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Fm 3mEdit

Note: This article presents Fm-3m as a crystallographic designation and describes its science in a neutral, scholarly manner.

Fm-3m is the Hermann–Mauguin symbol for a highly symmetric cubic space group that governs the arrangement of atoms in many solids with a face-centered cubic lattice. It is space group No. 225 in the international tables used by crystallographers and lies in the Oh family of point-group symmetries. This combination of a centric, face-centered lattice with full cubic symmetry underpins the crystal chemistry of a wide range of metals and ionic compounds, including common metals like copper and aluminum as well as well-known ionic structures such as the rock-salt and fluorite families. The group’s high symmetry makes it a convenient starting point for analyzing diffraction data and for modeling physical properties that depend on isotropic or near-isotropic environments around lattice sites. For concrete examples and historical use, see discussions of NaCl structure and related materials, as well as examples in copper and aluminium crystals.

Structure and symmetry

Fm-3m describes a face-centered cubic (fcc) lattice with the full octahedral symmetry of the Oh point group. The lattice, sometimes referred to in shorthand as a cubic lattice with F-centering, supports a rich set of symmetry operations, including fourfold rotations along the cube edges, threefold rotations along body diagonals, multiple mirror planes, and inversion. The Oh point group sits at the heart of this symmetry, and the space-group designation Fm-3m encodes how these symmetries are combined with translations of the underlying lattice.

  • Lattice type and crystal system: The lattice is face-centered cubic, which places Fm-3m squarely in the cubic crystal system. See face-centered cubic and cubic crystal system for broader context.

  • Symmetry content: The Oh point group provides the set of rotational and reflection symmetries that characterize the cubic environment around lattice sites. See Oh (point group) for a detailed account of these symmetries.

  • Space-group symbol and numbering: Fm-3m is the standard notation used in the International Tables for Crystallography to describe the combination of F-centering with m-3m symmetry. See space group for a general overview.

Wyckoff positions and site symmetry

In crystals described by Fm-3m, atoms typically occupy a small set of high-symmetry positions, known as Wyckoff positions. The most common ones for materials with this symmetry are:

  • 4a at (0, 0, 0)
  • 4b at (1/2, 1/2, 1/2)
  • 8c at (1/4, 1/4, 1/4)

These positions reflect the special symmetry constraints of the lattice and help explain why many materials with this space group exhibit straightforward, highly symmetric diffraction patterns. For a general treatment of how atoms occupy Wyckoff positions, see Wyckoff positions.

Diffraction and structure factors

The centering of the fcc lattice imposes distinctive selection rules on diffraction, producing systematic absences that crystallographers use to identify F-centered structures. In practical terms, reflections with certain h, k, l combinations are absent due to centering, while others are allowed under the Oh symmetry constraints. X-ray and neutron diffraction analyses routinely exploit these rules to determine whether a material adopts the Fm-3m description.

Materials associated with Fm-3m

The Fm-3m space group is a canonical description for many materials with very high cubic symmetry. It underlies:

  • Simple metals with an fcc lattice, such as copper, aluminium, silver, and gold.
  • Ionic crystals with rock-salt type arrangements, such as NaCl.
  • Fluorite-type structures, exemplified by materials like CaF2 (where fluoride ions occupy the 8c positions and calcium sits at the 4a position).

The ubiquity of Fm-3m in these materials makes it a foundational reference point for crystallography, materials science, and solid-state physics. Understanding the symmetry constraints helps interpret physical properties, diffraction data, and the ways atomic displacements or substitutions perturb the idealized structure.

Relationships and related concepts

  • space group: The broader framework for symmetry in crystals, of which Fm-3m is a specific instance.
  • crystal system: The cubic crystal system to which Fm-3m belongs.
  • cubic crystal system: A more detailed treatment of cubic symmetry, including Oh and related point groups.
  • Oh (point group): The octahedral point group that underpins the symmetry operations in Fm-3m.
  • Wyckoff positions: The standard sites in a space group where atoms can occupy with given site symmetries.
  • face-centered cubic: The lattice type associated with Fm-3m.
  • NaCl structure: A classic rock-salt structure, described by Fm-3m in many cases.
  • X-ray crystallography: The primary experimental method used to determine and validate the Fm-3m description in real materials.
  • rock-salt structure: A common arrangement in ionic solids closely related to the Fm-3m description in NaCl-type compounds.

See also