Find the lattice parameter given the diffraction angles (Bragg’s law)

1. The problem statement, all variables and given/known data

On a cubic crystal a set of diffraction lines (θ) was obtained with CuKα (1.54 Å) radiation:
13.70, 15.89, 22.75, 26.91, 28.25, 33.15, 36.62, 37.60 and 41.95 degrees.

What is the lattice parameter of this solid?

2. Relevant equations

nλ = 2dsin(θ)

1/d² = a² / (h²+k²+l²)

3. The attempt at a solution

The first think I believe I must find is the crystal lattice structure (bcc, fcc or sc). To achieve that I started by checking if it was bcc:

* When a structure is bcc the ratio between the sine squared of the first two diffraction angles will be 0.5 (sin²(θ1)/sin²(θ2)= 0.5. If it is fcc it will be 0.75. What about the cubic simple structure? Will the structure be cs if the result is different from both 0.5 and 0.75?

Using this condition I concluded that the structure was fcc.

To find the lattice parameter I need to modify Bragg’s law:

[itex]n\lambda = 2dsin(\theta) [/itex]

[itex]n\lambda = \frac{2a\cdot sin(\theta)}{\sqrt{h^{2}+k^{2}+l^{2}}}[/itex]

[itex]a = \frac{n\cdot \lambda \cdot \sqrt{h^{2}+k^{2}+l^{2}}}{2\cdot sin(\theta)} [/itex]

Since I know the planes allowed by fcc structures (h,k and l all odd or all even) I should be able to get the lattice’s parameter a. The first diffraction angle for fcc structures is the (1 1 1) plane.

Here is my main doubt. What n should I use? I can look at it in two different ways:
* n = 1, since this is the first diffraction angle for fcc structures.
* n = 3, since this is the third set of planes, after {1 0 0} and {1 1 0}.

The second answer seems to make more sense, however I couldn’t find a confirmation anywhere, which led me here. Also, is the method to find the type of structure which I used correct? Is there an easier way?

Kind regards.

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