CHAPTER 10Electrical ResistanceWhile the motion of the electric current through matter is equivalent to motion of matter through space, as brought out in the discussion in Chapter 9, the conditions under which each type of motion is encountered in our ordinary experience emphasize different aspects of the common fundamentals. In dealing with the motion of matter through extension space we are primarily concerned with the motions of individual objects. Newton’s laws of motion, which are the foundation stones of mechanics, deal with the application of forces to initiate or modify the motions of such objects, and with the transfer of motion from one object to another. Our interest in the electric current, on the other hand, is concerned mainly with the continuous aspects of the current flow, and the status of the individual objects that are involved is largely irrelevant. The mobility of the spatial units in the current flow also introduces some kinds of variability that are not present in the movement of matter through extension space. Consequently, there are behavior characteristics, or properties, of material structures that are peculiar to the relation between these structures and the moving electrons. Expressing this in another way, we may say that matter has some distinctive electrical properties. The basic property of this nature is resistance. As pointed out in Chapter 9, resistance is the only quantity participating in the fundamental relations of current flow that is not a familiar feature of the mechanical system of equations, the equations that deal with the motion of matter through extension space. Present-day ideas as to the origin of electrical resistance are summarized by one author in this manner:
As indicated in the preceding chapter, the development of the theory of the universe of motion arrives at a totally different concept of the nature of electrical resistance. The electrons, we find, are derived from the environment. It was brought out in Volume I that there are physical processes in operation which produce electrons in substantial quantities, and that, although the motions that constitute these electrons are, in many cases. absorbed by atomic structures, the opportunities for utilizing this type of motion in such structures are limited. It follows that there is always a large excess of free electrons in the material sector of the universe, most of which are uncharged. In this uncharged state the electrons cannot move with respect to extension space, because they are inherently rotating units of space, and the relation of space to space is not motion. In open space, therefore, each uncharged electron remains permanently in the same location with respect to the natural reference system, in the manner of a photon. In the context of the stationary spatial reference system the uncharged electron, like the photon, is carried outward at the speed of light by the progression of the natural reference system. All material aggregates are thus exposed to a flux of electrons similar to the continual bombardment by photons of radiation. Meanwhile there are other processes, to be discussed later, whereby electrons are returned to the environment. The electron population of a material aggregate such as the earth therefore stabilizes at an equilibrium level. These processes that determine the equilibrium electron concentration are independent of the nature of the atoms of matter and of the atomic volume. The concentration of electrons is therefore uniform in electrically isolated conductors where there is no current flow. It follows that the number of electrons involved in the thermal motion of atoms of matter is proportional to the atomic volume, and the energy of that motion is determined by the effective rotational factors of the atoms. The atomic volume and thermal energy therefore determine the resistance. Those substances whose rotational motion is entirely in time (Divisions I and II) have their thermal motion in space, in accordance with the general rule governing addition of motions, as set forth in Volume I. For these substances zero thermal motion corresponds to zero resistance, and the resistance increases with the temperature. This is due to the fact that the concentration of electrons (units of space) in the time component of the conductor is constant for any specific current magnitude, and the current therefore increases the thermal motion by a specific proportion. Such substances are conductors. Where there are two dimensions of rotation in space, as in many of the elements of Division IV, the thermal motion , which requires two open dimensions because of the finite diameters of the moving electrons, is necessarily in time. In this case, zero temperature corresponds to zero motion in time. Here the resistance is initially extremely high, but decreases with an increase in temperature. Substance of this kind are known as insulators or dielectrics. Where there is only one dimension of spatial rotation, as in Division III, the elements of greatest electric displacement, those closest to the electropositive divisions, are able to follow the positive pattern, and are conductors. The Division III elements of lower electric displacement follow a modified time motion pattern, with resistance decreasing from a high, but finite, level at zero temperature. These substances of intermediate characteristics are semiconductors. For the present we will be concerned primarily with the resistance of conductors, and will further limit the discussion to what may be called the “regular” pattern of conductor resistance. A limitation of this kind is necessary at the present stage of the investigation because the large element of uncertainty in the experimental information on the resistivity of the various conducting materials makes the clarification of the resistance relations a slow and difficult process. The early stages of the development of the Reciprocal System of theory, prior to the publication of the first edition of this work in 1959, which were very productive in the non-electrical areas, made relatively little progress in dealing with the electrical properties, largely because of conflicts between the theoretical deductions and some experimental results that have since been found to be incorrect. The increasing scope and accuracy of the experimental work in the intervening years has improved this situation very materially, but the basic problem still remains. Ideally it should be possible to deduce all of the pertinent information from theoretical premises alone, without reference to experimental determinations, but as a practical matter this is not feasible. A few steps can be, and have been, taken on a purely theoretical basis, particularly where the previous development of the theory has cast some important new light on the subject matter, but from the practical standpoint an extensive and detailed investigation in any area is possible only if the theoretical study and the checking of the theoretical conclusions against experimental and observational data go hand in hand. It follows that where empirical data are lacking, progress is difficult, and where they are seriously wrong, it is essentially impossible. Unfortunately, resistance measurements are subject to many factors that introduce uncertainty into the results. The purity of the specimen is particularly critical because of the great difference between the resistivities of conductors and dielectrics. Even a very small amount of a dielectric impurity can alter the resistance substantially. Conventional theory has no explanation for the magnitude of this effect. If the electrons move through the interstices between the atoms, as this theory contends, a few additional obstacles in the path should not contribute significantly to the resistance. But, as we saw in Chapter 9, the current moves through all of the atoms of the conductor, including the impurity atoms, and it increases the heat content of each atom in proportion to its resistance. The extremely high dielectric resistance results in a large contribution by each impurity atom, and even a very small number of such atoms therefore has a significant effect. Semiconducting elements are less effective as impurities, but they may still have resistivities thousands of times as great as those of the conductor metals. The resistance also varies under heat treatment, and careful annealing is required before reliable measurements can be made. The adequacy of this treatment in many, if not most, of the resistance determinations is questionable. For example, G. T. Meaden reports that the resistance of beryllium was lowered more than fifty percent by such treatment, and comments that “much earlier work was clearly based on unannealed specimens.”19 Other sources of uncertainty include changes in crystal structure or magnetic behavior that take place at different temperatures or pressures in different specimens, or under different conditions, often with substantial hysteresis effects. Ultimately, of course, it will be desirable to bring all of these variables within the scope of the theoretical treatment, but for the present our objective will have to be limited to deducing from the theory the nature and magnitude of the changes in resistance resulting from temperature and pressure variations in the absence of these complicating factors, and then to show that enough of the experimental results are in agreement with the theory to make it probable that the discrepancies, where they occur, are due to one of more of these factors that modify the normal values. Inasmuch as the electrical resistance is a product of the thermal motion, the energy of the electron motion is in equilibrium with the thermal energy. The resistance is therefore directly proportional to the effective thermal energy; that is, to the temperature. It follows that the increment of resistance per degree is a constant for each (unmodified) substance, a magnitude that is determined by the atomic characteristics. The curve representing the relation of the resistivity to the temperature, in application to a single atom, is thus linear. Like the curves representing the temperature variation of the other properties that we examined in the earlier chapters, and for the same reasons, the initial level of the resistivity curve is negative. From this initial level to the melting point the resistivity of an unmodified atom (one that has not undergone a structural rearrangement or other change that modifies the resistance relations) follows a single straight line, rather than a curve composed of two or more segments of different slopes, as in the specific heat and thermal expansion curves. This limitation to a single line is characteristic of the electron relations, and is due to the fact that the electron has only one rotational displacement unit, and therefore cannot shift to a multi-unit type of motion in the manner of the complex atomic structures. A somewhat similar change in the resistivity curve does occur, however, if the factors that determine the resistance are modified by some rearrangement of the kind mentioned earlier. As P. W. Bridgman commented in discussing some of his results, after a change of this nature has taken place, we are really dealing with a different substance. The curve for the modified atom is also a straight line, but it is not collinear with the curve of the unmodified atom. At the point of transition to the new form the resistivity of the individual atom abruptly changes to a different straight line relation. The resistivity of the aggregate follows a transition curve from one line to the other, as usual. At the lower end of the temperature range, the resistivity of the solid aggregate follows another transition curve of the same nature as those that we found in the curves representing the properties discussed earlier. The relation of the resistance to the temperature in this temperature range is currently regarded as exponential, but as we saw in other cases of the same kind, it is actually a probability curve that reflects the resistivity of the diminishing number of atoms that are still individually above the temperature at which the atomic resistivity reaches the zero level. The curve for the solid aggregate also diverges from the single atom curve at the upper end, due to the increasing proportion of liquid molecules in the solid aggregate. In this case, again, two values are required for a complete definition of the linear curve; either the coordinates of two points on the curve, or the slope of the curve and the location of one fixed point. A fixed point that is available from theoretical premises is the zero point temperature, the point at which the curve for the individual atom reaches the zero resistance level. The theoretical factors that determine this temperature are the same as those applying to the specific heat and thermal expansion curves, except that since the resistivity is an interaction between the atom and the electron it is effective only when the motions of both objects are directed outward. The theoretical zero point temperature normally applicable to resistivity is therefore twice that applicable to the properties previously considered. Up to this point the uncertainties in the experimental results have had no effect on the comparison of the theoretical conclusions with experience. It is conceded that the relation of resistivity to temperature is generally linear. with deviations from linearity in certain temperature ranges and under certain conditions. The only question at issue is whether these deviations are satisfactorily explained by the Reciprocal System of theory. When this question is considered in isolation, without taking into account the status of that system as a general physical theory, the answer is a matter of judgment, not a factual matter that can be resolved by comparison with observation. But we have now arrived at a place where the theory identifies some specific numerical values. Here agreement between theory and observation is a matter of objective fact, not one that calls for a judgment. But agreement within an acceptable margin can be expected only if (1) the experimental resistivities are reasonably accurate, (2) the zero point temperatures applicable to specific heat (which are being used as a base) were correctly determined, and (3) the theoretical calculation and the resistivity measurement refer to the same structure. Table 24 applies equation 7-1, with a doubled numerical constant, and the rotational factors from Table 22, to a determination of the temperatures of the zero levels of the resistance curves of the elements included in the study, and compares the results with the corresponding points on the empirical curves. The amount of uncertainty in the resistivity measurements is reflected in the fact that for 11 of these 40 elements there are two sets of experimental results that have been selected as the “best” values by different data compilers.20 In three other cases there are substantial differences in the experimental results at the higher temperatures, but the curves converge on the same value of the zero resistivity temperature. In a situation where uncertainties of this magnitude are prevalent, it can hardly be expected that there will be anywhere near a complete agreement between the theoretical and experimental values. Nevertheless, if we take the closer of the two “best” experimental results in the 11 two-value cases, the theoretical and experimental values agree within four degrees in 26 of the 40 elements, almost two-thirds of the total. The rare earth elements were not included in this study because the resistances of these elements, like so many of their other properties, follow a pattern differing in some respects from that of most other elements, including a transition to a new structural form at a relatively low temperature, accompanied by a major decrease in the slope of the resistivity curve. Because of this low temperature transition it is difficult to locate the zero point temperature from the empirical data, but in 9 of the 13 elements of this group for which sufficient data are available to enable an approximate identification of this temperature, it appears to be between 10 and 20 degrees K. The theoretical range for these elements, as indicated by the factors listed in Table 22, is from 12 to 20 degrees. Here again, then, the measured resistivities of two-thirds of the elements are at least approximately in agreement with the theoretical values. The existence of this amount of agreement, in spite of all of the influences tending to generate discrepancies, is about as good a confirmation of the validity of the theory, as a general proposition, as can be expected under the existing circumstances. Furthermore, it is not unlikely that there are alternate resistance patterns that result in explainable deviations from the calculated values, and some of the larger discrepancies may be thus accounted for when an investigation of broader scope in undertaken. Table 24: Temperature of Zero Resistance
For the second defining value of the resistivity curves we can use the temperature coefficient of resistivity, the slope of the curve, a magnitude that reflects the inherent resistivity of the conductor material. The temperature coefficient as given in the published physical tables is not the required value. This is merely a relative magnitude, the incremental change in resistivity relative to the resistivity at a reference temperature, usually 20 degrees C. What is needed for present purposes is the absolute coefficient, in microhm-centimeters per degree, or some similar unit. Some studies have been made in this area, and as might be expected, it has been found that the electric (one-dimensional) speed displacement is the principal determinant of the resistivity, in the sense that it is responsible for the greatest amount of variation. However, the effective quantity is not usually the normal electric displacement of the atoms of the element involved, as this value is generally modified by the way that the atom interacts with the electrons. The conclusions that have been reached as to the nature and magnitude of these modifications are still rather tentative, and there are major uncertainties in the empirical values against which the theoretical results would normally be checked to test their validity. The results of these studies have therefore been omitted from this volume, in conformity with the general policy of restricting the present publication to those results whose validity is firmly established. The experimental difficulties that introduce uncertainties into the correlations between the theoretical and experimental values of the resistivity do not play as large a role in the relative resistance under compression. The compression results therefore give us a more definite and unequivocal picture. Again, however, this initial exploration of the subject, as it appears in the context of the Reciprocal System of theory, will have to be confined to the “regular” pattern, the one followed by most of the metallic conductors. Because the movement of electrons (space) through matter is the inverse of the movement of matter through space, the inter-regional relations applicable to the effect of pressure on resistance are the inverse of those that apply to the change in volume under pressure. We found in Chapter 4 that the volume of a solid under compression conforms to the relation PV^{2}= k. By reason of the inverse nature of the electron movement, the corresponding equation for electrical resistance is
As in the compressibility equation, the symbol P in this expression refers to the total effective pressure. If we give the internal component of this total the designation P_{0}, as in the volume compressibility discussion, and limit the term P to the externally applied pressure, the equation becomes
The general situation with respect to the values of the internal pressure applicable to resistance is essentially the same as that encountered in the study of compressibility. Some elements maintain the same internal pressure throughout Bridgman’s entire pressure range, some undergo second order transitions to higher P_{0 }values, and others are subject to first order transitions, just as in the volume relations. However, the internal pressure applicable to resistance is not necessarily the same as that applicable to volume. In some substances, tungsten and platinum, for example, these internal pressures actually are identical at every point in the pressure range of Bridgman’s experiments. In another, and larger, class, the applicable values of P_{0 }are the same as in compression, but the transition from the lower to the higher pressure takes place at a different temperature. The values for nickel and iron illustrate this common pattern. The initial reduction in the volume of nickel took place on the basis of an internal pressure of 913 M kg/cm^{2}. Somewhere between an external pressure of 30 M kg/cm^{2}(Bridgman’s pressure limit on this element) and 100 M kg/cm^{2}(the initial point of later experiments at very high pressure) the internal pressure increased to 1370 M kg/cm^{2}(from azy factors 4-8-1 to 4-8-1¹/2). In the resistance measurements the same transition occurred, but it took place at a lower external pressure, between 10 and 20 M kg/cm^{2}. Iron has the same internal pressures in resistance as nickel, with the transition at a somewhat higher external pressure, between 40 and 50 kg/cm^{2}. But in compression this transition did not appear at all in Bridgman’s pressure range, and was evident only in the shock wave experiments carried to much higher pressures. Table 25 is a comparison of the internal pressures in resistance and compression for the elements included in the study. The symbol x following or preceding some of the values indicates that there is evidence of a transition to or from a different internal pressure, but the available data are not sufficient to define the alternate pressure level. Table 25: Internal Pressures in Resistance and Compression(Bridgman’s pressure range)
The amount of difference between the two columns of the table should not be surprising. The atomic rotations that determine the azy factors are the same in both cases, but the possible values of these factors have a substantial range of variation, and the influences that affect the values of these factors are not identical. In view of the participation of the electrons in the resistivity relations, and the large impurity effects, neither of which enters into the volume relations, some difference in the pressures at which the transitions take place can be considered normal. There is, at present, no explanation for those cases in which the internal pressures indicated by the results of the compression and resistance measurements are widely divergent, but differences in the specimens can certainly be suspected. Table 26 compares the relative resistances calculated from equation 10-2 with Bridgman's results on some typical elements. The data are presented in the same form as the compressibility tables in Chapter 4, to facilitate comparisons between the two sets of results. This includes showing the azy factors for each element rather than the internal pressures, but the corresponding pressures are available in Table 25. As in the compressibility tables, values above the transition pressures are calculated relative to an observed value as a reference level. The reference value utilized is indicated by the symbol R following the figure given in the “calculated” column. Table 26: Relative Resistance Under Compression
In those cases where the correct assignment of azy factors and internal pressures above the transition point is not definitely indicated by the corresponding compressibility values, the selections from among the possible values are necessarily based on the empirical measurements, and they are therefore subject to some degree of uncertainty. Agreement between the experimental and the semi-theoretical values in this resistance range therefore validates only the exponential relation in equation 10-2, and does not necessarily confirm the specific values that have been calculated. The theoretical results below the transition points, on the other hand, are quite firm, particularly where the indicated internal pressures are supported by the results of the compressibility measurements. On this basis, the extent of agreement between theory and observation in the values applicable to those elements that maintain the same internal pressures through the full 100.000 kg/cm^{2} pressure range of Bridgman’s measurements is an indication of the experimental accuracy. The accuracy thus indicated is consistent with the estimates made earlier on the basis of other criteria. Inasmuch as the difference in the form of the compressibility equation, pv^{2}= k (equation 4-4), and that of the pressure-resistance equation, p^{2}R = k (equation 10-1), is a requirement of the general reciprocal relation between space and time specified in the postulates of the Reciprocal System of theory, the joint verification of these two equations is a significant addition to the mass of evidence confirming the validity of this reciprocal relation, the cornerstone of the quantitative expression of the theory of the universe of motion. |