Correlations and anharmonic effects in solid helium

Abstract

A relationship between correlations and anharmonic effects in solid helium is investigated based on the Bogolyubov method. To this end, the internal energy of the system and the expression for the pressure derived with the use of one-and two-particle density matrices are considered for which the series of perturbation theory converge at any arbitrary temperature. The results obtained are compared with the available experimental data on diffraction of synchrotron radiation by helium crystals.Keywords: anharmonism, solid helium.In the last few decades, solid helium has become a subject of intensive research [1,2]. The reason is that high amplitudes of particle oscillations in these crystals gives rise to radically new effects observed in it [3,4]. In addition, these crystals are very promising for investigations of solid-liquid phase transitions [5]. In the last few years, the kinetic effects and cluster systems comprising atoms capable of forming quantum crystals have been intensively investigated [6,7]. In the middle 50s of the last century, the possibility appeared to investigate macroscopic systems of 3 He atoms. As a result, many works nowadays are devoted to binary 4 He and 3 He mixtures [8]. All this has led to a sharp increase in the number of articles devoted to quantum crystals. This is also favored by the rapid development of representations on liquid helium, since the modern theory of liquids, as a rule, uses the results obtained in the study of the ordered phase [3]. At the same time, a study of the crystal phase is of independent interest for the above-mentioned reasons.In spite of the fact that quantum crystals possess high degree of anharmonicity even at low temperatures, the notion of phonons is commonly used for their research. To this end, they are appropriately renormalized with allowance for short-range correlations [2]. However, this approach yields poorly converging series in the region of phase transition even for ideal gas crystals whose displacements from equilibrium positions are small [3]. To solve the problem of efficient allowance for the anharmonism, correlation expansions could be used; however, at low temperatures it is difficult to provide convergence of expansions within the framework of the standard approach going back to the Einstein model used as a main approximation. The present work studies a relationship between correlations and anharmonic terms used to describe solid helium crystals.We take advantage of the Bogolyubov method to construct statistical thermodynamics of quantum crystals. To this end, we consider an ordered system of N particles localized in the macroscopic volume V at the temperature T . The quantum equilibrium chain of the Bogolyubov equations can be written in the form [9]