VIBRATIONAL SPECTRUM

AND HEAT CAPACITY OF FACE-CENTERED

CUBIC CRYSTALS *)

As is known, the thermal properties of solids are semi-quantitatively explained by the thermal vibrations of atoms about their equilibrium positions. In a whole number of cases the simplest theory, proposed by Debye, proves inadequate. It is known, for example, that the heat capacities of silver and potassium chloride do not obey this theory. A more advanced theory, proposed by Born and Karman, has not yet been applied even once to the solution of such problems, because of the difficulties in solving the secular equation for the frequency spectrum of the lattice.

In the paper under review, for the first time an attempt is made to apply the theory of Born—Karman to a three-dimensional case and to carry the solution of this problem through to numbers comparable with experiment.

If \(N_1, N_2\), and \(N_3\) are the numbers of atoms along the three edges of the crystal, then the number of normal vibrations \(dn\), whose frequencies lie in the interval from \(\nu\) to \(\nu + d\nu\), is expressed by the formula

\[ dn = N_1 N_2 N_3 G(\nu)\, d\nu. \]

The function \(G(\nu)\) is determined with the aid of models. From the equation written down it follows that

\[ G(\nu)=\frac{1}{v_0}\frac{dv}{d\nu}, \]

where \(v_0\) and \(v\) are volumes in the space \(xyz\):

\[ x=\pi(-l/N_1 + m/N_2 + n/N_3);\qquad y=\pi(l/N_1 - m/N_2 + n/N_3); \]

\[ z=\pi(l/N_1 + m/N_2 - n/N_3). \]

Here \(l, m, n\) are integers.

) R. B. Leighton, Rev. Mod. Phys., 20*, 165 (1948).

To each point \(x, y, z\) there correspond three frequencies—the result of solving the secular equation. The total number of points in the \(xyz\) space is \(N_1N_2N_3\). The volume associated with them is \(v_0\), while \(v\) is the volume in the space bounded by the surface \(\nu=\mathrm{const}\).

It is not difficult to obtain the equations of the surfaces \(\nu=\mathrm{const}\) in the space under consideration. In view of the symmetry of the crystal, it is sufficient to study one eighth of the space enclosed between two planes of symmetry at \(y=0\) and \(x=y\). The lines of intersection of the surfaces \(\nu=\mathrm{const}\) with the planes of symmetry were drawn on graph paper and then, with the aid of pins, transferred to pieces of plasticine cut in the form of the part of phase space under consideration. On each piece a series of lines of intersection was marked for one of the frequency branches.

The blocks were cut into pieces so as to obtain an element of volume enclosed between two surfaces \(\nu=\mathrm{const}\). Extrapolation from the contours to the surfaces proved entirely possible; this is helped by the fact that these surfaces are perpendicular to the planes of symmetry. It should also be borne in mind that the symmetry of the crystal makes it possible to consider a relatively small solid angle of the \(xyz\) space.

By weighing and immersing the cut-out pieces in water, the author determined the function \(v(\nu)\), and then, by differentiation, the function \(G(\nu)\). The accuracy of such model calculations, according to the author, should be of the order of several percent.

The frequency spectrum obtained in this way was used by the author to calculate heat capacities. The author carried out, for silver, calculations of the Debye characteristic temperature as a function of temperature. Above \(7^\circ\) K the Born and Karman theory agrees excellently with experiment. Below this temperature there are anomalies which compel one to regard the model employed as incorrect.

A. K.