Comparative study of the requantization of the time-dependent mean field for the dynamics of nuclear pairing

To describe quantal collective phenomena, it is useful to requantize the time-dependent mean-field dynamics. We study the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory for the two-level pairing Hamiltonian, and compare results of different quantization methods. The one constructing microscopic wave functions, using the TDHFB trajectories fulfilling the Einstein-Brillouin-Keller quantization condition, turns out to be the most accurate. The method is based on the stationary-phase approximation to the path integral. We also examine the performance of the collective model which assumes that the pairing gap parameter is the collective coordinate. The applicability of the collective model is limited for the nuclear pairing with a small number of single-particle levels, because the pairing gap parameter represents only a half of the pairing collective space.

Figure 1

Energy contour plot for ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=8$ and $N=16$. The lines indicate the TDHFB trajectories fulfilling the EBK quantization condition of Eq. (3.7).

Figure 2

Excitation energies of $|0_2^{+}\rangle ,|0_3^{+}\rangle$, and $|0_4^{+}\rangle$ for $\mathrm{\Omega }=50$ systems with (a) $N=50$ and (b) $N=100$ as functions of the dimensionless parameter $x$ of Eq. (4.1).

Figure 3

Occupation probability in excited $0^{+}$ states as a function of $j$ for $\mathrm{\Omega }=50$ systems with (a) $N=50$ and (b) $N=100$. The upper and lower panels display the results for $x=0.5$ and $x=2$, respectively. The three vertical bars at each $j$ from the left to the right represent the squared components of the wave functions from exact, SPA, and CQ calculations, respectively. The left end of the horizontal axis at $j=j_{\min }$ corresponds to a component with $(n_1,n_2)=(N,0)$. The next at $j=j_{\min }+1$ corresponds to the one with $(n_1,n_2)=(N-2,2)$, and so on.

Figure 4

The strength of pair-addition transition $B(P_{\mathrm{ad}};k\to k^{\prime })$ for $\mathrm{\Omega }=50$ systems from $N=48$–50. Left panels, results of the CQ method; middle panels, FD; right panels, SPA. Dashed lines represent exact calculation. Upper panels show the intraband transitions of $|0_1^{+}\rangle \to |0_1^{+}\rangle$ and $|0_2^{+}\rangle \to |0_2^{+}\rangle$, while lower panels show the interband transition of $|0_1^{+}\rangle \to |0_2^{+}\rangle$ and $|0_2^{+}\rangle \to |0_1^{+}\rangle$.

Figure 5

The same as Fig. 4 but for $N=98\to 100$.

Figure 6

Excitation energies of $|0_2^{+}\rangle ,|0_3^{+}\rangle$, and $|0_4^{+}\rangle$ for $\mathrm{\Omega }=N=8$ systems as functions of $x$.

Figure 7

Occupation probability in excited $0^{+}$ states as a function of $j$ for $\mathrm{\Omega }=N=8$ systems. (a) and (b) Results for $x=0.5$ and $x=2$, respectively. See also the caption for Fig. 3.

Figure 8

The same as Fig. 4 but for $N=6\to 8$ with $\mathrm{\Omega }=8$.

Figure 9

The same as Fig. 6 but for $N=2\mathrm{\Omega }=16$.

Figure 10

The same as Fig. 7 but for $N=2\mathrm{\Omega }=16$.

Figure 11

The same as Fig. 4 but for $N=14\to 16$ with $\mathrm{\Omega }=8$.

Figure 12

Energy surfaces as functions of $j$, Eq. (2.15) with $\varphi =0$, for $x=0.5,2$, and 3.2. Dashed line is the pairing deformation parameter $\alpha$ of Eq. (4.2) as a function of $j$. (a) Mid-shell ($\mathrm{\Omega }=N=8$) and (b) closed-shell ($2\mathrm{\Omega }=N=16$).

Figure 13

Potential energy surface as functions of $\alpha$ with $x=2.4$. (a) $\mathrm{\Omega }=N=50$, (b) $2\mathrm{\Omega }=N=16$. Horizontal lines indicate energy spectra. Black lines are obtained from the potential energy surface, and green lines are obtained with the CQ method (Sec. 3b).