Journal of Photonic Materials and Technology 2015; 1(3): 46-61 Published online February 19, 2016 (http://www.sciencepublishinggroup.com/j/jpmt) doi: 10.11648/j.jmpt.20150103.11 ISSN: 2469-8423 (Print); ISSN: 2469-8431 (Online) Ab initio Theory of Elastic Properties of Rare-Gas Crystals Under High Pressure Elena P. Troitskaya 1 , Vladimir V. Rumyantsev 1, 2 , Ekaterina A. Pilipenko 1 , Ievgen Ie. Gorbenko 3 1 Galkin Institute for Physics & Engineering, Donetsk, Ukraine 2 Mediterranean Institute of Fundamental Physics, Marino, Rome, Italy 3 Department of Physics and Nanotechnology of Institute of Physics, Mathematics and Information Technology, Lugansk Taras Shevchenko National University, Lugansk, Ukraine Email address: vladimir.rumyantsev2011@yandex.ru (V. V. Rumyantsev), pilipenko.katerina@mail.ru (E. A. Pilipenko) To cite this article: Elena P. Troitskaya, Vladimir V. Rumyantsev, Ekaterina A. Pilipenko, Ievgen Ie. Gorbenko. Ab initio Theory of Elastic Properties of Rare- Gas Crystals Under High Pressure. Cauchy Relation. Journal of Photonic Materials and Technology. Vol. 1, No. 3, 2015, pp. 46-61. doi: 10.11648/j.jmpt.20150103.11 Abstract: The quantum mechanical model of deformable and polarizable atoms has been developed for the research of the elastic properties of rare-gas crystals Ne, Ar, Kr, and Xe over a wide range of pressure. It is shown that it is impossible to reproduce the observed deviation from the Cauchy relation () for Ne, Kr, Xe adequately taking into account the many-body interaction only. The individual dependence () for each of the crystals is the result of two competing interactions, namely, the many-body interaction and the electron-phonon interaction, which manifests itself in a quadrupole deformation of atoms electron shells due to displacements of the nuclei. The contributions of these interactions to Ne, Kr, and Xe compensated each other with high precision that provides δ with a positive value which is weakly dependent on pressure. In case of Ar the many- body interaction prevails. The compressed Ar has a negative deviation from the Cauchy relation the absolute value of which increases with the rise of pressure. The consideration of the quadrupole deformation is of great importance for heavy rare-gas crystals Kr and Xe. The represented ab initio calculated dependences of Birch elastic moduli   () and () are in good agreement with the experiment. Keywords: Rare-Gas Crystals, High Pressure, Many-Body Interaction, Quadrupole Deformation of the Atomic Electron Shells 1. Introduction Rare-gas crystals (RGS) are the simplest molecular crystals, that is why they are widely used as model objects when the theory is being tested. A large number of theoretical [1-9] and experimental [10-17] investigation of the elastic properties of RGS at high pressure are connected with their application as transmitting media in the diamond-anvil cells DAC [18]). In 2009 the article by Sasaki et al. [10] was published. This article and other ones [11-13] has finished the series of the particularly accurate measurements elastic properties of RGS under pressure. Shimizu et al. [12] have determined a large deviation from the Cauchy relation (CR) for Ar in the range of pressures up to 70 GPa. These measurements have proved the fact that interatomic interaction in face- centered cubic rare-gas crystals cannot be described within the framework of any models of the two-body potentials with the central interaction of atoms. Usually, for crystals with any type of chemical bond (metals, dielectrics, semiconductors) the main reason for the deviation from the CR is considered to be non-pair interactions in crystals. Many attempts have been made to reach an agreement with experiment for theoretical values δ calculated in the framework of phenomenological, as well as microscopic descriptions of the non-pair forces (see, e.g., 19, 1-4, 7, and references therein). Within the framework of different models of the three- body interaction in RGS it is possible to successfully describe fcc-hcp transition [20, 21], equation of state [22, 1], elastic properties including the negative deviation from the Cauchy relation in RGS in the wide range of pressure [1-4, 7, and others]. In the work [10] Sasaki et al. sum up and discuss how well the nowadays theory describes the experiment concerning the deviation from Cauchy relation. In the δ