Miller Indices Calculator (plane intercepts → (hkl) and cubic d-spacing)

Understanding the Miller indices calculator — concept and principle

This calculator turns the intercepts of a crystal plane on the three cell axes into Miller indices (hkl), and for a cubic lattice gives the spacing between neighbouring planes. It is used in materials science, solid-state physics, X-ray diffraction and crystallography courses.

How it is calculated

1. Write the intercepts in units of the cell edges a, b, c (∞ if the plane is parallel to an axis).
2. Take reciprocals (1/∞ = 0).
3. Clear fractions by multiplying by the smallest common denominator, and divide by any common factor.
4. Negative indices are written with a bar, for example (1 1̅ 0).

Cubic spacing: d = a ÷ √(h² + k² + l²)

• Intercepts may be entered as fractions such as 1/2; a is in pm.

The principle

A family of parallel planes in a crystal is fully described by how it cuts the unit cell. Using reciprocals turns planes parallel to an axis into a zero, and makes closely spaced planes have large indices. In a cubic lattice the reciprocal of the plane spacing grows with the length of the (hkl) vector, which gives d = a/√(h² + k² + l²). That spacing is what enters Bragg's law, nλ = 2d sin θ, in diffraction.

Worked example

Default inputs: intercepts 1/2, 1, ∞; a = 361.5 pm (copper).

1. Reciprocals: 2, 1, 0.
2. No fractions or common factors, so the indices are (2 1 0).
3. h² + k² + l² = 4 + 1 + 0 = 5.
4. d = 361.5 ÷ √5 = 161.67 pm = 1.6167 Å.

How to read the result

Low-index planes such as (100), (110) and (111) are the most widely spaced and usually the most densely packed with atoms. Higher indices mean closer spacing. In face-centred cubic metals such as copper, only planes with h, k and l all odd or all even give diffraction peaks, so (210) appears in the calculation but not in a powder pattern.

Common mistakes

• Forgetting to take reciprocals.
• Writing ∞ as an index instead of 0.
• Leaving a plane that passes through the origin; shift the origin to a neighbouring cell first.
• Using the cubic spacing formula for non-cubic lattices.

Frequently asked questions

What does a zero index mean?

The plane is parallel to that axis.

What do brackets mean in crystallography?

(hkl) is one plane family, {hkl} all equivalent planes, [uvw] a direction.

Why does (210) not appear in copper's powder pattern?

In face-centred cubic lattices, mixed odd and even indices give zero intensity.

How is the spacing used?

In Bragg's law, to predict diffraction angles.

Sources: Standard crystallographic convention for Miller indices and cubic interplanar spacing (e.g. W. D. Callister, Materials Science and Engineering: An Introduction)