Spin-orbit decomposition of ab initio nuclear wave functions

Although the modern shell-model picture of atomic nuclei is built from single-particle orbits with good total angular momentum $j$, leading to $j-j$ coupling, decades ago phenomenological models suggested that a simpler picture for $0p$-shell nuclides can be realized via coupling of the total spin $S$ and total orbital angular momentum $L$. I revisit this idea with large-basis, no-core shell-model calculations using modern ab initio two-body interactions and dissect the resulting wave functions into their component $L$- and $S$-components. Remarkably, there is broad agreement with calculations using the phenomenological Cohen-Kurath forces, despite a gap of nearly 50 years and six orders of magnitude in basis dimensions. I suggest that $L-S$ decomposition may be a useful tool for analyzing ab initio wave functions of light nuclei, for example, in the case of rotational bands.

Figure 1

Low-lying excitation spectrum of ${}^{11}\mathrm{B},$ comparing experiment, the Cohen-Kurath interaction, and the no-core shell model (NCSM) using a chiral two-body force evolved via SRG to $\lambda =2.0{\mathrm{fm}}^{-1}$. All states have $T=1/2$.

Figure 2

Decomposition of low-lying states of ${}^{11}\mathrm{B}$ into components of good $L$ (total orbital angular momentum), comparing wave functions computed from the Cohen-Kurath interaction (gray bars) and from the NCSM [hatched (red) bars]. All states have $T=1/2$.

Figure 3

Decomposition of low-lying states of ${}^{11}\mathrm{B}$ into components of good $S$ (total spin), comparing wave functions computed from the Cohen-Kurath interaction (gray bars) and from the NCSM [hatched (red) bars]. All states have $T=1/2$.

Figure 4

Low-lying excitation spectrum of ${}^{10}\mathrm{B},$ comparing experiment, the Cohen-Kurath interaction, and the no-core shell model (NCSM) using a chiral two-body force evolved via SRG to $\lambda =2.0{\mathrm{fm}}^{-1}$, with the harmonic oscillator basis frequency $\hslash \Omega =22$ MeV, for $N_{\max }=6$ and 8.

Figure 5

Decomposition of low-lying states of ${}^{10}\mathrm{B}$ into components of good $L$ (total orbital angular momentum), comparing wave functions computed from the Cohen-Kurath interaction (gray bars) and from the NCSM for both $N_{\max }=6$ [hatched (red) bars] and $N_{\max }=8$ [checked (blue) bars].

Figure 6

Low-lying excitation spectrum of ${}^{12}\mathrm{C},$ comparing experiment, the Cohen-Kurath interaction, and the no-core shell model (NCSM) using a chiral two-body force evolved via SRG to $\lambda =2.0{\mathrm{fm}}^{-1}$, with a harmonic oscillator basis frequency of $\hslash \Omega =22$ MeV.

Figure 7

Decomposition of low-lying states of ${}^{12}\mathrm{C}$ into components of good $L$ (total orbital angular momentum), comparing wave functions computed from the Cohen-Kurath interaction (gray bars) and from the NCSM [hatched (red) bars].

Figure 8

Decomposition of low-lying states of ${}^{12}\mathrm{C}$ into components of good $S$ (total intrinsic spin), comparing wave functions computed from the Cohen-Kurath interaction (gray bars) and from the NCSM [hatched (red) bars].

Figure 9

Low-lying excitation spectrum for ${}^9\mathrm{Be},$ plotting excitation energy versus $J(J+1)$ to better illustrate rotational bands. Shown are experimental data [(blue) diamonds] [48], phenomenological Cohen-Kurath calculations [(red) squares], and NCSM calculations (black circles); the latter were calculated with an SRG evolution parameter $\lambda =2{\mathrm{fm}}^{-1}$ and a harmonic oscillator basis frequency of $\Omega =20$ MeV. As a guide for the eye, I have added solid lines for the ground-state NCSM band and dashed lines for the excited-state NCSM band.

Figure 10

Decomposition of the ground-state rotation band for ${}^9\mathrm{Be},$ for NCSM calculation (a, b), with the same parameters as in Fig. 9, and for Cohen-Kurath calculation (c, d). In $L$-decomposition plots (a, c), for each of the members of the rotational band $3/2_1^{-},5/2_1^{-},7/2_1^{-}$, and $9/2_1^{-}$, I give the fraction of the wave function with $L=1$ [(red) circles], $L=2$ [(blue) squares], $L=3$ [(green) diamonds], and $L=4$ [(violet) triangles]. In $S$-decomposition plots (b, d), I give the fraction of the wave function with $S=1/2$ (solid black line), $S=3/2$ [dashed (red) line], and $S=5/2$ [dotted (blue) line].

Figure 11

Decomposition of the excited-state rotation band for ${}^9\mathrm{Be},$ for NCSM calculation (a, b), with the same parameters as in Fig. 9, and for Cohen-Kurath calculation (c, d). In $L$-decomposition plots (a, c), for each of the members of the rotational band $1/2_1^{-},3/2_2^{-},5/2_2^{-}$, and $7/2_3^{-}$, I give the fraction of the wave function with $L=1$ [(red) circles], $L=2$ [(blue) squares], $L=3$ [(green) diamonds], and $L=4$ [(violet) triangles]. In $S$-decomposition plots (b, d), I give the fraction of the wave function with $S=1/2$ (solid black line), $S=3/2$ [dashed (red) line], and $S=5/2$ [dotted (blue) line].

Figure 12

$L$-decomposition for selected ${}^{12}\mathrm{C}$ NCSM states as a function of the harmonic oscillator basis frequency $\Omega$. The states are (a) $0;0_1$, (b) $1;0_1$, (c) $0;0_2$, and (d) $1;1_1$. Shown are the fraction of the wave functions for $L=0$ (black circles), $L=1$ [(red) squares], and $L=2$ [(blue) diamonds].

Figure 13

$L$-decomposition for selected ${}^{12}\mathrm{C}$ NCSM states as a function of the SRG evolution parameter $\lambda$. The states are (a) $0;0_1$, (b) $1;0_1$, (c) $0;0_2$, and (d) $1;1_1$. Shown are the fraction of the wave functions for $L=0$ (black circles), $L=1$ [(red) squares], $L=2$ [(blue) diamonds], and $L=3$ [(violet) triangles].

Figure 14

$L$-decomposition for selected ${}^7\mathrm{Li}$ NCSM states as a function of the model-space truncation $N_{\max }$, with the SRG evolution parameter $\lambda$ fixed at 2 ${\mathrm{fm}}^{-1}$ and the harmonic oscillator basis frequency fixed at 22 MeV. All states have $T=1/2$. The states are (a) $7/2_1^{-}$, (b) $5/2_1^{-}$, (c) $7/2_2^{-}$, and (d) $5/2_2^{-}$. Shown are the fraction of the wave functions for $L=1$ [(red) squares], $L=2$ [(blue) diamonds], $L=3$ [(violet) upward triangles], and $L=4$ (black downward triangles).