We report improved precision measurements of the van der Waals potential strength
(C3) for Na atoms and a silicon-nitride
(SiNx) surface. We studied diffraction from nanofabricated gratings with a particular ``magic'' open fraction that allows us to determine
C3 without the need for separate measurements of the width of the grating openings. Therefore, finding the magic open fraction improves the precision of
C3 measurements. The same effect is demonstrated for a grating with an arbitrary open fraction by rotating it to a particular ``magic'' angle, yielding ${C}_{3}=3.42\ifmmode\pm\else\textpm\fi{}0.19\text{ }\text{eV}\text{ }{\text{\AA{}}}^{3}$ for Na and a
SiNx surface. This precision is sufficient to detect a change in
C3 due to a thin metal coating on the grating surface. We discuss the contribution to
C3 of core electrons and edge effects.
We analyze the van der Waals interaction of an atom with the internal surface of a metal or dielectric submicrometer-sized cylinder and derive closed analytical equations for the van der Waals surface potential and force in the electrostatic approximation. We show that for a concave cylindrical surface the van der Waals potential energy can be up to four times larger than for a flat surface. This result is qualitatively explained by a reduction in the effective distance between the atom and the concave surface compared to that for a flat surface. We also evaluate the shape of an atomic beam propagating inside a hollow dielectric submicrometer-sized cylinder and we propose a scheme that will enable a precise measurement of the van der Waals constant
C3.