The consequences of the reversal of direction (in the context of a fixed reference system) that takes place at unit distance were explained in a general way in Chapter 8 of Volume I. As brought out there, the most significant of these consequences is that establishment of an equilibrium between gravitation and the progression of the natural reference system becomes possible.
There is a location outside unit distance where the magnitudes of these two motions are equal: the distance that we are calling the gravitational limit. But this point of equality is not a point of equilibrium. On the contrary, it is a point of instability. If there is even a slight unbalance of forces one way or the other, the resulting motion accentuates the unbalance. A small inward movement, for instance, strengthens the inward force of gravitation, and thereby causes still further movement in the same direction. Similarly, if a small outward movement occurs, this weakens the gravitational force and causes further outward movement. Thus, even though the inward and outward motions are equal at the gravitational limit, this is actually nothing but a point of demarcation between inward and outward motion. It is not a point of equilibrium.
In the region inside unit distance, on the contrary, the effect of any change in position opposes the unbalanced forces that produced the change. If there is an excess gravitational force, an outward motion occurs which weakens gravitation and eliminates the unbalance. If the gravitational force is not adequate to maintain a balance, an inward motion takes place. This increases the gravitational effect and restores the equilibrium. Unless there is some intervention by external forces, atoms move gravitationally until they eventually come within unit distance of other atoms. Equilibrium is then established at positions within this inside region: the time region, as we have called it.
The condition in which a number of atoms occupy equilibrium positions of this kind in an aggregate is known as the solid state of matter. The distance between such positions is the inter-atomic distance, a distinctive feature of each particular material substance that we will examine in detail in the following chapter. Displacement of the equilibrium in either direction can be accomplished only by the application of a force of some kind, and a solid structure resists either an inward force, a compression, or an outward force, a tension. To the extent that resistance to tension operates to prevent separation of the atoms of a solid it is commonly known as the force of cohesion.
The conclusions with respect to the nature and origin of atomic cohesion that have been reached in this work replace a familiar theory, based on altogether different premises. This previously accepted hypothesis, the electrical theory of matter, has already had some consideration in the preceding volume, but since the new explanation of the nature of the cohesive force is basic to the present development, some more extensive comparisons of the two conflicting viewpoints will be in order before we proceed to develop the new theoretical structure in greater detail.
The electrical, or electronic, theory postulates that the atoms of solid matter are electrically charged, and that their cohesion is due to the attraction between unlike charges. The principal support for the theory comes from the behavior of ionic compounds in solution. A certain proportion of the molecules of such compounds split up, or dissociate, into oppositely charged components which are then called ions. The presence of the charges can be explained in either of two ways: (1) the charges were present, but undetectable, in the undissolved material, or (2) they were created in the solution process. The adherents of the electrical theory base it on explanation (1). At the time this explanation was originally formulated, electric charges were thought to be relatively permanent entities, and the conclusion with respect to their role in the solution process was therefore quite in keeping with contemporary scientific thought. In the meantime, however, it has been found that electric charges are easily created and easily destroyed, and are no more than a transient feature of matter. This cuts the ground from under the main support of the electrical theory, but the theory has persisted because of the lack of any available alternative.
Obviously some kind of a force must hold the solid aggregate together. Outside of the forces known to result directly from observable motion, there are only three kinds of force of which there has heretofore been any definite observational knowledge: gravitational, electric, and magnetic. The so-called “forces” which play various roles in present-day atomic physics are purely hypothetical. Of the three known forces, the only one that appears to be strong enough to account for the cohesion of solids is the electric force. The general tendency in scientific circles has therefore been to take the stand that cohesion must result from the operation of electrical forces, notwithstanding the lack of any corroboration of the conclusions reached on the basis of the solution process, and the existence of strong evidence against the validity of those conclusions.
One of the serious objections to this electrical theory of cohesion is that it is not actually a theory, but a patchwork collection of theories. A number of different explanations are advanced for what is, to all appearances, the same problem. In its basic form, the theory is applicable only to a restricted class of substances, the so-called “ionic” compounds. But the great majority of compounds are “non-ionic.” Where the hypothetical ions are clearly non-existent, an electrical force between ions cannot be called upon to explain the cohesion, so, as one of the general chemistry tests on the author’s shelves puts it, “A different theory was required to account for the formation of these compounds.” But this “different theory,” based on the weird concept of electrons “shared” by the interacting atoms, is still not adequate to deal with all of the non-ionic compounds, and a variety of additional explanations are called upon to fill the gaps.
In current chemical parlance the necessity of admitting that each of these different explanations is actually another theory of cohesion is avoided by calling them different types of “bonds” between the atoms. The hypothetical bonds are then described in terms of interaction of electrons, so that the theories are united in language, even though widely divergent in content. As noted in Chapter 19, Volume I, a half dozen or so different types of bonds have been postulated, together with “hybrid” bonds which combine features of the general types.
Even with all of this latitude for additional assumptions and hypotheses, some substances, notably the metals, cannot be accommodated within the theory by any expedient thus far devised. The metals admittedly do not contain oppositely charged components, if they contain any charged components at all, yet they are subject to cohesive forces that are indistinguishable from those of the ionic compounds. As one prominent physicist, V. F. Weisskopf, found it necessary to admit in the course of a lecture, “I must warn you I do not understand why metals hold together.” Weisskopf points out that scientists cannot even agree as to the manner in which the theory should be applied. Physicists give us one answer, he says, chemists another, but “neither of these answers is adequate to explain what a chemical bond is.”1
This is a significant point. The fact that the cohesion of metals is clearly due to something other than the attraction between unlike charges logically leads to a rather strong presumption that atomic cohesion in general is non-electrical. As long as some non-electrical explanation of the cohesion of metals has to be found, it is reasonable to expect that this explanation will be found applicable to other substances as well. Experience in dealing with cohesion of metals thus definitely foreshadows the kind of conclusions that have been reached in the development of the Reciprocal System of theory.
It should also be noted that the electrical theory is wholly ad hoc. Aside from what little support it can derive from extrapolation to the solid state of the conditions existing in solutions, there is no independent confirmation of any of the principal assumptions of the theory. No observational indication of the existence of electrical charges in ordinary matter can be detected, even in the most strongly ionic compounds. The existence of electrons as constituents of atoms is purely hypothetical. The assumption that the reluctance of the inert gases to enter into chemical compounds is an indication that their structure is a particularly stable one is wholly gratuitous. And even the originators of the idea of “sharing” electrons make no attempt to provide any meaningful explanation of what this means, or how it could be accomplished, if there actually were any electrons in the atomic structure. These are the assumptions on which the theory is based, and they are entirely without empirical support. Nor is there any solid basis for what little theoretical foundation the theory may claim, inasmuch as its theoretical ties are to the nuclear theory of atomic structure, which is itself entirely ad hoc.
But these points, serious as they are, can only be regarded as supplementary evidence, as there is one fatal weakness of the electrical theory that would demolish it even if nothing else of an adverse nature were known. This is our knowledge of the behavior of positive and negative electric charges when they are brought into close proximity. Such charges do not establish an equilibrium of the kind postulated in the theory; they destroy each other. There is no evidence which would indicate that the result of such contact is any different in a solid aggregate, nor is there even any plausible theory as to why any different outcome could be expected, or how it could be accomplished.
It is worth noting in this connection that while current physical theory portrays positive and negative charges as existing in a state of congenial companionship in the nuclear theory of the atom and in the electrical theory of matter, it turns around and gives us explanations of the behavior of antimatter in which these charges display the same violent antagonism that they demonstrate to actual observation. This is the kind of inconsistency that inevitably results when recalcitrant problems are “solved” by ad hoc assumptions that involve departures from established physical laws and principles.
In the context of the present situation in which the electrical theory is challenged by a new development, all of these deficiencies and contradictions that are inherent in the electrical theory become very significant. But the positive evidence in favor of the new theory is even more conclusive than the negative evidence against its predecessor. First, and probably the most important, is the fact that we are not replacing the electrical theory of matter with another “theory of matter.” The Reciprocal System is a complete general theory of the physical universe. It contains no hypotheses other than those relating to the nature of space and time, and it produces an explanation of the cohesion of solids in the same way that it derives logical and consistent explanations of other physical phenomena: simply by developing the consequences of the basic postulates. We therefore do not have to call upon any additional force of a hypothetical nature to account for the cohesion. The two forces that determine the course of events in the region outside unit distance also account for the existence of the inter-atomic equilibrium inside this distance.
It is significant that the new theory identifies both of these forces. One of the major defects of the electrical theory of cohesion is that it provides only one force, the hypothetical electrical force of attraction, whereas two forces are required to explain the observed situation. Originally it was assumed that the atoms are impenetrable, and that the electrical forces merely hold them in contact. Present-day knowledge of compressibility and other properties of solids has demolished this hypothesis, and it is now evident that there must be what Karl Darrow called an “antagonist,” in the statement quoted in Volume I, to counter the attractive force, whatever it may be, and produce an equilibrium. Physicists have heretofore been unable to find any such force, but the development of the Reciprocal System has now revealed the existence of a powerful and omnipresent force hitherto unknown to science. Here is the missing ingredient in the physical situation, the force that not only explains the cohesion of solid matter, but, as we saw in Volume I, supplies the answers to such seemingly far removed problems as the structure of star clusters and the recession of the galaxies.
One point that should be specifically noted is that it is this hitherto unknown force, the force due to the progression of the natural reference system, that holds the solid aggregate together, not gravitation, which acts in the opposite direction in the time region. The prevailing opinion that the force of gravitation is too weak to account for the cohesion is therefore irrelevant, whether it is correct or not.
Inasmuch as the new theoretical system applies the same general principles to an understanding of all of the inter-atomic and inter-molecular equilibria, it explains the cohesion of all substances by the same physical mechanism. It is no longer necessary to have one theory for ionic substances, several more for those that are non-ionic, and to leave the metals out in the cold without any applicable theory. The theoretical findings with respect to the nature of chemical combinations and the structure of molecules that were outlined in the preceding volume have made a major contribution to this simplification of the cohesion picture, as they have eliminated the need for different kinds of cohesive forces, or “bonds.” All that is now required of a theory of cohesion is that it supply an explanation of the inter-atomic equilibrium, and this is provided, for all solid substances under all conditions, by balancing the outward motion (force) of gravitation against the inward motion (force) of the progression of the natural reference system. Because of the asymmetry of the rotational patterns of the atoms of many elements, and the consequent anisotropy of the force distributions, the equilibrium locations vary not only between substances, but also between different orientations of the same substance. Such variations, however, affect only the magnitudes of the various properties of the atoms. The essential character of the inter-atomic equilibrium is always the same.
As indicated in the original discussion of gravitation, even though the various aggregates of matter do not actually exert gravitational forces on each other, the observable results of their gravitational motions are identical with those that would be produced if such forces did exist. The same is true of the results of the progression of the natural reference system. There is a considerable element of convenience in expressing these results in terms of force, on an “as if” basis, and this practice has already been followed to some extent in the previous volume. Now that we are ready to begin a quantitative evaluation of the inter-atomic relations, however, it is desirable to make it clear that the force concept is being used only for convenience. Although the quantitative discussion that follows, like the earlier qualitative discussion, will be carried on in terms of forces, what we will actually be dealing with are the inward and outward motions of each individual atom.
While the items that have been mentioned add up to a very impressive case in favor of the new theory of cohesion, the strongest confirmation of its validity comes from its ability to locate the point of equilibrium; that is to give us specific values of the inter-atomic distances. As will be demonstrated in Chapter 2, we are already able, by means of the newly established relations, to calculate the possible values of the inter-atomic distance for most of the simpler substances, and there do not appear to be any serious obstacles in the way of extending the calculations to more complex substances whenever the necessary time and effort can be applied to the task. Furthermore, this ability to determine the location of the point of equilibrium is not limited to the simple situation where only the two basic forces are involved. Chapters 4 and 5 will show that the same general principles can also be applied to an evaluation of the changes in the equilibrium distance that result from the application of heat or pressure to the solid aggregate.
Although, as stated in Volume I, the true magnitude of a unit of space is the same everywhere, the effective magnitude of a spatial unit in the time region is reduced by the inter-regional ratio. It is convenient to regard this reduced value, 1/156.44 of the natural unit, as the time region unit of space. The effective portion of a time region phenomenon may extend into one or more additional units, in which case the measured distance will exceed the time region unit, or the original single unit may not be fully effective, in which case the measured distance will be less than the time region unit. Thus the inter-atomic equilibrium may be reached either inside or outside the time region unit of distance, depending on where the outward rotational forces reach equality with the inward force of the progression of the natural reference system. Extension of the inter-atomic distance beyond one time region unit does not take the equilibrium system out of the time region, as the boundary of that region is at one full-sized natural unit of distance, not at one time region unit. So far as the inter-atomic force equilibrium is concerned, therefore, the time region unit of distance does not represent any kind of a critical magnitude.
As we saw in our examination of the composition of the magnetic neutral groups, however, the natural unit as it exists in the time region (the time region unit) is a critical magnitude from the orientation standpoint. An explanation of this difference can be derived from a consideration of the difference in the inherent nature of the two phenomena. Where the inter-atomic distance is less than one time region unit, the rotational forces are acting against the inward force of the progression of the reference system during only a portion of the unit progression. Similarly, where the inter-atomic distance is greater than one time region unit, the unit inward force is acting against only a portion of the greater-than-unit outward rotational forces. The variations in distance thus reflect differences in the magnitudes of the rotational forces. But the orientation effect has no magnitude. It either exists, or does not exist. As we have noted in the previous discussion, particularly in connection with the structure of the benzene molecule, this effect, if it exists, is the same regardless of whether it acts at short range or at long range. The essential requirement that it must meet is that it must be continuously effective. Otherwise, the orientation is destroyed during the off period. Where the rotational forces extend beyond one time region unit, so that the unit orientation effect is coincident with only a portion of the total rotational forces, the orienting effect is not continuous, and no orientation takes place.
In this chapter we are dealing mainly with what we are calling “rotational forces.” These are, of course, the same “as if” forces due to the scalar aspect of the atomic rotation that were called “gravitational” in some other contexts, the choice of language depending on whether it is the origin or the effect of the force that is being emphasized in the discussion. For a quantitative evaluation of the rotational forces we may use the general force equation, providing that we replace the usual terms of the equation with the appropriate time region terms. As explained in introducing the concept of the time region in Chapter 8 of Volume I, equivalent space 1/t replaces space in the time region, and velocity is therefore 1/t2 Energy, the one-dimensional equivalent of mass, which takes the place of mass in the time region expression of the force equation, because the three rotations of the atom act separately, rather than jointly, in this region, is the reciprocal of this expression, or t2. Acceleration is velocity divided by time: 1/t3. The time region equivalent of the equation F = ma is therefore F = Ea = t2 × 1/t3 = 1/t in each dimension.
At this point we will need to take note of the nature of the increments of speed displacement in the time region. In the outside region additions to the displacement proceed by units: first one unit, then another similar unit, yet another, and so on, the total up to any specific point being n units. There is no term with the value n. This value appears only as a total. The additions in the time region follow a different mathematical pattern, because in this case only one of the components of motion progresses, the other remaining fixed at the unit value. Here the displacement is 1/x, and the sequence is 1/1, 1/2, 1/3…1/n. The quantity 1/n is the final term, not the total. To obtain the total that corresponds to n in the outside region it is necessary to integrate the quantity 1/x from x = 1 to x = n. The result is ln (n), the natural logarithm of n.
Many readers of the first edition have asked why this total should be an integral rather than a summation. The answer is that we are dealing with a continuous quantity. As pointed out in the introductory chapters of the preceding volume, the motion of which the universe is constructed does not proceed in a succession of jumps. Even though it exists only in units, it is a continuous progression. A unit of this motion is a specific portion of this continuity. A series of units is a more extended segment of that continuity, and its magnitude is an integral. In dealing with the basic individual units of motion in the outside region it is possible to use the summation process, but only because in this case the sum is the same as the integral. To get the total of the 1/x series we must integrate.
To evaluate the rotational force we integrate the quantity 1/t from unity, the physical datum or zero level, to t:
If the quantity ln t is below unity in any dimension there is no effective outward force in that dimension, but the natural logarithm exceeds unity for all values of x above 2, and the atoms of all elements have a rotational displacement of 2 (equivalent to t = 3) or more in at least one dimension. Consequently, all have effective rotational forces.
The force computed from equation 1-1 is the inherent rotational force of the individual atom; that is, the one-dimensional force which it exerts against a single unit of force. The force between two (apparently) interacting atoms is:
|F = ln tA ln tB||
For a two-dimensional magnetic rotation this becomes
|F = ln² tA ln² tB||
As we found in Chapter 12, Volume. I, the equivalent of distance s in the time region is s², and the gravitational force in this region therefore varies inversely as the fourth power of the distance rather than the square. Applying this factor to the expression for the force of the two-dimensional rotation, together with the inter-regional ratio, the ratio of effective to total force derived in the same chapter, we obtain the effective force of the magnetic rotation of the atom:
|Fm = (0.006392)4 s-4 ln² tA ln² tB||
The distance factor does not apply to the force due to the progression of the natural reference system, as this force is omnipresent, and unlike the rotational force is not altered as the objects to which it is applied change their relative positions. At the point of equilibrium, therefore, the rotational force is equal to the unit force of the progression. Substituting unity for Fm in equation 1-4, and solving for the equilibrium distance, we obtain:
|s0 = 0.006392 ln½ tA ln½ tB||
The inter-atomic distances for those elements which have no electric rotation, the inert gas series, may be calculated directly from this equation. In the elements, however, tA = tB in most cases, and it will be convenient to express the equation in the simplified form:
|s0 = 0.006392 ln t||
The values thus calculated are in the neighborhood of 10-8 cm, and for convenience this quantity has been taken as a unit in which to express the inter-atomic and inter-molecular distances. When converted from natural units to this conventional unit, the Angstrom unit, symbol Å, equation 1-6 becomes:
|s0 = 2.914 ln t Å||
In applying this equation we encounter another of the questions with respect to terminology that inevitably arise in a basically new treatment of any subject. The significance of the quantity t as used in the foregoing discussion and in the equations is obvious from the context—it is the magnitude of the effective rotation—but the question is: What shall we call it? The basic quantity with which we are dealing, the rotational speed displacement, does not enter into the equations directly. The mathematical structure of these equations requires us to enter them with values that include the initial unit which constitutes the natural zero datum. Furthermore, each double vibrational unit rotates independently, and when the rotation extends to a second such unit the increment in the value of t is only one half unit per added unit of displacement. Under these circumstances, where the relation of the term t to the displacement is variable, it seems advisable to give this term a distinctive name, and we will therefore call it the specific rotation.
As brought out in the discussion of the general characteristics of the atomic rotation in Chapter 10, Volume I, the two magnetic displacements may be unequal, and in this event the speed distribution takes the form of a spheroid with the principal rotation effective in two dimensions and the subordinate rotation in one. The average effective value of the specific rotation under these conditions is (t12 t2)1/3. In this case we are dealing with the properties of a single entity, and the mathematical situation seems clear. But it is not so evident how we should arrive at the effective specific rotation where there is an interaction between two atoms whose individual rotations are different. As matters now stand it appears that the geometric mean of the two specific rotations is the correct quantity, and the values tabulated in Chapters 2 and 3 have been calculated on this basis. It should be noted, however, that this conclusion as to the mathematics of the combination is still somewhat tentative, and if further study shows that it must be modified in some, or all, applications, the calculated values will be subject to corresponding modifications. Any changes will be small in most cases, but they will be substantial where there is a large difference between the two components. The absence of major discrepancies between the calculated and observed distances in combinations of atoms with much different dimensions therefore gives some significant support to the use of the geometric mean pending further theoretical clarification.
The inter-atomic distances of four of the five inert gas elements for which experimental data are available follow the regular pattern. The values calculated for these elements are compared with the experimental distances in Table 1.
Table 1: Distances - Inert Gas Elements
Helium, which also belongs to the inert gas series, has some special characteristics due to its low rotational displacement, and will be discussed in connection with other elements affected by the same factors. The reason for the appearance of the 4½ value in the xenon rotation will also be explained shortly. The calculated distances are those which would prevail in the absence of compression and thermal expansion. A few of the experimental data have been extrapolated to this zero base by the investigators, but most of them are the actual observed values at atmospheric pressure and at temperatures which depend on the properties of the substances under examination. These values are not exactly comparable to the calculated distances. In general, however, the expansion and compression up to the temperature and pressure of observation are small. A comparison of the values in the last two columns of Table 1 and the similar tables in Chapters 2 and 3 therefore gives a good picture of the extent of agreement between the theoretical figures and the experimental results.
Another point about the distance correlations that needs to be taken into account is that there is a substantial amount of variation in the experimental results. If we were to take the closest of these measured values as the basis for comparison, the correlation would be very much better. One relatively recent determination of the xenon distance, for example, arrives at a value of 4.34, almost identical with the calculated distance. There are also reported values for the argon distance that agree more closely with the theoretical result. However, a general policy of using the closest values would introduce a bias that would tend to make the correlation look more favorable than the situation actually warrants. It has therefore been considered advisable to use empirical data from a recognized selection of preferred values. Except for those values identified by asterisks, all of the experimental distances shown in the tables are taken from the extensive compilation by Wyckoff.2 Of course, the use of these values selected on the basis of indirect criteria introduces a bias in the unfavorable direction, since, if the theoretical results are correct, every experimental error shows up as a discrepancy, but even with this negative bias the agreement between theory and observation is close enough to show that the theoretical determination of the inter-atomic distance is correct in principle, and to demonstrate that, with the exception of a relatively small number of uncertain cases, it is also correct in the detailed application.
Turning now to the elements which have electric as well as magnetic displacement, we note again that the electric rotation is one-dimensional and opposes the magnetic rotation. We may therefore obtain an expression for the effect of the electric rotational force on the magnetically rotating photon by inverting the one-dimensional force term of equation 1-2.
|Fe = 1/(ln t’A ln t’B)||
Inasmuch as the electric rotation is not an independent motion of the basic photon, but a rotation of the magnetically rotating structure in the reverse direction, combining the electric rotational force of equation 1-8 with the magnetic rotational force of equation 1-4 modifies the rotational terms (the functions of t) only, and leaves the remainder of equation 1-4 unchanged.
F = (0.006392)4
|(ln2 tA ln2 tB) / (s4 ln t’A ln t’B)||
Here again the effective rotational (outward) and natural reference system progression (inward) forces are necessarily equal at the equilibrium point. Since the force of the progression of the natural reference system is unity, we substitute this value for F in equation 1-9 and solve for s0, the equilibrium distance, as before.
s0 = (0.006392)
|(ln½ tA ln½ tB) / (ln¼t’A ln¼ t’B)||
Again simplifying for application to the elements, where A is generally equal to B,
|s0 = 0.006392 ln t / ln½ t’||
In Angstrom units this becomes
|s0 = 2.914 ln t / ln½ t’Å||
As already noted, when the rotation is extended to a second (double) vibrational unit, to vibration two, we may say, each added displacement unit adds only one half unit to the specific rotation. Inasmuch as 8 electric displacement units distributed three-dimensionally bring the rotation to a new zero point, and cause the rotational motion to revert to the translational status, the change to vibration two in the electric dimension must take place before the displacement reaches 8. Specific rotation 8 (displacement 7) is therefore followed by 8½, 9, 9½, etc. But the first effective rotational displacement unit is necessarily one-dimensional, and the linear equivalent of the 8-unit limit is 2 units. Thus this first unit has already reached the one-dimensional limit. The succeeding displacement units have the option of continuing on the one-dimensional basis and extending the rotation to vibration two rather than extending it into additional dimensions. The change to vibration two therefore may take place immediately after the first displacement unit. In this case specific rotation 2 (displacement 1) is followed by 2½, 3, 3½, etc. The lower value is commonly found where it first becomes possible; that is, displacement 2 normally corresponds to rotation 2½ rather than 3. The next element may take the intermediate value 3½, but beyond this point the higher vibration one value normally prevails.
In the first edition it was indicated that the one or two vibrational displacement units being rotated did not necessarily constitute the entire vibrational component of the basic photon, inasmuch as these one or two units are capable of being rotated independently of the remaining vibrational units, if any. Further consideration now leads to the conclusion that one or two units of a multi-unit photon frequency can, in fact, be set in rotation independently, as previously indicated, and that the original photon may have had an excess of vibrational units, but that in such an event the rotating portion of the photon begins moving inward, whereas the non-rotating portion continues moving outward by reason of the progression of the natural reference system. The two portions therefore separate, and the rotating portion retains no non-rotating vibrational component.
The general pattern of the magnetic rotational values is the same as that of the electric values. The tendency to substitute specific rotation 2½ for 3 applies to the magnetic as well as to the electric rotation, and in the lower group combinations (both elements and compounds) that follow the regular electropositive pattern the specific magnetic rotations are usually 2½-2½ or 3-2½, rather than 3-3. But the upper limit for specific magnetic rotation on a vibration one basis is 4 (three displacement units) instead of 8, as the two-dimensional rotation reaches the upper zero level at 4 displacement units in each dimension. Rotation 4½ therefore follows rotation 4 in the regular sequence, as we saw in the values given for xenon in Table 1. 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 5-4 or 5-4½ rotation occurs in some elements either in lieu of, or in combination with, the 4½-4 or 4½-4½ rotation.