23 A Particle Solution—The Inertia of Energy
253
In order to estimate the magnitude of the mass m o , we use a value for the line
tension that we get from the calculation of line energy for the crystal niob according
to F. Ackermann, H. Mugrabi and A. Seeger [90], here σ ≈ 3 · 10
−10 Nm. Together
with a lattice parameter of a ≈ 2, 86 · 10
−10 m, as well as an approximative value for
c o to sound velocity according to c o ≈ 4, 5 · 10
3 ms
−1 , we receive an estimate rate
for the dislocation mass m a of
m a ≈ 5 · 10
−27 kg .
(299)
The mass of the niob’s lattice atoms lies in the region of m Nb ≈ 154 · 10
−27 kg.
Hence, the mass m a of the dislocation chain on one lattice parameter a has around
3% of the mass m Nb of the niob atoms.
Lets us assume an estimate of four lattice parameters for the length L o , thus
L o ≈ 4 a, then the factor in (296) would result in f ≈ 0, 16. We would receive an
estimate rate for the kink mass m o ,
m o ≈ 0, 8 · 10
−27 kg ,
(300)
i.e. for the restmass m o of the object defined by our kink solution q
I . This would
make up around 0, 5% of the mass of surrounding lattice atoms.
Such a comparison between inertial masses of dislocations and the inertial masses
of the lattice atoms can easily lead to false conclusions. In fact, in order to be precise,
this should be: The inertia m a of a dislocation portion of the length a with respect to
the lattice totals around 3% of the inertia of niob atoms with respect to our ‘outside’
space, just as the inertia m o of the ‘body’ which we can, according to our previously
made calculations, call a kink takes up around 0, 5% of the inertia of niob atoms with
respect to our ‘outside’ space.
The difference in the corresponding energies is however, compared to this, far
larger. Our object with the mass m o possesses according to (297), if we once again
assume a value of c o ≈ 4, 5 · 10
3 ms
−1 , the stored-up energy of E o = m o c
2
o , thus
E o = 0, 8 · 10
−27
· 4, 5
2
· 10
6 kgm
2 s
−2 and therefore E o ≈ 1, 6 · 10
−20 Nm. However, for the stored-up energy of the niob atoms, the speed of light c L = 3 · 10
8 ms
−1
is valid. We therefore receive for the stored-up energy E Nb of a stationary niob atom
E Nb = m Nb · c
2
L ≈ 154 · 10
−27
· 9 · 10
16 kgm
2 s
−2 , thus E Nb ≈ 1, 4 · 10
−9 Nm. The
stored-up energy of our object, defined with respect to our crystal, with the mass
m o contains around 10
−9 % of the above value—this is an enormous difference!
1
It can also be proven that other solutions of the sine-Gordon equation can be seen
as particles or objects in a mechanical sense respectively using the above-described
fields. For example, this would apply to our field (117), the oscillating breather
1 There is also another important difference between the inertial mass of a kink with respect to the
lattice and the inertial mass of a niob atom to our physical space. For a niob atom, this inertial
mass is identical, according to the principle of equivalence of the General Theory of Relativity, to
its gravitational mass, which the niob atom underlies according to universal gravity. There is no
analogy for the internal observer’s particles of our infinite crystal for the property of ‘gravity’ of a
particle. We just use the everyday term of mass for this property.
253
In order to estimate the magnitude of the mass m o , we use a value for the line
tension that we get from the calculation of line energy for the crystal niob according
to F. Ackermann, H. Mugrabi and A. Seeger [90], here σ ≈ 3 · 10
−10 Nm. Together
with a lattice parameter of a ≈ 2, 86 · 10
−10 m, as well as an approximative value for
c o to sound velocity according to c o ≈ 4, 5 · 10
3 ms
−1 , we receive an estimate rate
for the dislocation mass m a of
m a ≈ 5 · 10
−27 kg .
(299)
The mass of the niob’s lattice atoms lies in the region of m Nb ≈ 154 · 10
−27 kg.
Hence, the mass m a of the dislocation chain on one lattice parameter a has around
3% of the mass m Nb of the niob atoms.
Lets us assume an estimate of four lattice parameters for the length L o , thus
L o ≈ 4 a, then the factor in (296) would result in f ≈ 0, 16. We would receive an
estimate rate for the kink mass m o ,
m o ≈ 0, 8 · 10
−27 kg ,
(300)
i.e. for the restmass m o of the object defined by our kink solution q
I . This would
make up around 0, 5% of the mass of surrounding lattice atoms.
Such a comparison between inertial masses of dislocations and the inertial masses
of the lattice atoms can easily lead to false conclusions. In fact, in order to be precise,
this should be: The inertia m a of a dislocation portion of the length a with respect to
the lattice totals around 3% of the inertia of niob atoms with respect to our ‘outside’
space, just as the inertia m o of the ‘body’ which we can, according to our previously
made calculations, call a kink takes up around 0, 5% of the inertia of niob atoms with
respect to our ‘outside’ space.
The difference in the corresponding energies is however, compared to this, far
larger. Our object with the mass m o possesses according to (297), if we once again
assume a value of c o ≈ 4, 5 · 10
3 ms
−1 , the stored-up energy of E o = m o c
2
o , thus
E o = 0, 8 · 10
−27
· 4, 5
2
· 10
6 kgm
2 s
−2 and therefore E o ≈ 1, 6 · 10
−20 Nm. However, for the stored-up energy of the niob atoms, the speed of light c L = 3 · 10
8 ms
−1
is valid. We therefore receive for the stored-up energy E Nb of a stationary niob atom
E Nb = m Nb · c
2
L ≈ 154 · 10
−27
· 9 · 10
16 kgm
2 s
−2 , thus E Nb ≈ 1, 4 · 10
−9 Nm. The
stored-up energy of our object, defined with respect to our crystal, with the mass
m o contains around 10
−9 % of the above value—this is an enormous difference!
1
It can also be proven that other solutions of the sine-Gordon equation can be seen
as particles or objects in a mechanical sense respectively using the above-described
fields. For example, this would apply to our field (117), the oscillating breather
1 There is also another important difference between the inertial mass of a kink with respect to the
lattice and the inertial mass of a niob atom to our physical space. For a niob atom, this inertial
mass is identical, according to the principle of equivalence of the General Theory of Relativity, to
its gravitational mass, which the niob atom underlies according to universal gravity. There is no
analogy for the internal observer’s particles of our infinite crystal for the property of ‘gravity’ of a
particle. We just use the everyday term of mass for this property.