Measuring the surface thickness of the weak charge density of nuclei

The present PREX-II and CREX experiments are measuring the rms radius of the weak charge density of ${}^{208}\mathrm{Pb}$ and ${}^{48}\mathrm{Ca}$. We discuss the feasibility of a new parity violating electron scattering experiment to measure the surface thickness of the weak charge density of a heavy nucleus. Once PREX-II and CREX have constrained weak radii, an additional parity violating measurement at a momentum transfer near $0.76{\mathrm{fm}}^{-1}$ for ${}^{208}\mathrm{Pb}$ or $1.28{\mathrm{fm}}^{-1}$ for ${}^{48}\mathrm{Ca}$ can determine the surface thickness.

Figure 1

Differential cross section including Coulomb distortions for 855 MeV electrons elastically scattered from ${}^{208}\mathrm{Pb}$ (solid black line) and ${}^{48}\mathrm{Ca}$ (dashed red line) versus scattering angle.

Figure 2

Parity violating asymmetry including Coulomb distortions for 855 MeV electrons elastically scattered from ${}^{208}\mathrm{Pb}$ (solid black line) and ${}^{48}\mathrm{Ca}$ (dashed red line) versus scattering angle. Symmetrized Fermi weak charge densities are used (see text).

Figure 3

Minus the weak charge density -${\rho }_W\left(r\right)$ of ${}^{208}\mathrm{Pb}$ versus radius $r$. The solid black line is from the FSU Gold relativistic mean field model [28] while the the long dashed blue line shows a Fermi function fit [Eq. (9)]. The short dashed red line shows the experimental charge density ${\rho }_{ch}\left(r\right)$ from Ref. [1].

Figure 4

As Fig. 3 except for ${}^{48}\mathrm{Ca}$.

Figure 5

Log derivative of the parity violating asymmetry with respect to the log of the surface thickness parameter $a$ (solid black line) or w.r.t the log of the weak radius $R_W$ (dashed red line) for ${}^{208}\mathrm{Pb}$ at 950 MeV versus scattering angle.

Figure 6

As Fig. 5 except for ${}^{48}\mathrm{Ca}$ at 2220 MeV.

Figure 7

As Fig. 5 except for ${}^{208}\mathrm{Pb}$ at 855 MeV.

Figure 8

As Fig. 5 except for ${}^{48}\mathrm{Ca}$ at 855 MeV.

Figure 9

Figures of merit ${\mathrm{FOM}}_a$ and ${\mathrm{FOM}}_R$ versus scattering angle $\theta$ to measure the weak surface thickness (solid black curve) or radius (dashed red curve) of ${}^{208}\mathrm{Pb}$ at 855 MeV.

Figure 10

Figures of merit ${\mathrm{FOM}}_a$ and ${\mathrm{FOM}}_R$ versus scattering angle $\theta$ to measure the weak surface thickness (solid black curve) or radius (dashed red curve) of ${}^{48}\mathrm{Ca}$ at 855 MeV.