Bruno Lanson
email: bruno.lanson@obs.ujf-grenoble.fr

Victor A Drits

Anne-Claire Gaillot

Ewen Silvester

Alain Plançon

Alain Manceau

Structure of heavy metal sorbed birnessite. Part 1: Results from X-ray diffraction

The structure of heavy-metal sorbed synthetic birnessites (MeBi) was studied by powder X-ray diffraction (XRD) using a trial-and-error fitting procedure to improve our understanding of the interactions between buserite/birnessite and environmentally important heavy metals (Me) including Pb, Cd, and Zn. MeBi samples were prepared at different surface coverages by equilibrating at pH 4 a Na-rich buserite (NaBu) suspension in the presence of the desired aqueous metal. Two main types of experimental XRD patterns were obtained as a function of the nature of Me cations sorbed from solution which exerts a strong control on layer stacking sequence, as well as on location and coordination of Me: 1) CdBi and PbBi samples correspond to a one-layer hexagonal (1H) structure, AbC b'A' C'b' AbC…, and 2) ZnBi exhibits a one-layer monoclinic (1M) structure in which adjacent layers are shifted by +a/3, AbC b'A'c' BcA c'B'a' CaB a'C'b' AbC.

Simulated XRD patterns shows that octahedral layers contain 0.833 Mn cations (Mn 4+

and Mn 3+ ) and 0.167 vacant octahedra; Mn 3+ interlayer and adsorbed Me interlayer compensate for the layer charge deficit. Mn 3+ interlayer is octahedrally coordinated in all samples and is located above or below vacant layer octahedra sharing three O layer with neighboring Mn layer octahedra to form a triple-corner surface complex ( VI TC sites). In ZnBi and CdBi samples, Me interlayer is also located in TC sites; all Cd is octahedrally coordinated whereas about 30% of Zn is tetrahedrally coordinated ( IV TC sites). In PbBi samples, all Pb is octahedrally coordinated, most of these cations (~75%) being located in TC sites. Additional Pb is located above or below empty tridentate cavities, sharing three edges with neighboring Mn layer octahedra ( VI TE sites).

Structural formulae calculated for each sample show that during the NaBu-to-MeBi structural transformation, interlayer Na + and Mn 2+ are replaced by Me and H + adsorbed from solution, whereas Mn 3+ interlayer resulting from the equilibration of NaBu at low pH is less affected. Sorption of divalent Me above and below vacant layer sites provides optimal conditions for local charge compensation in MeBi.

INTRODUCTION

Buserite and birnessite are the most common layered hydrous Mn oxides, or phyllomanganates, in natural environments. Birnessite is the main Mn-bearing phase in soils [START_REF] Taylor | The mineralogy and chemistry of manganese in some Australian soils[END_REF][START_REF] Chukhrov | Iron and manganese oxide minerals in soils[END_REF], and in marine nodules, coatings, and crusts [START_REF] Burns | Mineralogy of ferromanganese nodules[END_REF][START_REF] Glover | Characterization of a marine birnessite[END_REF][START_REF] Chukhrov | Birnessite characterization[END_REF][START_REF] Chukhrov | Crystal structure of birnessite from the Pacific ocean[END_REF][START_REF] Chukhrov | Supergenic manganese hydrous oxides[END_REF][START_REF] Drits | Manganese minerals of Fe-Mn nodules from the sediments of the central part of Pacific Ocean and their post-sedimentation transformation[END_REF]. Birnessite is also an essential component of some Mn-rich ore deposits [START_REF] Chukhrov | New data about mineralogy of Kertch ore deposits[END_REF][START_REF] Chukhrov | Supergenic manganese hydrous oxides[END_REF][START_REF] Post | Manganese oxide minerals: Crystal structures and economic and environmental significance[END_REF].

The structures of these minerals consist of layers of edge-sharing Mn-(O,OH) octahedra.

Buserite has a 10 Å periodicity along the c* axis with exchangeable cations and two layers of H 2 O molecules in its interlayer region to compensate for the negative layer charge which arises mostly from the coexistence of heterovalent Mn cations in the octahedral layers [START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF]. Partial dehydration of buserite leads to the formation of 7 Å birnessite with a single layer of H 2 O molecules and various interlayer cations (Burns andBurns 1977, 1978;[START_REF] Chukhrov | Birnessite characterization[END_REF][START_REF] Chukhrov | Supergenic manganese hydrous oxides[END_REF][START_REF] Cornell | Transformation of hausmannite into birnessite in alkaline media[END_REF]. Buserite and birnessite have remarkable cation exchange [START_REF] Healy | The effect of crystal structure on the surface properties of a series of manganese dioxides[END_REF][START_REF] Murray | The surface chemistry of hydrous manganese dioxide[END_REF][START_REF] Balistrieri | The surface chemistry of δMnO 2 in major ion seawater[END_REF][START_REF] Stumm | Chemistry of the solid-water interface and particle-water interface in natural systems[END_REF][START_REF] Goff | Synthesis, ion exchange and electrochemical properties of lamellar phyllomanganates of the birnessite group[END_REF], sorption [START_REF] Gray | Adsorption from aqueous solution by δ-manganese dioxide. II. Adsorption of some heavy metal cations[END_REF][START_REF] Catts | Adsorption of Cu, Pb and Zn by δMnO2: Applicability of the site binding-surface complexation model[END_REF][START_REF] Paterson | The influence of structure on Ba and K uptake by a synthetic phyllomangante[END_REF][START_REF] Tu | Transformations of synthetic birnessite as affected by pH and manganese concentration[END_REF][START_REF] Appelo | A consistent model for surface complexation on birnessite (δ-MnO2) and its application to a column experiment[END_REF], and redox properties [START_REF] Oscarson | Kinetics of oxidation of arsenite by various manganese dioxides[END_REF][START_REF] Stone | Reductive dissolution of manganese (III) and manganese (IV) oxides by organics[END_REF][START_REF] Stone | Kinetics and reaction stoichiometry in the reductive dissolution of manganese(IV) dioxide and cobalt(III) oxide by hydroquinone[END_REF][START_REF] Manceau | X-ray absorption spectroscopic study of the sorption of Cr(III) at the oxide-water interface[END_REF][START_REF] Bidoglio | X-ray absorption spectroscopy investigation of surface redox transformations of thallium and chromium on colloidal mineral oxides[END_REF][START_REF] Silvester | The mechanism of chromium(III) oxidation by Na-buserite[END_REF][START_REF] Manceau | Structural mechanism of Co 2+ oxidation by the phyllomanganate buserite[END_REF][START_REF] Pizzigallo | Oxidation of chloroanilines at metal oxide surfaces[END_REF][START_REF] Daus | Kinetics of the arsenite oxidation in seepage water from a tin mill tailings pond[END_REF][START_REF] Nico | Importance of Mn(III) availability on the rate of Cr(III) oxidation on δ-MnO 2[END_REF][START_REF] Chorover | Reaction of forest floor organic matter at goethite, birnessite and smectite surfaces[END_REF].

The ability of buserite and birnessite to fix trace elements (Co, Pb, Cd, Cu, Zn, …) ranks these minerals among the most efficient sorbents of heavy metals [START_REF] Mckenzie | The sorption of cobalt by manganese minerals in soils[END_REF](McKenzie , 1980;;[START_REF] Burns | Mineralogy of ferromanganese nodules[END_REF][START_REF] Bendell-Young | The relative importance of manganese and iron oxides and organic matter in the sorption of trace metals by surficial lake sediments[END_REF][START_REF] Post | Manganese oxide minerals: Crystal structures and economic and environmental significance[END_REF][START_REF] Manceau | Chemical forms of trace metals in soils by XAFS spectroscopy. I. Quantitative Zn speciation in smeltercontaminated soils[END_REF].

As a consequence, despite its low natural concentration, birnessite exerts an essential role in the partitioning of heavy metals (Me) in soils and in both marine and continental sediments [START_REF] Taylor | The association of trace elements with maganese minerals in Australian soils[END_REF][START_REF] Mckenzie | The sorption of cobalt by manganese minerals in soils[END_REF]McKenzie , 1980;;Manceau et al. 2000 and references therein). Because serious environmental problems may result from an excessive concentration of heavy metals in soils and aquatic systems it is essential to have a better understanding, at the atomic level, of the interactions between different heavy metals and phyllomanganates.

However, in nature, birnessite is typically disordered and occurs in a finely dispersed state, frequently mixed with other phases such as phyllosilicates and iron oxy-hydroxides. As a result, crystal chemical studies of heavy-metal bearing birnessites and buserites are very limited, and both the location of heavy metals in the phyllomanganate structure and the mechanism of their sorption remain poorly understood.

Monomineralic buserite and birnessite with a high degree of structural order are easily synthesized under various laboratory conditions (Giovanoli et al. 1970a(Giovanoli et al. , 1970b;;[START_REF] Strobel | Structural chemistry of phyllomanganates: Experimental evidence and structural models[END_REF]; [START_REF] Cornell | Transformation of hausmannite into birnessite in alkaline media[END_REF][START_REF] Ching | Sol-Gel synthesis of birnessite from KMnO 4 and simple sugars[END_REF][START_REF] Chen | The hydrothermal synthesis of sodium manganese oxide and a lithium vanadium oxide[END_REF][START_REF] Luo | Formation and transformation of mesoporous and layeredmanganese oxides in the presence of long-chain ammonium hydroxides[END_REF][START_REF] Kim | Preparation of layered MnO 2 via thermal decomposition of KMnO 4 and its electrochemical characterizations[END_REF]. Because these varieties may be considered as analogs of natural buserites/birnessites [START_REF] Manceau | Chemical forms of trace metals in soils by XAFS spectroscopy. I. Quantitative Zn speciation in smeltercontaminated soils[END_REF] the use of these synthetic varieties is preferable to determine the structural characteristics of heavy metal bearing birnessites as well as the reaction mechanisms of their adsorption. The present study aims at a better understanding of the interaction between synthetic buserite/birnessite structures and environmentally important Me such as Pb, Cd, and Zn. Special attention is paid to the structural modifications induced by the presence of sorbed Me cations in the birnessite structure.

Recent crystal chemical studies of synthetic buserite and birnessite by complementary diffraction and spectroscopic methods have provided important new insight on the structure of these compounds [START_REF] Chukhrov | Crystal structure of birnessite from the Pacific ocean[END_REF][START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF][START_REF] Manceau | X-ray absorption spectroscopic study of the sorption of Cr(III) at the oxide-water interface[END_REF][START_REF] Kuma | Crystal structures of synthetic 7 Å and 10 Å manganates substituted by mono-and divalent cations[END_REF][START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF]Drits et al. , 1998a;;[START_REF] Manceau | Structural mechanism of Co 2+ oxidation by the phyllomanganate buserite[END_REF][START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF][START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. This new information will serve as a basis for the present detailed crystal chemical studies of Me-sorbed birnessite (MeBi). Application of complementary techniques such as quantitative powder X-ray diffraction (XRD), selected area electron diffraction (SAED), energy dispersive analysis (EDA), chemical analysis, and powder as well as polarized EXAFS spectroscopy was also critical in solving the different aspects of structural chemistry of heavy metals in birnessite structures.

EXPERIMENTAL Experimental methods

NaBu used for this study was synthesized following the protocol of Giovanoli et al. (1970a). The solid phase was washed by centrifugation until the supernatant pH was 9-10.

The sorption of various Me cations was achieved by adding a Me(NO 3 ) n solution (Zn, Pb, Cd, and Cu) while equilibrating a NaBu suspension at pH 4. All samples were prepared in an argon-gas-saturated and constant ionic strength (0.1 mol.dm -3 NaNO 3 ) aqueous medium at 25°C as described by [START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF]. As NaBu was not pre-equilibrated at pH 4, NaBu-to-HBi transformation at low pH and Me sorption occurred simultaneously [START_REF] Manceau | Structural mechanism of Co 2+ oxidation by the phyllomanganate buserite[END_REF]). Variation of the initial Me concentrations in solution allowed to obtain different Me/Mn ratios in the solids. After several hours of equilibration, all samples were filtered and dried, yielding MeBi solids. Total Me, Na, and Mn concentrations were determined by inductively coupled plasma spectroscopy (ICP Perkin Elmer Optima 3300DV) after dissolution of the dried solids in NH 3 OHCl. Me/Mn ratios determined from these analyses are used hereafter to characterize the different MeBi samples.

Powder XRD patterns were recorded using a Bruker D5000 powder diffractometer equipped with a Kevex Si(Li) solid-state detector and Cu Kα radiation. Intensities were measured at a 0.04°2θ interval using a 30-50 s counting time per step.

Simulation of XRD patterns

The structural disorder induced by random and well-defined stacking faults decreases the efficiency of conventional X-ray methods of structural analysis, including the Rietveld method. Alternatively, the actual structure of defective layered compounds can be modeled by using the mathematical formalism described in detail by [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF] to simulate experimental XRD patterns. This trial-and-error simulation procedure was successfully used to propose crystal structure models for several natural and synthetic birnessites [START_REF] Chukhrov | Crystal structure of birnessite from the Pacific ocean[END_REF][START_REF] Manceau | Structural mechanism of Co 2+ oxidation by the phyllomanganate buserite[END_REF]Drits et al. 1998a;[START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. Details on the program used to simulate XRD patterns and on the fitting procedure are given by Drits et al. (1998a). Specifically, the lognormal coherent scattering domain size distribution along the c axis was quantified by two parameters: the average and maximum number of layers (Drits et al. 1998b). The agreement factors between experimental and calculated patterns were determined over the 35-65°2θ Cu Kα range (2.55-1.45 Å) using the usual R WP parameter. Calculations of XRD patterns were limited to 20l and 11l reflections (indexing based on an orthogonal base-centered unit cell) because these lines are the most sensitive to structural parameters of layered minerals, such as order/disorder in their stacking sequences [START_REF] Brindley | Order-disorder in clay mineral structures[END_REF]. For each sample, the background was assumed to decrease linearly over the angular range considered, and preferred orientation of particles was considered as a variable parameter modeled using the function described by [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF].

RESULTS

Cation composition from ICP data

Table 1 contains the Me/Mn and Na/Mn atomic ratios determined for all Me-sorbed birnessite samples. The two Pb-containing birnessite samples (PbBi 6 and 58) differ by their 0.006 and 0.058 Pb/Mn ratios, and both samples contain a significant amount of Na, indicating that the Pb 2+ and H + for Na + exchange was incomplete. Two of the four Zn-bearing birnessite specimens (ZnBi 122 and 128) have similar Zn/Mn ratios (0.122 and 0.128, respectively). A third ZnBi sample (ZnBi 69), which was prepared using a slower pH decrease rate, has a Zn/Mn ratio half that of these two ZnBi samples, and a fourth sample has a very low Zn/Mn ratio (0.008 -Table 1). In contrast to PbBi samples, the low Na/Mn values determined for these four ZnBi samples indicate that almost all Na was replaced by Zn and protons. The Cd-sorbed birnessite sample (CdBi 106) has a 0.106 Cd/Mn ratio and a low amount of Na (Na/Mn = 0.006).

Features of the experimental XRD patterns

Experimental XRD patterns of MeBi (Fig. 1) can be separated into three groups as a function of positions and profiles of their diffraction maxima. In the first group (Fig. 1a), XRD patterns exhibit reflections which can be indexed with a one-layer hexagonal (1H) unit cell (CdBi 106, PbBi 6 and 58 -Tables 2,3). ZnBi samples 69, 122, and 128 make up the second group of samples (Fig. 1b). Their XRD patterns can be indexed with a one-layer monoclinic (1M) unit cell with |c.cosβ| = a/3 (Tables 2,3). Sample ZnBi 8 defines the third group as its XRD pattern contains very weak and extremely broad maxima, the positions of which cannot be indexed using 1H or 1M unit cells. The high degree of stacking disorder which is responsible for this XRD pattern (see below) prevented us from refining a structure model for this sample. Experimental XRD patterns of the three groups also differ by their reflection profiles, which are increasingly broadened from PbBi 58 to CdBi 106, and then to ZnBi, the broadest and most asymmetrical profiles being observed for ZnBi 8 (Fig. 1).

Unit-cell parameters for samples of the first two groups are given in Table 2,and experimental and calculated d(hkl) values are compared in Table 3. Indexing was performed using a base-centered unit cell with a = b 3 , as the layers have hexagonal symmetry in both 1H and 1M varieties. However, within each group, experimental XRD patterns exhibit some specific features. For example, experimental and calculated d(hkl) values almost coincide for PbBi 6 and 58, whereas the difference between d cal (20l, 11l) and d exp (20l, 11l) exceeds the experimental error (± 0.001 Å) for CdBi 106, and ZnBi 122 and 128 (Table 3). The origin of this discrepancy is considered below.

Initial structure and composition of MeBi layers.

To build up initial structure models for the various MeBi samples, the NaBu-to-MeBi transformation was assumed to follow the same reaction pathways as the NaBu-to-HBi transformation [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF][START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF][START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. In addition, the NaBu-to-MeBi transformation involves partial replacement of Mn interlayer by Pb 2+ , Cd 2+ , or Zn 2+ cations sorbed from solution. All of these cations are larger than Mn 4+ , and it is realistic to assume that they do not migrate into the vacant layer sites as they are not oxidized during the process [START_REF] Manceau | Structure of heavy metal sorbed birnessite: Part 3. Results from powder and polarized EXAFS spectroscopy[END_REF].

Consequently, it was initially assumed that MeBi layers contain 0.833 Mn cations and 0.167 vacant layer sites regardless of the metal cation, as HBi layers. The similar values of a and b parameters determined for all MeBi samples including HBi (Table 2) support this hypothesis of a constant layer cation composition. By analogy with HBi, Mn interlayer and Me were initially assumed to be octahedrally coordinated in MeBi and Mn interlayer was assumed to be located either above or below vacant layer sites, sharing three O layer with neighboring Mn layer octahedra to form a triple-corner surface complex ( VI TC sites; [START_REF] Spadini | Structure and stability of Cd 2+ surface complexes on ferric oxides[END_REF]. Two hypotheses were considered regarding the location of divalent Me with respect to these vacant layer sites. In the first model, Me was located either above or below vacant layer sites, whereas it was located above and below vacant layer sites in the second model. XRD patterns were calculated for these two starting models and compared to experimental ones.

For these two models, the respective amounts of sorbed Me (N_Me) and Mn interlayer (N_Mn interlayer ) can be obtained from the Me/Mn ratios (ρ) determined from the chemical analysis (Table 1) as ρ = N_Me/(0.833+N_Mn interlayer ). As a consequence: N_Me = nρ/(1+nρ) and N_Mn interlayer = 0.167 -N_Me/n, (1), ( 2)

with n = 1, 2 for the first and second models, respectively.

Calculated N_Me values for the two models are listed in

AbC b'A' B'a' CaB a'C' A'c' BcA c'B' C'b' AbC… (c•cosβ = -0.333 a) ( 6 ) (5)
The first stacking sequence is similar to that of chalcophanite [START_REF] Post | Chalcophanite, ZnMn 3 O 7 -3H 2 O: New crystal-structure determinations[END_REF] and of the accessory phase present in HBi pH 5 [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF], whereas the second layer sequence is similar to that of high pH NaBi [START_REF] Post | Crystal structure determinations of synthetic sodium, magnesium, and potassium birnessite using TEM and the Rietveld method[END_REF][START_REF] Manceau | X-ray absorption spectroscopic study of the sorption of Cr(III) at the oxide-water interface[END_REF][START_REF] Lanson | Crystal structure determination of synthetic Na-rich birnessite: Evidence for a triclinic one-layer cell[END_REF].

For each sample, XRD patterns were calculated for initial models 1 and 2, and taking into account that PbBi and CdBi samples have an orthogonal layer stacking sequence, as HBi, and that ZnBi has a 1M layer stacking with +a/3 or -a/3 interlayer displacement. Comparison of experimental and calculated XRD patterns showed that initial model 2 provides a better agreement for ZnBi and CdBi whereas model 1 better fits PbBi samples (data not shown).

Therefore, henceforward structure models of PbBi correspond to model 1 in which Pb is located either above or below vacant layer octahedra, whereas in structural models of ZnBi and CdBi, Me is located above and below these sites. Figure 2 compares XRD patterns calculated for the initial layer model of ZnBi 122 with the two possible layer stacking sequences (±a/3). The intensity distributions calculated for 20l, 11l reflections differ dramatically for these two layer sequences, and experimental XRD patterns of ZnBi 122 and ZnBi 128 unambiguously correspond to the +a/3 stacking sequence (Eq. 5).

Stacking faults. The presence of well-defined and/or random stacking faults is common in natural and synthetic layered minerals, including birnessites (see [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF] for a definition of well-defined and random stacking faults). Figure 3a compares the experimental XRD pattern of ZnBi 122 with the one calculated for the initial 1M structural model free of well-defined stacking faults, and hereafter referred to as the defect-free model even though it contains random stacking faults. For this defect-free model, the calculated 20lbar,11(l+1), and 20l,11(l-1)bar reflections (l being positive) are significantly shifted in comparison with the experimental peak positions to lower and higher 2θ values, respectively (Fig. 3a -Table 3). The diffraction effects described by [START_REF] Drits | The nature of diffraction effects from illite and illitesmectite consisting of interstratified trans-vacant and cis-vacant 2:1 layers: A semiquantitative technique for determination of layer-type content[END_REF] for the random interstratification of layer fragments having the same thickness but different interlayer displacement provides an explanation for such a shift. Indeed, in such a case, the positions of the observed hkl reflections are located between neighboring hkl reflections of phases whose elementary components are interstratified. Similar diffraction effects have been observed and quantitatively interpreted for Ca-exchanged birnessite (Drits et al. 1998a) and

for HBi prepared at different pHs [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF].

Experimental 20l,11l exp reflections are located between calculated 20lbar,11(l+1) M and 20l',11l' H reflections (Fig. 3a). If one assumes that the 1M ZnBi 122 structure contains orthogonal layer pairs, positions of 20lbar,11(l+1) M , and 20l,11(l-1)bar M reflections should be shifted towards 20l,11l H maxima in comparison with the defect-free ZnBi 122 (Fig. 3a).

Accordingly, the best agreement between experimental and calculated 20l,11l reflection positions and profiles was obtained for a defective 1M ZnBi phase containing 12% and 15% of randomly interstratified orthogonal layer pairs in ZnBi 122 (Fig. 3b) and ZnBi 128, respectively.

In contrast to ZnBi samples, the initial agreement between experimental peak positions and those calculated for a defect-free 1H model was, at the first sight, satisfactory for the first group of samples (CdBi 106, PbBi 6 and 58). However, Figure 4a to fill the gap between experimental and calculated peak positions. The best agreement between these peak positions was obtained for a defective 1H CdBi model containing 8% of monoclinic layer pairs (Fig. 4b -R WP = 9.6%). For PbBi 6 and 58, calculation of XRD patterns shows that the amount of monoclinic layer pairs does not exceed 2-3%. The proportions of random and well-defined stacking faults providing the best agreement between experimental and calculated XRD patterns are listed in Table 4 as well as the optimum sizes of coherent scattering domains for each sample. One may note in this Table that in addition to the well-defined stacking faults, all samples contain a significant amount of random stacking faults (12-19% -Table 4).

Optimal structural parameters for 1H samples

Even though XRD patterns calculated for the initial 1H models of PbBi 6, 58 and CdBi 106 reproduce reasonably well their main diffraction features, several structural parameters were varied to improve the agreement between experimental and calculated patterns. Because of the high sensitivity of the intensity distribution to the amount of interlayer Me, and Mn, these parameters were analyzed with special care and it was found that their optimal values matched those set in the initial models on the basis of chemical analyses (Tables 1,5).

Additional structural parameters such as the number of vacant layer sites and the positions and content of H 2 O molecules coordinated or not to interlayer cations were adjusted. In et al. 2002), this additional interlayer position, whose symbolic notation is AbC a'B' B'c' AbC, was considered. In this position, Pb shares three edges with underlying Mn layer octahedra forming a triple edge-sharing surface complex ( VI TE site -Fig. 8; [START_REF] Spadini | Structure and stability of Cd 2+ surface complexes on ferric oxides[END_REF], as in quenselite (PbMnO 2 OH - [START_REF] Rouse | The crystal structure of quenselite[END_REF]. Adding this interlayer position, and constraining the overall interlayer composition to chemical analysis yielded an optimal fit to the data (0.112

Mn interlayer in VI TC sites, 0.044 and 0.010 Pb in VI TC, and VI TE sites, respectively -Fig. 7b -Tables 1, 5-7 -R WP = 9.2%, R EXP = 3.1%). One can note that a fit similar to the one described may be obtained (Fig. 7c -R WP = 9.1%) with a slightly different interlayer composition (0.123 Mn interlayer , 0.044 Pb in VI TC sites, and 0.010 Pb in VI TE sites) in which the amount of interlayer cations located in VI TC sites only is equal to the amount of vacant layer sites (0.167 per octahedron). In both cases, the size and shape of VI TC sites are similar to those described for PbBi6 (Fig. 6), whereas VI TE sites are less distorted (Fig. 8). In comparison with the Pb VI TC octahedra, the flattening of the The Cd z-coordinate and the x-, y-, and z-coordinates of H 2 O Cd in CdBi 106 had to be adjusted from the initial model to obtain an optimal agreement between experimental and calculated XRD patterns (Tables 5,6 -.Fig. 9 -R WP = 8.4%, R EXP = 4.1%). The optimal amounts of Cd (0.114) and Mn interlayer (0.108) are close to those derived from chemical analysis (0.100 and 0.116, respectively -Tables 1,5). In projection on the c axis, Cd is

Optimal structural parameters for 1M samples

In contrast to the previous models, no adjustment of the amount of Mn interlayer and Me from their initial values (Table 1) could provide a satisfactory fit to the experimental XRD patterns. Indeed, calculated 20lbar,11(l+1), and 20l,11(l-1)bar reflection intensities (l≤3)

have much stronger intensities than experimental ones when all interlayer cations (0.094

Mn interlayer and 0.116 Zn -Table 5) have octahedral coordination (Fig. 11a -R WP = 14.9%).

Because the coexistence of octahedrally ( VI Zn) and tetrahedrally ( IV Zn) coordinated Zn TC surface complexes was established in these samples from EXAFS spectroscopy (Manceau et al. 2002), the additional presence of IV Zn was introduced in the XRD simulations. Such a mixed tetrahedral and octahedral Zn coordination is common and has been described in a number of structures including hydrozincites, Zn 5 (OH) 6 (CO 3 ) 2 and Zn 5 (OH) 8 (NO 3 ) 2 (H 2 O) 2 [START_REF] Ghose | The crystal structure of hydrozincite, Zn 5 (OH) 6 (CO 3 ) 2[END_REF][START_REF] Stählin | The crystal structure of zinc hydroxide nitrate, Zn 5 (OH) 8 (NO 3 ) 2 (H 2 O) 2[END_REF] 1969). These solids exhibit similar structures with VI Zn building up a brucite-like layer and octahedral layer vacancies capped by IV Zn in the interlayer. In order to assess the sensitivity of XRD to the position of IV Zn, two alternative structural models were considered.

In the first, IV Zn is located either above or below vacant layer sites ( IV TC site -Fig. 10a), and the idealized local 1M ZnBi structure may be represented as:

AbC b'B' C'c' BcA c'C' A'a' CaB a'A' B'b' AbC… (7)
where a', b', and c' are the IV Zn sites for Zn, A', B', and C' the sites for H 2 O Zn which are located just above or below O layer (A, B, and C sites -Fig. 10a).

In the alternative model, IV Zn forms a TE complex as Pb in PbBi 58 (Fig. 10b). Its distribution in ZnBi interlayers may be described locally as :

AbC a'A' A'a' BcA b'B' B'b' CaB c'C' C'c' AbC… (8)
The best possible agreement between experimental and calculated XRD profiles (Fig. 11b -R WP = 10.3%, R EXP = 3.5%) was obtained for a model in which 0.098 Mn interlayer and 0.078

Zn per octahedron are octahedrally coordinated, and 0.040 Zn atoms are tetrahedrally coordinated at IV TC sites, in agreement with EXAFS results [START_REF] Manceau | Structure of heavy metal sorbed birnessite: Part 3. Results from powder and polarized EXAFS spectroscopy[END_REF]. The zcoordinate for IV Zn was calculated from the IV Zn-nearest Mn layer EXAFS distance (3. To illustrate the sensitivity of XRD patterns to the location of IV Zn, a XRD profile (Fig. 11c -R WP = 26.2%) was calculated for a model with the same interlayer cation composition (0.098 VI Mn interlayer , 0.078 VI Zn, and 0.040 IV Zn per octahedron) as that used for the optimum fit but with Zn located in the alternative IV TE site (Fig. 10b).

The same approach was applied to ZnBi 128 and led to a 0.100 VI Mn interlayer , 0.084 VI Zn, and 0.040 IV Zn cation interlayer composition, all cations being located in TC sites. Optimal parameters for the two ZnBi varieties are listed in Tables 2, 4, 5 and 6.

DISCUSSION

Layer stacking mode in MeBi as a function of the nature and structure of Me complex

The layer stacking mode in MeBi depends on the nature of cations sorbed from solution as illustrated by the 1H structure of PbBi and CdBi and the 1M structure of ZnBi. In these structures, interlayer cations are uniquely (Cd, Zn), or predominantly (Pb) located above/below vacant layer sites, as 25% of total Pb cations are located above and/or below empty tridentate cavities ( VI TE sites) in sample PbBi 58. In addition, about 30% of total Zn cations are tetrahedrally coordinated in ZnBi. To understand the reason for the formation of 1H or 1M MeBi from the initial NaBu, the interlayer structures of these two types of low-pH birnessite must be compared.

In the 1H structure (Fig. 12a), the distance between Me in VI TC sites and the nearest Mn layer of the adjacent layer is short (4.7-5.0 Å). In addition, their electrostatic repulsion is not screened by anions, as the three H 2 O Me form a distorted empty prism with the three closest O layer of the adjacent layer. On the contrary, the location of Me in TE sites prevents such a direct electrostatic interaction with Mn layer of the adjacent layer (Fig. 8). Because H 2 O interlayer -O layer distances are similar for both interlayer configurations (2.65-2.68 Å -Table 7), either location of Pb provides an equal opportunity for the formation of strong H-bonds between O layer of one layer and H 2 O Me of the adjacent one. The essential control exerted by these H-bonds on the layer stacking mode is evidenced by the prevalence of the 1H HBi phase at low pH [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. Predominance of Pb in VI TC sites is likely related to the formation of vacant layer octahedra during the NaBu-to-PbBi transformation, as location of Pb above and/or below these sites provides better local charge compensation than location of Pb in VI TE sites, the latter sharing only one edge with vacant octahedra.

In contrast to the 1H MeBi phase, distances between Mn layer and Me from adjacent layers are maximized in the 1M structure (Fig. 12b), and the electrostatic repulsion between these cations is thus minimized. However, H 2 O Me molecules form an empty octahedron with O layer atoms of the adjacent layer (Fig. 12b) and, accordingly, the H 2 O Zn -O layer distance in 1M ZnBi (~3.0 Å) is much longer than in the 1H structure, resulting in much weaker H-bonds as in the 1M HBi structure described by. Interestingly, the IV Zn complex serves to stabilize the 1M structure as H 2 O Zn coordinating IV Zn is located just above or below O layer of the adjacent layer (Fig. 10a). The resulting O layer -H 2 O Zn distance (2.37 Å -Table 7) is short, but plausible, and may always be increased by shifting H 2 O Zn in the ab plane to form an obtuse Zn-H 2 O Zn -O layer angle (Fig. 13). Thus, despite the +a/3 displacement between adjacent layers, which is a priori unfavorable to the formation of strong H-bonds between layers, the tetrahedral coordination of Zn offers this opportunity.

Structural formulae and crystal-chemical features of MeBi

According to [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF], the structural formula of synthetic HBi prepared at pH 4 is H + 0.320 Mn 3+ 0.123 Mn 2+ 0.043 (Mn 4+ 0.722 Mn 3+ 0.111 0.167 )O 2 .0.487H 2 O. Because the amount of vacant layer sites in all studied MeBi samples (0.167 per octahedron) is identical to that of HBi prepared at different pHs (5-2), it is logical to assume that the disproportionation and migration reactions occurring during the NaBu-to-MeBi transformation are similar to those described for the NaBu-to-HBi transformation by [START_REF] Drits | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 1. Results from X-ray diffraction and selected area electron diffraction[END_REF], [START_REF] Silvester | The structure of monoclinic Na-rich birnessite and hexagonal birnessite. Part 2. Results from Chemical Studies and EXAFS Spectroscopy[END_REF],

and [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. As a consequence, layer cation composition, and more especially the relative proportions of heterovalent Mn cations, are supposed to be identical in all MeBi samples. As a result, the following structural formulae were calculated from the content of Mn interlayer , Me, and H 2 O determined from the simulation of XRD patterns (Table 5), and from the Na content determined by chemical analysis ( The sum of Mn interlayer and TC Pb (0.166 per octahedron) in the two PbBi samples is similar to the amount of vacant layer sites (0.167 per octahedron), which suggest that these cations are located either above or below vacant layer sites in agreement with EXAFS results [START_REF] Manceau | Structure of heavy metal sorbed birnessite: Part 3. Results from powder and polarized EXAFS spectroscopy[END_REF]. Conversely, in 1H CdBi and in 1M ZnBi, the sum of interlayer cations is significantly greater than 0.167 per octahedron. For these samples the difference between the amounts of vacant layer sites (0.167) and of Mn 3+ interlayer is systematically about half the amount of Me. Consequently, most Cd and Zn cations are likely located above and below the same vacant layer sites, whereas Mn 3+ interlayer is located either above or below these sites. In addition, the amount of Mn interlayer in CdBi and ZnBi samples (0.098-0.108 atom per octahedron -Table 5) is similar to the amount of Mn 3+ interlayer in 1H HBi (0.111 atoms per octahedron). The similarity of the Mn interlayer contents suggest that during the NaBu transformation to CdBi and ZnBi, the main cation-exchange process occurs between interlayer Na and Mn 2+ , leaving the solid and Me and H + adsorbed from solution.

Layer charge compensation in MeBi

As mentioned before, all studied MeBi samples have the same layer cation composition and therefore the same high negative layer charge. As can be seen in Figure 14, O layer atoms of a vacant octahedron ideally receive 0.500 + 0.667 = 1.167 positive charges if coordinated to Mn 3+ and Mn 4+ (half-and-half circles) or 0.667 + 0.667 = 1.333 positive charges if coordinated to two Mn 4+ (open circles). In both cases these three O layer are strongly undersaturated as globally they receive only 4.000 (3 x 1.333 -Fig. 14a), 3.833

(2 x 1.333 + 1.167 -Fig. 14b), or 3.667 (2 x 1.167 + 1.333 -Fig. 14c) positive charges, and therefore require two additional positive charges on each side of a vacant layer site to achieve local charge compensation. As a first approximation, charge balance is satisfied for CdBi and ZnBi as vacant layer sites are capped on both sides by Cd and Zn, respectively. This charge distribution is idealized as Zn and Cd share their charge between O layer and H 2 O Me [START_REF] Manceau | Structure of heavy metal sorbed birnessite: Part 3. Results from powder and polarized EXAFS spectroscopy[END_REF]. In turn, H 2 O Me transfers part of this positive charge to O layer of the next layer through H-bonds. When a vacant layer site is capped by one Mn 3+ interlayer , local charge compensation requires the sorption of two protons on one side of this site and of (Mn 3+ OH -) 2+

on the other side. In this case, one H 2 O molecule is dissociated to permit this local charge compensation [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. This second mechanism, which occurs in 1H HBi, may also exist in PbBi because interlayer cations are located either above or below vacant layer sites, as Mn interlayer in HBi.

In PbBi 58, local charge compensation is improved by the coexistence of VI Pb in both TC and TE sites on the two sides of a vacant layer octahedron. As an illustration, Figure 15 shows a PbBi layer fragment in which such a vacant octahedron shares edges with two Mn 3+and four Mn 4+ -octahedra leading to O layer with contrasting degrees of undersaturation. As can be seen in this figure, location of Pb in both VI TE and VI TC sites optimally compensates the charge deficit of the most undersaturated O layer (half-and-half circles). Consistently, the interlayer cation compositions of 1H HBi (0.123 Mn 3+ , 0.044 Mn 2+ ) and PbBi 58 (0.123 Mn 3+ , 0.044 Pb 2+ in VI TC sites and 0.010 Pb 2+ in VI TE sites) suggest that during the NaBu-to-PbBi transformation Mn 2+ interlayer is replaced by Pb 2+ interlayer , both cations being located in TC sites. Additional Pb likely locates in VI TE sites to compensate the negative charge of most undersaturated O layer , and probably migrates towards these atoms as described by Manceau et al. (2002).

From the above considerations, it is clear that, except for protons, interlayer monovalent cations cannot provide local charge compensation to the permanent charge of birnessite resulting from vacant layer sites. This conclusion is supported by the low amount of Na in MeBi, in spite of its high concentration in solution.

Structural heterogeneity of MeBi

As can be seen in Table 4, MeBi structural heterogeneity strongly depends on the nature of Me as, for example, Pb and Cd dramatically increase phase and structural homogeneity of MeBi as compared with HBi pH4. Indeed, HBi pH4 consists of a physical mixture of a 1H phase containing 15% of monoclinic layers pairs and of a 1M phase containing 40% of orthogonal layer pairs [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. In contrast, the NaBu-to-PbBi transformation leads to the formation of unique 1H PbBi varieties containing only 2-3% of monoclinic layer pairs, even if the amount of Pb is low. A similar low content of monoclinic layer pairs (7%)

was observed for the 1H CdBi phase.

The origin of MeBi structural homogeneity may be explained in the light of the results obtained for low-pH HBi [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. These authors showed that almost defect-free 1H HBi is formed either when a slow enough reaction rate permits additional ±a/3 adjustments of adjacent layers leading to a defect-free layer stack (pH 5) or when the concentration of protons is high enough (pH 2) to stabilize the layer stacking through Hbonds.

By analogy with the origin of the well-defined stacking faults in the 1H HBi phase, one may assume that the presence of Pb and Cd cations in solution significantly decreases proton sorption because of the competition between Me and protons, thus decreasing the reaction rate. As a consequence, most adjacent layers in PbBi and CdBi have the opportunity to be additionally shifted by ±a/3 to build up a defect-free layer stack by forming strong H-bonds.

Both the low amount of interlayer protons and the significant proportion of Na interlayer present in 1H PbBi in comparison with HBi pH 4 support this hypothesis of a slow NaBu-to-MeBi transformation rate.

In contrast to 1H MeBi, ZnBi samples contain a significant amount of orthogonal layer pairs, as the proportion of well-defined stacking faults in 1M ZnBi 122 (12%) and 128 (15%)

samples is twice that in CdBi (7%). Moreover, the proportion of such stacking faults dramatically increases with decreasing content of Zn (see XRD pattern of ZnBi 8 -Fig. 1b).

This peculiar tendency to form defective layer stacking is likely induced by the coexistence of three competitive reactions during the NaBu-to-ZnBi transformation. The first reaction tends to form 1M layer pairs in which adjacent layers are shifted by +a/3 because the presence of IV Zn in TC sites permits the formation of strong H-bonds between adjacent layers. At high pH, such as those measured when the NaBu-to-ZnBi transformation begins, the presence of such IV Zn is frequent as shown, for example, by the formation of synthetic Zn-Ni hydroxy double salts, Ni 1-x Zn 2x (OH) 2 (CH 3 CO 2 ) 2x .nH 2 O in which IV Zn is located just above and below vacant octahedra of the Ni 1-x (OH) 2 brucite-like layer [START_REF] Choy | Intra-and interlayer structures of layered hydroxy double salts, Ni 1-x Zn 2x (OH) 2 (CH 3 CO 2 ) 2x .nH 2[END_REF]. The second reaction also tends to form such 1M layer pairs because only this interlayer configuration allows a chalcophanite-like distribution of vacant layer sites capped on both sides by VI Zn as discussed by [START_REF] Drits | Structural chemistry of heavy metal sorbed birnessite: Part 2. Results from electron diffraction[END_REF]. The third reaction tends to form orthogonal layer pairs because the presence of octahedrally coordinated Mn 3+ interlayer and Zn leads to the formation of strong Hbonds in such 1H layer pairs [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. This hypothesis is supported by the higher proportion of 1H layer pairs in ZnBi 8 as compared with ZnBi 122 (Fig. 1b) in agreement with their contrasting Zn content.

NaBu-to-MeBi structural transformation

As shown previously, the formation of strong H-bonds between adjacent layers, the minimization of cation electrostatic repulsion, and the nature of sorbed metal cations are key interdependent factors which control the layer stacking mode in MeBi structure. The low-pH equilibration of NaBu induces a modification of the layer stacking mode to allow the formation of strong H-bonds between adjacent layers [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF] because the -a/3 shift between adjacent layers of the NaBu idealized structure (Lanson, unpublished data) is unfavorable for such H-bonds. The energy necessary to transform 1M NaBu into 1H MeBi is minimized if the n+1 and n+2 layers are shifted by +a/3 and -a/3, respectively (Fig. 15 in [START_REF] Lanson | Structure of H-exchanged hexagonal birnessite and its mechanism of formation from Na-rich monoclinic buserite at low pH[END_REF]. A similar modification of the layer stacking is observed in PbBi and CdBi.

In contrast, the 1M ZnBi structure, in which adjacent layers are shifted with respect to each other by +a/3, may be obtained from the displacement of n+1 and n+2 layers of the initial NaBu crystal by -a/3 and +a/3, respectively. Because these two translations equally minimize the electrostatic repulsion of layer and interlayer cations, the origin of this different layer stacking modification, as compared with that observed for HBi, PbBi, and CdBi, remains unclear, as resulting H-bonds are weak. However, one may note that the interlayer displacement 1M ZnBi is similar to that in the chalcophanite structure [START_REF] Wadsley | The crystal structure of chalcophanite, ZnMn 3 O 7 .3H 2 O[END_REF][START_REF] Post | Chalcophanite, ZnMn 3 O 7 -3H 2 O: New crystal-structure determinations[END_REF]. Moreover, interpretation of SAED patterns of ZnBi 69, 122, and 128 samples has shown that the NaBu-to-ZnBi transformation is accompanied, for most particles, by the formation of vacant layer sites distributed as in chalcophanite with a A = B = b 7 = 7.536 Å, γ = 120° unit cell [START_REF] Drits | Structural chemistry of heavy metal sorbed birnessite: Part 2. Results from electron diffraction[END_REF]. These authors have shown that such a distribution of vacant layer sites capped by Zn on both sides is compatible only with the +a/3 displacement between adjacent layers. Therefore, it is likely that the main factor controlling the layer stacking mode in ZnBi is the tendency towards a chalcophanite-like distribution of Zn in interlayers. 4). All patterns as in Fig. 8b. .55 -3.55 3.55 -3.55 3.66 3.66 -3.66 -3.66 3.66 3.66 -3.66 -3.66 3.62 3.62 -3.62 -3.62 Occ

  the z-coordinate of H 2 O Zn was set to: z(H 2 O) = z(Zn) + d (Zn-H2O) = 3.70 Å, assuming that d (Zn-H2O) = 1.93 Å.

FIGURE CAPTIONS Figure 1 .

 CAPTIONS1 FIGURE CAPTIONS

Figure 2 .

 2 Figure 2. Comparison between experimental and calculated XRD patterns for ZnBi 122. Experimental data are shown as crosses. The XRD patterns calculated for the initial 1M layer model of ZnBi 122 and the two possible directions for the a/3 shift are shown on

Figure 3 .

 3 Figure 3. Comparison between experimental and calculated XRD patterns for ZnBi 122. (a) Patterns as for Figure 2 except for the gray line which represents the XRD pattern calculated for the initial layer model of ZnBi 122 with a 1H layer stacking mode.

Figure 4 .

 4 Figure 4. Comparison between experimental and calculated XRD patterns for CdBi 106 (top)

Figure 5 .

 5 Figure 5. Comparison between experimental and calculated XRD patterns for PbBi 6 (top)

Figure 6 .

 6 Figure 6. Schematic view of Pb site above/below vacant layer octahedra ( VI TC site). (x, y) coordinates of this position are (0, 0). (a) Projection on the ab plane. The upper surface of the lower layer is shown as light shaded triangles. O layer of this lower layer is shown as open circles. Dashed line outlines the ideal octahedron defined by the close packing of anions whereas irregular dashed line outlines the actual upper surface of the Pb octahedron, after further adjustment (arrows) of H 2 O positions to obtain realistic Pb-H 2 O interlayer and H 2 O interlayer -O layer distances. (b) Projection along the b axis. O layer is shown as large circles. Open symbols indicate atoms at y = 0, and solid symbols indicate atoms at y = ±1/2. Vacant layer sites are shown as open squares and Mn layer is not shown. Other patterns as Fig. 6a. Z-coordinates of Mn layer , O layer , Pb, and H 2 O interlayer are indicated.

Figure 7 .

 7 Figure 7. Comparison between experimental and calculated XRD patterns for PbBi 58 (top)

Figure 8 .

 8 Figure 8. Schematic view of Pb site above/below empty tridentate cavities ( VI TE site). (x, y) coordinates of this position are (-0.333, 0) above the lower layer and (0.333, 0) below this layer. All patterns as for Fig. 6. (a) Projection on the ab plane. Small solid circles represent Mn layer from the lower layer. (b) Projection along the b axis.

Figure 9 .

 9 Figure 9. Comparison between experimental and calculated XRD patterns for CdBi 106 (top)

Figure 10 .

 10 Figure 10. Possible positions of IV Zn in the interlayer space of ZnBi. Adjacent layers are

Figure 11 .

 11 Figure 11. Comparison between experimental and calculated XRD patterns for ZnBi 122

Figure 12 .

 12 Figure 12. Projection of the layer and interlayer structures of the MeBi structures along the b axis (modified from Lanson et al. 2000). Oxygen atoms are shown as large circles. Vacant layer sites are shown as open squares. Saturated O layer and protons are not shown. Open symbols indicate atoms at y = 0, and solid symbols indicate atoms at y = ±½. Bonds between very undersaturated O layer and H 2 O Me are outlined by dashed lines. These dashed lines also indicate possible H-bonds between these two species. (a) Idealized structural model for the 1H phase of MeBi. (b) Idealized structural model for the 1M phase of MeBi. The edges of the octahedra defined by very undersaturated O layer and H 2 O Me are too long to permit H-bonds.

Figure 13 .

 13 Figure 13. Schematic view of tetrahedrally coordinated Zn site vacant layer octahedra ( VI TC site). (x, y) coordinates of this position are (-0.333, 0) above the lower layer and (0.333, 0) below this layer. All patterns as for Figures 6, and 10. (a) Projection on the ab plane. (b) Projection along the b axis. Dashed line outlines the obtuse Zn-H 2 O interlayer -O layer angle resulting from the shift of H 2 O interlayer in the ab plane to increase the H 2 O interlayer -O layer distance.

Figure 14 .

 14 Figure 14. Idealized structural model for the MeBi phases. Distribution of undersaturated O layer within the upper surface of the lower layer (grey triangles) in projection on the abplane. Mn 3+ layer , Mn 4+ layer and vacant layer sites are shown as 3+, 4+, and open squares,

Figure 15 -

 15 Figure 15 -Idealized structural model for the PbBi 58 sample. (a) Projection on the ab plane. All patterns as in Fig. 14. The lower surface of the vacant layer site is shown as a dark shaded triangle outlined by a dashed line. O layer atoms defining this surface are outlined by a dashed line. Pb is located below this vacancy in a VI TC site to compensate the two most undersaturated O layer . Above this vacant site, Pb may equally be located in the VI TE site to compensate the two most undersaturated O layer (half and half circles). Small solid circles represent Mn layer from the lower layer. (b) Projection along the b axis.

  values have been determined by trial-and-error fitting of the experimental XRD patterns(Figs. 5, 7b, 9, and 11b). Proportions of well-defined stacking faults represented by 1M (±a/3) and 1H (no shift between adjacent layers) layer pairs are given for 1H and 1M varieties, respectively. Wr is the proportion of random stacking faults. The radius of the coherent scattering domains is given in the ab-plane, whereas the coherent scattering domain along the c*-axis is noted N and is expressed as a number of layers.

  values were determined by trial-and-error fitting of the experimental XRD patterns(Figs. 5, 7b, 9, and 11b). x, and y coordinates are expressed as fractions of the a and b parameters, respectively. Coordinates along the c* axis, ζ, are expressed in Å to emphasize the thickness of layer and interlayer polyhedra. * denotes occupancies given for either one of the two equivalent positions.

Note:

  All distances are given in Å. * indicates a very short distance which may be increased by a shift of Zn from its ideal position either in the ab plane or along the c* axis, † indicates a very short distance which may be increased by additional displacement of interlayer H 2 O molecules (Fig.13a).

Table 1

 1

	. Initially, coordinates of

  (R WP = 10.4%) shows that positions of 201,111 H and 202,112 H calculated for defect-free 1H CdBi are very slightly shifted towards lower 2θ values in comparison with the corresponding experimental reflections, most likely because of interstratification effects similar to those described for

ZnBi 122. A small amount of monoclinic layer pairs in which layers are shifted with respect to each other by +a/3 induces a displacement of 201,111 H and 202,112 H reflections towards 202,111bar M and 203,112bar M ones, respectively, that is towards higher 2θ values as needed

-

  particular, it was initially assumed that H 2 O Me formed regular triangles with edge lengths equal to the b parameter, but this assumption led to short distances between H 2 O Me and O layer of the next layer (H-bonds). This structural inconsistency was solved by shifting these H 2 O Me in the ab plane, their z-coordinate being adjusted to maintain realistic H 2 O Me -Me distances. Pb ones (2.59 Å -Table 7). As a result of the displacement of H 2 O Pb in the ab plane, the H 2 O Pb -O layer distances increase from 2.15 Å in the initial model up to 2.66 Å, a sound value for the formation of strong H-bonds (Table 7; Fig. 7a -R WP = 10.1%). Because SAED results suggest the presence of Pb located above/below empty tridentate cavities in some PbBi 58 crystals (Drits

	Fig. 6b).

The experimental XRD pattern of PbBi 6 was best reproduced by keeping the structural parameters of the initial model, and by adjusting the Pb z-coordinate and the x-, y-, and zcoordinates of H 2 O Pb (Tables

2, 4

, 5, and 6 -Fig.

5

-R WP = 10.0%, R EXP = 3.4%). As a result of the shift in the ab plane of H 2 O Pb from the initial anion close-packing position (Fig.

6a

), Pb has a distorted octahedral coordination. The H 2 O Pb -H 2 O Pb distance is increased up to 4.05 Å whereas the O layer -O layer distance is 2.84 Å (Fig.

6a

). Pb octahedra are also flattened, as the difference between Pb and H 2 O Pb coordinates in projection on the c axis (1.12 Å) is smaller than that between Pb and O layer ones (1.54 Å -Table

6

; Fig.

6b

); in parallel Pb-O layer interatomic distances (2.25 Å) are shorter than Pb-H 2 O

The experimental XRD pattern of PbBi 58 significantly differs from that calculated for the initial model 1 in which all Pb (0.056 atoms per octahedron) are located above or below layer vacant octahedra (Table

1

  located halfway from H 2 O Cd and bound O layer , but Cd-O layer distances (2.10 Å) are shorter than Cd-H 2 O Cd ones (2.35 Å) because H 2 O Cd was shifted in the ab plane from its ideal anion close-packing position. As a result of this shift, H 2 O Cd -H 2 O Cd distances are equal to 3.37 Å, whereas distances between H 2 O Cd and O layer of the next layer are equal to 2.68 Å, thus allowing the formation of strong H-bonds.

Table 1

 1

	):

Table 1 .

 1 Mn interlayer , Me, and Na contents calculated for the two initial structural models from the Me/Mn and Na/Mn ratios determined by chemical analysis

	Sample	Me/Mn	Na/Mn	N_Mn interlayer	N_Me	N_Na interlayer
				Model 1	Model 2	Model 1	Model 2
	ZnBi 8	0.008	0.017	0.159	0.159	0.008	0.016	0.017
	ZnBi 69	0.069	0.011	0.102	0.133	0.065	0.067	0.010
	ZnBi 122	0.122	0.007	0.058	0.110	0.109	0.115	0.006
	ZnBi 128	0.128	0.021	0.053	0.106	0.113	0.121	0.019
	PbBi 6	0.006	0.093	0.161	0.164	0.006	0.006	0.093
	PbBi 58	0.058	0.067	0.112	0.139	0.055	0.056	0.064
	CdBi 106	0.106	0.006	0.071	0.116	0.096	0.100	0.006
	CuBi 156	0.156	0.043	0.032	0.050	0.135	0.234	0.038
	Note: Relative error in the determination of the atomic ratios does not exceed 5%.
	N_Mn interlayer , N_Me, and N_Na interlayer are derived from the measured concentrations using
	equations 1-2 for both model 1 and model 2 (n = 1, 2, respectively).

Table 2 .

 2 Unit-cell parameters obtained from the indexing of d(hkl) values for the various samples.

	Sample	a	B	csinβ	β
	HBi pH4*	4.933(3) 2.848(2) 7.186(5)	90°Z
	nBi 122	4.930(3) 2.846(2) 7.070(5)	76.9°Z
	nBi 128	4.929(3) 2.846(2) 7.070(5)	76.9°P
	bBi 6	4.926(3) 2.844(2) 7.228(5)	90°P
	bBi 58	4.926(3) 2.844(2) 7.218(5)	90°C
	dBi 106	4.933(3) 2.848(2) 7.288(5)	90°*
	: from Lanson et al. (2000)

Table 3 .

 3 Indexing of the experimental XRD patterns of MeBi

	Sample	PbBi 6	PbBi 58	CdBi 106	ZnBi 122	ZnBi 128
		Exp.	Calc.	Exp.	Calc.	Exp.	Calc.	Exp.	Calc.	Exp.	Calc.
	001	7.210	7.228	7.201	7.218	7.285	7.288	7.097	7.070	7.092	7.070
	002	3.612	3.624	3.608	3.609	3.644	3.644	3.548	3.535	3.552	3.535
	110/200	2.461	2.463	2.463	2.463	2.469	2.466
	110/201							2.451	2.450	2.453	2.448
	111/200							2.401	2.402	2.399	2.401
	111/201	2.331	2.331	2.331	2.331	2.335	2.336
	11-1/202							2.242	2.237	2.243	2.235
	112/202	2.037	2.035	2.036	2.034	2.041	2.043
	11-2/203							1.919	1.913	1.919	1.912
	113/20-2							1.796	1.806	1.798	1.805
	004	1.809	1.807					1.777	1.768	1.777	1.768
	113/203	1.723	1.722	1.724	1.721	1.730	1.731
	11-3/204							1.615	1.608	1.616	1.608
	114/20-3							1.515	1.519	1.512	1.519
	114/204	1.459	1.457	1.454	1.456	1.472	1.466
	311/020							1.423	1.424	1.423	1.423
	310/020	1.421	1.422	1.422	1.422	1.425	1.424
	312/021							1.396	1.396	1.396	1.395
	311/021	1.395	1.395	1.398	1.395	1.400	1.398
	31-1/022							1.321	1.321	1.321	1.320
	312/022	1.323	1.323	1.321	1.323	1.327	1.326
	115/20-4							1.288	1.291	1.285	1.291
	115/205	1.246	1.247
	221							1.229	1.231	1.229	1.230
	220	1.230	1.231	1.231	1.231	1.234	1.233
	221	1.214	1.214			1.217	1.216

Note: d calc (hkl) values are calculated using the unit cell parameters listed in Table

2

. d exp (hkl) are measured experimentally.

Table 4 .

 4 Optimum structural parameters used for the simulation of X-ray diffraction profiles.

	Parameters defining layer stacking mode.
	Sample	Well-defined faults	Wr	Radius	Mean N	Max N
		Prop.	Displ.	Abund.	(Å)
	ZnBi 122	12%	no shift	12%	300	7	35
	ZnBi 128	15%	no shift	15%	225	7	35
	PbBi 6	2%	-a/3	15%	175	6	30
	PbBi 58	3%	-a/3	19%	200	5	25
	CdBi 106	8%	+a/3	14%	160	6	30
	Note: Optimal

Table 5 .

 5 Optimum structural parameters used for the simulation of X-ray diffraction profiles.Parameters defining the layer cation composition. When two amounts are given for H 2 O interlayer coordinating Me, the upper one refers to Me in VI TC sites, whereas the lower one refers to Me in IV TC (Zn) or VI TE (Pb) sites.

	Sample	Vacancies	(Mn + Me) inter.	Me/Mn	H 2 O/Mn	H 2 O coord. Me Add. H 2 O + Na
	ZnBi 122	0.167	0.216	0.127	0.294/0.098	0.234/0.078	0.180
						0.040/0.040
	ZnBi 128	0.167	0.224	0.133	0.300/0.100	0.252/0.084	0.150
						0.040/0.040
	PbBi 6	0.167	0.166	0.008	0.480/0.160	0.024/0.008	0.280
	PbBi 58	0.167	0.166	0.057	0.336/0.112	0.132/0.044	0.500
						0.030/0.010
	CdBi 106	0.167	0.220	0.121	0.324/0.108	0.342/0.114	0.180
	Note: Optimal values have been determined by trial-and-error fitting of the experimental XRD patterns (Figs.
	5, 7b, 9, and 11b).

Table 6 .

 6 Optimum structural parameters used for the simulation of X-ray diffraction profiles. -0.333 0.333 -0.333 0.333 -0.475 0.238 0.475 -0.238 -0.475 0.238 0.475 -0.238 -0.394 0.197 0.394 -0.197

	Sample						ZnBi 122 / ZnBi 128 / PbBi 6 / PbBi 58 / CdBi 106
	Mn layer	x						0
		y						0
		ζ						0
	Occ.						0.833
	O layer	x				0.333				-0.333
		y				0				0
		ζ				1.00				-1.00
	Occ.				1				1
	Sample		ZnBi 122	ZnBi 128	PbBi 6		PbBi 58		CdBi 106
	Mn interlayer	x	0	0	0	0	0	0	0	0	0	0
		y	0	0	0	0	0	0	0	0	0	0
		ζ	2.04 -2.04 2.04 -2.04	2.04	-2.04	2.04	-2.04	2.04	2.04
	Occ. 0.049 0.049 0.050 0.050	0.080	0.080	0.056	0.056	0.054	0.054
	H 2 O inter.	x	-0.333 0.333 -0.333 0.333	-0.333	0.333	-0.333	0.333	-0.333	0.333
		y	0	0	0	0	0	0	0	0	0	0
		ζ	3.13 -3.13 3.13 -3.13	3.13	-3.13	3.13	-3.13	3.13	-3.13
	Occ. 0.147 0.147 0.150 0.150	0.240	0.240	0.168	0.168	0.162	0.162
	Me	x	0	0	0	0	0	0	0	0	0	0
	TC sites	y	0	0	0	0	0	0	0	0	0	0
		ζ	2.21 -2.21 2.21 -2.21	2.54	-2.54	2.54	-2.54	2.31	-2.31
	Occ. 0.039 0.039 0.042 0.042	0.004	0.004	0.022	0.022	0.057	0.057

x

Table 7 .

 7 Typical inter-atomic distances calculated from the optimum atomic coordinates. Sample Mn layer -O layer Mn inter. -O layer Mn inter. -H 2 O Me inter. -O layer Me inter -H 2 O H2O inter -O layer Me inter -Mn layer

	ZnBi 122	1.925	1.946	1.973	2.04 ( VI TC site)	2.12 ( VI TC site)	3.01 ( VI TC site)	3.60 ( VI TC site)
					1.81* ( IV TC site)	1.93 ( IV TC site)	2.37 † ( IV TC site)	3.35 ( IV TC site)
	ZnBi 128	1.923	1.944	1.972	2.04 ( VI TC site)	2.12 ( VI TC site)	3.01 ( VI TC site)	3.60 ( VI TC site)
					1.92 ( IV TC site)	1.60 1 ( IV TC site)	2.37 1 ( IV TC site)	3.48 ( IV TC site)
	PbBi 6	1.923	1.944	1.971	2.25	2.59 2	2.66 2	3.81
	PbBi 58	1.923	1.944	1.971	2.25 ( VI TC site)	2.59 ( VI TC site)	2.65 2 ( VI TC site)	3.81 ( VI TC site)
					2.25 ( VI TE site)	2.60 ( VI TE site)	2.68 ( VI TE site)	3.02 ( IV TE site)
	CdBi 106	1.925	1.946	1.973	2.10	2.35	2.68	3.66

ACKNOWLEDGMENTS

VAD is grateful to the Environmental Geochemistry Group of the LGIT (Grenoble, France) and to the Russian Science Foundation for financial support. BL and AM acknowledge financial support from INSU/Géomatériaux, CNRS/ACI "Eau et Environnement" and CNRS/PICS709 programs. Céline Bartoli is thanked for the preparation of the CdBi sample. Constructive remarks and comments by AE David Bish and two anonymous reviewers helped to improved the original manuscript.


Maria Hellgren

Jacopo Baima

Anissa Acheche

Exciton-Peierls mechanism and universal many-body gaps in carbon nanotubes

Metallic" carbon nanotubes exhibit quasiparticle gaps when isolated from a screening environment. The gap-opening mechanism is expected to be of electronic origin but the precise nature is debated. In this work we show that hybrid density functional theory predicts a set of excitonic instabilities capable of opening gaps of the size found in experiment. The excitonic instabilities are coupled to vibrational modes, and, in particular, the modes associated with the Γ -E2g and K -A 1 Kohn anomalies of graphene, inducing Peierls lattice distortions with a strong electronphonon coupling. In the larger tubes, the longitudinal optical phonon mode becomes a purely electronic dimerization that is fully symmetry conserving in the zigzag and chiral tubes, but breaks the symmetry in the armchair tubes. The resulting gaps are universal (i.e. independent of chirality) and scale as 1/R with tube radius.

Low-dimensional carbon systems have been subject of intense research for the last two decades, both for their promise as building blocks in future nanoelectronic devices, and for their novel fundamental physical properties [1]. In reduced dimensions the electron-electron (e-e) interaction is expected to play an increasing role, driving phenomena such as edge magnetism [2], exciton condensation [3] and superconductivity [4,5]. Important examples are the quasi-one-dimensional carbon nanotubes (CNTs) that exhibit a wide range of features depending on their radius and chirality [6]. Within a single-particle description, some CNTs are predicted to be semiconducting while others metallic. However, in a pioneering experiment [7] it was demonstrated that the e-e interaction induces gaps in all "metallic" CNTs -if suspended in a clean environment. This observation was recently confirmed to be universal, i.e., independent of chirality [8].

The physical origin of the observed quasiparticle gaps has not yet been settled. First-principle calculations have mainly focused on Peierls transitions, driven by the electron-phonon interaction [9][10][11][12][13]. Local functionals in density functional theory (DFT) predict soft phonon modes and lattice instabilities associated with the K-A 1 Kohn anomaly of graphene [14]. The Peierls gaps are, however, an order of magnitude too small and scale as 1/R 3 with tube radius, in contrast to the approximate 1/R behaviour found in experiment. On the other hand, in Ref. [15] calculations were performed with an improved description of the e-e interaction via hybrid DFT, showing that the picture can change. If the long-range Coulomb interaction is accounted for, small armchair CNTs exhibit Peierls distortions already in the unit cell. Although this study points toward larger gaps they again appear to decay quickly with tube radius.

Purely electronic mechanisms such as excitonic instabilities, or electronic charge density waves, also driven by the long-range Coulomb interaction, have been shown to produce the correct scaling with tube radius [16][17][18][19][20][21]. Such an instability was recently studied with ab-initio quantum Monte Carlo (QMC) in the (3,3) armchair tube, and a stable charge-transfer electronic state that breaks the sublattice symmetry was found [21]. Calculated gaps were, however, more than an order of magnitude smaller than experimental observations. In addition, most theoretical studies have focused on the armchair family of CNTs. It is, therefore, not clear which theories can account for the weak chirality dependence.

In this work, we present a mechanism based on a unified picture of Peierls distortions and excitonic instabilities. We employ first-principle hybrid DFT that captures the long-range exchange interaction, crucial to describe both excitonic and lattice instabilities in suspended CNTs. We then show that the A 1 longitudinal optical phonon mode is unstable, leading to a Peierls distortion, or dimerization, that is fully symmetry conserving in the zigzag and chiral tubes, but breaks the mirror symmetry in the armchair tubes. The lattice distortion is shown to be coupled to a purely electronic instability of the same symmetry, allowing the dimerization to be formed spontaneously already at the electronic level. For the larger tubes the lattice distortion vanishes while the electronic instability remains and becomes degenerate within a multiplet of excitonic states. The corresponding gaps are universal and in good quantitative agreement with experiments.

To the nominally metallic CNTs belong the (n, n) armchair (A) tubes, the (3n, 0) zigzag (Z) tubes, and the (3l + n, n) chiral (C) tubes (n and l being integers). By applying periodic boundary conditions to the band structure of a rolled graphene sheet (zone-folding approximation) these tubes inherit the two Dirac points formed by the π and π * bands of graphene. The position of the Dirac points depend on the chirality, ranging from K = ±1/3 in A-tubes to the two degenerate Dirac points at Γ in the Z-tubes. In the case of A-tubes, the Dirac point is protected against symmetry-preserving perturbations, while in the Z-and C-tubes, different mechanisms such as curvature or strain, can open a small gap at the Fermi-level [6].

An atomic displacement along a phonon mode can also open a gap, provided the electron-phonon coupling (EPC) mixes valence and conduction states [9,13]. It is well known that in graphene the Γ -E 2g and K -A 1 phonon modes are significantly softened by the EPC, displaying a Kohn anomaly [14,22]. Within local approximations in DFT, it has been shown that the strong EPC carries over to metallic CNTs and is enhanced by confinement [13,23]. The degenerate Γ -E 2g mode splits into a longitudinal optical (LO) and a transverse optical (TO) mode, where the frequency of the LO mode is softened since the distortion along this pattern opens a gap [9]. Similarly, the K -A 1 mode of graphene turns into a 2k F mode in the A-and C-tubes, and a Γ mode (which breaks the rotational symmetry) in the Z-tubes. Depending on the strength of the EPC, the soft phonon frequency may result in a permanent lattice distortion with a gap, i.e., a Peierls transition. This has been shown to occur for different modes in some small CNTs. The strength of the mixing, is, however, sensitive to how the e-e interaction is treated, and, in particular, to the long-range part of the exchange interaction [22,24]. If strong enough, the orbitals could mix spontaneously, i. e., solely due to the e-e interaction, without the need for an atomic displacement. This would lead to an excitonic insulator state as formulated in Refs. [25,26].

Let us denote the valence (π) and conduction (π * ) orbitals of the CNT by ϕ v,k and ϕ c,k , where the index k is the momentum relative to the valley the orbital belongs to. The mixed orbitals (that must break the symmetry in the A-tubes, but not necessarily in the Z-and C-tubes) can be written as linear combinations of the unperturbed orbitals

φv,k = u k ϕ v,k + v k ϕ c,k (1) φc
,k = u * k ϕ c,k -v * k ϕ v,k . (2)
For a purely electronic instability, the new wavefunction is constructed as an anti-symmetrized product of the mixed valence orbitals, i.e.,

|Ψ = k φ † v,k |0 = k (u k + v k ϕ † c,k ϕ † v,k )|Φ (3)
where |0 , |Φ are the vacuum, and "high-symmetry" or "single-particle" ground states, respectively. The last equality is easily derived, and demonstrates the excitonic nature of the new ground state. In a single-particle description the symmetry of the Hamiltonian is always preserved. However, within a self-consistent field (SCF) method, such as Hartree-Fock (HF) or DFT, the effective self-consistent HF or Kohn-Sham potential may break the symmetry of the original Hamiltonian. In this way, a symmetry broken excitonic state can also be described as a single Slater determinant built up from the orbitals φv,k , generated by the symmetry broken Hamiltonian. However, in standard semi-local approximations within DFT, such as PBE, the long-range exchange interaction, responsible for the electron-hole coupling or excitonic effects, is not well described. By mixing in a fraction α of HF exchange (typically around 25%), electronic properties generally improve as compared to pure PBE, and excitonic effects are captured [27,28]. In the Supplemental Material we demonstrate how an excitonic instability can be captured with a hybrid functional [29].

Although the essential physics is described by the π and π * bands, in this work we carry out first-principle simulations that include all states of the CNTs. This is achieved with the CRYSTAL program [30,31], that uses atom-centered Gaussian basis functions, which allow for an efficient evaluation of HF exchange [START_REF]G * basis set[END_REF][START_REF] Demichelis | [END_REF]. Hybrid functionals have been shown to reasonably well reproduce the many-body enhancement of the EPC in graphene [22]. In Ref. [15] α was optimized to 30% for CNTs, using the experimentally determined gap and Peierls distortion in trans-polyacetylene (a 1D, sp 2 bonded system with similar electronic structure to CNTs). In this work we, therefore, use this fraction of exchange and denote the approximation by HYB30.

We start by analyzing the (10,10) A-tube (R = 0.68 nm), for which the semi-local PBE and the short-range hybrid HSE06 functionals stabilize a gapless symmetric state. With the HYB30 approximation we find this state to be unstable with respect to a distortion along the A 1u (LO) phonon mode (Fig. 1a). This distortion opens a gap of 124 meV (orange curve in Fig. 1e) and breaks horizontal and vertical mirror symmetries. A similar Peierls like distortion was found in the smaller tubes in Ref. [15], while within PBE a softened phonon mode is observed [9]. If we now fix the lattice at the symmetric configuration (i.e. the undistorted structure) but allow for the same symmetry to break in the SCF potential, we obtain an electronic broken symmetry state, or a bond density wave [19], following the same pattern as the Peierls distortion (see Fig. 1b). The dimerized bonding pattern thus occurs spontaneously, prior to the lattice distortion. In the Supplemental Material [29] we demonstrate that this corresponds to a particular orbital mixing, or an excitonic instability. The purely e-e interaction-induced gap accounts for as much as 80% of the gap in the (10,10) tube, as demonstrated in Fig. 1e.

A symmetry analysis [29,34] shows that it is also possible to mix the orbitals with a different phase of v k , breaking the sublattice symmetry. This results in a chargetransfer between the atoms, as in the "dual" charge density wave state of Ref. [19], and corresponds to the solution found by QMC [21]. This state is coupled to a radial optical (RO) phonon mode of A 2g symmetry, as displayed in Fig. 1c. As the A 1u (LO) and A 2g (RO) mixings differ only by the phase of v k and descend from a doublet of nearly degenerate excitonic states [29, 35,36], they are expected to be close in energy. Indeed, starting from the symmetric configuration and by selectively switching off symmetry constraints we could stabilize also the RO state, being only slightly higher in energy. This state is, however, only weakly coupled to the lattice and thus no further stabilization occurs. This can be understood from the fact that the corresponding out-of-plane optical (ZO) mode in graphene has no EPC. The dimerized LO state is therefore found to be the ground state in all (n, n) tubes with n ≥ 10.

In the Z-tubes the LO mode is a totally symmetric A 1g mode [9]. In Fig. 2 we present the results for the (15,0) tube, which has a similar radius (R = 0.58 nm) to the (10,10) tube discussed above. In this case the PBE and HSE06 functionals open small gaps of the order of a few meV. The HYB30 functional increases the gap substantially, up to 165 meV at relaxed geometry. It is often assumed that HF theory can only enhance single-particle gaps obtained at the PBE level. In the present case, such an enhancement is, however, expected to be much smaller [37]. We will now show that the HYB30 gaps are qualitatively different from the PBE gaps. the atoms along the A 1g (LO) pattern we find the PBE gaps to vanish, as expected from a single-particle effect. In contrast, if we use HYB30 only a minimum is reached, at around 110 meV (corresponding to 65% of the gap at relaxed geometry). To the left in Fig. 2b we show the dimerized charge density close to the Fermi energy. This is the A 1g (LO) dimerized pattern. Despite the absence of symmetry breaking, the physical mechanism is the same as in the A-tubes, i.e., a dimerization along the LO mode. The similarity to A-tubes is further emphasized by the fact that we also find a charge transfer solution at higher energy. In order to induce this state we have to remove the preference determined by the EPC and start from the minimum-gap geometry with the proper symmetry constraints switched off (see Fig. 2d).

The C-tubes have larger unit cells and less symmetry than the A -and Z-tubes, and exhibit only near LO and TO modes [13]. The near LO mode is a totally symmetric mode like in the Z-tubes. The C-and Z-tubes, thus, behave very similarly. In this work we have called the metal-insulator transition (MIT) excitonic due to the importance of long-range interactions, but we note that the excitonic insulator state is usually assumed to involve symmetry-breaking [38]. Symmetry conserving MITs of (16,4), (20,5), (24,6)].

electronic origin are often called Mott transitions.

We now extend the analysis to larger CNTs (R > 0.7 nm). In Fig. 3 we compare our computed gaps for the fully relaxed LO states in the A-, Z-and some Ctubes, with the low-temperature experiment performed by Deshpande et al. [7]. Our results match experiment rather well and in this range of radii the calculated gaps are close to independent of chirality. The presence of a lattice instability induces a small deviation from the excitonic 1/R behaviour [21] at low enough temperature. Around R = 1.5 nm, the lattice contribution vanishes and the gaps become purely electronic. The electronic gaps of the A-tubes (dashed orange line) follow a perfect 1/R behavior in the whole range, which allows us to extrapolate the results for R > 1.6 nm, indicated by the full orange line. The electronic states in the larger CNTs are highly degenerate, possibly infinitely degenerate (i.e. a complete phase invariance of the ground state). We note, however, that a strong EPC is likely to destroy super-transport properties associated with the excitonic ground state [39]. We also recall the approximate nature of the hybrid functional which tend to underestimate the lattice distortion [15]. The magnitude of the gaps varies as α 2 . To illustrate how a variation of α affects the gaps we have added an 'error-bar' in Fig. 3. Within a realistic range (0.2 < α < 0.4), the gaps are compatible with all experimental results presented so far [7,8]. We note that even on the lower bound we have a substantial gap-opening.

We now turn to the smaller CNTs for which lattice instabilities become increasingly important. The electronic states split in energy already by the e-e interaction, with 0 0.5 the A 1 -Peierls state always lower in energy. The splitting becomes larger the smaller is the tube, similarly to what is found for excitonic states in small semi-conducting CNTs [35,36,40,41]. For the small nanotubes we also find a structural relaxation along the A 2 (RO) phonon mode associated with the charge-transfer electronic state. We focus on the (3,3) tube, which is interesting in its own right as CNTs of this radius have been reported to superconduct up to 15 K [4]. In addition to the Γ-transitions we also study the 2k F -instabilities. As 2k F is incommensurate with the lattice we can, however, only study this instability approximately. We found that for the (3,3) tube a cell which is 7 times larger than the original cell approximates 2k F within 1.1%. By relaxing the density in this supercell we found, indeed, a symmetry broken solution of electronic origin [29]. At the electronic level this state is less stable than the ones at Γ, as shown in Fig. 4, which again follows the trend of the splitting between exciton binding energies [36]. However, the strength of the electron-lattice interaction in such a small tube is very large and a lattice instability is present even with local functionals [11,12]. As can be seen in Fig. 4, the interaction with the lattice increases the 2k F gap by a factor of 5 and inverts the stability order, turning the 2k F instability into the lowest state for the (3,3) tube. The role of the long-range e-e interaction is, however, still very important. In Tab. S1 of the Supplemental Material [29] we compare our results for the phonon frequencies to PBE data from Ref. [11]. We see that the instability with HYB30 is substantially stronger and broader in reciprocal space. With increasing radius the 2k F lattice contribution reduces, however, quickly, turning also the 2k F instability purely electronic and degenerate with the Γ-instabilities.

In conclusion, we have formulated a universal gapopening mechanism in metallic CNTs based on dimerization that takes into account both electronic and lattice degrees of freedom. In the Z -and C-tubes the gaps appear without any symmetry breaking, while in the A-tubes they are induced by a Peierls lattice instability that breaks the symmetry. We have shown that these seemingly different phenomena share the same origin in an electronic transition associated with the A 1 (LO) phonon mode. While the electron-lattice interaction is crucial in the smaller tubes, in the experimentally observed range of diameters the electronic transition dominates and yields gaps that can reproduce experimental findings.

This work was performed using HPC resources from GENCI-TGCC/CINES/IDRIS (Grant 2017-DARI A0010907625).

SUPPLEMENTAL MATERIAL FOR: EXCITON-PEIERLS MECHANISM AND UNIVERSAL MANY-BODY GAPS IN CARBON NANOTUBES

Below we present further details on the Hartree-Fock (HF) excitonic ground state, the symmetry of the CNT Bloch orbitals, and supplementary data on the 2k F instability in the (3,3) tube.

EXCITONIC INSTABILITIES WITHIN HARTREE-FOCK THEORY

The wave function of the excitonic ground-state is defined as a Slater determinant built up from the symmetry broken orbitals. These orbitals can be expressed as linear combinations of valence (v) and conduction (c) orbitals from a symmetry restricted calculation. Orbitals can be mixed with the same momentum, breaking a point group symmetry, or they can differ by momentum 2k F , breaking the translational symmetry. We can write the new valence orbitals as

φv,k = u k ϕ v,k + v k ϕ c,k , (4)
where k is the momentum relative to the valleys K or K and spin-indices have been suppressed (only spin-singlet states are considered). We then form a new Slater determinant

|Ψ = k φ † v,k |0 . (5)
To determine whether this new ansatz lowers the energy one should minimize the expectation value Ψ| Ĥ|Ψ with respect to the coefficients v k , u k . Ĥ is the many-body Hamiltonian given by

Ĥ = T + Vext + Ŵ (6)
where T is the kinetic energy operator, Vext is the external lattice potential and Ŵ is the interaction operator of the electron-electron interaction (v). The expectation value of Ŵ is the standard HF result

Ψ| Ŵ |Ψ = 1 2 kk φv,k φv,k |v| φv,k φv,k -φv,k φv,k |v| φv,k φv,k (7)
Assuming that the orbitals ϕ v,k , ϕ c,k are solutions to the HF equations of a symmetry restricted calculation we can write

Ψ| Ĥ|Ψ = k |u k | 2 ε v,k + k |v k | 2 ε c,k (8)
+ 1 2 kk |u k | 2 [|u k | 2 -2][ ϕ v,k ϕ v,k |v|ϕ v,k ϕ v,k -ϕ v,k ϕ v,k |v|ϕ v,k ϕ v,k ] (9)
+ 1 2 kk |v k | 2 |v k | 2 [ ϕ c,k ϕ c,k |v|ϕ c,k ϕ c,k -ϕ c,k ϕ c,k |v|ϕ c,k ϕ c,k ] (10)
+ kk |v k | 2 [|u k | 2 -1][ ϕ v,k ϕ c,k |v|ϕ v,k ϕ c,k -ϕ v,k ϕ c,k |v|ϕ c,k ϕ v,k ] (11)
+ 1 2 kk u * k u * k v k v k [ ϕ v,k ϕ v,k |v|ϕ c,k ϕ c,k -ϕ v,k ϕ v,k |v|ϕ c,k ϕ c,k ] + c.c. ( 12
) + 1 2 kk u * k v * k v k u k [ ϕ v,k ϕ c,k |v|ϕ c,k ϕ v,k -ϕ v,k ϕ c,k |v|ϕ v,k ϕ c,k ] + c.c. (13)
+ kk u * k |v k | 2 v k [ ϕ v,k ϕ c,k |v|ϕ c,k ϕ c,k -ϕ v,k ϕ c,k |v|ϕ c,k ϕ c,k ] + c.c. (14)
+ kk v * k u k [|u k | 2 -1][ ϕ c,k ϕ v,k |v|ϕ v,k ϕ v,k -ϕ c,k ϕ v,k |v|ϕ v,k ϕ v,k ] + c.c. (15)
A variation with respect to the expansion coefficients u k , v k yields the HF equations of the excitonic ground-state. [25] We see that the same expressions hold in the case of a hybrid functional by simply multiplying the Fock term with the fraction of exchange and by adding a single-particle PBE term. In model Hamiltonians [25,39] only terms which are invariant with respect to a k-independent phase of v k are kept (terms in Eqs. (7-10,12) above), but we see here that, in general, there are terms which are not invariant with respect to this phase. Indeed, for small CNTs we find a splitting at the electronic level between the A 1u (LO) and the A 2g (RO) states (two states that differ by a phase angle of π/2, see below). This splitting gets smaller with the size of the tube and eventually vanishes, indicating that for the larger tubes R > 1.6 nm the phase dependent terms vanish.

EXCITONIC BROKEN-SYMMETRY STATES IN A-TUBES

The symmetry group of (n, n) armchair nanotubes includes the following symmetry operations: [34] • an n-fold rotation around the axis of the tube

• infinite rototranslational symmetry, generated by a translation of a/2 combined with a rotation of π/n around the tube axis, which generates the periodicity

• a vertical and an horizontal mirror, σ v and σ h crossing at the center of each hexagon, plus a 180 degrees rotation U = σ v σ h . These symmetry operations are represented in Fig. 5. It should be noted that applying either operator σ h or U inverts the periodic direction. The atoms composing the nanotube can be divided into two sublattices A and B, with the first neighbours of each atom always belonging to a different sublattice. Applying either operator σ v or U exchanges the two sublattices.

Let us consider the crystalline orbitals of the symmetry-restricted problem in the highest valence band (π) ϕ v,k and the lowest conduction band (π * ) ϕ c,k . These are in the same representation of the rotational subgroup and have opposite parity Π = ±1 with respect to σ v (the valence band ϕ v,k has parity Π = -1 from Γ to K and Π = +1 from K to border zone, while the inverse is true for ϕ c,k ). [34] The wavefunction for a Γ exciton can be expressed as:

Ψ exc = k A k ϕ * v,k ϕ c,k . (16)
Due to the symmetry of the orbitals, Ψ exc is symmetric with respect to rotations and antisymmetric with respect to σ v . The relationship U = σ v σ h implies that the exciton wavefunction has opposite parity with respect to U and σ h . Moreover, the effect of symmetry operators U and σ h on the orbitals can be written as: [34]

σ h ϕ i,k = ϕ i,-k , U ϕ i,k = Πϕ i,-k (17)
with i = v, c. It is easy to verify that the parity of Ψ exc with respect to U is even if A k = -A -k (i.e. if the exciton wavefunction is antisymmetric with respect to exchanging the two valleys at K and -K) and odd if A k = A -k , as in the case of chiral semiconducting nanotubes. [36,40] These are often referred to as gerade and ungerade excitons. In semiconducting tubes, these excitons are slightly split in energy, and form a multiplet toghether with two excitons at 2K. Note that, if the two excitons are exactly degenerate, they generate a subspace of allowed excitations with no definite parity with respect to U and σ h . Similar symmetry considerations apply to the valence states φv,k in Eq. ( 4), which are obtained by mixing valence and conduction orbitals. If the e-e interaction is responsible for the mixing, these states can be interpreted as a spontaneous condensation of excitons, as mentioned in the main text. The charge density of the φv,k mixed state is:

n(r) = k | φv,k (r)| 2 = k |u k | 2 |ϕ v,k (r)| 2 + k |v k | 2 |ϕ c,k (r)| 2 + 2 k Re u * k v k ϕ * v,k (r)ϕ c,k (r) . (18)
The first and second terms are symmetric with respect to the full symmetry group. The third term resembles the exciton wavefunction, and the same symmetry considerations apply to it. In particular, it is asymmetric with respect to σ v and the symmetry properties with respect to σ h and U depend on the relationship between v k and v -k , assuming that u k is real and positive (equivalent to fixing the overall phase of the wavefunction). The relative phase between v k and v -k is then restricted only by time-reversal symmetry:

φv,-k = φ * v,k = u * k ϕ * v,k + v * k ϕ * c,k = u * k ϕ v,-k + v * k ϕ c,-k =⇒ v -k = v * k (19)
Thus if v k is purely real, v -k = v k and hence the third term of Eq. ( 18) is antisymmetric with respect to U and symmetric with respect to σ h (as in the case of the ungerade exciton). If v k instead is purely imaginary (i.e., having a phase e ±iπ/2 ), v -k = -v k the term has the same symmetries as the gerade exciton. The total charge density n(r) has no definite symmetry with respect to σ v whenever v k is finite, since it is the sum of two symmetric and one antisymmetric term. If v k is real, the symmetry U is also broken allowing a charge-transfer between the two sublattices (the A 2g (RO) solution in the main text). If v k is imaginary, σ h is instead broken but U is preserved and the sublattices are symmetry equivalent. Indeed, in this case all atoms in the CNT are still related to each other by symmetry operations, so no charge-transfer between atoms is possible. However, this state breaks the symmetry between the bonds connecting each atom to its neighbours along the periodic direction, resulting in a dimerization of the electronic ground state (the A 1u (LO) solution in the main text). In Fig. 6 we present the calculated orbital hybridization, i.e., the degree of mixing between the valence and conduction bands (Eq. ( 1) in the main text) given by

|v k | 2 = | ϕ c,k | φv,k | 2 . ( 20
)
The A 2g and A 1u electronic symmetry broken states have almost identical mixing and can thus only differ by a k-dependent phase factor v k → e iθ k v k , in agreement with the symmetry analysis presented above.

SUPPLEMENTARY DATA ON THE 2kF INSTABILITY

In Tab. I we have quantified the difference between the strenght of the 2k F Peierls distorsion in the (3,3) tube between a local functional (PBE) and the 30% hybrid that includes the long-range e-e interaction. Starting from the relaxed Γ geometry (the RO state for this tube) we calculate the hybrid phonon frequencies for the unstable mode at different values of k, achieved by using supercells of different sizes. The results are compared to PBE data from Ref. 11 taken at the same distance from 2k F . We see that the instability with the hybrid functional is substantially stronger and broader in reciprocal space, in addition to the fact that it is preceded by an electronic transition.

  FIG. 1. (a) and (c) Displacement pattern of the LO and RO modes in the A-tubes. (b) and (d) Density obtained from states within 50 meV below the top of the valence band. (e) Left: Band structure of the high-symmetry phase of the (10,10) tube. Right: A zoom around the Fermi-level. The gap of the LO mode is shown with (el. + dist.) and without (el.) lattice relaxation, and compared to that of the RO.

  FIG. 2. (a) and (c) Displacement pattern of the LO and RO modes in the Z-tubes. (b) and (d) Density obtained from states within 50 meV below the top of the valence band. (e) Left: PBE band structure of (15,0) tube. Right: A zoom around the Fermi-level. The gap of the LO mode is shown with (el. + dist.) and without (el.) lattice relaxation, and compared to that of the RO.

2 FIG. 3 .

 23 FIG.3. Exciton-Peierls gaps in comparison with experiment (black squares). The orange circles (blue squares) correspond to the fully relaxed states of the A(Z)-tubes. The dashed orange line is an interpolation of the electronic gaps of the A-tubes, extrapolated with a 1/R fit (full orange line). The green diamonds are C-tubes [(16,4),(20,5),(24,6)].

FIG. 5 .

 5 FIG. 5. Panel (a) shows the lattice of the (10,10) nanotube with symmetry operators indicated. Panel (b) and (c) show the density of the A1u and A2g purely electronic broken symmetry states. The symmetry that is preserved is indicated.

FIG. 6 .

 6 FIG. 6. (a)A zoom around the Fermi-level of the band structure of the (10,10) A-tube, comparing the symmetric solution with the A2g charge transfer solution (blue) and the A1u-LO one (green). In panel (b), the orbital hybridisation, i.e. the mixing between valence and conduction states of the high-symmetry phase is shown.

FIG. 7 .

 7 FIG. 7. Isosurface of the density obtained from the states close to the Fermi level in the 2kF excitonic ground state of the (3,3) nanotube, having the same symmetry as the 2kF Peierls instability.