Is the K-quantum number conserved in the order-to-chaos transition region ?

To study the order-to-chaos transition in nuclei we investigate the validity of the K-quantum number in the excited rapid rotating163Er nucleus, analyzing the variance and covariance of the spectrum fluctuations of γ -cascades feeding into lowK and highK bands. The data are compared to simulated spectra obtained using a microscopic cranked shell model. K-selection rules are found to be obeyed for decay along excited unresolved rotational bands of internal excitation energy up t 1.2 MeV and angular momenta 20 h̄ I 40h̄. At higher internal energy, from about 1.2 to 2.5 MeV, the selection rules found to be only partially valid.  2005 Elsevier B.V. All rights reserved. PACS: 21.10.Re; 21.60.-n; 23.20.Lv; 24.60.Lz; 25.70.Gh; 27.70.+q

The conditions under which K, the projection of aligned nucleonic angular momenta on the symmetry axis in deformed nuclei, is a good quantum number remain a topic of much current interest, as testified by the extensive experimental work on high-K isomers [1]. The study of nuclear states with high values of the K-quantum number is interesting both from the point of view of the decay out from such states but also in connection with their feeding, which allows to investigate the validity of the associated selection rules at higher excitation energies. In fact, as it was stated by B.Mottelson [2], the question of K-quantum number violation in thermally excited states is a key issue in the study of the transition between ordered and chaotic many-nucleon motion caused by the residual interaction and the high level density. This problem has been addressed by studying the γ-decay from neutron resonances at energy U ≈ 8 MeV [3,4] and by studying the γ transitions of quasi-continuum nature emitted by nuclei formed in fusion reaction, which are probing the energy region extending up to ≈ 4 MeV [5,6]. Since the violation or persistence of the K-quantum number depends on the thermal excitation energy, it becomes particularly interesting to focus the attention to where the order-to-chaos transition is predicted to take place, namely at U ≈ 1-2 MeV [7,8].
In this paper we present a new study of the warm rotational motion in the 163 Er nucleus based on the measurement of the γ-transitions forming quasi-continuum patterns in γ − γ spectra and populating specific configurations with different values of the K-quantum number. The data are analyzed with the fluctuation and covariance analysis technique [9,10,11]. This particular nucleus has been previously studied [5] and it represents a good case for further and more detailed investigations of the validity of selection rules in the order-to-chaos transition region. Two novelties are presented. First, a more detailed experimental investigation is carried out for the warmest part of the decay. Secondly and most important, a direct and rather realistic comparison between experiment and theory is made for the first time. This comparison is based on simulated spectra constructed using recent calculations on this specific nucleus [12].

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The experiment was carried out using the EUROBALL array at the IReS Laboratory (France), employing the reaction 18 O + 150 Nd, at E beam = 87, 93 MeV. The 150 Nd target was made of a stack of two thin foils for a total thickness of 740 µg/cm 2 . The corresponding maximum angular momentum reached in the reaction has been calculated to be 40 and 45h, respectively. Energy-dependent time gates on the Ge time signals were used to suppress background from neutrons. A total of ≈ 3×10 9 events of triple and higher Ge-folds were finally obtained, with 162,163 Er as main evaporation residua. The data have been sorted into a number of γ − γ matrices in coincidence with specific γ-transitions of the 163 Er nucleus [13]. First, a matrix collecting the entire decay flow of 163 Er (named total ) has been constructed by gating on the three cleanest low spin transitions. In addition, seven matrices gated by transitions belonging to the low-K (K=5/2) signature and parity configurations labeled A=(1/2,+), B=(-1/2,+), E=(1/2,-) and F=(-1/2,-), and by the high-K (K=19/2) bands labeled K1 (negative parity) and K2 and K4 (positive parity), as done in ref. [13], have been sorted together with their corresponding two-dimensional (2D) backgrounds. For each 2D spectrum all known peak-peak and peak-background coincidences have been subtracted using the Radware software [14]. The separately gated matrices have also been summed into one low-K (A+B+E+F) and one high-K (K1+K2+K4) matrix. Figure 1 (left column) shows example of cuts perpendicular to the E γ 1 = E γ 2 diagonal, 60 keV wide, in the total, low-K and high-K γ-γ matrices, at the average transition energy (E γ1 + E γ2 )/2 = 900 keV.
The fluctuations of counts in each channel of the selected measured 2D spectra, expressed as variance and covariance, are evaluated by the program STATFIT [10] and stored into 2D spectra. One additional option is applied: all pairs of resolved transitions are removed in the triangular sector E γ1 ≥ E γ2 with the proper intensity from the γ − γ spectra, before the fluctuations are extracted, since the fluctuations are severely affected by the low lying intense transitions [10]. Because each rotational E γ -cascade on the average contributes one count in each 4h 2 interval, the statistical moments are evaluated over sectors of 4h 2 × 4h 2 , corresponding to 60 keV×60 keV intervals for rare earth nuclei around 163 Er.
From the fluctuation spectra we first extract the effective number of decay paths, which eventually feed into the gate-selected band. The number of decay paths N (2) path having two γ transitions with energies lying in a chosen 60 keV ×60 keV window in the γ − γ coincidence spectrum is obtained from the simple expression where N is the number of events, while µ 1 and µ 2 are the first and second moments of the distribution of counts, all evaluated in a 4 h 2 × 4 h 2 sector of the γ − γ matrix.
The superscript (2) indicates that the extraction of the number of paths is based on first and second moments, while the P (2) factor corrects for the finite resolution of the detector system [9,10].
The number of paths obtained from the analysis of the first ridge of the 2D matrices gated by individual bands is found, in average, to be ≈ 10 for both the four low-K and the three high-K configurations. Adding together the number of paths relative to specific configurations in a similar way as described in ref. [11], a total number of In contrast to the results of the ridge analysis the number of paths obtained by analyzing the valley region is found to depend significantly on the nuclear configuration.
This result is shown in the top part of figure 2 together with the number of paths deduced from the total E γ1 × E γ2 spectrum. As the valley is probing the region in which the rotational bands are strongly mixed, this result intuitively suggests that the mixing process is indeed different for high-K and low-K states.
To provide a better understanding of the mixing of states with different K quantum numbers we have studied the correlations in fluctuations between the spectra associ-ated with low-K and high-K quantum numbers. These correlations are expressed by the covariance of counts, defined as [11] where M(A) and M(B) refer to spectra gated by transitions from two different bands, A and B. The sum is over a region spanning N ch channels (in this case 15 × 15) in a two-dimensional 60 keV×60 keV window, andM denotes an average spectrum, (which in our case is obtained by the routine STATFIT as a numerical smoothed 3rd order approximation to the 2D spectrum). To normalize the covariance and thereby determine the degree of correlation between the two spectra, the correlation coefficient Here, µ 2 denotes the second moment defined for the same region  Simulated γ − γ spectra of interest were constructed by a Monte Carlo code, successfully employed to study rotational damping in different region of mass and deformation [15,16]. The code is based on the levels and E2 transition probabilities microscopically calculated for the specific case of the 163 Er nucleus [12]. In addition, statistical E1 transitions are included in a more schematic way using the calculated level density and a GDR strength function corresponding to a prolate nucleus with quadrupole deformation β = 0.25 and rotating collectively. In addition, an exponential quenching factor that takes into account the difference in K-quantum number between the initial and final states has been used. Such factor is analogous to the one employed in the analysis of the E1 decay-out from isomeric states [17,18]. The number of paths on the ridge may be understood by comparing to the total number of discrete bands, as directly extracted from the band mixing calculations (BM) by counting the number of bands branching out to less than 2 states [20]. The good agreement between the number of paths extracted from data, from simulations and calculated from the band mixing calculations tells that the nucleus 163 Er contains about 45 discrete bands at low heat energies before damping sets in [20]. This is a 8 somewhat higher number than the typically 20 to 25 discrete bands obtained for 164,167,168 Yb [10,11], and can be attributed to the existence of the additional 20 high-K bands in 163 Er, which do not exist at such low heat energies in the other nuclei.
The smaller number of high-K gated paths in the valley, relatively to the low-K gated is due to a lower level density for high-K states, ≈ 3 times lower than for the low-K states [12]. Also, the rotational damping width has been measured to be ≈ 30% reduced for high-K states [21]. In schematic evaluations of the number of paths [10], the level density and the rotational damping width enter as quadratic terms, thus explaining roughly the factor of 10 separating the number of high-K and low-K gated paths.
Turning now to the correlation coefficient, the rather low value r = 0.2 typically obtained for the low-K versus low-K ridge analysis may be understood from the fact that at most one or two E1 transitions cool down from the excited unresolved bands around < U >≈ 0.6 MeV to the low lying resolved bands. A simple quantitative estimate of the correlation coefficient can be deduced from the ratio between path probabilities, as described in ref. [11]. In the case of bands with same parity and Further progress in the interesting topic of the order-to-chaos transition in nuclei will benefit from a better understanding of the thermal energy dependence of the K-mixing problem. For this purpose, future works focusing on high K-bands of larger internal energies should be made.
10 Figure 1: 60 keV wide projections perpendicular to the E γ 1 = E γ 2 diagonal of experimental and simulated 2D spectra of 163 Er (left and right panels, respectively), at the average transition energy < E γ >=900 keV. The spectra collect either the total γ-decay flow (panel a) and d)) or the γ-decay in coincidence with low-K (panel b) and e)) or high-K (panel c) and f)) specific configurations. In the simulation, a state is defined as low-K (high-K) if K≤ 8 (K> 8). The reduced intensity observed in the E γ 1 ≥ E γ 2 region of the spectra is due to the subtraction of all discrete lines known from the level scheme, in the case of the experimental data, and of the yrast and first excited bands, in the case of the simulation.