Shell-cluster transition in ⁴⁸Ti

Based on AQCM, we introduce shell model type (S-type) and cluster model type (C-type) wave functions. In both cases, the single-particle wave function has a Gaussian shape as in the Brink model [24]

where $\chi_{i}$ and $\eta_{i}$ are the spin and the isospin parts of the wave function, respectively. The parameter $\nu$ is a size parameter, and $\bm{\zeta}_{i}$ is the Gaussian center parameter. The total wave function $\Phi$ is the antisymmetrized product of these single-particle wave functions

where $\mathcal{A}$ is the antisymmetrizer and $A=48$ is the mass number.

These forty-eight single-particle wave functions consist of a ⁴⁰Ca core and eight valence nucleons. The ⁴⁰Ca core can be described as ten $\alpha$ clusters with small relative distances, which is consistent with the shell model description of ⁴⁰Ca due to the antisymmetrization effect. Each $\alpha$ cluster is defined as four nucleons (proton spin-up, proton spin-down, neutron spin-up, neutron spin-down) sharing a common value for the Gaussian center parameter $\bm{\zeta}_{i}$. The actual positions of the ten $\alpha$ clusters for the ⁴⁰Ca core are described in Ref. [12].

For the eight valence nucleons, we first introduce three $\alpha$ clusters (¹²C) around the ⁴⁰Ca core and then remove four protons afterwards. These three $\alpha$ clusters are introduced to have an equilateral triangular shape with a small relative distance around the ⁴⁰Ca core. The spin parts of the single particles in the three $\alpha$ clusters are also introduced with the equilateral triangular symmetry as described in Ref. [12]. These single-particle orbits in the three $\alpha$ clusters are excited to the $pf$-shell due to the antisymmetrization effect with the nucleons in the ⁴⁰Ca core. However, there is no spin-orbit contribution yet unless the $\alpha$ clusters are broken. Therefore, next, these single-particle orbits are transformed into the $f_{7/2}$ orbits of the $jj$-coupling shell model by giving the imaginary parts to the Gaussian center parameters based on the transformation of AQCM [11]

Here $\bm{R}_{i}$ represents the spatial location of the $i$th single particle, and $\bm{e}^{\rm spin}_{i}$ is a unit vector for the intrinsic spin. The imaginary parts of the Gaussian center parameters represent imparted momenta to the nucleons, and $\alpha$ clusters are broken in such a way that spin-up and spin-down nucleons are boosted in opposite directions and perform time-reversal motions. The parameter $\Lambda$ controls the breaking of the $\alpha$ clusters, and the ⁴⁸Ti wave function is constructed by removing four protons from the twelve nucleons around the ⁴⁰Ca core.

For the shell model wave function, S-type, to break clusters, the $\Lambda$ value is set to 1 for all the eight nucleons. In this way, the $jj$-coupling shell model wave function of ⁴⁸Ti [$(f_{7/2})^{2}$ for the protons and $(f_{7/2})^{6}$ for the neutrons around the ⁴⁰Ca core] is generated.

Next, we introduce the cluster model wave function, C-type. For the four nucleons (proton spin-up, proton spin-down, neutron spin-up, neutron spin-down) in the $f_{7/2}$ orbits around the ⁴⁰Ca core, we set $\Lambda=0$ in Eq. (3) and remove the imaginary part of the Gaussian center parameters; they are returned to an $\alpha$ cluster. This $\alpha$ cluster is separated from the rest (⁴⁴Ca) by the distance of $d$ fm. After setting all these Gaussian center parameters of S-type and C-type, the whole system is moved to satisfy the condition of $\sum_{i=1}^{48}\left<\bm{r}_{i}\right>=0$.

Once the model wave function $\Phi$ is set, the intrinsic density distribution $\tilde{\rho_{t}}(\bm{r})$ is obtained by calculating the expectation value of $\sum_{i\in t}\delta(\bm{r}_{i}-\bm{r})$,

where the summation is taken over protons ($t=p$) or neutrons ($t=n$). The center-of-mass wave function can be eliminated by using a Fourier transform [25] as

and we use $\rho^{\rm int}_{t}(\bm{r})$ as the intrinsic density free of center of mass motion. The density distribution in the laboratory frame is finally obtained by averaging the intrinsic density distribution over the angles [26] as

Proton-nucleus elastic scattering is one of the most direct ways of obtaining information on the density profile. We remark that the full density distribution can be obtained by measurements up to backward angles [27, 28]. As long as the nuclear surface density is of interest, only the cross sections at the forward angles, i.e., the cross section at the first peak in the proton-nucleus diffraction is needed to extract the “diffuseness” of the density distribution as prescribed in Ref. [29]. To relate the density profile to the reaction observables we employ a high-energy microscopic reaction theory, the Glauber model [30].

The differential cross section of the elastic scattering is given by

with the scattering amplitude of the nucleus-nucleus elastic scattering [31]

where $F_{C}(\theta)$ is the Rutherford scattering amplitude, $\bm{b}$ is the impact parameter vector, and $\eta$ is the Sommerfeld parameter. As the nuclear scattering occurs in several hundred MeV, relativistic kinematics is used for the wave number $k$.

Here, we treat proton- or $\alpha$-target nucleus ($xT$; $x=p$ or $\alpha$, T: target nucleus) system. The optical phase shift function $\chi_{xT}$ contains all the dynamical information for the $xT$ system within the Glauber model, but its evaluation involves multiple integration. For practical calculations, the optical limit approximation (OLA) [30, 31] is used to compute the optical phase shift function as

for a proton-nucleus system, where a single-particle coordinate is expressed by $\bm{r}=(\bm{s},z)$ with $z$ being the beam direction. Further, we evaluate the optical phase shift function for an $\alpha$-nucleus system with

where $\rho^{\alpha}$ is the intrinsic density distribution of the $\alpha$ particle with the $(0s)^{4}$ harmonic oscillator configuration and a size parameter reproducing the measured charge radius. The parameterization of the proton-proton (neutron-proton) profile function $\Gamma_{pp}=\Gamma_{nn}$ ($\Gamma_{pn}=\Gamma_{np}$) is given in Ref. [32]. Once the above inputs are set, the theory has no tunable parameter, and so, the resulting reaction observables are a direct reflection of the density profiles of the target nucleus. This model works well as shown, for example, in Refs. [33, 34], and its accuracy compared to those obtained by the full evaluation of the optical phase shift function was discussed in Refs. [35, 36, 37, 34].

Here we generate the shell-model-like (S-type) and cluster-model-like (C-type) configurations and show their properties. The C-type is further subdivided into C-type-1, C-type-2, and C-type-3 depending on the size parameter $\nu$ in Eq. (1) and the $\alpha$–⁴⁴Ca distance $d$. All these configurations reproduce the experimental charge radius of ⁴⁸Ti [38].

Figure 1 (a) and (b) show the point-proton and point-neutron density distributions of ⁴⁸Ti, respectively, as a function of $r$, the distance from the origin. Despite the fact that all these density distributions give the same charge radii, they have different density profiles. In the following subsection, we will discuss the proton-⁴⁸Ti elastic scattering, which is sensitive to the density in the region that the value is about half of the central one. This half-density region corresponds to $r\approx 3$ fm, which we call the surface region. Meanwhile, the $\alpha$ knockout reaction is sensitive to the wave function $r\mathop{\vbox{ \offinterlineskip\hbox{$>$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}5$ fm [6], which can be called the tail region. To quantify the density profiles at around the half-density region, it is convenient to evaluate the nuclear diffuseness for proton ($a_{p}$), neutron ($a_{n}$), and matter ($a_{m}$) density distributions  [29] by minimizing

with

Table 1 lists those calculated diffuseness values. The S-type configuration exhibits the smallest diffuseness values, i.e., the sharpest nuclear surface, and the nuclear surface becomes more diffused as the $\alpha$–⁴⁴Ca cluster structure develops. As we will see later, differences in these diffuseness values are actually reflected in the elastic scattering cross sections.

The difference in the density distributions becomes more visible when $r^{2n}$ ($n$: integer) is multiplied. Figure 2 shows the matter density distributions of ⁴⁸Ti ($\rho_{m}$) multiplied by (a) $r^{2}$ and (b) $r^{4}$. The integration of $4\pi r^{2}\rho_{m}$ over $r$ gives the particle number, and thus, in Fig. 2 (a), the four areas that the four lines create together with the horizontal axis are equal. We see some differences beyond the half-density radius, $r\mathop{\vbox{ \offinterlineskip\hbox{$>$}\hbox to7.7778pt{\hss\hbox{$\sim$}\hss}}}3$ fm in the $r^{2}\rho_{m}$ distribution. In Fig. 2 (b), the distribution of C-type-3 is significantly shifted to the larger $r$ side compared to the other three lines. This characteristic feature of C-type-3 shown in the $r^{4}\rho_{m}$ distribution stems from its large clustering, which affects the $\alpha$-nucleus elastic scattering cross section at the first diffraction peak, where the signature of the large $\alpha$ clustering is embedded, as we will discuss later.

These differences between the density distributions are reflected in the diffraction patterns of the proton-nucleus elastic scattering. Figure 3 shows the differential cross section for the proton-⁴⁸Ti elastic scattering. The proton incident energy is chosen to be 1000 MeV, and the experimental data (incident energy of 1044 MeV) are taken from Ref. [39]. Three of the four nuclear densities, except for C-type-3, are in reasonable agreement with the experimental result up to the second peak [Fig. 3 (a)], but for a more accurate comparison, we plot the cross sections in a linear scale in Fig. 3 (b) and (c). The angle and height of the first peak position reflect the size and diffuseness of the target nucleus, and Fig 3 (b) shows the experimental data around this region is best reproduced by the S-type configuration. This means that the ⁴⁸Ti nucleus has a density distribution close to the $jj$-coupling shell model picture around its surface region. Figure 3 (c) shows the cross sections around the second peak, and the agreement between the experimental data and the result of the S-type becomes even better than for the first peak region.

What regions of the density profiles are actually observed? To answer this question, it is intuitive to look at the radial dependence of the scattering amplitude, i.e., integrand of the second term of Eq. (8) at a specific scattering angle [29], which is explicitly written by

It is worthwhile to recall the relation

We set $\theta$ near the first and second diffraction peaks and compare $g$ with the different AQCM configurations as a function of the impact parameter $b$. Here we take S-type and C-type-3 configurations, where the most different results are expected.

Figure 4 displays the absolute difference of the real and imaginary parts of $g$ between S-type and C-type-3 configurations for the differential elastic scattering cross sections at around (a) the first peak ($\theta=7.8^{\circ}$) and (b) the second peak ($\theta=13^{\circ}$) positions. At the first peak position [Fig. 4 (a)], the absolute difference in $g$ is largest at the surface region $b\approx 4$ fm, which is consistent with our basis that the nuclear diffuseness is most reflected in the first diffraction peak [29], considering that the range of the nucleon-nucleon interaction is about 1 fm. At the second peak position [Fig. 4 (b)], the difference is also largest at the surface region. The second peak region includes the information of the density wider than the surface region, and there the picture of the $jj$-coupling shell model works well. Here, the contribution around the tail region ($b\approx 6$ fm) in Fig. 4 (b) is reduced from that in Fig. 4 (a) compared to the reduction of the surface region ($b\approx 4$ fm), and thus, the second peak reflects the difference of S-type and C-type-3 around the surface region more pronouncedly than the first peak.

From the analysis of the proton-nucleus elastic scattering, we have found that the ⁴⁸Ti nucleus has a $jj$-coupling shell model structure rather than the $\alpha+^{44}$Ca cluster structure. However, as discussed earlier, the $\alpha$ knockout reaction cross section, which is sensitive to the tail region of the wave function, is explained by the $\alpha+^{44}$Ca cluster structure with a large relative distance.

Indeed, we can deduce the vestige of the $\alpha$ clustering in the tail region of the wave function from the $\alpha$-nucleus elastic scattering cross section. Figure 5 shows the differential cross section of the $\alpha$ scattering on ⁴⁸Ti at 240 MeV. Here, (a) and (b) show the results in the logarithmic and linear scales, respectively. As can be recognized in Fig. 5 (b), the density of C-type-3 best reproduces the experimental data [41]. Figure 6 draws the absolute difference in $g$ between the S-type and C-type-3 configurations at around the first peak position of the $\alpha$-⁴⁸Ti elastic scattering cross sections ($\theta=7.2^{\circ}$). No difference appears at $b\lesssim 4$ fm because the $\alpha$-nucleus scattering is strongly absorptive. The difference is peaked at $b\approx 5$ fm, which corresponds to the sum of the matter radii of $\alpha$ and ⁴⁸Ti. The $\alpha$-nucleus scattering has no sensitivity to the inner region and is only sensitive to the outer region of the nuclear density, while the proton-nucleus scattering probes the density profile near the nuclear surface. Thus, it could be interpreted that while the surface region of ⁴⁸Ti is explained by the $jj$-coupling shell model configuration, the tail region is better explained by the $\alpha$ clustering configuration. This result suggests the possibility of a change in structure as a function of the distance from the center, from the $jj$-coupling shell model to the cluster model. We remark that a similar phenomenon has been discussed in ⁴⁴Ti, where the $\alpha$ cluster structure is completely broken in the region at small $\alpha$–⁴⁰Ca distances due to the strong spin-orbit contribution. However, with increasing relative distances, the $\alpha$ cluster structure appears beyond the interaction range of the spin-orbit interaction from the ⁴⁰Ca nucleus. Here, the tensor interaction plays a crucial role in the $\alpha$ clustering [46].