Finding unit cell dimensions of iron or copper

[CAF123]

Homework Statement

The research notes from an x-ray diffraction experiment were damaged and information was lost. The wavelength of the x-rays used in the experiment, and measurements of the three smallest Bragg angles (θ) from the sample were all that remained: they were 0.71Å, 10.1°, 14.4°, 17.7°. Using this information determine whether the samples was iron or copper, giving the steps in your reasoning.

Homework Equations

Bragg's Law: $n \lambda = 2 d \sin \theta$.
Distance between planes in a lattice structure.

The Attempt at a Solution

I know the distance between the Miller planes is, for a cubic unit cell, $d = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$. Using Bragg's Law, I can calculate the value of d given the angle and the wavelength. Since for each angle, I attain a different value for d, I take it then that the angles in question are the angles that the incident xray radiation make with 3 different families of planes. (e.g for angle 10.1, we have an intensity maximum (at n = 1) for a family of planes at some spacing d, for 14.4, this corresponds to another intensity maximum (at n=1) to another family of planes with a different spacing d.) At least, that is how I interpret the fact that the three angles give a different d.

If all three angles gave the same interplanar spacing, then (I think) I could conclude that the three angles correspond to the three smallest orders of intensity maxima for one family of planes. (e.g 10.1 for n =1, 14.4 for n =2 etc..). Are these correct interpretations?

I am not sure how to use what I have above to determine the unit cell dimensions. I know given a d, I can relate this to h,k,l and then I thought, since I was given three angles and three unknowns (h,k,l) I could solve, but when I tried this, I ended up with a contradiction.

Many thanks,

 

[anathema]

I would approach this problem by first understanding the basics of x-ray diffraction and Bragg's Law. I would also make sure to have a clear understanding of the difference between cubic and non-cubic unit cells and how to calculate the distance between planes in each type of unit cell.

Next, I would use the given information of the wavelength and the three smallest Bragg angles to calculate the corresponding interplanar spacing for each angle using Bragg's Law. From this, I would determine the different families of planes that are being diffracted at each angle.

Based on the given information, it seems that the three angles correspond to three different families of planes. If they were all the same, it would indicate that the sample has a cubic unit cell. However, since they are different, it suggests that the sample has a non-cubic unit cell.

To determine whether the sample is iron or copper, I would need to compare the calculated interplanar spacing values to the known values for these materials. Iron has a face-centered cubic (FCC) structure with a lattice constant of 2.866 Å, while copper has a body-centered cubic (BCC) structure with a lattice constant of 3.615 Å. By comparing the calculated values to these known values, I can determine which material has a closer match.

In order to determine the unit cell dimensions, I would need to use the calculated interplanar spacing values and the corresponding Miller indices for each angle to solve for the lattice parameters a, b, and c. This can be done by setting up a system of equations and solving for the unknowns. Once the lattice parameters are determined, the unit cell dimensions can be calculated.

In summary, to determine whether the sample is iron or copper, I would compare the calculated interplanar spacing values to the known values for these materials. To determine the unit cell dimensions, I would use the calculated interplanar spacing values and Miller indices to solve for the lattice parameters and then calculate the unit cell dimensions.