Q7 [3 +3 +3 + 3 + 5 pts]: Consider a one-dimensional chain of atoms with identical atomic mass m and alternative spring constants C & C2. The distance between neighboring atoms is a. C2 C C2 C C2 ... m m m m m m m .. Un Un+1 In+2 (a) What is the lattice periodicity of the atom chain? (b) Without doing any calculation, how many phonon branches are there for this 1D atomic chain and why? ) We now follow the procedure shown in our lectures to derive its phonon dispersion. Define un(t) as the displacement of the nth atom away from its equilibrium position. Here we only consider the nearest neighbor coupling. Write down equation of motion for un(t) following Newton's Second Law and Hooke's Law. (d) Write the equation of motion for un+t). (e) Assume a set of trial solutions un(t) = A el(kna-w), Untu(t) = B el(k(n=1)a-wt), Untz(t) = A ef(k(n=2)a-wt). Appy these trial solutions to the two equations of motion in (c) and (d) to get two new equations that relate frequency w and wavevector k. Q7 [3 +3 +3 + 3 + 5 pts]: Consider a one-dimensional chain of atoms with identical atomic mass m and alternative spring constants C & C2. The distance between neighboring atoms is a. C2 C C2 C C2 ... m m m m m m m .. Un Un+1 In+2 (a) What is the lattice periodicity of the atom chain? (b) Without doing any calculation, how many phonon branches are there for this 1D atomic chain and why? ) We now follow the procedure shown in our lectures to derive its phonon dispersion. Define un(t) as the displacement of the nth atom away from its equilibrium position. Here we only consider the nearest neighbor coupling. Write down equation of motion for un(t) following Newton's Second Law and Hooke's Law. (d) Write the equation of motion for un+t). (e) Assume a set of trial solutions un(t) = A el(kna-w), Untu(t) = B el(k(n=1)a-wt), Untz(t) = A ef(k(n=2)a-wt). Appy these trial solutions to the two equations of motion in (c) and (d) to get two new equations that relate frequency w and wavevector k