Can some body tell me about any theory, which had model a crystal structure and has shown local symmetry breaking with macroscopic symmetry remaining unchanged (average structure unchanged).
Alternatively, a model where local structure is different from average structure.
I realise this isn't exactly right, but order-disorder transitions in alloy systems have some features in common with what you're looking for.
Consider, for example, a 50:50 alloy of Cu and Zn: At low temperatures, an ordered B2 (CsCl) structure is formed. At high temperatures, the ordering is lost; each site has an equal probability of containing a Cu atom or a Zn atom, and the overall structure has A2 (bcc) symmetry. Of course, at the local level, it is not true that every unit cell has an A2 structure.
If X-ray diffraction is carried out while this alloy is heated through its transition temperature (approx. 200 ºC, if memory serves), some of the peaks in the B2 diffraction pattern will shrink and disappear as the systematic absences in the A2 pattern make themselves apparent.
@Philip: Actually in few material the observed physical properties contradicted with their crystal symmetry. Just take an example of Multiferroic YCrO3, it show ferroelectric property and its macroscopic crystal symmetry is centrosymmetric. In order to explain the orgion of its ferroelectric properties, it was proposed that the local symmetry of YCrO3 is noncentrosymmetric, while its with macroscopic symmetry is centrosymmetric. But I am not able to find any good theoretical model which has proposed local structure different from its average structure.
This type of materials are usually referred to as disordered crystalline materials. When we say 'average structural', we mean in the sense of Rietveld refinement results of an x-ray or neutron diffraction experiment. The fundamental flaw of these techniques are that they treat the atoms as independently vibrating entities and average out the local disorder. It is not uncommon that the average structure do not reflect the local structures in these type of materials.
One tool that solves the problem is pair distribution function analysis. It reconstructs the crystal not by knowing the average positions of atoms of a crystal but by knowing the correlation between atoms, i.e. pair atoms distance distribution. The pair distribution function contains both short-range order and long-range order information so you can have ideas about the local structures without worrying about breakdown of overall crystal symmetry. There are two common model tools that are used. Although I don't know exactly what your goal is, I figure the small box model might be appropriate for you where you can constraints the symmetry of certain atoms. Implementation of the PDF analysis isn't trivial, so you might want do a cost-and-benefit analysis before carrying it out.
I recently did a pair distribution analysis on the lithium garnet oxides. This system consists of complete ordered framework and completely disordered lithium ions. I was able to get local information about lithium while showing the overall symmetry maintain.
Hopefully this helps!
You should directly contact us at the Hulliger's group in Bern, switzerland.
Consider the rhombohedral-t0-cubic phase transition in the ferroelectric phase-change material GeTe that has attracted a lot of attention. The rhombohedral phase has a distorted NaCl type structure in which there are three short and three long Ge-Te bonds (Pierels distortion). At about 650 K the rhombohedral phase transforms to the cubic NaCl type phase in which the average structure has six equal Ge-Te bonds forming a perfect GeTe6 octahedron. But recent PDF analysis of the total diffraction data suggest that three short and three long Ge-Te bonds still survive in the local structure. The conventional refinement of the structure with the Bragg intensities (which neglects the diffuse scattering) yield higher symmetry in the average. The local symmetry is still rhombohedral. So the symmetry is already broken in the high temperature cubic phase but only locally.
The situation in YCrO3 is much less clear. The results you mention are reported by a single group and have not been reproduced by others. So it best not to give this example.
Syed, you probably found already the answer thru your second formulation of the problem: local structure vs average structure. From (X-ray) diffraction we get a model averaged over space and time, and it is quite common that the local or momentary structure is less symmetric.
Phase transitions (diffraction experiments in a temperature gradient) might be helpful to understand what happens. Taking Philip's example, the structure of higher order (Im3m or bcc, equal probability for Cu and Zn) would be observed as a time average at high temperature. Below phase transition, thermal motion gets frozen and for slow cooling rate a single crystal exhibiting the lower symmetry (Pm3m, CsCl) might form. For more rapid cooling, Pm3m domains might be small and out of phase (Cu-Cu instead of Cu-Zn contacts at domain boundary), so the higher symmetry would again be observed, this time due to space average.
My opinion is that one should be careful what do we see experimentally and what is described mathematically by group theory. Maybe our used 3D mathematic is not satisfying what is in fact true so that we have to describe this problem in higher dimensions (cf. incommunsurate or quasi crystals), or starting with the definition of super cells which allows the local reduction of symmetry. In other words, either your measurements of your property is OK (what I believe), or your structure description is higher symmetric than of your experimental phase. Both at the same time is from my point of view impossible.
You can maybe look into your diffraction pattern and look whether there are satellite reflections. This local symmetry might then be the true symmetry of the crystal. However, if there is disorder with only main reflections (considering the fact that there are no satellite reflections) you might look for phase transition at lower temperatures.
It is important to distinguish between local symmetry of a particular site and the crystal symmetry. Also what Tapan pointed out is very important
There is some more to say about the liquid state of GeTe. It seems that the liquid GeTe has also the local structure in which rhombohedral distortion and also the unequal three short and three long bond persists. So there is a local symmetry breaking even in the liquid state.
Although it is normal that there is a relationship between the local and crystal structure symmetry. In macromolecules (proteins, for example) could be different the symmetry of the crystal structure and the local [JP Glusker & K. N. Trueblood, "Crystal Structure Analysis. A Primer (1985) Oxford University Press and others]
The symmetry breaking is very important concept in condensed matter physics and has been discussed by Anderson, Goldstone and Nambu in varius publications and books. But please note that whenever symmetry breaking takes place there is always a memory of the high symmetric phase is left behind. The dynamic signature of the symmetry breaking is the Goldstone mode in ferromagnets. In most of the magnetic ordering transition, the symmetry of the paramagnetic phase is broken, the odered phase has always lower symmetry. At the antiferromagnetic transition in antiferromagnets the static signature of the symmetry breaking is the resulting domain formation. The number of domains equals the number of symmetry elements lost in the phase transition. The domains are the static memory of the broken symmetry. The nature attempts to restore the broken symmetry in a larger scale, although the symmetry is truely broken in the local scale. The same is true for ferroelectric transition where the number of domains also equals the number of broken symmetry elements. The local structure only breaks the true symmetry. Neither the ferromagnet nor the ferroelectric crystals have magnetization or electric polarization unless one rearranges the domains by applying either magnetic or electric field.
As demonstrated above, there are many possibilities all of which require careful investigation. Without making assumptions regarding the nature of your compound, the best approach depends on the nature of your problem and the crystallographic approaches open to you - single crystal or powder for example. I believe you will find the papers by Richard Welberry (some of which I had the privelege of working on) on the interpretation of diffuse scattering in X-ray diffraction instructive as the diffuse scatter is where the contribution from the parts of the structure which deviate from the mean appear - the data you seek lies between the Bragg peaks and if you have single crystals you can calculate suitable "synthetic precession images" from area detector data.
Experimentally you can see the difference between local and average structure using the pair distribution function (PDF) methods "total scattering technique). An example NaBi3O4 system. For more details see (The real structure of Na3BiO4 by electron microscopy HR-XRD and PDF analysis " Z. Kristallogr. 220,231-244 (2005).
@Alison& Ahmad: You are suggesting in principle essentially the same method of determining local symmetry and structure. The diffuse scattering is the diffraction signature of the local structure and contain the information of the deviation of the real structure from the average structure that can be determined from the intensities of the Bragg peaks only neglecting the diffuse scattering. Alison refers to single crystal study whereas Ahmad refers to the PDF analysis of the total scattering data from powder samples.
The single crystal study of the local structure is of course superior technique but this technique has some drawbacks as well. Firstly single crystal is not always available. Even if the single crystals are available, they may be twinned. Uless you have access to high energy synchrotron x-ray the Q (or two theta) range is not often sufficient with lab x-rays. We assume that the crystals are not perfect otherwise extinction effects may be troublesome. But the great advantage is the possibility of having directional information in reciprocal space. The PDF analysis of the total powder diffraction data do not have some of the drawbacks of the single crystal method but have no directional information in the reciprocal space. Extensive modelings are needed in both these techniques and there is no garantee that your model even if it fits your data is unique.
Let us now discuss the case of YCrO3. The real structure of YCrO3 is not orthorhombic but has a very low symmetry. It is reported to be monoclinic but recent single crystal neutron diffraction investigation suggests even lower symmetry, viz. triclinic P1 or P-1. It is extremely difficult to distinguish between the two space groups even from very good quality single crystal neutron and synchrotron data. (Forget about powder diffraction because this technique cannot distinguish centrosymmetric structure from the non-centrosymmetric one.) But all physical properties and neutron and synchrotron data suggest P-1. Ab initio calculations also suggest that the non-centrosymmetric structure has lower energy than the centrosymmetric one. The ferroelectricity is a surer test for non-centrosymmetric structure but the ferroelectricity is reported by a single group and has not been reproduced. So the situation is not yet clarified properly. The report of non-centrosymmetry in local structure is also doubtful so far. In such situation you do not expect to have any reliable model.
Sorry I meant YCrO3 and not YMnO3. Please replace Mn by Cr in the chemical formulae written in the above response.
I have now corrected these.
@Tapan: I can't edit comments of other, But you can edit your comment . RG provides us excess to edit our own comments at any time, but doesn't provide us excess to edits comments of other.
@Syed: I do not see any obvious way to edit my comment. How do I do it?
@Tapan: At the right upper side of our comment, if we put our cursor. A down triangle symbol appears. Just by clicking it, two option (i.e. Delete and Edit) are highlighted below it. You can use edit option by click it to editing you comment.
See the attached pic for clarity.
I have a lot of reading to do in order to be cogent with my comments. I see some "big players" participating and am immediately interested in the discussion. I do have a recommendation that has been in use since the 1980's, Bragg XRD Microscopy! It is also called XRD Rocking Curve Microscopy. Essential elements of this method were pioneered by Dr. Sigmund Weissmann (Rutgers) et al and Dr. Robert Green (JHU) et al. It is a method that uses the quantification of the deviation from IDEAL BRAGG CONDITION to correlate to the local VOXEL Nano structural order in 2D (topographically). Essentially, combining Ahmed's suggestion of "HR-XRD" and Alison's suggestion of "parts of the structure which deviate from the mean appear - the data you seek lies between the Bragg peaks". "HSR-HDR-XRD" (?) - High Spatial Resolution - High Dynamic Range XRD! Depending on the optics & geometry used, the depth of penetration (examination) could range from fractions of um to 10's of um. In this case of ZnSe (224) ACT, the examination depth is estimated to be 1.1um. If you observe the "Y-Omega" maps for various locations of the cursor on the wafer, you'll see crystal structure ranging from "near perfect" to "sub-grain" to "near glassy state". I'll put up detailed images as, when & if interest persists. This method does not require the Synchrotron. These images were acquired at
>….. see:" Comparative structural study of the disordered MCd6 quasicrystal approximants" C. Pay Gómez, S. Lidin, Phys…..
Panagiotis! I'd like to understand the implication of your thought process in crystallography. "geometric relations of the structure of the polyhedral from lines, triangles, tetrahedra, via the triangular face of a pyramidal model, whose structure is founded on a special “quadrature triangle”:"
All these appear to be critical components of Crystallography as well. Especially, when we Fourier Transform the real space (geometric shapes) in to reciprocal space by the interaction of the geometrical crystal lattice structures with the electromagnetic wave (X-ray).
"via the triangular face of a pyramidal model" this is usually what is happening when a certain set of (hkl) planes are co-operatively reflecting (diffracting) in XRD. Now when you include the expanded Ewald surface (volume) created by introducing an area (2D) detector besides oscillation about the Bragg condition, things get even more interesting.
So educate me as to how your concepts can alleviate the "pain" for those of us practicing Experimental & Applied X-ray (Neutron etc.) Crystallography. I tried looking through your methods and got lost last time.
Present focus being on "model a crystal structure" showing "local symmetry breaking with macroscopic symmetry remaining unchanged (average structure unchanged)." I presented XRD Microscopy as a means of verifying experimentally what one could hypothesis using theory in terms of local VOXEL "order" to "disorder" crystallographic "phase" states. Here is an example comparing the theoretical (Bruker Leptos) and experimental Bragg rocking curve profiles for (004) symmetric GaAs reflection:http://www.flickr.com/photos/85210325@N04/10502393183/
It may be prudent for you to originate another question if there is a feeling such matter to be "off topic". You may always include a link to that new query in this discussion as well making it convenient for those that wish to delve. Regards!
>.... I tried looking through your methods and got lost last time...
Panagiotis! If it is relevant to this topic of Crystallography then here, if not, post a new question regarding your interest. You are welcome to forward any helpful information to [email protected]
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