Relations between crystal structures

Fischer, W.; Koch, E.

doi:10.1107/97809553602060000532

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

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International Tables for Crystallography (2006). Vol. A. ch. 14.3, p. 873

Section 14.3.2. Relations between crystal structures

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

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

14.3.2. Relations between crystal structures

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Frequently, different crystal structures show the same geometrical arrangement for some of their atoms, even though their space groups do not belong to the same type. In these cases, the corresponding Wyckoff positions either belong to the same lattice complex or there exist close relationships between them, e.g. limiting-complex relations.

Examples

(1) The Fe atoms in pyrite FeS₂ occupy Wyckoff position $[4a\ 000\ .\bar{3}]$ . of $[Pa\bar{3}]$ (descriptive symbol F) that belongs to the invariant lattice complex $[Fm\bar{3}m\ a]$ . Accordingly, the Fe atoms in pyrite form a face-centred cubic lattice as do the Cu atoms in the element structure of copper.
(2) Cuprite Cu₂O crystallizes with symmetry $[Pn\bar{3}m]$ . The oxygen atoms occupy Wyckoff position $[2a\ 000\ \bar{4}3m]$ (descriptive symbol I) and the copper atoms position $[4b\ {1 \over 4} {1 \over 4} {1 \over 4}\ .\bar{3}m]$ (descriptive symbol $[{1 \over 4} {1 \over 4} {1 \over 4}\; F]$ ). Position 2a belongs to lattice complex $[Im\bar{3}m\ a]$ and position 4b to $[Fm\bar{3}m\ a]$ . Therefore, the O atoms form a body-centred cubic lattice like the W atoms in the structure of tungsten, and the copper atoms form a face-centred cubic lattice. The tungsten configuration is shifted by $[({1 \over 4} {1 \over 4} {1 \over 4})]$ with respect to the copper configuration.
(3) K₂NaAlF₆ (elpasolite, cf. Morss, 1974) and K₂PbNi(NO₂)₆ (cf. Takagi et al., 1975) crystallize with symmetry $[Fm\bar{3}m]$ and $[Fm\bar{3}]$ , respectively. $[\displaylines{&\hbox{K}_{2}\hbox{NaAlF}_{6}\hfill\cr &\quad\! \matrix{\hbox{Al}\hfill &4a \hfill &m\bar{3}m \hfill &000 \hfill &F\hfill\cr \hbox{Na} \hfill &4b \hfill &m\bar{3}m \hfill &{1 \over 2}{1 \over 2}{1 \over 2} \hfill &{1 \over 2}{1 \over 2}{1 \over 2} F\cr \hbox{K} \hfill &8c \hfill &\bar{4}3m \hfill &{1 \over 4}{1 \over 4}{1 \over 4} \hfill &{1 \over 4}{1 \over 4}{1 \over 4} P_{2}\cr \hbox{F} \hfill &24e \hfill &4m.m \hfill &x00 \hfill &F6z\hfill\cr & & &x = 0.219\hfill \cr}}]$ $[\displaylines{&\hbox{K}_{2}\hbox{PbNi}(\hbox{NO}_{2})_{6}\hfill\cr &\quad\! \matrix{\hbox{Ni} \hfill &4a \hfill &m\bar{3}. \hfill &000 \hfill &F\hfill\cr \hbox{Pb} \hfill &4b \hfill &m\bar{3}. \hfill &{1 \over 2}{1 \over 2}{1 \over 2} \hfill &{1 \over 2}{1 \over 2}{1 \over 2} F\hfill\cr \hbox{K} \hfill &8c \hfill &23. \hfill &{1 \over 4}{1 \over 4}{1 \over 4} \hfill &{1 \over 4}{1 \over 4}{1 \over 4} P_{2} \hfill\cr \hbox{N} \hfill &24e \hfill &mm2.. \hfill &x00 \hfill &F6z\hfill\cr & & &x = 0.1966\cr \hbox{O} \hfill &48h \hfill &m.. \hfill &0yz \hfill &F6z2x \hfill \cr}}]$ As the descriptive lattice-complex symbols for the various atomic positions show immediately, the two crystal structures are very similar. The only difference originates from the replacement of the fluorine atoms in elpasolite by NO₂ groups in K₂PbNi(NO₂)₆, which causes the symmetry reduction from $[Fm\bar{3}m]$ to $[Fm\bar{3}]$ .
(4) The crystal structure of CoU (Baenziger et al., 1950) may be interpreted as a slightly distorted CsCl (or β-brass, CuZn)-type structure. CsCl corresponds to Wyckoff positions 1a and 1b of $[Pm\bar{3}m]$ with descriptive symbol P and $[{1 \over 2}{1 \over 2}{1 \over 2}\ P]$ , respectively; Co and U both occupy Wyckoff position 8a .3. xxx of $[I2_{1}3]$ with for U and for Co. As the descriptive symbol $[2_{1}2_{1}..\ P_{2}Y^{*}1xxx]$ shows, this Wyckoff position belongs to a Weissenberg complex with two invariant limiting complexes, namely P ( $[Pm\bar{3}m\ a]$ ) and $[Y^{*}\ (I4_{1}32\ a)]$ . corresponds to $[P_{2}]$ , $[x = {1 \over 4}]$ to $[{1 \over 4}{1 \over 4}{1 \over 4}\ P_{2}]$ , $[x = {1 \over 8}]$ to $[^{+}Y^{*}]$ and $[x = {7 \over 8}]$ to $[^{-}Y^{*}]$ . Consequently, the uranium and cobalt atoms form approximately a $[P_{2}]$ and a $[{1 \over 4}{1 \over 4}{1 \over 4}\ P_{2}]$ configuration, respectively.

Publications by Hellner (1965, 1976a,b,c, 1977, 1979), Loeb (1970), Smirnova & Vasserman (1971), Sakamoto & Takahasi (1971), Niggli (1971), Fischer & Koch (1974), Hellner et al. (1981) and Hellner & Sowa (1985) refer to this aspect.

References

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International Tables for Crystallography (2006). Vol. A. ch. 14.3, p. 873