Motif Count Calculator

Formula Used

This calculator models a crystal as a repeated array of unit cells. A unit cell contains P lattice points. Each lattice point carries m motifs, and only a fraction f may be occupied.

• N_cells = nx × ny × nz
• Motifs per unit cell = P × m × f
• Total motifs = P × m × N_cells × f
• Total atoms = Total motifs × a

How to Use This Calculator

1. Select a lattice type that matches your structure.
2. Enter motifs per lattice point and atoms per motif.
3. Set nx, ny, nz to describe the crystal repetition.
4. Use occupancy fraction for vacancies or partial filling.
5. Press Calculate to display totals and exports.

Motif Count in Crystal Models

1) Purpose of this calculator

This tool estimates how many motifs exist in a repeated crystal built from identical unit cells. It is useful for planning atomistic simulations, checking basis definitions in crystallography, and validating supercell construction. You can work with common lattice choices or enter a custom lattice-point count for specialized unit cells.

2) What “motif” means here

A motif is the repeating basis attached to a lattice point. In many introductory models, the motif is a single atom and motifs per lattice point equals 1. In multi-atom bases, the motif can represent a group of atoms, ions, or molecules whose internal arrangement repeats throughout the lattice.

3) Lattice points per unit cell (P)

The lattice type sets P, the number of lattice points effectively contained in one unit cell. Typical values include SC = 1, BCC = 2, and FCC = 4. When your structural convention differs, choose “Custom” and enter the appropriate P for your chosen cell definition.

4) Motifs per lattice point (m)

The parameter m scales how many motifs are attached to each lattice point. For a single basis motif, set m = 1. For models where a point hosts multiple identical motifs (for example, layered repeats or multi-sublattice bookkeeping), increase m accordingly.

5) Occupancy fraction (f) and defects

Real crystals may include vacancies or partial site filling. The occupancy fraction f (0 < f ≤ 1) applies a uniform reduction to motif totals. For example, an FCC crystal with nx = ny = nz = 5 has 125 unit cells. With f = 1, total motifs are 4 × 1 × 125 = 500. With f = 0.90, the estimate becomes 450.

6) Supercell scaling (nx, ny, nz)

Replication factors nx, ny, and nz define the supercell size. The number of unit cells is N_cells = nx × ny × nz. This scaling is linear, so doubling one dimension doubles the total motifs. This is especially helpful when budgeting computational cost for molecular dynamics or Monte Carlo runs.

7) Converting motifs to atoms

If each motif contains multiple atoms, use a as atoms per motif to estimate total atoms: Total atoms = Total motifs × a. For a BCC example with nx = 4, ny = 3, nz = 2, m = 1, and f = 0.95, total motifs are 2 × 1 × 24 × 0.95 = 45.6. If a = 2, total atoms are 91.2.

8) Practical uses and reporting

Common use cases include comparing alternative unit-cell conventions, estimating how many basis units are needed for a target crystal volume, and producing clean calculation records. Use the export buttons to save CSV for lab notes or generate a PDF summary for reports, teaching materials, and design documentation.

Frequently Asked Questions

1) Why can the calculator output non-integer motifs?

Fractional occupancy f represents an average filling level, so totals can be non-integers. For discrete counts, interpret results as expected values across many similar cells or samples.

2) What should I enter for atoms per motif?

Enter the number of atoms contained in one motif (basis unit). Use 1 if you only want motif totals, or use the full basis size to estimate total atoms in the supercell.

3) How do I handle a known motifs-per-unit-cell value?

Set P to 1 using “Custom” and enter your known motifs-per-cell value as m, with f = 1. This reproduces the same total scaling with nx, ny, nz.

4) Does the lattice choice change geometry or only counting?

This calculator focuses on counting. The lattice choice supplies P for common conventions. Geometric properties like lattice parameters or angles are not used in the count.

5) What is a good way to model vacancies?

Use the occupancy fraction f to represent uniform vacancies. For example, f = 0.98 approximates a 2% vacancy level across all sites, producing an expected motif and atom total.

6) Can I use it for multi-sublattice crystals?

Yes. Represent each sublattice contribution by adjusting m and a to match your basis definition. If your unit cell convention changes P, use the “Custom” option for precise counting.

7) What do the CSV and PDF exports include?

Exports include all inputs, key outputs (unit cells, motifs per cell, total motifs, total atoms), and the formulas used. This makes it easy to paste results into notebooks or reports.