# Understanding Equivalent Wyckoff positions

I'm somewhat confused by the concept of Wyckoff positions in crystal structures. From Quantum Chemistry of Solids by Robert Evarestov, the definition of a Wyckoff position is "all the crystallographic orbits which have the same (not only isomorphic, but the same) site-symmetry group". I was okay with this, but he then went on to define equivalent Wyckoff positions as positions such that "equivalent sites have the same point-group symmetry and the same orientations of symmetry elements with respect to the lattice".

What I fail to understand is how the equivalent Wyckoff positions don't collapse to a single Wyckoff position, as they seem to meet the criteria of having the same site-symmetry. Is there a clear explanation of what distinguishes two Wyckoff positions enough to be equivalent, but not just a single position?

To clarify my question a little more, here is the specific example given in the text for the space group 136, also called P42/mnm or $D_{4h}^{14}$.

Pairs of Wyckoff positions a-b, f-g, and i-j have isomorphic site-symmetry groups ($D_{2h},C_{2v},\text{ and } C_s$ respectively). As is seen from the table, the pairs of points a-b and f-g not only have isomorphic, but also equivalent Wyckoff positions. The equivalent sites have the same point group symmetry and the same orientations of symmetry elements with respect to the lattice. I believe I have identified what my misunderstanding was. I was incorrectly conflating point group symmetry with site symmetry, which is a subgroup of the overall point group symmetry that leaves the given position unchanged. So, it is possible for two points to have equivalent site symmetries while not having the same site symmetry, e.g the points might both have mirror plane symmetry, but not the same mirror plane. The orbits from these generating points would not be conjugate and so they would be put into separate Wyckoff positions. However, depending on where you choose the origin to be, the different Wyckoff positions can interchange the generating points of their orbits, making them equivalent positions.

An explicit example of this is Rock salt $$\ce{NaCl}$$ (space group N225), where either $$\ce{Na}$$ or $$\ce{Cl}$$ can placed at Wyckoff position $$a$$ while the other is placed at $$b$$. Formally, equivalent Wyckoff positions are those which can be transformed into each other by applying the generators of the Euclidean normalizer. I recently found Symmetry Relations between Crystal Structures by Ulrich Müller and it has been very helpful in clarifying my understanding of space groups.

It seems to me that care has to be taken with the words 'equivalent' and 'same' in the definitions above. Equivalent positions do not mean the same position. In the case, say, of a 4 fold axis any point say in the first quadrant can be replicated to the other three quadrants and these are equivalent points but not the same point, clearly. In some cases such as with reflections in a plane, two sets of identical (or equivalent) points may be produced and so and the number of equivalent points is reduced. The number of equivalent sets is called the multiplicity. These have been called special positions when they lie on a symmetry element

In the Wyckoff notation equivalent positions are labelled consecutively $$a, b, c \dots$$ starting with $$a$$ for the highest symmetry. A position $$2f$$, for example, indicates two equivalent atoms at position given by symmetry $$f$$. For each space group a table of general and special positions can be made. This includes point symmetries, positions multiplicities and Wyckoff symbols. The table below shows examples. Table copied from page 75 of 'Introduction to Crystallography' by D. Sands, publ. Dover

• I've edited my question to include the specific example give of equivalent Wyckoff positions, which seems to differ from the definition you are using.
– Tyberius
Aug 6, 2018 at 16:33