Cluster form factor calculation in the<i>ab initio</i>no-core shell model

Abstract

We derive expressions for cluster overlap integrals or channel cluster form factors for ab initio no-core shell model (NCSM) wave functions. These are used to obtain the spectroscopic factors and can serve as a starting point for the description of low-energy nuclear reactions. We consider the composite system and the target nucleus to be described in the Slater determinant (SD) harmonic oscillator (HO) basis while the projectile eigenstate to be expanded in the Jacobi coordinate HO basis. This is the most practical case. The spurious center of mass components present in the SD bases are removed exactly. The calculated cluster overlap integrals are translationally invariant. As an illustration, we present results of cluster form factor calculations for $⟨^{5}\mathrm{He}\ensuremath{\mid}^{4}\mathrm{He}+n⟩$, $⟨^{5}\mathrm{He}\ensuremath{\mid}^{3}\mathrm{H}+d⟩$, $⟨^{6}\mathrm{Li}\ensuremath{\mid}^{4}\mathrm{He}+d⟩$, $⟨^{6}\mathrm{Be}\ensuremath{\mid}^{3}\mathrm{He}+^{3}\mathrm{He}⟩$, $⟨^{7}\mathrm{Li}\ensuremath{\mid}^{4}\mathrm{He}+^{3}\mathrm{H}⟩$, $⟨^{7}\mathrm{Li}\ensuremath{\mid}^{6}\mathrm{Li}+n⟩$, $⟨^{8}\mathrm{Be}\ensuremath{\mid}^{6}\mathrm{Li}+d⟩$, $⟨^{8}\mathrm{Be}\ensuremath{\mid}^{7}\mathrm{Li}+p⟩$, $⟨^{9}\mathrm{Li}\ensuremath{\mid}^{8}\mathrm{Li}+n⟩$, and $⟨^{13}\mathrm{C}\ensuremath{\mid}^{12}\mathrm{C}+n⟩$, with all the nuclei described by multi-$\ensuremath{\hbar}\ensuremath{\Omega}$ NCSM wave functions.