Thermal Radiation and Plancks Law

This Rayleigh-Jeans law matches the experimental results fairly well at low frequencies (in the infrared region) but is in strong disagreement with experiment at higher frequencies (in the ultraviolet region). 

In order to explain this discrepancy, M. Planck suggested in 1900 that each mode of the radiation field can only emit or absorb energy in discrete amounts qhv, which are integer multiples q of a minimum energy quantum hu. These energy quanta hi/ are called photons. Planck s constant h can be 

In thermal equilibrium the partition of the total energy into the different modes is governed by the Maxwell-Bolt2anann distribution, so that the probability p(q) that a mode contains the energy qhi/ is 

The thermal radiation field has the energy density p(i/)di/ within the frequency interval v to u+du, which is equal to the number n(i/)di/ of modes in the interval du times the mean energy W per mode. Using (2.7b, 12) one obtains 

This is Planck s famous radiation law (Fig.2.2) which predicts a spectral energy density of the thermal radiation that is fully consistent with experiments. The expression thermal radiation comes from the fact that the spectral energy distribution (2.13) is characteristic of a radiation field which is in thermal equilibrium with its surrounding (in Sect.2.1 the surrounding is determined by the cavity walls). 

In classical thermodynamics each degree of freedom of a system in thermal equilibrium at a temperature T has the mean energy kT/2, where k is the Boltzmann constant. Since classical oscillators have kinetic as well as potential energies, their 

The thermal radiation field described by its energy density p v) is isotropic. This means that through any transparent surface element dA of 

Let us consider thermal radiation in a certain cavity at a temperature T. By the term thermal radiation we mean that the radiation field is in thermal equilibrium with its surroundings, the power absorbed by the cavity walls, Fa (v), being equal to the emitted power, Pe v), for all the frequencies v. Under this condition, the superposition of the different electromagnetic waves in the cavity results in standing waves, as required by the stationary radiation field configuration. These standing waves are called cavity modes. 

By means of geometric arguments, it can be shown that the number of modes n v) per unit volume within a frequency interval d(v) is given by 

An Introduction to the Optical Spectroscopy of Inorganic Solids J. Garcfa Sol6, L. E. Bausd, and D. Jaque 2005 John Wiley Sons, Ltd ISBNs 0-470-86885-6 (HB) 0-470-86886-4 (PB) 

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