# Aren't unit cell and the relevant crystal similar?

"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?

• 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. mnh.si.edu/earth/text/2_2_1_4.html – MaxW Dec 22 '15 at 18:10
• The Bravais lattice and the unit cell combine to make a space group. See here: chemistry.stackexchange.com/questions/28507/… for more details. – Jon Custer Dec 22 '15 at 19:18
• 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
• 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
• @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 7:02

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.