# Paramagnetism

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## magnetism

magnetism, force of attraction or repulsion between various substances, especially those made of iron and certain other metals; ultimately it is due to the motion of electric charges.

### Magnetic Poles, Forces, and Fields

Any object that exhibits magnetic properties is called a magnet. Every magnet has two points, or poles, where most of its strength is concentrated; these are designated as a north-seeking pole, or north pole, and a south-seeking pole, or south pole, because a suspended magnet tends to orient itself along a north-south line. Since a magnet has two poles, it is sometimes called a magnetic dipole, being analogous to an electric dipole, composed of two opposite charges. The like poles of different magnets repel each other, and the unlike poles attract each other.

One remarkable property of magnets is that whenever a magnet is broken, a north pole will appear at one of the broken faces and a south pole at the other, such that each piece has its own north and south poles. It is impossible to isolate a single magnetic pole, regardless of how many times a magnet is broken or how small the fragments become. (The theoretical question as to the possible existence in any state of a single magnetic pole, called a monopole, is still considered open by physicists; experiments to date have failed to detect one.)

From his study of magnetism, C. A. Coulomb in the 18th cent. found that the magnetic forces between two poles followed an inverse-square law of the same form as that describing the forces between electric charges. The law states that the force of attraction or repulsion between two magnetic poles is directly proportional to the product of the strengths of the poles and inversely proportional to the square of the distance between them.

As with electric charges, the effect of this magnetic force acting at a distance is expressed in terms of a field of force. A magnetic pole sets up a field in the space around it that exerts a force on magnetic materials. The field can be visualized in terms of lines of induction (similar to the lines of force of an electric field). These imaginary lines indicate the direction of the field in a given region. By convention they originate at the north pole of a magnet and form loops that end at the south pole either of the same magnet or of some other nearby magnet (see also flux, magnetic). The lines are spaced so that the number per unit area is proportional to the field strength in a given area. Thus, the lines converge near the poles, where the field is strong, and spread out as their distance from the poles increases.

A picture of these lines of induction can be made by sprinkling iron filings on a piece of paper placed over a magnet. The individual pieces of iron become magnetized by entering a magnetic field, i.e., they act like tiny magnets, lining themselves up along the lines of induction. By using variously shaped magnets and various combinations of more than one magnet, representations of the field in these different situations can be obtained.

### Magnetic Materials

The term magnetism is derived from Magnesia, the name of a region in Asia Minor where lodestone, a naturally magnetic iron ore, was found in ancient times. Iron is not the only material that is easily magnetized when placed in a magnetic field; others include nickel and cobalt. Carbon steel was long the material commonly used for permanent magnets, but more recently other materials have been developed that are much more efficient as permanent magnets, including certain ferroceramics and Alnico, an alloy containing iron, aluminum, nickel, cobalt, and copper.

Materials that respond strongly to a magnetic field are called ferromagnetic [Lat. ferrum = iron]. The ability of a material to be magnetized or to strengthen the magnetic field in its vicinity is expressed by its magnetic permeability. Ferromagnetic materials have permeabilities of as much as 1,000 or more times that of free space (a vacuum). A number of materials are very weakly attracted by a magnetic field, having permeabilities slightly greater than that of free space; these materials are called paramagnetic. A few materials, such as bismuth and antimony, are repelled by a magnetic field, having permeabilities less than that of free space; these materials are called diamagnetic.

### The Basis of Magnetism

The electrical basis for the magnetic properties of matter has been verified down to the atomic level. Because the electron has both an electric charge and a spin, it can be called a charge in motion. This charge in motion gives rise to a tiny magnetic field. In the case of many atoms, all the electrons are paired within energy levels, according to the exclusion principle, so that the electrons in each pair have opposite (antiparallel) spins and their magnetic fields cancel. In some atoms, however, there are more electrons with spins in one direction than in the other, resulting in a net magnetic field for the atom as a whole; this situation exists in a paramagnetic substance. If such a material is placed in an external field, e.g., the field created by an electromagnet, the individual atoms will tend to align their fields with the external one. The alignment will not be complete, due to the disruptive effect of thermal vibrations. Because of this, a paramagnetic substance is only weakly attracted by a magnet.

In a ferromagnetic substance, there are also more electrons with spins in one direction than in the other. The individual magnetic fields of the atoms in a given region tend to line up in the same direction, so that they reinforce one another. Such a region is called a domain. In an unmagnetized sample, the domains are of different sizes and have different orientations. When an external magnetic field is applied, domains whose orientations are in the same general direction as the external field will grow at the expense of domains with other orientations. When the domains in all other directions have vanished, the remaining domains are rotated so that their direction is exactly the same as that of the external field. After this rotation is complete, no further magnetization can take place, no matter how strong the external field; a saturation point is said to have been reached. If the external field is then reduced to zero, it is found that the sample still retains some of its magnetism; this is known as hysteresis.

### Evolution of Electromagnetic Theory

The connections between magnetism and electricity were discovered in the early part of the 19th cent. In 1820 H. C. Oersted found that a wire carrying an electrical current deflects the needle of a magnetic compass because a magnetic field is created by the moving electric charges constituting the current. It was found that the lines of induction of the magnetic field surrounding the wire (or any other conductor) are circular. If the wire is bent into a coil, called a solenoid, the magnetic fields of the individual loops combine to produce a strong field through the core of the coil. This field can be increased manyfold by inserting a piece of soft iron or other ferromagnetic material into the core; the resulting arrangement constitutes an electromagnet.

Following Oersted's discovery the various magnetic effects of an electric current were extensively investigated by J. B. Biot, Félix Savart, and A. M. Ampère. Ampère showed in 1825 that not only does a current-carrying conductor exert a force on a magnet but magnets also exert forces on current-carrying conductors. In 1831 Michael Faraday and Joseph Henry independently discovered that it is possible to produce a current in a conductor by changing the magnetic field about it. The discovery of this effect, called electromagnetic induction, together with the discovery that an electric current produces a magnetic field, laid the foundation for the modern age of electricity. Both the electric generator, which makes electricity widely available, and the electric motor, which converts electricity to useful mechanical work, are based on these effects.

Another relationship between electricity and magnetism is that a regularly changing electric current in a conductor will create a changing magnetic field in the space about the conductor, which in turn gives rise to a changing electrical field. In this way regularly oscillating electric and magnetic fields can generate each other. These fields can be visualized as a single wave that is propagating through space. The formal theory underlying this electromagnetic radiation was developed by James Clerk Maxwell in the middle of the 19th cent. Maxwell showed that the speed of propagation of electromagnetic radiation is identical with that of light, thus revealing that light is intimately connected with electricity and magnetism.

### Bibliography

See D. Wagner, Introduction to the Theory of Magnetism (1972); D. J. Griffiths, Introduction to Electrodynamics (1981); R. T. Merritt, Our Magnetic Earth (2010).

## Paramagnetism

A property exhibited by substances which, when placed in a magnetic field, are magnetized parallel to the field to an extent proportional to the field (except at very low temperatures or in extremely large magnetic fields). Paramagnetic materials always have permeabilities greater than 1, but the values are in general not nearly so great as those of ferromagnetic materials. Paramagnetism is of two types, electronic and nuclear.

The following types of substances are paramagnetic:

1. All atoms and molecules which have an odd number of electrons. According to quantum mechanics, such a system cannot have a total spin equal to zero; therefore, each atom or molecule has a net magnetic moment which arises from the electron spin angular momentum. Examples are organic free radicals and gaseous nitric oxide.

2. All free atoms and ions with unfilled inner electron shells and many of these ions when in solids or in solution. Examples are transition, rare-earth, and actinide elements and many of their salts. This includes ferromagnetic and antiferromagnetic materials above their transition temperatures. For a discussion of these materials See Antiferromagnetism, Ferrimagnetism, Ferromagnetism

3. Several miscellaneous compounds including molecular oxygen and organic biradicals.

4. Metals. In this case, the paramagnetism arises from the magnetic moments associated with the spins of the conduction electrons and is called Pauli paramagnetism.

Relatively few substances are paramagnetic. Aside from the Pauli paramagnetism found in metals, the most important paramagnetic effects are found in the compounds of the transition and rare-earth elements which have partially filled 3d and 4f electron shells respectively.

Electronic paramagnetism arises in a substance if its atoms or molecules possess a net electronic magnetic moment. The magnetization arises because of the tendency of a magnetic field to orient the electronic magnetic moments parallel to itself.

Nuclear paramagnetism arises when there is a net magnetic moment due to the magnetic moments of the nuclei in a substance. Nuclear magnetic moments are about 103 times smaller than electron magnetic moments. As a result, nuclear paramagnetism produces effects 106 times smaller than electron paramagnetic or diamagnetic effects. See Diamagnetism, Magnetic resonance, Nuclear moments

McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Paramagnetism

the property of substances, placed in an external magnetic field, to be magnetized—that is, acquire a magnetic moment—in the direction of the field. Thus, the action of the magnetization J that arises within the paramagnetic substance is added to that of the external field. In this regard, paramagnetism is the opposite of diamagnetism, where the magnetic moment that arises in a substance under the action of a field is oriented against the direction of the external magnetic field strength H. Paramagnetic substances are therefore attracted to the poles of magnets (hence the term “paramagnetism”) and diamagnetic substances are repelled. Ferromagnetic and antifer-romagnetic substances also have paramagnets’ property of being magnetized along a field. When, however, there is no external field, the magnetization of paramagnetic substances is equal to zero and the substances lack a magnetic structure—mutual ordered orientation of the magnetic moments of their atoms. Ferromagnetic and antiferromagnetic substances, on the other hand, retain their magnetic structure when H = 0. The term “paramagnetism” was introduced in 1845 by M. Faraday, who divided all substances except ferromagnets into diamagnetic and paramagnetic substances.

Paramagnetism is characteristic of substances whose particles —atoms, molecules, ions, or atomic nuclei—have intrinsic magnetic moments that, in the absence of an external field, are oriented randomly, with the result that J = 0. When an external field is applied, the magnetic moments of paramagnets’ atoms are oriented mainly with the field. In weak fields, the magnetization of paramagnetic substances increases with field strength according to the law J = χH, where χ is the magnetic susceptibility of a mole of the substance; for paramagnets, χ is always positive and usually is of the order of magnitude 10–5–10–3. If the field is very strong, the magnetic moments of the paramagnetic particles are all oriented strictly with the field; that is, magnetic saturation is reached. For a constant field strength, an increase in the temperature T means an increase in the disorienting effect of the particles’ thermal motion and a decrease in magnetic susceptibility. In the simplest case, the susceptibility obeys Curie’s law χ = C/T, where C is the Curie constant, which depends on the nature of the substance. Deviations from Curie’s law are connected primarily with the interaction of particles (the effect of a crystal field) and are provided for by the Curie-Weiss law.

Paramagnetism is characteristic of many pure elements in the metallic state: alkali metals; alkaline-earth metals; some metals of the transition groups with an unfilled d or f shell—the groups of iron, palladium, platinum, the rare-earth elements, and the actinides. Alloys of these metals are also paramagnetic. In addition, paramagnetism is exhibited by salts of the iron group, salts of the group of rare-earth elements from Ce to Yb, salts of the actinides, and aqueous solutions thereof. Other substances characterized by paramagnetism include alkali-metal vapors; molecules of gases, such as O2 and NO; some organic molecules (biradicals); and a number of complex compounds. Ferromagnetic and antiferromagnetic substances become paramagnets at temperatures exceeding the Curie point or the Néel point, respectively, that is, the temperatures of the phase transition to the paramagnetic state.

The existence in atoms or ions of magnetic moments responsible for the paramagnetism of substances can be due to several factors: the orbital motion of the electrons about the nuclei (orbital paramagnetism), the spin angular momentum of the electrons (spin paramagnetism), and the magnetic moments of the atomic nuclei (nuclear paramagnetism). The magnetic moments of atoms, ions, and molecules are primarily a result of the spin and orbital moments of the electron shells. They are approximately 1,000 times greater than the magnetic moments of the nuclei. The paramagnetism of a metal consists primarily of the paramagnetism associated with the conduction electrons (Pauli paramagnetism) and the paramagnetism of the electron shells of the atoms (ions) of the metal’s crystal lattice. Since changes in temperature have practically no effect on the motion of conduction electrons in metals, the paramagnetism connected with conduction electrons is independent of temperature. It follows that the magnetic susceptibility of, for example, the alkali and alkaline earth metals is independent of temperature, since the electron shells of these metals’ ions lack a magnetic moment, and the metals’ paramagnetism is due exclusively to conduction electrons. For substances lacking the conduction electrons in which only the nucleus has a magnetic moment, such as the helium isotope He3, the paramagnetism is extremely small (x ~ 10–9–10–12) and can be observed only at extremely low temperatures (T < 0.1 °K).

According to the classical theory of P. Langevin (1906), the paramagnetic susceptibility of dielectrics is given by the formula χ = Nμa2/3kT, where N is the number of magnetic atoms in a mole of the substance, μa the magnetic moment of the atom, and k the Boltzmann constant. This formula was derived by the methods of statistical physics for a system of essentially nonin-teracting atoms located in a weak magnetic field or at high temperature—that is, when μaH << kT. The formula provides a theoretical explanation of the Curie law. In strong magnetic fields or at low temperatures, that is, when μaH << kT, the magnetization of paramagnetic dielectrics tends toward Nμa2, or toward saturation. The quantum theory of paramagnetism, which takes into account the space quantization of the moment μa (L. Brillouin, 1926), gives a similar expression for the susceptibility χ of dielectrics when μaH << kT: x = NJ (J + 1)μa2gj2/3kT, where j is the quantum number that determines the total angular momentum of the atom and gj is the Landé splitting factor. The paramagnetic susceptibility of semiconductors Xes owing to conduction electrons has, in the simplest case, an exponential dependence on the temperature T: Xes + AT½ exp(—ΔE/2kT), where A is a constant of the substance and ΔE is the width of the forbidden band of the semiconductor. The peculiarities of the individual structure of semiconductors strongly distort this dependence. In the simplest case, for metals —if we ignore Landau diamagnetism and the interaction of electrons—we have XeM = 3Nμe2/2E0, where E0 is the Fermi energy and μe is the magnetic moment of an electron; here ΞeM is independent of the temperature. In the absence of a strong interaction between nuclear spins and the electron shells of atoms, nuclear paramagnetism is characterized by the quantity Ξn = Nμn2/3KT; the electronic paramagnetic susceptibility is approximately 106 times larger than this quantity, since μe ~ 103μn.

Research on the paramagnetism of different substances and on electron paramagnetic resonance (the resonance absorption by paramagnets of the energy of an electromagnetic field) has made it possible to determine the magnetic moments of individual atoms, ions, molecules, and nuclei, to study the structure of complex molecules and molecular complexes, and to perform fine structural analysis of materials used in technology. Paramagnetic substances are made use of in physics to obtain temperatures below 1°K.

### REFERENCES

Vonsovskii, S. V. Magnetizm mikrochastits. Moscow, 1973.
Vonsovskii, S. V. Magnetizm. Moscow, 1971.
Dorfman, la. G. Magnitnye svoistva istroenie veshchestva. Moscow, 1955.
Abragam, A. Iadernyi magnetizm. Moscow, 1963. (Translated from English.)
Kittel, C. Vvedenie v fiziku tverdogo tela, 2nd ed. Moscow, 1963. (Translated from English.)