Equivalent point configurations, equivalent Wyckoff positions and equivalent descriptions of crystal structures

Koch, E.; Fischer, W.

doi:10.1107/97809553602060000535

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A. ch. 15.3, pp. 900-901

Section 15.3.2. Equivalent point configurations, equivalent Wyckoff positions and equivalent descriptions of crystal structures

E. Koch^a ^* and W. Fischer^a

^a Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: kochelke@mailer.uni-marburg.de

15.3.2. Equivalent point configurations, equivalent Wyckoff positions and equivalent descriptions of crystal structures

| top | pdf |

In the crystal structure of copper, all atoms are symmetrically equivalent with respect to space group $[Fm\overline{3}m]$ . The pattern of Cu atoms may be described equally well by Wyckoff position $[4a\ 000]$ or $[4b\ {1\over 2}{1\over 2}{1\over 2}]$ . The Euclidean normalizer of $[Fm\overline{3}m]$ gives the relation between the two descriptions.

Two point configurations (crystallographic orbits )¹ of a space group $[{\cal G}]$ are called Euclidean- or $[{\cal N}\!_{{\cal E}}]$ -equivalent (affine- or $[{\cal N}\!_{{\cal A}}]$ -equivalent) if they are mapped onto each other by the Euclidean (affine) normalizer of $[{\cal G}]$ .

Affine-equivalent point configurations play the same role with respect to the space-group symmetry, i.e. their points are embedded in the pattern of symmetry elements in the same way. Euclidean-equivalent point configurations are congruent and may be interchanged when passing from one description of a crystal structure to another.

Starting from any given point configuration of a space group $[{\cal G}]$ , one may derive all Euclidean-equivalent point configurations and –except for monoclinic and triclinic space groups – all affine-equivalent ones by successive application of the `additional generators' of the normalizer as given in Tables 15.2.1.3 and 15.2.1.4 .

Examples

(1) A point configuration $[F\overline{4}3m\ 16e\ xxx]$ with $[x_{1} = 0.10]$ may be visualized as a set of parallel tetrahedra arranged in a cubic face-centred lattice. The Euclidean and affine normalizer of $[F\overline{4}3m]$ is $[Im\overline{3}m]$ with $[a' = {1\over 2}a]$ (cf. Table 15.2.1.4 ). Since the index $[k_{g}]$ of $[{\cal G}]$ in $[{\cal K}({\cal G})]$ is 4, three additional equivalent point configurations exist, which follow from the original one by repeated application of the tabulated translation $[t({1\over 4} {1\over 4} {1\over 4}): 16e\ xxx]$ with $[x_{2} = 0.35]$ , $[x_{3} = 0.60]$ , $[x_{4} = 0.85]$ . $[{\cal L}({\cal G})]$ differs from $[{\cal K}({\cal G})]$ and an additional centre of symmetry is located at 000. Accordingly, the following four equivalent point configurations may be derived from the first four: 16e xxx with $[x_{5} = -0.10]$ , $[x_{6} = -0.35]$ , $[x_{7} = -0.60]$ , $[x_{8} = -0.85]$ . In this case, the index 8 of $[{\cal G}]$ in $[{\cal N}_{{\cal E}}({\cal G})]$ equals the number of Euclidean-equivalent point configurations.
(2) $[F\overline{4}3m\ 4a\ 000]$ represents a face-centred cubic lattice. The additional translations of $[{\cal K}(F\overline{4}3m)]$ generate three equivalent point configurations: $[4c\ {1\over 4}{1\over 4}{1\over 4},\ 4b\ {1\over 2}{1\over 2}{1\over 2}]$ and $[4d\ {3\over 4}{3\over 4}{3\over 4}]$ . Inversion through 000 maps 4a and 4b each onto itself and interchanges 4c and 4d. Therefore, here the number of equivalent point configurations is four, i.e. only half the index of $[{\cal G}]$ in $[{\cal N}\!_{{\cal E}}({\cal G})]$ .

The difference between the two examples is the following: The reference point 0.1, 0.1, 0.1 of the first example does not change its site symmetry .3m when passing from $[F\overline{4}3m]$ to $[Im\overline{3}m]$ . Point 000 of the second example, however, has site symmetry $[\overline{4}3m]$ in $[F\overline{4}3m]$ , but $[m\overline{3}m]$ in $[Im\overline{3}m]$ .

The following rule holds without exception: The number of point configurations equivalent to a given one is equal to the quotient $[i/i_{s}]$ , with i being the subgroup index of $[{\cal G}]$ in its Euclidean or affine normalizer and $[i_{s}]$ the subgroup index between the corresponding two site-symmetry groups of any point in the original point configuration.

As a necessary but not sufficient condition for $[i_{s} \neq 1]$ when referring to the Euclidean normalizer, the inherent symmetry (eigensymmetry) of the point configuration considered (i.e. the group of all motions that maps the point configuration onto itself) must be a proper supergroup of $[{\cal G}]$ . If $[{\cal D}]$ designates the intersection group of $[{\cal N}\!_{{\cal E}}({\cal G})]$ with the inherent symmetry of the point configuration, the number of Euclidean-equivalent point configurations equals the index of $[{\cal D}]$ in $[{\cal N}\!_{{\cal E}}({\cal G})]$ .

Example

The Euclidean and affine normalizer of $[P2_{1}3]$ is $[Ia\overline{3}d]$ with index 8. Point configuration 4a xxx with $[x_{1} = 0]$ forms a face-centred cubic lattice with inherent symmetry $[Fm\overline{3}m]$ . The reference point 000 has site symmetry .3. in $[P2_{1}3]$ but $[.\overline{3}.]$ in $[Ia\overline{3}d]$ . The number of equivalent point configurations, therefore, is $[i/i_{s} = 8/2 = 4]$ . One additional point configuration is generated by the translation $[t({1\over 2}{1\over 2}{1\over 2}): 4a\ xxx]$ with $[x_{2} = {1\over 2}]$ , the two others by applying the d-glide reflection $[y + {1\over 4}]$ , $[x + {1\over 4}]$ , $[z + {1\over 4}]$ to the first two point configurations: 4a xxx with $[x_{3} = {1\over 4}]$ and $[x_{4} = {3\over 4}]$ . The intersection group $[{\cal D}]$ of the inherent symmetry $[Fm\overline{3}m]$ with the normalizer $[Ia\overline{3}d]$ is $[Pa\overline{3}]$ . Its index 4 in $[Ia\overline{3}d]$ gives again the number of equivalent point configurations.

The set of equivalent point configurations is always infinite if the normalizer contains continuous translations but this set may be described by a finite number of subsets due to non-continuous translations.

Example

The Euclidean and affine normalizer of $[P6_{1}]$ is $[P^{1}622]$ (a, b, ɛc). With the aid of the `additional generators' given in Table 15.2.1.4 , one can calculate two subsets of point configurations that are equivalent to a given general point configuration 6a xyz with $[x = x_{0}]$ , $[y = y_{0}]$ , $[z = z_{0}]$ : 6a xyz with $[x_{0}, y_{0}, z_{0} + t]$ and $[y_{0}, x_{0}, {-z_{0}} + t]$ . If, however, the coordinates for the original point configuration are specialized, e.g. to $[x = y = x_{1}]$ , $[z = z_{1}]$ or to [x = y = 0] , $[z = z_{2}]$ , only one subset exists, namely $[x_{1}, x_{1}, z_{1} + t]$ or 0, 0, $[z_{2} + t]$ , respectively. The reduction of the number of subsets is a consequence of the enhancement of the site symmetry in the normalizer (.2. or 622, respectively), but the index $[i_{s}]$ , as introduced above, does not necessarily give the reduction factor for the number of subsets.

It has to be noticed that for most space groups with a Euclidean normalizer containing continuous translations the index $[i_{s}]$ is larger than 1 for all point configurations, i.e. the number of subsets of equivalent point configurations is necessarily reduced. The general Wyckoff position of such a space group does not belong to a characteristic type of Wyckoff sets (cf. Part 14 ) and the inherent symmetry of all corresponding point configurations is enhanced.

Example

The Euclidean and affine normalizer of P6 is $[P^{1}6/mmm]$ (a, b, ɛc). As a consequence of the continuous translations, the site symmetry of any point is at least m.. in $[P^{1}6/mmm]$ . With the aid of the `additional generators', one calculates four subsets of point configurations that are equivalent to a given general point configuration 6d xyz with $[x = x_{0}]$ , $[y = y_{0}]$ , $[z = z_{0}]$ : $[x_{0}, y_{0}, z_{0} + t]$ ; $[-x_{0}, {-y_{0}}, {-z_{0}} + t]$ ; $[y_{0}, x_{0}, z_{0} + t]$ ; $[-y_{0}, {-x_{0}}, {-z_{0}} + t]$ . The first two and the second two subsets coincide, however.

According to the above examples, Euclidean- (affine-) equivalent point configurations may or may not belong to the same Wyckoff position. Consequently, normalizers also define equivalence relations on Wyckoff positions:

Two Wyckoff positions of a space group $[{\cal G}]$ are called Euclidean- or $[{\cal N}\!_{{\cal E}}]$ -equivalent (affine- or $[{\cal N}\!_{{\cal A}}]$ -equivalent) if their point configurations are mapped onto each other by the Euclidean (affine) normalizer of $[{\cal G}]$ .

Euclidean-equivalent Wyckoff positions are important for the description or comparison of crystal structures in terms of atomic coordinates. Affine-equivalent Wyckoff positions result in Wyckoff sets (cf. Section 8.3.2 and Chapter 14.1 ) and form the necessary basis for the definition of lattice complexes. All site-symmetry groups corresponding to equivalent Wyckoff positions are conjugate in the respective normalizer.

Examples

The Euclidean and affine normalizer of $[I\overline{4}m2]$ is $[I4/mmm\ ({1\over 2}{\bf a} - {1\over 2}{\bf b}, {1\over 2}{\bf a} + {1\over 2}{\bf b}, {1\over 2}{\bf c})]$ . It maps the point configurations $[2a\ 000, 2b\ 00{1\over 2}, 2c\ 0{1\over 2}{1\over 4}]$ and $[2d\ 0{1\over 2}{3\over 4}]$ (body-centred tetragonal lattices) onto each other. Accordingly, Wyckoff positions a to d are affine-equivalent and together form a Wyckoff set. Analogous point configurations exist in subgroup $[P\overline{4}n2]$ of $[I\overline{4}m2]$ (again Wyckoff positions a to d). The Euclidean and affine normalizer of $[P\overline{4}n2]$ , however, is $[P4/mmm\ ({1\over 2}{\bf a} - {1\over 2}{\bf b}]$ , $[{1\over 2}{\bf a} + {1\over 2}{\bf b},{1\over 2}{\bf c})]$ , not containing $[t({1\over 2}0{1\over 4})]$ . Therefore, Wyckoff positions a and b form one Wyckoff set, c and d a different one. This is also reflected in the site-symmetry groups $[\overline{4}]$ .. and 2.22.

The existence of Euclidean-equivalent point configurations results in different but equivalent descriptions of crystal structures (exception: crystal structures with symmetry $[Im\overline{3}m]$ or $[Ia\overline{3}d]$ ). All such equivalent descriptions are derived by applying the additional generators of the Euclidean normalizer of the space group $[{\cal G}]$ to all point configurations of the original description. Since an adequate description of a crystal structure always displays the full symmetry group of that structure, the number of equivalent descriptions must equal the index of $[{\cal G}]$ in $[{\cal N}\!_{{\cal E}}({\cal G})]$ .

Example

Ag₃PO₄ crystallizes with symmetry $[P\overline{4}3n]$ (cf. Masse et al., 1976): P at 2a 000, Ag at $[6d\ {1\over 4} 0 {1\over 2}]$ and O at 8e xxx with [x = 0.1486] . $[{\cal N}\!_{{\cal E}}(P\overline{4}3n) = Im\overline{3}m]$ with index 4 gives rise to three additional equivalent descriptions: $[t({1\over 2} {1\over 2} {1\over 2})]$ yields P at 2a 000, Ag at $[6c\ {1\over 4} {1\over 2} 0]$ and O at 8e xxx with ; inversion through the origin results in P at 2a 000, Ag at $[6d\ {1\over 4} 0 {1\over 2}]$ , O at 8e xxx with [x = -0.1486] and in P at 2a 000, Ag at $[6c {1\over 4} {1\over 2} 0]$ and O at 8e xxx with . Although the phosphorus configuration is the same for all descriptions and the silver and oxygen atoms refer to only two configurations each, their combinations result in a total of four different equivalent descriptions of the structure.

If the Euclidean normalizer of a space group contains continuous translations, each crystal structure with that symmetry refers to an infinite set of equivalent descriptions. This set may be subdivided into a finite number of subsets in such a way that the descriptions of each subset vary according to the continuous translations. The number of these subsets is given by the product of the finite factors listed in the last column of Tables 15.2.1.3 and 15.2.1.4 .

Example

The tetragonal form of BaTiO₃ has been described in space group P4mm (cf. e.g. Buttner & Maslen, 1992): Ba at 1a 00z with [z = 0] , Ti at $[1b\ {1\over 2} {1\over 2}z]$ with [z = 0.482] , O1 at $[1b\ {1\over 2}{1\over 2}z]$ with [z = 0.016] , and O2 at $[2c\ {1\over 2}0z]$ with [z = 0.515] . $[{\cal N}\!_{{\cal E}}(P4mm) = P^{1}4/mmm]$ $[{({1\over 2}({\bf a}-{\bf b}), {1\over 2}({\bf a}+{\bf b}), \varepsilon{\bf c})}]$ gives rise to $[(2 \cdot \infty)\cdot 2 \cdot 1]$ equivalent descriptions of this structure. The continuous translation with vector (00t) yields a first infinite subset of equivalent descriptions: Ba at 1a 00z with [z = t] , Ti at $[1b\ {1\over 2}{1\over 2}z]$ with [z = 0.482 + t] , O1 at $[1b\ {1\over 2}{1\over 2}z]$ with [z = 0.016 + t] , and O2 at $[2c\ {1\over 2}0z]$ with [z = 0.515 + t] . The translation with vector $[({1\over 2} {1\over 2} 0)]$ generates a second infinite subset: Ba at $[1b\ {1\over 2} {1\over 2}z]$ with [z = t] , Ti at 1a 00z with [z = 0.482 + t] , O1 at 1a 00z with , and O2 at $[2c\ {1\over 2} 0z]$ with . Inversion through the origin causes two further infinite subsets of equivalent coordinate descriptions of BaTiO₃: first, Ba at 1a 00z with [z = t] , Ti at $[1b\ {1\over 2} {1\over 2} z]$ with [z = 0.518 + t] , O1 at $[1b\ {1\over 2} {1\over 2}z]$ with [z = -0.016 + t] , and O2 at $[2c\ {1\over 2} 0z]$ with [z = 0.485 + t] ; second, Ba at $[1b\ {1\over 2} {1\over 2}z]$ with [z = t] , Ti at 1a 00z with , O1 at 1a 00z with , and O2 at $[2c\ {1\over 2} 0z]$ with .

More details on Euclidean-equivalent point configurations and descriptions of crystal structures have been given by Fischer & Koch (1983).

References

Buttner, R. H. & Maslen, E. N. (1992). Structural parameters and electron difference density in BaTiO₃. Acta Cryst. B48, 764–769.Google Scholar

Fischer, W. & Koch, E. (1983). On the equivalence of point configurations due to Euclidean normalizers (Cheshire groups) of space groups. Acta Cryst. A39, 907–915.Google Scholar

Masse, R., Tordjman, I. & Durif, A. (1976). Affinement de la structure cristalline du monophosphate d'argent Ag₃PO₄. Existence d'une forme haute température. Z. Kristallogr. 144, 76–81.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 15.3, pp. 900-901