The 14 Bravais Lattices 3. Cubic 3. Tetragonal 3. Orthorhombic 3. Hexagonal 3. Monoclinic 3. Trigonal and Triclinic. The Role of Symmetry During this course we will focus on discussing crystals with a discrete translational symmetry, i.
Point Symmetry Besides the translational symmetry mentioned above we will now also make use of point symmetries , i. The Seven Crystal Systems First notice: The intention of the following listing is to give you an overview rather than making you feel required to learn them by heart!
The 14 Bravais Lattices So one classifies different lattices according to the shape of the parallelepiped spanned by its primitive translation vectors.
This idea leads to the 14 Bravais Lattices which are depicted below ordered by the crystal systems : Cubic There are three Bravais lattices with a cubic symmetry.
One is primitive and the other body centered. But I encourage you to think of a few examples of more complex arrangements, and consider which basic lattice you have. For example, you may notice that there is a centered-rectangular lattice, but not a centered-square lattice. Similarly, adding a centered atom to this lattice would give you back a rectangular lattice.
Because rectangles have higher symmetry I prefer to describe them as rectangular and centered-rectangular lattices. I bring up the 2D lattices because I want to give you an opportunity to play with them in your head.
The square 2D Bravais lattice completely tiles a space with squares. The vectors a and b are equal to each other and are at right angles. The hexagonal 2D Bravais lattice might also be described as rhombic. You may like to think of them as triangular, although it actually requires 2 triangles one up, the other down to maintain translational symmetry.
The vectors a and b are equal to each other and at an angle of The rectangular 2D Bravais lattice tiles a space with rectangles. The vectors a and b are a at right angles but have different magnitudes. Notice that another way to imagine this lattice is if you had a and b with the same length, but not at right angles, with another lattice point in the center.
I mention this just to help you think of the relationships between lattices, and how easily two seemingly-different lattices might end up being identical. The centered-rectangular 2D Bravais lattice tiles a space with rectangles that have an extra lattice point in the center.
The vectors a and b are at right angles and have different lengths; one extra point is at the center. From one perspective, this lattice is simpler than the original one I showed you, because the unit cell has less area and lattice points inside. Since this one is the simplest possible lattice, we call it the primitive lattice. However, this depiction of the primitive lattice does not show the full symmetry, so we tend to use the non-primitive, centered-rectangular depiction instead.
The rhomboidal 2D Bravais lattice tiles a space with rhomboids. Remember that Bravais lattices are the result of simplifying a basis to a single point. If the basis is multiple atoms or something more complex, like a pattern in a painting this may get trickier. You might think this is hexagonal, but the lattice seems to be shaped like a squished hexagon.
Additionally, the hexagonal lattice has an extra atom in the center of each hexagon. If you consider each yellow dot to be a lattice point, then this is not a Bravais Lattice. The bravais lattices are the simplest lattices. Instead, you could collapse multiple yellow dots into a single lattice point. With purple dots, you can see that the lattice has restored the point in the middle of the hexagon. Crystals and lattices can appear in many forms.
Any time you have some region which can be perfectly translated onto a copy of itself, you have a crystal. The region is the basis, and the possibilities for translation are the Bravais lattice. Take a look at this tessellation I made, inspired by one of M.
Can you identify a lattice and basis? You can see that this area perfectly repeats if you translate it up, down, left, or right. I drew the rhombus with a black star at each corner, but the same translational symmetry shows up no matter where I place the start point of the rhombus.
Remember that the hexagonal and rhomboidal lattices are the same. Finally, I drew a rectangular centered basis. Now we get into the Bravais lattices which are useful to materials science. There are 14 3D Bravais lattices. Remember that the Bravais lattices come by considering translational symmetry. Other symmetries, like reflection or inversion, are captured in point groups and space groups, not Bravais lattices.
For a more in-depth look at each kind of lattice, I have written a dedicated article for each one. You can find those linked at the top or bottom of this article.
Each lattice is a polyhedron with 6 faces, 12 edges, and 8 vertices. We can describe these polyhedrons using 3 vectors which correspond to 3 of the 12 edges because of the 4-sided nature of the polygon, there will be 3 sets of 4 matching edges. The cube is the highest-symmetry lattice shape. The lattices can have an extra lattice point on all the faces F , the top and bottom bases C , or the center I. By combining the 7 possible polyhedrons with 4 possible centerings and crossing off duplicates, you end up with 14 Bravais lattices.
As a crystal structure, simple cubic has a very low packing density, so it is rare. Polonium is one of the only materials that has a simple cubic crystal structure. There is an additional lattice point at the face of each side.
Nowadays, FCC is the more popular term for both the crystal structure and Bravais lattice. As a crystal structure, FCC has a very high packing density, so it is common. Nickel and copper are just two elements that have this crystal structure. There is an additional lattice point at the center of the cube. As a crystal structure, BCC has a high packing density, so it is common. Iron and tungsten are just two elements that have this crystal structure. If you come from a materials science background, you may be used to seeing the Hexagonal Close-Packed HCP crystal structure.
HCP is a crystal structure with a more complicated basis—the underlying Bravais lattice is Simple Hexagonal. The Hexagonal Bravais lattice is a hexagonal prism. The prism can also be constructed from a primitive cell which is a parallelepiped. As a crystal structure, simple hexagonal has a low packing density, so it is not very common.
Hexagonal hex. Describing Directions and Planes by Miller Indices. Working with lattices and crystals produces rather quickly the need to describe certain directions and planes in a simple and unambigous way. Stating that an elemental face-centered cubic crystal can be made by assigning one atom to any lattice point found on "that plane that runs somehow diagonally through the unit cell" just won't do it. So William Hallowes Miller invented a system with a lot of power for doing that in What we do is to describe any direction or any plane by three integer numbers , called Miller indices.
How to derive the Miller indices of a certain direction or plane is easy. Here is the recipe for directions in 2 dimensions for simplicity ; the figure below illustrates it: Start the desired direction from the origin. Express the direction as a vector given in integer multiples u, v, w of the base vectors. Make sure the three integers have the smallest possible value.
Negative integer values are written with a dash on top of the number instead of the conventional "-" sign. Getting Miller indices for planes is a bit more involved. Here is how it's done; the figure below gives examples: Put the origin not on the plane but on a neighboring plane. Find the intersection points h', k', and l' of the plane with the extended base vectors. Form the reciprocal values of h', k', and l' and call them h, k, and l. Triclinic lattice Intersections at 1, 1, 1 Indizes Miller Indices for Planes.
If you wonder why this slightly awkward procedure was adopted, the answer is easy: You can use the Miller indices directly in a lot of equations needed for calculating properties of crystals. From Lattice to Crystal. Any crystal can be made following this easy recipe: Pick a Bravais lattice Pick a base, a collection of atoms in a fixed spatial relation similar and often but not always identical to a molecule of the substance.
Put the base in the same way on any lattice point. The example above shows how to make a crystal of the diamond type. The base consists of two atoms. If the two atoms are of the same kind, e. If the atoms are different , e. This looks simple. It is not. It's the point where things get difficult and confusing.
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