Angle Between Crystal Planes Calculator

Calculator

The calculator treats each crystal plane as a reciprocal lattice vector.

Plane vector: g = h a* + k b* + l c*

Direct metric tensor:

G = [[a², ab cos gamma, ac cos beta], [ab cos gamma, b², bc cos alpha], [ac cos beta, bc cos alpha, c²]]

Reciprocal metric tensor: G* = inverse of G

Cosine formula: cos theta = (g1 · G* · g2) / sqrt((g1 · G* · g1)(g2 · G* · g2))

Plane spacing: d = 1 / sqrt(g · G* · g)

The calculator returns the acute angle between the two planes.

1. Select a crystal system, or keep General for manual entry.
2. Enter the Miller indices for the first plane.
3. Enter the Miller indices for the second plane.
4. Enter lattice constants a, b, and c.
5. Enter alpha, beta, and gamma in degrees.
6. Press Calculate Angle.
7. Review the result above the form and below the header.
8. Use CSV or PDF export for saving your result.

Crystal Plane Geometry

Crystallography studies ordered atoms. Planes describe repeated sheets inside a unit cell. Their angle helps compare cleavage, diffraction, slip, and growth directions. This calculator uses Miller indices and lattice data to measure that angular separation.

Why the Metric Matters

For simple cubic crystals, the method is familiar. The plane normal for (h k l) points along a reciprocal direction. The angle follows the dot product of two normals. When a crystal is not cubic, the cell edges and cell angles change the geometry. A direct dot product on h, k, and l can then be wrong.

General Cell Method

This tool handles a general unit cell. It builds the direct metric tensor from a, b, c, alpha, beta, and gamma. Then it inverts that tensor to create the reciprocal metric. Each plane is treated as a reciprocal vector. The calculator compares those two vectors and returns the angle in degrees and radians.

Useful Applications

The result is useful in materials science. It can check whether two faces are nearly parallel. It can support X ray diffraction interpretation. It also helps students see how lattice symmetry changes plane relationships. Cubic, tetragonal, orthorhombic, hexagonal, monoclinic, and triclinic cells can all be tested by changing the constants.

Input Rules

Good inputs matter. Use positive lattice lengths. Enter cell angles in degrees. Miller indices should be integers, but negative values are allowed. Do not enter zero for every index in one plane. The plane (0 0 0) has no valid orientation.

Result Details

The calculator also estimates each plane spacing from the reciprocal vector length. The d values are helpful reference checks. Smaller d spacing means closer parallel planes. The final table keeps both raw inputs and computed outputs together for easy review.

Export and Review

Use the preset menu for common systems. You can still edit every lattice value. This is helpful when a real sample has measured constants. After calculation, export the result to CSV for spreadsheets. You can also save a PDF summary for reports or lab notes. The example table gives quick test cases. Compare them with your result to confirm input format.

Accuracy Note

For best accuracy, keep enough decimal places in measured constants. Very small angle differences can be sensitive. Review rounded answers before using them in design decisions, grading, or final published lab calculations safely.