Einstein, black-body radiation

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

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. 

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. 

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. 

The Kinetics of Absorption and Emission of Radiation.—With Bohr s picture of the relation bet ween energy levels and discrete spectral lines in mind, Einstein gave a kinetic derivation of the law of black-body radiation, which is very instructive and which has had a great deal of influence. Einstein considered two particular stationary states of an atom, say the ith and jth (where for definiteness we assume that the ith lies above the jth), and the radiation which could be emitted and absorbed in going between those two states, radiation of frequency vi7, where... 

Einstein s derivation of the black-body radiation law is particularly important, for it gives us an insight into the kinetics of radiation processes. Being a kinetic method, it can be used even when we do not have thermal equilibrium. Thus if we know that radiation of a certain intensity is falling on atoms, we can find how many will be raised to the excited state per second, in terms of the coefficient Bn. But this means that we can find the absorptivity of matter made of these atoms, at this particular wave length. Conversely, from measurements of absorptivity, we can deduce experimental values of Bn. And from Eq. (2.8) we can find the rate of emission, or the emissive power, if we know the absorptiv-... 

Transition strengths can be given in terms of Einstein rate coefficients. For a pair of states j > and k > it is shown in elementary texts that these are related in a simple way. If one assumes that, for any pair of microstates i and j, the rate from i to j is equal to the rate from j to i one has the principle of detailed balance). Then, the relation between the coefficients is consistent with thermodynamics (Planck s black-body radiation law). 

Einstein has derived several useful relationships between the Ay-, Bjj-, and. fijj-coefficients by assuming thermodynamic equilibrium between the radiation and the atoms and comparing it with the equilibrium of a black-body radiator at the same temperature ... 

Although the corpuscular aspect of electromagnetic radiation, which was surmised by Newton in the seventeenth century, was used by Planck in 1900 to explain Wien s black body radiation law and by Einstein in 1905 to explain Lenard s photoelectric effect, its most spectacular demonstration was Compton s explanation in 1923 of the anomalous scattering of X-rays by bound electrons. 

The Bose-Einstein distribution (1.162) may be considered to recover the Planck law of black body radiation, i.e., the photon radiation modeling, by considering the following peculiarities ... 

The rate constant for fluorescence can be related to the dipole strength for absorption by a line of reasoning that Einstein developed in the period 1914-1917 [1]. Consider a set of atoms with ground-state wavefunction Pa and excited state wavefunction Pf,. Suppose that the atoms are enclosed in a box and are exposed only to the black-body radiation from the walls of the box. According to Eq. (4.8c), the rate at which the radiation causes upward transitions from Pa to Pb is... 

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. 

When considering the conditions of equilibrium between atomic particles and thermal radiation (black-body radiation), Einstein introduced another two elementary radiative processes whose rate depended on the radiation intensity. This was a... 

To compare this result with the Einstein coefficient B j, derived in Sect.2.3, we must take into account that the black-body radiation was isotropic whereas the EM wave (2.44) used in the derivation of (2.66) propagates into one direction. For randomly oriented atoms with the dipole... 

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]. 

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. 

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 quantum concept was introduced by Max Planck in 1900 to explain the distribution of energy radiated from a black body in thermal equilibrium with the surrounding. The idea that light travels as photons was originated by Einstein in 1905. 

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