Shell model states in the continuum

We suggest a method for calculating scattering phase shifts and energies and widths of resonances which utilizes only eigenenergies obtained in variational calculations with oscillator basis and their dependence on oscillator basis spacing $\hslash \mathrm{\Omega }$. We make use of simple expressions for the $S$ matrix at eigenstates of a finite (truncated) Hamiltonian matrix in the oscillator basis obtained in the HORSE ($J$-matrix) formalism of quantum scattering theory. The validity of the suggested approach is verified in calculations with model Woods-Saxon potentials and applied to calculations of $n\alpha$ resonances and nonresonant scattering using the no-core shell model.

Figure 1

The function $f_{\mathbb{N}\ell }\left(E\right)$ (symbols) for different $\mathbb{N}$ and $\ell$ and its low-energy approximations $f_{\ell }^{l.e.}\left(E\right)$ [see Eq. (23)] and $-2\sqrt{E/s}+\pi \ell /2$ [see Eq. (26)].

Figure 2

The lowest ${\frac{3}{2}}^{-}$ eigenstates $E_0$ of the model Hamiltonian with WSBG potential obtained with various $N_{max}$ and $\hslash \mathrm{\Omega }$ plotted as a function of the scaling parameter $s$.

Figure 3

The ${\frac{3}{2}}^{-}$ phase shifts obtained directly from the WSBG eigenstates $E_0$ using Eq. (19).

Figure 4

The lowest ${\frac{3}{2\phantom{{}_q}}}^{-}$ eigenenergies $E_0$ of the model WSBG Hamiltonian obtained with various $N_{max}$ (symbols) as functions of $\hslash \mathrm{\Omega }$ and their selection for the SS-HORSE analysis according to inequality $\mathrm{\Lambda }>385$ MeV/$c$. The shaded area shows the selected $E_0$ values. Solid lines are solutions of Eq. (51) for energies $E_0$ with parameters $a,b$, and $d$ obtained by the fit.

Figure 5

The ${\frac{3}{2\phantom{{}_q}}}^{-}$ WSBG eigenstates $E_0$ selected according to $\mathrm{\Lambda }>385$ MeV/$c$ plotted as a function of the scaling parameter $s$. The solid curve depicts solutions of Eq. (51) for energies $E_0$ with parameters $a,b$, and $d$ obtained by the fit with the respective selection of eigenstates.

Figure 6

The ${\frac{3}{2}}^{-}$ WSBG phase shifts obtained using Eq. (19) directly from eigenstates $E_0$ selected according to $\mathrm{\Lambda }>385$ MeV/$c$ (symbols). The solid curve depicts the phase shifts of Eq. (48) with parameters $a,b$, and $d$ obtained by the fit with the respective selection of eigenstates; the dashed curve is obtained by a numerical integration of the Schrödinger equation.

Figure 7

The lowest ${\frac{3}{2}}^{-}$ WSBG eigenstates $E_0$ (symbols) and their $N_{max}\le 6$ selection (shaded area). See Fig. 4 for more details.

Figure 8

Selected lowest ${\frac{3}{2}}^{-}$ WSBG eigenstates $E_0$ obtained with $N_{max}\le 6$ as a function of the scaling parameter $s$. See Fig. 5 for details.

Figure 9

The ${\frac{3}{2}}^{-}$ WSBG phase shifts generated by the selected eigenstates $E_0$ obtained with $N_{max}\le 6$. See Fig. 6 for details.

Figure 10

The lowest continuum ${\frac{1}{2}}^{+}\phantom{{}_{q_{\int }}}$ WSBG eigenstates $E_1$ (symbols) as functions of $\hslash \mathrm{\Omega }$ and their $\mathrm{\Lambda }>385$ MeV/$c$ selection (shaded area). Solid lines are solutions of Eq. (54) for energies $E_1$ with parameters $E_b,c,d$, and $f$ obtained by the fit with this selection of eigenstates.

Figure 11

The lowest continuum ${\frac{1}{2}}^{+}$ WSBG eigenstates $E_1$ as a function of the scaling parameter $s$. The solid curve depicts solutions of Eq. (54) for energies $E_1$ with parameters $E_b,c,d$, and $f$ obtained by the fit with the $\mathrm{\Lambda }>385$ MeV/$c$ selection of eigenstates.

Figure 12

The ${\frac{1}{2}}^{+}$ phase shifts obtained directly from the WSBG eigenstates $E_1$ using Eq. (19).

Figure 13

The ${\frac{1}{2}}^{+}$ WSBG phase shifts generated by the $\mathrm{\Lambda }>385$ MeV/$c$ selected eigenstates $E_1$ (symbols). The solid curve depicts the phase shifts of Eq. (47) with parameters $E_b,c,d$, and $f$ obtained by the fit with this selection of eigenstates; the dashed curve is obtained by a numerical integration of the Schrödinger equation.

Figure 14

The lowest continuum ${\frac{1}{2}}^{+}$ WSBG eigenstates $E_1$ (symbols) and their $N_{max}\le 5$ selection (shaded area). See Fig. 4 for more details.

Figure 15

The ${\frac{1}{2}}^{+}$ WSBG phase shifts generated by the selected eigenstates $E_1$ obtained with $N_{max}\le 6$. See Fig. 13 for details.

Figure 16

The same as Fig. 5 but in a larger scale. The dashed line corresponds to the resonance energy $E_r$; the shaded area shows the resonance width.

Figure 17

The lowest ${}^5\mathrm{He} {\frac{3}{2}}^{-}$ eigenstates $E_0$ obtained by the NCSM with various $N_{max}$ (symbols) as functions of $\hslash \mathrm{\Omega }$. The shaded area shows the $E_0$ values selected for the SS-HORSE analysis according to inequality $\mathrm{\Lambda }>600$ MeV/$c$. Solid lines are solutions of Eq. (51) for energies $E_0$ with parameters $a,b$, and $d$ obtained by the fit.

Figure 18

The lowest ${}^5\mathrm{He} {\frac{3}{2}}^{-}$ eigenstates $E_0$ as a function of the scaling parameter $s$.

Figure 19

The ${\frac{3}{2}}^{-} n\alpha$ phase shifts obtained directly from the ${}^5\mathrm{He}$ eigenstates $E_0$ using Eq. (19) and the phase shift analysis of experimental data of Refs. [52] (stars).

Figure 20

The ${}^5\mathrm{He} {\frac{3}{2}}^{-}$ eigenstates $E_0$ selected according to $\mathrm{\Lambda }>600$ MeV/$c$ plotted as a function of the scaling parameter $s$ (symbols). See Fig. 5 for other details.

Figure 21

The ${\frac{3}{2}}^{-} n\alpha$ phase shifts obtained using Eq. (19) directly from ${}^5\mathrm{He}$ eigenstates $E_0$ selected according to $\mathrm{\Lambda }>600$ MeV/$c$ (symbols). The solid curve depicts the phase shifts of Eq. (48) with parameters $a,b$, and $d$ obtained by the fit with the respective selection of eigenstates; stars and the dashed curve depict the phase shift analysis of experimental data of Refs. [52] and the fit by Eq. (48).

Figure 22

The lowest ${}^5\mathrm{He} {\frac{3}{2}}^{-}$ eigenstates $E_0$ (symbols) and their manual selection (shaded area). See Fig. 17 for more details.

Figure 23

Manually selected ${}^5\mathrm{He} {\frac{3}{2\phantom{{}_q}}}^{-}$ eigenstates $E_0$ plotted as a function of the scaling parameter $s$ (symbols). See Fig. 5 for other details.

Figure 24

The ${\frac{3}{2\phantom{{}_q}}}^{-} n\alpha$ phase shifts generated by the manually selected ${}^5\mathrm{He}$ eigenstates $E_0$. See Fig. 21 for details.

Figure 25

The lowest ${}^5\mathrm{He} {\frac{3}{2}}^{-}$ eigenstates $E_0$ (symbols) and their $N_{max}\le 4$ selection (shaded area). See Fig. 17 for more details.

Figure 26

Selected lowest ${}^5\mathrm{He} {\frac{3}{2}}^{-}$ eigenstates $E_0$ obtained in NCSM with $N_{max}\le 4$ as a function of the scaling parameter $s$. See Fig. 5 for other details.

Figure 27

The ${\frac{3}{2\phantom{{}_q}}}^{-} n\alpha$ phase shifts generated by the selected ${}^5\mathrm{He}$ eigenstates $E_0$ obtained in NCSM with $N_{max}\le 4$. See Fig. 21 for details.

Figure 28

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{-}$ eigenstates $E_0$ (symbols) and their $\mathrm{\Lambda }>600$ MeV/$c$ selection (shaded area). See Fig. 17 for more details.

Figure 29

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{-}$ eigenstates $E_0$ as a function of the scaling parameter $s$.

Figure 30

The ${\frac{1}{2\phantom{{}_q}}}^{-} n\alpha$ phase shifts obtained directly from the ${}^5\mathrm{He}$ eigenstates $E_0$ using Eq. (19). See Fig. 19 for other details.

Figure 31

The ${}^5\mathrm{He} {\frac{1}{2}}^{-}$ eigenstates $E_0$ selected according to $\mathrm{\Lambda }>600$ MeV/$c$ as a function of the scaling parameter $s$. See Fig. 5 for other details.

Figure 32

The ${\frac{1}{2\phantom{{}_q}}}^{-} n\alpha$ phase shifts generated by the $\mathrm{\Lambda }>600$ MeV/$c$ selected ${}^5\mathrm{He}$ eigenstates $E_0$. See Fig. 21 for details.

Figure 33

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{-}$ eigenstates $E_0$ (symbols) and their manual selection (shaded area). See Fig. 17 for more details.

Figure 34

Manually selected ${}^5\mathrm{He} {\frac{1}{2\phantom{{}_q}}}^{-}$ eigenstates $E_0$ plotted as a function of the scaling parameter $s$ (symbols). See Fig. 5 for other details.

Figure 35

The ${\frac{1}{2\phantom{{}_q}}}^{-} n\alpha$ phase shifts generated by the manually selected ${}^5\mathrm{He}$ eigenstates $E_0$. See Fig. 21 for details.

Figure 36

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{-}$ eigenstates $E_0$ (symbols) and their $N_{max}\le 4$ selection (shaded area). See Fig. 17 for more details.

Figure 37

Selected lowest ${}^5\mathrm{He} {\frac{1}{2}}^{-}$ eigenstates $E_0$ obtained in NCSM with $N_{max}\le 4$ as a function of the scaling parameter $s$. See Fig. 5 for other details.

Figure 38

The ${\frac{1}{2\phantom{{}_q}}}^{-} n\alpha$ phase shifts generated by the selected ${}^5\mathrm{He}$ eigenstates $E_0$ obtained in NCSM with $N_{max}\le 4$. The dash-dotted curve depicts the phase shifts obtained by the fit to all manually selected eigenstates. See Fig. 21 for other details.

Figure 39

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{-}$ eigenstates $E_0$ (symbols) and their $4\le N_{max}\le 6$ selection (shaded area). See Fig. 17 for more details.

Figure 40

Selected lowest ${}^5\mathrm{He} {\frac{1}{2}}^{-}$ eigenstates $E_0$ obtained in NCSM with $4\le N_{max}\le 6$ as a function of the scaling parameter $s$. See Fig. 5 for other details.

Figure 41

The ${\frac{1}{2}}^{-} n\alpha$ phase shifts generated by the selected ${}^5\mathrm{He}$ eigenstates $E_0$ obtained in NCSM with $4\le N_{max}\le 6$. See Fig. 21 for details.

Figure 42

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{+}\phantom{{}_{q_{\int }}}$ eigenstates $E_0$ (symbols) and their $\mathrm{\Lambda }>600$ MeV/$c$ selection (shaded area). See Fig. 10 for more details.

Figure 43

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{+}$ eigenstates $E_0$ as a function of the scaling parameter $s$.

Figure 44

The ${\frac{1}{2\phantom{{}_q}}}^{+} n\alpha$ phase shifts obtained directly from the ${}^5\mathrm{He}$ eigenstates $E_0$ using Eq. (19). See Fig. 19 for other details.

Figure 45

The ${}^5\mathrm{He} {\frac{1}{2}}^{+}$ eigenstates $E_0$ selected according to $\mathrm{\Lambda }>600$ MeV/$c$ as a function of the scaling parameter $s$. See Fig. 11 for other details.

Figure 46

The ${\frac{1}{2}}^{+} n\alpha$ phase shifts generated by the $\mathrm{\Lambda }>600$ MeV/$c$ selected ${}^5\mathrm{He}$ eigenstates $E_0$ (symbols). The solid curve depicts the phase shifts of Eq. (47) with parameters $E_b,c,d$, and $f$ obtained by the fit with this selection of eigenstates; stars and the dashed curve depict the phase shift analysis of experimental data of Refs. [52] and the fit by Eq. (47).

Figure 47

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{+}$ eigenstates $E_0$ (symbols) and their manual selection (shaded area). See Fig. 10 for more details.

Figure 48

Manually selected ${}^5\mathrm{He} {\frac{1}{2\phantom{{}_q}}}^{+}$ eigenstates $E_0$ plotted as a function of the scaling parameter $s$ (symbols). See Fig. 11 for details.

Figure 49

The ${\frac{1}{2\phantom{{}_q}}}^{+} n\alpha$ phase shifts generated by the manually selected ${}^5\mathrm{He}$ eigenstates $E_0$. See Fig. 46 for details.

Figure 50

The lowest ${}^5\mathrm{He} {\frac{1}{2}}^{+}$ eigenstates $E_0$ (symbols) and their $5\le N_{max}\le 7$ selection (shaded area). See Fig. 10 for more details.

Figure 51

Selected lowest ${}^5\mathrm{He} {\frac{1}{2}}^{+}$ eigenstates $E_0$ obtained in NCSM with $5\le N_{max}\le 7$ as a function of the scaling parameter $s$. See Fig. 11 for details.

Figure 52

The ${\frac{1}{2\phantom{{}_q}}}^{+} n\alpha$ phase shifts generated by the selected ${}^5\mathrm{He}$ eigenstates $E_0$ obtained in NCSM with $5\le N_{max}\le 7$. See Fig. 46 for details.