8 The sine-Gordon Equation of a Dislocation
71
Mathematically formulated, the condition for the exclusive gliding of a dislocation
is that the three vectors, the displacement vector δq of a dislocation, its Burgers
vector b and its line direction vector t are not allowed to set a different volume than
zero, in other words δq · (b × t) = 0 . We can fulfil this motion restriction from the
beginning, as it is done when possible, in mechanics by the choice of our variables
q as a transversal deflection, cf. also Günther [33]. Therefore we do not further need
to take this condition into consideration.
Here the assumption of a linear chain means that the sum of the forces F bα is
made up of two parts. Firstly the sum of the forces F βα which describe the elastic
interactions of the dislocation masses m α with its next neighbours and secondly the
sum of the forces F gα that the masses m g of the lattice atoms around the chain exert
on m α . Principally we would have to include all lattice atoms, but we will of course
again assume a continuous action and will thus only take the direct neighbours into
consideration. Therefore the following splitting of the sum of the forces is valid,
b =α
F bα =
β=α±1
F βα +
g
F gα .
(80)
In other words, the effective dislocation masses m α positioned at q α constitute such
a linear chain as was discussed in detail in Chap. 4, with the exception that this chain
is now embedded inside of a crystal and is defined exclusively relative to this crystal.
An ideal lattice always has the property as described in Fig. 7.7: On the lattice
we move k times in the x-direction to the next lattice point. We then take l ‘lattice
steps’ in the y-direction. After this we then move back in the opposite direction:
firstly k lattice steps in the negative x-direction and then l lattice steps in the negative
y-direction. If we have an ideal lattice, to be more precise if the lattice inside of the
traversed area is ideally structured we should arrive back at our starting location. This
is not the case when dealing with dislocations. As we have seen and characterised in
Fig. 7.8, a dislocation of the Burgers vector b, where the dislocation is perpendicular
to the traversed area (Burgers circuit) leads to the fact that we do not arrive at the
starting point, we arrive at a point at the distance b away from the starting point. One
could also say, the dislocation is a topological singularity of the crystal lattice (see
also the appendix, Chap. 26, Eqs. (367) and (368) and the Figs. 26.1 and 26.2). In the
light of this background we can emphasise our general dynamics of dislocations as
follows:
The physical property of the topological singularity ∗dislocation∗ inside of a crystal is that
the dislocation possesses, in a Newtonian sense, an inertial mass with respect to the lattice.
For those who are familiar with the experiments on plastic deformations, where
we discovered that the smallest element is the motion of a dislocation, see Chap. 26
for more details, our Eq. (79) will at first cause some irritation. Dislocations do not
seem to fulfil the inertia property of an already existing velocity so characteristic of
Newtonian mechanics. Here we wish to state that it is just as difficult to observe the
elementary laws of Newton’s mechanics in the motions of a plough ploughing a field.
In Chap. 21 we will discuss a more detailed description about the inertial properties
71
Mathematically formulated, the condition for the exclusive gliding of a dislocation
is that the three vectors, the displacement vector δq of a dislocation, its Burgers
vector b and its line direction vector t are not allowed to set a different volume than
zero, in other words δq · (b × t) = 0 . We can fulfil this motion restriction from the
beginning, as it is done when possible, in mechanics by the choice of our variables
q as a transversal deflection, cf. also Günther [33]. Therefore we do not further need
to take this condition into consideration.
Here the assumption of a linear chain means that the sum of the forces F bα is
made up of two parts. Firstly the sum of the forces F βα which describe the elastic
interactions of the dislocation masses m α with its next neighbours and secondly the
sum of the forces F gα that the masses m g of the lattice atoms around the chain exert
on m α . Principally we would have to include all lattice atoms, but we will of course
again assume a continuous action and will thus only take the direct neighbours into
consideration. Therefore the following splitting of the sum of the forces is valid,
b =α
F bα =
β=α±1
F βα +
g
F gα .
(80)
In other words, the effective dislocation masses m α positioned at q α constitute such
a linear chain as was discussed in detail in Chap. 4, with the exception that this chain
is now embedded inside of a crystal and is defined exclusively relative to this crystal.
An ideal lattice always has the property as described in Fig. 7.7: On the lattice
we move k times in the x-direction to the next lattice point. We then take l ‘lattice
steps’ in the y-direction. After this we then move back in the opposite direction:
firstly k lattice steps in the negative x-direction and then l lattice steps in the negative
y-direction. If we have an ideal lattice, to be more precise if the lattice inside of the
traversed area is ideally structured we should arrive back at our starting location. This
is not the case when dealing with dislocations. As we have seen and characterised in
Fig. 7.8, a dislocation of the Burgers vector b, where the dislocation is perpendicular
to the traversed area (Burgers circuit) leads to the fact that we do not arrive at the
starting point, we arrive at a point at the distance b away from the starting point. One
could also say, the dislocation is a topological singularity of the crystal lattice (see
also the appendix, Chap. 26, Eqs. (367) and (368) and the Figs. 26.1 and 26.2). In the
light of this background we can emphasise our general dynamics of dislocations as
follows:
The physical property of the topological singularity ∗dislocation∗ inside of a crystal is that
the dislocation possesses, in a Newtonian sense, an inertial mass with respect to the lattice.
For those who are familiar with the experiments on plastic deformations, where
we discovered that the smallest element is the motion of a dislocation, see Chap. 26
for more details, our Eq. (79) will at first cause some irritation. Dislocations do not
seem to fulfil the inertia property of an already existing velocity so characteristic of
Newtonian mechanics. Here we wish to state that it is just as difficult to observe the
elementary laws of Newton’s mechanics in the motions of a plough ploughing a field.
In Chap. 21 we will discuss a more detailed description about the inertial properties