Transmission resonance spectroscopy in the third minimum of ${}^{232}$Pa

The fission probability of ${}^{232}$Pa was measured as a function of the excitation energy in order to search for hyperdeformed (HD) transmission resonances using the ($d,pf$) transfer reaction on a radioactive ${}^{231}$Pa target. The experiment was performed at the Tandem accelerator of the Maier-Leibnitz Laboratory (MLL) at Garching using the ${}^{231}$Pa($d,pf$) reaction at a bombarding energy of $E_d=12$ MeV and with an energy resolution of $\Delta E=5.5$ keV. Two groups of transmission resonances have been observed at excitation energies of $E^{*}=5.7$ and 5.9 MeV. The fine structure of the resonance group at $E^{*}=5.7$ MeV could be interpreted as overlapping rotational bands with a rotational parameter characteristic to a HD nuclear shape (${\hslash }^2/2\Theta =2.10\pm 0.15$ keV). The fission barrier parameters of ${}^{232}$Pa have been determined by fitting talys 1.2 nuclear reaction code calculations to the overall structure of the fission probability. From the average level spacing of the $J=4$ states, the excitation energy of the ground state of the third minimum has been deduced to be $E_{\text{III}}={5.05}_{-0.10}^{+0.40}$ MeV.

Figure 1

The measured fission probability of ${}^{232}$Pa in the excitation energy range of $E^{*}=5.65$–6.2 MeV. Two resonance groups have been observed around $E^{*}=5.75$ and 5.9 MeV, respectively, in agreement with the results of a previous ($n,f$) experiment [15]. Below $E^{*}=5.82$ MeV a magnified scale of the $y$ axis was used for better visibility of the resonance structure.

Figure 2

The result of the Markov-chain peak searching algorithm applied to the histogram, which is shown in Fig. 1 ($E^{*}=5.7$–5.8 MeV). The continuous fission background stemming from the non-resonant tunneling process is subtracted. The positions of the identified resonances are indicated by arrows.

Figure 3

Excitation energy spectrum of ${}^{232}$Pa with statistical errors. The result of the fitting procedure with HD rotational bands is indicated by the solid line. The picket fence structure of the rotational bands together with the $K$ values of the bands are also shown. The quality of the fit (the reduced ${\chi }^2$ value) is ${\chi }^2/F=1.03$.

Figure 4

Excitation energy spectrum of ${}^{232}$Pa with the result of the fitting procedure (solid line) assuming SD rotational bands. The picket fence structure and the $K$ values of the bands are also indicated. The dashed lines represent the missing members of the bands. The quality of the fit (the reduced ${\chi }^2$ value) is ${\chi }^2/F=1.09$.

Figure 5

Experimental and calculated average level spacings of the $J=4$ states in the function of the excitation energy for ${}^{232}$Pa. Calculated level spacings are indicated by a dashed and a solid line for representing the values in the first and in the third potential minimum (with $E_{\text{III}}=5.05$ MeV), respectively. Within the calculation of the BSFG level densities, we used the same parametrization as in Ref. [22]. The error bar of the experimental point (circle) represents the upper and lower limits of the observed $J=4$ level spacings.

Figure 6

Experimental fission probability of ${}^{232}$Pa measured in the present experiment (below $E^{*}=6.2$ MeV) and in a previous, low-resolution measurement (above $E^{*}=6.2$ MeV) [26] together with the result of the talys 1.2 calculation (continuous line).

Figure 7

Triple-humped fission barrier of ${}^{232}$Pa as a result of the present study. The energy region of the observed HD resonances is indicated by horizontal dashed lines (and the hatched area). The curvature energies are $\hslash {\omega }_{A,\text{B1,B2}}=1.0$ MeV.

Figure 8

Calculated and experimental (a) inner ($E_A$) and (b) outer ($E_{\text{B}}$) barrier heights as a function of the atomic number ($Z$) and fissility parameter ($Z^2/A$), respectively, for actinide nuclei in the region of the “island of fission isomers” ($Z=88$–97). Clear tendencies can be seen for both barriers as illustrated by solid lines. The data points were taken from Refs. [28] (open circles), [8, 9] (full squares), [29] (open triangles), and [30, 31] (open stars). Present results on ${}^{232}$Pa are also shown (full triangles). The dashed line represents the tendency of empirical inner barrier heights, which were determined by using the double-humped fission barrier concept [28].