Solid State: Diamond lattice and scattering

In summary, the three scattering angles are consistent with a diamond lattice if the light has a wavelength of 330 nm. Laue's law states that delta(k) = G, and by conservation of energy, |k| = |k'|. To find G, one needs to find the lattice vectors of the primitive cell of diamond. These lattice vectors can be found by using the reciprocal lattice basis vectors, b_1 = (2 pi/a) (-xhat+yhat+zhat), b_1 = (2 pi/a) (xhat+yhat-zhat), and b_1 = (2 pi/a) (xhat-yhat+zhat). Once G
  • #1
barrinmw
7
0
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.
 
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  • #2
Maybe this question is more apt for the advanced physics homework forum.
 
  • #3
I can try there, I figured here because this is from an Intro to Solid State course.
 
  • #4
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.
 

1. What is a diamond lattice?

A diamond lattice is a type of crystal lattice structure in which the atoms are arranged in a repeating, three-dimensional pattern resembling a diamond. This structure is known for its high strength and hardness, making it a popular choice for industrial applications.

2. How is a diamond lattice formed?

A diamond lattice is formed through the process of crystallization, in which atoms are arranged in a regular pattern to form a crystal. In the case of a diamond lattice, carbon atoms are arranged in a tetrahedral structure, with each atom bonded to four neighboring atoms.

3. What is scattering in the context of solid state physics?

In solid state physics, scattering refers to the phenomenon of particles or waves being deflected or scattered when they encounter a material. This can occur due to interactions with the atoms or molecules in the material, and can provide valuable information about the properties of the material.

4. How is scattering used in the study of solid state materials?

Scattering is a commonly used technique in the study of solid state materials as it allows researchers to gain information about the structure and properties of a material without altering it. By analyzing the scattering patterns, scientists can determine the arrangement of atoms in a material and gain insight into its properties such as density, composition, and defects.

5. What are some applications of diamond lattice structures?

The unique properties of diamond lattice structures make them useful in a variety of applications. Some common uses include cutting and polishing tools, high-pressure experiments, and electronic devices such as transistors and diodes. Diamond lattice structures are also being explored for their potential in quantum computing and biomaterials.

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