Peierls Stress for an Idealized Crystal Model

Abstract

An analytical investigation is made of the geometry of an edge dislocation in a simple atomic model of an infinite crystal, and the effect on the dislocation of externally applied shear stress. This study indicates that in dislocation theory significant results may be obtained from the discrete model, on which accurate analysis is possible. Even the simple model used here seems to be sufficiently realistic to yield meaningful results.The width, $w$, of the dislocation, in this model, is proportional to ${\ensuremath{\gamma}}^{\ensuremath{-}1}$, where $\ensuremath{\gamma}$ is the shear strength of a perfect crystal. The Peierls-Nabarro (P-N) stress has a large cyclic variation as $\ensuremath{\gamma}$ is changed, but is proportional, in the average, to ${e}^{\ensuremath{-}w}$. This differs markedly from the result, ${\ensuremath{\sigma}}_{\mathrm{PN}}\ensuremath{\propto}{e}^{\ensuremath{-}4\ensuremath{\pi}w}$, found from the continuum analysis. The values of the P-N stress ranged from ${10}^{\ensuremath{-}3}$ to ${10}^{\ensuremath{-}5}$, in units of the shear modulus.In the analysis, the doubly infinite set of difference equations, derived from force balances on the atoms, is first reduced, by a type of Fourier series transform, to an infinite system of equations, which is then solved by the method of reduction. Numerical results were obtained from the IBM 1620 digital computer. The over-all error in the P-N stress for this model is believed to be less than ${10}^{\ensuremath{-}6}$.