larger than the width used for the resonance. The resultsof these calculations are compared to the data in Fig. 8. Thesensitivity of the calculation to the width of the energy binprecludes the extraction of spectroscopic factors in this case.

VI. CONCLUSIONS

Data are presented here for elastic scattering of 10Beon protons and deuterons, inelastic scattering on deuterons,and one-neutron transfer to the two bound states of 11Beat four energies ranging from Ed = 12 to 21.4 MeV, all ininverse kinematics. Additionally, transfer to the first resonanceat 1.78 MeV was measured for the higher three energies.Absolute cross sections, over the range of angles covered,were measured in all cases.

For the proton scattering data, no single potential couldreproduce the data at all four energies. However, the CH89 and

064612-8

0 10 20 30 40 50 θcm

0

10

20

30

40

dσ/dΩ

(mb/

sr)

Ed=21.4 MeV; 1/2-

(b)

0 10 20 30 40 50 θcm

0

10

20

30

40

dσ/dΩ

(mb/

sr)

Ed=21.4 MeV; 1/2+

(a)

10Be(d,p)11BeatEd=21.4MeV

K.T.Schmi_etal.,Phys.Rev.C88(2013)064612

½+ ½-

Testofthesingle-par$clecomponentofthemany-bodywavefunc$on

11Be(p,d)11BeatE/A=35.3MeV

5/2+

0 10 20 30 40 501

10

100

d𝜎/d𝜴(m

b/sr)

𝜃cm

5/2+

Whataboutbackground?

Θcm=6o

K. T. SCHMITT et al. PHYSICAL REVIEW C 88, 064612 (2013)

Strip Number2 4 6 8 10 12 14 16

Ene

rgy

(keV

)

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

10

210G.S.

0.32 MeV

1.78 MeV

FIG. 5. (Color online) Proton energy in keV in the laboratoryframe as a function of strip number for SIDAR at backward angles atan equivalent deuteron energy, Ed = 21.4 MeV.

The calculations using the original P-P global optical potentialare shown for comparison. The agreement with the elastic datais of similar quality for the new fitted potential compared toP-P; however, the fit provides much better agreement with thedata in the case of the inelastic scattering, except at the highestenergy, as described above.

IV. THE NEUTRON TRANSFER REACTION 10Be(d, p)

Angular distributions of protons emerging from (d,p)reactions with low orbital angular momentum transfer aretypically peaked at small center-of-mass angles. For this reasonmost of the data for the neutron-transfer channel comes fromthe SIDAR array, which was placed at backward angles inthe laboratory frame. Figure 5 shows the energies of protonsmeasured in SIDAR at Ed = 21.4 MeV as a function of angle,represented by the strip number, where strip 1 covers the largestlaboratory angles. Contours of constant Q value, indicating thepopulation of a discrete state, have been labeled.

The reaction Q value was calculated on an event-by-eventbasis. Figure 6 shows the Q-value spectrum at 140 in thelaboratory frame. The energy resolution in Q value for the(d,p) reaction was more than sufficient to resolve peaksfrom the population of the ground state and the first excitedstate at 0.320 MeV. The background was identified as fusionevaporation on 12C by measuring reactions on a carbontarget. It was then possible to account for this backgroundby fitting an exponential curve for the data at Ed = 12.0,15.0, and 18.0 MeV, and subtracting measured backgroundat Ed = 21.4 MeV. The background was significantly lessobtrusive for curve fitting at the higher energies as the energiesof protons from the (d,p) reaction increase more with beamenergy than do the energies of fusion-evaporation ejectiles.

The two bound states in 11Be were populated at each beamenergy. Data were extracted for the 1.78-MeV resonance fromthe three higher beam energy measurements. The peaks for thebound states were fitted with Gaussian curves. A Voigt profilewas used to fit the peak at 1.78 MeV, using the Gaussianwidths (associated with detector resolution) found for the

0

500

1000

1500

2000

2500

3000

3500

4000

4500

-6 -5 -4 -3 -2 -1 0Q-Value (MeV)

Cou

nts/

80 k

eV

Gro

und

Sta

te0.

320

MeV

1.78

MeV

2.69

MeV

3.41

MeV

FIG. 6. (Color online) Q-value spectrum for 10Be(d,p)11Be ininverse kinematics at 140 in the laboratory reference frame foran equivalent beam energy of Ed = 21.4 MeV. Background fromfusion evaporation on 12C was accounted for by subtracting datafrom reactions on a carbon target.

bound states in each spectrum and treating the natural width ofthe state as a free variable. For the data taken with ORRUBA,Gaussian curves were fitted to each peak, resulting in a Q-valueresolution of approximately 200 keV.

Angular distributions of protons emitted from the2H(10Be,p)11Be reaction to the ground and first excited stateare presented in Fig. 7. The curves show the results ofFR-ADWA calculations using the global optical potentialsCH89 and K-D as used for the elastic scattering channel above.The spectroscopic factors, S, were extracted for each state ateach energy by scaling the calculation to the data. The shape ofthe experimental angular distributions are well reproduced bycalculations using either CH89 or K-D; however, there is somevariance in the intensity of up to 13% between the calculations.

Cross-section data for population of the resonance at1.78 MeV could be extracted at the higher three energiesand are presented in Fig. 8. The protons emitted followingtransfer to the 1.78-MeV resonance at Ed = 12 MeV were toolow in energy to extract a reasonable angular distribution. Thetransfer to the resonance data is discussed more fully in thenext section.

V. TRANSFER ANALYSIS AND DISCUSSION

The conventional method of analyzing data from transferreactions using the DWBA has been shown to be particularlysensitive to the optical potentials used [34]. In the 1970s,Johnson and Soper [50] showed the importance of includingthe breakup channel for the deuteron in the theoreticaltreatment of (d,p) reactions, and by using a zero rangeformulated a method called the ADWA. The finite-rangeversion by Johnson and Tandy [51] (FR-ADWA) was recentlyapplied to a wide range of reactions by Nguyen, Nunes, andJohnson [41].

The transfer data presented are analyzed using FR-ADWA.In FR-ADWA, the reaction is treated as a three-body n + p +10Be problem, and thus neutron and proton optical potentialsare needed, along with the binding potentials for the deuteronand the final state in 11Be. For the nucleon global potentials we

064612-6