Reflectometry of X-ray Whispering Gallery Waves Propagating along Liquid Meniscuses

Abstract

X-ray reflectometry of whispering galleries (WGs), which propagate along meniscuses of deionized water or silica hydrosols (Ludox^® SM) enriched by Cs⁺ ions, was analyzed for the first time. The measurements were performed using the diffractometer with a moving tube-detector system. The X-ray beam rotation angle reached a maximum value of 4° on a silica hydrosol sample at CuK_α radiation, while for H₂O it is only 1°. The intensities of WG modes propagating near the surface of a concave meniscus were found from the rigorous solution of the respective Helmholtz equation and applying the Monte Carlo technique for the random surface. For analysis of intensities of the X-ray scattering we used a hybrid model of the liquid surface with the upper transition layer having sinusoidal ripples in which the concentration of levitating Cs⁺ ions has a maximum near the surface, derived from the experiment, while the concentration maximum for SiO₂ particles with sizes of ~6–15 nm is positioned at the depth of ~15 nm from the hydrosol’s surface. Our numerical results for reflectances are in a good agreement with the experimental data.