Réseau cristallin

Choose FCC, BCC or HCP and select Display. The Cell tab shows the conventional cell and the atom centers needed to count its contents. Select an atom by tapping its center or entering its number. Its position, fractional contribution and coordinates are shown beside the counting table. Identical projected positions are resolved by the nearer depth; use the number field to reach any overlapped center.

Cell convention. FCC and BCC use the conventional cube with edge a = 1. HCP uses the conventional hexagonal prism with side a = 1 and ideal c/a = √(8/3). This prism contains three primitive HCP cells. Its 17 displayed centers are not 17 complete atoms: fractional boundary contributions give six atoms per conventional prism. All coordinates are in units of a, with z along the stacking direction.

FCC has eight corners contributing ⅛ each and six face centers contributing ½ each: 8/8 + 6/2 = 4 atoms. BCC has eight corners at ⅛ and one body center: 8/8 + 1 = 2. HCP has twelve corners at ⅙, two base centers at ½ and three internal atoms: 12/6 + 2/2 + 3 = 6. At a hexagonal corner, the 120° interior angle gives one third in the basal plane and the end face contributes the remaining factor one half.

Neighbors and packing. The Neighbors tab shows the complete nearest shell around a fixed reference atom at the origin, using translated cells of the periodic lattice. It is not restricted to the centers displayed in one cell. FCC has 12 nearest neighbors, BCC 8, and HCP 12: six basal, three above and three below. The touching radius is half the nearest-center distance. Consequently r/a is √2/4, √3/4 and ½, respectively.

The counted atom total N, sphere volume 4πr³/3 and conventional-cell volume V give the packing fraction N·4πr³/(3V). Cubic V/a³ = 1; the HCP prism has V/a³ = 3√2. FCC and ideal HCP give π/(3√2), about 74.05%; BCC gives √3π/8, about 68.02%. These are ideal equal-sphere structures, not material-specific densities. Nonideal HCP ratios, different atom sizes, defects and real material properties are outside this model.

Drawing and scale. Small markers are schematic centers. Full touching spheres have their calculated projected radius; the drawing shows whole, translucent spheres rather than clipping fractions at cell boundaries. Their colors, the selected fraction and the counting table explain which portion belongs to the cell. The edges are the actual cell edges, not the bounding box of those spheres.

Both tabs and all lattice types share the same orthographic spatial scale and a viewing radius of 1.9a. The cell is centered at its geometric center; the neighbor shell at the origin. Azimuth and elevation change only the view, with z upward. Front is 0°/0°, top 0°/90°, and oblique 45°/25°. The axis triad follows the view. Overlapping spheres are intentionally translucent; this is not a complete surface-occlusion calculation. View and sphere changes do not recalculate the lattice. There is no animation or WebGL requirement, and ordinary wheel motion scrolls the page.

Changing lattice type clears the result and selects atom 1 of the next cell. Reset clears the result while keeping settings. Local tool-state storage can retain valid inputs, view, tab and selection. Returning within the app can preserve the computed model; reloading restores settings and waits for Display.

Sources: OpenStax Chemistry 2e, 10.6, on cubic cells, sharing and contact; Purdue University, Bodner Group: Structure of Metals, on structures and coordination; LibreTexts: Closest Packed Structures, including the note on the conventional HCP cell. No source illustrations or programs are copied.