Microscopic description of the scissors mode in odd-mass heavy deformed nuclei

Pseudo-SU(3) shell-model results are reported for M1 excitation strengths in 157-Gd, 163-Dy and 169-Tm in the energy range between 2 and 4 MeV. Non-zero pseudo-spin couplings between the configurations play a very important role in determining the M1 strength distribution, especially its rapidly changing fragmentation pattern which differs significantly from what has been found in neighboring even-even systems. The results suggest one should examine contributions from intruder levels.

The scissors mode in nuclei refers to a pictorial image of deformed proton and neutron distributions oscillating against one another [1]. A description of this mode within the framework of the IBM [2] led to its detection in 156 Gd using high-resolution inelastic electron scattering [3]. Systematic studies employing nuclear resonance fluorescence scattering (NRF) measurements [4] followed. The non-observation of these low-energy M1 excitations in inelastic proton scattering (IPS) [5] confirmed its orbital character [6]. Over the past two decades an impressive wealth of information about the scissors mode in even-even nuclei has been obtained and analyzed [7].
Low-energy M1 transitions in odd-mass nuclei were first observed in 163 Dy in 1993 [8]. Unambiguous spin and parity assignments of excited states in these nuclei are difficult to make due to the half-integer character of the angular momentum of the states [9]. Furthermore, the M1 strengths in odd-mass nuclei are highly fragmented. Since the intensities are far smaller than in even-even nuclei, their identification against the background [10], which is complicated by the presence of a small amount of impurities in the target [7], requires much higher experimental resolution [11].
Theoretical descriptions of scissors modes in odd-mass nuclei have been offered within the context of the IBFM [12,13], the particle-core-coupling model [14] and the QPNM [15]. While the different models agree in relating the presence of the uncoupled nucleon with the * Electronic address: vargas@ganil.fr † Electronic address: hirsch@nuclecu.unam.mx ‡ Electronic address: draayer@lsu.edu observed fragmentation, the detailed description of this mode, with a nearly flat spectrum in some nuclei and has well-defined peaks in others is still not understood. Recently, the interplay between the spin and orbital M1 channels was examined [16] in the energy range between 4-10 MeV [17].
In the present letter we analyze scissors-like M1 transitions in 157 Gd, 163 Dy and 169 Tm. These nuclei have been studied experimentally by a number of researchers [8,9,18]. A fully microscopic description of M1 transitions strengths between 2 and 4 MeV in these rareearth nuclei was carried out using the pseudo SU(3) shell model. Good qualitative descriptions of the fragmentation of the M1 transition strength is obtained by including, for the first time, states with pseudo-spin 1 (in addition toS = 0) and 3/2 (in addition toS = 1/2). For normal parity levels our findings suggest that while orbital couplings are important, in odd-even mass nuclei it is spin-flip type couplings that dominate M1 strengths in the low-energy domain. These spin-flip type transitions were also found to be essential for describing the rapidly changing fragmentation patterns found in neighboring odd-A nuclei. Freezing the unique parity orbitals, which is the usual assumption, prevents the theory from giving a quantitative description of the M1 strength, a result that is not surprising since intruder states have the largest l values and therefore contribute maximally to orbital-type M1 transitions.
The pseudo SU(3) model [19,20] capitalizes on the existence of pseudo-spin symmetry, which refers to the experimental fact that single-particle orbitals with j = l -1/2 and j = (l -2) + 1/2 in the shell η lie very close in energy and can therefore be labeled as pseudo-spin doublets with quantum numbersj = j,η = η -1, andl = l -1. Its origin has been traced back to the relativistic Dirac equation [21]. In the present version of the pseudo-SU(3) model, the intruder level with opposite parity in each major shell is removed from active consideration [22] and pseudo-orbital and pseudo-spin quantum numbers are assigned to the remaining single-particle states. This assumption represents the strongest limitation of the present model.
Many-particle states of n α active nucleons (α = p, n) in a given (N ) normal parity shell η N α are classified by the following group chain [23,24,25]: (1) where above each group the quantum numbers that characterize its irreducible representations (irreps) are given and γ α and K α are multiplicity labels of the indicated reductions.
The Hamiltonian used in the calculations includes spherical Nilsson single-particle terms for the protons and neutrons (H sp,π[ν] ), the quadrupole-quadrupole (Q ·Q) and pairing (H pair,π[ν] ) interactions, as well as three 'rotor-like' terms that are diagonal in the SU(3) basis: A detailed analysis of each term of this Hamiltonian and its parametrization can be found in [25]. The three free parameters a, b, A sym were fixed by the best reproduction of the low-energy spectra; no additional parameters enter into the theory -the calculated M1 transitions reported below were not fit to the data. A description of the low-energy spectra and B(E2) transition strengths in even-even nuclei [26] and oddmass heavy deformed nuclei [25,27] have been carried out using linear combinations of SU(3) coupled protonneutron irreps with largest C 2 values and pseudo-spin 0 and 1/2 (for even and odd number of nucleons, respectively), which are mixed by the single-particle terms in the Hamiltonian.
The large number of states that can decay through M1 transitions to the ground state in odd-mass nuclei, led us to enlarge the basis by including states with pseudo-spin 1 and 3/2. These configurations are necessary to describe excited rotational bands and to account for the strong fragmentation of the M1 strengths between 2 and 4 MeV in odd-mass nuclei.
The inclusion of configurations with pseudo-spin 1 and 3/2 in the Hilbert space allows for a description of highly excited rotational bands in odd-mass nuclei. This is illustrated in Ref. [28], where several rotational bands in 157 Gd, 163 Dy and 169 Tm are described, including both excitation energies and intra-and inter-band B(E2) transition strengths, and shown to be in close agreement with the experimental data. In contrast, when the configuration space was restricted to the most spatially symmetric configurations, those with pseudo-spin 0 and 1/2, it was only possible to describe in 163 Dy the first three lowenergy bands [27]. The pseudo-spin symmetry is still approximately preserved in the present case, with these three low-energy bands showing only a small amount of pseudo-spin 1 and 3/2 admixing into predominantly pseudo-spin 0 and 1/2 configurations, respectively.
The M1 transitions are mediated by the operator In Eq.     Figure 1. Experimental values taken from Ref. [8].
ergy of the M1 strengths is correct, but some transition strengths are overestimated by a factor 2 to 3. The ground state wave functions of the two nuclei with odd number of neutrons, 157 Gd and 163 Dy, have one important difference. In 163 Dy the ground state has only pseudo-spin 0 and 1/2 components, while 157 Gd has a 13% mixing with pseudo-spin 1 and 3/2 components. In the M1 transition matrix elements the presence of these components in the later case gives rise to interference and fragmentation, while its absence in the former nuclei is associated with few large M1 transitions. The odd proton number of 169 Tm allows orbital proton excitations between half-integer components, building up its large M1 summed transition strength.
Having analyzed the similarities and differences between the experimental data and the theoretical predictions, we proceed to discuss the spin and orbital contributions to the M1 transitions. In insert c) of each figure the M1 transition strengths calculated only with the spin operators, i.e. making g o π,ν = 0 in Eq. 3), is presented. Insert d) shows the M1 strength when only the orbital part of the operator (3) are included (g s π,ν = 0). In all cases the spin coupling is by far the dominant mode, but for 169 Tm the orbital contribution is also large.
In the case of 163 Dy, there is an almost null contribution from the orbital part of the transition operator (0.103 µ 2 ), which in fact interferes destructively with the spin channel (0.543 µ 2 ) to produce a summed M1  Figure 1. Experimental values were taken from Ref. [18].
strength of 0.483 µ 2 in the scissors energy region. The 'angle' between the orbital and spin channels, as defined by Fayache et al. [16] is 110 o for 163 Dy. For 157 Gd, this angle has a value of 83 o and for 169 Tm it is 96 o . From Table I it can be seen that below 2 MeV the spin transitions are clearly dominant. Nevertheless, it should be emphasized that contributions of the intruder sector have been neglected.
The pseudo SU(3) shell model for odd-mass nuclei has been shown to offer a qualitative microscopic description of the scissors mode and its fragmentation. In order to successfully reproduce the observed fragmentation of the M1 strength, it was necessary to use realistic values for the single particle energies and to enlarge the Hilbert space to include those pseudo SU(3) irreps with the largest C 2 values and pseudo-spin 1 and 3/2. This expansion of the model space allowed the T1 operator to connect the ground state with many excited states (|J f − J i | ≤ 1) in the energy range between 2 and 4 MeV. The transitions are dominated by spin couplings, but interference with the orbital mode is very important.
A fully quantitative treatment of the problem should take into account contribution from the intruder sector. Detailed studies of M1 transitions in other odd-mass nuclei are under investigation and should offer an opportunity to further apply and test the theory.
The authors thank S. Pittel, A. Frank and P. Van Isacker for constructive comments. This work was supported in part by CONACyT (México) and the US National Science Foundation.