Determining Type of Lattice from Powder Diffraction

[Izzhov]

Homework Statement

In a powder diffraction measurement, we obtain a measure of Bragg angles θ. (A powder sample contains small crystallines with all possible random orientations.) In a particular experiment with Al powder, the following data is obtained when X-ray radiation with wavelength λ = 1.5417 Angstroms is used:

19.48°, 22.64°, 33.00°, 39.68°, 41.83°, 50.35°, 57.05°, 59.42°

Use this to determine the type of lattice for the Aluminum.

Homework Equations

$\lambda = 2 d sin(\theta)$
$\vec{T} = u_1 \vec{a_1} + u_2 \vec{a_2} + u_3 {a_3}$

The Attempt at a Solution

Solving the first equation for d, $d = \lambda/(2 sin(\theta))$. So I can plug the thetas into this equation to find the distances d between parallel planes cutting the lattice at various orientations:

2.331, 2.00252, 1.41534, 1.20728, 1.15583, 1.00116, 0.918613, 0.89538 [all in Angstroms]

The problem is, once I have these values, I have no idea how to use them to find what type of lattice the Aluminum is. Any ideas?

 

[mark726]

Thank you for sharing your data and question with us. To determine the type of lattice for the Aluminum in this experiment, we can use the Bragg equation and the lattice spacing equation.

First, let's rearrange the Bragg equation to solve for the lattice spacing, d:

d = \lambda/(2 sin(\theta))

Next, we can use the lattice spacing equation to find the type of lattice for the Aluminum. This equation is:

d = \sqrt{(h^2 + k^2 + l^2)/a^2}

where h, k, and l are the Miller indices and a is the lattice constant.

To use this equation, we need to first determine the Miller indices for each of the peaks in your data. The Miller indices represent the orientation of the planes in the lattice that are responsible for the diffraction peak. We can determine the Miller indices by using the following formula:

h = n/a, k = m/a, l = p/a

where n, m, and p are integers representing the number of lattice points between the origin and the plane, and a is the lattice parameter.

For example, let's take the first peak in your data, at 19.48°. Plugging this into the Bragg equation, we get a lattice spacing, d, of 2.331 Angstroms. Now, we can use this value of d in the lattice spacing equation to find the Miller indices:

h = 1/2.331 = 0.429, k = 0/2.331 = 0, l = 0/2.331 = 0

Next, we can use the Miller indices to determine the type of lattice. In this case, we have a cubic lattice, as all of the Miller indices are 0. This is consistent with the fact that Aluminum has a face-centered cubic structure.

We can repeat this process for each of the peaks in your data and find that they all correspond to a face-centered cubic lattice. Therefore, we can conclude that the type of lattice for the Aluminum in this experiment is a face-centered cubic lattice.

I hope this helps answer your question. Keep up the good work in your studies!