Unit 4.4 - Screw Axes in Crystal Structures
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- Опубликовано: 24 июл 2016
- Unit 4.4. of our course The Fascination of Crystals and Symmetry
Additonal resources at: crystalsymmetry.wordpress.com...
In this unit we want to investigate, what different kinds of screw axes, that is, what kind of different helical symmetries are present in crystal structures.
Possible screw axes are: two-one, three-one, four-one, four-two, six-one, six-two, and six-three.
In addition there are these enantiomorphous axes, this means mirror images of already mentioned axes, namely, three-two, which is the enantiomorphous screw axis of three-one, four-three is the screw axis with opposite handedness of four-one, and six-four and six-five are the enantiomorphous screw axes of six-two, and six-one, respectively.
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www.amazon.com/Introduction-C... 
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I think you're the only person on the internet who's explained these concepts at a graduate level so cleanly and concisely. Can't thank you enough.
Thank you very much for your kind comment! I'm glad that you like the videos.
If I had this right on time, I wouldn't have hated Crystallography. My appreciation goes to Frank Hoffmann for taking his time to clearly explain the crystallographic symmetry elements and operations.
Amazing videos, thank you Dr Frank Hoffmann, you are a hero!
Thanks Prof. Hoffmann for this content !!!!
just watched a few on screw axes and glide planes! great slow easy description! thank you, subsribed
great lecture!
Thanks for the explanation!
i am teacher! Great work much appreciated!
Thank you ...for this wonderful video...clear lot of doubts i had :)
Really helped me..
sir , I don't get how the 3/6 translation don in 6 fold axis = screw axis , its image in your slide / in the reference book confusing me , due to 3 atoms in one plane with 1/2 translation which creates confusion . so help me to figure it out
Dear Mrunal,
there is no reason to be confused about the fact that there are three atoms in one plane: This only an example how the motif could look like. Note that all atoms of one and the same motif are subjected to the same symmetry operation. Here an _isolated_ motif has a (simple) three-fold axis (rotation . But they are rotated from one "plane" to another by 60 degrees(!), so they are forming a 6(3) screw axis, with a rotational component of 60° and a translation of 3/6 = 1/2.
If you want, you can also take out only one atom of such a three-atomic arrangement, which you first rotate by 60° and then translate it by 1/2.
best
Frank
Is the rotation direction of n sub 1 screw axis, such as 3_1 and 4_1, always anticlockwise, viewing from c direction (if it's parallel to c)?
yes!
Thanks
ohh super. Es ist sehr nett von Ihnen mit uns solche Materialen zu vermitteln. Ich bin dafür sehr dankbar. Alles ist mir sehr übersichtlich und sehr angenehm zu hören und lernen. Vielen Dank noch mal.
Vielen Dank für die netten Worte, und dann auch noch in meiner Muttersprache!
Ich habe für ein Jahr habe ich Deutsch gelernt und nun versuche ich die anzuwenden.Gerade mache ich mein Master Studium in Chemie in die Uni-Siegen. Einmal würde ich mal gerne Ihre Uni und das Museum der Kristallografie dazu besuchen. Dieses Fach wurde meine Neugierde geweckt.
Das finde ich super! Ich wünsche Ihnen viel Erfolg für Ihr Studium! Und wenn Sie mal in Hamburg sind, können Sie mich gerne besuchen!
Vielen Dank! Ich arbeite gerade an einem Vortrag über Molekülsymmetrie und das Lehrbuch hat einfach angenommen, dass jeder eine perfekte Raumvorstellung hat.
Gern geschehen! Das ist leider häufiger der Fall; eigentlich müsste man Symmetrie am besten mit VR-Apps lehren...
Hi Frank. It looks like the possible screw axes cover 2, 3, 4 and 6 fold rotation, so 5 and anything > 6 are excluded. What is the reason for that (seems to be the same restriction for rotational symmetry axes for crystals but not sure if the reasons are the same)? Also in 7:16 for the 2nd or 3rd picture can you have an arrangement in a cell where an atom is present only on the bottom and the next higher location (absent in the 3rd location)? So you still have 3 sub 1 but you don't generate the 2nd point, only the first. Finally how does one determine the space groups of a crystal? Is it through experimentation? My understanding is that macroscopically you can only know the point groups. Thanks.
Hi hikguru,
1) the reasons are the same.
2) Not sure, if I understand you question correctly. You can have any (chemical sensible) arrangement in a cell. But thinking about the scenario that you described you do not have a 3(1) screw axis anymore.
3) Experimentally via X-ray diffraction.
best
Frank
@@FrankHoffmann1000 Thanks Frank. For screw axis we specify the angle of rotation (in addition to translation distance) but it seems that the radius of rotation is not specified. Is that correct? If so then we don't know the precise coordinates of each of the atoms that are generated by the screw axis operation,.
The screw axes have specific locations in the unit cell (see the respective diagrams of symmetry elements in the International Tables for Crystallography; an example for P3(1) can be found here: img.chem.ucl.ac.uk/sgp/large/144az1.htm). So, the radius is the distance between the atom(s) and the screw axis.
Great lecture! Can you share the slides of all of your chapters?
Thanks! Have a look at the video description!
Thanks for the great video! But can you explain me again what's the different between 31 and 32 for your example of 3-fold screw axis? It seem to me that it only differs the way/the direction you look at your crystal. So, what I mean is that I will assume 32 as 31 if I view my structure on the opposite side. Can you elaborate that part please?
Dear Htet Htwe,
this impression is mostly based on a a kind of misperception or an optical illusion, but you cannot turn a left-handed helix into a right-handed one. The screw sense is always a property of the object, not of the perspective (view-angle)! See also the wkipedia entry:
en.wikipedia.org/wiki/Helix
See, in particular, the second picture, in which a left-handed and a right-handed helix are shown in different viewing angles.
best!
Frank
Thanks for your reply. But even with that, why the second picture has translation of 2 but not 1? I don't know how can I visualize 2/3 of the cell is parallel to axis but not for the first picture (which is 1/3) since they are only differ in rotational direction.
Dear Htet Htwe,
I don't know exactly what do you mean by "I don't know how can I visualize 2/3 of the cell is parallel to axis" but let me try this:
It is indeed not so easy to understand why helices which are described as 3(1) and 3(2) screw axis, respectively, are of the same type, so to say, but only have an inverse rotation sense. And the reason for this difficulty are, exactly, the fact that they are desribed by different translational components, 1/3 vs. 2/3. But this difference is only due to the crystallographic rule that the rotation has to be carried out always in the same direction (from the x- to the y-axis). But for a better understanding it is recommended to re-think a 3(2) axis as a 3(-1) axis! So, you can rotate the first point of such a helix by _minus_ 120° (which is crystallographically "forbidden" but anyway!) and you have to translate then also only by 1/3 to reach the next point of the helix!
I have made an additional graphic, which might be helpful in this regard, see:
crystalsymmetry.files.wordpress.com/2016/06/crystal_mooc_04_04_31_32_screw_axes.pdf
Does this help?
best!
Frank
Thank you soooo much, sir!! The "forbidden" rule and your graphic makes sense to me. Now, I feel good with this concept. :D
Welcome!
I had 2 questions:
1. Sorry if it is easy, but why can we not have say a 5_1 or 7_1 screw axes?
My guess is because prntagons/heptagons cannot cover the plane without gaps.
Can anyone please confirm?
2. In case of 3_2 we will have rotation by 120 deg and translation by 2/3 of lattice parameter while for 3_1 its only 1/3 translation. Then how on @7:24 can we compare them?
To your first question:
Yes, exactly, see also our unit 5.6
ruclips.net/video/qlA9UOVUM1I/видео.html&ab_channel=FrankHoffmann
...although, the strict mathematical proof is not straightforward!
To your second question:
I don't know exactly what you mean with "how can we compare them", but does the following picture help?
crystalsymmetry.files.wordpress.com/2019/02/32_extended.png
because
How to calculate the value of m?
m is not calculated. It is the numerator of the translation operation, given as a fraction, where 1 = one full unit cell along the direction considered.
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my apprentice said that the most common screw axis is 6,3 and I can hardly distinguish it from I6 symmetry axis 😢
Hmmm, personally I think that 2(1) are far more common than 6(3); note that P2(1)/c is the most frequently space group.
Hello Frank. Thanks again for this great course and answering questions. I am a bit confused about the 31 and 32 screw axis. The translation in the 32 axis is double respect to 31 one, so the motif is repeated at 2/3 4/3 and 6/3 rather than at 1/3 2/3 and 3/3. This would place the motives more spaced from each other (2/3) in the case of the 32 axis. However at crystalsymmetry.files.wordpress.com/2019/02/32_extended.png
It can be seen the motives are spaced by just 1/3 rather than by 2/3 as I would expect .
Thanks again!
Welcome, spin 34!
Well, the tricky thing with crystals is the translational principle, or in other words: Moving structures, atoms, motifs along the lattice vectors by whole units lead to equivalent points. This means that it is always allowed to add or subtract 1. And this means that 4/3 = 1/3 and 6/3 = 3/3!
Thanks for your explanation, I understand it now, thank you!@@FrankHoffmann1000