What this calculator does
Miller indices describe a crystal plane by the reciprocals of where it cuts the three axes, cleared of fractions. A plane cutting at 1, 1 and one half gives (1 1 2).
Reciprocals are used rather than the intercepts themselves because a plane parallel to an axis never cuts it, and an intercept of infinity is awkward to write down. Its reciprocal is zero, which is not, and that single convenience is why the notation took the form it did.
The formula
Each fractional intercept is inverted and the three reciprocals are multiplied up by their lowest common multiple to clear fractions. The interplanar spacing for a cubic lattice is the lattice constant divided by the square root of the sum of the squared indices, so planes with larger indices are more closely spaced.
| Term | Meaning |
|---|---|
| Miller indices (hkl) | The cleared reciprocals of the axial intercepts, in parentheses. |
| Interplanar spacing (d) | The perpendicular distance between adjacent parallel planes. |
| Bragg’s law | Relates d spacing to diffraction angle, which is how these are measured. |
| Parallel to an axis | Gives an index of zero, since the reciprocal of infinity is zero. |
The inputs explained
| Field | What to enter |
|---|---|
| x-axis intercept (fraction of a) | x-axis intercept as a fraction of the lattice constant. |
| y-axis intercept (fraction of b) | y-axis intercept. |
| z-axis intercept (fraction of c) | z-axis intercept. A small value means the plane cuts close to the origin on that axis and gives a large index. |
| Cubic lattice constant a (pm) | Cubic lattice constant in pm, used only for the spacing calculation. |
When to use it
Indexing a diffraction pattern
X-ray peaks correspond to specific planes, and the indices identify which.
Describing a crystal face
Growth faces, cleavage planes and surfaces are all specified by Miller indices.
Predicting cleavage
Crystals cleave along widely spaced planes, which are those with small indices.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What indices do these intercepts give?
A range of z-axis intercepts with x and y fixed at 1.
| z-axis intercept | Miller indices (hkl) | Interplanar spacing d | h²+k²+l² |
|---|---|---|---|
| 1 a | (1 1 1) | 230.94 pm | 3 |
| 0.5 a | (1 1 2) | 163.30 pm | 6 |
| 0.25 a | (1 1 4) | 94.28 pm | 18 |
Questions
Why use reciprocals?
Chiefly so that a plane parallel to an axis has a finite index. Such a plane never intercepts that axis, so its intercept is infinity, but the reciprocal is zero. Reciprocals also make the indices proportional to the spacing relationship used in diffraction.
What does a zero index mean?
That the plane is parallel to that axis. The (1 0 0) plane cuts the x-axis and runs parallel to both y and z, which makes it a face of the cubic cell. Zeros are common among the low-index planes that dominate crystal surfaces.
What do the brackets mean?
Round brackets denote a specific plane, curly braces a family of equivalent planes related by symmetry, square brackets a specific direction and angle brackets a family of directions. Mixing them up changes the meaning entirely.
Which planes have the widest spacing?
Those with the smallest indices. The (1 0 0) plane has a spacing equal to the lattice constant, while (1 1 1) has a smaller one. Widely spaced planes are the weakest-bonded and therefore the ones along which crystals cleave and grow.
For unit cell geometry, see the cubic unit cell calculator. For lattice energetics, see the lattice energy calculator.