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"A crystal is formed by a large number of repetitions of basic pattern of particles in space.

The basic structural unit which when repeated in three spatial directions generates the crystal structure is called unit cell."

Problem:

Assume I take some same perfect geometrical cubes and arrange them in all three dimensional directions, a larger box similar to the smaller ones will be introduced.

From this assumption and the information above it, I deduce that a unit cell should be similar to the relevant crystal. That means, the only difference would be lying between their size.

If this is the case, why consider the geometry of unit cell and not directly the respective crystal? We know in the assumption above; the fundamental cubical boxes would be determined by same geometrical information as would the derived cubical box.

That is; if the lengths of three axes of cube are represented by letters $a,b$ and $c$ and the angles between axes are represented by Greek letters $\alpha$, $\beta$ and $\gamma$ then for a cube,

$$a=b=c$$$$\alpha=\beta=\gamma=\pi^c$$

This information is general. It is for all size of cubes. Then, why to define a unit cell and think it simpler to handle with than the crystal?

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Man_Of_Wisdom
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    The unit cell doesn't really define the crystal shape. The 3D crystal shape depends on a multitude of factors. I think 2500 odd shapes have been detected for the 3D crystals of the mineral calcite. http://www.mnh.si.edu/earth/text/2_2_1_4.html – MaxW Dec 22 '15 at 18:10
  • But the unit cell for all those crystals is same. Right @MaxW? – Man_Of_Wisdom Dec 22 '15 at 18:16
  • There three calcium carbonate minerals. Calcite, aragonite and vaterite. – MaxW Dec 22 '15 at 18:36
  • Forget that (though that's true, too). Yes, unit cell is the same, so what? – Ivan Neretin Dec 22 '15 at 18:49
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    The Bravais lattice and the unit cell combine to make a space group. See here: http://chemistry.stackexchange.com/questions/28507/are-all-lattices-bravais-lattices/28510#28510 for more details. – Jon Custer Dec 22 '15 at 19:18
  • @IvanNeretin, the information I provided at the very beginning of my question is from my textbook. It is suggesting that unit cell is similar to the the relevant crystal. What about that? Do you have any explanation for it? Please tell me. – Man_Of_Wisdom Dec 22 '15 at 19:27
  • @Volunteers, the basic structural pattern which is being repeated throughout the crystal is the unit cell. This suggests that unit cell is similar to that crystal. – Man_Of_Wisdom Dec 22 '15 at 19:30
  • Well if I agree upon all the calcite crystals having same unit cells then it must be some other thing that is a unit cell. Because, in that case, all calcite crystals would become same (after combining their unit cells). – Man_Of_Wisdom Dec 22 '15 at 19:36
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    A crystal is not similar to its unit cell, nor does the textbook say so. See, you are made out of a large number of atoms; do you look like an atom? Certainly not! – Ivan Neretin Dec 22 '15 at 19:58
  • I think you are ignoring the CONTENT of the unit cells – Gerhard Dec 22 '15 at 20:06
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    The symmetry limitations (rotations, reflections and mirror images) of the unit cell also limit the crystals. So calcite won't ever form a perfect cube like pyrite can. But as you can see from the number of different calcite crystals the symmetry limitations don't prevent a great deal of variety in the 3D crystal shape. – MaxW Dec 22 '15 at 21:05
  • @IvanNeretin Draw a small square. Draw 3 more same boxes. Put them in all 2 dimensional directions over your paper. A larger box similar to the smaller box would appear. This example is still criticizing me. As far as a human composition is concerned, so there is no basic structural unit or a unit cell of human shape. You are considering an atom as a unit cell. However, unit cell is the least volume consuming repeating structure of particles in a crystal. An atom can not describe the repeating pattern of a human composition unless it is a sphere. – Man_Of_Wisdom Dec 23 '15 at 07:01
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    @Man_Of_Wisdom OK, a crystal may grow like that. But it may do otherwise as well. Draw a small square. Draw 4 more similar squares, and attach them to all sides of the initial square. What you see is not similar to the smaller box. Continue by adding more repeating unit cells to all sides of the new figure. This is still a crystal with the same unit cell, but it does not look quite like yours. – Ivan Neretin Dec 23 '15 at 07:02
  • According to my textbook, placing them in all three spatial directions would introduce the crystal structure. For a moment if I ignore it and do it as you just told me, how I am gonna know that in what way I should place them so as to get the crystal structure? If I do it my way, I am going to have similar larger unit cell (crystal structure) and if I do it yours way, I am going to have an unspecified geometrical shape maybe. – Man_Of_Wisdom Dec 23 '15 at 07:08
  • Think of unit cells as Lego blocks. As long as you attach new unit cells to similar faces of existing unit cells, you are doing it right. But the overall shape of the crystal can be just about anything. – Ivan Neretin Dec 23 '15 at 07:14
  • Along; I can see in my book that the crystals are drawn just beside their respective unit cells and they look exactly similar – Man_Of_Wisdom Dec 23 '15 at 07:15
  • Then your book does a poor job at explaining the crystal shape. Sure, sometimes a crystal may look like its unit cell. But more often than not, it will be more complicated. – Ivan Neretin Dec 23 '15 at 07:18
  • Then why does the crystals are specified by their unit cells? Unit cell fall into seven well known categories (Isometric-Triclinic). Following your example of 5 squares, the combinations of same number of unit cell may introduce several crystal structures/crystals. Unit cell are of 7 kinds and one kind may further introduce several crystal structure. Then, actually why define a unit cell? – Man_Of_Wisdom Dec 23 '15 at 07:24
  • Let us continue this discussion in chat. – Ivan Neretin Dec 23 '15 at 07:26

1 Answers1

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The statement from your book is correct, though the devil lies in the details. The point here is that the book is treating a theoretical crystal (i.e. one that is infinitely large). Let me elaborate and then answer your question.

A unit cell is simply a representation of a periodic structure. It is a means of conveying translational symmetry (and perhaps other types of symmetry as well, but let us focus on translational symmetry). If you have for instance a cubic unit cell, than a 3x3x3 grid of these unit cells is also a unit cell of that same lattice. If you extrapolate this idea, then the reasoning you pose in your question makes a lot of sense!

Let us now discuss the details. Assume that we have a theoretical crystal of infinite dimensions. What is then the unit cell? Can we define a unit cell which is also the infinite crystal? Every finite unit cell would obviously not be that crystal. This explains why we even use a unit cell in the first place: By defining the translational symmetry by means of the unit cell, we can define an infinite crystal.

In the field of solid state physics, it is attempted to keep things simple and well-defined and therefore a so-called primitive unit cell is introduced. This unit cell only contains one lattice point and you could see this as the smallest unit cell possible that is still able to convey the translational symmetry of the infinite crystal.

Your last question relates to why we want to use, for instance, the primitive unit cell or some other unit cell rather than the complete crystal? Let me answer this question by giving an example: If you cut a crystal, you will create a surface facet. Such surfaces also have translational symmetry and you can define these using Miller indices. Your textbook will probably treat these as well. Those Miller indices are defined by the reciprocal lattice vectors of your unit cell. You already touched upon on the fact that there exists an infinite set of unit cells that define that infinite crystal, yet the question arises which unit cell to use if we, for instance, want to use these Miller indices. In principle, any unit cell in the set would be valid. To avoid any kind of ambiguity, we tend to use the smallest unit cell possible.

A nice book treating these concepts is "Introduction to Solid State Physics" from Charles Kittel.

Ivo Filot
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