How to Draw Planes Using Miller Indices

Miller is the name of a material scientist. Indices is plural of index. Miller indices are the styles to designate the planes and directions in the unit cells and crystals.

Such designations become essential for investigations of various properties in different elements. In the most general way, Miller indices are expressed by (hkl) and no comma is used between them.

Here, symbols h, k and l indicate unknown integers. The kinds of brackets have special meanings which are as under:

  • (hkl)    denotes a plane,
  • [hkl]    denotes a direction,
  • {hkl}    denotes family of planes,
  • <hkl>  denotes family of directions.

Miller indices (hkl) are expressed as a reciprocal of intercepts p, q and r made by the plane on the three rectangular axes x, y and z respectively.

The intercepts are measured in terms of the dimensions of unit cell. These are the unit distances from the origin along the three axes. Thus:

h = 1/p ,  k = 1/q , l = 1/r

Where, p = intercept of the plane on x—axis,

q = intercept of the plane on y—axis, and

r = intercept of the plane on z—axis.

Reciprocal of these intercepts are then converted into whole numbers. This can be done by multiplying each reciprocal by a number obtained after taking LCM (lowest common multiplier) of denominator. This gives the Miller indices of required plane. The Miller indices are expressed by three smallest integers.

General Procedure to Find Miller Indices

The sequential procedure for finding Miller indices may be summarized as under:

  1. A system of three coordinate axes indicating their origin is chosen in an unit cell.
  2. The intercepts of the plane, whose Miller indices are desired, are found on the three coordinate axes. Let these are c1, c2 and c3 along x, y and z axes.
  3. They are expressed in terms of axial units that is c1 = pa, c2 = qb and c3 = rc. If the unit cell is cubical then a = b = c. Here the numerical parameters of the plane are taken as p, q and r.
  4. The ratio of the reciprocals is taken as (1/p , 1/q , 1/r).
  5. The reciprocals are converted into whole numbers by multiplying each with their LCM to get the smallest whole number. This gives the Miller indices (hkl) of the plane.

To make the procedure more clear, an illustration will be taken now.

Determination of  Miller Indices

A plane ABC in a cubical unit cell is shown in Figure. To determine Miller indices of plane ABC origin 0 and coordinate axes x, y and z are shown in the unit cell in which are OP = OQ = OR = a. The plane in Figure intercepts at A, B and C on x, y and z axes in such a way that

OA = (2a/3) along x-axis,

OB = (2a/5) along y-axis, and

OC = (la/3) along z-axis.

To obtain the required Miller indices, we proceed as under:

how to draw planes using miller indices

Here the factor 1/2 is the result of conversion of reciprocals to integers, and is usually omitted. We, therefore, conclude that Miller indices of plane ABC are

(hkl) = (356)

Miller indices of a plane in non-cubical unit cells may be determined in the same manner.

How to Draw Planes Using Miller Indices

In the above paragraph, we have determined Miller indices of an unknown plane. Now, we shall draw planes using miller indices (in unit cell). We have taken the case of a plane whose Miller indices are (111).

As h = 1, k = 1, l = 1;

therefore, p= 1/h = 1/1 ,

q = 1/k = 1/1 , and

r = 1/l = 1/1.

Therefore, intercepts are:

c1(on x-axis) = pa = 1 x a = 1,

c2(on y-axis) = qb = 1 x b = 1, and

c3(on z-axis) = rc = 1 x c = 1,

If the plane (111) is to be drawn in cubical unit cell, then a = b = c. Therefore intercepts c1 = c2 = c3 = a. Now a cubical unit cell is drawn showing origin 0 and coordinate axes x, y and z in Figure 2(a).

Significance of miller indices

Intercepts c1 = a = OP, c2 = a = OQ and c3 = a = OR are located. Points P, Q and R are joined. The hatched plane PQR represents the required Miller indices (111).

Plane Parallel to an Axis

Refer to above Figure (b). It is showing a plane whose Miller indices are (101). This plane intercepts x and z axes, and is parallel to y axis. A plane parallel to an axis is supposed to have its intercept at infinity.

In this case, h = 1, k = 0 and l = 1. Therefore, p = 1/1 , q = 1/0 i.e. tends to infinity, and r = 1/1 = 1. Hence, intercepts c1 = 1, c2 tends to infinity and c3 = 1.

Planes with Negative Indices

Refer to above Figure (c). It is showing origin at 0 and the coordinate axes x, y and z. Miller indices (1bar 00) of a plane is depicted in accordance with the above discussions.

If we have to draw a plane whose Miller indices are (1bar 00), then the origin 0 has to be shifted to 01. Now the coordinate system x, y1 and z1 is under consideration.

The direction 010 indicates negative side of x. Plane with Miller indices (1bar 00) will be one as shown in above figure. To indicate a minus sign, the symbol bar is marked on 1.

Family of Planes

Different planes in a cubical unit cell are shown in above Figure 2(d). Their Miller indices are (100), (010) and (001).

Miller indices of the plane (1bar00) is shown in above Figure(2c).

Similarly other planes in a cubical unit cell may be designated as (01bar0 ) and (001bar). Digits in all these planes are 1, 0 and 0 at different sequence. They belong to the same family, and are called family of planes. These planes can be designated by {100} only using a curly bracket.

The planes having Miller indices (123), (231), (321), (312), (1bar23), (12bar3), (213bar) etc. form a another family of planes. They can be designated by {123} which are the lowest integers.

Crystallographic Notation of Atomic Crystal Directions

Miller indices are also used to specify directions within a unit cell or a crystal. An unknown direction is designated by [hkl]. Here a square bracket is used to designate direction.

To determine the crystal directions, let us take a line passing through the origin and parallel to a given direction. A vector r passing through the origin 0 of the unit cell to a lattice point may be given by:

r = r1x + r2y + r3z     …….(equation 1)

Where, r1, r2 and r3 are integers.

The length of the projections of this line on the coordinate axes x, y and z are noted. These intercepts are reduced to smallest integers to obtain [hkl].

Miller indices planes

Consider vector r = OB in the above Figure 3(a) in which B is a lattice point. Here r1 = 1, r2 = 1 and r3 = 1 from the geometry of the cubical unit cell along x, y and z axes. Hence this direction is [111].

As an other case, consider point C located at a distance (2/3) of the dimension of cube along x-axis. The direction vector r = OC is parallel to direction z. In this case r1 = 2/3, r2 = 1 and r3 = 0. Therefore, its direction will be found as:

[hkl] = [2/3 10]  = 1/3[230]

i.e.

[hkl] = [230]

Miller indices of directions [100], [110] and [011] are also shown in the above Figure.

Alternate Method of Drawing a Direction

The direction [hkl] is perpendicular to the plane (hkl) in a cubic unit cell. Hence for drawing a direction, first draw a plane with Miller indices (hkl).

Now draw a line perpendicular to the above plane through the origin. This line is the required direction [hkl]. In the above Figure 3(b), a cubical unit cell is shown.

Consider origin at O in the coordinate system x, y and z. Our aim is to draw direction [001].

For that, we first draw a plane having Miller indices (001), and then direction [001] is drawn perpendicular to it while passing through the origin. This is shown by OA.

Direction with Negative Indices

Directions having negative Miller indices (with a bar sign) can be determined in the same way as discussed earlier for planes by shifting the origin. This is elaborated in the above Figure 3(b).

The original origin 0 is shifted to 01. Now a new coordinate system x1, y1 and z1 is considered. Here the original axis y coincides with the new axis y1.

The plane (01bar0 ) is drawn in the same way as discussed earlier. The direction having Miller indices [ 01bar0 ] is then drawn normal to it, as shown in this figure by 010.

Shifting of the origin is not necessarily required. Directions with negative indices can also be drawn without shifting the origin. This is self-explanatory from the above Figure 3(c). This diagram shows two consecutive unit cells along + z and —z directions.

Family of Directions

It is analogous to family of planes. Directions [130], [310], [013], [031], [ 1bar30], [ 3bar10] etc. can be represented by <130>.

Miller Bravais Indices

The system of Miller-Bravais indices is employed to designate planes and directions in a hexagonal unit cell. It is expressed by four digit notation (hkil). The indices use four axes system. The four axes are a1, a2, a3 and c.

Out of these, three axes a1, a2 and a3 are coplaner and lie on the basal plane; and are separated by 120° from each other. Fourth axis c is perpendicular to the basal plane. This system follows a rule given as

h + k= —i  ……….(equation 2)

draw planes using miller indices

In case of (1010), the values of h = 1, k = 0, i = —1 and l = 0. This plane intercepts at unit distance along a1 axis; is parallel to a2 axis; intercepts a3 axis at unit distance in negative direction of coordinate, and is parallel to c axis.

As indicated in prior paragraph, the parallel planes have intercepts at infinity, hence their indices are zero. So the equation (1) may be verified as:

1 + 0 = — (—1)

The plane (0001) is parallel to basal plane axes. It intercepts c-axis at unity. Therefore, its Miller-Bravais indices are (0001). The direction [21bar1bar0] is parallel to a1 axis. When resolved, its components along a2 and a3 axes are —1. Using the condition of equation (2), the index h is obtained as:

—(—1—1) = 2

This direction is parallel to c axis. Thus, the index [ 21bar1bar0] is the result.

Salient Features of Miller Indices

Some important features of Miller indices are as follows:

  • Miller indices of equally spaced parallel planes are the same.
  • A plane parallel to one of the coordinate axes has an intercept at infinity and so the Miller index of the plane is zero for that axis.
  • The plane passing through the origin is defined in terms of a parallel plane having non-zero intercept. Alternately, if the plane passes through the origin, the origin has to be shifted for indexing the plane.
  • Any two planes having Miller indices (h1 k1 l1) and (h2 k2 l2) will be perpendicular to each other if

h1h2 +k1k2 + l1l2 = 0

  • When Miller indices contain integers of more than one digit, the indices are separated by a comma for clarity, e.g. (3 4,12) or (4, 11, 17).
  • The direction [hkl] is normal to the plane having Miller indices (hkl) in a cubic system. This is generally not true for non-cubical crystal systems.
  • Planes with low index numbers have wide interplanar spacing as compared with those having high index numbers.
  • All members of family of planes are not necessarily parallel to one another. Similar is the case of crystal directions of a family.

Some Common Planes and Directions

The criteria of selecting origin and coordinate axes does not follow any hard and fast rule in an unit cell. Their selection is arbitrary.

Look at the origin 0; axes x, y and z chosen so far in various figures. Now we shall show some planes and directions in which the coordinate directions are of any arbitrary choice. These are shown in the following Figure 5.

how to draw planes using miller indices

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