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Why are there only 7 types of unit cells and 14 types of Bravais lattices?

I was reading about solid state chemistry for the first time and this limitation made no sense to me.

I tried to do the math and realized that there could be many more possibilities. Usually the standard unit cell is described on the basis of whether the sides are at 90 degrees to each other and whether the sides are equal or not. However, any angle other than 90 could be chosen as the reference angle. So why 90?

Is it because only these 14 systems are found in nature and there is no need to study others?

Moreover why are there only 4 sided shapes or hexagonal shapes? Why not pentagonal or trigonal?

Any reference from any site or textbook would be appreciated.

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  • $\begingroup$ Because three dimensional space is so constricted, maybe? If we could live in more dimensions there would be room for more choices. But, we can't. $\endgroup$ – Oscar Lanzi May 2 '18 at 1:20
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All quotes will be from Solid State Physics by Ashcroft and Mermin.

Bravais Lattice:

A fundamental concept in the description of any crystalline solid is that of the Bravais lattice, which specifies the periodic array in which the repeated units of the crystal are arranged. The units themselves may be single atoms, groups of atoms, molecules, ions, etc., but the Bravais lattice summarizes only the geometry of the underlying periodic structure, regardless of what the actual units may be."

Primitive Unit Cell:

A volume of space that, when translated through all the vectors in a Bravais lattice, just fills all of space without either overlapping itself or leaving voids is called a primitive cell or primitive unit cell of the lattice.

Unit Cell; Conventional Unit Cell:

One can fill space up with nonprimitive unit cells (known simply as unit cells or conventional unit cells). A unit cell is a region that just fills space without any overlapping when translated through some subset of the vectors of a Bravais lattice. The conventional unit cell is generally chosen to be bigger than the primitive cell and to have the required symmetry.

Crystal Structure:

A physical crystal can be described by giving its underlying Bravais lattice, together with a description of the arrangement of atoms, molecules, ions, etc. within a particular primitive cell.

So, one comes up with 14 Bravais lattices from symmetry considerations, divided into 7 crystal systems (cubic, tetragonal, orthorhombic,monoclinic, triclinic, trigonal, and hexagonal). This comes solely by enumerating the ways in which a periodic array of points can exist in 3 dimensions.

Now, what is on those points is a unit cell, which will itself have some symmetry. Thus, the combination of Bravais lattice and unit cell symmetry can again be enumerated and one comes up with 230 space groups.

Now for some of your related questions:

All cubic-related Bravais lattices will have 90 degree angles because they are based on cubic symmetry. The trigonal Bravais lattice has no 90 degree angles, but isn't talked about much in more basic textbooks because, well, it looks weird.

Why no pentagonal unit cells? Well, because you can't fill space with a 5-fold symmetric Bravais lattice. Quasicrystals, while they have 5-fold symmetry, are a tiling through space that does not obey the rules for a Bravais lattice.

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  • $\begingroup$ can you please explain the last line of the definition of Braviais lattices? Also another thing I asked ,"Is it because only these 14 systems are found in nature "? $\endgroup$ – Karan Singh Aug 20 '15 at 18:02
  • $\begingroup$ @KaranSingh - an example. Take a simple cubic Bravais lattice. The periodic structure is the set of points that make up the vertices of a bunch of cubes stacked up to fill space. If you look at it from any direction, that is what it looks like. Now, at each point place a sphere (unit cell). It really looks no different. OK, now place an ellipsoidal solid (aka American football) on each point, with the long axis of each aligned in the x-direction. You have the same Bravais lattice, but a different space group, because the unit on the lattice has reduced symmetry. $\endgroup$ – Jon Custer Aug 20 '15 at 18:12
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    $\begingroup$ I think the short answer is "these 14 systems are the only ones possible." You can prove using math that there cannot be another. $\endgroup$ – Geoff Hutchison Aug 20 '15 at 20:56
  • $\begingroup$ Indeed - group theory is a wonderful thing for chemists and physicists to learn. It certainly was a big part of Wigner's work. $\endgroup$ – Jon Custer Aug 20 '15 at 21:37
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    $\begingroup$ @KaranSingh All periodic structures have to obey the mathematical rules of group theory. It isn't about what exists in nature but about the logical constraints on symmetry. Having said that, nature and man have produced structures that don't play by those rules but these are strictly called aperiodic as they don't have strict long range repetition but are a little irregular. Some have close to 5-fold symmetry (exact 5-fold symmetry isn't possible in 3D). These are the 3D analog of 2D things like Penrose Tiles which break the rules of 2D tessellation. $\endgroup$ – matt_black Aug 21 '15 at 12:31
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If you want to follow the rule, that a crystall is formed by endless translational symmetry of a unit cell, then the only possibilites to start from are two structures: parallelepipeds and hexagonal prisms.

You can always, and only, except for hexagonal prisms, fill space by stacking parallelepipeds in all three directions. Rhombohedral, cubic, trigonal etc. are all special cases of the "triclinic" unit cell with higher symmetry, it is obvious that there are not endlessly more options that are more symmetric. Those make up for six of the seven crystal systems, and hexagonal is the special case making up the seventh.

The Bravais lattices come from unit cells which have an internal symmetry. You could go without these by describing them with one of the less symmetric crystal systems, but the rule is to assign the crystal system with highest symmetry. There are again not so many possibilities to have an internal symmetry, so this only makes 14 Bravais lattices out of the 7 crystal systems.

The paradigm is not to think of ways to make the system endlessly more complicated, but to start from the most odd system that is able to fill space, and think of the (limited) possibilities to make it more simple (i.e. symmetric).

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The crystal structure and symmetry depends upon lattice parameter a, b, c and angles $\alpha, \beta$ and $\gamma$. During repetition of these combination it repeats one of the 14 Bravais lattice. Hence, no further combination is possible. That is why we have only 7 crystal system and 14 Bravais lattices

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Bravais lattices have translational symmetry. As you mentioned all the possible translational symmetries are categorized into 14 types. Some of the lattices with extra symmetries (in addition to translational symmetry) are placed in certain types. For example, the cubic crystal system with three subcategories have the most degrees of symmetry. Any other lattice that doesn't have an extra symmetry is under Triclinic. Therefore, you can not have any other type. Because all the possible translational symmetries with extra point symmetries are categorized and the rest are in Triclinic type. Noting is left outside.

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There are point groups I.e. combination of symmetry elements. There could be many combination of symmetry elements possible. But only those are permitted which somehow incorporate well with translation symmetry. In three dimensional space, based on above restrictions,only 32 point groups are permissible. There 32 permissible point groups require 14 bravais lattices which are groups into seven systems.

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