Unit 2.4 - Bravais Lattices (I)
ฝัง
- เผยแพร่เมื่อ 11 ก.ค. 2016
- Unit 2.4 of the course The Fascination of Crystals and Symmetry
In this unit, we want to answer the question, if the smallest unit cell - the primitive one - is always the best choice or in which cases it might be better to choose a larger unit cell - and this will lead us to the so-called centerings and thereby to the 14 Bravais lattices.
Additonal resources at: crystalsymmetry.wordpress.com...
Direct link to the instructions for the optional assigment:
crystalsymmetry.files.wordpre...
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If you prefer books instead of videos, have a look at:
www.amazon.com/Introduction-C... - วิทยาศาสตร์และเทคโนโลยี
Thank you for taking the time to upload all of your different content. You're keeping our entire class afloat with your superior formulations and clear instructions. Keep up the good work!
Thank you for your kind compliment!
What you are engaging in is a sort of philanthropy - Phil = love; Anthropos = Humanity. So it is a love of humanity. I am enjoying myself going through the videos and by your generosity am able to download the script and slides as well. Grateful thanks for the knowledge imparted and materials shared.
Thank you very much for your kind comment - I am glad that you enjoy the videos. In a way philantrophy is a kind of self-interest: Hoping that other people will share their knowledge too, from which I can benefit :-)
@@FrankHoffmann1000 Glad to participate in such a project when I'm competent enough.
Thanks for the good structure and explanations! Helped me a lot during my Master :)
brilliant explanation.I just follow this lecture series to learn solid crystalline structure
Great! Excellent explanation of concepts, really helpful !
Herzlichen Dank für Ihre wunderschöne Tutorial, sie sind super hilfreich.
Sehr gern geschehen! Es freut mich, dass sie hilfreich sind! Ich bin neugierig: Wo haben Sie deutsch gelernt?
@@FrankHoffmann1000 In Deutschland😅 ich studiere eigentlich hier seit 2014 oder noch eher😊
@@xizhipang6250 Verstehe :-)
Thank you for the explanation. For some reason this is very hard for me since i am not a visual thinker. But this helped.
good job. great conceptualization.
Thank you very much!
I just needed a clarification. If we consider Chalcanthite with composition CuSO4.5H2O - can we view the Unit cell's 8 corners to be constituted of Cu atoms and the inner content or composition of the Unit Cell to be SO4.5H2O - thereby making the Motif to be CuSO4.5H2O? Would this be a correct visualisation?
This would add up to only one formula unit, but please note that the content of the unit cell comprises two formula units. So we have additional Cu, S, H2O positions. You can figure it out by looking at the respective CIF of Chalcanthite.
Thank you so much sir☺️
Hello Frank. At 5:30 for the centered unit cell would the lattice vectors be horizontal and vertical (from the left bottom corner to the right bottom corner and top left corner respectively)? If so the center lattice point in this UC would be half the sum of these vectors, but fractional coordinates to define any lattice points are forbidden (or so I thought). Can you explain where I am going wrong? Thanks.
Hi Hikguru,
"...but fractional coordinates to define any lattice points are forbidden (or so I thought)" - No, this is not forbidden; on the contrary, it is the _essence_ of centerings. How can you achieve a centering at all, if this should be forbidden?
@@FrankHoffmann1000 Yes, that's what I thought. But usually in standard texts it is stated that any point in a 2 D lattice can be represented by R=n1a1+n2a2 where n1, n2 are integers and a1, a2 are the primitive vectors. However, if I choose my a1 and a2 as the mutually perpendicular sides of the unit cell then the center point is 1/2(a1)+1/2(a2) i.e n1/n2 are not integers. However in a primitive UC the lattice vectors can be used to define any point in the lattice as R=n1a1+n2a2. Maybe I am getting confused on the difference between primitive and lattice vectors, perhaps the lattice vectors in the centered UC are not the same as primitive vectors.
@@hikguru I am sceptical :-) ..."that any point in a 2 D lattice can be represented by R=n1a1+n2a2 where n1, n2 are integers and a1, a2 are the primitive vectors." This cannot belong to the topic centering, this is the mathematical description of the primitive lattice itself, the description how you derive at all points of a primitive lattice (in my view an unnecessarily mathematical description).
And, exactly, the lattice vectors in the centered cell are not the same as the primitive vectors; the new lattice vectors are _linear combinations_ of the primitive ones.
@@FrankHoffmann1000 OK, starting to make more sense now. When standard texts for instance show the UC and primitive vectors of a BCC lattice they show 3 cubes (with the central lattice point) with all 3 cubes sharing a single common corner. The UC for the BCC is of course 1 of the cubes, but the primitive vectors are defined by a1=(a/2) (-x+y+z), a2=(a/2)(x-y+z) and a3=a/2(x+y-z) where a is the length of the cube that defines the UC and x, y and z are unit vectors along the x/y/z directions. In this scenario the 3 primitive vectors lie within 3 different UCs and are not within a single UC. In other words they don't scan a single UC the way lattice vectors (ax, ay and az) do. But the primitive vectors can be used to generate the lattice vectors, so a1+a2=az, a2+a3=ax etc. I should remember not to mix up primitive and lattice vectors! Thanks.
@@hikguru Makes indeed more sense! :-) best wishes!
Hello Frank. At the four minute mark you showed a trapezoidal cell as a possible unit cell. If I only look at this cell and not the lattice it represents, then I find that there is a plane of reflection, parallel to the longer side and passing through the center of the cell. However, this particular plane does not work as a reflection plane for the entire lattice. Does this mean that you can have a unit cell which has certain symmetry elements but these symmetry elements are not necessarily applicable to the lattice the cell represents? Similarly, you can have a symmetry element for the full lattice but that same element may not necessarily work a chosen unit cell. So I guess the unit cell should be such that it has the same symmetry elements as the lattice it represents and it should be as small as possible.If it meets those requirements it is the best choice. Please let me know if my understanding is correct. Thank you so much.
Hi Rahul,
first of all, I have to ask to which cell do you refer. In the whole video there is no trapezoidal unit cell shown. Do you mean the green parallelograms appearing at 2:56? If this is the case: Parallelograms have no reflection/mirror planes (or to be precise: lines, as we are in 2D).
Best wishes
Frank
Sorry my bad, I was referring to the green parallelograms and mistakenly referred to them as trapezoids. It appears that even though this parallelogram is a possible unit cell the reflection planes for the lattice do not act as the reflection planes for this unit cell. So is it correct to say that if the symmetry element (in this case the reflection plane) for the lattice and unit cell are different then it is not a good choice for the unit cell and we should select the cell which has the same symmetry elements as the lattice?
@@hikguru Completely correct! Cheers! 🙂
Hi Frank. Please explain more the word motif and in another way the word base as you mentioned or suggested for the meaning of one atom per unit cell and also conclusion of the need for the base ( I think the geometry here, you meant) for sequential adherence of unit cells to each other and smallest possible unit cell. Actually hard to remember if motif was used in material literature if it is stands for repeating feature and base as a geometry terms. Confused, help me. Thanks
Hi Mostafa,
here in this context I am using the terms base and motif as synonyms. And the motif is a chemical entity that is _represented_ by a lattice point; this might be a single atom, or an ion pair or a molecule or even two molecules... and so on.
See also unit 1.9:
th-cam.com/video/Z69LldxLL_U/w-d-xo.html&ab_channel=FrankHoffmann
@@FrankHoffmann1000 Hi Frank. Thanks indeed. Seems that although I am old but not very patient to review all your valuable lecture videos. I must complain about my shorts to my dear and lovely instructors of 70s i.e., Aziz Ahamdiyeh and Abaschian. I wish for you and your family a happy new year without Covid 19. Regards
@@mostafaamirjani8330 Thank you very much, Mostafa!
I also wish you and your family all the best!
Tq Sir.
Consider a square unit cell which has only one lattice point at it's centre. Is it a primitive unit cell or non-primitive unit cell?
It is a primitive unit cell. A completely equivalent cell would be obtained, if we translate the unit cell in such a way that the lattice point(s) are at the corner of the cell, giving again (4 x 1/4 =) 1 lattice point per unit cell.
@@FrankHoffmann1000 Thank you for your reply
good job..
I dont see any links for the instruction of the exercise . . . .
Sorry for the inconvenience:
crystalsymmetry.files.wordpress.com/2016/06/02_04_opt_assignment.pdf
the presentation is nice...
Thank you very much!
I have a lot of confusion
1) I studied that “Non-Bravais lattices have additional symmetry operations that cannot be generated from the symmetry operations of the 14 Bravais lattices”. But “The choice of lattice does not change the underlying symmetry”, this means A: “The non-bravais lattices which have additional symmetry operations cannot be expressed by bravais lattice”.
But there is a statement “All of the crystals belong to any one of the 14 Bravais lattices” which means B: “The crystals belonging to Non-bravais lattice can also be expressed by Bravais lattice”
A and B are contradicting.
2) What is the difference between, “translational vectors of a lattice” and “lattice vectors of a lattice”?
You are saying that, Bravais lattice are defined by their unit cells falling into one of the 7 crystal systems which may be primitive or non-primitive. My doubt is, if we want to consider primitive unit cell for all bravais lattices, do we need to consider the unit cell in such a way that it satisfies the “rules for best choice of unit cell” as much as possible? Are the translational vectors of the lattice same as lattice vectors of that unit cell?
1) Already your first statement is strange. Non-Bravais lattices do not exist. Your pattern can have an underlying lattice; then it belongs to one of the 14 Bravais lattices. Otherwise it is not a lattice. This also means that are no crystals that have a "non-Bravais lattice". (The confusion might be originating by the missing distiction between lattice and structure.)
2) "Translational vectors of the lattice" and "lattice vectors of the lattice" are synonyms.
Of course, every centred lattice also includes one with a primitive unit cell, but for the 7 centred lattices (of Bravais) these would be non-conventional ones, because they are not following the rules for the "best" choice of the unit cell.
I cant figure out the unit cell for the difficult example ! help !
Here you can find the solution:
1drv.ms/b/s!ArTbwWHXPrwehIxaQsAs1qbXNUDZgg
best wishes
Frank
Beautiful Moroccan Zellij.
Great videos, but they are much more interesting to watch at 1.5x the original speed.
Not all of us are native English speakers! It's okay the way it is.
try to define bravais lattice
What is non bravis lattice
Well, if Bravais lattices are the totality of all crystal (or translation) lattices in which the motifs are represented by lattice points and if you take the term lattice in a strict mathematical sense (a lattice is thing in which each point has an identical surrounding) then there are no other lattices than Bravais lattices.
Note that "Bravais lattices" is only a name for the 14 principally different lattices in 3D space.
@@FrankHoffmann1000 so in non bravais lattice each points has not identical surroundings. Is it right? And can I transform a non bravais lattice to a bravais lattice? If yes then how?
@@sayanjitb I am not familiar with the term or concept of a "non-Bravais lattice". I know that some people use this term for "lattice plus base", but in my opinion this is not meaningful, as the lattice point already represents the whole base.
So, in this sense all lattices are Bravais lattices; Bravais only classified them into certain types of centerings (i.e. to take non-primitive lattices into account).
0.39 i didnt understand how 1 motif came
I know this is quite late, but anyway: We can say that there's one motif per one unit cell - for each of the 7 primitive lattices, because if you look carefully, you'll notice that in the complete lattice arrangement, each of the motifs (which are at the corner points) are actually part of a total of 8 unit cells. So the contribution of any single motif to a unit cell will be 1/8, since only 1/8th of it is inside that unit cell. Thus, from the 8 motifs which are at the corner points of a unit cell, the total contribution will be 1/8 * 8 = 1 .