My Appendix: Chains of molecules, planes of scattering
In a very recent post, I mentioned an appendix to an article I wrote. I rather like it. The appendix grew out of a little document I put together. That document is longer, vaguer and a little different from the published appendix, and so I am putting it here. Now, the article was written in LaTeX, and this is a website, so I tried running htlatex on the file. It was very complicated:
$ htlatex planes $ firefox planes.html
And it worked. Next thing is to get it into WordPress… Easy enough to cut and paste the HTML code into the window here, but what about all the graphics that were turned into png files?Ah well…bit of manual fiddling. Equations and symbols seem to sit high, and some of the inline equations have been broken into a mix of graphics and characters… still, not too bad. The PDF version is available here.
Planes perpendicular to vectors
Say you have a vector in real space, expressed say in direct lattice terms, for
You may want the reciprocal plane(s) perpendicular to this vector.
Because correlations in a crystal collapse the scattering into features perpendicular to the direction of the correlation. In a normal, fully ordered three dimensions (3D) crystal, this collapsing happens in all three directions, so the scattered intensity coming off the atoms gets concentrated at points, the reciprocal lattice points, usually denoted hkl.
If you have only two dimensional ordering, the scattering is collapsed down in two directions but not the third, giving rise to rods or lines of scattering in reciprocal space (that is, in diffraction space). If there are only one dimensional correlations, the scattering collapses into sheets, that is, it is delocalised in two dimensions and only localised in one dimension (because there are only correlations in one dimension).
In diffuse scattering the crystal is typically long-range ordered in three dimensions, and the diffraction pattern shows nice Bragg peaks (hkl reflections). However, there can also be disorder, for example in the motions of the molecules or the chemical; substitution of one species of atom or molecule for another.
In a molecular crystal, one can sometimes identify a chain of molecules running through the crystal, and interactions within these chains are likely to be much stronger than those within. That tends to mean that the motions of the molecules along the direction of the chain (call that ‘longitudinal’ motion) is highly correlated, while it is not well correlated laterally.
In such a situation, the single crystal diffuse scattering will show ‘sheets’ of scattering perpendicular to the length of the chain.
Then we can say that
and these reciprocal vectors are defined in terms of the direct space vectors like this
and similarly for the other reciprocal vectors. The important thing for us to note is that this means is perpendicular to and . This is important when we go to take dot products later on. The bottom line here is basically the volume of the unit cell, and 2π is just a scalar, so from the point of view of defining the plane that we want, these are not important.
and since we have more variables than we need if we are to satisfy eq. 1, we can arbitrarily set qc⋆ = 0.
and this is useful because, to take the last term on the first line as an example, is perpendicular to ( × ) by the very nature of the cross product. This means that any terms with a repeated vector go to zero. Further, in the remaining terms the vector part is just of the form ⋅ which is the unit cell volume and a constant, which we can also factor out to be left with
which is nice and simple. This is not a surprise but still…
The next step is to find another vector in that plane. This is just , and if we use the same logic but, to make non-collinear with , we choose rb⋆ to be zero, we get an equation analogous to eq. 6. These can be summed up as
Now, in terephthalic acid (TPA), triclinic polymorph of form II, each molecule has a -COOH group at each end. These H-bond strongly with the groups on neighbouring molecules and you get strongly correlated chains of molecules running along the [-111] (direct space) direction. This then suggests that the planes of scattering perpendicular to these chains will extend in the directions
Now, does this work? Figure 1 is some data from TPA, diffuse scattering data measured on a synchrotron. It also shows the reciprocal axes and the white, two-ended arrows show the directions of the diffuse planes and by
counting Bragg spots it can be seen that these agree with the calculation above.
This means that we can ascribe these features to correlations in the displacements of the TPA molecules linked by the -COOH groups.