Deformed proton emitters

A large number of odd-Z elements has isotopes that are sufficiently proton-rich to be unbound with respect to the ground-state proton emission. To date, about 40 such proton emitters are experimentally known [23,24]. Proton emission from spherical nuclei is well understood in terms of the simplest WKB barrier penetration. Such an approach allows for a consistent identification of the single-particle orbit from which the emission occurs, because the barrier is strongly influenced by the centrifugal component on top of the standard Coulomb barrier. The success is undoubtedly due to the fact that no pre-formation factor has to be here taken into account, contrary to the physical situation pertaining to the $\alpha$-particle emission. The precision of the theoretical description allows to determine the proton spectroscopic factors by comparing the calculated and measured proton-decay life times, see Refs. [25,26] and references cited therein.

Recently, the focus of studying proton emitters, both in experiment and in theory, is on investigating such a process when it occurs from deformed nuclei. Here, one has to reconcile the picture of the barrier penetration, pertaining to the intrinsic frame (and the barrier height depending on the direction with respect to the shape principal axis), with the necessity of correctly describing the angular momentum content of orbitals, which defines the corresponding centrifugal barrier. In recent years, a number of theoretical tools have been developed by several groups in order to tackle the problem, and again it seems that the theoretical description gives the full account of experimental data. The method of choice here is the coupled-channel approach that takes into account the structure of collective states in the daughter nucleus [27,28,29] (the so-called non-adiabatic approach). Within the strong-coupling adiabatic approach the calculations were also performed by using the deformed resonance states [30,31], Green function methods [32], reaction theory [33], and time-dependent transmission calculations [34].

The challenge of achieving the best possible description of deformed proton emitters lies in a possibility of determining with a large precision the Nilsson quantum numbers of deformed single-particle states in nuclei which cannot be accessed by other spectroscopic methods. An example of such analysis is shown in the right panel of Fig. 2 [22]. Measured life times of protons emitted from the ground and isomeric states [22,35] provide a stringent test on the type of single-particle deformed orbitals occupied in ¹⁴¹Ho. Although the branching ratios to excited states in the daughter nucleus ¹⁴⁰Dy are not yet known, the yrast cascade up to 8⁺ was recently identified in this nucleus from a decay of a 20$\mu$s isomer [36]. On the other hand, the fine structure was already observed in the proton decay of ¹³¹Eu to the 0⁺ and 2⁺ states in ¹³⁰Sm, which allowed for an unambiguous determination of the corresponding Nilsson orbital in ¹³¹Eu [37,38,27]. Similar methods were also used for a study of odd-N proton emitting nuclei [39], where, in principle, the proton-neutron interaction effects could be studied. A possibility to perform a much more difficult experiment on the proton emission from oriented nuclei was discussed in Ref. [40].