The Mixed-Layer Structures of Ikunolite, Laitakarite, Jos é ite-B and Jos é ite-A

: We used high-angle annular dark ﬁeld scanning transmission electron microscopy (HAADF STEM) to image the crystal structures of four minerals in the Bi 4 X 3 isoseries (X = Te, Se, S), a subgroup of the tetradymite homologous series: ikunolite (Bi 4 S 3 ), laitakarite (Bi 4 Se 2 S), jos é ite-B (Bi 4 Te 2 S), and jos é ite-A (Bi 4 TeS 2 ). The four minerals are isostructural and interpretable in terms of regular stacking of seven-atom packages: [Bi–S–Bi–S–Bi–S–Bi], [Bi–Se–Bi–S–Bi–Se–Bi], [Bi–Te–Bi–S–Bi–Te–Bi], and [Bi–S–Bi–Te–Bi–S–Bi], respectively. The four phases are mixed-layer structures representing the Bi 2k Te 3 (k = 2) module within the tetradymite series. Diffraction patterns conﬁrm they are seven-fold superstructures of a rhombohedral subcell with c /3 = d~1.89–1.93 Å. Modulation along the d* interval matches calculations of reﬂection intensity using the fractional shift method for Bi 4 X 3 . Internal structures can be discerned by high-resolution HAADF STEM imaging and mapping. Paired bismuth atoms are positioned at the outside of each seven-atom layer, giving the minerals a modular structure that can also be considered as being composed of ﬁve-atom (X–Bi–X–Bi–X) and two-atom (Bi–Bi) sub-modules. The presence of mixed sites for substituting cations is shown, particularly for Pb. Moreover, Pb may be important in understanding the incorporation of Ag and Au in Bi– chalcogenides. Visualisation of crystal structures by HAADF STEM contributes to understanding relationships between phases in the tetradymite homologous series and will play an invaluable role in the characterization of potential additional members of the series.


Introduction
The tetradymite homologous series is an extended family of named minerals, unnamed natural phases, experimentally synthesized products, and predicted mixed-layer structures with Bi x X y stoichiometry (X = S, Se, or Te) that can be related to the tetradymite (Bi 2 Te 2 S) archetype [1,2]. To derive a systematic homology for the series and predict structural arrangements for any stoichiometry, Ciobanu et al. [2] presented a structural model drawing on earlier work by many authors ( [3][4][5][6][7][8], among others), in which layer stacks of different size are combined in various proportions.
This crystal-structural model in turn provides a crystal-structural framework for the series that can assist in the characterisation of new natural or synthetic phases without the requirement of single crystal diffraction studies, which are often impossible, or at least impractical, due to the characteristic nanoscale intergrowths and disordering among members of the series. Tetradymite and isostructural Bi 2 X 3 phases such as tellurobismuthite, paraguanajuatite, kawazulite, and skippenite are represented by a single five-atom X-Bi-X-Bi-X layer. A defining feature of Bi-richer members of the tetradymite homologous

Background: The Bi 4 S 3 -Bi 4 Se 3 -Bi 4 Te 3 Isoseries
There are currently five named minerals in the Bi 4 S 3 -Bi 4 Se 3 -Bi 4 Te 3 isoseries: ikunolite (Bi 4 S 3 ), laitakarite (Bi 4 Se 3 ), pilsenite (Bi 4 Te 3 ), joséite-A (Bi 4 TeS 2 ), and joséite-B (Bi 4 Te 2 S). Laitakarite has been defined as Bi 4 (Se, S) 3 [22] and indeed, most published data show the presence of up to one atom per formula unit (a.p.f.u.) S alongside Se ( Figure 8 in [1], and references therein). Based on compositional data from the Orijärvi deposit, southwest Finland, which showed a marked bimodal distribution [1], laitakarite was considered to most likely represent two distinct species: Bi 4 Se 3 and Bi 4 Se 2 S. The tellurian laitakarite mentioned in the same publication is probably an additional species with the formula Bi 4 Se 2 Te. Other predicted species in the Bi 4 X 3 isoseries include Bi 4 S 2 Se and Bi 4 S 2 Te. Natural ikunolite typically contains some Se, suggesting partial or complete solid solution series between Bi 4 S 3 and Bi 4 S 2 Se.
The structures of three minerals in the Bi 4 S 3 -Bi 4 Se 3 -Bi 4 Te 3 isoseries-pilsenite, laitakarite (Bi 4 Se 3 ), and ikunolite-are shown in Figure 1. Note that all structures are composed of a five-atom submodule [X-Bi-X-Bi-X] and a second submodule comprising Bi-Bi atom pairs. Such a five-and two-atom model was first introduced by Imamov and Semiletov [4] for phases in the Bi-Se, Bi-Te and Sb-Te systems, with the term "compositional polytypism" suggested to underline the structural modularity of phases with variable compositions. This model was followed by Shelimova et al. [7] who suggested the structural formula nBi 2 ·mBi 2 X 3 for describing homology in layered tetradymite-like compounds in the systems Bi-Te and GeTe-Bi 2 Te 3 .
This two-and five-atom model was questioned by Frangis et al. [6], who considered the component modules to be formed by the addition of M-X (instead of M-M layers) to the Bi 2 X 3 archetype. Based on HR TEM studies of compounds of the M 2+δ X 3 type, where M = Bi, Sb, Ge, X = Te, Se, and 0 ≤ δ ≤ 0.4, Frangis et al. [6] describe a continuous series of one-dimensional, interface-modulated structures based on the model of Van Landuyt et al. [3]. Lind and Lidin [8] applied superspace formalism to phases in the system Bi-Se and derived a general model in which all structures are included within the 4D group P: R3:m11, with cell parameters a~4.2 Å and c sub~5 .7 Å. Their study was based on X-ray diffraction of phases in the compositional range Bi 2 Se 3 -Bi 4 Se 3 , extrapolated to Bi 3 Se 2 (=Bi 4.5 Se 3 ). A saw-tooth displacive modulation was considered for the interpretation of these structures. This two-and five-atom model was questioned by Frangis et al. [6], who considered the component modules to be formed by the addition of M-X (instead of M-M layers) to the Bi2X3 archetype. Based on HR TEM studies of compounds of the M2+δX3 type, where M = Bi, Sb, Ge, X = Te, Se, and 0 ≤ δ ≤ 0.4, Frangis et al. [6] describe a continuous series of onedimensional, interface-modulated structures based on the model of Van Landuyt et al. [3]. Lind and Lidin [8] applied superspace formalism to phases in the system Bi-Se and derived a general model in which all structures are included within the 4D group P: R3:m1 ̅ 1, with cell parameters a ~4.2 Å and csub ~5.7 Å . Their study was based on X-ray diffraction of phases in the compositional range Bi2Se3-Bi4Se3, extrapolated to Bi3Se2 (=Bi4.5Se3). A sawtooth displacive modulation was considered for the interpretation of these structures. Figure 1. Ball-and-stick (A) and space filling (B) crystal models for pilsenite, laitakarite, and ikunolite viewed along the a axis. The three minerals are isostructural with slightly modified a and c dimensions. The two structural positions for Bi and chalcogen (X = Te, Se, S) are schematically marked at the top. Structures constructed in CrystalMaker ® (v10.5.7) using .cif files based on Yamana et al. [23] for pilsenite (Bi4Te3), Stasova [24] for laitakarite (Bi4Se3), and Kato [25] for ikunolite (Bi4S3).
Ciobanu et al. [2] considered the phases in the tetradymite series as mixed layer compounds with a general formula: S' (Bi2kX3)•L' (Bi2(k+1)X3) (k ≥ 1; X = chalcogen; S', L' = number of short and long modules, respectively). This working model was based on the HR-TEM study of multiple phases in an extended compositional range Bi2Te3-Bi8Te3, showing that they are all an N-fold superstructure of a rhombohedral subcell with c/3 = d~0.2 nm, where N is the number of layers in the stacking sequence [2]. Electron diffraction (ED) patterns, displaying the two brightest reflections about the middle of d*, are described by a monotonic decrease of two displacive modulations with an increase in Bi content. Such Figure 1. Ball-and-stick (A) and space filling (B) crystal models for pilsenite, laitakarite, and ikunolite viewed along the a axis. The three minerals are isostructural with slightly modified a and c dimensions. The two structural positions for Bi and chalcogen (X = Te, Se, S) are schematically marked at the top. Structures constructed in CrystalMaker ® (v10.5.7) using .cif files based on Yamana et al. [23] for pilsenite (Bi 4 Te 3 ), Stasova [24] for laitakarite (Bi 4 Se 3 ), and Kato [25] for ikunolite (Bi 4 S 3 ).
Ciobanu et al. [2] considered the phases in the tetradymite series as mixed layer compounds with a general formula: S' (Bi 2k X 3 )·L' (Bi 2(k+1) X 3 ) (k ≥ 1; X = chalcogen; S', L' = number of short and long modules, respectively). This working model was based on the HR-TEM study of multiple phases in an extended compositional range Bi 2 Te 3 -Bi 8 Te 3 , showing that they are all an N-fold superstructure of a rhombohedral subcell with c/3 = d~0.2 nm, where N is the number of layers in the stacking sequence [2]. Electron diffraction (ED) patterns, displaying the two brightest reflections about the middle of d*, are described by a monotonic decrease of two displacive modulations with an increase in Bi content. Such displacements were quantified by fractional shifts between reflections in the derived and basic structures [2]. Mixed-layer compounds are a class of minerals in which ED patterns display characteristic features [5], which, if linked to a homology in a defined series of structures, can be used to predict the stacking sequence and chemistry of discrete phases.
As the series progresses towards Bi-rich compositions, compounds in the Bi 4 X 3 isoseries represent the first structures defined by a single module (Bi 2k X 3 ; k = 2) in which single Bi-Bi pairs occur between the chalcogen-bearing, five-atom submodule, common to all phases in the series ( Figure 1B).

Electron Probe Microanalysis
Quantitative compositions were determined using a Cameca SX-Five electron probe microanalyzer (EPMA), equipped with five tuneable wavelength-dispersive spectrometers. The instrument runs PeakSite v6.5 software for microscope operation, and Probe for EPMA software (distributed by Probe Software Inc., Eugene, OR, USA) for all data acquisition and processing. Operating conditions utilized were 20 kV/20 nA with a focused beam.
The full list of elements analysed along with count times, nominal detection limits, and primary and interference standards are presented in Supplementary Materials, Tables S1-S3. Of particular note is the increased count time utilized for Au Lα for better detection limit. Matrix corrections of Armstrong-Love/Scott ϕ(ρz) [26] and Henke MACs were used for data reduction.
Traditional two-point backgrounds were acquired. Due to complex off-peak interferences in these sample matrices, the shared background function of Probe for EPMA was utilized. This function allows the collected background positions of elements on the same spectrometer be used for all elements on that spectrometer, allowing multipoint backgrounds to be applied to each element. However, in simple background regions, a traditional two-point linear fit was still used.
In addition, the first element acquired on each spectrometer (i.e., typically S, Te, Fe, Se, Zn) was analysed using the time dependent intensity (TDI) correction feature of Probe for EPMA (e.g., [27]). Using this method, the decay of X-ray counts over time is measured and modelled to return a t = 0 intercept, and from this a concentration is calculated. This minimizes the impact of element migration and can be reviewed, disabled, or enabled post analysis.

Nanoscale Analysis
Preparation of thinned (<100 nm) foils for TEM investigation from the polished blocks was performed using a FEI-Helios nanoLab dual-focused ion beam and scanning electron microscope (FIB-SEM), following procedures outlined by Ciobanu et al. [28]. Each TEM foil was attached to a copper grid.
Foils were analysed using high-angle annular dark field (HAADF) scanning-TEM (STEM) imaging and energy dispersive X-ray spectrometry (EDS)-STEM mapping using an ultra-high resolution, probe-corrected FEI Titan Themis S/TEM operated at 200 kV. This instrument was equipped with a X-FEG Schottky source and Super-X EDS geometry. The Super-X EDS detector provided geometrically symmetric EDS detection with an effective solid angle of 0.8 sr. Probe correction delivered sub-Ångstrom spatial resolution, and an inner collection angle greater than 50 mrad was used for HAADF imaging with a Fischione detector. Image acquisition was undertaken using FEI software, TIA (v4.15) and complementary imaging by a drift-corrected frame integration package (DCFI) included in the Velox (v. 2.13.0.1138) software. Various filters (radial Wiener, high-pass, average and Gaussian blur) were used to eliminate noise and/or enhance the images. EDS data acquisition and processing was carried out using Velox software. Indexing of diffraction patterns was conducted with WinWulff© (v1.6) (JCrystalSoft, Livermore, CA, USA) and publicly available data from the American Mineralogist Crystal Structure Database (http: //rruff.geo.arizona.edu/AMS/amcsd.php, accessed on 11 July 2021). Crystal structure models were generated in CrystalMaker ® (v10.5.7) and image simulations using STEM for xHREM TM (v4.1) software.
All instruments are housed at Adelaide Microscopy, The University of Adelaide.

Petrography
The four minerals discussed here are from specimens listed in Table 1. Background data on these samples was given in Ciobanu et al. [29], and for the laitakarite sample, also in Ciobanu et al. [30].
Petrographic aspects of the samples from which TEM foils were prepared are shown in Figures 2 and 3. The samples for ikunolite and joséite-B/-A were extracted from polished blocks prepared from hand specimens, whereas the laitakarite specimen comprises mounted fine flakes/platelets from a powdered laitakarite concentrate.
Ikunolite in epithermal veins from Ashio, Japan, forms euhedral, mm-sized grains within native bismuth ( Figure 2C). Ikunolite contains ubiquitous inclusions of an intermediate member of the lillianite-gustavite series ( Figure 2D). Coarser ikunolite grains in the specimen show evidence of deformation and local replacement by base metal sulphides. More abundant lillianite-and also cosalite-occur within ikunolite in the margin of the same polished block with irregular boundaries to ikunolite suggestive of replacement. Inclusions of lillianite-gustavite are often armoured by fine-grained electrum/gold at their contacts with the host ikunolite.
The two joséite species were investigated in a specimen of hedenbergite skarn from Hedley, B.C., Canada ( Figure 2E). Bismuth minerals form mm-scale patches ( Figure 2F,G), mainly comprising native bismuth, joséite-B, hedleyite and unnamed Bi 8 Te 3 [31], and are commonly associated with coarse scheelite ( Figure 2E), and thin slivers of molybdenite. Native gold is also present, associated with bismuth and molybdenite. Joséite-A is scarce in the specimen and occurs within much smaller patches along trails of retrograde alteration of skarn ( Figure 2H). In detail, the two grains of joséite-A sliced for nanoscale investigation ( Figure 3A,B) are hosted within chlorite-dominant alteration and are also fractured. It thus proved difficult to obtain high quality compositional data by electron probe and their composition below is obtained from STEM EDX. Cross-sectioning shows the association of joséite-A with native bismuth ( Figure 3C,D).

Compositional Data
Mean compositions of the four species are given in Tables 2-4. Laitakarite is defined by the presence of~1 a.p.f.u. sulphur, the absence of Te (0.14 a.p.f.u.), and by a modest Pb content ( Table 2).
Ikunolite features a mean Pb content approaching 7 wt.%, consistent minor Ag, and trace yet occasionally measurable concentrations of a range of other elements (Table 3). Lead contents correspond to approximately 0.3 a.p.f.u. Both joséite-B and joséite-A are close to stoichiometric, with low or absent Pb (Table 4).
Compositions are plotted in the Bi 4 S 3 -Bi 4 Se 3 -Bi 4 Te 3 ternary diagram confirming a good fit to stoichiometry ( Figure 4A) and in terms of Pb/total metals vs. Te/(Te + Se + S) ( Figure 4B).

Compositional Data
Mean compositions of the four species are given in Tables 2-4. Laitakarite is defined by the presence of ~1 a.p.f.u. sulphur, the absence of Te (0.14 a.p.f.u.), and by a modest Pb content ( Table 2).

Nanoscale Characterization
The data presented here for the four chalcogenides derives from one TEM foil for each species, apart from joséite-A for which two foils were prepared (Supplementary Materials, Figures S1 and S2). Laitakarite displays a fine-grained aggregate at the contact with native bismuth (2-µm-wide lamella in the centre of the foil) as well as sub-micron-scale acicular inclusions of galena and bohdanowiczite (Supplementary Materials, Figure S1). Joséite-B was studied from a foil cut across the boundary with adjacent hedleyite (Supplementary Materials, Figure S2). The latter is identified from EPMA data only (see [2]) as the stacking sequences could not be imaged due to grain orientation on [1]. Joséite-A is truncated at the base or on the side by chlorite and native bismuth (Supplementary Materials, Figure S2).  Compositions are plotted in the Bi4S3-Bi4Se3-Bi4Te3 ternary diagram confirming a good fit to stoichiometry ( Figure 4A) and in terms of Pb/total metals vs. Te/(Te + Se + S) ( Figure 4B).

Nanoscale Characterization
The data presented here for the four chalcogenides derives from one TEM foil for each species, apart from joséite-A for which two foils were prepared (Supplementary Materials, Figures S1 and S2). Laitakarite displays a fine-grained aggregate at the contact with native bismuth (2-µm-wide lamella in the centre of the foil) as well as sub-micron-scale acicular inclusions of galena and bohdanowiczite (Supplementary Materials, Figure S1). Joséite-B was studied from a foil cut across the boundary with adjacent hedleyite (Supplementary Materials, Figure S2). The latter is identified from EPMA data only (see [2]) as the stacking sequences could not be imaged due to grain orientation on [0001]. Joséite-A Chemical mapping of laitakarite at the contact with native bismuth shows that Ag and trace Pb is preferentially distributed into the native bismuth ( Figure 5A). Tiny inclusions of galena shown on this map become more abundant~150 nm away from the contact with native bismuth, forming sets of acicular needles of galena and bohdanowiczite (Supplementary Materials, Figure S1). STEM EDS maps of such associations show the presence of nanoscale galena at the margins between bohdanowiczite and host laitakarite ( Figure 5B). The presence of detectable sulphur within bohdanowiczite indicates this is probably an intermediate member of the matildite-bohdanowiczite solid solution (AgBiS 2 -AgBiSe 2 ).

Stacking Sequences in Bi 4 X 3 Phases
All four species show regular stacking sequences with repeats of~d~1.4 nm, representing the width of the seven-atom layer measured between atom arrays representing the Bi-Bi pairs (see below) ( Figure 6A-D). Such regular sequences occur with the same orientation across the length of each species in a given foil, and were imaged by tilting the specimen on a 2110 zone axis for each species. The high degree of layer-stack ordering is also revealed from the selected area electron diffraction (SAED) patterns obtained from each foil (Figure 7). HAADF STEM images of sequences of up to nine repeats ( Figure 6A-D) show the Bi-Bi pairs as closely-spaced double arrays of atoms, of variable intensity from one species to another, e.g., brightest in joséite-B relative to the other species. The interpretation of images is concordant with STEM simulations for each species. The Bi-Bi pairs are difficult to recognize in phases in the present study that show lattice distortion such as laitakarite and joséite-A ( Figure 6A,D). This is in sharp contrast with the layer sequences in five-atom layer phases, which are separated by Te-Te pairs with van der Waals bonds, readily recognizable on HAADF STEM images (e.g., [14]). Subtle differences also appear due to the presence of more than a single chalcogen (Se and S in laitakarite, Te and S in joséite-A and -B), as well as the relatively high Pb content of ikunolite. Laitakarite and ikunolite appear broadly similar, in that the central part of the layer stands out relative to the Bi-Bi pairs. STEM simulations ( Figure 6A-D) show a good match with the sequences for each of the four species.
The layer sequence in interface-modulated structures can be calculated from electron diffractions using the correlation between the displacement vector (q F *) and the rhombohedral subcell defined by the d* interval (d = 1/d* =~2 Å) along the c* axis in the tetradymite homologous series ([2,6]; Figure 7). The parameter q F * corresponds to the distance between two brighter satellites in the centre of d* and the layer stack is depicted by the number of divisions (i) within this interval. The smallest distance between any two reflections (d N *) across d* corresponds to the width of a given N layer type (N = number of atoms in the layer) and can be calculated from: (1) q F * = i × d N * = (i × d*)/N leading to: (2) d N = q F /i and N = (i × q F )/d The SAED patterns for the four species show a single layer stack (I = 1) with seven divisions across d* (Figure 7A), in this case d N * = q F * and d N = d × N =~1.4 nm for the seven-atom layer (d =~2 Å and N = 7). Measurements on the images and corresponding SAED (Figures 6 and 7A) show the d 7 width is in the range 1.33-1.35 Å, with corresponding values of d in the range 1.89-1.93 Å. Using c = 3 × d, the calculated c parameter is 40.5 Å for laitakarite and joséite-B, 39.6 Å for ikunolite, and 39.9 Å for joséite-A. These are close to data reported for species from the joséite-B isoseries with comparable compositions to those studied here ( [1], and references therein). The four species are typified by the same modulation along the d* interval with reflection intensity concordant with the variation of the sum of intensities for (N -i)/2 reflections calculated by Ciobanu et al. (Table 3 and Figure 9h in [2]) using the fractional shift method ( Figure 7B,C).
Minerals 2021, 11, x is truncated at the base or on the side by chlorite and native bismuth (Suppleme Materials, Figure S2).
Chemical mapping of laitakarite at the contact with native bismuth shows th and trace Pb is preferentially distributed into the native bismuth ( Figure 5A). Tiny sions of galena shown on this map become more abundant ~150 nm away from the c with native bismuth, forming sets of acicular needles of galena and bohdanowiczite plementary Materials, Figure S1). STEM EDS maps of such associations show the pre of nanoscale galena at the margins between bohdanowiczite and host laitakarite (F 5B). The presence of detectable sulphur within bohdanowiczite indicates this is pro an intermediate member of the matildite-bohdanowiczite solid solution (AgBiS BiSe2).

Stacking Sequences in Bi4X3 Phases
All four species show regular stacking sequences with repeats of ~d~1.4 nm, senting the width of the seven-atom layer measured between atom arrays repres the Bi-Bi pairs (see below) ( Figure 6A-D). Such regular sequences occur with the such as laitakarite and joséite-A ( Figure 6A,D). This is in sharp contrast with the layer sequences in five-atom layer phases, which are separated by Te-Te pairs with van der Waals bonds, readily recognizable on HAADF STEM images (e.g., [14]). Subtle differences also appear due to the presence of more than a single chalcogen (Se and S in laitakarite, Te and S in joséite-A and -B), as well as the relatively high Pb content of ikunolite. Laitakarite and ikunolite appear broadly similar, in that the central part of the layer stands out relative to the Bi-Bi pairs. STEM simulations ( Figure 6A-D) show a good match with the sequences for each of the four species.

Identity of Atoms in the Layers
High-resolution imaging of atomic arrangements within the seven-atom layers discussed here are shown in Figure 8. Variation in the intensity of the HAADF signal along the <0117> lattice direction shows differences between the Bi atoms in internal and external positions within the layers (Bi1 and Bi2, respectively). The signal profiles show lower intensity for Bi2 relative to Bi1 within each phase but with a greater difference in laitakarite, ikunolite and joséite-A, relative to joséite-B. Such Z-contrast differences are enhanced for species with chalcogens of lower atomic weight, such as S or Se, compared to Te. The inverse distributions of S and Te in joséite-B compared to joséite-A, i.e., Bi2-Te-Bi1-S-Bi1-Te-Bi2, and Bi2-S-Bi1-Te-Bi1-S-Bi2, respectively, are indicated by higher and lower signal intensities across the atom sequences ( Figure 8C,D). STEM simulations for the seven-atom sequences in each phase show good correlation with the images and signal variation across the profiles (insets to Figure 8A-D).
The distribution of Pb in ikunolite and S in joséite-A is shown on high-resolution maps of the layer sequences ( Figure 9). These show that Pb is associated with the Bi1 position, located centrally in the ikunolite layer ( Figure 9A). In joséite-A, the highest Bi concentration overlaps with the Bi2 locations on the margins of the layer, whereas S is highest in the central part ( Figure 9B). Sulphur should be placed between the Bi2 and Bi1 positions, flanking the margins of the central Bi1-Te-Bi1 segment of the layer, but this cannot be well discriminated on the maps presented here. The compositions calculated from several maps, including the one in Figure 9B, do however show a close fit to joséite-A stoichiometry, as shown in Table 4.    Insets outlined in yellow are STEM simulations showing the distribution and speciation of the seven-atoms across a single layer. The simulations were obtained using the atomic coordinates for structures (.cif files for the end-member tellurides shown in Figure 1) but with compositional modifications for cation mixed sites/chalcogen speciation within the seven-atom sequence according to EPMA data. Chalcogens that are visible on the STEM simulations are marked by short arrows. Note variation in signal intensity for Bi positions, whereby the Bi-Bi pairs (Bi2 position) have a lower intensity than the two 'internal' Bi atoms (Bi1) that alternate with chalcogens and which are better separated on the images. This effect is attenuated on the images/profile for joséite-B more than in the STEM simulation.

Nanoscale Inclusions and Lattice-Scale Defects
Inclusions of galena are common throughout the joséite-B specimen ( Figure 10). These are observed as sets of thin, 1-2 nm-wide needles, or wider, ~100 nm-thick slivers oriented parallel with the layer stacking in host joséite-B ( Figure 10A). Terminations of single needles of galena oriented on the [110] zone axis are accommodated across contacts between two layers of joséite-B ( Figure 10B). The narrowest PbS inclusions are imaged as single or double arrays of Pb atoms positioned in the centre of the tellurosulphide ( Figure  10C). In detail, the Pb (and adjacent S) atoms along <111> directions in galena can fit either along <011 ̅ 7> directions in the seven-layer telluride, or alternatively, as twins/kink planes in joséite-B ( Figure 10D). A single 'PbS' unit inserted within the telluride forms a nineatom sequence: Bi-Te-Bi-S-[Pb-S]-Bi-Te-Bi, giving the composition PbBi4Te2S2.
In contrast to joséite-B, which has a low Pb content (mean 0.2 wt.%), the species with the highest Pb concentration analysed here, ikunolite (mean 6.7 wt.%), contains no galena inclusions. Ikunolite is nonetheless characterised by numerous defects, which occur as  (Table 4) fits joséite-A stoichiometry closely.

Nanoscale Inclusions and Lattice-Scale Defects
Inclusions of galena are common throughout the joséite-B specimen ( Figure 10). These are observed as sets of thin, 1-2 nm-wide needles, or wider,~100 nm-thick slivers oriented parallel with the layer stacking in host joséite-B ( Figure 10A). Terminations of single needles of galena oriented on the [110] zone axis are accommodated across contacts between two layers of joséite-B ( Figure 10B). The narrowest PbS inclusions are imaged as single or double arrays of Pb atoms positioned in the centre of the tellurosulphide ( Figure 10C). In detail, the Pb (and adjacent S) atoms along <111> directions in galena can fit either along <0117> directions in the seven-layer telluride, or alternatively, as twins/kink planes in joséite-B ( Figure 10D). A single 'PbS' unit inserted within the telluride forms a nine-atom sequence: Bi-Te-Bi-S-[Pb-S]-Bi-Te-Bi, giving the composition PbBi 4 Te 2 S 2 .
In contrast to joséite-B, which has a low Pb content (mean 0.2 wt.%), the species with the highest Pb concentration analysed here, ikunolite (mean 6.7 wt.%), contains no galena inclusions. Ikunolite is nonetheless characterised by numerous defects, which occur as 'harmonic,' screw dislocations along <0110> directions in the Bi-chalcogenide ( Figure 11A,B). In detail, the atom displacements lead to dilational swells and kinks between and along the Bi layers ( Figure 11B) and these are associated with nucleation of inclusions in laitakarite ( Figure 11C,D). As in joséite-B, galena occurs as lamellae that are coherently stacked within the Bi-chalcogenide host, but in this case, as observed in ikunolite, abundant screw dislocations occur ( Figure 11C). The kinks along the galena are accommodated by defects whereby splitting and merging of atom arrays are observed, i.e., a Pb array connects with a double Bi-Bi atom array and vice versa ( Figure 11D). A comparison can be made with dislocations along 1/3<0111> directions in Bi 2 Te 3 nanowires, which have dissociated components leading to formation of seven-atom layers of the Bi 3 Te 4 type and accounting for changes in chemical composition [32]. 'harmonic,' screw dislocations along <011 ̅ 0> directions in the Bi-chalcogenide ( Figure  11A,B). In detail, the atom displacements lead to dilational swells and kinks between and along the Bi layers ( Figure 11B) and these are associated with nucleation of inclusions in laitakarite ( Figure 11C,D). As in joséite-B, galena occurs as lamellae that are coherently stacked within the Bi-chalcogenide host, but in this case, as observed in ikunolite, abundant screw dislocations occur ( Figure 11C). The kinks along the galena are accommodated by defects whereby splitting and merging of atom arrays are observed, i.e., a Pb array connects with a double Bi-Bi atom array and vice versa ( Figure 11D). A comparison can be made with dislocations along 1/3<0111> directions in Bi2Te3 nanowires, which have dissociated components leading to formation of seven-atom layers of the Bi3Te4 type and accounting for changes in chemical composition [32].  In the case discussed here, such lattice-scale defects can also accommodate more complex chemical changes, e.g., enrichment in Pb, Ag, Se, and S, coupled with relative depletion in Bi, leading to the formation of needles containing bohdanowiczite and galena ( Figure 12). Galena occurs as single-orientation lamellae whereas bohdanowiczite is polygranular ( Figure 13A-C). The Ag-Bi-selenide is identified from images and fast Fourier transform (FFT) patterns of bohdanowiczite tilted on the [2201] zone axis, which is very similar to the adjacent galena tilted on a [110] zone axis ( Figure 13D,E). In the case discussed here, such lattice-scale defects can also accommodate more com plex chemical changes, e.g., enrichment in Pb, Ag, Se, and S, coupled with relative deple tion in Bi, leading to the formation of needles containing bohdanowiczite and galena (Fig  ure 12). Galena occurs as single-orientation lamellae whereas bohdanowiczite is polygran ular (Figure 13A-C). The Ag-Bi-selenide is identified from images and fast Fourier trans form (FFT) patterns of bohdanowiczite tilted on the [22 ̅ 01] zone axis, which is very simila to the adjacent galena tilted on a [110] zone axis ( Figure 13D,E).       Figure 11. These are imaged on zone axes as marked on the FFT patterns. (D,E) crop of images in (B,C) and crystal models of the two phases. Overlays (dashed white lines) on the FFT patterns, images, and models highlight the rhombus motif which is common to both species for the respective zone axes.

Relationships with Other Phases
Our study shows that phases of the Bi 4 X 3 isoseries [1] represent the Bi 2k Te 3 (k = 2) module within the tetradymite series [2]. The ED patterns show they all are seven-fold superstructures of a rhombohedral subcell defined by the d* interval (d = 1/d* =~2 Å) along the c* axis with modulation matching the calculation of reflection intensity using the fractional shift method for Bi 4 X 3 compounds (Figure 7 [2]). In detail, measurements of d* and d 7 distances on ED patterns and images (Figures 6 and 7A) do show differences for each of the discussed phases. Although the individual crystal structures could be better refined if using dynamical diffraction refinement or precession electron diffraction to eliminate dynamical effects, e.g., [33][34][35], such dedicated methods were not used for the present study, which is focused on imaging the structures and correlating such imaging with generic homology in the tetradymite series.
Nonetheless, the ED patterns of the four species representing the Bi-X-Bi-X-Bi-X-Bi mixed-layer chalcogenide structure in the tetradymite series [2] will show similarities with those of analogous chalcogen-rich phases with the same number of atoms, i.e., X-Bi-X-Bi-X-Bi-X, such as the seven-atom layer member of the aleksite homologous series [2,13,36]. HAADF STEM imaging and STEM EDX mapping can, however, reveal the internal structure of the layers, confirming published structures in which Bi-Bi pairs are observed at the margins of each seven-atom layer (Figures 6-9). In the Bi 4 X 3 isoseries, HAADF STEM imaging reveals the tight Bi2-Bi2 bonding, in agreement with the metallic bonding stipulated from atomic structure determination and band structure calculations for Bi x Se y phases [37].
Species from the Bi 4 X 3 isoseries are therefore distinct from the five-atom layer phases of the Bi 2 X 3 isoseries (tetradymite, tellurobismutite, etc.) which are linked by van der Waal bonds between exterior chalcogen atoms, a feature shared by all homologues of the aleksite series [15,36], e.g., Te-Bi-S-Pb-S-Bi-Te in aleksite (PbBi 2 Te 2 S 2 ), the seven-atom member of the series.
It would be most interesting to use HAADF STEM imaging to assess the identity of mineral phases with Bi 3 X 4 stoichiometry, which can be thought of similarly, e.g., Te-Bi-Te-Pb-Te-Bi-Te in rucklidgeite, (Bi,Pb) 3 Te 4 , and with expansion to other phases with Bi 3 X 4 stoichiometry. Such species, if they have a simple seven-atom layer sequence, do not obey the homology of the tetradymite series Bi 2k X 3 , and should thus be formed by a combination of five-and seven-atom layers (see [2]). Additionally, such phases also have Pb-rich variants, suggesting a possible link to other series such as the aleksite series [36]. Rucklidgeite was defined as (Bi, Pb) 3 Te 4 [38] but it was later established that the Pb content is variable, up to one a.p.f.u., and probably not essential [39]. Both lead-bearing and lead-free members of a Bi 3 Te 4 -PbBi 2 Te 4 solid-solution series are described from several localities (see Figure 15 and extensive referencing in [1]). Cook and Ciobanu [40] documented both low-Pb rucklidgeite (0.02 Pb a.p.f.u.) and Pb-bearing rucklidgeite ((Pb 0.82-0.87 Bi 2.11-2.16 )S 3 (Te,Se,S) 4 ) in the same ore system. The same may apply to poubaite, PbBi 2 (Se,Te,S) 4 [41,42], a less common mineral. Cook et al. [1] raised the possibility of additional phases with a rucklidgeite-like stoichiometry based on published data for unnamed Bi 3 (Te,S) 4 [38,[43][44][45] and for a phase with the empirical composition Bi 3 (Te 1.67-2.21 Se 0.67-1.65 S 0.70-1.14 ) 4 [46].

Mixed Sites versus Inclusions: The Role and Location of Lead
Unlike other species in the tetradymite homologous series, all Bi 4 X 3 phases often, although not ubiquitously, contain several wt.% Pb [1]. Here we show that Pb can be incorporated into the crystal structure but can also form nanoscale inclusions of galena (Figures 9-12). Incorporation of Pb within a mixed site with Bi1 is inferred from the mapping of ikunolite, the species with highest Pb content analysed here (~0.3 a.p.f.u.). In contrast to the metal (cation) sites, the present data show very little tendency among chalcogens towards mixed site occupancy (Tables 2-4).
Although ikunolite also hosts the greatest concentration of other cations, particularly measurable Ag (~0.005 a.p.f.u.), the sum of other monovalent cations (including Ag and Au) is much lower than that required for charge balance compensation considering the substitution Pb 2+ + M + = Bi 3+ . A comparable problem occurs with presence of structural Pb within members of the Bi 3 X 4 isoseries, as introduced above. Based on the absence of nanoscale inclusions in the material analysed here, we are confident that the Pb measured in ikunolite is structurally bound.
Interestingly, five of the nine microprobe analyses that contain measurable Au (in the range 700-2290 ppm Au) were obtained from the location of the investigated foil. There is no direct correlation between Au and Ag concentrations, even though their sum is 0.014 a.p.f.u (0.006 a.p.f.u. Au and 0.008 a.p.f.u. Ag)-relatively high but still an order of magnitude below Pb measured in the same analyses (~0.3 a.p.f.u.). This implies no direct correlation between the presence of 'exotic' precious trace elements and Pb, at least not in the present data. Nonetheless, the EPMA measurements confirm measurable Au in ikunolite from Ashio at levels comparable with those reported from laser-ablation inductively coupled mass spectrometry [29]. If validated by other methods of atomic structural determination, the presence of heterovalent mixed cation sites in Bi-chalcogenides is also important for structural incorporation of Ag and Au, particularly in high-grade Au ores [14].
Galena occurs as nanoscale inclusions in both laitakarite and joséite-B (~0.14 and 0.01 a.p.f.u., respectively). This is despite the fact that ikunolite and laitakarite show widespread lattice-scale defects whereas joséite-B lacks such defects (Figures 10 and 11). We suggest that Pb behaviour is controlled by the local environment, e.g., ikunolite formation is buffered by high-Pb since it co-exists with, and also contains, ubiquitous inclusions of a Pb-Bi sulphosalt from the lillianite-gustavite solid solution ( Figure 2C,D). The abundant latticescale defects in ikunolite and laitakarite can be attributed to strain-induced deformation, either during vein re-opening at Ashio [47] or superimposed metamorphism at Orijärvi [48].
The formation of single and double PbS units through the centre of seven-atom layers in joséite-B ( Figure 10) suggests the possibility of an additional homologous series starting from joséite-B and extending to PbBi 4 Te 2 S 2 (nine-atom layer), Pb 2 Bi 4 Te 2 S 4 (eleven-atom layer) and so on, in a manner comparable the aleksite series ( [15,36]; see above).

Conclusions and Implications
The direct visualisation of structures by HAADF STEM and nanoscale chemical mapping of layer sequences unequivocally confirms the identity of the Bi 4 S 3 -Bi 4 Se 3 -Bi 4 Te 3 isoseries. Laitakarite, ikunolite, joséite-B, and joséite-A are mixed-layer structures representing the Bi 2k Te 3 (k = 2) module within the tetradymite series. SAED patterns confirm they are seven-fold superstructures of a rhombohedral subcell with c/3 = d~1.89-1.93 Å. Modulation along the d* interval matches calculations of reflection intensity using the fractional shift method for Bi 4 X 3 . The internal structure in each species can be discerned by high-resolution imaging. We confirm the presence of mixed sites for cations, particularly Pb. The role of Pb is important in understanding the incorporation of Ag and Au in Bi-chalcogenides, as well as the formation of other, parallel homologous series of Pb-Bi-chalcogenides, including those from the aleksite series. HAADF STEM imaging of crystal structures will continue to play a valuable role in understanding relationships between phases in the tetradymite series and for the characterization of potential additional members.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/min11090920/s1, Table S1: EPMA setup for analysis, Table S2: Interference corrections,  Table S3: Standard information, Figure S1: Secondary electron and HAADF STEM images showing aspects of the laitakarite foil, Figure S2: HAADF STEM images showing aspects of the four studied foils.
Author Contributions: N.J.C. and C.L.C. designed the research and wrote the paper. A.D.S. provided assistance with the operation of the Titan Themis S/TEM microscope and data acquisition/processing. B.P.W. provided assistance with EPMA data acquisition/processing. K.E. provided advice and guidance throughout. All authors participated in editing the manuscript. All authors have read and agreed to the published version of the manuscript.