Miller Indices Calculator Formula
Use this Miller Indices Calculator Formula to work through the same calculation as the main calculator page with clear steps, examples, and result context.
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What This Miller Indices Calculator Formula Helps You Do
This page focuses on the part of Miller indices most people actually need in calculations: turning h, k, and l into a cubic-crystal interplanar spacing. That makes it useful for quick homework checks, diffraction prep, and crystal-structure comparisons.
The result also shows the reciprocal-spacing term so you can see how the index sum changes the answer rather than only getting a single number.
How to Calculate Miller Indices Calculator Formula
- Enter the cubic lattice constant: Use a positive lattice constant in the same length unit you want for the spacing result.
- Enter h, k, and l: Use Miller indices for the plane of interest. At least one index must be non-zero.
- Apply the cubic d-spacing formula: The denominator uses the square root of h squared plus k squared plus l squared.
- Interpret the spacing: Higher-order planes with larger indices have smaller interplanar spacing.
Miller Indices Calculator Formula Formula
| Variable | Meaning | Unit |
|---|---|---|
| a | Lattice constant | pm, Å, or chosen length unit |
| h, k, l | Miller indices of the crystal plane | dimensionless integers |
| d | Interplanar spacing | same length unit as a |
Use the worked examples below to check how the formula behaves with real values. If the result looks unexpected, verify the unit assumptions and the meaning of each variable before interpreting the answer.
Worked Examples
- a: 400 pm
- hkl: (1 0 0)
Result: d = 400 pm.
A (100) plane in a cubic crystal has the same spacing as the lattice constant.
- a: 400 pm
- hkl: (1 1 0)
Result: d = 282.84 pm.
Adding a second non-zero index increases the denominator and reduces spacing.
- a: 400 pm
- hkl: (1 1 1)
Result: d = 230.94 pm.
The (111) plane is more closely spaced than (100) and (110) in the same cubic cell.
- a: 543 pm
- (200): h=2, k=0, l=0
- (211): h=2, k=1, l=1
Result: (211) gives the smaller d-spacing.
The larger h²+k²+l² sum produces the smaller interplanar distance.
How to Interpret Your Results
| Range | Meaning | Action |
|---|---|---|
| Low h² + k² + l² | Wider plane spacing. | Use these planes when you expect lower reciprocal spacing and larger d values. |
| Moderate h² + k² + l² | Intermediate plane spacing. | Compare planes within the same crystal by keeping the lattice constant fixed. |
| High h² + k² + l² | Tighter plane spacing. | Expect smaller d values and larger reciprocal spacing. |
Frequently Asked Questions
References
Last reviewed: March 2026