The energy dispersion relation for sc, bcc and fcc?

In summary, the task is to calculate the energy dispersion relation in the tight binding for simple cubic, base centered cubic, and face centered cubic crystals, without any given values. The formula to use is E(k) = alpha + beta * S * e^[ik(R-R')], where alpha is the Coulomb integral, beta is the exchange integral, and S is the sum over the nearest neighbors of atoms at position R. The sum depends on the crystal structure, and R' represents the positions of the nearest neighbors. The provided formula should be used for all three crystal structures, with only the number of nearest atoms changing. The position of R is not relevant.
  • #1
Rrrenata
2
0

Homework Statement


I need to calculate the energy dispersion relation in the tight binding for simple cubic, base centered cubic and face centered cubic crystals. There are no values given, they just need the result depending on the lattice constant a.

Homework Equations


E (k) = alpha + beta * S * e^[ik(R-R')],
for alpha = the Coulomb integral, beta = the exchange integral, and S = the sum over the nearest neighbors of atoms at position R.

The Attempt at a Solution


I can see how S depends on the crystal structure, but is that it? Should I just keep the formula and only change the number of the nearest atoms for the three different structures? Also, is it relevant where you fix R? Thank you!
 
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  • #2
Are R' the positions of those nearest neighbors?
I would expect that this exponential gets summed over, not a sum multiplied by an exponential.
 
  • #3
mfb said:
Are R' the positions of those nearest neighbors?
I would expect that this exponential gets summed over, not a sum multiplied by an exponential.
Oh my God, yes, it is the sum over the exponential and i am stupid.
 

1. What is an energy dispersion relation and why is it important in materials science?

An energy dispersion relation is a mathematical relationship that describes how the energy of a particle or system changes as a function of its momentum. In materials science, it is important because it helps us understand the behavior of electrons and other particles in different materials, which is crucial for developing new materials with specific properties.

2. How does the energy dispersion relation differ between sc, bcc, and fcc crystal structures?

The energy dispersion relation differs between these crystal structures because they have different arrangements of atoms, resulting in different energy levels and allowed energy states for electrons. For example, the simple cubic (sc) structure has a simple parabolic energy dispersion relation, while the body-centered cubic (bcc) and face-centered cubic (fcc) structures have more complex energy dispersions due to the presence of additional energy bands.

3. What factors influence the shape of the energy dispersion relation in these crystal structures?

The shape of the energy dispersion relation in these crystal structures is influenced by factors such as the type of bonding between atoms, the number of atoms in the unit cell, and the symmetry of the crystal structure. These factors affect the allowed energy states and contribute to the overall shape of the energy dispersion relation.

4. How does the energy dispersion relation affect the electrical and thermal conductivity of materials?

The energy dispersion relation has a direct impact on the electrical and thermal conductivity of materials. In materials with a linear energy dispersion relation (such as metals), electrons can easily move through the material, resulting in high electrical and thermal conductivity. In materials with a non-linear energy dispersion (such as semiconductors), electrons have a band gap between energy states, making it more difficult for them to move and reducing the material's conductivity.

5. Can the energy dispersion relation be experimentally measured?

Yes, the energy dispersion relation can be experimentally measured using techniques such as angle-resolved photoemission spectroscopy (ARPES) and electron energy loss spectroscopy (EELS). These techniques involve bombarding a material with electrons and measuring the energy and momentum of the emitted electrons to determine the shape of the energy dispersion relation.

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