2D Longitudinal Acoustic and Optical Phonon Simulation

Overview

This repository presents a two–dimensional lattice dynamics simulation of longitudinal acoustic (LA) and longitudinal optical (LO) phonon modes in a diatomic crystal with unequal atomic masses and a single nearest-neighbor harmonic spring constant.

The project combines:

• Analytical dispersion relations
• Real-space displacement formulation
• Time-dependent lattice animation
• Exportable GIF visualization

The simulation illustrates the physical distinction between acoustic and optical branches.

---

Lattice Geometry

We consider a two-dimensional Bravais lattice with lattice constant $a$.

Each unit cell contains two atoms with masses:

$$ M \quad \text{and} \quad m. $$

The equilibrium lattice positions are

$$ \mathbf{R}_{nm} = (n a, m a), $$

where $n,m \in \mathbb{Z}$.

The second atom is shifted row-wise by $a/2$, producing a staggered two-atom basis.

---

Harmonic Approximation

Atoms interact via a nearest-neighbor harmonic spring constant $K$.

Within the harmonic approximation, the equation of motion is

$$ M \ddot{\mathbf{u}}_M = -K \sum_{\text{n.n.}} (\mathbf{u}_M - \mathbf{u}_m), $$

$$ m \ddot{\mathbf{u}}_m = -K \sum_{\text{n.n.}} (\mathbf{u}_m - \mathbf{u}_M). $$

Assuming plane-wave solutions,

$$ \mathbf{u}(\mathbf{R}, t) = \mathbf{\hat{k}}, u_0 \cos(\mathbf{k}\cdot\mathbf{R} - \omega t), $$

where

• $\mathbf{k} = (k_x, k_y)$ is the wavevector
• $\mathbf{\hat{k}} = \dfrac{\mathbf{k}}{|\mathbf{k}|}$ is the longitudinal polarization
• $u_0$ is the amplitude

---

Dispersion Relation

The longitudinal phonon frequencies satisfy

$$ \omega^2 = \frac{A}{2} \pm \sqrt{ \left(\frac{A}{2}\right)^2 - B }. $$

where

$$ A = 2K\left(1-\cos(ka)\right) \left( \frac{1}{M} + \frac{1}{m} \right), $$

$$ B = \frac{ 4K^2 \left(1-\cos(ka)\right)^2 }{Mm}. $$

The two branches correspond to:

• $-$ Acoustic branch
• $+$ Optical branch

---

Long-Wavelength Limit

For small wavevector $k \to 0$:

Acoustic Mode

$$ \omega_{\text{ac}} \approx v_s k, $$

where $v_s$ is the sound velocity.

Optical Mode

$$ \omega_{\text{op}}(k \to 0) = \sqrt{ 2K \left( \frac{1}{M} + \frac{1}{m} \right) } $$

Thus:

• The acoustic branch is gapless.
• The optical branch has a finite frequency at the Brillouin zone center.

---

Atomic Displacements

The displacement fields for the two branches are:

Longitudinal Acoustic (LA)

Atoms move in phase:

$$ \mathbf{u}_M(\mathbf{R},t) = \mathbf{u}_m(\mathbf{R},t). $$

---

Longitudinal Optical (LO)

Atoms move out of phase:

$$ \mathbf{u}_M(\mathbf{R},t) = - \mathbf{u}_m(\mathbf{R},t). $$

These displacement relations are implemented directly in the time-dependent animation.

---

Numerical Implementation

The simulation is implemented in Python using:

• NumPy — numerical computation
• Matplotlib — visualization
• Pillow — GIF export

The atomic positions are updated according to

$$ \mathbf{R}(t) = \mathbf{R}_0 + \mathbf{u}(\mathbf{R}_0,t). $$

---