Density distributions of ${}^{11}$Li deduced from reaction cross-section measurements

We measured the reaction cross sections of the two-neutron halo nucleus ${}^{11}$Li with solid hydrogen and carbon targets at around 31 and 41 MeV/nucleon. The neutron density distribution of ${}^{11}$Li was deduced for the first time by the Glauber model calculation based on the optical limit approximation. The uncertainty of the matter density of ${}^{11}$Li was improved, compared with earlier measurements. The present root-mean-square radius of the proton distribution agrees with the previous one derived from an optical isotope shift measurement. The present root-mean-square radii reproduce theoretical calculations by the tensor optimized shell model by assuming core excitation. This consistency suggests the possibility that ${}^9$Li in ${}^{11}$Li is excited and the disappearance of the $N=8$ shell gap of ${}^{11}$Li is caused by correlations originating from the nucleon force, such as the tensor and the pairing.

Figure 1

Schematic view of the experimental setup around reaction targets.

Figure 2

TOF-$\Delta E$ two-dimensional scattering plot before the reaction target in the measurement using ${}^{11}$Li with an energy of 41 MeV/nucleon. The intensity is color coded.

Figure 3

$\Delta E$-$E$ two-dimensional scattering plots after (a) the SHT and (b) the carbon target in the measurement using ${}^{11}$Li at around 41 MeV/nucleon, where ${}^{11}$Li is selected before the reaction targets. The broken lines are the gates that are established for counting the noninteracting ${}^{11}$Li. The intensity is color coded.

Figure 4

$E$ distribution of the particles selected by the gate shown in Fig. 3a. The solid and dotted lines indicate the $E$ distributions in the case of experiments using the SHT and empty target, respectively. The arrow indicates the range of the subtraction to obtain the number of Li isotopes, which are located inside the gate.

Figure 5

Energy dependence of ${\sigma }_R$ of ${}^{11}$Li. Closed and open circles denote experimental data using a proton target and a carbon target, respectively. Solid lines are best-fit curves.

Figure 6

Density distributions of ${}^{11}$Li: (a) ${\rho }_n$, (b) ${\rho }_m$, and (c) ${\rho }_p$. Best-fit densities are indicated by the thick solid curves, and uncertainties are shown by shaded regions. The dashed lines indicate the upper and lower limits of ${\rho }_m$ deduced by the ${\sigma }_I$ experiment [3].

Figure 7

Root-mean-square (rms) neutron ($r_n$), matter ($r_m$), and proton ($r_p$) radii of Li isotopes obtained from experimental studies. The circles, squares, and triangles are $r_n$, $r_m$, and $r_p$, respectively. The closed symbols indicate present experimental results. The error bar of present $r_p$ indicates the uncertainty obtained by the method using present $r_n$ and $r_m$ with Eq. (2). The open circles indicate previous $r_n$ obtained from $r_m$ and $r_p$ with Eq. (2). The open squares indicate previous $r_m$ determined by the ${\sigma }_I$ experiment [3, 18]. The cross symbol indicates the previous $r_m$ determined by the proton elastic-scattering measurement [5]. The open triangles indicate previous $r_p$ converted from the rms charge radii obtained by the OIS measurement with Eq. (3) except for that of ${}^7$Li, which was determined by the electron scattering [9].

Figure 8

Root-mean-square (rms) neutron ($r_n$), matter ($r_m$), and proton ($r_p$) radii of ${}^{11}$Li. The closed circles indicate the present experimental results. The error bar of present $r_p$ indicates the uncertainty obtained by the method using present $r_n$ and $r_m$ with Eq. (2). The open circle and the cross symbol indicate the previous $r_m$ from the ${\sigma }_I$ measurement [3] and the proton elastic-scattering measurement [5], respectively. The diamond symbol indicates the previous $r_p$, which is converted from the rms charge radius obtained by the OIS measurement [9]. The open inverse triangle indicates the previous $r_n$ obtained from the previous $r_m$ reported in Ref. [3] and the previous $r_p$ with Eq. (2). The open and closed squares indicate the theoretical calculations by SVMC [6, 7] assuming an inert core and excited core, respectively. The open and closed triangles are the same as the squares, but for the theoretical calculations by TOSM [8].