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Modularity, poly­typism, topology, and complexity of crystal structures of inorganic compounds (Review)

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Abstract

The modular approach is a powerful tool in current inorganic crystal chemistry. It enables not only a more detailed analysis of the known structures and the determination of structural relationships between them, but also the prediction of potentially novel structures that can be applied in modern materials science. A large number of examples of compounds with modular structures allows us to state that structural modularity is a widely spread phenomenon among natural and synthetic compounds. The use of the formalism of OD theory makes it possible to analyze the symmetry of polytypes with different crystal structures. In this review, we collected new data published in the last 15 years about OD structures and phenomena of polytypism and modularity in inorganic compounds, as well as the topological approach to the analysis of crystal structures.

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Notes

  1. Strictly speaking, there are screw symmetry operations (screw turns with a translation along the axis direction) for ±2π/N angles, where N = 2, 3, 4 or 6. The screw axis is designated as Nr if its translational component is r/N relative to a, b, c0 vectors.

  2. Layer symmetry groups \(\mathbf{G}_{2}^{3}\) contain two translations (i.e. \(\mathbf{G}_{2}^{3}\supset {{T}_{2}}\)) and are derived from 17 planar groups \(\mathbf{G}_{2}^{2}\) (so-called wallpaper symmetry groups) by adding the mirror reflection relative to the xy coordinate plane.

  3. The equivalence of layers is designated by the symbol ::; e.g., the equivalence of the pair of (Lp, Lp+1) and (Lq, Lq+1) layers is designated as (Lp, Lp+1) :: (Lq, Lq+1).

  4. Not confuse with the number of formula units (Z).

  5. The number |G| of elements in G is called the order of group G.

  6. The layer position consistent with the position of the previous layer and VC-conditions.

  7. For OD structures with the number M > 1 of OD layers, the polar layers are designated as bi (or di depending on the type of orientation relative to the layer plane), and non-polar are marked Ai.

  8. Orthorhombic and monoclinic representatives of the lamprophyllite group [173] correspond to the same OD family consisting of two types of layers (category IV: non-polar layers are represented by HOH modules (A2 = L2n) and layers of A cations (A1 = L2n+1)) described by one groupoid \(\begin{matrix} \ P(1)2/m1 & & P(2/n){{2}_{1}}/m{{2}_{1}}m \\ & \ [0,s] \\ \end{matrix}\), where s ~ –0.09 [10].

  9. In universal algebra, groupoid m = ‹M,*› means a non-empty set M in which a binary operation * is given reflecting sets M×M in M. At present, the term magma [911] (or simply algebra [912]) is used to designate this groupoid.

  10. Morphisms are also shown as arrows φ: X → Y.

  11. In this case, pairs of identical (Lp,Lp+1,Lp+2) triplets are implied, e.g., (L1,L2,L3)::(L2,L3,L4).

  12. The antigorite polysomatic series belongs to the kaolinite-serpentine group, which contain two-layer ОТ modules [913]. The general formula for the representatives of the antigorite polysomatic series can be written as

    $$\{{{M}_{3(n-1)}}{{\psi }_{2(2n-3)}}(\text{S}{{\text{i}}_{2n}}{{\text{O}}_{5n}})\}$$
    (44)

    where n ≥ 2. The symmetry of representatives with even n is characterized by space group C2/m, and the symmetry of representatives with odd n is Pm [156]. Obviously, for n → ∞ the formula takes the form {M2Ψ4(Si2O5)}, which corresponds to the stoichiometry of minerals of the serpentine subgroup characterized by layered ОТ modules [150].

  13. This stoichiometry is obtained when formula (44) characterizing the TOT module stoichiometry [115] is added with a {MΨ2}fragment. In this case, according to T. Zoltai [113], the module stacking sequence can be written as R–0–0 for shattuckite and R–0–R–0–0 for plancheite. Thus, the formula of the TOT module in structures of the shattuckite-plancheite polysomatic series can be rewritten as:

    \(\{M_{(3n-1)}\psi_{2(n-1)}(\text{Si}_{2n}\text{O}_{(5n+1)})_2\} + \{M\psi_2\} = \{M_{3n}\psi_{2n}(\text{Si}_{2n}\text{O}_{(5n+1)})_2\}\).

  14. It was shown previously [170] that the XRD pattern of molybdophyllite was characterized by the presence of diffuse peaks, which indicated a partial disorder of TOT layers and the OD structure. Here the OD layer with the P321 symmetry with a1, a2, c0 (non-translational vector) is related to a, b, c translational vectors of the molybdophyllite structure as follows [170]: a1 = (a–b)/2; a2 = b; a1 = a2 = 9.373 Å; c0 = c sinβ = 13.944 Å. The OD grupoid has the form

    $$\begin{matrix} P2 & 2 & 2 & (3) & 1 & 1 & 1 \\ \{{{2}_{r}} & {{2}_{{{r}'}}} & {{2}_{{{r}''}}} & {{3}_{3}} & 1 & 1 & 1 \\\end{matrix},$$
    (50)

    where r + r′ + r′′ = 0 [ 170 ].

  15. The merotype series means that the structures of all representatives contains at least two fragments. One of them is constantly repeated, and another (second, third, fourth, etc.) can be unique for each structure of the series compounds. The plesiotype series is distinguished when all representatives are characterized by the same number of topologically identical modules that, nonetheless, can differ from each other by the chemical composition.

  16. 2M and 2O polytypes of lamprophyllite-related minerals can be described within one OD family [69] characterized by the presence of two types of non-polar OD layers and assigned to category IV [10]. The first layer corresponds to the HOH module and has the symmetry P(1)2/m1. The second OD layer with the symmetry P(2/n) 21/m 21/m corresponds to the interpackage space [10, 172]. In accordance with the ZNF ratio, only two MDO polytypes are possible: monoclinic (space group C2/m) and orthorhombic (space group Pnmn) [10, 73]. In the monoclinic polytype, the generating operation is the translation t = a0 + b/2 + 2sc (b ~ 7 Å, c ~ 5.4 Å; a0 ~ 10 Å). In the orthorhombic polytype, the generating operation is the glide reflection plane with the translational component a0+b/2 perpendicular to c. The respective OD grupoid has the form

    $$\begin{matrix} \ P(1)2/m1 & & P(2/n){{2}_{1}}/m{{2}_{1}}m \\ & \ [0,s] \\ \end{matrix}$$
    (54)

    where s takes values from –0.085 to –0.10 [10].

  17. In the schüllerite structure, the HOH module corresponds to linkage 3 [175] and is characterized by the symmetry [172, 173], which formally does not meet the IMA CNMNC criteria. Nonetheless, schüllerite often occurs in association with lamprophyllite and related minerals and also makes epitaxial intergrowths with them [182]. Schüllerite is characterized by the same structural features as the minerals related to lamprophyllite (coordination number 5 for L cations) and the absence of anionic groups and water molecules in the interpackage space. Furthermore, the general crystal chemical formula of schüllerite is close to that of minerals related to lamprophyllite and differes only in the presence of four symmetrically non-equivalent М sites in the octahedral О layer, (Z = 2): A2[M1M2M2′M3X2][L2O2(Si2O7)].

  18. All calculations were performed using the ToposPro program package [24]. In the calculations, we used the following significance criteria for working ions: Li+ (elementary channel radius, Rchan = 2.02 Å; elementary void radius Rsd = 1.38 Å); Na+ (Rchan = 2.16 Å; Rsd = 1.54 Å); Ag+ (Rchan = 2.20 Å; Rsd = 1.58 Å); Pb2+ (Rchan = 2.23 Å; Rsd = 1.62 Å); K+ (Rchan = 2.30 Å; Rsd = 1.70 Å); Rb+ (Rchan = 2.38 Å; Rsd = 1.78 Å); Cs+ (Rchan = 2.47 Å; Rsd = 1.88 Å).

  19. Taking into account a large number of publications devoted to various aspects of the synthesis, structure, properties, and applications of compounds forming a relatively small family considred here, in composing the bibliography to this section, we included only the most representative reviews and the most recent or bright examples illustrating the main text.

  20. Crystal structures of gainesite-group minerals are characterized by the presence of pentameric [BeP4O16]10– clusters.

  21. This transformation of the formula is highly justified because general formula (55) reflects only the stoichiometry between A and B cations and the relation between the number of B cations with the number of anions. Although the AxTa7O19 and AxNb7O19 compounds have the same stoichiometry, they differ in the stacking patterns of o- and p-modules: …|ppo|… (Fig. 46с) and …|pppopo|… (Fig. 46d) respectively.

  22. In the structure with the space group P63/mmc, the glaserite packing of atoms is violated due to the presence of alternative tetrahedra and ten-vertex polyhedra.

  23. For organic and metal-organic compounds with M = 1 OD layers the systematic studies of Prof. Berthold Stöger on the analysis of crystal structures, polytypism, and twinning [914-917] and some other works [41, 918] can also be mentioned.

  24. The tiling point symbol [600] is written as Aa.Bb...; the mentioned means that a angles belonging to the smallest A cycle converge in the network vertex; b angles belonging to the smallest B cycle, etc., with A < B < ... and a + b + ... = CN(CN – 1)/2.

  25. It should be noted that the maximum possible symmetry of the three-link wollastonite-type chain is described by cylindrical group pm2m.

  26. This symbol indicates that the repeating fragment of the chain contains two doubly connected 2Т tetrahedra and four three-connected 3T tetrahedra [596]. In the general form, the topological symbol of chains is based on the expression cTr = 1Tr2Tr3Tr4Tr, which shows possible types of the connection of tetrahedra (с = 1-4) in the repeating fragment and their number (r ≥ 0) [595, 596].

  27. This unusual decrease in the Rad(min) value for modeling the geological conditions such as increased temperatures of late hydrothermal solutions and pressures affect the cation diffusion.

  28. CBUs are large structural fragments occurring in two or more types of frameworks [619].

  29. Amount of information per atom in the structure is defined by the formula [90]:

    \({}^{str}{{I}_{G}}=-\sum\nolimits_{i=1}^{k}{{{p}_{i}}{{\log }_{2}}{{p}_{i}}},\)

    where р is the probability of a random selection of an atom from the i-th crystallographic orbit (pi = mi/v); mi is the multiplicity of the crystallographic orbit with respect to the reduced cell; v is the number of atoms in the reduced cell \((\nu =\sum\nolimits_{i=1}^{k}{{{m}_{i}}})\).

    To determine the total amount of information per cell the following formula is used:

    \({}^{struct}{{I}_{G,\text{total}}}=-\nu ({}^{struct}{{I}_{G}})=-\nu \sum\nolimits_{i=1}^{k}{{{p}_{i}}{{\log }_{2}}{{p}_{i}}}.\)

  30. According to S. V. Krivovichev’s classification [89, 90] based on the complexity parameter IG,total, structures are divided into very simple (0-20 bit/unit cell), simple (20-100 bit/unit cell), intermediate (100-500 bit/unit cell), complex (500-1000 bit/unit cell), and very complex (>1000 bit/unit cell).

  31. Cleavage is the ability of crystal of minerals and inorganic compounds to split with the formation of smooth planes (cleavage planes) characterized by the minimum cohesion in the direction perpendicular to the plane [920].

  32. The name is written in italics because the mineral was officially discredited [905].

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Funding

The work was supported by Russian Science Foundation grant No. 20-77-10065.

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Authors and Affiliations

  1. Laboratory of Arctic Mineralogy and Materials Science, Kola Science Center, Russian Academy of Sciences, Apatity, Russia

    S. M. Aksenov, D. O. Charkin, A. M. Banaru, S. N. Volkov & D. V. Deineko

  2. Geological Institute, Kola Science Center, Russian Academy of Sciences, Apatity, Russia

    S. M. Aksenov

  3. Faculty of Chemistry, Moscow State University, Moscow, Russia

    D. O. Charkin, A. M. Banaru, D. V. Deineko & A. N. Kuznetsov

  4. Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, Moscow, Russia

    D. A. Banaru

  5. Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, Moscow, Russia

    A. N. Kuznetsov

  6. Federal Research Center “Crystallography and Photonics”, Russian Academy of Sciences, Moscow, Russia

    R. K. Rastsvetaeva

  7. Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry, Russian Academy of Sciences, Chernogolovka, Russia

    N. V. Chukanov

  8. Faculty of Geology, Moscow State University, Moscow, Russia

    B. B. Shkurskii & N. A. Yamnova

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  1. S. M. Aksenov
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  3. A. M. Banaru
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  8. R. K. Rastsvetaeva
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  9. N. V. Chukanov
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  11. N. A. Yamnova
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Contributions

S. M. Aksenov. Doctor of Chemical Sciences, Head of the Laboratory of Arctic Mineralogy and Materials Science, Kola Science Center, Russian Academy of Sciences; senior researcher of the Laboratory for the Complex Analysis of Unique Ore-Bearing Systems, Geological Institute, Kola Science Center, Russian Academy of Sciences.

Area of scientific interests: crystal chemistry, single crystal X-ray diffraction analysis, modularity of crystal structures, mineralogy, topological analysis, polytypism, symmetry OD analysis.

E-mail: aks.crys@gmail.com

D. O. Charkin. Doctor of Chemical Sciences, associate professor of the Chair of Inorganic Chemistry, Department of Chemistry, Moscow State University; leading researcher of the Laboratory of Arctic Mineralogy and Materials Science, Kola Science Center, Russian Academy of Sciences. The author of section 3.5.

Area of scientific interests: inorganic chemistry, synthesis of new compounds, crystal chemistry of heavy elements, modular design of inorganic materials.

E-mail: d.o.charkin@gmail.com

A. M. Banaru. Candidate of Chemical Sciences, associate professor of the Chair of Physical Chemistry, Department of Chemistry, Moscow State University; leading researcher of the Laboratory of Arctic Mineralogy and Materials Science, Kola Science Center, Russian Academy of Sciences.

Area of scientific interests: mathematical crystallography, crystal chemistry of molecular crystals, topological analysis, theory of complexity of crystal structures.

E-mail: banaru@mail.ru

D. A. Banaru. Junior researcher, Vernadsky Institute of Geochemistry and Analytical Chemistry.

Area of scientific interests: molecular crystals, theory of complexity of crystal structures.

E-mail: banaru@geokhi.ru

C. N. Volkov. Candidate of Chemical Sciences, leading researcher of the Laboratory of Arctic Mineralogy and Materials Science, Kola Science Center, Russian Academy of Sciences.

Area of scientific interests: crystal chemistry of borates, inorganic materials science, single crystal X-ray diffraction analysis, aperiodic structures.

E-mail: s.n.volkov@inbox.ru

D. V. Deineko. Candidate of Chemical Sciences, associate professor of the Department of Chemistry, Moscow State University; leading researcher of the Laboratory of Arctic Mineralogy and Materials Science, Kola Science Center, Russian Academy of Sciences.

Area of scientific interests: inorganic materials science, optical properties of crystals, nanostructured materials, luminescence, bioceramics.

E-mail: deynekomsu@gmail.com

A. N. Kuznetsov. Corresponding Member of the Russian Academy of Sciences, Doctor of Chemical Sciences, leading researcher of the Chair of Inorganic Chemistry, Department of Chemistry, Moscow State University. The author of Section 3.5.

Area of scientific interests: inorganic chemistry, intermetallic compounds, DFT calculations, promising materials.

R. K. Rastsvetaeva. Doctor of Geological-Mineralogical Sciences, chief researcher of the Laboratory of X-Ray Methods of Analysis and Synchrotron Radiation, Federal Research Center of Crystallography and Photonics, Russian Academy of Sciences.

Area of scientific interests: general mineralogy and mineralogical crystallography, crystal chemistry of silicates, systematization of inorganic compounds, structural chemistry, single crystal X-ray diffraction analysis.

E-mail: rast.cryst@gmail.com

N. V. Chukanov. Doctor of Physical-Mathematical Sciences, chief researcher of the Laboratory of Kinetic Calorimetry, Federal Research Center of Physical and Medical Chemistry, Russian Academy of Sciences.

Area of scientific interests: infrared spectroscopy, general mineralogy, physics of minerals, crystal chemistry, systematization of natural compounds, energy-intensive compounds.

E-mail: nikchukanov@yandex.ru

B. B. Shkurskii. Candidate of Geological-Mineralogical Sciences, associate professor of the Chair of Petrology and Volcanology, Department of Geology, Moscow State University. The author of Sections 7.1-7.3.

Area of scientific interests: optical mineralogy, geometric crystallography, optical properties of crystals, general mineralogy, petrography.

E-mail: shkurskyBB@yandex.ru

N. A. Yamnova. Doctor of Geological-Mineralogical Sciences, senior researcher of the Chair of Crystallography and Crystal Chemistry, Department of Geology, Moscow State University.

Area of scientific interests: general crystal chemistry, polytypism, modular and polysomatic series, genetic mineralogy, single crystal X-ray diffraction analysis.

Corresponding author

Correspondence to S. M. Aksenov.

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Russian Text © The Author(s), 2023, published in Zhurnal Strukturnoi Khimii, 2023, Vol. 64, No. 10, 117102.https://doi.org/10.26902/JSC_id117102

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Aksenov, S.M., Charkin, D.O., Banaru, A.M. et al. Modularity, poly­typism, topology, and complexity of crystal structures of inorganic compounds (Review). J Struct Chem 64, 1797–2028 (2023). https://doi.org/10.1134/S0022476623100013

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