1
1
0
3.8396
Å
1
1
0
3.8396
Å
The Miller Indices and d-Spacing Calculator converts crystallographic plane intercepts into Miller indices (hkl) and computes the interplanar spacing for cubic crystal systems. Miller indices are the universal language for describing planes and directions in crystals, essential for interpreting X-ray diffraction patterns, understanding crystal growth, and predicting material anisotropy.
The d-spacing between parallel crystal planes determines the diffraction angles observed in XRD experiments through Bragg's law. This calculator bridges the gap between the geometric description of crystal planes and the experimentally observable diffraction geometry, making it an indispensable tool for crystallography students and researchers.
Miller indices are derived from plane intercepts in three steps:
1. Take the reciprocals of the intercepts (in units of lattice parameters):
$$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$$
2. Multiply by the smallest common factor to get integers
3. The resulting integers (h k l) are the Miller indices
For example, intercepts at 1a, 2b, 3c give reciprocals 1, 1/2, 1/3 → multiply by 6 → (6 3 2).
The d-spacing for a cubic system is:
$$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$$
For tetragonal systems: $$\frac{1}{d^2} = \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2}$$
An intercept of infinity (plane parallel to that axis) gives a reciprocal of 0. Negative intercepts produce negative Miller indices, denoted with a bar over the number (e.g., $$\bar{1}$$).
Low Miller index planes like (100), (110), and (111) are the most widely spaced and typically the most important for diffraction and crystal properties. The (111) plane in FCC crystals is the close-packed plane with the highest atomic density. In BCC crystals, (110) is the most densely packed plane. Higher index planes have smaller d-spacings and produce diffraction at larger angles.
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Results
Reciprocals: 1, 0.5, 0.333 → scaled to integers: (6 3 2), d = 5.43/√(36+9+4) = 5.43/7 = 0.776 Å
Inputs
Results
Equal intercepts give (111), d = 5.43/√3 = 3.135 Å, the most important silicon diffraction peak
Miller indices (hkl) are a set of three integers that describe the orientation of a crystal plane relative to the unit cell axes. They are obtained by taking the reciprocals of the plane's intercepts with the three crystallographic axes and clearing fractions. Negative indices are written with bars (e.g., (1̄10)) and indicate intercepts on the negative axis side.
An infinite intercept means the plane is parallel to that axis and never intersects it. The reciprocal of infinity is zero. For example, a plane parallel to the c-axis with intercepts a, b, ∞ has Miller indices (110). Enter a very large number or note that the corresponding index will be 0.
The d-spacing (interplanar distance) is the perpendicular distance between adjacent parallel planes of the same Miller indices. It determines the diffraction angle through Bragg's law: nλ = 2d sin θ. Larger d-spacings produce diffraction at smaller angles, and vice versa.
Low-index planes like (100), (110), and (111) have the largest d-spacings and highest atomic densities. They are preferred for crystal cleavage, slip (deformation), and growth. Silicon wafers are cut along (100) or (111) planes. Low-index planes also produce the strongest diffraction peaks.
Each XRD peak corresponds to a set of Miller indices (hkl). The peak position (2θ angle) is determined by d-spacing through Bragg's law. The peak intensity depends on the structure factor, which involves the positions and scattering factors of atoms in the unit cell. Systematic absences reveal the lattice type (e.g., FCC shows only all-odd or all-even hkl).
Miller indices (hkl) in parentheses describe planes, while [uvw] in square brackets describe directions. The direction [hkl] is perpendicular to the plane (hkl) in cubic systems but not necessarily in other crystal systems. Families of equivalent planes are denoted {hkl} and equivalent directions
For non-cubic systems, the d-spacing formula is more complex. Tetragonal: 1/d² = (h²+k²)/a² + l²/c². Hexagonal: 1/d² = 4(h²+hk+k²)/(3a²) + l²/c². Orthorhombic: 1/d² = h²/a² + k²/b² + l²/c². This calculator provides the cubic formula; use the appropriate formula for other systems.
Systematic absences are missing diffraction peaks caused by destructive interference due to lattice centering or glide/screw symmetry. FCC allows only peaks where h,k,l are all odd or all even. BCC allows only h+k+l = even. These rules help identify crystal structure from diffraction patterns.
Yes, negative indices indicate intercepts on the negative side of an axis. They are written with a bar: (1̄00) means the plane intercepts the negative x-axis at distance a. Crystallographically, (hkl) and (h̄k̄l̄) represent parallel planes on opposite sides of the origin.
Multiplicity is the number of equivalent planes in the {hkl} family. In cubic systems, {100} has multiplicity 6 (three pairs of parallel planes), {110} has 12, and {111} has 8. Higher multiplicity planes produce stronger diffraction peaks because more planes contribute to the reflection.
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