Chapter V
Interatomic Distances
Within the time
region the force which the atoms of matter exert upon each other because
of their rotational velocities acts in the same natural direction
as the gravitational force in the timespace region; that is, toward unity.
But this direction, toward unity, which is inward in the timespace region
and therefore makes the interatomic force which we term gravitation a
force of attraction, is outward in the time region, and the corresponding
interatomic force in this region is a force of repulsion, even though
it is merely gravitation in a different environment.
This reversal
of direction at the unit level makes possible the establishment of an
equilibrium in which the atoms of matter can maintain the same relative
positions in space indefinitely. Such an equilibrium cannot be established
in the timespace region because in this region the effect of a change
in the distance between the atoms is to accentuate any unbalance of forces.
Here the rotational force (gravitation) is directed inward and the spacetime
force outward. If the rotational force exceeds the force of the spacetime
progression an inward motion takes place, making the effective rotational
force still greater. Conversely, if the rotational force is the smaller
the resulting motion is outward, which further weakens the already inadequate
inward force. In either case there can be no establishment of equilibrium.
In the time region,
however, the effect of a change in relative position opposes the unbalanced
force which caused the change. If the rotational force is the greater
an outward motion takes place, weakening this rotational force and ultimately
reducing it to an equality with the spacetime force. Similarly if the
rotational force is the smaller the oppositely directed spacetime force
causes an inward motion. This strengthens the rotational force and again
produces an equilibrium. The separation between any two atoms under these
equilibrium conditions is the interatomic distance.
In order to calculate
these interatomic distances it will first be necessary to determine the
magnitudes of the corresponding interatomic forces. Since the interatomic
rotational force in the time region is merely a different aspect of gravitation
we may utilize the gravitational equation for its evaluation, providing
that we replace the spacetime region terms with the appropriate time
region terms. We have already noted that velocity in the time region is
1/t². Energy, the onedimensional equivalent of mass, which
will take the place of mass in the time region expression of the gravitational
equation because of the directional characteristics of the rotations in
this region, is the reciprocal of this expression, or t².
Acceleration is velocity divided by time, 1/t³. The time region
equivalent of the equation F = ma is therefore F
= Ea = t² x 1/t³ = 1/t in each dimension.
As previously
explained, the value 1/t applies only to the last of the t
units of time, whereas in calculating the effective rotational force we
will want the total. To obtain the latter we integrate 1/t from
unity to t, the initial point of the integration being taken at
unity rather than at zero because unit velocity is the natural datum,
the true physical zero.
The force computed
in this manner is the inherent rotational force of the individual atom;
that is, the force which it exerts against a single unit of force. The
force between two interacting atoms is then
F_{r}
= ln t_{A} ln t_{B}
(4)
The equivalent
distance s' between the two atoms will be measured in the timespace
region as s², because of the interregional relationship previously
discussed. The force at distance s' is therefore proportional to
(s²)² or s^{4} rather than to s².
On this basis the force at equivalent distance s'is
F_{r}
= ln t_{A} ln t_{B}) / s^{4}
(5)
To evaluate the
interatomic distance from this force equation we take advantage of the
fact that at the equilibrium point the force of the spacetime progression
and the component of the rotational force in the direction opposite to
that of the progression are necessarily equal. Since time is threedimensional
the rotational force in the time region is distributed threedimensionally.
The spacetime progression, however, is onedimensional and only that
portion of the rotational force in the dimension of the progression is
effective in the force equilibrium. It is therefore necessary to introduce
a factor into the equilibrium equation representing the ratio of effective
to total rotation. In determining this ratio we note that the first effective
unit of rotation (the first displacement unit) is equal to the spacetime
progression, since spacetime progresses at a unit rate. This one displacement
unit (two total units of rotation) therefore constitutes the time region
maximum if the units are disposed linearly. If these units are distributed
threedimensionally there can be two units of rotation in each dimension,
raising the allowable total to 2³ or 8. Only one of these 8 units,
the one displacement unit in the direction of the progression, is effective
in opposition to the spacetime force.
The same situation
prevails in each dimension of the twodimensional magnetic rotation except
that in this case there are two effective units per dimension, one for
each of the two rotational systems of the atom, and the ratio of effective
to total rotational units in each dimension is 1 to 4. It may be somewhat
confusing to speak of distributing the displacements in each spacetime
dimension in a threedimensional manner, but it should be remembered that
the three time region dimensions are dimensions of time, not of spacetime,
and the total time displacement of a rotation in any one spacetime dimension
may be disposed threedimensionally in the time region without in any
way affecting same situation the other spacetime dimensions. We will
encounter this again later in connection with other physical properties.
On this basis one unit of rotation out of every 4 x 4 x 8 = 128 is effective
against the spacetime force. This ratio is further modified by the initial
one unit negative level of the rotation due to the oppositely directed
motion of the basic oscillation, as the portion of the rotational force
required to overcome the negative initial level is not available to oppose
the force of the progression. This initial unit is distributed over three
dimensions and the onethird unit in the dimension parallel to the spacetime
progression is again distributed over the three dimensions of the time
region. The resultant is 1/9 unit in each magnetic dimension, a total
of 2/9 units. The electric rotation does not affect the initial level
since it is merely a secondary rotation of the existing magnetic rotational
structure. Other phenomena resulting from the rotational forces are similarly
affected by the presence of the oppositely directed basic oscillation
and we will encounter initial levels of one kind or another in a great
many of the physical properties which we will examine.
Because of this
negative initial level another 2/9 unit of displacement must be added
to each of the 128 units in order to obtain one full unit in opposition
to the spacetime progression. This increases the ratio of total to effective
units to 156.44 to 1. The onedimensional rotational force applicable
to each atom is therefore divided by 156.44 in setting up the equilibrium
equation. For the twodimensional magnetic rotation this factor becomes
(156.44)² and for two interacting magnetic rotations it increases
to (156.44)^{4}. Applying this factor to the square of the onedimensional
rotational force, equation 5, we obtain the effective magnetic rotational
force.
F_{m}
= (1/(156.44)^{4} s^{4}) ln² t_{A}
ln² t_{B}
(6)
The distance
factor does not apply to the spacetime force as this force is omnipresent
and unlike the rotational force is not altered as the objects to which
it is applied change their relative positions. At the point of equilibrium,
therefore, the rotational force is equal to the unit spacetime force.
Substituting unity for F_{m} in equation 6 and solving
for the equilibrium distance, we obtain
s_{o}
= (1/156.44) ln^{½} t_{A} ln^{½}
t_{B}
(7)
The interatomic
distances for those elements which have no electric rotation, the inert
gas series, may be calculated directly from this equation. In the elements,
however, t_{A} = t_{B} in most cases and
it will therefore be convenient to express the equation in the simplified
form
s_{o}
= 1/156.44 ln t
(8)
In cgs units
this is
s_{o}
= 2.914 X 10^{8} ln t cm
(9)
As brought
out in the discussion of the general characteristics of the atomic rotation,
the two magnetic displacements may be unequal and in this case the velocity
distribution takes the form of a spheroid with the principal rotation
effective in two dimensions and the subordinate rotation in one. The average
effective rotation under these conditions is (t_{1}² t_{2})^{1/3}
and this expression gives the equivalent value of t for use in
the rotational force equations. The interatomic distances for the inert
gases are as follows:
TABLE I
Atomic No.

Element

Magnetic Rotation

Interatomic Distance

Calculated

Observed

10

Neon

33

3.20

3.20

18

Argon

43

3.74

3.94

36

Krypton

44

4.04

4.02

54

Xenon

54, 4½4

4.36

4.41

Helium, which
also belongs to the inert gas series, has some special characteristics
due to its low rotational displacement and will be discussed in connection
with other elements affected by the same factors. The reason for the appearance
of the 4½ value in the xenon rotation will also be explained later.
Turning now to
the elements which have electric as well as magnetic displacement, we
note again that the electric rotation is onedimensional and opposes the
magnetic rotation. We may therefore obtain an expression for the effect
of the electric rotational force on the magnetically rotating photon by
inverting the onedimensional force term of equation 4.
F_{e}
= 1 / (ln t'_{A} ln t'_{B})
(10)
Because of the
fact that the electric rotation is not an independent motion of the basic
photon but a rotation of the magnetic velocities in the reverse direction,
combining the electric rotational force from equation 10 with the magnetic
rotational force of equation 6 modifies the rotational terms (the functions
of t) only and leaves the remainder of equation 6 unchanged.
F = (1/(156.44)^{4})
(ln² t_{A} ln² t_{B}) / (s^{4}
ln t'_{A} ln t_{B}
(11)
Here again the
effective rotational (outward) and spacetime (inward) forces are necessarily
equal at the equilibrium point. Since the spacetime force is unity we
substitute this unit value for F in equation 11 and solve for so,
the equilibrium distance.
s_{o}
= (1/156.44) (ln^{½} t_{A} ln^{½}
t_{B}) / (ln^{1/4} t'_{A} ln^{1/4}
t'_{B})
(12)
Again simplifying
for application to the elements, where A is generally equal to B,
s_{o}
= (1/156.44) (ln t) / (ln^{½} t')
(13)
In cgs units
this becomes
s_{o}
= 2.914 x 10^{8 ln t / ln½ t' cm}
^{ }^{ }^{ }^{ }
^{
(14)
} 