Science of Lattices and Crystals 

Before I get started: These two
modules give a first idea about lattices and crystals without all the formal stuff coming up here:


Bravais Lattices  
Any crystal lattice can be described
by giving a set of three base
vectors a_{1},
a_{2}, a_{3}. A
lattice is formed by generating an infinity of
translations vectors


If you have some lattice and move it by any translation vector you care to construct, you have exactly the same lattice once more. In other words: crystal lattices show a translation symmetry! For a long time, the words "crystal" and translation symmetry were seen as obvious synonyms  until the discovery of quasicrystals in 1982!  
Unfortunately, one and the same lattice can be defined by many different sets of vector triples as illustrated right below.  


It is far easier to use some
special lattices instead of just one
general type. The thing to do is to go for symmetries as the distinguishing
criterion. That's what Bravais did, showing that with 14
Bravais lattices all possible cases can
be represented. All material scientists know that "magic" number
14 but very few know how it is derived. I don't know details either but
I know it is an exercise in set theory. Note that a lattice is a mathematical construct, a succession of (infinitely small) mathematical points in space. A perfect drawing of such a lattice thus would show nothing at all. Instead of points, I use little blue spheres here. They are connected with lines but only to "guide the eye". These blue spheres are not representing atoms when a lattice is shown. More about figures to lattices and crystals in this link. 

If one wants to make a
crystal, one assigns a
socalled base of atoms to a lattice point. If that base happens
to consist of just one atom or element, we
make an element crystal. A schematic figure of such a crystal with one atom per
lattice point then looks exactly like the schematic representation of a lattice, causing no
end of confusion. It is nevertheless something completely different; see below. Note that mother nature has not made a cubic primitive element crystal. 

Bravais Lattices and Their Parameters  


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 facecentered 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 1839. 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:




Getting Miller indices for planes is
a bit more involved. Here is how it's done; the figure below gives examples:




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:




The example above shows how to make
a crystal of the diamond type. The base consists of two atoms. In the
coordinate system of the lattice unit cell (indicated by arrows), the two atoms
have the coordinates (0,0,0) and (¼,¼,¼). If the two atoms are of the same kind, e.g. silicon, (Si), germanium (Ge), or carbon (C), you get a silicon, germanium or diamond crystal. If the atoms are different, e.g. from group III or group V of the periodic table, you get most of the compound semiconductors like gallium arsenide (GaAs), or indium phosphide (InP). 

This looks simple. It is not. It's
the point where things get difficult and confusing. Ask yourself for any still
simple crystal: how many atoms are there to a lattice plane? How many atoms are
in a base? Below are three crystals, all have an fcc lattice. Different colors of the circles my or may not denote different atoms. Can you figure out the bases? If you can, you're ahead of my average thirdterm student. 



One last thought: Crystals in a general sense, meaning an arbitrary base arranged in a periodic way, can be found everywhere; here is an example:  


Periodic Table of the Elements
11.2.2 Metallurgy of Celtic Swords
Group 2 / IIA; Alkaline Earth Metals Group
Group 12 / IIB; Scandium Group
Group 12 / IIB; Titanium Group
Group 7 / VIIB; Manganese Group
Group 8  10 / VIIIB; Iron  Platinum Group
Group 16 / VIA; Chalkogenides or Oxygen Group
Group 3 / IIIB; Lanthanides or "Rare Earths"
Dislocation Science  1. The Basics
Phenomenological Modelling of Diffusion
Dislocation Science  2. The Reality
© H. Föll (Iron, Steel and Swords script)