Planck Theory of Black Body Radiation

Having developed the Planck distribution function for photons, we will now use it to obtain the Planck formula for the spectrum of black body radiation. 

Recall from Chapter 2 when solving the Schrodinger wave equation for electrons in a box, we looked for wave fimctions with the form ip x, y, z) = that satisfied 

Since for photons the energy E = cp, the allowed energy states are 

The lowest values of n fall within a sector of a sphere of radius n in Ux, Ux, Ux space. The density of states in n-space is given by 2 x 1/8 x Airn dn. The extra 2 accounts for the two possible polarization states for each wave. The density of states in energy 

Using Planck s distribution function, the radiation density in the cavity is given by 

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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. 

According to the Planck theory of black-body radiation. 

According to the Planck theory of black-body radiation, the radiant spectral emittance is given by the formula... 

Dunitz wrote of these equations Debye s paper, published only a few months after the discovery of X-ray diffraction by crystals, is remarkable for the physical intuition it showed at a time when almost nothing was known about the structure of solids at the atomic level. Ewald described how The temperature displacements of the atoms in a lattice are of the order of magnitude of the atomic distances The result is a factor of exponential form whose exponent contains besides the temperature the order of interference only [h,k,l, hence sin 9/M]. The importance of Debye s work, as stressed by Ewald,was in paving the way for the first immediate experimental proof of the existence of zero-point energy, and therewith of the quantum statistical foundation of Planck s theory of black-body radiation. ... 

In 1900 Max Planck devised a new theory of black-body radiation that agreed with experimental results. This theory is the first part of what we call the old quantum theory. The following statements are a simplified version of the assumptions that led to his result ... 

There was, however, an unresolved discrepancy in the theory of black-body radiation. In fact the solution of this apparently minor problem by Planck (1901) was to be... 

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 ... 

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. 

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. 

In the historical development of science, experimental progress in the accuracy of measurements have often brought about a refinement of theoretical models or even the introduction of new concepts [14.1]. Examples are A. Einstein s theory of special relativity based on the interferometric experiments of Michel son and Morley [14.2] M. Planck s introduction of quantum physics for the correct explanation of the measured spectral distribution of black-body radiation, the introduction of the concept of electron spin after the spectroscopic discovery of the fine structure in atomic spectra [14.3] or the test of quantum-electrodynamics by precision measurements of the Lamb shift [14.4]. 

For Quantum Mechanics, Planck s interest in the Second Law of Thermodynamics made him attempt to fit a formula to the spectrum of black- body radiation. The only formula he could find was one that would have resulted from an assumption that radiation is emitted in quanta of action. He treated his formula as an ad hoc temporary measure. Later on Bohr introduced yet another ad hoc temporary treatment in his atomic theory. Einstein s treatment of a particle-like photon in 1905 was another such hunch, sticking his neck.. It took till 1925 to find a new paradigm. The mutations here were wild guesses, completely unjustified by the existing theories. 

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. 

The Planck theory of blackbody radiation provides a first approximation to the spectral distribution, or intensity as a function of wavelength, for the sun. The black-body theory is based upon a "perfect" radiator with a uniform composition, and states that the spectral distribution of energy is a strong function of wavelength and is pro portional to the temperature (in units of absolute temperature, or Kelvin), and several fundamental constants. Spectral radiant exitance (radiant flux per unit area) is de fined as ... 

Bohr was following Planck s lead in departing from classical electromagnetic theory. In studying black-body radiation, which occurs at very short frequencies, Planck had found it necessary to introduce a constant, h, ako called the elementary quantum of action, to explain its discontinuous nature. Such radiation could be emitted or absorbed only in packets, or quanta, described by the formula hv, where V is the frequency of the radiation and h is Planck s constant. Bohr was su esting that the atom could Hkewise not be described adequately by the laws of classical mechanics but that it required a quantum description. 

Planck wanted to understand black body radiation. The black body may be modeled by a box, with a small hole (shown in Fig. 1.1). We heat the box up, wait for the system to reach a stationary state (at a fixed temperature), and see what kind of electromagnetic radiation (intensity as a function of frequency) comes out of the hole. In 1900, Rayleigh and Jeans tried to apply classical mechanics to this problem, and they calculated correctly that the hlack body would emit the electromagnetic radiation with a distribution of frequencies. However, the larger the frequency, the larger its intensity - an absurd conclusion, what is known as an ultraviolet catastrophe. Experiments contradicted theory (as shown in Fig. 1.1). 

With the solution of the hydrogen atom, the list of analytically solvable systems to be considered here is complete. Planck s quantum theory of light described black-body radiation, and now the simplicity of the spectrum of the hydrogen (and hydrogen-like ions) is adequately explained by quantum mechanics. We will find in the next chapter that although an exact analytic understanding of the behavior... 

The old quantum theory includes Planck s black-body radiation theory, Einstein s theory of the photoelectric effect, and Bohr s theory of the hydrogen atom. 

The old quantum theory consists of theories with arbitrary assumptions of quantization that were devised to explain phenomena that classical physics could not explain. The old quantum theory includes the black-body radiation theory of Planck, the photoelectric effect theory of Einstein, and the hydrogen atom theory of Bohr. 

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