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##

Space-group operations

The following facts are stated, their proof is beyond the scope of this
manuscript:

- The symmetry operations listed in Section 3.2 are elements of
space groups which leave the given point fixed, see definition
(D 3.2.1). Therefore, identity, inversion, rotations, rotoinversions,
and reflections are symmetry operations of space groups.
Moreover, the same restrictions for the possible angles of rotation and
rotoinversion of space-group operations hold as in Section 3.2.
This concerns also the rotations involved in screw rotations.
- It is always possible to choose a primitive basis,
see definition (D 1.5.2) and the remarks to it. Referred to a primitive
basis, all lattice vectors of the crystal are integer linear combinations
of the basis vectors. Each of these lattice vectors defines a (symmetry)
translation. The order of any translation
*T* is infinite
because there is no number such that
.
- Parallel to each rotation, screw-rotation, or rotoinversion axis
as well as parallel to the normal of each mirror or glide plane there
is a row of lattice vectors.
- Perpendicular to each rotation, screw-rotation, or rotoinversion
axis as well as parallel to each mirror or glide plane there is a
plane of lattice vectors.
- Let
be the rotation angle of a screw rotation,
then the screw rotation is called -fold. Note that the order of any
screw rotation is infinite. Let
**u** be the shortest lattice vector
in the direction of the corresponding screw axis, and
,
with and integer, be the screw vector of the
screw rotation by the angle . Then the symbol of the screw
rotation is .
Performing an -fold rotation -times results in the
identity mapping, *i.e.* the crystal has returned to its
original position. After screw rotations with
rotation angle
the crystal has its original
orientation but is shifted parallel to the screw axis by the lattice
vector **u**.

- Let
*W* be a glide reflection.
Then the glide vector is
parallel to the glide plane and is 1/2 of a lattice vector **t**.
Whereas twice the application of a reflection restores the
original position of the crystal, applying a glide reflection
twice results in a translation of the crystal with the translation
vector **t**. The order of any glide reflection is infinite. The
symbol of a glide reflection is in the plane and
, or in the space. The letter indicates the direction of the
glide vector **g** relative to the basis of the coordinate system.

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**Up:** Crystallographic symmetry
** Previous:** Crystallographic site-symmetry operations
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Crystallography**

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