Microscopic calculation of inelastic proton scattering off $^{18}$O, $^{10}$Be, $^{12}$Be, and $^{16}$C for study of neutron excitation in neutron-rich nuclei

TL;DR

This paper performs microscopic coupled-channel calculations to analyze inelastic proton scattering on neutron-rich nuclei, revealing the role of neutron excitations and the sensitivity of scattering cross sections to neutron transition densities.

Contribution

It introduces a microscopic approach combining antisymmetrized molecular dynamics and Melbourne g-matrix interactions to study neutron excitations in specific neutron-rich nuclei.

Findings

Reproduces elastic and inelastic scattering cross sections accurately.

Highlights the dominant neutron contribution in 2+1 states of studied nuclei.

Identifies the extended neutron transition density in 12Be and 16C.

Abstract

The microscopic coupled-channel calculation of inelastic proton scattering is performed for the study of neutron excitations in $2^+_1$ states of $^{18}$O, $^{10}$Be, $^{12}$Be, and $^{16}$C. Proton-nucleus potentials in the coupled-channel calculation are microscopically derived by folding the Melbourne $g$-matrix $NN$ interaction with matter and transition densities of target nuclei obtained by the structure model calculation of antisymmetrized molecular dynamics. The calculated result reasonably reproduces the elastic and inelastic proton scattering cross sections, and supports the dominant contribution of neutron in the $2^+_1$ excitation of $^{12}$Be and $^{16}$C as well as $^{18}$O. Sensitivity of the inelastic scattering cross sections to the neutron transition density is discussed. The exotic feature of the neutron transition density with the amplitude in the outer region in $^{12}$Be and $^{16}$C is focused.

Tables

Table 1. Table 1: Binding and excitation energies of 18 O, 10 Be, 12 Be, and 16 C. Theoretical values of 10 Be, 12 Be, and 16 C are taken from Refs.  Kanada-Enyo:1999bsw ; Kanada-Enyo:2003fhn ; Kanada-Enyo:2004tao , and experimental values are from Refs.  Tilley:2004zz ; Kelley:2017qgh ; Tilley:1993zz ; Tilley:1995zz . The band assignment ( K π superscript 𝐾 𝜋 K^{\pi} ) are given based on the AMD calculation.

	Energy (MeV)
	Band	AMD	exp
18O$\left(0_1^{+}\right)$	$K=0_1^{+}$	131.1	139.80
18O$\left(2_1^{+}\right)$	$K=0_1^{+}$	2.0	1.98
10Be$\left(0_1^{+}\right)$	$K=0_1^{+}$	61.1	64.98
10Be$\left(2_1^{+}\right)$	$K=0_1^{+}$	2.7	3.37
10Be$\left(2_2^{+}\right)$	$K=2^{+}$	6.8	5.96
10Be$\left(0_2^{+}\right)$	$K=0_2^{+}$	7.8	6.179
10Be$\left(2_3^{+}\right)$	$K=0_2^{+}$	9.0	7.54
12Be$\left(0_1^{+}\right)$	$K=0_1^{+}$	61.9	68.65
12Be$\left(2_1^{+}\right)$	$K=0_1^{+}$	1.8	2.11
12Be$\left(0_2^{+}\right)$	$K=0_2^{+}$	3.6	2.251
12Be$\left(2_2^{+}\right)$	$K=0_2^{+}$	4.6	$-$
16C$\left(0_1^{+}\right)$	$K=0_1^{+}$	102.6	110.75
16C$\left(2_1^{+}\right)$	$K=0_1^{+}$	2.4	1.77
16C$\left(2_2^{+}\right)$	$K=2^{+}$	7.8	3.99

Table 2. Table 2: Matter, proton, and neutron radii, and transition strengths of 18 O, 10 Be, 12 Be, and 16 C. Theoretical values of the AMD calculation for 10 Be, 12 Be, and 16 C are from Refs.  Kanada-Enyo:1999bsw ; Kanada-Enyo:2003fhn ; Kanada-Enyo:2004tao , and experimental values are taken from Refs.  Tilley:2004zz ; Kelley:2017qgh ; Tilley:1993zz ; Tilley:1995zz . The data of B λ = 2 ( p ) = B ​ ( E ​ 2 ) / e 2 subscript superscript 𝐵 𝑝 𝜆 2 𝐵 𝐸 2 superscript 𝑒 2 B^{(p)}_{\lambda=2}=B(E2)/e^{2} for 16 C are the values reported in Refs.  Ong:2007jb ; Wiedeking:2008zzb . ∗ The experimental values of B ​ ( E ​ 2 ) / e 2 𝐵 𝐸 2 superscript 𝑒 2 B(E2)/e^{2} of the mirror nuclei ( 18 Ne and 10 C) are shown for B λ = 2 ( n ) subscript superscript 𝐵 𝑛 𝜆 2 B^{(n)}_{\lambda=2} of 18 O and 10 Be.

	$R_p$ (fm)	$R_n$ (fm)	$R_m$ (fm)
18O$\left(0_1^{+}\right)$	2.75	2.88	2.82
exp	2.62	2.83${}^{\text{*mir}}$	2.61(8)
	$B_{\lambda =2}^{\left(p\right)}$ (fm4)	$B_{\lambda =2}^{\left(n\right)}$ (fm4)	$M_n/M_p$
18O$\left(2_1^{+}\to 0_1^{+}\right)$	0.7	18.6	5.4
exp	9.3(3)	50(5)${}^{\text{*mir}}$	2.3(2)${}^{\text{*mir}}$
	$R_p$ (fm)	$R_n$ (fm)	$R_m$ (fm)
10Be$\left(0_1^{+}\right)$	2.50	2.56	2.54
exp	2.17		2.30(2)
	$B_{\lambda =2}^{\left(p\right)}$ (fm4)	$B_{\lambda =2}^{\left(n\right)}$ (fm4)	$M_n/M_p$
10Be$\left(2_1^{+}\to 0_1^{+}\right)$	11.6	8.9	0.9
exp	10.2(1.0)	12.2(1.9)${}^{\text{*mir}}$	1.1(1)${}^{\text{*mir}}$
10Be$\left(2_2^{+}\to 0_1^{+}\right)$	0.2	3.2	3.9
10Be$\left(2_3^{+}\to 0_1^{+}\right)$	0.1	0.7	2.5
10Be$\left(2_3^{+}\to 0_2^{+}\right)$	34.5	118	1.8
	$R_p$ (fm)	$R_n$ (fm)	$R_m$ (fm)
12Be$\left(0_1^{+}\right)$	2.67	2.94	2.85
exp	2.39		2.59(6)
12Be$\left(0_2^{+}\right)$	2.56	2.84	2.75
	$B_{\lambda =2}^{\left(p\right)}$ (fm4)	$B_{\lambda =2}^{\left(n\right)}$ (fm4)	$M_n/M_p$
12Be$\left(2_1^{+}\to 0_1^{+}\right)$	14.4	51.1	1.9
exp	14.2(2.8)
12Be$\left(2_2^{+}\to 0_1^{+}\right)$	0.0	7.4	25.4
12Be$\left(2_2^{+}\to 0_2^{+}\right)$	7.5	9.0	1.1
	$R_p$ (fm)	$R_n$ (fm)	$R_m$ (fm)
16C$\left(0_1^{+}\right)$	2.58	2.85	2.75
exp			2.70(3)
	$B_{\lambda =2}^{\left(p\right)}$ (fm4)	$B_{\lambda =2}^{\left(n\right)}$ (fm4)	$M_n/M_p$
16C$\left(2_1^{+}\to 0_1^{+}\right)$	2.7	27.0	3.2
exp Ong:2007jb	2.6(9)
exp Wiedeking:2008zzb	4.15(73)
16C$\left(2_2^{+}\to 0_1^{+}\right)$	2.6	0.1	0.2

Equations

$${\rho }_p^{\text{tr}}(r)=(M_p^{\text{exp}}/M_p^{\text{cal}}){\rho }_p^{\text{tr-cal}}(r)$$

$$M_{n,p}\equiv \int r^{2+\lambda }{\rho }_{n,p}^{\text{tr}}(r)dr$$

$$B_{\lambda }^{(n),(p)}=\frac{1}{2J_i+1}|M_{n.p}{|}^2,$$

$$\sigma (h,h^{\prime })\propto |b_n^{(h,h^{\prime })}{M}_n+b_p^{(h,h^{\prime })}{M}_p{|}^2,$$

$$\sigma (p,p^{\prime })=|a_n^{(p,p^{\prime })}{(}^{A}Z,E)M_n+a_p^{(p,p^{\prime })}{(}^{A}Z,E)M_p{|}^2.$$

$$T=\frac{A}{A-1}{T}^{\prime },$$

$$\bigg{[}K+h+\frac{A-1}{A}\sum_{j=1}^{A}\tau_{j}-E\bigg{]}\Psi=0.$$

Full text

††preprint: KUNS-2768, NITEP 25

I Introduction

Shape difference in proton and neutron matter distributions in nuclei is one of the interesting phenomena in unstable nuclei. To discuss the difference between the neutron and proton deformation (or collectivity), the neutron and proton transition matrix elements in the ground-band $2^{+}_{1}\to 0^{+}_{1}$ transition have been investigated for a long time. In a naive expectation for ordinary nuclei with the same proton and neutron deformation, the ratio of the neutron transition matrix element ($M_{n}$) to the proton one ($M_{p}$) should be $N/Z$. However, the relation $M_{n}/M_{p}\approx Z/N$ is not satisfied even in stable nuclei with the proton or neutron shell closure as reported in Ref. Bernstein:1981fp . For instance, in 18O and 48Ca, the ratio becomes significantly larger than $N/Z$, which indicates the neutron dominance in the $2^{+}_{1}$ excitation because of the proton shell closure. The phenomena of the shape difference and/or the neutron dominance have been suggested also in unstable nuclei such as 10Be, 12Be, and 16C Kanada-Enyo:1996zsp ; Iwasaki:2000gh ; Kanada-Enyo:2004tao ; Kanada-Enyo:2004ere ; Sagawa:2004ut ; Jouanne:2005pb ; Takashina:2005bs ; Ong:2006rm ; Burvenich:2008zz ; Takashina:2008zza ; Elekes:2008zz ; Wiedeking:2008zzb ; Yao:2011zza ; Forssen:2011dr .

Experimental information of the proton part $M_{p}$ can be directly obtained from the $E2$ strength. By contrast, determination of the neutron part ($M_{n}$) is not easy even for stable nuclei. Instead of direct measurements, experiments of inelastic hadron scattering have been performed using such probes as $\alpha$, proton, neutron, and $\pi^{-}/\pi^{+}$. By combining the hadron scattering data with the electromagnetic data, $M_{n}$ and $M_{p}$ have been discussed based on reaction analysis (see Refs. Bernstein:1981fp ; Bernstein:1977wtr and references therein). For 18O, the neutron matrix element of the $2^{+}_{1}\to 0^{+}_{1}$ transition has been intensively investigated, and the anomalously large value of $M_{n}/M_{p}\approx 2$ has been reduced from the inelastic scattering data Bernstein:1977wtr ; Iversen:1978sc ; Grabmayr:1980qze ; Kelly:1986ysn consistently with $B(E2)$ of mirror transitions of 18Ne and 18O Bernstein:1979zza .

In study of unstable nuclei, the neutron collectivity, i.e., the $M_{n}/M_{p}$ ratio has been investigated extensively with the inelastic proton scattering experiments in inverse kinematics using radioactive ion beam Iwasaki:2000gh ; Jouanne:2005pb ; Ong:2006rm ; Jewell:1999jme ; Khan:2000rac ; Khan:2001gbs ; Scheit:2001ys ; Becheva:2006zz ; Elekes:2008zz ; Campbell:2007zz ; Elekes:2009zz ; Aoi:2010ah ; Michimasa:2014qca ; Riley:2014vnp ; Corsi:2015afa ; Cortes:2018izn . However, the reaction analysis still contains model ambiguities, for instance, in the proton-nucleus optical potentials, which are phenomenologically adjusted usually to elastic scattering cross sections but the applicability has not been well tested for inelastic scattering off exotic nuclei.

Recently, triggered by the complete microscopic folding model calculation by the Melbourne group Amos:2000 ; Karataglidis:2007yj , the microscopic description of proton-nucleus Min10 ; Toy13 ; Toyokawa:2015zxa ; Minomo:2017hjl and $\alpha$-nucleus Egashira:2014zda ; Toyokawa:2015zxa elastic scattering, without any free adjustable parameter and phenomenological parametrization, has been developed. Very recently, the framework was successfully applied to $\alpha$-nucleus inelastic processes Minomo:2016hgc ; Minomo:2017hjl ; Kanada-Enyo:2019prr ; Kanada-Enyo:2019qbp . One of the advantages of this approach is that, once reliable densities of target nuclei are given, there is no adjustable parameter in the reaction part. As for the structure part, proton and neutron matter and transition densities are obtained by microscopic structure model calculations, which describe characteristics of nuclear properties such as the cluster, deformation, and neutron skin structures in target nuclei.

In this paper, we investigate the inelastic proton scattering to the $2^{+}_{1}$ states of 18O, 10Be, 12Be, and 16C with the coupled-channel (CC) calculations of the microscopic single-folding model using the Melbourne $g$-matrix effective $NN$ interaction Amos:2000 . The proton and neutron matter and transition densities of the target nuclei are calculated with antisymmetrized molecular dynamics (AMD) KanadaEnyo:1995tb ; KanadaEnyo:1995ir ; KanadaEn'yo:2012bj . As test cases, we first show application to the proton scattering off $Z=N$ nuclei, 12C and 16O. Then, we apply the same method to the proton scattering off 18O, 10Be, 12Be, and 16C. The sensitivity of the $2^{+}_{1}$ cross sections to $M_{n}$ and $M_{p}$ is analyzed while focusing on the neutron-proton difference in the transition densities in 12Be and 16C.

The paper is organized as follows. The next section describes the present framework of the microscopic coupled-channel (MCC) calculation and that of the structure calculations for target nuclei. Results of 12C and 16O are shown in Sec. III, and results and discussions for the $N\neq Z$ case of 18O, 10Be, 12Be, and 16C are given in Sec. IV. Finally, a summary is given in Sec. V.

II Method

The present reaction calculation for the proton scattering is the MCC calculation of the single-folding model. As inputs from the structure calculations, the target densities are calculated with AMD combined with and without the cluster model of the generator coordinate method (GCM). The AMD and AMD+GCM calculations of target nuclei are the same as those of Refs. Kanada-Enyo:1999bsw ; Kanada-Enyo:2003fhn ; Kanada-Enyo:2004tao ; Kanada-Enyo:2019prr ; Kanada-Enyo:2019hrm ; Kanada-Enyo:2019qbp . The definitions of densities and form factors in the structure calculation are explained in Ref. Kanada-Enyo:2019prr . For details, the reader is referred to those references.

II.1 Microscopic coupled-channel calculation

The diagonal and coupling potentials for the nucleon-nucleus system are microscopically calculated by folding the Melbourne $g$-matrix $NN$ interaction Amos:2000 with the target densities described in Sec. II.2. The Melbourne $g$ matrix is obtained by solving a Bethe-Goldstone equation in a uniform nuclear matter at given incident energy; the Bonn-B potential Mac87 is adopted as a bare $NN$ interaction. In Ref. Amos:2000 , the Melbourne $g$-matrix interaction was constructed and applied to a systematic investigation on proton elastic and inelastic scattering off various stable nuclei and some neutron-rich nuclei at energies from 40 MeV to 300 MeV. The nonlocality coming from the exchange term was rigorously treated and the central, spin-orbit, and tensor contributions were taken into account. As a result, it was clearly shown that the microscopic calculation with the Melbourne $g$ matrix for the proton-nucleus scattering has predictive power for the proton-nucleus elastic and inelastic cross sections and spin observables. Later, the framework was applied also to proton inelastic scattering off 10C and 18O Karataglidis:2007yj .

In the present study, we adopt a simplified single-folding model described in Ref. Min10 . We employ the Brieva and Rook (BR) prescription Brieva:1977rsh ; Brieva:1977zz ; Brieva:1978fdf to localize the exchange terms. The validity of the BR localization for nucleon-nucleus scattering was confirmed in Refs. Min10 ; Hai16 and for nucleus-nucleus scattering in Ref. Hag06 . This simplified single-folding model has successfully been applied to nucleon-nucleus elastic scattering for various cases Min10 ; Toy13 ; Toyokawa:2015zxa ; Minomo:2017hjl . In this study, we extend the model to proton inelastic scattering in a similar manner to in our recent studies on $\alpha$ inelastic scattering Kanada-Enyo:2019prr ; Kanada-Enyo:2019qbp . To avoid complexity, we take into account only the central part of the proton-nucleus potential. The spin-orbit interaction is known to smear the dip structure of the diffraction pattern in general. Although at higher energies, say, above 150–200 MeV, it can somewhat affect the absolute amplitudes also near the peaks, such effect is expected to be minor in the energy region considered in this study. As in the previous studies including that by the Melbourne group Amos:2000 , the local density approximation is adopted to apply the $g$-matrix interaction to a finite nucleus.

The cross sections of the elastic and inelastic scattering are calculated by the CC calculations using the proton-nucleus potentials obtained with the AMD densities for 18O, 10Be, 12Be, and 16C, and the AMD+GCM densities for 12C and 16O. For 12C, we also use the densities of a 3$\alpha$-cluster model of the resonating group method (RGM) Kamimura:1981oxj .

It should be commented that a similar approach of the MCC calculation with the Jeukenne-Lejeune-Mahaux (JLM) interaction Jeukenne:1977zz has been applied to the proton inelastic scattering off 10Be and 12Be in the earlier work by Takashina et al. Takashina:2008zza . It was used also in continuum-discretized coupled-channels calculation for nucleon-induced breakup reactions of 6,7Li Mat11 ; Ich12 ; Guo19 and 11Li Mat17 . The JLM interaction is another kind of the $g$-matrix effective interaction that has only the direct term. This property allows one to implement it to reaction calculation with the minimal task. On the other hand, in general, phenomenological tuning of the JLM interaction is necessary to fit the scattering data.

II.2 Structure calculations for target nuclei

For the structure calculation of the target nuclei, we adopt the AMD wave functions obtained by the variation after projections (VAP). In the AMD+VAP method, the variation is performed for the spin-parity projected AMD wave function as done in Refs. KanadaEn'yo:1998rf ; Kanada-Enyo:2006rjf . The method was applied for the structure studies of 10Be, 12Be, and 16C in Refs. Kanada-Enyo:1999bsw ; Kanada-Enyo:2003fhn ; Kanada-Enyo:2004tao . In the present paper, the same method is applied to 18O to obtain the wave functions of the $0^{+}_{1}$ and $2^{+}_{1}$ states. For 12C and 16O, the AMD+VAP method is combined with the $3\alpha$- and ${}^{12}\textrm{C}+\alpha$-cluster GCM, respectively, as done in Refs. Kanada-Enyo:2019prr ; Kanada-Enyo:2019hrm ; Kanada-Enyo:2019qbp ; Kanada-Enyo:2015vwc ; Kanada-Enyo:2017ers . In this paper, we simply call the AMD+VAP “AMD” and that with the cluster GCM “AMD+GCM”.

The AMD wave functions used in this paper are in principle the same as those of Refs. KanadaEn'yo:1998rf ; Kanada-Enyo:2003fhn . We utilize the 10Be wave function for 10C by assuming the mirror symmetry. For 16C, the VAP(c) wave function of Ref. Kanada-Enyo:2004tao is adopted. The wave functions and transition densities of 12C and 16O are consistent with those of AMD+GCM used for the $\alpha$ scattering in Refs. Kanada-Enyo:2019prr ; Kanada-Enyo:2019qbp .

The neutron and proton matter and transition densities are calculated with the AMD and AMD+GCM wave functions. We denote the neutron and proton transition densities as $\rho^{\textrm{tr}}_{n}(r)$ and $\rho^{\textrm{tr}}_{p}(r)$, respectively. For $N=Z$ nuclei (12C and 16O), half of isoscalar density is used as the proton (neutron) density in the mirror symmetry assumption. For quantitative discussions of inelastic cross sections, we scale the original transition densities $\rho^{\textrm{tr-cal}}_{p}(r)$ to adjust the theoretical $B(E\lambda)$ values to the experimental data as

[TABLE]

Here the rank $\lambda$ ($\lambda>0$) transition matrix elements for the neutron and proton parts are defined as

[TABLE]

and related to the transition strengths as

[TABLE]

where $J_{i}$ is the angular momentum of the initial state. The $E2$ transition strength is given by the proton $\lambda=2$ transition strength as $B(E2)=e^{2}B^{(p)}_{\lambda=2}$.

The adopted states in the CC calculation for 10Be, 12Be, 16C, and 18O are 10Be$(0^{+}_{1,2},2^{+}_{1,2,3})$, 12Be$(0^{+}_{1,2},2^{+}_{1,2})$, 16C$(0^{+}_{1},2^{+}_{1,2})$, and 18O$(0^{+}_{1},2^{+}_{1})$. All $\lambda=0$ and $\lambda=2$ transitions between these states are taken into account. The experimental values of excitation energies are adopted as inputs of the CC calculation.

In the CC calculations for 12C and 16O, all the inputs from the structure part such as the adopted states, excitation energies, and transitions are the same as those used for the $\alpha$ scattering with the AMD+GCM wave functions in Refs. Kanada-Enyo:2019prr ; Kanada-Enyo:2019qbp .

As shown later, the CC effect gives only minor contribution to the inelastic scattering to the $2^{+}_{1}$ state at incident energies higher than 25 MeV, and the cross sections are approximately described by the one-step process of the distorted wave Born approximation (DWBA).

III Results of 12C and 16O

The $0^{+}_{1}$ and $2^{+}_{1}$ cross sections of 12C at incident energies $E_{p}=35$, 65, and 135 MeV are shown in Fig. 1, and those of 16O at $E_{p}=35$ and 135 MeV are shown in Fig. 2. In addition to the CC calculation, the one-step cross sections obtained by the DWBA calculation are also shown. The small difference between the CC and DWBA cross sections indicates that the CC effect is minor.

The AMD+GCM result of 12C is compared with experimental data and also with the calculation with the RGM density. From the electron scattering data, the RGM density is known to be good in quality and better than the AMD+GCM density Kamimura:1981oxj ; Kanada-Enyo:2019prr . As seen in Fig. 1, the present calculation with the AMD+GCM density reproduces well the elastic proton scattering cross sections of 12C at forward angles, but somewhat underestimates the third peak. A better result is obtained by the calculation with the RGM density, consistently with Ref. Minomo:2017hjl . The inelastic proton scattering cross sections of 12C$(2^{+}_{1})$ are described reasonably well with the AMD+GCM and RGM calculations except for the cross sections at $E_{p}=35$ MeV. The RGM density again gives a better agreement with the data at large angles. This result indicates that quality of the structure model densities can be tested by the detailed data of the proton scattering. For the proton scattering off 16O, the present calculation reproduces well the elastic and inelastic cross sections (Fig. 2). It should be commented that the $2^{+}_{1}$ state of 16O is not the ground-band member but belongs to the 12C+$\alpha$-cluster band built on the $0^{+}_{2}$ state. The present microscopic approach works well even for such the developed cluster state with structure much different from the ground state.

IV Results of $Z\neq N$ nuclei

IV.1 Structure properties

The theoretical and experimental values of structure properties of the target nuclei are listed in Tables 1 and 2. The energies are shown in Table 1, and radii and $\lambda=2$ transition strengths as well as the $M_{n}/M_{p}$ ratio are shown in Table 2. The structure calculation of 10Be, 12Be, and 16C are consistent with Refs. Kanada-Enyo:1999bsw ; Kanada-Enyo:2003fhn ; Kanada-Enyo:2004tao . We will describe detailed properties of the $0^{+}_{1}$ and $2^{+}_{1}$ states and the transition between them later.

IV.2 Results of 18O

In the $2^{+}_{1}\to 0^{+}_{1}$ transition of 18O, the significant $B^{(p)}_{\lambda=2}$ has been experimentally known but it is much underestimated by the AMD calculation meaning that the proton excitation from the $p$-shell closure is not sufficiently described in the theory. For the neutron part, the large $B^{(n)}_{\lambda=2}$ of the AMD calculation indicates the neutron dominance, which is qualitatively consistent with the mirror transition of 18Ne Bernstein:1979zza and the proton scattering experiment Kelly:1986ysn .

The calculated densities and form factors of 18O are shown in Figs. 3 and 4, respectively, together with the data measured by the electron scattering experiments. Here, the theoretical proton transition density $\rho^{\textrm{tr-cal}}_{p}(r)$ and form factors $F^{\textrm{cal}}(q)$ are scaled by the factor $M^{\textrm{exp}}_{p}/M^{\textrm{cal}}_{p}=3.88$ as $\rho^{\textrm{tr}}_{p}(r)=(M^{\textrm{exp}}_{p}/M^{\textrm{cal}}_{p})\rho^{\textrm{tr-cal}}_{p}(r)$ and $F(q)=(M^{\textrm{exp}}_{p}/M^{\textrm{cal}}_{p})F^{\textrm{cal}}(q)$ so as to fit the experimental $B(E2)$ value. After the scaling, the experimental data are reproduced well except for the small $r$ (large $q$) region.

For the neutron transition density $\rho^{\textrm{tr}}_{n}(r)$ of ${}^{18}\textrm{O}$, we tentatively assume the mirror symmetry and scale $\rho^{\textrm{tr-cal}}_{n}(r)$ with the scaling factor $M^{\textrm{exp}}_{p}(^{18}\textrm{Ne})/M^{\textrm{cal}}_{n}=1.72$ to prepare the default input $\rho^{\textrm{tr}}_{n}(r)$ in the reaction calculation. However, if we take into account the mirror symmetry breaking, another choice maybe possible, for example, about a 10% smaller value than $M^{\textrm{exp}}_{p}(^{18}\textrm{Ne})$ was theoretically recommended for $M_{n}(^{18}\textrm{O})$ in Ref. Kelly:1986ysn . In Figs. 3(b) and (c), the default $\rho^{\textrm{tr}}_{n}(r)$ is compared with the experimental estimation (denoted by $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$) of Ref. Kelly:1986ysn , which was reduced from the inelastic proton scattering at $E=135$ MeV/u by a model analysis. $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$ gives $B^{(n)}_{\lambda=2}=38$ fm4, which is slightly smaller than $B^{(n)}_{\lambda=2}=50$ fm4 of $\rho^{\textrm{tr}}_{n}(r)$ adjusted to $B(E2;^{18}\textrm{Ne})$. The 12% reduced transition density (0.88$\rho^{\textrm{tr}}_{n}(r)$) gives the same strengths ($B^{(n)}_{\lambda=2}=38$ fm4) as $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$, but it shows a different radial behavior from $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$. Compared with the theoretical transition density, $\rho^{\textrm{tr}(p,p^{\prime})}_{n,\textrm{exp}}(r)$ has the smaller amplitude at the nuclear surface ($r=3$–$4$ fm) and enhanced outer tail in $r\gtrsim 5$ fm region (see Fig. 3(c)). In the reaction analysis, we consider this difference between $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$ and the default $\rho^{\textrm{tr}}_{n}(r)$ as a model ambiguity from the neutron transition density.

We calculate the cross sections of the proton scattering at $E=24.5$ MeV/u, $35$ MeV/u, 43 MeV/u, and 135 MeV/u, and those of the neutron scattering at $E=24$ MeV/u. They are compared with the experimental data. The results are shown in Figs. 5 and 6. The calculation reproduces reasonably well the elastic and inelastic proton scattering cross sections in the wide range of $E=24$–$135$ MeV/u. It also reproduces well the neutron scattering cross sections at $E=24$ MeV/u. In comparison with the DWBA calculation shown in Fig. 7, one can see that the CC effect is minor in the $2^{+}_{1}$ cross sections.

Let us discuss the ambiguity from the proton and neutron transition densities. As shown previously, the (scaled) proton part $\rho^{\textrm{tr}}_{p}(r)$ used in the present calculation reproduces well the experimental data measured by the electron scattering, whereas the neutron part $\rho^{\textrm{tr}}_{n}(r)$ has the $r$ behavior different from the experimental one $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$. In order to see the effect of this difference in the neutron transition density to the inelastic cross sections, we perform the same reaction calculation using $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$ and $0.88\rho^{\textrm{tr}}_{n}(r)$. Figures 5(b) and 6(b) show respectively the proton and neutron scattering cross sections obtained with $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$ (light blue dashed lines) and that with $0.88\rho^{\textrm{tr}}_{n}(r)$ (blue dotted lines) in comparison with the default calculation (red solid lines) and experimental data. In the result of the proton scattering with $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$, the cross sections at forward angle slightly decrease to 70% of the default calculation, and the second and third peaks at the large angles are reduced further to $40$–$60\%$ of the default calculation. The reduction rate at large angels is larger than the naive expectation of $38/50\approx 75\%$ from the $B^{(n)}_{\lambda=2}$ ratio. It means that the outer tail amplitude of the neutron transition density gives relatively minor contribution to the proton scattering cross sections than the surface amplitude though it significantly enhances the $M_{n}$, i.e., $B^{(n)}_{\lambda=2}$. The calculation with $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$ obtains better agreement with the proton scattering data at least at $E=24.5$ MeV/u and 35 MeV/u suggesting that $\rho_{n,\textrm{exp}}^{\textrm{tr}(p,p^{\prime})}(r)$ may be favored rather than the default $\rho^{\textrm{tr}}_{n}(r)$ used in the present calculation. It indicates that the proton scattering is a sensitive probe for the neutron transition density. In contrast to the proton scattering, the neutron scattering cross sections are not so sensitive to the difference in the neutron transition densities as expected from the weaker $nn$ interactions than the $pn$ ones.

IV.3 Results of 10Be, 12Be, and 16C

The structure studies of 10Be, 12Be, and 16C with AMD have been done in Refs. Kanada-Enyo:1999bsw ; Kanada-Enyo:2003fhn ; Kanada-Enyo:2004tao . We here briefly describe the structure properties, in particular, of the ground bands in these nuclei.

In 10Be, the $M_{n}/M_{p}=0.9$ of the ground-band transition $2^{+}_{1}\to 0^{+}_{1}$ in the AMD calculation is smaller than $N/Z=1.5$ naively expected from the collective model and in reasonable agreement with the experimental value ($M_{n}/M_{p}=1.1$) reduced from $B(E2)$ of mirror transitions. It indicates that the neutron excitation is somewhat suppressed compared with the proton excitation in the $2^{+}_{1}$ state. In addition to the $2^{+}_{1}$ state of the ground $K^{\pi}=0^{+}$ band, the $2^{+}_{2}$ state of the side band ($K^{\pi}=2^{+}$) is obtained because of the triaxial deformation. In the higher energy region, the second $K^{\pi}=0^{+}$ band with the developed cluster structure appears.

In 12Be, the breaking of $N=8$ magicity is known in the ground state from various experimental observations such as the Gamov-Tellar transitions, inelastic scattering, and knock-out reactions Iwasaki:2000gh ; Suzuki:1997zza ; Navin:2000zz ; Pain:2005xw ; Imai:2009zza ; Meharchand:2012zz . The AMD calculation obtains the largely deformed ground band ($K^{\pi}=0^{+}_{1}$) with the dominant neutron $2\hbar\omega$ component. The ground-band transition, $2^{+}_{1}\to 0^{+}_{1}$, is strong because of the large deformation compared with the weaker transition in the $K^{\pi}=0^{+}_{2}$ band, which corresponds to the normal neutron $p$-shell closed configuration. In particular, the neutron transition is considerably strong because of the contribution of two $sd$-orbit neutrons. The values of the ratio $M_{n}/M_{p}=1.9$ ($M_{n}/M_{p}=1.1$) is obtained for the $K^{\pi}=0^{+}_{1}$($K^{\pi}=0^{+}_{2})$ band. The ratio of the ground band is as large as $N/Z=2$ because of the breaking of the $N=8$ magicity.

In the case of 16C, the AMD calculation predicted the weak proton transition in $2^{+}_{1}\to 0^{+}_{1}$ because of the $Z=6$ sub-shell closure. The observed $B(E2)$ values are consistent with the prediction. On the other hand, the neutron transition is significantly large because of the contribution of the $sd$-orbit neutrons, and results in the much larger ratio $M_{n}/M_{p}=3.2$ than $N/Z=1.67$, i.e., the dominant neutron contribution in the ground-band transition.

Figure 9 shows the neutron and proton matter densities of the ground state and the neutron and proton transition densities of the $2^{+}_{1}\to 0^{+}_{1}$ transition of 10Be, 12Be, and 16C. In 10Be, the proton and neutron transition densities have the peak amplitude at the same position at the nuclear surface and approximately satisfy the relation $\rho^{\textrm{tr}}_{n}(r)=(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$. By contrast, in 12Be and 16C, the neutron transition density shows the $r$ behavior quite different from the proton transition density. It has the peak amplitude in $r\approx 3$ fm region much outer than the proton transition density because of the contribution of the $sd$-orbit neutrons and no longer satisfies the relation $\rho^{\textrm{tr}}_{n}(r)=(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$. This is a different feature from 10Be, where the protons and neutrons in the same $p$ shell contribute to the $2^{+}_{1}$ excitation.

The proton scattering cross sections are calculated with the AMD densities. For 10Be, the theoretical proton and neutron transition densities are renormalized to fit the experimental transition strengths ($B^{(p),(n)}_{\lambda=2}$) by the scaling factors $M^{\textrm{exp}}_{p,n}/M^{\textrm{cal}}_{p,n}$ listed in Table 2. For 12Be and 16C, we use the original AMD transition densities, which reproduce well the experimental $B(E2)$ values. The calculated elastic and inelastic cross sections of 10Be at $E=60$ MeV/u, 12Be at $E=55$ MeV/u, and 10C at $E=45$ MeV/u are shown in Figs. 10 and  11. They are compared with the experimental data measured in inverse kinematics. In Fig. 12, the calculated cross sections of 16C at $E=33$ MeV/u are compared with the experimental data. The present calculation reproduces well the absolute amplitude of the $2^{+}_{1}$ cross sections as well as the elastic cross sections.

In Fig. 9(b) for 12Be, the DWBA calculation with the neutron transition density $\rho^{\textrm{tr}}_{n}(r)=(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$ is also shown. This calculation corresponds to the case with the collective model transition density. Compared with the result using the original AMD transition density, the cross sections somewhat increase and the peak and dip positions slightly shift toward larger angles.

For the neutron transition in 12Be and 16C, there is no data from the mirror nuclei. The good reproduction of the inelastic cross sections supports reliability of the neutron transition densities adopted in the present calculation, that is, the dominant neutron contributions as $M_{n}/M_{p}\approx 2$ and $M_{n}/M_{p}\approx 3$ for 12Be and 16C, respectively. This result is qualitatively consistent with those in Refs. Ong:2006rm ; Takashina:2008zza . It should be stressed again that phenomenological adjustable parameters were needed in the reaction models of Refs. Ong:2006rm ; Takashina:2008zza , but not in the present model. For further detailed discussion of the transition densities, higher quality data are required.

IV.4 Discussions

We discuss how one can link the inelastic proton scattering cross sections with the neutron transition matrix element $M_{n}$ of the $0^{+}_{1}\to 2^{+}_{1}$ transition. The experimental studies of Refs. Iwasaki:2000gh ; Ong:2006rm have discussed the neutron matrix elements of 12Be and 16C with the reaction analysis of the proton scattering data, and concluded the significant neutron contribution in the $0^{+}_{1}\to 2^{+}_{1}$ transition. According to the model analysis in Refs. Iwasaki:2000gh ; Ong:2006rm using the Bernstein prescription Bernstein:1981fp , $B^{(n)}_{\lambda=2}=17$ fm4 of 12Be is obtained by the reaction analysis using $B(E2)=14.2(2.8)$ $e^{2}$fm4, and $B^{(n)}_{\lambda=2}=25$ fm4 of 16C is reduced using the updated data of $B(E2)=2.6(9)$ $e^{2}$fm4 Ong:2006rm . The value of 16C is consistent with our value of $B^{(n)}_{\lambda=2}=27.0$ fm4, but the value of 12Be is much smaller than $B^{(n)}_{\lambda=2}=51.1$ fm4 of the present calculation. In the theoretical study of the proton scattering of 12Be with a MCC calculation using the same AMD densities Takashina:2008zza , the slightly smaller value $B^{(n)}_{\lambda=2}=37$ fm4 was favored to reproduce the inelastic cross sections.

The reaction analysis with the Bernstein prescription usually assumes the simple collective model transition densities given by the derivative of the matter density, and follows the relation of inelastic hadron ($h,h^{\prime}$) scattering cross sections with the transition matrix elements as

[TABLE]

where $b^{(h,h^{\prime})}_{n}$ and $b^{(h,h^{\prime})}_{p}$ are the neutron and proton field strengths of the external field from the hadron probe. For the proton scattering, $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}$ depends on the energy. A standard value $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}\approx 3$ at $E=10$–50 MeV/u, which was obtained from the data of various ordinary nuclei, is often used. The Bernstein prescription has been widely used for the inelastic proton scattering, but it greatly relies on the reaction model, which contains ambiguity such as the proton-nucleus optical potentials and the transition densities. The ansatz of Eq. (4) means the linear relation of the squared cross section with the neutron ($M_{n}$) and proton ($M_{p}$) transition matrix elements. The ratio $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}$ indicates the sensitivity of the cross sections to the neutron transition matrix element ($M_{n}$) relative to the proton part ($M_{p}$), and is supposed to be system independent. The linear relation could be valid only if the relation $\rho^{\textrm{tr}}_{n}(r)\propto\rho^{\textrm{tr}}_{p}(r)$ is satisfied.

However, it is not the case with 12Be and 16C, for which the neutron transition density has the outer amplitude than the proton part. Such the exotic behavior of the neutron transition density may give non-trivial effects on the relation between the cross sections and the transition matrix elements ($M_{n}$ and $M_{p}$). To see this effect we microscopically derive the ratio $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}$ within the present MCC approach and discuss how the sensitivity of the cross section to $M_{n}$ changes depending on the system as well as on the incident energy.

Here we assume that the AMD calculation gives correct $r$ dependence of the $\rho^{\textrm{tr}}_{n}(r)$ and $\rho^{\textrm{tr}}_{p}(r)$ but contains ambiguity of the overall factor in each of the neutron and proton parts. By artificially changing the overall factor of $\rho^{\textrm{tr}}_{n}(r)$ or $\rho^{\textrm{tr}}_{p}(r)$, we calculate the integrated cross sections and reduce the coefficients in the relation

[TABLE]

Here $a^{(p,p^{\prime})}_{n}$ and $a^{(p,p^{\prime})}_{p}$ are the system- and energy-dependent parameters determined from the calculated cross sections. The ratio $a^{(p,p^{\prime})}_{n}/a^{(p,p^{\prime})}_{p}$ is nothing but the ratio $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}$ to be discussed.

In Fig. 13, we show the obtained result of $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}=a^{(p,p^{\prime})}_{n}/a^{(p,p^{\prime})}_{p}$ for each systems, 18O, 10Be, 12Be, and 16C, with the default transition densities by open squares, crosses, filled circles, and filled triangles, respectively. For 12Be and 16C, we also show the result for the naive case of $\rho^{\textrm{tr}}_{n}(r)\propto\rho^{\textrm{tr}}_{p}(r)$ by open circles and triangles, respectively, which are obtained by the MCC calculation using $\rho^{\textrm{tr}}_{n}(r)=(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$. In all the cases, the calculated values of $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}$ show similar energy dependence, i.e., decreasing with the increase of $E$. This energy dependence mainly comes from the energy dependence of the effective $g$-matrix $NN$ interactions. In fact, if the nucleon-nucleus optical potentials fixed at $E=60$ MeV/u is used, $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}$ varies slightly from 1.90 (1.64) to 1.89 (1.57) in the energy range $30$–120 MeV/u for the proton scattering off 10Be (12Be).

At each energy, almost the same values of $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}$ are obtained for 18O and 10Be, and also by the calculation with the $\rho^{\textrm{tr}}_{n}(r)=(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$ assumption for 12Be and 16C. These values can be regarded as standard values for ordinary systems with $\rho^{\textrm{tr}}_{n}(r)\approx(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$. However, in the exotic case $\rho^{\textrm{tr}}_{n}(r)\neq(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$ of 12Be and 16C with the default transition densities, the values of $b^{(p,p^{\prime})}_{n}/b^{(p,p^{\prime})}_{p}$ significantly deviate from the standard values: they are smaller than the standard values by about 0.3 indicating that the sensitivity of the cross sections to $M_{n}$ is by about $15\%$ weaker than the ordinary case of $\rho^{\textrm{tr}}_{n}(r)=(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$. The reason for the weaker sensitivity of the cross sections is that the outer amplitude of the neutron transition density significantly contributes to $M_{n}$ but does not so much to the cross sections.

This result may suggest a possible modification of the phenomenological reaction analysis. For simplicity, let us suppose that there is no ambiguity in the reaction model except for the neutron transition density $\rho^{\textrm{tr}}_{n}(r)$, and other inputs are so reliable that the model can properly reproduce the cross sections for the ordinary case. If one performs an inconsistent analysis using $\rho^{\textrm{tr}}_{n}(r)=(M_{n}/M_{p})\rho^{\textrm{tr}}_{p}(r)$ for the exotic case, one could draw an underestimated value of $M_{n}$ from the observed cross sections.

V Summary

We investigated the proton inelastic scattering off 18O, 10Be, 12Be, and 16C to the $2^{+}_{1}$ states with the microscopic coupled-channel calculation. The proton-nucleus potentials are microscopically derived by folding the Melbourne $g$-matrix $NN$ interaction with the AMD densities of the target nuclei. The calculated result reasonably reproduces the elastic and inelastic proton scattering cross sections, and supports the dominant neutron contribution in the $2^{+}_{1}$ excitation of 12Be and 16C. In order to discuss the detailed behavior of transition densities, further high quality data of the differential cross sections are required.

The sensitivity of the inelastic scattering cross sections to the neutron transition density was discussed. A particular attention was paid on the exotic systems such as 12Be and 16C that the neutron transition density has the remarkable amplitude in the outer region than the proton part. This outer amplitude of the neutron transition density significantly contributes to the neutron matrix element $M_{n}$. However, its contribution to the inelastic cross sections is quite modest because the reaction process considered has no strong selectivity for the outer region. This result indicates that a phenomenological analysis with the Bernstein prescription is no longer valid. Our finding will suggest that a phenomenological analysis with collective model transition densities can result in an undershooting of $M_{n}$ for such exotic systems.

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 partly supported by Grants-in-Aid of the Japan Society for the Promotion of Science (Grant Nos. JP18K03617, JP16K05352, and 18H05407) and by the grant for the RCNP joint research project.

Appendix A Resummation factor in the folding model calculation

According to the multiple scattering theory for nucleon-nucleus scattering constructed by Kerman, McManus, and Thaler Ker59 , the transition matrix $T$ is given by

[TABLE]

where $A$ is the mass number of the nucleus and $T^{\prime}$ is the transition matrix corresponding to the Schrödinger equation

[TABLE]

$K$ is the kinetic energy operator, $h$ is the internal Hamiltonian of the nucleus, $E$ is the total energy, and $\Psi$ is the total wave function. $\tau_{j}$ is the effective interaction between the incident nucleon and a nucleon inside the nucleus, which is approximated to the Melbourne $g$-matrix $NN$ interaction in this study. The two factors, $A/(A-1)$ in Eq. (6) and $(A-1)/A$ in Eq. (7) appear as a result of the resummation of the $NN$ collisions originally written in terms of a bare $NN$ interaction. Although these resummation factors usually do not play a role, for nucleon scattering off a light nucleus especially at low energies, these may slightly modify the result as shown in Ref. Minomo:2017hjl . These factors are taken into account in all the calculations shown in the present calculation.