Neutron Transfer Reactions for Deformed Nuclei Using Sturmian Basis States

We study the spin-parity distribution P(J$^π$,E) of $^{156}$Gd excited states above the neutron separation energy $S_{n}=8.536$ MeV \cite{156Gd_list_of_levels} that are expected to be populated via the 1-step neutron pickup reaction $^{157}$Gd($^{3}$He,$^{4}$He)$^{156}$Gd. In analogy with the rotor plus particle model \cite{Bohr&Mottelson-II}, we view excited states in $^{156}$Gd as rotational excitations built on intrinsic states consisting of a neutron hole in the $^{157}$Gd core, that is, a neutron removal from a deformed Woods-Saxon type single-particle state \cite{Woods-Saxon:1954} in $^{157}$Gd. The particle-core interaction usually dominated by a Coriolis coupling are accounted via first order perturbation theory \cite{VGGueorguiev:07062002}. The reaction cross section to each excited state in $^{156}$Gd is calculated as coherent contribution using a standard reaction code \cite{CHUCK} based on spherical basis states. The spectroscopic factor associated with each state is the expansion coefficient of the deformed neutron state in a spherical Sturmian basis along with the spherical form factors \cite{VGGueorguiev:07062002}. The total cross section, as a function of the excitation energy, is generated using Lorentzian smearing distribution function. Our calculations show that, within the assumptions and computational modeling, the reaction $^{3}$He+$^{157}$Gd $\rightarrow$ $^{4}$He+$^{156}$Gd$^{\star}$ has a smooth formation probability P(J$^π$,E) within the energy range relevant to the desired reaction $^{155}$Gd+n $\rightarrow$ $^{156}$Gd$^{\star}$. The formation probability P(J$^π$,E) resembles a Gaussian distribution with centroids and widths that differ for positive and negative parity states.