Unit 4.7 - Space Group Pmm2 (II)
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- เผยแพร่เมื่อ 2 ต.ค. 2024
- Unit 4.7 of our course The Fascination of Crystals and Symmetry
Additonal resources at: crystalsymmetr...
In this unit, we will look at the remaining parts of the description of the space group Pmm2 in the International Tables.
We will explain in which way the positions are listed, what special and general positions, multiplicities and Wyckoff letters are, and what 'site symmetry' means.
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If you prefer books instead of videos, have a look at:
www.amazon.com...
Why doesn't my university have a professor like you?
@Oakley Leroy Thank you Oakley! I’ll take a look.
Hi Frank, Liquor Lady again, thank you for the great videos! When people ask what do the letters mean, eg. Pnma, I would answer as the following, but please correct me if I use any terms not correctly. My answer: Pnma is a space group belonging to the orthorhombic crystal system. "P" stands for primitive unit cell. "nma" are symmetry elements along a, b, c viewing directions. "n" stands for diagonal plane for a translational symmetry along, "m" is a mirror plane, and "a" is the a plane for another translational symmetry.
Hi Liquor Lady,
Hi!
Almost correct - let me add the following:
"n" stands for a glide plane being oriented perpendicular to the a-direction with translation componenets along the diagonal, which is in this case 1/2 along b and 1/2 along c
"m" ist indeed a mirror plane perpendicular to the b-direction
"a" stands for a glide plane perpendicular to the c-direction with a translation of 1/2 along the a-direction.
I also have a question: What's your favorite liquor?
Danke schön! So many.... favorite may be whisky and beer, you are welcome to check on my channel too. I happened to start to study your videos with this account.
Thank you very much for your videos! You explain everything really well which is of a great help to me. I can´t express how grateful I am to finally learn how it all works after so many attempts in vain.
Thank you very much for your kind comment - I am glad that you like the videos.
Hi Frank. May I ask a question? Is there a easy way to find special positions? I have a problem with finding special positions of P222 and P21('1' is subscript.). I can find special positions in 2D space group but there are too many special positons in 3D space groups. Thanks in advance!
Hi!
No, there is no easy way to derive all possible special positions. This is tedious work... Fortunately, this work was already done by Mr. Wyckoff. You can lookup the special positions either in the International Tables for Crystallography or at the Bilbao Crystallographic Server:
www.cryst.ehu.es/cryst/get_wp.html
@@FrankHoffmann1000 Thank you Frank!
Thank you so much for your videos. Those are being really helpful for my method course X-Ray diffraction in Augsburg :D. I want to ask only one question related to de special positions. Why when we apply the symmetry operations of the space group we should add +1. I don´t have that clear. I realized with these videos I am understanding better all related to crystallography
Hi, thanks for your nice comment :-)
Concerning your question: You don't have to add +1. But adding (subtracting) whole numbers to the coordinates of a certain position leads to completely equivalent locations. The addition plus one to the y coordinate was only done to make more clear that the coordination transformation (0.5, -y, z) leads to an position which is related to the first coordinate by symmetry (here the two-fold axis of rotation along the c-direction). Or in other words: if you take a certain coordinate like here in the example 0.3 you may think: well, the coordination transformation will lead to a position outside the original cell (which is right...). To translate the newly generated position to the same, original cell we added here +1.
@@FrankHoffmann1000 Oh I got it !!! Yeah it´s only for leading the point inside the unit cell that we are considerating. Thank you so much for this detailed explanaition. I want to thank for all these videos. Greetings from Augsburg Uni !
you are the fucking master of the universe
You saved my grades! My crystallography Prof (who coincidentally is also German...) just made things more confusing than it needs to be...
I am glad that you find these videos helpful!
I have a question about the symmetry element, in space group 137 we have 2mm. why is there an extra dot when we only have three viewing directions? Thank you for excellent video!
Very good question - the International Tables state that the site-symmetry is actually expressed as a so-called _oriented_ site-symmetry symbol, which was introduced, I think, by Donnay and Turell. The '2' is along the first viewing direction (along [001]), the two symbols 'm' refer _together_ to the second viewing direction (along [100] _and_ along [010] because a = b) and the dot corresponds to the third viewing direction. The text/type setting in the International Tables, which give no optical guidance for these site-symmetry symbols not very convincing!
Hi Frank. May I ask a question? Why in some space groups there are no special positions? For example, I can't find special positions on International Tables for Crystallography for space group P212121 ("1" is subscript). I think there can be some positions in on the symmetry elements (the screw axis) but why there are no special positions for the space group? Thanks in advance
Yes, of course, feel free to ask any question!
The imporant point here is that screw axes (as well as glide planes) involve a second operation, a translation component that will move the atoms around! So, there are only special positions, if the site/atom is located at a _point_ symmetry element. So, even if an atom is located on a point falling together with the axis of a screw axis, meaning that it remained unchanged during the first part of the symmetry operation, it's position will change with the second part of it.
@@FrankHoffmann1000 I understood. Thank you very much, sir.
Beautifully explained. Thanks
that's wonderfull, thank you! :)
Thank You very much for the video lectures, I'm following them thoroughly for my research in structural biology.
how do we know if the wyckoff site are occupied?
Only by inspecting the concrete crystal structure of a given sample, usually solved by single-crystal X-ray diffraction experiments.
so helpful, thank you :)
Hi Frank,
Sorry to bother you again. I have a question regarding the multiplicity on special positions (for instance at 4:16 min). The multiplicity decreases when an atom falls precisely on a symmetry element, but only if the symmetry element does not have translation component (inversion center, mirror plane and fold axis). Is this correct? Or it applies to any symmetry element (including screw axis and glide planes).
Thanks again and thank you again for all the videos.
Always welcome! You are completely correct (only if the symmetry element does not have translation component)!
Thank you, sir!
Thank you professor for such a helpful videos because I was actually struggling in this topic. I went through lot of videos but I couldn't understand but your lectures are very interesting and easily understandable. I am finding difficulty in solving special positions problems but I didn't understand 1 thing here that why did uh choose y=0.3 as x axis was 1/2. Can you clear my this doubt. And I am struggling in how to write general and special positions when we have this diagram? Because we have so many exercises to solve but I couldn't find?
Please help me through this, if possible?
y = 0.3 was an arbitrary choice. The special position 2h in this space group is 1/2, y, z. So we are free to choose any y (and also z) position, so I decided to choose y = 0.3, but you can of course choose any other value, except y = 1/2 (because this would mean that this position is not 2h any more but 1d!).
Concerning you difficulties to derive the coordinates of general and special positions: The most important thing is to, make yourself clear, where the symmetry elements are located. If we choose again this space group you see that there are horizontal mirrors running through x = 0 (=1) and x = 1/2 and vertical mirrors at y = 0 (=1) and y =1/2. Furthermore there is for instance one 2-fold rotational axis at x = 1/2 and y = 1/2 (here you can choose an arbitrary z-position).
Awesome Videos !!!
Thank you so much. How we chose the letter in the Wychoff position notation?
Welcome!
I am not sure, if I correctly understand your question. You have to look, where your atom is located; in this example: if it is at 0,0,z it has multiplicity 1 and gets the Wyckoff letter a, if it is at 0.5, 0.5, z it also has multiplicity 1, but the letter would be d.
best wishes
Frank
Well explained and very clear! Thank you! I just have a question: why some point groups indicate the symbol (and the existence) of a screw axis but it doesn't appear into the projection graph? (like for example in IT n° 113). Other times I can see the symbol in the projection plane but not in the point group...
Dear Guiseppe,
thank you for your kind words!
I do not know exactly, if I understand your question correctly, but let's try this:
First of all, we have strictly to distinguish between point and space groups. As point groups do not contain symmetry elements with a translational component screw axes never appear in point groups.
Okey, but let's take your example of space group no. 113: P-42(1)m. There is a 2(1)-screw axis parallel to the _a_ axis. And if we look at the projection diagram, in which the symmetry elements are shown, we don't see such ellipses with two hooklets, however, we see at the border of the diagramm arrows with half arrow heads - these are indeed the symbols for the 2(1) screw axes which are running along a (and of course b, because a = b in the tetragonal system). The projection plane is the (a,b)-plane.
See also the following PDF:
crystalsymmetry.files.wordpress.com/2017/09/it_sg_113.pdf
Then you stated that sometimes you can see the symbol in the projection plane but not in the point group: as stated above, this is correct because screw axes never appear in point group symbols (only in space group symbols).
Hope this helps!
best,
Frank
Dear Dr. Hoffmann,
thank you again for your explanation!
Actually, I didn't pay attention to the little arrows at the border! Yes, I used the word point group in a wrong way, I actually meant Spece Group (I'm sorry).
What is not clear to me now is why sometimes space groups don't specify the presence of screw axis. For example; the space group R32 (n°155) indicates the presence of two rotation axis as symmetry operators, but looking at the projection plane we can see there are also screw axis. I guess, in this case, specifying only those two axes is enough because they involve the presence of screw axes. Am I right?
I hope to have been clear,
Sincerely
Giuseppe
No problem! I am glad that the explanation helped!
best!
Frank
I had troubles with my connection and only half message has been sent. Hope you could reply to the second part added in the previous message.
Thank you
Sorry, maybe I overlooked the second part of your answer!
Your guess is completely correct. In the space group symbol only the so-called "generators" are specified. And they are called "generators", because they generate or automatically imply in many cases the existence of other symmetry elements!
Greetings!
Frank
Hello! i want to thank u for this video and i would like to ask u to put a video explaining how to obtain atomic positions from wyckoff one's, and what mean, for examples, (x,x,x) or (x,-x,y), (x+1/2, y-....) etc... i can't understand them. Thank's in advance!
Hello!
I do not know, if I understand your question correctly. But let's try the following:
a) You asked how to obtain atomic positions from wyckoff one's. I think there is a slight misconception. Strictly speaking, there are no Wyckoff positions, but only Wyckoff letters. Each Wyckoff letter represents a set of atomic coordinates. These sets of coordinates are listed (from the top the bottom) for each space group from the most general position (highest multiplicity) to the most special position (lowest multiplicity).
b) You asked, what the meaning of these coordinates are, for instance (x,x,x) or (x+1/2, y-...). Well, first of all, I think it is easier, if we stick to the given example of the video, i.e. space group Pmm2. Furthermore, please remember that you can read the atomic coordinates or atomic cordinate sets as coordinate transformations, which represent a certain symmetry operation. The impact of a symmetry element/symmetry operation on the atomic coordinates is now dependent on the exact location of an atom. Let's look at the sets of coordinates of Wyckoff letter a: hmmm, only one coordinate is listed: (0,0,z). So, what does it mean? Well, if any atom of a concrete crystal structure sits on a position (0,0,z), whereby z can have an arbitrary value, then nothing happens - no more positions will be generated. Why? Because this atom is located at three symmetry elements simultanouesly (mm2). Now, let's look at the set of coordinates with the Wyxckoff letter e: there is a set of two coordinates (x,0,z) and (-x,0,z). So, what does this mean? If an atom is located at a position (x,0,z) there must be a second one at the position (-x,0,z). And the reason is clear: this atom is located on _one_ mirror plane, but there are two in the space group Pmm2. The second mirror plane "converts" or translates" one atom from (x,0,z) to (-x,0,z).
c) If a coordination triple is called for instance (x,x,x) this means simply a coordinate in which x = y = z, say x,y,z = 0.137, 0.137, 0.137. If one coordinate has the values (x,-x,z) then y = -x, for instance 0.584, -0.584, 0.722.
best
Frank
Thanks a lot sir, but what i didnt understand is by what can we replace the x, or y or z? they are unknown!! can we choose any value???
Well, if the coordinates are unknown we have to use the placeholders x,y,z! This means that these tables represent _generalised_ structures or coordinates for a particular space group. But _if_ a partcular structure is known and therefore the space group and also the coordinate triple of all atoms then you will see that these coodinate relations are valid for all atoms.
Assume that a structure solution has found that an atom A of a structure with space group Pmm2 is located at the coordinate x,y,z = 0.128, 0.177, 0.245 then you know that autoamtically also an atom A' (from the same type) must be located at -x,-y, z, i.e. -0.128, -0.177, 0,245.
clear?
Frank
Look, for example, i have a spinel structure (AM2X4) belongs to the Fd3m space group; i found that the atom A has 8a Wyckoff letter, the M 16d, the X 32e, So first, how can we attribute these letters to the different atoms A, M and X, and second: I found that 32e correspond to the first (x,x,x) position and the x position is fixed to 0.385!!! Why this exacte value?? from where is it extracted?..... i hope you understand me and excuse my english. Thanks in advance sir.
i'm sorry no! :( what is placeholders?
Where is the lecture 4.8 ?
4.8 doesn't exist. The next lecture is unit 5.1
th-cam.com/video/C608SqhzK2A/w-d-xo.html
Dear Professor, I have confusion with m (mirror) part, so what is it means exactly ? Please help me to understand.
Please, can you rephrase your question? Your question is much to unspecified and I have no idea what help at what part you need.
@@FrankHoffmann1000 Like in Pm-3m, m is mirror plane but what is it means exactly ? Hope it could work now. Thanking you
@@shivamkansara1968 I think it is advisable that you watch the unit concerning the nomenclature of space groups (4.5) first.
In short: in Pmm2 we have a primtive lattice, then a mirror plane is present perpendicular to the a-direction, another mirror plane is present perpendicular to the b-direction and, finally, there is a 2-fold axis of rotation along the c-direction.
@@FrankHoffmann1000 Thanking you for replying. I have watched your 4.5 lecture so my confusion is in second example for choosing space group. As I understood - we are taking n glide plane (n) for diagonal plane, so why we took m over here. Please make me clear.
@@shivamkansara1968 Sorry, I think I don't understand your question at all. What are you talking about? Glide planes? Where are glide planes in Pmm2?
I took Pmm2 as an example to illustrate some further concepts of space groups. Pmm2 is one of the 230 space groups. In Pmm2 there is simply no glide plane present, so the question why we took m (mirror) here instead of n (glide plane) is not meaningful.
It seems that you have slight problems explaining exactly where your understanding fails. Please try to explain it in detail, otherwise I can't help you.
You are the best!!! May I ask a question? For a random crystal, is it possible that all atoms are in general position of its space group? Or there is at least one atom at every spacial position (Wyckoff position)? Thanks!!
Yes, this is very often the case for crystals composed of organic molecules that all atoms are at general positions. And also the opposite is true: It is possible that all atoms of a crystal structure are at special positions.
@@FrankHoffmann1000 Thanks Frank. May I ask another one. Is it possible that two same type atoms at the same special position? say the special position is (0,y,z),and I have two Na atoms at (0,0,z) and (0,1/2,z). Would that make these two atoms identical since they are at the same special position?
@@qizhang6336 No, they are not at the same Wyckoff position (in the space group Pmm2) and they are not identical: Your atom at 0,0,z belongs to the Wyckoff position 1a (0,0,z) - the multiplicity is only one. Your second atom (0,1/2,z) belongs to the Wyckoff position 2g (0,y,z), meaning that there is a symmetry-related copy at 0,-y,z = 0,-1/2,z.
@@FrankHoffmann1000 ohhh let's say it's (0,0.23,z) and (0,0.123,z). I mean an arbitrary in y. Since they both belong to 2g in Pmm2. Are these two atoms identical?. Thanks for the replay!
Oh I see! Why should they be _identical_? They have _different_ fractional coordinates.
Thank you mr 😊
welcome :-)
i love u
Dear Frank,
could you please explain in more details the importance of Wyckoff positions? I understand, that you can "build" a crystal if you know in which Wyckoff position a particular atom lies, e.g. as explained here cbc-wb01x.chemistry.ohio-state.edu/~woodward/ch754/sym_itc.htm for the case of Sr2AlTaO6.
The thing I dont understand is how do I find the positions of the atoms which are not explicitly specified by Wyckoff position, e.g. according to the previous webpage
for Sr2AlTaO6 (F m -3 m),oxygen atoms should lie in the wyckoff position 24 e: [x,0,0]. How do I know the x position for the particular atom? Also I found e.g. for CuFe2S3 that there can be several different sets of [x,y,z], which makes it even harder to grasp.
Dear Sego,
I had a quick look onto the mentioned webpage. There is one particular sentence, which is - in the narrower sense - wrong. It is stated (somewhere in the middle of the webpage): "The Wyckoff positions tell us where the atoms in a crystal can be found." This is wrong, because the atom positions can only be found by X-ray or neutron diffraction experiments!
So, in a way it is the other way round: The Wyckoff positions are independent of any concrete crystal structure! They were derived theoretically by inspecting the principally different locations which are available in a unit cell, by inspecting the symmetry framework, by inspecting at which (fixed!) positions the symmetry elements are located for a given space group. And only if we know the atomic positions from experiment, we then know at which Wyckoff position it is located!
From top to down the symmetry rises. The first entry is always the general position, which means a general place within the unit cell which is not located at a symmetry element. In the example of the space group Pm it has a multiplicity of two; lofgically, because the only symmetry elements in this spacegroup are two mirror planes, and if an atom is not located on one of these mirrors, then it will be doubled by this mirror. But if an atom of a structure is on Wyckoff site 2c or not can only be derived by experimental data.
The next entry is 1b. The coordinates are given as x, 1/2, z. So, this means, _if_ an atom is found at, for instance, x,y,z = 0.22, 1/2, 0.521, then we can conclude: oh, yes, it is on site 1b, and this is at a mirror (the mirror halfway up along the b direction in this case). And if we found another atom at 0.176, 1/2, 0.895, then we again can conclude: this must be at this mirror, too....
Concerning your example of Sr2AlTaO6: Note, the example begins with the fractional coordinates of the experimentally derived crystal structure! Then we can look on which wyckoff positions they are located. 24e for oxygen: x,0,0. You ask: How can we know x? And the answer is given in the table: well, a crystal structure analysis has determined x = 0.25 in that particular case! If the oxygen atom were found at x = 0.174, it would be also, of course, Wyckoff 24e! So, Wyckoff tells you - again in the narrower sense - nothing about a concrete position of an atom of a concrete crystal structure.
Does this help?
Frank
PS: What do you mean by different sets [xyz]?
Dear Frank,
thank you for your explanation. Up to now, according to the examples i found onlinne I had the impression, that Wyckoffs are used together with some additional info (coordinates unspecified by Wyckoff) to generate the crystal structure for particular crystal. Now I see, if I understand you correctly, that it is the other way, the crystal structure can be examined experimentally and the coordinates compared with tables to see which Wyckoff sites are populated. However this leads me to the question about the importance of Wyckoff site - how do I further work with the knowledge that some atom resides on particular Wyckoff site? Obviously it tells me if the atom lies e.g. on a mirror plane or rotation axis, but is this knowledge somehow usable further e.g. in electron diffraction patterns (this is actually my area of interest)?
Regarding different sets [xyz], I had in mind that the crystal of the same composition can have different sets of [x,y,z] position e.g. for cubanite - rruff.geo.arizona.edu/AMS/minerals/Cubanite , which is given by the conditions in which it was grown. In light of your explanation, can it be said, that the crystal of the same composition e.g.CuFe2S3 but grown under different conditions, must have atoms placed at the same Wyckoff site, with the growth conditions influencing only the unspecified coordinates. This could be seen for different entries in the above mentioned website.
Sego
Dear Sego,
exactly, Wyckoff positions are sometimes used to complete a structure; this can be done, for example, if you have only partial structural information but also the results of the elemental analysis, i.e. the stoichiometry. Then for the remaining atom sort only certain _multiplicities_ are conceivable, which you can find in the overview on the Wyckoff positions. Together with some general chemical knowledge (bond length, ionic radii, typical coordination numbers for certain elements etc.), you then can sometimes infer, how the structure must be completed.
I don't know other importance areas with regard to Wyckoff positions.
Once again to the different sets [xyz]: These are actually the same sets for a given condition within the _experimental errors_! Early X-ray structure analysis methods were relatively unprecise. Modern techniques and structure determinations are much more reliable. And, of course, the structure will change with changing temperature or pressure.
best
Frank
Dear Frank, thank you for your explanations and mainly for this great learning course. It provided me with explanations which were in other books very well hidden. :-)