Transverse wobbling. F. Dönau 1 and S. Frauendorf 2 1 XXX 2 Department of Physics, University of Notre Dame, South Bend, Indiana 46556

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1 Transverse wobbling F. Dönau and S. Frauendorf XXX Department of Physics, University of Notre Dame, South Bend, Indiana PACS numbers:..re, 3..Lv, 7.7.+q II. I. INTRODUCTION TRANSVERSE AND LONGITUDINAL WOBBLING Although the first description of the collective wobbling mode was based on even-even nuclei, all examples to date have been observed in odd-a nuclei. In fact the unpaired, highly-aligned i 3/ quasiproton has proven to play a pivotal role for the observation of wobbling in lutetium and tantalum nuclei. As first suggested in Ref. [7], the i 3/ quasiproton drives the nuclear shape toward a larger deformation than the other quasiparticle states located near the Fermi surface. At these larger deformations, a significant gap in the quasineutron energy levels opens at N = 94 and stabilizes an asymmetric shape with γ. Following the initial discovery of the first wobbling structure in 63 Lu [8], particle-rotor model calculations were primarily used to describe features of the exotic wobbling mode, see Refs. [9 ]. Random-phase approximation calculations were also able to reproduce experimental results of the wobbling bands, see Refs. [3 6]. In particular, the unusually large transition strength ratios B(E) out /B(E) in could be described in both models. However, the previous particle-rotor model phonon description of the wobbling energy has not been able to reproduce a systematic feature of the known wobbling bands. Figure displays the energy splitting between the wobbling band and the πi 3/ sequence upon which the wobbling structure is based. In all four cases where the wobbling energies are known, it is found that the energy splitting reduces with spin. That is, the wobbling band is becoming lower in energy, with respect to the πi 3/ band, as spin increases. As outlined below, with the previous assumptions in the particle-rotor calculations, the wobbling sequence should actually increase in energy with spin. However, we propose a different set of assumptions that can not only reproduce the same large B(E) out /B(E) in ratios, but also describe the reduction in energy splitting between the wobbling and πi 3/ sequences. A. Simple, transverse, and longitudinal Wobblers Bohr and Mottelson [] described the motion of a triaxial nucleus and pointed out the presence of a wobbling sequence for an even-even system. Starting from the Hamiltonian of a rigid triaxial rotor they derive the energies for the wobbling excitations as the small amplitude limit of the nutation of the angular momentum vector ( H = A 3 I(I + ) + ˆn + ) ω w, () where the wobbling phonon energy ω w is equal to ω w = I[(A 3 A )(A A 3 )] /, () and the A k parameters are related to the moments of inertia ( k ) about the three principle axes as A k = k. (3) Here, it is assumed that the 3-axis is the axis with the maximal moment of inertia. However, to account for the presence of a highly-aligned unpaired quasiparticle, the triaxial rotor Hamiltonian must be replaced by the triaxial particle rotor Hamiltionian H = A 3 (Ĵ3 ĵ ) + A (Ĵ ĵ ) + A (Ĵ j ), (4) where j k is the angular momentum associated with the unpaired quasiparticle and Ĵk is the total angular momentum about the respective axis.. Semiclassical analysis The consequences of the presence of the unpaired quasiparticle can be best understood in terms of the classical motion of the angular momentum vector. Its orbits are determined by the conservation of angular momentum = = I(I + ) (5) FIG. : Energy splitting between the ˆn = wobbling band and the πi 3/ ˆn = sequence in 63,65,67 Lu and 67 Ta. and energy E = A 3 ( 3 j) + A + A, (6)

2 where for simplification we assume that unpaired quasiparticle is rigidly aligned with axis 3. The classical orbit of ~ is the intersection of the angular momentum sphere (5) with the energy ellipsoid (6). Let us first consider the case of a simple triaxial rotor without the additional quasiparticle, i. e. j = in equation (6). Fig. illustrates three types of orbits for a given angular momentum, which is the radius of the angular momentum sphere (5)). We assume for the rotational constants the ratios A = 6A3 and A = 3A3. The size of the energy ellipsoid increases with the energy E. The yrast line corresponds to uniform rotation about the 3-axis with the maximal moment of inertia. The upper panel shows an orbit just above the yrast line, which represents the harmonic wobbling motion as discussed by Bohr and Mottelson. The middle panel shows the orbit called separatrix. It has the energy of the unstable uniform rotation about the -axis with the intermediate moment of inertia. The frequency of this orbit is zero, because it takes infinite long time to get to or to depart from the point of the labile equilibrium (uniform rotation about the -axis). The orbits with larger energy than the separatrix revolve the -axis. The lower panel shows one example. Quantum mechanically one has to take into account the invariance of the rotor with respect to rotations by π/ about its principal axes. It has the consequence that the states corresponding to revolution about the 3-axis have a signature quantum number α = I+even number, which is fixed by the quantum number I of the angular momentum. The signature alternates between and, starting with for the yrast line. Above the separatrix, there are two classical orbits with the same energy, which revolve the positive and negative -half axes. Quantum mechanically they make an even and odd linear combination, which have signature and. The rotational appear as I = sequences above the separatrix and as I + sequences below. In order to describe the motion of ~ we introduce the canonical variables and φ, q = cosφ, = sinφ, =, (7) %vspace-3cm where φ is the angle of the projection of ~ onto the - plane. The phase space for the one-dimensional motion on the angular momentum sphere is π φ < π and. Fig. 3 shows several orbits in phase space. Orbit corresponds to wobbling about the 3-axis. Orbit 4 is the separatrix. Orbit 7 corresponds to wobbling about the -axis. According to classical mechanics, the period of the orbit T = πds/de, where S is the phase space area enclosed by the orbit. The orbits in Fig 3 are calculated for energies that increase by the same amount. As seen, the difference S is maximal near the separatrix, which means the period T has a maximum and the frequency ω = π/t has a minimum. In classical mechanics the energy increases continouusly, and the frequency of the separatrix goes to zero, as mentioned FIG. : Angular momentum sphere and energy ellipsoid for a simple triaxial rotor with the rotational constants A = 6A3 and A = 3A3. The intersection line is the classical orbit

3 FIG. 3: Classical orbits of the angular momentum vector for a simple triaxial rotor with the rotational constants A = 6A 3 and A = 3A 3. The unit of angular momentum is j. The angular momentum is = j. The energy difference between the orbits is.4, where the energy unit is A 3j. above. In quantum mechanics the increase of the energy is discrete, such that S = h, i. e. the energy distance between adjacent levels has a minimum at the separatrix. A stable minimum is defined at the point where 3 = about which small oscillations describe the wobbling motion. The derivation of the energy of the wobbling mode () is given in []. Figs and XXX in Bohr and Mottelson [] shows the quantal spectrum of a triaxial rotor. Below the separatrix the ond sees the I = wobbling bands, which come together the when approaching the seperatrix and merge into i = sequences above the separatrix. In the presence of the odd quasiparticle one must distinguish between the case of its angular momentum j being aligned with the axis of the largest moment of inertia, which we refer to as longitudinal, and the case of j being perpendicular to this axis, which we refer to as transverse. The longitudinal case is similar to the simple rotor, only that the energy ellipsoid is shifted by j upwards. The yrast line corresponds to uniform rotation about the 3-axis and the lowest excited states to wobbling about this axis, similar to the upper panel of Fig.. The orbits are shown in Fig. 4. The wobbling bands have alternating signature ±j, starting with at the yrast line. For the transverse transverse rotor we assume that the -axis has the largest moment of inertia, and the -axis the smallest. We use the same ratios as for the simple rotor, i. e. the rotational constants A = 6A and A 3 = 3A. As illustrated in Fig. 5, the yrast line consists of two pieces. At low angular momentum it corresponds to rotation about the 3-axis. The energy ellipsoid touches the angular momentum sphere at the point 3 = on the 3-axis. The yrast energy is E = A 3 ( j). The low en- FIG. 4: Classical orbits of the angular momentum vector for a longitudinal triaxial rotor with the rotational constants A = 6A 3 and A = 3A 3. The angular momentum Ī = j. The unit of angular momentum is j. The energy difference between the orbits is.3, where the energy unit is A 3j Energy S FIG. 5: Energy of the yrast line and the separatrix (S) for the transverse rotor with the rotational constants A = 6A and A 3 = 3A. The unit of angular momentum is j, and the energy unit is A 3j. ergy orbits represent wobbling about the 3-axis. Fig. 6 displays the intersection of the lowest orbit in Fig. 7, which shows the orbits in phase space. At the critical angular momentum c = ja 3 /(A 3 A ) the rotational axis of the yrast line flips to the direction of the point =, = c, 3 = c, where the energy ellipsoid touches the angular momentum sphere from inside. The upper panel of Fig. 8 displays the intersection of the energy ellipsoid with angular momentum sphere for a slightly higher energy. In Fig. 9, it is the first orbit enclosing the touching point. At E = A 3 ( j), there is the separatrix, which is illustrated in the middle panel of Fig. 8. Above the separatrix the orbits revolve about the 3-axis as shown in the lower panel of Fig. 8.

4 4 Φ FIG. 6: Angular momentum sphere and energy ellipsoid for a transverse triaxial rotor with the rotational constants A = 6A and A 3 = 3A. The intersection line is the classical orbit of the angular momentum vector relative to the body fixed frame. The figure correspond to the lowest energy orbit in Fig. 7. For < c, the yrast line is E = A 3 ( j). It continues as separatrix for > c. As discussed above for the simple rotor, the classical frequency of the separatrix is zero. This means that the frequency of the wobbling mode with infinitesimal amplitude goes to zero at = c, where uniform rotation about the 3-axis becomes unstable, and the new branch of the yrast line starts. That is, the wobbling frequency decreases with such that it reaches zero at c. Quantum mechanically, the yrast staes have signature j for < c and the first wobbling state has signature j. It encloses the fixed area h in phase space, which means its energy decreases with. It merges with the yrast with yrast line, which becomes a I = sequence for > c. In case of the longitudinal rotor, there is no bifurcation of the yrast line, which is reflected by an increase of the wobbling frequency with.. Harmonic wobbling Now we consider small amplitude wobbling vibrations about the 3-axis. The angular momentum of the odd particle is assumed to be firmly aligned with the 3-axis and can be considered a a number. Then the Hamiltonian becomes H = A 3 (Ĵ3 j) + A Ĵ + A Ĵ, (8) FIG. 7: Classical orbits of the angular momentum vector for a transverse triaxial rotor with the rotational constants A = 6A and A 3 = 3A. The angular momentum =.5j. The unit of angular momentum is j. The energy increases from the top to the bottom. The energy difference between the orbits is.3, where the energy unit is A 3jI. The separatrix in the lower part of the figure is additionally placed at its energy. where j is a number. Using Ĵ 3 = (Ĵ + Ĵ with = I(I + ), the Hamiltonian becomes H = A 3 ( j) + ), (9) (A Ā3())Ĵ + (A Ā3())Ĵ, () ( Ā 3 () = A 3 j ) () The Hamiltonian has the form of the simple triaxial rotor Hamiltonian, except that A 3 is replaced by the -dependent rotational constant Ā3() = A 3 (( j/). Therefore, one has only to replace A 3 by Ā3() in the expressions derived by Bohr and Mottelson []. (The - axis has been exchanged by the 3-axis.) The wobbling frequency becomes ω w = [(A Ā3())(A Ā3())] /, ( Ā 3 () = A 3 j ), = ( + ). () The E-transition probabilities are B(E, n, I n, I ± ) = 5 6π e Q, (3) B(E, n, I n, I ) = 5 n 6π e ( 3Q x Q y), (4) B(E, n, I n +, I + ) = 5 n + 6π e ( 3Q x Q y), (5)

5 Φ FIG. 9: Classical orbits of the angular momentum vector for a transverse triaxial rotor with the rotational constants A = 6A and A3 = 3A. The angular momentum I = 3j. The unit of angular momentum is j. The energy difference between the orbits is.4, where the energy unit is A3 I. The dot indicates the yrast orbit, from where energy increases. The separatrix in the lower part of the figure is additionally placed at its energy. where / α x ± (6) = (α β )/ y α = A + A A 3 (), β = (A A )), (7) and Q and Q are the quadrupole moments of the triaxial density relative to the 3-axis. The harmonic approximation is only valid as long as << and <<. The component has the largest amplitude, which is reached at its classical turning point where =. WIth the energy of the first wobbling excitation of 3/~ωw one finds for the turning point A A 3 = 3 A A 3 / <<, (8) Eqn. () can be rewritten as j ~ωw = FIG. 8: Angular momentum sphere and energy ellipsoid for a transverse triaxial rotor with the rotational constants A = 6A and A3 = 3A. The intersection line is the classical orbit of the angular momentum vector relative to the body fixed frame. The three panels correspond to the orbit with the smallest energy, the separatrix, and the next orbit in Fig j / + (9). j Within the previous particle-rotor model studies [] the assumption of A3 < A, A ( >, ) was made, which is the longitudinal wobbler in our terminology. It is apparent that with this assumption both parenthesis by which in Eqn. (9) is multiplied are always positive, such that the wobbling energy will increase with spin. Thus, the wobbling band must increase in energy with spin relative to the πi3/ band; exactly the opposite of the trend observed in Fig.. This is the reason why previous studies could not properly reproduce the reduction in energy splitting between the two bands.

6 6 We suggest that the observed wobbling excitations are of the transverse type, i. e. A < A 3, A 3 < A ( > 3, 3 > ). Within Eqn. (9), the factor + ( 3 / )/j decreases with. The wobbling energy will decrease for sufficiently large, because it will be zero for c = j /( ), which is the previously discussed critical angular momentum where the separatrix bifurcates from the yrast line. There the yrast and the wobbling bands will merge into a single I = sequence, which is based based upon tilted rotational axis of the upper part of the yrast line of the transverse rotor. Clearly with ω w reducing with spin, the energy splitting between the wobbling and πi 3/ bands will decrease, as is experimentally found in Fig.. Tanabe and Sugawara-Tanabe [, ] considered the less restrictive approximation that the odd particle is not rigidly fixed to the core. Its angular momentum may execute small amplitude oscillations. They find a moderate coupling between the two oscillators, such that the lowest states may be classified as being predominantly a wobbling mode of the core or an excitation of the odd particle. Our discussion above concerns only the first type, the wobbling modes. 3. Microscopic calculations In the considered case, the i 3/ quasiproton is a nearly perfect particle (chemical potential in the low part of the intruder shell), which aligns with the short axis. This orientation minimizes the energy of the attractive shortrange interaction between the orbital and the triaxial core, because it correspond to maximal overlap. In the case of a quasiparticle that is a nearly perfect hole (chemical potential in the high part of the intruder shell) the preferred orientation is the long axis. Qualitatively, it is expected that the medium axis has the largest moment of inertia, because the deviation of the density distribution from rotational symmetry is for this axis larger than for the other axes. Hence, the transverse wobbler is the typical case. This is also the case for the odd quasiparticle being of hole type. Previous studies [9 ] assumed A 3 < A, A, i. e. the case of the longitudinal wobbler. Accordingly, they found that the wobbling frequency increased with I, in contrast to experiment. Ref. [] used a common scaling factor for all moments of inertia which increased as the moment of inertia of the yrast line. The corresponding decrease of the scale of A k results in a proportional decrease of ω w. We studied the transverse wobbling scenario for 63 Lu. This nucleus was chosen as it is was the first case of wobbling ever identified and it is perhaps the best example of this phenomenon. The ratios between the moments of inertia for classical irrotational or rotational flow cannot not be considered as quantitatively acceptable. For this reason we calculated them microscopically by means of cranking the Z = 7, N = 9 core about its three principal axes. Using the TAC code [] with a typical TSD of ε =.39 γ = 7 o and zero pairing, we found 3 / =7 MeV, / =77 MeV, and / =3 MeV for the short, medium, and long axis, respectively The ratios differ qualitatively from the ones assumed in the previous work [, ]. As expected, the moment of inertia is largest for the medium axis, i. e. the the wobbler is transverse. We multiplied the calculated moments of inertia by a common I-depend factor, which was chosen such that We diagonalized the particle-rotor Hamiltonian (4) numerically using the code by [? ] assuming the same deformation for the potential and no pairing. A scaling factor was multiplied to all moments of inertia. Fig.?? shows the experimental wobbling frequency determined as ω w (I) = E (I) (E (I ) + E (I + ))/, where E (I) are the yrast energies and E (I) the energies of the one phonon wobbling band. For the two phonon band ω w = E (I) E (I) are shown. The particle-rotor values calculated in the same way. The small difference between 3 and 3 In addition, the B(E) out /B(E) in ratios for the ˆn = The theoretical values are shown in Fig.?? along with the experimental values from Ref. [8]. One may see a satisfactory agreement between the two lending further credence to the tilted wobbling description. It is interesting to note that a decline in the B(E) out /B(E) in ratios with spin is predicted by both the tilted wobbling assumption and the previous assumptions made in the particle-rotor calculations (see Fig. 5 in Ref. [8]). However, experimental values measured in 63 Lu [8], 65 Lu [8], 67 Lu [9], and 67 Ta [? ] remain constant with spin in each case. Constant quadrupole deformation was assumed in both cases, and as measured in 63 Lu, there appears to be a decrease in the quadrupole moment with spin []. The B(E) in would reduce in this case, which may account for the discrepancy observed between the models and experiment. [] A. Bohr and B. R. 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