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1994-09-23
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Some basic information on geometrical crystallography
This is a short overview over some crystallographic definitions and facts
that might be helpful when using XTAL4POV. It is by no means a complete
introduction to geometrical crystallography, but gives only a few hints.
If you are interested in getting deeper into this, I highly recommend
reading a good textbook on elementary crystallography.
Well, here we go. First, a crystal is a solid with the unique property of
being built up of a lattice of so-called unit cells, which is strictly
periodic in all three dimensions of space. This 3-dimensional periodicity
imposes severe restrictions on the possible symmetries and shapes. I want to
emphasize here, that the description of crystal symmetry is not a matter of
empiric observations, though most of it was found out that way, but is one
of the subjects of Group Theory, which is *REALLY* hardcore mathematics. Only
very few crystallographers, if any, are able to deal with the complete Group
Theory, most of us are happy, if we understand the part we have to use. So
you can be sure that the numbers you will read in the following are correct,
and if I tell you there are exactly 32 crystal classes or point groups, as the
mathematicians call them, you will never have the chance to find a 33rd,
because this number is proven theoretically.
When we want to decribe something in 3D-space, we use a coordinate system,
and for convenience we use a so-called cartesian coordinate system, where
three axes are perpendicular to each other and have the same scale. Un-
fortunately, crystal symmetry forces us to use other coordinates, too, and
in fact, there are 7 different coordinate systems, the Crystal Systems, to
be used in crystallography. These are:
1. The Cubic (or Isometric) System
Three orthogonal axes with identical scales, just as the well-known
cartesian system.
2. The Hexagonal System
Two axes with the same scale and with an angle of 120° between them, and
perpendicular to both of them a third one, the main axis, which has a
different scale. This axis is conventionally called c-axis.
3. The Trigonal System
Here we have a little problem. We can describe a trigonal crystal with
a hexagonal coordinate system as above, so an extra system would not be
necessary, but we can as well use the so-called rhombohedral coordinate
system with three axes of identical scale and identical angles between
them, which are not orthogonal to each other. This is quite complicated,
and therefore I don't use this system here, but describe trigonal crys-
tals in terms of the hexagonal system.
4. The Tetragonal System
Three orthogonal axes, but one axis, the main axis or c-axis, has a
different scale than the two others.
5. The Orthorhombic System
Three orthogonal axes with a different scale for each.
6. The Monoclinic System
Three axes with different scales, and one of the angles is >= 90°.
Usually this is the angle between the a-axis and the c-axis, so this
angle is called beta, and the b-axis is orthogonal to the others and
is therefore called unique.
7. The Triclinic (or Anorthic) System
Three axes with different scales and non-orthogonal angles between them.
OK, now we know how to describe the overall symmetry of a crystal and can go
into the details. The symmetry of a crystal is described by a set of symmetry
elements, namely rotation axes, roto-inversion axes and/or mirror planes in
certain directions. These directions are the only directions where symmetry
elements can occur, and they are different for the different crystal systems.
So, what symmetry elements do we have?
Rotation Axes:
1. The Onefold Axis - Symbol 1
Well, this is not really a useful symmetry element, for it means that if
we turn a crystal around an angle of 360°, we will not be able to discern
this new position from the original one. Of course not, and therefore
this symmetry element is usually never mentioned, except in the triclinic
system, where it can be the only symmetry element present.
2. The Twofold Axis - Symbol 2
If we turn a crystal 180° around this axis, we cannot see any difference
to the original position, and two such rotations brings us back to the
origin, hence the name.
3. The Threefold Axis - Symbol 3
Here we can rotate 3 times with an angle of 120° without seeing any
difference between the positions.
4. The Fourfold Axis - Symbol 4
4 rotations with an angle of 90°.
5. The Sixfold Axis - Symbol 6
Finally we have an axis, around which we can rotate 6 times with steps of
60°.
A word to the usage of these symbols: We always choose the highest possible
symmetry, so for example, a sixfold axis contains of course twofold and
threefold axes, but we don't mention them, because we know they must be there
when we see a sixfold axis.
Roto-Inversion Axes:
1. The Center of Symmetry or Center of Inversion - Symbol -1
Well, actually the symbol is 1-bar, a 1 with a bar over it, but how do
this in plain ASCII? The same applies to the other symbols. But what
does this symbol mean? Not so easy to explain without a picture. Imagine
a cartesian coordinate system with a center of inversion in the origin.
Then a point with the coordinates (x,y,z) will be projected onto another
point with the coordinates (-x,-y,-z). You see what I mean? Well, at
least try it. Usually, a -1 is never mentioned in the crystal class sym-
bol, it is present or absent depending on the other symmetry elements.
The only exception is again the triclinic system, where it can be the
only symmetry element.
2. The Mirror Plane - Symbol m
I know you expected the symbol -2 in this place, didn't you? You've got
it, for -2 and m are identical, and we prefer m for some good reasons,
the explanation of which will go far beyond the scope of this text. A
mirror plane just acts as a mirror. No, it's not the same as a twofold
axis: Put your hands on the table, side by side, and put a vertical
mirror between them. Then you will realize, that the mirror image of one
hand has the same orientation as the other hand. There is no rotation
that can achieve this.
3. The Threefold Inversion Axis - Symbol -3
How shall I describe it without a picture? Well, rotate the crystal
around 120° and then invert it through its center! Easy enough, don't
you think so? It's indeed really hard to explain, and perhaps it will
help when I tell you that for example an octahedron has such -3 -axes
right through the centres of its triangular faces. Take some cardboard
and build an octahedron, I know this will help.
4. The Fourfold Inversion Axis - Symbol -4
Same as above, but rotation angle 90°. Example: a tetrahedron has -4
axes right through the middle of opposite edges. Build one and try!
5. The Sixfold Inversion Axis - Symbol -6
Rotation of 60° and inversion. No example readily available. Use your
brain and try to imagine how it works.
Hard stuff? Not really, the problem is that I can't show pictures and give
you crystal models. But anyway, if you really have problems with this, and I
know there are a lot of people who have, don't mind, because the program
handles this for you. Just stick to the instructions and play around, and
you'll be able to render perfect crystals.
So far we have the crystal systems and the various symmetry elements, and now
we have to put them together to get the actual crystal classes. Remember: The
basis of this is mathematics, and therefore you can be sure, that the above-
mentioned 7 systems and 10 symmetry elements are definitely all we have. Due
to the restrictions caused by threedimensional periodicity, we can't either
have an eighth crystal system or an eleventh symmetry element, e.g. a five-
fold axis. It's strictly prohibited!!!
So now we know what tools we have. But how to use them? Well, as I
mentioned above, there exists a specific set of possible symmetry-bearing
directions for each crystal system, which will be described in the following.
But first keep in mind that all symmetry elements intersect in exactly one
point in the center of the crystal, that's why the mathematicians speak of
point groups.
1. Triclinic system
Here we have no specific directions, the center of the crystal either is a
center of symmetry, or it is not.
2. Monoclinic system
Here we have one symmetry-bearing direction, and this is the abovementioned
unique axis, usually called the b-axis or direction [010]. In this direction
we can have a 2 axis, or a m plane perpendicular to it, or both together.
Please note that we describe an axis by it's direction, and a mirror plane by
the direction of it's normal. Furthermore, if there is a mirror plane perpen-
dicular to an axis, we use a / to indicate this in the symbol, e.g. a mirror
plane perpendicular to a 2-fold axis would give the symbol 2/m (pronounced
two over m).
3. Orthorhombic system
Here we have three possible directions of symmetry identical with the axes of
the system. In all three directions we can have 2-fold axes and/or mirror
planes, so there is always a symbol consisting of three parts, indicating
the symmetry elements in x, y and z, respectively.
4. Trigonal, tetragonal and hexagonal system
Now things get a little more complicated. As we know, in the tetragonal
system, for example, the symmetry in x and y must be the same, so if we know
the symmetry in one of them, we aren't interested in the symmetry of the
other, because we know it already. So we look at a direction just in the
middle between the two axes. The same applies to the other two systems.
For the all three systems, the first part of the symbol describes the symmetry
in the z-direction, or [001], which can be in the tetragonal system 4, -4
and, if present, m, in the hexagonal system it can be 6, -6 and m, and finally
in the trigonal system 3, -3, and m. The second part of the point group
symbol describes the symmetry in direction [100], the a-axis, which can be
2 and/or m. Finally, in the third part of the symbol, we find the symmetry
of [110] for the tetragonal system, or [120] for the hexagonal system. The
trigonal system doesn't have a third part. The third part of the symbol can
be 2 and/or m.
5. Cubic system
Well, here things seem to go completely mad. The first direction is [100],
the direction of the axes (yes, of all three axes, because they have the same
symmetry). Here we can have 4, -4, 2, and m. The second direction is [111],
which is the direction of the space diagonal between two opposite corners,
where we can find 3 or -3, but never a mirror plane. The third direction,
finally, goes parallel to the diagonals of the faces (symbol [110]), and can
have 2, m, or nothing at all.
Now let's have a look at the symbols we can construct with these informa-
tions. In crystallography, we use two kinds of symbols, namely the full
symbols containing the complete information, and the short symbols,
containing only the necessary parts of information. One very important
information is whether a crystal has a center of symmetry or not, so it is
included here.
Crystal Class Symbols (Hermann-Mauguin-Symbols)
System Full Short Center
Triclinic 1 1 no
-1 -1 yes
Monoclinic 2 2 no
m m no
2/m 2/m yes
Orthorhombic 222 222 no
mm2 mm2 no
2/m 2/m 2/m mmm yes
Tetragonal -4 -4 no
4 4 no
422 422 no
-42m -42m no
4/m 4/m yes
4mm 4mm no
4/m 2/m 2/m 4/mmm yes
Trigonal 3 3 no
32 32 no
-3 -3 yes
3m 3m no
-3 2/m -3m yes
Hexagonal -6 -6 no
-6m2 -6m2 no
6 6 no
622 622 no
6/m 6/m yes
6mm 6mm no
6/m 2/m 2/m 6/mmm yes
Cubic 23 23 no
432 432 no
2/m -3 m-3 yes
-43m -43m no
4/m -3 2/m m-3m yes
Finally, here's a list of all possible crystal shapes. There are two
different kinds of shapes, namely closed and open ones. A closed shape is
one that is on all sides surrounded by faces, and thus is finite, while an
open shape is infinite in at least one direction, and thus cannot occur as
the only form of a crystal, but is always combined with other forms to make
a closed convex polyhedron. The occurence of h,k,l in the Miller-indices
means, they can be replaced by any number as long as the resulting symbol
is different from other symbols in the list.
One word about those funny curly braces around {hkl}: In crystallography, we
have to deal with single directions, denoted by square brackets [uvw], with
complete sets of symmetry equivalent directions, denoted by angle brackets
<uvw>, with single planes, enclosed in parentheses (hkl), and with complete
sets of symmetry equivalent planes, which are marked by curly braces {hkl}.
Triclinic System
Possible Shapes:
1 Pinacoids of different orientations (open)
2 Pedions of different orientations (open)
Occurrence of shapes in crystal classes:
Crystal Class Shape No.
{001} {100} {010} {hk0} {h0l} {0kl} {hkl}
-1 Pinacoid of appropriate orientation
1 Pedion of appropriate orientation
_______________________________________________________________
Monoclinic System
Possible Shapes:
No. Name Indices Rem.
1 mcl. Pinacoid { 0 0 1 } open
2 mcl. Pinacoid { 1 0 0 } open
3 mcl. Pinacoid { 0 1 0 } open
4 mcl. Prism { h k 0 } open
5 mcl. Pinacoid { h 0 l } open
6 mcl. Prism { 0 k l } open
7 mcl. Hemipyramid { h k l } open
8 mcl. Pedion { 0 0 1 } open
9 mcl. Pedion { 1 0 0 } open
10 mcl. Doma { h k 0 } open
11 mcl. Pedion { h 0 l } open
12 mcl. Doma { 0 k l } open
13 mcl. Pyramid-Doma { h k l } open
14 mcl. Pedion { 0 1 0 } open
15 mcl. Sphenoid { h k 0 } open
16 mcl. Sphenoid { 0 k l } open
17 mcl. Pyramid-Sphenoid { h k l } open
Occurrence of shapes in crystal classes:
Crystal Class Shape No.
{001} {100} {010} {hk0} {h0l} {0kl} {hkl}
2/m 1 2 3 4 5 6 7
m 8 9 3 10 11 12 13
2 1 2 14 15 5 16 17
---------------------------------------------------------------
Orthorhombic System
Possible Shapes:
No. Name Indices Rem.
1 o'rhomb. Pinacoid { 0 0 1 } open
2 o'rhomb. Pinacoid { 1 0 0 } open
3 o'rhomb. Pinacoid { 0 1 0 } open
4 o'rhomb. Prism { h k 0 } open
5 o'rhomb. Prism { h 0 l } open
6 o'rhomb. Prism { 0 k l } open
7 o'rhomb. Bipyramid { h k l } closed
8 o'rhomb. Pedion { 0 0 1 } open
9 o'rhomb. Doma { h 0 l } open
10 o'rhomb. Doma { 0 k l } open
11 o'rhomb. Pyramid { h k l } open
12 o'rhomb. Bisphenoid { h k l } closed
Occurrence of shapes in crystal classes:
Crystal Class Shape No.
{001} {100} {010} {hk0} {h0l} {0kl} {hkl}
2/m 2/m 2/m 1 2 3 4 5 6 7
m m 2 8 2 3 4 9 10 11
2 2 2 1 2 3 4 5 6 12
---------------------------------------------------------------
Tetragonal System
Possible Shapes:
No. Name Indices Rem.
1 tetr. Base-Pinacoid { 0 0 1 } open
2 tetr. Prism II { 1 0 0 } open
3 tetr. Prism I { 1 1 0 } open
4 ditetr. Prism { h k 0 } open
5 tetr. Bipyramid II { h 0 l } closed
6 tetr. Bipyramid I { h h l } closed
7 ditetr. Bipyramid { h k l } closed
8 tetr. Pedion { 0 0 1 } open
9 tetr. Pyramid II { h 0 l } open
10 tetr. Pyramid I { h h l } open
11 ditetr. Pyramid { h k l } open
12 tetr. Prism III { h k 0 } open
13 tetr. Bipyramid III { h k l } closed
14 tetr. Bisphenoid I { h h l } closed
15 tetr. Scalenohedron { h k l } closed
16 tetr. Trapezohedron { h k l } closed
17 tetr. Pyramid III { h k l } open
18 tetr. Bisphenoid II { h 0 l } closed
19 tetr. Bisphenoid III { h k l } closed
Occurrence of shapes in crystal classes:
Crystal Class Shape No.
{001} {100} {110} {hk0} {h0l} {hhl} {hkl}
4/m 2/m 2/m 1 2 3 4 5 6 7
4 m m 8 2 3 4 9 10 11
4/m 1 2 3 12 5 6 13
-4 2 m 1 2 3 4 5 14 15
4 2 2 1 2 3 4 5 6 16
4 8 2 3 12 9 10 17
-4 1 2 3 12 18 14 19
---------------------------------------------------------------
Trigonal System
Possible Shapes:
No. Name Indices Rem.
1 hex. Base-Pinacoid { 0 0 1 } open
2 hex. Prism I { 1 0 0 } open
3 hex. Prism II { 1 1 0 } open
4 dihex. Prism { h k 0 } open
5 trig. Rhombohedron I { h 0 l } closed
6 hex. Bipyramid II { h h l } closed
7 trig. Scalenohedron { h k l } closed
8 hex. Base-Pedion { 0 0 1 } open
9 trig. Prism I { 1 0 0 } open
10 ditrig. Prism I { h k 0 } open
11 trig. Pyramid I { h 0 l } open
12 hex. Pyramid II { h h l } open
13 ditrig. Pyramid { h k l } open
14 hex. Prism III { h k 0 } open
15 trig. Rhombohedron II { h h l } closed
16 trig. Rhombohedron III { h k l } closed
17 trig. Prism II { 1 1 0 } open
18 ditrig. Prism II { h k 0 } open
19 trig. Bipyramid II { h h l } closed
20 trig. Trapezohedron { h k l } closed
21 trig. Prism III { h k 0 } open
22 trig. Pyramid II { h h l } open
23 trig. Pyramid III { h k l } open
Occurrence of shapes in crystal classes:
Crystal Class Shape No.
{001} {100} {110} {hk0} {h0l} {hhl} {hkl}
-3 2/m 1 2 3 4 5 6 7
3 m 8 9 3 10 11 12 13
-3 1 2 3 14 5 15 16
3 2 1 2 17 18 5 19 20
3 8 9 17 21 11 22 23
---------------------------------------------------------------
Hexagonal System
Possible shapes:
No. Name Indices Rem.
1 hex. Base-Pinacoid { 0 0 1 } open
2 hex. Prism I { 1 0 0 } open
3 hex. Prism II { 1 1 0 } open
4 dihex. Prism { h k 0 } open
5 hex. Bipyramid I { h 0 l } closed
6 hex. Bipyramid II { h h l } closed
7 dihex. Bipyramid { h k l } closed
8 hex. Base-Pedion { 0 0 1 } open
9 hex. Pyramid I { h 0 l } open
10 hex. Pyramid II { h h l } open
11 dihex. Pyramid { h k l } open
12 hex. Prism III { h k 0 } open
13 hex. Bipyramid III { h k l } closed
14 hex. Trapezohedron { h k l } closed
15 hex. Pyramid III { h k l } open
16 trig. Prism I { 1 0 0 } open
17 ditrig. Prism { h k 0 } open
18 trig. Bipyramid I { h 0 l } closed
19 ditrig. Bipyramid { h k l } closed
20 trig. Prism II { 1 1 0 } open
21 trig. Prism III { h k 0 } open
22 trig. Bipyramid II { h h l } closed
23 trig. Bipyramid III { h k l } closed
Occurrence of shapes in crystal classes:
Crystal Class Shape No.
{001} {100} {110} {hk0} {h0l} {hhl} {hkl}
6/m 2/m 2/m 1 2 3 4 5 6 7
6 m m 8 2 3 4 9 10 11
6/m 1 2 3 12 5 6 13
6 2 2 1 2 3 4 5 6 14
6 8 2 3 12 9 10 15
-6 m 2 1 16 3 17 18 6 19
-6 1 16 20 21 18 22 23
---------------------------------------------------------------
Cubic System
Possible shapes (all shapes are closed):
No. Name Indices
1 Cube { 1 0 0 }
2 Rhomb-Dodecahedron { 1 1 0 }
3 Tetrakis-Hexahedron { h k 0 }
4 Octahedron { 1 1 1 }
5 Triakis-Octahedron { h h l }
6 Icosi-Tetrahedron { h l l }
7 Hexakis-Octahedron { h k l }
8 Tetrahedron { 1 1 1 }
9 Deltoid-Dodecahedron { h h l }
10 Triakis-Tetrahedron { h l l }
11 Hexakis-Tetrahedron { h k l }
12 Pentagon-Dodecahedron { h k 0 }
13 Dis-Dodecahedron { h k l }
14 Pentagon-Icositetrahedron { h k l }
15 Tetrahedral Pentagon-Dodecahedron { h k l }
Occurrence of shapes in crystal classes:
Crystal Class Shape No.
{100} {110} {hk0} {111} {hhl} {hll} {hkl}
4/m -3 2/m 1 2 3 4 5 6 7
-4 3 m 1 2 3 8 9 10 11
2/m -3 1 2 12 4 5 6 13
4 3 2 1 2 3 4 5 6 14
2 3 1 2 12 8 9 10 15
---------------------------------------------------------------
I hope, this quite abbreviated introduction is useful (and free of
substantial errors). I tried to put in everything one needs to under-
stand what XTAL4POV does, and I tried to write it down in a way even a
complete novice to crystallography hopefully can understand. I know this
is not a masterpiece of literature, but I wrote this down from scratch today
to get the program released as soon as possible.