Chapter 8
The sine-GORDON Equation of a
Dislocation
The distances between the points of a lattice make up a characteristic length for the
properties of a dislocation. We have already observed this in our elementary considerations of the critical shear stress in the last chapter. This length is about 10
−8 cm.
Hence, we find ourselves, measured in lattice units inside a small crystal (with a linear
dimension in the cm-region) infinitely far away from its surface. For the following,
we will use as a good approximation for this situation an infinitely extended crystal.
The starting point of our considerations will be the geometrically simplest, and out
of reasons of symmetry, the physically simplest form of a straight, never-ending
infinitely long dislocation line in a crystal. We have seen that the displacement of the
straight dislocation line as a whole is a somewhat suitable model for plastic deformations. However, does the dislocation really get displaced as a whole, rigid as a stick,
or is the displacement process of a dislocation line by one lattice position, compare
with Fig. 7.10, generally more complex? Are there other neighbouring physical stabile geometric forms existing alongside the straight dislocation with its energetically
preferred line form, thus allowing them to exist force free inside of a crystal as for an
arbitrary time, where these forms are composed partially of straight lines, dents and
edges? One must also think of those line forms that can only exist whilst participating
in characteristic motions. Such lines are called dynamically stabile.
The mathematical condition for all these sorts of dislocations in the proximity
of a straight dislocation line is made up of one equation, the sine-Gordon equation.
The sine-Gordon equation has a long history. It describes the energetically possible,
physically stabile forms of a dislocation in the neighbourhood of a straight dislocation
line. It was a long hard trek until the uniform and mathematically simple condition
was discovered for this. After years of research by L. Prandtl [78], U. Dehlinger
[10] and J. Frenkel and T. Kontorova [25], A. Seeger [87] in 1949 (cf. also A. Seeger
[88] and P. Schiller) and later F. C. Frank [22] and J. H. van der Merwe managed
to formulate the sine-Gordon equation for a dislocation. Ten years later this equation
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_8
67
The sine-GORDON Equation of a
Dislocation
The distances between the points of a lattice make up a characteristic length for the
properties of a dislocation. We have already observed this in our elementary considerations of the critical shear stress in the last chapter. This length is about 10
−8 cm.
Hence, we find ourselves, measured in lattice units inside a small crystal (with a linear
dimension in the cm-region) infinitely far away from its surface. For the following,
we will use as a good approximation for this situation an infinitely extended crystal.
The starting point of our considerations will be the geometrically simplest, and out
of reasons of symmetry, the physically simplest form of a straight, never-ending
infinitely long dislocation line in a crystal. We have seen that the displacement of the
straight dislocation line as a whole is a somewhat suitable model for plastic deformations. However, does the dislocation really get displaced as a whole, rigid as a stick,
or is the displacement process of a dislocation line by one lattice position, compare
with Fig. 7.10, generally more complex? Are there other neighbouring physical stabile geometric forms existing alongside the straight dislocation with its energetically
preferred line form, thus allowing them to exist force free inside of a crystal as for an
arbitrary time, where these forms are composed partially of straight lines, dents and
edges? One must also think of those line forms that can only exist whilst participating
in characteristic motions. Such lines are called dynamically stabile.
The mathematical condition for all these sorts of dislocations in the proximity
of a straight dislocation line is made up of one equation, the sine-Gordon equation.
The sine-Gordon equation has a long history. It describes the energetically possible,
physically stabile forms of a dislocation in the neighbourhood of a straight dislocation
line. It was a long hard trek until the uniform and mathematically simple condition
was discovered for this. After years of research by L. Prandtl [78], U. Dehlinger
[10] and J. Frenkel and T. Kontorova [25], A. Seeger [87] in 1949 (cf. also A. Seeger
[88] and P. Schiller) and later F. C. Frank [22] and J. H. van der Merwe managed
to formulate the sine-Gordon equation for a dislocation. Ten years later this equation
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_8
67