Select Density & APF or Interstitial sites with the buttons at the top right. Drag in the 3D view to rotate the unit cell; press Reset view to return to the starting orientation.
Density & APF: choose a crystal structure and an element, or move the atomic radius and atomic weight sliders to make a custom element. Atoms are colored by where they sit in the unit cell, which sets the fraction of each atom that belongs to the cell. The right panel shows the plane in which the atoms touch; the table below it calculates the lattice parameter, cell volume, atomic packing factor, and density, and compares the density with the measured value.
Interstitial sites: choose an FCC or BCC host, an octahedral or tetrahedral site, and an interstitial atom. Click any site in the 3D view (or press Next site) to show its coordinating host atoms and coordination polyhedron. The right panel is a planar section through the site: the dashed circle is the largest sphere that fits without moving the host atoms, and the filled circle is the interstitial atom.
Atoms are rigid spheres of radius \(R\) that touch along the close-packed direction of each structure. The touching condition gives the lattice parameter \(a\):
$$\text{SC: } a = 2R \qquad \text{BCC: } \sqrt{3}\,a = 4R \qquad \text{FCC: } \sqrt{2}\,a = 4R$$ $$\text{HCP: } a = 2R, \qquad c = \left(\tfrac{c}{a}\right) a, \qquad \left(\tfrac{c}{a}\right)_{\text{ideal}} = \sqrt{8/3} = 1.633$$An atom at a cube corner is shared by 8 cells, on an edge by 4, on a face by 2, and inside the cell by 1. In the HCP hexagonal prism, corner atoms are shared by 6 cells.
$$n = N_{\text{inside}} + \tfrac{1}{2}N_{\text{face}} + \tfrac{1}{4}N_{\text{edge}} + \tfrac{1}{8}N_{\text{corner}}$$This gives \(n\) = 1 (SC), 2 (BCC), 4 (FCC), and 6 (HCP).
where \(A\) is the atomic weight (g/mol) and \(N_A = 6.022\times10^{23}\) atoms/mol. The APF is 0.52 for SC, 0.68 for BCC, and 0.74 for FCC and ideal HCP, independent of \(R\). For HCP with a nonideal ratio, \(\text{APF} = 0.7405\,(1.633)/(c/a)\).
An interstitial site is a hole between host atoms. The largest sphere of radius \(r\) that fits without displacing the host atoms touches its nearest neighbors, so \(R + r\) equals the distance from the site center to the nearest host-atom center.
$$\text{FCC octahedral: } 2R + 2r = a = 2\sqrt{2}R \;\Rightarrow\; r = (\sqrt{2}-1)R = 0.414R$$ $$\text{FCC tetrahedral: } R + r = \tfrac{\sqrt{3}}{4}a \;\Rightarrow\; r = \left(\sqrt{3/2}-1\right)R = 0.225R$$ $$\text{BCC octahedral: } R + r = \tfrac{a}{2} \;\Rightarrow\; r = \left(2/\sqrt{3}-1\right)R = 0.155R$$ $$\text{BCC tetrahedral: } R + r = \tfrac{\sqrt{5}}{4}a \;\Rightarrow\; r = \left(\sqrt{5/3}-1\right)R = 0.291R$$| Site | Location in cell | Sites/cell | Sites/host atom | CN |
|---|---|---|---|---|
| FCC octahedral | body center, edge centers | 1 + 12(¼) = 4 | 1 | 6 |
| FCC tetrahedral | (¼,¼,¼) type | 8 | 2 | 4 |
| BCC octahedral | face centers, edge centers | 6(½) + 12(¼) = 6 | 3 | 6 |
| BCC tetrahedral | (½,¼,0) type | 24(½) = 12 | 6 | 4 |
BCC octahedral sites are distorted: two host atoms are at \(a/2\) and four at \(a/\sqrt{2}\). An interstitial atom pushes only the two close atoms apart, which produces a tetragonal distortion.
Although FCC packs more densely than BCC, its largest hole (octahedral, 0.414\(R\)) is larger than the largest BCC hole (tetrahedral, 0.291\(R\)). This is why carbon (\(r\) ≈ 0.071 nm) is far more soluble in FCC austenite (up to 2.14 wt%) than in BCC ferrite (0.022 wt%).
Atomic radii are hard-sphere values from lattice parameters at room temperature, except γ-Fe, which is stable above 912 °C.
| \(a\) | lattice parameter: edge length of the cubic cell, or edge of the basal hexagon for HCP (nm) |
| \(A\) | atomic weight (g/mol) |
| APF | atomic packing factor: fraction of the cell volume occupied by hard-sphere atoms |
| BCC | body-centered cubic structure |
| \(c\) | height of the HCP hexagonal prism (nm) |
| \(c/a\) | axial ratio of the HCP cell |
| CN | coordination number: number of nearest-neighbor atoms that touch an atom or surround an interstitial site |
| coordinating atom | host atom that is a nearest neighbor of the selected interstitial site |
| FCC | face-centered cubic structure |
| HCP | hexagonal close-packed structure |
| host atom | atom of the parent crystal (solvent) that forms the lattice |
| interstitial atom | small atom (C, N, O, H) that sits in a hole between host atoms |
| interstitial site | hole between host atoms in which a small atom can sit |
| \(n\) | number of atoms per unit cell |
| \(N_A\) | Avogadro's number, \(6.022\times10^{23}\) atoms/mol |
| \(N_{\text{corner}}, N_{\text{edge}}, N_{\text{face}}, N_{\text{inside}}\) | numbers of atoms at corners, on edges, on faces, and inside the cell |
| octahedral site | interstitial site surrounded by 6 host atoms at the corners of an octahedron |
| \(r\) | radius of the interstitial atom (nm) |
| \(r_{\text{max}}\) | radius of the largest sphere that fits in a site without displacing the host atoms (nm) |
| \(R\) | atomic radius of the host atom (nm) |
| SC | simple cubic structure |
| tetrahedral site | interstitial site surrounded by 4 host atoms at the corners of a tetrahedron |
| unit cell | smallest repeating volume that generates the crystal by translation |
| \(V_C\) | volume of the unit cell (nm3 or cm3) |
| \((hkl)\) | Miller indices of a crystal plane, such as (001) or (110) |
| \(\rho\) | density (g/cm3) |
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.