Radiation Transport

Type of physical science: Radiation Transport, Classical physics

Field of study: Electromagnetism

Radiation transport studies the transfer of energy by electromagnetic radiation and is often called "radiation transfer." This process mainly determines the energy balance of Earth's atmosphere, is critically important in Earth's interior, and strongly influences the structure of stars. Practical applications include heating systems, lighting, lasers, and the understanding of global warming.

Overview

A simple illustration of radiation transport is light reaching Earth from the Sun, the passage of electromagnetic waves through seemingly empty space and sky. Even this simple radiation transport involves substantial interaction of radiation with matter. To escape the Sun, radiation must pass through a caldron of seething gases at the Sun's surface, then travel the tenuous solar wind, and finally bridge Earth's hardy atmosphere. Even on the clearest day, the ozone layer high in the atmosphere blocks much of the Sun's ultraviolet radiation, heating that layer of the atmosphere. Enough near-ultraviolet radiation remains to tan sunbathers at the beach, while visible and infrared rays warm the sand. Earth, its atmosphere, and its life drink in energy in the form of heat from the Sun, 93 million miles--or 8 minutes of light travel--away.

Radiation transport, conduction, and convection are the three means by which heat energy moves from one place to another. Heat conducts along a metal frying pan handle, while the swirling smoke from a cigarette end convects heat upward. Of these three, only radiation transport requires no physical contact between the energy source and the receiver. High temperature strongly favors radiation transport over conduction and convection in moving energy, even in dense solids. Thus, radiation transport dominates energy transfer in portions of many stellar interiors and deep within Earth's mantle.

Matter must be present at the radiation source and the reception point for radiation transport to take place. Emission of radiation by atoms or molecules occurs at the source, and the inverse process of absorption takes place at the receiver. If matter gets in the way of the radiation, as it normally does, scattering takes place along with intermediate emission and absorption. Radiation transport thus involves a complicated exchange of radiation energy with atoms and molecules by emission, absorption, and scattering.

Atoms exist in a variety of discrete energy states (while this discussion refers primarily to atoms, everything said applies to molecules as well). At low temperatures, most atoms are in their lowest energy state, but heating matter drives many atoms to higher-energy, excited states. An excited atom may lose its energy by emission of electromagnetic radiation to a lower state. The lost energy appears as a photon of radiation, which carries the energy away at the speed of light. The photon travels until it hits an atom, nearby or far away, in an energy state that allows absorption; the photon then deposits its energy, leaving the atom excited. The emitting and absorbing atoms need not be of the same type, but the energy differences in their upper and lower states and the photon energy must match to conserve energy. Strong radiation absorbers appear black at the absorption frequency and are the strongest emitters.

Absorption of a photon by the electron inside a single atom in its lowest state typically requires that the photon have high energy, often placing the photon in the ultraviolet. In molecules, photon absorption may also take place by the component atoms themselves. Such molecular absorption requires very low energy, and the absorption falls in that part of the infrared, far from the visible. Thus many materials, such as glass and water, are transparent in the visible but are black in the far infrared.

At times, the atom (or molecule) re-emits the photon before the absorption is complete. The photon continues on its way, scattering in a new direction. Scattering is, thus, intimately connected to absorption and emission. When photon and atom energies do not quite match, scattering dominates absorption.

Scattering normally slows the forward progress of photons and produces a refractive index in the scattering material, given by the ratio of the speed of light in vacuum to that in the material. As a consequence, part of the incident light reflects at the material surface, and the remaining light bends, or refracts, as it enters the surface.

The refractive index of water is 1.33, and reflection and refraction at the surface of water droplets produce light scattering in the atmosphere. At sea level, the atmosphere itself has a refractive index of 1.00029, just barely above unity. However, the very slight bending of sunlight by the atmosphere is great enough to allow the Sun's image to linger several minutes after the Sun has dipped below the horizon at sunset.

The laws of radiation take their simplest form when a body is in equilibrium so that its constituents possess a single temperature, uniform throughout, and when its emission properties are uniform throughout its radiation spectrum. The electromagnetic energy radiated by such an equilibrium body--for example, the Sun's outer layer, or an incandescing filament--follows a formula derived by the German physicist Max Planck (1858-1947), based on the photon as the unit of electromagnetic energy. Planck's radiation formula is often referred to as the "black body law."

A black body absorbs all the energy that falls on it. Since emission and absorption are proportional to each other, a black body emits the maximum energy possible at any frequency in the electromagnetic spectrum. Planck's law demonstrates that an equilibrium body radiates a total energy, across the entire spectrum, at a rate proportional to the fourth power of its temperature, to the area of the body, and to its frequency-independent emissivity (this is summarized in a related equation, the Stefan-Boltzmann law). The emissivity of a body is the ratio of its emission, and consequently absorption, to that of a black body. An additional consequence of Planck's law requires the product of the temperature of an equilibrium body and wavelength at its radiation peak to be constant independent of the emissivity (Wien's law).

The Sun's surface temperature is near 5,800 Kelvins, or about 10,000 degrees Fahrenheit, and is well represented as a black body. At this temperature, the Sun radiates its peak energy in the yellow-green spectrum at about 0.5 micrometers (1 micrometer is one millionth of a meter), matching the peak response of the human eye. The total power radiated is 640 watts per square centimeter, or 64 million watts for each square meter. Cut the temperature of any black body in half, and the radiation emitted for each unit area drops by one-sixteenth, while the peak wavelength doubles.

The surface of the star Antares and the filament in an incandescent lamp both have a temperature about half that of the Sun's surface. Antares is a black body, but its copious radiation obscures the blackness. However, an unmounted filament is gray. A gray body absorbs some fraction of the radiation falling on it given by its emissivity, which is typically one-third that of a hot filament. When lit, the filament, half the temperature of the Sun, emits one-third as much as a black body (133 watts per square centimeter) and requires about a 0.6-square-centimeter area to give off 80 watts of radiation (as in a typical 100-watt lightbulb). The extra 20 watts goes to conduction and convection. The peak wavelength of the gray-body filament radiation occurs at 1 micrometer, in the near-infrared, producing heat felt from a nearby incandescent lamp.

The most extreme departure from black-body, or gray-body, equilibrium radiation takes place when atoms radiate to give off a line emission. The photons cover a very narrow frequency range emitted from a single excited state. In effect, the body is black for frequencies within the line but is transparent outside the line. In an extended medium such as a hot gas or plasma, the line emission broadens as it moves through the medium. If the unexcited atoms of the material are in the lowest energy state, they all can absorb the line radiation, which then finds itself strongly imprisoned. The radiation progress slows considerably as a result. The radiation transport may still be handled without great difficulties, despite the line broadening, if there is a single temperature that describes absorption and emission in the medium.

When equilibrium does not apply, the equations of radiation transport may become very complicated and can tax the largest computers. In simplified terms, the basic radiation-transport equations used give the radiation at any location as a consequence of two parts. The first part is radiation directly from a source, but with account taken of the absorption and scattering along the path from the source. The second part arises from all the indirect radiation, re-emitted after absorption and scattering through the path to the source, once more allowing for absorption and scattering along all intermediate paths to the location. In the most general case, the absorption and emission may depend very strongly on frequency and may depend on other variables, some of which may be poorly known.

Applications

An important application of radiation transport is the warming of Earth by the Sun. It is possible to estimate Earth's temperature by balancing the radiation received from the Sun against that radiated by Earth into space. For simplicity, assume that Earth has no atmosphere, that its rotation keeps its temperature uniform, and that its emissivity is independent of wavelength. The radiant energy leaving the Sun's spherical surface area, proportional to the fourth power of its temperature, spreads out until it covers the whole area of a sphere at Earth's orbital radius. At Earth's orbit, the solar power falls to about 1,370 watts per square meter, the "solar constant." The disc of Earth absorbs the solar energy and re-radiates into cold space at a rate proportional to the fourth power of Earth's temperature and to its global surface area. Balancing Earth's absorption and radiation gives the ratio of Earth-to-Sun temperature as one-fourth power of the ratio of Sun surface area to four times the total orbital area at Earth's radius from the Sun. The factor of four is the ratio of emitting spherical area to absorbing disc area for Earth. If Earth's emissivity is the same at the absorption and emission wavelengths, its value does not influence Earth's temperature, since higher absorption means higher emission. Taking the Sun's radius ,000 miles), temperature ,Kelvins), and distance (93 million miles) gives the temperature of Earth as 280 Kelvins (7 degrees Celsius, or 45 degrees Fahrenheit).

This simple approximation is not bad. The average temperature of Earth is 288 Kelvins (15 degrees Celsius, or 59 degrees Fahrenheit). On the Kelvin absolute-temperature scale, the agreement appears good. However, small changes in temperatures produce important effects on Earth, and these changes are more easily noted in the more familiar Fahrenheit and Celsius scales. A temperature difference of 14 degrees Fahrenheit produces large differences in comfort in a house, in survival in a blizzard, or in ocean levels as polar ice caps melt or freeze, for example.

Radiation transport plays an important role in the physical processes within the deep Earth, despite the rocks there. The temperature in the solid mantle of Earth, from about 500 to 1,700 miles below the surface, varies from 1,800 Kelvins to nearly 4,000 Kelvins. The strong radiation emitted at these temperatures is in the near and middle infrared, where many rocks are transparent. As a result, radiative heat transport competes effectively with conduction through the solid rock, and with convection, in determining the temperature of this portion of Earth's interior.

Stars similar to the Sun are heated at their cores by thermonuclear furnaces above 10 million Kelvins. The heat from these furnaces flows from their cores to their much cooler surfaces and is then emitted into space. At the huge temperatures of the core and intermediate interior, radiation transport dominates the energy flow. As with the Sun, convection then brings the energy the remaining distance to the luminous surface.

Back on Earth, the applications of radiation transport are numerous. Such diverse examples as fireplaces, thermal panels, Thermos bottles, and the night sky illustrate the simplest applications. A normal fireplace warms a room by the infrared radiation emitted from the burning logs, the flames, and the hot walls, while the convection heat goes up the chimney. Heat panels are often designed to maximize infrared, which radiation transports to heat the room. A vacuum Thermos bottle, on the other hand, suppresses heat transport in (or out of) the bottle. This type of bottle is an evacuated and silvered, double-walled glass container. The vacuum eliminates heat conduction and convection between the inner and outer walls. Thin metallic films line the outside of the inner wall and the inside of the outer wall. Being reflective, these films have low emissivity, sharply reducing the radiative heat transport between the walls. When present in the night sky, clouds reflect and absorb far-infrared radiation leaving the planet's surface. The clouds prevent the radiation from escaping into space and re-radiate an additional portion back to the surface, maintaining surface warmth in the winter. Remove the night clouds, and Earth chills.

Examples of narrow-line radiation transport appear in fluorescent lamps and in lasers. A fluorescent lamp produces white light from a phosphor coating lining the inside of the lamp envelope. The phosphor is excited by a single narrow ultraviolet line at 0.254 micrometers, at an efficiency of about 60 percent. The free electrons, which carry the electrical current, are extremely hot and are in approximate equilibrium at nearly 12,000 Kelvins--about twice the temperature of the Sun's surface. These light electrons, accelerated by the mercury discharge's electrical field, readily bounce off the much heavier atoms of the discharge and deliver little energy directly to the atoms. These electrons, however, readily transfer their energy to the electrons bound inside the atoms, exciting the ultraviolet energy levels in the mercury atoms. Interestingly, the peak wavelength of a true black body at the electron temperature would fall at the ultraviolet wavelength. Since this line radiates to the lowest energy level of mercury, it is strongly trapped. As the radiation moves toward the wall, the line width broadens; depending on details of operation, the line has its lifetime extended from thirty to fifty times.

Lasers are line emitters that operate as far as possible from equilibrium. The population of atoms in the upper radiation energy state must be greater than that of the lower state, which is impossible in equilibrium. When laser radiation falls on an atom populating the upper state, it cannot be absorbed, since there is no electron in the lower level to take up the energy. Instead, the radiation produces stimulated emission of another photon at the same frequency. The process amplifies, rather than absorbs, radiation that is transported along the axis of the laser, resulting in the characteristic narrow, brilliant beam associated with laser light.

Context

The scientific understanding of radiation transport began in 1791, when Pierre Provost introduced the theory of exchange. This theory postulated that all bodies radiate energy and that equilibrium occurs by an exchange or transport of this heat energy between the bodies to achieve a common temperature. Based on experiments by others, the Austrian physicist Josef Stefan in 1879 postulated that equilibrium radiation was proportional to the fourth power of temperature. In 1884, another Austrian physicist, Ludwig Boltzmann, gave a theoretical derivation of this relation. In 1898, W. Wien deduced that wavelength and absolute temperature should be constant for equilibrium radiation. Finally, the German physicist Max Planck gave his derivation of black-body radiation on December 14, 1900, erecting a satisfactory theoretical basis for equilibrium radiation and, at the same time, initiating quantum theory. In 1917, Albert Einstein showed that stimulated emission must occur (in addition to absorption and normal emission--and, by implication, scattering), in order for equilibrium to take place. The present laws of radiation transport are guided by Planck's law and Einstein's relations.

Radiation transport plays its most pressing role for life on Earth in the realm of global warming. The simple calculation above showed the Sun's dominant role in determining Earth's temperature. Yet that simple calculation ignored several factors that are crucial in determining a precise Earth temperature. The atmosphere is, of course, present, and human influence on the atmosphere strongly affects radiation transport through the atmosphere and any global warming that might result.

The emissivity of the planet differs widely between the visible, near-infrared wavelengths, where absorption of solar energy takes place, and the far-infrared wavelengths at which Earth emits. Of the solar radiation falling onto Earth, the surface absorbs 50 percent, while the atmosphere and its clouds absorb another 20 percent, and the remaining 30 percent is scattered and reflected into space; this last value known as the "global albedo." Water and other molecules, such as ice, carbon dioxide, ammonia, dust, and many products of manufacturing and automobile exhaust, are often black in the far infrared. In the atmosphere, they absorb surface radiation, trap it, and return a large fraction back to Earth, producing the "greenhouse effect." Once deep-infrared radiation reaches the top of the atmosphere, it radiates substantially as a black body, but it does so at the temperature of the high atmosphere, which is much lower than that of Earth's surface.

In order to make precise predictions of rates of global warming, realistic computer projections require accurate calculations of radiation, conduction, and convection transport through the atmosphere, along with complicated chemical analyses. These calculations are needed on scales as small as a hundred miles of Earth's surface and throughout the full Earth-solar radiation spectrum. In addition, the influence of other dimly understood factors must be incorporated into such calculations. The task is awesome, but many observers believe that the mission is imperative.

Principal terms

ELECTROMAGNETISM: Waves, composed of both electric and magnetic fields, that travel at the speed of light; in order of increasing frequency, the electromagnetic spectrum consists of radio waves, television waves, infrared waves, visible light, ultraviolet waves, X rays, and gamma rays

HEAT: Energy that flows from a hot body to a cold body; heat may flow by atom contact, in conduction or convection, or by radiation transport of electromagnetic waves

PHOTON: The smallest unit of electomagnetic energy; photon energy increases with frequency

RADIATION: In this context, electromagnetic energy liberated from a source; common sources include atoms and molecules in hot solids and gases

TEMPERATURE: The measurement of how hot or cold a body is; common temperature scales are the Fahrenheit, Celsius, and Kelvin scales

Bibliography

"Physics and the Environment." Physics Today, November, 1994. This special issue includes five articles and an introduction on the environment. One article discusses the nitrogen cycle and its impact on the greenhouse effect, while another covers the effect of clouds on radiative transport in the atmosphere. Semitechnical, but quite readable.

Siegel, R., and J. R. Howell. Thermal Radiation Heat Transfer. 3d ed. Washington, D.C.: Hemisphere, 1992. A very thorough text, primarily for technical use. However, chapters 14 and 15 cover nonequilibrium radiation transport in detail.

"Tropospheric Processes." Science, May 16, 1997. This issue carries nine articles on processes in the upper atmosphere influencing the greenhouse effect on Earth's climate. One short article covers clouds, radiative processes, and climate, while another discusses transport processes in the upper atmosphere. Contains an excellent short summary of global warming and a fine article on computer modeling of the greenhouse effect.

White, Harvey E. Modern College Physics. 5th ed. New York: Van Nostrand Reinhold, 1966. Chapter 20 of this undergraduate text covers heat transport by conduction, convection, and radiation and related atmospheric effects. Elementary and quite readable, with only a few simple equations and numerous fine illustrations.