Evidence for and Against Chiral Doublet Bands in Nuclei

9

Evidence for and against Chiral doublet bands in Nuclei

J.H. HAMILTON1, S.J. ZHU1,2,3, Y.X. LUO1,4, A.V. RAMAYYA1, J.O. RASMUSSEN4, S. FRAUENDORF5, J.K. HWANG1, J.Y. ZHANG6,

P.M. GORE1 AND E.F. JONES1

1Physics Department, Vanderbilt University, Nashville, TN 37235, USA

2Joint Institute for Heavy Ion Research, Oak Ridge, TN 37831, USA

3Physics Dept., Tsinghua University, Beijing 100084, People’s Republic of China

4Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

5Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA

6Department of Physics, University of Tennessee, Knoxville, TN 37996, USA

Prompt g-g-g coincidence studies of neutron-rich nuclei with A = 99-113, populated in the spontaneous fission of 252Cf, were carried out at Gammasphere. New band structures and significant extension of previously known bands were observed in 99,101Y, 101,105Nb, 105,107,109Tc, 111,113Rh. One quasi-particle plus triaxial rotor calculations were carried out. These nuclei show a smooth evolution from axial symmetry in Y to maximum triaxiality in Rh. In 106Mo and 110,112Ru, pairs of DI = 1 doublet bands are established to high spins. Since these nuclei exhibit evidence for triaxial shapes, their close energy differences for states of the same spins, their constant and equal moments of inertia with spin and their very similar E2/M1 branching ratios for each band support their interpretation as chiral bands. If these are not chiral bands, then a new symmetry is indicated.

1. Introduction

The formation of right- and left-handed, chiral, structures has long been of interest in molecules, especially bio-molecules and in elementary particles. Chirality in molecules is related to the geometry of how atoms are attached in a molecule as illustrated in Fig. 1. By selecting a bond to CH3 CH4, then H, I, and CH3 form right-handed and left-handed screws related to each other by mirror reflection. The DNA double-helix and other complex bio-molecules are chiral. However, of two DNA enantiomers with the same binding energy, only one is synthesized.

Nuclei with only two types of particles and a few rather simple shapes were long thought to be achiral. Then Frauendorf and his colleagues [1,2,3] pointed out that in well deformed triaxial nuclei, when there are substantial components of the angular momentum along all three axis, chiral doublet bands can occur as illustrated in Fig. 2.

In Figure 2, the axis of rotation J is out of the three planes spanned by the long, intermediate and short axes. The operation TRy(p) 1, where Ry(p) is the rotation operator and T is the time reversal operator. “ It changes the chirality of the long, intermediate and short axes with respect to the axis of rotation J. Since the left- and right-handed solutions have the same energy, they give rise to two

degenerate DI = 1 bands. They are linear combinations of the left- and right- handed configurations, which restore the spontaneously broken TRy(p) symmetry.” [1] When the total angular momentum is not aligned with any of the three axes of a well-deformed, triaxial nucleus, chiral symmetry breaking can occur. An example is shown in Fig. 3, where one has a high-j proton particle with its angular momentum aligned along the short axis, a high-j neutron hole with its angular momentum aligned along the long axis, and the rotational angular momentum aligned along the intermediate axis.

Chiral symmetry breaking in nuclei gives rise to two sets of DI=1 bands of the same parity. The two bands with the same spin in each set should be nearly degenerate in energy. As shown in Fig. 2, this chiral breaking in triaxial nuclei has a dynamic nature, depending on the rotation of the nucleus, rather than being geometric in nature, as in complex molecules. The characteristics which are necessary to produce chirality in nuclei are significant triaxial deformation, and significant angular momentum along the three axes deformation axis [1-3].

These predictions of DI=1 chiral doublet bands in nuclei led to a flurry of activity both experimentally and theoretically. The first report of chiral bands was in 134Pr [4]. A number of cases were reported in this region from A = 128 to 140. and then around A=105. It was reported that “ the best chiral properties observed to date ” are in 104Rh [5]. These results were followed by the first report of chiral vibrational doublet bands in an even-even nuclei 106Mo [6].

Very recently Petrache et al. [7] have discussed the possible misinterpretation of nearly degenerate pairs of bands as chiral partners in nuclei. In particular, they discuss, in detail, the first reported case of chiral doublet degenerate bands in 134Pr. This nucleus has now been observed to higher spin and the B(E2) values for the two bands extracted from lifetime measurements [8]. Now the degeneracy of the 15+ and 16+ states, which are only 36 and 44 keV apart are related to multiple band crossings [7]. So both the doublet bands are complex as shown in Fig. 4 [7]. More important, they emphasize that the ratio of 2.0(4) for the E2 transition moments for the two bands indicates they have different intrinsic structure since the ratio should be one if the two sets of bands are chiral. They also note the present data for the two sets of bands in 136Pm are not in favor of their reported chiral character [7]. They emphasize that chiral doublet bands should have identical or very similar energies, spin alignments, shapes and electromagnetic transition probabilities.

At nearly the same time, Meng et al. [9] have applied an adiabatic and configuration fixed constrained triaxial relativistic mean field (RMF) approach. They predicted the existence of at least two sets of chiral doublet bands in 106Rh built on two different close-lying shapes which do not mix. They suggest a search for a second set of chiral doublet bands in addition to the pair of chiral doublets reported in 106Rh [12]. Their calculations indicate that 102,108,110Rh and 108-112Ag are candidates in which to observe two sets of chiral doublets.

In this paper we will look at our report of chiral doublets in 106Mo [6] in the light of further branching ratio analysis and theoretical calculations of branching ratios for these bands to see if they are only accidentally degenerate from the coupling to similar energy bands in 105Mo or chiral.

Evidence for triaxial shapes around A = 110-112 neutron-rich odd A nuclei are found. Then we will present new results for 110,112Ru where DI = 1 doublet bands provide further insight into the existence of chiral bands in even - even nuclei in this region. These new doublet bands either demonstrate a new type of chiral vibrational bands to help establish the general nature of chiral symmetry breaking in nuclei or they point to some new symmetry not seen before.

2. Evidence for Triaxiality in A=99-113 Neutron-Rich Nuclei

From g-g-g data in the spontaneous fission of 252Cf (see Ref. [10] for experimental details) the p5/2+[422] bands in 99,101Y and 101,105Nb [10] and the 7/2+[413] bands in 105,107,109Tc [11] and 111,113Ru [12] were identified and observed to higher spins, up to 23/2+ to 33/2+. In Fig. 5 are shown the signature splittings, S(I) for ground bands in 99,101Y, 101Nb and 105,-109Tc where:

S(I) = [E(I) - E(I-1)][I(I+1) - (I-2)(I-1)]/[E(I) - E(I-2)][I(I+1) - (I-1)I] - 1

One quasi-particle plus triaxial rotor calculations [10-12] reproduce exceedingly well the observed strong splittings for the values of g shown in Fig. 6, where g = 0 corresponds to prolate symmetric and g = 30o to maximum triaxiality. Note the smooth decrease in very strong quadrupole deformation in 99Y to still well-deformed 113Rh and smooth evolution from g ~ 0 in Y to g ~ 30o in 111,113Rh nuclei. One and two phonon g vibrational bands are seen in 104,106Mo and 110,112Ru with very strong staggering in the one phonon g band in 112Ru . All these and other data on the odd- A Mo and Ru isotopes (for example [13]) clearly indicate definite triaxial shapes in these neutron-rich Mo and Ru isotopes.

3. Possible Chiral Bands in 106Mo and 110,112Ru

As noted above 106Mo lies in a region where nuclei tend to take a triaxial shape. Our new DI = 1 doublet bands have all the properties expected for chiral doublets are shown in Fig. 7.

The two-quasi proton states lie at higher energy than the two-quasi neutron states so bands 4 and 5 are interpreted as two-quasi neutron excitations. The lowest configuration corresponds to the excitation of a neutron from the highest h11/2 band to two close-lying mixed d5/2 - g7/2 positive parity bands. Our measured lifetimes of less than 8 ns for both band heads exclude a symmetric shape where K is a good quantum number since significant retardation would occur for DK > 3. The levels in 110,112Ru were likewise significantly extended in our work. These nuclei lie in the region of maximum triaxiality. The new DI = 1 doublet bands 4 -5 and 6-7 in 110Ru are shown in Fig. 8.

To aid in an interpretation, we carried out 3D-Tilted Axis Cranking (TAC) calculations. The TAC calculations yielded b2 = 0.31 and g = 31o for 106Mo.

While the TAC calculations give a d5/2 - g7/2 neutron hole strongly aligned with the long axis and a h11/2 neutron lying in the short intermediate plane, the

microscopic TAC calculations cannot be reduced to the simple picture discussed above for odd-odd nuclei. Chirality comes about from the interplay of the neutrons in the open shell.

As a test for chiral bands Varman, et al. [5] pointed out that S(I) = 1/(2J1) = (E(I) - E(I-1))/2I, where J1 is the kinetic moment of inertia, should be constant with I for chiral bands. Plots of J1 for 104Rh, 106Mo and 110Ru are shown in Fig. 9. Note J1 for bands 4 and 5 in 106Mo are more constant and more equal than found for this reported “ best ”case of chiral bands in 104Rh [5]. They are likewise equally constant for 110Ru and nearly so for 112Ru. The TAC calculations also predict a constant J1. There is no evidence for any band crossing in these bands as found in 134Pr. The ground bands in 110,112Ru are crossed above 10+. The J's for 106Mo and 110,112Ru are significantly more constant than for 134Pr [8] shown in Fig. 4.

In Fig. 10 is shown the energy differences between the levels with the

same spin in 106Mo, 110Ru, 104Ru and 134Pr. Note 106Mo and 110Ru have similar small energy differences as expected for chiral bands. The sharp decrease in 104Rh and 112Ru may indicate that these nuclei are becoming more chiral rotors at high spin. This crossing behavior is not understood at present.

Table I. Branching ratios of the in band E2 cross over and the related M1(+E2)
transitions in doublet bands 4-5 and 6–7 in 110,112Ru and bands 4 - 5
in 106Mo.
110Ru 112Ru 106Mo
spin 4-5 6-7 4-5 6-7 4 5
12 >4.4(6) 6.4(9) >4.0(6) 5.1(8) >6.0 >4.8
11 9.2(11) 11.4(15) 7.0(11) 5.6(9) 4.7 >3.9
10 5.9(6) 4.9(5) 5.7(8) 7.0(11) 6.8 8.5
9 6.9(6) 7.9(7) 10.4(17) 8.3(13) 8.3 5.8
8 3.3(3) 3.4(3) 4.2(4) 6.4 2.7
7 2.6(3) 3.2 1.3

As a further test for these bands being chiral doublet or being accidentally degenerate from the coupling of say an h11/2 neutron to two different

neutron bands in 105Mo, and 109,111Ru, the ratios of the E2 to M1 + (E2) strengthens within the two sets of doublet bands in 106Mo, 110Ru and 112Ru were carefully extracted, as given in Table 1. These ratios for each spin state are in very good agreement to indicate they have very similar structures as required for chiral doublets.

Then calculations were carried out for various quasi-particle configurations with negative parity in 106Mo, and 110,112Ru for both axial shapes and g=30o. In all cases, the B(E2)/B(M1) ratios of the two lowest bands differ typically by one order of magnitude. The TAC calculations are consistent with the findings in the framework of the triaxial rotor plus two quasi-particle model. The clear

disagreement of the B(E2)/B(M1) ratios based on various quasi-particle configurations with the experimental data are strong evidence these doublet bands do not arise from the couplings of different quasi-particle configurations which are just accidentally degenerate.

4. Summary

In summary, bands -(4) and -(5) in 106Mo are proposed as the first chiral vibrational bands in an even-even nucleus. The TAC calculations strongly support this interpretation. The calculations indicate a mechanism of generating chirality that is quite different from the previous examples for chirality.

The doublet bands in 110Ru are very similar in every way to those in 106Mo and are most easily interpreted as chiral vibrational bands although the TAC calculations there are less clear. The doublet bands in 112Ru have a different nature in which the two bands may start out as chiral vibrations, become more chiral rotor at intermediate spin and go back to chiral vibrations. In conclusion, the best interpretation of these DI=1 doublet bands in 106Mo and 110,112Ru is that they are chiral bands. If this is not correct, then these nuclei are revealing some new symmetry not seen before.

Acknowledgement

The work at Vanderbilt, and Notre Dame was supported by the U.S. DOE Grants DE-FG-05-88ER40407 and DF-FG02-95ER40934 and at Lawrence Berkeley by DOE Contract DE-AC03-76SF00098. The work at Tsinghua was supported by the Major State Basic Research Development Program Contract

G2000077405, the National Natural Science Foundation of China Grant 10375032, and the Special Program of Higher Education Science Foundation Grant 20030003090. The Joint Institute for Heavy Ion Research is supported by its members, Vanderbilt, University of Tennessee and Oak Ridge National Lab. and the U.S. DOE.

References

1.  V. Dimitrov and S. Frauendorf, Proc. of the Third Int. Conf. on Fission and

Properties of Neutron-Rich Nuclei, eds. J. H. Hamilton, et al., (World

Scientific, Singapore, 2003) 93.

2. V. I. Dimitrov et al., Phys. Rev. Lett. 84, (2000) 5732.