68
8 The sine-Gordon Equation of a Dislocation
was discussed whilst searching for field theoretic models for elementary particles,
see T. H. R. Skyrme [84] [91] (who gave the equation its name) and also the survey
by C. Rebbi [80] and G. Soliani. The attractiveness of the sine-Gordon equation in
various areas of physics has, up to today not diminished.
The sine-Gordon equation is similar to the wave equation, and it has been shown
how we arrived at this equation. We will therefore use the experiences we made with
the wave equation in order to help us develop the sine-Gordon equation.
Our physical starting point is the Newtonian equations (53) and (57), in which we
assume those forces are included that can describe the motions of the masses of a
crystal lattice. In the simplest case, these masses are positioned in the ‘proximity’ of
ideal lattice points where they oscillate.
An ideal crystal is attributed a mathematically well defined, unbounded space
lattice. If all the points in a mathematical lattice are occupied with atoms or groups
of atoms, we get an ideal crystal. If, however, the space lattice points of one half
plane, for example the plane that ends on the x-axis at the coordinate origin, i.e.
y = 0 , z < 0, are not occupied by atoms, we get an ideal crystal with a gap between
two half planes. If both half planes are joint together, we receive the projection shown
in Fig. 7.8 the lattice plane of a crystal with a straight dislocation line pointing directly
out towards the reader, also see Kröner [47, 48].
A dislocation is thus a one-dimensional line of disturbance in a crystal. This line
of disturbance only exists in relation to the surrounding lattice. If we remove the
lattice, we also remove the line of disturbance. The motion of a dislocation line
is defined alone by the motion of its surrounding lattice atoms and can therefore
be exclusively described as motion relative to a lattice. We will choose the centres
of inertia q α = q α (t) of the lattice atoms surrounding the dislocation line as the
coordinates for the motion of this line. In fact, ‘all’ lattice atoms have to be considered,
but only those in near proximity are in fact affected. In Fig. 7.8, we have marked a
dislocation coordinate with a cross. This is the geometrical centre of the dislocation.
The index α counts the lattice planes perpendicular to the drawing plane. The q α
do not necessarily have to lie directly in the planes shown in Fig. 7.8. In Chap. 5,
we showed that the centres of inertia q α of Newtonian mass distributions principally
satisfy the equations of Newtonian type (57). The dislocation coordinates q α also
follow the following equation, which will be explained below in more detail,
d
dt
m α
d
dt
q α
=
b =α
F b α with F b α = −F α b .
(76)
A strict calculation of the coefficients m α and the interaction forces F αb out of the
system of the Newtonian equations (53) or (57) should now be made. We will not try
this here, and only at the end of this chapter, we will think about an approximation for
the magnitude of these inertial masses m α . Instead of this, we will concern ourselves
with the physical interpretation of these quantities and their role in the development
of the sine-Gordon equation, and then, we will develop the sine-Gordon equation
strictly out of Eq. (76).
8 The sine-Gordon Equation of a Dislocation
was discussed whilst searching for field theoretic models for elementary particles,
see T. H. R. Skyrme [84] [91] (who gave the equation its name) and also the survey
by C. Rebbi [80] and G. Soliani. The attractiveness of the sine-Gordon equation in
various areas of physics has, up to today not diminished.
The sine-Gordon equation is similar to the wave equation, and it has been shown
how we arrived at this equation. We will therefore use the experiences we made with
the wave equation in order to help us develop the sine-Gordon equation.
Our physical starting point is the Newtonian equations (53) and (57), in which we
assume those forces are included that can describe the motions of the masses of a
crystal lattice. In the simplest case, these masses are positioned in the ‘proximity’ of
ideal lattice points where they oscillate.
An ideal crystal is attributed a mathematically well defined, unbounded space
lattice. If all the points in a mathematical lattice are occupied with atoms or groups
of atoms, we get an ideal crystal. If, however, the space lattice points of one half
plane, for example the plane that ends on the x-axis at the coordinate origin, i.e.
y = 0 , z < 0, are not occupied by atoms, we get an ideal crystal with a gap between
two half planes. If both half planes are joint together, we receive the projection shown
in Fig. 7.8 the lattice plane of a crystal with a straight dislocation line pointing directly
out towards the reader, also see Kröner [47, 48].
A dislocation is thus a one-dimensional line of disturbance in a crystal. This line
of disturbance only exists in relation to the surrounding lattice. If we remove the
lattice, we also remove the line of disturbance. The motion of a dislocation line
is defined alone by the motion of its surrounding lattice atoms and can therefore
be exclusively described as motion relative to a lattice. We will choose the centres
of inertia q α = q α (t) of the lattice atoms surrounding the dislocation line as the
coordinates for the motion of this line. In fact, ‘all’ lattice atoms have to be considered,
but only those in near proximity are in fact affected. In Fig. 7.8, we have marked a
dislocation coordinate with a cross. This is the geometrical centre of the dislocation.
The index α counts the lattice planes perpendicular to the drawing plane. The q α
do not necessarily have to lie directly in the planes shown in Fig. 7.8. In Chap. 5,
we showed that the centres of inertia q α of Newtonian mass distributions principally
satisfy the equations of Newtonian type (57). The dislocation coordinates q α also
follow the following equation, which will be explained below in more detail,
d
dt
m α
d
dt
q α
=
b =α
F b α with F b α = −F α b .
(76)
A strict calculation of the coefficients m α and the interaction forces F αb out of the
system of the Newtonian equations (53) or (57) should now be made. We will not try
this here, and only at the end of this chapter, we will think about an approximation for
the magnitude of these inertial masses m α . Instead of this, we will concern ourselves
with the physical interpretation of these quantities and their role in the development
of the sine-Gordon equation, and then, we will develop the sine-Gordon equation
strictly out of Eq. (76).