Planck, blackbody

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

Figure 4.24. The Planck distribution law spectral radiance of blackbody radiation as a function of temperature and wavelength. (After Touloukian and DeWitt (1972). Plenum Press.)...

Figure 4.26. Special radiance versus wavelength for CaAljSi Og glass shocked to 48 GPa and 84 GPa. Best-fitting Planck blackbody curves are shown in relation to the radiance data. (After Boslough and Ahrens (1984). the American Geophysical Union.)...

As seen in Eq. (17-1), the total radiation from a blackbody is dependent on the fourth power of ifs absolute temperature. The frequency of the maximum intensity of this radiation is also related to temperature through Wien s displacement law (derived from Planck s law) ... 

Total heat transfer consists of radiation at different frequencies. The distribution of radiation energy in a spectrum and its dependency on temperature is determined from Planck s law of radiation. M ,and are the spectral radiation intensities for a blackbody ... 

A hundred years ago it was generally supposed that all the properties of light could be explained in terms of its wave nature. A series of investigations carried out between 1900 and 1910 by Max Planck (1858-1947) (blackbody radiation) and Albert Einstein (1879-1955) (photoelectric effect) discredited that notion. Today we consider light to be generated as a stream of particles called photons, whose energy E is given by the equation... 

Although historians of science have studied the breakthrough that led to quantum mechanics, nobody can be exactly sure what was in Planck s orderly, disciplined mind when he devised the equation that revolutionized physics. He tackled the blackbody problem in several ways, but nothing worked. Finally, he tried an idea that was contradictory to all established concepts at the time What if energy was not continuous What if blackbodies absorbed and emitted it in little chunks He wrote down his equation ... 

When Planck used this relationship to calculate the spectrum of blackbody radiation, he came up with a result that agreed perfectly with experiment. More importantly, he had discovered quantum mechanics. Energy emitted by a blackbody is not continuous. Instead, it comes in tiny, irreducible packets or quanta (a word coined by Planck himself) that are proportional to the frequency of the oscillator that generated the radiation. 

Bohr knew of the work of Planck and Einstein. What if the energies of electrons in an atom were not continuous What if they could only take on certain values What if atoms were quantized, just as blackbody oscillators and light that struck a metal plate were. The challenge was how to apply quantum ideas to the atom. 

The constant h and the hypothesis that energy is quantized in integral multiples of hv had previously been introduced by M. Planck (1900) in his study of blackbody radiation. In terms of the angular frequency a> deflned in equation (1.2), the energy E of a photon is... 

Blackbody radiation law proposed by J. Stefan form the basis for the radiation thermometers, with Planck s law. The radiation law is strongly nonlinear in temperature, since it depends on T4... 

PLA/MMT nanocomposites, 20 311 Planar cameras, 21 277 Planar cavity surface-emitting laser (PCSEL) diodes, 22 178 Planar diodes, 19 163 Planarization, dielectrics for, 22 192 Planck s blackbody radiation law, 14 662, 663 24 452... 

This is Planck s famous radiation law, which predicts a spectral energy density, p , of the thermal radiation that is fully consistent with the experiments. Figure 2.1 shows the spectral distribution of the energy density p for two different temperatures. As deduced from Equation (2.2), the thermal radiation (also called blackbody radiation) from different bodies at a given temperature shows the same spectral shape. In expression (2.2), represents the energy per unit time per unit area per frequency interval emitted from a blackbody at temperature T. Upon integration over all frequencies, the total energy flux (in units of W m ) - that is, Atot = /o°° Pv Av - yields... 

Let us now assume that our two-level system is placed in a blackbody cavity whose walls are kept at a constant temperature T. Once thermal equilibrium has been reached, we can consider that our system is immersed in a thermal cavity where an electromagnetic energy density has been estabhshed. The spectral distribution Pa of this energy density is given by Planck s formula ... 

Radiation thermometry (visual, photoelectric, or photodiode) 500-50,000 Spectral intensity I at wavelength A Planck s radiation law, related to Boltzmann factor for radiation quanta Needs blackbody conditions or well-defined emittance... 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

Spectroscopy was to prove indispensable in unlocking the structure of atoms, particulary their electronic stmcture— but those developments would depend on other, later researchers. Max Planck s analysis of blackbody radiation and Bohr s theory of the hydrogen spectrum are just two examples. 

Spectral distribution of blackbody radiation. The family of curves is called the Planck distribution after Max Planck, who derived the law governing blackbody radiation. Note that both axes are logarithmic. 

The exitance (power per unit area per unit wavelength) from a blackbody (Box 20-1) is given by the Planck distribution ... 

Planck distribution Equation giving the spectral distribution of blackbody radiation ... 

Wien displacement law Approximate formula for the wavelength, X, of maximum blackbody emission Xmax - T hc/5k — 2.878 X 10" 3 m K, where T is temperature in kelvins, h is Planck s constant, c is the speed of light, and k is Boltzmann s constant. Valid for T > 100 K. working electrode One at which the reaction of interest occurs. 

The spectral emissivity, f.>. is defined as the ratio of the emission at wavelength /. of the object to that of an ideal blackbody at the same temperature and wavelength. When ty is unity, the foregoing equation becomes the Planck radiation equation for a black body. 

Blackbody radiation sources are accurate radiant energy standards of known flux and spectral distribulion. They are used for calibrating other infrared sources, detectors, and optical systems. The radiating properties of a blackbody source are described by Planck s law. Energy distribution... 

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