First microscopic coupled-channel calculation of $\alpha$ inelastic cross sections on $^{16}$O

TL;DR

This paper presents the first microscopic coupled-channel calculation of alpha inelastic scattering cross sections on $^{16}$O, using microscopically derived potentials and detailed nuclear structure models, successfully reproducing experimental data.

Contribution

It introduces a novel microscopic coupled-channel approach for alpha inelastic scattering on $^{16}$O, integrating detailed nuclear structure and interaction models.

Findings

Reproduces observed elastic and inelastic cross sections at multiple energies.

Demonstrates the significance of coupled-channel effects on cross sections.

Provides insights into nuclear structure and reaction mechanisms.

Abstract

The $\alpha$ inelastic scattering on $^{16}$O is investigated with the coupled-channel calculation using the $\alpha$-nucleus coupled-channel potentials, which are microscopically derived by folding the the Melbourne $g$-matrix $NN$ interaction with the $^{16}$O and $\alpha$ densities. The matter and transition densities of $^{16}$O are calculated by a microscopic structure model of the variation after the spin-parity projections combined with the generator coordinate method of $^{12}$C+$\alpha$ in the framework of the antisymmetrized molecular dynamics. The calculation reproduces the observed elastic and inelastic cross sections at incident energies of $E_\alpha$=104 MeV, 130 MeV, 146 MeV, and 386 MeV. The coupled-channel effect on the cross sections is also discussed.

Figures

Tables

Table 1. Table 1: Excitation energies E x subscript 𝐸 𝑥 E_{x} (MeV) and rms matter radii R 𝑅 R (fm) of 16 O calculated with the VAP+GCM. The experimental values of the excitation energies from Ref.  Tilley:1993zz are also shown. The experimental data R = 2.55 𝑅 2.55 R=2.55  fm of the rms radius of the ground state is deduced from the experimental charge radius measured by the electron scattering Angeli2013 . We assign the fourth 0 + superscript 0 0^{+} , third 0 + superscript 0 0^{+} , third 2 + superscript 2 2^{+} , and the second 2 + superscript 2 2^{+} states obtained by the VAP+GCM to the experimental 0 3 + subscript superscript 0 3 0^{+}_{3} , 0 4 + subscript superscript 0 4 0^{+}_{4} , 2 2 + subscript superscript 2 2 2^{+}_{2} , and 2 3 + subscript superscript 2 3 2^{+}_{3} states, which we label as 0 3 , IV + subscript superscript 0 3 IV 0^{+}_{3,\textrm{IV}} , 0 4 , III + subscript superscript 0 4 III 0^{+}_{4,\textrm{III}} , 2 2 , III + subscript superscript 2 2 III 2^{+}_{2,\textrm{III}} , and 2 3 , II + subscript superscript 2 3 II 2^{+}_{3,\textrm{II}} , respectively.

	exp	VAP+GCM
	$E_x$ (MeV)	$E_x$ (MeV)	$R$ (fm)
$0_1^{+}$	0	0.0	2.73
$0_2^{+}$	6.0494	9.7	3.29
$0_{3,\text{IV}}^{+}$	12.049	15.3	3.53
$0_{4,\text{III}}^{+}$	13.6	13.6	3.64
$0_5^{+}$	14.01	18.3	3.53
$2_1^{+}$	6.917	10.8	3.27
$2_{2,\text{III}}^{+}$	9.846	14.7	3.37
$2_{3,\text{II}}^{+}$	11.52	14.0	3.51
$2_4^{+}$	13.04	16.1	3.55
$2_5^{+}$	14.926	16.9	3.26
$2_6^{+}$	15.26	18.9	3.56
$2_7^{+}$	16.44	20.3	3.86
$2_8^{+}$	16.93	20.6	3.38
$4_1^{+}$	10.356	13.7	3.34
$4_2^{+}$	11.097	14.5	3.18
$4_3^{+}$	13.869	16.6	3.63
$1_1^{-}$	7.1169	9.4	2.87
$1_2^{-}$	9.585	12.1	3.58
$3_1^{-}$	6.1299	7.6	2.78
$3_2^{-}$	11.600	13.4	3.65

Table 2. Table 2: The transition strengths B ​ ( E ​ λ ) 𝐵 𝐸 𝜆 B(E\lambda) of 16 O calculated with the VAP+GCM and the experimental data from Refs.  Tilley:1993zz ; Buti:1986zz . For the IS dipole transition strengths of the 1 − → 0 + → superscript 1 superscript 0 1^{-}\to 0^{+} transitions, the values of B ​ ( IS ​ 1 ) / 4 𝐵 IS 1 4 B(\textrm{IS}1)/4 are shown. The scaling factor f tr = B exp ​ ( E ​ λ ) / B cal ​ ( E ​ λ ) subscript 𝑓 tr subscript 𝐵 exp 𝐸 𝜆 subscript 𝐵 cal 𝐸 𝜆 f_{\textrm{tr}}=\sqrt{B_{\textrm{exp}}(E\lambda)/B_{\textrm{cal}}(E\lambda)} determined by the ratio of the experimental value B exp ​ ( E ​ λ ) subscript 𝐵 exp 𝐸 𝜆 B_{\textrm{exp}}(E\lambda) to the calculated value B cal ​ ( E ​ λ ) subscript 𝐵 cal 𝐸 𝜆 B_{\textrm{cal}}(E\lambda) for each transition is also shown. For the transitions with no experimental data of B ​ ( E ​ λ ) 𝐵 𝐸 𝜆 B(E\lambda) , f tr = 1 subscript 𝑓 tr 1 f_{\textrm{tr}}=1 is used. The units are e 2 superscript 𝑒 2 e^{2} fm 2λ for B ​ ( E ​ λ ) 𝐵 𝐸 𝜆 B(E\lambda) ( λ ≠ 0 𝜆 0 \lambda\neq 0 ), e 2 superscript 𝑒 2 e^{2} fm 4 for B ​ ( E ​ 0 ) 𝐵 𝐸 0 B(E0) , and fm 6 for B ​ ( IS ​ 1 ) 𝐵 IS 1 B(\textrm{IS}1) . a The calculated B ( E 2 : 2 2 + → 0 2 + ) B(E2:2^{+}_{2}\to 0^{+}_{2}) is too small, and therefore, is not adjusted to the experimental value but f tr = 1 subscript 𝑓 tr 1 f_{\textrm{tr}}=1 is used. b For the scaling factor of the 1 1 − → 0 1 + → subscript superscript 1 1 subscript superscript 0 1 1^{-}_{1}\to 0^{+}_{1} transition, we use f tr = 1 subscript 𝑓 tr 1 f_{\textrm{tr}}=1 in the default CC calculation, and also use the modified value f tr = 1.3 subscript 𝑓 tr 1.3 f_{\textrm{tr}}=1.3 , which are phenomenologically adjusted so as to reproduce the charge form factors and α 𝛼 \alpha scattering cross sections.

	exp		VAP+GCM
	$B\left(E\lambda \right)$ Tilley:1993zz	$(e,e^{\prime })$ Buti:1986zz	$B\left(E\lambda \right)$	$f_{\text{tr}}$
$E2:2_1^{+}\to 0_1^{+}$	7.42 $\left(0.24\right)$	7.79	3.05	1.56
$E2:2_1^{+}\to 0_2^{+}$	65 $\left(7\right)$		140	0.68
$E2:2_2^{+}\to 0_1^{+}$	0.07 $\left(0.01\right)$	0.05	0.29	0.51
$E2:2_2^{+}\to 0_2^{+}$	2.87 $\left(0.72\right)$		0.02	1a
$E2:2_3^{+}\to 0_1^{+}$	3.59 $\left(1.20\right)$	3.40	2.39	1.23
$E2:2_3^{+}\to 0_2^{+}$	7.42 $\left(1.20\right)$		43.7	0.41
$E2:4_1^{+}\to 2_1^{+}$	156 $\left(14\right)$			1
$E2:4_2^{+}\to 2_1^{+}$	2.39 $\left(0.72\right)$			1
$E2:1_1^{-}\to 3_1^{-}$	50 $\left(12\right)$		33.7	1.22
$E2:1_2^{-}\to 3_1^{-}$			1.0	1
$E0:0_2^{+}\to 0_1^{+}$	12.6	11.8	12.0	1.03
$E0:0_3^{+}\to 0_1^{+}$	16.2	14.2	16.7	0.99
$E0:0_4^{+}\to 0_1^{+}$			10.7	1
$E0:0_5^{+}\to 0_1^{+}$	10.9		9.0	1.10
$E3:3_1^{-}\to 0_1^{+}$	205 $\left(11\right)$		207	0.99
$E3:3_1^{-}\to 0_2^{+}$				1
$E3:3_2^{-}\to 0_1^{+}$				1
$E3:3_2^{-}\to 0_2^{+}$				1
$E4:4_1^{-}\to 0_1^{+}$	378 $\left(133\right)$	420	345	1.10
$E4:4_1^{-}\to 0_2^{+}$				1
$E4:4_2^{-}\to 0_1^{+}$		372	71	2.30
$E4:4_2^{-}\to 0_2^{+}$				1
IS1:$1_1^{-}\to 0_1^{+}$			13.8	1(1.3)b
IS1:$1_1^{-}\to 0_2^{+}$			74	1
IS1:$1_2^{-}\to 0_1^{+}$			0.2	1
IS1:$1_2^{-}\to 0_2^{+}$			745	1

Full text

††preprint: KUNS-2758, NITEP 17

I Introduction

The $\alpha$ scattering has been used for study of isoscalar (IS) monopole and dipole excitations in nuclei. The inelastic cross sections have been analyzed by reaction model calculations to determine the strength functions in a wide range of excitation energy covering the giant resonances Harakeh-textbook . The $\alpha$ scattering is also a good tool to probe cluster states because these have generally strong IS monopole and dipole transition strengths and can be populated by the $\alpha$ scattering reaction Kawabata:2005ta ; Yamada:2011ri ; Chiba:2015khu . Indeed, the $(\alpha,\alpha^{\prime})$ reaction experiments have been intensively performed to investigate cluster structures of excited states in light nuclei such as 12C and 16O recently.

For the study of cluster structures in 12C, the 12C($\alpha,\alpha^{\prime}$) reaction has been investigated with reaction models John:2003ke ; Ohkubo:2004fu ; Takashina:2006pc ; Khoa:2007as ; Takashina:2008zz ; Itoh:2011zz ; Ito:2018opr ; Adachi:2018pql , but many of the reaction calculations encountered the overshooting problem of the $0^{+}_{2}$ cross sections. Also for $sd$-shell nuclei such as 16O, $\alpha$ scattering experiments have reported the similar overshooting problem of the $0^{+}$ cross sections in the reaction model analysis Adachi:2018pql . Recently, Minomo and one of the authors (K. O.) have carried out microscopic coupled-channel calculation and succeeded in reproducing the $0^{+}_{2}$ cross sections of the 12C($\alpha,\alpha^{\prime}$) reaction with no adjustable parameter Minomo:2016hgc . In the study, $\alpha$-nucleus CC potentials are constructed by folding the Melbourne $g$-matrix effective $NN$ interaction Amos:2000 by a phenomenological matter density of $\alpha$ and the matter and transition densities of 12C obtained with the resonating group method Kamimura:1981oxj . In our previous paper Kanada-En'yo:a12C , we have applied the $g$-matrix folding model to the same reaction and reproduced the cross sections of the $0^{+}_{2,3}$, $1^{-}_{1}$, $2^{+}_{1,2}$, and $3^{-}_{1}$ states of 12C with the transition density obtained by the antisymmetrized molecular dynamics (AMD) KanadaEnyo:1995tb ; KanadaEnyo:1995ir ; KanadaEn'yo:1998rf ; KanadaEn'yo:2001qw ; KanadaEn'yo:2012bj , which is a microscopic structure model beyond the cluster models. These works indicate that, if reliable transition densities are available from structure model calculations, the approach of the $g$-matrix folding model can be a useful tool to investigate cluster states by the ($\alpha,\alpha^{\prime}$) reaction.

In the structure studies of 16O, a variety of cluster structures such as the $4\alpha$-tetrahedral, 12C+$\alpha$, and a $4\alpha$-cluster gas state have been suggested by the cluster models 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 ; Horiuchi:2014yua ; Bijker:2014tka ; Bijker:2016bpb . Recently, the experimental studies of 16O have been performed by the 16O($\alpha,\alpha^{\prime}$) reaction Wakasa:2006nt ; Adachi:2018pql . In Ref. Wakasa:2006nt , the $0^{+}_{4}$ state at 13.6 MeV has been discussed in relation with the $4\alpha$-gas state with the reaction model analysis using phenomenological CC potentials. In the study, the $\alpha$-scattering cross sections are naively assumed to scale the IS monopole strengths.

However, no microscopic CC calculation of the 16O$(\alpha,\alpha^{\prime})$ reaction was performed so far, mainly because of the theoretical difficulty of microscopic structure models in description of 16O. For instance, a well-known problem is that microscopic cluster models largely overshoot the excitation energy of the $K^{\pi}=0^{+}_{2}$ band. Recently, one of the authors (Y. K-E.) investigated the cluster structures of 16O Kanada-En'yo:2013dma ; Kanada-Enyo:2017ers ; Kanada-Enyo:2019hrm with the AMD. She has performed the variation after spin-parity projections (VAP) combined with the generator coordinate method (GCM) of the 12C+$\alpha$ cluster in the AMD framework, which we called the VAP+GCM. The VAP+GCM calculation qualitatively described the energy spectra of 16O and obtained the $0^{+}_{2}$, $2^{+}_{1}$, $4^{+}_{1}$, $1^{-}_{2}$, and $3^{-}_{2}$ states in the 12C+$\alpha$ bands, and also the $3^{-}_{1}$ and $4^{+}_{2}$ states in the $4\alpha$-tetrahedral ground band. Moreover, it predicts the $4\alpha$-gas state as the $0^{+}_{5}$ state near the $4\alpha$ threshold energy.

In this paper, we apply the $g$-matrix folding model to the 16O$(\alpha,\alpha^{\prime})$ reaction using the matter and transition densities calculated with the VAP+GCM in a similar way to in our previous work on the 12C$(\alpha,\alpha^{\prime})$ reaction Kanada-En'yo:a12C . The present work is the first microscopic CC calculation of the 16O$(\alpha,\alpha^{\prime})$ reaction that is based on the microscopic $\alpha$-nucleus CC potentials derived with the $g$-matrix folding model. The calculated cross sections are compared with the observed data at incident energies of $E_{\alpha}=104$ MeV, 130 MeV, 146 MeV, and 386 (400) MeV Adachi:2018pql ; Wakasa:2006nt ; Hauser:1969hkb ; Knopfle1975 ; Harakeh:1976rtd . The IS monopole and dipole transitions to the $0^{+}_{2,3,4,5}$ and $1^{-}_{1}$ states are focused. We try to answer the following questions. Can the microscopic reaction calculation describe the $\alpha$ scattering cross sections? Does the overshooting problem of the monopole strength exist? Is the scaling law of the $\alpha$ scattering cross sections and the IS monopole transition strength satisfied?

The paper is organized as follows. Section II describes the structure calculation of 16O with the VAP+GCM, and Sec. III discusses the 16O$(\alpha,\alpha^{\prime})$ scattering investigated with the microscopic CC calculation. Finally, a summary is given in Sec. IV.

II Structure calculation of 16O with VAP+GCM

II.1 Wave functions of 16O

The wave functions of 16O are those obtained by the variation after spin-parity projections (VAP) KanadaEn'yo:1998rf combined with the 12C+$\alpha$ GCM in the AMD framework, which we called the VAP+GCM Kanada-Enyo:2017ers . As shown in Ref. Kanada-Enyo:2019hrm , the VAP+GCM calculation reasonably reproduces the energy spectra and transition strengths of 16O, and obtains various cluster structures such as the $4\alpha$ and 12C+$\alpha$ cluster structures. For the details of the formulation of the structure calculation and the resulting structures and band assignments in 16O, the reader is referred to Refs. Kanada-En'yo:2013dma ; Kanada-Enyo:2017ers ; Kanada-Enyo:2019hrm . Using the VAP+GCM wave functions, the transition strengths, matter and transition densities, and form factors are calculated. The definitions of these quantities are given in Refs. Kanada-En'yo:a12C ; Kanada-Enyo:2019hrm .

II.2 Excitation energies and radii

The excitation energies and radii of 16O from the VAP+GCM calculation and the experimental data are listed in Table 1. We assign the fourth $0^{+}$($0^{+}_{\textrm{IV}})$, the third $0^{+}$($0^{+}_{\textrm{III}})$, the third $2^{+}$($2^{+}_{\textrm{III}})$, and the second $2^{+}$($2^{+}_{\textrm{II}})$ states in the theoretical spectra to the experimental levels of the $0^{+}_{3}$(12.05 MeV), $0^{+}_{4}$(13.6 MeV), $2^{+}_{2}$(9.85 MeV), and $2^{+}_{3}$(11.52 MeV) states, respectively, because the VAP+GCM calculation gives incorrect ordering of the $K^{\pi}=0^{+}_{3}$ and $0^{+}_{4}$ bands; for the detailed discussion, see Ref. Kanada-En'yo:2013dma . The nuclear sizes of the $3^{-}_{1}$ and $1^{-}_{1}$ states are comparable to the ground state and relatively smaller than those of other excited states because the $3^{-}_{1}$ is the $4\alpha$-tetrahedral state in the ground band and the $1^{-}_{1}$ is the vibration mode on the tetrahedral ground band. The $4^{+}_{2}$ state is also regarded as the $4\alpha$-tetrahedral band but its size is slightly larger than those of the other two in the ground band because of the mixing with the $4^{+}_{1}$ state in the 12C+$\alpha$ cluster band. Other states are developed cluster states and have relatively larger radii than those of the ground-band states. The density distribution of the $0^{+}_{1,2,3,4,5}$, $1^{-}_{1,2}$, $2^{+}_{1,2,3}$, $3^{-}_{1,2}$, and $4^{+}_{1,2}$ states obtained by the VAP+GCM is shown in Fig. 1. The $0^{+}_{2,3,4,5}$, $1^{-}_{2}$, $2^{+}_{1,2,3}$, $3^{-}_{2}$, and $4^{+}_{1}$ states tend to have the slightly enhanced surface density in the range of $r=4$–5 fm because of the developed cluster structures.

II.3 Transition strengths, transition densities, and charge form factors of 16O

In Table 2, the transition strengths, $B(E\lambda)$, of 16O calculated with the VAP+GCM are shown compared with the experimental data. For the IS dipole transition strengths of the $1^{-}\to 0^{+}$ transitions, the values of $B(\textrm{IS}1)/4$ are shown. The energy levels and the major $E2$ transitions are shown in Fig. 2. In the figure, the energy levels are connected by (green) dashed and (red) solid lines for the case of remarkable $E2$ transitions of $50<B(E2)<100$ $e^{2}$fm4 and $B(E2)>100$ $e^{2}$fm4, respectively. Rather strong $E2$ transitions are found for developed cluster states. Some of those remarkable $\lambda=2$ transitions give significant CC effects to the $\alpha$ scattering cross sections as discussed later. The $K^{\pi}=0^{+}_{2}$ band of the 12C+$\alpha$ cluster is composed of the $0^{+}_{2}$, $2^{+}_{1}$, and $4^{+}_{1}$ states, and its parity-partner $K^{\pi}=0^{-}$ band is constructed by the $1^{-}_{2}$ and $3^{-}_{2}$ states. The $4\alpha$-gas state is obtained as the $0^{+}_{5}$ state.

To reduce ambiguity from the structure model calculation in application of the theoretical transition density to the reaction calculation, we scale the calculated result as $\rho^{\textrm{(tr)}}(r)\to f_{\textrm{tr}}\rho^{\textrm{(tr)}}(r)$ to fit the observed $B_{\textrm{exp}}(E\lambda)$. The scaling factor $f_{\textrm{tr}}=\sqrt{B_{\textrm{exp}}(E\lambda)/B_{\textrm{cal}}(E\lambda)}$ introduced here is defined by the square root of the ratio of the experimental $B(E\lambda)$ value to the theoretical one. The adopted value of $f_{\textrm{tr}}$ for each transition is shown in Table 2. For the transitions for which experimental data of $B(E\lambda)$ do not exist, we take $f_{\textrm{tr}}=1$ and use the original transition density. For the $1^{-}_{1}\to 0^{+}_{1}$ transition, $B(\textrm{IS1})$ is unknown, but the charge form factors are available from the $(e,e^{\prime})$ reaction data. For this transition, we use $f_{\textrm{tr}}=1$ in the default CC calculation, and also test a modified value $f_{\textrm{tr}}=1.3$ of the $0^{+}_{1}\to 1^{-}_{1}$ transition density, which consistently reproduces the charge form factors and $\alpha$ scattering cross sections. Figure 3 shows the scaled transition density $f_{\textrm{tr}}\rho^{\textrm{(tr)}}(r)$ for transitions from the ground state.

We show in Fig. 4 the charge form factors of 16O calculated with the VAP+GCM and the experimental data measured by the electron scattering Buti:1986zz . The calculated form factors are scaled by multiplying $f^{2}_{\textrm{tr}}$ consistently with the scaled transition density. The experimental data are reasonably reproduced by the scaled form factors of the VAP+GCM. In the $E2$ form factors of the $2^{+}$ states (Fig. 4(d)), a clear difference can be seen in the $0^{+}_{1}\to 2^{+}_{2}$ from the other $E2$ transitions of $0^{+}_{1}\to 2^{+}_{1,3}$ because the corresponding transition density of $0^{+}_{1}\to 2^{+}_{2}$ shows the different behavior that it is the compact spatial distribution with no nodal structure (see Fig. 3(b)). In the $E0$ form factors of $0^{+}$ states, the $0^{+}_{1}\to 0^{+}_{2}$ transition has the dip at the smallest momentum transfer ($q^{2}$) corresponding to the broadest distribution of the transition density.

III $\alpha$ scattering

III.1 Coupled-channel calculation

Using the matter and transition densities calculated with the VAP+GCM as the input from the structure calculation, we perform the CC calculation of 16O($\alpha,\alpha^{\prime})$ with the $g$-matrix folding model in the same way as in our previous work Kanada-En'yo:a12C . The $\alpha$-16O CC potentials are constructed by folding the Melbourne $g$-matrix $NN$ interaction Amos:2000 with the densities of $\alpha$ and 16O in the approximation of an extended version of the nucleon-nucleus folding (NAF) model Egashira:2014zda . For the $\alpha$ density, we adopt the one-range Gaussian distribution given in Ref. Satchler:1979ni .

In the default CC calculation of the cross sections of the $0^{+}$, $1^{-}$, $2^{+}$, and $3^{-}$ states, we adopt the $0^{+}_{1,2,3,4,5}$, $2^{+}_{1,2,3,4}$, $1^{-}_{1,2}$, and $3^{-}_{1,2}$ states with the $\lambda\leq 3$ transitions in the target 16O nucleus. The scaled transition density $f_{\textrm{tr}}\rho^{\textrm{(tr)}}(r)$ is used. For the excitation energies of 16O, we use the experimental values listed in Table 1. In the calculation of the $4^{+}$ cross sections, we adopt $0^{+}_{1,2,3,4}$, $2^{+}_{1,2,3,4,5,6,7,8}$, and $4^{+}_{1,2,3}$ states with the $\lambda=0,2,4$ transitions. For the cross sections to the $0^{+}_{5}$ state, we also perform the CC calculation using the $0^{+}_{1,2,3,4,5}$ and $2^{+}_{1,2,3,4,5,6,7,8}$ states with the $\lambda=0,2$ transitions to take into account the strong $E2$ transition between the $0^{+}_{5}$ and $2^{+}_{8}$ states and compare the result with the default CC calculation.

III.2 $\alpha$ scattering cross sections

The $\alpha$ scattering cross sections at the incident energies of $E_{\alpha}=104$ MeV, 130 MeV, 146 MeV, and 386 MeV are shown in Figs. 5, 6, and 7, respectively. The cross sections calculated by the DWBA are also shown for comparison.

The elastic cross sections are well reproduced by the present calculation except at large scattering angles for $E_{\alpha}$=104–146 MeV. For the $\lambda=2$ and $\lambda=3$ transitions to the $2^{+}_{1}$, $2^{+}_{3}$, and $3^{-}_{1}$ states, the calculated cross sections are in good agreement with the experimental data. These states are strongly populated in the direct transitions, and the cross sections are dominantly described by the DWBA calculation. For these states, the CC effect is minor, in particular, at $E_{\alpha}$=386 MeV, but not negligible in the cross sections at the relatively low incident energies of $E_{\alpha}$=104–146 MeV. For the $2^{+}_{2}$, the calculation reproduces the absolute amplitude of the cross sections at forward angles but does not satisfactorily describe the diffraction pattern of the observed data. For the $4^{+}_{1}$ state, the present calculation predicts a very weak population and much underestimates the experimental cross sections.

For the IS dipole excitation to the $1^{-}_{1}$ state, the experimental cross sections are somewhat underestimated by the default CC calculation (solid lines of Fig. 7) with the original $1^{-}_{1}\to 0^{+}_{1}$ transition density, but successfully reproduced by the calculation with the modified $0^{+}_{1}\to 1^{-}_{1}$ transition density scaled by the factor of $f_{\textrm{tr}}=1.3$, which reproduces the charge form factors of this transition. In comparison with the DWBA calculation, one can see the significant CC effect in the $1^{-}_{1}$ cross sections. Namely, the absolute amplitude of the cross sections is drastically reduced and the dip positions are slightly shifted to forward angles. This CC effect, which is mainly through the $3^{-}_{1}$ state, is essential to reproduce the first dip position of the experimental cross sections at $E_{\alpha}=104$ MeV and 130 MeV.

For the monopole excitations, the calculated $0^{+}_{3}$ and $0^{+}_{4}$ cross sections are in good agreement with the observed data. The present CC calculation describes well not only the diffraction pattern but also the absolute amplitude in the wide range of the incident energies of $E_{\alpha}=104$–386 MeV, and there is no overshooting problem of the $0^{+}$ cross sections.

In comparison with the DWBA calculation, one can see how the CC effect contributes to the monopole transitions in the $\alpha$ scattering. In the $0^{+}_{3}$ and $0^{+}_{4}$ cross sections, the CC effect is not so large but not negligible, in particular, at the low incident energies, $E_{\alpha}=104$–146 MeV. By contrast, the CC effect gives the drastic change of the $0^{+}_{2}$ cross sections mainly because of the strong in-band $E2$ transition with the $2^{-}_{1}$ state in the 12C+$\alpha$-cluster band built on the $0^{+}_{2}$ state. At $E_{\alpha}=104$–146 MeV, the peak amplitude of the $0^{+}_{2}$ cross sections is largely reduced by about a factor of three from the result of the DWBA calculation. Even at $E_{\alpha}=386$ MeV, the peak amplitude of the CC result is smaller by a factor of two than of DWBA. Also for the $0^{+}_{5}$ cross sections, the CC effect is found to be of importance, because of the strong $E2$ transitions between the $0^{+}_{5}$ and $2^{+}_{8}$ states with the developed cluster structure. Although the CC effect seems to be not so strong in the default CC calculation without the coupling with the higher $2^{+}$ states (the (red) solid lines in Fig. 5), the CC calculation with the $0^{+}_{1,2,3,4,5}$ and $2^{+}_{1,2,\ldots,8}$ states shows the drastic CC effect of the significant reduction of the $0^{+}_{5}$ cross sections at the low incident energies of $E_{\alpha}=104$–146 MeV (the (blue) long-dashed lines in Fig. 5). The CC effect in the $0^{+}_{5}$ cross sections becomes weak at the relatively high incident energy of $E_{\alpha}=386$ MeV.

In the experimental studies of the monopole transitions, the $\alpha$ scattering cross sections have been used to deduce the monopole strengths based on the reaction model analysis mainly with the DWBA calculation by naively expecting the scaling law of the $\alpha$-scattering cross sections and the electric monopole transition strength, $B(E0)$. However, the present analysis of the $\alpha$ scattering indicates that the scaling law is not necessarily valid for the cluster states. Firstly, the amplitude of the $0^{+}$ cross sections can be significantly affected by the CC effect mainly through the strong $\lambda=2$ transitions between the developed cluster states. Secondly, the scaling law is not satisfied even in the one-step process of the DWBA cross sections because of the difference in the matter and transition densities between excited $0^{+}$ states. These results indicate that, for study of the monopole transitions by means of the $(\alpha,\alpha^{\prime})$ reaction, it is necessary to analyze the $\alpha$ scattering cross sections with microscopic reaction models considering such the CC effect and density profiles. Nevertheless, we should remark that $0^{+}$ cluster states with significant monopole strengths are strongly populated by the $(\alpha,\alpha^{\prime})$ reaction, meaning that it is still a good probe for the cluster states and can be useful for qualitative discussion even though the scaling law is not quantitatively valid.

IV Summary

The $\alpha$ inelastic scattering cross sections on 16O was investigated by the folding model with the Melbourne $g$-matrix $NN$ interaction. This is the first microscopic CC calculation of the 16O($\alpha,\alpha^{\prime})$ reaction that is based on the $\alpha$-nucleus CC potentials microscopically derived with the $g$-matrix $NN$ interactions and the matter and transition densities of the target 16O nucleus calculated with the microscopic structure model. As for the structure model, we employed the VAP+GCM in the framework of the AMD, which is the microscopic approach beyond the cluster models. In the application to the reaction calculation, the calculated transition density is scaled to fit the experimental transition strengths to reduce the ambiguity of the structure model.

The calculation reproduces well the observed cross sections of the $0^{+}_{2,3,4}$, $2^{+}_{1}$, $1^{-}_{1}$, and $3^{-}_{1}$ states as well as the elastic cross sections at incident energies of $E_{\alpha}=104$ MeV, 130 MeV, 146 MeV, and 386 MeV. In the $0^{+}$ cross sections, there is no overshooting problem. In comparison with the DWBA calculation, the significant CC effect was found in the $0^{+}_{2}$, $0^{+}_{5}$, and $1^{-}_{1}$ cross sections because of the strong $\lambda=2$ coupling between excited states that have developed cluster structures. We clarified that the scaling law of the $\alpha$-scattering cross sections and $B(E0)$ is not necessarily satisfied for the cluster states because of the significant CC effect through the strong $\lambda=2$ transitions between the developed cluster states. Nevertheless, it should be remarked that the $(\alpha,\alpha^{\prime})$ reaction can be used for qualitative discussion on the cluster states because $0^{+}$ cluster states with significant monopole strengths are strongly populated by the $(\alpha,\alpha^{\prime})$ reaction.

It is suggested that the microscopic reaction calculation is needed in the quantitative analysis of the $\alpha$ scattering cross sections. The present $g$-matrix folding model was proved to be applicable to describe the $\alpha$ scattering cross sections. This approach is a promising tool to extract information on cluster structures of excited states in other nuclei by the $(\alpha,\alpha^{\prime})$ reaction.

Acknowledgements.

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 in part by Grants-in-Aid of the Japan Society for the Promotion of Science (Grant Nos. JP18K03617, and JP16K05352) and by the grant for the RCNP joint research project.