Nuclear moments of inertia and wobbling motions in triaxial superdeformed nuclei
Abstract
The wobbling motion excited on triaxial superdeformed nuclei is studied in terms of the cranked shell model plus random phase approximation. Firstly, by calculating at a low rotational frequency the -dependence of the three moments of inertia associated with the wobbling motion, the mechanism of the appearance of the wobbling motion in positive- nuclei is clarified theoretically — the rotational alignment of the quasiparticle(s) is the essential condition. This indicates that the wobbling motion is a collective motion that is sensitive to the single-particle alignment. Secondly, we prove that the observed unexpected rotational-frequency dependence of the wobbling frequency is an outcome of the rotational-frequency dependent dynamical moments of inertia.
pacs:
21.10.Re, 21.60.Jz, 23.20.Lv, 27.70.+qI Introduction
Deformation of the nuclear shape from spherical symmetric one has long been one of the most important issues in nuclear structure physics. Among them, searches for evidences of the triaxial ( or ) one have been pursued long time, for example, the even-odd energy staggering in the low-spin part of the bands A. S. Davydov and G. F. Filippov (1958), the signature dependence of the energy spectra and the transition rates in medium-spin odd-odd and odd- nuclei Bengtsson et al. (1984); Hamamoto and Mottelson (1983); Matsuzaki (1990a), properties of the isomers Narimatsu et al. (1996); Tajima and Onishi (1989), and so on. But their results have not been conclusive; making a clear distinction between the static and the dynamic (vibrational) ones has not been successful up to now. Theoretically, appearance of the wobbling motion, which is well-known in classical mechanics of asymmetric tops L. D. Landau and E. M. Lifshitz (1960) and whose quantum analog was discussed in terms of a rotor model about thirty years ago Bohr and Mottelson (1975), is a decisive evidence of static triaxial deformations. Subsequently its microscopic descriptions were developed by several authors Janssen and I. N. Mikhailov (1979); Marshalek (1979). Since the small-amplitude wobbling mode carries the same quantum numbers, parity and signature , as the odd-spin members of the band, Ref.I. N. Mikhailov and Janssen (1978) anticipated that it would appear as a high-spin continuation of the band, but it has not been resolved that in what nuclei, at what spins, and with what wobbling modes would be observed.
Shimizu and Matsuyanagi Y. R. Shimizu and Matsuyanagi (1984) and Onishi Onishi (1986) performed extensive numerical calculations for normally-deformed Er isotopes with relatively small . Matsuzaki Matsuzaki (1990b), Shimizu and Matsuzaki Y. R. Shimizu and Matsuzaki (1995), and Horibata and Onishi Horibata and Onishi (1996) also studied Os with relatively large negative- but their correspondence to experimental information has not been very clear.
These studies indicate the necessity of high-spin states in stably and strongly -deformed nuclei. Bengtsson studied high-spin states around Hf Bengtsson and found systematic existence of the TSD (triaxial super- or strongly deformed) states with and . This confirmed the discussion on the shell gap at in Ref.Schmitz et al. (1993), the work in which the yrast TSD band in Lu was reported; and in 2000 an excited TSD band was observed in this nucleus and from the strengths of the interband transition rates this was unambiguously identified with the wobbling motion S. W. Ødegård et al. (2001). This data was analyzed by using a particle-rotor model Hamamoto (2002) and the transition rates were reproduced well. Subsequently TSD bands were found in some Lu and Hf isotopes and wobbling excitations were observed also in Lu Schönwaßer et al. (2003); Amro et al. (2003). A close look at these data, however, tells us that the sign of their -deformation seems to contradict to an irrotational motion and that the unexpected behavior of the wobbling frequency has not been explained yet.
Thus in the preceding Rapid Communication Matsuzaki et al. (2002) we presented an answer to these problems. In the present paper, after summarizing the discussion there we extend numerical analyses to elucidate it. An emphasis is put on the behavior of the calculated dynamic moments of inertia.
Ii Wobbling motion in terms of the random phase approximation
We start from a one-body Hamiltonian in the rotating frame,
(1) | |||
(2) | |||
(3) |
In Eq.(2), and 2 stand for neutron and proton, respectively, and chemical potentials are determined so as to give correct average particle numbers . The oscillator frequencies in Eq.(3) are expressed by the quadrupole deformation parameters and in the usual way. They are treated as parameters as well as pairing gaps . The orbital angular momentum in Eq.(3) is defined in the singly-stretched coordinates , with 1 – 3 denoting – , and the corresponding momenta. By diagonalizing at each , we obtain quasiparticle (QP) orbitals and the nuclear yrast (0QP) state. Since conserves parity and signature , nuclear states can be labeled by them. Nuclear states with QP excitations are obtained by exchanging the QP energy and wave functions such as
(4) |
where denotes the signature partner of .
We perform the random phase approximation (RPA) to the residual pairing plus doubly-stretched quadrupole-quadrupole () interaction between QPs. Since we are interested in the wobbling motion that has a definite quantum number, , only two components out of five of the interaction are relevant. They are given by
(5) |
where the doubly-stretched quadrupole operators are defined by
(6) |
and those with good signature are
(7) |
The residual pairing interaction does not contribute because is an operator with . The equation of motion,
(8) |
for the eigenmode
(9) |
leads to a pair of coupled equations for the transition amplitudes
(10) |
Then, by assuming , this can be cast Marshalek (1979) into the form
(11) |
which is independent of s. This expression proves that the spurious (Nambu–Goldstone) mode given by the first factor and all nomal modes given by the second are decoupled from each other. Here as usual and the detailed expressions of are given in Refs.Marshalek (1979); Matsuzaki (1990b); Y. R. Shimizu and Matsuzaki (1995). Among normal modes, one obtains
(12) |
by putting . Note that this gives a real excitation only when the right-hand side is positive and it is non-trivial whether a collective solution appears or not. Evidently this coincides with the form derived by Bohr and Mottelson in a rotor model Bohr and Mottelson (1975) and known in classical mechanics L. D. Landau and E. M. Lifshitz (1960), aside from the crucial feature that the moments of inertia are -dependent in the present case.
One drawback in our formulation is that our tends to be larger than corresponding experimental values because of the spurious velocity dependence of the Nilsson potential as discussed in Refs.S. -I. Kinouchi ; Nakatsukasa et al. (1996). A remedy for this was discussed there but that for has not been devised yet. Therefore we assume for the present a similar discussion holds for the latter and accordingly the ratio which actually determines is more reliable than their absolute magnitudes.
Interband electric quadrupole transitions between the -th excited band and the yrast are given as
(13) |
in terms of
(14) |
They will be abbreviated to later for simplicity. In-band ones are given as
(15) |
in terms of
(16) |
and assumed to be common to all bands. They will be abbreviated to . Here we adopted a high-spin approximation Marshalek (1977). The transition quadrupole moment is extracted from by the usual rotor-model prescription.
To compare collectivities of these two types of transitions, we introduce a pair of deformation parameters
(17) |
Then it is evident that the in-band one is expressed as
(18) |
As for the interband ones, by expanding by s and s, where runs both normal modes and the Nambu–Goldstone mode , we obtain from a kind of sum rule
(19) |
Consecutively introducing the ratios of the dynamic to static deformations,
(20) |
the sum rule above reads
(21) |
The dynamic amplitudes describe shape fluctuations associated with the vibrational motion in the uniformly rotating frame. Transformation to the body-fixed (Principal-Axis) frame Marshalek (1979) turns the shape fluctuation into the fluctuation of the angular momentum vector, i.e., the wobbling motion. This transformation relates the ratios, and , to the moments of inertia Y. R. Shimizu and Matsuzaki (1995):
(22) |
where is a real amplitude that relates the dynamic amplitude and the moment of inertia, is the sign of (so for wobbling-like RPA solutions), and
(23) |
Thus, the interband is rewritten as,
(24) |
which coincides with the formula given by the rotor model Bohr and Mottelson (1975), except for the appearance of the amplitude and sign . Substituting the ratios, and , into Eq.(21), one finds that the amplitudes should satisfy
(25) |
This form of sum rule clearly indicates that the amplitude is a microscopic correction factor quantifying the collectivity of the wobbling motion, for which means the full-collectivity and reproduces the results of the macroscopic rotor model in both the energy and the interband values.
Iii Numerical calculation and Discussion
iii.1 Summary of the preceding study
Since the first experimental confirmation of the wobbling excitation in Lu S. W. Ødegård et al. (2001), has been widely accepted as the shape of the TSD states in this region. This is predominantly because the calculated energy minimum for is deeper than that for Bengtsson according to the shape driving effect of the aligned quasiparticle. The recent precise measurements of Schönwaßer et al. (2002) also support this. On the other hand, the sign of -deformation leads different consequences on moments of inertia, which are directly connected to the excitation energy of the wobbling mode through the wobbling frequency formula Bohr and Mottelson (1975), c.f. Eq.(12). Since the RPA is a microscopic formalism, no distinction between the collective rotation and the single-particle degrees of freedom has been made.
Therefore, the moments of inertia calculated in our RPA formalism in sect.II are those for rotational motions of the whole system. In contrast, the macroscopic irrotational-like moments of inertia are often used in the particle-rotor calculations, where for and they lead to an imaginary wobbling frequency . It is, however, noted that the moments of inertia of the particle-rotor model are those of the rotor and no effect of the single-particle alignments is included, so that they do not necessarily correspond to those calculated in our RPA formalism.
In the preceding paper Matsuzaki et al. (2002) we have performed microscopic RPA calculations without dividing the system artificially into the rotor and particles. That work proved that for the calculated moment of inertia, , the contribution from the aligned QP(s), with being the aligned angular momentum, is superimposed on an irrotational-like moment of inertia () of the “core”. Consequently the total is larger than , which makes wobbling excitation in nuclei possible.
The second consequence of the formulation adopted in Ref.Matsuzaki et al. (2002) is that the three moments of inertia are automatically -dependent even when the mean-field parameters are fixed constant. This is essential in order to explain the observed -dependence of — decreasing as increases. Otherwise is proportional to .
Another important feature of the data is that the interband values between the wobbling and the yrast TSD bands are surprisingly large. Our RPA wave function gave extremely collective that gathered 0.6 – 0.8 in the sum rule (Eq.(25)) but the result accounted for only about one half of the measured one.
To elucidate these findings more, in the following we extend our numerical analyses putting a special emphasis on the -dependence of the moments of inertia in subsect.III.2. Dependence on other parameters is also studied in detail. Features in common and different between even-even and odd- nuclei are also pointed out. In subsect.III.3, we discuss -dependence. In subsect.III.4, characteristics of are discussed. Calculations are performed in five major shells; 3 – 7 for neutrons and 2 – 6 for protons. The strengths and in Eq.(3) are taken from Ref.Bengtsson and Ragnarsson (1985).
iii.2 Dependence on the mean-field parameters , , and
iii.2.1 The even-even nucleus Hf
Hafnium-168 is the first even-even nucleus in which TSD bands were observed Amro et al. (2001). In this nucleus three TSD bands were observed but interband -rays connecting them have not been observed yet. This means that the character of the excited bands has not been established, although we expect at least one of them is wobbling excitation. An important feature of the data is that the average transition quadrupole moment was determined as eb. This imposes a moderate constraint on the shape. Referring to the weak parameter dependence discussed later, we choose , , and 0.3 MeV, which reproduce the observed , as a typical mean-field parameter set.
First we study the dependence of various quantities on and other mean-field parameters at 0.25 MeV. Around this frequency the alignment that is essential for making wobbling excitation in nuclei possible is completed and therefore the wobbling motion is expected to emerge above this frequency (see Fig.7 shown later).
Figure 1 shows dependence on calculated with keeping and 0.3 MeV. Figure 1(a) graphs the calculated excitation energy in the rotating frame, . As comes close to 0 (symmetric about the axis) and (symmetric about the axis), approaches 0, see Eq.(12). We did not obtain any low-lying RPA solutions at around whereas a collective solution appears again for .
Figure 1(b) shows the calculated moments of inertia. Their -dependence resembles the irrotational, the so-called -reversed, and the rigid-body moments of inertia, in , , and , respectively. These model moments of inertia are given by
(26) |
(27) |
and
(28) |
where 1 – 3 denote the – principal axes, the irrotational mass parameter, the rigid moment of inertia in the spherical limit, and is a deformation parameter like . The -reversed moment of inertia was introduced to describe positive- rotations in the particle-rotor model Hamamoto and Mottelson (1983) but its physical meaning has not been very clear; in particular, it does not fulfill the quantum-mechanical requirement that the rotations about the symmetry axis should be forbidden. We have clarified in the preceding paper Matsuzaki et al. (2002) that the contributions from aligned quasiparticles superimposed on irrotational-like moments of inertia () can realize and this is the very reason why the wobbling excitation (see Eq.(12)) appears in positive- nuclei. We also discussed that multiple alignments could eventually lead to a rigid-body-like moment of inertia. Figure 1(c) indicates that, in the present calculation in which configuration is specified as the adiabatic quasiparticle vacuum at each , two protons align for as mentioned above while they have not fully aligned for at this . In other words, these figures cover the both regions in which the alignment is necessary () and that is not necessary () for obtaining wobbling excitations. This aligned angular momentum determines the overall -dependence of in Fig.1(b). As for the neutron part, corresponding to the disappearance of the solution at around , the expectation value of the neutron angular momentum, , drops around this region.
To look at this more closely, we investigate the Nilsson single-particle diagram at . Figure 2(a) graphs neutron single-particle energies for with , while Fig.2(b) for with . The chemical potential that gives correct neutron number for at = 0.25 MeV is also drawn in the latter. This figure clearly shows that with this a shell gap exists for at . And by comparing this with Fig.1 we see that the dropping of is a consequence of the deoccupation of the orbital that is [651 1/2] at (hereafter simply referred to as the [651 1/2] orbital even at ) originating from the mixed (-) spherical shell. Figure 2(b) also explains the reason why the wobbling excitation revives at around again; the occupation of other oblate-favoring orbitals such as [503 7/2] makes it possible and leads to a rigid-body-like behavior of the moments of inertia. Figures 2(c) and (d) are corresponding ones for protons. This indicates that the proton shell gap is robuster.
Figure 1(d) graphs the quadrupole transition amplitudes () associated with the wobbling mode. ( corresponds to in Ref.Y. R. Shimizu and Matsuzaki (1995).) This shows that their relative sign changes with that of as discussed in Refs.Matsuzaki (1990b); Y. R. Shimizu and Matsuzaki (1995). This feature can be understood as: is the -vibrational region because the component is dominant (see also and in Fig.1(b)), and the mixing of the component due to triaxiality and rotation gives rise to the character of the wobbling motion. This relative sign leads to a selection rule of the interband transition probabilities Y. R. Shimizu and Matsuzaki (1995). In the present case we obtain for , and typically their ratio to the in-band ones is .
Figure 3 shows dependence on calculated with keeping and 0.3 MeV. The steep rises at around in Figs.3(a) and (b) indicate the necessity of the (the [660 1/2] orbital in Fig.2(c)) alignment for the appearance of the wobbling mode although the critical value of itself is frequency dependent. Aside from this, is almost constant in the calculated range. The slight increase at around stems from the occupation of the [651 1/2] orbital. We have confirmed that in this case the alignment at around seen in Fig.3(b) does not affect visibly since in this case is almost the same as although its reason is not clear. Figure 3(c) graphs . This figure indicates that the chosen shape and reproduces the measured .
Figure 4(a) shows dependence on the pairing gaps. Since we do not have detailed information about the gaps, we assume for simplicity. This figure shows that the dependence on the gaps is weak unless they are too large. Since the static pairing gap is expected to be small, say, 0.6 MeV, in the observed frequency range, is not sensitive to the value of . This is a striking contrast to the and vibrations; it is well known that pairing gaps are indispensable for them. Here we note that the behavior of the correlates well with presented in Fig.4(b).
iii.2.2 The odd- nucleus Lu
Next we study Lu in a way similar to the preceding Hf case. We choose and 0.3 MeV as representative mean-field parameters as above. As for , however, we examined various possibilities because has not been measured in this nucleus. Since the sensitive -dependence through the occupation of the [651 1/2] orbital appears only at 0.4 MeV and therefore the “band-head” properties do not depend on qualitatively, first we discuss them adopting in order to look at the difference between the even-even and the odd- cases.
Figure 5 shows dependence on at 0.25 MeV with keeping and 0.3 MeV constant. Figure 5(a) graphs . In the region, the solution is quite similar to the Hf case. In the region, that for is quite similar again but for its character is completely different. In this region the presented solution is the lowest in energy and becoming collective gradually as decreases. The largeness of corresponds to that of in Fig.5(b). Comparison of Figs.5(c) and 1(c) certifies that the alignment of the quasiparticle(s) is almost complete for whereas less for . This produces quantitative even-odd differences as explained below.
Having confirmed that these features are independent of and except that we did not obtain any low-lying solutions for in the small- cases, we look into underlying unperturbed 2QP energies to see the even-odd difference. In Fig.6 we present the energies of the lowest states which represent the biggest difference. In the yrast configuration, and in the usual notation are occupied in the even- case, the lowest 2QP state of signature with respect to this is (where denotes the conjugate state, see Eq.(4)). In the odd- case in which is occupied, the lowest one is . Since both and decrease as decreases, this 2QP state becomes the dominant component in the lowest-energy RPA solution. Note here that the sum corresponds to the signature splitting between and when they are seen from the usual even-even vacuum. Since both and are of character, the resulting RPA solution can not have the collectivity as shown in Fig.5(d). According to the relation Y. R. Shimizu and Matsuzaki (1995),
(29) |
in Fig.5(b) becomes small for . These discussion serves to exclude the possibility of for the TSDs that support collective wobbling excitations in the odd- cases, whereas the even-odd difference in is merely quantitative.
iii.3 Dependence on the rotational frequency
iii.3.1 Hf and Hf
The analyses above indicate that the chosen mean-field parameters are reasonable, and therefore we proceed to study -dependence with keeping these parameters constant. Figure 7 shows the result for Hf. These figures indicate again the alignment that makes larger than is indispensable for the formation of the wobbling excitation. At around 0.45 MeV the alignment occurs. In contrast to the low-frequency case reported in Fig.3, in the present case its effect on is visible as a small bump. Although the character of the observed excited TSD bands has not been resolved, some anomaly is seen at around this in one of them Amro et al. (2001). We suggest this is related to the alignment since this is the only alignable orbital in this frequency region of this shape. However we note that in Lu an interaction with a normal deformed state at around this frequency is discussed in Ref.Amro et al. (2003).
We performed calculations also for . In that case, however, wobbling excitation exists only at small because is small as seen from Fig.1(b).
Very recently TSD bands were observed in another even-even nucleus, Hf M. K. Djongolov et al. (2003). It is not trivial if a similar band structure is observed in the nucleus with six neutrons more since the existence of the TSD states depends on the shell gap. Multiple TSD bands were observed but connecting -rays have not been resolved also in this nucleus. We performed a calculation adopting and suggested in Ref.M. K. Djongolov et al. (2003) and 0.3 MeV. The result is presented in Fig.8. The most striking difference from the case of Hf above is that decreases steadily as increases after the alignment is completed. This is because the alignment that causes the small bump in the Hf case shifts to very low due to the larger neutron number.
iii.3.2 Lu
The wobbling excitation was first observed experimentally in Lu S. W. Ødegård et al. (2001), later it was also observed in Lu Schönwaßer et al. (2003) and Lu Amro et al. (2003). The characteristic features common to these isotopes are 1) decreases as increases contrary to the consequence of calculations adopting constant moments of inertia, and 2) is large — typically around 0.2.
Here we concentrate on the isotone of Hf discussed above, that is, Lu in order to see the even-odd difference. A comparison of Figs.7 and 9 proves that all the differences are due to the fact that the number of the aligned quasiparticle is less by one: 1) The alignment at around = 0.2 MeV is absent, and 2) the crossing occurs at around = 0.55 MeV, which is proper to the configuration. Figure 9(a) shows that our calculation does not reproduce the data, although in each frequency range in which the configuration is the same decreases at high as in the cases of the even-even nuclei presented above. This result might indicate that there is room for improving the mean field. The in Fig.9(c) is larger than the experimentally deduced value by about 20 – 30 %. This is due to the spurious velocity dependence of the Nilsson potential mentioned in Sec.II.
iii.4 Interband transitions
Compared to the excitation energy, the interband values relative to the in-band ones have been measured in only few cases. In Fig.10, we report calculated ratios for (wobbling on yrast TSD) (yrast TSD) transitions in Hf and Lu. The measured ones are also included for the latter.
The first point is the magnitude of the larger () ones. Apparently, the calculated values are smaller by factor 2 – 3. The measured interband values amount almost to the macroscopic rotor value. In the RPA calculations, as summarized in sect.II, the value is reduced by a factor (see Eq. (24)): Only the case with the full-strength the rotor value is recovered. Although the obtained RPA wobbling solutions are extremely collective in comparison with the usual low-lying collective vibrations, like the - or -vibrations, for which typically 0.3 – 0.4, this factor is still 0.6 – 0.8. This is the main reason why the calculated values are a factor 2 – 3 off the measured ones. As is well-known, giant resonances also carry considerable amount of quadrupole strengths, so it seems difficult for the microscopic correction factor to be unity; it is not impossible, however, because the “sum rule” discussed in sect.II is not the sum of positive-definite terms. In the RPA formalism, the reduction factor for the value, Eq. (24), comes from the fact that the wobbling motion is composed of the coherent motion of two-quasiparticles, and reflects the microscopic structure of collective RPA solutions. The measurement that the value suffers almost no reduction may be a challenge to the microscopic RPA theory in the case of the wobbling motion. Calculated ratios for Hf are slightly smaller than those for Hf in Fig.10(a).
The second point is the staggering, that is, the difference between . We clarified Y. R. Shimizu and Matsuzaki (1995) its unique correspondence to the sign of as mentioned in subsect.III.2; that holds for both even-even and odd- systems. Recently this staggering was discussed from a different point of view R. F. Casten et al. (2003); but it looks to apply only to cases.
Iv Conclusion
The nuclear wobbling motion, which is a firm evidence of stable triaxial deformations, was identified experimentally in the triaxial superdeformed odd- Lu isotopes. In principle, wobbling excitation is possible both in and nuclei. Every information, theoretical and experimental, suggests for these bands. According to the wobbling frequency formula Bohr and Mottelson (1975), c.f. Eq.(12), its excitation in nuclei rotating principally about the axis requires , although irrotational-like model moments of inertia give for . To solve this puzzle, we studied the nuclear wobbling motion, in particular, the three moments of inertia associated with it in terms of the cranked shell model plus random phase approximation. This makes it possible to calculate the moments of inertia of the whole system including the effect of aligned quasiparticle(s). The results indicate that the -dependence of the calculated moment of inertia is basically irrotational-like ( for ) if aligned quasiparticle(s) ( in the present case) does not exist. But once it is excited, it produces an additional contribution, , and consequently can lead to . This is the very reason why wobbling excitation exists in nuclei. In this sense, the wobbling motion is a collective motion that is sensitive to the single-particle alignments.
The resulting moment of inertia for resembles the -reversed one, i.e., the irrotational moment of inertia but with and being interchanged. That for , where single-particle angular momenta dominate, is rigid-body-like. That for is irrotational-like except for odd- nuclei with where a specific 2QP state determines the lowest RPA solution.
Having studied qualitative features of the three moments of inertia at a low rotational frequency, we calculated wobbling bands up to high . Experimentally they were confirmed only in odd- Lu isotopes as mentioned above. The most characteristic feature of the data is that decreases as increases. This obviously excludes constant moments of inertia. In our calculation three moments of inertia are automatically -dependent even when mean field parameters are fixed constant. It should be stressed that the wobbling-like solution in our RPA calculations is insensitive to the mean-field parameters, especially to the pairing gaps, as is shown in subsect.III.2.1. This distinguishes the wobbling-like solution from the usual collective vibrations, which are sensitive to the pairing correlations. Thus, our microscopic RPA calculation confirms that the observed band is associated with a new type of collective excitation, although comparisons to the observed excitation energy indicate that there is room for improving the calculation.
As for the interband transition rates, our calculation accounted for only about one half or less of the measured ones, even though the wobbling-like solution is extremely collective compared to the usual vibrational modes. This issue is independent of the details of choosing parameters. This confronts microscopic theories with a big challenge.
Acknowledgements.
We thank G. B. Hagemann for providing us with some experimental information prior to publication. This work was supported in part by the Grant-in-Aid for scientific research from the Japan Ministry of Education, Science and Culture (Nos. 13640281 and 14540269).References
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