A unified description of the structure and electromagnetic breakup of ^{\bf 11}Be
aa r X i v : . [ nu c l - t h ] J a n A unified description of the structure and electromagnetic breakup of Be M. Dan, ∗ R. Chatterjee, † and M. Kimura
2, 3, 4, ‡ Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247 667, India Department of Physics, Hokkaido University, 060-0810 Sapporo, Japan Nuclear Reaction Data Centre, Faculty of Science, Hokkaido University, 060-0810 Sapporo, Japan Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan
We study both the static properties of Be and its reaction dynamics during electromagneticbreakup under a unified framework. A many-body approach - the antisymmetrized molecular dy-namics (AMD) is used to describe the structure of the neutron-halo nucleus, Be. The same AMDwave function is then adapted as an input to the fully quantum theory of Coulomb breakup underthe aegis of the finite range distorted wave Born approximation theory. The calculated observablesare also compared with those obtained with a phenomenological Woods-Saxon potential model wavefunction. The experimental core-valence neutron relative energy spectrum and dipole response alongwith other observables are well described by our calculations.
I. INTRODUCTION
Since the discovery of halo nuclei [1], several observa-tions in their study have shown unconventional results,which were contrary to traditional nuclear structure es-timations. For example, unlike the case of stable nucleiwhere the matter radius generally follow the charge ra-dius, the matter radius of Be was found to be largerthan its charge radius. The full width at half-maximum(FWHM) of the parallel momentum distribution (PMD)of stable nuclei ( ≈
140 MeV/c ) is much higher comparedto that from a halo nucleus ( ≈
40 MeV/c). For , , Bebreaking up on a heavy target (Au), the FWHM of thePMD of the charged fragment are 191.13 MeV/c, 43.23MeV/c and 88.93 MeV/c, respectively [2].It is generally considered that a large neutron to pro-ton ratio results in a sharp decrease of the one-neutronseparation energy, and consequently an extension in theneutron wave function far outside the nuclear mean fieldis observed [3]. This extension directly affects the staticproperties of the system. The root mean square mat-ter radius of Be and Be are 2.30 ± ± Be.However more analysis are required for a consistent pic-ture. The comparison between charge and matter radiiis significant for the nuclei with different distribution ofneutron and proton halo.Tanihata et. al. [1] have shown that there is a notableincrease in interaction cross section for drip line nuclei ∗ [email protected] † [email protected] ‡ [email protected] compared to the neighbouring isotopes of light elements.There is an increase in one neutron removal cross section( σ n ) for Be as compared to Be, while breaking up ona Pb target. The average one neutron removal cross sec-tion for Be (beam energy ranging from 37 - 70 MeV/u)is 0.126 ± Be (beam energyranging from 17 - 66 MeV/u), is 2.16 ± Be is an order of magnitude higher than thatof Be. The unusually large reaction cross section of ahalo nuclei, compared to its isobars, is a consequence ofthe matter radius significantly deviating from the usual A / dependence expected for stable nuclei [11].The analysis of an external nuclear or electromagneticfield response by a nucleus is one of the key elements tounderstand the characteristics of a many-body nuclearsystem [12]. At present, there are several discussionsabout the way giant dipole resonance strength evolvesfrom stable to weakly bound exotic nuclei in extremeneutron to proton ratios. In general, the presence of col-lective soft-dipole resonance is expected to occur in heav-ier neutron-rich structures at excitation energies lowerthan the giant dipole resonance [13, 14]. Such a modemay arise when loosely bound valence neutrons vibrateagainst the residual core. In the literature, it is oftenreferred to as pygmy resonance. In electromagnetic dis-sociation experiments ( e.g. [15, 16]), a prominent low-lying dipole strength was observed in light halo nucleus.Their presence is justified by two arguments: first is dueto the coherent vibration of two halo neutrons againstthe charge core ( e.g. He [17] and Li [18]) and secondis due to the non-resonant breakup of one neutron halonucleus ( e.g. Be [15] and C [16]) into the continuum[12, 19, 20]. It is also reported that astrophysical aspectssuch as abundance pattern in the r-process nucleosyn-thesis could also be related to the presence of low-lyingdipole strength present in neutron-rich nuclei [21, 22].In this article, we aim to combine nuclear structureand reaction models to discuss both the static and dy-namical properties of a neutron-halo nucleus, Be. Forthis purpose, we use the antisymmetrized molecular dy-namics (AMD) [23–25] to calculate the static propertiessuch as one neutron separation energy, charge and mat-ter radii. The AMD wave function is also used as an in-put to the fully quantum mechanical Coulomb breakuptheory of finite range distorted wave Born approxima-tion (FRDWBA) to calculate several reaction observablesin the breakup of Be on a heavy target (
Pb) suchas triple differential cross section, neutron energy distri-bution, parallel momentum distribution, relative energyspectrum, and dipole response of Be. The results arealso compared with the available experimental data, andalso with those obtained from a phenomenological wavefunction derived using a Woods-Saxon (WS) potentialwhose depth is adjusted to fit the one neutron separa-tion energy of Be.In the following section, a brief description of the FRD-WBA theory and details of the AMD framework are pre-sented. The results and analysis from our calculationshave been discussed in section 3, wherein we present thestatic properties of Be followed by calculations of var-ious reaction observables in during its electromagneticbreakup on a heavy target. The conclusions of our workappear in section 4.
II. FORMULATIONA. Framework of FRDWBA
If we assume a projectile ‘a’ ( Be), consisting of sub-structures ‘b’ ( Be) and ‘ c ’ (neutron) to breakup in thepure Coulomb field of a heavy target ‘t’ ( Pb). Then,the triple differential cross section for the process a + t −→ b + c + t can be written as, d σdE b d Ω b d Ω c = 2 π ~ v at ρ ( E b , Ω b , Ω c ) X l,m | β lm | , (1)where, v at is the relative velocity between the a - t sys-tem in the entrance channel, ρ ( E b , Ω b , Ω c ) is the threebody final state phase space factor [26]. The reducedtransition amplitude in the post form FRDWBA, β lm ,for the breakup process is given by [27], β lm ( q b , q c ; q a ) = D ζ ( − ) b ( q b , r ) ζ ( − ) c ( q c , r c ) (cid:12)(cid:12)(cid:12) V bc ( r ) × (cid:12)(cid:12)(cid:12) φ lma ( r ) ζ (+) a ( q a , r i ) E . (2) ζ i ’s ( i = a, b, c ) are the pure Coulomb distorted wavesof the appropriate particles with respect to the target and q i s are the associated Jacobi wave vectors. The positionvectors are shown in Fig. 1 with r = r i - α r and r c = γ r + δ r i , where α , γ and δ are the mass factors: α =m c /(m c + m b ); γ = m t /(m b + m t ); γ = (1 - αδ ) and m i s( i = b, c, t ) are the masses of the appropriate particles. φ lma ( r ) is the ground state wave function of the projec-tile ( a ), which is an eigenfunction of the two-body boundstate potential V bc ( r ), with l and m as the orbital an-gular momentum and its projection, respectively. Other FIG. 1. The three body Jacobi coordinate system. The cor-responding position vectors are denoted by r ’s. reaction observables such as relative energy spectrum,neutron energy distribution, and parallel momentum dis-tribution can be calculated by suitably integrating Eq.(1). The dipole strength distribution dB ( E /dE can beobtained [28] from the relative energy spectra dσ/dE rel using dσdE rel = 16 π ~ c n E dB ( E dE , (3)where, n E is the virtual photon number for electricdipole transition. For more details of the theory one isreferred to [27].The main input to this theory is the wave function φ lma ( r ) = u l (r ) Y lm ( ˆr ), or more specifically the boundstate radial wave function u l (r ). In this work, we testtwo different approaches to calculate this primary struc-ture input to the theory. An ordinary option is to calcu-late u l (r ) from the core-valence neutron interaction withthe WS form whose depth is adjusted to reproduce theone-neutron separation energy with fixed radius and dif-fuseness parameters. An alternative and more sophisti-cated approach would be to use u l (r ) derived from a mi-croscopic many-body wave function of the AMD, whichis discussed in details in the next section. B. Model for the structure calculation
The framework of AMD and the method to calculatevalence neutron wave function (overlap amplitude) arebriefly explained. For more details, readers are directedto Refs. [23–25].
1. Framework of AMD
The Hamiltonian used in this study is given as, H = A X i =1 t ( i ) + A X i
2. The generator coordinate method and AMD plusresonating group method
To describe the ground and excited states, we performthe angular momentum projection and the generator co-ordinate method. The optimized wave functions Φ πint ( β ) are projected to the eigenstates of the total angular mo-mentum,Φ JπMK ( β ) = P JMK Φ πint ( β )= 2 J + 18 π Z d Ω D J ∗ MK (Ω) R (Ω)Φ πint ( β ) , (9)where, P JMK , D JMK (Ω) and R (Ω) denote the angular mo-mentum projector, the Wigner D function and the rota-tion operator, respectively. The integrals over three Eulerangles Ω are evaluated numerically. Then, we superposethe wave functions with different quadrupole deformation β and projection of angular momentum K (GCM),Ψ JπMα = J X K = − J N X i =1 e Kiα Φ JπMK ( β i ) , (10)where, N is a number of the basis wave functions to besuperposed. The coefficients e Kiα and eigenenergy E Jπα are obtained by solving the Hill-Wheeler equation [33], X K ′ i ′ H JπKiK ′ i ′ e K ′ i ′ α = E Jπα X K ′ i ′ N JπKiK ′ i ′ e K ′ i ′ α , (11) H JπKiK ′ i ′ = h Φ JπMK ( β i ) | H | Φ JπMK ′ ( β i ′ ) i , (12) N JπKiK ′ i ′ = h Φ JπMK ( β i ) | Φ JπMK ′ ( β i ′ ) i . (13) Be
12 fm f m n FIG. 2. A schematic illustration of the basis wave functionfor the Be + n system used in the AMD+RGM method. As explained later, the basis wave functions Φ
JπMK ( β i )generated by the energy variation are not sufficient todescribe the neutron halo of Be. To incorporate withthe proper asymptotics of the halo wave function, we haveintroduced a set of wave functions as additional basis. Asschematically illustrated in Fig. 2, we have generated the Be + n wave functions by placing the Be and n onthe grid points within the 12 fm ×
12 fm size with 1 fmintervals. These wave functions may be represented as,Φ( ξ i , χ n ) = A (cid:26) Φ Be ( − ξ i ) ϕ n ( 1011 ξ i , χ n ) (cid:27) . (14)Here, the wave function of Be is the intrinsic wave func-tion [Eq. (6)] obtained by the energy variation withoutconstraint on the deformation parameter β , and the va-lence neutron is described by a Gaussian wave packet[Eq. (7)] placed at 10 / ξ i . Because Φ Be has approx-imate axial and reflection symmetry, the relative coor-dinate ξ i between Be and the valence neutron can berestricted within the first quadrant of the xy -plane wherethe y -axis is the symmetry axis of Φ Be . Consequently,we have generated 13 × × × neutron spin), which aresuperposed after the angular momentum projection,Ψ JπMα = J X K = − J (cid:26) N X i =1 e Kiα Φ JπMK ( β i )+ X i =1 X χ n = ↑ , ↓ f Kiχ n α P JMK P π Φ( ξ i , χ n ) (cid:27) . (15)The coefficients e Kiα , f Kiχ n α and eigenenergy E Jπα aredetermined by solving the Hill-Wheeler equation again.We note that this method named AMD plus resonatinggroup method (AMD+RGM) has already been used todescribe neutron halo of Ne [34–36].
3. Calculation of the valence neutron wave function
We have extracted the valence neutron wave function(overlap amplitude) from the microscopic wave functionsof Be and Be. For this purpose, firstly, we calculatethe overlap between the wave functions of Be and Be. ψ ( r ) = √ n Ψ J π M ( Be) | Ψ / + M ′ ( Be) o . (16)For simplicity, we assume that the wave functionsof , Be are described by the parity- and angular-momentum-projected wave functions given by Eq. (9).Then, Eq. (16) reads, ψ ( r ) = X jl C / M ′ JM,jM ′ − M u jl ( r ) /r [ Y l (ˆ r ) ⊗ χ ] jM ′ − M , (17)where, the overlap amplitude u jl ( r ) is defined as, u jl ( r ) = X k C / K ′ JK ′ − k,jk X p =1 ( − ) p rϕ ( p ) jlk ( r ) × J + 18 π Z d Ω D J ∗ KK ′ − k (Ω) det B ( p ) (Ω) . (18) Here, ϕ ( p ) jlk is the multi-pole decomposition of the single-particle wave packet [Eq. (7)], ϕ p ( r ) = X jlk ϕ ( p ) jlk ( r )[ Y l (ˆ r ) ⊗ χ ] jk . (19)det B ( p ) (Ω) is the determinant of the sub-matrix B ( p ) formed by removing p th column from B (Ω). And B (Ω)is the (10 × Beand Be, that is defined as B (Ω) ij = < φ i | R (Ω) | ϕ j where φ i and ϕ j are being the single-particle wave packets of Be and Be, respectively. Once the overlap amplitudeis calculated, its integral yields the spectroscopic factor, S jl = Z ∞ dr | u jl ( r ) | . (20)The derivation of the above formulae is explained inRef. [37]. It is straightforward to extend them to theGCM and AMD+RGM wave functions given by Eqs. (10)and (15). III. RESULTS AND DISCUSSIONSA. Static properties of Be Fig. 3 shows the density profile of the Be and Beintrinsic wave functions [Eq. (6)] which are obtainedby the energy variation and are the dominant compo-nents of the ground state. As clearly seen, both nu- -4 422 00 -2-2 x [fm] y [f m ] -4 422 00 -2-2 x [fm] y [f m ] -4 422 00 -2-2 x [fm] y [f m ] -4 422 00 -2-2 x [fm] y [f m ] r [fm -3 ] Be (proton) Be (proton) Be (neutron) Be (neutron)
FIG. 3. The proton and neutron density distributions calcu-lated from the intrinsic wave functions which are the domi-nant component of the ground state. Upper (lower) panelsshow the proton (neutron) densities. clei have dumbbell-shaped proton density distributionswhich indicate the pronounced α + α clustering. The va-lence neutrons (two valence neutrons of Be and threeof Be) occupy the so-called “molecular orbits” whichare the single-particle orbits formed around α + α clus-ter core [38–41]. In both nuclei, two valence neutronsoccupy the π -orbit, and the last valence neutron of Be s × s s s p s s × p p p , p d +- - -+ r [fm]0 2 4 6 s i ng l e - p a r ti c l e e n e r gy FIG. 4. Schematic figure showing the behavior of the molecu-lar orbits of α + α cluster system as function of the inter-clusterdistance. The orbits plotted with red lines show the molec-ular orbits ( π - and σ -orbits) relevant to the low-lying statesof Be and Be. This figure is reconstructed from Fig. 1 ofRef. [39]. occupies the σ -orbit. These molecular-orbit configura-tions are often denoted as π ( Be) and π σ ( Be). Asdiscussed in Refs. [38–45], the π -orbit reduces the α + α clustering, while the σ -orbit enhances it. This featureoriginates in the single-particle energies of the molecu-lar orbits as function of the inter-cluster distance illus-trated in Fig. 4. The single-particle energy of the π -orbit( σ -orbit) decreases (increases) as function of the inter-cluster distance. As a result, the π -orbit ( σ -orbit) favorsweaker (stronger) α + α clustering. Since, Be has anadditional neutron in the σ -orbit, it manifests more pro-nounced α + α clustering than Be as seen in its protondensity distribution (Fig. 3) and larger quadrupole de-formation parameter (Table I). These characteristics of Be and Be qualitatively agree with those discussedin the preceding studies [42–46]. We also note that the σ -orbit is a linear combination of the spherical sd -shells,and hence, the valence neutron wave function (overlapamplitude) of Be should be, in general, an admixtureof the l = 0 and 2 components.Fig. 5 shows the excitation spectra of Be and Beobtained by the GCM calculations (denoted as AMDin the figure). Contradictory to the observation, theadopted effective interaction does not bound Be, al-though it gives the correct order of the 1 / ± doublet andother excited states of Be. The shortage of the bindingenergy is due to the insufficient description of the asymp-totics of the valence neutron wave function by the AMDframework. The s -wave valence neutron wave functionshown in Fig. 6 (black dotted line) decays at short dis-tance and does not show halo nature, reflecting the factthat the AMD framework approximates the valence neu-tron wave function by a single Gaussian. As a result,neither of the binding nor large matter radius of Beare reproduced [4].To overcome this problem, we have performed theAMD+ RGM calculation which superposes the Gaus-sian wave packets to describe the asymptotics of neu-tron halo. Fig. 6 shows how the AMD+RGM drasti- + + + + + + + + AMD AMD e n e r gy [ M e V ] Be Be AMD+RGM expt. expt.-1213
FIG. 5. The calculated and observed excitation spectra of Be and Be. The energy is relative to the ground state of Be. cally improves the results. The asymptotics of the s -wave( Be(0 + ) ⊗ s / channel) is greatly extended toward out-side of the core nucleus, and the neutron distributionradius is considerably increased compared to that cal-culated by AMD (Table I). On the contrary, the asymp-totics of the d -wave ( Be(2 + ) ⊗ d / and Be(2 + ) ⊗ d / channels) do not change significantly. This may be due tothe centrifugal barrier in these channels which preventsthe long-ranged stretched asymptotics. Thanks to theimproved asymptotics, the kinetic energy of the halo or-bit is reduced and the calculated one-neutron separationenergy is now comparable with the observed value.An interesting side effect brought about by the AMD+RGM is the reduction of the core deformation and the de-coupling between the core and valence neutron. As listedin Table I, the quadrupole deformation of the proton dis-tribution decreases in the AMD+RGM result ( β p =0.65)compared to that of AMD ( β p =0.70). This implies thatthe coupling between the core ( Be) and the valence neu-tron is weakened, and the core polarization decreases. In-deed, in the AMD+RGM result, the spectroscopic factorof the s -wave increases, while that of the d -wave decreasescompared to the AMD results. Thus, the AMD+RGMframework brings about a remarkable improvement ofthe neutron wave function and offers a reasonable de-scription of the neutron halo of Be with the deformedcore nucleus Be. It is also noted that the overlapamplitudes obtained by the AMD+RGM (Fig. 6) lookconsistent with those obtained by an ab-initio calcula-tions [9, 47, 48].
TABLE I. The calculated and observed one-neutron separation energy ( S n ), charge and point nucleon distribution radii ( p h r ch i and p h r m i ). The second and third rows shows the results obtained by AMD and AMD+RGM calculations, respectively. Thecalculated point neutron distribution radii ( p h r n i ) and the quadrupole deformation parameter for proton and neutron ( β p and β n ) are also tabulated. The proton finite size effect is taken into account for the charge radii, while it is not for the neutronand matter radii. AMD(+RGM) expt. S n p h r ch i p h r m i p h r n i β p β n S n p h r ch i p h r m i (MeV) (fm) (fm) (fm) (MeV) (fm) (fm) Be 6.21 2.43 2.35 2 .
42 0 .
60 0 .
56 6.81 2 . . Be -0.22 2 .
54 2 .
59 2 .
71 0 .
70 0.69 0.50 2 . .
55 2 .
71 2.89 0 .
65 0.62 -0.4 50 10 15-0.30.3-0.20.2-0.10.100.4 [ a r b . un it ] r [fm] AMDAMD+RGMAMDAMD+RGM AMDAMD+RGMtail correction
FIG. 6. The overlap amplitudes of the valence neutron of Becalculated by the AMD and AMD+RGM. The amplitude isarbitrary scaled for the presentation.
For later use, we further improved the asymptotics ofthe valence neutron wave function to be consistent withthe observed one-neutron separation energy, S n = 0 . s -wave overlap function u ( r ) calculated fromAMD+RGM is smoothly connected to the exact asymp-totics of A exp( − κr ) where κ = √ µS n / ~ and µ is beingthe reduced mass for Be + n system. The amplitude A and the matching distance a are determined from thecondition, ddr u ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r = a = ddr A exp( − κr ) (cid:12)(cid:12)(cid:12)(cid:12) r = a . (21)The overlap function with the tail correction, thus ob-tained, is shown by the dashed line in Fig. 6. This overlapfunction is used as an input to calculate various reactionobservables in the breakup of Be on a heavy target inthe subsequent sections.
1. Wave function inputs for reaction calculations
The bound state, single-particle radial wave functionof Be has been built from a Be(0 + ) ⊗ s / ν configu-ration with a one neutron separation energy ( S n ) of 0.50MeV. It is also known that the contribution from the d -wave configuration is an order of magnitude lower thanthe s -wave and so any admixture (along with a low spec-troscopic factor, cf. Table II) would not be perceptiblein reaction observables (see e.g. Refs. [15, 49, 50]).The bound state radial wave function of Be was con-structed in two ways. The first by considering a Woods-Saxon potential of 70.99 MeV, radius and diffuseness pa-rameters as 1.15 fm and 0.50 fm, respectively, so as to re-produce the one-neutron separation energy of 0.50 MeV.The other is by using the overlap wave function obtainedby the AMD + RGM framework with tail correction, asdescribed earlier. In all subsequent sections we refer tothis wave function as the AMD wave function itself. Thetwo wave functions are compared in Fig. 7. The solid anddashed lines show the phenomenological WS and the mi-croscopic AMD wave functions, respectively.
2. Matter and charge radii in the cluster model
It will be interesting to calculate the matter and chargeradii in the cluster model [53], with the phenomenolog-ical WS and AMD wave functions as inputs and com-pare them with those obtained in the previous section(Table I). The mean square matter radius h r i A a andcharge radius h r ch i A a of a dicluster nucleus of mass num-ber A a and charge Z a (consisting of subclusters A b , Z b and A c , Z c ) can be written as [53] h r i A a = A b A a h r i A b + A c A a h r i A c + A b ∗ A c A a h R i , (22) TABLE II. The calculated spectroscopic factors which are obtained from the integral of the overlap amplitudes shown in Fig. 6.channel AMD AMD+RGM expt. Be(0 +1 ) ⊗ s / Be(2 +1 ) ⊗ d / Be(2 +1 ) ⊗ d / r (fm) -0.6-0.300.30.6 u l (r ) AMDWS0 2 4 6-0.500.5
FIG. 7. The normalized bound state radial wave functions of Be in WS (solid line) and AMD (dashed line) models. Theinset shows the wave functions in the nuclear interior. and h r ch i A a = Z b Z a h r i A b + Z c Z a h r i A c + h R i Z a (cid:18) Z b . (cid:18) A c A a (cid:19) + Z c . (cid:18) A b A a (cid:19) (cid:19) , (23)respectively. In Eqs. (22) and (23), h R i = h u l ( r ) | r | u l ( r ) i . In our case, b and c are the core ( Be) and the valenceneutron of the projectile respectively. If we neglect thesecond term of Eq. (22), then the mean square matterradius can be written as h r i A a = A b A a h r i A b + A b ∗ A c A a h R i . (24)Furthermore, given that Z c = 0, the mean square chargeradius simplifies to h r ch i A a = Z b Z a h r i A b + h R i Z a (cid:18) Z b . (cid:18) A c A a (cid:19) (cid:19) . (25)For our calculations, we have used the size of thecharged core, p h r i A b = 2.28 fm. [55]. Evidently, the TABLE III. Root mean square matter ( p h r i A a ) and charge( p h r ch i A a ) radii of Be in the cluster model.WS (fm) AMD (fm) expt. (fm) p h r i A a p h r ch i A a WS and the AMD wave functions directly enter the cal-culation of the radii only through h R i term in Eqs. (24- 25). Thus any difference between the phenomenolog-ical WS wave function and the microscopic AMD willbe reflected in h R i . We have also used a spectroscopicfactor of 0.82 (in Table II), obtained in the structuralcalculations shown earlier.In Table 3, we show that our calculated root meansquare matter and charge radii using both the WS andAMD wave functions compare well with the availabledata. The matter radius calculated from other mod-els such as relativistic mean field model (2.52 fm [6]),Glauber model (2.76(0.03) fm [7]), and fermionic molec-ular dynamics model (2.80 fm [8]) also agree well withour results. Similarly, the charge radius deduced fromthe fermionic molecular dynamics (2.38 fm [8]) and nocore shell model (2.37(11) fm [9]) agree well with the WSand AMD results. B. Reaction observables
1. Triple differential cross section
In Fig. 8 we plot the triple differential cross sections inthe breakup of Be on
Au target at a beam energyof 44 MeV/nucleon as a function of the neutron energy(E n ), for four different combinations of the neutron angle( θ n ) and the angle of the charged fragment ( θ b ). Giventhat the spectroscopic factor of the s-wave is close tounity [3, 15, 51, 52], our subsequent breakup calculationsalso take it as unity.As expected the cross sections are indeed larger atsmall scattering angles as the breakup is Coulomb dom-inated. We observe that the calculations obtained fromboth the WS and AMD wave functions are similar. Giventhat triple differential cross sections are exclusive reac-tion observables, this similarity in results builds up anexpectation that other reaction observables may also notbe too different while using these wave functions. Thisis because other inclusive reaction observables can beobtained from the triple differential cross section afterperforming suitable integrations over various kinematicparameters.
20 30 40 50 60 70040080012001600
WSAMD
20 30 40 50 60 7005010015020020 30 40 50 60 70010203040 20 30 40 50 60 700102030 θ n = θ b = 1 o θ n =10 o , θ b = 1 o θ n = θ b = 1 o d σ / d E n d Ω n d Ω b ( m b / M e V . s r ) θ n = θ b = 10 o θ n =1 o , θ b = 10 o E n (MeV) FIG. 8. Triple differential cross section for the breakup of Beon
Au at a beam energy of 44 MeV/nucleon. The solidand dashed lines correspond to WS and AMD calculations,respectively. For more details see text.
2. Neutron energy distribution
We now calculate the neutron energy distribution inthe breakup of Be on
Au and compare it with ex-isting experimental data (solid circles) [56], at θ n = 1 ◦ .Incidentally, the beam energy for the same experimentaldata was not unique and was in the range of 36.9 to 44.1MeV/u. To take care of this variation in our calculation,in Fig. 9 we have shown the neutron energy distributionusing the WS (solid line) and AMD (dashed line) wavefunctions, calculated at a series of beam energies rangingfrom 37 - 44 MeV/u. In the same figure [Fig. 9(h), dot-dashed line], we have also plotted the average of all theWS results performed at different beam energies.With the progress in current radioactive ion beam facil-ities, it would indeed be interesting to perform these ex-clusive measurements as that could constrain the spreadin the data seen in Fig. 9.
3. Parallel momentum distribution
We now turn our attention to the PMD of the chargedfragment in the breakup of Be on
Ta at a beam en-ergy of 63 MeV/u. The width of this distribution is alsoa measure of the size of the nucleus in coordinate space.From a statistical model calculation [57], the width canbe given by ∆ = ∆ (cid:2) A b ( A a − A b ) /A a ), where A a and A b are the mass numbers of the projectile and fragment,
024 024024 024024 0240 10 20 30 40 50 60 70024 0 10 20 30 40 50 60 7002437 MeV/u 38 MeV/u39 MeV/u 40 MeV/u41 MeV/ u 42 MeV/u43 MeV/u 44 MeV/u E n (MeV) d σ / d E n d Ω n ( b / M e V . s r) (a)(c)(e)(g) (b)(d)(f)(h) FIG. 9. Neutron energy distributions in the Coulomb breakupof Be on a gold target at beam energies ranging from 37to 44 MeV/nucleon, calculated using WS (solid line) andAMD (dashed line) wave functions. The experimental dataare shown by solid circles are from Ref. [56]. In figure (h),the dot-dashed line shows the average of all the WS resultsperformed at different beam energies. respectively and ∆ ( ≈
80 MeV/c) has a constant valueand it is also known to be independent of target massand beam energy. This approximation suggests that thewidth of the Be distribution in the breakup of Becould be approximately 80 MeV/c. However, the experi-mental FWHM of the PMD for the s -wave configurationhas been found to be 43 . ± . Be) fragmentsemitted during the elastic Coulomb breakup of Be on
Ta at a beam energy of 63 MeV/u, in the rest frame ofthe projectile. The solid and the dashed lines correspondto our calculations with WS and AMD wave functions,respectively and are normalized to the peak of the data.The FWHM for both the theoretical (WS and AMD) cal-culations are 42 MeV/c which is in good agreement withthe experimental value of 43 . ± .
4. Relative energy spectrum and dipole response
We now continue our efforts in testing the WS and theAMD wave functions by calculating the relative energyspectrum and the dipole strength distribution of Be ona heavy target. In Fig. 11, we have shown the relativeenergy spectrum in the elastic Coulomb breakup of Beon a
Pb target at beam energy of 72 MeV/u (upperpanel) and dipole strength distribution (lower panel) andcompare our results with the available experimental data[15]. The solid and the dashed lines represent FRDWBAcalculations using the WS and the AMD wave functions,respectively, while the experimental data are shown bysolid circles. We see that both the WS and AMD re- -100 -50 0 50 100 p z (MeV/c) d σ / dp z ( a r b . un it s ) Expt.AMD WS
FIG. 10. The parallel momentum ( p z ) distribution of Be inthe elastic Coulomb breakup of Be on
Ta at 63 MeV/uin the rest frame of the projectile. The solid and dashed linescorrespond to WS and AMD calculations, respectively andthe experimental data, shown by solid circles, are from Ref.[58]. sults are able to reproduce the peak positions of the rela-tive energy spectra and the dipole distribution quite well.Contributions at higher relative energies are dominatedby the nuclear breakup [27].The total one neutron removal cross sections ( σ n ), ob-tained by integrating the relative energy spectrum withthe WS and AMD wave functions, are 1.76 b and 1.51 b,respectively and the corresponding experimental value is1.8 ± B ( E
1) value for Be determined ex-perimentally at a beam energy of 72 MeV/u is 1.3 ± e f m [15] and the theoretical values obtained by in-tegrating the lower panel of Fig. 11 for WS and AMDcalculations are 1.17 and 0.97 e f m , respectively. Weemphasize that our post form FRDWBA considers thebreakup process to be a one-step process [27] and hencethe agreement of our calculations with the experimentaldata is a direct proof that the enhanced dipole strengthat low energies is due to the breakup of nucleus into thecontinuum and not because of any soft dipole resonance.A similar conclusion was also reached by the authors of[52].Interestingly, in an extreme single-particle model, pro-vided other excited bound states do not contribute to thedipole transition, the total B ( E
1) is known to be propor-tional to the mean square radius of the valence nucleon( h r ν i ) [19, 20, 59–61], via B ( E
1) = (3/4 π )( Z a e /A a ) h r ν i . Under this approximation, using the theoretical es-timates of B ( E
1) obtained earlier, the root mean square E rel [MeV] d σ / d E r e l [ b / M e v ] Expt.WSAMD
72 MeV/u E rel (MeV) d B ( E ) / d E r e l ( e f m / M e V ) Expt.WSAMD
72 MeV/u
FIG. 11. The relative energy spectra in the elastic Coulombbreakup of Be on
Pb at 72 MeV/u (upper panel) anddipole strength distribution (lower panel). Calculations withthe WS and the AMD wave functions are shown by solid anddashed lines, respectively. The solid circles show the experi-mental data from Ref. [15]. radius of the valence neutron ( p h r ν i ) with the WS andthe AMD wave functions turns out to be 6.08 fm and5.54 fm, respectively. These do compare well with exper-imental estimate of 6.4 ± IV. CONCLUSIONS
We have investigated the static properties and reactionobservables of Be breaking in the presence of Coulombfield of
Pb. For our theoretical calculations we havecombined the AMD framework and the fully quantummechanical FRDWBA model.We have used the AMD to describe the structure andthe valence neutron wave function of Be. To incorpo-rate the long-ranged asymptotics of the halo wave func-tion, an extended AMD framework named AMD+RGMhas also been adopted. The AMD+RGM has drasticallyimproved the asymptotics of halo wave function, andplausibly described various properties such as charge andneutron distribution radii, one neutron separation energyand excitation spectrum.The valence neutron wave function calculated byAMD+RGM was used as an input of the FRDWBAmodel to describe the dynamical properties of Be. Theadvantage of the method over other first-order pertur-bative theories is that it requires only the ground statewave function of the projectile as an input and includesall orders of the electromagnetic interaction between thefragments and the target.Apart from the many body AMD, a phenomenologicalWoods-Saxon wave functions has also been used for thepurpose of comparison. We have shown that the staticproperties calculated with these wave functions, mainly the matter and charge radii of Be, are in good agree-ment with the available data. This gave us the confidenceto calculate several reaction observables in the breakupof Be on heavy targets. Several observables like thetriple differential cross section, neutron energy distri-bution, parallel momentum distribution, relative energyspectrum and dipole strength distribution are presentedand compared with experimental data, wherever avail-able. The upshot of this method is that the same inputis used to calculate various exclusive and inclusive ob-servables. Given the validity of our method in the lowmass region, it would also be interesting to extend ourcalculations to the deformed medium mass region of thenuclear chart where experimental data are scarce.
ACKNOWLEDGMENT
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