What do miller indices describe

Spacing of Planes 4. Miller indices is a system of notation of planes within a crystal of space lattice. They are based on the intercepts of plane with the three crystal axes, i. The intercepts are measured in terms of the edge lengths or dimensions of the unit cell which are unit distances from the origin along three axes. Procedure for finding miller indices:. The Miller indices of a crystal plane are determined as follows: Refer to Fig. Find the intercepts of the plane along the axes x, y, z The intercepts are measured as multiples of the fundamental vector.

The factor that results in converting the reciprocals of integers may be indicated outside the brackets, but it is usually omitted.

The directions in space are represented by square brackets [ ]. The commas inside the square brackets are used separately and not combined. Negative indices are represented by putting a bar over digit, e. The general way of representing the indices of a direction of a line is [x y z].

The indices of a plane are represented by a small bracket, h, k I. The following procedure is adopted for sketching any direction:. Now through the origin, draw a line normal to the sketched plane, which will give the required direction. Some of the important features of Miller indices particularly for the cubic system are detailed below:. All equally spaced parallel planes with a particular orientation have same index number h k I.

Miller indices do not only define particular plane but a set of parallel planes. It is the ratio of indices which is only of importance. The planes and are the same. All the parallel equidistant planes have the same Miller indices. Thus the Miller indices define a set of parallel planes.

A plane parallel to one of the coordinate axes has an intercept of infinity. If the Miller indices of two planes have the same ratio i. When the integers used in the Miller indices contain more than one digit, the indices must be separated by commas for clarity, e.

The crystal directions of a family are not necessarily parallel to one another. Similarly, not all members of a family of planes are parallel to one another. By changing the signs of all the indices of a crystal direction, we obtain the antiparallel or opposite direction. By changing the signs of all the indices of a plane, we obtain a plane located at the same distance on the other side of the origin. The normal to the plane with indices hkl is the direction [hkl]. All you need to do is take this point and properly format it.

If you remember, the format for directions is a square bracket. If you wanted to talk about the family of directions, use angle brackets. In other words,. Reciprocal space means you take the inverse of whatever point you were thinking of. In a cubic system, it turns out that the direction will always be perpendicular to the plane. For example the direction is perpendicular to the plane. Directional families are the set of identical directions or planes.

These families are identical because of symmetry. Imagine that I handed you a cube and asked you to draw the. By now, I hope you could do this easily! However, if I gave the same cube to someone else, they would probably draw a different , because they chose a different origin or a different initial rotation. In this way, we can say that and belong to the same directional family.

The only way to distinguish between the two is to define a consistent rotational frame of reference. This means that any material property which is true along will also be true of or any other direction in the family.

To find the different directional families, find all the permutations that can replace with a negative version, such as or. If the lattice vectors are the same length and have the same angle between them, you can also change the order, such as or. Here is a list of the individual directions in the directional families , ,.

If two directions belong to the same directional family, their corresponding planes will also belong to the same planar family. Since the cubic lattice has the most symmetry, there are the most number of identical directions in each directional family. Imagine, however, that you had a tetragonal crystal that was longer in the direction than the direction. In this case, they would NOT belong to the same family. However, the family would only include and.

Identifying directional families becomes especially confusing if the lattice parameters h, k, and l are not perpendicular to each other. This was the main motivation for creating Miller-Bravais indices, which only apply to hexagonal crystals and convert the 3-term hkl values into 4-term hkil values.

This conversion is a bit complex, but allows you to identify hexagonal directional families just based on the numerical value of the index. Additionally, the way I used x-, y- and z-axes is technically incorrect.

In the cubic system, they are the same, but they are not the same in other crystal systems. In most practical cases, you will just need to understand the meaning of basic indices such as , , , and. Now you know how to read and write Miller indices! For a quick review of notation:. Practice 1. Draw the , , and directions in a cubic crystal.

First, define an origin. In this case, you need the origin to intersect along the indicated direction, so you can move the origin the back left corner. Alternatively, you could simply translate that vector so it intersects with the back-left corner. Thus, the direction is. Practice 3. Write the position of the point, and the Miller Index for the direction from the origin to the point.

Assume the origin is at the back left corner. Thus, the point is at location. The direction would be identical, except that we prefer to avoid fractions. Remember that directions extend infinitely, so we can easily multiply the value by 6, which is a common denominator. Saturday, August 6. Step 1: The intercepts are 2,3 and 2 on the three axes. In the above plane, the intercept along X axis is 1 unit. A plane passing through the origin is defined in terms of a parallel plane having non zero intercepts.

If h k l is the Miller indices of a crystal plane then the intercepts made by the plane with the crystallographic axes are given as. Step 5: By writing them in parenthesis we get 4 2 4.

Therefore the Miller indices of the given plane is 4 2 4 or 2 1 2. The intercepts are 2, - 3 and 4. Step 1: The intercepts are 2, -3 and 4 along the 3 axes. Step 2: The reciprocals are. Step 3: The least common denominator is Step 4: Hence the Miller indices for the plane is 6,4,3. The direction [h k l] is perpendicular to the plane h k l.

The relation between the interplanar distance and the interatomic distance is given by,. Interplanar spacing. We know the perpendicular distance between the origin and the plane is 1 2 1. The perpendicular distance between the planes 1 2 1 and 2 1 2 are,.

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