How does x-ray diffraction from different Bragg planes add up?

In summary, the conversation discusses an algorithm for creating a powder diffraction spectrum in Mathematica. The speaker has successfully run the algorithm but has questions about the intensity of peaks at certain angles and whether certain factors should be taken into account. The conversation also mentions the use of Rietveld fitting to account for these factors and optimize the parameters for a more accurate diffraction pattern.
  • #1
SadScholar
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I'm writing a little bit of Mathematica code that should be able to make a reasonable powder diffraction spectrum. The algorithm is like this:
Take Bravais lattice and basis. Compute reciprocal vectors.
Compute structure factor (and its square magnitude)
Have triple nested loop that creates ordered triplets of Miller indices. {(h1,k1,l1),(h2,k2,l2)...}
For each of these triplets, compute the momentum transfer vector G (in Kittel. It's K in A&M. Sometimes it's called Q or Δk in other texts). Compute the angle of the peak from Bragg's law and the intensity from the structure factor's square.

OK, this algorithm runs. I look through the list of angles. They all compare perfectly with reference literature. Here's where the real question comes in. I have this list of Miller indices, and each one comes with an angle and an intensity. Sometimes, many many miller indices come at the same Bragg angle, because the MAGNITUDE of G is the same. Should I just add the intensity (structure factor squared) of these so that they make a superpeak at that angle? Was I wrong to square the structure factor in the first place? Should I instead leave it unsquared and then square it only after I've added together all of the structure factors at a given angle? I tried both of these things, and they both yield nonsense in terms of peak intensity. I'm getting anomalously high peaks at high angle. The atomic form factor should be preventing that from happening. Any insight on this would be greatly appreciated.

PS: Debye-Waller factor is not the answer. I'm not at high enough momentum transfer for that to account for the massive discrepancies I see.
 
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  • #2
In a powder, different G vectors correspond to different orientations, i.e. the diffracted intensity comes from different grains within the powder. There add incoherently, therefore you have to add the intensities.

There are more "correction" factors that take into account the polarization, geometry, etc.

http://reference.iucr.org/dictionary/Lorentz–polarization_correction
 
  • #3
Thanks for this. Does Rietveld fitting take this kind of thing into account?
 
  • #4
It looks like I lied. I took the Debye-Waller factor into account, and the pattern showed up perfectly.
 
  • #5
Common Rietveld fitting codes take all this into account. The Debye-Waller factor can even be asymmetric, giving a "thermal ellipsoid".

The codes calculate a "theoretical" x-ray diffraction pattern from the sample and experimental input parameters, and then optimize the parameters to best match the calculated pattern to the experimental data. How exactly the pattern is calculated, and how you can constrain and limit parameters varies from code to code, but the fundamentals you point out above should be taken into account in any good Rietveld code.
 

1. How does x-ray diffraction work?

X-ray diffraction is a technique used to study the atomic and molecular structure of materials. When x-rays are directed at a crystal, they interact with the atoms and produce a diffraction pattern, which can be used to determine the arrangement of the atoms within the crystal.

2. What is the Bragg equation and how is it used in x-ray diffraction?

The Bragg equation, named after physicist William Lawrence Bragg, describes the relationship between the angle of incidence of x-rays and the resulting diffraction pattern. It is used to calculate the spacing between atomic planes within a crystal, which is essential for determining the crystal structure.

3. What are Bragg planes in x-ray diffraction?

Bragg planes are imaginary planes within a crystal that have a specific distance between them. When x-rays interact with these planes at a specific angle, they produce constructive interference, resulting in a diffraction pattern.

4. How does x-ray diffraction from different Bragg planes add up?

When x-rays interact with a crystal, they are diffracted by multiple Bragg planes, resulting in a complex diffraction pattern. The intensity of each diffraction spot is determined by the number of x-rays diffracted by each set of planes. The overall pattern is the sum of all these individual diffraction spots.

5. What information can be obtained from x-ray diffraction from different Bragg planes?

X-ray diffraction from different Bragg planes can provide information about the crystal structure, such as the unit cell dimensions, atomic arrangement, and lattice symmetry. It can also be used to identify the type of crystal present, as different crystal structures produce unique diffraction patterns.

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