Miller Indices Calculator for Cubic System

In a cubic crystal system, we consider the following three types:

1. Face-Centered Cubic (FCC)
2. Body-Centered Cubic (BCC)
3. Simple Cubic (SC) (Rare in natural elements)

Here’s a list of elements with their lattice constants (a) in Å for the cubic system (use values from the table below in the calculator)

Face-Centered Cubic (FCC)

Body-Centered Cubic (BCC)

Simple Cubic (SC)

Polonium (Po) → 4.40 Å (The only naturally occurring element with a simple cubic structure)

What are Miller Indices?

Miller Indices are a set of three numbers (h, k, l) that describe the orientation of a specific plane of atoms within a crystal lattice.

Think of a crystal as a highly organized arrangement of atoms, like a neatly stacked pile of identical boxes. Miller Indices help us label and identify these “layers” or planes of atoms.

Why do we need them?

Understanding how atoms are arranged in crystals is crucial in materials science. Miller Indices provide a standardized way to talk about specific planes within a crystal, so scientists can easily communicate and compare their findings.

These planes are important because they influence many material properties, like how a crystal breaks, how it interacts with light, and so on.

How do we find them? (Steps)

1. Imagine the plane: Visualize the plane of atoms you want to describe within the crystal.
2. Intercepts with axes: Imagine three axes (like x, y, and z) running through the crystal. See where your plane intersects these axes. Let’s say it intersects the x-axis at 2, the y-axis at 3, and the z-axis at 4. These are called the intercepts.
3. Reciprocals: Take the reciprocal of each intercept. So, 1/2, 1/3, and 1/4.
4. Clear fractions: Multiply all the reciprocals by the smallest number that turns them into whole numbers. In our example, the smallest number is 12. So, we get (1/2)*12 = 6, (1/3)*12 = 4, and (1/4)*12 = 3.
5. Miller Indices: These whole numbers are your Miller Indices! In this case, (hkl) = (643).

What do the numbers mean?

The numbers (h, k, l) represent the reciprocals of the intercepts, after clearing fractions. They tell you how many “steps” you need to take along each axis to get to the next parallel plane. Smaller indices indicate planes that are farther apart, while larger indices correspond to planes that are closer together

Example

Let’s say a plane intercepts the x-axis at 1, the y-axis at 2, and the z-axis at infinity (meaning it never intersects the z-axis).

• Intercepts: 1, 2, ∞
• Reciprocals: 1, 1/2, 0
• Clear fractions (multiply by 2): 2, 1, 0
• Miller Indices: (210)

Important Notes

Parentheses: Miller Indices are written in parentheses: (hkl).

No commas: There are no commas between the numbers.

Negative numbers: A negative number is represented with a bar over the number, like (1̅20).

Zero: A zero indicates that the plane is parallel to that axis (it never intersects).

Family of planes: Sometimes, we use curly braces {hkl} to represent a family of equivalent planes.