## Chapter XXXIII
Radiation and Refraction
The role of radiation
in the atomic transformation processes is generally confined to carrying
off part of the excess energy released by the reaction. It is possible,
however, for the radiation to supply the energy required for initiation
of such a process and there is a general class of reactions such as
Be^{9}
+ *y* -> Be^{8} + n¹
8-1 + *y* -> 8-0 + 0-1
Another rotational
effect which can be produced by radiation is the creation of a positron-electron
pair by two oppositely directed photons: the inverse of the neutralization
process previously mentioned.
A more general
result of the interaction between radiation and the rotational systems
is the conversion of radiant energy into translational energy or vice
versa. As the simplest of all space-time phenomena, radiation is one of
the major constituents of the physical universe and every material and
sub-material unit is subjected to a ceaseless bombardment by the omnipresent
photons. Some of this radiation is *reflected* and merely undergoes
a change in direction without losing its identity as radiation. Some is
able to pass through certain material substances in much the same manner
as through open space, in which case we say that these substances are
transparent to the particular radiation. Another portion is *absorbed*
and in this process the radiation is transformed into a different type
of motion.
We will look
first at the situation which results when the radiation strikes electrons
within the material aggregate rather than colliding with atoms of matter.
The electron already possesses some translatory velocity due to energy
interchange with the thermal motion of the matter in which it is located.
A portion of the energy absorbed from the radiation is utilized in bringing
this velocity up to the ionizing level. On being ionized the electron
acquires the ability to move freely in space and the balance of the radiant
energy becomes kinetic energy of translation in the time-space region,
which means that the newly charged electron is ejected from the material
aggregate and moves off into space. This emission of electrons on exposure
to radiation is known as the *photoelectric effect*.
If we call the
energy of the photon *E* and the ionizing energy *k*, the maximum
kinetic energy of the ejected electron is *E - k*. Energy, *E*,
is *t/s*, but in the time-space region where velocity is below the
unit value, the effective value of *s* in primary processes is unity,
hence *E* = *t*. The unit value of *s* has a similar effect
on frequency (velocity) *s/t*, reducing it to *1/t*. The conversion
factor which relates frequency to energy is therefore *t* divided
by *1/t* or *t²*. Since the interchange in the photoelectric
effect is across the boundary between the time-space region and the time
region it is also necessary to introduce the dimensional factor 3 and
the regional ratio, 156.44. We then have
*E* = 3
/ 156.44 *t²v* (139)
In cgs units
this becomes
*E* = (3
/ 156.44) × ((0.1521×10^{-15)²/6.670×108) v = 6.648×10-27
v ergs (140)}
^{ }^{
The coefficient
of this equation is Planck's Constant, commonly designated by
the letter h. It is ordinarily considered to have the dimensions
ergs x seconds, but these dimensions have no actual physical significance.
In reality this is merely a conversion constant relating velocity s/t
to energy t/s, and it has the dimensions t²/s²
since
s/t x t²/s²
= t/s.
These reversing
dimensions t²/s² then reduce numerically to t³/1
because of the unit value of s. The latest experimental values
of the constant are in the neighborhood of 6.624 x 10-27. The
amount of difference between this and the calculated value indicates the
possibility of a secondary effect similar to those entering into the mass
relationships, but this is one of the "fine structure" details which has
not yet been investigated.
Substituting
the value hv for E in the expression E - k, we have
the Einstein equation for the maximum kinetic energy of the photo-electrons.
Emax
= hv - k (141)
If the absorption
of energy from the radiation happens to take place under such conditions
that none of it is lost to other electrons or to matter while the electron
is escaping from the material aggregate, the kinetic energy will be the
full amount indicated by equation 141. More often, however, there are
contacts on the way out in which some of the energy is transferred to
other units, and the actual kinetic energies therefore range from the
maximum down to zero. It is, of course, also possible that the loss of
energy on the way may be sufficient to reduce the total below the ionizing
level, in which case the charge is given up and no photoelectron appears.
In this case the entire energy of the photon becomes thermal energy of
the material aggregate.
The term k,
the energy required for ionization of the electron and escape from the
material aggregate, is a characteristic of the substance in which the
electron is situated and it is known as the work function. No
detailed study of this quantity has been undertaken, but it is obvious
from mere inspection that it is a simple function of the rotation in the
dimension in which the electron escapes, and it can be expressed empirically
as
k = 2.25
n½ electron-volts (142)
The full rotation,
including the initial level, is effective in the magnetic dimensions but
the value of n in the electric dimension is the displacement only.
Normally the escape takes place in the minimum dimension, the path of
least resistance, but the experimental conditions may be such as to compel
the electrons to leave by way of one of the other dimensions, in which
case the effective value of the work function is higher. The minimum theoretical
value for beryllium, for example, is 3.2, which corresponds to the electric
displacement 2, and the experimental results normally ran-e from 3.10
to 3.30, but the value 3.92 has also been obtained. We might be inclined
to dismiss this as an error except that it agrees with the theoretical
value 3.9 for an escape in the primary magnetic dimension. Similarly,
some of the results for iron agree with the secondary magnetic value,
3.9, while a number of others are reasonably close to the primary magnetic
figure, 4.5. There are many other elements for which the experimental
results are grouped around two of the theoretically possible values and
it seems apparent that in some instances the experimental conditions must
inhibit escape in the minimum dimension.
Table CIX shows
how the observed work functions of the elements compare with the theoretical
values derived from the atomic rotations. In this table the letters c
and a refer to the electric and magnetic dimensions respectively.
The entry of the rotation into the second space-time unit in the 4A and
4B groups reduces the value of n by one-half in some instances,
and these modified values are identified in the table by the symbols a*
and c*. It should also be noted that where the value 3 appears
as the magnetic rotation of one of the higher group elements this is the
inverse of the actual rotation, 5.
If the radiation
strikes an atom of matter rather than an electron and impinges in such
a manner that it is absorbed instead of being reflected, the full amount
of radiant energy becomes thermal energy. This action is easily visualized,
but the nature of the reverse process, the emission of radiant energy
by the vibrating atom, is not self-evident and requires some further explanation.
Here again the Principle of Inversion is the controlling factor. From
this principle we find that the thermal motion of the atoms of matter
is in equilibrium with a similar vibratory motion of the space units in
which the atoms are located. Since the positions of the material atoms
are determined by the gravitational forces, the atomic thermal motion
likewise occupies gravitational positions and cannot progress with space-time.
The coexisting space motion, on the other hand, is not affected by gravitation
and as space-time progresses it carries this vibrational motion of the
space units along as radiation. In order to restore the equilibrium required
by the Principle of Inversion, motion is then transferred from the atoms
to the new space units with which they are now associated.
Table CIX
Work Functions
n
Calculated
Observed
Li
1
c
2.25
2.28
2.42
2.49
Be
2
c
3.2
3.10
3.17
3.30
3
a
3.9
3.92
B
3
c
3.9
4.4
4.6
C
4
c
4.5
4.34
4.81
Na
1
c
2.25
2.06
2.25
2.26
2.28
2.29
2.47
Mg
2
c
3.2
2.74
3.0
3
a
3.9
3.58
3.59
3.62
3.68
3.78
Al
3
c
3.9
2.98
3.38
3.43
4.08
4.20
4.36
Si
3
a
3.9
3.59
4
c
4.5
4.2
K
1
c
2.25
2.0
2.12
2.24
2.26
Ca
2
c
3.2
2.42
2.71
2.76
3.20
3.21
Ti
3
a
3.9
3.95
4.14
4.17
V
3
a
3.9
3.77
4.44
Cr
4
a
4.5
4.37
4.38
4.60
Mn
3
a
3.9
3.76
4.14
Fe
3
a
3.9
3.91
3.92
4.23
4
a
4.5
4.40
4.44
4.63
4.72
4.77
Co
3
a
3.9
3.90
4.12
4.21
4.25
4
a
4.5
4.40
Ni
3
a
3.9
3.67
4.06
4.12
4
a
4.5
4.32
4.61
4.87
4.96
5.01
Cu
3
3.9
3.85
4.07
4.18
4
a
4.5
4.26
4.46
4.55
4.86
5.61
Zn
2
c
3.2
3.08
3.28
3.32
3.40
3
a
3.9
3.60
3.66
3.89
4.24
4.28
4.31
Ga
3
c
3.9
3.80
4.12
Ge
4
c
4.5
4.29
4.50
4.73
4.80
As
5
c
5.0
5.11
Se
4
a
4.5
4.42
4.62
5.11
Rb
1
c
2.25
2.09
2.16
Sr
2
c
3.2
2.06
2.24
2.74
Zr
3
a
3.9
3.60
3.73
3.91
4
a
4.5
4.12
4.3
4.5
Nb
3
a
3.9
3.96
4.01
Mo
3
a
3.9
4.08
4.15
4.20
4
a
4.5
4.32
4.34
4.44
4.48
Ru
4
a
4.5
4.52
Rh
4
a
4.5
4.57
4.58
4.80
4.92
Pd
4
a
4.5
4.49
5
a
5.0
4.92
4.97
4.99
5.11
Ag
4
a
4.5
4.08
4.33
4.47
4.73
4.75
Cd
3
a
3.9
3.68
3.73
3.94
4.00
4.07
4.1r@
Sn
3
a
3.9
3.62
3.87
4.09
4
a
4.5
4.21
4.38
4.50
Sb
3
a
3.9
4.01
4.14
Te
4
a
4.5
4.70
4.76
Cs
I
c
2.25
1.81
1.9
1.96
2.48
2.52
Ba
I
c
2.25
1.73
2.11
2.39
La
2
a*
3.2
3.3
Ce
2
a*
3.2
2.6
2.84
Pr
2
a*
3.2
2.7
Nd
2
a*
3.2
3.3
Sm
2
a*
3.2
3.2
Hf
2
a*
3.2
3.53
Ta
3
a
3.9
3.96
4.12
4.16
4.19
4.25
W
4
a
4.5
4.25
4.35
4.50
4.52
4.54
4.58
4.60
Re
5
a
5.0
5.0
5.1
Os
4
a
4.5
4.55
Ir
4
a
4.5
4.57
Pt
4
a
4.5
4.09
4.52
5
a
5.0
5.08
5.32
5.36
4.73
4.82
4.86
Au
4
a
4.5
4.0
4.46
4.58
Hg
4
a
4.5
4.50
4.52
4.53
Tl
3
a
3.9
3.68
3.84
Pb
3
a
3.9
3.94
3.97
4.14
Bi
4
a
4.5
4.14
4.17
4.31
4.46
Th
2
a*
3.2
3.35
3.38
3.46
3.47
3.57
U
2
a*
3.2
3.27
3.63
4
a
4.5
4.32
These space units
in turn carry the vibratory motion forward as radiation and the whole
process is repeated over and over again indefinitely.
This situation
is quite similar to that which we encountered in our examination of electrolysis.
We found that the ions in the electrolyte are unable to carry their charges
into the anode in response to the differential forces which are operative
in this direction, since matter cannot penetrate matter except under special
conditions, but the associated space motion is not subject to this limitation
and the space units leave the ions and move into the anode, taking with
them the rotational motion of the ionic charges.
In the interior
of a solid or liquid aggregate the radiation from any one atom is promptly
reabsorbed by its neighbors and no net change in the total thermal energy
of the aggregate results. The radiation from the outside surface is, however,
lost to the surroundings and to the extent that this loss is not counterbalanced
by incoming radiation from other sources the temperature of the aggregate
falls.
The rate at which
energy is radiated from a surface depends on two factors: the number of
space units leaving the surface per unit of time, and the energy carried
by each unit. The first of these factors is simply the velocity of the
space-time progression. From the Principle of Equivalence we may deduce
that it can be expressed as one unit of radiation per unit of area per
unit of time, but the space-time progression is not effective in the direction
of the basic linear frequency, 1/9 of the total, which reduces the effective
rate to 8/9 unit of radiation per unit of area per unit of time. From
equation 72 the atomic thermal energy is proportional to the fourth power
of the temperature. The energy of the associated space unit is the same
value multiplied by the reversing dimensions, t²/s²,
or (0.3334 x 10-10)² in cgs units. The radiation rate
can then be expressed as
Erad
= (8/9) x ((0.3334 x 10-10)² x 5.001 X 10-4
ergs) / ((2.914 x 10-8 cm)² x 0.1521 x 10-15
sec x (510.16 deg)4)
= 5.655 x 10-5
ergs / sec cm² deg4
<143>
This is the Radiation
Constant or Stefan-Boltzmann Constant.
The frequency
or vibrational velocity of the escaping space unit, the photon, is determined
by the characteristics of the atomic vibrational motion from which the
motion of the space unit was derived. At zero temperature the thermal
vibration period is infinite and the equivalent thermal velocity 1/t²
is 1/&inf; or zero. The addition of thermal energy, which is space
displacement, is equivalent to reducing the time displacement. The vibration
period and the corresponding equivalent thermal velocity therefore decrease
with increasing temperature up to a limit of t = 1, at which point
the molecule has reached the boundary of the time region and is ready
to make the transition into the time-space region. Since it is the thermal
velocity (with some modifications to be considered later) which is radiated,
the distribution of radiated frequencies or spectrum emitted
by time region structures comprises all values of 1/t² from
1/&inf; to a maximum which depends on the temperature, the absolute
maximum being 1/1. Because of the small interval between these values
of 1/t² in the solid and the modifications to which the frequencies
are subjected in escaping from the dense solid or liquid structures the
actual distribution of the frequencies is essentially continuous, and
we observe a continuous spectrum.
At the unit level,
the boundary between the time region and the time-space region, there
is a directional reversal and further additions of motion of the same
nature, motion which is equivalent to thermal energy, go into the reverse
velocity component of a compound velocity: velocity of a velocity (mass).
The total equivalent thermal energy in the two regions is the sum of the
regional components, but since the velocity in the time-space region is
inversely directed, the resultant velocity (frequency) of the radiation
is the difference between the time region and the time-space region velocity-energy
components.
As we have found,
velocity in the time region is in equilibrium with energy in the time-space
region. The latter in turn is proportional to the square of the time-space
region velocity. Where the time displacement is b, the velocity
is 1/b and the energy per mass unit is equal to 1/b².
This is the time-space contribution to the radiation frequency and since
it is in the opposite direction from the time region component 1/a²
the resultant is 1/a² - 1/b². In order to maintain
a positive value of the resultant it is necessary that b exceed a by at
least one unit, and the minimum value of b is therefore 2.
It will be noted
that the velocity interval for the normal range of temperatures in the
time-space (gas) region is relatively large; that is, the difference between
1/2² and 1/3² involves a reduction of about 55 percent. Furthermore,
there is comparatively little interference on the way out of a gas aggregate.
Instead of a continuous spectrum the gas therefore has a line spectrum:
a regular succession of discrete frequencies resulting from the various
possible values of the displacements a and b.
In the case of
hydrogen there are no modifying factors and the frequencies can be obtained
directly from the expression 1/a² - 1/b², utilizing
the value of unit frequency previously employed, which we will designate
R, in accordance with the usual practice.
vH
= R(1/a² - 1/b²) (144)
Placing a equal
to 1 and assigning the successive values 2, 3, 4, etc. to the displacement
b, we obtain the Lyman series of spectral lines. Although
the time region displacement a can reach unit value before gas
motion starts, this is not mandatory and other series based on higher
values of a also occur, becoming less and less probable as a
increases beyond 2. When a = 2 and b has the values 3, 4,
5, etc. the well-known Balmer series, the first spectral
series to be identified, is the result. With a = 3 we obtain the
Paschen series, a = 4 gives us the Brackett series,
and so on.
The frequencies
for ionized helium can also be calculated from equation 144 by introducing
the factor 4.
vHe
= 4R(1/a² - 1/b²) (145)
In this case
the normal one unit of space in the expressions 1/a² and 1/b²
has been increased to two by the addition of one unit of rotational space
vibration. The velocity 1/a², which is actually (1/a)²,
now becomes (2/a)² or 4(1/a)². A similar change
takes place in the b component. To generalize, we may say that
ionization increases the 2 spectral frequencies by the factor e²,
where e is the total ionization based on the normal state as unity.
The generalized equation for the hydrogen type spectrum is therefore
v = Re²
(1/a² - 1/b²) (146)
When we examine
the velocity relations of other elements we find that this hydrogen type
spectrum is not characteristic of the normal atom but represents a special
case in which the effects of all motion other than thermal are eliminated
by the absence of any rotational motion with a force integral exceeding
unity. In the normal atom effective rotational motions do exist and the
radiation frequencies are modified very substantially by these coexisting
velocities.
In order to understand
why this should be the case we need only to recognize that the velocity
which detaches itself from the atom as radiation is necessarily the absolute
velocity relative to space-time, not the thermal velocity alone. This
is similar to the situation which exists when a projectile is fired from
a rapidly moving airplane. The force of the explosion imparts a certain
definite velocity to the projectile irrespective of the motion of the
plane but the velocity relative to the earth's surface, which in this
case is analogous to the velocity relative to space-time, is not this
explosion-generated velocity but the resultant of the latter plus or minus
the effective component of the velocity of the plane, In order to determine
the velocity which will be radiated we must likewise modify the purely
thermal velocity by adding or subtracting the effective velocity components
of the other motions of the atom.
In the earlier
stages of this present project a considerable amount of work was done
toward the development of theoretical methods of calculation of the individual
values of the spectral terms, on the assumption that this would open up
some avenues of approach to the solution of the general problems of structure.
It ultimately became apparent that the spectra contain too much information;
each individual term is a composite of the eff ects of all of the diff
erent motions in the particular atomic system and in order to sort out
the various components it is practically essential to determine the nature
and general order of magnitude of each item in advance. Instead of using
the spectral relationships as an aid in the study and analysis of the
general questions of atomic structure, it became evident that these general
structural principles would have to be used-to calculate the spectral
terms. For this reason no recent work has been done on the spectra and
since it has not appeared advisable to delay this presentation long enough
to bring the previous work up to date, this material will be omitted.
It may be mentioned,
however, that even a preliminary analysis is sufficient to indicate that
the numerical values of the spectral terms conform to the general relationship
that would be expected on theoretical grounds; that is, each term of the
spectral combinations is one of the terms of equation 146 (the thermal
motion) plus or minus the effective components of the other motions of
the atom, including the rotation, the basic linear vibration, the rotational
vibration, the secondary motion of the associated space unit, any electric
or magnetic motion that may be present, etc. The splitting of the
various terms under certain conditions is obviously due to the fact that
the directions of these other motions are not necessarily fixed with reference
to the direction of the thermal motion and the corresponding frequency
increments may be either plus or minus.
In our original
examination of the phenomenon of radiation we noted that the photons have
no translatory motion of their own and are stationary with respect to
space-time. They are, however, carried along by the progression of space-time
itself and therefore have an apparent velocity equal to the space-time
ratio, which in the absence of displacement is unity. This unit velocity,
as we have seen, is the condition of rest in the physical universe: the
datum from which all physical activity starts. Where time or space displacements
are present so that the space-time ratio is no longer unity the apparent
velocity of the radiation is modified accordingly. Since matter is a time
displacement, the space-time ratio involved in the passage of radiation
through matter or matter through radiation is not unity but a modified
value resulting from the addition of the time displacement to the time
component of the original unit velocity. Addition of more time to the
ratio s/t decreases the numerical value and the apparent velocity
of radiation in matter is therefore less than unity.
One of the important
consequences of the velocity change is a bending of the path of the radiation
in passing from space to matter or from one material medium to another.
The amount of this bending or refraction is measured by the ratio
of the sine of the angle of incidence to the sine of the angle of refraction,
or the equivalent ratio of the velocities in the two media. It is called
the index of refraction and is represented by the letter n.
It will be noted
that on this basis the index of refraction relative to a vacuum is equal
to the total time associated with unit space in the motion of the radiation.
For present purposes we will be interested in the time displacement rather
than in the total time and since the time per unit space in undisplaced
space-time is unity, the displacement is n - 1. The displacement
due to the presence of any specific atom or quantity of matter is independent
of temperature but there is a temperature variation in the refractive
index due to the accompanying change in density. We may eliminate this
effect by dividing each displacement by the corresponding density, obtaining
a temperature-independent quantity (n - 1)/d.
The refractive
displacement is the sum of two components, one due to the motion of matter
through radiation (the apparent translatory motion of the radiation) and
the other due to the vibratory motion of the radiation through matter.
In the first of these we have a simple motion at unit velocity in the
time region. We have previously determined that the three-dimensional
distribution of motion in the time region reduces the component parallel
to one-dimensional time-space region motion to 1/8 of the total, and the
vibratory nature of the motion of matter in the time region introduces
an additional factor of 1/2. We therefore find the displacement on a time-space
region basis to be 1/16 of the effective time region displacement units.
If the magnetic
rotational displacement is unity the refractive displacement is also unity,
but where the rotational displacement is n the radiation travels
through only one of the n displacement units and the effective
refractive displacement is 1/n. If we represent the average value
of 1/n as kr, the refractive displacement due
to the translatory motion is
(n-1)/d
(translation) = kr/16 (147)
When the density
is expressed in g/cm³ rather than in natural units, equation 147
must be multiplied by 10.53, the coefficient of the volume equation 53.
We then have
(n-1)/d
(translation) = .6583 kr (148)
In the second
refractive component, that due to the vibratory motion of the radiation,
we are not dealing with unit velocity but with a lower velocity (frequency),
and the refractive effect is reduced by the ratio v/1. This component
is also modified by the geometric relationship between the path of the
radiation and the structure of the material medium through which it passes.
We will call this modifying factor the vibration factor, Fv.
The vibrational refraction is then
(n-1)/d
(vibration) = .6583 v Fvkr (149)
Adding equations
148 and 149,
(n-1)/d
= .6583 kr (1 + v Fv) (150)
The wavelength
most commonly used for refraction measurements is that of the sodium D
line, 5893 X 10-8 cm. This is equivalent to 0.0774 natural units of frequency.
Substituting this value for v in equation 150 we obtain
(n-1) Id = .6583
k, (1 + .0774 F,) (151)
Unless otherwise
specified, the symbol n will refer to nd wherever
it is used in the subsequent paragraphs.
The value of
Fv applicable to a number of the most common organic
series is .75. For convenience in dealing with these compounds we may
simplify equation 151 to
(n-1)/d
= .6965 kr (152)
This equation
shows that evaluation of the refractive index for compounds of this class
is merely a matter of determining the refraction constant kr.
As mentioned in the discussion of diamagnetic susceptibility, the refraction
constant is the reciprocal of the effective magnetic rotational displacement:
the total displacement minus the initial level. The situation is, however,
complicated to some extent by a variability in the initial levels, especially
those of the two most common elements in these compounds: carbon and hydrogen.
In Table CX the variable factor is shown in the column headed "Dev.,"
the numerical values there listed being the total deviation from the normal
intial levels of the component elements in the particular group of compounds,
as shown in the group sub-heading. The deviations are expressed in 1/9
units and the figure as given indicates the number of hydrogen atoms (or
equivalent rotational mass units) in which the initial level has been
shifted 1/9 unit, upward unless otherwise indicated.
In the acids,
for example, the rotational displacement of the oxygen atoms and the carbon
atom in the CO group is 2, while that of the hydrogen atoms and the remaining
carbon atoms is 1. The normal initial level is 2/9 in all cases, and the
normal refraction factors of the individual mass units are therefore .389
for the displacement 2 atoms and .778 for those of displacement 1. All
of the acids from acetic to enanthic inclusive have normal initial levels
and the differences in the individual refraction factors are due entirely
to a higher proportion of the .778 units as the size of the molecule increases.
The normal initial level of the hydrogen in the corresponding hydrocarbons,
however, is only 1/9 and when the chain becomes long enough to free some
of the hydrocarbon groups at the positive end of the molecule from the
influence of the acid radical at the negative end, these groups revert
to their normal initial levels as hydrocarbons, beginning with the CH3
end group and moving inward. In caprylic acid the three hydrogen atoms
in the end group have made the change, those in the adjoining CH2
group do likewise in pelargonic acid, and as the length of the molecule
increases still further the hydrogen in additional CH2 units
follows suit.
Table CX
Index of Refraction
(n-1)/d
Compound
Dev.
kr
0.697 kr
Observed
120 KR
Observed
ACIDS
O -.389 CO -.389 C -.778 H -.778
Acetic acid
0
.511
.356
.354
.356
61.3
63
64
Propionic
0
.564
.393
.391
.393
67.7
67.5
69.5
Butyric
0
.600
.418
.415
.417
72.0
72
73.5
Valeric
0
.625
.436
.434
75.0
76
76.5
Caproic
0
.644
.449
.448
77.3
79
79.5
Enanthic
0
.659
.459
.458
79.1
81.5
82.5
Caprylic
3
.675
.470
.472
81.0
83
83.5
Pelargonic
5
.687
.479
.478
82.5
84
Capric
7
.697
.486
.485
83.6
Hendecanoic
9
.705
.491
.491
84.6
88
Lauric
11
.713
.496
.500
85.5
98
Myristic
15
.724
.505
.502
86.9
Palmitic
19
.733
.511
.511
88.0
90
Stearic
23
.741
.516
.514
88.9
91
ESTERS
O -.399 CO -.389 C -.778 H -.778
Methyl formate
0
.511
.356
.353
61.3
61.5
Ethyl
0
.564
.393
.390
.392
67.7
68
68.5
Propyl
0
.600
.418
.417
.419
72.0
73
74
Butyl
0
.625
.436
.437
75.0
76
77
Amyl
0
.644
.449
.447
.452
77.3
79.5
Hexyl
3
.664
.462
.463
79.7
81.5
Octyl
5
.687
.479
.479
82.5
Isopropyl
0
.600
.418
.419
72.0
74
Isobutyl
0
.625
.436
.437
.438
75.0
74.5
77.5
Isoamyl
0
.644
.449
.449
77.3
78.5
Methyl
acetate
-3
.556
.387
.385
.389
66.7
65.5
68
Ethyl
-3
.593
.413
.413
.417
71.1
70
71.5
Propyl
0
.625
.436
.433
.434
75.0
73
75.5
Butyl
0
.644
.449
.447
.448
77.3
76
77
Amyl
0
.659
.459
.456
.461
79.1
80
Hexyl
3
.675
.470
.470
81.0
81
Heptyl
3
.685
.477
.478
82.1
84
Isopropyl
0
.625
.436
.433
75.0
74
Isobutyl
0
.644
.449
.447
.448
77.3
77
79
Isoamyl
0
.659
.459
.458
.459
79.1
79
80
Methyl propionate
-3
.593
.413
.412
71.1
71
Ethyl
-3
.619
.431
.430
.432
74.3
72.5
74.5
Propyl
0
.644
.449
.447
77.3
76
Butyl
0
.659
.459
.458
79.1
77.5
Methyl butyrate
-3
.619
.431
.431
74.3
73.5
Ethyl
0
.644
.449
.447
77.3
76
77
Propyl
0
.659
.459
.458
79.1
78
79
Butyl
0
.671
.467
.463
.467
80.5
80.5
Amyl
3
.685
.477
.477
82.1
82
C -.778 H -.889
Propane
5
.934
.581
.582
99.0
Butane
3
.820
.572
98.4
Pentane
3
.818
.570
.570
98.1
100
Hexane
3
.816
.568
.568
.569
97.9
101
Heptane
3
.814
.567
.567
.568
97.7
98
Octane
3
.813
.567
.5655
97.6
100
Nonane
3
.812
.566
.565
97.5
Decane
3
.812
.566
.5645
97.4
100
Undecane
3
.811
.565
.566
97.4
100
Dodecane
0
.807
.563
.563
96.9
101
Tridecane
0
.807
.562
.575
96.9
Tetradecane
0
.807
.562
96.9
Pentadecane
0
.807
.562
.5605
96.9
Hexadecane
0
.807
.562
.561
96.8
Heptadecane
0
.907
.562
.562
96.8
98
Octadecane
0
.807
.562
.562
96.8
96
2-Methyl propane
5
.827
.576
.577
99.2
2-Methyl butane
5
.823
.573
.573
98.7
100
2-Methyl pentane
3
.816
.568
.566
97.9
99
2-Methyl hexane
3
.814
.567
.567
97.7
99
2-Methyl heptane
3
.813
.567
.5655
97.6
100
If the compound
has side branches or more than one main branch (as in the ethers, diamines,
etc.) the normal sequence of deviations, 3, 5, 7... n, may
be modified to some such order as 3, 6, 8 n. The exact point in the series
at which any particular change takes place varies to some extent between
the related groups of compounds because of the geometric characteristics
of the individual molecules.
In some cases
small negative deviations from the normal initial levels are indicated.
The explanation is obvious in such families as the paraffins where the
normal initial level of the hydrogen atoms is only 1/9 unit. The - 3 deviation
in 2, 7 - Dimethyl octane, for example, simply means that three hydrogen
atoms take the 2/9 initial level. The reason for the negative deviations
in some of the esters and a few other compounds in which the normal initial
levels are already at the theoretical 2/9 maximum is not entirely clear,
but the carbon atoms in the negative components of these compounds are
on the borderline between rotation one and rotation two and the negative
deviations are probably connected with the rotational state of the carbon
atoms.
From equation
150 it is apparent that the magnitude of (n-1)/d varies
with the frequency of the radiation, and that the difference between the
values corresponding to any two specified frequencies is likewise variable
because of differences in the rotational characteristics of the various
compounds, as expressed by the factors kr and Fv.
The variability in the refractive index resulting from a change in the
frequency is the dispersion. It is generally measured as the
difference between the refractive index for a wavelength of 6563 X 10-8
cm (the C line) and that for a wavelength of 4861 x 10-11 cm
(the F line). The corresponding frequencies in natural units are .0695
and .0938 respectively, and the frequency difference F - C is .0243. Substituting
this value in equation 149 and multiplying by 104 to conform
to the usual units for expressing dispersion we obtain the general dispersion
equation,
F - C
= 160 Fvkr (153)
For the class
of compounds thus far examined, those in which the value of Fv
is constant at .75, equation 153 reduces to
F - C
= 120 kr (154)
Dispersions have
been calculated for all of the compounds of Table CX and these values
are listed in the table, together with the corresponding experimental
figures where the latter are available. The value of kr
is so nearly the same for all of the paraffins that all of these compounds
have dispersions in the neighborhood of 100. In the other organic families
to which equation 154 is applicable the presence of elements of effective
rotational displacement two lowers the refraction constant and reduces
the dispersion accordingly. This effect is, of course, most noticeable
in the lower members of each series and the dispersion rises toward the
hydrocarbon values as the molecule lengthens.
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