1. Lattice Vibrations, Part I
Solid State Physics
355

2. 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.

3. 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

4. Normal Modes x1 x2 x3 x4 x5 u1 u2 u3 u4 u5
Atoms are bound to each other with bonds that have particular directions and strengths. These bonds may be modeled as ideal springs, as long as the displacements from the equilibrium positions of the atoms is small.
We’re going to consider a classical model consisting of masses connected by ideal springs. We can write expressions for the forces, apply Newton’s 2nd law and generate a set of differential equations, one for each atom. Our goal is to find solutions for which the atomic displacements have the same frequency.
Quantum mechanics can be used to describe atomic motions. Both approaches lead to the same result.

5. 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.
The forces tend to restore the atoms to their equilibrium separations. If u1 > u2, the force on atom 1 is in the negative x direction.
Net Forces on these atoms...

6. Normal Modes
Mr. Newton...
To find normal mode solutions, assume that each displacement has the same
sinusoidal dependence in time.
The complex notation is used for later convenience. Of course, displacements are real quantities, so we only really use the real part.
The second derivative of u with respect to t is just
so we can then write...

7. Normal Modes
Of course, one possible solution is u = o for all of the atoms...boring!
Other solutions only exist if the determinant formed from the coefficients is zero. The determinant will allow us to find the normal mode frequencies.
The bottom equation is satisfied by three values of omega squared.

8. Normal Modes
Only two equations can be considered independent and the displacements are not completely determined by these equations. However, any two ratios of displacements can be evaluated.
This is all easily extended to the two and three dimensional cases following the same procedure as we did here.
Before we can go further, we need to define a useful construction called the Brillouin Zone.

9. u represents the displacement of atoms on plane n
q is the wavevector
a is the spacing between planes of atoms and will depend on the particular value of q
Longitudinal Wave

10. Transverse Wave

11. Traveling wave solutions
Dispersion Relation

12. Dispersion Relation q
0.6
The boundary of the first Brillouin zone lies at q = +/- pi/a
The slope of w versus q is 0 at the boundary.
The dashed line represents the continuum limit (interatomic distance much smaller than wavelength).
The special significance of the phonon wavevectors that lie on the boundary will be developed shortly.

13. First Brillouin Zone
What range of q’s is physically significant for elastic waves?
qa varies between –pi and +pi, so q ranges from -pi/a to +pi/a, which is the first Brillouin zone of the linear lattice
The range  to + for the phase qa covers all possible values of the exponential. So, only values in the first Brillouin zone are significant.

14. 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 .
The wave represented by the solid curve conveys no information not given by the dashed curve. Only wavelengths longer than 2a are needed to represent the motion.
We need both positive and negative values of q because waves can propagate to the left or to the right.
We can treat values of q outside this zone by subtracting multiples of 2pi/a to give an equivalent wavevector inside the first Brillouin zone

15. 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.
This situation is equivalent to what happens in Bragg reflection of x-rays. When the Bragg condition is satisfied a traveling wave cannot propagate in the lattice, but through successive reflections back and forth, a standing wave is set up.
The critical value at plus/minus pi/a found here satisfies the Bragg condition: theta = pi/s, d = a q = 2pi/lambda, n=1, so lambda = 2a. With x-rays it is possible to have n greater than 1 because the amplitude of the wave has meaning in between the atoms, whereas the displacement amplitude of an elastic wave only has meaning at the atoms themselves.

16. Group Velocity
The transmission velocity of a wave packet is the group velocity, defined as
The gradient of the frequency with respect to q gives the velocity for the transmission of energy through the medium.
Note that the group velocity is zero at the zone boundary and we expect zero transmission at the boundary.
In separate papers, Sommerfeld and Brillouin wrote that, in anomalous dispersion, the group velocity cannot be the signal velocity.1 Indeed, in anomalous dispersion, the group velocity goes through both negative and positive infinite values. It also goes through values greater than the speed of light2 (as does the phase velocity).

17. Group Velocity

18. 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.

19. 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.
The result for the speed of sound is the same as the result in the continuum theory of elastic waves

20. Force Constants and integrate
In metals the effective forces may be quite long range...interactions have been found that extend more than 20 atomic planes.
We can generalize the dispersion relation to p nearest planes as shown.
r is some integer
The integral vanishes except for p = r. So, the force constant at range pa is
for a structure that has a monatomic basis.

21. Diatomic Coupled Harmonic Oscillators
Consider a cubic crystal where atoms of mass m1 and m2 lie on planes that are interleaved.
It is not essential that the masses be different, but if the two atoms of the basis are on two non-equivalent sites, then either the masses are different or the force constants are different.
As before, assume only nearest-neighbor interactions and the force constants are all the same.

22. Diatomic Coupled Harmonic 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. 
The lower curve, corresponding to the minus sign, is the acoustic branch. q