Blackbody Radiation

FIELDS OF STUDY: Quantum Physics; Thermodynamics

ABSTRACT: Blackbodies are ideal physical objects that absorb all frequencies of electromagnetic radiation from any source. As a result of this total absorption, the only radiation a blackbody emits is thermal radiation generated by its temperature. This radiation, called blackbody radiation, is emitted at full efficiency across the entire spectrum. Discoveries about the nature of blackbody radiation ultimately led to the development of quantum mechanics in the early twentieth century.

principal terms

• continuous energy levels: the idea that there are limitless levels of energy between each point on a continuum.
• Planck’s law: a mathematical description of the amount of radiation emitted at different frequencies by a blackbody at a given temperature.
• quantum mechanics: the branch of physics that deals with phenomena on a subatomic scale.
• Stefan-Boltzmann law: a mathematical description of the total radiant energy emitted by a blackbody, relating it to the temperature of the blackbody raised to the fourth power.
• ultraviolet catastrophe: the erroneous prediction, based on the laws of classical physics, that a blackbody would emit an infinite amount of energy at short wavelengths, starting around the ultraviolet region.

A Few Questions Remain

At the end of the nineteenth century, scientists believed that only a few unexplained phenomena stood between their current understanding of physics and a complete understanding of how the universe works. All they needed, they believed, was to tie up those loose ends. Then, they would be in a position to account for everything they could observe around them.

Scientists had good reason for this belief. Isaac Newton’s (1642–1727) three laws of motion accounted for the behavior of objects at rest and in motion. James Clerk Maxwell’s (1831–79) work with electromagnetism had tied together two fields of physics to explain behavior in a significant area. Once they cleared up the rest, they could use their knowledge to look at new problems as they arose.

The problem was that these scientists were looking at phenomena that took place at the readily observable level—the things they could observe with the naked eye or with optical microscopes. However, the discovery of subatomic particles and subsequent study of how those particles interact led to the realization that the laws that govern the very large do not govern the very small. In fact, the same element can behave in different ways depending on the scale on which its behavior is being observed.

Among these unexplained phenomena were questions about the nature of light, or visible electromagnetic radiation. Scientists knew that the color of the light emitted by an object changes based on the temperature of that object. Cooler fires burn with a reddish flame, while hotter fires burn blue or white; an iron rod glows red when heated, then changes colors as it cools. They also knew that all objects at thermodynamic equilibrium emit exactly as much, or as little, radiation as they absorb. In other words, good absorbers are good emitters, and poor absorbers are poor emitters.

German physicist Gustav Kirchhoff (1824–87) and chemist Robert Bunsen (1811–99) discovered in 1859 that different substances absorb radiation at particular frequencies and then reemit it at those same frequencies. This is the principle behind spectroscopy. Kirchhoff then considered under what conditions an object would emit radiation based solely on its temperature rather than its chemical composition. In 1860, he introduced the concept of the blackbody, an idealized body that would absorb all incident radiation, reflecting none. This would cause the interior of the blackbody to heat up, which in turn would cause it to emit thermal radiation. This thermal radiation, which came to be known as blackbody radiation, would be emitted across all wavelengths and frequencies. The peak intensity would depend solely on the temperature of the blackbody.

Measuring Blackbody Radiation

No genuine blackbody exists in nature, although some objects, such as the sun and other stars, come close. To approximate such a body, Kirchhoff proposed using an opaque box or cavity with a single small hole in the side. This cavity would be a near approximation of a perfect absorber. Any radiation that entered through the hole would be mostly absorbed by the interior walls, with very little escaping. Similarly, if the interior of the cavity were heated, the radiation emitted from the hole would approximate a perfect emitter. With these parameters in place, Kirchhoff challenged his colleagues to measure the energy curve of this temperature-based radiation.

Using experimental data, Slovenian physicist Josef Stefan (1835–93) determined in 1879 that the energy radiated by a blackbody is proportional to the temperature of the body raised to the power of four. Five years later, one of Stefan’s former doctoral students, Austrian physicist Ludwig Boltzmann (1844–1906), mathematically derived the same relationship based on the laws of thermodynamics. This relationship, now known as the Stefan-Boltzmann law, can be expressed as

L = AσT⁴

where L is the luminosity of the blackbody in watts (W), or joules per second (J/s); A is the surface area in meters squared (m²); T is the temperature in kelvins (K); and σ is the Stefan-Boltzmann constant, equal to 5.670373 × 10⁻⁸ W/m²·K⁴. It can also be written more simply as

E = σT⁴

where E is the total energy output in joules per second per meter squared (J/s·m²).

The Ultraviolet Catastrophe

In 1905, English physicists Lord Rayleigh (1842–1919) and James Jeans (1877–1946) published the Rayleigh-Jeans law, which Rayleigh had first derived five years earlier. This law, based on classical physics equations, was meant to calculate how much radiation a blackbody would emit at a given frequency or wavelength based on its temperature. (Frequency and wavelength are inversely proportional; the higher the frequency, the shorter the wavelength.) However, although the law was accurate at lower frequencies, it predicted that the radiation output would increase exponentially at higher frequencies. That is, blackbodies would emit infinite amounts of high-frequency radiation. This was obviously not the case, or else any thermal radiation emitted by a body at thermodynamic equilibrium would reduce anything or anyone nearby to ash. The discrepancy was later dubbed the ultraviolet catastrophe, as the errors began around the ultraviolet range.

The source of the catastrophe was the assumption that electromagnetic radiation was emitted at continuous energy levels. In other words, it was assumed that light acted as a wave under all conditions and at all frequencies. Scientists had long debated whether light was a particle or a wave—in other words, whether it traveled in discrete units or as a continuous stream of energy. By the start of the twentieth century, the wave theory had largely prevailed. This was due in part to Maxwell’s electromagnetic equations, which worked on the assumption that different forms of light (infrared, visible, ultraviolet) were simply different frequencies of the same electromagnetic waves. The ultraviolet catastrophe was one indication that the wave theory of light may not be the whole truth.

German physicist Max Planck (1858–1947) found a solution while addressing a similar issue raised by fellow physicist Wilhelm Wien (1864–1928). Several years before Rayleigh and Jeans, Wien had derived an equation to describe the complete thermal radiation spectrum. However, Wien’s equation had the opposite problem of Rayleigh and Jeans’s: it diverged at lower frequencies, starting around the infrared range. Planck revised the equation and added a constant based on experimental data. This constant had the effect of quantizing energy—that is, treating it as discrete packets rather than continuous waves. Planck published his revised equation, known as Planck’s law, in 1900, shortly after Rayleigh began working on what would become the Rayleigh-Jeans law. Planck’s law resolved the problems of Wien’s equation and, it was later realized, of the Rayleigh-Jeans law as well. However, it raised new questions about physics as a whole.

A New Branch of Physics

Planck "backed into" his discovery of energy quantization while trying to reconcile Wien’s equation with reality. However, he was not convinced that energy was truly quantized. He believed that the constant was merely a mathematical convenience that did not reflect reality. Shortly thereafter, in 1905, Albert Einstein (1879–1955) introduced the concept of wave-particle duality—the idea that light can behave as both a particle and a wave.

What Planck did not appreciate at the time was that he was describing the phenomenon that, when understood, would change the very nature of physics. The idea that something such as energy could behave one way at one level and another at a different level would become the basis of a new branch of physics. Quantum mechanics would answer the unresolved questions of classical physics while introducing a host of new questions about the behavior of particles at small scale. It would throw the traditional view of the universe on its head and usher in a new way of looking at the world.

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