Quantitative description of the $^{20}$Ne($p$,$p\alpha$)$^{16}$O cross section as a means of probing the surface $\alpha$ amplitude

TL;DR

This study uses a theoretical framework to analyze the $^{20}$Ne($p$,$peta$)$^{16}$O reaction, demonstrating its effectiveness in probing the surface $eta$ amplitude of $eta$ cluster wave functions through knockout cross sections.

Contribution

It provides a quantitative description linking the $eta$ cluster wave function to the knockout cross section, emphasizing the reaction's peripherality as a probe for surface $eta$ amplitudes.

Findings

The framework reproduces experimental cross sections at 101.5 MeV.

Surface $eta$ amplitude is effectively probed by the ($p$,$peta$) cross section.

Reaction peripherality reflects the surface region of the cluster wave function.

Abstract

The proton-induced $\alpha$ knockout reaction has been utilized for decades to investigate the $\alpha$ cluster states of nuclei, of the ground state in particular. However, even in recent years, it is reported that the deduced $\alpha$ spectroscopic factors from $\alpha$ knockout experiments and reaction analyses with a phenomenological $\alpha$ cluster wave function diverge depending on the kinematical condition of the reaction. In the present study we examine the theoretical description of the $^{20}$Ne($p$,$p\alpha$)$^{16}$O cross section based on the antisymmetrized molecular dynamics and the distorted wave impulse approximation by comparing with existing experimental data. We also investigate the correspondence between the $\alpha$ cluster wave function and the $\alpha$ knockout cross section. The existing $^{20}$Ne($p$,$p\alpha$)$^{16}$O data at 101.5 MeV is well reproduced by the present framework. Due to the peripherality of the reaction, the surface region of the cluster wave function is selectively reflected to the knockout cross section. A quantitatively reliable $\alpha$ cluster wave function, $p$-$\alpha$ cross section, and distorting potentials between scattering particles, $\alpha$-$^{16}$O in particular, are crucial for the quantitative description of the ($p$,$p\alpha$) cross section. Due to the peripherality of the reaction, the ($p$,$p\alpha$) cross section is a good probe for the surface $\alpha$ amplitude.

Tables

Table 1. Table 1: Coefficients for the BB wave functions. The resulting S α subscript 𝑆 𝛼 S_{\alpha} are also shown.

	original	smaller peak	shorter tail
$C_1$	0.67	0.55	0.69
$C_2$	0.17	0.17	0.09
$S_{\alpha }$	0.24	0.18	0.21

Equations

$$y(R)=\sqrt{\frac{20!}{16!4!4\pi }}⟨\frac{\delta (r-R)}{r^2}{\Phi }_{\alpha }{\Phi }_{{}^{16}\mathrm{O}}⟩{\Psi }_{{}^{20}\mathrm{Ne}}.$$

$$S_{\alpha }={\int }_0^{\infty }{R}^{2}dR|y(R){|}^2.$$

$$\frac{d^3\sigma }{d{E}_1^{\mathrm{L}}d{\Omega }_1^{\mathrm{L}}d{\Omega }_2^{\mathrm{L}}}$$

$$F_{\mathrm{kin}}$$

$$C_0$$

$${\bar{T}}_{{\mathbf{K}}_i}$$

$$F_{{\mathbf{K}}_i}(\mathbf{R})$$

$$\hat{H}$$

$${\delta }^{\mathrm{Tr}}(R)$$

$$\int dR\mathrm{Re}[{\delta }^{\mathrm{Tr}}(R)]$$

$$\int dR\mathrm{Im}[{\delta }^{\mathrm{Tr}}(R)]$$

Full text

The proton-induced $\alpha$ knockout reaction has been utilized for decades to investigate the $\alpha$ cluster formation in the ground state of nucleus. However, even today, the theoretical description of the reaction is not precise enough for the quantitative study. For example, the $\alpha$ spectroscopic factors reduced from $\alpha$ knockout experiments with reaction analyses using phenomenological $\alpha$ cluster wave functions disagree with those given by a structure theory. In some cases they also scatter depending on the kinematical condition of the experiment. This suggests that the theoretical description of the $\alpha$ knockout reaction is insufficient from a quantitative viewpoint.

Purpose

We show that the distorted wave impulse approximation can describe 20Ne($p$,$p\alpha$)16O reaction quantitatively if reliable inputs are used; the optical potential, the $p$-$\alpha$ cross section, and the $\alpha$ cluster wave function. We also investigate the relationship between the $\alpha$ cluster wave function and the $\alpha$ knockout cross section.

Method

The 20Ne($p$,$p\alpha$)16O reaction is described by the distorted wave impulse approximation. An input of the calculation, the $\alpha$-16O cluster wave function, is obtained by the antisymmetrized molecular dynamics and the Laplace expansion method.

Results

In contrast to the previous work, the 20Ne($p$,$p\alpha$)16O data at 101.5 MeV is successfully reproduced by the present framework without any free adjustable parameters. It is also found that the knockout cross section is sensitive to the surface region of the cluster wave function because of the peripherality of the reaction.

Conclusions

Using a reliable $\alpha$ cluster wave function, $p$-$\alpha$ cross section, and distorting potentials, it is found that the 20Ne($p$,$p\alpha$)16O cross section is quantitatively reproduced by the present framework. This success demonstrates that the proton-induced $\alpha$ knockout reaction is a quantitative probe for the $\alpha$ clustering.

††preprint: NITEP 22

I Introduction

As it is schematically illustrated in the Ikeda diagram Ikeda et al. (1968), the $\alpha$ particle is expected to emerge as a subunit in nuclear systems, in the light mass region in particular, reflecting the fact that the $\alpha$ particle is a tightly bound system of four nucleons with the large first excitation energy compared to neighboring nuclei. Various cluster models have been developed so far, and in recent decades microscopic cluster models based on the nucleon degrees of freedom with fermionic quantum statistics are available, as reviewed in a recently published article Freer et al. (2018). Among them, the antisymmetrized molecular dynamics (AMD) has been applied to many systems and succeeded in describing cluster structures  Kanada-En’yo et al. (1995); Kanada-En’yo (1998); Kanada-En’yo and Horiuchi (2001); Kimura (2004); Kanada-En’yo et al. (2012); Kimura et al. (2016); Chiba et al. (2016); Kanada-En’yo (2016a, b); Chiba and Kimura (2017). One of the questions remaining today is to what extent $\alpha$ cluster state exists in the nuclear ground states that lie far below the $\alpha$ threshold.

From a nuclear reaction point of view, the proton-induced $\alpha$ knockout reaction, ($p$,$p\alpha$), is considered to be a good probe for the $\alpha$ cluster state in the ground state of a target nucleus. Much effort has been made on the ($p$,$p\alpha$) reaction studies Roos et al. (1977); Nadasen et al. (1980); Carey et al. (1984); Wang et al. (1985); Nadasen et al. (1989); Yoshimura et al. (1998); Neveling et al. (2008); Mabiala et al. (2009); Yoshida et al. (2016); Lyu et al. (2018); Yoshida et al. (2018), but even today quantitative understanding of the ($p$,$p\alpha$) cross section and its relation with the $\alpha$ cluster wave function in the ground state of the target nucleus have not yet fully established. For example, in Ref.Carey et al. (1984) it is reported that the $\alpha$ spectroscopic factor ($S_{\alpha}$) of the $\alpha$-16O cluster state in the ground state of 20Ne deduced from the 20Ne($p$,$p\alpha$)16O experiment is 2.4–3.0 times as large as those given by structure theories. It is also reported in Ref. Mabiala et al. (2009) that $S_{\alpha}$ of 12C from the 12C($p$,$p\alpha$)8Be reaction at 100 MeV has large uncertainty of $S_{\alpha}=0.19$–$1.68$ depending on the kinematics of the experiment, while $S_{\alpha}$ should be determined purely from the structure of 12C and should not depend on the kinematical condition of the reaction. Those uncertainties may arise from the ambiguities in the reaction analyses, namely, the optical potentials, the $\alpha$ cluster wave function, and the $p$-$\alpha$ differential cross section and its energy and angular dependence.

Considering the above-mentioned situation, a qualitative description of the ($p$,$p\alpha$) reaction is still challenging. In the present study, we aim to reproduce the 20Ne($p$,$p\alpha$)16O cross section data Carey et al. (1984) within the distorted wave impulse approximation (DWIA) framework Chant and Roos (1983); Wakasa et al. (2017). Since the target and the residue of this reaction are typical stable nuclei and the distorting potentials of them are well known, the description of the reaction will be free from ambiguities of the optical potential. In addition, in the experiment the emission angles of the $p$ and $\alpha$ are fixed and the corresponding $p$-$\alpha$ binary scattering angle is very limited. This may also help to reduce the ambiguity arising from the angular dependence of the $p$-$\alpha$ cross section required in the DWIA calculation. As for the $\alpha$ cluster wave function, the $\alpha+^{16}$O reduced width amplitude (RWA) of the 20Ne ground state obtained by the AMD framework Chiba and Kimura (2017) is adopted as an input of the DWIA calculation.

In Sec. II, the theoretical description of AMD and DWIA for the $\alpha+^{16}$O cluster state and the 20Ne($p$,$p\alpha$)16O reaction, respectively, are introduced. In Sec. III, numerical input for the calculations is explained, and a comparison between the obtained 20Ne($p$,$p\alpha$)16O cross section and the experimental data are shown. The probed region of the reaction is also investigated by applying several Brink-Bloch wave functions Brink of the $\alpha$-16O cluster state to the reaction calculation. Finally, a summary is given in Sec. IV.

II Theoretical framework

II.1 Reduced width amplitude

The RWA is the probability amplitude to find the clusters at intercluster distance $R$, and is defined as the overlap between the ground state wave function of ${}^{20}{\rm Ne}$ and the reference wave function for the $\alpha$+${}^{16}{\rm O}$ clustering,

[TABLE]

The integral of the RWA called $\alpha$ spectroscopic factor $S_{\alpha}$ is often used as a measure of the clustering,

[TABLE]

In the definition of the RWA given by Eq. (1), the bra vector represents the reference cluster state, in which the $\alpha$ and ${}^{16}{\rm O}$ clusters are coupled to angular momentum zero with the inter-cluster distance $R$. The ground state wave functions of the $\alpha$ and ${}^{16}{\rm O}$ clusters ($\Phi_{\alpha}$ and $\Phi_{{}^{16}{\rm O}}$) are the harmonic oscillator wave functions of double closed shell configurations. The ket state $\Psi_{{}^{20}{\rm Ne}}$ is the ground state wave function of ${}^{20}{\rm Ne}$ calculated by AMD. The AMD wave function is a superposition of parity and angular momentum projected Slater determinants, and it was shown that the known properties of the 20Ne ground and excited states such as the radius, energies, electromagnetic transitions, and $\alpha$ decay widths are reasonably described. Once the ground state wave function is obtained, Eq. (1) is calculated by using the Laplace expansion method Chiba and Kimura (2017). For more details of the calculation, readers are directed to Refs. Kimura (2004); Chiba et al. (2016). It should be noted that other cluster models yield similar RWA and reasonably reproduce the observed decay widths of the excited states Nemoto and Bandō (1972); Matsuse et al. (1975); Fujiwara (1979); Kanada-En’yo et al. (2014); Kimura (2004). The use of AMD in the present study is aimed to extend our framework to the investigation of the $\alpha$ clustering in unstable nuclei in the future.

II.2 Distorted wave impulse approximation

We employ the DWIA framework in the present study. Details of DWIA for the description of the knockout reaction can be found in a recent review paper Wakasa et al. (2017). The incident and emitted protons are labeled as particles 0 and 1, respectively. The wave number and its solid angle, the total and kinetic energies of particle $i$ ($=0,1,\alpha$) are represented by $\bm{K}_{i}$, $\Omega_{i}$, $E_{i}$, and $T_{i}$, respectively. Quantities with (without) the superscript L are evaluated in the laboratory (center-of-mass) frame.

The triple differential cross section (TDX) of the $A$($p$,$p\alpha$)$B$ reaction within the so-called factorized form of the DWIA framework is given by

[TABLE]

It is essentially a product of the absolute square of the reduced transition matrix $\bar{T}_{\bm{K}_{i}}$ and the $p$-$\alpha$ two-body differential cross section $d\sigma_{p\alpha}/d\Omega_{p\alpha}$ at a given scattering angle (energy) $\theta_{p\alpha}$ ($T_{p\alpha}$). To relate an off-the-energy-shell $p$-$\alpha$ scattering in the ($p,p\alpha$) three-body kinematics to an on-shell observable, the final-state prescription of the on-shell approximation is adopted; $T_{p\alpha}$ is determined by the $p$-$\alpha$ relative momentum in the final state. The kinematical factor (or also referred to as the phase space factor) $F_{\mathrm{kin}}$ and a constant $C_{0}$ are defined by

[TABLE]

where $\mu_{p\alpha}$ is the reduced mass of the $p$-$\alpha$ binary system and $J_{\mathrm{L}}$ is the Jacobian from the center-of-mass frame to the L frame. The reduced transition matrix is given by

[TABLE]

where $A_{\alpha}=4$, $A=20$, and $\chi_{i,{\bm{K}_{i}}}$ is a distorted wave between particle $i$ and $A$ when $i=0$, otherwise between $i$ and $B$. The outgoing and incoming boundary conditions of the scattering waves are specified by the superscripts $(+)$ and $(-)$, respectively. Note that the $S_{\alpha}$ does not appear explicitly in Eqs. (3)–(7) because it is already taken into account in the RWA.

III Results and discussion

III.1 Numerical inputs

The RWA is calculated by AMD and the Laplace expansion method Chiba and Kimura (2017). The Hamiltonian used in the AMD calculation is given by

[TABLE]

where $\hat{t}_{i},\hat{t}_{\mathrm{c.m.}},\hat{v}_{NN}$, and $\hat{v}_{\mathrm{Coul}}$ being the nucleon and the center-of-mass kinetic energies, effective nucleon-nucleon interaction, and the Coulomb interaction, respectively. As for $\hat{v}_{\mathrm{NN}}$, the Gogny D1S interaction Berger et al. (1991) is adopted. The obtained RWA Chiba and Kimura (2017) (AMD-RWA) is shown in Fig. 1, and its $\alpha$ spectroscopic factor is $S_{\alpha}=0.26$.

The kinematical condition of the 20Ne($p$,$p\alpha$)16O reaction is shown in Fig. 2. In the experiment Carey et al. (1984) the so-called energy sharing distribution was measured; the emitted angle of $p$ ($\alpha$) is fixed at $-70^{\circ}$ ($46.3^{\circ}$), and the proton emission energy $T_{p}$ is varied in the range of $40$–$75$ MeV. By the energy conservation, $T_{\alpha}$ is ranging from $55$ to $21$ MeV accordingly.

The incident energy is set to 101.5 MeV and all the scattering particles are kept in the same plane. With such a kinematical setup with no recoil, i.e., the reaction residue being at rest in the final state, is achieved at around $T_{p}=67$ MeV ($T_{\alpha}=30$ MeV). The TDX has a peak at this condition reflecting the fact that the struck $\alpha$ is bound by an $s$ orbit. For the optical potential of the incident and emitted protons, the EDAD1 optical potential of the Dirac phenomenology Hama et al. (1990); Cooper et al. (1993) is adopted. The highly reliable optical model parametrization by F. Michel et al. Michel et al. (1983) is adopted for the $\alpha$-16O scattering in the final state. As shown in Fig. 1 of Ref. Michel et al. (1983), this parametrization of the $\alpha$-16O optical potential provides excellent agreement with the $\alpha$-16O elastic scattering data up to very backward angles, ranging 30–150 MeV $\alpha$ incident energies.

Since $\theta_{p}$ and $\theta_{\alpha}$ are fixed in the present ($p$,$p\alpha$) kinematics, the required $p$-$\alpha$ differential cross section lies in the very limited range of $\theta_{p\alpha}=84^{\circ}$–$86^{\circ}$ in the $p$-$\alpha$ center-of-mass frame and $E_{p\alpha}=75$–$100$ MeV in the $p$-$\alpha$ laboratory frame. Its energy and angular dependence are obtained by the microscopic single-folding model Toyokawa et al. (2013) with a phenomenological $\alpha$ density and the Melbourne nucleon-nucleon g-matrix interaction Amos et al. (2000). As shown in the dashed line in Fig. 3, the calculated $p$-$\alpha$ differential cross section at 85 MeV deviates from the experimental data Votta et al. (1974) by about a factor of 2 at around $\theta_{p\alpha}=84^{\circ}$–$86^{\circ}$. In the following analysis the cross section is tuned by a factor of 2.0 (dotted line) to correctly reproduce the $p$-$\alpha$ cross section data. It should be noted that this correction is valid thanks to the limited range of required $\theta_{p\alpha}$, which is the outcome of the kinematical condition in which $\theta_{p}$ and $\theta_{\alpha}$ are fixed in the 20Ne($p$,$p\alpha$)16O experiment.

III.2 $\alpha$ cluster wave function and 20Ne($p$,$p\alpha$)16O cross section

In Fig. 4 the energy sharing distribution calculated by DWIA with the AMD-RWA is shown.

The present framework well reproduces both the height and distribution of the experimental data Carey et al. (1984) without any additional adjustment. Thus, the original problem, the inconsistency between the reaction analysis and the nuclear structure calculations, is resolved in the present approach. It should be stressed that the quantitative reproduction of the cross section data is guaranteed by the sophisticated RWA in 20Ne, by the proper $p$-$\alpha$ cross section, and by the use of an appropriate $\alpha$-16O optical potential. In particular, $\alpha$-16O optical potential suggested in Ref. Michel et al. (1983) was essential for this success of quantitative description. This result indicates that once the proper ingredients mentioned above are adopted, the ($p$,$p\alpha$) reaction can be a quantitative probe for the $\alpha$ amplitude in the ground state of target nuclei.

Next, we investigate the relationship between the $\alpha$ cluster wave function and the $\alpha$ knockout cross section. Three different types of cluster wave functions shown in Fig. 5 are considered as artificial input data.

They are constructed by superposing two Brink-Bloch (BB) wave functions Brink of the $\alpha$-16O cluster. The inter-cluster distances of the two BB wave functions are fixed at 3.0 fm and 5.5 fm, whereas the amplitude, $C_{1}$ and $C_{2}$, of these are varied by hand. In Table 1, we show $C_{i}$ ($i=1$ or $2$) and the resulting $S_{\alpha}$ of the three RWAs.

As shown in Fig. 5, $C_{i}$ of one labeled “original” (dashed) is determined so as to reproduce the largest peak and the tail behavior of the AMD-RWA as much as possible, while for “smaller peak” (dotted) and “shorter tail” (dot-dashed), they are tuned to reduce the peak and tail region of the original RWA, respectively.

To investigate how these RWAs contribute to the TDX at the peak of the energy sharing distribution, it is useful to consider the transition matrix density (TMD) defined by Wakasa et al. (2017)

[TABLE]

From Eq. (6) it has the following properties:

[TABLE]

Thus, one may regard $\mathrm{Re}[\delta^{\mathrm{Tr}}(R)]$ as a “radial distribution of the cross section” and its integrated value is proportional to the TDX. In Fig. 6 $\mathrm{Re}[\delta^{\mathrm{Tr}}(R)]$ at the recoilless condition ($T_{p}=67$ MeV) is shown.

It is clearly seen that the internal region of the RWAs are suppressed by the absorption effect and the surface region contributes to the ($p$,$p\alpha$) cross section. In Fig. 7 the energy sharing distributions with the RWAs of Fig. 5 are shown. The peak height of the energy sharing distribution is reduced significantly in the shorter tail case (dot-dashed), while it is similar to the original one in the smaller peak case (dotted), because of the peripherality of the reaction.

Another response of the difference in the RWAs is the width of the energy sharing distribution. Since the RWA with smaller peak (shorter tail) has narrower (wider) momentum distribution, its energy sharing distribution has a narrower (wider) width accordingly. From these results it is shown that the ($p$,$p\alpha$) reaction is a quantitative probe for the $\alpha$ cluster state, putting emphasis on the $\alpha$ amplitude around the nuclear surface.

IV Summary

The 20Ne($p$,$p\alpha$)16O reaction at 101.5 MeV was investigated within the DWIA framework. AMD was adopted to describe the $\alpha$-16O cluster state of 20Ne and its RWA was obtained by the Laplace expansion method. It was found that by using reliable $p$-$\alpha$ differential cross section, distorting potentials as well as the AMD-RWA, the present DWIA calculation quantitatively reproduces the experimental data without any additional correction or scaling in describing the ($p$,$p\alpha$) cross section. Thus, the inconsistency between the 20Ne($p$,$p\alpha$)16O reaction analysis and the nuclear structure calculations is clearly resolved in the present approach.

By the analyses using the BB wave functions with different spatial distributions and corresponding $\alpha$ spectroscopic factors, it was shown that the surface region of the target is selectively probed by the 20Ne($p$,$p\alpha$)16O reaction. Therefore the ($p$,$p\alpha$) reaction is a good probe for the surface amplitude of the cluster wave function, which should be directly related to the $\alpha$ clustering of interest. One may also deduce the momentum distribution of the $\alpha$ cluster wave function from the width of the energy sharing distribution. For the momentum distribution of the $\alpha$ cluster to be extracted, the ($p$,$p\alpha$) cross section should be measured very precisely down to the tail region.

ACKNOWLEDGMENTS

This work was supported in part by Grants-in-Aid of the Japan Society for the Promotion of Science (Grants No. JP18K03617, JP16K05352, JP19K03859, and JP16K05339), the Hattori Hokokai Foundation Grant-in-Aid for Technological and Engineering Research, and by the grant for the RCNP joint research project.