Question

What are the miller indices if the plane intersect the crystal axis at \(2 \mathrm{a}, \mathrm{b}, 2 \mathrm{c} ?\)

Short answer

(1, 2, 1)

Step-by-step solution

Identify the Intercepts

The given plane intersects the crystal axes at 2a, b, and 2c. These are the intercepts along the x, y, and z axes, respectively.

Take Reciprocals of the Intercepts

Calculate the reciprocals of the intercepts. For the intercepts 2a, b, and 2c, the reciprocals are 1/2, 1, and 1/2 respectively.

Clear the Fractions

To express these reciprocals as integers, multiply each by the least common multiple (LCM) of the denominators. In this case, the LCM is 2. Multiplying the reciprocals by 2 gives: (1/2)*2 = 1, 1*2 = 2, and (1/2)*2 = 1.

Write the Miller Indices

The integers obtained in step 3 are the Miller indices of the plane. Thus, the Miller indices are (1, 2, 1).

Key concepts

Crystallography

Crystallography is the study of crystals, which are solid materials whose atoms are arranged in highly ordered, repeating patterns.
These repeating patterns are known as crystal lattices.

Crystallography helps us understand the structure of materials at the atomic level, which is essential in fields like chemistry, physics, and materials science.

• Crystals are classified into different systems based on their symmetry and lattice parameters.
• Each crystal system has its own unique set of lattice parameters, which are distances between atoms along the crystal axes.


One of the key tasks in crystallography is to determine the orientation of different crystal planes within the crystal lattice, which is where Miller indices come into play.

Reciprocal Lattice

The reciprocal lattice is a mathematical construct used in crystallography to make it easier to visualize and calculate properties related to the crystal lattice.
While the crystal lattice describes the arrangement of atoms in real space, the reciprocal lattice describes the arrangement of wave vectors that correspond to the periodicity of the crystal.

• It is particularly useful in the study of diffraction patterns, such as those obtained in X-ray crystallography.
• In the context of Miller indices, the reciprocal lattice helps by making calculations involving plane spacings simpler.

When calculating Miller indices, taking the reciprocals of the intercepts of a crystal plane can be seen as a step into the world of the reciprocal lattice.
This step and the following mathematical manipulations help in uniquely identifying the orientation of crystal planes.

Crystal Planes

Crystal planes are flat, two-dimensional slices through a crystal lattice.
They are defined by their intercepts with the x, y, and z crystal axes.

Miller indices are a notation system used to uniquely identify these crystal planes.

• To find the Miller indices of a plane, you first determine the intercepts of the plane with the three crystal axes.
• Next, you take the reciprocals of these intercepts.
• Finally, you clear any fractions by multiplying by the least common multiple (LCM) to get a set of integers.

In the given exercise, the plane intersects at 2a, b, and 2c.
Taking the reciprocals of 2a (1/2), b (1), and 2c (1/2), and then clearing the fractions by multiplying by the LCM (2) gives us the Miller indices (1, 2, 1).

Miller indices help in understanding the orientation and properties of different planes and are essential in analyzing crystal structures and their behaviors.