Unit 1.8 - The Seven Crystal Systems
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- เผยแพร่เมื่อ 4 ก.ค. 2016
- Unit 1.8 of the course The Fascination of Crystals and Symmetry
A unit cell is characterized in the first instance by its so-called metric. These are the six lattice parameters: The length of the edges, a, b, and c and the angles between these edges or faces: alpha defines the angle between edge b and c, beta is the angle between a and c, and finally gamma is the angle between a and b.
Furthermore, the unit cell contains all symmetry elements of the crystal.
All the crystals of the world can be classified into not more than 7 crystal systems. Regarding the metric of the unit cells there are only 7 different unit cells, in principle!
- triclinic
- monoclinic
- orthorhombic
- tetragonal,
- trigonal
- hexagonal
- and finally cubic
However, the the classification according to crystal systems is not based upon the metric, instead it is based on the symmetry. This unit will explain these relationships.
Additonal resources at: crystalsymmetry.wordpress.com...
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If you prefer books instead of videos, have a look at:
www.amazon.com/Introduction-C... - วิทยาศาสตร์และเทคโนโลยี
your videos are the most vividly explained videos about crystallography on internet
By far one of the best lecture series available online. Looking forward to more.
I was so frustrated with this topic since morning, have been searching and browsing through a lot of videos on the topic but none were satisfactory enough until I came across this sensei! Thank you so much!
I guess I am kind of off topic but do anyone know of a good place to stream newly released series online ?
@Chandler Khalid I would suggest Flixzone. Just google for it :)
@Chandler Khalid i watch on flixzone. Just search on google for it =)
@Khalid Flynn Yup, I've been using FlixZone for years myself :D
@Khalid Flynn Thanks, I went there and it seems to work =) I really appreciate it!
This is no doubt the best video series on crystallography I've ever seen. I'm soooooo happy that I found this. My most sincere gratefulness to you, Frank.
Thank you very much, Shuyi!
Thanks for the explanation. Now I can sleep peacefully
Thank you so much for your lectures. I am a 2nd year PhD student and I did not understand any of this and did not find text books useful either. Your lectures are a God sent.
Thank you so much for these videos! Your explanations and diagrams are so clear and helpful.
Thanks very much for your kind comment!
and of course yes good point, it can be the one with the restrictions placed and all above with fewer restrictions, makes sense! thank you!
A) Unit cell attributes--
1. Metric-- 6 parameters
1.1. 3 lattice parameters
1.2. 3 angles bw the lattices
2. Symmetry aspect.. it defines the min size of unit cell.
3. Chemical composition-- of the unit cell must be the same of the compound
B) Classification of unit cells-- 7 systems-- Inc order of symmetry (tmotthc)
1. Triclinic -- no limitations
2. Monoclinic-- any length+ 2 angles 90°
3. Orthoclinic-- any length+ all angles 90°
4. Tetragonal-- 2 equal sides+ all angles 90°
5. Trigonal-- 2 equal sides+ 2_90°+ 1_120°
6. Hexagonal-- a=b + 2_90°+ 1_120°
7. Cubic-- a=b=c + all 90°
Trigonal+ hexagonal belongs to the same hexagonal crystal (coincidental cell parameters due to symmetry)
C) The crystal systems are not defined by lengths and angles but symmetry
D) VIMs -- Symmetry determined the metric not the other way round.
E) Exercise
Thanks, that's very helpful.
Thanks alot for the explanation.
Really helpful. Thanks.
This is so great, thanks a lot for the videos, keep it up!
Welcome! Thanks for your compliment - we will try :-)
U are way more better than my chemistry teacher👍👍👍
Hello from Korea!
First of all, thanks for the lectures, I recommend your channel to all of my viewers when they ask me about crystallography.
I have a question about coincide triclinic system, which has a=b=c, A=B=C=90`.
How can I define such conditioned crystals to a single system?
Maybe should I use TEM to see the reciprocal patterns?
Hello to Korea!
Thank you very much for promoting my channel :-)
In TEM pictures you don't see the reciprocal lattice.
Usually, you have to do a single-crystal structure analysis (based on diffraction), either with X-rays or neutrons or electrons. One of the first steps is then to establish the space group that will give then the answer to the question to which crystal system the crystal belongs.
Best
Frank
thank you!!! this is so helpful for my studying
you are the best if only i could have liked this video million of times
I am just starting to learn this subject. Thanks you for this introduction to the crystal systems. It was most helpful. How is it that an octahedron and a dodechedron crystal are classified as having a cubic structure?
Well, first of all a crystal with a _regular_ dodecahedral outer shape does not exist, because it would have 5-fold rotational symmetry - that's 'forbidden'. But you are right that the non-regular pyritohedron belongs to the cubic crystal system and the octahedron as well.
That is because they have cubic symmetry although they don't look really like a cube. The systematization or classification according to crystal systems is based on the presence of symmetry elements that you can find on a crystal. I would recommend to watch the other units as well, in particular chapter 3 - then this concept will become clear, I believe.
best
Frank
Hello! Thanks for the lessons, they are really easy to follow and to understand.
I'm looking for the accompanying quizzes but i can't find them. Can you tell me where can i find them?
Dear Even,
the quizzes were embedded in th eoroginal course which ran on th eiversity platform. They are not embedded here on TH-cam. However, the quizz for this unit comprised two further excercises:
(a) To which of the different crystal systems could a crystal with the following metric of the unit cell belong to? (multiple answers possible)
a = 10 Å, b = 10 Å, c = 10 Å
α = 90°, β = 90°, γ = 90°
(b) To which of the different crystal systems could a crystal with the following metric of the unit cell belong to? (multiple answers possible)
a = 10 Å,b = 10 Å,c = 12 Å
α = 90°, β = 90°, γ = 120°
best
Frank
Apparently these definitions vary a bit. For example, on page 59 of Structure of Materials, 2ed., by Graef, trigonals (aka rhombohedrals) are such that a = b = c.
The problem is that there is no rhombohedral crystal system. A lattice can be rhombohedral and a unit cell, too, but all crystals with a (primitive) rhombohedral unit cell belong to the _trigonal_ crystal system. And because there is a second kind of trigonal crystals for which the unit cell is _not_ rhombohedral one shouldn't use the definition by De Graef. For more details I would like to refer to unit 2.9:
th-cam.com/video/Ks1TNkG6tXY/w-d-xo.html
Sir, can you please answer me ?...
The question is
(Which type of crystal (triclinic or monoclinic) has three unequal axes and its Beta (B)angle less or greater than 90 degree
Is it true, that a crystal with certain set of symmetries can be generated only by unit cell of a crystal system, which itself exhibits this set of symmetries? I assume so according to this unit, but let us consider a gedankexperiment: we have a growing crystal belonging to the cubic crystal system, but due to external conditions (local temperature, pressure) the crystal will grow in six directions faster than in the rest. Hence the resulting crystal would have six fold rotational symmetry even though the cubic crystal system does not exhibit this symmetry. How should I view this?
Dear Sego,
interestingly, this process or this kind of crystal growth you imagine in your gedankenexperiment is impossible. There is always a relationship between the symmetry of the unit cell and the symmetry of the external shape (morphology) of a crystal, which is called correspondence principle. This means that a crystal with cubic symmetry also must exhibit a shape of cubic symmetry; indeed it must not be a cube, but can also be an octahedron or every other form which exhibits cubic symmetry. Of course, there are always some imperfections in the real world (defects etc.) which slighty violate the perfect symmetry, however, apart from these imperfections you cannot change the (internal) symmetry by applying whatsover crystal growth conditions.
best
Frank
thank you very much for you answer
my oh my, the table at 6:59 is what I had already memorized...I will have to take this up with my professor...Indeed they are the suppositions that I have met so far, however, are there some texts that also disapprove of this?
yes, in particular the book "crystal structure determination" by Werner Massa:
www.springer.com/de/book/9783540206446
best!
Dear Prof. Hoffmann! Could you please give the example of real structure triclinic system with a = b = c α = β =γ= 90°?
I don't know an example for the triclinic case. As I mentioned in the lecture, it is indeed not very likely that a crystal belonging to the triclinic crystal system shows that metric, but the point here is that it is mathematically not forbidden, it cannot be excluded. If you like, there is an interesting example for the monoclinic crystal system with an angle beta of 90°, see here:
crystalsymmetry.wordpress.com/2015/03/02/the-monoclinic-crystal-system-and-the-skew-angle-beta/
Thank you so much!
We are pleased that you find the explanations useful!
Crystals !!! Yeah !!! The entire earth (as a solid rock) underground, is a MASSIVE stockpile of minerals !!! Since all of us were one year old we were looking at minerals .... we just did not understand what we looking at until we took mineralogy !!!
Hallo Herr Hoffmann, ich höre gerade an meiner Uni die Vorlesung "Anorganische Strukturchemie II" im Chemie Master, die sich auch mit diesen Themen beschäftigt. Uns wurde auch die falsche Definition der 7 Kristallsysteme gelehrt. Auf meinen Hinweis, dass diese Definition falsch ist (wie in ihrem Lehrbuch geschrieben) bekam ich nur die Antwort der Dozentin, dass es dazu unterschiedliche Meinungen gibt und sie die IUPAC-Nomenklatur lehrt. Gibt es denn irgendwelche Beispielverbindungen, die z.B. im trikilinen Kristallsystem kristallisieren und die Gitterparameter a=b=c und alpha = beta = gamma = 90° haben? Von der IUPAC habe ich übrigens keine Definition der Kristallsysteme gefunden - gibt es dazu noch andere gute Quellen, mit denen ich meine Dozentin überzeugen könnte?
Moin!
Es gibt mindestens zwei beeindruckende Beispiele, die die Dozentin vielleicht noch einmal ins Grübeln bringen könnte, beide aus dem monoklinen Kristallsystem :-)
a) crystalsymmetry.wordpress.com/2015/03/02/the-monoclinic-crystal-system-and-the-skew-angle-beta/
b)
crystalsymmetry.wordpress.com/2015/03/04/follow-up-monoclinic-symmetry-with-almost-perfect-cubic-metric/
Was die Dozentin meint, wenn sie sagt, sie würde sich an die IUPAC halten, weiss ich auch nicht - denn bei der IUPAC findet man tatsächlich keine Definition der Kristallsysteme. Die International Union of Crystallography hat einen Eintrag, aber auch in dieser Definition kommen keine Gitterparameter vor:
dictionary.iucr.org/Crystal_system
Aber die Übersicht über die Kristallklassen enthält, so wie es sein soll, Symmetrieangaben, keine über Gitterparameter:
dictionary.iucr.org/Arithmetic_crystal_class
Und schliesslich findet man die korrekte Version dieser tabellarischen Kristallsystem-Übersicht auf Seite 15 der ITA (5th Edition)! (Auf Wunsch kann ich die senden.)
Ich fände es mutig, wenn die Dozentin sich gegen die Internationalen Tabellen stellen würde...
Beste Grüße!
Frank
sir in many text books the entries in the row of trigonal crystal has been given as (a = b= c) and interfacial angles are( α= β =γ ≠90 degree) but in your video it is (a = b≠ c) and interfacial angles are( α= β =90 γ 120 degree) plz clarify it which one is correct?? I am confused now
Dear Umendra, hopefully you will never find these restrictions for the trigonal crystal system! Apparently, you are talking the restrictions of the rhombohedron or rhombohedral lattices, but please note, there is no rhombohedral crystal system. For clarifying the relationships between these two, have a look at unit 2.9:
th-cam.com/video/Ks1TNkG6tXY/w-d-xo.html
best
Frank
I had a question in my mind. If we fix two angles = 90, as in case of monoclinic crystal system, then how can third angle be anything other than 90?
Dear Harshit,
oh, this is indeed possible. Perhaps, have a look at Unit 2.4
th-cam.com/video/s4rRMmToGBQ/w-d-xo.html
in whioch all the 7 primitive lattices in their general form are shown.
The marked angle (with an arc) for the monoclinic system can be varied without influencing the other two orthogonal angles. Simply push the two side faces in the one or other direction. This will not change the angles of the top/bottom faces nor the angle of the left and right face.
OK?
best
Frank
Yeah, got it. I thought that the orientation can't be changed in space.
Helpful
danke shun !!
Grüezi Herr Prof. Hoffmann
Ich und meine Freunde besuchen momentan eine Vorlesung in diesem Bereich und haben uns folgende Frage gestellt: Die unit cell muss definitionsgemäss die Symmetrieelemente des Kristalls besitzen, doch im Falle von einem Hexagon ist die Unit cell trotzdem ein Rhomboid, der ja offensichtlich die C6 Achse nicht in sich trägt. Wie ist dies mit der Definition vereinbar?
Unit 2.8 erklärt dieses Problem, danke!
hi! please help me! .... how do you determine the values of a, b and c of a monoclinic system?.... I have a pattern diffraction
Dear Brisa,
well, usually the diffractometer software automatically determines the cell parameters. Or do you have such an old diffractometer that uses still a photographic film as the detector unit?
In principle, all cell parameters are determined by the reciprocal values a*, b*, c* and alpha*, beta* and gamma*, not only for monoclinic crystal structures. You can find the relevant equations in any textbook on X-ray crystallography, for instance, here:
www.springer.com/de/book/9783540206446
However you also need some device parameters such as the distance from the sample to the detector etc.
best
Frank
Thank you for your wonderful lecture. I've really learned a lot of fundamental concepts from your video. I have tiny questions that are unclear to me. When we mention a unit cell, do we use the name of its crystal system? For example, as in your example, even if the unit cell parameters look like tetragonal (a = b = 12 Å, c= 12.1Å, α = β = γ = 90°), if its crystal system is triclinic, should we call it triclinic unit cell"? Or the unit cell is "tetragonal", but the crystal system is "triclinic"? / Another thing in my mind is that I can imagine a unit cell having a = b = c, α = β = 90°, and γ ≠ 90°. But I cannot find this metric, so I don't know how to call this unit cell. But now I understand that its crystal system is based on restrictions that you showed in the video. I would appreciate it if you would explain these points.
Hi!
To your first question/example: The classification according to crystal systems is based on the symmetry of the crystal. This means that also the smallest repeat unit of the crystal, the unit cell, has that symmetry. So a triclinic unit cell means it has triclinic symmetry. The lattice parameter are only the metric of the unit cell.
To your second question: Yes, it is possible that a unic cell has the metric you mentioned, a = b = c, α = β = 90°, and γ ≠ 90°. You don't find this restriction, because none of the possible 7 kind of the symmetries leads to such a restriction, but there is at least one restriction that covers your metric (triclinic). If you would redefine you axes in such a way that instead α = β = 90°, and γ ≠ 90° --> α = γ = 90°, and β ≠ 90° then also the monoclinic system is possible.
Best
Frank
Teşekkürler from Turkey
Rica ederim
I have a question. In 3D we choose parallelopiped as a unit cell. why we always choose this? and if we choose just parallelopiped as our unit cell. then why 7 crystal systems have different unit cells. they all must have one unit cell which is parallelopied. Please explain
The answer to your first question is, because then we are able to describe all unit cells of all crystals in a uniform way. The point that you may have missed is that the concrete shape of a parallelepiped is not fixed. You're making a false equalization between parallelepiped and unit cell. All unit cells are indeed parallelepipeds, regardless of their different metric that they may have (as a result of their different symmetry). Remember that parallelepiped only means that you have 6 faces of which each two are congruent and lie in parallel planes.
@@FrankHoffmann1000 Thanks a lot. I got it. :)
It is more or less parallelepiped but with different parameters. For example cubic unit cell etc can all be generalized to parallelepiped I guess
smarter than the textbook, i like that ;)
Extremely urgent question!
So a crystal system is actually same as a unit cell?
No, of course not. A unit cell is a unit cell (see unit 1.7), and the crystal systems are a way to categorize all unit cells into certain categories according to the symmetry of the different unit cells.
A shark is an animal, a rabbit is an animal, a duck is an animal. All are animals. But they belong to different _kinds_ of animals: fishes, mammalians, birds... according to their different features. Does this analogy help?
@@FrankHoffmann1000Thank u so much for the quick response!
i understand unit cells but i am so confused by this term "Crystal systems." Some books have this topic named as "7 different types of unit cells" but in some other books and videos they use the heading "7 different types of crystal systems."
I mean, what are crystal systems? Aren't the 7 "crystal systems" the 7 types of unit cells? If yes, then why the term "crystal systems." And if not , than what do people actually mean when they say "crystal systems?" Are we categorizing "crystals" or "unit cells" in "crystal systems?"
Different crystal systems differ in their unit cells is what i got from this video but what exactly is the idea of a crystal system is still unclear.
@@blzKrg Well, you shouldn't take it philosophical :-)
Yes, in a way the "7 crystal systems" are categories for the "different type of unit cells", but the question is, what is the criterion for this category? And I already answered: The criterion is the symmetry! But also not the _whole_ symmetry (this would be the space group).
When you say, your crystal belongs to, for instance, the orthorhombic crystal system, you say something about the _symmetry_ of the lattice or unit cell, but nothing, for instance, about the actual length of the unit cell vectors nor something about the chemical content of the unit cell, nor something about the crystal class. But if you know that your crystal has orthorhombic symmetry, you know at least that all angles of your unit cell must be 90°.
The symmetry specifications of crystals are made in an hierarchical way: First, the crystal system, then the crystal class, then the space group.
"Aren't the 7 "crystal systems" the 7 types of unit cells? If yes, then why the term "crystal systems.""
Well, imagine there are 7 different types of animals. Then, the (name of the) animal is of course not the term of the type! A monkey is a) a monkey, but also b) (now the type:) a mammalian. To say, why does the term mammalian exist, is not a good question, or?
And with "monkey" you haven't said what kind of monkey... A baboon, a chimpanzee? So a specific monkey is on the one hand more than a (general) monkey and on the other hand less: it belongs also to the class of mammalians.
@@FrankHoffmann1000 oh...i somewhat get it now.
Thanks professor.
I do not understand how a triclinic and a cube can have the same dimensions and angles but still be different systems.
Dear EDY el O,
if you think of one _particular_ crystal structure with its specific (and known) atom coordinates of that particular compound, then you are right: this compound belongs to _one_ of the seven crystal systems and not to two (or more) different crystal systems simultaneously.
The statement was rather that we cannot say to which crystal system a crystal compound belongs as long as we do not know its symmetry, because the classification according to the crystal systems is based upon the symmetry. It is not based on the metric, i.e. the cell lengths and angles.
So, the next step is to consider, in which way a given set of symmetry elements determines specific values of the metric: For instance, if a crystal structure has three mirror planes, each two of them being perpendicular to each other, then it follows immediately that all angles have to be (necessarily or mathematically) 90 degrees, because you cannot construct a structure that has three perpendicular mirror planes but has angles unequal to 90 degrees.
However, if we have only information about the metric then in many cases we cannot say _unambiguously_ to which crystal system the compound belongs. This would be only clear, if we obtained the full information on the structure and therefore all symmetry elements.
This means, conversely, that it is, for instance, not impossible to find a crystal compound that has angles of say three times 90 degrees, but has only a center of inversion as its sole symmetry element. Then, the compound belongs to the triclinic system, even despite all angles are 90 degree.
Do not know if this helps - if not, do not hesitate to ask further questions.
best
Frank
It helps a lot. Thank you very much.