QB_PH_SEC7_019: Crystal Structure & Lattice Dynamics

For a three-dimensional crystalline solid containing $p$ atoms in its primitive unit cell, which of the following statements is/are CORRECT regarding its vibrational phonon branches?

Correct Answer

### Step-by-Step Multi-Select Evaluation: 1. **Total Modes in 3D Lattice**: A crystal with $p$ atoms per unit cell in 3 dimensions possesses $3p$ total phonon branches for every allowed wavevector $\vec{k}$. 2. **Acoustic Branches (Statement A & C)**: - As $\vec{k} \to 0$, all $p$ basis atoms in each unit cell move in phase (rigid translation of the crystal). - This corresponds to continuous translational symmetry, requiring $\omega \to 0$ at $k = 0$. - There are 3 orthogonal acoustic modes (1 longitudinal acoustic LA, 2 transverse acoustic TA). (Statements A and C are CORRECT) 3. **Optical Branches (Statement B)**: - The remaining modes involve out-of-phase relative displacements of basis atoms against one another within the unit cell. - Number of optical branches $= 3p - 3$. (Statement B is CORRECT) 4. **Monoatomic Evaluation (Statement D)**: - For $p = 1$: $3(1) - 3 = 0$ optical branches. There are 3 acoustic branches and NO optical branches. (Statement D is INCORRECT) 5. **Conclusion**: Options A, B, and C are correct.