A LOGARITHMIC SCALE FOR THE NATURAL SPACING OF SPECTRAL DETAIL$^{\ast}$
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Date
1952
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Ohio State University
Abstract
Because of the indecision about selecting a single standard from among the current spectral scales, a critical study was made to determine the basis that would give natural spacing to the spectral detail and thereby gain such a distinct advantage as to promise a permanent solution of this problem. The conclusions are drastically different from those toward which popular conceptions had been leading and in general confirm and extend some ideas reported by $Shurcliff,^{1}$ the potentialities of which apparently were not widely recognized. The quality necessary for a natural distribution of the spectral detail, which has been lacking and is sought here, may be defined as constant graphical resolution. The contest between the wavelength and frequency scales seems virtually undecidable, because each bears a reciprocal relation to the other and therefore the dispersions are not fundamentally different except in direction. Each gives unnatural spacing that can only be increasingly unsatisfactory as progress extends the spectral scopes and connects the different fields; moreover, each has the intrinsic defect of continuously distorting the detail towards a condition of infinite crowding in one direction and of infinite dispersion in the other. In this case, as in those of hydrogen ions and radiation intensities, a distinction must be made between the dual purposes and vectoral relations of the primary units, which apply to numerical evaluations with the spacing relations of a highly dispersing rate, and of secondary units, which apply to the spacing relations at a deaccelerated rate. The deacceleration must be logarithmically based in order to change between the infinite limits at a constant rate, to be graphed in proportion to the chief effects, and to accord with common geometric methods and the equivalent simpler forms of mathematics. Another consideration is that halving and doubling processes are the most frequent in this case, as in other natural phenomena, besides being the most convenient mechanically; therefore a base of 2 for the logarithmic deacceleration is desirable. Since our common numerical scale is based on a cycle of 10, instead of on one of the simple powers of 2, as it presumably should have been, the best acceleration accord between the natural needs and the numerical system is obtained by introducing the factor $\log_{10} 10/log_{10}^{2}$, so that doubling is recognizable by occurring at definite repeating points on the 10 cycle. In consideration of the need for wieldiness of numbers, it is desirable to select one constant that spreads the doubling of the primary units to cycles of 100 of the logarithmic units, and another constant that places the scale on a basis whereby most of the natural waves can be expressed in positive figures and to an exactness in the thousands place by four figures before the decimal. By introducing these concepts into the equation of the straight line, using $\log_{2} k_{\lambda}$ as an entity, the fundamental equation for the new units, thought of as universal units and designated Usu, becomes $$Usu = k_{1} \log_{2} k_{2} \lambda + k_{3}$$ Because lateral shifting of the scale to a selected orientation point would offer insufficient advantage to justify the additional complexity when the numerical system has an unnatural base of 10, this constant may be dropped; the practical equation then becomes $$Usu = 332.19280 \log_{10} 10^{9} \mu, or\ 332.193 (9 + \log ^10\mu)$$ The validity of the equation has been tested by using spectral data in the range from 0.1 to $38 \mu$ on the same grid, and no evidence of unnatural spacing trends was found. Also, the equation gives convenient values to the units between such extremes as cosmic rays $(0.0005 {\AA}, 564 Usu)$ and radio waves (1500 kc, 5747 Usu). Since the quality of naturalness inherently holds advantages beyond the ability of being fully estimated or appreciated, a positive decision about the truth of the contention that the scale is natural should determine its acceptance, at least in regard to the fundamental principles. Some of the advantages (certain ones of which are also considered to be criteria of naturalness successfully met) are: The scope is infinite, and the applicability covers all known radiations; constant geometric and graphical resolution is attained; relations to instrumental factors become more natural and often simpler; the bands tend to be in series and systems which have recognizable features and from which mathematical relations are evident; and (as with a slide rule) the calculation becomes largely a simple mechanical manipulation of distances, except for additions and subtractions, which must depend on a supplementary nonlinear scale.
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Author Institution: Bureau of Agricultural and Industrial Chemistry, Agricultural Research Center