  We report improved precision measurements of the Van der Waals potential
strength ($C_3$) for Na atoms and a silicon-nitride (SiN$_x$) surface. We
studied diffraction from nano-fabricated gratings with a particular "magic"
open-fraction that allows us to determine $C_3$ without the need for separate
measurements of the width of the grating openings. Therefore, finding the magic
open-fraction improves the precision of $C_3$ measurements. The same effect is
demonstrated for a grating with an arbitrary open-fraction by rotating it to a
particular "magic" angle, yielding $C_3=3.42\pm 0.19 \textrm{eV}
\mathring{\textrm{A}}^3$ for Na and a SiN$_x$ surface. This precision is
sufficient to detect a change in $C_3$ due to a thin metal coating on the
grating surface. We discuss the contribution to $C_3$ of core electrons and
edge effects.
