8 The sine-Gordon Equation of a Dislocation
79
During this period of time, the crystal block in our figure has plastically moved one
lattice position. The rigid motion of the crystal block of the mass M can be attributed
to an average velocity V ,
V =
a
T
=
a
L
dq
dt
,
with a kinetic energy E according to
E =
M
2
=
1
2
M
a
L
2
dq
dt
2
=
1
2
m
L
a
3
a
L
2
dq
dt
2
=
1
2
m
L
a
dq
dt
2
.
The kinetic energy of the crystal block is assigned to the kinetic energy E L of the
dislocation. In fact, the complete crystal block of the mass M, however, does not
glide rigidly so that
E L <
1
2
m
L
a
dq
dt
2
.
Hence, for the kinetic energy E a of this dislocation of the length a of a lattice
parameter, we find that
E a <
1
2
m
dq
dt
2
.
The kinetic energy of an inertial mass m stands on the right side of this inequality.
This mass moves through the crystal with the velocity of the dislocation dq/dt ; m
is the mass of the surrounding lattice atoms. Therefore, the inertial mass m α of a
dislocation of the length a of a lattice parameter has to be smaller, i.e.
m α < in conformity with the estimation that we will make for a dislocation mass in Chap. 23
on the basis of the determination of line tension.