In fact, the answer is relatively easy to get from crystallography once the structure has been solved. Solve in P1 (often even easier than in P-1) and then just look whether the atoms you have found actually show -1 symmetry. "PLATON"s "ADDSYM" option can do this.

Modern refinement programs automatically fix the origin in P1, that is no longer a problem. What could happen is that the structure refinement becomes unstable because the two halves of the structure try to explain the diffraction effects together. What can happen is that "one atom shifts left, the other one shifts right and the thermal ellipsoids go berserk". If that happens, you actually have a good indication that the structure may be centrosymmetric....

@Paolo, "resonant scattering" is a modern term that is used for what people called "anomalous scattering" in Bijvoet's times. In many circles "resonant" is preferred over "anomalous". As we understand the process by which the difference is generated, "anomalous" is no longer an appropriate description.

I am not sure how to distinguish it by crystallographic methods, however if it is centrosymmetric it can not be ferroelectric, so if you could measure your sample's polarization( at the T range you want to find out whether it is P1 or P-1) and see that it is ferroelectric then it should be non-centrosymmetric.

That is right. But the ferroelectric polarization can be too week to observe. There is some report of electric polarization in the sample I am interested in (YCrO3) but by only one group and has not been reproduced.

It will be extremely difficult to distinguish between two by just crystallographic method, if there are no systematic absences. But if your sample has Raman and IR lattice modes, you may try to use spectroscopy. In centrosymmetric structure modes are either Raman-active or IR-active, but not both. In non-centrosymmetric structure you will have modes, which are both IR and Raman active. Once you know your answer from spectroscopy, you shall refine structure more easily.

International Tables, Vol. B, pag. 285

It is not resonant scattering. It's anomalous scattering that breaks Friedel law and allows the diffraction pattern to show the true symmetry of the crystal .....

The starting point should be with space group P-1, unless you are working with a chiral molecule, in this case the choice of P1 is mandatory. If you try to refine a centrosymmetric structure in P1 you should observe high values of (coordinates or thermal factors) in the correlation matrix. This leads to unreliable bond distances or distortions in phenyl groups for example).

In P1 the (x,y,z) coordinates of one atom must be fixed (i.e. those of a heavy atom of a metal complex) but this is usually fixed by the refinement program. In other cases (monoclinic) the P21 vs P21/m can be distinguished by an analysis of Patterson map.

The E2-1 statistic can give at least a hint.

As far as choice of origin is concerned: in P-1 the inversion center is the origin. In P1 the origin floats in all three dimensions. In order to keep the refinement stable "floating origin restraints" (not constraints) are applied. The sum of coordinates of all atoms are restraint to be constant within a very small standard uncertainty (in the case of P1 this is done separately for x, y, and z). This keeps the molecule from shifting around.

In fact, the answer is relatively easy to get from crystallography once the structure has been solved. Solve in P1 (often even easier than in P-1) and then just look whether the atoms you have found actually show -1 symmetry. "PLATON"s "ADDSYM" option can do this.

Modern refinement programs automatically fix the origin in P1, that is no longer a problem. What could happen is that the structure refinement becomes unstable because the two halves of the structure try to explain the diffraction effects together. What can happen is that "one atom shifts left, the other one shifts right and the thermal ellipsoids go berserk". If that happens, you actually have a good indication that the structure may be centrosymmetric....

@Paolo, "resonant scattering" is a modern term that is used for what people called "anomalous scattering" in Bijvoet's times. In many circles "resonant" is preferred over "anomalous". As we understand the process by which the difference is generated, "anomalous" is no longer an appropriate description.

Paolo has give the correct answer. Since Friedel Law, al difracted data in reciprocal space will be centrosimetric, so it's complicated to differenciate by routine techniques...

@Rob, I can understand your point, but resonant scattering is in my view, that can be naive, referred to diffraction in resonant conditions, i.e. at the exact energy where absorption takes place. Concerning the breaking of Friedel law, you can have it whenever the impinging X-ray beam has an energy close to an absorption edge, that is close to the binding energy of a core level. In this case, the dispersion corrections to f0 cannot be neglected. Thus, anomalous scattering and resonant scattering are not exactly the same process. You can have the breaking of Friedel law even if not in exact resonant conditions ....

I think Rob has nailed a key thing remembering my crystallography: you can usually tell in refinement if you have missing symmetry, since certain refinement parameters start to go crazy. If it were me I'd:

0. I'd try to collect a full sphere of data, so that I'm armed when faced with the problem (hey, disk is cheap these days and most diffractometers can do this in reasonable time. What do you gain by not doing this?)

1. But when solving/refining assume it's P-1 unless I have orthogonal evidence to tell me otherwise.

2. If I have some other evidence to suggest its not P-1 (e.g. known chirality) I'd think about a P1 refinement, but I'd look carefully at the correlation matrix between parameters (particularly the ADPs but others too.). If the refinement program is telling me that these parameters are strongly correlated I'd be really suspicious since the system is screaming.

You have to consider the physics of P1 versus P-1. Inversion centres are energetically quite favourable. Kitiagorodski realised this a long long time ago ...

Yep full agree with Rob.

I also used PLATON software for similar proposes (check for higher simmetry) and is very powerfull.

One advise, the refinements in P1 or P-1 is quite complicated because of the big amount of parameters to refine... So if your refinement is not good enough, these softwares may drive you to wrong assignements...

On a slightly different note it is possible to distinguish between the two structure types experimentally. A simple approach is to use a Nd:YAG (1064 nm) laser and observe the second-harmonic (SH) generation of light at 532 nm. The SH process is symmetry forbidden in a crystal structure with a center of inversion and therefore if observed would confirm the presence of a non-centrosymmetric structure type.

I think Peter's 2nd answer, Rob's and Paolo's comments and Jason's suggestions sum up quite a nice reply to the original question of how to solve the apparently intrinsecally unsolvable Friedel law of diffraction. On the other hand, most crystallographic packages offer intensity statistics giving some figure of merits as regards centricity/acentricity (Wilson plot and the like) which can serve as a first check; when the answer is unambiguous, there is no real need to have further doubt.

There is a nice ruler of thumb: "It is never P1" (like Dr. House says: "it's never lupus"). Unless your molecule is chiral and the sample enantiopure, this rule of thumb holds surprisingly well and I think that many if not most small-molecule structures reported in space group P1 should actually be in P-1.

"rule" not "ruler" of course.

Well my structure is not molecular. YCrO3 is a distorted perovsite structure. So I have no chiral molecule. Since the substance may be ferroelectric so it likely to be P1 rather than P-1. But electric polarization although reported by a group has not been reproduced. So I must solve the problem crystallographically. We have collected a very large number of intensity data from the time-of-flight neutron diffraction on SXD at ISIS. But I am amazed to get so many helpful answers from crystallographers! And I thought that crystallography (except for biological crystallography) is already dead!

Note for for Ashkan: SHG is useless because it probes only the surface. I am interested in the bulk structure and not the surface or skin structure.

For Cole: I am surprised to hear that "Inversion centres are energetically quite favourable. Kitiagorodski realised this a long long time ago ...". We know however that there are a lot of ferroelectric materials that can not have inversion centre. Kitiagorodski must have known that! Where did he write all these? In his book? Can you please give me the exact Reference?

For Cole again: Kitiagorodski may have written this about molecular crystals only. This inot certainly true for a large number of crystals that have low temperature ferroelectric ground state.

Small distortions present some extra challenges. Usually the crystal will be twinned and the useful reflections are weak. There has been a lot of progress in the tools for exploring group-subgroup relations and symmetry modes (eg: at the bilbao crystallographic server or stokes' isotropy code). These tools also help you find the twin operators, which correspond to the symmetry that has been removed from the parent structure. There may be a monoclinic (or higher) approximation for your structure from the sub-group graph. Being able to easily transform your structure into the various different space-group candidates is also very useful. Once the refinement is finished you can decompose your P1 structure into the various distortion mode components and see how large the acentric displacements are in comparison to the others.

In terms of refinement it is sometimes easy to find a "false" minimum where some atomic displacements are reversed and stuck. Even worse, this can be correlated with a twin fraction. While it is best to overcome that experimentally by de-twinning the crystal, this is easier said than done. Computationally you can check for such problems by repeating the refinement from randomised starting co-ordinates and seeing which models keep coming back.

For magnetite we had sort out the twinning experimentally to be able to resolve the weaker reflections. The "refine in P1, apply Platon" recipe was suggested for that problem too, but we did not get it to work with the badly twinned data we had at the time.

Some years ago the structure of a quiral molecule in P1 was refused in a famous crystallographic journal because it was Z=2 and there was a heavy atom which induced confusion to the testing programs. Finally, I repeated the measurement at low T to convince the editors. Furthermore, anomalous scatering may be a good test of non-centrosymmetry.

The problem mentioned may be even more complicated because of the possible presence of twinning. Even in the case that the oxide crystallize (locally) in P1, the real crystal may probably have domains and behave as non-polar. I've seen similar things where some results were different from different batches of crystals, just because the domain structure was different.

I would like to propose another purely crystallographic solution for this problem. The traditional way of solving a structure is to fix first the space group symmetry and then solve and refine the structure. Since the use of the dual space methods (charge flipping for example), the process is inversed, i.e. the structure is solved first and then the symmetry is deduced from the solution. The freely available Superflip program will solve the structure first and then propose the best possible space group symmetry. Apparently you have a complete set of data and this should be of great help for your problem.

I am so happy to get expert opinions of really good crystallographers! So despite the fact that German (may be also other European) universities are abolishing the crystallography departments, crystallography is still very much living and kicking!

With this assurance I shall now ask my colleagues to enlighten me about "Flack parameter" that can tackle the problem of centrosymmetry or non-centrosymmetry. Is this any good for practical purpose of distinguishing between them? Has anyone ever used such analysis and is it recommended?

Tapan : The Flack parameter is normally used for X-ray data where the scattering factors have a small imaginary component that makes F(h)!=F(-h). With neutron data it is unusual to get this contrast unless one of your elements is highly absorbing.

On inversion centers: you are correct that Kitiagorodski was considering the packing of molecular systems in his original book.

I'm sure that there will be specific examples which counter his general observation out there in the 'big wide world'. In a sense though this backs up what many folk here have said: Specific evidence to back up a case for a less likely space group over a more likely one is generally a good thing.

Tapan: the correct answer in my opinion is to go to to have a talk with your neighbours at the ESRF. Francois de Bergevin or Carsten Detlefs. They could help you to plan a resonant scattering experiment at the Cr-K edge. If I understand properly your question, your problem is to determine if Cr sits on an inversion center or not in the YCrO3 perovskite. Thus, if not you should have dipolar-quadrupolar scattering in the resonant experiment. This is forbidden if the inversion symmetry does exist. The dipolar-quadrupolar channel can be distinguished by performing a full polarisation analysis of the diffracted intensities. Hope this can help ....

Paolo: That is a good idea. It This may as well give the polar displacement of Cr atom we are drying to determine. Yes ESRF guys have determined such polar displacement in TbMnO3 but I guess that was done on ID20. You know definitely that ESRF has killed this very useful beam line. I don't know at which beam line we can do such resonance experiments. May be on XMAS but in order to get beam time on XMAS I must find a British collaborator I guess. I shall check that. Thank you.

For Jon: Thanks for your explanation of Fluck parameter. So they exploit anomalous scattering for the X-ray intensities. For reactor neutrons only a few elements like In, Cd etc. is useful but for spallation source neutron like ISIS where we collected the data you can get resonances with many elements. In fact some of the resonance linewidths are used to monitor temperature in high pressure experiments in PEARL.

For Jason: Yes I think so also. But then if you have a non-centrosymmetric molecule you may produce non-centrosymmetric crystal structure. Am I right?

For Gervais:This is high crystallography for me. I never heard of "charge flipping", "dual space method"and "Superflip" program! But I can guess what they are. I have to do some reading and homework. Thanks a lot.

For All: Thanks everyone. I am so impressed by all these discussions and advices from enthusiastic colleagues!

The statistic R and wR values should answer this, try to find out the lower error values, also see the GOOF, e.g. in Shelx and related software it should lay ca. 1 of magnitude for the best answer. Besides check residual electronic density in the .lst file before final refinement in both space groups, the best results should provide the lower residual density. Finally revise your obtained solution with an ellipsoid able viewer, the best solution should also provide the best thermal factors for your dispersers. but if all this fails please follow the clues that Prof. Ghigna already provided to you, they have more deeper physical and geometrical basis than those provided in my past lines. Hope this help.

If the molecule you are crystallizing is non-enantiomorphic (such as a natural amino-acid, all L, a peptide or a protein, or a nucleic acid) then you eliminate the center of symmetry, mirror planes and glide planes from the list of possible space groups (only 230 space groups left for such molecules).

So the first thing to do is to check what type of molecule you have crystallised. Only worry if you have (say) a molecule with an inversion center, or a racemic mixture (for the latter, there are cases where the crystallisation process selects only one of the enantiomers in the crystal).

HTH.

Your argument is of course valid for molecular crystals with well defined molecules. I am however looking at YCrO3 which has a infinite lattice. So your symmetry argument is not valid here.

Dear Tapan,

There are some a bit more difficult but much more direct investigations to check the presence of central symmetry. Just for checking it is enough if you make a CBED measurement, and check the symmetry of the Kikuchi lines, or use X-ray over the absorption edge of Cr, and check either the transmission, or the angle dependent fluorescent image. Both shows Kossel lines. If the Kossel or Kikuchi lines are sum of a symmetric and asymmetric part corresponding the imaginary and real part of the structure factor thus the existence of the symmetric part shows the lack of central symmetry. (in this case the origin is always at the emitting atom). To find the exact position of the Cr, I would propose atomic resolution holographic measurement with X-ray or electron (unfortunately it is not working with neutrons since the Cr is a bad absorber).

Thanks for the suggestions of these experiments. I know so little about these X-ray techniques. I guess all these can perhaps be in in SLS at PSI. Will be interested to help me if you know all these techniques?

The most of the measurements were done at DASY, Los Alamos, and Japan, but I can help on the basics and in anything which needs no specific X ray physics knowledge. The connection between Kossel lines, holography and the phase of the structure factor (without using the dinamical theory of diffraction :) is given in Stefano Marchesini's work:

PHYSICAL REVIEW B 66, 09411 (2002)