The Continuous Emission Spectrum

James Richard Fromm

All objects will emit light, whether visible or not, when they are given sufficient energy in the form of heat. The emission of this light, or electromagnetic radiation, is one of the ways in which objects can lose energy to their lower-temperature surroundings. The light is emitted as a continuous spectrum of all frequencies, but the distribution of the emitted energy depends on the temperature of the object which is radiating. As the temperature of the emitting object increases, the distribution shifts toward emission of higher-energy (higher-frequency) light. Moderately warm objects, such as our bodies, emit light in the low-energy infrared region of the electromagnetic spectrum. Hotter objects, such as the tungsten filament of an electric lamp, the sun, or an open fire emit light which contains more of the blue, violet, and ultraviolet region of the spectrum at still higher energies.

An ideal or perfect radiation source, which is called a black body, emits a continuous spectrum of light of all frequencies. The spectral distribution, the distribution of energy among the frequencies of the spectrum, of a black body, depends only upon the temperature of the body. The higher the temperature of the source the greater the proportion of light in the higher-energy region of the spectrum as shown on the left-hand side of the Figure below.

Real bodies behave in a way similar but not identical to ideal black bodies. Our sun emits radiation very similar to that of a black body at 6000 K; the much hotter blue-white star Sirius emits radiation very similar to that of a black body at 11000 K. These temperatures are believed to be the actual surface temperatures of the stars.

Experimentally, a nearly ideal black body is not difficult to construct. A hollow object such as a furnace with a small hole in it will, when heated, emit light from the internal cavity through the hole. If the hole is small the walls can be held at a constant temperature T which will also be the temperature of the interior of the cavity. It is found that the power, or energy/time quotient, of the radiation emitted from the hole, P, depends only upon the temperature of the cavity. The form of this dependence is given by the Stefan-Boltzmann equation:

P = E/t = sT4

where T is the absolute temperature and s, the Stefan-Boltzmann constant more commonly symbolized by the Greek lower-case sigma, has the value 5.67051(19) x 10-8 J m-2s-1K-4. This relationship is obeyed regardless of the size or material of the cavity.

The Stefan-Boltzmann constant is now known to be a combination of more fundamental constants, 8(pi)5k4/60h3c2, and so the Stefan-Boltzmann law could be written as:

P = E/t = [8(pi)5/60h3c2](kT)4

The significance of the law remains the same: the energy or the power emitted increases as the fourth power of the absolute temperature.

As indicated earlier, the light emitted by a black body is a continuous spectrum of all wavelengths. The energy emitted at each wavelength can be measured. A typical plot is shown on the right-hand side of the Figure above; both the shape and the location of the wavelength maximum depend upon the temperature.

It is expected that the intensity or power of radiation should increase with increasing frequency. For a collection or assembly of oscillators of frequency f (more commonly symbolized by Greek lower-case nu), the energy emitted will be 8(pi)f 2E/c3 when each oscillator has energy E. It does not matter whether the oscillators, which are the sources of electromagnetic radiation, fill the cavity or line its walls; the result will remain the same. By 1900, Rayleigh and Jeans had postulated that the energy of an oscillator would be simply kT, where k is the Boltzmann constant and T is the absolute temperature. This gives the total energy E(f) emitted at frequency f in the form of the Rayleigh-Jeans equation:

E(f) = [8(pi)f 2/c3]kT

The Rayleigh-Jeans equation does in fact fit the observed spectrum of black-body radiation, but only at lower frequencies. As the frequencies increase and wavelengths shorten toward the ultraviolet region of the electromagnetic spectrum, the prediction of the Rayleigh-Jeans equation rapidly deviates from experimental observation -- and also from any other reasonable theory, for at very short wavelengths the predicted energy emission of a black body approaches infinity! This absurdity became known as the "ultraviolet catastrophe", and appears as the dashed line in the above Figure.

Copyright 1997 James R. Fromm