Blackbody Radiation

The wave theory of light, which Maxwell’s equations captured so well, became the dominant light theory in the 1800’s (surpassing Newton’s corpuscular theory, which had failed in a number of situations). The first major challenge to the theory came in explaining thermal radiation, which is the type of electromagnetic radiation emitted by objects because of their temperature.

Testing Thermal Radiation

An apparatus can be set up to detect the radiation from an object maintained at temperature T₁. (Since a warm body gives off radiation in all directions, some sort of shielding must be put in place so the radiation being examined is in a narrow beam.) Placing a dispersive medium (i.e. a prism) between the body and the detector, the wavelengths (lambda) of the radiation disperse at an angle (theta). The detector, since it’s not a geometric point, measures a range delta-theta which corresponds to a range delta-lambda, though in an ideal set-up this range is relatively small.

If I represents the total intensity of the electromagnetic radiation at all wavelengths, then that intensity over an interval delta-lambda (between the limits of lambda and delta-lamba) is:

delta-I = R(lambda) delta-lambda

R(lambda) is the radiancy, or intensity per unit wavelength interval. In calculus notation, the delta-values reduce to their limit of zero and the equation becomes:

dI = R(lambda) dlambda

The experiment outlined above detects dI, and therefore R(lambda) can be determined for any desired wavelength.

Radiancy, Temperature, and Wavelength

Performing the experiment for a number of different temperatures, we obtain a range of radiancy vs. wavelength curves, which yield significant results:

1. The total intensity radiated over all wavelengths (i.e. the area under the R(lambda) curve) increases as the temperature increases.

   This is certainly intuitive and, in fact, we find that if we take the integral of the intensity equation above, we obtain a value that is proportional to the fourth power of the temperature. Specifically, the proportionality comes from Stefan’s law and is determined by the Stefan-Boltzmann constant (sigma) in the form:

   I = sigma T⁴

2. The value of the wavelength lambda_max at which the radiancy reaches its maximum decreases as the temperature increases.

   The experiments show that the maximum wavelength is inversely proportional to the temperature. In fact, we have found that if you multiply lambda_max and the temperature, you obtain a constant, in what is known as Wein’s displacement law:

   lambda_max T = 2.898 x 10^-3 mK

Blackbody Radiation

The above description involved a bit of cheating. Light is reflected off objects, so the experiment described runs into the problem of what is actually being tested. To simplify the situation, scientists looked at a blackbody, which is to say an object that does not reflect any light.

Consider a metal box with a small hole in it. If light hits the hole, it will enter the box, and there’s little chance of it bouncing back out. Therefore, in this case, the hole, not the box itself, is the blackbody. The radiation detected outside the hole will be a sample of the radiation inside the box, so some analysis is required to understand what’s happening inside the box.

1. The box is filled with electromagnetic standing waves. If the walls are metal, the radiation bounces around inside the box with the electric field stopping at each wall, creating a node at each wall.
2. The number of standing waves with wavelengths between lambda and dlambda is

   N(lambda) dlambda = (8pi V / lambda⁴) dlambda

   where V is the volume of the box. This can be proven by regular analysis of standing waves and expanding it to three dimensions.
3. Each individual wave contributes an energy kT to the radiation in the box. From classical thermodynamics, we know that the radiation in the box is in thermal equilibrium with the walls at temperature T. Radiation is absorbed and quickly reemitted by the walls, which creates oscillations in the frequency of the radiation. The mean thermal kinetic energy of an oscillating atom is 0.5kT. Since these are simple harmonic oscillators, the mean kinetic energy is equal to the mean potential energy, so the total energy is kT.
4. The radiance is related to the energy density (energy per unit volume) u(lambda) in the relationship

   R(lambda) = (c / 4) u(lambda)

   This is obtained by determining the amount of radiation passing through an element of surface area within the cavity.

Graph of Wavelength vs. Radiancy in blackbody radiation.

Failure of Classical Physics

Throwing all of this together (i.e. energy density is standing waves per volume times energy per standing wave), we get:

u(lambda) = (8pi / lambda⁴) kT

R(lambda) = (8pi / lambda⁴) kT (c / 4) (known as the Rayleigh-Jeans formula)

Unfortunately, the Rayleigh-Jeans formula fails horribly to predict the actual results of the experiments. Notice that the radiancy in this equation is inversely proportional to the fourth power of the wavelength, which indicates that at short wavelength (i.e. near 0), the radiancy will approach infinity. (The Rayleigh-Jeans formula is the purple curve in the graph to the right.)

The data (the other three curves in the graph) actually show a maximum radiancy, and below the lambda_max at this point, the radiancy falls off, approaching 0 as lambda approaches 0.

This failure is called the ultraviolet catastrophe, and by 1900 it had created serious problems for classical physics because it called into question the basic concepts of thermodynamics and electromagnetics that were involved in reaching that equation. (At longer wavelengths, the Rayleigh-Jeans formula is closer to the observed data.)

Planck’s Theory

In 1900, the German physicist Max Planck proposed a bold and innovative resolution to the ultraviolet catastrophe. He reasoned that the problem was that the formula predicted low-wavelength (and, therefore, high-frequency) radiancy much too high. Planck proposed that if there were a way to limit the high-frequency oscillations in the atoms, the corresponding radiancy of high-frequency (again, low-wavelength) waves would also be reduced, which would match the experimental results.

Planck suggested that an atom can absorb or reemit energy only in discrete bundles (quanta). If the energy of these quanta are proportional to the radiation frequency, then at large frequencies the energy would similarly become large. Since no standing wave could have an energy greater than kT, this put an effective cap on the high-frequency radiancy, thus solving the ultraviolet catastrophe.

Each oscillator could emit or absorb energy only in quantities that are integer multiples of the quanta of energy (epsilon):

E = n epsilon, where the number of quanta, n = 1, 2, 3, . . .

The energy of each quanta is described by the frequency (nu):

epsilon = h nu

where h is a proportionality constant that came to be known as Planck’s constant. Using this reinterpretation of the nature of energy, Planck found the following (unattractive and scary) equation for the radiancy:

(c / 4)(8pi / lambda⁴)((hc / lambda)(1 / (e^{hc/lambda kT} – 1)))

The average energy kT is replaced by a relationship involving an inverse proportion of the natural exponential e, and Planck’s constant shows up in a couple of places. This correction to the equation, it turns out, fits the data perfectly, even if it isn’t as pretty as the Rayleigh-Jeans formula.

Consequences

Planck’s solution to the ultraviolet catastrophe is considered the starting point of quantum physics. Five years later, Einstein would build on this quantum theory to explain the photoelectric effect, by introducing his photon theory. While Planck introduced the idea of quanta to fix problems in one specific experiment, Einstein went further to define it as a fundamental property of the electromagnetic field. Planck, and most physicists, were slow to accept this interpretation until there was overwhelming evidence to do so.