Quantum Mech Homework Help Urgent: FCC, SC Lattice, Graphene Heat Capac

In summary, the conversation is about two physics problems. The first problem involves calculating the equilibrium nearest neighbor distance and total static energy for an ionic crystal with different lattice structures. The second problem involves finding the 2D density of states and heat capacity for graphene using the Debye approximation. The speaker has solved the second problem and has provided their work, but is asking for help with the first problem. They mention a Forum Rules and forum specifically for homework help and request assistance from others.
  • #1
alal
4
0
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.

Bye!
 
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  • #2
you should read the Forum Rules, there is a forum specifically to homeworks, and finally you have to show us your work before we help.
 
  • #3
I have edited my post! Added my work! Please look through and try to help.
 
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  • #4
Anybody to help?
 

1. What is the difference between the FCC and SC lattices in quantum mechanics?

The FCC (face-centered cubic) and SC (simple cubic) lattices are two common types of crystal lattices used in studying the properties of materials in quantum mechanics. The main difference between them lies in their arrangement of atoms. In an FCC lattice, atoms are arranged in a face-centered cubic structure, meaning that there are atoms at each corner of a cube and one in the center of each face. In a SC lattice, atoms are arranged in a simple cubic structure, with one atom at each corner of a cube. This results in a more tightly packed structure in the FCC lattice, allowing for higher densities and stronger bonds between atoms.

2. How does the heat capacity of graphene differ from other materials?

The heat capacity of a material is a measure of its ability to store heat energy. In the case of graphene, a single layer of carbon atoms arranged in a hexagonal lattice, its heat capacity is much lower than that of most other materials. This is because graphene has a unique electronic structure that allows it to dissipate heat more quickly, making it a promising material for heat management applications. Additionally, the heat capacity of graphene is highly anisotropic, meaning it varies with the direction of heat flow, which is not typically seen in other materials.

3. How does the FCC lattice structure affect the properties of materials?

The FCC lattice structure has a significant impact on the properties of materials, particularly in terms of their strength and density. Due to its tightly packed arrangement of atoms, the FCC lattice is able to provide strong bonds between atoms, resulting in materials with high strength and rigidity. In addition, the FCC lattice allows for a high density of atoms, which can also contribute to the strength and other properties of a material. This lattice structure is commonly found in metals and alloys, making them strong and durable materials.

4. How is the concept of heat capacity related to quantum mechanics?

In quantum mechanics, heat capacity is a measure of the energy required to increase the temperature of a material by a certain amount. At the atomic level, heat is transferred through the vibrations of atoms and the energy levels of these vibrations are quantized, meaning they can only have certain discrete values. Therefore, the heat capacity of a material is influenced by the quantum nature of these vibrations. In graphene, for example, the unique electronic structure and vibrational modes of its atoms result in a lower heat capacity compared to other materials.

5. How can I apply my understanding of lattice structures and heat capacity to real-world applications?

The concepts of lattice structures and heat capacity have a wide range of applications in various fields, including materials science, engineering, and thermal management. For example, understanding the properties of different lattice structures can help in designing stronger and more durable materials for use in construction, manufacturing, and other industries. Additionally, the study of heat capacity can inform the development of better thermal insulators, heat sinks, and other materials used in electronics and other devices that require efficient heat management.

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