Phonon Brillouin-Zone Folding

Theory

In a monoatomic nearest-neighbor harmonic chain with lattice spacing a, spring constant K, and mass m, the phonon dispersion relation giving the angular frequency ω for a given wavevector k is:

ω(k) = 2√(K/m)|sin(ka/2)|

The lattice structure introduces a fundamental periodicity. Since the positions of the atoms are discrete (at x_n = n a), wavevectors differing by an integer multiple of the reciprocal lattice vector (2π)/a yield identical displacement patterns at all lattice sites.

This equivalence means all physically distinct states are fully described within a single interval of wavevectors of width (2π)/a, typically chosen symmetrically around zero as [-(π)/a, (π)/a), known as theFirst Brillouin Zone. The process of mapping an arbitrary wavevectork back into this interval is called zone folding.

In the long-wavelength limit (k → 0), the dispersion relation is linear: ω(k) ≈ v_s |k|, where v_s = a√(K/m) is theacoustic speed (the speed of sound). However, one must not identify this continuum linear approximation with the full nonlinear dispersion, particularly near the zone boundaries (k = ± (π)/a) where the group velocity drops to zero.

The group velocity of a wave packet is given by v_g = (dω)/dk. For the positive-frequency branch, the group velocity is:

v_g = a√(K/m)sign(sin(ka/2))cos(ka/2)

At k = 0, the absolute value in the dispersion branch has a cusp with no two-sided derivative, making the group velocity formally unavailable at that exact point.

Worked Example

Consider a harmonic chain with normalized inputs where we fold a raw wavevector back into the first Brillouin zone and evaluate its frequency and group velocity. Let spacinga = 1, spring constant K = 1, mass m = 1, and let the initial raw wavevector ratio be given by ka/π = 0.5.

1. First, find the reduced wavevector ratio. Since ka/π = 0.5 is already within the interval [-1, 1), it maps to itself: reduced ka/π = 0.5.
2. Calculate the acoustic speed v_s = a√(K/m) = 1 × √(1/1) = 1.0.
3. Evaluate the angular frequency:

   ω(k) = 2√(K/m) * |sin(ka/2)| = 2 * 1 * |sin(0.5π/2)| = 2 * sin(π/4) = 2 * (√2/2) = √2 ≈ 1.414213562373095

4. Determine the group velocity v_g:

   v_g = a√(K/m) * sign(sin(ka/2)) * cos(ka/2) = 1 * 1 * sign(sin(π/4)) * cos(π/4) = 1 * 1 * (√2/2) = √2/2 ≈ 0.7071067811865476

Because the raw ka/π = 0.5 is not an integer or half-integer, we can also see its exact neighbor phase: cosine(0.5π) = 0.0, sine(0.5π) = 1.0.

Experiment

You can interactively explore zone folding and the resulting dispersion properties in the Phonon Brillouin-Zone Folding module.

1. Try selecting a "Raw ka over pi" of 3. Notice that the Reduced ka/π wraps back to -1. Consequently, the actual discrete site displacement pattern remains completely identical, which confirms that unwrapped, shorter wavelengths are not uniquely observable on the lattice.
2. Adjust the "Spring constant" to 4 to see the direct scaling effect on the acoustic speed and dispersion curve steepness, resulting in a higher angular frequency.

Limitations

This model assumes a strictly one-dimensional, monoatomic chain with only nearest-neighbor harmonic interactions. Real crystals are typically three-dimensional, feature polyatomic bases leading to multiple optical and acoustic branches, and contain anharmonic effects which allow for phonon-phonon scattering. Furthermore, next-nearest neighbor interactions would change the shape of the dispersion curve away from the simple sine-absolute function.

References

We rely on foundational external academic resources for correct derivations of the phonon dispersion relations in monoatomic one-dimensional harmonic chains.

• Phonon Applet - University of Cologne:
  https://ph2.uni-koeln.de/lehre/applets-solid-state-physics/phonons
• Kittel, C. (2004). Introduction to Solid State Physics. Wiley.