Fermions

Type of physical science: Fermions, Elementary particles, Particles, elementary, Elementary particle (high-energy) physics

Field of study: Systematics

Fermions are elementary particles that have odd-half-integer intrinsic angular momentum, or spin. As a consequence of this peculiar angular momentum, fermions obey the Pauli exclusion principle, which states that no two fermions can exist in the same quantum state at the same time. Many of the properties of ordinary matter arise because of this rule. Collections of fermions are described by Fermi-Dirac statistics.

Overview

The classification scheme for elementary particles consists of two main branches, fermions and bosons. Fermions are particles that have odd-half-integer values of spin. All the known fundamental matter particles are classified as fermions. In the standard model of fundamental particles and interactions, the fundamental fermions consist of the six quarks and their antiparticles and the six leptons and their antiparticles.

In contrast, bosons are particles that have integer values of spin. The fundamental bosons include the photon, the intermediate-vector bosons (W⁺, W^-, Z^o), the gluons, and the graviton. While the intermediate-vector bosons and the gluons have mass and as such are considered as matter, the fundamental bosons are generally thought to be the mediators of the four fundamental forces rather than the constituents of ordinary matter.

It is important to understand that the word "particle" can have different meanings depending upon the context. From a historical or classical point of view, a particle is any object localized to a finite region of space. With the invention of the atomic hypothesis by the Greeks in the fifth century b.c.e. and its modern formulation by the English chemist John Dalton in the nineteenth century, the word "particle" came to be synonymous with the atom. Finally, in the late nineteenth and twentieth centuries, the discovery of subatomic particles led to a further evolution in the meaning. For example, the earth is considered to be a particle that is made up of other particles, atoms, which are in turn made of smaller particles yet, the quarks and leptons. Many physicists believe that only the fundamental fermions and bosons are the "true" particles of the universe, because they have no observed internal structure. All other "particles"--the baryons, mesons, nuclei, atoms, molecules, microscopic objects, and macroscopic objects--contain observable internal structure and are not considered fundamental.

When collections of fundamental fermions are studied and classified, the resulting systems can be considered as fermionic or bosonic depending on the total spin. For example, the proton is a composite particle consisting of three quarks, two up and one down, each having 1/2 unit of spin. The proton is classified as a fermion since its total spin is 1/2 unit. The pion, sometimes called the "pi meson," was discovered by British physicists C. Lates, H. Murhead, G. Occhialini, and C. Powell in 1947. It is also a composite particle consisting of two quarks, an up and an antidown, each having 1/2 unit of spin. The pion is classified as a boson, since its total spin is zero units.

Another example of this classification scheme comes from nuclei, bound states of protons and neutrons. The nucleus of helium-3, for example, is considered a fermion because the two protons and one neutron that constitute it have a total spin of 3/2 units. In contrast, the nucleus of helium-4 is considered a boson, since the spin of the two protons and two neutrons combine to give a total spin of 2 units. In general, any nuclei with an odd number of nucleons is considered a fermion, while one with an even number of nucleons is classified as a boson.

Superconductivity provides an interesting example of the classification of physical systems based on the total spin of their constituents. The phenomenon of superconductivity was discovered in 1911 by the Dutch physicist Heike Kamerlingh Onnes after he liquefied helium in 1908. Kamerlingh Onnes observed that the electrical resistance of a frozen sample of the metal mercury suddenly dropped to zero when cooled below a temperature of 4.2 Kelvin (-268.95 Celsius). In an ordinary conductor, the electric current is carried by electrons, a fermionic system. However, in the accepted theoretical description of superconductivity, proposed by John Bardeen, Leon Cooper, and John Schrieffer in 1957, the superconducting current is carried by bosonic particles called "Cooper pairs." A Cooper pair is a bound state of two electrons with equal and opposite linear and spin angular momenta. Since the spin of the electrons, themselves fermions, is equal and opposite, the total spin of the Cooper pair is equal to zero units and thus is classified as a boson.

Collections of atoms and molecules can also be classified as fermions or bosons in a similar way. For example, many compounds containing transition metals and rare-earth metals display magnetic effects. Transition metals are elements that contain d-shell electrons, and rare-earth metals contain f-shell electrons. Magnetic effects are not limited to the elemental metals themselves; they also occur in compounds containing these elements. The phenomena of paramagnetism and antiferromagnetism provide illustrative examples of this classification scheme. Paramagnetism is a magnetic effect that arises when the magnetic moments resulting from the spin and orbital motion of atomic electrons in a compound align with an externally applied magnetic field. Paramagnetism can be considered a fermionic effect, since it occurs only in compounds containing atoms with unpaired valence electrons. The total spin must be nonzero in order for paramagnetism to be observed. Antiferromagnetism, on the other hand, is a magnetic effect that arises in compounds that have permanent magnetic moments. It is a collective phenomenon that occurs when the permanent magnetic moments of neighboring ions in a compound have opposite, antiparallel ordering. Since this ordering results in a total spin of zero units for the system, antiferromagnets are classified as bosonic.

Quantum mechanics is the fundamental theory that describes the behavior of microscopic objects. Basic elements of this theory were proposed by Max Planck, Albert Einstein, Niels Bohr, Louis de Broglie and others in the first twenty years of the twentieth century. Erwin Schrödinger, Werner Heisenberg, and Paul Dirac established the major part of its mathematical formalism from about 1925 to 1930. In this same period, the Austrian physicist Wolfgang Pauli discovered the exclusion principle, often called the "Pauli exclusion principle." This discovery provided a clear theoretical distinction between fermions and bosons.

The basis of the exclusion principle is the idea of indistinguishability. In classical physics, two particles can always be distinguished by continual observations along their separate paths of motion. This holds true even if the two particles have identical attributes, such as mass, color, and electric charge, for example. Classically, the distinguishability between identical particles is built into Newtonian dynamics; the initial conditions and the laws of motion uniquely determine the path of motion. This concept, called "determinism," is one of the key features of the Newtonian worldview. Classically speaking, initial conditions can be determined to an arbitrary accuracy enabling the trajectories of identical particles to be distinguished.

However, in the microscopic realm, identical particles are indistinguishable. For example, every electron has the same mass, electric charge, magnetic moment, and other physical characteristics. In addition, the measurement of properties of microscopic particles also includes another feature not found in the classical realm. The uncertainty principle, postulated by German physicist Werner Heisenberg in 1927, quantifies the disturbance of a system or object being observed by the act of measurement. For example, in trying to determine the location of an electron by illuminating it with light, an example given by Heisenberg in his original paper, the unavoidable collisions between the photons of light and the electron limit the accuracy to which the electron's position can be determined. Attempts to minimize this disruption by decreasing the energy, and hence the unwanted influence of the photons, result in less-accurate information about the location of the electron. In the quantum realm, measurement cannot be made to an arbitrary level of accuracy as it can in the classical realm, thus limiting the information that can be known.

When considering a collection of identical particles, their indistinguishability must be accounted for in the wave function of the system. Pauli discovered, for example, that two electrons at the same location must have different quantum numbers. In the language of quantum mechanics, this statement means that the probability of finding two electrons with the same quantum numbers at the same location is equal to zero. Pauli recast this statement into a mathematical form that could be applied to a collection of any number of identical quantum particles: The total wave function of such a collection must be antisymmetric upon the exchange of any pair of coordinates. He also proved that this statement applies in general to fermions and not only to electrons. Since the exclusion principle applies only to fermions, it provides a clear distinction between fermionic and bosonic matter.

Applications

One of the most important examples of the Pauli exclusion principle applied to fermions concerns the shell structure of atoms and the periodic table. The wave functions and the energy levels for the electrons in a multielectron atom can be determined by solving the Schrödinger equation. However, since this equation can be solved exactly only for the simplest atom, hydrogen, approximations must be used. One of the most common methods involves the approximation that the various electrons occupy "hydrogenlike" states characterized by the hydrogen-type quantum numbers: n, l, ml, and ms. The labels n, l, ml, and ms represent the principal, orbital-angular-momentum, orbital-magnetic, and spin-magnetic quantum numbers, respectively. Solutions to the Schrödinger equation show that the energy of a particular electron depends on both the quantum numbers n and l. In a multielectron atom, electrons with the same value of n are said to occupy the same shell, and those with the same values of n and l are said to occupy the same subshell.

Using the standard spectroscopic notation, the ordering of the energy levels in a multielectron atom is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, and so on. In this notation, the value of the orbital angular momentum quantum number is given by the code l = 0, 1, 2, 3, . . . denoted by the letters s, p, d, f, . . . respectively. For example, the state given by 4s means that the principal quantum number n and the orbital angular momentum quantum number l have the values four and zero, respectively. Note that the ordering of energy levels is not sequential, since the 4s level is in between the 3p and 3d levels. This is a consequence of the electrostatic interactions between the individual electrons in a multielectron atom.

When the states of a multielectron atom are populated, the Pauli exclusion principle must be obeyed, since electrons are fermions. This means that each electron must occupy a distinct state denoted by a unique set of quantum numbers (n, l, ml, ms). For example, in the 1s state (n = 1, l = 0), the solution to the Schrödinger equation requires that the quantum numbers ml and ms have the values ml = 0 and ms = +1/2 or -1/2. The lowest energy state for two electrons consists of one electron in the state ,0,0,+1/2) and the other in the state ,0,0,-1/2). This is the ground-state electronic configuration for helium. For the ground state of lithium, for example, the third electron must occupy one of the two available 2s levels. In a similar way, the configuration of electrons in more complex multielectron atoms is constructed by placing the electrons into individual hydrogen-like states, satisfying the Pauli exclusion principle.

The periodic table, developed independently by Russian scientist Dmitri Mendeleev and German scientist Lothar Meyer, is a graphic display of the shell structure of multielectron atoms. The table is organized by rows of elements--called "periods"--with increasing atomic numbers, and by columns--called "groups"--of elements with similar chemical properties. The number of elements in the various groups is directly related to the number of subshells in a particular shell, a fact that follows from the Schrödinger equation and the Pauli exclusion principle applied to the fermionic electrons.

Another example of application of the Pauli exclusion principle to collections of fermions is Fermi-Dirac statistics. Statistical mechanics is a physical theory in which the properties of macroscopic systems are predicted in terms of the statistical behavior of the large number of constituent particles. For example, the properties of a mole of a gas, containing approximately 102³ molecules, can be deduced by considering the motions and interactions between the individual molecules. The important problem of a noninteracting ideal gas was studied by James Clerk Maxwell in the 1860's from the Newtonian perspective.

When statistical theories are applied using quantum-mechanical techniques via which energy and other properties are quantized, the resulting process is called "quantum statistics." Fermi-Dirac statistics are quantum statistics applied to a system composed of fermions. Important applications of Fermi-Dirac statistics arise in condensed-matter physics. Besides the phenomena of paramagnetism in solids and current conduction in metals, Fermi-Dirac statistics are used in theoretical studies of diamagnetism, the de Haas-van Alphen effect, specific heats of metals, and thermionic and photoelectric emission in metals. In addition, Fermi-Dirac statistics are used in astrophysics to study the thermodynamics of white-dwarf stars and other astronomical systems.

Context

The terms "fermion" and "boson" constitute the elements of a classification scheme. Their present use follows from experimental advances in atomic, nuclear, and particle physics in the late nineteenth and twentieth centuries. The electrical and magnetic properties of matter were known from antiquity. Electric and magnetic phenomena were also extensively studied by many European scientists in the seventeenth and eigthteenth centuries. The fields of electricity and magnetism were unified by the theoretical work of the English physicist James Clerk Maxwell in the 1860's, leading others to consider whether the electrical behavior of matter was caused by an electrified atom.

The first experimental evidence for an atom made up of electrified subatomic parts came from English physicist J. J. Thomson's studies of cathode rays in 1897. He concluded that the particles composing these rays were electrically charged. The particles composing cathode rays are now called "electrons." In 1911, another English physicist, Ernest Rutherford, performed a set of experiments that yielded information about the geometry of the atom. He discovered that most of the mass of the atom is concentrated in a small core, the nucleus. The planetary model of the atom, developed after Thomson's and Rutherford's work, views the positive charge as contained in the small, massive nucleus, with the negatively charged electrons orbiting about the nucleus. The neutron, the other particle found in the nucleus of the atom, was discovered by James Chadwick in 1932.

While the subatomic structure of the atom was determined by the 1930's, other discoveries led physicists to think that their understanding of the microworld was far from complete. The theoretical prediction of the neutrino, positron, and the pion and the experimental detection of the muon in 1937 led physicists to question whether the proton, neutron, and electron were in fact the fundamental building blocks of matter. In subsequent decades, more and more particles were found, ultimately making up a particle "zoo" containing hundreds of species.

Physicists turned their attention to classifying the particles, with the divisions into fermions and bosons being one of the most important. Fermions are named after the Italian physicist Enrico Fermi, best known for his work on statistical physics and the Manhattan Project during World War II. Bosons are named after the Indian physicist Satyendra Nath Bose, who with Albert Einstein formulated a quantum-statistical theory of photons. Other classification schemes group subatomic particles based on other physical properties such as mass, electric charge, size, and composition. Although these other schemes have their individual strengths and weaknesses, the fermion-and-boson scheme contains an additional feature in that it can be applied to fundamental particles, collections of fundamental particles, composite particles, and collections of composite particles.

Bibliography

Blakemore, J. S. Solid State Physics. 2d ed. Cambridge, England: Cambridge University Press, 1988. A college-level text for advanced undergraduates in physics and related areas in physical science and engineering. While the book covers the whole range of topics in the subject, its treatment of quantum statistical mechanics, specifically Fermi-Dirac statistics, is noteworthy. Extensive applications of this theory to electrons in metals is given. A short bibliography is also included.

Coughlan, G. D., and J. E. Dodd. The Ideas of Particle Physics: An Introduction for Scientists. 2d ed. Cambridge, England: Cambridge University Press, 1991. An excellent text on the subject that bridges the gap between college-level textbooks and popular accounts. While the book is not exclusively about fermions, it provides the interested reader a thorough introduction to the modern classification scheme of elementary particles and background about the theories composing the standard model. This book also includes an account of research of current interest in particle physics, a dictionary of terms, and an extensive bibliography.

Krane, Kenneth. Modern Physics. 2d ed. New York: John Wiley & Sons, 1996. A sophomore-level college text on the major theoretical and experimental aspects of twentieth century physics. The coverage of quantum mechanics, elementary particles, and statistical physics is particularly accessible to the beginning student.

Pathria, R. K. Statistical Mechanics. Oxford, England: Pergamon Press, 1972. The standard graduate-level text on the subject. This text provides the theoretical framework for both the classical and quantum-mechanical approaches. Fermi-Dirac statistics are extensively discussed, as are some varied applications. Also includes a bibliography with many references to the original literature.

Stevens, Charles F. The Six Core Theories of Modern Physics. Cambridge, Mass.: MIT Press, 1995. Presents a brief, self-contained summary of the main subjects that constitute the foundations of modern physics. The text is aimed at advanced undergraduate and first-year graduate students. Of particular interest in this context are the chapters on quantum mechanics, statistical physics, and quantum field theory.