Miller Indices Calculator
ChemistryCalculate interplanar d-spacing from Miller indices (hkl) for cubic crystal systems. Find Bragg diffraction angle θ and lattice plane geometry.
1
1
0
4.05 Å
Å
1.541 Å
Å
1
Interplanar Spacing (d)
2.864
Bragg Angle (θ)
15.604
2θ (XRD peak position)
31.207
Plane Family
(1 1 0)
What is a Miller Indices?
The Miller Indices Calculator computes the interplanar d-spacing (Å) and Bragg diffraction angle (2θ) for any (hkl) crystal plane in a cubic system. Enter the Miller indices, lattice constant, X-ray wavelength, and diffraction order to get d-spacing and the XRD peak position.
Miller indices (hkl) define crystal planes by the reciprocals of their fractional intercepts with the crystallographic axes. The d-spacing formula for cubic systems is d = a / √(h²+k²+l²), and Bragg's law (nλ = 2d sinθ) relates d-spacing to the diffraction angle. The 2θ positions of XRD peaks form the crystal's unique diffraction fingerprint — used for phase identification, lattice constant measurement, and crystal structure determination.
For computing the lattice constant and atomic packing from unit cell geometry, the Cubic Cell Calculator provides the geometric relationships between lattice constant a and atomic radius. For computing molar mass and density using the lattice constant from XRD, combine the Cubic Cell Calculator with the Molar Mass Calculator.
How to use this Miller Indices calculator
- Enter Miller Indices (h, k, l) — integers identifying the crystal plane. For the most common XRD peaks, try (111), (200), (220), (311) for FCC; (110), (200), (211) for BCC.
- Enter Lattice Constant (a, Å) — from literature or previous XRD measurement. Al: 4.05 Å; Fe: 2.87 Å; Cu: 3.61 Å; Si: 5.43 Å.
- Enter X-Ray Wavelength (λ, Å) — Cu Kα = 1.5406 Å (most common); Mo Kα = 0.7107 Å (for smaller unit cells); synchrotron (varies).
- Enter Diffraction Order (n) — usually 1. Higher orders (n=2,3) give the same peak positions as (2h, 2k, 2l) planes.
- Read 2θ (°) — the expected peak position in your XRD diffractogram.
Formula & Methodology
d-spacing (cubic system):d_hkl = a / √(h² + k² + l²)Bragg's Law:nλ = 2d sinθ sinθ = nλ / (2d) θ = arcsin(sinθ) [valid only if |sinθ| ≤ 1] 2θ = 2 × θWorked example — Silicon (100) wafer, Cu Kα radiation: Silicon: a = 5.430 Å, cubic diamond structure (FCC-based with 2-atom basis). Most intense peak: (111) plane.d(111) = 5.430 / √(1²+1²+1²) = 5.430 / 1.732 = 3.135 Å Bragg condition: sinθ = 1 × 1.5406 / (2 × 3.135) = 1.5406/6.270 = 0.2457 θ = arcsin(0.2457) = 14.22° 2θ = 28.44°This matches the known Si (111) XRD peak at 2θ = 28.44° with Cu Kα radiation — the reference peak used for XRD instrument calibration at BARC Mumbai and CSIR-NML Jamshedpur. Silicon single-crystal wafers from the planned Tata-PSMC fab in Dholera will be characterised by this exact XRD signature to confirm crystalline perfection before device fabrication.
Frequently Asked Questions
Miller indices (hkl) are a notation system for describing crystal planes in a lattice. They are defined as: take the intercepts of the plane with the three crystallographic axes (x=a/h, y=a/k, z=a/l), take the reciprocals, and convert to the smallest integers. For a plane that cuts the x-axis at 1, and is parallel to y and z (intercepts ∞): reciprocals (1, 0, 0) → Miller indices (100). For a plane cutting x at 1, y at 1, z at ∞: indices (110). For x at 1, y at 1, z at 1: indices (111). Negative intercepts are written with a bar: (1̄00) for a plane cutting −x. Curly brackets {hkl} denote the family of equivalent planes.
Enter the three Miller indices h, k, l (integers, can be negative). Enter the Lattice Constant (a, in Å) for your cubic crystal. Enter the X-Ray Wavelength (λ, default 1.5406 Å = Cu Kα radiation, the most common XRD source). Enter Diffraction Order (n, default 1). The calculator returns d-spacing (Å), Bragg angle θ (°), XRD peak position 2θ (°), and a formatted plane label. Default: (110) plane of aluminium (a=4.05 Å) with Cu Kα radiation.
For cubic crystal systems (a = b = c, all angles 90°), the interplanar spacing is: d_hkl = a / √(h² + k² + l²). This is derived from the general triclinic formula simplified for cubic symmetry. For common planes: (100): d = a. (110): d = a/√2 ≈ 0.707a. (111): d = a/√3 ≈ 0.577a. (200): d = a/2. (220): d = a/(2√2) ≈ 0.354a. For non-cubic systems, the formula differs: tetragonal: 1/d² = (h²+k²)/a² + l²/c²; hexagonal: 1/d² = (4/3)(h²+hk+k²)/a² + l²/c².
Bragg's law (1913, William and Lawrence Bragg, Nobel Prize Physics 1915): nλ = 2d sinθ, where n = diffraction order (integer, usually 1), λ = X-ray wavelength (Å), d = interplanar spacing (Å), θ = Bragg angle (half the diffraction angle). At θ: X-rays reflected from successive crystal planes interfere constructively — giving a peak in the diffraction pattern at angle 2θ. The 2θ peak positions in an XRD diffractogram are the 'fingerprint' of a crystal structure — used to identify phases, measure lattice constants, and quantify crystal size (Scherrer equation: L = Kλ/(β cosθ), where β = peak full-width at half maximum).
Cu Kα radiation (λ = 1.5406 Å) is the standard for laboratory XRD (Rigaku, Bruker, PANalytical — instruments at IIT Bombay, BARC, CSIR-CECRI): Aluminium FCC (a=4.05 Å): (111) 2θ=38.5°; (200) 2θ=44.7°; (220) 2θ=65.1°. Iron BCC (a=2.87 Å): (110) 2θ=44.7°; (200) 2θ=65.0°; (211) 2θ=82.3°. Silicon diamond cubic (a=5.43 Å): (111) 2θ=28.4°; (220) 2θ=47.3°; (311) 2θ=56.1°. Quartz (SiO₂, hexagonal): d(100)=4.26 Å → 2θ=20.8°; d(011)=3.34 Å → 2θ=26.6° (the characteristic quartz peak). Indian XRD applications: iron ore characterisation (NMDC), alumina/silica analysis in SAIL steel plants, zirconia ceramics at CSIR-CIMFR.
Not all (hkl) reflections are observed in XRD — systematic absences depend on the lattice type: SC (Primitive): all hkl peaks allowed. BCC: only peaks where h+k+l = even are observed (e.g., (110), (200), (211) but not (100), (111), (210)). FCC: only peaks where h,k,l are all odd or all even (e.g., (111), (200), (220), (311) but not (100), (110), (210)). These selection rules arise from the destructive interference of X-rays scattered by atoms at the lattice points versus the body-centre or face-centre positions. The pattern of present vs absent peaks identifies the lattice type — a key step in crystal structure determination.
Silicon wafer orientation is specified by Miller indices: (100) wafers: used for most logic devices (CMOS, DRAM); preferential etch along {100} planes with KOH → pyramidal etch pits for MEMS fabrication. (111) wafers: used for bipolar transistors (epitaxial layers); also for MEMS pressure sensors. (110) wafers: higher electron/hole mobility, used for some high-speed devices. India's planned semiconductor fabs (Tata Electronics PSMC fab in Dholera, Gujarat; ISMC fab) will use (100) silicon wafers. XRD Laue diffraction is used to precisely align wafer orientation before cutting — the flat (notch) on commercial wafers indicates the reference direction.
The Scherrer equation gives the crystallite size (L) from XRD peak broadening: L = K × λ / (β × cosθ), where K = Scherrer constant ≈ 0.9, λ = wavelength, β = peak width (FWHM in radians), θ = Bragg angle. The 2θ and θ values used in the Scherrer equation come from the Miller indices-based Bragg calculation (this calculator). For nanoparticles: L < 100 nm causes measurable broadening. Indian nanoparticle research (silver nanoparticles from Ayurvedic Swarna Bhasma at 5–50 nm, TiO₂ photocatalysts at CSIR labs, ZnO nanoparticles) routinely uses Scherrer analysis on the (111) or (200) XRD peaks.
d-spacing physically represents the perpendicular distance between adjacent parallel planes of atoms in the crystal. For the (111) planes of an FCC metal: neighbouring parallel (111) planes are separated by d = a/√3. For larger d-spacing (lower hkl indices): planes are further apart, lower diffraction angle (2θ), easier to see in diffraction. The (100) planes of NaCl (d = 2.82 Å) give a 2θ ≈ 31.7° peak with Cu Kα. Real-space d-spacings in materials range from 0.5 Å (small unit cell, high angle diffraction) to >100 Å (large biomolecules, protein crystallography with synchrotron radiation at DESY/APS — Indian crystallography groups at IISc, TIFR, NCBS).
Industrial XRD applications in India: (1) Steel quality control: phase identification (Fe₃C carbide, martensite, austenite, ferrite) at SAIL Bhilai and Tata Steel Jamshedpur. (2) Cement industry: clinker phase analysis (alite/belite ratio, free lime) at ACC, Ambuja, UltraTech Cement. (3) Pharmaceutical polymorphism: detecting active ingredient crystal form (amorphous vs crystalline, polymorph identification) at Cipla, Aurobindo — critical because different polymorphs have different dissolution rates. CDSCO requires API polymorph characterisation. (4) Mining/ore characterisation: NMDC iron ore, NALCO bauxite. (5) Zeolite catalysts: pore structure at IOCL R&D, Reliance petrochemicals for FCC (fluid catalytic cracking) catalyst QC.