Advanced Crystal Structure Calculator

Crystal structure calculator

This form uses a responsive three-column grid on large screens, two columns on medium screens, and one column on mobile devices.

Crystal structure

Primary input mode

Atomic mass (g/mol)

Example: Cu = 63.546, Fe = 55.845, Mg = 24.305.

Atomic radius r (Å)

Lattice parameter a (Å)

c/a ratio for HCP

Ideal HCP ratio is about 1.633.

Miller index h

Miller index k

Miller index l

X-ray wavelength λ (Å)

Diffraction order n

Decimal precision

Repeated cells in x

Repeated cells in y

Repeated cells in z

Structure	Radius (Å)	Atomic Mass	Plane	Lattice a (Å)	Density (g/cm³)	APF (%)
FCC	1.28	63.546	(111)	3.6204	8.8947	74.0480
BCC	1.24	55.845	(110)	2.8637	7.8977	68.0175
SC	1.60	39.098	(100)	3.2000	1.9813	52.3599
HCP	1.44	24.305	(101)	2.8800	2.3894	74.0480

Formula used

Density: ρ = (Z × M) / (N_A × V)

Packing efficiency: APF = (volume of atoms in cell) / (unit cell volume)

Cubic plane spacing: d_hkl = a / √(h² + k² + l²)

Hexagonal plane spacing: 1 / d² = (4/3)(h² + hk + k²) / a² + l² / c²

Bragg law: nλ = 2d sinθ

Common radius relations: SC → a = 2r, BCC → a = 4r/√3, FCC → a = 2√2r, Diamond → a = 8r/√3, HCP → a = 2r

Here, Z is atoms per unit cell, M is molar mass, N_A is Avogadro's number, V is unit cell volume, and d is interplanar spacing.

How to use this calculator

Select the crystal structure that matches your material.
Choose whether atomic radius or lattice parameter a is your main input.
Enter atomic mass to calculate unit cell mass and density.
For HCP, keep or adjust the c/a ratio.
Enter Miller indices to compute interplanar spacing for a chosen plane.
Add wavelength and diffraction order to estimate Bragg angle.
Set repeated cells in x, y, and z to count atoms in a larger block.
Press the calculate button to display the result above the form.
Use the CSV button for spreadsheet export or the PDF button for a report copy.

FAQs

1. What does this calculator solve?

It computes lattice parameters, atomic radius, density, packing efficiency, coordination number, unit cell volume, d-spacing, and Bragg diffraction angles for several common crystal structures.

2. Which structures are supported?

The file supports simple cubic, body-centered cubic, face-centered cubic, hexagonal close-packed, and diamond cubic structures. These cover many introductory and intermediate solid-state problems.

3. When should I use radius mode?

Use radius mode when atomic radius is known from a handbook, textbook, or material reference. The calculator then derives the lattice parameter from the correct geometric relation.

4. When should I use lattice mode?

Use lattice mode when X-ray data or reference tables already provide the lattice constant. The calculator then back-calculates the effective atomic radius using the selected structure.

5. Why does Bragg angle sometimes fail?

A real angle exists only when nλ/(2d) is less than or equal to one. If the ratio exceeds one, the selected order and wavelength cannot satisfy Bragg's law.

6. Why is HCP handled differently?

HCP needs both a and c dimensions, so the calculator uses the c/a ratio. Its volume and plane-spacing equations differ from cubic structures because the geometry is hexagonal.

7. What does packing efficiency mean?

Packing efficiency shows how much of the unit cell volume is occupied by atoms. Higher values mean less empty space and usually tighter atomic packing.

8. Can I save the result?

Yes. Use the CSV button to export the current calculation as tabular data. Use the PDF button to create a shareable report of the visible results section.