# Structure Factor for non-Integer miller indices

#### NanoMagno

Hi all.

I am a PhD student in a condensed matter group.

Consider: I observe superlattice reflections due to ferrimagnetic order that requires one cell parameter to be multiplied by M, the next by N and the third by O .

In other words, the magnetic order is described by a magnetic unit cell that is Ma x Nb x Oc

This leads to (H/M K/N L/O) reflections, where H,K,L are integers.

In order to calculate the structure factor for reflection (H/M K/N L/O) I construct the Ma x Nb x Oc cell and perform the sum:

$$F \left( \textbf{Q} \right) = \sum_{i}{f0_{i} \left( \textbf{Q} \right) \exp{ \left( i \textbf{Q} \cdot{} \textbf{r}_{i}\right) } } = \sum_{i}{f0_{i} \left( \textbf{Q} \right) \exp{ \left[ 2\pi i \left( hx + ky + lz\right) \right]} }$$

Over the now much larger basis of atoms.

I divide by the ratio of the volumes in order to compare with any intensity calculated directly from the a x b x c cell (for example to compare with previously calculated nuclear scattering).

To me this is fine. My thesis supervisor is concerned based upon the following:

If you take the Ma x Nb x Oc cell but make it ferromagnetic the satellites become forbidden.

When I calculate F(H K L) corresponding to a non-integer peak I get zero (yay).

However, the non-integer reflection is also forbidden in the original cell, and my supervisor insists that if I substitute non-integer values for H, K and L into the structure factor for the a x b x c cell with its limited basis I should be able to demonstrate it is forbidden.

I believe that the structure factor calculation as done above is only valid for integer H, K and L due to the fundamental relation between the atomic basis and the real space (and reciprocal space) translation vectors.

I have attached something I gave to my supervisor; however, I think it is still unclear as I merely demonstrate a fairly simple example (the symmetry of the real system is much worse than simple cubic!).

If you are still reading and haven't lost interest: can anyone see what I mean and what I SHOULD be saying?

Thanks in advance for anything people can come up with.

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