Rational Intercepts and Miller Indices


The tendency of crystal faces to follow simple, rational orientations through the crystal lattice has resulted in two laws:

LAW OF HUAY: Crystal faces make simple rational intercepts on crystal axes.

LAW OF BRAVAIS: Common crystal faces are parallel to lattice planes that have high lattice node density.


Because crystal faces have predictable orientations relative to the crystal lattice, it is possible to develop a handy shorthand system to describe the orientation of crystal faces and crystallographic planes.  This notation is referred to as the Miller index (all crystals systems except hexagonal) or Miller-Bravais index (hexagonal crystal system).

The definition of a crystal lattice is an imaginative pattern of points or nodes in three dimensions in which every point (or node) has an environment that is identical to that of any other point or node.

A lattice therefore has no specific origin as it can be shifted parallel to itself or translated.

In such a lattice, three coordinate axes are selected parallel to the unit cell edges of the lattice.  We call these axes x, y, and z or a, b, and c (crystallographic axes).

Note that the coordinate axes are not necessarily at right angles to each other, and the lengths of the unit cell in the x, y, and z direction are not necessarily the same (dependent upon the crystal system).

Once a coordinate system has been established for a specific crystal lattice, then the orientation of any plane made up of nodes in the lattice can be uniquely described in terms of the intercepts of such a plane on the coordinate axes.  These intercepts are designated as either a whole integer of the unit cell dimension or a rational fraction of the cell along each axis.

Face intercepts are strictly relative values based on unit cell dimensions and do not indicate actual cutting lengths.

These intercept values can then be used to calculate the Miller index.

A Miller index is a series of coprime integers that are inversely proportional to the intercepts of the crystal face or crystallographic planes with the edges of the unit cell.  It describes the orientation of a crystal face with respect to the axes, but not the absolute size or location of the face.

The general form of the Miller index is (h, k, l) where h, k, and l are integers related to the unit cell along the a, b, c crystal axes.

We use (h, k, i, l) for hexagonal crystals.

For convenience it is easiest to reference the crystal face or crystallographic plane to a single unit cell.  So, obviously, if we want to define the position of a plane in absolute terms we need to know the size of the unit cell.

Miller indices are generally used for the definition of the orientation of a plane in a lattice.  Since there are a family of planes that are parallel to this orientation it is only necessary to define one plane.

Miller indices are the reciprocal of the intercepts of a plane such that the first number reflects the intersection on a, the second on b, and the third on c.


Technique 1: If you know the actual dimensions of the crystal and the unit cell dimensions or axial ratios.

Radius ratio for a, b, and c

    a                  b                    c
1.09 A°        1 A°            0.59 A°

Actual dimension of crystal along a, b, and c

    a                  b                    c
3.82 cm        3.50 cm        2.07 cm

Divide the radius ratio values by the dimensions

     a                    b                    c
1.09/3.82        1/3.50        0.59/2.07


0.29            0.29            0.29

Divide through by the smallest number

1            1            1        = (111)


Technique 2: If you know the intercept values of the plane relative to the unit cell dimensions given as a fraction.


The Recipe:

a. Determine the intercepts of the plane of interest with the crystallographic axes using unit cell dimensions as the units.  Record the intercept values as some whole integer or fractional value of the cell dimensions.

b. Invert each intercept so that the value for a, b, and c become 1/a, 1/b, and 1/c.

c. Multiply all terms by the lowest common denominator.

d. Put parentheses around the numbers.


Scenario 1: plane that intersects one axis and is parallel to the other two.

a    b    c

1   ∞   ∞   plane intersects a
or
∞   1   ∞   plane intersects b
or
∞   ∞   1   plane intersects c

Take the reciprocal of the intercepts and clear the denominator if necessary.

1/1    1/∞    1/∞ which gives    1    0    0

Miller index = (100) for a plane that plane intersects a

Miller index = (010) for a plane that plane intersects b

Miller index = (001) for a plane that plane intersects c

Scenario 2: A plane intersects two axes and is parallel to a third.

Determine the values for the intercepts:

a        b        c

2        ∞        3

Invert these values

1/2        1/∞        1/3

Clear the denominator so that it is equal to 1 by multiplying x 6

3        0        2

Miller index for the plane = (302)

Note that the index not only indicates which axes the plane intersects but the relative positions.  Larger values are closer to the origin of the a-b-c coordinate system.

Scenario 3: a plane intersects all three axes at different or similar unit cell dimensions along a, b, and c.


a        b        c

1        2        1

1/1        1/2        1/1        = (212)


a        b        c

1        1        1

1/1        1/1        1/1        = (111)


a        b        c

1/2        1        1/2

2/1        1        2/1        = (212)



Technique 3: If you do not know the number of unit cell intercepts of a specific plane along coordinate axes.

In a crystal or crystal model you can determine the general orientation of a plane with respect to its crystallographic axes, but you will only be able to estimate the relative values for the intercept.  How can you do this?


On the basis of the overall symmetry of the crystal or model, locate the coordinate axes or define these axes (a, b, c).

Identify the largest faces on the crystal that cuts a, b, and c and assign it the intercept values of 1a, 1b, and 1c with an index value of (111).  This face is called the unit face.  Note that the length 1a, 1b, and 1c do not have to be equal.

Once the unit face is defined (if possible) then the orientations of all other faces can be determined relative to it.





If it is impossible to define the absolute values for crystal faces then we use a general Miller index that can represent any number of faces.  In this method we use h, k, or l instead of numerical values.


For example, instead of (100) we use (h00) to indicate a series of faces that are oriented in space perpendicular to axes a.


Example:

(h00)
(0k0)
(00l)
(hk0)
(h0l)
(0kl)
(hkl)


Form Numbers

You can also use the Miller index of each specific crystal face that is part of a form to label all faces of that form.

For referring to a specific crystal form it is conventional to use only one of the form’s several Miller indices, namely the Miller index symbol with positive values.

For example, a side or b pinacoid consists of two parallel faces with indices of (010) and (010).  The form symbol for this pinacoid is {010} because the two faces are related by symmetry.

Another example is the octahedron that has eight faces:

(111), (111), (111), (111), (111), (111), (111), (111)

The form symbol for the octahedron is {111} because using operations of symmetry can generate all other faces.