page_title: Frenkel–Kontorova model
section_title: The Peierls–Nabarro potential
hierarchical_section_title: Frenkel–Kontorova model / The Peierls–Nabarro potential
caption_attribution_description: English: Stationary configuration for the FK model for a single kink. Top image corresponds to a stable configuration. Bottom image corresponds to an unstable configuration
context_page_description: The Frenkel–Kontorova model, also known as the FK model, is a fundamental model of low-dimensional nonlinear physics.
  The generalized FK model describes a chain of classical particles with nearest neighbor interactions and subjected to a periodic on-site substrate potential. In its original and simplest form the interactions are taken to be harmonic and the potential to be sinusoidal with a periodicity commensurate with the equilibrium distance of the particles. Different choices for the interaction and substrate potentials and inclusion of a driving force may describe a wide range of different physical situations.
  Originally introduced by Yakov Frenkel and Tatiana Kontorova in 1938 to describe the structure and dynamics of a crystal lattice near a dislocation core the FK model has become one of the standard models in condensed matter physics due to its applicability to describe many physical phenomena.
  Physical phenomena which can be modeled by FK model include dislocations, the dynamics of adsorbate layers on surfaces, crowdions, domain walls in magnetically ordered structures, long Josephson junctions, hydrogen-bonded chains, and DNA type chains.
context_section_description: In the preceding section the excitations of the FK model were derived by considering the model in a continuum-limit approximation. Since the properties of kinks are only modified slightly by the discreteness of the primary model, the SG equation can adequately describe most features and dynamics of the system.
  
  The discrete lattice does, however, influence the kink motion in a unique way with the existence of the Peierls–Nabarro (PN) potential . Here,  is the position of the kink's center. The existence of the PN potential is due to the lack of translational invariance in a discrete chain. In the continuum limit the system is invariant for any translation of the kink along the chain. For a discrete chain only those translations that are an integer multiple of the lattice spacing  leave the system invariant. The PN barrier, , is the smallest energy barrier for a kink to overcome so that it can move through the lattice. The value of the PN barrier is the difference between the kink's potential energy for a stable and unstable stationary configuration.  The stationary configurations are shown schematically in figure 2.
caption: Stationary configuration for the FK model for a single kink. Top image corresponds to a stable configuration. Bottom image corresponds to an unstable configuration
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