Basic principles

In accord with the previous theoretical considerations (Krivovichev, Reference Krivovichev2012, Reference Krivovichev2013), the structural complexity is estimated as the amounts of structural Shannon information per atom (^strI _G) and per unit cell (^strI _G,total) calculated according to the following equations:

(1)$$^{str} I_G = {-}\sum {_{{ i} = 1}^k } {\it{p}}_i{\rm lo}{\rm g}_ 2{\it{p}}_i\,{\rm ( bits/atom) }$$

(2)$$^{{\it str}} I_{G, total} = v \times ^{str}I_G = {-}v \times \sum {_{{\it i} = 1}^k } {\it p}_i{\rm lo}{\rm g}_2p_i\,( {\rm bits/cell}) $$

where k is the number of different crystallographic sites (crystallographic orbits) in the structure and p_i is the random choice probability for an atom from the ith crystallographic orbit, that is:

(3)$$p_i = m_i/ v$$

where m_i is a multiplicity of a crystallographic orbit (i.e. the number of atoms of a specific Wyckoff site in the reduced unit cell), and v is the total number of atoms in the reduced unit cell.

By analogy, the chemical complexity of a mineral can be estimated from its end-member formula extracted from the International Mineralogical Association List of Minerals (Pasero, Reference Pasero2021) and, in some cases, corrected by Krivovichev (Reference Krivovichev2021b). It should be emphasised that, within the context of this study, it is implicitly understood that each mineral has a unique end-member formula (Hawthorne et al., Reference Hawthorne, Mills, Hatert and Rumsey2021; Hawthorne, Reference Hawthorne2021). Thus, for the mineral with the end-member formula E⁽¹⁾_{c 1}E⁽²⁾_{c 2}…E^(k)_ck, where E⁽ⁱ⁾ is an ith chemical element in the formula and c_i is its integer coefficient, the chemical information per atom (^chemI _G) and per formula unit, f.u. (^chemI _G,total) can be calculated as follows (Siidra et al., Reference Siidra, Zenko and Krivovichev2014; Krivovichev et al., Reference Krivovichev, Krivovichev and Hazen2018a, Reference Krivovichev, Charykova and Krivovichev2020a):

(4)$$^{chem} I_{G} = {-}\sum {_{i = 1}^k } \,{p}_i{\rm log}_{2} \,p_i\,( {\rm bits/atom}) $$

(5)$$^{{\rm chem}}I_{G, \;total} = e \times ^{chem}I_{G} = {-}e \times \sum {_{i = 1}^k} \,{p}_i{\rm log}_{2}{p_i}\,( {\rm bits/f}{\rm .u}.) $$

where k is the number of different elements in the formula and p_i is the random-choice probability for an atom of the ith element, that is:

(6)$$p_i = c_i/ e$$

where c_i is the coefficient of the ith element in the formula and e is the total number of atoms in the chemical formula:

(7)$$e = \sum {_{i = 1}^k } c{_i}$$

Krivovichev et al. (Reference Krivovichev, Krivovichev and Hazen2018a) demonstrated that, on average, structural and chemical complexities correlate with each other, which agrees well with the intuitive feeling that chemically complex species generally possess complex crystal structures. Though there are exceptions of this empirical rule (Krivovichev, Reference Krivovichev2013), it is a statistically valid observation.

New developments: Hornfeck extension

In some cases, the information-based measures proposed by Krivovichev (Reference Krivovichev2012, Reference Krivovichev2013) do not possess enough discriminative power to distinguish between the crystal structures that are thought to be different in terms of their symmetry or thermodynamic stability. For instance, α- and β-quartz have the same structural complexity parameters, 0.918 bits/atom and 8.265 bits/cell, which is counter-intuitive, as the α → β transition is associated with the increase of symmetry from trigonal to hexagonal. In order to overcome these difficulties, Hornfeck (Reference Hornfeck2020) proposed to extend the information-based complexity measures by introducing chemical, combinatorial and coordinational parameters. In the framework of the Hornfeck's approach, the complexity parameters ^strI _G and ^strI _G,total defined by equations (1) and (2), respectively, are classified as combinatorial. The coordinational measures are defined as follows (in order to be consistent with the definitions given above, the Hornfeck notations of complexity measures are modified).

Each crystallographic (Wyckoff) site in a crystal structure has a finite number of degrees of freedom depending upon its site symmetry. For instance, the site located on the inversion centre has zero degrees of freedom, whereas one located on the mirror plane has two degrees of freedom. The number of degrees is called by Hornfeck (Reference Hornfeck2020) an arity, by analogy with mathematics, where arity means the number of arguments taken by a function. The arity may be equal to 0, 1, 2, and 3. Let the arity of the ith crystallographic site be denoted as a_i. Then

(8)$$A = \sum {_{i = 1}^k } a_i$$

is the sum of arities for a crystal structure consisting of k crystallographic orbits (k different Wyckoff sites). The information amount of the distribution of arities in the crystal structure is defined as (d.f. = degree of freedom)

(9)$$I_A = {-}\sum {_{{i} = 1}^{k} } \,{p}_i^a {\rm log} _{2}\,p_i^a \;( {\rm bits/d}{\rm .f}{ ) }$$

(10)$$I_{A, total} = A \times I_A = {-}A \times \sum {_{{i} = 1}^{k} } \,{p}_i^a \,{\rm lo}{\rm g}_2p_i^a \quad( {\rm bits/cell}) $$

where

(11)$$p_i^a = a_i/A$$

For instance, for α-quartz (space group P3₂21), a ₁(Si) = 1 (the Si atom is located on the twofold axis) and a ₂(O) = 3 (general site). Therefore, A = 4, I_A = 0.811 bits/d.f., I_A,total = 3.245 bits/cell. In contrast, for β-quartz (space group P6₂22), a ₁(Si) = 0, a ₂(O) = 1 (general site), A = 1, I_A = 0.000bits/d.f., I_A,total = 0.000 bits/cell (Banaru et al., Reference Banaru, Aksenov and Krivovichev2021).

The I_A and I_A,total parameters are identified by Hornfeck (Reference Hornfeck2020) as coordinational complexity. The configurational complexity I_GA can be calculated as a combination of combinatorial and coordinational complexities via the formulas:

(12)$$I_{GA} = I( {v, \;A} ) + ( {I_{G, total} + I_{A, total}} ) /( {v + A} ) $$

where

(13)$$I( {v, \;A} ) = \displaystyle{v \over {v + A}}\,{\rm log_2}\;\;\displaystyle{{v + A} \over v} + \displaystyle{A \over {v + A}}{\rm log_2}\,\displaystyle{{v + A} \over A}$$

The dimension of the I_GA parameters is in bits per degree of freedom per atom (bits/d.f.; each atom is considered as an additional degree of freedom). The total configurational information amount per reduced unit cell is calculated as

(14)$$I_{GA, total} = ( {v + A} ) \times I_{GA}$$

The application of the equations (9 to 14) to α- and β-quartz results in different configurational complexities, 1.775 and 1.296 bits/d.f., respectively, and 15.978 and 12.955 bits/cell, respectively. The relative configurational simplicity of β-quartz corresponds to its thermodynamic nature of a high-temperature polymorph (see below).

The analogous calculations for sillimanite and andalusite that, otherwise, have the same structural complexity (Krivovichev, Reference Krivovichev2013) provide the values of 3.564 bits/d.f. and 160.366 bits/cell, and 3.638 bits/d.f. and 167.350 bits/cell, respectively. The lower complexity of sillimanite agrees well with its nature as a high-temperature Al₂SiO₅ polymorph.

Hornfeck (Reference Hornfeck2020) also provided measures for the chemical complexities of crystalline compounds based upon the atomic numbers of their constituent elements. Combined with combinatorial and coordinational complexities, the Hornfeck chemical complexity parameters provide a measure for a crystallographic complexity. The latter may be considered as numerical characteristics of crystalline compounds, and therefore is of use for digital crystal-structure classification purposes. In this review, however, we use the structural and chemical complexity parameters as defined by Krivovichev (Reference Krivovichev2013) and Krivovichev et al. (Reference Krivovichev, Krivovichev and Hazen2018a), respectively.

Complexity, configurational entropy, and entropy of mixing

The configurational entropy of crystalline solids has received considerable attention recently due to the development of high-entropy materials (see: Miracle and Senkov (Reference Miracle and Senkov2017), McCormack and Navrotsky (Reference McCormack and Navrotsky2021), Dippo and Vecchio (Reference Dippo and Vecchio2021), and references therein). In all cases, the authors refer to configurational entropy as to the entropy of mixing, in conformity with geochemical and mineralogical textbooks (Putnis, Reference Putnis1992; Ottonello, Reference Ottonello2000). However, it is evident that the configurational entropy as an entropy of an atomic configuration is not restricted to mixing entropy only, as a complex crystal structure with many crystallographic orbits (= many different Wyckoff sites) should obviously have lower configuration entropies compared to the isochemical structure with few crystallographic orbits. Thus, it makes sense to separate configurational entropy (S _cfg) into configurational entropy sensu stricto (S _cfg,ss) and configurational entropy of mixing (S _cfg,mix):

(15)$$S_{{\rm cfg}} = S_{{\rm cfg, ss}} + S_{{\rm cfg, mix}}$$

In the case of ideal mixing and a ‘single-lattice model’ (i.e. when the solid consists of one lattice or one crystallographic orbit), the value of S _cfg,mix is calculated as follows:

(16)$$S_{{\rm cfg, mix}} = {-}R\sum {_{\,j = 1}^N } \,f_j{\rm ln}\,f_j$$

where R is a gas constant (~8.3145 J⋅K^–1⋅mol^–1), N is the number of different chemical elements occupying the ‘lattice’ under consideration, and f_j is the concentration of the jth element (note that vacancy can also be considered as an entropic species) so that

(17)$$\sum {_{\,j = 1}^N } \,f_j = 1$$

For the ‘sublattice-model’ (i.e. when the crystal structure consists of several ‘sublattices’ (= several crystallographic orbits), the formula (16) transforms into (Miracle and Senkov, Reference Miracle and Senkov2017; Dippo and Vecchio, Reference Dippo and Vecchio2021):

(18)$$S_{{\rm cfg, mix}} = {-}R\Big( \sum\nolimits_{x = 1}^x {a^x} \,\sum\nolimits_{\,j = 1}^N {\,f_j} \,{\rm ln}f_j\Big) /\sum\nolimits_{x = 1}^x {a^x} $$

where x is the number of ‘sublattices’ (= the number k of crystallographic orbits in our notations above), and a^x is the number of sites in the xth ‘sublattice’ (= multiplicity m_i of an ith crystallographic orbit relative to reduced unit cell).

Taking into account that a^x = m_i and x = k and applying the equation (3), the formula (18) transforms into:

(19)$$S_{{\rm cfg, mix}} = {-}R\Big( \sum\nolimits_{i = 1}^k {\,p_i} \,\sum\nolimits_{\,j = 1}^N {\,f_j} \,{\rm ln}f_j\Big)$$

According to the value of S _cfg,mix, materials with structural disorder are classified into low-entropy (S _cfg,mix < 1R), medium-entropy (1R < S _cfg,mix < 1.5R), and high-entropy (S _cfg,mix > 1.5R) materials (Dippo and Vecchio, Reference Dippo and Vecchio2021).

In order to derive the expression for the estimates of configurational entropy sensu stricto (i.e. the entropy associated with structural complexity, assuming that more complex structures have lower entropies), Krivovichev (Reference Krivovichev2016) suggested the following reasoning. The entropy is defined as

(20)$$S = k_B{\rm ln}W$$

where k _B is the Boltzmann constant (~1.38064⋅10²³ J⋅K^–1) and W is the number of microstates that realise the given macrostate. For the crystal structure consisting of one crystallographic orbit, the S _cfg,ss is maximal with atomic entropy equal to

(21)$$S_{{\rm cfg, ss}}^{max} /N = k_B{\rm ln}( {N\ndash 1} ) $$

where N is the number of atoms that may exchange their positions locally at a given temperature (Krivovichev (Reference Krivovichev2016) assumed that this value is specific for a particular compound; the exchange of positions implies realisation of different microstates and the difference of atoms of the same element and the same crystallographic orbit). For the molar entropy, the equation (21) transforms into

(22)$$S_{cfg, ss}^{{\rm max}} = R{\rm ln}( {N\ndash 1} ) $$

Let us now suppose that, due to an unspecified reason, the symmetry of the initial crystal structure is reduced, and the single crystallographic orbit splits into k crystallographic orbits. Obviously, the crystal structure becomes more complex and its configurational entropy sensu stricto should decrease. Krivovichev (Reference Krivovichev2016) demonstrated that the amount by which the maximal value S _cfg,ss^max is decreased, ΔS _cfg,ss, is expressed as the following:

(23)$$\Delta S_{cfg, ss}/N = k_B^{str} I_G{\rm ln}2$$

Then the configuration entropy sensu stricto of the resulting low-symmetry structure, S _cfg,ss, can be calculated as

(24)$$S_{{\rm cfg, ss}} = S_{{\rm cfg, ss}}^{{\rm max}} \ndash \Delta S_{{\rm cfg, ss}} = R[ {{\rm ln}( {N\ndash 1} ) \ndash^{str}I_G{\rm ln}2 } ] $$

Through the combination of equations (1), (15), (19) and (24), one deduces that

(25)$${S_{{\rm cfg, ss}} = R[ {\rm ln}( N-1) -{\rm ln}2\sum\nolimits_{i = 1}^k {\,p_i} ( {\rm lo}{\rm g}_2p_i-\sum\nolimits_{\,j = 1}^N {\,f_j} \,{\rm log_2}f_j) ] $$

or

(26)$$S_{{\rm cfg, ss}} = R[ { ln( {N\ndash 1} ) \ndash {\rm ln}2^{{\rm str}}I_{G , mix} } ] $$

where

(27)$$^{str} I_{G, mix} = {-}\sum\nolimits_{i = 1}^k {\,p_i} \,{\rm lo}{\rm g}_2p_i-\sum\nolimits_{\,j = 1}^N {\,f_j} \,{\rm lo}{\rm g}_2f_j) $$

is a new measure of structural complexity that takes into account contributions from complexity sensu stricto and structural disorder that by definition decreases structural complexity. Equation (27) provides a procedure to account for the influence of chemical substitutions and vacancies upon the overall structural complexity, and the parameter ^strI_{G ,mix} can therefore be considered as a measure of crystal chemical complexity (or the degree of atomic order) that takes into account not only structural architecture, but also its chemical nature. As many important rock-forming minerals are complex solid solutions and can be considered as low-, medium- or high-entropy materials (McCormack and Navrotsky, Reference McCormack and Navrotsky2021), this measure might be useful for mineralogical and geochemical implications.

Equation (27) can be re-written as

(28)$$^{str} I_{G, {\rm mix}} = {-}\sum\nolimits_{i = 1}^k {\,p_i} {\rm lo}{\rm g}_2p_i + \sum\nolimits_{i = 1}^k {\,p_i} \,\sum\nolimits_{\,j = 1}^N {\,f_j} {\rm lo}{\rm g}_2f_j = ^{str}\!\!I_G-I_{{\rm mix}}$$

where

(29)$$I_{{\rm mix}} = {-}\sum\nolimits_{i = 1}^k {\,p_i} \sum\nolimits_{\,j = 1}^N {\,f_i} {\rm lo}{\rm g}_2f_j = S_{{\rm cfg, mix}}/( R\,{\rm ln}2) $$

is a contribution of mixing to the overall crystal-chemical complexity. It is obvious that, as ^strI_G ≥ 0 and I _mix ≥ 0, the case is possible when ^strI_G < I _mix and ^strI_{G ,mix} < 0. This means that, in contrast to the measure of structural complexity ^strI_G, the parameter of crystal chemical complexity may be negative and provide a positive contribution to the total configurational entropy of a crystalline solid.

Kaußler and Kieslich (Reference Kaußler and Kieslich2021) recently suggested the following method to calculate the atomic structural complexity parameter (denoted here as ^KKI_G) in order to account for the presence of chemical substitutions (or partially occupied sites) in a crystal structure:

(30)$${\rm ^{\rm KK}}I_G = {-}\sum\nolimits_{\,j = 1}^N {\sum\nolimits_{i = 1}^k {\,f_jp_i} } \,{\rm lo}{\rm g}_2( f_j\,p_i) \quad( {\rm bits/site}) $$

where f_j is the occupancy of the ith site by jth element.

Taking into account that, for a given site,

(31)$$\sum\nolimits_{\,j = 1}^N {\,f_j} = 1$$

one may re-group the equation (30) to get

(32)$${\rm ^{\rm KK}}I_G = {-}\sum\nolimits_{i = 1}^k {\,p_i} {\rm lo}{\rm g}_2p_i-\sum\nolimits_{i = 1}^k {\,p_i} \sum\nolimits_{\,j = 1}^N {\,p_i} f_j{\rm lo}{\rm g}_2f_j\,( {\rm bits/site}) $$

or

(33)$$^{{\rm KK}}I _G = ^{{\it str}}\!\!I_G + I_{{\rm mix}}$$

Through the comparison of equation (33) with equation (28), it is clear that the application of the formula (30) to solid solutions leads to the results exactly opposite to those provided by the formula (27). According to eq. (30), the presence of chemical substitutions (including substitutions by vacancies) results in an increase of the overall structural complexity, i.e. the increase in the atomic order, which is counter-intuitive, as disorder decreases the degree of order (= structural complexity). In order to show this, let us consider the forsterite–fayalite solid solution, Mg₂SiO₄–Fe₂SiO₄. The crystal structure of forsterite–fayalite (space group Pbnm) contains two M sites (4a and 4c), one Si site (4c) and three O sites (4c, 4c and 8d). For the sake of convenience, we suppose that, in the Mg₂SiO₄–Fe₂SiO₄ solid solution, Fe distributes equally between two M sites. The application of the formulae (1) and (2) provides ^strI_G = 2.522. The application of the formulae (29) and (28) provides for the composition Mg_2(1–x)Fe_2xSiO₄:

(34)$$I_{{\rm mix}} = \ndash ( {2/7} ) [ {x{\rm lo}{\rm g}_2x + ( {1 \ndash x} ) {\rm lo}{\rm g}_2( {1 \ndash x} ) } ] $$

(35)$$^{{\rm str}} I_{{G} {, {\rm mix}}} = 2.522 + ( {2/7} ) [ {x{\rm lo}{\rm g}_2x + ( {1 \ndash x} ) {\rm lo}{\rm g}_2( {1 \ndash x} ) } ] }$$

In contrast, equation (30) provides:

(36)$$^{{\rm KK}}I _G = 2.522 \,\ndash \,( {2/7} ) [ {x{\rm lo}{\rm g}_2x + ( {1 \ndash x} ) {\rm lo}{\rm g}_2( {1 \ndash x} ) } ] $$

Graphs corresponding to the behaviour of structural complexity indices versus the content x (the average Fe content in the M sites) are shown in Fig. 1. The value x = 0.5 corresponds to the state of maximal disorder (minimal degree of order) and has the minimal value of ^strI_{G ,mix}, which agrees with general understanding of relations between order and complexity, whereas the ^KKI_G value is maximal for x = 0.5, which contradicts those relations.

Fig. 1. Atomic structural complexities calculated using different approaches versus x value for the fayalite–forsterite Mg_2(1–x)Fe_2xSiO₄ solid solution.

The value of ^strI_{G ,mix} allows one to distinguish mineral species that have the same space group, similar unit-cell parameters, and differ in the degree of disorder only. Sanidine (Sa), orthoclase (Or) and microcline are the polymorphs of KAlSi₃O₈ with a different degree of Al–Si order at the tetrahedral sites. According to the current nomenclature (see, e.g. Krivovichev, Reference Krivovichev2020a), in K feldspars, Al preferentially occurs at the T₁ site. If x _Al(T1) is the content of Al at the T₁ site, then x _Al(T1) = 0.5 for orthoclase and 0.25 for sanidine (in microcline, all Al is at the T₁0 site with x = 1). The behaviour of ^strI_{G ,mix} and entropy of mixing versus x _Al(T1) is shown in Fig. 2. The crystal chemical complexity (= the degree of order) decreases with the increasing entropy of mixing.

Fig. 2. The behaviour of ^strI_{G ,mix} and entropy of mixing versus the x _Al(T1) content of Al at the T₁ site for the orthoclase (Or) – sanidine (Sa) solid solution.

Ladder diagrams

The complexity of a crystal structure is a sum of contributions of different factors. In order to estimate quantitatively the amount of information generated by a particular structural element, Krivovichev (Reference Krivovichev2018) suggested the use of ladder diagrams for visualising different information amounts arising from specific levels of hierarchical organisation of a crystal structure. For oxysalt minerals in particular, the following levels can be recognised: (1) topological information (TI) reflecting the amount of information with respect to the topological structure of basic structural units (clusters, chains, ribbons, layers and frameworks); (2) structural information (SI) arising from the symmetry reduction of structural units due to their distortions induced by their interactions with their local environments (in some cases, structural information equals topological information); (3) packing information (PI) due to the appearance of several identical structural units in the reduced unit cell (for example, the cell may contain two identical layers that may or may not be symmetrically related); (4) information arising from the interstitial structure (IS) except for H atoms; and (5) information due to the presence of H atoms responsible for the formation of a hydrogen bonding (HB) system. The calculation of different contributions proceeds via sequential calculations starting from the determination of the highest symmetry of structural units present in the structure. This first step is probably the most challenging and corresponds to finding the highest symmetry embedding of the chemical bonding net in 3D Euclidean space. In principle, it can be generated by the Systre software program (Delgado-Friedrichs et al., Reference Delgado-Friedrichs, Hyde, O'Keeffe and Yaghi2017). To find the most symmetric embedding of a net, the algorithm of Systre software uses barycentric placement in which each vertex of the net is placed into the centre of gravity of its adjacent vertices (all vertices have the same weight). The resulting net is called stable if no two vertices collide. Stable nets have the maximal achievable crystallographic symmetry (Delgado-Friedrichs and O'Keeffe, Reference Delgado-Friedrichs and O'Keeffe2003). In many cases, the situation does not need application of Systre software, as the highest symmetry embedding may be determined from analysis of the topological structure of structural units by visual inspection. Figure 3 shows ladder diagrams for the four CaAl₂Si₂O₈ polymorphs: anorthite (space group P $\bar{1}$), high-pressure or high-temperature anorthite (space group I $\bar{1}$), dmisteinbergite and svyatoslavite (Krivovichev, Reference Krivovichev2020a). Both room- and high-temperature anorthites are based upon the feldspar-type tetrahedral framework, which is rather low in complexity (blue colour). The major contribution to the structural complexity is due to the distortion of the tetrahedral framework induced by its interaction to the interstitial Ca²⁺ cations (green colour). The specific contribution of Ca²⁺ cations is quite small (yellow colour). The tetrahedral networks in svyatoslavite and dmisteinbergite are topologically simpler than the feldspar framework, which is typical for metastable polymorphs (see below).

Fig. 3. The ladder diagrams for the Ca[Al₂Si₂O₈] polymorphs with tetrahedron topologies: 1 – anorthite I $\bar{1}$; 2 – anorthite P $\bar{1}$; 3 – dmisteinbergite; 4 – svyatoslavite. After Krivovichev (Reference Krivovichev, Charykova and Krivovichev2020a).

For some interesting applications of ladder diagrams to the analysis of uranyl minerals and inorganic compounds see Gurzhiy and Plášil (Reference Gurzhiy and Plášil2019) and Gurzhiy et al. (Reference Gurzhiy, Kuporev, Kovrugin, Murashko, Kasatkin and Plášil2019 and Reference Gurzhiy, Kalashnikova, Kuporev and Plášil2021).

Database on structural and chemical complexity of minerals

The present review is based upon the updated database of structural and chemical complexity parameters of minerals calculated by means of the equations (1–7). For the crystal structures of hydrated minerals, where the positions of H atoms could not be determined, the H-correction procedure was used as first described by Pankova et al. (Reference Pankova, Gorelova, Krivovichev and Pekov2018a) through introduction of surrogate H sites with corresponding multiplicities. No positional disorder was taken into account. For the calculation of chemical complexities, the end-member mineral chemical formulae were used as approved by the International Mineralogical Association (IMA; Pasero, Reference Pasero2021). The structural complexity parameters for mineral species have recently been partially incorporated into the Global Earth Mineral Inventory (GEMI) online database (Prabhu et al., Reference Prabhu, Morrison, Eleish, Zhong, Huang, Golden, Perry, Hummer, Ralph, Runyon, Fontaine, Krivovichev, Downs, Hazen and Fox2021; see below).

Average complexity for all minerals

Krivovichev (Reference Krivovichev2013) provided the following average structural complexity values (arithmetic means) for the mineral kingdom: 3.23(2) bits per atom and 228(6) bits per reduced unit cell, which corresponds to the average 70.6 atoms per cell (based upon 3949 structure reports). For October 2021, the revised values (arithmetic means) are 3.54(2) bits/atom and 345(10) bits/cell, respectively, corresponding to the average of 97.5 atoms per cell (based upon 4443 structure reports). The difference between the 2013 and 2021 values is due to the H-correction of the 2013 database and a high number of very complex minerals that have been discovered since 2013 (in particular, polyoxometalates; see below). The average chemical complexities (arithmetic means) are 1.63(1) bits per atom and 63(1) bits per formula, which account for the average number of 38.7 atoms per formula (based upon 5455 chemical formulae). The latter value is rather high, keeping in mind that most rock-forming minerals are rather simple in their chemical compositions.

Complexity distributions

The distributions of atomic information amounts, ^chemI_G and ^strI_G, versus the number of mineral species (N) are shown in Fig. 4a and b, respectively. Both distributions fit the normal modes, with the goodness-of-fit R ² values of 0.970 and 0.974, respectively. In contrast, the distributions of total complexities, ^chemI_G,total and ^strI_G,total, are heavily right-skewed (Fig. 5a and b, respectively) and correspond to log-normal distributions.

Fig. 4. Distributions of the atomic (a) chemical and (b) structural complexities versus the number of minerals (N).

Fig. 5. Distributions of the total (a) chemical and (b) structural complexities versus the number of minerals (N).

The N− ^chemI_G,total and N− ^strI_G,total can be described by the following exponential functions:

(37)$$N = 5285.19 \times {\rm exp}( { \ndash^{chem}I_{G, total}/52.28} ) ( {R^2 = 0.998} ) $$

(38)$$N = 1988.16 \times exp( { \ndash^{str}I_{G, total}/218.62} ) ( {R^2 = 0.998} ) $$

The distributions N− log(^chemI _G,total) and N − log(^strI _G,total) for the representative sampling of all minerals are symmetrical and fit the log-normal distribution with the goodness-of-fit R ² values of 0.954 and 0.974, respectively.

For a log-normal distribution, the most precise method for estimating the parameters μ* and σ* of the statistical population relies on the log transformation. The mean and empirical standard deviation of the logarithms of the data are calculated and then back-transformed (Sachs Reference Sachs1997; Limpert et al., Reference Limpert, Stahel and Abbt2001) as follows:

(39)$$\bar{X}^{\rm \ast } = {\rm antilg}( {{\bar{X}}_{{\rm log}x_i}} ) = {\rm antilg}\left({\displaystyle{1 \over n}\sum\nolimits_{i = 1}^n {{\rm log}( x_i) } } \right) = ( {\Pi_{i = 1}^n x_i} ) $$

(40)$$s^\ast{ = } \;{\rm antilg}( {\rm\sigma}_{{\rm log}x_i}) = {\rm antilg}\sqrt {\left({\displaystyle{1 \over {n-1}}\sum\nolimits_{i = 1}^n {[ {\rm log}( x_i) -{\rm log}( {{\bar{X}}^\ast } ) } \,] ^2} \right)} $$

where $\bar{X}^\ast$ is the geometric mean of the data; s* is back-transformed empirical standard deviation of the logarithms of the data.

The mean ($\bar{X}_{\rm log\it x_i}$) and empirical standard deviation $( \rm\sigma_{\rm log \it x_i}$) of the logarithms of the data and their estimators ($\bar{X}^\ast$and s*) for all minerals are given in Table 1. Note that the geometric means are substantially lower than the arithmetic means given above.

Table 1. Average chemical and structural complexities for minerals.*

* n – number of minerals taken into account; $\bar{X}$ = arithmetic mean; σ = standard deviation; ${\rm \sigma} _{\bar{X}}$ = standard error of mean; $\bar{X}_{logx_i}$ = arithmetic mean; $\rm \sigma _{\rm log\it x_i}$= standard deviation; ${\rm \sigma}_{\bar{X}}$ = standard error of mean; $\bar{X}^\ast{ = } 10 ^\wedge \bar{X}_{\rm log \it x_i}$; $s^\ast{ = } 10 ^\wedge \rm \sigma _{\rm log\it x_i}$.

The log-normal character of the total structural and chemical complexity distributions is the result of the log-normal distributions of the numbers of atoms per reduced unit cell and per formula unit (Table 1).

Krivovichev (Reference Krivovichev2012, Reference Krivovichev2013) mentioned that the atomic structural complexity (^strI_G) is symmetry-sensitive, whereas the number of atoms per cell (v) is a size-sensitive parameter, while ^strI_G,total combines both symmetry and size as two sides of complexity. In some sense, the ^strI_G and v quantities can be considered as intensive and extensive parameters, respectively. The normal distribution of ^strI_G seems to be quite natural and reflects the influence of many mutually unrelated parameters upon the symmetry of crystals. The nature of the log-normal distribution of the v value deserves special attention.

Log-normal distributions are of high importance in many areas of science (Limpert et al., Reference Limpert, Stahel and Abbt2001) and, in particular, are used to describe species abundance distributions (SADs) in ecology. According to this model, there are few species with high and low abundances and many with intermediate values. It is, however, unclear how the ecological models can be used to understand the behaviour of mineral systems, where mineral species do not compete for food, territory and other resources. A possible explanation for the log-normal distribution of the number of atoms per formula or per cell is the balance between the need to accommodate different elements in the same cell (most minerals contain from four to five different chemical elements (Krivovichev and Charykova, Reference Krivovichev and Charykova2013; Krivovichev et al., Reference Krivovichev, Charykova and Krivovichev2018b) and the tendency of the crystal structures to be as simple as possible.

Average complexity for mineral classes

Arithmetic means for chemical and structural complexities of different mineral classes are provided in Table 2. It can be seen that the O-free (anoxic) mineral species are both chemically and structurally simpler than the mineral species containing oxygen as a mineral-forming component. Among O-bearing species, oxides are the simplest (taking into account the large number of binary oxides with relative chemical simplicity), whereas sulfates and borates are the most complex. The latter observation is expected, as the majority of sulfates and borates crystallise at low temperatures and are very frequently hydrated. Next in complexity are silicates, arsenates and phosphates that have different modes of origin, including both low- and high-temperature environments.

Table 2. Average structural and chemical complexities for different mineral classes.*

* N = number of minerals taken into account; $\bar{X}$ = arithmetic mean; $\rm\sigma _{\bar{X}}$ = standard error of arithmetic mean.

The higher complexity of O-bearing minerals agrees well with the fact that a large number of them have a secondary origin and form at the expense of primary O-free minerals (e.g. during oxidation of sulfide mineral deposits; Krivovichev et al., Reference Krivovichev, Krivovichev and Charykova2019b, Reference Krivovichev, Krivovichev and Charykova2020b). The difference in complexity between anoxic and O-bearing minerals also plays a crucial role in the complexity changes during mineral evolution, i.e. with the formation of a large suite of complex oxysalt minerals as a result of the Great Oxidation Event (Hazen et al., Reference Hazen, Papineau, Bleeker, Downs, Ferry, McCoy, Sverjensky and Yang2008; see below).

Twenty most complex minerals: the 2021 list

Krivovichev (Reference Krivovichev2013) provided the following list of twenty most structurally complex minerals known at the time (I_G,total in bits per cell): paulingite-(Ca) (6767), fantappièite (5948), sacrofanite (5317), mendeleevite-(Ce) (3399), bouazzerite (3035), megacyclite (2951), vandendriesscheite (2835), giuseppettite (2723), stilpnomelane (2484), stavelotite-(La) (2411), rogermitchellite (2321), parsettensite (2310), apjohnite (2305), antigorite (m = 17 polysome) (2250), tounkite (2188), tschörtnerite (2132), farneseite (2094), kircherite (2053), bannisterite (2031) and mutinaite (2025). The updated 2021 list of twenty most structurally complex minerals is given in Table 3 (the difference between the 2013 and 2021 parameters for particular minerals is due to the fact that the 2013 data did not take into account contributions from H atoms). Only one representative of a given structure type is listed: for example, only one member of the voltaite group is given, whereas other mineral species isotypic to voltaite (Majzlan et al., Reference Majzlan, Schlicht, Wierzbicka-Wieczorek, Giester, Pöllmann, Brömme, Doyle, Buth and Bender Koch2013) are omitted; mendeleevite-(Nd) is omitted, etc. Only eight species from the 2013 list remain in the 2021 list. The main reason is the discoveries of seven very complex minerals since 2013 (ewingite, morrisonite, vanarsite, paddlewheelite, gauthierite, rowleyite and meerschautite), the crystal-structure solution of ilmajokite, and the inclusion of the previously neglected or H-corrected species (voltaite, alfredstelznerite, manitobaite, parsettensite and postite).

Table 3. Twenty most structurally complex minerals, their space groups and complexity characteristics.

References: (1) Olds et al. (Reference Olds, Plášil, Kampf, Simonetti, Sadergaski, Chen and Burns2017a); (2) Kampf et al. (Reference Kampf, Hughes, Nash and Marty2016); (3) Zolotarev et al. (Reference Zolotarev, Krivovichev, Cámara, Bindi, Zhitova, Hawthorne and Sokolova2020); (4) Gordon et al. (Reference Gordon, Samson and Kamb1966); (5) Cámara et al. (Reference Cámara, Bellatreccia, Della Ventura, Mottan, Bindi, Gunter and Sebastiani2010); (6) Brugger et al. (Reference Brugger, Meisser, Krivovichev, Armbruster and Favreau2007); (7) Olds et al. (Reference Olds, Plášil, Kampf, Dal Bo and Burns2018); (8) Effenberger et al. (Reference Effenberger, Giester, Krause and Bernhardt1998); (9) Burns, Reference Burns1997); (10) Sokolova et al. (Reference Sokolova, Hawthorne, Pautov, Agakhanov and Karpenko2011); (11) Cooper et al. (Reference Cooper, Hawthorne, Galliski and Márquez-Zavalía2010); (12) Olds et al. (Reference Olds, Plášil, Kampf, Škoda, Burns, Čejka, Bourgoin and Boulliard2017b); (13) Kampf et al. (Reference Kampf, Cooper, Nash, Cerling, Marty, Hummer, Celestian, Rose and Trebisky2017); (14) Biagioni et al. (Reference Biagioni, Moëlo, Orlandi and Stanley2016); (15) Mereiter, Reference Mereiter1972); (16) Guggenheim and Eggleton, Reference Guggenheim and Eggleton1994); (17) Tait et al. (Reference Tait, Ercit, Abdu, Černý and Hawthorn2011); (18) Kampf et al. (Reference Kampf, Hughes, Marty and Nash2012); and (19) McDonald and Chao (Reference McDonald and Chao2010).

Ewingite, Mg₈Ca₈[(UO₂)₂₄(CO₃)₃₀O₄(OH)₁₂(H₂O)₈](H₂O)₁₃₀, is the Earth's most complex mineral reported so far with an estimated complexity of 23478 bits/cell. The mineral was found as golden-yellow crystals on a damp wall of the old Plavno mine of the Jáchymov ore district, western Bohemia, Czech Republic (Olds et al., Reference Olds, Plášil, Kampf, Simonetti, Sadergaski, Chen and Burns2017a), where it crystallised from low-temperature uranyl-bearing aqueous solutions. Its crystal structure contains a 54-nuclear (24U + 30C) uranyl carbonate cluster [(UO₂)₂₄(CO₃)₃₀O₄(OH)₁₂(H₂O)₈]^32– shown in Fig. 6a. Its skeletal representation (Fig. 6b), where each node corresponds either to U (yellow) or to C (grey) centre emphasises the presence of four U₃ triangles (with U–U distances shorter than 4 Å) that correspond to the trimers of three ((UO₂)O₅) pentagonal bipyramids sharing the same equatorial O atom. Two other building units are the [(UO₂)(CO₃)₃] and [(UO₂)(CO₃)₂(H₂O)₂] hexagonal bipyramids. The visual complexity of the cluster architecture can further be reduced by leaving only U atoms and the addition of the U–U links corresponding to the U–U distances in between 4 and 6.2 Å (Fig. 6c). The resulting graph can be considered as consisting of four U₃ trimers (U–U < 4 Å) and six U₄ dihedra of two edge-sharing U₃ triangles (U–U = 4.0–6.2 Å). The centres of the trimers and the dihedra (denoted by red and blue circles in Fig. 6c) form a tetrahedron and an octahedron, respectively, with six tetrahedral edges in correspondence to six octahedral vertices (Fig. 6d). Such a relation between the tetrahedron and octahedron graphs is known in graph theory as an edge-to-vertex duality.

Fig. 6. The [(UO₂)₂₄(CO₃)₃₀O₄(OH)₁₂(H₂O)₈] cluster in: (a) ewingite; (b) its skeletal representation; (c) the U core (the U^…U contacts shorter than 4 Å are shown as thick black lines; those between 4 and 6.2 Å as thin black lines; red and blue points indicate centres of the U₃ trimers and midpoints of the shared U^…U edges of the U₄ dihedra, respectively); and (d) the arrangements of the red and blue circles that correspond to the intersection of tetrahedron and octahedron, respectively. After Krivovichev (Reference Krivovichev, Charykova and Krivovichev2020a).

Morrisonite, Ca₁₁[As³⁺V⁴⁺₂V⁵⁺₁₀As⁵⁺₆O₅₁]₂⋅78H₂O, has 13,558 bits of Shannon information per unit cell and, after ewingite, is the second most complex mineral known so far. The vanarsite-group minerals (including morrisonite itself; Kampf et al., Reference Kampf, Hughes, Nash and Marty2016) contain a unique and previously unknown type of the V–As polyoxometalate cluster with the composition [As³⁺V₁₂As⁵⁺₆O₅₁]. The cluster consists of twelve (VO₆) octahedra forming a wheel- or corona-shaped unit centred by an As³⁺ cation in trigonal-pyramidal coordination (due to the stereoactivity of a lone-electron pair) and surrounded by six (As⁵⁺O₄) tetrahedra (Fig. 7). The same polyoxometalate cluster occurs in the crystal structure of vanarsite, NaCa₁₂[As³⁺₂V⁵⁺₁₇V⁴⁺₇As⁵⁺₁₂O₁₀₂]⋅78H₂O, but with a different mode of packing of polyoxometalate units. The vanarsite-group minerals form from low-temperature aqueous solutions with V and As derived from the oxidation of primary unoxidised phases.

Fig. 7. The [As³⁺V⁴⁺₂V⁵⁺₁₀As⁵⁺₆O₅₁] cluster in morrisonite in (a) polyhedral and (b) skeletal representations. Adapted from Krivovichev (Reference Krivovichev2020a).

Zolotarev et al. (Reference Zolotarev, Krivovichev, Cámara, Bindi, Zhitova, Hawthorne and Sokolova2020) reported the crystal structure of ilmajokite, Na₁₁KBaCe₂[Ti₁₂Si_37.5O₉₄(OH)₃₁]·29H₂O, the third most complex mineral, which was first reported from a hydrothermal vein in Karnasurt Mountain, Lovozero, Kola Peninsula, Russia (Bussen et al., Reference Bussen, Gannibal, Goiko, Mer'kov and Nedorezova1972). The crystal structure is based on a 3D titanosilicate framework consisting of trigonal prismatic titanosilicate (TPTS) clusters centred by Ce³⁺ in [9]-coordination (Fig. 8). There are two symmetry-independent clusters within the unit cell. In each cluster, three [Ti₂O₁₀] dimers of edge-sharing TiO₆ octahedra in parallel orientation form a trigonal prism centred by Ce³⁺ cations. The triple-dimer titanate structure is surrounded by SiO₄ tetrahedra to form a TPTS cluster. Four adjacent TPTS clusters are linked into four-membered rings within the (010) plane and linked into ribbons parallel to [$\bar{1}$01]. The ribbons are organised into layers parallel to (010) and modulated with a modulation wavelength of 32.91 Å and an amplitude of 13.89 Å. The layers are further linked into a titanosilicate framework via additional SiO₄ tetrahedra. The Na⁺, K⁺, Ba²⁺ and H₂O groups occur in the framework cavities and have different occupancies and coordination environments.

Fig. 8. Trigonal-prismatic titanosilicate (TPTS) clusters in the crystal structure of ilmajokite shown in (a, c) polyhedral and (b, d) nodal representations. The numbering scheme corresponds to the numbering of Si and Ti atoms from the experiment. The Ce-centred Ti₆ trigonal prism is highlighted in yellow. Legend: Si tetrahedra = yellow, Ti octahedra = blue; H₂O molecules, Na, K, Ba and Ce atoms are shown as red, light-blue, green, brown, and orange spheres, respectively (after Zolotarev et al., Reference Zolotarev, Krivovichev, Cámara, Bindi, Zhitova, Hawthorne and Sokolova2020).

Complexity-generating mechanisms in minerals

The list of twenty most structurally complex minerals known so far given in Table 3 reveals the following most important complexity-generating mechanisms in minerals:

1. (1) the presence of large (sometimes nanometre-scale) clusters such as polyoxometalates or related finite-cluster structures (Krivovichev, Reference Krivovichev2020b) isolated from each other; such multinuclear atomic units possess a large number of atoms with different topological functions; the examples are ewingite, morrisonite, vanarsite, bouazzerite and postite; as a rule, natural polyoxometalate minerals are also highly hydrated, with the exception of arsmirandite and lehmannite, which are anhydrous polyoxocuprates formed in volcanic fumaroles (Britvin et al., Reference Britvin, Pekov, Yapaskurt, Koshlyakova, Göttlicher, Krivovichev, Turchkova and Sidorov2020);

2. (2) the presence of large clusters linked together to form three-dimensional frameworks; the examples are ilmajokite and paddlewheelite (both minerals are also highly hydrated, which contributes greatly to their structural complexities);

3. (3) the formation of complex three-dimensional modular frameworks formed by cages of different sizes and topologies; this complexity type corresponds to paulingite-group minerals, fantappièite and sacrofanite (members of the sodalite–cancrinite ABC series (Bonaccorsi and Nazzareni, Reference Bonaccorsi and Nazzareni2015; Chukanov et al., Reference Chukanov, Aksenov and Rastsvetaeva2021), tschörtnerite, mendeleevite-(Ce) and rowleyite;

4. (4) the formation of complex layers with different combinations of modules (chains or rings); this type is characteristic for layered uranyl minerals (such as vandendriesscheite and gauthierite) and layered silicates (parsettensite);

5. (5) a high hydration state in salts with complex heteropolyhedral units (alfredstelznerite and voltaite-group minerals);

6. (6) the formation of ordered superstructures of relatively simple structure types; the examples are meerschautite (which is the only sulfide species in the list and was described by Biagioni et al. (Reference Biagioni, Moëlo, Orlandi and Stanley2016) as an expanded derivative of owyheeite) and manitobaite (that has a fivefold superstructure relative to alluaudite (Tait et al., Reference Tait, Ercit, Abdu, Černý and Hawthorn2011)); for further discussion on the relations between superstructures and complexity see Krivovichev et al. (Reference Krivovichev, Panikorovskii, Zolotarev, Bocharov, Kasatkin and Škoda2019a, Reference Krivovichev, Panikorovskii, Yakovenchuk, Selivanova and Ivanyuk2021), Kornyakov et al. (Reference Kornyakov, Vladimirova, Siidra and Krivovichev2021) and Kornyakov and Krivovichev (Reference Kornyakov and Krivovichev2021).

The Fedorov–Groth statistical law

Fedorov (Reference Fedorov1913, Reference Fedorov1914) and Groth (Reference Groth1921) pointed out that, in general, symmetry of crystalline compounds correlates with their chemical complexity. Less complex compounds (e.g. native elements) tend, on average, to have higher symmetries than compounds consisting of two elements, etc. This empirical observation, known in Russian literature as a Fedorov–Groth law (Shafranovskii, Reference Shafranovskii1973), was recently analysed by Krivovichev and Krivovichev (Reference Krivovichev and Krivovichev2020) using modern crystallographic data for minerals and the concepts of structural and chemical complexity of crystals formulated above. They have shown that there is a strong correlation (Fig. 9; confidence level >0.99) between the atomic chemical (^chemI_G) and structural (^strI_G) complexities, and the point-group order |P_G| taken as a measure of symmetry (|P_G| may be equal to 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48). The relations between these parameters are described by the following exponential functions:

(41)$$\eqalign{&^{{\rm chem}} I_G = 1.122 + 0.634 \times {\rm exp} \cr &\quad [ -\vert P_G\vert /30.956] , \;{\rm bits/atom\ }( {R^2 = 0.979} )} $$

(42)$$ \eqalign{& ^{{\rm str}} I_G = 1.580 + 3.532 \times {\rm exp} \,\cr &\quad [ -\vert {\rm P}_G\vert /10.718] , \;{\rm bits/atom\ }( {R^2 = 0.987} )} $$

Fig. 9. The plot of average atomic chemical (^chemI _G) and structural (^strI _G) complexities of minerals versus the order of the point group, |P_G|. After Krivovichev and Krivovichev (Reference Krivovichev and Krivovichev2020).

No good correlations have been obtained for the amounts of chemical and structural information per formula and per reduced unit cell, versus the |P_G| values, which is not surprising, as the point-group symmetry of a system is not a function of its size.

The observed strong correlation between the amount of chemical information per atom, ^chemI _G, and the order of the point group, |P_G|, indicates that the Fedorov–Groth law is indeed valid and has a statistical meaning. Chemical simplicity measured as an amount of Shannon information per atom on average corresponds to higher symmetry measured as an order of the point group of a mineral.

Complexity and crystal systems

The distribution of minerals among the crystal systems and classes has been extensively discussed in the past, especially in the Russian mineralogical literature (Povarennykh, Reference Povarennykh1966; Shafranovskii and Feklichev, Reference Shafranovskii and Feklichev1982; Shafranovskii, Reference Shafranovskii1983a, Reference Shafranovskii1983b, Reference Shafranovskii1985, Reference Shafranovskii1987, Reference Shafranovskii1988; Dolivo-Dobrovolsky, Reference Dolivo-Dobrovolskii1987, Reference Dolivo-Dobrovolskii1988; Urusov, Reference Urusov2002, Reference Urusov2007; Filatov, Reference Filatov2020). Table 4 provides the modern analysis of the problem that shows that the monoclinic system is most common (>35% of all minerals), followed by orthorhombic (~20%), trigonal (~11.5%) and triclinic (~10.6%) systems (we distinguish between trigonal and hexagonal systems as having threefold and sixfold rotation or inversion axes, respectively). In their analysis of the 1981 data, Shafranovskii and Feklichev (Reference Shafranovskii and Feklichev1982) reported the following sequence of crystal systems: monoclinic (30.8%), orthorhombic (23.4%), cubic (11.2%), hexagonal (9.0%), trigonal (8.7%), tetragonal (8.5%) and triclinic (8.4%). Shafranovskii (Reference Shafranovskii1983a, Reference Shafranovskii1983b, Reference Shafranovskii1987, Reference Shafranovskii1988) suggested that the quantitative distribution of minerals among different crystal systems can be considered as a statistical ‘law of nature’, similar to the Fedorov–Groth law (see above), however Dolivo-Dobrovolsky (Reference Dolivo-Dobrovolskii1988) challenged this view and showed that the number of cubic and triclinic minerals is decreasing and increasing with the historical time, respectively. Urusov (Reference Urusov2002, Reference Urusov2007) supported this view by noticing that most of the rare minerals discovered within last few decades are complex and possess low symmetry. Filatov (Reference Filatov2020) explained the high abundance of monoclinic minerals by the ability of the monoclinic system to adapt complex atomic arrangements, but did not provide any quantitative metrics to support his conclusion. Moreover, Filatov (Reference Filatov2020) based his analysis on old statistical data (e.g. he used 30.5 and 7.4% for the monoclinic and triclinic systems, respectively, which do not correspond to the current situation described in Table 4).

Table 4. Distribution of mineral species among different crystal systems and their average atomic complexities.

The analysis of atomic chemical and structural complexities for different crystal systems shows general agreement with the Fedorov–Groth law with the exception of the trigonal system, which has relatively high values compared to tetragonal and hexagonal systems. The slightly anomalous behavior of the trigonal system is due to the fact that, surprisingly enough, this system is characteristic for many very complex minerals, including, e.g. the members of the cancrinite–sodalite supergroup.

Complexity and space groups

A list of most common space groups observed in minerals according to data for 2020, along with their average atomic structural complexities is shown in Table 5. As the ^strI _G parameter is symmetry-sensitive, it is not surprising that it correlates with the degree of symmetry described by corresponding space groups. Shafranovskii and Feklichev (Reference Shafranovskii and Feklichev1982) analysed the relative abundance of different space groups in minerals based upon the 1981 data and reported the following ten space groups taken in the order of their abundance: P2₁/c – Pnma – P $\bar{1}$ – C2/m – C2/c – R $\bar{3}$m – P2₁/m – Fd $\bar{3}$m – Fm $\bar{3}$m – P6₃/mmc. Comparison with the data given in Table 5 indicates the number of monoclinic minerals increased significantly since 1981 relative to the numbers of orthorhombic, trigonal and cubic species. Analysis of Table 3 reveals that, among twenty most structurally complex minerals, four crystallise in the P2₁/c space group, whereas all others correspond to unique space groups.

Table 5. Ten most common space groups for minerals.*

* n _min = number of minerals crystallising in the given space group

Urusov and Nadezhina (Reference Urusov and Nadezhina2006) listed 54 space groups that are ‘empty’ of minerals, i.e. have no mineral species crystallised with those symmetries. Our analysis (Table 6) shows that: (1) several ‘empty’ space groups were missed in the compilation and (2) several ‘empty’ space groups have been filled by new mineral species discovered since 2006. For instance, the Pm $\bar{3}$ space group, which was ‘empty’ in 2006, was ‘filled’ by the discovery of mendeleevite-(Ce) (Sokolova et al., Reference Sokolova, Hawthorne, Pautov, Agakhanov and Karpenko2011). It is remarkable that the previous compilation of ‘empty’ mineralogical space groups given by Povarennykh (Reference Povarennykh1966) listed 112 groups, i.e. slightly less than half of all possible space groups. In 2021, the number of ‘empty’ space groups (45) is less than 20% of the total number of possible space groups.

Table 6. ‘Empty’ space groups (45) in structural mineralogy.*

* The space-group number is given in brackets

** Absent in the list of ‘empty’ groups by Urusov and Nadezhina (Reference Urusov and Nadezhina2006)

*** ‘empty’ group for all inorganic crystal structures

Recently Hummer (Reference Hummer2021) investigated the distribution of mineral species among 32 point groups and demonstrated that the abundance of minerals belonging to each point group approximately obeys a power law with respect to group order, i.e. corresponds to fractal behaviour.

It is interesting that the order of space-group abundance in inorganic compounds (i.e. both minerals and synthetic compounds) differs from that observed in minerals and corresponds to the following sequence (Urusov and Nadezhina, Reference Urusov and Nadezhina2009): Pnma (7.4%), P2₁/c (7.2%), Fm $\bar{3}$m (5.6%), Fd $\bar{3}$m (5.1%), P $\bar{1}$ (4.0%), I4/mmm (4.0%), C2/c (3.8%), P6₃/mmc (3.4%), C2/m (3.4%) and Pm $\bar{3}$m (3.0%). These data show that the total kingdom of inorganic crystal structures is, on average, higher in symmetry relative to the mineral kingdom, which is rather remarkable. One may speculate that this tendency is due to the limited chemical diversity of natural systems compared to synthetic ones, which provides a broad avenue for synthetic chemical exploration of chemically simple, but naturally unstable or geochemically unusual combination of elements. In fact, synthetic chemists rarely study systems of more than three or four elements. Even though they consider combinations of elements not found in Nature, they very rarely attempt to study compounds in a system with seven or eight elements, as are often found in Nature. Therefore, the trend of fewer elements leading to higher average symmetry is reflected in synthetic inorganic chemistry. High temperatures of synthesis (compared to all of the low-temperature weathering/alteration minerals) may also play a role. We may hypothesise that the average chemical simplicity of synthetic systems results in their higher symmetry, according to the Fedorov–Groth law, which is statistically valid for large numbers of observations (see above).

For organic compounds, the P2₁/c group is the most common (46%), followed by P $\bar{1}$ (20%), C2/c (7%), Pbca (6%), P2₁2₁2₁ (5%), P2₁ (3%), Pna2₁ (2%) and Pca2₁ (2%) (Rekis, 2020). The space-group distribution pattern here is drastically different from that observed in minerals and inorganic compounds, though the leading role of the P2₁/c group is again remarkable.

As far as we know, there is no comprehensive explanation for the space-group distribution in crystals. Hellner and Sowa (Reference Hellner and Sowa1985) and Dolivo-Dobrovolsky (Reference Dolivo-Dobrovolskii1988) pointed out that space groups with large percentages of characteristic orbits (i.e. orbits that possess symmetry of the generating space groups and not higher) have higher abundance. However, as was mentioned by Urusov and Nadezhina (Reference Urusov and Nadezhina2009), this cannot explain the predominance of the P2₁/c space group that contains only one kind of characteristic orbit, whereas all its orbits of other kinds are non-characteristic.

Structural complexity in Data-Driven Discovery

Hazen (Reference Hazen2014) and Hazen et al. (Reference Hazen, Downs, Eleish, Fox, Gagné, Golden, Grew, Hummer, Hystad, Krivovichev, Li, Liu, Ma, Morrison, Pan, Pires, Prabhu, Ralph, Runyon and Zhong2019) advocated Data-Driven Discovery as a new direction in the science of mineralogy that deals with the analysis of large volumes of data on chemical composition, crystal structure, physical properties and geological origins of minerals that have been accumulated in scientific literature over several last hundred years. The implementation of various numerical techniques (e.g. network analysis (Morrison et al., Reference Morrison, Liu, Eleish, Prabhu, Li, Ralph, Downs, Golden, Fox, Hummer, Meyer and Hazen2017, Reference Morrison, Buongiorno, Downs, Eleish, Fox, Giovannelli, Golden, Hummer, Hystad, Kellogg, Kreylos, Krivovichev, Liu, Merdith, Prabhu, Ralph, Runyon, Zahirovic and Hazen2020)) would provide us with new insights into distribution and diversity of minerals through space and time. Structural and chemical complexity are important characteristics of minerals reflecting their formation and evolution in geological environments. Incorporation of complexity data into modern mineralogical databases (such as GEMI (Prabhu et al., Reference Prabhu, Morrison, Eleish, Zhong, Huang, Golden, Perry, Hummer, Ralph, Runyon, Fontaine, Krivovichev, Downs, Hazen and Fox2021)) provides additional possibilities to extract novel and interesting regularities, due to the general relations between complexity and configurational entropy of crystalline solids (Krivovichev, Reference Krivovichev2016).

Structural complexity and thermodynamics

The relations between structural complexity and temperature and pressure have been discussed briefly by Krivovichev (Reference Krivovichev2012, Reference Krivovichev2013) and is beyond the topic of the current review. However, several general points can be outlined here. As the structural complexity ^strI_{G ,mix}, defined by equations (27) and (29) as consisting of the complexity of atomic architecture and positional disorder, is related to configurational entropy, and the latter is related to vibrational entropy, it can be expected that structural complexity decreases with increasing temperature and increases with increasing pressure. The first trend is confirmed by numerous data, including recent results on phase transitions in minerals (e.g. Avdonceva et al., Reference Avdontceva, Krzhizhanovskaya, Krivovichev and Yakovenchuk2015a, Reference Avdontceva, Zolotarev and Krivovichev2015b). Increasing temperature also generally leads to an increase in structural disorder. Filatov (Reference Filatov2011) emphasised the principle that increasing symmetry is often correlated with an increase of temperature. However, the principle of decreasing complexity with increasing temperature includes symmetry as one of its aspects and better describes the behaviour of solids under heating. For instance, Ismagilova et al. (Reference Ismagilova, Zhitova, Krivovichev, Sergeeva, Nuzhdaev, Anikin, Krzhizhanovskaya, Nazarova, Kupchinenko and Zolotarev2021) noticed that there are two natural Cu₃(VO₄)₂ polymorphs, pseudolyonsite and mcbirneyite, that correspond to high- and low-temperature modifications, respectively, which conforms well with their densities, 4.71 and 4.48 g^.cm^–3, respectively. However, pseudolyonsite has higher (monoclinic) symmetry than mcbirneyite (triclinic polymorph), which is in obvious contradiction with the rule emphasised by Filatov (Reference Filatov2011). In contrast, structural complexity calculations show that triclinic mcbirneyite is structurally simpler (36 bits/cell) than monoclinic pseudolyonsite (72 bits/cell), in agreement with the observation that, generally, complexity decreases upon heating. However, this empirically derived tendency may be violated as, for example, was shown for BiSeO₃Cl (Aliev et al., Reference Aliev, Kovrugin, Colmont, Terryn, Huvé, Siidra, Krivovichev and Mentré2014). The low- and high-temperature modifications, α- and γ- BiSeO₃Cl, respectively, have the same complexity, 62 bits/cell. However, the intermediate β-phase is very complex with ^strI_G,total = 4706 bits/cell. Aliev et al. (Reference Aliev, Kovrugin, Colmont, Terryn, Huvé, Siidra, Krivovichev and Mentré2014) showed that the extreme complexity of this phase is due to its transitional character, as its structure can be considered a ‘frozen’ transitional state between the low- and high-temperature forms. In general, transitional crystalline states existing on the border between stability fields of simple phases, may possess extraordinarily high complexity (Gurzhiy et al., Reference Gurzhiy, Tyumentseva, Krivovichev, Krivovichev and Tananaev2016).

Hazen and Navrotsky (Reference Hazen and Navrotsky1996) analysed the influence of pressure upon order–disorder reactions and showed that, in the majority of cases, increasing pressure is associated with decreasing positional disorder, though there are exceptions to this rule (Deng et al., Reference Deng, Kang, Croft, Li, Shen, Zhao, Yu, Jin, Kotliar, Liu, Tyson, Tappero and Greenblatt2020). The ordering reactions mean that structural complexity is increasing. Krivovichev (Reference Krivovichev2021a) recently reviewed the crystal chemistry of high-pressure silicates and showed that, in many cases, structural complexity increases with pressure, but only when coordination numbers of atoms remain constant during phase transitions from low- to high-pressure modification. However, even then the situation is not straightforward, as can be seen from the high-pressure phase transitions of coesite, SiO₂. Bykova et al. (Reference Bykova, Bykov, Černok, Tidholm, Simak, Hellman, Belov, Abrikosov, Liermann, Hanfland, Prakapenka, Prescher, Dubrovinskaia and Dubrovinsky2018) reported five coesite polymorphs, numbered from I to V with increasing pressure. The coesite-I → coesite-II → coesite-III phase transitions are displacive and do not modify the topology of the framework; Si atoms remain in tetrahedral coordination. Along the pathway, the symmetry is decreasing (C2/c → P2₁/n → P $\bar{1}$), which corresponds to the increasing number of symmetrically independent sites and the increasing structural information per atom (2.752 → 4.585 → 5.198 bits/atom). However, the total structural information does not behave similarly (66 → 440 → 374 bits/cell). The coesite-IV and V polymorphs contain tetra-, penta- and hexa-coordinated Si and have the same complexity, 4.585 bits/atom and 220 bits/cell. The change in coordination number may modify the complexity behaviour unpredictably. The most probable reason is that the increase in pressure results in the increase of vibrational entropy, which is not a direct function of configurational entropy and structural complexity described by equations (27) and (28).

Structural complexity and crystallization

Krivovichev (Reference Krivovichev2013) provided an overview of relations between structural complexity and metastable crystallisation based upon the Goldsmith's simplexity principle that states that simple structures crystallise more easily. This idea was supported by many subsequent observations, including metastable feldspar polymorphs (Zolotarev et al., Reference Zolotarev, Krivovichev, Panikorovskii, Gurzhiy, Bocharov and Rassomakhin2019; Krivovichev, Reference Krivovichev and Krivovichev2020, and references therein), perovskite-group minerals (Zaitsev et al., Reference Zaitsev, Zhitova, Spratt, Zolotarev and Krivovichev2017), uranyl minerals (Plášil et al., Reference Plášil2018b, Reference Plášil2018c), sulfates (Plášil et al., Reference Plášil, Petříček and Majzlan2017; Majzlan et al., Reference Majzlan, Dachs, Benisek, Plášil and Sejkora2018), phosphates and arsenates (Krivovichev et al., Reference Krivovichev, Zolotarev and Popova2016; Krivovichev, Reference Krivovichev2017; Kolitsch et al., Reference Kolitsch, Weil, Kovrugin and Krivovichev2020), Cu₂(OH)₃Cl polymorphs (Krivovichev et al., Reference Krivovichev, Hawthorne and Williams2017), borates (Grew et al., Reference Grew, Krivovichev, Hazen and Hystad2016), borosilicates (Cempírek et al., Reference Cempírek, Grew, Kampf, Ma, Novák, Gadas, Škoda, Vašinová-Galiová, Pezzotta, Groat and Krivovichev2016), oxalates (Huskić et al., Reference Huskić, Novendra, Lim, Topić, Titi, Pekov, Krivovichev, Navrotsky, Kitagawa and Friščić2019), etc. In his review on metastable-mineral formation in oxidation zones and mine wastes, Majzlan (Reference Majzlan2020) recently noted that “a quantitative link…” between structural complexity and metastability “…is missing”. Indeed, no equation could be written that would directly relate structural information and the ease of crystallisation, as complexity is not the only factor that defines the sequence of crystallisation. An example of non-linear behaviour is given by TiO₂ polymorphs. It is usually assumed that, under ambient conditions, anatase and brookite are metastable, whereas rutile is thermodynamically stable (Ranade et al., Reference Ranade, Navrotsky, Zhang, Banfield, Elder, Zaban, Borse, Kulkarni, Doran and Whitfield2002). The structural complexity parameters of the three phases do not conform to Goldsmith's principle, as anatase and rutile have the same complexity (0.918 bits/atom and 5.5 bits/cell), whereas brookite is the most complex among these TiO₂ polymorphs (1.585 bits/atom and 38 bits/cell). This example shows that the correlation between structural complexity and metastability is not straightforward and outlines an empirical tendency rather than rigorous regularity. The detailed account of the relations between complexity and metastability will be published elsewhere.

Structural complexity and modularity

Modularity is an important structural principle for minerals and inorganic compounds, as many mineral structures can be considered as built up from modules (blocks) derived from simple archetype structures and recombined in a particular way (Ferraris et al., Reference Ferraris, Makovicky and Merlino2004). As one could imagine an infinite number of modular combinations, it could be expected that the number of possible derivative structures is infinite. However, only a few combinations are observed in Nature, and those are usually the simplest possible. Pankova et al. (Reference Pankova, Krivovichev, Pekov, Grew and Yapaskurt2018b) showed that, in the kurchatovite family, only the two simplest variations are realised in Nature that can be generated according to the given structural principles of modular combinations. The principle of simplicity in modular series and its relation to abundances of structural architectures will be discussed in detail elsewhere.