Planck black-body radiation

This is the key relationship and is the Planck black-body radiation equation. Integration of this equation over all wavelengths yields the total radiant exitance of a blackbody. 

In this chapter we consider classical and quantum mechanical descriptions of electromagnetic radiation. We develop expressions for the energy density and irradiance of light passing through a homogeneous medium, and we discuss the Planck black-body radiation law and linear and circular polarization. Readers anxious to get on to the interactions of light with matter can skip ahead to Chap. 4 and return to the present chapter as the need arises. 

In his papers of 1914—1917, Einstein s actual line of reasoning was the reverse of the argument presented here. Einstein began with the assumption that an excited system can decay spontaneously as well as by stimulated emission. He also assumed that the relative populations of the ground and excited states follow the Boltzmann distribution (Eq. 5.8). With these assumptions, he obtained a simple derivation of the Planck black-body radiation law (Eqs. 3.41 and 5.10) and went on to show that absorption of light transfers momentum to the absorber. 

Much of the radiation with which we are familiar in everyday life is of thermal origin, arising by definition from matter in thermal equilibrium. In an ideal atomic gas in thermal equilibrium, for example, the upward versus downward transitions of bound electrons between energy levels in individual atoms are in close balance due to the exchange of energy between particles via collisions and the absorption and emission of radiation. The velocities of particles in an ideal thermal gas follow the well-known Maxwellian distribution, and the collective continuous spectrum of the radiating particles is described by the familiar Planck black-body radiation curve with its characteristic temperature-dependent profile and maximum. 

Fig. 2 illustrates two characteristics of Planck black-body radiation. It can be seen that as a source is heated, the is shifted to higher frequencies, and the intensity at each frequency is increased in a nonlinear manner. Intuitively, one would assume that as the source temperature is increased, the sensitivity would also be increased. Two factors counteract the benefits of increased radiation from a hotter source. As the source temperature is increased, the intensity of multiple orders of radiation emanating from the dispersive element is increased disproportionately. 

Fig. 2. The Effect of Temperature Elevation on the Graph of Planck Black-body Radiation.

The explanation of the hydrogen atom spectmm and the photoelectric effect, together with other anomalous observations such as the behaviour of the molar heat capacity Q of a solid at temperatures close to 0 K and the frequency distribution of black body radiation, originated with Planck. In 1900 he proposed that the microscopic oscillators, of which a black body is made up, have an oscillation frequency v related to the energy E of the emitted radiation by... 

Historical Background.—Relativistic quantum mechanics had its beginning in 1900 with Planck s formulation of the law of black body radiation. Perhaps its inception should be attributed more accurately to Einstein (1905) who ascribed to electromagnetic radiation a corpuscular character the photons. He endowed the photons with an energy and momentum hv and hv/c, respectively, if the frequency of the radiation is v. These assignments of energy and momentum for these zero rest mass particles were consistent with the postulates of relativity. It is to be noted that zero rest mass particles can only be understood within the framework of relativistic dynamics. 

Studies of black-body radiation led to Planck s hypothesis of the quantization of electromagnetic radiation. The photoelectric effect provides evidence of the particulate nature of electromagnetic radiation. 

The Stefan-Boltzmann Law and Wien s Law for black body radiation have been unified into Planck s Law for black body radiation, from which Planck s constant was first introduced. Planck s analysis of the spectral distribution of black body radiation led him to an understanding of the quantisation of energy and radiation and the role of the photon in the theory of radiation. The precise law relates the intensity of the radiation at all wavelengths with the temperature and has the form ... 

The radiation density p(v) is given by Planck s black body radiation law Bnhv3... 

The early years of the twentieth century saw giant advances in man s understanding of nature which must be mentioned in any synopsis of the scientific history of this era. Thus, in 1901, M. Planck (NLP 1918 ) published his first paper on the black-body radiation law which ushered in the era of quantum mechanics. In 1905, A. Einstein (NLP 1918 ) published his Anna Mirabilis Papers on the photo effect, on Brownian motion, and on the theory of special relativity and the equivalence of matter and energy. 

From Figure 7.10 it is seen that spontaneous emission according to the Planck theory of Black body radiation as well as Einstein s work starts to dominate above 10 Hz at 300K, this corresponds to the infrared range of the electromagnetic spectram. Note, that if the temperature increases the zero crossing point moves into the visual and UV range. 

On the other hand, from Planck s derivation of energy density for a black body radiation at temperature T (Sec. 1.4), we know that... 

LI The Planck Distribution of Black-body Radiation. The Planck relationship between the energy of the photon and the frequency of monochromatic light leads to the equation of distribution of the intensity of light as a function of frequency (or wavelength)... 

Figure 2.10 Examples of the intensity versus wavelength or frequency distribution of black-body radiation according to the Planck equation. Each distribution corresponds to a temperature T of the radiating body...

The most familiar way of representing black body radiation is to use the Planck radiation law for the energy density, or the square of the electric field, p(v). Explicitly4... 

In 1900 Max Planck proposed a solution to the problem of black-body radiation described above. He suggested that when electromagnetic radiation interacts with matter, energy can only be absorbed or emitted in certain discrete amounts, called quanta. Planck s theory will not be described here, as it is highly technical. In any case, Planck s proposal was timid compared with the theory that followed. He supposed that quanta were only important in absorption and emission of radiation, but that otherwise the wave theory did not need to be modified. It was Einstein who took a more radical step in 1905 (the year in which he published his first paper on the theory of relativity and on several other unrelated topics). Einstein s analysis of the photoelectric effect is crucial, and has led to a complete change in the way we think of light and other radiation. 

In 1920 de Broglie returned to his studies later he stated that his attraction to theoretical physics was. . . the mystery in which the structure of matter and of radiation was becoming more and more enveloped as the strange concept of the quantum, introduced by Max Planck in 1900 in his researches into black-body radiation, daily penetrated further into the whole of physics (quoted by Heathcote, pp. 289-290). 

W. Nernst and F. A. Lindemann, Berl. Ber., 1911, p. 494, discuss the deviations from Einstein s result. P. Ehrenfest, Welche Rolle spielt die Lichtquantenhypothese in der Theorie der W rme-strahlung Ann. d. Phys., 36 (1911), 91, studies the possibility of a generalization of Planck s assumption in the field of black-body radiation. 

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