Unit 4.6 - International Tables and The Space Group Pmm2

Unit 4.6 of our course The Fascination of Crystals and Symmetry
Additonal resources at: crystalsymmetr...
In this unit we will have a closer look at the "International Tables for Crystallography", the crystallographer's bible.
In this book all of the 17 plane groups and all 230 space groups are listed systematically, their short and full symbols are given,
together with an unique group number. Furthermore, the crystal system and the respective crystallographic point group, to which the space group belongs, is indicated.
One particular important and very useful part of these tables are the diagrams: for each of the space groups, you can find diagrams, in which the symmetry elements are depicted and positioned at their correct position within the unit cell. In addition to that, a so-called general position diagram is displayed, which will be explained for the space group Pmm2 as a first example.
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If you prefer books instead of videos, have a look at:
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[@DamyHaoo]

Frank you're the best. You are creating a whole generation of good crystalllographers. Keep your work men!

[@FrankHoffmann1000]

Thank you very much Damy! That's very kind. I will try my best to do so!

[@whysoscared435]

As A student of Chemistry doing their bachelors and masters theses in inorganic solid state chemistry, I am very glad to have found your videos, thank you!

[@barnaliwaugh2368]

Thanks for elaborating. My interest in crystallography has increased many fold than before.

[@FrankHoffmann1000]

Thanks for your nice comment! Have fun with crystallography!

[@shimjaejin810]

A really good voice and an even better lesson, thank you very much!

[@GAUTAMSHARMA-xu7bl]

Beautiful explanation...........................I loved it....................................................

[@qianzhang2031]

i hope i could become a professor like you, that is, to teach the student selflessly, efficienctly, and you are the ideal professor in my mind！

[@dhoonygo]

I'm very very impressed your high quality lectures. Thank you very much. It helps a lot. But I wonder you have any plan for lectures about X-ray diffraction. I hope to see you there.

[@FrankHoffmann1000]

Dear Donghoon, thank you very much for your kind comment! No, I haven't any concrete plan for lectures concerning X-ray diffraction, but the higher the number of people who ask for such lectures the more likely it is that I will realize them. best! Frank

[@sasukeuchiha998]

A lot easier than looking at the guide.

[@HotChick_Gardener_NorthTexas]

you re the best, you got my subscribe. thank you so much

[@chrisharrison3180]

Hi Frank - I am picking up the course again after some time away and have some further questions which built on a previous question. I am still not so sure what the difference between the following are: (1) Crystal Classes, Crystal Systems, Crystal Groups - are these all the same thing? If not what is the difference between them? (2) Point Groups vs Wallpaper Groups - are point groups just 2D symmetry systems? Or is this wallpaper groups? (3) Crystal Groups vs Space Groups vs Point Groups - and (in addition to (2))....from what you said previously, the crystal groups don't consider the internal symmetry whilst the space groups do. Do you mean we are not considering the atomic structure and rather the macroscopic geometry? Many thanks again for all your help, Chris

[@FrankHoffmann1000]

Hi Chris! (1) (a) No, they are not the same. The symmetry of crystals can be regarded on different levels. The most basic level is what we call crystal systems, there are only seven of them (triclinic, monoclinic, orthorhombic...). Here you specify the (maximum possible) symmetry of the underlying _lattice_. And because lattices are in principle infinitely extended they have translational symmetry. (1) (b) "Crystal groups" are not a proper term in crystallography. You can call the types of symmetry that occur in crystal shapes/forms either crystallographic point groups or crystal classes (these are the same). There are only 32 different types of symmetry concerning the outer shape of crystals. (2) Wallpaper groups are the 2D equivalents of (3D) space groups. They are also called plane groups because they describe patterns of the plane. (3) Yes, we are considering the macroscopic symmetry of the outer shape of crystals. We disregard not only fthe atomic structure but also the translational symmetry of the lattice. best regards Frank

[@matchem9178]

Thanks a lot!!

[@hb8540]

Hi Frank, do you have any plan to talk about XRD in details, including refinement etc. ?

[@FrankHoffmann1000]

Hi Houcine, not in the near future, but the more numerous the requests, the higher the probability that I will realize a KZbin course on that topic eventually. best wishes Frank

[@chenchen2889]

Hi Frank, thanks so much for your lesson! Just have a question, how do I know which plane is the standard projection projected to? It's not indicated in the graph

[@FrankHoffmann1000]

Hi ChenChen, you are welcome! In the orthorhombic crystal system the standard projection is along the c-direction, i.e. onto the (a,b)-plane. In the monoclinic crystal system the standard projection is along the b-direction, in the tetragonal, the tri-, and hexagonal system it is along the c-direction. In case of doubt, you have to look at the International Tables. Best Frank

[@chenchen2889]

@@FrankHoffmann1000 I see! Thank you so much, Frank! Have a great day!

[@kaptensnall]

Very good work! ty

[@sanjaydubey8929]

I am a bit confused with this terminology that one lattice point per cell in the cube. As cube has 8 vertexes to make single cube which can be single cell so the "cell has 8 lattice point" Why 1 in cube?? I am new to this area and your correction will assist me

[@FrankHoffmann1000]

Dear Sanjay, I would suggest to watch Unit 2.5 "Bravais Lattices (II)", kzbin.info/www/bejne/pWfUfKWVhL9krNk in which this issue is explained. The key to understand the concept is that a lattice point at the corner belongs to eight unit cells simultaneously. best regards Frank

[@sanjaydubey8929]

that video was quite helpful to understand!

[@jeoh93]

Hi Frank, why do we assume that there are mirror planes at t/2?

[@FrankHoffmann1000]

Hi Jeoh, I am not quite sure, what t/2 means, but probably a/2 and/or b/2. Well, in this particular case for this space group it is a consequence of the presence of the mirror plane at a = 0 (and therefore a = 1) and/or b = 0 (and therefore b = 1). If you have "two" mirror planes at the borders of your unit cell along the a direction (and/or b...) there _must_ be also a mirror plane at the center (horizontally and/or vertically). I am confident that you are able to derive this necessity: try to construct an arrangement of objects that obey the symmetry framework of the two mirrors at the borders (expand your drawing/sketch to 2 or 3 unit cells) of the cell and you will see that "automatically" the mirror at the center will "appear". best wishes! Frank

[@dhoonygo]

Sir, I have a question. Now I've studied more about space group, and I found P2/c. Since it only has 2 fold in one direction, it is monoclinic so each lattice point has 2-fold symmetry. it's okay so far but from the diagram, which indicates symmetry and general pole, when describing diagram why is projection structure outlined by inversion centers instead of 2-fold lattice point? Is there any reason? Or am I missing something? P2/c link : img.chem.ucl.ac.uk/sgp/large/013ay1.htm

[@FrankHoffmann1000]

Dear Donghoon, yes, the diagram shows where to find the symmetry elements within the unit cell (here a prjection along the b-axis is shown) and it shows the general positions (not poles), meaning locations of atoms, which are not lying on a symmetry element. We see indeed all the black ellipses, indicating the 2-fold axes of rotations. What does probably need a little more attention is the fat-black angled arrow at the lower left side of the diagram, pointing with its arrow head along the c-direction. So, we do not have only 2-fold rotational symmetry! We have additionally a glide plane _c_ perpendicular to the b-axis. Therefore, this glide plane lies in the plane of projection. This is the reason why such an additional angled arrow os placed beside the diagram, there is hardly no other way to visualize it within the borders of the unit cell. Now you ask, why there are _addional_ (not instead as you suggested!) inversions centers: Because the presence of a 2-fold axis of rotation plus a glide plane _c_ perpendicular to that direction of this 2-fold axis automatically generates this additional inversion center. This is very often the case that the simultaneous presence of certain symmetry elements imply the presence of further symmetry elements. However, in the space group symbol only the so-called 'generators' are given, please have also a look at unit 4.5 at minute 4:20. best! Frank

[@dhoonygo]

Thank you so much for answering! So now I fully understand what the diagram presents. but I have an additional question related to that(sorry to bother!). When drawing diagram, first of all I think determining starting point is needed. starting point I say means origin which is located lower left. And from certain starting point I can draw lattice, put general position near it, and put others derived by generator In P2/c diagram, starting point is inversion center. But I think starting point could have been 2-fold rotation so general position could have started near 2-fold rotation... right?

[@FrankHoffmann1000]

Dear Donghoon, in principle you are right! Actually, there is no such starting point or a specific symmetry element you have to start with. Or to turn this the other way round: it does not matter with which symmetry element you start. The result is always the same, independent of the choice of the sequence of the applied symmetry elements. This holds also for positioning of the first atom at a general position. The only requirement is that it must not lie on a symmetry element. best regards! Frank

[@dhoonygo]

Frank Hoffmann Yes exactly! Thank you so much again!

[@yg.8185]

Can you please explain it the way you would explain to someone with no background? I have no background in materials science and engineering, and I come from a biomedical engineering. This video is way too confusing to me as I have a million questions to ask. 1) Where those names come from? 2) Can you explain this in a more 3D view 3) What does Pmm2 stands for? Ect...

[@FrankHoffmann1000]

I believe that all the questions you have indicated will be answered, if you watch the entire course, not just a single lesson - - for instance the nomenclature of space groups is considered in unit 4.5 - in this unit there are also more 3D examples given; the crystallographic viewing directions are explained again (unit 3.5 and 3.6 are also recommended for this purpose), but part of the purpose of unit 4.6. is to learn to read the International Tables for Crystallography and those are 2D... I am sorry, but learning crystallography can neither be done by watching just one 10 minute video nor is it possible to explain all relevant background knowledge that is needed to understand a particular advanced content in a comment on KZbin.

[@natasharedding134]

Is there a video about the special positions of the space groups?

[@FrankHoffmann1000]

Yes, the following video in this chapter (unit 4.7) describes also the special positions: kzbin.info/www/bejne/d5ukaYRofreshrM best Frank

[@finite_elements]

Hello, I would like to ask a question; what are the principal axes of a crystal belonging to the trigonal system?

[@FrankHoffmann1000]

Hello, the viewing directions are first along the c direction and then, secondly, along the a direction. best Frank

[@finite_elements]

Hello, Thank you very much. I would like to ask a precise question: if we have the components of the thermal expansion tensor of a trigonal crystal (alpha11=alpha22 and alpha33) these values correspond to the measured along the X and Z axes respectively? Best Idir

[@FrankHoffmann1000]

Hello Idir, unfortunately, I don't know much about the physics of crystals and these tensor things... I would assume that these values correspond to x, y, and z. And I would assume that the tensor has to reflect the syymetry of the crystal, or crystal system to be precise, meaning for trigonal crystals that x and y or a and have equal values, while c or z should be different. This means that the aspect ratio c/a will change. But don't take that for granted.... Best, Frank

[@finite_elements]

Hello, Thank you very much. Best, Idir

[@finite_elements]

Hello Frank, Thank you very much for your answer. Best, Idir