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Lattice Vibrations, Part I. Solid State Physics 355. Introduction. Unlike the static lattice model , which deals with average positions of atoms in a crystal, lattice dynamics extends the concept of crystal lattice to an array of atoms with finite masses that are capable of motion.
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Lattice Vibrations, Part I Solid State Physics 355
Introduction • Unlike the static lattice model, which deals with average positions of atoms in a crystal, lattice dynamics extends the concept of crystal lattice to an array of atoms with finite masses that are capable of motion. • This motion is not random but is a superposition of vibrations of atoms around their equilibrium sites due to interactions with neighboring atoms. • A collective vibration of atoms in the crystal forms a wave of allowed wavelengths and amplitudes.
Applications • • Lattice contribution to specific heat • • Lattice contribution to thermal conductivity • • Elastic properties • • Structural phase transitions • Particle Scattering Effects: electrons, photons, neutrons, etc. • • BCS theory of superconductivity
Normal Modes x1 x2 x3 x4 x5 u1 u2 u3 u4 u5
Consider this simplified system... x1 x2 x3 u1 u2 u3 Suppose that only nearest-neighbor interactions are significant, then the force of atom 2 on atom 1 is proportional to the difference in the displacements of those atoms from their equilibrium positions. Net Forces on these atoms...
Normal Modes Mr. Newton... To find normal mode solutions, assume that each displacement has the same sinusoidal dependence in time.
Traveling wave solutions Dispersion Relation
0.6 q Dispersion Relation
First Brillouin Zone What range of q’s is physically significant for elastic waves? The range to + for the phase qa covers all possible values of the exponential. So, only values in the first Brillouin zone are significant.
First Brillouin Zone There is no point in saying that two adjacent atoms are out of phase by more than . A relative phase of 1.2 is physically the same as a phase of 0.8 .
First Brillouin Zone At the boundaries q = ± /a, the solution Does not represent a traveling wave, but rather a standing wave. At the zone boundaries, we have Alternate atoms oscillate in opposite phases and the wave can move neither left nor right.
Group Velocity The transmission velocity of a wave packet is the group velocity, defined as
Phase Velocity • The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity. The phase velocity is given in terms of the wave's angular frequency ω and wave vector k by • Note that the phase velocity is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate.
Long Wavelength Limit When qa << 1, we can expand so the dispersion relation becomes The result is that the frequency is directly proportional to the wavevector in the long wavelength limit. This means that the velocity of sound in the solid is independent of frequency.
Force Constants and integrate The integral vanishes except for p = r. So, the force constant at range pa is for a structure that has a monatomic basis.
Diatomic CoupledHarmonic Oscillators For each q value there are two values of ω. These “branches” are referred to as “acoustic” and “optical” branches. Only one branch behaves like sound waves ( ω/q → const. For q→0). For the optical branch, the atoms are oscillating in antiphase. In an ionic crystal, these charge oscillations (magnetic dipole moment) couple to electromagnetic radiation (optical waves). Definition: All branches that have a frequency at q = 0 are optical. q
