by the compression, in such a manner that the product λθ of wave-length and the absolute temperature should remain constant. According to this relation, which is known as Wien’s Displacement Law, the frequency corresponding to the maximum ordinate of the energy curve of the normal spectrum of full radiation should vary directly (or the wave-length inversely) as the absolute temperature, a result previously obtained by H. F. Weber (1888). Paschen, and Lummer and Pringsheim verified this relation by observing with a bolometer the intensity at different points in the spectrum produced by a fluorite prism. The intensities were corrected and reduced to a wave-length scale with the aid of Paschen’s results on the dispersion formula of fluorite (Wied. Ann., 1894, 53, p. 301). The curves in fig. 7 illustrate results obtained by Lummer and Pringsheim (Ber. deut. phys. Ges., 1899, 1, p. 34) at three different temperatures, namely 1377°, 1087° and 836° absolute, plotted on a wave-length base with a scale of microns (μ) or millionths of a metre. The wave-lengths Oa, Ob, Oc, corresponding to the maximum ordinates of each curve, vary inversely as the absolute temperatures given. The constant value of the product λθ at the maximum point is found to be 2920. Thus for a temperature of 1000° Abs. the maximum is at wave-length 2.92 μ; at 2000° the maximum is at 1.46 μ.
44. Form of the Curve representing the Distribution of Energy in the Spectrum.—Assuming Wien’s displacement law, it follows that the form of the curve representing the distribution of energy in the spectrum of full radiation should be the same for different temperatures with the maximum displaced in proportion to the absolute temperature, and with the total area increased in proportion to the fourth power of the absolute temperature. Observations taken with a bolometer along the length of a normal or wave-length spectrum, would give the form of the curve plotted on a wave-length base. The height of the ordinate at each point would represent the energy included between given limits of wave-length, depending on the width of the bolometer strip and the slit. Supposing that the bolometer strip had a width corresponding to .01 μ, and were placed at 1.0 μ in the spectrum of radiation at 2000° Abs., it would receive the energy corresponding to wave-lengths between 1.00 and 1.01 μ. At a temperature of 1000° Abs. the corresponding part of the energy, by Wien’s displacement law, would lie between the limits 2.00 and 2.02 μ, and the total energy between these limits would be 16 times smaller. But the bolometer strip placed at 2.0 μ would now receive only half of the energy, or the energy in a band .01 μ wide, and the deflection would be 32 times less. Corresponding ordinates of the curves at different temperatures will therefore vary as the fifth power of the temperature, when the curves are plotted on a wave-length base. The maximum ordinates in the curves already given are found to vary as the fifth powers of the corresponding temperatures. The equation representing the distribution of energy on a wave-length base must be of the form
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| Fig. 7.—Distribution of energy in the spectrum of a black body. |
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| Fig. 8.—Distribution of energy in the spectrum of full radiation at 2000° Abs. according to formulae of Planck & Wien. |
where F (λθ) represents some function of the product of the wave-length and temperature, which remains constant for corresponding wave-lengths when θ is changed. If the curves were plotted on a frequency base, owing to the change of scale, the maximum ordinates would vary as the cube of the temperature instead of the fifth power, but the form of the function F would remain unaltered. Reasoning on the analogy of the distribution of velocities among the particles of a gas on the kinetic theory, which is a very similar problem, Wien was led to assume that the function F should be of the form e−c/λθ, where e is the base of Napierian logarithms, and c is a constant having the value 14,600 if the wave-length is measured in microns μ. This expression was found by Paschen to give a very good approximation to the form of the curve obtained experimentally for those portions of the visible and infra-red spectrum where observations could be most accurately made. The formula was tested in two ways: (1) by plotting the curves of distribution of energy in the spectrum for constant temperatures as illustrated in fig. 7; (2) by plotting the energy corresponding to a given wave-length as a function of the temperature. Both methods gave very good agreement with Wien’s formula for values of the product λθ not much exceeding 3000. A method of isolating rays of great wave-length by successive reflection was devised by H. Rubens and E. F. Nichols (Wied. Ann., 1897, 60, p. 418). They found that quartz and fluorite possessed the property of selective reflection for rays of wave-length 8.8μ and 24μ to 32μ respectively, so that after four to six reflections these rays could be isolated from a source at any temperature in a state of considerable purity. The residual impurity at any stage could be estimated by interposing a thin plate of quartz or fluorite which completely reflected or absorbed the residual rays, but allowed the impurity to pass. H. Beckmann, under the direction of Rubens, investigated the variation with temperature of the residual rays reflected from fluorite employing sources from −80° to 600° C., and found the results could not be represented by Wien’s formula unless the constant c were taken as 26,000 in place of 14,600. In their first series of observations extending to 6μ O. R. Lummer and E. Pringsheim (Deut. phys. Ges., 1899, 1, p. 34) found systematic deviations indicating an increase in the value of the constant c for long waves and high temperatures. In a theoretical discussion of the subject, Lord Rayleigh (Phil. Mag., 1900, 49, p. 539) pointed out that Wien’s law would lead to a limiting value Cλ−5, of the radiation corresponding to any particular wave-length when the temperature increased to infinity, whereas according to his view the radiation of great wave-length should ultimately increase in direct proportion to the temperature. Lummer and Pringsheim (Deut. phys. Ges., 1900, 2, p. 163) extended the range of their observations to 18 μ by employing a prism of sylvine in place of fluorite. They found deviations from Wien’s formula increasing to nearly 50% at 18μ, where, however, the observations were very difficult on account of the smallness of the energy to be measured. Rubens and F. Kurlbaum (Ann. Phys., 1901, 4, p. 649) extended the residual reflection method to a temperature range from −190° to 1500° C., and employed the rays reflected from quartz 8.8μ, and rocksalt 51μ, in addition to those from fluorite. It appeared from these researches that the rays of great wave-length from a source at a high temperature tended to vary in the limit directly as the absolute temperature of the source, as suggested by Lord Rayleigh, and could not be represented by Wien’s formula with any value of the constant c. The simplest type of formula satisfying the required conditions is that proposed by Max Planck (Ann. Phys., 1901, 4, p. 553) namely,
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| Fig. 9.—Variation of energy of radiation corresponding to wave-length 30μ, with temperature of source. |
which agrees with Wien’s formula when θ is small, where Wien’s formula is known to be satisfactory, but approaches the limiting
