Modulated materials with electron diffraction

Editor's Notes

• #2: Before explaining how to treat commensurately and incommensurately modulated materials, I need to give you a first idea already simply of what is a modulated material, and then what does commensurate and incommensurate mean, so that you will know what is the goal of this lecture.
• #3: If you have a simple structure, all your unit cellscontain the sameatomson the samepositions. Ifyou have a modulatedstructure, itmeansthatfor neighbouring cells the structure is very close to an average structure, butfromonecell to the next somethingsmallchanges. Butnotsomuchthatyoudon’trecognizeanymorethat average pattern. Most often, butnotexclusively, the smallchangesthatoccur are eitheratomsbeingreplacedbyotheratoms, orbeingremoved, and in this case youcall the modulationanoccupationalmodulation. Or the atomsare slightly shiftedfromonecell to the next and thenyoucallit a displacivemodulation.Very many structures are modulated versions of well known structures, which prevents that you have to start from nothing, as soon as you can figure out the relation between the average cell and the modulated structure.
• #4: The periodicity of this modulation canbeeither a wholenumber of times the periodicity of the basic pattern, and ifso, youcall the modulationcommensurate. Butitcouldalsobethat the periodicity of the basic pattern does not fit an integer number of times in the periodicity of the modulation and thenyou have anincommensuratemodulation.All termswillbeexplained in more detail stillduringthis lecture with pictures and figures and schemes, in whichyouwillseehow to treat the electrondiffractionpatterns of suchmodulatedstructuresshouldyou meet one.
• #5: Let’s take itstep by step. Supposeyou have a verysimplestructure,consistingfrom a stacking of the planesshownhere. The a and b axes are lying in the plane as shown, the c-axis is perpendicular to the plane.
• #6: If youwouldtry to predictwhat the electrondiffractionpatternwill look like, youcalculate the reciprocalcell parameters, as shownearlierthis week, soyouknowwhere are the reciprocallatticepoints, and youalso have to calculate the structure factors, soyouwillknow the intensity of each reflection. Now throughout this lecture the axes a and b will keep referring to that simple cell with all blue atoms, which we call the basic cell. If I take new axes for the modulated structure, I will call them a’ and b’. The purple indices also will remain, throughout the lecture, the indices using the parameters of the first simple structure with only A atoms, the basic structure.
• #7: In the case shown, you have only one atom A and it is at 000, so all reflections have the same structure factor, thus the same intensity.
• #8: Ifyounowreplaceeachsecondatomalong the b-directionbyanotheratom of type B, youwill have a periodicity along the b-direction that is twice larger.
• #9: This meansthatyouwillgetelectrondiffractionpatternswherethe reflections along the b* axis are twicecloser to eachother.
• #10: In the notationwith g, the large G gives all vectors to reflections of the basiccell, small g gives all reflections in the modulatedcell. So you now have the same reflections as in the basic cell, which we will call the basic reflections,plus extra reflections at a half b* off from each basic reflection. These extra reflections are called the satellites.
• #11: Ifyounow want to describethisnewstructure, youcan start from zero in determining the structure of thisnewmaterial, oryoucansay: youknow the basicstructure, which was the onewithonly A atoms and we willnowconsider the newstructure as beinga modulated version of that basicstructure.
• #12: In this case you can choose to describe it using a supercell...
• #13: ...or you can choose for a modulation vector.
• #14: If you choose to use a supercell, in the case shown your supercell has a twice larger b’-parameter in direct space, and a twice smaller reciprocal b’ parameter. Your indices for the reflections will change so your original 010 will now be 020. The indices in the supercell will be put in green throughout this lecture.
• #15: Or if you choose to use a modulation vector, you keep the parameters of the basic cell, and describe the extra reflections you have relative to that cell with a modulation vector q and use 4 indices instead of 3, which I will explain in detail after ten minutes.
• #16: You must have noticedthat in the theoreticalelectrondiffractionpattern, the extra reflectionsare weaker than the basic reflections. This is in general like this and you can easily show it using again the calculation of the structure factor:
• #17: ... you now have anatom A at 000 and anatom B at 0 ½ 0. Sothisgives the scattering factor of atom A, fA, + fBtimes e to the power pi i k.
• #18: If k is even, e to the power of pi i k equals 1, if k is odd, it equals -1. So you get as the structure factor the sum of the scattering factor for atoms A and B for all reflections with k even, which corresponds exactly to the basic reflections and you get the difference between the two for all reflections with k odd, which are the extra reflections due to the modulation. This indeed means that half of the reflections, exactly those which are the extra reflections, are weaker.
• #19: So for the simple case where the modulation is along one of the main axes, you can use a supercell where the length of that axis is multiplied by n in direct space and you will have a corresponding reciprocal axis that is n times shorter to be able to index all reflections.
• #20: For alternatingatoms A and B along the B-axis, youmultiply b by 2, in reciprocalspace b* is twotimesshorter.
• #21: Ifyou have oneotheratomeachthreeatoms, youwill have extra reflections at 1/3 of the originalspacingbetweenyourreflections. In direct spaceyour b parameter is multipliedby 3, the reciprocal b basic vector is threetimesshorter, allowingyou to again index all extra reflections.
• #22: The sameifyou have a fourfoldsupercell, youwill have extra reflection at ¼ of the spacing of yourbasiccell. In direct spaceyou have a new b’ parameter that is 4 times b and in reciprocalspaceyourbasic vector b’* is 4 timesshorterthanb*.But off course this is only the simplest of simple examples, more as an introduction, to make sure you have all the concepts and connections correctly in your mind.
• #23: Whatif the modulation is *not* lyingnicelyalongoneof the basis vectors? For example in thestructureon the right: if all atomswere blue, itwouldbe the samestructure as thaton the left. Soagain the subcell is the same, given by those same axes a and b.
• #24: Now the blue atoms are lyingon the planes (110). The d-spacingbetween equivalent planeshas become three times larger.
• #25: Since the reflectionscorresponding to a certain set of planes are lyingalong the perpendicular to that set of planes, and have a distancethat is the inverse of the d-spacing...
• #26: ... you will now get extra reflections at 1/3 of the distance to the 110 reflection.So in terms of the diffraction vectors, you have reflections at all positions of the basic reflections, given by big G, plus extra reflections at m times 1/3 1/3 0, in which m is an integer.
• #27: You see here the same pattern again, but at a larger scale. If youwouldindex these extra reflections using the cell parameters of the basiccell, thatwould give you 1/31/30 and 2/32/30 and soon. Youcanagain fix these non-integer indices bytaking a supercell.
• #28: You could off course again multiply all cell parameters with a factor like 3 here. You then get the green indices, where the basic 100 becomes the supercell reflection 300, and the basic reflection 010 becomes the supercell reflection 030. This will make the extra reflections 110 and 220, but you see that you have a lot of extinct reflections in this case: 100, 200, 120, 210 etc. And actually there will be no conventional reflection conditions corresponding to these extinctions.
• #29: A better way to construct the supercell is tochoose the twoshortestreciprocalvectors, assign to them indices 100 and 010, and calculatefromthat the cell parameters thatcorrespond to thatchoice. Thenyoutry to index with those cell parameters all other reflections. In the case shown here, the shortest vectors are those along 110 and -120.
• #30: If you take those as basis vectors, you can index all reflections in this zone.You also have to be able to index also all other zones you might have of the same material with this same choice. It is a matter of collecting enough zones to be able to get a 3D view of the whole reciprocal lattice and then deciding which are the shortest vectors, those will allow you to index all others. If you have several possibilities, then the same rules apply of course as for other unit cells: choose the one that is has the smaller volume with a higher symmetry.
• #31: Then when you have found the new basis vectors, write down the relation to the basis vectors of the basic cell, you can do this in the form of one matrix P. In reciprocal space the relation is that P times the reciprocal basis vectors of your supercell gives the reciprocal basis vectors of your basic cell. For example, on the electron diffraction pattern you can see that if you take two times a’* minus 1 time b’*, you will have a*, so the first line in your transformation matrix will be 2-10. You also see that one time a’* plus one time b’* gives you b*, so the second line in your matrix is 110. If we just keep the vector perpendicular to this reciprocal plane the same, the last line will be 001.
• #32: This transformation matrix willthenimmediately also give you the relationin *direct space* betweenyourbasiccelland supercell. In direct space the basis vectors of the supercell are given by the basis vectors of your basic cell times P. Don’t mix it up.
• #33: So for the current case this gives that a prime equals 2a plus b and b prime equals minus a plus b, and you get the unit cell indicated in green, which makes sense on sight.
• #34: The advantage of having the transformation matrix is that all atoms of the basiccellcannowbeeasilytransformedusingthissame matrix P, soyou at leastalready have the average positions. So now you knowwhat the cell parameters and average atom coordinates are, butyoudon’tknowfromelectrondiffractionalonewhat is the origin of the modulation. Anoccupationalmodulationcan look the same in reciprocalspace as a diplacivemodulation. Soyouneed to combine itwith either prior knowledge (ifyouonpurposereplacedsomeatoms in the compositionby others you could expect a compositional modulation), or combine withrealspace images to seewhat is goingon in projection at least, or pick a model and refine with XRPD or NPD, thatcan often refine the structureonceyouknowwhat are the cell parameters and space group from electron diffraction. That is off course, only in the cases where XRPD or NPD pick up the extra reflections, which is not always the case.
• #35: Sowhenyou have extra reflectionsat indices that are given by a rational number with a small denominator, like, one half, onethird, 2/3 etc, youcantake a supercell which will multiply the necessary reflections by at least that denominator n so the denominator will be gone and all indices will be integers.
• #36: But what if you have extra reflections with indices whichcannotbe written as a ratio of twonumberswith a low denominator? For examplereflections at 0.458, which is equal to 229/500. To have a correct supercell, you have to take a supercellthat is 500 timesbiggerthan the originalcell. This is immense, and forrefinementsafterwardswillbepracticallyimpossible.Youcouldapproximate the number, butforeachapproximationyoutake, youwillget a different structure, a different spacegroup. For refinements the peakswillnotbe at the exact right positions and youwillknowit is not correct anyway.For such cases youcantake a different approach and use a modulation vector, whilekeeping the cell parameters of the basiccell.
• #37: Thisapproachuses the cell parameters of the basiccell, a*, b* and c*, which give the purple indices in this lecture, and ads a vector to one of the extra reflections as a fourthreciprocal vector to beused to index all otherreflections. This vector is called the modulation vector q, and is a linearcombination of the threebasicvectors, withcomponents alfa, beta and gamma. The basicreflections are thengivenbythose indices that have 0q, so hkl0, while all satellites are givenby hklm with m different from 0 .
• #38: Takeagain the simplest case. The diffraction pattern is indexed using the cell parameters of the basiccell. Yousee extrareflections at ½ b* away from the basic reflections.
• #39: Using the modulation vector, youwillwritethis as … and the modulation vector willbe ½ b*, so the beta component is ½.
• #40: The indices using this modulation vector are in brown colour. The indices will be 0001 for the first extra reflection, 0100 for the firstbasicreflection. The next extra reflectionwillbe 0101. 100 becomes 1000 and next to ityou have 1001.
• #41: If we return to the case that gave a problem, where the extra reflections were at 0.458b* off of the basic reflections, you can do exactly the same, butnowyourmodulation vector is 0.458b* and yourbetavalue is 0.458. Youuse the samebasiccell.
• #42: The first extra reflection is still 0001, basic 010 still becomes 0100, the only difference is that there is an extra reflection here which is at –q off from 0100. So it is called 010-1. In the previous pattern the 0001 and 010-1 overlapped because the beta component was exactly one half.
• #43: Anadvantage is thatnow all these structureswith different periodicities of alternations of the atoms are describedwith the same unit cell, the samespacegroup, which we willcall a superspacegroup in case of more than 3 indices, and the only parameter thatchangesfromonestructure to the other is the modulation vector.
• #44: Youcanalso do thisforthisexample, where the extra reflections are lyingalong 110, againyou keep the samesubcell, and youjustchange the modulation vector. You wil have q being one third of a* plus one third of b*. So alfa and beta are both one third.
• #45: As youcansee, withthismethodnotonly the relationbetween the subcell and the modulatedstructure is clear, butalso the relationbetween all modulatedstructuresderivedfrom the same onesubcell. Ifyouusesupercells, this is not the case, becauseeachsupercellis completely different and will have a different space group and everything.Another advantage is that you can use it for incommensurately modulated crystals. In theory this means that at least one of your components alfa, beta or gamma is an irrational number, a neverending number which simply can*not* be written as a ratio of two integers. Off course, in practice, the precision of the instruments is finite, so all experimentally measured values will have an end, but as you can see with 0.458, even short numbers can give troubles for a supercell because the denominator becomes very large. So in practice cases like the 0.458b* case are also treated as incommensurate, where using the correct supercell would give you an enormous cell to deal with.
• #46: Ifyoucannot imagine whatwouldbesuchincommensuratematerial, here are two schemes of typical cases.The firstone is anoccupationallymodulatedincommensuratestructure. Itconsists of analternation of twoatoms as in many of ourearlierexamples, butthere is no translation symmetry, the pattern does not repeat itself. As youcansee we have two green, three blue, three green, two blue, two green, there willnotbe a unit cellyoucanchoose as a repeated unit, no matter over howlargean area youwill look. The secondexample is a displacive incommensurately modulated structure, whereone of the atoms, the dark blue onemakesa sinusoidal wave aroundits average position, but the periodicity of the wave does not match the periodicity of the average structure. Again you can not choose a unit cell that repeats itself. However, if these modulations do give sharp extra reflections, then they are not random or defects and it turns out that they are periodical, just not in 3D space but in higher dimensional space.
• #62: Sonowifyougetanelectrondiffractionpatternthatyourecognize as beingfromanincommensuratelymodulatedmaterial, what do you do: first separate the basicreflectionsfrom the satellites. Almost always you can do this on sight, as the satellites are usually weaker.
• #63: For example in the part of a diffraction pattern shown here, these reflections circled in purple are the basic reflections. They are definitely brighter than those in between, which are the satellites. The basic reflections should still form a regular 3D lattice and if you have several choices, go for the one with the highest symmetry and lowest volume.
• #64: Whatalsocan help in recognizing the satellites is thatsometimestheychangeposition as a function of compositionortemperaturewhile the basicreflections do not.
• #65: As soon as youknowwhichreflections are satellites, you have to find a modulation vectorwhichwillallowyou to index all satellites. There are oftenseveralpossibilities. Youcan split them in anirrational and a rational part.
• #66: For example, the possiblemodulation vector q on the leftsideconsists of alfatimes a* plus one time b*, alfa beinganirrationalnumber, and one of coursebeing the rational part. Withthismodulation vector thisfirstreflectionwillbe 0001 thisone 0002 – 0003 thenstartingfromthisbasicreflection 240-1, 240-2, 240-3.Youwillbeable to define a set of reflectionconditionswith these indices, whichyoucan translate intopossible superspace groups, justlike in the 3D case.Oryoucantake the othermodulation vector on the right side, in which the beta component one has been left out and youonly have thatsame alfa component. Withthismodulation vector youalso have vectorsstartingfromextinctbasicreflections, butthis is noproblem. Alsoyou do notseeforexample 0001, butyousee 0002, yousee 0101 and 0103 butnot 0102. Sonowyou have different reflectionconditions, and thischoice of modulation vector willgiveyou a different superspace group.
• #67: The choice of modulation vector is notcompletely free. Itcanbe proven thatonlyspecific modulation vectors canbecombinedwithcertainchoices of spacegroupsfor the basicstructure. This is a consequence of the conditionthat the point group of the subcellshouldtransformbasicreflectionsintobasicreflections and satellitesintosatellites. In the lecturenotesyoucanfind a completelyworked out example of such condition if you want.
• #68: This condition leads to the followingcombinationswhich shows that the possibleirrationalcomponentsyoucan have, have to correspondwith the crystalclassyou’vechosen, forexampleifyou have an alfa ànd a beta, youcannot have orthorhombic, but at the most monoclinic.
• #69: And on the other hand, the rational component needs to correspondwith the centering, so in totalyouget these possibilitiesformodulationsvectors per crystalclass.
• #70: As a last statement, nowthatyouknowwhat aremodulated materials, what are the advantages and disadvantages of usingelectrondiffraction? Wellcompared to bulk diffractiontechniquessuch as X raypowderdiffraction and neutron powderdiffraction, electrondiffractionallowsto more easily determine the periodicity and modulation vector because of the factthatyoucan look along separate directions, italsoallowsyou to more easilydetermine the reflectionconditionsfor the samereason and it is better in picking up satellitesdueto modulations of lightatoms. On the otherhand X ray powderdifraction and neutron powder diffraction are better at refiningthe structures. As we saw in the other lectures during this school, it is alreadypossible to solvestructuresquantitativelyfromelectrondiffraction, butrefinementfromelectron diffraction data is notquite up to the level of refinement with the bulk diffractiontechniquesyet.
• #71: Sowhatshouldyourememberfromthislecture: what are commensurate modulations, so that’s when you have a rational ratio between the periodicity of the subcell and the modulation, and how to analyseelectrondiffractionpatternsof such commensurately modulatedmaterialsbyway of a supercell and byway of a modulation vector. Alsowhatare incommensurately modulatedmaterials, which is when you have an irrational ratio between the two periodicites, and that in such case you cànmake a commensurateapproximationbutit’sreallynot the right way to treatthem, what youshould do is use the modulation vector, and treatthem in superspacewithsuperspacegroups. With these approaches, you keep the links with the basic structure or parent structure, which is not only useful for you to make it easier to solve the structure, but which will also make it easier for other people to understand the structure and to understand its relation to other, existing materials.