Consistency of nucleon-transfer sum rules in well-deformed nuclei

Nucleon-transfer sum rules have been assessed via a consistent reanalysis of cross-section data from neutron-adding ($d,p$) and -removing ($d,t$) reactions on well-deformed isotopes of Gd, Dy, Er, Yb, and W, with $92\le N\le 108$, studied at the Niels Bohr Institute in the 1960s and 1970s. These are complemented by new measurements of cross sections using the ($d,p$), ($d,t$), and ($p,d$) reactions on a subset of these nuclei. The sum rules, defined in a Nilsson-model framework, are remarkably consistent. A single overall normalization is used in the analysis, which appears to be sensitive to assumptions about the reaction mechanism, and in the case of sums using the ($d,t$) reaction, differs from values determined from reactions on spherical systems.

Figure 1

The normalized summed strength, defined in the text, for Nilsson orbitals as determined from ($d,p$) (lighter shade) and ($d,t$) (darker shade) reaction data from NBI on isotopes of Gd [12], Dy [13], Er [14], Yb [15], and W [16]. The blue bars (left) are for the $j^{\pi }=1/2^{-}$ member of the $1/2^{-}\left[521\right]$ band, and the orange bars (right) are for the $j^{\pi }=7/2^{-}$ member of $5/2^{-}\left[512\right]$ band. The gray band is the rms spread.

Figure 2

The occupancy, derived from the average of the adding and removing strength as determined from the NBI data [12, 13, 14, 15, 16], as a function of neutron number, $N$.

Figure 3

(a) Excitation-energy spectra for the ${}^{170}\mathrm{Er}(d,p){}^{171}\mathrm{Er}$ reaction at 12 MeV for ${\theta }_{\mathrm{lab}}={90}^{\circ }$. States belonging to the $5/2^{-}$[512] (blue lines, text) and $1/2^{-}$[521] (black lines, text) Nilsson configurations are labeled by their energy and spin. The states with their spin value given in a rectangle are those of interest in this work. The lines connect the states in a given band, highlighting their similarities in each isotope. (b) The same as in panel (a) for the ${}^{174}\mathrm{Yb}(d,p){}^{175}\mathrm{Yb}$ reaction at 12 MeV for ${\theta }_{\mathrm{lab}}={90}^{\circ }$. (c) Yields for both the ${}^{170}\mathrm{Er}(p,d){}^{169}\mathrm{Er}$ reaction at 18 MeV for ${\theta }_{\mathrm{lab}}={17}^{\circ }$ (black histogram) and the ${}^{170}\mathrm{Er}(d,t){}^{169}\mathrm{Er}$ reaction at 12 MeV for ${\theta }_{\mathrm{lab}}={90}^{\circ }$ (red histogram). (d) The same reactions as in panel (c) on the ${}^{174}\mathrm{Yb}$ target.

Figure 4

Information similar to that given in Fig. 1, comparing the new MLL data to the NBI data, labeled by the source of the data and by reaction, with the sum being ($d,p$) $[+n]$ and either ($d,t$) or ($p,d$) $[-n]$ as indicated. The ($p,d$) data are also hatched to distinguish them from the ($d,t$) data.