Self-consistent quasiparticle formulation of a multiphonon method and its application to the neutron-rich ${}^{20}\text{O}$ nucleus

A Bogoliubov quasiparticle formulation of an equation-of-motion phonon method, suited for open-shell nuclei, is derived. Like its particle-hole version, it consists of deriving a set of equations of motions whose iterative solution generates an orthonormal basis of $n$-phonon states ($n=0,1,2,...$), built of quasiparticle Tamm-Dancoff phonons, which simplifies the solution of the eigenvalue problem. The method is applied to the open-shell neutron-rich ${}^{20}\text{O}$ for illustrative purposes. A Hartree-Fock-Bogoliubov canonical basis, derived from an intrinsic two-body optimized chiral Hamiltonian, is used to derive and solve the eigenvalue equations in a space encompassing a truncated two-phonon basis. The spurious admixtures induced by the violation of the particle number and the center-of-mass motion are eliminated to a large extent by a Gram-Schmidt orthogonalization procedure. The calculation takes into account the Pauli principle, is self-consistent, and is parameter free except for the energy cutoff used to truncate the two-phonon basis, which induces an increasing depression of the ground state through its strong coupling to the quasiparticle vacuum. Such a cutoff is fixed so as to reproduce the first $1^{-}$ level. The two-phonon states are shown to enhance the level density of the low-energy spectrum, consistently with the data, and to induce a fragmentation of the $E1$ strength which, while accounting for the very low $E1$ transitions, is not sufficient to reproduce the experimental cross section in the intermediate energy region. This and other discrepancies suggest the need of including the three-phonon states. These are also expected to offset the action of the two phonons on the quasiparticle vacuum and, therefore, free the calculation from any parameter.

Figure 1

Theoretical versus experimental level scheme of ${}^{20}\text{O}$.

Figure 2

TDA (a) and EMPM (b) $E1$ strength distribution in ${}^{20}\text{O}$.

Figure 3

Theoretical versus experimental [27] $E1$ reduced strengths of the two lowest levels in ${}^{20}\text{O}$.

Figure 4

Theoretical versus experimental $E1$ cross section in ${}^{20}\text{O}$. The contribution of the lowest two levels in the interval $5–7$ MeV, measured in Ref. [27], is not included to make a consistent comparison with the data of Ref. [25]. We have used a Lorentz width $\mathrm{\Delta }=1$ MeV.