Solid State: Diamond lattice and scattering

[barrinmw]

I have the following homework question I am working on.

I am given three scattering angles: 42.8, 73.2, 89. (in degrees) without the wavelength of the light used. I am to show that these are consistent with a diamond lattice.

I started with Laue's Law: delta(k) = G and according to the professor in this instance, I am only worried about magnitudes.

By conservation of energy I know |k| = |k'|

This led me to |G| = 2 |k| Sin[theta / 2] where |k| = 2 pi / lambda

Now, if I take the ratios of |G_1|, |G_2|, |G_3| I get:

|G_2| / |G_1| = 1.63; |G_3| / |G_2| = 1.68

To get G, I started with the lattice vectors of the primitive cell of diamond which I believe are the same lattice vectors of the primitive cell of an FCC lattice.

So a_1 = (1/2) a (yhat + zhat); a_2 = (1/2) a (xhat + zhat); a_3 = (1/2) a (xhat + yhat)

I form the reciprocal lattice basis vectors from these.

b_1 = (2 pi / a) (-xhat + yhat + zhat); b_1 = (2 pi / a) (xhat + yhat - zhat); b_1 = (2 pi / a) (xhat - yhat + zhat)

Now one problem is, that I don't know how to construct the G's from this since I don't know how to find the coefficients for the diamond lattice. I know that G = v_1 * b_1 + v_2 * b_2 + v_2 * b_2 but in the end I know that |G| should equal (2 pi / a) Sqrt( v_1^2 + v_2^2 + v_3^2)

Any help would be appreciated, once I get this I can answer the next part of the question where he gives me the wavelength of the x-rays and show that "a" is that for carbon diamond lattice.

 

[DrDu]

Maybe this question is more apt for the advanced physics homework forum.

 

[barrinmw]

I can try there, I figured here because this is from an Intro to Solid State course.

 

[M Quack]

You are doing fine. I get the same value for G_1/G_2, but not for G_3/G_2.

Since you are using a primitive unit cell for the diamond lattice, your v_1, v_2 and v_3 can be any integer, 0,+/-1, +/-2,...

Try using an Exel spreadsheet or python program to calculate the first few G and their ratios.