Short-range correlations and the $3s_{1/2}$ wave function in ${}^{206}\mathrm{Pb}$

The charge-density difference between ${}^{206}\mathrm{Pb}$ and ${}^{205}\mathrm{Tl}$, measured by elastic electron scattering, offers a unique opportunity to look for effects of short-range correlations on a shell-model wave function of a single proton. The measured difference is very similar to the charge density due to a proton in a $3s_{1/2}$ orbit. If there is a potential whose $3s_{1/2}$ wave function yields the measured difference between the charge distributions, no effect of short-range correlations is evident. To check this point, we look for a potential whose $3s_{1/2}$ wave function yields the measured data. We developed a novel method to obtain the potential directly from the density and its first and second derivatives. Fits to parametrized potentials were also carried out. The $3s_{1/2}$ wave functions of the potentials determined here reproduce fairly well the experimental data within the quoted errors. To detect possible effects of two-body correlations on the $3s_{1/2}$ shell-model wave function, more accurate measurements are required.

Figure 1

(a) The experimental difference $\mathrm{\Delta }{\rho }_c\left(r\right)$ between ${}^{206}\mathrm{Pb}$ and ${}^{205}\mathrm{Tl}$ charge distributions (solid line). The dashed line is for $\mathrm{\Delta }{\rho }_{Rc}\left(r\right)$, the data after rearrangement correction. The dotted lines indicate the experimental uncertainty. (b) Similar to (a) for $R_P^2\left(r\right)=4\pi r^2\mathrm{\Delta }{\rho }_p\left(r\right)$ where $\mathrm{\Delta }{\rho }_p\left(r\right)$ is derived from the experimental $\mathrm{\Delta }{\rho }_c\left(r\right)$. The dashed line is for $R_{Rp}^2\left(r\right)$ related to $\mathrm{\Delta }{\rho }_{Rp}\left(r\right)$ similarly obtained from $\mathrm{\Delta }{\rho }_{Rc}\left(r\right)$.

Figure 2

Potentials fitted to data in Fig. 1. The $V_F\left(r\right)$ potential (solid line), the $V_{\mathrm{FR}}\left(r\right)$ version including rearrangement (dashed line), and the fitted $V_{\mathrm{WSF}}\left(r\right)$ potential (double-dotted-dashed line). Also shown is the conventional Woods-Saxon $V_{\mathrm{WS}}\left(r\right)$ potential (dashed-dotted line).

Figure 3

Experimental values of (a) $R_c^2\left(r\right)=4\pi r^2\mathrm{\Delta }{\rho }_c\left(r\right)$ and (b) $\mathrm{\Delta }{\rho }_c\left(r\right)$ plotted between dotted lines of error limits. They are compared to calculated charge distributions due to the $3s_{1/2}$ wave functions of the fitted $V_F\left(r\right)$ potential (solid lines), the fitted Woods-Saxon $V_{\mathrm{WSF}}\left(r\right)$ potential (double-dotted-dashed lines), and the conventional $V_{\mathrm{WS}}\left(r\right)$ potential (dashed-dotted lines).

Figure 4

Calculated charge densities of a proton in the (a) $1s_{1/2}$, (b) $2s_{1/2}$, and (c) $3s_{1/2}$ orbits in the $V_F\left(r\right)$ potential (solid lines) and the conventional $V_{\mathrm{WS}}\left(r\right)$ potential (double-dotted-dashed lines).