7 - Lecture notes for GEOL3010


Crystal Structure

Review on Bravais lattice types Why, for example do we not have a base-centered tetragonal lattice?

See figure below. Full lines delineate a cell that is C-centered. But the dashed lines show that the centered arrangement is not really a new arrangement and that a primitive cell can be found. In the case below, the primitive cell is chosen because its volume is smaller. Therefore, there is no need to create a C-centered tetragonal lattice type.

In any lattice, a unit cell can be chosen in an infinite number of ways. A unit cell therefore, does not "exist" as an entity in the a lattice. Unit cells are mental constructs of our mind chosen for convenience, simply to conform to the symmetry elements that can be found in the lattice.

In fact any of the 14 Bravais lattice types may be referred to a primitive cell.

Another example can bee seen in the Face-centered (F) isometric lattice (shown below in solid lines). This can also be represented by a rhombohedral lattice (R) where, a = 60

Why then do F-centered lattices appear in the list of 14 Bravais lattices? (i.e., why not eliminate the F-centered cell and use the R cell?)

The answer lies in the fact that when a = 60 for an R type lattice, then the unit cell can be described as a lattice with three 4-fold axes, which places it into the more highly symmetrical isometric class. Only one independent variable is needed to describe the unit cell. If a is not equal to 60, then the lattice type would not be F-centered isometric and R is chosen. We do it for convenience.

We use non-primitive lattices because it allows us to translate points by using simple operators. These are applied to a point of origin, in the directions of the crystallographic axes to generate the lattice points in 3-D space.


More on lattice directions and zone axis

We can describe the direction through a lattice by a system of zone axis indices. See figure below. Directions are defined by the intercept that a vector from the origin makes with unit cell edge. The point on the unit cell edge is defined by it fractional coordinates.

Steps include (1) listing the fractional intercepts, (2) then clearing the fractions. The integer values that result, u, v, and w are then closed with square brackets [uvw] and they define an axis within the lattice. With exception to the isometric system, it is important to note that the [uvw] direction is not perpendicular to corresponding planes with the same (hkl) integer values.

Mathematically, the [uvw] direction in a lattice with a , b and c lattice parameters is parallel to the vector ua + vb + wc.

The zone axis is a concept that describes the relationship between a direction in the lattice and a set of planes (e.g., crystal faces) that parallel the vector direction. This is most easily visualized by comparing the [uvw] vector and the intersection of two planes (e.g., the edge defined by the intesection of two crystal faces.)

If the Miller indices are known for two intersecting crystal faces, then the zone can be readily determined. The technique uses a simple form of matrix algebra that effectively multiplies two vectors (recall that Miller indicies are a type of vector notation).

 

Suppose you want to know the zone (direction in the crystal) that results from the intersection of two faces. Follow these steps.

  1. Write the integer Miller indice values for the first plane twice in a row.
  2. Write the integer Miller indice values for the second plance twice in row directly below.
  3. Disregard the first and last numbers in each row.
  4. Cross multiply each pair of adjacent columns
  5. Subtract the product of the upper right to lower left operation from the product of the upper left to lower right operation.
  6. Reduce the three integer values to the smallest integer.

 h1

k1

l1

h1

k1

l1

h2

k2

l2

h2

k2

l2

u = k1 * l2 - l1 * k2

v = l1 * h2 - h1 * l2

w = h1 * k2 - k1 * h2

 

Here's an example for the intersection of the (100) and (010) faces.

1

0

0

1

0

0

 0

1

0

0

1

0

u = (0 * 0) - (0 * 1) = 0

v = (0 * 0) - (1 * 0) = 0

w = (1 * 1) - (0 * 0) = 1

[uvw] = [001]


Mineral properties and their relationship to lattice directions

Some properties are independent of direction in the lattice. This includes the scalar properties of density and heat capacity.

Some properties are directionally dependent depending upon the size of the crystal. This includes the effects of surface free energy. When a crystal is large, then the effect of surface free energy relative to the bulk crystal is small. When crystals are small, then surface free energy is important and directional dependence needs to be considered. 

Vector dependent properties can be described as anisotropic. If a property is independent of direction, then it is referred to being isotropic. Minerals with isometric symmetry generally exhibit isotopic properties.

Common sense tells us that the higher the symmetry of the lattice (e.g. isometric) then the more isotropic the properties are going to be.  Conversely, the lower the internal symmetry of a mineral, then the more anisotopic its properties will be. Each property of the same mineral must be considered seperately. For example, elastic properties in isometric crystals are generally anisotropic. But, thermal conductivity properties in isometric crystals are generally isotropic.

Properties that are directionally dependent include:

  1. Hardness (e.g., kyanite)
  2. Thermal conductivity
  3. Electrical conductivity
  4. Thermal expansion
  5. Diffusion of ions through a crystal
  6. External form of crystal (i.e., growth rate, which is in part related to node density)
  7. Dissolutiuon rates
  8. Diffraction of X-rays
  9. Speed of light or optical properties. (you will hear more about this in a few weeks!)