Chapter VI
Atomic Bonds
Inter-atomic distances for most of the elements in the first half of each rotational group can be calculated directly from equation 14, using the rotational values corresponding to the displacements previously determined for each individual element. Many elements, however, have different rotations which take the place of these normal values under certain conditions. The occurrence of these alternate rotations is largely dependent upon the position of the element within the rotational group and in preparation for the ensuing discussion of this factor it will be advisable for convenient reference to set up a classification according to position.
Within each of the rotational groups the most probable electric displacement for the elements in the first half of the group is in time, while for those in the latter half of the group it is in space. We will distinguish between these two divisions of the group by applying the term electropositive to those elements with probable electric time displacement and the term electronegative to those with probable electric space displacement. It should be understood, however, that this distinction is being drawn on the basis of the most probable situation in the electric dimension considered independently. Because of the conditions prevailing elsewhere in the environment of the atom an electronegative element often acts in an electropositive capacity, but this does not affect the classification as herein described.
There are also important differences between the behavior of the first four members of each series of positive or negative elements and that of the elements with higher rotational displacements. We will therefore divide each series into a lower division and an upper division so that those elements with similar general characteristics can be treated together. The classification will be based on the magnitude of the displacement, the lower division in each case including the elements with displacements from 1 to 4 and the upper division comprising those with displacements of 4 or over. The elements with displacement 4 belong to both divisions as they are capable of acting either as the highest members of the lower divisions or as the lowest members of the upper divisions. It should be recognized that in the electronegative series the members of the lower divisions have the higher net time displacement (higher atomic number).
For convenience these divisions within each rotational group will be numbered in the order of increasing atomic number as follows:
| Division I | Lower | electropositive |
| Division II | Upper | |
| Division III | Upper | electronegative |
| Division IV | Lower |
Another item which needs to be explained before resuming the calculation of inter-atomic distances is the relation between displacement and rotation. It has been shown that the elements constitute a continuous series in which the successive members differ by the equivalent of one unit of electric time displacement. The gravitational force in the time-space region is a function of the total three-dimensional rotation; that is, of the net total rotational time displacement. In the time region, however, the displacements in the electric dimension are opposite in time direction from those in the magnetic dimensions and the inter-atomic force is a function of the rotations in the different dimensions separately, as indicated in the force equations which have been developed.
In the simplest rotational combinations the basic rotating unit is a unit vibrational displacement, and unit rotational displacement applied to this unit vibration results in one unit of rotation. As long as the rotation is based entirely on this unit linear displacement, which for brevity we will call vibration one, the specific rotation, the quantity which enters into the force equations, is equal to the rotational displacement plus one unit (the physical zero value). When the rotation of this single vibrational unit reaches the time region maximum the rotational motion must be extended to vibration two (two units of linear displacement) if further additions of rotational displacement are to be made.
As previously indicated, the maximum time region rotation of a single displacement unit is two linear units or eight units distributed three-dimensionally. The change to vibration two therefore may take place after the first unit of displacement, if the rotational units are disposed linearly, and must take place before the addition of the eighth displacement unit. After the change to vibration two there are two units of vibrational displacement to be rotated and hence each added unit of rotational displacement corresponds to only one-half unit of specific rotation. As in the other time region phenomena which have been discussed, the higher displacement is in addition to and not in lieu of the lower displacement. The succession of rotation values is therefore either 1, 2, 2˝, 3, 3˝, etc., or 1, 2,… 7, 8, 8˝, 9, etc. The lower value is very commonly found where it first becomes possible; that is, displacement 2 normally corresponds to rotation 2˝ rather than 3. The next element may take the intermediate value 3˝ but beyond this point the higher value normally prevails.
The independent rotation of the different vibrational displacements may be visualized by means of a mechanical analogy. Let us consider an object with a translatory motion in some specific direction. If we rotate this object around the line of motion as an axis it is clear that this rotation will not interfere with the translatory motion. Furthermore, if the object is jointed so that it is capable of rotating by parts it is entirely possible for one or more parts to be rotating and the remainder not rotating, while the ensemble moves forward in translation unaffected by the nature of the rotation.
The references which have been made to the rotation of one or two units of vibrational displacement do not imply that these are necessarily the full vibrational displacements of the photons. Each unit of frequency is independent from the standpoint of motion in the opposite space-time component and the first unit of vibrational space displacement can be rotated in time irrespective of the number of additional units of displacement present, just as a single one of the individual parts described in the preceding paragraph can be rotated around the line of motion without regard to the total number of parts which make up the object as a whole. Similarly the second unit of vibrational displacement can be rotated if a second unit exists, no matter how many more of these displacement units may be included in the vibration as a whole. The change to a higher vibration level may affect either the electric or the magnetic rotation or both, since these rotations are not only independent of the total magnitude of the basic vibrational displacement but are also independent of each other, except to the extent that probability considerations are effective.
The general pattern of the magnetic rotational values is the same as that of the electric values but the upper limit for rotation on a vibration one basis is 4 rather than 8, for reasons previously discussed. Rotation 4˝ therefore follows rotation 4 in the regular progression. It is possible to reach rotation 5 in one dimension, however, without bringing the magnetic rotation as a whole up to the 5 level, and a 5-4 rotation therefore occurs in some elements either instead of or in combination with the 4˝-4 rotation.
Combinations of this kind frequently occur where there are alternate rotations of nearly equal probability. In the inter-atomic distance tables the magnetic combinations are indicated by notations such as 5-4, 4˝-4, and the electric combinations by two-figure designations (6-10, etc.). In each case the effective value of t is taken as the geometric mean of the values applicable to the two components.
We may now resume the calculation of inter-atomic distances. Table II lists the values computed for the elements in Division I of all groups except Group 2A, together with the corresponding experimental values. Data for the elements of Group 2A will be presented later as the members of the lower rotational groups have some special characteristics which will need further consideration.
Table II
| Group | Atomic Number |
Element | Magnetic Rotation |
Electric Rotation |
Inter-atomic Distance | |
|---|---|---|---|---|---|---|
| Calculated | Observed | |||||
| 2B | 11 | Sodium | 3-2˝, 3-3 | 2 | 3.72 | 3.72 |
| 12 | Magnesium | 3-2˝ | 2˝ | 3.16 | 3.20 | |
| 13 | Aluminum | 3-2˝ | 3 | 2.85 | 2.86 | |
| 3A | 19 | Potassium | 4-3 | 2 | 4.49 | 4.50 |
| 20 | Calcium | 4-3 | 2˝ | 3.95 | 3.98 | |
| 21 | Scandium | 4-3 | 4 | 3.18 | 3.21 | |
| 22 | Titanium | 4-3 | 5 | 2.95 | 2.92 | |
| 3B | 37 | Rubidium | 4-4 | 2 | 4.85 | 4.86 |
| 38 | Strontium | 4-4 | 2˝ | 4.27 | 4.28 | |
| 39 | Yttrium | 4-4 | 3˝ | 3.62 | 3.66 | |
| 40 | Zirconium | 4-4 | 5 | 3.18 | 3.23 | |
| 4A | 5˝ | Cesium | 5-4, 4˝-4 | 2 | 5.23 | 5.24 |
| 56 | Barium | 5-4 | 3 | 4.26 | 4.34 | |
| 57 | Lanthanum | 5-4, 4˝-4 | 4 | 3.70 | 3.74 | |
| 58 | Cerium | 5-4 | 5 | 3.52 | 3.55 | |
| 4B | 90 | Thorium | 4˝-5 | 5 | 3.52 | 3.56 |
The values calculated for the inter-atomic distance in Table II and the other similar tabulations included herein are those which would prevail in the absence of compression and thermal expansion. Some of the experimental data have been extrapolated to this zero base by the investigators but others are the actual observed values at atmospheric pressure and at different temperatures, depending on the properties of the substances involved, and the latter are not exactly comparable to the calculated figures. In general, however, the expansion and compression up to the temperature and pressure of observation are small and a comparison of the values in the last two columns gives a reasonably good idea of the extent of agreement between the theoretical figures and the experimental results. The effects of compression and thermal expansion will be examined in detail later.
Most of the elements of Division II also assume equilibrium positions on the basis of the same type of force relations as those of Division I but an alternate type of equilibrium can exist in this division and some of the Division II elements form structures of this second variety, either in addition or in preference to the regular electropositive structures. The factor which makes this alternate type of equilibrium possible is the ability of the atom to reorient itself with reference to the space-time zero point. We have already seen that eight units of time or space displacement, when distributed three-dimensionally, constitute a full space-time unit. Applying this to rotation, addition of eight units of displacement completes a full cycle and returns to the starting point. These eight rotational displacement units or any multiple of eight therefore constitute the equivalent of no displacement at all. Any intermediate displacement can be described in either of two ways: as x units in the positive direction from zero, or as 8-x units in the negative direction from the equivalent of zero.
It is possible on this basis to change a positive (time) displacement x to a negative (space) displacement 8-x merely by reorientation with reference to the space-time zero point. This does not involve any alteration in the rotation of the atom. It is simply a change in the relationship between the atomic positions and the space-time units in which they are situated. Consequently an atom may establish an equilibrium on the 8-x basis with one adjoining atom and yet maintain the normal electropositive equilibrium with other atoms in different directions.
For reasons which will be developed later, an equilibrium between two negative 8-x displacements is not possible, and the 8-x displacement, where it occurs in Division II, is in equilibrium with a positive displacement x. Let us consider the nature of this equilibrium. In the inter-atomic relationships previously examined all displacement has been in time. As long as this situation prevails there are no limitations on the combinations. The principles which have been established are valid for unequal as well as equal rotations and any atom with rotational time displacement x can establish equilibrium with any other atom having time displacement y, the resulting effective displacement being the geometric mean (xy)˝. However, if one displacement is in time and the other in space the rotations are no longer concurrent. Being in opposite space-time directions they are additive and the resulting effective displacement is the sum, x + y.
This expression x + y represents x units of space (or time) in association with y units of time (or space). But a ratio of units of space to units of time is a velocity and any velocity, other than a velocity of unity (zero displacement), is obviously incompatible with the establishment of equilibrium. It therefore becomes apparent that there is a rigid limitation on inter-atomic combinations of this character: the displacements x and y must be the same or equal. A displacement x is equal to a displacement -x and a time displacement x can therefore establish equilibrium with a space displacement x. A time displacement x is the same as a space displacement 8n - x if the atom is located in the appropriate position in the space-time unit, and hence a combination of these displacements also meets the equilibrium requirements. Except for a different type of zero point shift which will be discussed in a subsequent section all of the equilibrium relationships between space and time displacements follow one or another of these patterns.
The cohesive force which is exerted between the atoms of a molecule is commonly termed the chemical bond, and the variation in the magnitude of the inter-atomic force under different conditions is ascribed to the existence of several kinds of bonds. From the explanation in the preceding paragraphs it is apparent that the different “bonds” are merely the products of different rotational orientations, but the use of this term has an element of convenience, particularly in view of its general acceptance, and it will therefore be adopted in this work with the understanding that as herein used it refers to the net resultant of a particular orientation of the interacting rotational forces.
The regular bond of the electropositive elements on which the structures listed in Table II are based is a direct combination of two positive displacements. In the Division I elements the two displacements are equal and if we call their magnitude x the resultant relative displacement which, according to the principles previously stated, is the geometric mean of the two displacements, is also x. This we will call the positive bond.
When we turn to Division II, the upper electropositive division, we find that the positive bond is still very common, but since these elements are closer to the midpoint of the group negative rotations are more probable than in Division I and many of these elements form the alternate type of combination in which a positive electric displacement x is in equilibrium with a negative (space) displacement 8-x. As we have found, the resultant displacement is the sum of the two or 8. This is the neutral value, a full rotational cycle which returns to the equivalent of zero, and we will therefore call this the neutral bond. The rotation corresponding to displacement 8 is 10, as the complete space-time unit includes not only an initial time unit at one end but also an initial space unit at the other.
Table III
| Group | Atomic Number |
Element | Magnetic Rotation |
Electric Rotation |
Band | Inter-atomic Distance | |
|---|---|---|---|---|---|---|---|
| Calculated | Observed | ||||||
| 3A | 23 | Vanadium | 4-3 | 6-10 | Comb. | 2.62 | 2.63 |
| 24 | Chromium | 4-3 | 7 | Pos. | 2.68 | 2.72 | |
| 24 | Chromium | 4-3 | 10 | Neut. | 2.46 | 2.49 | |
| 25 | Manganese | 4-3 | 8 | Pos. | 2.59 | 2.56 | |
| 26 | Iron | 4-3 | 81 | Pos. | 2.56 | 2.57 | |
| 26 | Iron | 4-3 | 10 | Neut. | 2.46 | 2.48 | |
| 27 | Cobalt | 4-3 | 9 | Pos. | 2.52 | 2.515 | |
| 27 | Cobalt | 4-3 | 10 | Neut. | 2.46 | 2.50 | |
| 3B | 41 | Niobium | 4-4 | 6-10 | Comb. | 2.93 | 2.85 |
| 42 | Molybdenum | 4-4, 4-5 | 10 | Neut. | 2.73 | 2.72 | |
| 43 | Technetium | 4-4, 4-5 | 10 | Neut. | 2.73 | 2.735 | |
| 44 | Ruthenium | 4-4, 4-5 | 10 | Neut. | 2.73 | 2.765 | |
| 45 | Rhodium | 4-4 | 10 | Neut. | 2.66 | 2.68 | |
| 4A | 59 | Praseodymium | 5-4˝, 5-5 | 5 | Neut. | 3.61 | 3.64 |
| 60 | Neodymium | 5-4, 5-5 | 5 | Neut. | 3.61 | 3.62 | |
| 62 | Samarium | 5-4, 5-5 | 5 | Neut. | 3.61 | 3.59 | |
| 63 | Europium | 5-4, 5-5 | 1-5 | Comb. | 4.06 | 4.08 | |
| 64 | Gadolinium | 5-4, 5-5 | 5 | Neut. | 3.61 | 3.59 | |
| 65 | Terbium | 5-4 | 5 | Neut. | 3.52 | 3.54 | |
| 66 | Dysprosium | 5-4 | 5 | Neut. | 3.52 | 3.54 | |
| 67 | Holmium | 5-4 | 5 | Neut. | 3.52 | 3.52 | |
| 68 | Erbium | 5-4 | 5 | Neut. | 3.52 | 3.50 | |
| 69 | Thulium | 5-4 | 5 | Neut. | 3.52 | 3.48 | |
| 70 | Ytterbium | 5-4, 4˝-4 | 1-5 | Comb. | 3.87 | 3.87 | |
| 4B | 92 | Uranium | 4˝-5 | 10 | Neut. | 2.95 | 2.97 |
| 92 | Uranium | 4˝-4˝ | 10 | Neut. | 2.88 | 2.85 | |
If the magnetic rotation extends to vibration two the specific electric rotation corresponding to displacement 8 may be reduced by one-half, in which case it becomes 5 instead of 10. This is the prevailing rotation in the Division II elements of Group 4A.
Inter-atomic distances based on the neutral bond can be calculated from equation 14 by substituting the relative electric rotation 10 or 5 for the normal rotation values used in evaluating the positive bond distances. The change in bond type does not affect the magnetic rotation since the magnetic displacement is always in time. Values obtained for the inter-atomic distances of both positive and neutral bond structures of Division II elements are listed in Table III.
In the elements which have been discussed thus far the most probable bond in any one individual dimension is applicable to all dimensions and the force system of the atom is isotropic. It follows that any aggregate of atoms of these elements has a structure in which the constituents are arranged in one of the geometrical patterns possible for equal forces: an isometric crystal. All of the electropositive elements crystallize in isometric forms and except for a few which apparently have quite complex structures each of the crystals of these elements belongs to one or another of three types, the face-centered cube, the body-centered cube, or the hexagonal close-packed structure.
We now turn to the other major subdivision of matter, the electronegative elements, those whose normal electric displacement is in space. Here the force system is not necessarily isotropic since the most probable bond in one or two dimensions may be the negative bond, a direct combination of two electric space displacements, but it is not possible to have negative bonds in three dimensions and wherever such bonds exist the atomic forces are anisotropic. The controlling factor in this situation is the necessity for a net rotational displacement in time in order to establish equilibrium in space. Negative orientation in three dimensions is obviously incompatible with this requirement but if the negative displacement is restricted to one dimension we have fixed atomic positions in two dimensions with a fixed average position in the third because of the net electric time displacement of the atom as a whole. This results in a crystalline structure which is difficult to distinguish from one with fixed positions in all dimensions. Such crystals are not usually isometric, however, as the inter-atomic distance in the odd dimension is generally different from that in the other two. Where the distances in all dimensions do happen to coincide we will find on further investigation that the space symmetry is not an indication of force symmetry.
If the negative displacement is very small, as in the lower Division IV elements, it is possible to have negative orientation in two dimensions as long as the positive displacement in the third dimension exceeds the sum of these two negative components so that the net resultant is still positive. Here the relative positions of the atoms are fixed in one dimension only, but the average positions in the other two dimensions are constant by reason of the net three-dimensional time displacement. The aggregate consequently retains most of the external characteristics of a crystal but when the internal structure is examined the atoms appear to be distributed at random rather than in the orderly arrangement of the crystal. In reality there is just as much order as in the crystalline structure but part of the order is in time rather than in space and there are no fixed equilibrium positions in space. This phase of matter we identify as the glassy or vitreous form to distinguish it from the crystalline form.
The term “state” is frequently used in this connection instead of “form” but the physical state of matter has an altogether different meaning based on another kind of differentiation and it seems advisable to confine the use of this term to the one application. Both glasses and crystals are in the solid state, the essential characteristics of which will be discussed in detail later.
In beginning a consideration of the structures of the individual electronegative elements it will be desirable to start with Division III. The possible equilibrium states in this division are analogous to those of Division II. The negative bond is comparable to the positive bond of the electropositive divisions. As in Division II there is an alternate bond in which the negative displacement x combines with the inverse orientation 8-x. While this is just the reverse of the Division II neutral bond in which the normal displacement is positive and the inverse is negative, the net result is exactly the same and this Division III combination will also be considered a neutral bond.
Where two or more alternate structures are possible the actual form which the crystal will take is a matter of probability. Low displacements are, of course, more probable than high displacements. Electric displacement in time is likewise more probable than electric displacement in space since the former conforms to the space-time direction of the rotational motion as a whole. In Division I both of these factors operate in the same direction. The positive bond, which is based entirely on time displacement and hence is inherently more probable than the neutral bond, also has the lower displacement. All structures in this division therefore take the positive bond. In Division II the margin of probability is narrow. Here the positive displacement is higher than the inverse 8-x displacement and this operates against the greater inherent probability of the time displacement. As a result both types of structures are encountered in this division, together with a combination of the two.
In Division III the greater probability of the time displacement is sufficient to keep the borderline elements of Groups 3A and 3B on the Division II basis, the electropositive preference extending as far as copper and silver. The higher Division III elements of Group 4A are beyond the range of positive bond structures and all of these elements crystallize on the basis of the neutral bond. Decreasing negative displacement in each group increases the probability of the negative bond and these bonds make their appearance in the vicinity of displacement 7. In Groups 3A and 3B the remaining elements of this division have the characteristic asymmetric electronegative structures utilizing both negative and neutral bonds. The negative bond is rare in Group 4A and in this group the neutral bond structures continue to predominate throughout Division III. Inter-atomic distances for the Division III elements are listed in Table IV.
As we pass from Division III to Division IV the magnitude of the inverse displacement 8-x increases and neutral bond structures become correspondingly less probable, although still very much in evidence. Negative bonds in crystal structures also become increasingly rare, not because of any decrease in probability but because they are likely to exist in two dimensions if they occur at all in these Division IV structures and this means a glassy or vitreous aggregate rather than a crystal. There is, however, a different type of combination which makes its appearance here where the inherently more probable bonds are excluded for one reason or another. Thus far we have examined the positive and negative bonds, in which one normal displacement is in equilibrium with another normal displacement, and the neutral bond, in which the normal displacement x combines with the inverse displacement 8-x. Now we complete the picture with a bond in which two of the 8-x inverse displacements are combined.
Table IV
| Group | Atomic Number |
Element | Magnetic Rotation |
Electric Rotation |
Bond | Inter-atomic Distance | |
|---|---|---|---|---|---|---|---|
| Calculated | Observed | ||||||
| 3A | 28 | Nickel | 4-3 | 9˝ | Pos. | 2.49 | 2.49 |
| 28 | Nickel | 4-3 | 10 | Neut. | 2.46 | 2.485 | |
| 29 | Copper | 4-3 | 8-10 | Comb. | 2.53 | 2.55 | |
| 30 | Zinc | 4-4 | 7 | Neg. | 2.90 | 2.915 | |
| 30 | Zinc | 4-4 | 10 | Neut. | 2.66 | 2.66 | |
| 31 | Gallium | 4-3 | 6 | Neg. | 2.79 | 2.80 | |
| 31 | Gallium | 4-3 | 10 | Neut. | 2.46 | 2.44 | |
| 3B | 46 | Palladium | 4-4, 4-5 | 10 | Neut. | 2.73 | 2.74 |
| 47 | Silver | 5-4, 4-5 | 10 | Neut. | 2.87 | 2.88 | |
| 48 | Cadmium | 5-4 | 7 | Neg. | 3.20 | 3.26 | |
| 48 | Cadmium | 5-4 | 10 | Neut. | 2.94 | 2.97 | |
| 49 | Indium | 5-4 | 6 | Neg. | 3.33 | 3.33 | |
| 49 | Indium | 5-4 | 5-10 | Neut. | 3.22 | 3.24 | |
| 50 | Tin | 5-4 | 5-10 | Neut. | 3.22 | 3.15 | |
| 50 | Tin | 5-4 | 10 | Neut. | 2.94 | 3.02 | |
| 4A | 71 | Lutecium | 5-4, 4˝-4 | 5 | Neut. | 3.43 | 3.47 |
| 72 | Hafnium | 5-4, 4˝-4 | 5-10 | Neut. | 3.14 | 3.20 | |
| 73 | Tantalum | 4˝-4 | 10 | Neut. | 2.80 | 2.85 | |
| 74 | Tungsten | 4-4˝ | 10 | Neut. | 2.73 | 2.73 | |
| 75 | Rhenium | 4-4˝ | 10 | Neut. | 2.73 | 2.75 | |
| 76 | Osmium | 4-4˝ | 10 | Neut. | 2.73 | 2.72 | |
| 77 | Iridium | 4-4˝ | 10 | Neut. | 2.73 | 2.71 | |
| 78 | Platinum | 4-4˝ | 10 | Neut. | 2.73 | 2.77 | |
| 79 | Gold | 4˝-4˝ | 10 | Neut. | 2.88 | 2.88 | |
| 80 | Mercury | 4˝-4˝ | 7-10 | Comb. | 3.00 | 3.00 | |
| 80 | Mercury | 4˝-4˝ | 5 | Neut. | 3.44 | 3.46 | |
| 81 | Thallium | 4˝-4˝ | 5 | Neut. | 3.44 | 3.42 | |
| 82 | Lead | 4˝-4˝ | 5 | Neut. | 3.44 | 3.48 | |
This bond was not encountered previously as it has a very low probability because of its high effective displacement and where more probable structures can be formed the existence of a crystal based on a bond of low probability is precluded. In Division IV the positive bond is impossible and the low probability of the 8-x combination competes only with the neutral bond, the probability of which is likewise low for the displacement values in this division. It was mentioned in the discussion of the Division II structures that a bond based on a direct combination of two 8-x displacements in that division is not possible. The 8-x displacement in Divisions I and II is negative and like the negative bond an 8-x combination would be confined to a subordinate role in one or two dimensions of an asymmetric structure. Such a crystal cannot compete with the high probability of the symmetrical electropositive structures and therefore does not exist. In the electronegative divisions, however, the 8-x displacement is positive and there are no limitations upon it other than those arising from the high effective displacement.
The effective displacement of this secondary positive bond, as we will call it, is even greater than might be expected from the magnitude of the quantity 8-x as the change of zero points for two oppositely directed motions is also oppositely directed and the new zero points are 16 units apart. The resultant displacement is 16 - 2x and the corresponding rotation is 18 - 2x. The numerical values of the latter expression range from 10 to 16 and because of the low probability of such high rotations the secondary positive bond is limited to one or one and one-half dimensions in spite of its positive direction. In the upper elements of Division IV the other dimensions take the neutral bond but the probability of this bond decreases as the value of 8-x approaches its upper limit and in some of the lower elements of the division there is no electric rotational force at all in one or more dimensions; that is, the specific rotation is unity. For example, the crystal of iodine has one dimension based on the secondary positive bond with rotation 16, the inter-atomic distance being 2.68 Å. A second dimension has unit rotation and an inter-atomic distance of 4.46 Å. The third dimension combines these two forces with a resultant distance of 3.46 Å, midway between the other two values. A similar combination bond with unit rotation as one of the components was shown for two of the rare earth elements in Table III.
A special type of structure occurs only in those electronegative elements which have a rotational displacement of four units. This rotation is on the borderline between Divisions III and IV where the neutral bond and the secondary positive bond are about equally probable. Under similar conditions other elements crystallize in hexagonal or tetragonal structures, utilizing the different bonds in different directions. For these displacement 4 elements, however, the two bonds have the same relative rotation: 10. The inter-atomic distance in these crystals is therefore the same in all dimensions and the crystal is isometric, even though the rotational forces are quite different in character. The molecular arrangement in this crystal pattern, the diamond structure, indicates the true nature of the force equilibrium. Outwardly this crystal cannot be distinguished from the isotropic cubic crystals but the analogous body-centered cubic structure has an atom at each corner of the cube as well as one in the center, whereas the diamond structure leaves alternate corners open to accommodate the abnormal projection of forces in the secondary positive dimensions. Inter-atomic distances for the Division IV elements are listed in Table V.
Up to this point no consideration has been given to the elements of atomic number below 10 as the rotational forces of these elements are subject to certain special influences which make it advisable to discuss them separately. One of these causes of deviation from the normal behavior is the small size of the rotational group. In the larger groups the four divisions are distinct and except for some overlapping each has its own characteristic force combinations. In an S-element group, however, the second series of four elements which would normally constitute Division II is actually in the Division IV position. As a result these four elements have to a certain extent the properties of both divisions.
Table V
| Group | Atomic Number |
Element | Magnetic Rotation |
Electric Rotation |
Bond | Inter-atomic Distance | |
|---|---|---|---|---|---|---|---|
| Calculated | Observed | ||||||
| 2B | 14 | Silicon | 3-3 | 5-10 | Dia. | 2.31 | 2.345 |
| 15 | Phosphorus | 3-3, 3-4 | 10 | Neut. | 2.19 | 2.18 | |
| 16 | Sulfur | 3-3 | 14 | S.P. | 1.97 | 1.92 | |
| 16 | Sulfur | 3-3, 3-4 | 1 | Zero | 3.33 | 3.30 | |
| 17 | Chlorine | 3-3 | 16 | S.P. | 1.92 | 1.98 | |
| 17 | Chlorine | 3-3 | 1-16 | Comb. | 2.48 | 2.52 | |
| 3A | 32 | Germanium | 4-3 | 10 | Dia. | 2.46 | 2.435 |
| 33 | Arsenic | 4-3 | 12 | S.P. | 2.37 | 2.44 | |
| 33 | Arsenic | 4-3 | 10 | Neut. | 2.46 | 2.50 | |
| 34 | Selenium | 4-3 | 14 | S.P. | 2.30 | 2.32 | |
| 34 | Selenium | 3-4 | 1 | Zero | 3.46 | 3.49 | |
| 35 | Bromine | 4-3 | 16 | S.P. | 2.25 | 2.28 | |
| 35 | Bromine | 3-4 | 1 | Zero | 3.46 | 3.30 | |
| 3B | 50 | Tin | 4-4, 5-4 | 10 | Dia. | 2.80 | 2.80 |
| 51 | Antimony | 5-4 | 12 | S.P. | 2.83 | 2.87 | |
| 52 | Tellurium | 5-4 | 14 | S.P. | 2.745 | 2.76 | |
| 53 | Iodine | 5-4 | 16 | S.P. | 2.68 | 2.70 | |
| 53 | Iodine | 5-4 | 1 | Zero | 4.46 | 4.35 | |
| 53 | Iodine | 5-4 | 1-16 | Comb. | 3.46 | 3.54 | |
| 4A | 83 | Bismuth | 4˝-4˝ | 5 | Neut. | 3.44 | 3.42 |
| 83 | Bismuth | 4˝-4˝ | 5-10 | Neut. | 3.15 | 3.11 | |
| 84 | Polonium | 4˝-4˝ | 5 | Neut. | 3.44 | 3.34 | |
A second influence which affects the crystal structures of the lower group elements is the inactivation of the rotational forces in certain dimensions. It was previously noted that a magnetic rotation of two units produces no effects in the positive direction. The reason for this is revealed by equation 3 which tells us that the rotational force in the time region is ln t. The value of this expression for t = 2 is 0.693, which is less than the space-time force 1.00. The net effective force of rotation 2 is therefore below the minimum value for action in the positive direction. In order to produce an active force the rotation must be high enough to make ln t greater than unity. This is accomplished at rotation 3. In some crystals the effective forces of the higher rotational combinations are also reduced to two dimensions, but this is a geometrical effect resulting from the nature of the force equilibrium and it appears only in the more complex structures, whereas the inactivity of the rotation 2 force is an inherent property.
The normal magnetic rotation of the 1B group, which includes only the two elements hydrogen and helium, and the 2A group of eight elements beginning with lithium is 3-2. Where the rotation value 2 applies to the subordinate rotation one dimension is inactive; where it applies to the principal rotation two dimensions are inactive. This reduces the force exerted by each atom to 2/3 of the normal amount for one inactive dimension and to 1/3 for two inactive dimensions. Inter-atomic distance is proportional to the square root of the product of the two forces involved, which means that the reduction in distance is also 1/3 per inactive dimension.
Since the electric rotation is not a simple motion but a reverse rotation of the magnetic rotational system, the limitations to which the basic rotation is subject are not applicable. The electric rotational displacement merely modifies the magnetic rotation and the low value of the rotation 2 integral makes itself apparent by an inter-atomic distance which is greater than that which would prevail if there were no electric displacement at all (unit rotation). Table VI gives the force constants and the inter-atomic distances for the elements of the lower groups.
Table VI
| Group | Atomic Number |
Element | Active Dim. |
Magnetic Rotation |
Electric Rotation |
Bond | Inter-atomic Distance | |
|---|---|---|---|---|---|---|---|---|
| Calculated | Observed | |||||||
| 1B | 1 | Hydrogen | 1 | 3-3 | 10 | Neut. | 0.70 | 0.74 |
| 2 | Helium | 1 | 3-3 | 1 | Zero | 1.07 | 1.09 | |
| 2A | 3 | Lithium | 3 | 2-3, 3-2 | 2 | Pos. | 3.06 | 3.03 |
| 4 | Beryllium | 2 | 3-3 | 2˝ | Pos. | 2.25 | 2.27 | |
| 6 | Carbon | 2 | 3-3 | 5-10 | Dia. | 1.54 | 1.54 | |
| 6 | Carbon | 2 | 3-3 | 10 | Neut. | 1.41 | 1.42 | |
| 6 | Carbon | 3 | 3-3 | 1 | Zero | 3.20 | 3.40 | |
| 7 | Nitrogen | ii | 3-3 | 10 | Neut. | 1.06 | 1.09 | |
| 9 | Fluorine | 2 | 3-3 | 10 | Neut. | 1.41 | 1.46 | |
In the preceding pages the mathematical relations governing the inter-atomic force systems were developed on a comprehensive basis but the subsequent examination of specific cases dealt only with the forces exerted between like atoms. We are now ready to begin a study of the forces between unlike atoms. The general principles developed in the discussion of the structures of the elements will, of course, apply to this situation as well, but the existence of differences between the components of the system will introduce some new factors into the calculations.
Looking first at combinations of electropositive elements, let us consider an element with electric rotation t1, in equilibrium with an element having electric rotation t2. Here the two forces are identical in character and concurrent, the kind of a force combination that we have called the positive bond. The resultant, according to the principles previously set forth, is ( t1t2 )˝, the geometric mean of the two constituent rotations. If the two elements have different magnetic rotations the resultant in each magnetic dimension is also the geometric mean of the individual rotations, since the magnetic rotations are always in time and they combine in the same manner as the electric time displacements; that is, the magnetic equivalent of the positive bond.
Since the properties of matter are determined by the nature and magnitude of the inter-atomic forces, the properties of a combination of this kind are in general intermediate between the properties of the components. This type of association between elements is called a mixture if the combination is irregular and incomplete or an alloy if it is uniform and fully effective.
There is no inherent limitation on the composition of mixtures or alloys. Any of the electropositive elements can enter into such combinations and they can mix in any proportions, except to the extent that geometrical considerations intervene. Many of the electronegative elements, particularly those of Division III, follow the same pattern to some degree by reorientation on an 8-x basis. When we turn to combinations involving other than positive bonds the situation is considerably different. These other bonds are based on the establishment of equilibrium through the balancing of opposing forces at neutral space-time values and this requires that the components have certain definite relations with respect to each other. Combinations of this kind therefore take place in definite proportions, each atom of one component being associated with a specific number of atoms of the other component or components. Such a combination is called a chemical compound.
In addition to the constant proportions of their components, compounds also differ from mixtures or alloys in that their properties are not necessarily intermediate between those of the components but may be of an altogether different character, since the resultant of a force equilibrium of this kind may differ widely from any of the force arrangements of the individual elements.
The simplest type of bond in chemical compounds is an equilibrium between electropositive and electronegative rotations of the same magnitude. Here the basis of the equilibrium is a space-time ratio of unity, the time displacement of one component being equal to the space displacement of the other. As indicated in the discussion of the structures of the elements, the resultant of a combination of space and time displacements is the sum of the two, in this case 2x. The corresponding rotation is 2 (x + 1), as it includes two initial unites, one of space and one of time.
Because of the fundamental character of this bond and the important part which it plays in the world of chemical compounds we will call it the normal bond. Inter-atomic distances for the normal bond structures can be calculated in the same manner as before, applying the mean magnetic rotation and the effective electric rotation 2(x + 1) to equation 14. If the active dimensions are not the same in both components the full rotational force of the more active component is effective in its excess dimensions. For example, the value of ln t for 3-3 magnetic rotation is 1.099 in three dimensions or 0.7324 in two dimensions. If this two-dimensional rotation is combined with a three-dimensional magnetic rotation x, the resultant is (0.7324 x)˝, the geometric mean of the individual values, in two dimensions, and x in the third. The average value for all three dimensions is (0.7324 x)1/3.
When this bond unites one electropositive atom with each electronegative atom the resulting structure is usually a simple cube with the atoms of each element occupying alternate corners of the cube. This is called the Sodium Chloride structure, after the most familiar member of the family of compounds crystallizing in this form. Table VII gives the inter-atomic distances for the common NaCl type crystals.
From this tabulation it can be seen that the special rotational characteristics which certain of the elements possess in the elemental aggregates carry over into their compounds. The elements of the lower groups have inactive force dimensions in these crystals just as in the structures previously examined. The second element in each group also shows the same preference for vibration two rotation that we encountered in studying the structures of the elements. As in the latter this preference extends to some of the following elements and in such series of compounds as CaO, ScN, TiC, one component keeps the vibration two status throughout the series and the resulting effective rotations are 5˝, 7, 8˝, rather than 6, 8, 10.
Except for certain types of crystals which are essentially interchangeable, the structures of the elements are determined almost entirely by the nature of the bonds. In compounds there is another equally active factor: the relative proportions of the components. Where two atoms of one kind form a normal bond compound with one atom of another, the unequal proportions make the NaCl arrangement impossible and instead we find the Calcium Fluoride structure. Inter-atomic distances for a number of common CaF2 type crystals are listed in Table VIII.
Table VII
| Compound | Magnetic Rotations |
Electric Rotation |
Internal Distance | ||
|---|---|---|---|---|---|
| Calculated | Observed | ||||
| LiH | 3-3(2) | 3-3(2) | 3 | 2.04 | 2.05 |
| LiF | 3-3(2) | 3-3(2) | 3 | 2.04 | 2.005 |
| LiCl | 3-3(2) | 4-3, 3-4 | 4 | 2.57 | 2.57 |
| LiBr | 3-3(2) | 4-4 | 4 | 2.77 | 2.745 |
| LiI | 3-3(2) | 5-4 | 4 | 2.96 | 3.00 |
| NaF | 3-2˝ | 3-3(2) | 4 | 2.27 | 2.31 |
| NaCl | 3-2˝ | 4-3, 3-4 | 4 | 2.78 | 2.81 |
| NaBr | 3-2˝ | 4-4 | 4 | 2.95 | 2.97 |
| NaI | 3-3 | 5-4 | 4 | 3.21 | 3.23 |
| MgO | 3-3 | 3-3(2) | 6 | 2.09 | 2.10 |
| MgS | 3-3 | 4-3, 3-4 | 6 | 2.54 | 2.595 |
| MgSe | 3-3 | 4-4 | 6 | 2.68 | 2.725 |
| KF | 4-3 | 3-3(2) | 4 | 2.63 | 2.665 |
| KCl | 4-3 | 4-3, 3-4 | 4 | 3.11 | 3.14 |
| KBr | 4-3 | 4-4 | 4 | 3.30 | 3.29 |
| KI | 4-3 | 5-4 | 4 | 3.47 | 3.525 |
| CaO | 4-3 | 3-3(2) | 5˝ | 2.38 | 2.40 |
| CaS | 4-3 | 4-3, 3-4 | 5˝ | 2.81 | 2.84 |
| CaSe | 4-3 | 4-4 | 5˝ | 2.98 | 2.955 |
| CaTe | 4-3 | 5-4 | 5˝ | 3.13 | 3.17 |
| ScN | 4-3 | 3-3(2) | 7 | 2.23 | 2.22 |
| TiC | 4-3 | 3-3(2) | 8˝ | 2.12 | 2.145 |
| RbF | 4-4 | 3-3(2) | 4 | 2.77 | 2.815 |
| RbCl | 4-4 | 4-3, 3-4 | 4 | 3.24 | 3.285 |
| RbBr | 4-4 | 4-4 | 4 | 3.43 | 3.43 |
| RbI | 4-4 | 5-4 | 4 | 3.61 | 3.645 |
| SrO | 4-4 | 3-3(2) | 5˝ | 2.50 | 2.55 |
| SrS | 4-4 | 4-3, 3-4 | 5˝ | 2.92 | 2.935 |
| SrSe | 4-4 | 4-4 | 5˝ | 3.10 | 3.115 |
| SrTe | 4-4 | 5-4 | 5˝ | 3.26 | 3.24 |
| CsF | 5-4 | 3-3(2) | 4 | 2.96 | 2.97 |
| BaO | 5-4 | 3-3(2) | 5˝ | 2.68 | 2.735 |
| BaS | 5-4 | 4-3, 3-4 | 5˝ | 3.07 | 3.085 |
| BaSe | 5-4 | 4-4 | 5˝ | 3.26 | 3.28 |
| BaTe | 5-4 | 5-4 | 5˝ | 3.42 | 3.47 |
| LaN | 5-4 | 3-3(2) | 6 | 2.61 | 2.63 |
| LaP | 5-4 | 4-3, 3-4 | 6 | 2.99 | 3.01 |
| LaAs | 5-4 | 4-4 | 7 | 3.04 | 3.06 |
| LaSb | 5-4 | 5-4 | 7 | 3.20 | 3.24 |
| LaBi | 5-4 | 5-5 | 7 | 3.28 | 3.28 |
In spite of the difference in structure the inter-atomic distances in the CaF3 crystals are normally identical with the NaCl distances for the same positive component unless there is a secondary combination between the atoms of the two-atom component. Thus Na2S has the same inter-atomic distance as NaCl, CaF2 the same as CaO, and so on. The equilibrium distance is independent of the nature of the negative component because the atom is essentially a time structure and the magnitude of the inter-atomic forces is primarily a function of the net time displacement. Where space displacement enters into the situation it does so only as a modifier or neutralizer of the time displacement. The number of negative atoms that are required in combination with each positive atom to accomplish this result is immaterial from a force standpoint. For this reason the question of combining power or valence does not need to be taken into account in the present discussion and will be given separate consideration later.
Table VIII
| Compound | Magnetic Rotations |
Electric Rotation |
Inter-atomic Distance | ||
|---|---|---|---|---|---|
| Calculated | Observed | ||||
| Li2O | 3-3(2) | 3-3(2) | 3 | 2.04 | 2.00 |
| Li2S | 3-3(2) | 3-4 | 4 | 2.50 | 2.47 |
| Li2Se | 3-3(2) | 4-3, 3-4 | 4 | 2.57 | 2.60 |
| Li2Te | 3-3(2) | 4-5 | 4 | 2.87 | 2.82 |
| Be2C | 3-3(2) | 3-3(2) | 3˝ | 1.91 | 1.875 |
| Na2O | 3-2˝ | 3-3(2) | 3˝ | 2.40 | 2.405 |
| Na2S | 3-2˝ | 4-3, 3-4 | 4 | 2.78 | 2.83 |
| Na2Se | 3-2˝ | 4-4 | 4 | 2.95 | 2.95 |
| Na2Te | 3-3 | 5-4 | 4 | 3.21 | 3.17 |
| Mg2Si | 3-3 | 4-3 | 5 | 2.73 | 2.77 |
| Mg2Ge | 3-3 | 4-4 | 5˝ | 2.76 | 2.76 |
| Mg2Sn | 3-3 | 5-4 | 5˝ | 2.90 | 2.93 |
| Mg2Pb | 3-3 | 5-5 | 6 | 2.895 | 2.93 |
| K2O | 4-3 | 3-3(2) | 3˝ | 2.78 | 2.79 |
| K2S | 4-3 | 4-3 | 4 | 3.175 | 3.20 |
| K2Se | 4-3 | 4-4 | 4 | 3.30 | 3.32 |
| K2Te | 4-3 | 5-4 | 4 | 3.47 | 3.53 |
| CaF2 | 4-3 | 3-3(2) | 5˝ | 2.38 | 2.37 |
| Rb2O | 4-4 | 3-3(2) | 3˝ | 2.93 | 2.92 |
| Rb2S | 4-4 | 4-3 | 4 | 3.30 | 3.32 |
| SrF2 | 4-4 | 3-3(2) | 5˝ | 2.50 | 2.51 |
| SrCl2 | 4-4 | 4-3 | 5˝ | 2.98 | 3.03 |
| BaF2 | 5-4 | 3-3(2) | 5˝ | 2.68 | 2.69 |
| BaCl2 | 5-4 | 4-3 | 5˝ | 3.13 | 3.18 |
(See Appendix B for description of material omitted from this edition.)