Cubic Cell Calculator - Lattice Constant, Volume, APF
Use this cubic cell calculator to compute the lattice constant a, the unit cell volume V, the atoms per cell, and the atomic packing factor for simple, body-centered, or face-centered cubic crystals.
Cubic Cell Calculator
Results
What Is the Cubic Cell Calculator?
This calculator solves the geometry of the three cubic Bravais lattices (SC, BCC, FCC) from the atomic radius or the measured lattice constant. Enter an atomic radius and pick a lattice type, and it returns the lattice constant a, unit cell volume V, atoms per cell, and atomic packing factor.
- • Homework and exam preparation: Confirm the lattice constant for aluminum (FCC), iron (BCC), or any element from a tabulated atomic radius.
- • X-ray diffraction interpretation: Turn a measured lattice parameter from a powder pattern into the atomic radius that produced it.
- • Materials science comparisons: Compare packing efficiency between SC, BCC, and FCC for a given atomic species.
- • Crystallography intuition: See how the radius, lattice constant, and atoms-per-cell combine to give the APF values 0.5236, 0.6802, and 0.7405.
In crystallography a solid is described by a unit cell, the smallest repeating block that tiles through space. Every cubic cell is a cube, so the only lattice parameter is the edge length a. The three cubic Bravais lattices differ in how many extra atoms sit inside that cube, and the geometry of those extras fixes a in terms of the atomic radius r.
When the same crystal is studied by X-ray diffraction, the Braggs Law Calculator handles the wavelength, d spacing, and diffraction angle that go with the lattice constant this calculator returns.
How the Cubic Cell Calculator Works
The calculator implements the three cubic lattice relations as a pure function. Inputs are the lattice type, solve direction, numeric input, and length unit. Everything converts to angstroms, the geometric relation is applied, and the result converts back to the chosen output unit.
- r: Atomic radius, the radius of the sphere used to model one atom.
- a: Lattice constant, the cube edge length.
- V: Unit cell volume, V = a^3 in cubic units of the chosen length.
- n_atoms: Atoms per cell: 1 for SC, 2 for BCC, 4 for FCC.
- APF: Atomic packing factor = n_atoms * (4/3) * pi * r^3 / a^3.
For simple cubic each corner atom touches its six neighbors along the edges, so the edge equals two atomic radii: a = 2r. For body-centered cubic the interior atom touches the corner atoms along the body diagonal of length sqrt(3) * a, giving a = 4r / sqrt(3). For face-centered cubic each face-center atom touches the corner atoms along the face diagonal of length sqrt(2) * a, giving a = 4r / sqrt(2).
Once a is known, V = a^3, the atoms-per-cell constant is read from a lookup (1, 2, 4), and APF = sphere volume / cell volume. The three APF values are pi / 6 for SC, pi * sqrt(3) / 8 for BCC, and pi / (3 * sqrt(2)) for FCC - approximately 0.5236, 0.6802, and 0.7405.
Aluminum (FCC), r = 1.43 angstrom
FCC lattice, atomic radius r = 1.43 angstrom
a = 4 * 1.43 / sqrt(2) = 4.045 angstrom, V = 66.17 angstrom^3, atoms per cell = 4
a = 4.045 angstrom, V = 66.17 angstrom^3, atoms per cell = 4, APF = 0.7405, packing efficiency = 74.05 percent
Matches the tabulated FCC lattice constant for aluminum; APF stays at the theoretical maximum 0.7405.
Iron (BCC), r = 1.24 angstrom
BCC lattice, atomic radius r = 1.24 angstrom
a = 4 * 1.24 / sqrt(3) = 2.864 angstrom, V = 23.48 angstrom^3, atoms per cell = 2
a = 2.864 angstrom, V = 23.48 angstrom^3, atoms per cell = 2, APF = 0.6802, packing efficiency = 68.02 percent
BCC iron has only 68 percent of its cell volume filled; a phase change to FCC increases the density.
According to Wikipedia - Cubic crystal system, the cubic crystal system has three Bravais lattices - simple cubic, body-centered cubic, and face-centered cubic - all with three equal edges of length a and three equal 90-degree angles.
Once the lattice constant and atomic mass are known, the Cube Density Calculator converts the unit cell volume into a mass density using the atoms-per-cell count.
Key Concepts Explained
Four ideas carry the calculation: the unit cell, the Bravais lattice, the geometric relation between radius and cube edge, and the atomic packing factor.
Unit cell and basis
A crystal is described by a unit cell, the smallest block that tiles space by translation, and a basis, the atoms placed at every lattice point. For a monoatomic cubic crystal the basis is a single atom.
Bravais lattice and Miller indices
The Bravais lattice is the infinite array of translation vectors that maps one cell onto the next. The cubic family contributes three Bravais lattices (SC, BCC, FCC); Miller indices h, k, l label the lattice planes.
Lattice constant vs atomic radius
The lattice constant a is the cube edge length. The atomic radius r is the radius of the model sphere. Their relation depends on the lattice type because contact geometry differs along edges, face diagonals, and body diagonals.
Atomic packing factor
APF is the ratio of the volume of atom spheres inside one cell to the volume of the cell. SC gives pi / 6, BCC gives pi * sqrt(3) / 8, FCC gives pi / (3 * sqrt(2)). FCC is the theoretical maximum for equal spheres.
For ionic crystals that form in the cubic system, the Percent Ionic Character Calculator covers the bonding side of the same chemistry toolkit.
How to Use This Calculator
Run the calculator in any of its six combinations in under a minute.
- 1 Choose the lattice type: Open the cubic lattice type dropdown and pick SC, BCC, or FCC.
- 2 Pick the solve direction: Use 'a from r' when you know the radius; switch to 'r from a' when the lattice constant is measured.
- 3 Enter the input value: Type the numeric value in the chosen length unit. Aluminum is 1.43 angstrom (FCC), iron is 1.24 angstrom (BCC), polonium is 1.67 angstrom (SC).
- 4 Pick the length unit: Select angstrom, picometer, or nanometer. Input and output use the same unit.
- 5 Read the lattice constant: The primary output shows a in the chosen unit. V = a^3 and the atoms-per-cell constant follow.
- 6 Read the packing factor: The last two rows give the APF as a fraction and a percentage. SC is 0.5236, BCC is 0.6802, FCC is 0.7405 regardless of the radius.
Open with the default FCC settings and atomic radius at 1.43 angstrom. The result panel shows a = 4.045 angstrom, V = 66.17 angstrom^3, atoms per cell = 4, APF = 0.7405, and packing efficiency = 74.05 percent. Switch to BCC and the same radius returns a = 3.302 angstrom with APF = 0.6802.
When the same atomic radius is being estimated from first principles, the Bohr Model Calculator handles the hydrogen-like atomic structure that underlies the size of the atom.
Benefits of Using This Calculator
A focused calculator gives more reliable answers than rebuilding the geometric relations from scratch.
- • Solves both directions: Switch between 'a from r' and 'r from a'.
- • Returns four derived quantities: Alongside the lattice constant the calculator gives V = a^3, the atoms per cell, the APF, and the packing efficiency percentage.
- • Three length units: Input and output stay in the same angstrom, picometer, or nanometer scale, with internal conversion handled automatically.
- • APF guard rail: SC, BCC, and FCC have fixed APF values, so the APF row is a built-in sanity check: if it changes with the radius, the lattice type is wrong.
- • Worked aluminum and iron examples: The default values reproduce the textbook FCC aluminum and BCC iron lattice constants used in introductory chemistry.
- • Classroom and lab ready: Plain numeric inputs with explicit units fit into exam preparation and homework checks.
The calculator replaces a multi-step calculation with one read-and-write step.
When the cubic cell result feeds into a wave equation such as the phonon dispersion of the same crystal, the Harmonic Wave Equation Calculator handles the wavelength and frequency bookkeeping.
Factors That Affect Your Results
Two factors set the geometric factor the calculator applies, and three mark where the simple model stops describing a real crystal.
Choice of cubic lattice type
SC, BCC, and FCC use geometric factors of 2, 4/sqrt(3), and 4/sqrt(2). Picking the wrong lattice type scales the lattice constant by up to about 1.41.
Atomic radius of the species
Different reference tables report atomic radii that can differ by 1 to 5 percent, so the lattice constant for the same lattice type can shift accordingly.
Temperature and zero-point motion
Real crystals expand as temperature rises, so the measured lattice constant at room temperature is larger than the cold-crystal value, typically by a fraction of a percent.
Polyatomic and alloy crystals
The simple SC/BCC/FCC formulas assume one atomic species per lattice point. Alloys and ionic crystals need an averaged or effective radius.
Defects and grain boundaries
Vacancies, interstitials, and grain boundaries change the measured density and effective packing fraction without changing the geometric relation between r and a for the perfect lattice.
- • The geometric relation a = f(type) * r assumes atoms are equal spheres in contact along the cube edges, face diagonals, or body diagonals. Real atoms are not perfectly spherical.
- • The atomic packing factor is the maximum only for equal spheres. Mixed-species or non-spherical packings can differ, and the value is only meaningful for the closest-packed direction.
- • This tool does not include hexagonal close-packed (HCP), which has the same APF as FCC but a different cell shape. Use a separate HCP calculator when the lattice is not cubic.
Sanity check: compare the computed APF to the theoretical value. SC always returns 0.5236, BCC 0.6802, FCC 0.7405 regardless of the atomic radius entered.
According to Wikipedia - Atomic packing factor, the APF of simple cubic is pi/6 (about 0.5236), of body-centered cubic is pi * sqrt(3) / 8 (about 0.6802), and of face-centered cubic is pi * sqrt(2) / 6 (about 0.7405).
When the same atomic species is being characterized spectroscopically, the Rydberg Equation Calculator handles the line spectrum that determines which element is in the cubic cell.
Frequently Asked Questions
Q: What are the three types of cubic unit cells?
A: A cubic unit cell comes in three Bravais variants. Simple cubic (SC) places one atom at each corner. Body-centered cubic (BCC) adds a single atom in the middle. Face-centered cubic (FCC) adds an atom at the center of each face. The corner atoms are shared between cells, so the actual atom counts per cell are 1 for SC, 2 for BCC, and 4 for FCC.
Q: How do I calculate the lattice constant from the atomic radius?
A: The lattice constant a follows from the contact geometry of the lattice type. For SC use a = 2r. For BCC use a = 4r / sqrt(3), because the body diagonal of length sqrt(3) * a equals 4r. For FCC use a = 4r / sqrt(2), because each face diagonal of length sqrt(2) * a equals 4r.
Q: What is the atomic packing factor of FCC and why is it the highest?
A: The APF of face-centered cubic is pi * sqrt(2) / 6, approximately 0.7405. FCC is the cubic lattice that lets equal spheres touch along the face diagonals with no spare space in the third dimension either. It is one of two closest-packed structures (the other is hexagonal close-packed) and reaches the theoretical maximum APF for any packing of equal spheres.
Q: Why does FCC have 4 atoms while BCC has 2?
A: Atoms on a face are shared between two cells, so each face-centered atom contributes half an atom. Six face centers give 6 * 1/2 = 3 atoms from the faces. Adding the eight corner atoms (each contributing 1/8) gives 8 * 1/8 + 3 = 4 atoms. For BCC, the eight corners give 1 atom and the single body-centered atom is entirely inside the cell, giving 1 + 1 = 2 atoms.
Q: What is the difference between lattice parameter and lattice constant?
A: Lattice parameter and lattice constant are used interchangeably in most crystallography texts to mean the length a of the cube edge. For a cubic cell there is only one independent lattice parameter because the three edges are equal and the three angles are fixed at 90 degrees.
Q: Which elements crystallize in the simple cubic lattice?
A: Simple cubic is rare because it is the least dense cubic structure with an APF of only 0.5236. Polonium is the best-known element that crystallizes in simple cubic at room temperature. Most metals adopt BCC (alkali metals, iron, chromium, tungsten) or FCC (aluminum, copper, nickel, silver, gold, lead) instead.