Hi Lukas,

what exactly do you mean by "unit cell parameters calculated" ? What do you mean by "variations"? Are these calculations the result of a Rietveld refinement ?

Keep in mind that the unit cell parameters alpha, beta and gamma are all fixed by symmetry for the crystal systems orthorhombic, tetragonal, trigonal, hexagonal, and cubic. For these systems the uncertainty of the angles is set to zero.

Hi. Thanks for your answer. Yes, I am doing the Rietveld refinement. By variations I've meant the larger differences in the obtained angle unit cell values for triclinic and monoclinic crystals when compared to the literature data. I'm pretty sure I've seen an explanation of this (related to the degree of freedom of the relatively low-symmetry, that is, triclinic and monoclinic, crystals) somewhere and I just wanted to find the reference.

Cheers,

Luke Kruszewski

I am not sure whether I correctly understand your question since also your assumption does not sound reasonable to me. It is clear that angle variations only appear for rhombohedral, monoclinic and triclinic phases since in all other lattices the angles are fix. Here already becomes obvious for rhombohedral that this is not really true since if you use the hexagonal description of the same lattice all parameters are fixed as well. The free parameter alpha for a rhombohedral description is the result of the c/a ratio of the hexagonal lattice.

Anyway, I believe (!) that the variation of experimentally determined lattice parameters is not bigger for lower symmetric phases. Of course you can constrain lattice parameters by the introduction of higher symmetry, but it might turn out that you simply do not noticed the lower symmetry. A typical example is that a triclinic phase can also have a cubic metric, i.e. a=b=c and alpha=beta=gamma=90°. It is often wrong to describe that triclinic means that all lattice parameters have to have different length and angles are not equal to 90°. The correct condition is that they only CAN be different. Of course, the majority will use this opportunity to show different lengths and angles but in fact the lattice parameters are the result of the structure and from my opinion there is not reason for molecules or atoms to have a higher accuracy and precision in their positioning if the entire arrangement shows a higher symmetry. I think that our subjective introduction of constraints cannot be interpreted as fundamental law of nature.



Hi. Thanks for your answer. Yes, the angles in trigonal & hexagonal crystals are also fixed. I do observe larger variations in the calculated unit cell values for angles in the triclinic and monoclinic crystals. This is especially true for the triclinic one: the difference between the calculated values for some angles in alunogen, which is triclinic, may reach ca. 1.5% when compared to the extreme-low or extreme-high values given in the literature. Although 1.5% is not much, it is much higher than the usual unit cell parameters difference I get, that is in the 0.0n-0.n0 % range.

As for the constrains, I try not to constrain too much parameters to avoid the overparametrization and false parameter interdependencies (and false positive fit residuals).

Indeed, some triclinic crystals have some of their angles very close to 90 deg.....

By the "degree of freedom" (which I might understand wrongly) I've meant that the lower number of symmetry elements gives more atomic positions which are not special positions, and which is also responsible for the usually larger amount of reflections seen for the triclinic and monoclinic crystals.

Cheers,

Luke Kruszewski

I am really not the expert on this field since I don't do this anymore for a long time, however, I see this from the point of view of a lattice projection (what I am doing recently).

Forget symmetry (or better: take the highest of an orthonormal description) and start with the fractional coordinates of a structure description...and use the lattice description which defines a deformation tensor (crystal matrix or other terms are used) and enables to transfere this "orthonormal" description into a Cartesian coordinates in a triclinic cell. There is practically only the uncertainty how exact you can describe this tensor since the atomic positions which are exclusively responsible for the symmetry are only used for the intensity and not for the peak positions in your diffraction patterns. in other words, the peaks only give you an educated guess about a possible symmetry, but they ar no proof.

From my point of view only the error becomes bigger since you have a higher number of undependent coeffcients in your tensor. But the error is not primarily related to the symmetry. It is your experimental technique. You can also have a very strange distribution of atoms...and your symmetry is cubic...although I have to agree that the opposite is more likely. In so far this might be a rule but I don't think it is a law. A rule enables exclusions...a law does not. Therefore it is called Friedels rule but Braggs law. (The term "Friedels law" is from my point of view wrong!)

Hello Gert! Many thanks for your effort, again!

I only use TOPAS software because this is the only one I have it; TOPAS uses STRUCTURE "files" = full structural information (site coordinates, initial unit cell parameters, space group, possible crystal size and strain influences, preferred orientation correction, etc.), so it makes the matrices etc. itself.

This independent coefficients is, I think, exactly what I likely mixed up with "degrees of freedom". The refinement procedure must have to have a lot to do with interdependences (of the parameters). My symmetry is not cubic - the minerals I take to the refinements are (almost) always exactly known structures; in case of alunogen it is triclinic. I will now go to read about Mr. Friedel....

Have a nice week!

Best regards,

Lukasz Kruszewski

Friedels rule was only an example... It won't help you solving your problem. Smaller lattice parameter deviations for high-symmetric phases only follows since you insert constrains so that deviations cannot described by different lattice parameters anymore but by e.g. the introduction of lattice defects in order to explain e.g. a broadening of peaks.

It is like a measurement in an optical goniometer. You could measure each plane normal without any crystal alignement. However, first you are trying to find one zone axis very accurately...and then you start your angle measurement for each plane normal. The introduction of a fundamental crystallographic law helps you to improve accuracy and precision of your measurements. The difference to your problem is that the zone law is a real law whereas your assumption of a high-symmetric phase is your assumption. It apparently improves your results but follows from your (possibly incorrect) assumption.

Hi again, Gert. I think the higher amount of variable (refined) parameters for the low-symmetry crystals is the most simple answer to these deviations.... Plus, of course, the diadochy substitutions in my samples...

Cheers,

Ł. Kruszewski