The properties of nanocrystalline materials are critically dependent on the atomic structure of the constituent grains. The tentative model of the materials assumes that the particles constitute a two-phase, core/surface-shell system where the atomic positions at the surface are different than those in the bulk. A conventional diffraction techniques and standard methods of the diffraction data analysis are insufficient to detect small differences between the actual atomic positions in nanograins and those in a regular (unambiguously defined) crystallographic phase; a standard elaboration of the diffraction measurements may readily lead to erroneous interpretations of the experimental results. We have developed a new method of evaluation of nano-powder diffraction data based on the analysis of the shifts of the Bragg reflections from their perfect-lattice positions measured in a large diffraction vector range (Q larger than about 15A-1), we introduced a new variable, "Apparent Lattice Parameter" (ALP), instead of the commonly used average value (lattice constant). The ALP quantities calculated from the actual positions of each individual Bragg reflection plotted versus Q-vector show characteristic features that were used to evaluate the synchrotron X-ray diffraction data of GaN, SiC, and diamond nanocrystals. The analysis of these data shows a strong evidence of the presence of a strained surface shell and of a considerable internal pressure in the nanoparticles. The model of a nanocrystal that combines core and surface-shell phases was found to be consistent with in situ high-pressure (up to 40 GPa) diffraction experiments performed in a diamond anvil cell (DAC). The results of these experiments indicate, that the compressibilities of the surface layers are different than those in the core of the nanograins.