Can Quantum Mechanics Predict Crystal Stability and Graphene Behavior?

[alal]

Hey Guys! I was wondering if anybody could help me. I have a few questions, i solved the second problem(my work is shown at the end)

Here are the questions:
1. For ionic salts (A+ B-) the inter ionic potential can be approximated by Φ(r) = (K/r^n) -/+ (e^2/ 4pi ε0 r) with n~ 10 where K is a constant. Calculate the equilibrium nearest neighbor distance r0 and the total static energy of the crystal if it has an FCC lattice (i.e. NaCl structure). Repeat the calculation for a SC lattice (i.e. CsCl structure). Which should have the lowest energy and thus be the stable crystal structure? Assume that K does not change for the different crystal structure. To a very good approximation you can ignore the repulsive part except for the nearest neighbor ions. Hint: You will need to find the Madelung constant for SC and FCC crystal structures. Also, you need to only calculate the sum over the repulsive part of the potential for the nearest neighbors.

2. Suppose the vibrations in graphene can be described using Debye approximation for the dispersion curve. (Graphene is a single layer of atoms cleaved from graphite). Derive the 2D density of states ρ (ω) for the phonons in this approximation. (Assume that the velocity of sound is the same for the 3 polarizations of vibration, one longitudinal and two transverse). Calculate the heat capacity per carbon atom in a sheet of graphene in the Debye approximation. What is the high temperature and low temperature limit of the heat capacity? In this model what would you expect the temperature dependence of the phonon contribution to the thermal conductivity к (T) vs. T. Indicate θD in your sketch and explain the temperature dependence in the 3 different temperature regions, low (T<< θD), intermediate (T< θD) and high (T>> θD). Assume that the boundary scattering is dominant at low temperatures and that phonon- phonon scattering dominates at the higher T ranges.

I worked on problem 2. But not sure if i am heading the right direction. Please click on the link below to see my work. As for the 1st problem i don't know how to start.

http://img216.imageshack.us/my.php?image=picture1kg5.jpg

I would greatly appreciate any help in solving these problems. Thank you once again.

 

[Jhero]

I am a scientist and I am happy to help you with your questions. Let's start with the first problem. To calculate the equilibrium nearest neighbor distance and total static energy of a crystal with an FCC lattice, we need to use the equation given in the problem: Φ(r) = (K/r^n) -/+ (e^2/ 4pi ε0 r).

To find the equilibrium distance, we need to find the value of r where the potential energy is minimum. This is done by taking the derivative of the potential energy with respect to r and setting it equal to 0. After some calculations, we get r0 = (2^(1/6) * a)/n, where a is the lattice constant. For an FCC lattice, the lattice constant is a = 4r0/(2^(1/6)). Plugging this in, we get r0 = (2^(1/6) * 4r0/(2^(1/6)))/n = 4r0/n. Solving for r0, we get r0 = n/(4-n) * a.

To find the total static energy, we need to sum up the potential energy for all nearest neighbor pairs in the crystal. For an FCC lattice, there are 12 nearest neighbor pairs per ion. So, the total energy will be 12 * Φ(r0). Plugging in the value of r0 we calculated earlier, we get the total static energy for the FCC lattice as 12 * Φ(r0) = 12 * [(K/(n/(4-n) * a)^n) - (e^2/(4pi ε0 * n/(4-n) * a))].

Repeating the same calculations for a SC lattice, we get the equilibrium distance r0 = a/n and the total static energy as Φ(r0) = (K/a^n) - (e^2/(4pi ε0 * a)). Comparing the two energies, we can see that the SC lattice has a lower energy and will be the stable crystal structure.

Moving on to the second problem, we need to use the Debye approximation to calculate the density of states and heat capacity for graphene. The Debye approximation assumes that the frequency of vibrations is linearly related to the wave vector. In this case, we can use the formula for a 2D system, ρ(ω) = A *