Thermal Properties Of Solids

Type of physical science: Condensed matter physics

Field of study: Solids

The thermal properties of solids, including thermal expansion, specific heat, and thermal conductivity, provide information about the chemical bonding forces that hold the crystal structure together and the allowed vibrational and electronic energy levels of the material.

Overview

Heat energy is energy that is randomly distributed. When a quantity of gas is heated, the molecules move more quickly and their kinetic energy increases. The average kinetic energy of the gas molecules is proportional to the temperature. If the gas is maintained under conditions of constant pressure, then it will expand. If one wall of a container of gas is heated, then gas molecules, on the average, gain kinetic energy when they collide with the wall and share the increased kinetic energy with other molecules through further collisions. When a solid is heated, the same things occur: Temperature increases, the solid expands, and the increased energy is spread throughout the material, but at different rates. The differences between solid and gas thermal behavior are attributable to the strong chemical bonding between atoms in a solid.

The structure of a solid is determined by the chemical bonds that form between the component atoms. These bonds involve the redistribution of electrons between atoms by either transfer (ionic bonds), localized sharing (covalent bonds), or nonlocalized sharing (metallic bonds). Regardless of type, each chemical bond is characterized by an equilibrium bond distance, such that additional force is required either to separate the atoms or to push them closer together.

In this respect, the bond may be treated as a microscopic spring which supplies a restoring force when the atoms are separated from their equilibrium distance. Unlike an ideal spring or harmonic oscillator, the chemical bond supplies a slightly smaller restoring force when the atoms are separated than when the atoms are brought together by the same distance. Physicists are able to explain most of the thermal properties of nonmetallic solids using a simple model in which the atoms are treated as point masses connected by nearly perfect springs. For metallic solids, this model must be supplemented by a consideration of the motion of the mobile conduction electrons as described by quantum mechanics.

The "spring" force between atoms provides a bit more resistance to compression than to expansion. Therefore, as the average energy of vibration of the atoms in a solid increases, the atoms tend to spend more time at greater separation and the solid expands with increasing temperature. The total change in the length of a solid object with an increase in temperature is proportional to the original length and to the number of degrees of temperature increase. The proportionality constant, called the coefficient of linear expansion, is a characteristic of the substance of which the object is made and is typically a few millionths per degree Celsius. A 1-meter-long rod of aluminum will expand by only 0.0025 millimeter for each degree Celsius increase in temperature.

The specific heat of a substance is the amount of energy required to raise the temperature of a specified amount by 1 degree Celsius. For the solid form of the chemical elements, the specific heat at room temperature is about 25 joules per degree Celsius per mole of atoms, where a mole equals Avogadro's number (the number of atoms in a gram molecular weight of an element). This result, known as the law of Dulong and Petit, is consistent with the behavior expected from the equipartition theorem of classical statistical mechanics. This theorem states that, in any system for which the energy is the sum of terms involving the squares of displacements and velocities, the average energy per atom is one-half of Boltzmann's constant, k (1.38 x 10^-23 joules/Kelvin), multiplied by the absolute temperature, T, for each component of velocity or displacement which contributes to the total energy.

For any collection of masses connected by springs, the total energy is the sum of the kinetic energy for each of the masses, which depends on the sum of the squares of the three vector components of its velocity squares, and the potential energy, which can be expressed as a sum of the squares of three displacement components per mass. The expected specific heat for this collection of masses equals 3 x kT x the number of masses. Multiplying this result by Avogadro's number yields a predicted specific heat of 25.3 joules per degree Celsius per mole of atoms. At low temperatures, however, the specific heats of solids become much smaller, approaching zero at absolute zero temperature. The low-temperature specific heats of solids can only be understood using quantum mechanics.

In classical physics, the overall vibrational motion of a collection of masses connected by springs is most easily described using the concept of normal modes of vibration. Each normal mode is a pattern of motion in which all the masses vibrate around their equilibrium positions with the same frequency. The most general vibrational motion of the system involves each mass executing a motion which is a combination of all the normal-mode motions. Although quantum, rather than classical, mechanics must be used to describe the vibration of atoms in a crystal or other solids, the concept of normal modes is still useful. Quantum mechanics requires, at absolute zero temperature, each normal mode of the crystal to have a vibrational energy of exactly one-half of the value of Planck's constant, h (6.63 x 10^-23 joules per second), multiplied by the frequency, f, of the normal mode. As the temperature increases, the energy of each normal mode increases, though not continuously, as would be the case in classical mechanics, but in equally sized jumps of hf, that is, Planck's constant multiplied by the normal-mode frequency. Since vibrational energy increases in equally sized packages, or quanta, it is often useful to treat the vibrational energy as present in the form of energy-carrying particles called phonons, which are analogous to the photons that carry light energy. The number of quanta or phonons in each normal mode is determined by the size of the quantum and the temperature. At low temperatures, the energy classically available for each normal mode, equal to kT, is much less than the quantum hf for most of the normal modes. Therefore, only the lowest-frequency modes contribute to the specific heat, which is then much smaller than the Dulong and Petit value.

Solid-state physics explains the difference between metals and nonmetals with the band theory of solids. In the band theory, the allowed energy levels for electron motion in a solid are formed from the valence, or outermost, shells of the atoms from which the solid is formed. The energy levels, each of which can hold only two electrons as a result of the Pauli exclusion principle, are grouped into bands which are separated in energy by well-defined gaps. Nonmetals are those substances in which the energy bands are either completely filled or completely empty.

Since a substantial amount of energy is required to promote an electron from a filled band to an empty one in an insulating material, much more than kT, the electrons in such materials make no contribution to the specific heat. In a metal, however, electrons only partially fill the highest energy band, called the conduction band. The term "fermi level" is used to describe the highest electron energy level to be filled in the conduction band at absolute zero temperature. At nonzero temperatures, electrons lying within an energy kT of the conduction band can absorb energy to reach higher energy states in the same band. Thus, only a small fraction of the electrons in a metal makes a contribution to the specific heat.

When one part of a solid is warmer than another, heat flows from the warmer part to the cooler part. The thermal conductivity of a material is a ratio of the amount of heat energy passing through a layer of the material per unit area per second to the temperature difference across the layer. In nonmetallic solids, the thermal conductivity is a consequence of the interactions of phonons. In fact, one can describe the distribution of vibrational energy by treating the phonons as particles of definite energy and momentum that can collide with one another like gas molecules. Thermal conductivities are generally higher for metals than for nonmetals because the conduction electrons, which move very rapidly, also carry heat energy.

Diamond, a nonmetallic substance, is unusual because its thermal conductivity is higher than that of many metals.

Applications

The simplest and most obvious thermal property of matter is thermal expansion.

Although solids do not usually change their lengths enough with small changes in temperature to be useful as thermometers, builders must allow for the thermal ex pansion that will occur on hot days when they design roadways, sidewalks, or stretches of railroad track. Many devices for temperature control make use of a bimetallic strip, in which strips of metal that expand at different rates are joined back to back. Changes in temperature cause the curvature of such a strip to change, and the strip can be included in an electrical circuit so that a heating or cooling element is turned off when the temperature goes above or below a preset value. The coefficient of linear expansion of a solid is the fractional change in length of the solid per degree increase in temperature. Values of this coefficient have been tabulated for numerous materials for use by applied scientists and engineers.

Physicists have devoted much effort to measuring and computing the frequencies of the normal modes of different solids because of the importance of these modes for thermal and other properties of solids. The simplest cases of normal modes are crystals of a single chemical element in which each atom is bonded equally strongly to several near neighbors. In such materials, each normal mode is wavelike in character and can be described by wave vector, frequency, and direction of polarization. In directions perpendicular to the wave vector, all the atoms in the crystal are vibrating in phase, that is, reaching their maximum displacements at the same time. The phase of the atomic motions changes along the direction of the wave vector, so that atoms separated by one-half of a wavelength (equal to 3.14 divided by the magnitude of the wave vector) are vibrating out of phase with one another. The wavelengths for vibrational normal modes range from the spacing between neighboring atoms to the size of the crystal. Since shorter wavelengths normally involve the greatest stretching and compression of the bonds between atoms, they have the highest frequencies and the highest phonon energies. The direction of polarization indicates whether the normal-mode motions are of a longitudinal or transverse type.

The vibrational properties of polyatomic solids are more complex. For polyatomic solids, the crystalline structure is built up of unit cells which contain at least one formula unit of the chemical composition of the solid. Thus, for sodium chloride, the unit cell includes one sodium ion and one chloride ion, and the normal modes of the system can be divided into two types or branches: those in which neighboring sodium and chloride ions are moving together in the same direction and those in which the nearest-neighbor ions are moving in opposite directions. Normal modes of the former type are said to constitute the acoustical branch of the vibrational spectrum and are comparable to the normal modes in a monatomic crystal, as the longest wavelength modes have the lowest energies. The term "acoustical modes" is related to the fact that these modes are principally responsible for the motion of sound through the crystal.

Those modes in which atoms or ions in the same unit cell move in opposite directions, stretching and compressing the bonds between them, are called optical modes because they are most easily excited by light or infrared radiation. The lowest frequencies for optical modes are almost always above the highest frequencies for acoustic modes.

Measurement of the specific heats of solids over the range of very low temperatures provides general information about the energies of the normal modes in a solid. The most complete experimental information about phonon frequencies is obtained from inelastic neutron-scattering experiments, in which the energy loss and change of direction of neutrons scattered by a solid sample are measured. By combining careful neutron-scattering experiments with the results of computer calculations for assumed interatomic force constants and other information, physicists are able to develop a detailed picture of the forces between atoms and the solid state.

Context

The fact that the symmetrical shapes of pure crystals might reflect an underlying atomic structure was well known to mineralogists and chemists by the early nineteenth century.

In 1819, the French scientists Pierre-Louis Dulong and Alexis-Therese Petit reported their experimental discovery that the specific heat per mole of any solid chemical element was three times the universal gas constant (equal to Boltzmann's constant multiplied by Avogadro's number). In 1907, the German physicist Albert Einstein applied the then-new quantum theory to the specific heat of solids. He treated the atoms in a solid as a collection of independent harmonic oscillators and was able to derive the law of Dulong and Petit and to provide a qualitatively, but not quantitatively, correct explanation for the decrease in the specific heat of a solid with decreasing temperature. The Dutch physicist Peter J. W. Debye adopted a model for solid-state vibrations with a range of normal-mode frequencies, based on the vibrational properties of a continuous medium. Debye's calculation, which also made use of quantum theory, predicted a specific heat which was proportional to the cube of the absolute temperature at low temperatures, in agreement with observation.

The electron was identified as a constituent of normal matter by the English physicist Joseph John Thomson in 1897. This discovery stimulated an intense period of activity by physicists, aimed at explaining the physical and chemical properties of matter in terms of the properties of the constituent electrons. By 1910, the German physicist Paul Drude and the Dutch physicist Hendrik Antoon Lorentz had developed theories of the electrical and thermal conductivity of metals by treating the conduction electrons as free particles much in the manner of the kinetic theory of gases. Although the Drude-Lorentz theory was able to account for a number of experimental observations, including the relationship between thermal and electrical conductivity, it had a number of unacceptable features, including the prediction of specific heats much larger than observed. The development by a number of physicists of the quantum mechanical band theory of solids, beginning about 1930, led to an improved understanding of the electronic contribution to specific heat and thermal conductivity.

Experimental confirmation of the atomic structure of crystals began with the discovery by the German physicist Max von Laue that the diffraction pattern for X rays passing through a solid provided detailed structural information. While the elastic scattering of X rays--that is, the scattering of X rays without energy loss--yields information about the average position of atoms in a crystal, the inelastic scattering of X rays and neutrons provides information about the energies, wave vectors, and polarizations of phonon modes. Scattering studies, together with computer calculations of the vibrational behavior of atoms in a crystal lattice, have become standard tools in the investigation of the vibrational properties of solids.

Some of the most interesting consequences of lattice vibrations involve the interaction of electrons and phonons in a metal. Generally, the effect of the electron-phonon interaction is the scattering of electrons, resulting in an increase in electrical resistance as the temperature and the number of phonons increase. In certain materials and at low-enough temperatures, the coupling between electrons and phonons can, in effect, lead to the pairing of electrons, which then travel through the material without further scattering. The formation of such electron pairs by electron-phonon coupling is the basis for the generally accepted theory of superconductivity, which was first proposed by the American physicists John Bardeen, Leon N Cooper, and John Robert Schrieffer in 1957.

Principal terms

ACOUSTIC MODES: in a polyatomic crystal, those normal modes in which atoms in the same unit cell vibrate in phase with one another

DEBYE MODEL: a model for the vibrational properties of solids in which the solid is treated as a continuous medium, neglecting atomic structure

EINSTEIN MODEL: a model for the vibrational properties of solids in which the solid is treated as a collection of independent harmonic oscillators

EQUIPARTITION THEOREM: one of the fundamental results of classical statistical mechanics requiring that, at thermal equilibrium, all independent modes of vibration in an oscillating system have the same average energy

NORMAL MODE OF VIBRATION: for a system of coupled harmonic oscillators, any pattern of motion in which each mass oscillates around its equilibrium position with a common frequency

OPTICAL MODE: in a polyatomic crystal, those normal modes in which atoms in the same unit cell vibrate out of phase with one another

PHONON: a quantum of vibrational energy, which is the minimum energy that can be absorbed or released by a vibrational mode

SPECIFIC HEAT: the amount of energy required to raise the temperature of a fixed amount of a substance by 1 degree Celsius, generally expressed in joules per mole per degree Celsius

THERMAL CONDUCTIVITY: the ratio of amount of heat energy passing through a layer of a material per unit area per second to the temperature difference across the layer

UNIT CELL: the smallest unit of a crystal, from which the entire crystal can be built by repetition

Bibliography

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. THE FEYNMAN LECTURES ON PHYSICS. Reading, Mass.: Addison-Wesley, 1963. This comprehensive set of lectures by one of the leading theoretical physicists of the mid-twentieth century is an attempt to convey both modern and classical physics to beginning university students. Thermodynamics and the thermal and other properties of crystalline materials are discussed at several points throughout the lectures.

Holden, Alan. THE NATURE OF SOLIDS. New York: Columbia University Press, 1965. This book is unusual because it attempts to present the basic concepts of solid-state physics through well-selected diagrams and without mathematical equations. Specific heats, crystal structures, crystal vibrations, and electronic energy bands are discussed in some detail.

Kittel, Charles. INTRODUCTION TO SOLID STATE PHYSICS. New York: John Wiley & Sons, 1986. One of the standard introductions to the physics of solids. Includes extensive summaries of the properties of solids and detailed discussions of the vibrational and electronic states of solids and the thermal properties of metals, insulators, and semiconductors.

Moore, Walter J. SEVEN SOLID STATES. New York: W. A. Benjamin, 1967. This brief book is intended to be supplementary to introductory chemistry texts. Provides additional materials on solid-state chemistry, a topic sometimes neglected in these general works. Moore's approach is to discuss a variety of specific materials, rather than solids in general. At various points, Moore discusses heat capacity, vibrational energy levels, and energy bands for electrons.

Rosenberg, H. M. THE SOLID STATE. Oxford, England: Oxford University Press, 1988. This introductory treatment is shorter than most and provides a more detailed treatment of imperfect solids than is customary. Although technical in places, emphasis is placed on the basic ideas used by physicists to explain the properties of solids.

Weller, Paul F. SOLID STATE CHEMISTRY AND PHYSICS. New York: Marcel Dekker, 1973. This two-volume set is a compilation of introductory articles on various aspects of solid-state physics and chemistry. The first three chapters provide an excellent overview of the structure and energy levels of crystals.

Thermal Properties of Matter