Unit 2.8 - Three Aspects of the Hexagonal Crystal System
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- เผยแพร่เมื่อ 30 ก.ย. 2024
- Unit 2.8 of the course The Fascination of Crystals and Symmetry
Additonal resources at: crystalsymmetr...
The hexagonal crystal system is a bit challenging to manage, a little less clear than the others. In this unit we want to answer three questions:
(1) In which way are hexagonal unit cells assembled correctly?
(2) To how many unit cells does an atom on the corner of a hexagonal primitive cell belong?
(3) And where is the 6-fold axis of rotation located? At the center of the cell or at the corners?
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You are right about the 6-fold axis is kinda tricky because with the primitive cell given in the example i wouldn't think at all about that 6-fold rotational axis but just a 2-fold. Thanks Frank great video.
Hello Frank! You have nooo idea how helpful your videos are!! but i have one doubt are lattice systems and crystal systems one and the same.. because i was interested about hexagonal crystal systems and so after this lecture i went on my own to do some research and found out this article in wikipedia that said this----
"In crystallography, the hexagonal crystal family is one of the 6 crystal families, which includes 2 crystal systems (hexagonal and trigonal) and 2 lattice systems (hexagonal and rhombohedral)."
Frank, dude, this really confused me cause I even remember you saying that trigonal is a part of hexagonal due to same metrices but whats up with rhomohedral??? I truly hope you'll help me!!!!!!
Hello Erza,
once again no contradiction :-)
The trigonal crystal _system_ is part of the hexagonal crystal _family_ , right. A rhombohedral _crystal system_ doesn't exist. The rhombohedral _lattice_ belongs to the trigonal crystal system.
I also strongly recommend to watch unit 2.9!
ok?
Frank
Please, do not write ›8!‹, 40320 is a bit too much :P
I am glad that you didn’t understand it wrongly as a mathematical operation :-)
Dear sir,
How many lattice point the smallest possible hexagon contain , and how to calculate it's area?
Probably the first part of your question asks about a hexagonal unit cell? (A hexagon is only a geometrical shape, while a hexagonal unit cell is a 3D body, a parallelepiped.) Then the answer is: As all other primtive unit cells, too, it comprises exactly one lattice point.
I don't understand the second part of your question, what do you mean by "area"? The area of the basal face of a hexagonal unit cell? Then it is A = 1/2 * sqrt(3) * a^2.
@@FrankHoffmann1000 thanks I got my answer
how exactly does one construct a primitive unit cell, is there any source from which I can look it up, also does the word 'primitive' have any special meaning in this context, i just started a course on condense matter and this stuff feels kinda confusing. Your video is pretty nice tho
There is no real construction plan for unit cells. The choice of the unit cell, i.e. the choice of the lattice vectors, is not unambiguously determined. Primitive means that lattice points are only at the corner of the unit cell, implying that this unit cell comprises exactly one lattice point.
in a previous section you mentioned that the unit cell should already include the complete symmetry, which is not the case for the primitive cell here?In particular you get a new symmetry if you build the crystal with the primitive cell.
Is that the reason why you should take the centered unit cell for a hexagonal lattice?
Dear Thomas,
I admit that I do not understand your question. Would you be so kind to reformulate it? Centerings do not occur in this unit. The possible R centering of crystals belonging to the trigonal crystal system is subject of the following unit 2.9
You can also write me an email (perhaps with some drawings for clarification?): kohaerenz@gmail.com
best!
Frank
Hello Frank, thanks for such a nice video! I had one doubt: When you rotate the unit cell by keeping the centre of rotation at the centre of the unit cell, the unit cell does not coincide with itself. Then you shift the centre of rotation to one of the lattice points in the unit cell and rotate the unit cell. However, the unit cell is again not coinciding with itself (although it is coinciding with the lattice points of the whole lattice structure). So does that mean, for symmetry operation, it just has to coincide with the lattice points and those lattice points can belong to the unit cell which is rotated (or any other operation) or to any other lattice point in the whole lattice structure?
Hello Akshat!
Yes, exactly! Because tto be a symmetry operation the final configuration only has to be indistinguishable from the initial configuration - and this is the case here.
Hallo, Frank. I was a bit confused by the unit cell in hexagonal. I was asking myself what is the difference then between trigonal and hexagonal crystal systems. Then I learned that trigonal has a 3-fold symmetry and hexagonal has a 6-fold one. Is it reasonable to understand better like this: hexagonal consists of three trigonal, as a overall unit to fulfill this 6-fold symmetry. Even though the unit cell is a trigonal shape, but this hexagonal shape is the real although imaginative system. This 60 degree rotation you showed at 6’14” can not be counted as a symmetry in trigonal system, am I kinda right?
Not at 6’14” but 6’40”
Hello Jasmine,
I am not sure, if I correctly understand your question. The hexagonal unit cell does not consist of three trigonal ones. You might be confused by the lines between the circles show in diagram at 6'40''. So the hexagonal unit cell is also not a trigonal shape. The projection of the hexagonal unit cell onto the (a,b)-plane is shown correctly at 7'43'' again. It might be the case that it is confusing that _one_ particular hexagonal unit cell doesn't look like a hexagon (did you mean that by 'imaginative system'?) - this is the reason why you always find illustrations where the cell is slightly extended in order to view the hexagonal surrounding (see 4'08'') of each lattice point.
You are, however, completely right with your last statement. In a trigonal system the maximum rotation symmetry is of order 3, rotation by 120°. But this cannot be seen in a simple trigonal unit cell comprising only the lattice points and which looks exactly the same as the primitive hexagonal unit cell (see also unit 2.9) - it is the _motif_ which made this impossible.
best!
Frank
Frank: "Remember? A lattice is..." Is what? (I feel dumb, but I think I missed it.)
FRancisco, please do not feel dumb! I only had the feeling that I told this sentence a few times before, so that you might think, Frank, please do not repeat yourself so many times. "A lattice is infinite, and remember, a lattice is defined by the circumstance that every point of a lattice has the same surrounding." :-)
Frank, thanks for the reply!
Is that the reason why you shouldn't take the centered unit cell for a hexagonal lattice?
Well, rather because it does not make sense to construct centered cells for structures belonging to the hexagonal crystal system. But it might be the reason why some of the trigonal crystal structures look like hexagonal ones.
Did you say what is translation rule in previous lecture? My brain is bleeding :(
Yes, some lectures are hard to understand without the preceding units :-) The transational aspect of the crystal structure or the unit cell is partly explained in some other lectures, I would recommend in particular Unit 1.7 to 1.9 of chapter 1.
Best!
Frank
Frank Hoffmann thank you very much! I’ve just finished 3 chapters. You save my life 🙏🏼
Does the unit cell have the same symmetry as that of the crystal system?
Hi Dawn,
this is a little hard to answer because two categories are mixed in your question, but we can say the following: the best unit cell (best in the sense of the criteria that we should apply when we choose unit cells) reflects, or has the same symmetry of its underlying lattice. The symmetry of a lattice is an intrinsic property and determines the belonging to the one or the other crystal system - and then, the unit cell should have, of course, the same symmetry.
Does this answer your question?
Thanks. That helps.
why the hexagonal system has 4 crystallization axes? please answer me
What do you mean by "crystallization axes"? The unit cell of the hexagonal system is - as all other crystal systems - characterized by only three axes.
i mean that the hexagonal system and trigonal system have 4 crystallographic axes ( a1, a2, a3, and c ) , why ?
?
There are two options:
a) forget about this fourth, and completely redundant axis; you don't need it. Once again: to define unit cells or crystallographic axes systems - independent of the concrete crystal system - three axes (and three angles) are completely sufficient. In this way, there is no such "fourth axis".
b) If you otherwise desparately wish to know, why in some neighboring disciplines of crystallography, for instance mineralogy and metallurgy a fourth axis is defined, I would point to the paragraph "Case of hexagonal and rhombohedralstructures" of the English wikipedia entry for Miller indices:
en.wikipedia.org/wiki/Miller_index
In short, the fourth axis "a3", which encloses an angle of 120° with both the axis "a1" (a) as well as "a2" (b), is _symmetry equivalent_ with a1 or a2; this means that also no additional symmetry elements can occur in this direction. The only rational to introduce such a fourh axis - and hence also a fourth index [--> Bravais-Miller indices (h, k, i, l) with i = -(h + k)] is to point to the symmetry relationships of directions in these lattices, for instance that the direction [1 0 0] is equivalent to [-1-1 0].
one more question is trigonal lattice becoming hexagonal due to addition of motif in a certain way? and another one, can we make hexagonal a trigonal lattice??
Hi Erza,
yes and no, so to say.
If we look at the primtive hexagonal and trigonal _lattice_ then we find no difference, they are identical; this means that the trigonal lattice has not a three-fold rotational symmetry but a six-fold one. This sounds in a way weird, but the _lattice_ is a mathematical construct abstracting from the motif, and, thus, can have _higher_ but _not lower_ symmetry than the a specific crystal within the crystal system.
If we have a crystal that belongs either to the trigonal or hexagonal crystal system depends on the motif: If we substitute the lattice points with a certain motif that has only three-fold rotational symmetry then we have a trigonal crystal system.
Now, it should be clear that a trigonal lattice cannot become a hexagonal lattice by adding a motif, allright?
best
Frank
@@FrankHoffmann1000 thanks frank for answering all my queries!!
Welcome!
Mr Frank thank you so so much .