Planck’s Curve

Parallel to efforts to understand the inner workings of the atom was research in the late-1800’s to understand how matter emits energy. The energy an object emits as a function of the wavelength of light is called its continuous spectrum. The field of science that studies spectrum is called spectroscopy. One of the primary results from the field of spectroscopy was the discovery that the spectrum of an object changes with temperature.

The Austrian physicist Josef Stefan found in 1879 that the total radiation energy per unit time emitted by a heated surface per unit area increases as the fourth power of its absolute temperature T (Kelvin scale). This means that the Sun’s surface, which is at T = 6,000 K, radiates per unit area (6,000/300)4 = 204 = 160,000 times more electromagnetic energy than does the same area of the Earth’s surface, which is taken to be T = 300 K. In 1889 another Austrian physicist, Ludwig Boltzmann, used the second law of thermodynamics to derive this temperature dependence for an ideal substance that emits and absorbs all frequencies. Such an object that absorbs light of all colours looks black, and so was called a blackbody.

The wavelength or frequency distribution of blackbody radiation was studied in the 1890s by Wilhelm Wien of Germany. It was his idea to use as a good approximation for the ideal blackbody an oven with a small hole. Any radiation that enters the small hole is scattered and reflected from the inner walls of the oven so often that nearly all incoming radiation is absorbed and the chance of some of it finding its way out of the hole again can be made exceedingly small. The radiation coming out of this hole is then very close to the equilibrium blackbody electromagnetic radiation corresponding to the oven temperature. Wien found that the radiative energy ‘dW’ per wavelength interval ‘d’ has a maximum at a certain wavelength ‘m’ and that the maximum shifts to shorter wavelengths as the temperature ‘T’ is increased, as illustrated in the figure below.

Wien’s law of the shift of the radiative power maximum to higher frequencies as the temperature is raised expresses in a quantitative form commonplace observations. Warm objects emit infrared radiation, which is felt by the skin; near T = 950 K a dull red glow can be observed; and the color brightens to orange and yellow as the temperature is raised. The tungsten filament of a light bulb is T = 2,500 K hot and emits bright light, yet the peak of its spectrum is still in the infrared according to Wien’s law. The peak shifts to the visible yellow when the temperature is T = 6,000 K, like that of the Sun’s surface.

Both these relationships were synthesized by physicist Max Planck into what is called Planck’s curve. All objects emit energy in the shape of Planck’s curve, where the amount and the peak energy vary only as the temperature of the body.

The problem with Planck’s curve is that it does not agree with Rutherford’s model of the atom. Atoms absorb and emit light through a process called scattering. Photons fly near the atoms and are deflected. Sometimes their motion pushes the atom (where push means electromagnetic forces), and the photon loses energy (i.e. becomes redder). Sometimes the atom pushes the photon, and the photon gains energy (i.e. becomes bluer). However, given the nature of atoms, the photons should receive more energy than the atoms, so there should be more and more blue photons, but clearly the Planck curve drops off at short wavelengths. This is called the UV catastrophe, which was resolved by quantum physics.