a special position

A special position is a point that DOES lie on a symmetry element — right on a rotation axis, on a mirror, at an inversion centre, or on several at once. Because at least one operation maps the point straight back onto itself, that operation makes no new copy, so a special position always has higher site symmetry and lower multiplicity than a general one.

The pinning shows up in the coordinates. A general point is free to be anywhere, (x, y, z), but a special point has some coordinates locked by the symmetry it sits on. An inversion centre forces a fixed point such as (0, 0, 0). A point on a two-fold axis running along b is stuck at (0, y, 1/4) — x and z fixed, only y free. On a mirror perpendicular to c a point is (x, y, 0). Fewer free coordinates means fewer numbers to determine when solving the structure, and a definite, symmetric coordination geometry forced on whatever atom sits there.

High-symmetry ions — a metal cation at the centre of an octahedron, say — very often take special positions, which is why heavy atoms in inorganic crystals so frequently land on tidy coordinates like (0,0,0) or (1/4,1/4,1/4). The flip side is a classic beginner trap: if you place an atom on a special position, you must reduce its occupancy or count so you do not multiply it more than the symmetry intends, and if you place it a hair off a special position by accident, the symmetry will breed a spurious near-neighbour.