Hermann-Mauguin Notation

Hermann-Mauguin notation is the standard system used in mineralogy and crystallography to describe the point group symmetry of a crystal. ^[1] The notation is used to specifically identify each of the 32 point groups and appears in the mineral descriptions in every major mineralogy reference text.

How It Works

The key insight behind Hermann-Mauguin notation is that the symmetry elements in a crystal are not independent - the presence of some operations automatically implies the presence of others, so not all of them need to be listed explicitly. ^[1] The notation lists only what is necessary to fully specify the symmetry, with implied elements understood.

Consider an orthorhombic mineral with centre symmetry (i), three 2-fold rotation axes (3A2), and three mirror planes (3m). In full symbolic notation that would be written: i, 3A2, 3m. In Hermann-Mauguin notation the same crystal is written 2/m 2/m 2/m, which means three 2-fold axes each perpendicular to a mirror plane. ^[1] The centre symmetry does not need to be stated separately because three mirrors at right angles to each other require inversion - it is implied by the other elements present. ^[1]

Reading the Notation

The order in which symbols appear in a Hermann-Mauguin designation is not arbitrary - it is specific to each crystal system and refers to particular symmetry directions defined by the crystal axes of that system. ^[1] For example, in the orthorhombic system, the first symbol refers to the a axis, the second to the b axis, and the third to the c axis. In the tetragonal system, the first symbol refers to the c axis, the second to the a and b axes, and the third to directions that bisect the angles between a and b. Learning to read the notation therefore requires knowing the axis conventions for each crystal system - but once those conventions are understood, a Hermann-Mauguin symbol gives a complete and unambiguous description of a crystal’s point group symmetry in a compact form.

The Isometric System: Three-Part Symbols

The isometric system uses a distinct three-part symbol structure that differs from every other crystal system. Where other systems use their symbol positions to refer to specific axes or planes, the isometric system uses its three positions for three qualitatively different categories of symmetry direction - axes, body diagonals, and edge diagonals - because the cubic unit cell has three geometrically independent sets of equivalent directions that need to be described separately.

The first position refers to the symmetry operations associated with the three crystallographic axes themselves, which are the three mutually perpendicular edges of the cubic unit cell. The second position refers to the three body-diagonal directions [111], which run from corner to corner through the centre of the unit cell. The third position refers to the six face-diagonal directions [110], each running from the midpoint of one edge to the midpoint of the opposite parallel edge. ^[1] Reading an isometric symbol like 4/m 3̄ 2/m therefore tells you: the crystallographic axes have 4-fold rotation axes with perpendicular mirrors; the body diagonals [111] have 3-fold rotoinversion axes; and the face-diagonal directions [110] have 2-fold axes with perpendicular mirrors. Each position addresses a geometrically distinct set of directions that together specify the full cubic symmetry.

References

1. Nesse, W. D. (2018). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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