Microscopic and nonadiabatic Schrödinger equation derived from the generator coordinate method based on zero- and two-quasiparticle states

A new approach called the Schrödinger collective intrinsic model (SCIM) has been developed to achieve a microscopic description of the coupling between collective and intrinsic excitations. The derivation of the SCIM proceeds in two steps. The first step is based on a generalization of the symmetric moment expansion of the equations derived in the framework of the generator coordinate method (GCM), when both Hartree-Fock+BCS (HF+BCS) states and two-quasi-particle excitations are taken into account as basis states. The second step consists in reducing the generalized Hill and Wheeler equation to a simpler form to extract a Schrödinger-like equation. The validity of the approach is discussed by means of results obtained for the overlap kernel between HF+BCS states and two-quasiparticle excitations at different deformations.

Figure 1

HF+BCS potential energy along the symmetric fission barrier in ${}^{236}$U.

Figure 2

Energy of the selected 2-qp excitations around $q=70$ b as a function of deformation. Letters are used to differentiate excitations with same $K^{\pi }$.

Figure 3

Energy of the single-quasiparticle excitations around $q=30$ b for $K^{\pi }=1/2^{+}$.

Figure 4

Comparison between the overlaps $N_{00}(\overline{q}+s/2,\overline{q}-s/2)$ calculated for different $\overline{q}$ as functions of $s$ for $20\text{b}\le \overline{q}\le 100\text{b}$ (a) and $100\text{b}\le \overline{q}\le 350\text{b}$ (b).

Figure 5

Overlap kernel $N_{ii}(\overline{q}+s/2,\overline{q}-s/2)$ for HF+BCS solutions and different 2-qp excitations not involved in repulsions [labeled $K^{\pi }\left(i\right)$] at $\overline{q}=60$ b, plotted as functions of $s$ (a). Overlap kernel $N_{ii}(\overline{q}+s/2,\overline{q}-s/2)$ for HF+BCS solutions and different 2-qp excitations involved in repulsions [labeled by $K^{\pi }\left(i\right)$, and $q_r$ the deformation where the repulsion occurs] at $\overline{q}=50$ b, plotted as functions of $s$ (b). Letters are used to differentiate the same $K^{\pi }$ excitations in both figures.

Figure 6

Widths of the overlap extracted from the moments of zero, second, and fourth orders, as functions of the elongation, shown with the black circle, red cross, and blue triangle, respectively. Full lines correspond to the widths from $N_{00}^{\left(n\right)}$ terms and circles to diagonal terms $N_{ii}^{\left(n\right)}$, $i\ne 0$. (a) Results for a large range of deformation $[0,350\text{b}]$. (b) Widths from $N_{00}^{\left(n\right)}$ in the region $[0,110\text{b}]$.

Figure 7

Overlaps $N_{0i}(\overline{q}+s/2,\overline{q}-s/2)$ (a) and $N_{ij}(\overline{q}+s/2,\overline{q}-s/2)$ (b) at $\overline{q}=30$ b for different excitations labeled by $K^{\pi }$ as a function of $s$. Letters are used to differentiate same $K^{\pi }$ excitations.

Figure 8

Nondiagonal term $N_{ij}\left(\overline{q}\right)$ built with excitations involved in a repulsion at $q_r=218$ b.

Figure 9

$N_{00}^{\left(0\right)}\left(\overline{q}\right)$ (a) and its first and second derivatives (b) as functions of the deformation.

Figure 10

The normalized second-order moment ${\overline{N}}^{\left(2\right)}\left(\overline{q}\right)$ (a) and its first (black solid circles) and second (red solid triangles) derivatives (b) of the nonexcited overlap along the deformation.

Figure 11

Comparison between $C^{\left(2\right)}(N^{\left(2\right)},N^{\left(0\right)})$ and first iterated $C^{\left(2\right)}({\overline{N}}^{\left(2\right)},1+\alpha )$ values as a function of the deformation.

Figure 12

$A$ and $C$ coefficients in Eq. (33) as function of the deformation.