# User:TedJanssen

### From Online Dictionary of Crystallography

Synonyms: window, atomic domain

Domaine d'acceptation, fenetre (Fr.)

When an aperiodic crystalline point set is obtained by the intersection method, as the intersection of a periodic array of finite, disjoint components in superspace and the physical
space, then there is, for each point , a component in the higher-dimensional
unit cell. These components are called *acceptance domains*, *atomic surfaces*, *atomic domains*, or
*windows* in the literature. The positions of atoms in aperiodic crystals (or vertices in the case of a tiling)
are the intersection of the atomic surfaces with the physical space. This construction of the
points is called the *intersection method*.

**History**

The term *window* comes from an alternative construction. One considers a 'tube'
in *n*-dimensional superspace
that is the product of the *m*-dimensional physical space *V* and a transversal finite (*n-m*)-dimensional object *O*:
*V times O*. Then the points of the *n*-dimensional lattice, corresponding
to the periodic embedding of the structure in *V*,
that lie inside the tube are projected
on *V* (See Fig.File:BO02F20.jpg). These are the points of the aperiodic structure in *V*.
The construction is called the *cut-and-project method*. The object *O*
is the *window*, or *acceptance domain*. The latter expresses the fact that only
lattice points inside the tube are projected. The result is the same as attaching a copy
of *O* to each lattice point, and considering the intersection of this
periodic array with *V*. Actually, the latter construction is more special, in the
sense that the atomic surfaces in this are by definition flat, whereas the intersection
method allows arbitrarily shaped ones.

See also: *atomic surface*.