Miller index, d-spacing and plane angle calculator
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KEY POINTS
For a cubic crystal the interplanar spacing is d = a/√(h²+k²+l²). For silicon (a = 5.431 Å) the (111) spacing is 3.136 Å and its Bragg angle with Cu Kα radiation is 2θ = 28.44°. The (100) and (111) planes meet at 54.74°, the slope angle of windows etched into (100) wafers with KOH.
Interplanar spacing, diffraction angle and angle between planes
Enter negative indices with a minus sign, for example (1 1 −1). The default wavelength is Cu Kα1 (1.5406 Å).
Main diffraction planes of Silicon (Si)
| Plane | Spacing | 2θ | Selection rule |
|---|---|---|---|
| (111) | 3.1356 Å | 28.44° | Allowed |
| (220) | 1.9201 Å | 47.30° | Allowed |
| (311) | 1.6375 Å | 56.12° | Allowed |
| (400) | 1.3578 Å | 69.13° | Allowed |
| (331) | 1.2460 Å | 76.38° | Allowed |
| (422) | 1.1086 Å | 88.03° | Allowed |
| (511) | 1.0452 Å | 94.95° | Allowed |
| (440) | 0.9601 Å | 106.71° | Allowed |
| (200) | 2.7155 Å | 32.96° | Forbidden (all even with h+k+l = 4n+2) |
| (222) | 1.5678 Å | 58.86° | Forbidden (all even with h+k+l = 4n+2) |
Common plane angles in silicon
| Pair | Angle | Where it matters |
|---|---|---|
| (100) and (111) | 54.74° | Sidewall angle in KOH anisotropic etching |
| (100) and (110) | 45.00° | Relation to the <110> notch direction of (100) wafers |
| (110) and (111) | 35.26° | Vertical trench etching in (110) wafers |
| (111) and (11-1) | 70.53° | Between {111} planes (cleavage planes) |
Miller indices and semiconductor processing
Miller indices describe the orientation of a crystal plane; in a cubic crystal the normal to the (hkl) plane points in the [hkl] direction. CMOS logic and memory are built on (100) wafers, which have few interface states and high electron mobility, and the notch (or orientation flat) marks the <110> direction. Transistor channels are normally laid out along this <110> direction.
Orientation also governs wet etching and cleaving. Alkaline etchants such as KOH and TMAH barely etch (111) planes, so a window opened in a (100) wafer becomes a V-groove or pyramidal pit bounded by (111) planes with 54.74° slopes, which is used to make MEMS diaphragms. Silicon cleaves easily along {111} planes, so crystal orientation also shows up in dicing and in how wafers break.
FAQ
Why is the (200) reflection missing for silicon?
In the diamond structure, planes with all-even indices diffract only when h+k+l is a multiple of 4. For (200) the sum is 2, so the reflection is forbidden.
Why do KOH-etched grooves in (100) silicon have 54.74° walls?
Alkaline etchants barely attack (111) planes, so etching stops on them, and the angle between (100) and (111) planes is 54.74°.
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Last updated: October 23, 2026