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In the general case, where the contributions from the two parts of the unit cell
have different amplitudes, *F*(*hkl*) can be represented as the sum of two vectors
of different lengths (= amplitude) and different directions (= phase) as in Fig.
1.

The phase of *F*(*hkl*) can take on any value between 0 and 360. The
special case that will result in a restriction of
the
possible value of the phase of *F*(*hkl*). This is called **phase
restriction**. If
the phase of the first contribution in (7) is and **ht** = 0, then
the phase of the second contribution becomes . As is clear from Fig. 2
the only possible phase values for the sum of these two contributions are 0 and
180, or if expressed in radians 0 and . If , the phase of *F*(*hkl*) becomes 0, and if the phase of *F*(*hkl*) will be 180. The phase of
*F**hkl* is denoted here in order to distinguish it from the phase of
the contributions.

We say (*hkl*) has a phase restriction of 0 ( 180).

All reflections in centrosymmetric space groups have phase restriction 0 (180). Most reflections in non-centrosymmetric space groups lack phase restriction, but some special reflections have phase restriction. The phase is not necessarily restricted to 0 ( 180) - such phase restrictions as 45, 60, 90 and so on exist. All phase restrictions are 180 or modulo 180. In order to clarify this an example is worked out in some detail.

What is the phase restriction of (*h*01) in space group *P*3_{1}21 (No 152)? The
equivalent positions are:

The rotation matrices and translation vectors are:

While the equivalent positions are derived from the rotation matrices and translation vectors through

The reflections equivalent to (*hkl*) are:

and so on, giving (

Phase restrictions occur if and only if (*hkl*) **R**_{i} = (-*h*, -*k*, -*l*),
that is when the Friedel pair of a reflection is generated by any
**R**_{i}.
If only **R**_{5} creates a Friedel pair of (*h*01).

These results are introduced into (7). If the first summation over half the
atoms in the unit cell gives a contribution to the structure factor of amplitude
|*F*| and phase , the other half of the atoms in the unit cell will give
a contribution of amplitude |*F*| but with a phase exp
(2*i***ht**)(). The value exp (2*i***ht**)
is short for exp (2*i*[*ht _{1}* +

Reflections with phase restrictions are more often very strong or very weak than
general reflections. This is due to the fact that the two contributions are
either both large or both small in the case of a phase restricted reflection,
whereas in the general case their amplitudes are independent. On the basis of
their probability to take on extreme amplitude values all reflections are sorted
into two categories: acentric or centric reflections. Reflections without phase
restriction are called acentric and reflections with phase restrictions are
called centric. The probability distribution of centric and acentric
reflections is so different that it is often possible to distinguish between the
space groups *P*1 and only from intensity data. The concept
centric should not be mixed up with centrosymmetric or centred (also spelled
centered). While centrosymmetric refers to a space group, centric refers only
to single reflections. Although all reflections in centrosymmetric space groups
are centric, not all reflections in non-centrosymmetric space groups are
acentric.

**Copyright © 1981, 1998 International Union of
Crystallography**