Assigning <PERSON> indices to <PERSON> diffraction angles
I am a bit confused with the procedure to index diffraction peaks obtained in X-ray crystallography using the powder method. The incident wavelength $\lambda$ is known and the lattice constant $a$ is to be determined. This process is based on the relation \begin{equation} h^2 + k^2 + l^2 = \frac{4a^2}{\lambda^2} \sin^2 \theta = C \sin^2 \theta \end{equation} and comprises the following steps:
1. From the observed angles $\theta_i$, compute $\sin^2 \theta_i$
2. Use the above relation and find $C$ through trial and error
3. Round $C \sin^2 \theta_i$ to an integer $I$
4. Break $I$ into $h^2 + k^2 + l^2$. The triplet $(h,k,l)$ corresponds to the <PERSON> indices of the diffraction peak.
My problem is in step 2. 474
For instance, if I consider a sample with diffraction angles $2 \theta_i = 43.72$°, $63.55$° and $80.32$° (iron in the $\delta$ phase at $1565$K, from Problem 9.8 in <PERSON>, F. Problems in Solid State Physics, p.134), then I get respectively $\sin^2 \theta_i \approx 0.139, 0.277$ and $0.416$. To find $C$, I computed the first values of $n/\sin^2 \theta_i$ for $n = 1, 2, 3, ...$
I find that \begin{equation} \frac{1}{\sin^2 \theta_1} \approx \frac{2}{\sin^2 \theta_2} \approx \frac{3}{\sin^2 \theta_3} \approx 7.213 \equiv C. \end{equation} From this, I can deduce that $\theta_1$ corresponds to $n = h^2 + k^2 + l^2 = 1$, hence to the family of planes $(1,0,0)$. Similarly, $\theta_2$ corresponds to $(1,1,0)$ and $\theta_3$ to $(1,1,1)$. As all the reflexes are present, I identify the structure with a simple cubic structure, whose lattice constant is determined using the first equation, now that we have the <PERSON> indices. This seems fine.
The problem is that I also get \begin{equation} \frac{2}{\sin^2 \theta_1} \approx \frac{4}{\sin^2 \theta_2} \approx \frac{6}{\sin^2 \theta_3} \approx 14.426 \equiv C, \end{equation} from which I could deduce that $\theta_1$ corresponds to $n= 2$, hence to the family $(1,1,0)$, while $\theta_2$ corresponds to $(2,0,0)$ and $\theta_3$ to $(2,1,1)$. This time, as the selection rule for the diffraction is "$h+k+l$ even", I do not conclude that the structure is simple cubic but rather body-centered cubic.
How can one find the correct crystal structure, since it seems that the ordering of the index of the diffraction peaks is not uniquely determined? I suspect that it might be due to some of the peaks not having the same order, but I do not see how to determine the order of a peak from the given data. Any help in clarifying this would be greatly appreciated! 402