Coupling of Dipole Mode to γ -unstable Quadrupole Oscillations

arXiv:nucl-th/0007052v1 24 Jul 2000 Coupling of Dipole Mode to γ-unstable Quadrupole Oscillations C.E. Alonso1, M.V. Andr´es1, J.M. Arias1, E.G. Lanza1,2, and A. Vitturi3 1Departamento de F´ısica At´omica, Molecular y Nuclear, Facultad de F´ısica Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain 2INFN, Sezione di Catania, I-95129 Catania, Italy 3Dipartimento di Fisica and INFN, Padova, Italy (June 7, 2017) Abstract The coupling of the high-lying dipole mode to the low-lying quadrupole modes for the case of deformed γ-unstable nuclei is studied. Results from the geometrical model are compared to those obtained within the dipole boson model. Consistent results are obtained in both models. The dipole boson model is treated within the intrinsic frame, with subsequent projection onto the laboratory frame. As an application, calculations of photonuclear crosssections in γ-unstable nuclei are presented. PACS numbers: 21.60.Ev, 21.60.Fw, 21.10.Re 1 I. INTRODUCTION Low-lying quadrupole oscillations in nuclei have been studied extensively with all kinds of nuclear structure models. Among the more popular are the geometrical model [1] and the interacting boson model (IBM) [2]. Although rather different in their microscopic interpretation and formulation, both of them have been extremely successful in describing low-lying energy levels associated to quadrupole oscillations. In three different situations both models provide analytical results that can be identified in the geometrical model as those corresponding to spherical [1], well quadrupole deformed [1] and deformed γ-unstable shapes [3]. The corresponding situations occur in IBM for the U(5) [4], SU(3) [5] and O(6) [6] limits respectively. Connection between both models has been established by exploring the geometrical content of the IBM by using coherent states [7–9]. High-lying dipole states are studied in the context of the geometrical model by using the dynamic collective model within the hydrodynamical approach [10,11]. In the IBM context these states can be studied with the so called dipole boson model [12–15], which includes a dipole pboson into the usual IBM space (quadrupole sand dbosons). The connection between both approaches, especially for the case of well deformed nuclei, has already been discussed [15]. In this paper we are interested in studying the coupling of the dipole mode to a γunstable quadrupole form. The results of the geometrical model will be presented and the study within the dipole boson model will be performed both in the laboratory frame and in the intrinsic frame projecting afterwards onto the laboratory frame. The paper is structured as follows. In Section 2 the coupling of the dipole mode to a γ-unstable rotor is analysed within the geometrical model. In Section 3 the coupling of the dipole mode to an O(6) quadrupole deformed nuclei is investigated in the interacting boson model. This is done in two steps, first we study the problem in the intrinsic frame and then the projection to the laboratory frame is performed. Comparisons of exact results and those obtained in the intrinsic frame plus projection for different sets of parameters are 2 presented and discussed. In Section 4 examples of calculations for photonuclear cross-section are presented, and the different approximations compared. Finally, in Section 5 the paper is summarised and the main conclusions assessed. II. COUPLING THE DIPOLE MODE TO A γ-UNSTABLE ROTOR IN THE GEOMETRICAL MODEL The interaction between dipole and quadrupole oscillations has been studied in the geometrical model. The coupling to the quadrupole degree of freedom splits the energy of the dipole mode, and within the linear coupling in the quadrupole amplitude [1] the energies of the three resulting dipole resonances as seen from the intrinsic frame are E1µ(β, γ) = ǫ1"1 + 2 151/2κ1 C1 βcos γ−2(µ+ 2)π 3!# ;µ=−1,0,1,(1) where ǫ1is the unperturbed dipole energy, C1is the restoring force parameter, and κ1is the dipole–quadrupole coupling coefficient. A plot of these energies with respect to the unperturbed value ǫ1is given in Fig. 1 as a function of the asymmetry parameter γ, for a fixed value of the quadrupole deformation parameter β A detailed structure of the dipole absorption line can be obtained from a numerical diagonalization of the coupled dipole–quadrupole hamiltonian. A simple model in the strong coupling approximation for extracting the dipole strength distribution from the intrinsic values is provided by the procedure of averaging the three contributions over the different quadrupole βand γdeformations, according to the ground state distribution. The relative strength of the dipole transition probability B(E1; 0 →1− n) per unit of energy is in this way proportional to P(E) = 1 3 1 X µ=−1Z Z φ2 0(β, γ)δ(E−E1µ(β, γ)) β4dβ|sin 3γ|dγ , (2) where φ0(β, γ) is the wave function of the ground state resulting from the quadrupole energy surface. The specific case of γ-unstable rotor occurs when the intrinsic energy surface is 3 isotropic in the γvariable, while displaying a sharp minimum for deformation β0. Assuming the extreme situation of a very sharp minimum, we can in first approximation keep fixed the value of β0and only average over the γvariable. In this adiabatic picture one obtains the probability distribution shown in Fig. 2. The distribution is symmetric around ǫ1, with three maxima, two at the edges (at energies ǫ1±∆) and one in the center (at energy ǫ1), with ∆ = ǫ1s2 15 κ1 C1 β0,(3) and with zeros occurring at energies ǫ1±∆ 2. The resulting total splitting 2∆ is therefore directly proportional to the quadrupole deformation β0, as in the case of a static quadrupole deformation. Note that although the probability at the edges is 3 times the one at the center, the total probability is anyway exactly equally divided into the three bumps. It may be worth noticing that, aside from the broadening of the three peaks, the predicted situation in the γ-unstable case is actually rather similar to the prediction associated with a static quadrupole deformed system with γequal to 30 degrees. This similarity of the two situations is a feature that also appears for other observables and has been discussed elsewhere [16,17]. III. COUPLING THE DIPOLE MODE TO AN O(6) QUADRUPOLE DEFORMED NUCLEUS IN IBM To include the dipole degrees of freedom in the usual IBM, one negative parity dipole p–boson (l= 1) has to be included in addition to the positive parity quadrupole s– (l= 0) and d– (l= 2) bosons. For a boson system with Nquadrupole bosons plus 1 dipole boson, the hamiltonian can be written as H=Hsd +Hp+Vsd−p,(4) where Hsd is the usual IBM hamiltonian for quadrupole degrees of freedom, Hpis the dipole hamiltonian given by 4 Hp=ǫ1ˆnp,(5) and Vsd−pis the interaction between quadrupole and dipole degrees of freedom. The operator ˆnpis the p-boson number operator, which in this case is 0 or 1, and ǫ1is the unperturbed dipole energy. For the interaction a simple quadrupole-quadrupole form is assumed Vsd−p=−κ Q(sd)·q(p),(6) where Q(sd)is the usual IBM quadrupole operator Q(sd) µ=s†˜ d+d†s(2) µ+χd†˜ d(2) µ,(7) and q(p) µ=p†˜p(2) µ.(8) The operators ˜γm= (−1)mγ−m(where γstands for s,dor pbosons) are introduced so as to have operators with the appropriate properties under spatial rotations. The simplest form for the dipole operator is Dµ=ξp†+ ˜pµ,(9) where ξis an overall normalization parameter, which will be assumed equal to unity in the applications. A. The coupling in the laboratory frame First the laboratory IBM calculation is presented. The quadrupole s−dhamiltonian for the O(6) limit can be written in the form Hsd[O(6)] = b C2[O(5)] + c C2[O(3)] + d C2[O(6)] (10) where C2stands for the quadratic Casimir operator of the indicated algebra as defined in Ref. [2]. Energies associated with this s−dhamiltonian can be obtained analytically to be 5 E= 2b τ(τ+ 3) + 2c J(J+ 1) + 2d σ(σ+ 4) ,(11) where σ,τand Jare the labels of the irreducible representations of the algebras O(6), O(5) and O(3) respectively. The coupling to the dipole oscillations and the calculation of transition probabilities is done numerically by using the computer codes GDR and GDRT, respectively [18]. The states belonging to the σ=Nrepresentation are favoured in energy and the relative position of the states belonging to other representations is governed by the parameter d. In the O(6) limit these representations are not connected by the quadrupole sd operator if χ= 0 is taken1in Eq. (7), and they therefore remain unmixed (and characterized by a good value of σ) also with the inclusion of the dipole-quadrupole coupling. Consequently the dipole operator will only connect the ground state with states of the σ=Nrepresentation, which will therefore be the only ones contributing to the B(E1; 0+ 1→1− n) distribution. This implies that, as far as the dipole distribution is concerned, the calculation is completely insensitive to the value of the parameter d. For the other parameters, in order to favour the comparison with preceding results obtained within the simple scheme based on the geometrical model, the parameter chas been put equal to zero in order to minimize the splitting due to the angular momentum, a feature not accounted for by the unprojected intrinsic state. The results obtained for the energies of dipole states and the corresponding B(E1; 0+ 1→1− n) are shown in Fig. 3 (left panels) for different values of the parameter bin front of the O(5) Casimir operator. At variance with the previous distribution displayed in Fig. 2 we have now a discrete distribution due to the finite number of bosons and consequently finite number of states. Note that the pure SU(3) limit with χ=−√7/2 gives a dipole strength distribution with only two lines, since only two 1−states can be obtained by coupling the dipole boson with the ground state rotational band. In the O(6) limit there are instead (N+ 1) 1−states connected to the ground state by the dipole operator, namely all those originated by the 0+ 1With this choice the quadrupole operator is a generator of the O(6) algebra. 6 and 2+states appearing in the σ=Nband. We can however compare the envelope of this discrete distribution with the continuous distribution given by the geometrical model in its simplest adiabatic version. In order to facilitate comparisons an appropriate discretization of Fig. 2 for the case of N=15 bosons, an unperturbed dipole energy ǫ1= 15 MeV, and a dipole-quadrupole coupling κ=0.2 MeV (same parameters as in Fig. 3) is presented in Fig. 4. In the limit of very small values of b, namely in strong-coupling situations, the splitting due to the different values of τwithin each O(6) representation is small compared to the coupling. As a consequence the patterns obtained in Figs. 3 and 4 are similar, since the mixing between the different states induced by the quadrupole-dipole interaction is large. At the other extreme, for values of blarge with respect to the dipole-quadrupole coupling the pattern observed in Fig. 3 becomes very asymmetric, tending to concentrate the transition strength onto the lowest state, washing out any resemblance with the one obtained in the geometrical model, Fig. 4, and giving rise to a situation close to the pure spherical case. In fact in this weak-coupling limit, because of the large τsplitting, only the lowest 1−states have large overlaps with the initial s–d ground state. For a better understanding of the situation we will study next the problem within the intrinsic frame in the IBM. B. The coupling in the intrinsic frame Let us now study the problem of the coupling of quadrupole and dipole degrees of freedom in the intrinsic frame, within the IBM description. The basic idea of the intrinsic-frame formalism in this case is to consider that the pure quadrupole states of the ground “band” are globally described by a boson condensate of the form |gi=1 √N!(Γ† g)N|0i,(12) where the basic boson is given by Γ† g=1 √1 + β2"s†+βcos γd† 0+1 √2βsin γ(d† 2+d† −2)#,(13) 7 βand γbeing obtained by minimizing the energy surface. In the case of SU(3) one has β=√2 and γ= 0, and the intrinsic state is fully representative of the ground-state rotational band. In the case of O(6), instead, one has β= 1, and the intrinsic state, which gives rise to a γ-independent energy surface, is associated with all states of the “band” corresponding to the σ=Nrepresentation. For the dipole part, the corresponding building blocks are p† 0,p† 1and p† −1. Thus, the three states |gpµiprovide with an intrinsic basis where the dipole-quadrupole states can be studied. In this basis, the pure sd and pparts of the hamiltonian are diagonal and the only non-diagonal contributions come from the interaction term. The matrix elements needed in order 1/N are hgpµ|Q(sd)·q(p)|gpµi=X i=0,±2hg|Q(sd) i|gihpµ|q(p) −i|pµi,(14) where hg|Q(sd) i|gi ≡ Q(sd) i(β, γ) have already been calculated [19], Q(sd) 0(β, γ) = N 1 + β2 2βcos γ−s2 7χβ2cos 2γ ,(15) Q(sd) 2(β, γ) = Q(sd) −2(β, γ) = N 1 + β2 √2βsin γ+s1 7χβ2sin 2γ .(16) The Q(sd)matrix elements not specified are zero. The matrix elements of q(p)can be easily computed to be hp1|q(p) 0|p1i=hp−1|q(p) 0|p−1i=1 √6,(17) hp0|q(p) 0|p0i=−2 √6,(18) hp1|q(p) 2|p−1i=hp−1|q(p) −2|p1i= 1 .(19) All the remaining relevant matrix elements for our study are zero. The parameter χis the structure constant in the quadrupole operator Q(sd). The SU(3) case (studied in Ref. [15]) is obtained for χ=−√7 2and the O(6) limit corresponds to χ= 0. The dipole energies are obtained by diagonalising the interaction Vsd−pin the basis |gpµi. They are analytically given by 8 λ0=ǫ1+κ2 √6Q(sd) 0(β, γ), λ+=ǫ1−κ"1 √6Q(sd) 0(β, γ) + Q(sd) 2(β, γ)#,(20) λ−=ǫ1−κ"1 √6Q(sd) 0(β, γ)−Q(sd) 2(β, γ)#, which provide the dependence of the dipole energies on the IBM deformation parameters (β, γ). With the choice χ=−√7 2, corresponding to the SU(3) limit, Eqs. (20) become equivalent to Eqs. (9b) in Ref. [15] with terms linear and quadratic in β. The corresponding energies are shown in Fig. 5a. The O(6) limit corresponds instead to χ= 0. In this case Eqs. (20) have only a linear term in β, and the corresponding energies are shown in Fig. 5b. These latter energies are equal to those obtained in the preceding section for the geometrical model, once the correspondence is made between deformation parameters and coupling strengths in the two models, such that ∆IBM = 2s2 3 βIBM 1 + β2 IBM κN = ∆ = ǫ1s2 15 κ1 C1 β0.(21) As in the case of the geometrical model, a simple way of obtaining the dipole strength distribution from the IBM intrinsic state is to average the intrinsic energies over γ. In analogy to the intrinsic energies, the resulting distribution will show, mutatis mutandis, a pattern identical to the one shown in Fig. 2 for the geometrical model. C. Projection from the intrinsic frame to the laboratory We have seen so far that the straightforward use of the intrinsic state in the IBM without any projection technique gives results similar to the ones obtained in the geometrical model. 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[22] Reference Input Parameter Library for Theoretical Calculation of Nuclear Reactions (RIPL Handbook), Chapter 6: Gamma-Ray Strength Functions. http://www-nds.iaea.or.at/ripl/ 17 FIGURES −1 −0.5 0 0.5 1 π/6 γ (rad) π/3 (E1µ−ε1)/∆ FIG. 1. Energies of the dipole resonances for an ellipsoidal nucleus with quadrupole deformation β0as a function of the shape parameter γin the geometrical model. Energies are given with respect to the unperturbed energy ǫ1in units of ∆ (q2 15ǫ1β0κ1/C1). 18 −1 −0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 (E−ε1)/∆ P(E) ∆ FIG. 2. Probability per unit energy for absorption of a quantum E, in units of 1/∆, as a function of the energy in the geometrical model for a γ-unstable nucleus. Energies are given with respect to the unperturbed energy ǫ1in units of ∆. 19 10 12 14 16 18 20 0 0.2 0.4 0 0.2 0.4 0 0.4 0.8 0 0.5 1 1.5 2 12 14 16 18 20 En (MeV) B(E1;01+ → 1n−) Lab b=0.15 MeV Lab Lab Lab b=0.015 MeV b=0.0015 MeV b=0.00015 MeV b=0.15 MeV b=0.015 MeV b=0.0015 MeV b=0.00015 MeV Intr. + Proj. Intr. Intr. Intr. Proj. Proj. Proj. Proj. + + + FIG. 3. Distribution of B(E1) strength obtained in the IBM dipole-boson model. The calculations have been performed in the case of N= 15 bosons, with an O(6) hamiltonian with parameters c=0 and different values of b, an unperturbed dipole energy ǫ1= 15 MeV and a dipole-quadrupole coupling κ= 0.2 MeV. Left panels present the exact calculation in the laboratory system and right ones the corresponding calculations in the intrinsic frame plus projection on τ. The continuous lines just joint the extremes of the bars. 20 10 12 14 16 18 20 En (MeV) 0 0.2 0.4 B(E1; 01+ → 1n−) Geometrical (adiabatic) FIG. 4. Distribution of B(E1) strength obtained in the adiabatic picture of the geometrical model or, equivalently, in the unprojected IBM dipole-boson model. The bar diagram is obtained by discretizing the continuous distribution of Fig. 2 into 16 states (appropriate to the case N=15 presented in Fig. 3) and summing the contribution of each discrete state in each bin (full line just joint the extremes of the bars). The calculations have been performed for the case of unperturbed dipole energy ǫ1= 15 MeV and dipole-quadrupole coupling κ=0.2 MeV (same parameters as the ones in Fig.3). The relation between κand κ1 C1β0is given in Eq. (21). 21 −1 −0.5 0 0.5 1 1.5 SU(3) O(6) π/6 π/3 γ (rad) π/6 π/3 (E1µ−ε1)/∆IBM (a) (b) FIG. 5. Energies of the dipole resonances obtained in the intrinsic frame from an IBM dipole-boson model as a function of the shape shape parameter γ. Fig. (a) refers to the case of SU(3) and Fig. (b) to the case of O(6). Energies are given with respect to the unperturbed energy ǫ1in units of ∆IBM 22 10 12 14 16 18 E (MeV) 0 1 2 σ (arbitrary units) 0 1 2 3 σ (arbitrary units) 12 14 16 18 20 E (MeV) a) Lab b) Lab a) Intr. + Proj. b) Intr. + Proj. FIG. 6. Photonuclear cross-sections for two selected cases from Fig. 3. Left panels give the laboratory results and right panels the intrinsic plus projection on τresults. Upper panels correspond to the case in which the dipole widths are assumed to have a power dependence law on the energy with Γn=0.026 E1.91 n(MeV). Lower panels represent equivalent cases but taking the dipole widths as a constant equal to 2.5 MeV. Full lines give the result for the case b=0.00015 MeV (almost τ degenerate case) while dashed lines are for b=0.15 MeV. 23