Updated analysis of the ${}^{170}\mathrm{Er}(p,t){}^{168}\mathrm{Er}$ reaction data

More than 200 states up to 4.1 MeV excitation have been populated in ${}^{168}\mathrm{Er}$ with the ${}^{170}\mathrm{Er}(p,t)$ reaction at 25 MeV incident energy. About 80 of these states, with $0^{+}$ and $2^{+}$ assignments, were reported in a previous publication [D. Bucurescu et al., Phys. Rev. C 73, 064309 (2006)]. The present work considerably enriches the knowledge of this nucleus. A multistep coupled-channels analysis of the angular distributions is now presented for all the states observed in this experiment. Spin and parity values between $0^{+}$ and $7^{-}$ are newly assigned for more than 100 states. For the states already reported in the ENSDF database with $J^{\pi }$ values there is a good agreement with our values. The ${}^{168}\mathrm{Er}$ nucleus remains one of the best experimentally known nuclei for states with low and medium spins below 4 MeV excitation energy, representing a challenge for future microscopic structure model calculations aiming to disentangle the contributions of different excitation degrees of freedom.

Figure 1

Coupling schemes used in the chuck3 coupled-channels calculations. (a) General coupling scheme for a $(p,t)$ reaction, involving seven channels (states); see text for comments. The seven channels are numbered 1,2,...,7 for reference in the text. (b) A simplified scheme highlighting the sequential transfer for the ${}^{170}\mathrm{Er}(p,t){}^{168}\mathrm{Er}$ reaction. For reasons explained in the text, channel 2 is not considered here. Note that this scheme is not purely sequential, but it includes some other transitions that may compete with the sequential ones (simultaneous direct $1\to 4$ and two-step $1\to 3\to 4$ transfers). (c) Simultaneous transfer schemes for ${}^{170}\mathrm{Er}(p,t){}^{168}\mathrm{Er}$, with the notation names of [7]. See Table 1 for the assignment of coupling schemes to particular levels.

Figure 2

Comparison of the experimental data for the $2_1^{+}$ and $4_1^{+}$ states of ${}^{168}$ with chuck3 calculations. Graphs (a) and (c): Including sequential transfer [scheme (b) of Fig. 1], with two states, 1/$2^{-}$ and 7/$2^{-}$, in ${}^{169}\mathrm{Er}$. The $S^{1/2}$ values chosen for the $(p,d)$ transitions to these states are 1.0 and 0.5, respectively. Note that the label “Sequential” means only that sequential transfers are included (see the legend of Fig. 1). Graphs (b) and (d): Using simultaneous transfer [scheme m1a of Fig. 1]. The factors used to approximately normalize the calculated curves to the experimental points are (a) 1.0, (b) 0.5, (c) 0.6, (d) 0.4. Cross sections for different reaction paths are also shown, to give an idea about the magnitude of the different amplitudes that add coherently to provide the full cross section. See text for more details.

Figure 3

Angular distributions of ${}^{168}\mathrm{Er}$ states with (a) $2^{+}$ and (b) $4^{+}{J}^{\pi }$ assignments. Calculations with chuck3 are shown by curves normalized to the experimental data in order to reasonably describe the observed shapes. The energies from Table 1 given for each of the states were rounded off to the nearest keV of the more precise energy given in Table 1. The continuous (red) curves are one-step DWBA calculations, while the dashed (blue) ones correspond to coupled-channels calculations with the coupling schemes specified in Table 1. A label “E” or “(E)” in the lower-left corner of a graph denotes a firm or tentative $J^{\pi }$ assignment, respectively, as adopted in the ENSDF database [18] for that state (see Table 1). For the states with energy within parentheses our spin assignment is tentative (see also Table 1). Here and in Figs. 5 and 6, for some levels that are considered to be doublets, dotted lines show the angular distributions for each of the states. Since these are incoherent, in this case the shapes should be added algebraically, with weights determined by a fit. For the 3295 keV state in graph (b) we tentatively propose a doublet of $1^{-}$ and $4^{+}$ states, on the basis of a combination of the two one-step calculations shown. Note that in some cases, for angular distributions with relatively large difference from the one-step DWBA prediction (e.g., the 3683 or 1411 keV state), we show the cross sections separately calculated for the three coupled-channels paths (Fig. 1): One-step process, $1\to 4$ (continuous red line); two-step $1\to 3\to 4$ (dashed black line) and two-step $1\to 2\to 4$ (dash-dotted violet line), as an indication of the magnitude of the three transfer amplitudes that add coherently to produce the total cross section for the final state.

Figure 4

The same as Fig. 3, but for the states with (a) $6^{+}$ and (b) $1^{-}{J}^{\pi }$ assignments. Note that in some cases, for angular distributions with a relatively large difference from the one-step DWBA prediction (e.g., the 549 or 2238 keV state), we show the cross sections separately calculated for the three coupled-channels paths (Fig. 1): One-step process, $1\to 4$ (continuous red line); two-step $1\to 3\to 4$ (dashed black line) and two-step $1\to 2\to 4$ (dash-dotted violet line), as an indication of the magnitude of the three transfer amplitudes that add up coherently to produce the total cross section for the final state.

Figure 5

The same as Fig. 3, but for the states with $3^{-}$ and $5^{-}{J}^{\pi }$ assignments. The 2002 keV state is assumed to be a doublet, fitted by adding one-step DWBA calculations for a $3^{-}$ state and the known $5^{-}$ states (see Table 1). For the 3113 keV state we show the cross sections separately calculated for the three coupled-channels paths (Fig. 1): One-step process, $1\to 4$ (continuous red line); two-step $1\to 3\to 4$ (dashed black line) and two-step $1\to 2\to 4$ (dash-dotted violet line), as an indication of the magnitude of the three transfer amplitudes that add up coherently to produce the total cross section for the final state.

Figure 6

Similar to Figs. 3, 4, and 5, for states of other $J^{\pi }$ values from this study. (a) Reanalysis of states with excitation energies from 2800 to 3929 keV previously assigned as $0^{+}$ (Fig. 3 of [4]). The two graphs following the 3929 keV state (second column) refer to the 2392 and 3715 keV states which had only tentative assignments [4]. Two other states also tentatively assigned as $0^{+}$ (2644 and 3065 keV) are now assigned as $1^{-}$ (see Fig. 3). For the remaining three states assigned as $\left(0^{+}\right)$, 3117 keV is one of a set of seven states which, due to their relatively large forward angle peaking were proposed as “possible $0^{+}$ states” (Fig. 4 of Ref. [4]), while 3380 and 3768 keV are newly assigned. Analysis of the other states from 3117 to 3697 keV proposed as “possible $0^{+}$ states” in [4] showed two of the states from this category (3313 and 3581 keV) could not be assigned $J^{\pi }$ values [see part (b) of Fig. 6], while the remaining four states were assigned as follows: 3147 and 3697 as $3^{-}$ (Fig. 4), 3529 and 3683 keV as $4^{+}$ and $2^{+}$, respectively (Fig. 2). For the 3929 keV state we show the cross sections separately calculated for the three coupled-channels paths (Fig. 1): One-step process, $1\to 4$ (continuous red line); two-step $1\to 3\to 4$ (dashed black line) and two-step $1\to 2\to 4$ (dash-dotted violet line), as an indication of the magnitude of the three transfer amplitudes that add up coherently to produce the total cross section for the final state. (b) Analysis of states with other $J^{\pi }$ values. The 1618 keV and 2969 keV states are assumed as doublets, as their angular distributions could be fitted by a combination of two one-step DWBA curves (see Table 1).

Figure 7

(a) Some cases from our analysis where more than one $J^{\pi }$ could be assigned to a certain state. For the first four states one of our possible assignments (marked with bold red number) coincides with the value adopted by ENSDF on the basis of other data. See also the discussion in the text. (b) Remaining states for which $J^{\pi }$ values could not be assigned in this study.

Figure 8

Comparison between experimental states and) states calculated by spdf-IBM-1. Experimental states: Solid (dashed) longer lines denote firm (tentative assignment of states observed in this experiment, and shorter magenta lines with the same convention are states listed in ENSDF but not observed in our experiment. The spdf-IBM-1 calculated states: Dashed-dotted lines denote the two-octupole states.

Figure 9

Collective bands in ${}^{168}\mathrm{Er}$ observed in this $(p,t)$ reaction experiment. The band assignments are described in the text. The states with firm (tentative) spin assignment (Table 1) are represented by circles ($\times$'s). The two known lowest $8^{+}$ states are shown, although they were not observed in this experiment. The bands with $K\ne 0$, the states of unnatural parity, most of them not observed in this experiment, have been added (when known [18]) as small triangles. For each identified band, the straight line shows the fit with the rotational formula, and its moment of inertia, MoI (in units ${\hslash }^2{\mathrm{MeV}}^{-1}$), is given in the upper part of the line. The letters next to the MoI values have the following meaning. The capital (red) letters indicate the known bands labeled by that letter in [18]. The bands proposed on the basis of the present data are labeled by small (blue) letters (see the text for their identification). Note that the ground state band (gsb) was shifted up by 500 keV to make the figure more readable.