Geos 306, Fall 2007, Lecture 9
Crystallography II

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Lattices

• Suppose we define an origin, O, in a crystal that has the basis vectors {a, b, c}. The location of the origin is arbitrary. Then O + a is another point that is translationally equivalent to O. Being translationally equivalent means that everything about the two points is identical in every respect except that they are distinct points, located in different places. But if you were tiny, the size of an atom, and were situated at O, or at O+a, you could not tell the difference.
• The points O+b and O+c are also equivalent to O and O+a.
• In fact, all the points, O+ua+vb+wc, {where u,v,w are integers}, are translationally equivalent. The set of all these points is called a lattice.
• The faces of a crystal are parallel to planes of lattice points, while the edges of a crystal are parallel to lines of lattice points.

• We can say that translation maps a lattice into self-coincidence. This means that if we translated the lattice by, say, a, then it would lie exactly on top of itself in such a way that you could not tell that the lattice had been moved.

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Symmetry

• The operations that map an object into self-coincidence are called symmetry operations. A crystal is solid matter that possesses atomic scale translational symmetry. Aside from the liquid core and a small amount of molten mantle, most of the Earth is crystalline.

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Rotations

• Also found in crystals are other symmetry operations that map a lattice into self coincidence, for instance, rotations. In the illustration of the lattice above, not only is translation a symmetry operation but so are rotations of 0=360°, 60°, 120°, and 180°.
• The only rotations that can map a lattice into self-coincidence are those with turn angles of 0, 60, 90, 120, 180°. We call these symmetry operations 1-fold, 6-fold, 4-fold, 3-fold and 2-fold rotations, respectively. The name comes from the fact that a n-fold rotation has a turn angle of (360/n)°.
• These rotation are defined such that a counter-clockwise rotation is considered positive (right-hand rule). A clockwise rotation is designated as an inverse n-fold rotation and is denoted n^-1. This is because an n-fold rotation followed by an inverse n-fold rotation takes you back to where you started, i.e. n n^-1 = 1.
• In the lattice illustrated above, not only is there a 6-fold rotation axis, but there are also 2-fold, 3-fold and 1-fold rotation axes. The set of all symmetry operations that map an object into self-coincidence, while not moving the origin, is called its point group. The translation symmetry operation is not an element of a point group, because it does not leave any point fixed.
• All crystals can be characterized by the symmetry in their point group, for instance, the crystal form, because planes of lattice points define faces of a crystal. The symmetry of the point group is present in the symmetry of all of the physical properties of a crystal. For instance, the optical properties of a crystal match the point group symmetry. So we will spend some time examining the point group symmetries of crystals.
• Based upon the 1-, 2-, 3-, 4-, and 6-fold rotations described above, we can define 5 different point groups, generated by 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold rotations:
  1 = {1}
  2 = {1,2}
  3 = {1,3,3^-1}
  4 = {1,4,2,4^-1}
  6 = {1,6,3,2,3^-1,6^-1}
  

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Inversion

• The only symmetry rotations that map a crystal lattice into self-coincidence are 1-, 2-, 3-, 4-, and 6-fold rotations. The reason that only these rotations exist is related to the translational symmetry of the lattice.
• There is yet another type of symmetry operation that maps a lattice onto itself besides the translations and rotations, and it is called an inversion. The symbol for an inversion is i. If O marks the origin, then an inversion takes any point, P, and maps it through the origin to a new point, P', that is exactly the same distance away from the origin, (i.e. OP = OP') such that POP' are on the same line.

• In a mathematical sense, the inversion takes a point located at {x,y,z} and maps it to {-x,-y,-z}. Thus an object has an inversion as a symmetry element if every point located at some {x,y,z} is equivalent to another one located at {-x,-y,-z}.

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Roto-Inversion

• In addition to translations, rotations and inversions, there is one final type of symmetry element called a roto-inversion. This symmetry element is a hybrid mixture of a rotation that is followed by an inversion.
• A well known example of a roto-inversion is a mirror, m. This is made from a 2-fold rotation followed by an inversion.
• Based upon the rotations described above, and the roto-inversion hybrid just described, we can define some new symmetry elements: (Note that the line is supposed to represent a bar over the number)
  1i = i
  2i = m
  3i = `3
  3<sup>-1</sup>i = `3^-1
  4i = `4
  4<sup>-1</sup>i = `4^-1
  6i = `6
  6<sup>-1</sup>i = `6^-1
  

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Point Groups

• The symmetry of every mineral can be defined by the set of all symmetry elements that map it into self-coincidence. There are 32 possible sets of symmetry elements, known as the crystallographic point groups. Here is a table that summarizes the crystallographic point groups.
• The symmetry of a mineral also defines the constraints on the cell parameters, so the point group is also related to crystal system.




G	GUGi	HUG\Hi	System
1	`1	-	Triclinic
2	2/m	m	Monoclinic
3	`3	-	Hexagonal
4	4/m	`4	Tetragonal
6	6/m	`6	Hexagonal
222	2/m2/m2/m	2mm	Orthorhombic
322	`32/m2/m	3mm	Hexagonal
422	4/m2/m2/m	4mm and `42m	Tetragonal
622	6/m2/m2/m	6mm and `62m	Hexagonal
332	`3`3 2/m	-	Cubic
432	4/m`3 2/m	`4 3 m	Cubic

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Reading:
Wenk and Bulakh, Chapter 4
Klein, Chapter 5
Nesse, Chapter 2