## Lattice and Crystal - Simple View | |||||||||||||

What is a Lattice? |
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A lattice
is a mathematical thing. It is simply a periodic arrangement of mathematical points in space,
extending to infinity in all directions. Here is a picture of a part
of a lattice (It's hard to draw infinitely large pictures): | |||||||||||||

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If you don't see much you have to look harder. Mathematical points, after all, are infinitely small. | |||||||||||||

OK - you can't find your good glasses, I understand. So let me help you by showing
that picture of a periodic arrangement of points in space once more but with some of the points
now symbolized by little spheres: | |||||||||||||

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Not all that great either? So let me help you once more by introducing some lines and colors to guide the eye: | |||||||||||||

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Now we "see" it - except we see
all kinds of things that are not really "there" and we do not see things that actually
are there, like the "inside" points or the points out there up to to infinity. There is just no way to produce a "correct" picture
of a lattice or of a crystal that is of any use to mere humans. More about the problems with
drawings of lattices / crystal in this link. Of course,
if you are not a mere human but a mathematician, you don't need pictures. You actually hate
them (to the extent you are capable of having emotions) because they are so imperfect. A few
simple and beautiful equations are so much better. | |||||||||||||

What even we imperfect humans gather from the the
figure above is that all the essential information about the lattice shown is contained in
the three arrows or vectors in the left-hand corner. Three arbitrary
arrows with respect to lengths and mutual orientation can describe any lattice whatsoever
- just repeat them. However, if we want to go beyond this general case and try to differentiate
with respect to some special lattices, we find that there
are 13 special cases. Together with the general and least symmetric case, we have a grand
total of 14 Bravais lattices, as they are called. The
link gives the in-depth view of lattices and crystals. | |||||||||||||

Some special cases emerge if we define some
conditions for the three vectors defining a lattice. We might, for example, demand that they
all must have exactly the same length and should be at right angles to each
other. This is actually the definition of the sides of a cube and the "cubic primitive" Bravais lattice results. aFrom a purists point of view, you do not have to do this. It's like classifying humans, for example, into "friends, Romans, countrymen". It doesn't' change a thing with respect to the real humans being around, but it makes life easier. Here are the three most important Bravais lattices: |
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What is a Crystal? | ||||||||||

A crystal results when you put exactly the same arrangement of atoms, called a "base",
at every lattice point of some Bravais lattice. In the most simple case you put just one atom on a lattice point. | ||||||||||

If we do that with the three
lattices above, I don't have to redraw the figure. We can now take the blue spheres to symbolize atoms, and - Bingo! - we have a schematic drawing of some
crystals. Of course, in a correct drawing the circles symbolizing
atoms should touch each other. But if we draw it this way, the figures will become utterly
confusing as illustrated in this link. Note that putting atoms on any lattice point of the hexagonal Bravais lattice does not
produce an hexagonally close-packed crystal. We need two
(identical) atoms in the base as illustrated below: | ||||||||||

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The blue circles symbolize the lattice points of the
hexagonal Bravais lattice and some atom. The pink circles
symbolize only the atom and for hexagonally close-packed elements it is same kind as the blue ones;
Cobalt (Co), or Zinc (Zn), for example. There are two atoms
in the base. | ||||||||||

In the case
of the face-centered cubic Bravais lattice, putting one
atom on each lattice point does produce a close-packed crystal. | ||||||||||

However, nobody can keep me from putting two or three atoms in the base,
for example like this: | ||||||||||

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You figure out the the base. What we get this way are
many kinds of different crystals for the same lattice, depending on what groups of atoms or
bases we put on a lattice point. Note that the two crystals above, while made from a close-packed lattice, are not close-packed
crystals!Note also that I could have claimed, for example, that the blue atoms are iron, and the green or red atoms are carbon. On paper a lot is possible. Mother nature, however, does not give a damn about my or your claims; she just will not comply. Not everything you can draw on a piece of paper will be possible. Only combinations of atoms that "want" to crystallize in some specific way will be found as real crystals. While it is easy to analyze an existing crystal,
it is not so easy to predict how a bunch of atoms will crystallize. | ||||||||||

In the examples given so far, we had the base of atoms put on a lattice point in some special way. There are, in principle, other ways too. If we pick some alternative, we get yet another crystal for the same lattice. Here is a simple example: | ||||||||||

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Same lattice,
same base - but different crystals. Note once more: The fact that you can draw this, does not mean that you can make it. Typically, only one of the many possibilities will be realized with real atoms at a given temperature. | ||||||||||

Now you have at least a vague idea, why there are 230 different combinations (or
"point groups") of bases and Bravais lattices as claimed in the main text. | ||||||||||

In the Hyperscript are several modules related to this topic and many examples for lattices and crystals. Go and find them yourself! | ||||||||||

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"

Transmission Electron Microscopes

Producing "Nirvana" Silicon or Nearly Perfect Silicon Single Crystals

© H. Föll (Iron, Steel and Swords script)