Debye theory of solids

Type of physical science: Debye Theory of Solids, Solids, Crystals, Condensed matter physics

Field of study: Solids

Simple solids can be modeled as a collection of atoms oscillating about their fixed positions in the crystal lattice. Peter Debye used this model to explain the properties of monoatomic solids.

Overview

One of the principle aims of modern science is to explain experimentally measurable properties of materials in terms of atomic and molecular models. The Debye theory of solids is one such theory; it aims to explain the thermal properties of solids in terms of the motions of the atoms that make up the solids. The model discussed here will assume that one is dealing with monoatomic solids such as solid forms of the elements. It should be noted that the actual theory is mathematical, beginning with some simple assumptions and deriving a connection between the properties of the model and measurable properties of the solid. Of necessity, such a theory involves quantum mechanics, statistical mechanics, and somewhat elaborate mathematical operations. What is of interest is the nature of the model and the success it has in explaining observable facts.

At the beginning of the twentieth century, Walter Nernst was making significant measurements of the thermal properties of solids at temperatures approaching absolute zero. In particular, it was observed that the molar heat capacity of monoatomic solids such as most metals fell off sharply toward zero as the temperature of the solid approached absolute zero. The equipartition of energy theory predicted the constant room temperature value but could not account for the drop-off with temperature. In the context of new ideas about quantization, attempts were made to model solids and to explain heat capacities in terms of quantized vibrations.

The first attempt to model a solid was made by Albert Einstein. Since the Einstein theory of solids uses some of the ideas of the Debye theory, it will be treated first. In the Einstein model, the atoms are assumed to be located at definite positions in the solid, and they form a regular arrangement in three-dimensional space known as a lattice. Although each atom occupies a fixed position on this lattice, it can oscillate about this fixed position. The manner in which a given atom oscillates or vibrates depends upon the spacing between it and adjacent atoms and upon the manner in which the atoms interact with one another. The model does not consider the details of these interactions but simply assumes that since any given atom is surrounded on all sides by other atoms, it experiences forces upon it that are the same in every direction. Finally, the model assumes that the oscillations of the atoms are relatively small. These assumptions allow some simplification of the mathematical treatment of the model.

The motion of each atom can be described in terms of the frequency of its oscillation. Since one is dealing with a model in which the moving objects are of atomic sizes, it is necessary to solve the problem using quantum mechanical theories. The only important consequence of this assumption is that the energy associated with a vibrating atom cannot vary in a continuous way but can only have discrete values given by the formula E = (n + ½)hv, where v is the frequency of the vibration, E is the energy associated with the vibration, h is Planck's constant, and n is a number that can be any integer value. It should be pointed out that the thermal properties of a solid will be determined by how energy is incorporated into the vibrational motions of the atoms.

How do the atoms vibrate? Because of the simplicity of the assumptions, each atom is considered to vibrate independently of all the others. Furthermore, since all atoms in the solid are the same, and since they all have the same surroundings (except for a negligible number of atoms on the surface of the solid), all atoms vibrate in the same way. That is, all atoms possess the same vibrational frequency. Atoms will differ in vibrational energy only through the value of the constant n describing their motion.

Given that all atoms are independent and are vibrating at the same frequency, the connection between the frequency and experimentally determinable thermal properties can be derived. The resulting equations can be used to compare experiment with theory. This will be done after the discussion of the Debye model.

The main deficiency in the Einstein model lies in the assumption that all the atoms are vibrating independently. The Debye model eliminates this assumption, recognizing that the oscillatory motions of the atoms must be treated in terms of groups of atoms and that the motions cannot be described in terms of a single vibrational frequency. The details of the analysis are buried in a mathematical procedure known technically as a "normal mode analysis." The consequences of such an analysis are as follows:

Since the atoms are all simultaneously interacting with one another, the oscillatory motion of one atom will have an effect on that of the other atoms in its immediate surroundings. Furthermore, by successive coupling of motions of adjacent atoms, the motion of any given atom is coupled to that of all the atoms in the solid. This coupling leads to a wide variety of kinds of vibrations. At one extreme, there is a kind of oscillatory motion in which all pairs of adjacent atoms are oscillating relative to one another, all pairs throughout the entire solid moving in concert. This motion can be described incompletely by looking at a single pair. Since this motion involves only the interactions between the two atoms in the pair, it is characterized by a high energy of interaction and a high vibrational frequency. At the other extreme, the oscillations involve large groups of atoms. In effect, these oscillations represent elastic waves in which the bulk material is set into vibratory motion. These sorts of motions are described as "low-frequency motions."

The Debye model for solids is a significant improvement over the Einstein model. The Debye model takes into account that atoms are not independent of one another and that their relative motions are coupled. The Debye model allows for a range of vibrational frequencies, while the Einstein model allows only for a single vibrational frequency.

A detailed mathematical treatment of both models shows that there are 3N vibrations possible, where N is the number of atoms in the solid. Since N is of the order of Avogadro's number, 6.02 × 10²³, there are clearly a large number of vibrations. In the Einstein model, all 3N vibrations have the same frequency, but this is not the case in the Debye model, according to which vibrations can range from low to high frequencies. How are the frequencies chosen in the Debye case? Since the low frequencies correspond to vibrations or oscillations that are bulk vibrations, it is possible to use the theory of elastic materials to count the number of vibrations within a given frequency range. This theory is a "bulk material" theory—that is, it does not take into account the atomic structure explicitly. For low-frequency vibrations, it is found that the number of vibrations within a given frequency range is proportional to the frequency at the center of the range. This result, which is accurate for low frequencies, was adopted in the Debye model for all frequencies, low and high.

In trying to understand how this physical model is related to observable properties, consider what happens when energy is added to a solid by allowing heat to flow into the solid. The process of heat flow causes the temperature of the solid to rise. Now temperature is taken as a manifestation of molecular and atomic motions. In the two models discussed here, the atomic motions in question are the oscillatory motions of the atoms. Recall the equation connecting energy and frequency, E = (n + ½)hv. This equation gives the energy for one of the many possible vibrations. Now, as thermal energy (heat) flows into the solid, the energy associated with the vibrations will increase. This can occur only by causing the value of n in the equation to increase in steps of one (n can only have integer values). As more heat flows into the solid, more of the vibrations become excited, and the total energy of the solid increases.

The physical property of materials—whether they be solids, liquids, or gases—that best describes the material's response to heat flows is the "heat capacity" of the material. When heat flows into a material, the material experiences an increase in temperature. Clearly, one would expect that the temperature increase and the heat flow would be closely related. In fact, all other things being equal, one would expect that doubling the heat flow would approximately double the temperature increase. These two processes are connected in the heat capacity, which is defined as the ratio of the heat flow to the temperature rise caused by the heat flow. The heat capacity depends upon the amount of material present. A given heat flow into a small mass causes a large temperature rise, while the same heat flow into a large mass of the same material causes a small temperature rise. The mass of material is taken into account in the "molar heat capacity" and in the "specific heat," the latter being the heat capacity of one gram of material.

Applications

The Debye model of solids is an atomic explanation of the behavior of solids and is of limited direct practical applicability. However, the Debye theory has implied consequences wherever chemical thermodynamics touches our lives. To see this, briefly consider the uses of chemical thermodynamics.

One important use of chemical thermodynamics is to employ the measured properties of chemicals to make predictions of the outcome of hypothetical chemical reactions. For example, one might be interested in producing a precursor chemical for use in synthesizing plastics from an oil-derived substance. It is possible, using the thermodynamic properties of the chemicals involved, to predict accurately the composition of reaction-product mixtures in order to decide whether a given synthetic route would produce enough of the desired material. Key to the success of such a calculation is the availability of accurate thermodynamic data for all the substances involved. One of the most important thermodynamic properties of chemicals is their entropy. The determination of a material's entropy depends upon the measurement of its heat capacity over a range of temperatures, starting with a temperature as close to absolute zero as is experimentally possible. Cooling materials to temperatures near absolute zero is, however, a difficult task demanding specialized equipment and techniques. The Debye theory is used to aid in interpreting heat-capacity data obtained at these very low temperatures.

Using the Debye model, one can derive an equation connecting the heat capacity of a solid material to the vibrational frequencies that are part of the model. In particular, one assumes that the number of vibrations within a given frequency range is proportional to the square of the frequency at the center of the range and that the allowed frequencies vary up to a maximum cutoff frequency, corresponding roughly to the vibrations of individual pairs of atoms. The equations connecting the heat capacity (a measurable property) with the model are complicated and not especially informative. However, these equations take on a peculiar form at low temperatures, the very conditions that are of interest to scientists making entropy measurements. The Debye model predicts that the heat capacity of monoatomic solids at temperatures approaching absolute zero becomes proportional to the cube of the absolute temperature. This relationship has come to be known as the Debye T-cubed law and it is typically valid between absolute zero and 10 to 40 Kelvins.

Here is how the T-cubed law can be used: Entropy values of chemical reactants and products (the quantities to be used in calculating the product distribution of the chemical reactions) are obtained from thermodynamic equations. These equations use the values of the heat capacities of these materials over the range of temperatures starting at absolute zero up to the temperature of the proposed chemical reaction—from room temperature to several hundred degrees. Because the calculations demand heat-capacity data starting at absolute zero, it is essential that measurements be made on the materials as close to absolute zero as possible. The minimum temperature needed can be raised (reducing the experimental difficulties and expenses) by using the T-cubed law. One measures the heat capacity of the solid down to a range of temperatures over which the law is valid—that is, down to about 10 to 30 Kelvins. The heat-capacity data at the lowest temperatures is used to obtain the proportionality constant in the T-cubed law. Once the constant is in hand, it is possible to use the T-cubed law to calculate the heat capacity from the lowest temperature at which it is measured experimentally on down to absolute zero. In this way, heat-capacity information starting at absolute zero is obtained, allowing accurate determination of the entropy of the substance along with all other needed thermodynamic data.

Context

The measurement of heat capacities played an important part in the development of the atomic theory during the nineteenth century. The law of Dulong-Petit gave rise to the proper assignment of atomic weights. As shown above, the Debye model gave an equation for the heat capacities of monoatomic solids. Without going into the details, it is possible to show that the Debye equation for heat capacities at high temperatures gives the correct experimental value, since at high temperatures all atomic solids tend toward the same value. However, this is a rather weak test of the theory. Some solids, such as diamond, do not follow the law of Dulong-Petit at room temperature but have heat capacities significantly below the expected value. By choosing the correct value for the maximum or cutoff frequency for diamond, the Debye model accounts for the behavior of diamond correctly. Finally, at low temperature, the Debye model gives the correct dependence upon temperature.

Of greater interest is the use of quantum theory in the derivation of the equations connecting experimental quantities with theoretical quantities. The inclusion of quantum mechanics in the mathematics is contained in the integer n in the equation connecting energy and frequency given above. In brief, it represents the fact that the energy contained in a given vibration cannot increase continuously but can increase only in steps or jumps, a behavior known as quantization. The quantum theory was being developed at the beginning of the twentieth century; thus, both the Einstein model and Debye model were among the early applications of quantum ideas to the explanation of the behavior of materials. A mathematical derivation of the equations giving the heat capacity based on classical Newtonian physics, for example, predicts that the heat capacity should have the Dulong-Petit value at all temperatures. Classical physics could not account for the falloff in the value of the heat capacity as the temperature decreased. By incorporating quantum ideas into the theories, both the Einstein and Debye models successfully accounted for the falloff of heat capacity with temperature. This result not only allowed the application for low-temperature measurements but also provided a needed illustration of the validity of quantum theory at the very time when the new and strange ideas were first being accepted.

A fundamental assumption of the Debye theory is that the number of vibrations of a given frequency can be obtained from a consideration of the bulk elastic behavior of solids. The Debye model can be applied to other measurable phenomena such as the melting points of solids, the behavior of solids under the application of high pressures, and the expansion of solids as a result of increases in temperature. Since many of these properties reflect the elastic properties of materials, it is expected that there will be a connection between heat capacity data and other data describing the elastic properties of materials. By means of the Debye model, these connections have been shown to exist, pointing to a common underlying mechanism for all these phenomena.

For all of its success, the Debye model is not completely accurate. The model needs some experimental data incorporated into it before making further comparisons with experiment. Even then, significant discrepancies exist. Other, more mathematically demanding models give better agreement with experiment. During a lecture on his theories delivered during his last years, Debye made mention of how his theories agree with experiment only up to a certain extent. He was interested in the basic underlying mechanisms (in this case, the quantized oscillation of atoms) that help to explain why matter behaves as it does.

Principal terms

ABSOLUTE ZERO: The lowest temperature that can be approached by any experimental method; the temperature at which all molecular motion ceases (excepting residual vibrational motion)

AVOGADRO'S NUMBER: The number of particles, atoms, or molecules in one mole of materials; equal to 6.02 × 102³

DULONG-PETIT LAW: An empirical law that states that the product of the atomic weight of an element times its specific heat equals a value of about 6.2

ELASTIC PROPERTY: Any property of a material that reflects the fact that matter can undergo compression and expansion

ENTROPY: A thermodynamic property of substances that is a measure of the amount of order or disorder that exists within the substance

FREQUENCY: The number of oscillations occurring during a unit of time, normally measured in units of oscillations per second

HEAT CAPACITY: The ratio of the amount of heat or thermal energy introduced into a substance to the temperature change caused by that introduction of heat; normally measured in calories per degree or joules per degree

NORMAL MODE: One of a number of simple oscillatory motions that a system composed of a large number of particles and undergoing complex motions can adopt

QUANTIZATION: A phenomenon in which certain measurable properties of atomic systems, such as energy, can have only certain discrete values

QUANTUM MECHANICS: A branch of physics that treats the motions of atomic-sized systems and that introduces concepts such as discrete energy states, indeterminacy, and probabilities

VIBRATION: A motion that a system can undergo that repeats itself periodically and is characterized by the frequency with which the motion repeats itself

Bibliography

Brown, Laurie M, et al., eds. Twentieth Century Physics. Vol. 2. New York: Institute of Physics Publishing, 1995. In the chapter entitled "Vibration and Spin Waves in Crystals," the Debye model is discussed in the context of work on low-temperature thermodynamics as well as in the context of Plank's quantization.

Debye, P. J. W. The Collected Papers of Peter J. W. Debye. New York: Interscience, 1954. Contains an English translation of the one paper that Debye wrote in 1912 on his model of solids. While the paper is technical in nature, there is sufficient text to illustrate Debye's clear style of writing.

Gopal, E. S. R. Specific Heats at Low Temperatures. London: Heywood Books, 1966. Chapter 2 contains some details about the Einstein and Debye models. It gives several comparisons of the theories with experiment.

Hill, Terrell L. An Introduction to Statistical Mechanics. Reading, Mass.: Addison-Wesley, 1960. A typical mathematical treatment of both the Einstein and the Debye models intended for chemists and physicists. However, the chapter on solids contains much textual information and criticisms of the theories to allow the general reader to follow the flavor of the analysis.

Lanczos, Cornelius. The Einstein Decade. New York: Academic Press, 1974. Gives a brief account of the 1907 paper of Einstein in which the Einstein model is discussed. Helps to place the Debye model in its historical context.