Frenkel–Kontorova model

StabFK

Frenkel–Kontorova model (FK model) is a theoretical model used to describe the behavior of a chain of atoms or molecules constrained between two surfaces, which can also be applied to study various phenomena in condensed matter physics, such as dislocations, adsorption of atoms on crystal surfaces, and charge density waves. The model was first introduced by Yakov Frenkel and Tatiana Kontorova in the 1930s to explain the properties of dislocations in solid crystals.

Overview

The FK model consists of a one-dimensional array of particles (atoms or molecules) connected by springs (representing the interatomic forces) that lie in a periodic potential. This setup models the atoms in a crystal lattice subjected to an external periodic potential. The balance between the spring forces and the external potential leads to a rich variety of physical phenomena, making the FK model a powerful tool in theoretical physics.

Mathematical Formulation

The Hamiltonian of the FK model is given by:

\[ H = \sum_{i=1}^{N} \left[ \frac{p_i^2}{2m} + \frac{1}{2}K(x_{i+1} - x_i - a)^2 + V(x_i) \right] \]

where \(N\) is the number of particles, \(p_i\) is the momentum of the \(i\)-th particle, \(m\) is the mass of the particles, \(K\) is the spring constant, \(x_i\) is the position of the \(i\)-th particle, \(a\) is the natural length of the spring (equilibrium distance between particles), and \(V(x_i)\) is the external periodic potential.

Applications

The FK model has been used to study a wide range of physical systems and phenomena, including:

- Dislocations in crystals: The model provides insights into the motion and behavior of dislocations, which are important for understanding the mechanical properties of materials. - Adsorption of atoms on surfaces: The FK model can describe how atoms adsorb on crystal surfaces, which is relevant for surface science and catalysis. - Charge density waves: The model helps in understanding the formation and dynamics of charge density waves in quasi-one-dimensional conductors. - Friction at the atomic scale: The FK model has been applied to study the fundamentals of friction between atomic surfaces, contributing to the field of nanotribology.

Extensions and Variations

Over the years, several extensions and variations of the FK model have been developed to address specific physical situations and to include additional effects, such as:

- The inclusion of thermal fluctuations to study temperature effects on the system's behavior. - Modifications to the potential to represent different types of interactions or to study systems with more complex geometries. - Coupling with other degrees of freedom, such as electronic or magnetic, to explore more complex coupled phenomena.

Conclusion

The Frenkel–Kontorova model remains a fundamental tool in theoretical and computational physics for studying the properties and dynamics of a wide range of systems at the atomic and molecular scales. Its simplicity and versatility have made it a cornerstone in the study of condensed matter physics and materials science.