Structural complexity of natural uranyl sulfates

Uranyl sulfates, including those occurring in Nature (∼40 known members), possess particularly interesting structures. They exhibit a great dimensional and topological diversity of structures: from those based upon clusters of polyhedra to layered structures. There is also a great variability in the type of linkages between U and S polyhedra. From the point of view of complexity of those structures (measured as the amount of Shannon information per unit cell), most of the natural uranyl sulfates are intermediate (300–500 bits per cell) to complex (500–1000 bits per cell) with some exceptions, which can be considered as very complex structures (>1000 bits per cell). These exceptions are minerals alwilkinsite-(Y) (1685.95 bits per cell), sejkoraite-(Y) (1859.72 bits per cell), and natrozippeite (2528.63 bits per cell). The complexity of these structures is due to an extensive hydrogen bonding network which is crucial for the stability of these mineral structures. The hydrogen bonds help to propagate the charge from the highly charged interlayer cations (such as Y³⁺) or to link a high number of interlayer sites (i.e. five independent Na sites in the monoclinic natrozippeite) occupied by monovalent cations (Na⁺). The concept of informational ladder diagrams was applied to the structures of uranyl sulfates in order to quantify the particular contributions to the overall informational complexity and identifying the most contributing sources (topology, real symmetry, interlayer bonding).

1. Introduction

Naturally occurring uranyl sulfates are important phases both from the mineralogical and the environmental point of view. Assemblages of uranyl sulfate minerals are common in oxidized parts of uranium deposits worldwide. They form via the oxidation–hydration weathering of primary uranium minerals, mainly uraninite, which interact with acid solutions derived from the decomposition of sulfides, such as pyrite or chalcopyrite (Finch & Murakami, 1999; Krivovichev & Plášil, 2013; Plášil, 2014). The alteration of primary uranium ores under the low pH generates highly mobile acid solutions, containing dissolved UO₂²⁺ as UO₂–SO₄ aqueous complexes, which yields a considerable environmental impact, especially around the old mining sites (Fernandes et al., 1995; Brugger et al., 2003; Johnson, 2003; Johnson & Hallberg, 2005). Uranyl sulfates show a great structural diversity (Krivovichev, 2010; Krivovichev & Plášil, 2013) arising from combinatorial and topological possibilities of linkage of the basic structural elements, uranyl coordination polyhedra and sulfate tetrahedra, which leads to a large variety of topological and geometrical isomers. Herein, we would like to characterize and quantify the topological and structural complexity parameters for one of the most representative groups of supergene uranium minerals – uranyl sulfates. We have used the approach recently developed by S. V. Krivovichev (Krivovichev, 2012, 2013, 2014, 2018), which allows the crystal structures to be characterized in terms of the information content.

2. Natural uranyl sulfates and their crystal structures

Hexavalent uranium in the solid state is most frequently present as a (nearly) linear uranyl ion, UO₂²⁺ (Ur), in which the bond-valence requirements of strongly bonded O atoms are almost satisfied (Fig. 1). Therefore, the bonding of additional elements takes place via ligands arranged in the equatorial planes of uranyl coordination polyhedra. In the case of uranyl sulfates, it goes exclusively about the uranyl pentagonal bipyramid, UO₂Φ₇ (where Φ = O^2–, OH^–, H₂O). Sulfur is tetrahedrally coordinated as an SO₄ tetrahedron and the linkage between U⁶⁺ and S⁶⁺ polyhedra is usually monodentate (with exceptions, see below). Polymerization of basic units leads to chain (10) or sheet (23) structures, which prevail over structures. There are fewer structures based upon isolated clusters of polyhedra (5). To date, 38 well characterized uranyl sulfates are known from Nature. Their overview and basic structural and chemical properties are reported in Table 1 as well as displayed in Fig. 2.

Table 1
Overview of natural uranyl sulfates and their basic characteristics

Structural units	Mineral	Chemical formula	Space group	Reference
Clusters	Belakovskiite	Na7(UO2)(SO4)4(SO3OH)(H2O)3		Kampf et al. (2014)
	Bluelizardite	Na7(UO2)(SO4)4Cl(H2O)2	C2/c	Plášil et al. (2014a)
	Klaprothite	Na6(UO2)(SO4)4(H2O)4	P21/c	Kampf et al. (2017b)
	Ottohahnite	Na6(UO2)2(SO4)5(H2O)8.5		Kampf et al. (2017b)
	Péligotite	Na6(UO2)(SO4)4(H2O)4		Kampf et al. (2017b)
Chains	Adolfpateraite	K(UO2)(SO4)(OH)(H2O)	P21/c	Plášil et al. (2012a)
	Alwilkinsite-(Y)	Y[(UO2)3(SO4)2O(OH)3](H2O)14	P212121	Kampf et al. (2017c)
	Bobcookite	NaAl(UO2)2(SO4)4(H2O)18		Kampf et al. (2015b)
	Fermiite	Na4(UO2)(SO4)3(H2O)3	Pmn21	Kampf et al. (2015c)
	Meisserite	Na5(UO2)(SO4)3(SO3OH)(H2O)		Plášil et al. (2013b)
	Oppenheimerite	Na2(UO2)(SO4)2(H2O)3		Kampf et al. (2015c)
	Shumwayite	[(UO2)(SO4)(H2O)2]2(H2O)	P21/c	Kampf et al. (2017a)
	Svornostite	K2Mg[(UO2)(SO4)2]2(H2O)8	Pmn21	Plášil et al. (2015a)
	Uranopilite	(UO2)6(SO4)O2(OH)6(H2O)14		Burns (2001)
	Wetherillite	Na2Mg(UO2)2(SO4)4(H2O)18	P21/c	Kampf et al. (2015b)
Sheets	Ammonio­mathesiusite	(NH4)5(UO2)4(SO4)4(VO5)(H2O)4	P4/n	Kampf et al. (2018a)
	Ammonio­zippeite	(NH4)2[(UO2)2(SO4)O2](H2O)	Ccmb	Kampf et al. (2018b)
	Beshtauite	(NH4)2(UO2)(SO4)2(H2O)2	P21/c	Pekov et al. (2014)
	Deliensite	Fe[(UO2)2(SO4)2(OH)2](H2O)7	Pnn2	Plášil et al. (2012b)
	Feynmannite	Na(UO2)(SO4)(OH)(H2O)3.5	P2/n	Kampf et al. (2018c)
	Geschieberite	K2(UO2)(SO4)2(H2O)2	Pna21	Plášil et al. (2015b)
	Greenlizardite	(NH4)Na(UO2)2(SO4)2(OH)2(H2O)4		Kampf et al. (2017e)
	Johannite	Cu(UO2)2(SO4)2(OH)2(H2O)8		Mereiter (1982)
	Leydetite	Fe(UO2)(SO4)2(H2O)11	C2/c	Plášil et al. (2013a)
	Magnesioleydetite	Mg(UO2)(SO4)2(H2O)11	C2/c	Kampf et al. (2018d)
	Marécottite	Mg3[(UO2)4O3(OH)(SO4)2]2(H2O)28		Plášil & Škoda (2015)
	Mathesiusite	K5(UO2)4(SO4)4(VO5)(H2O)4	P4/n	Plášil et al. (2014b)
	Meitnerite	(NH4)(UO2)(SO4)(OH)(H2O)2		Kampf et al. (2018e)
	Magnesiozippeite	Mg[(UO2)2O2(SO4)](H2O)3.5	C2/m	Plášil et al. (2013c)
	Natrozippeite	Na5[(UO2)8(SO4)4O5(OH)3](H2O)12	P21/n	Burns et al. (2003)
	Plášilite	Na(UO2)(SO4)(OH)(H2O)2	P21/c	Kampf et al. (2015a)
	Plavnoite	K0.8Mn0.6[(UO2)2O2(SO4)](H2O)3.5	C2/c	Plášil et al. (2017)
	Pseudojohannite	Cu3(OH)2[(UO2)4O4(SO4)2](H2O)12		Plášil et al. (2012c)
	Rabejacite	Ca2[(UO2)4O4(SO4)2](H2O)8		Plášil et al. (2014c)
	Rietveldite	Fe(UO2)(SO4)2(H2O)5	Pmn21	Kampf et al. (2017d)
	Sejkoraite-(Y)	Y3(OH)2[(UO2)8O7(OH)(SO4)4](H2O)24		Plášil et al. (2011a)
	Strassmannite	Al(UO2)(SO4)2F(H2O)16	C2/c	Kampf et al. (2018d)
	Zippeite	K2[(UO2)4O2(SO4)2(OH)2](H2O)4	C2/m	Plášil et al. (2011b)

Figure 1 Graph representation of the uranyl pentagonal bipyramid, UO7, with displayed bond-valence sums (in valence units, vu) at particular O sites (vertices of the graph).

Figure 2 Structural units of natural uranyl sulfates. Color scheme: UO7 is yellow, SO4 is orange, the H atoms are displayed as black-and-white ball and stick.

3. Complexity calculations

In order to evaluate the influence of various crystal-chemical factors (such as dimensionality of the uranium-bearing units and their hydration state) on the structure and symmetry of uranyl sulfate complexes and on the structural architecture of minerals in general, the structural and topological complexity was studied in terms of the information-based approach developed by Krivovichev (2012, 2013, 2014) and recently used by Cempírek et al. (2016), Gurzhiy et al. (2016, 2017, 2018a, b), Krivovichev et al. (2016, 2017), Majzlan et al. (2018) and Plášil (2018a,b). The structural complexity is quantitatively estimated as a Shannon information content per atom (I_G) and per unit cell (I_G,total). The amount of Shannon information reflects diversity and relative proportion of different objects, e.g. the number and relative proportion of different sites in an elementary unit cell of a crystal structure. For a crystal structure, the calculation involves the use of the following equations (Krivovichev, 2012, 2013, 2014):

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

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 ν is the total number of atoms in the reduced unit cell.

The reliable correlation of complexity parameters is possible only for compounds with the same or very close chemical composition (e.g. polymorphs), whereas changes in hydration state could significantly change the complexity values. Analyzing the stability parameters of the minerals and possible mineral evolution trends, it is more reasonable to analyze topological and structural complexity of uranyl sulfate structural units as backbones of crystalline phases, whereas additional cations and water molecules occupying interlayer space can be excluded from the detailed consideration.

4. Topological and structural complexity

The information-based complexity parameters for the U-bearing units in the structures of uranyl sulfate minerals are given in Table 2. Calculations have been performed in several steps. Firstly, the structural complexity of the structural units has been analyzed, taking into account their real layer (RL) or rod group (RG) symmetries. Secondly, the topological complexity (according to the maximal symmetry group) has been calculated. Complexity parameters for the whole structures have been calculated using the TOPOS (Blatov et al., 2014) package and are given in Table 2 for comparison. It should be taken into account that all calculations were based on the original CIF files from the structural databases (Inorganic Crystal Structure Database and American Mineralogist Crystal Structure Database) and respective publications. In addition, positions of all H atoms have been assigned manually (if these data were not reported in the original entries) considering the distribution of hydrogen-bonding system, which include traditional ranges of bond lengths and angles within the D—H⋯A atoms (here, D is a donor and A is an acceptor; both are O atoms).

Table 2
Natural uranyl sulfates and their complexity measures

Mineral	Space group	Structural complexity of the U–S unit				Topological complexity of the U–S unit				Complexity parameters of the crystal structure
		Layer or rod symmetry group	ν	IG	IG,total	Layer or rod symmetry group	ν	IG	IG,total	ν	IG	IG,total
Zippeite topology
Ammonio­zippeite	Ccmb	pbab	52	2.777	144.423	pbab	52	2.777	144.423	104	4.085	424.846
Magnesiozippeite	C2/m	p2/a	52	3.700	192.423	p2/a	26	2.777	72.211	98	4.839	474.242
Plavnoite	C2/c	p2/n	52	3.700	192.423					104	4.777	496.846
Pseudojohannite			26	3.700	96.211					69	5.123	353.488
Rabejacite			26	3.700	96.211					55	4.800	263.975
Marécottite			54	4.755	256.764		54	4.755	256.764	141	6.161	868.677
Natrozippeite	P21/n	pn	110	5.781	635.950	pn	110	5.781	635.950	384	6.585	2528.626
Sejkoraite-(Y)			106	5.728	607.160		106	5.728	607.160	264	7.044	1859.720
Zippeite	C2/m	p2/a	30	2.974	89.207	p2/a	30	2.974	89.207	44	3.732	164.215
Leydetite topology
Geschieberite	Pna21	pn	32	4.000	128.000	c2mm	16	2.875	46.000	84	4.392	368.955
Leydetite	C2/c	c2	32	3.125	100.000					94	4.618	434.131
Magnesioleydetite	C2/c	c2	32	3.125	100.000					94	4.618	434.131
Strassmannite	C2/c	c2	32	3.125	100.000					130	5.069	658.908
Wetherillite	P21/c	pc	32	4.000	128.000	p21am	32	3.500	112.000	166	5.387	894.257
Phosphuranylite topology
Deliensite	Pnn2	pn	40	4.322	172.877	pmmn	40	3.122	124.877	168	5.392	905.909
Feynmannite	P2/n	p2/n	40	3.322	132.877					88	4.505	396.430
Greenlizardite	P1	p1	20	4.322	86.439	cmm2	20	3.122	62.439	76	6.248	474.842
Johannite			20	3.322	66.439	c2/m	20	3.122	62.439	45	4.514	203.133
Meitnerite			20	3.322	66.439					42	4.392	184.477
Plášilite	P21/c		20	3.322	66.439					68	4.087	277.947
Mathesiusite topology
Ammonio­mathesiusite	P4/n	p4/n	76	3.353	254.842	p4/n	76	3.353	254.842	150	4.309	646.323
Mathesiusite	P4/n	p4/n	76	3.353	254.842	p4/n	76	3.353	254.842	110	3.890	427.950
Isolated clusters
Belakovskiite		p1	26	4.700	122.211	pmm2	26	3.470	90.211	91	5.519	502.209
Bluelizardite	C2/c	p2	46	4.524	208.084	pmm2	46	3.915	180.084	148	5.236	774.999
Klaprothite	P21/c	p1	23	4.524	104.042	pm	23	3.915	90.042	164	5.382	882.639
Péligotite		p1	23	4.524	104.042					82	5.382	441.319
Ottohahnite			62	4.954	307.160		62	4.954	307.160	126	6.000	768.000
Chains of polyhedra
Adolfpateraite	P21/c	21	26	3.700	96.211	mc21	26	3.085	80.211	56	3.807	213.212
Fermiite	Pmn21	m	36	4.503	162.117	a21/m or ama2	36	3.503	126.117	124	5.212	646.320
Meisserite			36	4.170	150.117					64	5.031	322.000
Bobcookite			32	4.000	128.000	amam	32	2.875	92.000	82	5.382	441.319
Oppenheimerite			32	4.000	128.000					48	4.585	220.078
Rietveldite	Pmn21	21	32	4.000	128.000					116	5.065	587.526
Svornostite	Pmn21	21	32	4.000	128.000					106	4.898	519.160
Shumwayite	P21/c	a	28	3.807	106.606	ama2	28	3.236	90.606	124	4.954	614.320
Dense chains of polyhedra
Alwilkinsite-(Y)	P212121	21	52	4.700	244.423	amam	52	3.547	184.423	276	6.109	1685.953
Uranopilite		1	55	5.781	317.975	a112	55	4.800	263.975	158	6.304	995.997
Rare layers
Beshtauite	P21/c	p21/a	64	4.000	256.000	pbam	64	3.500	224.000	116	4.858	563.526

The complexity calculations have been undertaken for 38 uranyl sulfate minerals with well defined crystal structures. The most frequent value of structural complexity of the entire structure (including H atoms) is between 500 and 600 bits per cell (with an average of 629.49 bits per cell and a median of 485.84 bits per cell). The distribution of the complexity values is asymmetric, showing a positive skewness (= 2.431) (Fig. 3). The majority of uranyl sulfate structures should be considered as intermediate (100–500 bits per cell) to complex (< 1000 bits per cell). Nevertheless, there are three exceptions, alwilkinsite-(Y) (Kampf et al., 2017c), sejkoraite-(Y) (Plášil et al., 2011a) and natrozippeite (Burns et al., 2003), that have very complex structures (> 1000 bits per cell) (Table 2). These structures are highly hydrated, either containing highly charged metal cations (Y³⁺ and REE³⁺) or higher amounts of interstitial metal cations (cf. natrozippeite contains five Na and eight U, while magnesiozippeite contains one Mg and two U). Topological and structural complexities are described and discussed more in detail in the following text.

Figure 3 Histogram showing distribution of structural complexity values for 38 natural uranyl sulfates.

Crystal structures of belakovskiite, bluelizardite, klaprothite, péligotite and ottohahnite are based on the isolated uranyl sulfate structural units [Figs. 4(a)–4(d)]. The structural unit of belakovskiite contains: one U, four S, two uranyl O, 16 sulfate O, one water O and two H orbits, all with multiplicities of 1 (taking into account its p1 structural symmetry). In total, there are 26 orbits with onefold multiplicity (26 atoms). Structural complexity parameters for the uranyl sulfate unit in the structure of belakovskiite is calculated as I_G = 4.700 bits per atom, I_G,total = 122.211 bits per cell. In order to calculate the topological complexity parameters for the same unit, its maximal symmetry should be taken into account. The maximal symmetry of this topology is pmm2 (Fig. 4a), i.e. it is higher than the real symmetry. It is possible if sulfate tetrahedra have a mirror arrangement relative to the plane passing through the water molecule and Ur, and if two non-shared O atoms of sulfate tetrahedra are arranged one above another along the c direction, so the mirror plane would pass through the equatorial planes of uranyl bipyramids. In this case, the ν (number of atoms) is 26, but the distribution of orbits will change into another scheme: one U orbit with multiplicity of 1; one S orbit with multiplicity of 4; one O_Ur orbit with multiplicity of 2; six O_S orbits with multiplicities of 2, 2, 2, 2, 4, 4; one O_H2O orbit with multiplicity of 1; one H orbit with the multiplicity of 2. In total, there are two orbits with multiplicities of 1, six orbits with multiplicities of 2 and 3 orbits with multiplicities of 4. The topological complexity parameters for the uranyl sulfate units in the structure of belakovskiite are: I_G = 3.470 bits per atom, I_G,total = 90.211 bits per cell. Therefore, the structural information for the units in the structure of belakovskiite is higher than the topological information by a factor of ∼ 1.35. The topological symmetry of uranyl sulfate isolated units in bluelizardite is also pmm2 (Fig. 4b) and the structural information is higher than the topological information by the factor of ∼ 1.15. Isolated uranyl sulfate complexes in the structures of klaprothite and péligotite have rather low topological symmetry pm (Fig. 5c) due to the presence of three non-shared vertices at the sulfate tetrahedra trans-arranged relative to the edge-shared tetrahedra; thus, it is impossible for two mirror planes to be passed through them. The largest isolated complex among the uranyl sulfate minerals has been observed in the structure of ottohahnite (Fig. 4d). Due to the presence of two three-connected sulfate tetrahedra, two geometrical isomers could be possible. If the fourth non-shared vertices were to arrange up and down relative to the equatorial planes of the uranyl polyhedra as it is in the structure of ottohahnite, the topological symmetry would be . If both vertices were arranged in one direction, topological symmetry would be p2, which in fact wouldn't affect the complexity parameters, since the number of orbits and their multiplicity of 2 will be the same in both cases.

Figure 4 Lower dimensional structural units (0D and 1D) and their respective highest symmetry groups for (a) belakovskiite, (b) bluelizardite, (c) klaprothite/péligotite, (d) ottohahnite, (e) uranopilite, (f) alwilkinsite-(Y), (g) adolfpateraite, (h) fermiite/meisserite, (i) bobcookite/oppenheimerite/rietveldite, (j) shumwayite. Color scheme as in Fig. 3.

Figure 5 Structural sheets, symmetry elements and the respective layer group symmetries for (a) geschieberite (real LG), (b) the highest LG for the same isomer, (c) the highest LG for wetherillite and (d) the highest LG for beshtauite. Color scheme as in Fig. 3.

The symmetry of the infinite chains is considered in terms of rod group theory. The structural RG symmetry of all ten minerals, where structures are based on the rarefied (adolfpateraite, fermiite, bobcookite and shumwayite structural types) and dense [alwilkinsite-(Y), uranopilite] 1D uranyl sulfate complexes, is rather low and varies from to [Figs. 4(e)–4(j)]. At the same time, the presence of one- and two-connected sulfate tetrahedra allows much higher orthorhombic topological symmetry to be obtained. It is of interest that the maximal symmetry for topologically quite similar chains in the structures of fermiite [, Fig. 4(h)] and bobcookite [, Fig. 4(i)], in which the water molecule is replaced by the one-connected sulfate tetrahedron, differs as it is impossible to have more than two planes passing through the additional three non-shared tetrahedral vertices. It should be also noted that the lowest topological symmetry [, Fig. 4(f)] among these 1D complexes has the chain in the structure of uranopilite.

The crystal structures of geschieberite, leydetite, magnesioleydetite, strassmannite and wetherillite are based upon the [(UO₂)(SO₄)₂(H₂O)]²⁻ uranyl sulfate layers of the same topological type [Figs. 5(a) and 5(b)]. Being two-connected, all sulfate tetrahedra could rotate around this edge that result in various possible arrangements, and thus different LGs of symmetry. But all varieties are finally reduced to the maximal c2mm LG. Fig. 5(a) illustrates the real pn symmetry of the uranyl sulfate layer in the structure of geschieberite. In total, there are 16 orbits with the twofold multiplicity (32 atoms in the reduced cell). Structural complexity parameters for the uranyl sulfate layer in the structure of geschieberite is calculated as I_G = 4.000 bits per atom, I_G,total = 128.000 bits per cell. The highest symmetry could be obtained if both unshared vertices of the sulfate tetrahedra were arranged one above another along the c direction, so that the mirror plane would pass through the equatorial planes of uranyl bipyramids, resulting in twice reduced base-centered cell. In this case, the ν (number of atoms) is 16, and the distribution of orbits is as follows: one U orbit with multiplicity of 1; one S orbit with multiplicity of 2; one O_Ur orbit with multiplicity of 2; three O_S orbits with multiplicities of 2, 2 and 4; one O_H2O orbit with multiplicity of 1; one H orbit with multiplicity of 2. In total, there are two orbits with multiplicities of 1, five orbits with multiplicities of 2 and one orbit with the multiplicity of 4. The topological complexity parameters for the uranyl sulfate layer in the structures of geschieberite, magnesioleydetite and strassmannite are: I_G = 2.875 bits per atom, I_G,total = 46.000 bits per cell. Another interesting point is the symmetry of uranyl sulfate layer in the structure of wetherillite, which is topologically identical to the structures described above. However, these layered complexes are distinct because of the different arrangement of H₂O molecules (Fig. 5c) (see also Krivovichev et al., 2005; Krivovichev, 2008). Although the overall topology is the same, the current geometrical isomer will have another topological symmetry p2₁am with twice the number of atoms in the reduced cell. The ideal topology for the geometrical isomers could be different (for details see references cited above).

More complex topology of the uranyl sulfate layer in the structure of beshtauite, with elongated 12-membered voids, results in an increased dimensionality of the reduced cell. But two-connected way of tetrahedra arrangement still keeps the high maximal pbam symmetry (Fig. 5d).

Crystal structures of deliensite, feynmannite, greenlizardite, johannite and meitnerite are based on the layered uranyl sulfate complexes of the same topological type. This type of topology has three-connected sulfate tetrahedra with the fourth non-shared vertex arranged up or down relative to the plane of the layer. This variability gives rise to geometric isomers with various orientations of the sulfate polyhedra (Krivovichev & Burns, 2003; Krivovichev et al., 2005; Gurzhiy et al., 2015). To identify and classify the isomers of this topological type, their orientation matrices should be determined. According to this approach, the symbols u (up) or d (down) are assigned to each sulfate tetrahedron. As the result, three different geometric isomers have been determined, which has their individual maximal symmetry group. Johannite and meitnerite have the dudu sequence of vertices orientation and c2/m LG symmetry (Fig. 6a). Layers in greenlizardite have the uuuu sequence of vertices orientation and cmm2 LG symmetry (Fig. 6b). Deliensite and feynmannite have the uddu orientation matrix and pmmn LG symmetry (Fig. 6c).

Figure 6 Structural sheets, symmetry elements, and the respective highest layer group symmetries for (a) johannite, (b) greenlizardite, (c) deliensite and (d) mathesiusite. Color scheme as in Fig. 3; blue polyhedra = VO5 pyramids.

The presence of (VO₅) pyramids within the uranyl sulfate layers in the structures of mathesiusite and ammonio­mathesiusite stabilizes the LG with the highest p4/n structural and topological symmetries (Fig. 6d) among the natural uranyl sulfates (Table 2). Although the symmetry is high, there is one U orbit with multiplicity of 8, one S orbit with multiplicity of 8, one V orbit with multiplicity of 2, one O orbit with multiplicity of 2 and seven O orbits with multiplicities of 8, which results in ν = 76 and rather high complexity parameters for the uranyl sulfate layers in the structures of mathesiusite and ammonio­mathesiusite: I_G = 3.353 bits per atom, I_G,total = 254.842 bits per cell.

Zippeite-group minerals can be considered as the most representative group among naturally occurring uranyl sulfates. Additionally, they represent also the trickiest ones due to a variable OH content of the layers and it has to be considered in topological complexity calculations as further discussed in detail. Among the hydroxyl-free layers, two groups of minerals could be separated due to geometrical isomerism that is caused by the shift of U-polyhedra chains along the chain direction. The first isomer group contains magnesiozippeite, plavnoite and pseudo­johannite. They have the smallest reduced cell of the p2/a LG symmetry, which is defined with the topological complexity parameters: ν = 26, I_G = 2.777 bits per atom, I_G,total = 72.211 bits per cell. Further on, Figs. 7(a) and 7(b) show the real LG symmetry in the structures of magnesiozippeite and pseudojohannite, and the highest topological symmetry for hydroxyl-free layers of the currently considered isomer is shown in Fig. 7(c). The second geometrical isomer is represented by the structure of ammonio­zippeite and its structural symmetry pbab is equal to the topological (Fig. 7f): ν = 52, I_G = 2.777 bits per atom, I_G,total = 144.423 bits per cell. While the values of information content per atom are equal for both isomers, information content per unit cell is twice as high for the second type due to the twice larger cell, it doubles the amount of atoms in the reduced cell. An appearance of hydroxyl groups strongly affects and reduces the symmetry of the layered complexes. Statistical distribution of H atoms in the structure of zippeite results in a `hydroxyl-full', electroneutral [(UO₂)₂(SO₄)(OH)₂]⁰ layer in contrast to the hydroxyl-free ones [(UO₂)₂O₂(SO₄)]^2–. It appears that occupation of all O atoms shared between three uranyl polyhedra by H atoms will result in the same highest p2/a LG symmetry (Fig. 7d), but slightly higher complexity parameters: v = 30, I_G = 2.974 bits per atom, I_G,total = 89.207 bits per cell. All intermediate options of hydration could result in structurally and topologically different layers. For instance, Fig. 7(e) shows the uranyl sulfate layer of symmetry in the structure of sejkoraite-(Y). The four times larger reduced cell is a clear structural response to the U:OH ratio equal to 8:1.

Figure 7 Structural sheets, symmetry elements and the respective layer group symmetries for zippeite-group minerals: (a) magnesiozippeite (real LG), (b) pseudojohannite (real LG), (c) holosymmetrical OH-free structure, (d) zippeite (the highest LG), (e) sejkoraite-(Y) (real LG) and (f) ammonio­zippeite (the highest LG).

5. Ladders of information

A concept of informational ladders has been introduced quite recently by Krivovichev (2018). This method allows the quantitative estimation of various contributions to the complexity of the whole structure. We have applied this concept in order to evaluate and distinguish informational sources contributing to the complexity of natural uranyl sulfates. Let us consider a few low-dimensional, 0D and 1D structures, including minerals belakovskiite (0D), klaprothite (0D), péligotite (0D), ottohahnite (0D), alwilkinsite-(Y) (1D) and uranopilite (1D). We considered three distinct categories contributing to the information content of the entire structure: topological complexity (TOPO), complexity of the desymmetrized structural units (STRU) and the complexity of an interlayer complex, including a hydrogen bonding network (INT). Results of the analysis are given in Fig. 8. Among the above-chosen representatives, the highest topological complexity has the structural unit in ottohahnite [Figs. 2 and 4(d)], which is so far the most complex isolated uranyl-anion cluster unit (> 300 bits per cell). Interestingly, péligotite and uranopilite show a similar portion of information increase while desymmetrization takes place (STR/TOPO in Fig. 8). Nevertheless, the highest jump in the information content is related to the contribution of the interlayer complex of these minerals. We can distinguish here several factors that contribute to the complexity of the interlayer. Among them the hydration state, i.e. the amount of molecular H₂O, is the most crucial factor. The H₂O content is closely connected with the presence of highly charged elements [such as Y³⁺ in the case of alwilkinsite-(Y)], or due to presence of a large number of interlayer metal cation sites (such as Na⁺ in the cases of ottohahnite and klaprothite). The remarkable informational content in alwilkinsite-(Y) is the highest one among the selected group (see INT/SU in Fig. 8), and is a result of the incorporation of highly charged, trivalent yttrium cations.

Figure 8 Ladder diagrams showing information growth for the crystal structures of the selected uranyl sulfate minerals (0D and 1D structures) due to contributions from different sources. TOPO = topological information of the structural unit; STR = structural information resulting from desymmetrization of the basic structural unit; INT = information increase due to complexity of interlayer complex including hydrogen bonding network. Insets STR/TOPO and INT/STR represent an alternative graphical expression of the information increase.

The extensive role of the interstitial H₂O network and its contribution to the overall complexity of the mineral structures containing highly charged cations is well documented in Fig. 9. Among the studied uranyl sulfates it is mostly related to alwilkinsite-(Y), Y[(UO₂)₃(SO₄)₂O(OH)₃](H₂O)₁₄ (Kampf et al., 2017c), bobcookite, NaAl(UO₂)₂(SO₄)₄(H₂O)₁₈ (Kampf et al., 2015a), leydetite, Fe(UO₂)(SO₄)₂(H₂O)₁₁ (Plášil et al., 2013a), magnesioleydetite, Mg(UO₂)(SO₄)₂(H₂O)₁₁ (Kampf et al., 2018d) and strassmannite, Al(UO₂)(SO₄)₂F(H₂O)₁₆ (Kampf et al., 2018d).

Figure 9 Information diagram for the crystal structures of natural uranyl sulfates listed in Table 2 showing the informational contribution from distinct sources (topology of structural units = TOPO, desymmetrization = STRU, interstitial complex = INT).

As it was mentioned above mathesiusite, ammonio­mathesiusite and zippeite possess structures with the highest content of information with regard to their topological complexity (Fig. 9). In the cases of mathesiusite and ammonio­mathesiusite, it is due to incorporation of VO₅ polyhedra of the high point-group symmetry (C_4v). In the case of zippeite, it is due to distribution of the OH⁻ groups within the uranyl sulfate sheet.

6. Conclusions

The complexity of the uranyl sulfate structural units varies significantly and it is highly dependent on the relative density of the layer (chain or cluster), which in turn depends on the connectivity of U and S polyhedra: the more connectivity within the layer, the higher density is; and the larger complexity parameters attributed to such structures.

The two-connected arrangement of the sulfate tetrahedra makes the rotation of these groups, which should make the structure less stable. In contrast, the three-connected arrangement has fewer degrees of freedom, and thus it is more stable. The distribution of the complexity parameters confirms this observation. The majority of considered minerals have the real symmetry of the U-bearing layers, chains and isolated clusters much lower than the topological symmetry, which means that the complexity of the structural units is determined by the cations and H₂O molecules arranged in the interlayer space.