In principle, the complete phonon dispersion relations can be measured by coherent inelastic neutron scattering, or by inelastic X-ray scattering, provided the phonon peaks are well resolved, and their intensities are sufficient. In contrast, the polarization vectors may differ by a phase factor in going from one Brillouin zone to another zone. In other words, the phonon dispersion relations in every Brillouin zone of the crystal are the same. A strong coupling of displacements between neighboring atoms causes a large phonon dispersion, that is, a large k dependence of ω k, j. Local atomic vibrations lead usually to flat phonon dispersion relations. Optic modes could have similar longitudinal and transverse characters. Other phonon dispersion relations, being of nonzero frequency at Γ point ω ( k = Γ, j ) ≠ 0, are called optic. In some high-symmetry crystals, and along high-symmetry directions, the atomic vibrations are either polarized along the propagation wave vector k (longitudinal acoustic modes, LA), or perpendicular to k (transverse acoustic modes, TA). The exact relationship between the slope and elastic constants depends on the crystal point group. Close to k = Γ, the acoustic branches show a linear dependence, that is, ω ∼ k, and the slope is described by a combination of the elastic constants of a crystal. The phonon branches starting at ω ( k = Γ, j ) = 0 are called acoustic phonon dispersion curves. At the Γ point, which corresponds to k = 0, or to a reciprocal lattice vector k = b *, there are always three phonons with zero frequency, ω ( k = Γ, j ) = 0. Vibrational amplitudes are much larger than in real crystals.Ĭonventionally, the phonon dispersion relations are drawn along crystal high-symmetry axis, but other directions might also be shown. Animations of normal-mode vibrations of selected phonons can be viewed at. In this example the force constants for A and B atoms are assumed to be the same, while the masses are different, M A = 1.5 M B. Schematic representation of phonon dispersion relations of a primitive cubic crystal AB, with atoms A and B at positions (0,0,0) and (1/2, 1/2, 1/2). The quantity k n turns out to be the wavenumber of the phonon, i.e. The potential energy of the entire lattice is the sum of all pairwise potential energies multiplied by a factor of 1/2 to compensate for double counting: 1 2 ∑ i ≠ j V ( r i − r j ) The forces between each pair of atoms may be characterized by a potential energy function V that depends on the distance of separation of the atoms. Magnetic and gravitational forces are generally negligible. These forces may be Van der Waals forces, covalent bonds, electrostatic attractions, and others, all of which are ultimately due to the electric force. Since the lattice is rigid, the atoms must be exerting forces on one another to keep each atom near its equilibrium position. N is a large number, say of the order of 10 23, or on the order of the Avogadro number for a typical sample of a solid. These particles may be atoms or molecules. The equations in this section do not use axioms of quantum mechanics but instead use relations for which there exists a direct correspondence in classical mechanics.įor example: a rigid regular, crystalline (not amorphous) lattice is composed of N particles. While normal modes are wave-like phenomena in classical mechanics, phonons have particle-like properties too, in a way related to the wave–particle duality of quantum mechanics. Normal modes are important because any arbitrary lattice vibration can be considered to be a superposition of these elementary vibration modes (cf. In classical mechanics this designates a normal mode of vibration. 2.5 Interpretation of phonons using second quantization techniquesĪ phonon is the quantum mechanical description of an elementary vibrational motion in which a lattice of atoms or molecules uniformly oscillates at a single frequency.

Phonon dispersio dahed vertical