A3 HANDWRITTEN THEN BOX THE FINAL ANSWERS (DO NOT USE AI)

III. DIATOMIC HARMONIC CHAIN [20 pts] Consider a one-dimensional crystal with a diatomic ba- sis of atoms with mass M, and M2. The separation be- tween like atoms (lattice constant ) is a. 10 pts Obtain the dispersion relation w(K). 10 pts Derive the leading terms of the low energy dis- persion relations for optical and acoustic phonons (Ka