Question

Question 31: Linear Chain with different spring constants) U2,+1 A linear chain with two different spring c onstants Let us analyse the vibrational spectrum for a 1D row of atoms with two different chemical bonds, which is of the general type: The conmesponding physical situation is shown in the Figure above. Note that: 1. The equilibrium positions of atoms are x2n na and x2n+i-na +b. The local displacements are denoted by uanUan+1 2. The atom of mass m, which vibrates around x2n, interacts with another atom of mass m, which is vibrating around x,n+1. The coresponding force constant is ?? 3. The atom of mass m, which vibrates around x2n interacts with another atom of mass m, which is vibrating around x2n-1 The c force constant is ?2 (a) Use the Lagrangian method to show that: (b) Use the Ansatz: in (a) and show that: (c) Solve (b) and show that: (d) What is ? at k 0, and which ?, corresponds to the acoustic branch, and which co-corresponds to the optical branch? (e) For small k we find that ka | 2???2 Calculate the corresponding sound velocity (in ) using m-3 8-102632 and a

Answer 1

¿kb - mua 2 - mua 2mm

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