33 Radiation and Refraction

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

Be9 + y → Be8 + 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 = 6.648×10-27v 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 × 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 × 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 6.624×10-27. 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

a

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 4.1?
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 1 c 2.25 1.81 1.9 1.96 2.48 2.52
Ba 1 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 (0.3334×10-10)2 in cgs units. The radiation rate can then be expressed as

Erad = (8/9) × ((0.3334×10-10)2 × 5.001 × 10-4 ergs) / ((2.914×10-8 cm)2 × 0.1521×10-15 sec × (510.16 deg)4)

= 5.655×10-5 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/∞ 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/∞ 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/a2and 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 effects of all of the different 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 ½. 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) = 0.6583 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) = 0.6583 v Fvkr

(149)

Adding equations 148 and 149,

(n-1)/d = 0.6583 kr (1 + v Fv)

(150)

The wavelength most commonly used for refraction measurements is that of the sodium D line, 5893×10-8 cm. This is equivalent to 0.0774 natural units of frequency. Substituting this value for v in equation 150 we obtain

(n-1) /d = 0.6583 kv (1 + 0.0774 Fv)

(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 0.75. For convenience in dealing with these compounds we may simplify equation 151 to

(n-1)/d = 0.6965 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 initial 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 0.778 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   .511 .356 .354 .356 61.3 63 64
Propionic   .564 .393 .391 .393 67.7 67.5 69.5
Butyric   .600 .418 .415 .417 72.0 72 73.5
Valeric   .625 .436 .434   75.0 76 76.5
Caproic   .644 .449 .448   77.3 79 79.5
Enanthic   .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   .511 .356 .353   61.3 61.5
Ethyl   .564 .393 .390 .392 67.7 68 68.5
Propyl   .600 .418 .417 .419 72.0 73 74
Butyl   .625 .436 .437   75.0 76 77
Amyl   .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   .600 .418 .419   72.0 74
Isobutyl   .625 .436 .437 .438 75.0 74.5 77.5
Isoamyl   .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   .625 .436 .433 .434 75.0 73 75.5
Butyl   .644 .449 .447 .448 77.3 76 77
Amyl   .659 .459 .456 .461 79.1 80
Hexyl 3 .675 .470 .470   81.0 81
Heptyl 3 .685 .477 .478   82.1 84
Isopropyl   .625 .436 .433   75.0 74
Isobutyl   .644 .449 .447 .448 77.3 77 79
Isoamyl   .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   .644 .449 .447   77.3 76
Butyl   .659 .459 .458   79.1 77.5
Methyl butyrate -3 .619 .431 .431   74.3 73.5
Ethyl   .644 .449 .447   77.3 76 77
Propyl   .659 .459 .458   79.1 78 79
Butyl   .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   .807 .563 .563   96.9 101
Tridecane   .807 .562 .575   96.9
Tetradecane   .807 .562     96.9
Pentadecane   .807 .562 .5605   96.9
Hexadecane   .807 .562 .561   96.8
Heptadecane   .907 .562 .562   96.8 98
Octadecane   .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, 7n, 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 6563×10-8 cm (the C line) and that for a wavelength of 4861×10-11 cm (the F line). The corresponding frequencies in natural units are 0.0695 and 0.0938 respectively, and the frequency difference F - C is 0.0243. 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.