The Planck Radiation Law

let us find the number of overtone vibrations in the range dv. We follow Chap. XIV in detail and for that reason can omit a great deal of calculation. In Sec. 2 of that chapter, we found the number of overtones in the range dv, in a problem of elastic vibration, in which the velocity of longitudinal waves was Vi, that of transverse waves vt. From Eqs. (2.20) and (2.21) of that chapter, the number of overtones of longitudinal vibration in the range dv, in a container of volume F, was 

There was an upper, limiting frequency for the elastic vibrations, but as we have just stated there is not for optical vibrations. In our optical case, we can take over Eq. (2.2) without change. Light waves are only 

In Chap. IX, Sec. 5, we found the average energy of a linear oscillator of frequency v in the quantum theory, and found that it was 

If we assume the expression (2.5), without the zero-point energy, for the average energy of a standing wave, and take Eq. (2.3) for the number of standing waves in volume V and frequency range dv9 we can at once derive uV) and we have 

As Lord Rayleigh pointed out, the classical expression for radiation 

Given that c = vX, show that Eqs. (1-3) and 1-4) are equivalent. Exercise 1-2 

The first computer project is devoted to solving Eq. (1-5) for a iteratively. When a has been determined, the remaining constants can be substituted into 

Procedure. One approach to the problem is to select a value for a that is obviously too small and to increment it iteratively until the equation is satisfied. This is the method of program WIEN, where the initial value of x is taken as 1 (clearly, -E i 1 as you can show with a hand calculator). 

In Program QWIEN (written in QBASIC, Appendix A), x is initialized at 1 and incremented by 0.1 in line 3, which is given the statement number 10 for future reference. Be careful to differentiate between a statement number like 10 x = x -E. 1 and the product 10 times r which is 10%. A number a is calculated for x= 1.1 that is obviously too small so (a — 1) is less than 0 and the IF statement in line 5 sends conhol back to the statement numbered 10, which increments a by 0.1 again. This continues until (a — 1) 0, whereupon control exits from the loop and prints the result for a and a . 

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This is known as the Planck radiation law. Figure A2.2.3 shows this spectral density fiinction. The surface temperature of a hot body such as a star can be estimated by approximating it by a black body and measuring the frequency at which the maximum emission of radiant energy occurs. It can be shown that the maximum of the Planck spectral density occurs at 2.82. So a measurement of yields an estimate of the... 

The fixed points in the lTS-90 are given in Tabie 11.39. Platinum resistance thermometers are recommended for use between 14 K and 1235 K (the freezing point of silver), calibrated against the fixed points. Below 14 K either the vapor pressure of helium or a constant-volume gas thermometer is to be used. Above 1235 K radiometry is to be used in conjunction with the Planck radiation law,... 

The integral of the temperature gradient of the spectral power density from wavelength Xl to X2, is readily calculable using the Planck radiation law (5). Constant emissivity is assumed for equation 3. 

The ITS 90 was adopted by the Comite International des Poids et Mesures in September 1989 [14-16], The ITS 90 extends from 0.65 K to the highest temperatures, practicably measurable in terms of the Planck radiation law using monochromatic radiation. The defining fixed points of the ITS 90 are mostly phase transition temperatures of pure substances given in Table 8.2. 

Above the freezing point of silver, Tgq is defined in terms of a defining fixed point and the Planck radiation law, and optical pyrometers are frequently used as temperature probes. The Comite Consultatif de Thermometrie gives a thorough discussion of the different techniques for approximation of the international temperature scale of 1990 [2, 4],... 

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... 

The ITS-90 scale extends from 0.65 K to the highest temperature measurable with the Planck radiation law (—6000 K). Several defining ranges and subranges are used, and some of these overlap. Below —25 K, the measurements are based on vapor pressure or gas thermometry. Between 13.8 K and 1235 K, Tg is determined with a platinum resistance thermometer, and this is by far the most important standard thermometer used in physical chemistry. Above 1235 K, an optical pyrometer is the standard measrrremerrt instmment. The procedtrres used for different ranges are sttmmarized below. 

Range Above 1234.93 K. Above the freezing point of silver, an optical pyrometer is nsed to measnre the emitted radiant flux (radiant excitance per unit wavelength interval) of a blackbody at wavelength A. The defining equation is the Planck radiation law in the form... 

Optical Pyrometers. The optical pyrometer can be used for the determination of temperatures above 900 K, where blackbody radiation in the visible part of the spectrum is of sufficient intensity to be measured accurately. The blackbody emitted radiation intensity at a given wavelength A in equilibrium with matter at temperature Tis given by the Planck radiation law,... 

In 1900, Max Planck (1858-1947) discovered a formula (now often called the Planck radiation law) that modeled curves like those shown in Figure 24-21 nearly perfectly. He followed this discoveiy by developing a theory that made two bold assumptions regarding the oscillating atoms or molecules in blackbody radiators. He assumed (1) that these species could have only discrete energies and (2) that they could absorb or emit energy in discrete units, or quanta. These assumptions, which are implicit in Equation 24-3, laid the foundation for the development of quantum theory and eventually won him the Nobel Prize in Physics in 1918. 

In principle, any device that has one or more physical properties uniquely related to temperature in a reproducible way can be used as a thermometer. Such a device is usually classified as either a primary or secondary thermometer. If the relation between the temperature and the measured physical quantity is described by an exact physical law, the thermometer is referred to as a primary thermometer otherwise, it is called a secondary thermometer. Examples of primary thermometers include special low-pressure gas thermometers that behave according to the ideal gas law and some radiation-sensitive thermometers that are based upon the Planck radiation law. Resistance thermometers, thermocouples, and liquid-in-glass thermometers all belong to the category of secondary thermometers. Ideally, a primary thermometer is capable of measuring the thermodynamic temperature directly, whereas a secondary thermometer requires a calibration prior to use. Furthermore, even with an exact calibration at fixed points, temperatures measured by a secondary thermometer still do not quite match the thermodynamic temperature these readings are calculated from interpolation formulae, so there are differences between these readings and the true thermodynamic temperatures. Of course, the better the thermometer and its calibration, the smaller the deviation would be. 

Describe the ultraviolet catastrophe and its empirical resolution by the Planck radiation law. 

Pyrometry When the temperatures of interest are in a suitable range, pyrometry is the best technique for measuring the temperature of the Knudsen cell vapor sources in KEMS instruments. Some of the major advantages are that it is a noncontact technique, with the pyrometer placed outside the furnace and vacuum chamber. Also, one pyrometer can be used to measure temperature at multiple locations, which improves the consistency of calibration. The key advantage is that pyrometry, as stated in the Temperature Measurement section, is based on the Planck radiation law, which in ratio form defines ITS-90 at all temperatures above the Ag fixed point (1234.93 K).Thus,pyrometry is the standard method for realizing thermodynamic temperature through the use of Equation 48.10 ... 

Most practicing physicists have learned what little they know of the history of this period by reading textbooks written after the quantum revolution. Often texts and teaehers treat the Planck radiation law, the Einstein photoelectric equatiorr, the Bohr atom and the Compton effect in one sequence assuming that this provides an adequate background for understanding E = hv and p = hv/c [de Broglie s equation]. This can leave a student with less than total respect for the physicists who took so long to see the obvious necessity for this form of quantization (p. 95). 

A useful approximation to the Planck radiation law can be obtained in the low-frequency limit where hv is small compared with kt (hv < kT). This condition is generally met over the full range of planetary temperatures and at radio wavelengths. It leads to the... 

For problems that deal with the energy budget of the planets, it is necessary to know the total amount of power radiated over all frequencies. Once again the concept of the blackbody is useful. The Planck radiation law can be integrated over all frequencies and solid angles to obtain the relationship known as the Stefan-Boltzmann law, given by R = aT, (8)... 

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