Unit Cells
The Unit Cell is the smallest volume that can be used to generate the entire
crystal structure using ONLY TRANSLATION. It is usually a parallelepiped with
non-orthogonal axes.
We use fractional coordinates to describe positions in the unit cells. These are
fractions of a, b, and c (the axes) and these coordinates are very much like x,
y, and z of cartesian coordinates. It is much easier to talk about fractions
than actual lengths-- i.e. if a=12.234, b=15.698, and c=9.045, then a/2 is more
convenient than 6.117! We can always get "real" distances by
multiplying the true axis length by the fractions.
In fractional coordinates .75a=1.75a=2.75, etc. since one can always add 1 for
the translations. -.25a is also equivalent since -1 can also be added.
Unit cell determination is also somewhat arbitrary, since one can usually define
several unit cells for the same arrangement. Often a particular cell is chosen
for symmetry reasons and the convention is to choose the cell with the highest
symmetry.
Asymmetric Unit
The asymmetric unit is the smallest part of the crystal structure required to
create the entire crystal structure. Now, one can use symmetry (space group) AND
translations, as opposed to the unit cell, which only allows translations. The
asymmetric unit may consist of whole or portions of molecules, depending on
symmetry. (Benzene example in class)
Z
Z is the number of "molecules" in the cell. It is very dependent on
the symmetry of the crystal, not necessarily that of the molecules. Increasing
the unit cell axis or changing from a primative cell to a centered
cell can change Z, i.e. in class we saw a primative cell (1 fish in the
cell, Z=1) and a centered cell (2 fish in the cell, Z=2).
Nomenclature
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The unit cell is defined with axes a, b, and c.
The angles between the axes are called a, b,
and g. |
A lattice is a connection of equivalent points.
These points are merely a geometrical construct and are not equivalent to the
structure! One can overlay the lattice onto the structure in many ways and we
use a lattice as cartesian coordinates are used to describe positions.
A primative unit cell is a unit
cell which has lattice points only at the corners. As such, each lattice has
only one lattice point total-- in 2-dimensions each lattice point at the corner
is shared by 4 cells. so they each contribute 1/4 of a point (1/4 x 4 = 1). In
3-D each corner is shared by 8 cells and each contributes 1/8 of a point (1/8 x
8 = 1).
A centered unit cell is a unit cell with lattice
points inside the cell as well as at the corners. In class we saw an example of
a centered cell where a fish eye resided in the middle of the cell and fish eyes
were also at the corners. We will discuss face- and body-centered cells later
on.
d-spacing
d-spacing is the perpendicular distance between planes. Before we discuss
d-spacing, let's look at Miller indices.
Miller Indices are a convenient way to discuss
planes and, ultimately, faces of crystals. They are just names for planes
determined by using the lattice, just as one defines a plane in any mathematical
course. However, these planes are RECIPROCAL PLANES! Crystallography loves
reciprocals, so just take it in stride.
To determine the Miller Indices of a plane, first find the intercepts of the
plane with the axes. Let's say it intercepts halfway up the edges of each: a/2,
b/2 c/2. Now drop the axes names and take the reciprocal to get 2 2 2. Now
multiply or divide to get the lowest possible integer values, here 1 1 1. So
this plane is the 111 plane. If a plane only intercepts one axis (i.e. it is
parallel to the other axes), let's say the b axis, it is the 010 plane. You may
see these called hkl planes. Later on we will discuss this more.
Now that we know how to define planes, we can determine d-spacing...
d-spacing is just the perpendicular distance between
planes and is very useful for determining the unit cell. The d(hkl) is the
distance between 2 hkl planes (remember that we divided to get the lowest
integer values so 100 and 200 are both called 100). Therefore, d(100) should be
the length of a, d(010) should be b, and d(001) should be c. We'll actually be
doing this in lab next week.
Miller indices can also be negative, it just depends on your definition of the
axes.
There is a handout of Escher drawings.
We drew unit cells on Escher drawings in class. Perhaps now is a good time to
check out the Escher
drawing web site.
Point Groups
You've likely seen these in D33, but this is a refresher.
We represent symmetry operations on paper using symbols. They are somewhat
logical and I will not post them here. See the International Tables, Volume A.
When you combine the symmetry operations with lattices you get 14 and only 14
lattice possibilities (determined mathematically long before X-rays were
discovered). These lattices are called Bravais Lattices.
Bravais Lattices
Primative Bravais Lattices (7 total)
| triclinic |
a,b,c |
a,b,g |
| monoclinic(1st setting) |
a,b,c |
a=b=90,g |
| monoclinic(2nd/standard setting) |
a,b,c |
a=g=90,b |
| orthorhombic |
a,b,c |
a=b=g=90 |
| trigonal |
a=b=c |
a=b=g= not 90 |
| tetragonal |
a=b,c |
a=b=g=90 |
| hexagonal |
a=b,c |
a=b=90,g=120 |
| cubic |
a=b=c |
a=b=g=90 |
Traditionally we use the 2nd setting for monoclinic since it was the one that
originally appeared in the International Tables. However, it was decided that
the setting with c unique rather than b should be the 1st setting since that
follows the system used for high symmetry systems (tetragonal, hexagonal).
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