Excitation of multiple giant dipole resonances from spherical to deformed nuclei

a r X i v :n u c l -t h /9910055v 1 20 O c t 1999Excitation of multiple giant dipole resonances:

from spherical to deformed nuclei

M.V.Andr′e s a ,https://www.wendangku.net/doc/6610591093.html,nza a ,b ,P.Van Isacker c ,C.Volpe d and

F.Catara b

a Departamento

de F′?sica At′o mica,Molecular y Nuclear,Universidad de Sevilla,Apdo 1065,41080Sevilla,Spain b Dipartimento

di Fisica dell’Universit`a and INFN,Sezione di Catania,I-95129Catania,Italy c GANIL,BP 5027,F-14076Caen Cedex 5,France d Groupe de Physique Th′e orique,Institut de Physique Nucl′e aire,F-91406Orsay Cedex,France

Nuclear multiphonons both at low and at high energies have attracted much interest lately [1–4].At low energy,the controversy centers around the collec-tive character of multiphonon states.While in vibrational nuclei ?rm experi-mental evidence now exists for states with triple quadrupole phonon character (see,e.g.,[5]),in deformed nuclei the collective character of double βand γvibrations is still a matter of acrimonious debate.At high energy,the study of multiphonons has pointed out the limitations of the small-amplitude approx-imation for vibrational collective motion and of the linear approximation for the external exciting ?eld.These assumptions are routinely made for the ?rst phonon but recent studies of double phonons [6–8]have shown the importance of anharmonicities and non-linearities in the excitation of large-amplitude vi-brations.Similarly,anharmonicities and non-linearities in nuclear vibrations at low energy have been shown to play a crucial role in the calculation of the heavy-ion fusion cross sections at energies close to the Coulomb barrier [9,10].

In this letter yet another aspect of phonon excitations at high energy is investi-

gated,namely the modi?cation of the excitation of the giant dipole resonance (GDR)away from shell closure as a result of deformation.Already at the

level of the single GDR the strength function shows a splitting into two peaks associated with the two di?erent frequencies of vibration along with or perpen-

dicular to the axis of axial symmetry.The question under scrutiny here is how

deformation in?uences the E1strength to the double GDR(DGDR)and what its e?ect is on the Coulomb excitation cross section.The proposed approach

makes use of the interacting boson model(IBM)[11]for the description of the

low-energy collective levels.These are coupled to the GDR excitations mod-elled as p bosons.The advantage of using the IBM is that the multiphonon

states are obtained as exact eigensolutions of the hamiltonian and,therefore, no folding procedure is required to obtain the double phonon states,as is

for instance the case in[12,13].The folding procedure is only approximately

correct for vibrational or well-deformed nuclei(in the latter case the folding must be done in the intrinsic frame assuming additivity of intrinsic phonons)

but there is no simple recipe in the intermediate case of transitional nuclei. Another advantage of the IBM is that calculations are quick and this enables

an easy estimate of the excitation cross section to the triple GDR concerning

which experiments are currently planned[14].

A general and appropriate basis to discuss the problem of multiple GDR ex-citations is of the form

|αL L×nR;JM J .(1) The multiple GDR is built on a low-energy nuclear state characterised by αL L where L is the angular momentum of the state andαL any other label.

The multiplicity of the GDR is indicated by n(i.e.,n=1for a single GDR,

n=2for a double GDR,etc.)and its angular momentum by R.The single resonance is approximated as a p boson;the allowed angular momenta of the

multiple GDR are R=n,n?2,...,0or1.The basis(1)can be referred to

as a weak-coupling basis in the sense that the angular momenta L and R are good quantum numbers and are coupled to total angular momentum J with

projection M J.The weak-coupling basis arises naturally when the interaction between the GDR and the low-lying states is weak or is of dipole type?L·?R.

More general nuclear interactions are not necessarily diagonal in the basis(1).

Speci?cally,the interaction predominantly responsible for the splitting of the GDR in deformed nuclei is of quadrupole type?Q L·?Q R and is not diagonal in the weak-coupling basis(1).From the analogous problem in the particle–core

coupling model[15]it is known that the diagonalisation of?Q L·?Q R in the basis(1)gives rise to a strong-coupling basis with the quantum numbers[16]

|αL×n;αJ K J JM J .(2)

The angular momenta L and R no longer are conserved quantities and are replaced by K J,the projection of the total angular momentum J on the axis of axial symmetry.The choice of basis depends on the competition between various terms in the nuclear https://www.wendangku.net/doc/6610591093.html,rge splittings in L or R induce the weak-coupling basis(1);if,in contrast,these are small in comparison with the quadrupole coupling between the low-energy states and the GDR,such as is the case in well-deformed nuclei,the strong-coupling basis(2)is obtained. The above remarks are rather general and model independent.Low-energy nuclear states and their coupling with the GDR can be modelled in several di?erent ways and a convenient one is in the context of the IBM[11,17].In deformed nuclei the problem was worked out analytically for a single GDR by Rowe and Iachello[18],while numerical results were presented for the single GDR in[19–22]and later for the double GDR in[23].In this letter analytical results for the energies and E1transitions in spherical and deformed nuclei are generalised to the multiple GDR,and numerical results are presented for intermediate cases.From these results the Coulomb-excitation cross sections are calculated taking into account the dynamics of the collision,which was lacking in previous IBM treatments of the single and the double GDR.

It is assumed in the IBM that collective nuclear states can be described in terms of N s and d bosons where N is half the number of valence nucleons[11]. The dynamical algebra of the model is U(6)in the sense that a single of its representations(namely the symmetric one,[N])is assumed to contain all low-energy collective nuclear states.To this space are coupled the multiple GDR excitations.Assuming that a single GDR is described by a p boson,multiple GDR excitations are represented by the direct sum(0,0)⊕(1,0)⊕(2,0)⊕···of symmetric representations(n,0)of U(3).The dynamical algebra of the coupled system is thus U(6)?U(3)with the proviso that several U(3)representations must be taken to build the model space.

The model hamiltonian has the generic form[17]

?H=?H

+?H p+?V sd?p.(3)

sd

The?rst term is the usual IBM hamiltonian[11]which gives an adequate description of the low-energy spectrum of spherical,deformed and transitional nuclei.The second term in(3)governs the multiple GDR spectrum and is of the form

?H

=?p?n+αp?n(?n+3)+βp?R2,(4)

p

where?n is the p-boson number operator and?R the associated angular mo-mentum operator.The coe?cient?p represents the unperturbed single GDR

energy.The second term in(4)induces a diagonal anharmonicity in the excita-tion energy of the multiple GDR.The interaction term in(3),?nally,acquires the form

?V

sd?p =α0?n d?n+2α1?L·?R+2

√

4

α2,

In the SU(3)strong-coupling limit states are labelled by

|[N](λsd,μsd)×(n,0);(λ,μ)K J JM J .(8) The labels(λsd,μsd)are associated with the SU(3)algebra of the s and d bosons.They characterise the band structure of the low-energy spectrum;for example,the ground band has(λsd,μsd)=(2N,0).The energy eigenvalues of the states(8)are

E=(αsd?α2)[λsd(λsd+3)+μsd(μsd+3)+λsdμsd]+

?p n+(αp?α2)n(n+3)+

α2[λ(λ+3)+μ(μ+3)+λμ]+(α1?3

The splitting of the GDR comes about because of the fourth term in(9).The (GDR)n excitation splits into n+1peaks corresponding to(λ,μ)values (λ,μ)=(2N+n,0),(2N+n?2,1),...,(2N?n,n),(10)

where2N≥n is assumed.The energies of the di?erent peaks are found from(9).The values of K J and J allowed for a given(λ,μ)representation are given by Elliott’s rule[11,24].

The Coulomb excitation of GDRs occurs predominantly through E1.In the context of the present model an E1excitation corresponds to the creation of a p boson(annihilation in case of E1de-excitation)and thus the electric multipole operator M(E1μ)[15]is parametrised asζ(p?μ+?pμ).The calculation of E1transition probabilities requires the matrix elements of p?in the basis(6) or(8)which can be done by standard group-theoretical techniques.Analytical expressions are found in the two limiting cases.

(i)Weak-coupling limit.For the GDR excitations built on the0+ground state results up to the DGDR are shown in Fig.1a.Generally,for the B(E1)values between multipole GDR excitations built on the0+ground state one recovers the independent-quanta result[15]

B(E1;n d=0×(n,0)R i=J i;J i→n d=0×(n?1,0)R f=J f;J f) f

B(E1;αnJ i→α(n?1)J f)=nB(E1;αn=1→αn=0),(11) =

f

whereαdenotes all other quantum numbers that cannot change.

(ii)Strong-coupling rotational limit.Results up to the DGDR are shown in Fig.1b.Although all individual B(E1)s are known,for simplicity of presen-tation only the summed strengths K′L′B(E1;(λ,μ)KL→(λ′,μ′)K′L′)are shown.In the limit of large boson number N one recovers harmonic results that have a simple geometric interpretation.For example,the B(E1)values from the0+ground state to the two1?GDRs are1and2,respectively,the?rst associated with an oscillation along the axis of symmetry(say the z direction) and the second with oscillations in the x and y directions.For the single-to-double GDR excitation one?nds B(E1)values which are,for N→∞,2,2,1 and3.The large-N results can be generalised to(GDR)n.

Coulomb excitation in heavy-ion collisions is usually described by treating the relative motion classically while the internal degrees of freedom of the colliding nuclei are accounted for quantum mechanically.The operator responsible for the excitation depends on time through the relative distance.For relativistic collisions its expression is as in equation(35)of[7]where each term of the

multiple expansion of the external?eld factorizes into two elements.The?rst

depends on the collision properties,the second on the structure of the nucleus being excited.In the present study only contributions from the M(E1)matrix

elements are considered and those are calculated within the model described above.The solution of the time-dependent Schr¨o dinger equation leads to a

set of coupled di?erential equations for the probability amplitudes to excite

the(GDR)n states.For each impact parameter b these equations are inte-grated along the appropriate classical trajectory.For each(GDR)n state the

total inelastic cross section is then obtained by integrating the correspond-

ing probability over all impact parameters,starting from a mininum value b min=1.34[A1/31+A1/32?0.75(A?1/31+A?1/32)]fm[26].

The above formalism will now be applied to the relativistic Coulomb excitation

of the single and double GDRs in238U and in the chain of even isotopes148Sm

to154Sm.The former is an example of a well-deformed nucleus while the samar-ium isotopes exhibit a change from vibrational to deformed as A increases.The

description of such structural changes requires the use of a transitional IBM hamiltonian and hence the following analysis is not con?ned to any of the

previously discussed analytical limits but always involves a numerical diago-

nalisation.For238U,a consistent-Q[27]hamiltonian?H sd=κ?Qχ·?Qχ+κ′?L·?L is used withκ=?16keV,κ′=1.5keV andχ=?0.72.These parameters yield

an adequate description of the ground–gamma band splitting,of the moments

of inertia and of the E2transitions from gamma to ground band.The?H sd hamiltonian for the Sm isotopes is taken as in[21].The additional parameters ?p,α0,α2andζin?H p,?V sd?p and in the E1operator are given in Table1. They have been chosen as to reproduce the observed photoabsorption cross section[28,29]to the?rst GDR.Agreement is obtained if to each eigenstate

is associated a spreading widthΓi=0.007E2.5i in238U andΓi=0.029E1.81

i in

the Sm isotopes(with E i andΓi in MeV).To reproduce the photoabsorption cross section of a well-deformed nucleus one uses the fact that the energy split-ting of the GDR is very sensitive to the parameterα2.Then,onceα2is?xed, one variesα0to sligthly change the contribution of each peak to the energy-weigthed sum rule and?p to shift the energy of the dipole states.Finally,the parameterζis obtained from a global normalisation.

Recently,an experiment was done for238U+208Pb at0.5GeV/A(the data analysis is in progress[30]).The result of the corresponding calculation is shown in Fig.2.The full line corresponds to the total cross section obtained by smoothing the cross section to each discrete state with a Lorentzian having a width ofΓ1=2.5MeV andΓ2=

√

perturbation theory.In fact,if we use perturbation theory then the result for theσGDR increases up to4.2b.The two approaches give similar results only for large impact parameters.Some di?erence comes also from the di?erent B(E1)distribution.Figure2also shows the contributions associated with the 0+(dashed line)and2+(dot-dashed line)component of the DGDR.The1+ component does not appear in the?gure since it is extremely small.After substraction of the long single GDR tail,the three peaks,expected from(10), are clearly visible.(If the convolution of the cross section were done with a Γ2=2Γ1,the three peaks are still visible though less evident).Therefore,an exclusive experiment in coincidence withγ–γdecay might conceivably give a direct signature of the excitation of the DGDR.

The present results di?er from those of Ponomarev et al.[13]who study the same reaction with the particle–phonon model,using second-order perturba-tion theory to calculate the cross section.In ref.[13]the DGDR cross section appears as a structureless peak while here it does not.The reason is that the calculated cross section to the single GDR in ref.[13]shows three peaks in the energy region11–15MeV.As a result,since the DGDR states are constructed as products of two single GDR states,one expects six peaks in the DGDR energy region which eventually smear out the cross section to the DGDR. In our case we have only two peaks,which are present in the experimental data,because we have?xed the parameters of our hamiltonian by?tting the photoabsorption cross section.

The results of the calculated inelastic cross sections for the reactions208Pb +A Sm at0.5GeV/A are shown in Fig.3.The cross sections to each dis-crete state are shown as well as the ones obtained by the same smoothing procedure described above.In the transition from spherical to deformed one observes a continous evolution in the shape of the cross section resulting in the deformed case in a clear splitting in two peaks of the GDR and,corre-spondingly,three bumps in the DGDR energy region.This e?ect is due to the increase in separation between the two main components of the single GDR and the concentration of the small components in a more narrow energy range which in turn results from the increasing coupling of the GDR with quadrupole modes.

In summary,energy and E1properties of multiple giant dipole resonances have been studied in the context of the interacting boson model.The model has been applied to238U and to transitional samarium isotopes for which Coulomb excitation cross sections have been calculated in the reaction with208Pb at 0.5GeV/A.The example of the samarium isotopes shows how the excitation cross section is modi?ed when going from spherical to deformed nuclei.The calculation shows also that exclusive experiments on a well-deformed nucleus like238U could give a direct signature of the existence of the double giant dipole resonance.

This work has been supported by the Spanish DGICyT under contract PB95-0533-A,by an agreement between IN2P3(France)and CICYT(Spain)and by an agreement between INFN(Italy)and CICYT(Spain).E.G.L.is a Marie Curie Fellow with contract ERBFMBICT983090within the TMR program of the European Community.

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0+

0+

3

3e

e u ?

?10(2N,0)0+

(2N ?1,1)1?

(2N,1)1+

2N +1

e

e u ?

?4N N +1e

e u ??2N

2N

e e u ??6N ?3

3α2(keV)ζ(e fm)

Fig.2.Coulomb excitation of the single and double GDR in238U in the238U(0.5 GeV/A)+208Pb reaction.The dashed line is the contribution from the0+com-ponent of the two-phonon state while the2+one is represented by the dot–dashed line.The results corresponding to the1+component are too small to be seen.

Fig.3.Coulomb excitation cross sections of single and double GDR in several samar-ium isotopes in the 208Pb (0.5GeV/A)+A Sm reaction.The bars correspond to the cross sections to the discrete states and the full line corresponds to the convolution of these cross sections by a lorentzian of width Γ1=2.5MeV for the one-phonon states and Γ2=√