Crystals are described by planes and directions of atoms, and Miller indices are the standard shorthand for naming them. They decide how silicon wafers are cut ((100) and (111) wafers), how X-ray diffraction is interpreted, and why properties such as etching rate and carrier mobility depend on direction. “Find the Miller indices of a plane with intercepts …” and “interplanar spacing of a cubic crystal” are common problems in physical electronics, materials science, BCS and university exams. This post gives a step-by-step method, a rotating 3D animation, solved examples and MCQs.
- Miller indices (hkl) = smallest integers proportional to the reciprocals of the intercepts of a plane on the x, y, z axes, in units of lattice constants.
- A plane parallel to an axis has intercept ∞ on that axis, so its index is 0.
- A negative index is written with a bar: (1̄10).
- Notation: (hkl) = one plane; {hkl} = family of equivalent planes; [uvw] = direction; ⟨uvw⟩ = family of directions.
- Cubic crystal: dhkl = a / √(h² + k² + l²), and direction [hkl] is perpendicular to plane (hkl).
1. Why Do We Need Miller Indices?
A crystal is a regular 3D arrangement of atoms built by repeating a unit cell. Through this lattice we can draw many sets of parallel planes, each with a different density of atoms. Physical properties are often anisotropic, i.e. they depend on the plane or direction. For example, silicon oxidizes and etches at different rates on (100) and (111) faces, and MOSFETs are usually built on (100) wafers because of their lower interface-state density. Miller indices, introduced by W. H. Miller in 1839, give each plane a compact, unambiguous label.
2. Step-by-Step Method
- Choose the origin at a lattice point that is not on the plane. Shift the origin to a neighbouring corner if the plane passes through it.
- Find the intercepts of the plane on the x, y and z axes as multiples of the lattice constants a, b, c. Use ∞ if the plane is parallel to an axis.
- Take reciprocals of the intercepts (1/∞ = 0).
- Clear fractions: multiply by the LCM to get the smallest whole numbers.
- Enclose in parentheses (hkl), with no commas, and put a bar over negative numbers.
3. Important Cubic Planes (3D Animation)
| Plane | Intercepts (x, y, z) | Reciprocals | Description |
|---|---|---|---|
| (100) | 1, ∞, ∞ | 1, 0, 0 | A cube face, perpendicular to x |
| (110) | 1, 1, ∞ | 1, 1, 0 | Diagonal plane, parallel to z |
| (111) | 1, 1, 1 | 1, 1, 1 | Cuts all three axes equally; triangular section |
| (200) | ½, ∞, ∞ | 2, 0, 0 | Parallel to (100) at half the spacing |
In an FCC crystal (Cu, Al, Au), the {111} planes are the most densely packed planes. In BCC crystals (Fe at room temperature), the {110} planes are the most densely packed. Slip in metals usually occurs on these close-packed planes.
4. Solved Examples
Example 1. A plane cuts the axes at 2a, 3b and ∞c. Find its Miller indices.
Intercepts 2, 3, ∞ → reciprocals ½, ⅓, 0 → × 6 → (3 2 0)Example 2. A plane has intercepts a, −b, c/2.
1, −1, ½ → reciprocals 1, −1, 2 → (1 1̄ 2)Example 3 (reverse problem). Find the intercepts of the (123) plane.
Reciprocals of 1, 2, 3 → 1, ½, ⅓ → the plane cuts x at a, y at b/2, z at c/3Example 4. The lattice constant of a cubic crystal is a = 4 Å. Find d100, d110 and d111.
d100 = 4/√1 = 4 Å, d110 = 4/√2 = 2.83 Å, d111 = 4/√3 = 2.31 ÅExample 5. For a = 3.6 Å and Cu-Kα X-rays (λ = 1.54 Å), find the first-order Bragg angle for the (111) planes.
d111 = 3.6/√3 = 2.078 Å; sin θ = λ/2d = 1.54/4.157 = 0.370 ⟹ θ ≈ 21.7°5. Notation: Planes, Families and Directions
| Symbol | Meaning | Example |
|---|---|---|
| (hkl) | A specific plane (or set of parallel planes) | (111) |
| {hkl} | Family of equivalent planes | {100} = (100), (010), (001), (1̄00), (01̄0), (001̄) |
| [uvw] | A specific direction | [110] |
| ⟨uvw⟩ | Family of equivalent directions | ⟨111⟩ = all 8 body diagonals |
| (hkil) | Miller–Bravais indices for hexagonal crystals, with i = −(h + k) | (0001) basal plane |
To find a direction [uvw], subtract the tail coordinates from the head coordinates and reduce to the smallest integers. For example, from (0,0,0) to (1,1,1) is [111].
6. Interplanar Spacing and Bragg’s Law
Cubic: dhkl = a / √(h² + k² + l²) Bragg: 2d sin θ = nλHigher Miller indices mean more closely spaced planes, and each plane contains fewer atoms per unit area. The angle between two planes in a cubic crystal is
cos φ = (h1h2 + k1k2 + l1l2) / [√(h1² + k1² + l1²) √(h2² + k2² + l2²)]Example: between (100) and (111): cos φ = 1/√3 → φ = 54.7°. This is the angle of the sloped side walls that KOH etching produces on (100) silicon, a well-known MEMS fact.
7. Important Features
- All parallel planes have the same Miller indices; (hkl) and (h̄k̄l̄) describe the same set of planes.
- A plane passing through the origin is described by a parallel plane after shifting the origin.
- Miller indices are always integers with no common factor (for planes; (200) is used in diffraction to mean planes at half the (100) spacing).
- In cubic crystals only, direction [hkl] is normal to plane (hkl).
- A zero index means the plane is parallel to that axis; a large index means the plane cuts that axis close to the origin.
8. Practice MCQs
1. A plane parallel to the y- and z-axes and cutting the x-axis at a has Miller indices: (a) (010) (b) (100) (c) (001) (d) (111)
2. Intercepts 2a, 2b, ∞ give Miller indices: (a) (220) (b) (110) (c) (001) (d) (112)
3. Interplanar spacing of (111) in a cubic crystal with a = 3 Å is: (a) 1.73 Å (b) 3 Å (c) 5.2 Å (d) 1 Å
4. The notation {100} represents: (a) A single plane (b) A direction (c) A family of equivalent planes (d) A family of directions
5. In a cubic crystal, the direction [hkl] is ____ to the plane (hkl). (a) Parallel (b) Perpendicular (c) At 45° (d) At 54.7°
6. The most densely packed planes in FCC are: (a) {100} (b) {110} (c) {111} (d) {210}
7. Intercepts a, b/2, c/3 give: (a) (123) (b) (321) (c) (632) (d) (236)
8. Bragg’s law is: (a) nλ = d sin θ (b) nλ = 2d sin θ (c) λ = 2d cos θ (d) nλ = d/θ
9. Angle between (100) and (111) planes of a cubic crystal: (a) 45° (b) 54.7° (c) 60° (d) 90°
10. Miller–Bravais indices (hkil) are used for: (a) Cubic (b) Tetragonal (c) Hexagonal (d) Triclinic crystals
9. বাংলায় দ্রুত রিভিশন
- মিলার ইনডেক্স: অক্ষগুলোর ওপর তলের ছেদাংশের (intercept) বিপরীত রাশির ক্ষুদ্রতম পূর্ণসংখ্যা অনুপাত (hkl)।
- ধাপ: ছেদাংশ → বিপরীত রাশি → ভগ্নাংশ দূর → (hkl); অক্ষের সমান্তরাল হলে সূচক ০।
- ঋণাত্মক সূচকের ওপর বার চিহ্ন; {hkl} = সমতুল্য তলের পরিবার, [uvw] = দিক।
- ঘনক কেলাসে d = a/√(h² + k² + l²); [hkl] দিক (hkl) তলের লম্ব।
- FCC-তে {111} এবং BCC-তে {110} তল সবচেয়ে ঘন সন্নিবিষ্ট।
- ব্র্যাগের সূত্র: 2d sin θ = nλ।
Related Posts
- Unit Cell and Its Types (SC, BCC, FCC)
- Bravais Lattices
- What Is a Crystal?
- Types of Solids
- Semiconductor Surface Effects
References: Charles Kittel, Introduction to Solid State Physics; William D. Callister, Materials Science and Engineering; S. O. Pillai, Solid State Physics. Images and animations: original work by Totalecer.