Phonon Dispersion

Phonons are quantized lattice vibrations — the collective oscillations of atoms in a crystal. In a 1D monatomic chain with atomic mass M, spring constant K, and lattice constant a, the dispersion relation is:

$\omega(k) = 2\sqrt{\frac{K}{M}} \left|\sin\left(\frac{ka}{2}\right)\right|$

This is an acoustic branch: ω → 0 as k → 0 (long-wavelength sound waves).

Group and Phase Velocity

The group velocity (energy transport speed) is:

$v_g = \frac{d\omega}{dk} = a\sqrt{\frac{K}{M}}\cos\left(\frac{ka}{2}\right)$

The phase velocity is:

$v_{ph} = \frac{\omega}{k}$

Brillouin Zone Boundary

At the zone boundary k = π/a, the sine equals 1, giving the maximum frequency:

$\omega_{max} = 2\sqrt{\frac{K}{M}}$

At this point, cos(π/2) = 0, so v_g = 0 — the wave becomes a standing wave with no net energy transport.

Key Insight

Near k = 0, the dispersion is linear: ω ≈ a√(K/M) · k, so the group velocity equals the speed of sound. As k increases toward the zone boundary, the dispersion flattens and the group velocity decreases to zero.

Your Task

Implement three functions:

import math

def phonon_frequency_rad_s(k_m, K_N_m, M_kg, a_m):
    # omega = 2*sqrt(K/M) * |sin(k*a/2)|
    pass

def group_velocity_m_s(k_m, K_N_m, M_kg, a_m):
    # v_g = a * sqrt(K/M) * cos(k*a/2)
    pass

def zone_boundary_frequency_rad_s(K_N_m, M_kg):
    # omega_max = 2 * sqrt(K/M)
    pass