Fundamental Constants of Nature

Type of physical science: Astronomy; Astrophysics; Atomic physics; Classical physics; Condensed matter physics; Elementary particle (high-energy) physics; Nuclear physics; Relativity

The fundamental constants of nature, such as the velocity of light and the electron charge and mass, determine the behavior of nature's varied phenomena. Armed with the values of the fundamental constants and the natural laws from which they derive, scientists and engineers can calculate the course of events.

Overview

The fundamental constants of nature obtain their values from measurements that scientists make using physical laws that govern nature. Given the fundamental constants, scientists can calculate the imprint of the laws of nature on the march of events. Some of the fundamental constants include the gravitational constant; the three electromagnetic constants, permittivity and permeability of space and light velocity; the two electron constants, its charge and mass; Planck's constant; and finally, Boltzmann's constant.

Measurements require specification of the scales, or units, that enter into an experiment. Velocity is measured by the distance moved by an object over time, and measurement of the fundamental constant, velocity of light, requires specified distance units and time units. Some distance units are inches, meters, kilometers, and miles, while time units include seconds, hours, days, and years. The velocity (speed) of light has a value very near 300,000 kilometers per second. For practical reasons of communication, international scientists have agreed among themselves on a common set of units. This widely accepted Systeme International d'Unites, or SI, consists of the meter for length, second for time, and the kilogram for mass. The meter is slightly larger than 39 inches, while the kilogram is about 2.2 pounds of mass. These three units are supplemented by the ampere for electrical current, the Kelvin for absolute temperature, and the candela for light luminosity supplement. Electrical current is the flow of electrical charge, and 1 ampere corresponds to an electrical charge of 1 coulomb flowing in one second. The Kelvin absolute temperature scale sets its zero at the coldest temperature possible so that water freezes at about 273 Kelvins. Finally, the candela measures the light emitted by a source per unit of solid angle. When lit, a 100-watt incandescent lamp draws a current of about 0.9 ampere, emits an averaged 135 candela, and has a filament temperature near 2,800 Kelvins.

The oldest law used in the discussion of fundamental constants is Sir Isaac Newton's law of gravity. The gravitational constant derives from Newton's law and measurements of the attractive force between two known masses separated by a fixed distance. The experiments are inherently difficult and are illustrated by the torsion balance experiment of the English physicist and chemist Henry Cavendish (1731-1810).

In the torsion balance, a pair of equal, small masses hang on opposite ends of a rod suspended at its center from a thin fiber. The two masses counterbalance each other. A second pair of large masses is moved near the small masses. These large masses hang counterbalanced along the same axis as the small masses. The small masses move toward the heavy ones and the back torque of the fiber holding the light masses offsets this movement. The small masses settle into a new position, and the change determines the attractive gravitational force, once the torque of the fiber is calibrated. The gravitational constant derives directly from the gravitational force just measured and from the known masses and distances. Improvements on the original torsion balance include the use of an oscillating torsion balance. The accuracy of the gravitational constant is better than one part in about ten thousand, but, because of the difficulty in its measurement, it is the least precise of the constants discussed.

James Clerk Maxwell (1831-1879) codified the set of laws that unified electricity and magnetism and predicted a value for light velocity. Electrical charge is the quantity of electricity in a substance and may be positive or negative. Like charges repel, while opposite charges attract. The electrical forces set up by the charges produce electrical fields that permeate empty space. The flow of electrical charge is electrical current; such currents set up magnetic fields. As Maxwell knew, the laws governing electrical and magnetic forces resemble Newton's law of gravity. Maxwell found that light was electricity and magnetism in motion. Because of Maxwell's efforts, it is known that light is one form of an electromagnetic wave, an oscillation of electric and magnetic fields coupled in space. These waves include radio, television, radar, light, infrared, ultraviolet, X rays, and γ rays, in order of higher frequency or oscillations per second. All electromagnetic waves have the same speed as light.

In principle, measurements of electrical and magnetic forces give the electric and magnetic constants by a manner similar to measurement of the gravitational constant. The electric and magnetic constants are the permittivity (dielectric constant) and permeability of space, respectively. Nevertheless, scientists simplified their task of obtaining these two constants and, as a result, obtained exactly accurate values. There is freedom in choosing the units that define electric and magnetic quantities, primarily the electrical current. By defining this unit properly, it is possible to obtain one electromagnetic constant by fiat. The constant chosen is the permeability of space. Rather than attempting to measure the remaining permittivity in a manner similar to the gravitational constant, scientists appealed to Maxwell's genius to get a much more accurate value for this constant. Maxwell gave a fascinating relation between the velocity of light and the two constants; the square of light speed is the reciprocal of the product of permeability and permittivity. Since the speed of light has, historically, been measured with extraordinary precision, the permittivity simply is calculated from light speed. Nevertheless, in 1983, scientists defined the speed of light as exactly 299,792.458 kilometers per second.

The speed of any wave is the product of its frequency (oscillations per second) and wavelength (distance per oscillation). Accurate measurement of frequency requires precise timing, while measuring wavelength requires precise distances. Initial measurements of the speed of light consisted in finding the time that a light pulse took to travel a fixed distance. As precision increased in measuring times and distances, highly accurate values for light speed came by measuring both frequency and wavelength and multiplying these to get the velocity.

By the 1980's, scientists measured electromagnetic frequency in the visible and infrared so precisely that it was logical to define the velocity of light. Armed with a defined velocity of light and a precisely measured frequency, scientists calculate the companion wavelength precisely. They then mark off distances by counting the number of precise wavelengths that fit the desired distance or by measuring the precise time that light takes to travel the distance. Thus, in 1983, light velocity replaced the original definition of the meter. Up to that time, the meter was the length of a platinum-iridium bar in the vaults of the International Bureau of Weights and Measures near Paris. With the definition of the velocity of light, all three electromagnetic quantities are now exact.

The evolution in measurement techniques also shows up in the electron constants. The electron is an extremely light, negatively charged fundamental particle. It gives shape and activity to all atoms and molecules and is the agent responsible for most chemical reactions. In addition, its light mass makes the electron the main carrier of electrical current in everyday materials. The charge on all electrons is the same, a condition that scientists use to fix "gauge" requirements on the modern laws of nature. This same charge, or its multiples, appears on all atomic particles, either positive or negative. Subatomic quarks, however, have charges that are multiples of one-third the electron charge. All electrons at rest have the same mass. While no laws are known that fix the value of the electron's charge and mass, applying Newton's law of gravity and Maxwell's laws of electromagnetism determined these values initially.

The classical measurement of the electron charge was made by the American physicist Robert Andrews Millikan (1868-1953) in 1906. Millikan sprayed minute oil drops from an atomizer into the region between two metal plates, charged by high-voltage batteries so that the upper plate was positive and the lower one negative. He produced electrons in the same region by exposure to X rays. At times, the oil drops picked up one, two, or several electrons. He then illuminated the oil drops and followed one with a low-powered microscope. Without any charge, the oil droplet fell under gravity, and its speed through the air determined the mass of the droplet.

If the droplet picked up an electron, the electrical force on the droplet's electron caused by the charged plates opposed gravity. The droplet reversed its fall, and its upward velocity provided the difference between the electrical and gravitational force. Since Millikan knew the gravitational force in the fall, he could find the electrical force and the electron's charge. By switching off the electrical charge on the plates, he could allow the droplet to fall and repeat the experiment. Millikan obtained the first accurate value for the electron charge and discovered that it was negative and came in multiples as extra electrons crowded onto the droplets.

The English physicist Sir Joseph John Thomson (1856-1940) made one of the first accurate measurements that led to finding the mass of the electron. In 1897, he measured the magnetic deflection of an electron beam in the vacuum of a cathode ray tube, the predecessor of the television tube. The deflection, beam length, and magnetic field determine the ratio of electron charge to mass, a ratio that is ubiquitous in electronic applications. Combined with Millikan's later value for the electron charge, Thomson's measurement yields the electron mass.

The electron is very light with a mass slightly less than 1/1,836 of the proton mass, even though the positive proton has the same charge as the electron.

The modern, extremely precise determination of the electronic charge and mass is derived by very different measurements. Quantum mechanics dictates that the minute electron spins around. The spinning electron charge is equivalent to a current, and that current generates a magnetic field. An electron is both charge and magnet. Maxwell's classical theory gives a value for the spinning electron's magnetism, but quantum mechanics comes into play to alter magnetism slightly. The slight alteration can be measured with extraordinary precision. The electron charge can be calculated from the measured alteration. Since the basic effect is electromagnetic, the exact values of the velocity of light and the permittivity of space enter the calculation, and since quantum mechanics plays a role, Planck's constant is introduced.

The German physicist Max Planck (1858-1947) introduced his constant in 1900 to describe the radiation emitted by hot, black bodies. The data forced Planck to propose that energy came in packets, or quanta. The energy of the packet is Planck's constant times the frequency of the packet. Quantum theory thus ascribes a frequency and wave character to objects--such as electrons and atoms--that are normally thought of as matter and gives energy packets to pure waves--such as light--calling them photons.

By fitting the radiation data, Planck's theory gave an accurate value for his constant.

Later measurements of quantum phenomena produced prodigious improvements in the value.

Superconductors are large quantum systems in which electron waves pair together to allow current flow without resistance and without energy loss. Current will even flow between two superconductors separated by thin insulating regions. When the superconductors experience a voltage difference V between them, Planck's quanta appears. Their frequency f times Planck's constant equals the energy created when transferring the electron pair across potential difference (hf = 2eV). This superconducting quantum effect is the Josephson effect (named for Brian D. Josephson) and is measured with extreme precision. In fact, the U.S. National Bureau of Standards (now, National Institute of Standards and Technology) adopted the ratio 2e/h = 483,593.420 gigahertz per volt in 1972 to set its standard voltage. This ratio and the electron anomalous magnetic moment fix both the electron charge and Planck's constant.

The particles of electromagnetism and light, electrons and photons, often interact in large objects, such as a heated body, where the temperature controls the statistical behavior of the mixture. Ludwig Boltzmann (1844-1906), an Austrian physicist, described the behavior of these large groupings and developed Boltzmann's constant. Boltzmann's constant multiplied by absolute temperature is energy. Classical physics ascribed half this energy, on average, to each possible pattern of particle motion or wave oscillation in the large interaction, as long as each pattern had a continuous energy content. Planck used Boltzmann's statistics but quantized the energy content to derive the black body energy emission. Measurement of black body emission, use of light velocity, and Planck's constant give Boltzmann's constant. The measurement is difficult and the value of Boltzmann's constant is the second least precise constant, after the gravitational constant.

Applications

As one might expect, the use of the fundamental constants appears everywhere in science and technology. The table summarizes the values of the constants and their units. Powers of ten notation is used to give the value. The number 10⁸ is ten raised to the power eight and is the same as the number one followed by eight zeros or 100 million. The negative power means divide into one, so that 10^-6 is the same as 1 divided by 10⁶ or one millionth. The gravitational constant is slightly uncertain in its last digit, but the other constants are exact to the four figures given.

The gravitational constant appears constantly in detailed calculations involving galaxies, stars, planets, and natural and artificial satellites. The space program cannot run without it. It influences everything above, on, and inside the earth, often through its companion constant, the acceleration caused by earth gravity. The permittivity of space is evident in electrical calculations and effects. Since electrical forces hold matter together, materials science needs it.

Permeability controls magnetic forces between conductors and magnets. The velocity (speed) of light governs all electromagnetic radiation, from the radar used by a policeman to check a driver's speed to the light from a distant quasar seen perhaps 12 billion years after its initial appearance.

Electronic devices almost always move electrons through materials or empty spaces inside devices. The low ratio of electron charge to mass allows electrons to gain high speeds with relatively low voltages. The electrons appear through dense matter in an electronic computer but approach a small fraction of the velocity of light in a television set.

Context

The interplay of the physical constants weaves together the separate and incomplete picture that scientists have of nature. Not only do the fundamental constants allow scientists to calculate accurately with the specific laws from which they derive but also they form a network with which scientists seek to obtain a more coherent picture of nature. In addition, the interplay of the constants supplies surprising predictions about nature and raises questions.

When atoms emit or absorb light, the effect is both electromagnetic and quantum. In 1917, Albert Einstein, using Planck's formula for the radiation from a black body, deduced that atoms must be capable of amplifying light. The ratio of the amplified to normal emission of light is porportional to the third power of the electromagnetic wavelength divided by Planck's constant. With this deduction, Einstein laid the basis for the eventual appearance of lasers.

Quantum waves give stability to atoms. Otherwise, negative electrons, as particles circling the positive nuclear protons that hold them, would spiral rapidly down into the nucleus and the atoms would disappear quickly. Quantum provides stability by demanding that electron waves have lengths that match the distance around the nucleus. A minimum atom size appears and no collapse takes place. That size is the Bohr radius, named for the Danish physicist Niels Bohr (1885-1962). The radius must combine the electromagnetic constants, the electron's constants, and Planck's quantum constant: Planck's quantum constant: e₀h/πm_ee², with π as the mathematician's constant 3.14159265. The diameter of hydrogen computes to slightly more than one hundred-millionth of a centimeter; most atoms are somewhat larger.

Physicists are attempting to determine if it is possible to unite gravity with quantum in the same way that electromagnetism was united with quantum theory. The search for the theory continues, but Planck gave physicists a start. He combined the gravitational constant with the velocity of light and his own constant to obtain a ratio with the units of distance. The distance is the Planck length and is given by (hG/2πc³)^1/2. The value is 1.616 x 10^-35 meters, an extraordinarily small distance and perhaps the distance where quantum unites with gravity. Einstein's theory of gravity states that gravity is a warp in space. If the argument concerning stability of the atom is a guide, this measurement also may be the distance at which quantum will give stability to the warps of gravity. Then, paradoxically, the emptiness of space might never penetrate the realms beneath Planck's distance.

Principal terms

BOLTZMANN'S CONSTANT k sub B: characterizes the statistical and thermal properties of matter and of electromagnetic waves

ELECTRON CHARGE e: the quantity of electricity in the electron; all atomic and nuclear charges come in positive and negative multiples of the electron's charge

ELECTRON MASS me: a fixed quantity of matter in the stationary electron

GRAVITATIONAL CONSTANT G: the constant in Newton's law of gravity that determines the gravitational attraction of two masses separated from each other

PERMEABILITY OF SPACE μ 0: determines magnetic attraction or repulsion between two separated sections of electrical charge flow or current

PERMITTIVITY OF SPACE ε 0: determines the electrical attraction or repulsion between two electrical charges separated from each other

PLANCK'S CONSTANT h: marks the quantum domain of nature

VELOCITY OF LIGHT c: the speed of electromagnetic waves, including light, through empty space

Bibliography

Baker, Adolph. MODERN PHYSICS AND ANTIPHYSICS. Reading, Mass.: Addison-Wesley, 1970. A readable account of several topics in physics, including particles and antiparticles. The material serves as very interesting background to some of the physical constants. Light and quantum mechanics are explored for the general reader in chapter 5 and chapters 12 through 15. Questions along with selected answers appear in the appendix.

Cohen, E. Richard, and Barry N. Taylor. "The Fundamental Physical Constants." PHYSICS TODAY 43 (August, 1989): BG9. This short article gives a concise review of the values, at the time of its writing, of a complete list of the fundamental constants, along with the error in these values. The article presents a review of the status in determining the constants. Cohen and Taylor are long-standing authorities in the compilation and analysis of data on the fundamental constants.

Taylor, Barry N. "Fundamental Constants." In ENCYCLOPEDIA OF PHYSICS, edited by Rita G. Lerner and George L. Trigg. Reading, Mass.: Addison-Wesley, 1981. This is a definitive reference on the evaluation of the fundamental constants by one of the evaluators. Although the constants have been revised since the publication of the article, the evaluation technique is essentially the same, and the alterations in value will not be noticed in most contexts. Technical reading.

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 an understanding of the origins of the universe. Chapter 3 discusses black body radiation and its importance in the cosmic background radiation. There are many useful pictures, diagrams, and tables, along with a glossary of terms. The mathematical rigor is available in the appendix. Easy to understand.

White, Harvey E. MODERN COLLEGE PHYSICS. New York: Van Nostrand Reinhold, 1966. An introductory physics text. The index should be reviewed to find the constant or related law of interest. White gives a short, readable account of the subject, often with a brief sketch of the prominent scientist involved. The history is brief, the physics clear, and the equations simple.

Values of the fundamental physical constants

Charges and Currents

Electrons and Atoms