8 The sine-Gordon Equation of a Dislocation
69
The quantity m α describes per definition the inertia during a (plastic) displacement of the dislocation coordinates q α relative to the lattice. In 1964 Kröner [50]
showed the necessity of introducing such a resistance of dislocations against an acceleration relative to the lattice, the inertia of dislocations with respect to this lattice, an
effective mass of dislocations.
In 1939, J. Frenkel [25] and T. Kontorova using U. Dehlinger [10] decisive
preparations from 1929 wrote down the system of differential equations for the
deflection q i of a one-dimensional row of lattice atoms, which were directly next to
the geometrical centre of the dislocation line. For the Newtonian equations (55) of
the deflections q i of the lattice atoms laying transversal to the dislocation line the
authors found
m i
d
2 q i
dt 2 = σ q i−1 − 2σq i + σ q i+1 − D sin
2π
a
q i
Frenkel–Kontorova
equation
(77)
with the mass of the lattice atoms m i and the two quantities σ and D calculated
from the parameters of the lattice (see A. Seeger [86]). Equation (77) is identical
with the Eq. (31) of a linear chain of elastic coupled masses, which are additionally
under the influence of an external force −D sin
2π
a
q i
. The force on the atom with
the number i and the deflection q i originates from the influence of the surrounding
lattice and takes into consideration that when a deflection of the size of a multiple
lattice parameter takes place, q i = n a, then periodically, the same force is produced.
We notice that the sine function is not in the least prescribed, compare also with (74)
and can only boast the property of simplicity.
1
Due to the fact that the q i are the supposed transversal deflections of the chain,
the quantity σ contains the base tension of this atom row as we have seen in Chap. 6.
An important result of the investigations by Dehlinger, Frenkel and Kontorova is
that the rows of lattice atoms in near proximity to a dislocation line show a special
base tension. In the years 1948/49, A. Seeger [87] was able to derive the sine-Gordon
equation out of Eq. (77). This requires the conclusion from the first three terms on the
right side of (77) to the term E x d
2 s/ds
2 in the continuum approximation of the
chain with q i (t) −→ s(x, t), in the same way as we did it in Chap. 4 when dealing
with a linear chain and its longitudinal deflections (see Eqs. (63)–(68)) and also later
with the transversal deflections.
The Frenkel–Kontorova Eq. (77) and also Seeger’s sine-Gordon equation (cf.
Seeger [87, 88]) were originally derived and considered only for lattice atoms. Later,
it was A. Seeger who also applied the equation on the dislocation line itself. Here, we
wish to remind the reader of the fundamental importance that the conclusion from
the lattice atom masses m i in (77) has on the relative inertial masses m α of a
dislocation in (76). Only the inertial masses m α of a dislocation can principally
1 The choice of the (non-linear) sine function moved J. Rubinstein to name the equation the
sine-Gordon equation in order to remind people of its linear approximation, the Klein-Gordon
equation.
69
The quantity m α describes per definition the inertia during a (plastic) displacement of the dislocation coordinates q α relative to the lattice. In 1964 Kröner [50]
showed the necessity of introducing such a resistance of dislocations against an acceleration relative to the lattice, the inertia of dislocations with respect to this lattice, an
effective mass of dislocations.
In 1939, J. Frenkel [25] and T. Kontorova using U. Dehlinger [10] decisive
preparations from 1929 wrote down the system of differential equations for the
deflection q i of a one-dimensional row of lattice atoms, which were directly next to
the geometrical centre of the dislocation line. For the Newtonian equations (55) of
the deflections q i of the lattice atoms laying transversal to the dislocation line the
authors found
m i
d
2 q i
dt 2 = σ q i−1 − 2σq i + σ q i+1 − D sin
2π
a
q i
Frenkel–Kontorova
equation
(77)
with the mass of the lattice atoms m i and the two quantities σ and D calculated
from the parameters of the lattice (see A. Seeger [86]). Equation (77) is identical
with the Eq. (31) of a linear chain of elastic coupled masses, which are additionally
under the influence of an external force −D sin
2π
a
q i
. The force on the atom with
the number i and the deflection q i originates from the influence of the surrounding
lattice and takes into consideration that when a deflection of the size of a multiple
lattice parameter takes place, q i = n a, then periodically, the same force is produced.
We notice that the sine function is not in the least prescribed, compare also with (74)
and can only boast the property of simplicity.
1
Due to the fact that the q i are the supposed transversal deflections of the chain,
the quantity σ contains the base tension of this atom row as we have seen in Chap. 6.
An important result of the investigations by Dehlinger, Frenkel and Kontorova is
that the rows of lattice atoms in near proximity to a dislocation line show a special
base tension. In the years 1948/49, A. Seeger [87] was able to derive the sine-Gordon
equation out of Eq. (77). This requires the conclusion from the first three terms on the
right side of (77) to the term E x d
2 s/ds
2 in the continuum approximation of the
chain with q i (t) −→ s(x, t), in the same way as we did it in Chap. 4 when dealing
with a linear chain and its longitudinal deflections (see Eqs. (63)–(68)) and also later
with the transversal deflections.
The Frenkel–Kontorova Eq. (77) and also Seeger’s sine-Gordon equation (cf.
Seeger [87, 88]) were originally derived and considered only for lattice atoms. Later,
it was A. Seeger who also applied the equation on the dislocation line itself. Here, we
wish to remind the reader of the fundamental importance that the conclusion from
the lattice atom masses m i in (77) has on the relative inertial masses m α of a
dislocation in (76). Only the inertial masses m α of a dislocation can principally
1 The choice of the (non-linear) sine function moved J. Rubinstein to name the equation the
sine-Gordon equation in order to remind people of its linear approximation, the Klein-Gordon
equation.