Planck’s formula

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 ... 

This expression can be compared with Planck s formula - Equations (A3.2) or (2.2) - to obtain the following two relations among the Einstein coefficients ... 

In 1917 Einstein [18] wrote a paper on the dualistic nature of light in which he discusses emission without excitation from external causes, in other words stimulated emission and also spontaneous absorption and emission. He derives Planck s formula but also discusses the recoil of molecules when they emit photons. It is the latter discussion that Einstein regarded as the most significant aspect of the paper If a radiation bundle has the effect that a molecule struck by it absorbs or emits a quantity of energy hv in the form of radiation (ingoing radiation), then a momentum hvlc is always transferred to the molecule. For an absorption of energy, this takes place in the direction of propagation of the radiation bundle for an emission, in the opposite direction. ... 

Moreover, a solid is termed a gray body if its monochromatic emissivity ex is independent of the wavelength. The monochromatic-emissive power of a blackbody at a specific temperature and wavelength is given by Planck s formula... 

Prove that the Stefan-Boltzmann law for thermal radiation given in Eq. (1.59) can be derived by using Planck s formula given in Eq. (1.62). Also show that... 

An element in a thermally radiative environment absorbs, reflects, refracts, diffracts, and transmits incoming radiative heat fluxes as well as emits its own radiative heat flux. Most solid materials in gas-solid flows, including particles and pipe walls, can be reasonably approximated as gray bodies so that absorption and emission can be readily calculated from Stefan-Boltzmann s law (Eq. (1.59)) for total thermal radiation or from Planck s formula (Eq. (1.62)) for monochromatic radiation. Other means of transport of radiative... 

The major selling point of standard cosmology is the observed isotropic microwave background radiation, with black-body spectrum. In a closed universe it needs no explanation. Radiation, which accumulates in any closed cavity, tends, by definition, to an equilibrium wavelength distribution according to Planck s formula (Figure 2.5). 

This is reminiscent of Planck s formula for the energy of a photon. It comes as no surprise then that the quantum theory of radiation has the structure of an assembly of oscillators, with each oscillator representing a mode of electromagnetic waves of a specified frequency. 

The maximum in Planck s formula for the emission of blackbody radiation can be shown to occur at a wavelength 4max = 0.20 hc/kT The radiation from the surface of the sun approximates that of a blackbody with A ,ax = 465 nm. What is the approximate surface temperature of the sun ... 

A year or two later (1907) Einstein showed that Planck s formula for the mean energy of an oscillator,... 

As a matter of fact, Planck s formula, to which reference will be made in a moment, reduces to this expression for the simple case chosen, viz identity of one son (Cl ) in the two salts, and identity of concentration of the salts)... 

Planck s formula has been extended by K R Johnson (Ann der Physik, 14, 995, 1904) to the case in which the valency of the ions is not the same... 

The applicability of Planck s formula has been investigated by Fales and Vosburgh (Joum Amer Chern Soc, 40, 1291 (1918)) using cells of the type— ... 

M. Planck, Theorie der Warmestrahlung, Formula (64). The factor 2 in Planck s formula arises from his considering a perfectly reflecting element, while we have a perfectly absorbing one. 

This agrees with Planck s formula (5) if we put W0=hv. This last relation can be established with the help of Wien s displacement law, which can be deduced from thermo-dynamical considerations combined with the Doppler principle. Wien s law states that the density of radiation must depend on the temperature and frequency in the following way ... 

By consideration of Planck s formula W0=Av, Einstein was led to interpret phenomena of another type in terms of the quantum theory, thus giving rise to a new conception of this equation which has proved very fruitful. The phenomenon in question is the photoelectric effect. If light of frequency v falls on a metallic surface,4 electrons are set free and it is found that the intensity of the light influences... 

In quantum physics, Planck s formula (E = hoS) relates the wave pulsation to the energy-per-corpuscle. 

Here h — 6.626 x Js) is Planck s constant and kB(= 1-381 x 10 JK ) is Boltzmann s constant. We shall not present the derivatives of Planck s formula (which requires statistical mechanics) our focus will be on thermodynamic aspects of radiation. Finally, we note that total energy of thermal radiation is... 

Planck was able to achieve agreement with experimental data by choosing a value of h approximately equal to the presently accepted value, 6.62608 x 10 Js. Planck s formula agrees with the Stefan-Boltzmann law and with Wien s law, which states that the wavelength at the maximum in the spectral radiant emittance curve is inversely proportional to the absolute temperature. 

Note the long wavelength agreement of the Rayleigh-Jeans function with experiment, does Planck s formula satisfy this condition ... 

One additional amazing confirmation is that we can integrate Planck s formula over all frequencies to obtain the total emissive power to compare to the Stefan-Boltzmann law... 

The photon energy mode density can be expressed as the product of Equations 3.28 and 3.32 giving the well-known Planck s formula (Planck s black body radiation distribution law) ... 

Forcing B21/B12 = 1 (which has some basis when the temperature-dependent terms are considered to cancel) [36] brings Equation 3.38 in line with Planck s formula given in Equation 3.33 (but multiplied with (hv) since in Equation 3.33 (pv) is defined as number of modes with energy (hv). Equating Equations 3.33 and 3.38 leads to a relationship between the Einstein s A21 (associated with spontaneous emission) and B21 (associated with stimulated emission) coefficients as... 

Planck s formula a hypothesis of quanta—intensity of light from wave and quantum points of view... 

Planck s formula well describes limiting transitions. So, at hco < kT the exhibitor in the denominator of function (6.6.14) can be decomposed in a series and be limited by two terms. This leads to Rayleigh-Jeans formula which describes very well experiment in this area of frequencies. In contrast, at hm kT the unit in the denominator can be neglected and functions r and/fall according to the exponent that is found by experiment. 

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