How Do You Determine the Dynamical Matrix for a BCC Lattice?

[Gogsey]

Hi,

So I have to find the dynamical matricx of a bcc lattice considering only nearest neighbours. Its like if the nearest neighbours were all attached to the atom in question by springs.

So first we need the equations of motion for the lattice. There are 8 nearest neighbours, so the equations of motion, which there should be 3 for each dimension, should have 8 terms due to the forces from the 8 neighbours. So we should also have a 3x3 matrix with each element in the matrix being a sum of 8 terms.

I'm just having a little difficulty in picturing what's going on.

So I oriented the x and y-axis so the directly point along 2 of the 4 directions of the neighbours. And the distance between the atoms is root(3)a/2. So I found that, with a little algebra and geometry, that there are 4 coefficients in these equations. I found that they are root(3)/2, root(3/2), 3/2 and 3, but I'm not sure if this is what I should be doing.

These are the coefficients of the forces, or magnitudes of the forces, in the specified direction of each atom, if we orient them as mentioned above. I won't write out the full equations since that would be very messy and difficult to do since there are 3 equations with 8 terms from each neighbour, plus the ma term.. Am I doing it right?

 

[tomcat81]

Hi,

So I have to find the dynamical matricx of a bcc lattice considering only nearest neighbours. Its like if the nearest neighbours were all attached to the atom in question by springs.

So first we need the equations of motion for the lattice. There are 8 nearest neighbours, so the equations of motion, which there should be 3 for each dimension, should have 8 terms due to the forces from the 8 neighbours. So we should also have a 3x3 matrix with each element in the matrix being a sum of 8 terms.

I'm just having a little difficulty in picturing what's going on.

So I oriented the x and y-axis so the directly point along 2 of the 4 directions of the neighbours. And the distance between the atoms is root(3)a/2. So I found that, with a little algebra and geometry, that there are 4 coefficients in these equations. I found that they are root(3)/2, root(3/2), 3/2 and 3, but I'm not sure if this is what I should be doing.

These are the coefficients of the forces, or magnitudes of the forces, in the specified direction of each atom, if we orient them as mentioned above. I won't write out the full equations since that would be very messy and difficult to do since there are 3 equations with 8 terms from each neighbour, plus the ma term.. Am I doing it right?

I'm confused myself. How does one go from the equations of motion to a 3x3 matrix assuming that the BCC lattice has a basis of two atoms? If we take the components x,y,z as if it were a 3-D crystal, would that mean you end up with six equations (3 eqns for each basis)?

Aschroft Mermin is way over my head. Kittel (the required text) doesn't even cover the dynamical matrix approach in three dimensions.

 

[Gogsey]

Turns out there should be 9 equation, but 6 of them are 0.

Rge matris elements are, movement of an atom in x directio, what is the restoring force in x-direction.

This is the same for xy,xz,yx,yy,yz,zx,zy,zz.

xx, yy, zz are down the diagonal of the matrix. the others would be in there positions, but they should be all zero I think.