Proposal of a directly measurable parameter quantifying the halo nature of one-neutron nuclei

We propose a measurable parameter $\mathcal{H}$ quantifying the halo nature of one-neutron halo nuclei $\left(a\right)$ and investigate the properties of $\mathcal{H}$, assuming the core + neutron $(c+n)$ model for $a$. The parameter $\mathcal{H}$ is defined by $\mathcal{H}=[{\sigma }_{\mathrm{abs}}\left(a\right)-{\sigma }_{\mathrm{abs}}\left(c\right)]/{\sigma }_{\mathrm{abs}}\left(n\right)$ with directly measurable absorption cross sections ${\sigma }_{\mathrm{abs}}$ of $a$, $c$, and $n$ scattering at the same incident energy per nucleon. It varies with the one-neutron separation energy $S_n$ in a range of $0\le \mathcal{H}\le 1$, and the halo structure is most developed when $\mathcal{H}=1$. This situation is realized only for $s$-wave halo nuclei in the $S_n=0$ limit. We consider ${}^{11}\mathrm{Be}$ and ${}^{15,19}\mathrm{C}$ as $s$-wave halo nuclei, ${}^{31}\mathrm{Ne}$ and ${}^{37}\mathrm{Mg}$ as $p$-wave halo nuclei, and ${}^{17}\mathrm{C}$ as an example of $d$-wave nonhalo nuclei. For each of halo nuclei, the value of $\mathcal{H}$ is deduced at a measured $S_n$ from measured total reaction cross sections for $c$, $n$, and $a$ scattering at intermediate and high incident energies where projectile breakup effects are negligible. The location of the resulting $(S_n,\mathcal{H})$ is plotted in the $S_n-\mathcal{H}$ plane. The empirical values of $\mathcal{H}$ at the measured $S_n$ are extrapolated to small $S_n$ with model calculations based on the eikonal + adiabatic approximation. In the $S_n-\mathcal{H}$ plane, the model lines are well separated into three groups of $s$-wave halo, $p$-wave halo, and $d$-wave nonhalo particularly in the vicinity of $S_n=0$, and the $s$-wave halo lines are always above the other lines, since only the $s$-wave halo lines can reach a point $(S_n,\mathcal{H})=(0,1)$ independently of the concrete form of the interaction between $c$ and $n$. The relation among the three kinds of lines may be universal for any halo nucleus with small $S_n$. The point $(S_n,\mathcal{H})=(0,1)$ can be regarded as a scale-invariant point in the sense that the $z$-integrated neutron density characterizing halo structure is invariant under the scale transformation there.

Figure 1

Three-body system.

Figure 2

Total reaction cross sections for $p+{}^{12}\mathrm{C}$ scattering as a function of $E_{\mathrm{in}}$.

Figure 3

Location of halo nuclei in the parameter $S_n-\mathcal{H}$ plane. Experimental or empirical data on $S_n$ are taken from Refs. [35, 36, 37]. See the text for the derivation of $\mathcal{H}$.

Figure 4

Root mean square (rms) matter radii of core and halo nuclei. The open squares stand for the results of the present model, while the closed circles, closed triangles, and closed squares denote the experimental data deduced from ${\sigma }_{\mathrm{R}}$ and ${\sigma }_{\mathrm{I}}$ with the optical-limit Glauber model based on harmonic-oscillator projectile densities (OL-HO), the optical-limit Glauber model based on few-body model densities (OL-FB) [9, 10], and the double-folding model based on Woods-Saxon model densities (DF-WS) [15]. For ${}^{12}\mathrm{C}$, the closed circle represents the phenomenological density [33] determined from electron scattering.

Figure 5

Behavior of $\mathcal{H}$ in the vicinity of $S_n=0$. A logarithmic scale is taken for the horizontal axis. The theoretical results are shown by a solid (dotted) line for ${}^{11}\mathrm{Be}\left({}^{19}\mathrm{C}\right)$, by a dashed (long and short dashed) line for ${}^{31}\mathrm{Ne}\left({}^{37}\mathrm{Mg}\right)$, by a long and two short dashed line for ${}^{17}\mathrm{C}$ and by a short and two long dashed line for ${}^{15}\mathrm{C}$. See Fig. 3 for the experimental data.

Figure 6

Same as Fig. 5 but for the probability parameter $P$ defined by Eq. (8).

Figure 7

Halo parameter $\mathcal{H}$ as a function of $E_{\mathrm{in}}/A_{\mathrm{P}}$ for ${}^{15}\mathrm{C}+{}^{12}\mathrm{C}$ scattering. The empirical value of $\mathcal{H}$ is taken from Fig. 3. The model results are shown by open squares at $E_{\mathrm{in}}/A_{\mathrm{P}}=83$ and 960 MeV. Here, the normalization factors are $(f_c,f_N)=(0.54,1)$ for $E_{\mathrm{in}}/A_{\mathrm{P}}=960$ MeV and $(f_c,f_N)=(0.60,0.60)$ for $E_{\mathrm{in}}/A_{\mathrm{P}}=83$ MeV. The model results are extrapolated to other $E_{\mathrm{in}}$ by assuming that the normalization factors $f_c$ and $f_N$ do not depend on $E_{\mathrm{in}}$. The solid (dashed) line stands for the model result of $(f_c,f_N)=(0.54,1)[(f_c,f_N)=(0.60,0.60)]$.

Figure 8

$b$ dependence of $z$-integrated projectile densities ${}^{11}\mathrm{Be}$, ${}^{17}\mathrm{C}$, and ${}^{31}\mathrm{Ne}$. The solid, dashed, and dotted lines denote the $b$ dependence for ${}^{11}\mathrm{Be}$, ${}^{17}\mathrm{C}$, and ${}^{31}\mathrm{Ne}$, respectively.

Figure 9

Test of the Glauber formula for in the $S_n-\mathcal{H}$ plane. Exact results (thick lines) are compared with approximate results (thin lines) based on the Glauber formula (GF) of Eq. (4). The solid, dashed, and dotted lines indicate the cases of ${}^{11}\mathrm{Be}$, ${}^{17}\mathrm{C}$, and ${}^{31}\mathrm{Ne}$, respectively.