Interplanar force constants

We now show that these differences are due to the distinction between the first and the second nearest neighbor force constants of the four elements. Let ft i and ft2 be the first and second nearest neighbor force constants. And ft i(hkl) and ft2(hkl) be their respective force constants in the direction, and ft(hkl), the interplanar force constants for (hkl) planes. Then, the following force constant relationships can be derived for a bee structure as illustrated in Figs. 2, 3. 

The crystal lattice vibration and the force coefficients are the subject of Chapter 12. We describe the experimental dispersion curves and conclusions that follow from their examination. The interplanar force constants are introduced. Group velocity of lattice waves is computed and discussed. It allows one to make conclusions about the interatomic bonding strength. Energy of atomic displacements during lattice vibration (that is propagation of phonons) is related to electron structure of metals. 

Fourier Transformation of Dispersion Curves interplanar Force Constants j 181... 

Table 12.3 The interplanar force constants (N m ) for metals with a bcc crystal lattice. Oscillation mode is L[111]. The values of Fp were calculated by the author in accordance with (12.21) proceeding from the experimental dependences r — J [9].

Table 12.3 shows the values of the calculated interplanar force constants for five metals with the body-centered cubic crystal lattice. For simple metals sodium and potassium the first interplanar coefficients are of the order of several N m h These coefficients of niobium and molybdenum make tens of N m. The F3 coefficient is maximal. 

Two examples of such chains are given in Fig. 5.3.1. The atomic planes are connected by interplanar force constants k and, for symmetric choices of propagation direction, the equations of motion of the "linear chain" lead to a simple secular equation 2 x 2, whatever the range of the forces different sets k are then needed for different choices of polarization and of propagation direction. For choices less symmetric than k [100] or k ... 

The above picture of the "linear chain" is not new. The new ingredient is the faculty to determine the planar force constants ab initio, independently of any phenomenological model for the interactions. By switching from interatomic to interplanar force constants we have achieved two goals 1) The ab initio evaluation of k requires supercells which are large only in one dimension -and wRich thus become feasible. 2) Moreover, the k fall off with distance faster than the corresponding ( k V ). This property will be particularly useful in polar crystal where the k include all electrostatic interactions. 

Table 5.1 Interplanar force constants k defined by eq. (5.3.1) as obtained from the ab initio self-consistent calculations in Refs. 39 and 10 the anharmonic contributions of the lowest order are eliminated. The transverse [100] force constants were determined on sextupled supercells, all the others on the quadrupled ones the last decimal place is not guaranteed. The labels c-c, a-a on the even-n forces correspond to cation-cation nd anion-anion interactions. All force constants in 10 dyn/cm. 

The interplanar force constants defined by eq. (5.3.1) are summarized in Tab. 5.1. As explained in Section 5.4, care was taken to remove from them the anharmonicity of the lowest order allowed by symmetry only the values used for calculation of phonon dispersion are quoted. Translational invariance of the supercell implies that the restoring force on the displaced plane -k is given by °... 

Recent Rahman et al. have explained the disagreement in the large 0 region in terms of the geometrical inward relaxation of the surface which in turn produces an enhancement of the interplanar force constant between the surface and the next to the last plane. 

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