OF SOLIDS' CARL W. GARLAND Massachusetts Institute of Technology, Cambridge
THE
purpose of this paper is to review briefly the present state of the theory of the lattice heat capacity of crystalline solids; to show how Co,the heat capacity at constant volume, can be obtained from a knowledge of the frequency spectrum of normal modes of lattice vibrations; and to point out the role that the elastic properties of solids play in determining this frequency spectrum. We shall be concerned with the ideal, macrocrystalline case. Such special phenomena as particle size effects, phase transitions, and electronic specific heats will not be considered. The variation of lattice heat capacity with temperature from room temperature to liquid hydrogen or liquid helium temperatures has been extensively studied experimentally for many years. Although most solids have the same qualitative temperature dependence for C,, a striking illustration of the wide range of quantitative differences is offered by comparing two extreme cases, lead and diamond. At 50°K., C,, is 5.055, and 0.005 cal. deg.3 mole-' for lead (I) and diamond (2) respectively. I n a crude way this shows
' Presented as part of the Symposium on Recent Developments in the Solid State before the Division of Chemical Education at the 129th Meeting of the American Chemical Society, Dallas, April, 1956.
VOLUME 34, NO. 12, DECEMBER, 1957
the effect on C , in changing from a soft, compressible solid like lead t o a very hard crystal such as diamond. In lead the forces between atoms in the lattice are weak, and many of the lattice vibrations lie at low frequencies which contrihute greatly to the heat capacity at low temperatures; in diamond the atomic forces are strong, so that most lattice modes lie a t high frequencies and contribute little a t low temperatures. This can he handled mathematically by expressing the heat capacity a t constant volume in terms of the normal modes of the crystal. Einstein (3) proposed that C, = k
Z ( h ~ , / k T ) ~ e h " * / k T / ( e h h h / kT 1)%
(1)
where vr is the frequency of the ith normal mode. Einstein calculated heat capacities in rough agreement with experiment on the assumption that the atoms or molecules in the lattice are independent and that all normal modes have the same frequency. This was an important advance over the classical Dulong-Petit theory which predicted a constant C , independent of temperature. Further improvement is achieved by using a more realistic distribution of normal modes. I n a macro crystal t,here are of the order of loz3normal modes in a small interval from zero to the largest frequency. These modes are so closely spaced in fre-
quency that one can consider the distribution as continuous and define a function g(v) such that g(v)dv is the number of normal modes in the interval v to v dv. I n terms of this distribution function
+
where v, is the largest frequency. If the distribution function g(v) (often called the frequency spectrum) is known, the heat capacity as a function of temperature is completely known from equation (2). In what follows, we shall be concerned with determining g(v). DEBYE THEORY
The calculation of specific heats was greatly improved by Debye (4) who treated a monatomic solid as a continuous elastic medium. This model should certainly hold well a t very low frequencies (low temperatures) where the wave length is much longer than the atomic dimensions of the lattice. The significant fact of the Debye theory is the use of this model over the entire frequency range. The solid is also assumed to be isotropic; that is, the frequency is independent of the direction of wave propagation. With these assumptions, g(v) is proportional to vZ where the proportionality constant is determined by the elastic properties of the material. The last major approximation is in choosing the largest frequency for the lattice vibrations. A single cut-off frequency for both longitudinal and transverse waves is found by requiring that the total number of vibrations be 3N for a lattice of N atoms; that is,
The mathematical details which follow from the assumptions of this model are given in many standard texts (6). The heat capacity is given in terms of one parameter OD, the so-called Debye temperature. This parameter, which can be calculated from the elastic constants of the solid, should theoretically be a constant independent of temperature. At low temperatures the theory predicts that C. varies as T a . I n general the agreement of the Debye result with experiment is quite good; in fact, for some time deviations from the theory were discussed in terms of " anomalies." I t is now recognized that there are important inadequacies in the model, but it should be emphasized that the Debye treatment is simple and perhaps the best one-parameter theory available. The most striking exceptions to Debye behamor are seen in the case of highly anisotropic solids such as graphite (6) where the low temperature heat capacity is found to vary as T2. Such an effect has been handled by using a Debye-like two-dimensional continuum theory first proposed by Tarassov but the results are not highly satifactory. An excellent discussion of the specific heat of lamellar crystals is given by Newel1 (7). A more common deviation from the Debye theory arises from the lack of constancy of the 8,. Although in some cases the experimental data may be fitted satisfactorily using a constant value for B,, such an empirical value often disagrees with the value theoretically calculated from the elastic properties.
More serious is the fact that much experimental data cannot be fitted using a constant value of 8,; the disagreement usually is most serious in the low temperature region from about 5' to 50°K. This sort of deviation can be somewhat improved by using a common cut-off wave length for all waves (and therefore a different cut-off frequency for longitudinal and transverse waves). The reasons for this choice, which has a sounder theoretical basis than the Debye cut-off frequency, are well stated by Brillouin (8). Since the Debye theory uses a model which is physically unrealistic except a t the lowest temperatures, it is not surprising that disagreement with experiment should arise. It should be noted that at sufficiently low temperatures a macroscopic three-dimensional crystal will have a C. with a T3dependence. This has been observed even in graphite below 2°K. The important point is that this "true" Debye region is a t very low temperatures, usually in the liquid helium range. LATTICE DYNAMICS
About the same time that the Debye theory was proposed, Born and von Karman suggested a molecular approach to the problem of finding the frequency spectrum, g(v), which was based on a lattice model. Although this was a sounder theoretical approach based on a realistic model and inherently capable of giving better results, it is much more complex than the Debye treatment and has been little used until recent years. With the development of mathematical techniques and the availability of high speed computers for numerical work, there has been a revival of interest in the Born theory of lattice dynamics. The theory and several applications of it are very fully stated in a book by Born (9). Recent work has been mostly on bodycentered and face-centered metals, graphite and alkali halides. One starts with an ideal crystalline solid of known crystallographic structure. A knowledge of the forces between atoms in the lattice is needed. Often central forces are used in which the atomic force constants are a function of distance alone. I n this case there is one force constant for each type of neighbor interaction which is considered. Except in the case of ionic salts where long range electrostatic terms are important, the forces decrease rapidly with distance, and neighbors beyond the third nearest neighbor have been neglected. More general tensor forces between atoms can be used a t the expense of making the problem more complex. (As an example, in the case of a nonideal packed hexagonal close-packed lattice using only the twelve nearest neighbors, there are two central force constants and seven tensor force constants.) With the structure, the mass of the atoms, and forces known, the total kinetic and potential energy for a lattice with N unit cells is written in terms of the displacements of the atoms from their equilibrium positions. The potential energy is usually written as quadratic in these displacements which assumes harmonic or Hooke's law forces. Some work has been done using cubic terms (anharmonic terms) but this effect can be neglected in discussing the low temperature heat capacity. The Lagrange equations of motion are found making use of Born's cyclic boundary conditions. These boundary conditions are equivalent JOURNAL OF CHEMICAL EDUCATION
to studying a crystal of N unit cells imbedded in an infinite crystal, which will be satisfactory as long as N is large. From the equations of motion one obtains a secular equation for the frequency of the normal modes. For a material with n atoms per unit cell, there is a secular determinant of order 3n for each of N points in reciprocal space which define the direction and wave length of lattice normal modes. This gives 3 n N normal modes as required. Thus both "optical" and "acoustical" branches of the frequency spectrum are obtained. Although this description is given in terms of classical mechanics, Montroll (10) has pointed out that the same set of normal modes would be obtained from the solution of the Schrodinger equation. As an illustration of the secular equation which results from the lattice theory, the secular determinant for a monatomic face-centered lattire is given below (W,
cu and ell - c12 are independent of electron gas contribution, so that only two force constants can be derived. In hexagonal close-packed metals there are five elastic constants which give four atomic force constants. I n the past, the elastic constants of solids have been measured by a variety of static methods in which the deformations produced by time constant forces are determined. More recently, a dynamic resonance method has been employed using a "composite oscillator" in which the test specimen is affixed to a quartz crystal of the same cross section and the enciting frequency varied until resonance is obtained. Using this technique, Sutton (12) has measured the elastic constants of aluminum from 63' to 770°K. An even more powerful technique is now available. The adiabatic elastic constants of a crystalline substance may be calculated quite simply from a knowledge of the
where Si = sin (rkta), Ci = cos (rkta), P = y/a; and a is the nearest neighbor force constant, y is the secondnearest neighbor force constant, M is the mass of the atom. Points in reciprocal space are indexed by k ~k2, , k3. I n principle, the problem is completely solved since one can solve equations (4) to give the frequencies of the normal modes and then obtain the frequency spect.rum. I n practice, the problem is very complex since there are of the order of loz3equations (4) for a macrocrystal. Also it is very difficult, if a t all possible, to express p(v) in an analytical form as does the Debye theory. Current research in lattice dynamics is concerned with determining accurate values for the atomic force constants especially a t low temperatures, and developing techniques for obtaining the frequency spectrum from the secular equation. Each of these fields will be discussed briefly.
velocities of propagation of ultrasonic waves along various crystallographic axes. The development of pulsing circuits has created a convenient method for measuring these velocities. A short (about one microsecond) high frequency (about 10 Mc./sec.) pulse is applied to a piezoelectric crystal cemented to a known crystal face of the test sample. The elastic wave so generated travels the length of the sample, reflects from a parallel face, returns and is partially reconverted into an electrical pulse by the transmitting crystal. By knowing the sample length and measuring the time delay between successive echo pulses, the velocity is determined. Measurements have been made from 4'K. to room temperature by Overton (IS) on copper, and other face-centered cubic metals are heing studied by him. Measurements are in progress in this laboratory on hexagonal close-packed metals. A similar method has been used by McSkimmin (14) on silicon and germanium a t low temperatures. A comparison of the central force atomic force constants obtained from elastic constants and the tensor force constants obtained from thermal diffuse X-ray scattering is given by Jacobsen (15) in a recent room temperature X-ray study of copper.
ATOMIC FORCE CONSTANTS
Atomic force constants can he determined experimentally from either thermal diffuse X-ray scattering or from the elastic constants. The X-ray technique is more powerful in the sense that general tensor force constants can he obtained, but it is experimentally quite difficult and capable of less accuracy than the measurement of elastic constants. From elastic constants, usually only central force constants can he cbtained due to the small number of independent elastic constants for a given lattice type. However, these elastic constants can be measured with excellent precision down t o liquid helium temperatures, and the atomic force constants are readily obtained from them. For a face-centered cubic metal there are three elastic constants, en, el*, era,in terms of which the atomic force constants for nearest and next nearest neighbors, a and y respectively, are a = aeu, r = '1, a(cll - ell - c4.) (5) where a is the dimension of the cubic unit cell. Only VOLUME 34, NO. 12, DECEMBER, 1957
FREQUENCY SPECTRUM
There are three lines of attack currently being investigated for obtaining the frequency spectrum of lattice normal modes, g(v), from the secular equation. The most fundamental of these is the analytic approach aimed at expressing g(v) in a closed analytic form, as can be done in the Debye theory. I n this case the mathematical difficulties are formidable; so far the most complex lattice which has been done analytically is the simple cubic lattice (16). The next approach is t o express g(v) as a power series in v or as a Legendre polynomial expansion. The coefficients in this series are determined from the moments of the frequency spectrum which can be found from the secular equ* tion. This technique is due to Montroll (10, 17) who
applied it to the simple cubic and body-centered cubic lattices. The moment calculation has also been made for a face-centered cubic lattice (18). The third approach is a numerical one. Leighton (19) obtained g(v) for a face-centered cubic lattice by the unique method of modeling constant frequency surfaces in plaster of Paris and measuring the volume enclosed between successive surfaces. More recently, high speed computers have been used to solve the secular equation with given force constants for a large number (several thousand) of representative points, and the distribution of frequencies obtained has been assumed to be the true distribution. The disadvantage of a numerical solution is that the calculation must be completely repeated if one wishes to change the force constants or study a different material of the same structure, whereas the analytic or moment method gives g(v) as an explicit function of force constants. It is also possible to make some quite general statements about the form of the frequency spectrum from topological arguments. Van Hove (20) first introduced this concept and pointed out that predictions can be made for a crystal with harmonic forces as a consequence of the periodic structure alone. He showed that for a three-dimensional crystal g(u) is continuous, dg/dv has at least two infinite discontinuities and takes thevalue - m at the maximum frequency. Rosenstock (21) has recently applied this idea in greater detail to cubic lattices and located many of the singularities in the frequency spectrum. Figure 1 is taken from Rosen6
quency spectrums resulting from lattice dynamics are t o be found in the references (15,17, and 18). Jacobson (15) has a particularly interesting comparison of the frequency spectrum obtained from a Debye, central force, and general tensor force treatment. CONCLUSION
We have seen that the specific heat of a solid may be calculated from a knowledge of the frequency spectrum, g(v), which depends on the atomic force constants of the lattice. These constants can be related to the elastic properties of the material which are readily measurable. Inadequacies of the simple Debye modeP can be overcome by use of Born's theory of lattice dynamics which gives a quite different looking frequency spectrum At this time the most significant tests of the calculation of specific heats by the Born theory have been made on copper (16) and silver (18, 19) where the lattice dynamical result has shown better agreement with experiment than the Debye result. Lattice dynamics provides a method of getting information about the type and range of forces in solids by allowing one to vary the type of forces assumed and the number of neighbors included and then test the t.heoretical result with experiment. Lattice dynamics and the frequency spectrum are also very useful in predicting and correlating many other effects such as the coefficient of thermal expansion, Raman crystal scnttering, X-ray thermal diffuse scattering, optical absorption, and lattice transport properties such as thermal and electrical conductivity. LITERATURE CITED
5
(1) LMEADS,P. F., W. R. F o ~ s r ~ r r xA ,N D W. F. GIAUQUE, J. Am. Chem. So
