By N N Greenwood; T C Gibb
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The magnitude of the frictional force depends on the type of the dislocation and to a considerable degree, on the model of the dislocation core. To study the role ofthe discrete structure of the crystal lattice, Peierls [29J suggested a simple model which now has his name. The Peierls model for the edge dislocation is, in essence, the following. Let the atoms of the undistorted crystal form a simple tetragonal lattice with a period b along the x-axis and with a period a along the y-axis (fig. 30).
L We consider a plane circular prismatic dislocation which has the Burgers vector b perpendicular to the loop plane (along the z-axis). Such a dislocation is acted upon by the following moment of forces: where S is the area of the loop. 2 The force between two edge dislocations is not, generally speaking, radial and, besides, has essential angular dependence even in an isotropic medium. We assume the glide planes of the dislocations to be parallel. Let the z-x-plane be parallel to the glide planes and the z-axis be parallel to the dislocation lines.
L A special method for the solution of this system was suggested by Eshelby, Frank and Nabarro . It appears that if we introduce a polynomial of degree n which has X k as its zeros, this polynomial satisfies a certain differential equation of the second order with variable coefficients. g. if the position of the first dislocation is fixed* and the external stress is uniform, the coordinates of the remaining dislocations in the set will be the zeros ofthe derivative of the n-degree Laguerre polynomial.
Mössbauer spectroscopy by N N Greenwood; T C Gibb