Nonideal Frenkel-Kontorova model of two-level systems in glasses

A boundary between two perfectly packed noncrystalline clusters is modeled by a crystal grain boundary of incommensurate orientation in two dimensions. A Frenkel-Kontorova-type model is obtained with a mixture of springs of two different equilibrium lengths. For the case of a dilute solution of one of the components, the perturbation due to a single length defect is calculated in a quasispin representation. The same is done for a so-called coupling defect, related to mobile impurities in superionic conductors. The quasidegeneracy of two neighboring positions of a vacancy, with its surroundings symmetric up to second neighbors, is lifted by the presence of a randomly placed length defect. The result is a set of two-level systems of a density of states which diverges at zero energy splitting as ${\left(\Delta E\right)}^{-1\mathrm{}}$. For real glasses the model is used to propose a sequence of distinct stages of quenching.