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:
|Any crystal lattice can be described by giving a set of three base vectors
a1, a2, a
3. A lattice is formed by generating an infinity of translations
|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 quasi-crystals 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 so-called 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 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 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
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 third-term 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
4.2.2 Being Iron
History of Carbon
11.2.2 Metallurgy of Celtic Swords
Group 1 / IA; Alkali Group
Group 2 / IIA; Alkaline Earth Metals Group
Group 12 / IIB; Scandium Group
Group 12 / IIB; Titanium Group
Group 5 / VB; Vandium Group
Group VIB; Chromium Group
Group 7 / VIIB; Manganese Group
Group 8 - 10 / VIIIB; Iron - Platinum Group
Group 11 / IB; Copper Group
Group 12 / IIB; Zinc Group
Group 13 / IIIA;
Group 14 / IVA; Carbon Group
Group 15 / VA; Nitrogen Group
Group 16 / VIA; Chalkogenides or Oxygen Group
Group 18 / VII; Noble Gases
Group 1/ I; Hydrogen
Group 3 / IIIB; Lanthanides or "Rare Earths"
Group 17 / VIIA; Halogens
Alloying Elements in Detail
Dislocation Science - 1. The Basics
Lattice and Crystal
Phenomenological Modelling of Diffusion
Beer and Conquering The World
Science of Deformation
Dislocation Science - 2. The Reality
Pictures of Grain Boundaries
Phase Boundary - Advanced
© H. Föll (Iron, Steel and Swords script)