9 Natural Measuring-Rods and Clocks
85
L o = x 2 − x 1 = 2
a/2
∂q
I
o
∂x
x=0
=
a
2a
π
1
λ o
e
0
1 + e 0
,
L o = π λ o .
(99)
Using this, we have defined a natural unit of measure L o out of the internal geometry
of the kink line. We can use this as a measuring-rod in our continuum.
This unit of measure L o has, according to Eq. (91), the length of a few Ångstroms
which is equivalent to a few lattice parameters. Referring to the lattice would however
be a step backwards which is no longer necessary. The unit of length L o is our natural
‘ruler’ which can be reproduced at any time by the internal geometry of the kink lines
of our continuum.
The decisive point is as follows: Using L o we can measure any other length.
Here, L o compares less with the Parisian standard metre, where careful preparations
have to be taken to ensure its consistency, but it compares more with the wavelength
of the yellow sodium line, for example, which is identically reproduced by nature.
Let us recapitulate: The length X of an object can be determined by seeing how
often our measuring-rod L o fits on to the object, for example X = 27L o . It is obvious
the coefficient of measure x for the length X increases if the scale of the measuringrod used as our unit decreases.
Having introduced a standard of length, our measuring-rod, we can now do geometry, in other words we can describe the positions and distances of ‘objects’ from
each other. This fact will play an important role later on in Chap. 12. We however
want more. We also want to describe the processes of motion of these objects. We do
not just want to know where an object is, and we also want to know when the object is
there. We want to describe ‘events’ as one would say. An event is characterised by the
location x where it takes place and by the time t when it happened. Therefore, we
still need a clock. Once again we wish to measure the time in our crystal without the
help of outside instruments. In order to achieve this, we need to refer to the internal
structures that the crystal itself possesses. It is the sine-Gordon equation that once
again delivers us a suitable period of time, a unit to measure time with. To find this,
we have to observe the so-called localised breather solution q
III
o which represents a
time and space-dependent solution of the sine-Gordon equation (95); see A. Seeger
[86] cf. also H. Günther [34],
q
III
o = q
III
o (x, t) = 4 arctan
sin
t
√
2
cosh
x
√
2
.
(100)
The actual displacement q
III
o of the dislocation as a solution of (88) is according to
(96) once again q
III
o (x, t) =
a
2π
q
III
o
x
λ o
,
c o t
λ o
, thus
85
L o = x 2 − x 1 = 2
a/2
∂q
I
o
∂x
x=0
=
a
2a
π
1
λ o
e
0
1 + e 0
,
L o = π λ o .
(99)
Using this, we have defined a natural unit of measure L o out of the internal geometry
of the kink line. We can use this as a measuring-rod in our continuum.
This unit of measure L o has, according to Eq. (91), the length of a few Ångstroms
which is equivalent to a few lattice parameters. Referring to the lattice would however
be a step backwards which is no longer necessary. The unit of length L o is our natural
‘ruler’ which can be reproduced at any time by the internal geometry of the kink lines
of our continuum.
The decisive point is as follows: Using L o we can measure any other length.
Here, L o compares less with the Parisian standard metre, where careful preparations
have to be taken to ensure its consistency, but it compares more with the wavelength
of the yellow sodium line, for example, which is identically reproduced by nature.
Let us recapitulate: The length X of an object can be determined by seeing how
often our measuring-rod L o fits on to the object, for example X = 27L o . It is obvious
the coefficient of measure x for the length X increases if the scale of the measuringrod used as our unit decreases.
Having introduced a standard of length, our measuring-rod, we can now do geometry, in other words we can describe the positions and distances of ‘objects’ from
each other. This fact will play an important role later on in Chap. 12. We however
want more. We also want to describe the processes of motion of these objects. We do
not just want to know where an object is, and we also want to know when the object is
there. We want to describe ‘events’ as one would say. An event is characterised by the
location x where it takes place and by the time t when it happened. Therefore, we
still need a clock. Once again we wish to measure the time in our crystal without the
help of outside instruments. In order to achieve this, we need to refer to the internal
structures that the crystal itself possesses. It is the sine-Gordon equation that once
again delivers us a suitable period of time, a unit to measure time with. To find this,
we have to observe the so-called localised breather solution q
III
o which represents a
time and space-dependent solution of the sine-Gordon equation (95); see A. Seeger
[86] cf. also H. Günther [34],
q
III
o = q
III
o (x, t) = 4 arctan
sin
t
√
2
cosh
x
√
2
.
(100)
The actual displacement q
III
o of the dislocation as a solution of (88) is according to
(96) once again q
III
o (x, t) =
a
2π
q
III
o
x
λ o
,
c o t
λ o
, thus