Anharmonic double-γ vibrations in nuclei and their description in the interacting boson model

Anharmonic double- ␥ vibrations in nuclei and their description in the interacting boson model J. E. Garcı ´a-Ramos,1C. E. Alonso,1J. M. Arias,1and P. Van Isacker2 1Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain 2Grand Acce ´le ´rateur National d’Ions Lourds, B.P. 5027, F-14076 Caen Cedex 5, France 共Received 17 November 1999; published 10 March 2000兲 Double- ␥ vibrations in deformed nuclei are studied in the context of the interacting boson model with special reference to their anharmonic character. It is shown that large anharmonicities can be obtained with interactions that are 共at least兲of three-body nature between the bosons. As an example the ␥ vibrations of the nucleus 68 166Er98 are studied in detail. PACS number共s兲: 21.60.Fw, 21.10.Re, 21.60.Ev, 27.70.⫹q Nuclear quadrupole shape oscillations can be of two types: ␤ or ␥ vibrations 关1兴. The ␤ vibration preserves axial symmetry and a one-quantum excitation gives rise to a K ⫽0 band where Kis the projection of the angular momentum on the axis of symmetry of the nucleus. A ␥ vibration breaks axial symmetry and leads to a K⫽2 band. Although their existence has been conjectured a long time ago 关2兴, the observation and interpretation of ␤ -vibrational K ␲ ⫽0⫹bands is still fraught with questions and difficulties. In contrast, ␥ -vibrational K ␲ ⫽2⫹bands are systematically observed in deformed nuclei and their properties are correspondingly better understood. Since single- ␥ vibrations are so well established, it is natural to search for double- ␥ vibrations and to examine their harmonic nature 共i.e., whether they occur at twice the energy of the single vibration兲. Two intrinsic K⫽2 quanta can be combined parallel or antiparallel and hence lead to two bands: one with K⫽0 and another with K⫽4. The experimental identification of double- ␥ vibrations in deformed nuclei is difficult since they are expected to lie above the pairing gap and to mix with two-quasiparticle excitations, resulting in fragmentation and a corresponding reduction in the collectivity of the states. During the past few years, however, a steady improvement of experimental techniques has allowed the measurement of low-spin states in the energy region of interest 关3–5兴. This possibility has reopened the old debate on the existence of two-phonon ( ␤ or ␥ ) vibrational states and their properties. Experiments have been reported recently pointing out the existence of double- ␥ vibrations in several deformed nuclei with a wide range of anharmonicities 关6–11兴. In particular, in Refs. 关7,8兴the first observation of the K ␲ ⫽0⫹and K ␲ ⫽4⫹double- ␥ states in one nucleus, 166Er, is reported. They are observed at 1.949 MeV and 2.029 MeV, respectively. This information is of great interest since it provides a stringent test of nuclear models; for instance, the quasiphonon nuclear model 共QPNM兲predicts no K ␲ ⫽0⫹two-phonon state below 2.5 MeV in 166Er 关12兴. Several calculations of two-phonon states, using either phenomenological or microscopic models, are available, particularly for 166Er and 168Er 关12–17兴. One of the models employed is the interacting boson model 共IBM兲关18兴.Inthe simplest version of this model, referred to as IBM-1, an even-even nucleus with nvalence nucleons is treated as a system of N⫽n/2 bosons with l⫽0(sbosons兲or l⫽2(d bosons兲. In the usual formulation of the model only up to two-body interactions between the bosons are taken. What are the predictions of IBM with regard to twophonon states in deformed nuclei and their 共an兲harmonic nature? It was pointed out some time ago by Bohr and Mottelson 关13兴that the IBM-1 is unable to accommodate large anharmonicities, as observed for instance in 168Er. Subsequently, it was shown that these can be described but require gbosons with l⫽4 in addition to the sand dbosons (sdg-IBM) 关16兴. More recently, we reported a study of twophonon states in IBM-1 treated in the intrinsic frame 关19兴 and showed that the IBM-1 is a harmonic model in the limit of large boson number. Anharmonicities can only exist for finite boson number and they are always small if only up to two-body interactions are considered. It was also suggested that anharmonicity in the model is linked to triaxiality. Since it is known that IBM-1 with only up to two-body interactions cannot give rise to a stable triaxial minimum, the model’s capability for describing anharmonicities depends on the inclusion in the Hamiltonian of higher-order interactions, some of which are known to induce triaxial shapes 关20,21兴. In this article the relation between three-body interactions in IBM-1 and the anharmonicity of ␥ vibrations in deformed nuclei is investigated. Although the analysis presented is not exhaustive, it is shown that anharmonic behavior can be obtained with reasonable three-body interactions. As an example, the energy and E2 transition properties of the ␥ vibrations of the nucleus 166Er are studied in detail. In addition, the nature of the 02 ⫹state in the same nucleus, which has been the subject of an intense debate in the past few years 关22–25兴, is briefly discussed. The Hamiltonian adopted in the following includes a quadrupole-quadrupole term, a rotational L ˆ2term, and threebody interactions between the dbosons, H ˆ⫽⫺ ␬ Q ˆ•Q ˆ⫹ ␬ ⬘L ˆ•L ˆ⫹兺 kl ␪ l„共d†⫻d†兲(k)⫻d†…(l) ⫻„共d ˜ ⫻d ˜ 兲(k)⫻d ˜ …(l),共1兲 where •denotes scalar product, d ˜ ␮ ⫽(⫺1) ␮ d⫺ ␮ ,Q ˆis the boson quadrupole operator, and L ˆis the angular momentum operator: PHYSICAL REVIEW C, VOLUME 61, 047305 0556-2813/2000/61共4兲/047305共4兲/$15.00 ©2000 The American Physical Society61 047305-1 Q ˆ⫽s†d ˜ ⫹d†s ˜ ⫹ ␹ 共d†⫻d ˜ 兲(2),L ˆ⫽ 冑 10共d†⫻d ˜ 兲(1).共2兲 Five independent three-body d-boson interactions exist which have l⫽0, 2, 3, 4, and 6. Interactions with the same l but different kare not independent but differ by a normalization factor only 关20兴. The combinations (k,l)⫽(2,0), 共0,2兲, 共2,3兲,共2,4兲, and 共4,6兲are chosen here. The Hamiltonian 共1兲is certainly not the most general that can be considered. Notably, a vibrational term ⑀ dn ˆdwhich dominates in spherical nuclei is omitted since it is thought of lesser importance in the deformed nuclei considered here. It is clear that the inclusion of such additional terms might improve the quality of detailed fits to particular nuclei such as the one for 166Er presented below. Finally, of all possible three-body interactions only those between the dbosons are retained here since these are most crucial for obtaining a stable triaxial minimum 关20兴. For the discussion of anharmonicities of ␥ vibrations it is convenient to define the following ratios of excitation energies: R0 ␥ ⬅Ex共0 ␥␥ ⫹兲 Ex共2 ␥ ⫹兲⫺Ex共21 ⫹兲,R4 ␥ ⬅Ex共4 ␥␥ ⫹兲⫺Ex共41 ⫹兲 Ex共2 ␥ ⫹兲⫺Ex共21 ⫹兲,共3兲 where 0 ␥␥ ⫹and 4 ␥␥ ⫹are the band heads of the K ␲ ⫽0⫹and K ␲ ⫽4⫹double- ␥ bands, respectively. It should be noted that the quantities RK ␥ do not depend upon the L ˆ2term in the Hamiltonian; if a single three-body term is included they depend on two parameters, ␹ and the ratio ␪ l/ ␬ .Inthe present work the identification of the states 0 ␥␥ ⫹and 4 ␥␥ ⫹is based on the B(E2) values for decaying to the single gamma state. In Fig. 1 the quantities RK ␥ are plotted 共for N⫽15 bosons兲as a function of the quadrupole parameter ␹ 关varying between its SU共3兲and O共6兲values ⫺1 2 冑 7 and 0兴in the absence of three-body interactions. The ratio R4 ␥ remains about constant and of the order 1.8; R0 ␥ shoots up for small 兩 ␹ 兩 . Close to the O共6兲limit the concept of a ␥ vibration is not well defined and so nothing is plotted for 兩 ␹ 兩 ⬍0.15. The value of ␹ is constrained by E2 transition probabilities and in deformed rare-earth nuclei it ranges typically between ⫺0.4 and ⫺0.7 关26兴. From Fig. 1 it is clear that no substantial anharmonicity occurs in the ␥ vibration for these values of ␹ . In Fig. 2 the influence of the various three-body interactions is shown for a typical value of ␹ ( ␹ ⫽⫺0.5) and for N⫽15 bosons. It is seen that ␥ -vibrational anharmonic behavior is obtained which can be different for the K ␲ ⫽0⫹ and K ␲ ⫽4⫹bands 共e.g., positive for the former while negative for the latter兲. Care has been taken to plot results only up to values of ␪ lthat do not drastically alter the character of rotational spectrum; beyond these values, the three-body interaction, being of highest order in the Hamiltonian 共1兲, becomes dominant. Also shown in Fig. 2 are the ratios RK ␥ as observed in 166Er 关7,8兴,R0 ␥ ⫽2.76 and R4 ␥ ⫽2.50. This simple analysis shows that, on purely phenomenological grounds, the appropriate three-body interaction with the correct anharmonic character for the K ␲ ⫽0⫹and K ␲ ⫽4⫹bands in 166Er, has l⫽4. Figure 3 shows the experimental spectrum of 166Er 关7,8兴 and compares it to the eigenspectrum of Hamiltonian 共1兲 with an l⫽4 three-body interaction. The parameters are ␬ FIG. 1. The ratios RK ␥ 共as defined in the text兲as a function of ␹ . The Hamiltonian 共1兲is used with ␪ l⫽0; the boson number is N ⫽15. FIG. 2. The ratios RK ␥ 共as defined in the text兲as a function of ␪ l/ ␬ for different l. The Hamiltonian 共1兲is used with ␹ ⫽⫺0.5; the boson number is N⫽15. The dashed lines give the experimental values for the corresponding ratios in 166Er. BRIEF REPORTS PHYSICAL REVIEW C 61 047305 047305-2 ⫽23.8 keV, ␹ ⫽⫺0.55, ␬ ⬘⫽⫺1.9 keV, and ␪ 4 ⫽93.9 keV, with boson number N⫽15. With these values the calculated excitation energies of the double- ␥ band heads are 1926 keV and 1972 keV for the K ␲ ⫽0⫹and K ␲ ⫽4⫹ levels, respectively, leading to the ratios R0 ␥ ⫽2.82 and R4 ␥ ⫽2.45, in excellent agreement with observation. Note, however, that although all ␥ -band heads are well reproduced by the calculation, problems arise for the moments of inertia, in particular of the ␥ band. An extensive survey of combinations of cubic d-boson interactions has shown that it is difficult to substantially improve upon this fit although it is of course near-impossible to do an exhaustive search of the complex parameter space of all three-body interactions. In contrast, exploratory searches with simple quartic Hamiltonians quickly yield the correct result with respect to both band-head energies and moments of inertia. For the calculation of E2 transition probabilities the consistent-Q formalism 共CQF兲关27兴is adopted by using the E2 transition operator T ˆ共E2兲⫽eeffQ ˆ,共4兲 where Q ˆis the boson quadrupole operator used in the Hamiltonian 共1兲and eeff is a boson effective charge, determined from the observed B(E2;21 ⫹ →01 ⫹) value. It should be noted that the inclusion of three-body terms in the Hamiltonian would allow the use of a two-body E2 operator. However, we have not tried to do that in order to keep the calculation TABLE I. Observed and calculated B(E2) values and ratios for 166Er. The E2 operator 共4兲is used with eeff 2⫽(1.83)2W.u. and ␹ ⫽⫺0.55. B(E2) value or ratio Observed Calculated B(E2;21 ⫹ →01 ⫹) (W.u.) 214⫾10a214 B(E2;41 ⫹ →21 ⫹) (W.u.) 311⫾10a304 B(E2;2 ␥ ⫹ →01 ⫹) (W.u.) 5.5⫾0.4a5.3 B(E2;02 ⫹ →21 ⫹)/B(E2;2 ␥ ⫹ →01 ⫹)0.49⫾0.19b0.21 B(E2;02 ⫹ →2 ␥ ⫹)/B(E2;2 ␥ ⫹ →01 ⫹)0.44⫾0.13b6.2 B(E2;0 ␥␥ ⫹ →2 ␥ ⫹)/B(E2;2 ␥ ⫹ →01 ⫹)3.8⫾1.3c(2.2⫹1.1⫺0.7d兲3.2 B(E2;4 ␥␥ ⫹ →2 ␥ ⫹)/B(E2;2 ␥ ⫹ →01 ⫹)1.3⫾0.4c(0.9⫾0.3d兲2.5 aFrom Ref. 关28兴. bFrom Ref. 关22兴. cFrom Ref. 关8兴. dFrom Ref. 关7兴. FIG. 3. Experimental 共a兲and calculated 共b兲spectrum for 166Er. The theoretical results are obtained with the Hamiltonian 共1兲 with ␬ ⫽23.8 keV, ␹ ⫽⫺0.55, ␬ ⬘⫽⫺1.9 keV, and ␪ 4⫽93.9 keV. The boson number is N ⫽15. BRIEF REPORTS PHYSICAL REVIEW C 61 047305 047305-3 on the anharmonicity of the double-gamma excitation as simple as possible. In Table I the observed B(E2) values and ratios concerning the ␥ vibrational band heads in 166Er are summarized and compared to the theoretical results obtained with eeff 2⫽(1.83)2Weisskopf units 共W.u.兲. A good overall agreement is found but for the decay of the 02 ⫹state: the B(E2;02 ⫹ →2 ␥ ⫹) value is overpredicted by more than an order of magnitude while the B(E2;02 ⫹ →21 ⫹) value is too small by a factor two. This casts doubt on the interpretation of the 02 ⫹observed at 1460 keV as the ␤ -band head. Previous interpretations of this state are contradictory: it is considered as the ␤ -band head in 关1兴but as a two-quasiparticle state in 关22兴while Casten and von Brentano 关23兴claim it is a collective phonon excitation built on the ␥ band. Other 0⫹ states are found in 166Er at slightly higher energy 关22兴but none has the decay pattern in agreement with the present calculation. A possible explanation is that collective strength is fragmented through mixing with two-quasiparticle states which are absent from the IBM-1 model space. In summary, three-body interactions in the IBM-1 can account for a wide variety of ␥ -vibrational anharmonicities in nuclei such as for instance those observed in 166Er but not without substantially changing the moments of inertia of various bands. The knowledge of the two double- ␥ vibrational bands (K ␲ ⫽0⫹and K ␲ ⫽4⫹) in a single nucleus provides a stringent test of nuclear models and, specifically, of the type and strength of three-body interactions in IBM-1. More experiments on double- ␥ vibrations are thus called for since they should provide essential information concerning the systematic behavior of these states and hence the interactions involved. From the theoretical side, a systematic analysis of all three-body interactions and not just those between the dbosons seems in order. 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