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Interplanar Spacing (Cubic) Calculator

Relates the unit cell dimension and Miller indices to the distance between lattice planes.

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Interplanar Spacing

Formula first

Overview

This formula defines the perpendicular distance between adjacent parallel planes in a cubic crystal lattice, designated by their Miller indices. It serves as a fundamental geometric relationship in crystallography, linking the unit cell dimension to the observable diffraction patterns of minerals.

Symbols

Variables

= Interplanar Spacing, a = Lattice Parameter, h = Miller Index h, k = Miller Index k, l = Miller Index l

Interplanar Spacing
The perpendicular distance between parallel planes of atoms
Lattice Parameter
The edge length of the cubic unit cell
Miller Index h
Reciprocal intercept on the crystallographic x-axis
Miller Index k
Reciprocal intercept on the crystallographic y-axis
Miller Index l
Reciprocal intercept on the crystallographic z-axis

Apply it well

When To Use

When to use: Apply this equation exclusively to minerals in the isometric (cubic) system where all crystal axes are equal in length and perpendicular to one another. It is typically used during X-ray diffraction analysis to calculate d-spacing from known lattice parameters or vice versa.

Why it matters: Understanding interplanar spacing is vital for mineral identification and interpreting atomic arrangements within a crystal. It allows geoscientists to relate the macroscopic physical properties of a mineral, such as cleavage and hardness, to its internal microscopic structure.

Avoid these traps

Common Mistakes

  • Applying this cubic-specific formula to minerals in non-cubic crystal systems.
  • Neglecting to square the h, k, and l values before summing them.

One free problem

Practice Problem

A sample of Halite (NaCl) has a cubic lattice constant of 5.64 Å. Calculate the interplanar spacing for the (200) plane.

Lattice Parameter5.64 The edge length of the cubic unit cell
Miller Index h2 Reciprocal intercept on the crystallographic x-axis
Miller Index k0 Reciprocal intercept on the crystallographic y-axis
Miller Index l0 Reciprocal intercept on the crystallographic z-axis

Solve for:

Hint: Plug the indices into the denominator: √(2² + 0² + 0²) simplifies to 2.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Charles Kittel, Introduction to Solid State Physics
  2. William D. Callister Jr., David G. Rethwisch, Materials Science and Engineering: An Introduction
  3. Wikipedia: Miller index
  4. Elements of X-ray Diffraction (3rd ed.) by B.D. Cullity and S.R. Stock
  5. Introduction to Mineralogy by William D. Nesse (4th ed., Oxford University Press)
  6. Introduction to Solid State Physics by Charles Kittel (8th ed., Wiley)
  7. Wikipedia article "Diamond
  8. Wikipedia article "Halite