Parameterization of Deformed Nuclei for Glauber Modeling in Relativistic Heavy Ion Collisions

The density distributions of large nuclei are typically modeled with a Woods-Saxon distribution characterized by a radius $R_{0}$ and skin depth $a$. Deformation parameters $\beta$ are then introduced to describe non-spherical nuclei using an expansion in spherical harmonics $R_{0}(1+\beta_2Y^0_2+\beta_4Y^0_4)$. But when a nucleus is non-spherical, the $R_{0}$ and $a$ inferred from electron scattering experiments that integrate over all nuclear orientations cannot be used directly as the parameters in the Woods-Saxon distribution. In addition, the $\beta_2$ values typically derived from the reduced electric quadrupole transition probability B(E2)$\uparrow$ are not directly related to the $\beta_2$ values used in the spherical harmonic expansion. B(E2)$\uparrow$ is more accurately related to the intrinsic quadrupole moment $Q_{0}$ than to $\beta_2$. One can however calculate $Q_0$ for a given $\beta_2$ and then derive B(E2)$\uparrow$ from $Q_0$. In this paper we calculate and tabulate the $R_0$, $a$, and $\beta_2$ values that when used in a Woods-Saxon distribution, will give results consistent with electron scattering data. We then present calculations of the eccentricity $\varepsilon_2$ and $\varepsilon_3$ with the new and old parameters. We demonstrate that $\varepsilon_3$ is particularly sensitive to $a$ and argue that using the incorrect value of $a$ has important implications for the extraction of $\eta/s$ from the QGP created in Heavy Ion collisions.