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I am given the Laue interference patterns (in reciprocal space) of two crystal structures, being either simple cubic (sc), body-centered cubic (bcc) or face-centered cubic (fcc), with the task to link each pattern to its structure. The size of the points is proportional to the intensity of the Bragg diffraction. Black dots correspond to non-observed diffraction points.

Here are the two diffraction patterns I am supposed to identify

I know that a connection between the intensity of the diffracted beam and the crystal structure is provided by the geometrical structure factor $S_{\boldsymbol{K}}$ \begin{equation} I(\boldsymbol{K}) \propto |S_{\boldsymbol{K}}|^2, \end{equation} with $S_{\boldsymbol{K}}$ defined as (mono-atomic basis) \begin{equation} S_{\boldsymbol{K}} = \sum_{j=1}^n e^{i \boldsymbol{K} \cdot \boldsymbol{d}_j}. \end{equation} The index $j$ spans all atoms in the basis of the Bravais lattice (lattice constant $a$) and $\boldsymbol{d}_j$ is the vector to the $j$th atom. For the lattices considered in this exercise and with $\boldsymbol{K} = \frac{2\pi}{a} (n_1 \hat{x} + n_2 \hat{y} + n_3 \hat{z})$, one gets \begin{equation} \text{SC: } S_{\boldsymbol{K}} = 1, \quad \text{BCC: } S_{\boldsymbol{K}} = \left\{ \begin{array}{ll} 2 & n_1 + n_2 + n_3 \text{ even} \\ 0 & n_1 + n_2 + n_3 \text{ odd} \end{array} \right., \quad \text{FCC: } S_{\boldsymbol{K}} = \left\{ \begin{array}{ll} 4 & n_1,n_2,n_3 \text{ all even/odd} \\ 0 & n_1,n_2,n_3 \text{ mixed even/odd} \end{array} \right. \end{equation}

As some points do not appear in the diffraction pattern, one can readily exclude the SC lattice (and hence answers 1 and 2). If I'm not mistaken, the origin of the reciprocal lattice is the central dot (biggest one). Then the point $(1,1, n_3)$, which appears in image a) but not in image b), should help decide between answers 3 and 4.

However, as the diffraction patterns I am given are 2-dimensional, if I assume (may I?) that $n_3$ is even, then I am left with $S_{(1,1,n_3)}\neq 0$ for the BCC lattice only, wich means that a) corresponds to the BCC structure and b) to the FCC one. On the other hand, by the same reasoning, I could choose $n_3$ to be odd, such that a) would be the FCC structure and b) the BCC one. This degree of freedom is somewhat confusing.

Could someone enlighten me on the way to answer this problem properly?

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1 Answer 1

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The easiest way to solve this problem is to note that the reciprocal lattice of FCC is a BCC lattice and vice versa. Thus, we can see that the right lattice (BCC -- in reciprocal space, as you pointed out) must correspond to an FCC lattice in "real" space.

Likewise, though I find this much trickier to realize, the left reciprocal lattice is an FCC lattice, meaning it corresponds to a BCC lattice.

I visualize that the left image is a "projection" of the FCC lattice by picturing the unit cell (not a primitive cell) held in two opposite corners colinear to the incoming beam. I don't think this way of thinking is physical, but I find it helpful.

An FCC cell with the two corners marked in red. Incoming beam in blue.

Regarding your Ansatz, I believe the point where you assume $n_3$ to be either odd or even is the sticking point. You would have to take one of the structures and then find if there would be an atom in the unit cell at a certain $n_3$. Then, with this found index, you could check if we get the correct diffraction pattern or not (using the structure factor).

Cheers!

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