Bi 8 Te 3 , the 11-Atom Layer Member of the Tetradymite Homologous Series

: Bi 8 Te 3 is a member of the tetradymite homologous series, previously shown to be compositionally and structurally distinct from hedleyite, Bi 7 Te 3 , yet inadequately characterized structurally. The phase is identiﬁed in a sample from the Hedley district, British Columbia, Canada. Compositions are documented by electron probe microanalysis and structures are directly imaged using high-angle annular dark ﬁeld (HAADF) scanning transmission electron microscopy (STEM). Results conﬁrm that Bi 8 Te 3 has an 11-atom layer structure, in which three Bi-Bi pairs are placed adjacent to the ﬁve-atom sequence (Te-Bi-Te-Bi-Te). Bi 8 Te 3 has trigonal symmetry (space group R 3 m ) with unit cell dimensions of a = ~4.4 Å and c = ~63 Å calculated from measurements on representative electron diffraction patterns. The model is assessed by STEM simulations and EDS mapping, all displaying good agreement with the HAADF STEM imaging. Lattice-scale intergrowths are documented in phases replacing Bi 8 Te 3 , accounting for the rarity of this phase in nature. These results support prior predictions of crystal structures in the tetradymite homologous series from theoretical modeling and indicate that other phases are likely to exist for future discovery. Tetradymite homologues are mixed-layer compounds derived as one-dimensional superstructures of a basic rhombohedral sub-cell. Each member of the series has a discrete stoichiometric composition and unique crystal structure.

This approach was introduced as a working model based on a high-resolution transmission electron microscopy (HR-TEM) study of phases in the extended compositional range Bi 2 Te 3 -Bi 8 Te 3 [2]. This study showed that each phase is a N-fold superstructure of a rhombohedral sub-cell with c/3 = d~0.2 nm, where N is the number of layers in the stacking sequence. 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. Such displacements are quantifiable by fractional shifts between reflections in the derived and basic structures [2].
Two polished blocks were analyzed: (1) H163b, containing patches of bismuth minerals within skarn, and (2) H1-H2, a mounted chip with lamellae of Bi 8 Te 3 within joséite-B. The same lamellae in H1-H2 were also analyzed by [2] using conventional HR TEM. The nanoscale study was carried out on three foils prepared from the two polished blocks.

Electron Probe Microanalysis
Quantitative compositions were determined using a Cameca SX-Five electron probe microanalyzer (EPMA), equipped with five tunable 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 analyzed along with count times, nominal detection limits, and primary and interference standards are presented in the Supplementary Materials, Tables S1 and S2. Matrix corrections of Armstrong-Love/Scott ϕ(ρz) [16] 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 2-point linear fit was still used.
In addition, the first elements acquired on each spectrometer were analyzed using the Time-Dependent Intensity (TDI) correction feature of Probe for EPMA (e.g., [17]). Using this method, the decay of X-ray counts over time is measured and modeled to return a t = 0 intercept, and from this a concentration is calculated, minimizing the impact of element migration.

Nanoscale Analysis
Thinned (<100 nm) foils for TEM investigation were prepared from polished blocks using a FEI-Helios nanoLab dual-focused ion beam and scanning electron microscope (FIB-SEM), as outlined by Ciobanu et al. [18]. Each TEM foil was attached to a copper grid.
Foils were analyzed 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 is equipped with a X-FEG Schottky source and Super-X EDS geometry. The Super-X EDS detector provides 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 the 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) and publicly available data from the American Mineralogist Crystal Structure Database (http://rruff. geo.arizona.edu/AMS/amcsd.php, accessed between April and August, 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.

Specimen Petrography
Bi 8 Te 3 occurs within disseminations of bismuth minerals (dominantly joséite-B and native bismuth, with subordinate hedleyite and traces of joséite-A) in a hedenbergite skarn (Figures 1 and 2). Micron-to nano-scale compositional and structural characterization of joséite-A and -B is described by Cook et al. [13]. In reflected light, Bi 8 Te 3 displays high reflectance, anisotropy, and is indistinguishable from hedleyite in air. It is, however, brighter than joséite-B and has a distinct grey color relative to native bismuth ( Figure 1B). Unnamed Bi 8 Te 3 is also identified as distinct within much smaller patches of bismuth minerals within the same polished block using highmagnification backscatter electron (BSE) imaging ( Figure 2) and subsequent nanoscale analysis (see below). The smaller bismuth mineral patches occur along micron-scale trails following brecciation of gangue minerals ( Figure 2A) and are often associated with lamellae of molybdenite ( Figure 2B). In detail, each of the multi-component patches display complex relationships among coexisting phases, with narrow slivers of native bismuth along mutual contacts between joséite-B and Bi 8 Te 3 ( Figure 2C). FIB-SEM cross-sectioning during extraction of a S/TEM sample from one of these patches shows one of the sub-µmsized bismuth inclusions embedded within Bi 8 Te 3 , close to the boundary with joséite-B ( Figure 2D-F).
The Bi 8 Te 3 phase was also analyzed in a mounted chip of material from the same locality and previously documented by Ciobanu et al. [2] (Supplementary Materials, Figure S1). In this case, the Bi 8 Te 3 occurs as lamellae of a few µm in width within joséite-B.

Chemical Composition
Electron probe microanalytical data for Bi 8 Te 3 and associated hedleyite are provided in Table 1. The data show that the two phases have distinct compositions. Both phases contain minor S and Sb but no detectable Se.

Nanoscale Characterization
Nanoscale characterization of the Bi 8 Te 3 phase was carried out on three S/TEM foils: foil #1 ( Figure 3) and foils #2 and #3 (Supplementary Materials, Figure S2). These were obtained from the specimens shown in Figure 2 and Supplementary Materials, Figure S1. Attempts to analyze the stacking sequences within Bi 8 Te 3 and hedleyite from the larger Bi-mineral patch in Figure 1B were unsuccessful due to the [0001] orientation of the two phases, prohibiting access to zone axes of interest for the stacking sequences within either phase (tilting > 30 • , beyond the capability of the double-tilt holder of the instrument).
However, bright field (BF) TEM imaging for the stacking sequence and ED pattern typical of hedleyite from Hedley was shown in [2].   Structural data for native bismuth from [19]: space group R3m, a = 4.533 Å, b = 11.797 Å.
The Bi 8 Te 3 phase is well-exposed in the middle part of foil #1 (Figure 3), which was obtained from the location shown in Figure 2D. This foil also exposes grain contacts with joséite-B and native bismuth. Bi 8 Te 3 exhibits a funnel-shaped morphology in the crosssection ( Figure 3A) and displays ordered stacking sequences across a distance of >10 µm, as confirmed by imaging of the upper part of the foil. Sets of defects are noted in Bi 8 Te 3 on both sides of the native bismuth ( Figure 3B). Changes in the layer orientation across the boundary contacts to joséite-B ( Figure 3C,D) indicate non-epitaxial growth. Joséite-B displays strong stacking disorder on one side of the foil ( Figure 3C) and regular 7-atom layer sequences on the other ( Figure 3D). The simple dumbbell motif of native bismuth is shown on the [2110] zone axis ( Figure 3E) for the purpose of comparison with Bi 8 Te 3 (see below).

Stacking Sequences and Crystal Structure of the Bi 8 Te 3 Phase
Thin bands of Bi 8 Te 3 are also exposed at the top of each of the other two foils studied here (Supplementary Materials, Figure S2). These show regular stacking sequences with a repeat of~2.1 nm ( Figure 4). In detail, each repeat shows brighter and darker slabs corresponding to 5-and 6-atom arrays, respectively, for each interval ( Figure 4A). These correspond to (i) the chalcogen-bearing, Te-Bi-Te-Bi-Te five-atom array (hereafter called mod5, following [2]), and (ii) three Bi-Bi dumbbell pairs (hereafter called 3 × mod2), together forming the 11-atom layer. A profile across the length of this sequence ( Figure 4B) shows a 'harmonic' variation in the HAADF signal which can be broadly associated with the intensity variation from low (mod5) to high (3 × mod2). However, bright field (BF) TEM imaging for the stacking sequence and ED pattern typical of hedleyite from Hedley was shown in [2].
The Bi8Te3 phase is well-exposed in the middle part of foil #1 (Figure 3), which was obtained from the location shown in Figure 2D. This foil also exposes grain contacts with joséite-B and native bismuth. Bi8Te3 exhibits a funnel-shaped morphology in the crosssection ( Figure 3A) and displays ordered stacking sequences across a distance of >10 µm, as confirmed by imaging of the upper part of the foil. Sets of defects are noted in Bi8Te3 on both sides of the native bismuth ( Figure 3B). Changes in the layer orientation across the boundary contacts to joséite-B ( Figure 3C,D) indicate non-epitaxial growth. Joséite-B displays strong stacking disorder on one side of the foil ( Figure 3C) and regular 7-atom layer sequences on the other ( Figure 3D). The simple dumbbell motif of native bismuth is shown on the [21 1 0] zone axis ( Figure 3E) for the purpose of comparison with Bi8Te3 (see below).

Stacking Sequences and Crystal Structure of the Bi8Te3 Phase
Thin bands of Bi8Te3 are also exposed at the top of each of the other two foils studied here (Supplementary Materials, Figure S2). These show regular stacking sequences with a repeat of ~2.1 nm (Figure 4). In detail, each repeat shows brighter and darker slabs corresponding to 5-and 6-atom arrays, respectively, for each interval ( Figure 4A). These correspond to (i) the chalcogen-bearing, Te-Bi-Te-Bi-Te five-atom array (hereafter called mod5, following [2]), and (ii) three Bi-Bi dumbbell pairs (hereafter called 3 × mod2), together forming the 11-atom layer. A profile across the length of this sequence ( Figure 4B) shows a 'harmonic' variation in the HAADF signal which can be broadly associated with the intensity variation from low (mod5) to high (3 × mod2).  The~2.1 nm distance represents a good approximation of the 11-atom layer width (d 11 ). Considering the 3R symmetry, space group R3m of this phase [2], and using measured d 11 to calculate c (=3 × d 11 = 63 Å), we have built a crystal structure for the Bi 8 Te 3 phase using Crystal Maker software ( Figure 5). The a parameter (~4.4 Å) is relatively constant across all members of the series. The asymmetric unit cell comprises six unique atom positions, two for Te, and four for Bi, with distribution as shown in Figure 5A. The crystal structure of Bi 8 Te 3 is shown on two projections ( Figure 5B,C), both of which are relevant for the definition of stacking sequences for phases in the tetradymite group. This model is in good agreement with incremental layer expansion within the group via addition of n × mod2 (Bi-Bi pairs) to the mod5 slab common to all structures [1,2,13]. The ~2.1 nm distance represents a good approximation of the 11-atom layer width (d11). Considering the 3R symmetry, space group R3 m of this phase [2], and using measured d11 to calculate c (= 3 × d11 = 63 Å), we have built a crystal structure for the Bi8Te3 phase using Crystal Maker software ( Figure 5). The a parameter (~4.4 Å) is relatively constant across all members of the series. The asymmetric unit cell comprises six unique atom positions, two for Te, and four for Bi, with distribution as shown in Figure 5A. The crystal structure of Bi8Te3 is shown on two projections ( Figure 5B,C), both of which are relevant for the definition of stacking sequences for phases in the tetradymite group. This model is in good agreement with incremental layer expansion within the group via addition of n × mod2 (Bi-Bi pairs) to the mod5 slab common to all structures [1,2,13]. Assessment of the Bi8Te3 crystal structure was carried out using high-resolution HAADF STEM imaging and simulation tilting the specimen on the [21 1 0] zone axis (Figure 6A). Simulations were performed using the crystallographic information file (.cif) obtained for the model presented here (Supplementary Materials, .cif file). There is an excellent match between the STEM simulation and the image (compare upper and lower parts of Figure 6A) obtained from the upper part of foil #1. Assessment of the Bi 8 Te 3 crystal structure was carried out using high-resolution HAADF STEM imaging and simulation tilting the specimen on the [2110] zone axis ( Figure 6A). Simulations were performed using the crystallographic information file (.cif) obtained for the model presented here (Supplementary Materials, .cif file). There is an excellent match between the STEM simulation and the image (compare upper and lower parts of Figure 6A) obtained from the upper part of foil #1. shown underneath the SAED are drawn schematically to reflect intensity variation. The 11-atom layer has 11 equal intervals with d11* = qF* (qF = modulation vector). These reflections display modulation in agreement with a displacement vector (qF*) typical for the mixed-layer compounds in the tetradymite series and related series (see [2,6]). This is shown by the intensity variation matching the calculated values for sum intensities for this phase given in [2], i.e., intensity reflections from 0000 to 000.15 along 1/2d* are: 1.0506, 0.9063, 0.5956, 0.5059, 0.7450, and 1.0208 (see Table 3 in Ciobanu et al. [2]). Additional explanation is provided in the text.
The layer sequence in mixed-layer compounds with interface modulated structures can be calculated from electron diffractions using the correlation between the displacement vector (qF*) and the rhombohedral sub-cell defined by the d* interval (d = 1/d* = ~2 Å) along the c* axis in the tetradymite homologous series ( [2,6]; Figure 6B). The qF* parameter corresponds to the distance between two brighter satellites in the center of d*, and the layer stack is defined by the number of divisions (i) within this interval. The smallest distance between any two reflections (dN*) 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) qF* = i x dN* = (i × d*)/N, leading to: (2) dN = qF/i and N = (i × qF)/d The selected area electron diffraction (SAED) shows a single-layer stack (i = 1) with 11 divisions across d* ( Figure 6B). In this case, dN* = qF* and dN = d × N = ~2.1 nm for the 11-atom layer (d = ~1.9 Å and N = 11). Therefore, the results calculated from measured values for d11 have a good fit with one another. The measured value of da (3.8 Å; see Figure  6B) was used to calculate a by the function a = da/cos30ᵒ, where a is 4.4 Å.
Indexing of the 11-atom layer supercell is marked along the d* interval (drawing at the bottom of Figure 6B). The modulation with respect to intensity of reflections along d* is concordant with the variation of sum of intensities for (N-i)/2 reflections calculated by shown underneath the SAED are drawn schematically to reflect intensity variation. The 11-atom layer has 11 equal intervals with d 11 * = q F * (q F = modulation vector). These reflections display modulation in agreement with a displacement vector (q F *) typical for the mixed-layer compounds in the tetradymite series and related series (see [2,6]). This is shown by the intensity variation matching the calculated values for sum intensities for this phase given in [2], i.e., intensity reflections from 0000 to 000.15 along 1/2d* are: 1.0506, 0.9063, 0.5956, 0.5059, 0.7450, and 1.0208 (see Table 3 in Ciobanu et al. [2]). Additional explanation is provided in the text.
The layer sequence in mixed-layer compounds with interface modulated structures can be calculated from electron diffractions using the correlation between the displacement vector (q F *) and the rhombohedral sub-cell defined by the d* interval (d = 1/d* =~2 Å) along the c* axis in the tetradymite homologous series ([2,6]; Figure 6B). The q F * parameter corresponds to the distance between two brighter satellites in the center of d*, and the layer stack is defined 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 selected area electron diffraction (SAED) shows a single-layer stack (i = 1) with 11 divisions across d* ( Figure 6B). In this case, d N * = q F * and d N = d × N =~2.1 nm for the 11-atom layer (d =~1.9 Å and N = 11). Therefore, the results calculated from measured values for d 11 have a good fit with one another. The measured value of d a (3.8 Å; see Figure 6B) was used to calculate a by the function a = d a /cos30·, where a is 4.4 Å.
Indexing of the 11-atom layer supercell is marked along the d* interval (drawing at the bottom of Figure 6B). The modulation with respect to intensity of reflections along d* is concordant with the variation of 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 of van Landuyt et al. [4].

Atom Identity within the 11-Atom Layer
A clearer separation between the mod5 and mod2 slabs within the 11-atom layer is evident from high-resolution HAADF STEM images ( Figure 7A). In this structure, the chalcogen-bearing mod5 slab is smaller than the 3 × mod2 (6-atom), Bi-only slab, and the two are well-separated as darker and brighter strips on the image. Ciobanu et al. (Table 3 and Figure 9h in [2]) using the fractional shift method of van Landuyt et al. [4].

Atom Identity within the 11-Atom Layer
A clearer separation between the mod5 and mod2 slabs within the 11-atom layer is evident from high-resolution HAADF STEM images ( Figure 7A). In this structure, the chalcogen-bearing mod5 slab is smaller than the 3 × mod2 (6-atom), Bi-only slab, and the two are well-separated as darker and brighter strips on the image.  Figure 7B,C). This sequence is replicated by variation in size and intensity along a profile encompassing the mod5 slab at the center of two 3 × mod2 (6-atom) slabs ( Figure 7D,E).
The relative variations in the HAADF signal interpreted as Te and Bi atoms along the profile in Figure 7E (lower and higher intensity, respectively) show a very good match  Figure 7B,C). This sequence is replicated by variation in size and intensity along a profile encompassing the mod5 slab at the center of two 3 × mod2 (6-atom) slabs ( Figure 7D,E).
The relative variations in the HAADF signal interpreted as Te and Bi atoms along the profile in Figure 7E (lower and higher intensity, respectively) show a very good match with the structure and STEM simulation ( Figure 7B,C). This interpretation is also confirmed by high-resolution EDS mapping across three 11-layer repeats (Figure 8). These show that the distribution of Te and Bi reproduces the 5-and 6-atom slabs in Bi 8 Te 3 .
Minerals 2021, 11, x FOR PEER REVIEW 12 of 19 with the structure and STEM simulation ( Figure 7B,C). This interpretation is also confirmed by high-resolution EDS mapping across three 11-layer repeats ( Figure 8). These show that the distribution of Te and Bi reproduces the 5-and 6-atom slabs in Bi8Te3.

Stacking Disorder among Bi-rich Layers
Disordered stacking sequences observed in the sulphotelluride enclosing the Bi8Te3 phase were studied in closer detail from area 7 in foil #1 (sample H163b; Figure 3A,C).

Stacking Disorder among Bi-Rich Layers
Disordered stacking sequences observed in the sulphotelluride enclosing the Bi 8 Te 3 phase were studied in closer detail from area 7 in foil #1 (sample H163b; Figure 3A,C).
High-resolution images of the disordered sequences ( Figure 10) show dark lines clearly separating individual layer units with central arrays of single or double Bi-Bi dumbbells ( Figure 10A). Such dark lines are attributable to sulfur occurring in the middle part of joséite-B (7-atom layer [13]). The sequence can be identified using d 7 and d 9 layer widths of~13.5 and~17 Å. The 9-atom layer is irregularly distributed, and this stacking disorder can be recognized on Fast Fourier Transform (FFT) patterns obtained from the images ( Figure 10B). As shown above, single 11-atom units also occur alongside the 9-and 7-atom units ( Figure 10C). The single-, double-, and triple-arrays of Bi-atom pairs are separated by the chalcogen-bearing mod5 slabs marked as overlays in Figure 10C. The different layers are distinguished in the figure by their asymmetric rather than centered arrangements of the double-Bi (n × mod2) arrays relative to the mod5 slab. In stacking sequences involving different chalcogens (e.g., S and Te in joséite-B), layer units can be readily recognized using the centered approach even if the double-Bi rows are slightly distorted and more difficult to count. Unit widths are, however, effectively identical irrespective of which method is considered, as shown in Figure 10D.  Figure 3A) comprising dominantly 7-and 9-atom layers. This sequence has an average composition of Bi4.67X3, with the composition of individual frames (1-4) calculated from the layer sequence observed in each. One 11-layer unit is shown in frame 4. Abbreviation: L-layer.
High-resolution images of the disordered sequences ( Figure 10) show dark lines clearly separating individual layer units with central arrays of single or double Bi-Bi dumbbells ( Figure 10A). Such dark lines are attributable to sulfur occurring in the middle part of joséite-B (7-atom layer [13]). The sequence can be identified using d7 and d9 layer widths of ~13.5 and ~17 Å. The 9-atom layer is irregularly distributed, and this stacking disorder can be recognized on Fast Fourier Transform (FFT) patterns obtained from the images ( Figure 10B). As shown above, single 11-atom units also occur alongside the 9-and 7-atom units ( Figure 10C). The single-, double-, and triple-arrays of Bi-atom pairs are separated by the chalcogen-bearing mod5 slabs marked as overlays in Figure 10C. The differ-  Figure 3A) comprising dominantly 7-and 9-atom layers. This sequence has an average composition of Bi 4 . 67 X 3 , with the composition of individual frames (1-4) calculated from the layer sequence observed in each. One 11-layer unit is shown in frame 4. Abbreviation: L-layer.
ily recognized using the centered approach even if the double-Bi rows are slightly distorted and more difficult to count. Unit widths are, however, effectively identical irrespective of which method is considered, as shown in Figure 10D.

The Bi-Rich End of the Tetradymite Series
This study is focused on one of the closest natural species known towards the Bi-rich side of the tetradymite homologous series. It thus complements existing compositional and structural data obtained from electron diffractions provided previously for the Bi 8 Te 3 phase from the same locality [2], but adds HR HAADF STEM imaging and a crystal structure model. Other studies have addressed a generalized structural model for synthetic compounds analogous to the tetradymite series but without extending the model to the Bi-rich side of the system. Among these, the studies of Frangis et al. [6] and Lind and Lidin [9] are relevant for the discussion here. Based on HR TEM studies of compounds of 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 structures using the fractional shift method of van Landuyt et al. [4]. Lind and Lidin [9] later introduced a general structural model for Bi-Se phases using a super-space formalism based on X-ray diffraction study 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 ), but thus not including the Bi-rich side of the system.
The structural model built for the Bi 8 Te 3 phase (space group R3m; a = 4.4 Å, c = 63 Å; Figure 3; Supplemental Materials, .cif file) was assessed by measurements of cell parameters from electron diffractions and confirmed by direct atomic-scale imaging and STEM simulations (Figures 6 and 7). STEM EDS mapping is in agreement with the atom speciation considered for this structure (Figure 8). The Bi 8 X 3 phase represents the k = 4 structure within the series described by the general formula Bi 2k X 3 , where k is an integer value ≥ 1 [2].
The data presented here support the homology model based on fractional shift theory proposed for phases in the tetradymite series (whereby the number of layers within each unit constrains modulation along the d interval, representing the c subcell common to all phases) [2]. An alternative model for homology in the tetradymite series is provided as nBi 2 ·mBi 2 X 3 by Shelimova et al. [8]. This correlates with the Bi 2k X 3 modules of [2] by the relationship: n/m = k−1. Ciobanu et al. [2] draw attention to the fact that although intuitive in terms of imaging, the generalized formula nBi 2 ·mBi 2 X 3 does not account for the q F modulation underpinning homology within the series. For example, native bismuth ( Figure 3E), which displays identical imagery in terms of the Bi-only mod2 slabs of tetradymite species, would be part of the tetradymite series with k→∞ if we consider the model of Shelimova et al. [8] rather than the homology proposed by [2].
Whichever model is best-suited to describe the tetradymite series, recognition of layered compounds within a homologous series allows new structures to be accurately predicted from compositional data and the specific characteristics of electron diffractions, an intrinsic feature of mixed-layer compounds [21]. The data presented here further emphasize that Z-contrast imaging techniques such as HAADF STEM are optimally suited for characterization of mixed-layer compounds [13,20,[22][23][24][25].

Relationships between Bi 8 Te 3 , Hedleyite, and Other Species with Higher Bi/X > 1 Ratios
Remarkably, the Bi 8 Te 3 phase is very well-ordered over a distance of >10 µm, confirming the compositional data presented here and in [2] for the Hedley material. Like all other phases in the series representing single-layer stacks, this should be more stable than those species formed by combinations of two types of modules, i.e., S (Bi 2k X 3 )·L (Bi 2(k+1 )X 3 , where k ≥ 1, X = chalcogen, and S and L are the number of short and long modules, respectively. Paradoxically, one such phase, hedleyite, with a 9.11 stacking sequence (k = 3, S = 9, and L = 11), has been the most commonly reported phase at the Bi-rich end of the series. As initially defined [26] from the type locality (Good Hope claim, Hedley, B.C., Canada), hedleyite has the chemical formula Bi 7 Te 3 . Subsequent work questioned the validity of this formula, suggesting that Bi 2+x Te 1−x (x = 0.13-0.19) represents a more appropriate formula [27]. Structural data for hedleyite [26] indicate unit cell dimensions of a = 4.4733 (20) Å and c = 17.805(11) Å, Z = 3.
The literature contains several prior references to unnamed Bi 8 Te 3 (e.g., [14,15,28]), or to microprobe data which were ascribed to hedleyite, yet where Bi is clearly in excess of stoichiometry (e.g., [29,30]). Some authors have chosen to assume that compositions closer to Bi 8 Te 3 than Bi 7 Te 3 were hedleyite (e.g., [28]), or referred specifically to 'bismuthian hedleyite' (see review in [1]). The Bi 8 Te 3 phase was, however, clearly shown to be a distinct phase, different from and coexisting with hedleyite, in the example presented by Cabral and Corrêa-Neto [15]. Likewise, we show here the co-existence of the two species in the same area ( Figure 1B), defined by their distinct compositions ( Table 1).
The scarcity of both phases could be related to a decreasing probability of maintaining a regular stacking sequence during growth with a larger number of layers involved. However, the relative scarcity of Bi 8 Te 3 relative to hedleyite or other associated phases typical of high-grade gold ores (e.g., [31]) is the chance of preservation during deposit evolution or over a protracted geological history. Interaction with late, S-bearing fluids can lead to replacement of Bi 8 Te 3 by hedleyite + joséite-B, the most common association in Au skarns such as Hedley [32]. The lack of epitaxial relationships and the change in layer orientation across the boundary between Bi 8 Te 3 and disordered joséite-B/Bi 4 . 6 X 3 ( Figure 3A,C) is evidence that these phases did not form at the same time. One example of relict Bi 8 Te 3 would be the lamellae preserved within joséite-B (Supplementary Materials, Figure S2).
In contrast, 11-atom layer units, epitaxial with 7-and 9-atom layers within the disordered sequences of slabs with an average composition of~Bi 4 . 67 X 3 (Figures 9 and 10), are more likely part of a newly formed assemblage. Such sequences may show some degree of stack ordering if analyzed over larger intervals, as for example the phases Bi 4 . 5 X 3 and Bi 5 X 3 representing both distinct polysome slabs and a combination thereof (~Bi 4 . 63 X 3 ) [2]. Note that some of the disordered stacks presented here (Figure 9) have the same compositions as (quasi)ordered sequences shown in [2], e.g., the [777.9] layer stack for Bi 4 . 5 X 3 , or 7.9 for Bi 5 X 3 .
It is likely that stacking sequences involving layer units of different size will be more disordered than those composed of a single module type (e.g., 7-, 9-, or 11-atoms), thus explaining the deviation from ideal stoichiometry in some Bi-rich compounds such as hedleyite (see above). Nonetheless, stacking disorder induced to accommodate compositional variation during cycles of growth is likely to be far more common in nature. The data here draw attention to the fact that tetradymite series specimens with compositions in the range between those of single-layer structures require assessment by S/TEM or X-ray diffraction methods before they can be considered as distinct phases. On the other hand, the mutual relationships between layers across and within a stacking sequence can be suggestive of primary versus secondary origin, if analyzed at the nanoscale.

Prospects for Other Phases in the Tetradymite Homologous Series
Based on observed crystal structures and theoretical arguments, Ciobanu et al. [2] predicted an extended family of single-module phases with incremental k increase with compositions from Bi 2 Te 3 (k = 1, 5-atom layer) to Bi 14 Te 3 (k = 7, 17-atom layer), each with distinct structures defined by different c parameters.
Whether phases that share a structure with Bi 8 Te 3 , but with compositions including S and/or Se (i.e., Bi 8 S 3 , Bi 8 Se 3 , Bi 8 Te 2 S, Bi 8 TeS 2 , Bi 8 Se 2 S, Bi 8 SeS 2 , Bi 8 Se 2 Te, and Bi 8 Te 2 Se), also exist in nature is unknown at present. We note that no reports of unnamed phases of S-or Se-bearing analogues of Bi 8 Te 3 (or indeed, Bi 7 Te 3 ) have been published, with the single exception of unnamed Bi 2 . 243 S 0 . 742 Se 0 . 113 (=Bi 7 . 87 (S 2 . 6 Se 0 . 4 ) 3 ) mentioned by Fuksová et al. [37]. The present study shows that the 11-atom layer structure may accommodate chalcogens other than Te, e.g., S, as suggested by the HR-STEM images ( Figure 10C).

Conclusions and Implications
Bi 8 Te 3 is a new member of the tetradymite homologous series and is compositionally and structurally distinct from hedleyite (Bi 7 Te 3 ). HAADF STEM imaging showed that Bi 8 Te 3 has an 11-atom layer structure, in which three Bi-Bi pairs (3 × mod2) are placed adjacent to a 5-atom sequence (mod5, Te-Bi-Te-Bi-Te). This is an 11-fold superstructure of a rhombohedral sub-cell with d~1.9 Å, and the trigonal symmetry (space group R3m) for the unit cell, a is~4.4 Å and c is~63 Å (=3 × 11 × d, or 3 × d 11 ), as calculated from d* a , d*, and d* 11 , measured from electron diffraction patterns. STEM simulations based on the crystal structure model matched the images and showed the distribution of the 11 atoms along <0.1.1.11> directions. Intensity profiles and STEM EDS mapping showed a very good match with assumed atom speciation within the structure.
Lattice-scale intergrowths are documented as epitaxial growth of single 11-atom layer units within a strongly disordered sequence of 7-and 9-atom layer units of average composition, Bi 4 . 67 X 3 . Disordered sequences such as this, replacing Bi 8 Te 3 , likely account for the rarity of this phase in nature and show how compositional non-stoichiometry, although not represented by a discrete phase, is nevertheless interpretable in terms of layer stacks.
Results support predictions of crystal structures from theoretical modeling of the series and indicate that multiple phases likely exist but are yet to be discovered and named. Each has a discrete stoichiometric composition and unique crystal structure. These types of modular structures can be predicted from their basic principles as mixed-layer compounds derived as one-dimensional superstructures of a basic rhombohedral sub-cell. Although their stabilities are unknown, there is likely a continuous range of compositions and compounds extending towards native bismuth.