Hermann-Mauguin notation
o In geometry, Hermann–Mauguin notation represents the symmetry
elements in point groups, plane groups, and space groups –
International notation
o It is named after the German crystallographer Carl Hermann (who
introduced it in 1928) and the French mineralogist Charles-Victor
Mauguin (who modified it in 1931)
Hermann-Mauguin notation
o Since crystals exhibited only 2, 3, 4, and 6 fold symmetry, there are only 32 possible
combinations of symmetry elements, and therefore only 32 possible classes of
crystals
H-M Code
o H-M symbols consist of some combination of numerals 1, 2, 3, 4, and 6; numerals
with a bar-over 1ത , 3ത , 4ത , 𝑎𝑛𝑑 6ത ; the letter m; and a slash, /.
o Numerals 1, 2, 3, 4, and 6 indicate an axis of rotational symmetry (usually referred to
as an axis of rotation)
o For example, 3 indicates that a rotation of 120o brings the crystal into coincidence
with its original position in space and this occurs 3 times in a rotation of 360o
o It is a 3-fold axis of rotational symmetry
o Similarly, 2, 4, and 6 indicate axes of 2-fold, 4-fold, or 6-fold rotation, and A 1
symbolizes "no symmetry".
Hermann-Mauguin notation
o Numerals with a bar-over show an axis of rotatory inversion, usually called an axis
of inversion
o A 4ത means that a rotation of 90o followed by an inversion through the center brings
the crystal to occupy the same space as at the start
o Four such operations brings the crystal to its original position (Operationally, turn
crystal 90o to the right, then invert top to bottom clockwise. Four times returns the
crystal to its original position)
o In reality, all points on the surface of a crystal rotate n degrees and invert through the
center (somewhat like turning the crystal inside-out) but this is physically
impossible.
o Thus operationally, we follow a single point, such as a corner. 1ത is one complete
rotation plus an inversion through the center and merely indicates a center of
symmetry.
Hermann-Mauguin notation
o Letter m, indicates a plane of symmetry, usually referred to as a mirror plane
o A mirror plane divides a body (crystal) such that the half on one side of the plane is
the mirror image of the half on the opposite side
o A slash, /, means "perpendicular to". Thus, 2/m means "a 2-fold axis of rotation
perpendicular to a mirror plane"
o Symbols in succession means "parallel to". 3m (two elements: same as 3m) denotes a
3-fold axis parallel to 3 mirror planes.
o Given a 3-fold axis parallel to (coincident with) one mirror plane, the 3-fold
symmetry requires that there be three mirror planes at 120o angles to each other and
with each of the three planes divided by the axis
o The axis of rotation lies in the three mirror planes
Hermann-Mauguin notation
o Letter m, indicates a plane of symmetry, usually referred to as a mirror plane
o A mirror plane divides a body (crystal) such that the half on one side of the plane is
the mirror image of the half on the opposite side
o A slash, /, means "perpendicular to". Thus, 2/m means "a 2-fold axis of rotation
perpendicular to a mirror plane"
o Symbols in succession means "parallel to". 3m (two elements: same as 3m) denotes a
3-fold axis parallel to 3 mirror planes.
o Given a 3-fold axis parallel to (coincident with) one mirror plane, the 3-fold
symmetry requires that there be three mirror planes at 120o angles to each other and
with each of the three planes divided by the axis
o The axis of rotation lies in the three mirror planes
