Miller indices are sets of numbers, usually three, that indicate the orientation of planes or sets of planes of atoms in a lattice, depending on the type of parentheses/brackets used. For instance, the numbers representing a single plane are enclosed in round brackets (1 1 1), while those for a family of planes are encased in curly brackets {1 1 1}.
A unit cell in a crystal lattice consists of a set of crystallographic axes, x, y, and z. It is the intersection of these axes with any plane that determines the Miller index for that plane.
In fact, the reciprocal of these intercepts provides the Miller index (h k l) for the given plane.
h = 1/x
k = 1/y
l = 1/z
All such planes that belong to this family will form a set of planes altogether and will be represented with curly brackets {h k l}.
In addition to planes, Miller Indices are also used to specify directions. A direction is represented with the help of square brackets [u v w], while a family of directions with pointy brackets <u v w>.
By default, Miller indices are written without commas or any other symbols between them.
Additionally, for the hexagonal system, hcp, with four crystallographic axes, a set of four numbers is used. The first three axes are related to each other through a mathematical relationship.
In some cases, the Miller indices are multiplied or divided by a common number, to simplify them. As an example, the index (1 ½ ¾) will be expressed as (4 2 3), by multiplying the three values with the common value of 4.
If an intercept has a negative value on the axis, the negative sign is represented by a bar above the number, when written in the form of Miller indices.
Parallel planes are considered equivalent, so they have the same Miller index values. For example, a plane intersecting at 1 1 1 is equivalent to that intersecting at 2 2 2, so both of the planes will have the simplified Miller index of (1 1 1).
Identification of Miller Indices
In order to identify the Miller index of a plane, the first step is to specify the origin point on the unit cell. The origin will have coordinates of 0 with respect to all of the axes. It is, by default, chosen as the back left corner of the unit cell.
The directional vectors then arise from this origin point on the crystallographic axes, intercepting the given plane.
Afterward, the intercept coordinates are written in fractional form, divided by the dimensions of the original unit cell.
Finally, reciprocals are taken of the fractional intercepts to generate the Miller indices. The reciprocals are then changed into the smallest integers possible if needed.
The Origin
An origin is a point of reference having the coordinates (0,0,0). It can be defined at any point in the crystal. By convention, it is taken as the back left corner of the cell. However, in the identification of Miller indices for planes, it is, at times, easier to shift it elsewhere in the unit lattice. However, for a given plane or family of planes, at a given instance, only a single point at any corner can be the origin.
Example 1
The plane shown in red intercepts the x-axis at a length of 1a (where a is the length, width, or height of the cube). However, this plane runs parallel to y and z axes and so does not intersect these two axes at any point, so the y and z intercepts are ∞ each. Once we have the intercepts, all we need to do is write these intercepts in fractional form by dividing them with the lengths of the cell (x/a, y/a, z/a).
For an orthorhombic crystal lattice, the unit cell dimensions are a, b, and c, for the x, y, and z axes respectively. So the fractions would look like x/a, y/b, z/c.
For this case, of a cube, the specified plane will have the fraction of a/a, ∞/a, ∞/a, giving coordinates of 1, ∞, ∞ when solved (anything divided by infinity is infinity).
Finally, all that remains to be done is to take the reciprocal of these coordinates to give the miller index of this plane. 1/1, 1/∞, 1/∞ will result in the Miller index (1 0 0) for this plane.
Example 2
The plane shown in blue intercepts the y-axis at a length of 1a. However, this plane runs parallel to x and z axes and so does not intersect these two axes at any point, so the x and z intercepts are ∞ each: (∞, a, ∞).
Writing these in fraction form (∞/a, a/a, ∞/a) gives the coordinates (∞, 1, ∞).
Taking the reciprocal of these coordinates results in the Miller index of (0, 1, 0) for this plane.
Example 3
The 3 intercepts for the x, y, and z axes, respectively, are (∞, ∞, a).
This results in the coordinates (∞, ∞, 1).
So this plane has the Miller index of (0 0 1).
Example 4
In this example, since the bottom left back corner lies on the plane, we need to shift the origin elsewhere. Taking the bottom left front corner as the origin this time, the vector in the x direction will be reversed. The intercepts now will be (-a, ∞, ∞).
The coordinates (-1, ∞, ∞) result after taking the reciprocal.
This plane will have a Miller index of (-1 0 0).
(Note that the negative sign will be written over 1 as given in the figure).
Family of Planes – Example
The four planes mentioned in the examples given above, (1 0 0), (0 1 0), (0 0 1), (-1 0 0), along with the planes (0 –1 0) and (0 0 –1) are all included in the family of planes {1 0 0}.
Example 5
In this particular example, the plane is at distance a from the point of origin in both the x and y axes. Hence, the intercepts for the x, y, and z axes, respectively, are (a, a, ∞).
This results in the coordinates (1, 1, ∞).
So this plane has the Miller index of (1 1 0).
Example 6
This plane is intercepted by the x-axis at a distance a, while the y-axis intercepts the plane at a distance of a/2, with the z axis running parallel to the plane. So, the intercepts for the x, y, and z axes, respectively, are (a, a/2, ∞).
This results in the coordinates (1, 1/2, ∞).
The plane has the Miller index of (1 2 0).
Example 7
In this example, the plane is at distance a from the point of origin for all the three axes. Hence, the intercepts for the x, y, and z axes, respectively, are (a, a, a).
This results in the coordinates (1, 1, 1).
This plane will have a Miller index of (1 1 1).
Concepts Berg
Miller Indices represent the orientation of planes and directions in a crystal lattice.
Reciprocals of the intercepts on crystallographic axes provide the Miller Indices: h = 1/x, k = 1/y, l = 1/z.
Planes with equivalent Miller Indices belong to the same family, sharing parallel orientations.
“∞” represents infinite intercepts, indicating a plane is parallel to a particular axis without intersecting it.
The origin is typically the back left corner of the unit cell, but it can be shifted for convenience in identifying specific planes.
No, Miller Indices can also be used to specify crystallographic directions in a lattice.
Single planes: (h k l) with round brackets. Families of planes: {h k l} with curly brackets.
They can be multiplied or divided by a common value to simplify the indices while maintaining the plane’s specific orientation.
The Miller Index is (1 1 1) for such a plane in a cubic lattice.
A negative intercept is denoted with a bar above the number when writing Miller Indices.
References
- Miller Indices (hkl) (chem.libretexts.org)
- Miller Indices (vedantu.com)
