Isoscalar dipole excitations in $^{16}$O

TL;DR

This paper investigates isoscalar monopole and dipole excitations in $^{16}$O using advanced nuclear modeling, revealing significant low-energy transition strengths and novel dipole modes related to cluster structures, aligning well with experimental data.

Contribution

The study introduces a detailed analysis of low-energy isoscalar dipole modes in $^{16}$O, highlighting the coexistence of compressive and vortical modes with cluster structures, using the shifted basis antisymmetrized molecular dynamics method.

Findings

Significant low-energy IS monopole and dipole transition strengths below giant resonances.

Identification of vortical dipole mode in the $1^-_1$ state.

The $1^-_1$ state accounts for 5 ext% of the energy-weighted sum rule, matching experimental observations.

Abstract

Isoscalar (IS) monopole and dipole excitations in $^{16}$O were investigated by the method of shifted basis antisymmetrized molecular dynamics combined with the generator coordinate method. Significant strengths of the IS monopole and dipole transitions were obtained in the low-energy region below the giant resonances. In addition to the compressive mode, which mainly contributes to the high-energy strengths for the IS dipole giant resonance, we obtained a variety of low-energy dipole modes such as the vortical dipole mode in the $1^-_1$ state of the vibrating tetrahedral $4\alpha$ and the $^{12}$C+$\alpha$ cluster structure in the $1^-_2$ state. The $1^-_1$ state contributes to the significant low-energy strength of the IS dipole transition as 5\% of the energy-weighted sum rule, which describes well the experimental data observed by the $\alpha$ inelastic scattering.

Tables

Table 1. Table 1: Properties of 0 + superscript 0 0^{+} states; the binding energy (B.E.), excitation energies ( E x subscript 𝐸 𝑥 E_{x} ), rms matter radii ( R 𝑅 R ), and the IS0 matrix elements ( M ​ ( E ​ 0 ) 𝑀 𝐸 0 M(E0) ). The present result (sAMD+GCM) and the VAP and VAP+GCM values from Ref.  Kanada-Enyo:2017ers are shown compared with the experimental data Tilley:1993zz . The experimental value of the rms radius of the ground state is deduced from the experimental charge radius measured by the electron scattering Angeli2013 .

	VAP	VAP	sAMD	exp
		+GCM	+GCM
B.E. (MeV)	123.0	123.5	125.6	127.62
$E_x\left(0_2^{+}\right)\text{(MeV)}$	13.1	9.7	11.6	6.05
$E_x\left(0_3^{+}\right)\text{(MeV)}$		15.3	18.6	12.05
$E_x\left(0_4^{+}\right)\text{(MeV)}$		13.6	15.5	13.6
$E_x\left(0_5^{+}\right)\text{(MeV)}$		18.3	20.6	14.01
$R\left(0_1^{+}\right)\text{(fm)}$	2.69	2.73	2.72	2.55
$R\left(0_2^{+}\right)\text{(fm)}$	2.96	3.29	3.16
$R\left(0_3^{+}\right)\text{(fm)}$		3.53	3.45
$R\left(0_4^{+}\right)\text{(fm)}$		3.64	3.21
$R\left(0_5^{+}\right)\text{(fm)}$		3.53	3.36
$M\left(E0;0_1^{+}\to 0_2^{+}\right)$ ($e$ fm2)	1.8	3.5	3.8	3.55(0.21)
$M\left(E0;0_1^{+}\to 0_3^{+}\right)$ ($e$ fm2)		3.3	3.9	4.03(0.09)
$M\left(E0;0_1^{+}\to 0_4^{+}\right)$ ($e$ fm2)		4.1	4.1
$M\left(E0;0_1^{+}\to 0_5^{+}\right)$ ($e$ fm2)		3.0	3.2	3.3(0.7)

Table 2. Table 2: Properties of the 1 1 − subscript superscript 1 1 1^{-}_{1} and 1 2 − subscript superscript 1 2 1^{-}_{2} states, excitation energies, rms radii, the IS1 strengths, and the EWSR ratio P IS1 subscript 𝑃 IS1 P_{\textrm{IS1}} of the energy-weighted IS1 strengths. The present result of the sAMD+GCM and those of the VAP and VAP+GCM calculations from Ref.  Kanada-Enyo:2017ers are shown compared with the experimental data Tilley:1993zz . The experimental data of the EWSR ratio P IS1 ​ ( 1 1 − ) subscript 𝑃 IS1 subscript superscript 1 1 P_{\textrm{IS1}}(1^{-}_{1}) is the value from Ref.  Harakeh:1981zz of α 𝛼 \alpha inelastic scattering analysis.

	VAP	VAP	sAMD	exp
		+GCM	+GCM
$E_x\left(1_1^{-}\right)\text{(MeV)}$	10.3	9.4	9.6	7.12
$E_x\left(1_2^{-}\right)\text{(MeV)}$	17.0	12.1	14.4	9.59
$R\left(1_1^{-}\right)\text{(fm)}$	2.76	2.87	2.80
$R\left(1_2^{-}\right)\text{(fm)}$	2.96	3.58	3.37
$B\left(\text{IS1};0_1^{+}\to 1_1^{-}\right)$	124.5	165.5	169.8
$P_{\text{IS1}}\left(1_1^{-}\right)$	0.042	0.048	0.051	0.42a
$B\left(\text{IS1};0_1^{+}\to 1_2^{-}\right)$	7.9	2.9	10.2
$P_{\text{IS1}}\left(1_2^{-}\right)$	0.0044	0.0011	0.0045

Equations

$${\Phi }_{\mathrm{AMD}}(\mathbf{Z})$$

$${\phi }_i$$

$${\varphi }_{{\mathbf{X}}_i}({\mathbf{r}}_j)$$

$${\chi }_i$$

$$\delta \frac{⟨\Phi |H|\Phi ⟩}{⟨\Phi |\Phi ⟩}=0,$$

$$\rho (\mathbf{r})$$

$$\mathbf{j}(\mathbf{r})$$

$${\rho }_{0\to f}^{\text{(tr)}}(\mathbf{r})=⟨f|\rho (\mathbf{r})|0⟩,$$

$$\delta \mathbf{j}(\mathbf{r})=⟨f|\mathbf{j}(\mathbf{r})|0⟩.$$

$${\rho }_{0\to f}^{\text{(tr)}}(\mathbf{r})$$

$$\times \sum _{\mu }{Y}_{\lambda \mu }^{\ast }(\hat{\mathbf{r}})(J_i{M}_i\lambda \mu |J_f{M}_f),$$

$$F(q)=\frac{\sqrt{4\pi }}{Z}\frac{1}{\sqrt{2J_i+1}}\int dr{r}^2{j}_{\lambda }(qr){\rho }_{\lambda ;0\to f}^{\text{ch}}(r),$$

$$M_{\text{IS0}}\equiv$$

$$M_{\text{IS1}}(\mu )\equiv$$

$$B(\text{IS}\lambda )=\frac{1}{2J_i+1}{\left|⟨J_k^{\pi }||M_{\text{IS}\lambda }||0_1^{+}⟩\right|}^2,$$

$$⟨J_k^{\pi }||M_{\text{IS}\lambda }||0_1^{+}⟩=\sqrt{4\pi }\int dr{r}^2{r}^{\lambda +2}{\rho }_{\lambda ;0\to f}^{\text{(tr)}}(\mathbf{r})$$

$$⟨J_k^{\pi }||M_{\text{IS}\lambda }||0_1^{+}⟩=\int dr{r}^2{r}^{\lambda +2}{\rho }_{\lambda ;0\to f}^{\text{(tr)}}(\mathbf{r})$$

$$\sum _k(E_k-E_0)B(\text{IS0};0_1^{+}\to 0_k^{+})=\frac{2{\hslash }^{2}A}{m}⟨r^2⟩$$

$$\sum _k(E_k-E_0)B(\text{IS1};0_1^{+}\to 1_k^{-})$$

$$=\frac{3{\hslash }^{2}A}{32m\pi }\left(11⟨r^4⟩-\frac{25}{3}⟨r^2{⟩}^2-10ϵ⟨r^2⟩\right),$$

$$M_{\text{CD}}(\mu )$$

$$M_{\text{TD}}(\mu )$$

$$⟨1_k^{-}|M_{\text{CD}}(\mu )|0_1^{+}⟩=$$

$$⟨1_k^{-}|M_{\text{CD}}(\mu )|0_1^{+}⟩$$

$$\tilde{B}(\text{CD,TD};0_1^{+}\to 1_k^{-})\equiv {\left(\frac{10\hslash c}{E}\right)}^2{\left|⟨1_k^{-}||M_{\text{CD,TD}}||0_1^{+}⟩\right|}^2.$$

Full text

††preprint: KUNS-2751

I Introduction

In the past decades, low-energy monopole and dipole excitations have been attracting great interests (see, for example, reviews in Refs. Harakeh-textbook ; Paar:2007bk ; aumann-rev ; Savran:2013bha ; Bracco:2015hca and references therein). A central issue is possible appearance of new excitation modes decoupled from collective vibration modes corresponding the giant resonances (GR). In experiments with $\alpha$ inelastic scattering extensively performed for study of isoscalar (IS) monopole and dipole excitations, significant low-energy strengths with the fraction of several percentages of the energy weighted sum rule have been observed in various stable nuclei such as 16O, 40Ca, and 208Pb Harakeh:1981zz ; Decowski:1981pcz ; Poelhekken:1992gvp . The questions to be answered are what is the origin of these IS low-energy dipole (LED) strengths and how the dipole modes come down to the energy much lower than the IS giant dipole resonances (GDR).

In order to understand the IS LED strengths, the vortical dipole (VD) mode (called also the torus or toroidal mode) has been studied firstly with hydrodynamical models semenko81 ; Ravenhall:1987thb , and later with microscopic approaches Paar:2007bk ; Vretenar:2001te ; Ryezayeva:2002zz ; Papakonstantinou:2010ja ; Kvasil:2011yk ; Repko:2012rj ; Kvasil:2013yca ; Nesterenko:2016qiw ; Nesterenko:2017rcc . The VD mode is characterized by the nuclear vorticity and has a unique feature different from the standard IS dipole mode so-called compressive dipole (CD) in the IS GDR. Since the nuclear density is conserved in the VD mode, its energy can be lower than the IS GDR involving compression of nuclear density. As a measure of the nuclear vorticity in the dipole excitations, the toroidal dipole (TD) operator has been introduced semenko81 ; Dubovik75 . The TD operator is given by the rotational component (a curl term) of the transition current density and the counter part of the compressive dipole (CD) operator with the irrotational component (a divergence term) of the transition current density, and has been proved to be a good probe for the low-energy VD mode Kvasil:2011yk .

In light nuclei, also cluster states may contribute to the low-energy IS monopole (IS0) and dipole (IS1) transition strengths because the IS0 and IS1 operators contain higher order $r^{\lambda+2}$ terms and can excite not only the compressive vibration modes but also the inter-cluster motion in the cluster states as pointed out by Yamada et al. Yamada:2011ri and Chiba et al. Chiba:2015khu . Indeed, the low-energy IS monopole strengths in 16O have been described well by cluster states with a semi-microscopic $4\alpha$-cluster model Yamada:2011ri . It is an important issue to clarify the IS dipole excitations in 16O, in particular, the cluster and vortcal aspects of the low-energy modes.

Theoretical calculations with cluster models have been performed for 16O and suggested a variety of cluster structures such as the tetrahedral $4\alpha$ and 12C+$\alpha$ structures Yamada:2011ri ; wheeler37 ; dennison54 ; brink70 ; Suzuki:1976zz ; Suzuki:1976zz2 ; fujiwara80 ; Libert-Heinemann:1980ktg ; bauhoff84 ; Descouvemont:1987uvu ; Descouvemont:1991zz ; Descouvemont:1993zza ; Fukatsu92 ; Funaki:2008gb ; Funaki:2010px ; Kanada-En'yo:2013dma ; Horiuchi:2014yua . However, there have been no microscopic calculation that successfully describes the energy spectra of 16O. Recently, we applied a microscopic model of the antisymmetrized molecular dynamics (AMD) KanadaEnyo:1995tb ; KanadaEnyo:1995ir ; KanadaEn'yo:2001qw ; KanadaEn'yo:2012bj to 16O, and obtained reasonable reproduction of the energy spectra of 16O such as $0^{+}_{2}$, $2^{+}_{1}$, $4^{+}_{1}$, $1^{-}_{2}$, and $3^{-}_{2}$ states in the positive- and negative-parity bands with the 12C+$\alpha$ structure and $3^{-}_{1}$ and $4^{+}_{2}$ states in the ground band with the tetrahedral $4\alpha$ structure Kanada-En'yo:2013dma ; Kanada-Enyo:2017ers .

Our aim is to investigate the IS dipole excitations in 16O. Main interest are properties of the IS LED modes such as the cluster and vortical aspects. For this aim, we apply the method of the shifted basis AMD (sAMD) Kanada-Enyo:2015knx ; Kanada-Enyo:2015vwc ; Kimura:2016heo combined with the cluster generator coordinate method (GCM). The sAMD+GCM has been recently constructed to describe both the single-particle excitation and large amplitude cluster mode. This method has been applied to 12C to discuss the cluster, vortical, and compressive IS dipole modes, and proved to be a powerful approach for the IS monopole and dipole excitations in a wide energy range including the low-energy states and high-energy GRs. Kanada-Enyo:2015vwc ; Kanada-Enyo:2017fps .

In our previous work of 16OKanada-Enyo:2017ers , we have investigated the cluster states with variation after spin-parity projections (VAP) Kanada-Enyo:1998onp combined with the 12C+$\alpha$-cluster GCM, which we called the VAP+GCM, but not the IS GDR because the sAMD bases have not been adopted in the previous work. The great advantages of the present sAMD+GCM are that it describes both the low-energy cluster state and the GDR in a unified framework owing to inclusion of one-particle and one-hole (1p-1h) excitations in the sAMD bases, and is suitable to discuss details of the IS dipole excitations. In this paper, we show the IS monopole and dipole strength functions in 16O in a wide energy range covering the low-lying vortical and cluster modes, and also the high-energy compressive vibration modes of the GRs. For detailed analysis of the monopole and dipole transitions, we calculate the form factors and transition densities and compare them with experimental data measured by the electron scattering. We discuss the vortical and cluster aspectes of the IS LED states and clarify properties of the IS dipole excitations.

The paper is organized as follows. The formulation of the sAMD+GCM for 16O is explained in Sec. II. Section III shows the calculated results and discusses the properties of the IS monopole and dipole modes. Finally, the paper is summarized in section IV. In appendix sections, the definitions of the transition operators, densities, and strengths are given.

II Formulation

In order to calculate the IS monopole and dipole excitations in 16O, we combine the sAMD with the previous VAP+GCM model Kanada-Enyo:2017ers . Namely, we prepare the sAMD wave functions and combine them with the the basis wave functions adopted in the previous VAP+GCM calculation. We call the present calculation “sAMD+GCM”. In this section, we explain the framework and procedure of the present calculations of 16O. For details of the VAP+GCM and the sAMD, the reader is referred to Refs. Kanada-Enyo:2015vwc ; Kanada-Enyo:2017fps ; Kanada-Enyo:2017ers ; Kanada-Enyo:2017uzz and references therein.

II.1 VAP+GCM with AMD wave functions

An AMD wave function is given by a Slater determinant of single-particle Gaussian wave functions,

[TABLE]

where ${\cal{A}}$ is the antisymmetrizer, $\phi_{{\boldsymbol{X}}_{i}}$, $\chi_{i}$, and $\tau_{i}$ are the spatial, spin, and isospin functions of the $i$th single-particle wave function, respectively. The isospin part is fixed to be up (proton) or down (neutron). $\nu$ is the width parameter, which is fixed to be $\nu=0.19$ fm*-2* used in the previous calculation. The condition $\sum_{i=1,\ldots,A}\boldsymbol{X}_{i}/A=0$ is always kept and the contribution of the center of mass motion is exactly removed from the total system. The AMD wave function is specified by the set of variational parameters ${\boldsymbol{Z}}\equiv\{{\boldsymbol{X}}_{1},\ldots,{\boldsymbol{X}}_{A},\xi_{1},\ldots,\xi_{A}\}$ for the centroids of single-nucleon Gaussian wave packets and nucleon-spin orientations, which are determined by the energy variation.

It should be stressed that, in the AMD model, the existence of any clusters is not a priori assumed because Gaussian centroids, ${\boldsymbol{X}}_{1},\ldots,{\boldsymbol{X}}_{A}$, of all single-nucleon wave packets are independently treated as variational parameters. Nevertheless, the model wave function can describe various cluster wave functions, and also shell-model wave functions because of the antisymmetrization of Gaussian wave packets.

To obtain the AMD wave function optimized for the $J^{\pi}$ state, the VAP is performed with respect to the variation of $\boldsymbol{Z}$ by

[TABLE]

for the $J^{\pi}$-projected AMD wave function $\Phi=P^{J\pi}_{MK}\Phi_{\textrm{AMD}}(\boldsymbol{Z})$, where $P^{J\pi}_{MK}$ is the spin-parity projection operator. For the AMD wave function $\Phi^{{}^{16}\textrm{O}}_{\textrm{AMD}}(\boldsymbol{Z})$ of 16O, we perform the VAP with $J^{\pi}_{k}=0^{+}_{1,2}$, $2^{+}_{1}$, $4^{+}_{1,2}$, $1^{-}_{1}$, $2^{-}_{1}$, $3^{-}_{1}$, and $5^{-}_{1}$, and obtain nine configurations of $\Phi^{{}^{16}\textrm{O}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta})$ with the parameters $\boldsymbol{Z}^{\rm opt}_{\beta}$ optimized for each $\beta=J^{\pi}_{k}$ state. In the simple VAP calculation, we superpose the nine configurations.

In the GCM calculation, we adopt the 12C+$\alpha$ cluster wave functions, where the angular momentum projection and internal excitations of the sub system 12C-cluster are considered. We first perform the VAP calculation of the subsystem 12C for three states 12C($0^{+}_{1})$, 12C($0^{+}_{2})$, and 12C($1^{-}_{1})$. Using the obtained 12C-cluster wave functions, the 12C+$\alpha$ wave function is constructed as done in Ref. Kanada-Enyo:2017ers . The relative distance $d$ between 12C and $\alpha$ clusters is treated as a generator coordinate. The angular-momentum projection of the subsystem 12C is also practically performed by taking into account rotation of the 12C-cluster.

II.2 sAMD+GCM: combination of sAMD with VAP+GCM

In addition to the VAP and 12C+$\alpha$ wave functions, the sAMD wave functions are also superposed to describe 1p-1h excitations on the ground state. Starting from the ground state wave function $\Phi^{{}^{16}\textrm{O}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=0^{+}_{1}})$ obtained by the VAP, we consider small variations of single-particle wave functions by shifting the Gaussian centroid of each single-particle wave function, ${\boldsymbol{X}}_{i}\rightarrow{\boldsymbol{X}}_{i}+\epsilon{\boldsymbol{e}}_{\sigma}$ (the spatial position parameters), of $\boldsymbol{Z}^{\rm opt}_{\beta=0^{+}_{1}}$ in the AMD wave function. Here $\epsilon$ is an enough small constant, ${\boldsymbol{e}}_{\sigma}$ ($\sigma=1,\ldots,8$) are unit vectors for 8 directions. Spin non-flip and flip states and recoil effects are taken into account as explained in Ref. Kanada-Enyo:2015vwc . Consequently, totally $16A=256$ bases of the shifted AMD wave functions are superposed in addition to the VAP and 12C+$\alpha$ wave functions in the sAMD+GCM calculation of $0^{+}$ and $1^{-}$ states.

In the present sAMD+GCM calculation, we use the $\Phi^{{}^{12}\textrm{C}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=0^{+}_{1}})$+$\alpha$ configuration with the inter-cluster distances of $d=\{1.2,2.4,\ldots,7.2$ fm} and $\Phi^{{}^{12}\textrm{C}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=0^{+}_{2},1^{-}_{1}})$+$\alpha$ configurations with $d=\{1.2,2.4,\ldots,4.8$ fm} to save the computational cost. ($\Phi^{{}^{12}\textrm{C}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=0^{+}_{1},0^{+}_{2},1^{-}_{1}})$+$\alpha$ with $d=\{1.2,2.4,\ldots,8.4$ fm} are used in Ref. Kanada-En'yo:2013dma , and $\Phi^{{}^{12}\textrm{C}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=0^{+}_{1},0^{+}_{2}})$+$\alpha$ with $d=\{1.2,2.4,\ldots,8.4$ fm} and $\Phi^{{}^{12}\textrm{C}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=1^{-}_{1}})$+$\alpha$ with $d=\{1.2,2.4,\ldots,4.8$ fm} are used in Ref. Kanada-Enyo:2017ers .)

The IS0 and IS1 transition strengths are calculated with the $0^{+}$ and $1^{-}$ states obtained by the sAMD+GCM. The form factors and transition densities are also calculated with these operators. As for the IS dipole excitations, transition strengths of the CD and TD operators are also calculated. The definitions of the operators, matrix elements, strengths, form factors, and transition densities are given in appendixes.

III Results

III.1 Structure properties of low-energy levels of $0^{+}$ and $1^{-}$ states

The sAMD+GCM result of the binding energy, root-mean-square (rms) matter radii, and excitation energies of low-lying $0^{+}$ states are listed in Table 1, and those of the $1^{-}_{1}$ and $1^{-}_{2}$ states are shown in Table 2. For comparison, values calculated with the VAP (without the 12C+$\alpha$ nor sAMD bases) and those of the VAP+GCM (without the sAMD bases) are also shown in the tables. These corresponds to the VAP and VAP+GCM calculations presented in the previous paper Kanada-Enyo:2017ers .

Various cluster states are obtained in the excited $0^{+}$ levels in $E\lesssim 20$ MeV. Compared the sAMD+GCM and VAP+GCM, there is no essential difference between the two calculations for these states, because the developed cluster states are dominantly contributed by the GCM bases but not by the sAMD bases. It is not the case for the ground state, but the sAMD+GCM obtains 2 MeV energy gain of the $0^{+}_{1}$ state compared with the VAP+GCM meaning that the sAMD bases efficiently improve the ground state correlations. Because of this additional energy gain of the ground state, the relative energy position of the excited $0^{+}$ states are raised up by about 2 MeV in the sAMD+GCM. As a result, the agreement with the experimental energy spectra in the sAMD+GCM is not as good as the VAP+GCM, but it is much better than the preceding microscopic cluster model calculations. We note that, the calculated fourth $0^{+}$ state with the 12C$(2^{+}_{1})$+$\alpha$ cluster structure should be assigned to the experimental $0^{+}_{3}$ state, because the sAMD+GCM and VAP+GCM calculations eventually give the opposite ordering of the $0^{+}_{3}$ and $0^{+}_{4}$ states as discussed in the previous paper.

In the calculated $1^{-}$ levels, the $1^{-}_{1}$ and $1^{-}_{2}$ states are obtained in $E<15$ MeV. The higher state ($1^{-}_{2}$) is the well developed cluster state and regarded as the band-head state of the $K^{\pi}=0^{-}$ 12C+$\alpha$ band, which is the parity doublet of the $K^{\pi}=0^{+}_{2}$ 12C+$\alpha$ band built on the band-head $0^{+}_{2}$ state. The lower state ($1^{-}_{1}$) has the small rms radius comparable to that of the ground state and shows less prominent cluster structure than the $1^{-}_{2}$ and $0^{+}_{2,3,4,5}$ states. Comparing with the VAP+GCM, the sAMD+GCM gives the smaller radius of the $1^{-}_{1}$ state. Moreover, the excitation energy of the $1^{-}_{1}$ state is almost same between the sAMD+GCM and VAP+GCM calculations indicating that the sAMD bases describe additional correlations contributing the size shrinkage and the 2 MeV energy gain comparable to that of the ground state.

III.2 Cluster structures of low-lying states

Cluster aspects of the low-lying states have been investigated in Ref. Kanada-Enyo:2017ers . We here briefly review the cluster structures of the $0^{+}_{1}$, $0^{+}_{2}$, $1^{-}_{1}$, and $1^{-}_{2}$ states following the discussions in the previous paper based on the analysis of the intrinsic wave functions, $\Phi^{{}^{16}\textrm{O}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=J^{\pi}_{k}})$, obtained by the VAP calculation.

Figure 1 shows the intrinsic density distribution of the $0^{+}_{1}$, $0^{+}_{2}$, and $1^{-}_{2}$ states. The $0^{+}_{1}$ state shows the tetrahedral $4\alpha$ cluster structure, in which three $\alpha$s form the triangle shape on the $X$-$Y$ plane and the last $\alpha$ cluster is sitting on the $Z$(vertical) axis (Fig. 1(a)). Its cluster development is not so remarkable as seen in the compact density distribution. The $1^{-}_{1}$ state also has a tetrahedral $4\alpha$ clustering with a compact density distribution similar to the $0^{+}_{1}$ state, but the orientation of the triangle $3\alpha$ part is somewhat tilted from the $0^{+}_{1}$. This tilting motion of the triangle $3\alpha$ produces the dipole excitation with $K^{\pi}=1^{-}$ in the $1^{-}_{1}$. This mode is similar to the vibration mode of the tetrahedral $4\alpha$ discussed by the algebraic $4\alpha$ cluster model Bijker:2014tka ; Bijker:2016bpb . However, the $0^{+}_{1}$ and $1^{-}_{1}$ states obtained in the present calculation are not the equilateral tetrahedral states but the prolately deformed one with the $3\alpha+\alpha$ configuration and contain the $\alpha$ breaking component.

The $0^{+}_{2}$ state has the developed 12C+$\alpha$ cluster structure, in which 4$\alpha$ clusters are arranged in a planar-like configuration. Because of the remarkably developed 12C+$\alpha$ clustering, the $0^{+}_{2}$ state shows a largely deformed intrinsic density compared with the $0^{+}_{1}$. The developed 12C+$\alpha$ clustering constructs the $K^{\pi}=0^{+}$ band and the parity-partner $K^{\pi}=0^{-}$ band starting from the the band-head $1^{-}_{2}$ state.

We should note that, even though the $0^{+}_{1}$, $0^{+}_{2}$, $1^{-}_{1}$, and $1^{-}_{2}$ show the formation of four $\alpha$ clusters, the clusters are not necessarily the ideal $\alpha$ clusters with the $(0s)^{4}$ configuration but contain the $\alpha$-cluster breaking because of the spin-orbit interaction. We can evaluate the $\alpha$-cluster breaking component from the expectation value of the squared proton spin $\langle\boldsymbol{S}_{p}^{2}\rangle$ because it measures the $S=1$ mixing induced by the the $\alpha$ breaking. The values calculated with the VAP are $\langle\boldsymbol{S}_{p}^{2}\rangle=0.07$, 0.71, 0.35, and 0.79 for the $0^{+}_{1}$, $0^{+}_{2}$, $1^{-}_{1}$, and $1^{-}_{2}$, respectively, indicating the slight breaking in the $0^{+}_{1}$ and the significant $\alpha$ breaking in the $0^{+}_{2}$, $1^{-}_{1}$, and $1^{-}_{2}$.

It should be also commented that, these VAP configurations couple with other configurations such as the 12C-cluster rotation and 1p-1h excitations in the sAMD+GCM calculation, but they still give significant contributions and roughly describe main properties of the $0^{+}_{1}$, $0^{+}_{2}$, $1^{-}_{1}$, and $1^{-}_{2}$ states.

III.3 Transition strengths

The calculated IS0 and IS1 transition strengths to the $0^{+}_{2,3,4,5}$ and $1^{-}_{1,2}$ states are listed in Tables 1 and 2. Here the strengths $B(E0)=B(\textrm{IS0})/4$ are compared with the experimental data. The observed $B(E0)$ of the $0^{+}_{2}$, $0^{+}_{3}$, and $0^{+}_{5}$ state are reproduced well by the sAMD+GCM calculation. In the dipole excitations, the remarkably large $B(\textrm{IS1})$ is obtained for the $1^{-}_{1}$ with the energy weighted sum rule ratio of 5%, where as the much weaker IS1 transition is obtained for the $1^{-}_{2}$ state in the 12C+$\alpha$ band. The relatively weak IS1 transition to the cluster state seems to contradict the naive expectation that the compressive operator could excite cluster states, but it is not true the case of the $1^{-}_{2}$ state. As mentioned previously, the $1^{-}_{2}$ state in the 12C+$\alpha$ band has the planar-like configuration and shows the different orientation of the triangle 12C-cluster from the initial $0^{+}_{1}$. Therefore, the $0^{+}_{1}$ to $1^{-}_{2}$ excitation involves not only the inter-cluster excitation but also the 12C-cluster rotation, which can not be directly excited by the IS1 operator.

The IS0 and IS1 strength functions up to up to $E=60$ MeV are shown in Fig. 2. The energy weighted sum rule ratios calculated with the sAMD+GCM are plotted. In the IS0 strength function, a large fraction of the strengths are distributed in $E\leq 40$ MeV. The cluster states significantly contribute to the lower part of the strengths in $E\leq 20$ MeV, which are not clearly separated from the GMR strengths. On the other hand, in the IS1 strength function, the $1^{-}_{1}$ state contributes to the significant low-energy strength separated from the IS GDR peak around 40 MeV.

III.4 Form factors and transition densities

Figure 3 shows the calculated elastic and inelastic form factors of the IS0 and IS1 transitions from the ground state to the $0^{+}_{1,2,3}$ and $1^{-}_{1,2}$ states in comparison with experimental data observed by electron scattering Buti:1986zz .

The calculated form factors of the $0^{+}_{1,2,3}$ states are in good agreement with the experimental data in the low-momentum region. In the shape of the observed inelastic form factors, a difference can be seen between the $0^{+}_{2}$ and $0^{+}_{3}$ states. The form factor of the $0^{+}_{2}$ drops off at the smaller transfer momentum $q$ than the $0^{+}_{3}$ reflecting the broader radius dependence of the transition density of the $0^{+}_{2}$. This trend is qualitatively described in the present calculation and understood by the difference in the cluster structures between the $0^{+}_{2}$ and $0^{+}_{3}$ states: the dominant 12C$(0^{+}_{1})$+$\alpha$ component in the $0^{+}_{2}$ state and the 12C$(2^{+}_{1})$+$\alpha$ component in the $0^{+}_{3}$ state.

For the dipole transition to the $1^{-}_{1}$, the magnitude and shape of the experimental form factor are nicely reproduced by the present calculation. Compared to the $1^{-}_{1}$, the calculated IS1 transition to the $1^{-}_{2}$ is quite weak. At the maximum peak, the form factor of the $0^{+}_{1}\to 1^{-}_{2}$ transition is about two orders less than that of the $0^{+}_{1}\to 1^{-}_{1}$ transition. Moreover, the shape of the form factor is different between the $1^{-}_{1}$ and $1^{-}_{2}$ states because of the structure difference. The form factor of the $1^{-}_{2}$ in the 12C+$\alpha$ band shows two peak structure with a dip at $q\sim 2$ fm*-2*, which can not be seen in the form factor of the $1^{-}_{1}$ state with the compact tetrahedral $4\alpha$.

For further discussions of the IS0 and IS1 transitions, we show the transition densities for the $0^{+}$ and $1^{-}$ states with $B(\textrm{IS0})>10$ fm4 and $B(\textrm{IS1})>10$ fm6 in Fig. 4. The transition density of the $0^{+}$ states in Fig. 4(a) shows qualitatively similar behavior with one node around $r=2.5-3.0$ fm, but one can see a quantitative difference between the $0^{+}_{2}$ and high-energy $0^{+}$ states. The transition density in the $0^{+}_{2}$ state is expanded outward and its node is located at the largest position $r\sim 3$ fm due to the developed 12C$(0^{+}_{1})$+$\alpha$ cluster structure. Conversely, the transition density of higher states in $E>22~{}\textrm{MeV}$ is contracted inward. This trend can be understood by the character of small amplitude vibration in the high-energy monopole excitations. The transition density for other $0^{+}$ states in $15<E<22$ MeV shows the intermediate feature.

Compared with the monopole transitions, the IS1 transition density sensitively reflects different characters of dipole excitations. In particular, one can see clear differences in the transition density between the $1^{-}_{1}$, $1^{-}_{2}$, and high-energy GDR. The transition density in the $1^{-}_{1}$ with the compact $4\alpha$ structure shows the most contracted distribution with a node at $r<3$ fm and the surface peak at $r\sim 4$ fm. On the other hand, in the $1^{-}_{2}$ state assigned to the 12C$+\alpha$ band, the transition density has two nodes and shows the broadly stretched distribution with the surface peak at $r\sim 5$ fm. In the high-energy GDR transition, which are contributed by the $1^{-}$ states in $40<E<44~{}\textrm{MeV}$, the transition density shows the intermediate feature with one node at $r\sim 3.5$ fm and the surface peak at $4\lesssim r\lesssim 4.5~{}\textrm{fm}$. In $15<E<30~{}\textrm{MeV}$, most of the $1^{-}$ states have the GDR-like transition density but a few states show the $1^{-}_{1}$-like contracted behavior.

III.5 Vortical nature of dipole excitations

In order to clarify properties of the LED and GDR states, we calculate the transition strengths with the CD and TD operators. Note that the CD strength, which is in principle equivalent to the IS1 strength, is sensitive to the compression dipole mode, whereas the TD strength can probe the nuclear vorticity in the dipole excitation.

The calculated CD and TD strength functions are shown in Fig. 5. In the CD transitions, we obtain the significant strength below 10 MeV for the $1^{-}_{1}$ state and the huge peak around $E=40$ MeV for the IS GDR. In contract to the CD strength, there is no remarkable TD strength in the high-energy region for the IS GDR. From this result, it is concluded that the IS GDR do not have the vortical feature but is the normal compressive mode. Instead, the TD strength is concentrated on the $1^{-}_{1}$ state probing the vortical nature. The $1^{-}_{2}$ in the 12C+$\alpha$ band has the weak CD and TD transitions because this state is the inter-cluster excitation involving the 12C-cluster rotation and is weakly excited by the CD and TD operators.

The present result indicates quite different characters of the dipole excitations between the $1^{-}_{1}$, $1^{-}_{2}$, and IS GDR states: the strong CD and TD transitions in the $1^{-}_{1}$, weak CD and TD transitions in the $1^{-}_{2}$, and strong CD but weak TD transitions in the IS GDR. In particular, one of the prominent features of the $1^{-}_{1}$ is the strong TD strength. In the analysis of the intrinsic wave functions, we find that the TD strength in the $0^{+}_{1}\to 1^{-}_{1}$ is contributed by the dominant $K=1$ component of the prolately deformed $3\alpha$+$\alpha$ structure of the $1^{-}_{1}$. On the other hand, the CD strength in the $0^{+}_{1}\to 1^{-}_{1}$ is mainly contributed by the $K=0$ component. In the $4\alpha$ structure, the $K=1$ and $K=0$ components have large overlap and mixes to each other because of the bosonic symmetry of $\alpha$ clusters. This is a unique feature of the dipole excitation in 16O, in which the $1^{-}_{1}$ state has the strong TD and CD strengths.

To illustrate the vortical and compressive natures of the $1^{-}_{1}$, we show in Fig. 6 the transition current density of the $0^{+}_{1}\to 1^{-}_{1}$ transition in the intrinsic frame calculated using the wave functions $\Phi^{{}^{16}\textrm{O}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=0^{+}_{1}})$ and $\Phi^{{}^{16}\textrm{O}}_{\textrm{AMD}}(\boldsymbol{Z}^{\rm opt}_{\beta=1^{-}_{1}})$ obtained by the VAP. Here, the transition current density before the $K$ and parity projections at the $Y=0$ and $X=0$ planes, (c) (d) that after the $K$ projection before the parity projection, and (e) (f) that after the $K$ and parity projections are shown. The nuclear matter density of the $0^{+}_{1}$ and $1^{-}_{1}$ states are also shown by solid and dashed lines, respectively. Note that, the parity (axial) symmetry is broken in the intrinsic states before the parity projection ($K$ projection) but it is restored after the projection.

In the transition current density before the $K$ and parity projections, a vortex is created at the lower part by the tilting motion of the triangle $3\alpha$ in the tetrahedral $4\alpha$ configuration as seen in Fig. 6(a) and (b). After the $K=1$ projection, where the nuclear current is averaged around the $Z$-axis, a $K=1$ vortex appears clearly at the lower part of Fig. 6(c). Then, after the parity projection, the vortical current is duplicated and two vortexes appear in the lower and upper parts. The $K=1$ vortexes aligned along the prolate deformation is the feature of the $K=1$ VD mode in the prolately deformed system. This mode differs from the torus-shape vortex, which has been originally proposed in the $K=0$ dipole excitation (obviously, the torus current is allowed only in the $K=0$ dipole excitation because of the mathematical condition.) The geometrical shape of the current in the $K=1$ VD mode is described in detail in our previous paper Shikata:2019wdx .

Let us turn to the nuclear current in the $K=0$ component shown in Fig. 6(d) and (f) before and after the parity projection, respectively. The $0^{+}_{1}\to 1^{-}_{1}$ excitation also contains the relative motion between the last $\alpha$ cluster and the $3\alpha$. In the $K=0$ component, this corresponds to the $L=1$ excitation of the $3\alpha$-$\alpha$ relative distance. The relative oscillation of the last $\alpha$ cluster against the $3\alpha$ induces the compressive nuclear current as seen in Fig. 6(d) and (f) and contributes to the significant CD strength in the $0^{+}_{1}\to 1^{-}_{1}$ transition.

Strictly speaking, it is not be able to uniquely define the intrinsic frame for physical states with eigenvalues of angular momentum, but in the present case that the system has the prolate deformation because of the tetrahedral $3\alpha+\alpha$ configuration, the discussion in the “intrinsic” frame can be useful to get the intuitive understanding.

IV Summary and outlook

The IS monopole and dipole excitations in 16O were investigated with the sAMD+GCM. The significant IS0 and IS1 transition strengths were obtained in the low-energy region in addition to the GRs. The $1^{-}_{1}$ state contributes to the significant low-energy strength of the IS1 transition with 5% of the energy-weighted sum rule, which describes well the experimental data observed by $\alpha$ inelastic scattering. The calculated form factors of the inelastic transitions to the $0^{+}_{2}$, $0^{+}_{3}$, and $1^{-}_{1}$ states reproduce the experimental electron scattering form factors. The transition densities were also analyzed.

The different characters of the dipole excitations were found in the $1^{-}_{1}$, $1^{-}_{2}$, and IS GDR: the strong CD and TD transitions in the $1^{-}_{1}$, the weak CD and TD transitions in the $1^{-}_{2}$, and the strong CD but weak TD transitions in the IS GDR. Cluster and vortical aspects of the low-energy dipole states were investigated. In conclusion, we regard the $1^{-}_{1}$ as the vortical vibration mode with the tetrahedral $4\alpha$ structure, the $1^{-}_{2}$ as the 12C+$\alpha$ cluster mode, and the IS GDR as the collective vibration of the compressive dipole mode.

Acknowledgements.

The authors would like to thank the Dr. Nesterenko and Dr. Chiba for fruitful discussions. The computational calculations of this work were performed by using the supercomputer in the Yukawa Institute for theoretical physics, Kyoto University. This work was supported by JSPS KAKENHI Grant Nos. 18K03617 and 18J20926.

Appendix A Transition densities

The density and current density operators for the nuclear matter are defined as

[TABLE]

Here, $\boldsymbol{j}(\boldsymbol{r})$ includes only the convection term of the nuclear current but not the spin term of magnetization. The transition density and current density for the $|0\rangle\to|f\rangle$ transition are given as

[TABLE]

The $\lambda$th transition density is obtained from the multipole decomposition of the transition density,

[TABLE]

where $J_{i}$ and $M_{i}$ ($J_{f}$ and $M_{f}$) are the spin quantum numbers of the initial $|0\rangle$ (final $|f\rangle$) state. The $\lambda$th multipole component of the so-called longitudinal form factor is related to the Fourier-Bessel transform of the transition charge density $\rho^{\textrm{ch}}_{\lambda;0\to f}(r)$ by

[TABLE]

where $\rho^{\textrm{ch}}_{\lambda;0\to f}(r)$ is calculated by taking into account the proton charge radius and assuming the mirror symmetry.

Appendix B IS monopole and dipole operators and transition strengths

The standard compressive-type IS$\lambda$ operators of the IS monopole and dipole excitations are defined as

[TABLE]

The IS0 and IS1 transition strengths for $|0^{+}_{1}\rangle\to|J^{\pi}_{k}\rangle$ are given by the reduced matrix elements as

[TABLE]

where the angular momentum of the initial state is $J_{i}$ and that of the final state is $J^{\pi}_{k}=0^{+}_{k}$ and $1^{-}_{k}$ for $\lambda=0$ and 1, respectively. The reduced matrix elements are related to the transition densities as

[TABLE]

for the IS0 transition and

[TABLE]

for the IS1 transition.

The energy-weighted sum rule of the IS0 operator is

[TABLE]

with the mean square radius $\langle r^{2}\rangle=\langle 0^{+}_{1}|\sum_{i}r^{2}_{i}|0^{+}_{1}\rangle/A$ of the ground state. For the IS1 operator, we use the following energy-weighted sum rule from Ref. Harakeh:1981zz ,

[TABLE]

where $\langle r^{4}\rangle=\langle 0^{+}_{1}|\sum_{i}r^{4}_{i}|0^{+}_{1}\rangle/A$ and $\epsilon=(4/{\cal E}_{2}+5/{\cal E}_{0})\hbar^{2}/3mA$. Here ${\cal E}_{2}$ and ${\cal E}_{0}$ are the IS GQR and GMR energies, for which the empirical values of ${\cal E}_{2}=63A^{-1/3}$ MeV and ${\cal E}_{0}=80A^{-1/3}$ MeV are used, respectively.

Appendix C CD and TD strengths

In the analysis of isoscalor dipole excitations, the CD and TD operators are used as done in Refs. Kanada-Enyo:2015knx ; Kanada-Enyo:2017fps . The former (CD) corresponds to the standard IS1 operator and sensitive to the compressive dipole excitations, and the latter (TD) has been proved to be as a good measure of the nuclear vorticity in the dipole excitations as discussed in Ref. Kvasil:2011yk . They are defined as

[TABLE]

where $\boldsymbol{Y}_{\lambda L\mu}$ is the vector spherical harmonics.

The matrix elements of these IS dipole operators for the $|0^{+}_{1}\rangle\to|1^{-}_{k}\rangle$ transitions are given as

[TABLE]

Using the continuity equation, the CD matrix element is related to the matrix element of the standard IS1 operator $M_{\textrm{IS1}}$ as

[TABLE]

The CD and TD strengths, which are scaled with the factor $\left(\frac{10\hbar c}{E}\right)^{2}$, are defined as

[TABLE]