The 1%-accurate calculations of the van der Waals interaction between an atom and a cavity wall are performed in the separation region from 3 nm to 150 nm. The cases of metastable He * and Na atoms near the metal, semiconductor or dielectric walls are considered. Different approximations to the description of wall material and atomic dynamic polarizability are carefully compared. The smooth transition to the Casimir-Polder interaction is verified. It is shown that to obtain accurate results for the atom-wall van der Waals interaction at shortest separations with an error less than 1% one should use the complete optical tabulated data for the complex refraction index of the wall material and the accurate dynamic polarizability of an atom. The obtained results may be useful for the theoretical interpretation of recent experiments on quantum reflection and Bose-Einstein condensation of ultracold atoms on or near surfaces of different nature.

We find the lateral projection of the Casimir force for a configuration of a sphere above a corrugated plate. This force tends to change the sphere position in the direction of a nearest corrugation maximum. The probability distribution describing different positions of a sphere above a corrugated plate is suggested which is fitted well with experimental data demonstrating the nontrivial boundary dependence of the Casimir force.12.20. Ds, 12.20.Fv, 61.16.Ch, 03.70.+k Considerable recent attention has been focused on the Casimir effect [1,2]. On the theoretical side much work was done to investigate the corrections to the Casimir force due to the finite conductivity of the boundary metal [3][4][5][6], nonzero temperature [7][8][9], and surface roughness [10][11][12]. In the experimental field the new precision measurements of the Casimir force between metallic surfaces of a plane disk and a spherical lens (or a sphere) were performed [13][14][15][16]. The experimental results correlate well with the theoretical expressions taking into account all the corrections mentioned above. This provided a way to obtain new stronger constraints on the constants of Yukawa-type additional terms to Newtonian gravitational law predicted by the unified gauge theories, supersymmetry, supergravity, and string theory [17][18][19][20][21]. The Casimir effect has assumed a new meaning as a tool for investigation of fundamental interactions and their unification. Because of this, the detailed analyses of the fit of the theory to the data takes on great significance.In Ref.[22] the Casimir force between an aluminum coated plate with sinusoidal corrugations and a large sphere was measured using an atomic force microscope. It was concluded that the measured force shows significant deviation from the perturbative theory which takes into account the periodic corrugation of the plate in the surface separation. In the absence of corrugations the same theory shows good agreement with the measured Casimir force. These together were considered in [22] to represent the nontrivial boundary dependence of the Casimir force and until recently has no theoretical explanation (in line with [22] dependence of this kind is to be expected due to diffractive effects associated with corrugated surface).Here we present the perturbative calculation for both vertical and lateral Casimir force acting in the configuration of a sphere situated above a corrugated plate (note that the lateral force arises due to the absence of translational symmetry on a plate with corrugations). Our study revealed that the lateral force acts upon the sphere in such a way that it tends to change its position in the direction of a nearest maximum point of the vertical Casimir force (which coincides with the maximum point of corrugations). In consequence of this, the assumption made in [22] that the locations of the sphere above different points of a corrugated surface are equally probable can be violated. As indicated below, the diverse assumptions on the probability distribution ...

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Within a real-space renormalization-group framework, the spin-i Heisenberg ferromagnet in the presence of an Ising-like anisotropy on a self-dual hierarchical lattice is discussed. The results are exact for this lattice but can also be considered as a quite satisfactory approximation for the simple square one. The controversial point on how T c vanishes in the isotropic Heisenberg limit is analyzed: Quite strong evidence is presented favoring a continuous function of anisotropy. The crossover from the isotropic Heisenberg model to the pure Ising one is exhibited.