**Problem 1**
Consider a linear chain in which alternate ions have masses M_{1} and M_{2}. Ions have a harmonic interaction with their nearest neighbors only, with spring constant K.

a) Show that the dispersion relation for the normal modes of elastic vibration (the phonons) is

ω_{±}^{2}(k) = (K/[M_{1}M_{2}])(M_{1} + M_{2} ± [M_{1}^{2} + M_{2}^{2} + 2M_{1}M_{2}cos 2ka]^{1/2})

where a is the spacing between ions and k is a wavevector in the interval (-π/2a, π/2a).

b) Consider the frequencies and the nature of the normal modes (the *eigenvectors* of the dynamical matrix) in the limits |ka|<<1 and |ka|~ π/2a.

c) Sketch the dispersion relations.

d) Consider the frequencies and the nature of the normal modes when M_{1}>>M_{2}.

e) Determine the dispersion relation when M_{1} − M_{2} → 0, and compare with that of the monatomic linear chain.

**Problem 2**
Consider a three-dimensional monatomic Bravais lattice in which each ion only interacts with its nearest neighbors. Assume that the interaction between neighboring ions is given by a harmonic potential,

φ(**r**_{i}−**r**_{j}) = (1/2)K(|**r**_{i}−**r**_{j}| − d)^{2},

where d is the equilibrium spacing between the atoms and K is the spring constant. Show that the frequencies of the three normal modes for each wavevector **k** are given by,

ω_{s}(**k**) = [&lambda_{s}(**k**)/M]^{1/2},

where M is the ion mass and the λ_{s}(**k**), s = 1,2,3, are the three eignevalues of the 3×3 matrix,

D_{μν}(**k**) = 2K∑_{R≠0} sin^{2}(**k**⋅**R**/2)**e**_{μ}(**R**)**e**_{ν}(**R**),

where the sum is over only the nearest neighbors of the point **R**=0, and **e**_{μ}(**R**) is the μ^{th} component of the unit vector in the direction of **R**. (You may find Eqs (22.59) and (22.11) of Ashcroft and Mermin a good place to start).

**Problem 3**
Consider the model of the previous problem and apply it to compute the normal modes of an fcc lattice, where the 12 nearest neighbor vectors are given by

(a/2)(±**e**_{x}±**e**_{y}), (a/2)(±**e**_{y}±**e**_{z}), (a/2)(±**e**_{z}±**e**_{x}),

where **e**_{μ} is the unit vector in direction μ.

a) Show that when **k** is in the (100) direction, i.e. **k** = (k, 0, 0), then one normal mode is longitudinal with frequency

ω_{L} = 2 [2K/M]^{1/2} sin (ka/4),

and the other two modes are transverse and degenerate with frequency

ω_{T} = 2 [K/M]^{1/2} sin (ka/4).

b) Next consider **k** to be in the (111) direction, i.e. **k** = (κ, κ, κ), |**k**|=(3)^{1/2}κ. Show that one normal mode is longitudinal with frequency

ω_{L} = 2 [2K/M]^{1/2} sin (κa/2)

and the other two modes are transverse and degenerate with frequency

ω_{T} = [2K/M]^{1/2} sin (κa/2)

c) Finally, consider **k** to be in the (110) direction, i.e. **k** = (κ, κ, 0), |**k**|=(2)^{1/2}κ. Show that one normal mode is longitudinal with frequency

ω_{L} = 2 [(K/M)(sin^{2} (κa/4) + sin^{2} (κa/2))]^{1/2}

one is transverse and polarized along the z-axis with frequency

ω_{T1} = 2 [2K/M]^{1/2} sin (κa/4)

and the third is transverse and polarized perpendicular to the z-axis with frequency

ω_{T2} = 2 [K/M]^{1/2} sin (κa/4).

d) Make three sketches of the dispersion relations in the above three directions, for **k** extending from the origin to the edge of the 1st Brillouin zone. Note that your results look quite similar to the measured dispersion relations of aluminum, shown in Fig. (22.13) of Ashcroft and Mermin.