## 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:- Crystal Models. You learn how you have to look at all these schematic figures of crystals and why you can't take them at face value.
- Lattice and Crystal - Simple View. The basics without math.
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Bravais Lattices |
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Any crystal lattice can be described by giving a set of three base vectors
, a_{1}, a_{2}. A lattice is formed by generating an infinity of a
_{3}translations
vectors
= uTa_{1} + va_{2} + wa_{3}u, v, w, = integers. The end points of all possible translations vectors define the lattice as
a periodic sequence of points in space. | ||||

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. | ||||

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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.
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Bravais Lattices and Their Parameters | |||||||||||||||||||||||||||||||||||||||||||||||

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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:- 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.
- Write the direction as [u v w] or <u v w> (we won't concern us here with the subtleties involved in using two kinds of brackets).
- Negative integer values are written with a dash on top of the number instead of the conventional
"-" sign. (not possible in simple HTML)
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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. If there is none, the value is ¥ .
- Form the reciprocal values of h', k', and l' and call them h, k, and l. If, for example, h' = ¥, you have h = (1/h') = 0.
- The Miller indices of the plane to be indexed then are {hkl} or (hkl).
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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.
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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 third-term
student. | |||||||||||

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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: | |||||||||||

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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)