74
8 The sine-Gordon Equation of a Dislocation
The second term on the right-hand side of this equation is the force resulting from
out fore mentioned lattice potential (78). For ∂τ (x, t)/∂x we once again use our
assumption of linear elasticity (85) and finally arrive at
∂
2
∂x 2 q(x, t) −
1
c 2
o
∂
2
∂t 2 q(x, t) =
D
σ
sin
2π
a
q(x, t)
,
c o =
σ
ρ o
.
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
sine–Gordon equation
(88)
This is already the famous sine-Gordon equation of a dislocation.
A particular property of this equation is its non-linearity based on the sine function
on the right hand side. Didn’t we just make the base assumption (68) of the linear
theory of elasticity? Well yes, but only for the one dimensional theory of elasticity
of a linear chain. This linear chain finds itself under the influence of the non-linear
potential U of the surrounding lattice according to (78). This produces the nonlinearity in (88).
Because the relative deflection ∂q/∂x is without dimension, the modulus σ has
according to (85) the dimension of tension τ , it is therefore a force, an energy per unit
of length. The modulus of elasticity must be equated to the tension of equilibrium of
the chain when the deflection is perpendicular to the chain as seen in Chap. 6. Hence
σ is nothing else than the line tension in the string model of a dislocation, see Seeger
[88] and P. Schiller. A dislocation however cannot in comparison to a string end
inside of a crystal. Another difference is that a dislocation cannot release its energy
by just cutting through it, whereas a string can. From this point of view, the string
model is not sufficient for the dislocation. Furthermore, the quantity L ρ o = m is in
the case of a dislocation not just simply the mass of the lattice atoms on the length
L—because there are no atoms on this length!—it is in fact the effective mass that
should be calculated by inertia measurements of dislocations as a whole with respect
to the lattice.
Our considerations are not sufficient to be able to numerically calculate the inertia
of a dislocation. The determination of the inertia of a dislocation is according to the
second equation (88) dependent on the line tension σ of a dislocation and the velocity
c o . The quantity c o is the critical signal velocity of the sine-Gordon equation. Its
relationship to the sound velocity will be discussed later (see Chap. 12). Furthermore,
there are elementary lines of thought that lead to an explanation of the quantity of
the line tension σ of a dislocation. We will delve further into this in Chap. 23, and
there we will make an estimation that the mass m a of a dislocation of the length a
of a lattice parameter in one of our considered examples will be around 3% of the
mass of the surrounding lattice atoms, see Eqs. (295) and (299).
Such a statement can easily be misunderstood. The calculated mass m a of the
dislocation section is its inertia with respect to the crystal lattice; in other words, m a
is the ratio of the acting force (the configuration force to be exact) to the produced
acceleration of this dislocation section relative to the lattice. The significant difference
of this dislocation mass m a to the mass of the atoms of the lattice becomes clear as
8 The sine-Gordon Equation of a Dislocation
The second term on the right-hand side of this equation is the force resulting from
out fore mentioned lattice potential (78). For ∂τ (x, t)/∂x we once again use our
assumption of linear elasticity (85) and finally arrive at
∂
2
∂x 2 q(x, t) −
1
c 2
o
∂
2
∂t 2 q(x, t) =
D
σ
sin
2π
a
q(x, t)
,
c o =
σ
ρ o
.
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
sine–Gordon equation
(88)
This is already the famous sine-Gordon equation of a dislocation.
A particular property of this equation is its non-linearity based on the sine function
on the right hand side. Didn’t we just make the base assumption (68) of the linear
theory of elasticity? Well yes, but only for the one dimensional theory of elasticity
of a linear chain. This linear chain finds itself under the influence of the non-linear
potential U of the surrounding lattice according to (78). This produces the nonlinearity in (88).
Because the relative deflection ∂q/∂x is without dimension, the modulus σ has
according to (85) the dimension of tension τ , it is therefore a force, an energy per unit
of length. The modulus of elasticity must be equated to the tension of equilibrium of
the chain when the deflection is perpendicular to the chain as seen in Chap. 6. Hence
σ is nothing else than the line tension in the string model of a dislocation, see Seeger
[88] and P. Schiller. A dislocation however cannot in comparison to a string end
inside of a crystal. Another difference is that a dislocation cannot release its energy
by just cutting through it, whereas a string can. From this point of view, the string
model is not sufficient for the dislocation. Furthermore, the quantity L ρ o = m is in
the case of a dislocation not just simply the mass of the lattice atoms on the length
L—because there are no atoms on this length!—it is in fact the effective mass that
should be calculated by inertia measurements of dislocations as a whole with respect
to the lattice.
Our considerations are not sufficient to be able to numerically calculate the inertia
of a dislocation. The determination of the inertia of a dislocation is according to the
second equation (88) dependent on the line tension σ of a dislocation and the velocity
c o . The quantity c o is the critical signal velocity of the sine-Gordon equation. Its
relationship to the sound velocity will be discussed later (see Chap. 12). Furthermore,
there are elementary lines of thought that lead to an explanation of the quantity of
the line tension σ of a dislocation. We will delve further into this in Chap. 23, and
there we will make an estimation that the mass m a of a dislocation of the length a
of a lattice parameter in one of our considered examples will be around 3% of the
mass of the surrounding lattice atoms, see Eqs. (295) and (299).
Such a statement can easily be misunderstood. The calculated mass m a of the
dislocation section is its inertia with respect to the crystal lattice; in other words, m a
is the ratio of the acting force (the configuration force to be exact) to the produced
acceleration of this dislocation section relative to the lattice. The significant difference
of this dislocation mass m a to the mass of the atoms of the lattice becomes clear as