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In analogy with equivalent positions, and as an effect of these, there are
equivalent reflections. Two equivalent reflections, **h** and
**h**, always have the same amplitudes, i.e. |*F*(**h**)|
= |*F*(**h**)|, but their phases may differ. The phases are,
however, related to each other in an easily deduced way. The difference in
phase between two equivalent reflections is called **phase shift**.
How the phase shift arises and how great it is will be shown now.

Two reflections **h** and **h** are equivalent if there
exists an **R**_{i} such that .Due to Friedel's law **h** is always equivalent to -**h**. The
phases of two equivalent reflections are related as:

*Proof* :

The proof is carried out for a 2-fold symmetry for the sake of simplicity. A
similar strategy can be used for higher symmetries. If **x** and
**Rx** + **t** are equivalent positions, then **R**^{-1}
(**x** - **t**) is also an equivalent position, since just as well
as **x** gives rise to **x**, **x** gives
rise to **x**, by the same symmetry operation:

(8) |

(9) |

(10) |

(11) |

(7) |

The phase shift is called and is equal to
exp (**ht**).
Note the minus sign!

(12) |

Since the phase shift depends on the translation vector, all phases of equivalent reflections derived through symmetry operations with translation vector = 0, are equal. An example of equivalent reflections with different phases will be given. Derive the phases of all reflections equivalent to (

If the first reflection, (3 0 1), is +60, then the second becomes

In a similar way the phases of the other reflections are 60 and .

Note that the fifth reflection () also is the Friedel pair of
(3 0 1).
Due to Friedel's law the phase of any reflection must be minus that of its
Friedel pair. In all cases where a symmetry operation generates an equivalent
reflection which is also its Friedel mate, we have two indications of the phase
value. If the phase of (*hkl*) is then the phase of (-*h* -*k* -*l*) is
due to Friedel's law, and the phase is due to the
phase shift. We now have a system of equations:

with the solution ,i.e. (modulo 180 since of course is modulo 360). This alternative way of deriving phase restrictions is clearly more relevant than that of section 4.

In a similar way the systematically absent reflections can be shown to be
exactly those reflections which have two contradictory phase indications. In
*P*2_{1} (0 *k* 0)-reflections with *k* odd are extinct. The equivalent positions
of *P*2_{1} are (*x*, *y*, *z*) and (). The reflections
(*hkl*) and (-*h*, *k*, -*l*) are equivalent and the phase shift is *k*/2. A
reflection like (0 3 0) is thus equivalent to itself, but the equivalent
reflection generated has a phase differing from the original one by
180. The phase of (0 3 0) is at the same time and , which of course is only possible if the amplitude of the
reflection is 0!

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