Emergence of rotational bands in ab initio no-core configuration interaction calculations of the Be isotopes

The emergence of rotational bands is observed in no-core configuration interaction (NCCI) calculations for the $\mathrm{Be}$ isotopes ($7\le A\le 12$), as evidenced by rotational patterns for excitation energies, electromagnetic moments, and electromagnetic transitions. Yrast and low-lying excited bands are found. The results indicate well-developed rotational structure in NCCI calculations, using the JISP16 realistic nucleon-nucleon interaction within finite, computationally accessible configuration spaces.

Figure 1

Rotational predictions for (a) electric quadrupole moments and (b) electric quadrupole transition reduced matrix elements, within a rotational band, normalized to the intrinsic quadrupole moment $Q_0$, following from Eq. (4). Predictions are shown for bands with $0\le K\le 5/2$, as indicated. The staggering induced in transitions within a $K=1/2$ band, taking $\langle {\varphi }_K|Q_{2,2K}|{\varphi }_{\overline{K}}\rangle /\langle {\varphi }_K|Q_{2,0}|{\varphi }_K\rangle =+0.1$ for illustration, is indicated by the dotted lines.

Figure 2

Rotational predictions for each of the terms contributing, in Eqs. (10) and (11), to (a) magnetic dipole moments and (b) magnetic dipole transition reduced matrix elements, within a rotational band: the core rotor term for dipole moments (dashed line), direct term (solid lines, $K\ge 1/2$ only, as indicated), and cross term (dotted lines, $K=1/2$ only). For purposes of comparison, these terms are shown with equal coefficients, i.e., with $a_1=a_2=a_3={\mu }_N$. Predictions are shown for bands with $0\le K\le 5/2$.

Figure 3

Energy eigenvalues obtained for states in the natural (left) and unnatural (right) parity spaces of the odd-mass $\mathrm{Be}$ isotopes ($7\le A\le 11)$. Energies are plotted with respect to an angular momentum axis, which is scaled to be linear in $J(J+1)$, to facilitate identification of rotational energy patterns. Solid symbols indicate candidate yrast band members, and shaded symbols indicate candidate excited band members. The lines indicate the corresponding best fits for rotational energies (see text). Where quadrupole transition strengths indicate significant fragmentation, more than one state of a given $J$ may be indicated as a band member. The vertical dashed lines indicate the maximal angular momentum accessible within the lowest harmonic oscillator configuration (or valence space).

Figure 4

Energy eigenvalues obtained for states in the natural parity spaces of the even-mass $\mathrm{Be}$ isotopes ($8\le A\le 12)$. See Fig. 3 caption for discussion of the plot contents and labeling.

Figure 5

Dipole and quadrupole matrix element observables for the ${}^7\mathrm{Be}$ natural parity yrast band: (a) quadrupole ($E2$) moments, (b) quadrupole transition reduced matrix elements, (c) dipole ($M1$) moments, and (d) dipole transition reduced matrix elements. For quadrupole moments and matrix elements, values calculated using both proton and neutron operators are shown. Similarly, for dipole moments and matrix elements, values calculated using all four dipole terms (i.e., proton/neutron and orbital/spin contributions) are shown. Quadrupole moments and matrix elements are normalized to $Q_0$, obtained from the lowest band member's quadrupole moment. The curves indicate the rotational values for the quadrupole moments and matrix elements (normalized to $Q_0$) and for the dipole moments and matrix elements (using band best-fit parameters obtained from the dipole moments). The vertical dashed lines indicate the maximal angular momentum accessible within the lowest harmonic oscillator configuration (or valence space).

Figure 6

Dipole and quadrupole matrix element observables for the ${}^9\mathrm{Be}$ natural parity yrast band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 7

Dipole and quadrupole matrix element observables for the ${}^9\mathrm{Be}$ natural parity excited band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 8

Dipole and quadrupole matrix element observables for the ${}^9\mathrm{Be}$ unnatural parity yrast band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 9

Dipole and quadrupole matrix element observables for the ${}^{11}\mathrm{Be}$ natural parity yrast band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 10

Dipole and quadrupole matrix element observables for the ${}^{11}\mathrm{Be}$ natural parity excited band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 11

Dipole and quadrupole matrix element observables for the ${}^{11}\mathrm{Be}$ unnatural parity yrast band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 12

Dipole and quadrupole matrix element observables for the ${}^8\mathrm{Be}$ natural parity yrast band. See Fig. 5 caption for discussion of the plot contents and labeling. Proton and neutron observables are identical owing to isospin symmetry for ${}^8\mathrm{Be}$.

Figure 13

Dipole and quadrupole matrix element observables for the ${}^{10}\mathrm{Be}$ natural parity yrast band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 14

Dipole and quadrupole matrix element observables for the ${}^{10}\mathrm{Be}$ natural parity excited band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 15

Dipole and quadrupole matrix element observables for the ${}^{12}\mathrm{Be}$ natural parity excited band. See Fig. 5 caption for discussion of the plot contents and labeling.

Figure 16

Dependence of calculated energies for ${}^9\mathrm{Be}$ on the basis truncation $N_{\max }$: absolute energies $E$ (top) and excitation energies $E_x$ (bottom). Low-lying states in the natural (left) and unnatural (right) parity spaces are shown. Calculations are for $6\le N_{\max }\le 10$ for natural parity or $7\le N_{\max }\le 11$ for unnatural parity (calculations of increasing $N_{\max }$ are indicated by pluses, crosses, and diamonds, respectively). Excitation energies are taken separately for each parity. See Fig. 3 caption for discussion of other aspects of the plot contents and labeling.

Figure 17

Dependence of calculated electromagnetic moments for ${}^9\mathrm{Be}$ on the basis truncation $N_{\max }$: quadrupole moments (top) and dipole moments (bottom) of the natural (left) and unnatural (right) parity yrast bands. Calculations are for $6\le N_{\max }\le 10$ for natural parity, or $7\le N_{\max }\le 11$ for unnatural parity (calculations of increasing $N_{\max }$ are indicated by pluses, crosses, and circles, respectively). See Fig. 5 caption for discussion of other aspects of the plot contents and labeling.

Figure 18

Intrinsic quadrupole moments plotted as $e{Q}_{0,p}$ vs $e{Q}_{0,n}$ for all bands considered in the present work. Bands are distinguished as natural parity yrast (circles, solid), natural parity excited (circles, shaded), and unnatural parity yrast (diamonds). Data points are labeled by the mass number of the $\mathrm{Be}$ isotope. Values are shown for successive $N_{\max }$ values ($6\le N_{\max }\le 10$ for natural parity or $7\le N_{\max }\le 11$ for unnatural parity) to provide an indication of convergence (the larger symbols indicate higher $N_{\max }$ values) and of the stability of the ratio $Q_{0,p}/Q_{0,n}$. The line $Q_{0,p}/Q_{0,n}=1$ is marked (dashed diagonal line), as is the range of Weisskopf estimates as $A$ ranges from 7 to 12 (shaded bands).

Figure 19

Band energy parameters for all bands considered in the present work, for the odd-mass $\mathrm{Be}$ isotopes (left) and even-mass $\mathrm{Be}$ isotopes (right). Parameters considered are the rotational energy constant $A$ (top), the Coriolis decoupling parameter $a$ (middle), and the band excitation energy $E_x$ (bottom), which is defined as the band energy $E_0$ for the given band [see Eqs. (2) and (3)] relative to that of the natural parity yrast band. Bands are distinguished as natural parity yrast (solid circles), natural parity excited (shaded circles), and unnatural parity yrast (diamonds). Values are shown for successive $N_{\max }$ values ($6\le N_{\max }\le 10$ for natural parity or $7\le N_{\max }\le 11$ for unnatural parity) to provide an indication of convergence (the larger symbols indicate higher $N_{\max }$ values). The parameter values extracted from experimental bands are indicated by horizontal lines. The $K^P$ assignments for the bands are indicated at bottom.