Planck frequency distribution

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... 

The radiant flux can be determined as a function of frequency from Planck s distribution law for emission ... 

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...

To find the maximum frequency at a given temperature, we will need to take a derivative of Planck s law with respect to frequency and set it equal to 0 dp/dv = 0. At x values, - 1) e, so at high frequencies, - 1) Ra g-hv/kT Rearranging Planck s distribution... 

The frequency distribution p = dp/dv of a black-body radiator is given by Planck s law (Equation 2.6) ... 

Here, n is an integer, is the frequency of the vibration, and h is known as Planck s constant This quantization of energy, as it is known, was first postulated by Max Planck in 1900 as a key part of his theory to explain the frequency distribution of radiation emitted by a black body. It is found that energy is quantized whenever a particle is confined to a small space because of the need to match the wavefunction of the particle to the space available. This applies just as much to electrons travelling around an atomic nucleus as it does to atoms vibrating in a solid. 

At the end of the nineteenth century, classical thermodynamics faced the challenge of determining the exact functional form of m(v, T) or /(v, T). All the deductions based on the principles that were known at that time did not agree with experimental measurements of m(v). This fundamental problem remained unsolved until Max Planck (1858-1947) introduced the revolutionary quantum hypothesis. With the quantum hypothesis, according to which matter absorbed and emitted radiation in discrete bundles or quanta, Planck was able to derive the following expression which agreed well with the observed frequency distribution m(v) ... 

Maxwell s equations describe the propagation of electromagnetic radiation as waves within the framework of classical physics however, they do not describe emission phenomena. The search for the law that defines the energy distribution of radiation from a small hole in a large isothermal cavity gave rise to quantum theory. The function that describes the frequency distribution of blackbody radiation was the first result of that new theory (Planck, 1900,1901). 

In the previous section we discussed light and matter at equilibrium in a two-level quantum system. For the remainder of this section we will be interested in light and matter which are not at equilibrium. In particular, laser light is completely different from the thennal radiation described at the end of the previous section. In the first place, only one, or a small number of states of the field are occupied, in contrast with the Planck distribution of occupation numbers in thennal radiation. Second, the field state can have a precise phase-, in thennal radiation this phase is assumed to be random. If multiple field states are occupied in a laser they can have a precise phase relationship, something which is achieved in lasers by a teclmique called mode-locking Multiple frequencies with a precise phase relation give rise to laser pulses in time. Nanosecond experiments... 

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 ... 

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... 

A space entirely surrounded by material walls of sufficient thickness to be impenetrable to radiation is traversed in all directions by waves of every possible frequency. Unit volume contains a definite amount of radiant energy —the radiation density—determined only by the temperature of the walls, and distributed among the different frequencies in accordance with Planck s law. 

Nn = numbers of emitters per unit volume of the light source P(v) = probability density distribution function of emission per unit time and per unit frequency h = Planck s constant , = spherical coordinates... 

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)... 

Planck s constant was discovered as part of the solution to a nineteenth century conundrum in physics, known as the black-body problem. The challenge was to model the wavelength distribution of radiation emitted through the aperture in a closed cavity at various temperatures6. The standard equations of statistical thermodynamics failed to produce the observed spectrum, unless it was assumed that the energy of radiation with frequency v was an integral multiple of an elementary energy quantum hv. 

If theory will not predict observed fact, then another approach is needed. Max Planck (1858-1947), a German physicist, set about to provide a mathematical description of the actual distribution of frequencies in the observed... 

Planck constant — To describe the spectral distribution of energy of black body radiation -> Planck made the ad hoc assumption that the radiant energy could exist only in discrete quanta which were proportional to the frequency E = hu with h = 6.62 6 0 6 93(11) x 10 - 34 Js. Before 2003 the accepted value was 6.6260755(40) x 10-34 Js = 4.1356692(12) x 10-15 eV s. The quantity h later was referred to as Planck s constant. 

---