Session SJ - Mini-symposium: Triaxiality in Nuclei.
FOCUS session, Saturday morning, October 20
Lokelani III, Outrigger Wailea Resort

[SJ.001] Triaxial nuclei at high spin: Chiral wobblers

The orientation of an axial nucleus is specified by one angle whereas two angles are needed for a triaxial nucleus. Recently evidence for the coupling of this new degree of freedom with the ones represented by nucleons on high-j orbitals has been found at high spin. The triaxial density may execute a wobbling motion relative to the angular momentum vector of

a high-j orbital, which leads to two rotational \Delta I=2 bands with characteristic decay pattern. The angular momenta of high-j particles and holes may combine in two different ways with the angular momentum of the remaining nucleons forming the triaxial core. The two arrangements, which have opposite chirality, give rise to two degenerate rotational \Delta I=1 bands. Tunneling between the left- and right-handed configurations causes an energetic split of the two bands.

The angular momentum of the core nucleons may execute a wobbling motion with respect to the plane spanned by the high-j particles and holes. Such a chiral wobbler or chiral vibrator appears as two \Delta I=1 bands connected by characteristic electromagnetic transitions.

note

[SJ.002] Systematic investigation of triaxiality in odd-odd ^128-134Cs

Bands built on the \pih_11/2\nuh_11/2 configuration in four odd-odd _55Cs isotopes, namely ^128-134Cs, have been systematically investigated using (HI,xn) reactions. In ^128-132Cs, nearly degenerate (\leq300 keV) \DeltaI=1 partner bands, which are linked to the previously observed positive-parity bands by M1/E2 transitions, have been observed. Both partner band members in each isotope have been identified as being based on the \pih_11/2\nuh_11/2 configuration. They are interpreted in terms of chirality attained in the body-fixed frame; the angular momenta of the valence proton, valence neutron, and the triaxial core rotation are mutually perpendicular leading to either a right- or left-handed system. For the N=79 ^134Cs isotope being close to the N=82 closed shell no collectivity was observed. Experimental results will be compared to lab-frame calculations using phenomenological core-particle-hole coupling based on the Kerman-Klein-Dönau-Frauendorf method. A rigid triaxial rotor with an irrotational flow moment of inertia was used for the core. In addition, a variable moment of inertia was incorporated, which yielded good energy-level fits to the yrast bands in the even-even Xe core nuclei. Experimentally, the energy displacement between the partner bands, B(M1)/B(E2) ratios, and S(I)=[E(I)-E(I-1)]/2I values show unique characteristics; these will be presented as sensitive comparisons between the experiment results and the model calculations.

[SJ.003] Testing the limits of chiral structure in the mass 130 region

In the mass A\sim130 region, pairs of bands with the same spins and parities and similar excitation energies in some odd-odd N=73, 75, and 77 isotones have been suggested as candidate bands for chiral symmetry breaking(See, for example, K. Starosta et al., Phys. Rev. Lett. 86, 971 (2001).). As chiral structures are predicted to occur only for a limited range of particle numbers, it is important to test the limits of this region of chirality. Here, we address the N=77 isotones ^136Pr, ^138Pm, and ^140Eu. High-spin states in the ^136Pr, ^138Pm, and ^140Eu nuclei were populated with the ^122Sn(^19F,5n), ^116Cd(^27Al,5n), and ^92Mo(^51V,2pn) reactions, respectively. The data are being analyzed in a search for chiral symmetry breaking, and new high-spin levels have been found for ^140Eu. Work supported by the U.S. DOE grants DE-FG02-91ER-40609, DE-FG05-96ER-40983, and DE-FG02-88ER-40417 and by DFG grant Pi393/1-1.

[SJ.004] Signature inversion / chiral-twin bands in odd-odd Pr nuclei?

Over the past few years, sufficient data have been accumulated to enable a meaningful study of the systematic trends of the signature inversion (inversion point shift in spin with increasing proton and neutron numbers in a chain of isotones / isotopes as well as the magnitude of odd-even staggering). Our aim is to understand these systematic features within the framework of particle rotor model including both a residual pn interaction and a \gamma deformation.

Signature inversion is present in the bands of odd-odd nuclei , ^120-130Cs, ^124-132La, ^126-134Pr and ^132-136Pm and having an yrast structure built on \pi h_11/2\nu h_11/2 orbitals. Pr isotopes seem to indicate an inversion decreasing for smaller neutron numbers, trend that is opposite for the Cs nuclei(J.F. Smith et al., Phys. Lett B 406, 7 (1997)). Why?

A question that remains to be answered is if there is any relation of signature inversion to chiral twin bands (two ''look alike positive parity'' bands proposed for as in ). The lower band has signature inversion all the way up. Could these effects be related to triaxiality? Can one trust an apparent conclusion suggested by L.L. Riedinger( L.L. Riedinger, talk presented at \mathitHigh) \mathitSpin \mathitPhysics \mathit2001, Warsaw, Poland, February, 2001, to be published in Acta Phys. Pol.: ''signature inversion in an odd-odd band of two quasiparticles pointed along different axes is always associated with the formation of chiral twin bands''?

[SJ.005] Triaxial Deformation in the Monte-Carlo Shell Model

We present recent results of the Monte Carlo Shell Model (MCSM) which has been introduced to solve shell-model problems with large dimensions. The main topic is the triaxial deformation in medium-heavy nuclei. Since the dimension of the many-body Hilbert space is so huge that conventional shell model calculations cannot be used. On the other hand, the triaxial deformation occurs usually in transitional nuclei where mean-field approaches are not adequate. We introduced an extended version of the MCSM with condensed-pair basis vectors, in order to incorporate pairing correlations well. We will discuss the triaxially deformed states of Xe and Ba on the microscopic basis, without introducing any special apparatus to bring about triaxiality. The interaction is of the form of pairing plus quadrupole-quadrupole. The energy levels and E2 transition quantities obtained by the MCSM show distinct characteristics of the triaxial deformation and agree well with experiments.

[SJ.006] Relativistic mean field description of tilted axis rotation

Recently a number of evidence that show the appearance of tilted axis rotation have been obtained. One example is the so-called shears bands in weakly deformed nuclei, in which almost perpendicular coupling of the proton and neutron angular momenta results in the total spin tilted from the principal axes. Up to now, many shears bands have been observed in several mass regions. Another example is the chiral doublet bands in triaxially deformed nuclei. As a result of the breaking of `chiral symmetry', almost degenerated twin bands are expected to be developed. Such doublet bands were recently discovered.

As for the mean field approaches to such tilted axis rotation, only studies based on simple mean field Hamiltonian such as the P+QQ model have been done. Those based on more sophisticated models, such as the Skyrme Hartree Fock or the Relativistic Mean Field (RMF) models are still missing and strongly desired. In this work, we for the first time apply the RMF model to the tilted axis rotation. The generalization of this model to the one dimensionally rotating frame was first done by Munich group, and extensive studies of the superdeformed bands have been performed. We extend this model to the three dimensionally rotating frame, which is a straightforward work. Several numerical results on the tilted bands in the A \sim 80 and 130 regions are reported and discussed.

[SJ.007] Universal view of nuclear E2 properties provided by a triaxial rotor model

The quadrupole properties of the yrast states of even-even nuclei are shown to widely exhibit rigid rotor values. This is an experimental fact which has not, to our knowledge, been recognized before. We use for our compilation all evaluated nuclear structure data currently available from the National Nuclear Data Center [1]. This apparently universal feature of even-even nuclei is interpreted using a triaxial rotor model with moments of inertia fitted to the data. This is opposed to the usual method [2] of using ``irrotational flow'' moments of inertia, which are smaller than the experimental numbers by about a factor of five. For a broad range of nuclei the ground-state triaxiality is extracted with our method making use of three experimental matrix elements. The yrast electromagnetic properties are well described by this model for most even-even nuclei within all mass and deformation regions and are related to the average triaxiality of the ground-state. Work supported in part by U.S. Dept. of Energy grant DE-FG02-96ER40958.

[1] National Nuclear Data Center, Brookhaven National Laboratory, online data base, http://www.nndc.bnl.gov

[2] A.S. Davydov and G.F. Filippov, Nucl. Phys. 8, 237 (1958).

[SJ.008] One- and two-phonon \gamma-vibrational motions in deformed nuclei

The intrinsic E2 matrix elements of the one-phonon \gamma-vibrational excitation for many deformed nuclei have been deduced from the measured interband E2 matrix elements between the K=2 and 0 bands after correcting for the first order angular momentum dependence of the coupling between the rotation and intrinsic motions. The strong correlation of the triaxiality derived from those intrinsic E2 matrix elements and from the excitation energies of the K=2, \gamma-vibrational states provides a quantitative measure of triaxial quadrupole deformation of the nuclear shape. The intrinsic E2 matrix elements of the two-phonon \gamma-vibrational excitation have been deduced from the measured interband E2 matrix elements between the K=4 and 2 bands. The integrity of the two-phonon K=4, \gamma-vibrational states has been measured by comparing the derived intrinsic E2 matrix element of two-phonon to one-phonon excitations using the harmonic two-phonon strength as a yardstick. It is found that the two-phonon \gamma-vibrational strength is highly fragmented for most deformed nuclei with a few exceptions. C.Y. Wu and D. Cline, Phys. Rev. C 54, 2356 (1996). C.Y. Wu and D. Cline, Phys. Lett. B 382, 214 (1996). C.Y. Wu et al., Nucl. Phys. A 607, 178 (1996). C.Y. Wu et al., Phys. Rev. C 64, 014307 (2001). This work was supported by National Science Foundation.

[SJ.009] Use of E2 matrix elements to determine the centroids and fluctuation widths for triaxial quadrupole collective motion

Measured E2 properties are a sensitive and unambiguous probe of the collective shape parameters for quadrupole collective motion in nuclei. Collective motion produces strong correlations of the measured E2 matrix elements that can be related to the E2 properties in the principal axis frame of the rotating nucleus. By analogy with Bohr's quadrupole shape parameters (\beta,\gamma), the instantaneous principal axis frame E2 tensor can be expressed in terms of two parameters, Q,\delta where E2(2,0)=Q\cos\delta, and E2(2,\pm 2)=\fracQ\sqrt2\sin\delta. The E2 properties can be used to extract the E2 triaxiality parameter \delta which can be related to \gamma by use of a geometrical collective model. The 1965 measurement [1] of the Q_2^+ state in ^114Cd provoked considerable interest in collective triaxial deformation in nuclei and stimulated measurement of Q_2^+ values in many other nuclei in order to probe the centroid of the E2 triaxial deformation. The heavy-ion Coulomb excitation experimental technique, plus the Coulomb excitation least-squares search code GOSIA, made it possible to measure rather complete sets of E2 matrix elements adding a new dimension to the study of triaxiality in nuclear collective motion [2]. This development also made it possible to exploit the rotational invariant technique [3-6] to extract directly from the measured E2 matrix elements, the expectation values of the centroids and fluctuation widths of principal axis E2 parameters for any state. The usefulness, range of validity, and results of this technique for determining the centroids and fluctuation widths for the triaxiality degree of freedom \delta in a range of nuclei will be presented. \ The completeness required is a disadvantage of the rotational invariant technique. A comparison will be made of the use of the full rotational invariant technique with results obtained using restricted E2 data in conjunction with model-dependent analyses or truncation schemes. [1] J. de Boer et al, Phys. Rev. Lett. 14 (1965) 564; [2] D. Cline Ann. Rev. Nucl. Part. Sci 36 (1986) 683; [3] D. Cline, Proc. Orsay Colloquium on Intermediate Nuclei Ed Foucher et al, Inst. Phys. Nucl. (1971) P 4; [4] D. Cline, Journ of the Phys. Soc. of Japan, 34 (1972) 377; [5] D. Cline and C. Flaum, Proc. Int. Conf. on Nucl, Struct. Studies using Electron Scattering and Photoreaction, Sendai. Ed K. Shoda, H. Ui, Tohoku Univ. Vol 5 (1972) p61; [6] K. Kumar Phys. Rev. Lett. 28 (1972) 249; Work supported by the N.S.F.

[SJ.010] Evidence for the Wobbling Mode in Nuclei

The nucleus ^163Lu has been populated through the reaction ^139La(^29Si,5n)^163Lu with a beam energy of 152 MeV and gamma--ray spectroscopy was performed using the EUROBALL IV setup in

Strasbourg. The electromagnetic properties of several connecting transitions between

two presumably triaxial, strongly deformed (TSD) bands have been studied

and spin and parity of the excited band was firmly established. Evidence is presented for the assignment of the excited TSD band as a wobbling mode built on the yrast TSD band, based on comparisons with new

calculations in which an aligned particle is coupled to a strongly deformed triaxial rotor. The wobbling mode is uniquely related to triaxiality in nuclei.

[SJ.011] Algebraic approach to the wobbling motion of the superdeformed odd mass nucleus

The wobbling motion found recently in the superdeformed states for an odd mass nucleus ^163Lu is quite interesting from nuclear structure point of view (S. W. Ødegård et al, Nucl. Phys. A682)(2001)427c.. It was almost thirty years ago that we applied the Holstein-Primakoff (HP) boson expansion method to describe the quantum states of the wobbling motion of triaxial rotor (K. Tanabe and K. Sugawara-Tanabe, Phys. Lett. 34B)(1971)575; K. Sugawara-Tanabe and K. Tanabe, Nucl. Phys. A208(1973)317; and K. Tanabe, J. Math. Phys. 14(1973)618.. We have shown that, if the z-axis is chosen as quantization axis instead of the x-axis, and contributions are collected up to the second order in the HP expansion, a quite accurate algebraic expression for the triaxial rotor level is obtained. It continuously goes to a well-known rotational level formula in the symmetric limit (\cal J_x=\cal J_y). Thus, this method is able to avoid divergence of the transformation coefficents in the axially symmetric limit as seen in the Bohr and Mottelson's formula. Our attempt is to extend this formalism to the double wobbling motion of the odd mass nucleus with a valence nucleon in an orbital of definite j, and discuss its levels and transition rates as functions of I.

[SJ.012] Wobbling motion and gamma vibration at high-spin

We report tilted-axis cranked HFB solutions which possibly show wobbling motion coupled to gamma deformation at very high-spin (J\simeq 60\hbar) in ^164Hf. This nucleus is known to have triaxial super-deformed (TRS) state at very high-spin (J\simeq 40\hbar) by the systematic survey through one-dimensional cranking calculations (R.Bengtsson, private communications)..

A discussion is also given through the comparison between classical and nuclear wobbling motions, for their similarities and differences.

Based on these states, further investigation by means of GCM (Generator Coordinate Method) calculations will be presented in order to study the excited structures produced by the wobbling motion.

A possibility of anharmonicity in the wobbling phonon is then discussed from a viewpoint of large amplitude wobbling motion.

Part S of program listing