Black body radiation

Type of physical science: Classical physics

Field of study: Statistical mechanics

The radiation emanating from a body that is fully and uniformly absorbing across the electromagnetic spectrum body and that is at equilibrium at its temperature is called black body radiation. The theoretical description of black body radiation by Max Planck laid the basis for classical quantum mechanics.

Overview

Black body radiation is the electromagnetic radiation emitted by a black body caused solely by temperature. At low temperatures, such as in a room, a black body appears black, as the name implies. All the external radiation is absorbed by the body or object and, at low temperatures, the radiation from the object is too small to be noticed by the eye. When the black body is heated, however, the object will begin to glow and so lose its black appearance. Thus, paradoxically, if the black body can withstand a high temperature without burning or decomposing, its color changes toward white as its temperature increases. The black body still absorbs all the radiation that falls on it, but the internal radiation emitted by the body at high temperatures gives its appearance some shade of white. The internal radiation, and the radiation emitted through the surface of the black body, constitute black body radiation.

The sun is an example of an excellent black body, but one would not judge it so by the blinding yellow-white appearance resulting from its very hot surface. On Earth, the sun's black body radiation gives warmth, gives light, and fuels life in and around humans.

The energy source for the radiation emitted by a black body is the mechanism that maintains the temperature of the black body. The sun's energy source is a giant nuclear furnace burning at the sun's center, more than 600,000 kilometers below its surface, that slowly converts the huge hydrogen stores of the sun to helium. The energy released by the hydrogen conversion works its way to the sun's surface, where the temperature has dropped tremendously to somewhat less than 6,000 Kelvins, still nearly twenty times as hot as the earth's surface.

In determining the properties of black body radiation, it is necessary to specify the temperature of the object by some absolute temperature scale. Temperature gives a measure of the equilibrium energy that may flow from a hot body to a colder one. Since there is a minimum energy (zero) that the coldest object has available to transfer, there is a coldest temperature achievable on any temperature scale. The coldest temperature is -273 degrees Celsius. Absolute temperature scales, such as the Kelvin scale, reflect the zero energy available at the coldest possible temperature and set their zero at that temperature. Thus, Kelvin zero is at -273 degrees Celsius. Its degrees, however, increase in step with the Celsius scale, and so Celsius converts to Kelvins by adding 273.

Throughout the black body, the energy source that maintains the temperature must feed energy to electromagnetic waves and to thermal motion of the atoms and molecules of the body.

This electromagnetic energy bathes the whole of the black body, including all of its empty spaces, and escapes through the outer surfaces of the body as black body emission. No object can absorb more energy than a black body of the same surface area, since the black body absorbs all the energy falling on its surface. A law of thermodynamics then demands that no body can emit more energy for a given surface area and time than a black body at the same temperature.

Practical bodies are not completely black and do not emit as strongly as an ideal black body at the same temperature. A special case is the gray body that absorbs a constant fraction of the energy falling on its surface uniformly across the energy spectrum. The absorption fraction is termed the "emissivity of the grey body," since the gray body will emit that fraction of the radiation from a black body that has the same temperature and surface area as the gray body.

Metals have emissivities much less than unity, since they reflect radiation well. Many metals have constant emissivities through the visible and so their surfaces appear metal-gray. If the emissivity varies across the visible spectrum, however, the appearance of the surface takes on a colored appearance. For example, the emissivity of aluminum paint is substantially constant at 0.30 throughout the visible and is metallic gray to the eye, while gold, on the other hand, has a low emissivity that varies strongly across the visible spectrum producing the distinctive gold color.

Roughening the surface of a material increases the emissivity. Indeed, an experimental method for producing a black body surface is to drill a small hole in the surface of a material, such as tungsten, which will withstand a high temperature and to use the empty opening as the area of the black body. If the hole is sufficiently deep, most of the radiation falling through the opening will make many collisions with the walls and is highly absorbed before it has a chance to exit the opening. The absorption of the opening is nearly complete; therefore, it behaves as the surface of a black body. The walls supply the temperature and thermal radiation that fills the "empty" hole and escapes through the opening, which now approximates the surface of a true solid black body.

In the theory of a black body, the idea of an empty hole reaches its extreme. The black body is the empty space inside an enclosure or oven whose walls are fixed at one temperature. A small hole may be allowed in the enclosure and the area of the hole becomes the emitting area of the black body, while its volume is the entire interior volume. It turns out to be rather simple, then, to determine some properties expected of the ideal black body. Classical physicists knew that each pattern or mode of oscillation of a wave should carry an energy proportional to the temperature of the wave. It turns out that a mode occupies a volume comparable to a wavelength on a side, so that the energy content per unit volume should be proportional to temperature divided by wavelength cubed or temperature times frequency cubed. A common measurable quantity in black body studies is the energy content per unit volume and per unit frequency, which then is proportional to temperature times frequency squared. This expectation holds for long wavelength black body radiation; classical physicists were able to calculate the proportionality constant exactly. The simple picture, however, is inadequate for short wavelength black body radiation, and the defect pointed the way, in time, to the discovery of quantum theory.

The spectrum of radiation from a black body is uninterrupted throughout the electromagnetic spectrum and has a maximum at one wavelength that decreases as the temperature of the black body increases. The relation describing the shift down in wavelength as the temperature increases was given by the 1911 Nobel laureate in Physics Wilhelm Wien (1864-1928). The relation bears his name as the Wien displacement law. Doubling the temperature of a black body halves the wavelength at which the maximum black body radiation occurs.

Wavelengths of visible light are commonly given in micrometers, which are millionths of a meter. The visible spectrum spans the range from about 0.4 micrometer, violet, through blue, green, yellow, and somewhat beyond the deep red at 0.7 micrometer. Wien's law states that a black, or gray, body at 1,000 Kelvins has its peak near 3 micrometers. The radiation from the sun's surface--nearly 6,000 Kelvins--then peaks close to 0.5 micrometer, in the green, where evolution has given the eye its greatest sensitivity to detect the light of the sun. The tungsten filament in a high-intensity incandescent lamp has a temperature near 3,000 Kelvins, and its peak radiation is near 1.0 micrometers, outside the visible in the near infrared. One sees mainly the red wavelength tail of this radiation, and this red light from the incandescent lamp gives a pleasant appearance to skin tones. Unfortunately, the main radiation is in the unseen infrared, producing heat and low light efficiency. A star with a temperature near 12,000 Kelvins has its peak in output in the invisible ultraviolet near 0.25 micrometer. The visible blue tail of its black body radiation produces a distinctive bluish hue in its starlight.

At any wavelength, the energy emitted by a black body increases as the temperature increases. The total radiation content of black body radiation across the full electromagnetic spectrum rises very strongly with the temperature of the black body. In fact, the energy increases with the fourth power of the temperature, a relation that is known as the Stefan-Boltzmann law.

A small, 1 square centimeter black body--half the area of a dime--held at 1,000 Kelvins, emits almost 6 watts with most of the radiation in the infrared. A black body at 3,000 Kelvins of an incandescent lamp would radiate 81 times as much power, for the same square centimeter, or about 400 watts. A black body at 6,000 Kelvins, near the temperature of the sun's surface, would radiate 1,296 times more strongly, emitting more than 7,000 watts for each square centimeter.

Earth has an absolute temperature near 300 Kelvins. It maintains that temperature by intercepting about four ten-billionths of the sun's enormous output and eventually radiating the absorbed energy back into the depths of "empty" space. The peak in the re-radiated energy is far into the infrared at about 10 micrometers. The empty space into which it radiates has a very cold temperature, somewhat below 3 Kelvins, and an emission seen as a cosmic background of microwave noise with a peak wavelength just above 1 millimeter. The emptiness of the cosmos is an immense and frigid black body in which at least one species of intelligent life prospers on Earth, drawing its energies from one of the myriad hot spots that form the stars of the universe.

Applications

The principles behind black body radiation have an extremely wide range of applications in fields as disparate as lighting, heating, astronomy, temperature measurements, and lasers.

A simple method to construct a light source is to heat a material to incandescence.

Tungsten has one of the highest melting temperatures of any practical material. Since it is a metal, it can be conveniently heated by passing an electrical current through a long, thin strand of tungsten. In an incandescent lamp, the strand is coiled once, and often a second time, to form a compact filament. The coiling increases the effective emissivity of the reflective metal to about 0.4 in the visible. Plugged into a wall socket, the filament heats to nearly 3,000 Kelvins and, with a surface area designed to half a square centimeter, radiates about 80 watts. In a 100-watt lamp, additional heat losses, by convection through the gas within the bulb and by conduction along the base holding the filament, use up the remaining watts.

A radiant electrical heater works on a similar principle to an incandescent lamp. Visible light is not required, so the temperature of the metal coils is greatly reduced, placing the heating rays farther into the infrared. A very substantial increase in coil area compensates the reduction in total energy radiated for each parcel of area at the lower temperature.

While practical black or gray body temperatures do not exceed about 3,000 Kelvins for solid bodies on Earth, most stars are huge balls of gases whose outer temperatures may greatly exceed this limit. The color of the black body spectrum of a star gives a direct estimate of the star's elevated temperature, while the total power--energy per time--received on Earth many years after its emission from the star tells the distance to the star, if the size of the star can be estimated.

The star's color classifies its temperature and many internal properties. A simple scheme has evolved among astronomers that labels stars by a letter in the sequence O, B, A, F, G, K, and M, which are determined by color. Red M stars, such as Betelgeuse, are the coolest, less than 3,500 Kelvins, while the yellow-white sun is a G star on the high side of the temperature range--between 5,000 and 6,000 Kelvins. Blue-white Rigel is in the B class, whose temperatures vary from 11,000 to 25,000 Kelvins. Most normal stars in this letter sequence have a simple relation between class, which gives their temperature, and size. Therefore, the class of these main sequence stars and the amount of light received on Earth may be used to tell astronomers the distance to the star.

On Earth, the spectrum of a body is often used, with the laws of black body radiation, to determine quite accurately the temperature of the hot material. This combination of light and law is the basis of optical pyrometry. With visible light, the lowest practical temperature limit is about 873 Kelvins below which the visible surface radiation becomes quite weak. Infrared temperature measurements, however, can determine very small temperature increases above the surroundings and can be used, for example, to measure heat radiation leaks from homes or buildings.

While lasers are as far removed from black bodies as any radiation source can be, the basic equations that govern their operation were uncovered by Albert Einstein (1879-1955) in the early part of the twentieth century from the law of black body emission.

Context

Scientists knew the basic experimental characteristics of the radiation emitted from a black body by the end of the nineteenth century. The search for the fundamental equation that described these characteristics was to lead to two great revolutions of twentieth century science: quantum theory, early in the twentieth century, and lasers, six decades into the twentieth century.

On December 14, 1900, Max Planck (1858-1947) announced a derivation of the law of black body radiation, which now bears his name. In order to complete his derivation, Planck was forced to introduce the revolutionary idea that the emission and absorption of electromagnetic energy could take place only in multiples of a discrete quantity. The idea was so revolutionary that Planck resisted its use for a considerable time before finally announcing his derivation; even then, he did not associate the multiples directly with the electromagnetic waves. That association was made by Einstein, 1921 Nobel Prize winner in Physics and father of relativity. These packets, or quanta, of energy are universally named photons and have an energy hf, where h is called Planck's constant and f is the frequency of the radiation. With the introduction of this constant into his formula, Planck described the experimental data on black bodies exactly. In 1918, Planck won the Nobel Prize in Physics for this monumental work.

The connection of waves with quantized energy and momentum (both quantities associated more intuitively with particles) naturally prompts the question whether particles, such as electrons, protons, and neutrons, are themselves waves, which they are. In the hands of Niels Bohr (1885-1962), Erwin Schrodinger (1887-1961), Paul Adrien Maurice Dirac (1902-1984), and others, ideas about waves, particles, and quantized energy, prompted by the study of black body radiation, produced the full quantum theory.

In 1917, Einstein announced a derivation that showed that Planck's formula for black body radiation demanded that the radiation field must be able to stimulate radiation from excited atoms. Excited atoms radiate quite spontaneously even if there is no radiation present. Rather naturally, this process is called spontaneous emission. Einstein noted that Planck's law required that any radiation present, at the spontaneous frequency, must induce the excited atoms to emit their radiation more rapidly than at the spontaneous rate. The more radiation that bathes the atoms, the faster the stimulated emission.

When the quantum theory is applied to atoms, the result is that only certain sharply defined, or quantized, energy states occur. Where calculations can be made, the energy difference between these discrete states is in full agreement with the quantized photon energies observed as emitted from various atoms. The possibility of stimulated emission induced by radiation implies that the quantized photons can be amplified when they pass excited atoms--in discrete high-energy states--that are ready to emit at the frequency of the incoming photons. The amplification will take place as long as atoms populate fewer lower-energy states than upper states, preventing absorption of the stimulated photons from outweighing their production. This condition of more atoms in a higher-energy state than a lower one cannot occur at equilibrium, which characterizes black bodies, but it happens very often in practice when processes prevent equilibrium from being reached. Yet, if excited atoms are placed between mirrors aimed at each other, the excited atoms can be pumped in a nonequilibrium manner so that they maintain more upper states than lower ones, while the amplification takes place; a laser is then produced. This process is extraordinarily simple. Many after Einstein noted that black body radiation demanded stimulation emission; Theodore Maiman put that knowledge to practical use. In 1960, he silvered the ends of a 5-centimeter long ruby crystal, illuminated the crystal with the intense light from a flashlamp encircling it, and observed the brilliant red light of the first laser.

Principal terms

ELECTROMAGNETIC SPECTRUM: the range of radiant, coupled electric and magnetic waves traveling at the speed of light; examples, with decreasing wavelength, include radio, radar, infrared, visible light, ultraviolet, X rays, and γ rays

EQUILIBRIUM: radiant equilibrium occurs when matter and radiant energy interact at a single, uniform temperature that is constant in time

QUANTUM MECHANICS: classical quantum mechanics ascribes both a wave and particle nature to all particles and waves

RADIATION: emission of energy as waves from a source; electromagnetic radiation travels at the speed of light, which is the product of wavelength and frequency

TEMPERATURE: a measure of random, equilibrium energy content of a material or wave; the lowest temperature is zero in an absolute temperature scale

Bibliography

Baker, Adolph. MODERN PHYSICS AND ANTIPHYSICS. Reading, Mass.: Addison-Wesley, 1970. A readable account of several topics in up-to-date physics, including particles and antiparticles. The wave-particle duality, prompted by Planck's study of black body radiation, is still an up-to-date mystery that is explored for the general reader in chapters 12 through 15. Contains an appendix, and there are questions along with selected answers.

Eisberg, Robert Martin. FUNDAMENTALS OF MODERN PHYSICS. New York: Wiley, 1961. An upper-level physics textbook. Chapter 2 gives a thorough exposition of black body radiation. Necessary details are given, with the necessary mathematics.

Feynman, Richard P. QED: THE STRANGE THEORY OF LIGHT AND MATTER. Princeton, N.J.: Princeton University Press, 1985. A readable account of the nature of light, photons, quantum theory, and of the unresolved mystery in their behavior. The behavior is explained beautifully and the mystery, in as simple an event as the reflection of light, is described clearly by a pioneer of modern quantum theory.

Rusk, Rogers D. INTRODUCTION TO ATOMIC AND NUCLEAR PHYSICS. New York: Appleton-Century-Croft, 1958. An introductory physics text. Chapter 12 covers radiation, while sections 12-1 to 12-6 review black body radiation in some detail and present its importance in the formulation of quantum theory. There are some equations but most are understandable.

Weinberg, Steven. THE FIRST THREE MINUTES: A MODERN VIEW OF THE ORIGIN OF THE UNIVERSE. New York: Basic Books, 1977. A well-written account of the understanding of the origins of the universe in popular language. Chapter 3 discusses black body radiation and its importance in cosmic background radiation. There are many useful pictures, diagrams, and tables, along with a glossary of terms.

The Effect of Electric and Magnetic Fields on Quantum Systems

Optical Properties of Solids

Thermal Properties of Matter