The Boguslawski Melting Model

The anharmonic vibrator, whose expression of potential energy contains second and third powers of coordinates, is treated on the basis of dynamical procedure, which presents the state of motion by means of mean position and mean amplitude of vibration. The divergent statistical integral comes here not into consideration. The free energy is represented through mean atomic displacement and developed in power series, retaining fourth degree. The graphs show that at certain temperature, the minimum in free energy disappears, and the atom escapes from the potential pit. A simple atomic model that represents this phenomenon is proposed and the influence of model dimension and pressure on melting temperature will be presented.

Model of Cubic Cell for Description of Some Phase Transitions in Crystals

The purpose of the research is to develop a dynamical theory of phase transitions in crystalline structures, when except for temperature, the pressure is acting. So, the phase diagram temperature-pressure (dimensions) must be constructed. In general case, it is a complicated question, which can be solved for simple models of crystal, as three atomic models, introduced in the work of Frenkel [1]. In this model, three identical atoms are placed on the straight line and interact with the forces, which can be described by the expression, given in the article of Lennard-Jones [2]. Such simple model may have success, when the crystalline structure is simple, which consists of one type of atoms, for example: carbon. The model was generalized to cubic cell model with a moving atom in the inner part of the cell. The rigorous calculation of phase diagram for transition graphite-diamond shows some similarity with results of numerous experimental investigations (which are not discussed here). So, the way of phase diagram calculation may attract attention.

To the Question of Validity Grüneisen Solid State Equation

The first justified theory of solid state was proposed by Grüneisen in the year 1912 and was based on the virial theorem. The forces of interaction between two atoms were assumed as changing with distance between them according to inverse power laws. But only virial theorem is insufficient to deduce the equation of state, so this author has introduced some relations, which are correct, when the forces linearly depend on displacement of atoms. But with such law of interaction the phase transitions cannot take place. Debye received Grüneisen equation in another way. He deduced the expression for thermocapacity, using Plank formula for energy of harmonic vibrator. Taking into account the dependence of atomic vibration frequency from distance between atoms, when the forces of interaction are anharmonic, he received the equation of state, which in classical limit turns to Grüneisen equation. The question, formulated by Debye is—How can we come to phase transitions, when Plank formula for harmonic vibrator was used? Debye solved this question not perfectly, because he was born to small anharmonicity. In the presented work a chain of atoms is considered, and their movement is analysed by means of relations, equivalent to virial theorem and theorem of Lucas (disappearing of mean force). Both are the results of variation principle of Hamilton. The Grüneisen equation for low temperature (not very low, where quantum expression for energy is essential) was obtained, and a family of isotherms and isobars are drown, which show the existence of spinodals, where phase transitions occur. So, Grüneisen equation is an equation of state for low temperatures.

Dynamical Examinations of Vibrator Models, Describing Some Non Elastic Properties of Crystals

On the base of a vibrator atomic model the mechanical and thermal properties of the object are analyzed. The potential energy of the vibrator is represented by means of positive term with coordinate deflection in second power and negative term with deflection in fourth power. With the use of dynamical procedure of calculation, which permits to calculate mean deflection and root mean square amplitude of vibrations, the dependence of applied force from mean amplitude and temperature is calculated. This dependence shows a maximum (or minimum, when the direction of force is reversed), the height of which diminishes with rising temperature. When the force reaches the value of the maximum, the object does not elastic counteract to the force, and gliding begins. It is also considered a vibrator with positive term, containing the deflection in second power and a term, where the deflection treats in third power (Boguslawski vibrator). Exact calculations of the dependence of the force from the temperature in adiabatic process, where the entropy is maintained constant, shows that it is represented by means of a curve with a maximum, so that stretching leads to cooling till the point of maximum is reached.