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:

[tex]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]} }[/tex]

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.

 

[BigJDubs]

My feeling is that I need to explain it in terms of the fact that a non-integer peak is only forbidden in the original cell due to the limited basis, and that when we go to the Ma x Nb x Oc cell with its larger basis the non-integer peak becomes allowed.

 

[charng]

Hello,

As a fellow scientist, I understand your concern and your supervisor's concern about the structure factor for non-integer Miller indices. While your approach of constructing a larger magnetic unit cell and calculating the structure factor using the sum of exponential terms is valid for the observed superlattice reflections, it may not be applicable for the forbidden reflections.

As you have correctly pointed out, the fundamental relation between the atomic basis and the real space (and reciprocal space) translation vectors is only valid for integer Miller indices. Therefore, substituting non-integer values for H, K, and L into the structure factor for the original a x b x c cell may not accurately demonstrate the forbidden nature of the non-integer reflections.

In order to address your supervisor's concerns, it may be helpful to consider the symmetry of the real system and how it affects the calculation of the structure factor. You could also explore alternative methods for calculating the structure factor for non-integer reflections, such as using dynamical diffraction theory or considering the effects of magnetic moments on the scattering.

Overall, I believe your approach is valid for the observed superlattice reflections, but further investigation may be needed to accurately demonstrate the forbidden nature of the non-integer reflections in the original cell. I hope this helps and best of luck with your research.