Given the (22) equation, prove that (23) and (24) are the frequencies of the optical and acoustic branch The homogeneous linear equations have a solution only if the determinant ofthe coefficients of the unknowns u, v vanishes:or2C-M₁w²-C[1 + exp(iKa)]² = 20 (₁ + M₂)112Cw² =-C[1 + exp(ika)]2C-M₂w²M₁M₂w¹ - 2C(M₁ + M₂)w² + 2C²(1 - cos Ka) = 0.(22)We can solve this equation exactly for w², but it is simpler to examine thelimiting cases Ka < 1 and Ka = ±π at the zone boundary. For small Ka wehave cos Ka 1-K²a² + ..., and the two roots areC--K²a²M₁ + M₂=0,(optical branch);(21)(acoustical branch).(23)(24)

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- M₁M₂w¹ – 2C(M₁ + M₂)² + 2C²(1 — cos Ka) = 0. (22) We can solve this equation exactly for w², but it is simpler to examine the limiting cases Ka < 1 and Ka = ±π at the zone boundary. For small Ka we have cos Ka 1-K²a² + ..., and the two roots are 1 w² = 2C - ( 1₁ + M₂) C -K²a² M₁ + M₂ (optical branch); (acoustical branch). (23) (24)

- M₁M₂w¹ – 2C(M₁ + M₂)² + 2C²(1 — cos Ka) = 0. (22) We can solve this equation exactly for w², but it is simpler to examine the limiting cases Ka < 1 and Ka = ±π at the zone boundary. For small Ka we have cos Ka 1-K²a² + ..., and the two roots are 1 w² = 2C - ( 1₁ + M₂) C -K²a² M₁ + M₂ (optical branch); (acoustical branch). (23) (24)