text:
Hans Kristian Nilsen
email: hans.k.nilsen@uia.no

Anne Vegusdal
email: anne.vegusdal@uia.no

Mathematics at the enterprise: Industry, university and school working together to facilitate learning

Keywords: Diversity, relevance, authenticity, out-of-school contexts, enterprise

Based on a pilot study, this paper reflects upon how the industry, university and school can work together to facilitate learning of mathematics. Through a project called MathEUS (Mathematics at the Enterprise, University and School) a modern energy recovery enterprise, Returkraft, the University of Agder and a school collaborated to offer pupils (in 8 th and 9 th grade) an opportunity to engage with mathematics contextualized at an enterprise. Through comparing the 2015 edition with our re-design in 2016, we discuss the outcome of the project. In particular, we focus on the factors necessary for such a project to succeed. Findings suggest that mathematics contextualized at an enterprise as an isolated event has limited value. Further, findings seem to indicate that working together with the enterprise over time, by including a pre-project, strengthens pupils' experience of mathematics as a relevant subject.

Introduction

Diversity in mathematics education relates to political, cultural and linguistic aspects. In this paper, the issue of diversity will be addressed in terms of different sites where mathematics education potentially could take place. [START_REF] Meaney | Learners in transition between contexts[END_REF] emphasize the importance of transitions between different contexts and that such transitions "can be a fairly minor issue for learners if they perceive similarities in what knowledge is valued and how learners and others should interact together and with the mathematical content" (p. 169). Throughout the decades, a growing body of research has been carried out, focusing on the learning of mathematics in different contexts (Masinglia, Davidenko, & Prus-Wisniowska, 1996). By linking mathematical tasks to an enterprise, we hope that the pupils experience mathematics as important, also outside the classroom, and in turn this influences their motivation for learning mathematics at school. In Norway, there has been an increased focus on how mathematics could be taught in ways that is perceived by pupils as relevant and more related to their own reality and experiences (Det Kongelige Kunnskapsdepartement, 2011). Diversity in teaching methods and arenas of learning has been important issues in this debate (Norges offentlige utredninger 2015, p. 8). Pupils in Norway are mixed together heterogeneously until grade 10 and with the curriculum focus being academic, offering pupils few perspectives on, for example, vocational and professional practice. In the attempt to diversify such experiences, we found that the processes going on at the energy recovery enterprise, Returkraft, bear the potential of visualizing mathematics at a working place. Further, Mathematics at the Enterprise, University and School (MathEUS), in some sense should be regarded as a response to the identified need of moving the teaching of mathematics in a more practical and vocationally-oriented direction. The last two years (2015 and 2016) funding from Regional Research Fund Agder enabled research to be carried out on the MathEUS project. In this paper, we pose the following research questions: 1) Which factors are of importance in the collaboration between school and industry for the pupils to experience the mathematics involved as relevant and authentic? 2) What might prevent the pupils for experiencing the project as relevant and authentic? Our discussions focus on the differences in outcome between the first year and the redesigned project run in the second year.

Theoretical background

The idea that learning is situated within a certain context and practice was in many ways coined by [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF] and in mathematics education, research has shown that the environment in which activities takes place is of great importance for the outcome [START_REF] Nunes | Street mathematics and school mathematics[END_REF]. In the MathEUS project, mathematics is contextualized and linked to the activities at an energy recovery enterprise, Returkraft. Situated in this case alludes both to the physical environment at Returkraft (which surrounded the pupils while they were doing their activities) and to the mathematical content, which was consequently linked to the enterprise. By establishing this link between content and context, it is our aim that the mathematical tasks and activities provided appear more relevant and authentic to the pupils. When applying the concept of relevance, we draw on the work of Hernandez-Martinez and Vos (submitted) where the main point is that "it involves a judgment of value that is made by a person involved in the activity" (p. 26) and that the "judgment is connected to the motive or object that drives the activity" (p. 26). [START_REF] Nyabanyaba | Whither relevance?: Mathematics teachers' discussion of the use of 'reallife' contexts in school mathematics[END_REF] and [START_REF] Dalby | The relevance of mathematics: The case of functional mathematics for vocational students[END_REF] point to the complexity of the relevance concept in mathematics by listing a number of different ways mathematics could be conceived of "relevant", depending on contextual, content-related and affective factors. Authentic has to do with the nature of the activity and has several somewhat distinct definitions. In the case of MathEUS, we find the definition from [START_REF] Gulikers | The surplus value of an authentic learning environment[END_REF] to be useful, "An authentic learning environment provides a context that reflects the way knowledge and skills will be used in real life" (p. 509). Here both the contextual aspect and the use value are emphasized, which is fully in line with the purpose of the MathEUS project. To achieve the goal of authentic tasks, considered relevant to the pupils, we aimed also to overcome some of the typical challenges that guided tours of enterprises often entail. Recent research, both national and international, shows that the "classical" excursion day, where pupils use a day to visit an enterprise in terms of a guided tour, is often disconnected to what they normally do at school and leads to some didactical challenges. The pupils easily become passive, engaging in artificial and low-quality tasks (like memorizing and duplicating information posters) due to the lack of preparatory and follow-up work [START_REF] Dewitt | A short review of school field trips: Key findings from the past and implications for the future[END_REF][START_REF] Remmen | Implementation of guidelines for effective fieldwork designs: Exploring learning activities, learning processes, and student engagement in the classroom and the field[END_REF].

Methods

In 2015, two schools were participating, each with one secondary class of 9 th grade pupils (14-15 years old).

The implementation of the MathEUS project

The schools' visiting day at Returkraft took place in March on two different days, one day for each class. Before the day of visit, student teachers completing a master degree at the university had designed and prepared mathematical tasks for the pupils. After a guided tour at Returkraft, the student teachers constructed the tasks over a period of two weeks, and during this period staff from Returkraft was available for helping them with information concerning the enterprise. Teachers from the participating schools were also available for commenting on their suggestions. The premises for these tasks were that they should in some way be linked to the enterprise at Returkraft. These tasks were given to the pupils on their day of visit, and they were given approximately two hours to work with the tasks in groups of four to six pupils. The tasks were solved in a 'classroom' at Returkraft. Following an evaluation of MathEUS as it had been carried out in 2015, we decided to make some changes in an attempt to improve the project (see also the section "results and discussion"). We wanted to strengthen the quality of the tasks so that the pupils experience them as more relevant. In addition we wanted the pupils' experience with Returkraft and the mathematics they engaged with to be more than just an isolated event. Aiming towards this, we invited the class to take part in a preproject (lasting for six weeks), with a representative for Returkraft coming to their class early in the semester to initiate the following assignment: "How can we improve peoples' habits, when it comes to sorting their garbage for recycling?" To solve this assignment, the pupils developed questionnaires and went out to different geographical locations to interview people about their sorting habits. Due to the lack of responses during oral interviews the pupils posted the questionnaires on social media, and the number of response escalated rapidly. Their empirical data were treated in spread sheets, resulting in different diagrams, and the results were presented at the university in front of an audience consisting of both the student teachers and employees from Returkraft. In turn, this pre-project served as an important source for the student teachers when they designed the tasks for the pupils. In 2016, one secondary class of 8 th grade pupils (13-14 years old) participate in the project.

Methods and data collection

We (the authors of this paper) were responsible for carrying out research and at the same time we planned the implementation of the project together with the participating schools and employees from Returkraft. We also provided the framework conditions for the student teachers' elaborations of the tasks. The pupils, the student teachers, the teachers (at the participating schools) and the employees at Returkraft all served as informants for our research. In this paper, we mainly focus on data from the pupils. Conducting research at the same time as we are responsible for carrying out the project situates us within the domain of action research. [START_REF] Bryman | Social research methods[END_REF] describes this as "an approach in which the action researcher and client collaborate in the diagnosis of a problem and in the development of a solution based on the diagnosis" (p. 709). We conceive of this definition to be in line with our agenda, where we use data from our experiences to discuss, develop and improve the forthcoming editions of the project.

To measure parts of the outcome of the MathEUS project, we developed a web-based questionnaire which the pupils should answer after the project was carried out. Except from minor changes and some reformulations, we used the same questionnaires both years of research, for the sake of comparison. The questions were mainly statements that the pupils were asked to scale from 1 to 6, based on their "degree of agreement" and in total there was 34 statements to be evaluated. Mainly these statements aimed to inform our research questions mentioned earlier in terms of measuring the pupils' own experience of their learning outcome, their motivation and beliefs and whether they consider the project meaningful and relevant. In addition, some open ended questions were posed where the pupils had the possibility to comment on these aspects. Since two school classes were involved in 2015, and just one in 2016, the sample size varied and in 2015 there was 44 respondents to the questionnaires, while 23 responded in 2016. But in 2016 we collected some additional data by including a semi-structured interview with three of the pupils. These pupils were randomly selected among those who volunteered when we visited the class. In this interview, we went deeper into the pupils' own experiences related to the project, and to the content of the tasks and activities provided for them by the student teachers. Both in 2015 and in 2016, interviews involving representatives from the three collaborators (the enterprise, the university and the schools) were conducted, mainly focusing on the outcome of the project. Both years we made observations and voice recorded all the group work sessions at Returkraft. As part of our research, we also conducted interviews with some of the student teachers responsible for designing the tasks.

Results and discussion

Since the scale on our questionnaires ranges from 1 to 6, we sort the results in two halves for the sake of overview. The "low achieving" half ranges from 1 to 3 and the "high achieving" half ranges from 4 to 6. Figure one gives an overview of some of the questions we consider as most significant. One can observe from the first columns in the diagrams (1a) that the majority of the pupils both years felt that they understood the mathematics being taught. About half the pupils felt that they discovered new aspects of mathematics in the 2016 edition of MathEUS, a slight improvement compared to 2015 (1b). An improvement related to the experience of mathematics used in a real-life setting is visible (1c), while the experience of mathematics as being meaningful is almost unchanged (1d). Finally, there is a clear improvement on almost 20 per cent related to the appreciation of mathematics as a consequence of the MathEUS project (1e). Since the number of pupils forming the basis of these results is limited and varying one should be careful to draw strong conclusions only from this data set. But since statements in the questionnaires consequently were higher rated in 2016 compared to the year before, a positive trend could be suggested. In addition to the scaling of different statements, there was also a possibility for the pupils to add comments and give some additional justifications for their scaling. In 2015, about half the pupils wrote critical comments, especially related to the question of whether or whether not their expectations had been fulfilled. "I thought they would try to make this fun", "it was extremely boring, tiring and difficult maths" and "I thought we would learn more. The tasks were just like a typical maths tests", serve as examples of such comments. Despite that 43 out of 44 pupils in 2015 expressed themselves positively about learning mathematics at a working place, about half of the pupils responded in negative terms related to the question about the fulfilment of expectations, in line with the previous quotations. These comments mainly implied that a negative experience with the content of the tasks. Even though the tasks were designed with the purpose of being relevant, in terms of creating a link between school mathematics and the enterprise, the pupils did not seem to perceive this the same way. [START_REF] Dalby | The relevance of mathematics: The case of functional mathematics for vocational students[END_REF] emphasizes that even though a realistic context is provided, the impact on pupils can vary since "the context itself is often no more than a metaphor to illustrate an aspect of pure mathematics rather than authentic use of a scenario as a source of mathematics" (p. 90). In the aftermath of MathEUS in 2015, we saw that several of the studentdesigned tasks could represent examples of contextualized tasks, but with little authenticity.

Excerpt 1: Excerpt from task, 2015

The task above serves as an example of a task where the context was reduced to serve only as a metaphor, rather than an authentic use of Returkraft as a source of mathematics.

When revising MathEUS 2015, we wanted to improve especially on two aspects: 1) Make the content of the tasks more relevant and authentic for the pupils and 2) Engage the involved pupils and student teachers earlier in the process, so that the visit to Returkraft became something more than just an isolated event. In this process, we build on DeWitt and Storksdieck's ( 2008) "boarder visions on field-trips" (p. 181) and their research-based conclusions that field trips and out-of-school context ought to be embedded in teaching, more holistically. Even though their focus was on natural sciences, we found these ideas to be relevant for mathematics and in the 2016 edition of MathEUS we included a pre-project (as described in the "methods" section). Due to the pre-project, the student teachers designed tasks were also used earlier in the process.

From the web-based questionnaires, illustrated in figure 1, there are some indications that pupils' experiences in general were more positive in 2016 compared to 2015. Interviews with some of the pupils, also involved statements that could be interpreted as having to do with relevance and authenticity.

Interviewer: Earlier in the interview, you said something about "mathematics in practice"…was this something which characterized the tasks at Returkraft, or did you feel that you just as well could have done this at school?

[…]

Pupil: Since we had been to Returkraft and the content was about Returkraft, and the numbers used were the same as we had learned about…the numbers weren't just made up, in a way.

This pupil emphasizes the experience of working with "real" tasks, in terms of pointing to a close connection between the context and the content of the tasks. Since the student teachers had followed their pre-project and had access to all their results and data, they were also able to build on pupils' own data from the pre-project when designing the tasks, and in this case provided a more authentic context.

Excerpt 2: Excerpt from task, 2016

By using numbers from the pupils own pre-project, this task illustrates how one student teacher applied these data to discuss statistical issues. Based on observations from the pupils engaging with this task, we see this as an example of successfully creating a link between the context and the mathematics involved. Pupils judged this as "relevant" since an authentic connection between the mathematics involved and the context is provided in terms of linking the task to something familiar, namely their pre-project.

From this relatively brief analysis, we suggest that pupils' experience of relevance, and the authenticity of tasks provided in an out-of-school context not only depends on the tasks themselves, but also on the pupils' possibilities to relate to the content. For the context to "reflect the way knowledge and skills will be used in real life" (Gulikers et al., 2015, p. 509) a preparatory phase, were pupils are offered the possibility to become familiar with the context seemed to be advantageous.

Conclusions

Providing out-of-school contexts like excursions and field trips are, unfortunately, often being carried out as isolated, stand-alone events [START_REF] Dewitt | A short review of school field trips: Key findings from the past and implications for the future[END_REF][START_REF] Remmen | Implementation of guidelines for effective fieldwork designs: Exploring learning activities, learning processes, and student engagement in the classroom and the field[END_REF]. There are few reasons to believe that this is different in mathematics. Our findings tend to show that if mathematics in out-of-school contexts is treated like stand-alone activities, the outcome when it comes to authenticity and pupils' experiences of relevance is limited. Pupils' experience of relevance changed in a positive way when a preparatory phase in terms of a pre-project was implemented. [START_REF] Nyabanyaba | Whither relevance?: Mathematics teachers' discussion of the use of 'reallife' contexts in school mathematics[END_REF] argues that pupils and teachers' conceptions of relevance often differ. In line with this, we suggest that even though mathematics in out-of-school contexts through excursions days at enterprises as stand-alone activities is carried out with good intentions from teachers, such activities ought to be more substantiated in teaching. Authenticity entails reflection on knowledge and how it could be used in real life (Gulikers et al. 2015) and for such a reflection to happen, [START_REF] Meaney | Learners in transition between contexts[END_REF] point to the necessity of being able to make transitions between these different contexts. From our experience a pre-project substantiates the out-of-school context and better prepare the pupils for the subsequent visit to the enterprise. Hence, the connection between the mathematical content and the context provided are perceived by pupils as more relevant. "Making connections between mathematics and life that appear authentic and convincing for students" (Dalby, 2014, p. 91) serves as core criteria in the question of relevance and learning, also when it comes to the subject of mathematics.

Implications and possibilities

From a societal and political perspective, this paper addresses issues concerning the contextual nature of mathematics teaching and learning. One aspect is the pupils' outcome of experiencing mathematics in an out-of-school setting, but in a boarder perspective, MathEUS also contributes to strengthen the bond between school and industry, which according to [START_REF] Sjaastad | Naeringsliv i skolen: Erfaringer med partnerskapsavtaler mellom skoler og bedrifter og synspunkter på organiseringen av prosjektet [Industry in school: Experiences with partnerships agreements between schools and enterprises and view on the organization of the project[END_REF], is "too weak" (p. 16).

The results and reflections in the wake of the two previous editions of MathEUS serve as a basis for further developing our collaboration strategies. We found that by involving the enterprise at an early stage, pupils to a larger extent were able to make connections between the context and the mathematical content. The pre-project also entailed interdisciplinary activities, where mathematics became an important contributor and ICT was applied by the pupils as a crucial tool in the process of visualizing their findings. The interplay between mathematics and natural sciences in particular played a significant role in the pupils' activities, both during the pre-project and at their visiting day at Returkraft. The topic concerning recycling of garbage also links our project to local and global environmental issues. People's attitudes towards recycling of garbage, were a valuable outcome also for the enterprise, as Returkraft aims to reach out to the public with messages concerning the importance of such issues. These are all, from our point of view, valuable synergistic effects worth looking deeper into. Task-design is a topic of research on its own, and the process of developing tasks and activities within this particular context could also be worth mentioning as potential, forthcoming research.

Figure 1 :

 1 Figure 1: Pupils scaling (in per cent) in 2015 compared to 2016

Proceedings of CERME10

Acknowledgments

In the process of developing MathEUS, the Norwegian Centre for Science Education has played a vital role as contributor both scientifically and technically. We want to thank them especially for their professional and didactical input which eventually led to the pre-project, and for assisting us in the development and in the treatment of questionnaires. Kristiansand municipality, educational centre and Node Southern Cooperate should also be mentioned for their support both pedagogically and administratively. Finally, our gratitude goes to the enterprise Returkraft, which has opened the doors and met us in all our requests.


Anne-Claire Gaillot

Victor A Drits

Alain Plançon

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

Structure of Synthetic K-rich Birnessites Obtained by High-Temperature Decomposition of KMnO 4 . III. Phase and Structural Heterogeneities

Synthetic K-rich birnessites (KBi) were prepared from the thermal decomposition of a fine-grained KMnO 4 powder heated in air atmosphere at temperatures ranging from 200-1000°C. The qualitative analysis of powder X-ray diffraction (XRD) patterns reveals a complex range of structural transformations from one metastable phase to the other, often through intermediate mixed-layer structures (MLSs). Phase and structural heterogeneities of KBi samples synthesized at 700°C, 800°C and 1000°C (referred to as KBi 7 , KBi 8h and KBi 10h ) have been studied in details by chemical and thermal analysis and by simulation of K + 0.27 (Mn 4+ 0.77 Mn 3+ 0.21 0.02 )O 2 •0.53 H 2 O for KBi 10h , are in agreement with the main crystal chemical features of the phases prevailing in these samples. When heated to 350°C, the 2O polytype presents a hexagonal layer symmetry with a b parameter (2.894 Å) which is significantly increased as compared to that determined at room temperature (2.850 Å). Both modifications arise from random orientation of elongated Mn 3+ octahedra along directions forming n60° angles with the a axis. The main factors responsible for the phase and structural heterogeneity of the KBi samples are discussed.

the experimental powder XRD patterns. Two-layer orthogonal (2O), and hexagonal (2H) as well as three-layer rhombohedral (3R) polytypes were identified in these samples. The 2O structure consists of vacancy-free layers and their orthogonal symmetry is linked to the high content of layer Mn 3+ cations and to the unique azimuthal orientation of Mn 3+ octahedra which are elongated because of Jahn-Teller distortion. In the 2H and 3R polytypes, the layers have a hexagonal symmetry as they contain only Mn 4+ -and vacant octahedra. As a result, their interlayers have a heterogeneous cation composition, because of the migration of Mn 3+ from the layers to the interlayers. In addition to the periodic KBi polytypes, KBi 7 and KBi 8h contain MLSs in which layer pairs of the 2H polytype are interstratified at random with those of the 3R or of the 2O polytype. Interstratification of incommensurate 2O and 2H structural fragments leads to peculiar diffraction effects and represents a new type of structural disorder in birnessites. The increase of temperature from 700°C to 1000°C is associated with the replacement of 3R/2H, 2H, and 2O/2H mixed-layered structures by the more stable 2O polytype. KBi 10h consists of a mixture of a minor 2H phase with three 2O varieties having slightly different layer unit-cell parameters. This phase heterogeneity results from the partial disorder in the orientation of Mn 3+ octahedra. The average structural formulae, K + 0.265 Mn 3+ 0.145 (Mn 4+ 0.825 0.175 )O 2 •0.68 H 2 O for KBi 7 and KBi 8h , and

Introduction

Layers building up hydrous manganese oxides such as birnessite consist of edge-sharing MnO 6 octahedra. Hydrated exchangeable cations are present in the interlayer space to compensate for the layer charge deficit arising from the presence within layers of Mn 3+ cations and/or of vacant layer octahedra. [1][2][3][4][5][6] Birnessite is easily synthesized under laboratory conditions and some of the synthetic birnessite species have commonly been used as analogs of the natural species to determine the structural mechanism of heavy metal sorption or to investigate the structural modification of birnessite as a function of pH (e.g.). [5][6][7][START_REF] Manceau | Synchrotron X-ray methods in clay science[END_REF][START_REF] Manceau | [END_REF][10][11][12][13][14][15][16] Simultaneously, synthetic birnessites have drawn a special attention as a potential cathode material for secondary batteries following their intercalation with lithium cations. [17][18][19][20][21] Because of its apparent lamellar framework stability, K-rich birnessite (hereafter referred to as KBi) represents a promising candidate as lithium battery cathode material. This birnessite variety can be obtained from the thermal decomposition of KMnO 4 , 21 or of a mixture of MnO, KNO 3 and LiOH, 22 or from the Mn 7+ reduction under hydrothermal conditions. 19,20 Because of their influence on reactivity it is essential to have a detailed understanding of the structural and chemical features of these birnessite species. However, structural studies of birnessite are most often impaired as these compounds occur normally in a finely dispersed state. In addition, structural variety of birnessite is extreme as these species may differ from each other by their layer symmetry, their layer stacking, or the content and distribution of heterovalent Mn cations and layer vacancies within layers (e.g.). Furthermore, structural and chemical defects, such as the interstratification of different layer types and stacking faults, are frequent in these compounds. [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF] As a result, only a few publications were devoted to the structural study of KBi varieties. According to Chen et al. 19 KBi variety synthesized under hydrothermal conditions presents a three-layer rhombohedral (3R) structure in which octahedral layers are shifted with respect to each other by -a/3 (assuming an orthogonal basecentered unit cell) leading to the prismatic coordination of interlayer K coordinated to layer oxygen atoms (O layer ). Kim et al. 21,22 synthesized new high temperature KBi and K-Li birnessite-like varieties with a two-layer polytype, in which adjacent layers are rotated with respect to each other by 180° around the c axis passing through the layer Mn (Mn layer ) cations.

Even though the main idealized structural and chemical properties of high temperature KBi have been determined, 21 crystal chemistry of this new polytype and significant details of its structure remain poorly understood.

In the two companion articles [START_REF] Gaillot | [END_REF]25 it is shown that KBi obtained from thermal decomposition of KMnO 4 at 800°C and 1000°C have different unit cells and supercells, different sources of the layer charge deficit and specific distribution of heterovalent Mn cations. However, the two KBi samples described in these articles [START_REF] Gaillot | [END_REF]25 have a high structural perfection and do not contain (KBi 10o ) or contain only a few stacking faults (KBi 8o ). In contrast, the present article deals with phase, structural and chemical heterogeneity of KBi varieties obtained at different temperatures and assesses the origin of such variability.

Specific structural and chemical heterogeneities are described for KBi and the original methodology used to reveal these structural defects is detailed.

Main Structural Features of KBi 8o and KBi 10o

KBi samples with a high degree of structural order were obtained from thermal decomposition of KMnO 4 at 800°C (KBi 8o ) and 1000°C (KBi 10o ) and their structure refined from a single crystal [START_REF] Gaillot | [END_REF] (KBi 8o ) or using the Rietveld method on a powder XRD pattern (KBi 10o ). 25 As the present description of KBi structures relies for a large part on the structures refined on these two samples, the main features of the refined models will be summarized.

Both samples have a two-layer unit cell in which adjacent layers are rotated with respect to each other by 180° around the c axis passing through the Mn layer cations. As a result of this specific layer stacking mode, O layer atoms from adjacent layers define in the interlayer region prismatic cavities located above or below empty tridentate cavities, sharing three edges with neighboring Mn layer octahedra (TE sites).

However, extensive differences exist between the structure models refined for samples KBi 8o and KBi 10o . The unit cell of sample KBi 8o is hexagonal with a = 2.840(1) Å, and c = 14.03(1) Å, and space group P6 3 /mmc. The hexagonal symmetry of the layer results from the sole presence of Mn 4+ in the octahedral layers, the presence of 0.12 vacant layer sites per octahedron being responsible for the layer charge deficit. [START_REF] Gaillot | [END_REF] This layer charge deficit is compensated for 1) by the presence of 0.08 interlayer Mn 3+ 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), and 2) by the presence of 0.24 interlayer K + in prismatic cavities located above or below empty tridentate cavities, sharing three edges with neighboring Mn layer octahedra ( VI TE sites).

Contrastingly, the amount of Mn 3+ cations is much higher in the structure of KBi 10o (0.25 Mn 3+ per octahedron) than in that of KBi 8o (0.08 Mn 3+ per octahedron). The systematic presence of these cations in the octahedral sites of vacancy-free layers accounts for the layer charge deficit, which is compensated by the sole presence of interlayer K + cations. 25 To minimize steric strains within the octahedral layers of KBi 10o , heterovalent Mn layer cations are segregated in Mn 3+ -and Mn 4+ -rich rows parallel to [010], and all Mn 3+ -octahedra, which are elongated because of Jahn-Teller distortion, present a unique azimuthal orientation. These two features lead to the departure from the hexagonal layer symmetry, the unit cell of sample KBi 10o being orthogonal with a = 5.155 Å, b = 2.846 Å, c = 14.09 Å, and space group Cmcm.

In addition, the regular alternation along the a axis of each Mn 3+ -rich row with two Mn 4+rows, emphasized by the associated distribution of interlayer cations, gives rise to a supercell with A = 3a.

Experimental

KBi samples were prepared from the thermal decomposition of fine-grained KMnO 4 powder (particle size < 50 µm) in air following the modified procedure of Kim et al. 21 In the companion papers [START_REF] Gaillot | [END_REF]25 a flat crucible covered by a thin layer of KMnO 4 powder (100-180mg/cm 2 ) was used for the pyrolysis to form homogeneous and highly periodic structures.

In the present article high-temperature (800-1000°C) pyrolysis were performed with crucibles containing a thick layer of KMnO 4 (Table 1). At lower temperature (200-700°C), pyrolysis was performed using a flat crucible covered by a thin layer of KMnO 4 (Table 1).

Heterogeneous KBi samples synthesized at 200°C, 400°C, 600°C, 700°C, 800°C and 1000°C

for the present study will hereafter be referred to as KBi 2 , KBi 4 , KBi 6 , KBi 7 , KBi 8h , and KBi 10h . Details on the processing of pyrolysis products, as well as on the thermal and chemical analyses are given by Gaillot et al. [START_REF] Gaillot | [END_REF] The mean oxidation degree of manganese in birnessite was determined by potentiometric titration using (NH 4 ) 2 Fe(SO 4 ) Mohr salt and sodium pyrophosphate. 26,27 Knowing the mean oxidation state, 2x, and the atomic K/Mn ratio determined by ICP, y , it is possible to calculate an average structural formula for a given KBi sample using the following equation [START_REF] Gaillot | [END_REF] :

K + 2y/w (Mn 4+ (4x-6)/w Mn 3+ (8-4x)/w 1-2/w )O 2 (1) where w = (2x + y) /2.

The amount of interlayer water may be introduced in this structural formula using the DTA and DTG data.

Powder XRD patterns were recorded using a Bruker D5000 powder diffractometer equipped with a Kevex Si(Li) solid detector and CuKα 1+2 radiation. Intensities were recorded from 5 to 90° at a 0.04° 2θ interval using a 40 sec counting time per step. Full widths at half maximum intensity (FWHM) were determined for diffraction maxima using the standard EVA program available from Bruker. A rotating sample holder was used for room temperature data collections to minimize preferential orientation effects. A TTK450 Anton Paar chamber was used to record XRD patterns from samples heated in situ.

Simulation of Powder XRD Patterns.

In the present study, samples with both high phase heterogeneity and high proportion of structural defects have been investigated. As compared to the KBi 8o and KBi 10o samples described in the two companion papers [START_REF] Gaillot | [END_REF]25 the combination of these two intrinsic features impairs the ability to find a suitable single crystal and would dramatically reduce the representativeness of this crystal. These features also strongly reduce the efficiency of usual structure refinement methods including the Rietveld method. One of the most effective ways to determine the actual structure of defective layered compounds is the calculation of XRD patterns using the mathematical formalism described in detail by Drits and Tchoubar. [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF] This trial-and-error procedure has been used successfully to determine the crystal-chemical structure of different natural and synthetic birnessites consisting of commensurate layers in the ab plane. [12][13][14][15][16]28 Details on the program used to calculate XRD patterns, and on the fitting procedure are given by Plançon 29 and Drits et al., 13 respectively. For birnessites in which partially incommensurate layers are interstratified, the mathematical formalism described by Plançon 29 was used. The fit quality was estimated over the 34°-56° 2θ CuKα range using the usual R wp parameter. XRD patterns calculations were restricted to 20l and 11l reflections (indexing based on an orthogonal base-centered unit cell) because these lines are most sensitive to structural parameters of layered minerals, including order/disorder and stacking sequences. [START_REF] Brindley | In Crystal Structures of Clay Minerals and their X-ray Identification[END_REF] For each sample, the background was assumed to be linearly decreasing over the considered angular range, and preferred orientation of particles was considered as a variable parameter.

Results

Structural Formulae of KBi Samples. K/Mn atomic ratios determined for KBi 7, KBi 8h

and KBi 10h are similar (0.270, 0.274 and 0.271, respectively) whereas the mean oxidation degree of Mn is lower for KBi 10h (3.78) than for KBi 7 and KBi 8h (3.87 and 3.85 -Table 2).

From these analytical results, and the experimentally determined structural water weight losses (Table 2) the following formulae may be deduced from Equation (1): 1).

K + 0.
However, the profiles of these basal reflections, as well as the number, the position, the FWHM, the profiles and the relative intensities of hkl reflections differ significantly from one pattern to the other (Figure 1). These differences are indicative of the structure contrast between the different samples. For example, KBi 2 possesses a turbostratic structure as its samples described in the present paper, the former ones will hereafter be referred to as KBi 10o and KBi 8o . [START_REF] Gaillot | [END_REF]25 Indexation of the Experimental Powder XRD Patterns. All XRD patterns were indexed using base-centered orthogonal unit cells independently of their hexagonal (a = b√3)

or orthogonal (a > b√3) layer symmetry. Therefore, reflections located in the 34°-70° 2θ

CuKα range have 20l,11l indices and 31l,02l ones in the 64°-70° 2θ CuKα range.

KBi 10h . In the 34°-56° 2θ range, two sets of non-basal reflections may be distinguished 3). 4). Such a systematic disagreement between calculated and experimental d-values could possibly indicate a physical mixture of the 2H and 2O birnessite varieties. In such case, the observed discrepancy between the experimental and calculated peak positions may result from the overlap of the 2H and 2O reflections which usually sandwich experimental maxima. However, the positions of 20l lines determined experimentally for the 2O variety should strictly coincide with those calculated for this polytype. On the contrary, these experimental positions are systematically shifted towards the positions calculated for the 11l,20l reflections of the 2H variety. As a consequence, the systematic disagreement between the experimental and calculated peak positions most likely results from the coexistence within KBi 8h coherent scattering domains (CSDs) of layer fragments from both 2H and 2O varieties. In this case, specific diffraction effects should result from the interstratification of partially incommensurate layers, shifting 11l and 20l reflections, changing their relative intensity and modifying their profiles as described by Drits et al., 13,[START_REF] Drits | [END_REF] and Lanson et al. 14 In addition to these structural varieties, KBi 8h sample contains a 3-layer rhombohedral (3R) birnessite-like polytype as indicated by the presence of three weak peaks at 2.412 Å, 2.144 Å, and 1.817 Å (Figure 2) which correspond respectively to 112, 115 and 118 reflections of this 3R phase (see description of KBi 7 ). 5).

(

FWHM of 20l and 11l

Reflections as a Function of l. For three-dimensional (3D) periodic structures, peak broadening has two main sources. [START_REF] Klug | X-ray diffraction procedures for polycrystalline and amorphous materials[END_REF] Crystal-size broadening originates from the small CSD size and leads to an increase of the FWHM values proportional to 1/cosθ. Strains or fluctuations of the unit cell parameters represent the second possible source of peak broadening and lead to a FWHM increase proportional to tanθ. After correction for either of these factors, FWHM of the reflections with identical hk values should be similar. Accordingly, for the defect-free KBi 10o sample, FWHM values of 20l and11l reflections are almost independent of l or slightly increase with l after correction by cosθ (Figure 4). For KBi 10h on the other hand, FWHM values dramatically decrease with l for 20l reflections. Such a decrease of peak breadth with l should not be observed for strictly periodic structures.

Simulation of KBi 10h Powder XRD Pattern.

In agreement with the model proposed by Kim et al. 21 and further refined by Gaillot et al. [START_REF] Gaillot | [END_REF] , it is assumed that the two-layer periodicity of the KBi 10h structure results from the rotation of adjacent layers by 180° around the c axis passing through Mn layer . As determined for different synthetic birnessites 5,[13][14][15][16]21 including KBi 8o and KBi 10o [START_REF] Gaillot | [END_REF]25 it is assumed that the thickness of octahedral layers is 2.00 Å, and that Mn cations are located in the center of MnO 6 octahedra. As in sample KBi 10o , 25 O layer atoms are shifted along the a axis as compared to the ideal anion close packing site (0.341 a Vs.

0.333 a -Table 6) and K + cations are located above or below the empty tridentate cavity. This site is actually split, each of the split positions being shifted in the ab plane from the center of the prismatic cavity defined by O layer of adjacent layers towards one of its faces. [START_REF] Gaillot | [END_REF]25 Similarly, interlayer H 2 O molecules are located in the middle of the interlayer space in-between two O layer from adjacent layers but slightly shifted along the a axis towards the nearest Mn layer (Figure 5). Optimal coordinates of the layer and interlayer sites leading to the fits shown in Figure 6 are listed in Table 6 together with their occupancies, whereas selected interatomic distances are given in Table 7.

As compared to the experimental XRD pattern collected for KBi 10h , the one calculated for a defect-free 2O structure with a 1 = 5.101 Å, b 1 = 2.850 Å, c = 14.240 Å exhibits 20l maxima which are sharper and more intense than experimental ones in contrast to 11l reflections (Figure 6a). To increase the FWHM of 20l reflections and to decrease their intensity, KBi 10 is assumed to integrate several 2O modifications having slightly different a and b parameters but keeping the same c parameter. To keep profiles and intensities of 11l reflections these unit-cell parameters are related by the following relation:

2 2 2 1 1 ) 110 ( 1 i i b a d + = (4)
The position of 11l reflections is unchanged for each 2O modifications whose a and b parameters verify Equation ( 4). On the other hand, 20l reflections of these modifications are slightly shifted to higher or lower 2θ values by decreasing or increasing their a parameter, and increasing or decreasing their b parameter, respectively. Figure 6b Accordingly, all attempts to fit the experimental XRD patterns as a physical mixture of periodic 2H, 2O and 3R polytypes failed. Alternatively, KBi 8h and KBi 7 samples can be described a mixture of elementary phases each consisting of interstratified fragments of 2H, 2O and 3R polytypes. Such interstratification may shift 11l and 20l reflections, change their relative intensity and modify their profiles as described by Drits et al., 13,[START_REF] Drits | [END_REF] and Lanson et al. 14 Structure of Elementary 2O, 2H, and 3R Fragments. Atomic positions and occupancies of the various sites in the interstratified 2O and 2H fragments are assumed to be similar to those refined for KBi 10o using the Rietveld method 25 and for KBi 8o from a single crystal. [START_REF] Gaillot | [END_REF] Occupancies of these sites are set according to KBi 8h structural formulae (Eq. 3). However, this formula should be modified to account for the migration of Mn 3+ cations from layer to interlayer in the 2H structure and for the resulting presence of vacant octahedra [START_REF] Gaillot | [END_REF] :

K + 0.265 Mn 3+ 0.145 (Mn 4+ 0.825 0.175 )O 2 •0.68 H 2 O KBi 7 , KBi 8h , (5)
However, the contrasting layer symmetry in the 2H and 2O varieties most likely results from different contents of Mn 3+ cations in the KBi structure. [START_REF] Gaillot | [END_REF]25 From the two structure models proposed by these authors, the Mn 3+ content is indeed much lower in the 2H variety than in the 2O one. In turn, these contrasting Mn 3+ contents give rise to different ways to minimize the steric strains arising from the elongation of Mn 3+ octahedra, and lead in the first case to the departure of Mn 3+ cations from layer to the interlayer (2H -lower Mn 3+ content), and in the second case to an ordered distribution of Mn 3+ cations within vacancy-free layers (2O -higher Mn 3+ content). As a consequence, the proposed structural formula, which is calculated from the mean chemical parameters determined on the bulk KBi 8h sample, averages the relative contributions of 2H and 2O varieties, which are likely closer to the models proposed by Gaillot et al. [START_REF] Gaillot | [END_REF]25 In the 2H variety, the site for interlayer H 2 O is split to provide an octahedral coordination to interlayer Mn 3+ cations on the one hand and, on the other hand, to ensure Hbonds with O layer of adjacent octahedral Mn layers as in the 2O variety. In both 2O and 2H polytypes, the K interlayer site is split as described for KBi 10h (Figure 5). Atomic positions and 8. For the sake of simplicity, atomic positions and occupancies in the layers and interlayers of the 3R polytype are assumed to be identical to those in the 2H one.

Diffraction Effects Resulting from the Interstratification of Elementary 2O, 2H, and 3R

Fragments. 2O and 2H fragments consists of two octahedral layers rotated with respect to each other by 180° around the c axis passing through Mn layer site, and will be hereafter referred to as A and B fragments, respectively. These fragments differ by their a parameter (5.060 Å and 4.945 Å, respectively) whereas they have the same b parameter (2.855 Å). As a result, interstratified 2O/2H structures consist of partly incommensurate A and B fragments. In addition, in such interstratified structures, the interlayer structure depends on the sequence of structural fragments. For example in an AB sub-sequence, the first and third interlayers are identical to those in the 2O and 2H polytypes, respectively, whereas the second interlayer has an intermediate structure corresponding to a "mixed" 2O and 2H interlayer (Figure 7).

As mentioned before, in the 3R polytype octahedral layers are shifted with respect to the previous one by -a/3 along the a axis. Such layer pairs are hereafter referred to as C fragments and are assumed to have layer and interlayer structures similar to those in B fragments (Table 8). B and C fragments are stacked without displacement in the ab plane in BB and BC subsequences whereas they are shifted with respect to each other by -a/3 along the a axis in CB and CC sub-sequences (Figure 7b).

The comparison of KBi 8h experimental pattern with that calculated for a randomly interstratified 2O/2H (60:40) mixed-layered structure (MLS -Figure 8b) shows that such interstratification induces a significant alteration of 20l reflection profiles. With increasing l values the 20l maxima appear as shoulders on the low angle side of corresponding 11l reflections (Figure 8b). 11a pleads for a realistic description of KBi 8h structural heterogeneity. However, this quality relies on a significant number of adjusted parameters and the actual presence of each of the four elementary phases needs to be assessed. The respective contributions of these different MLSs to the diffracted intensity are shown on Figures 9b-d which compare the experimental KBi 8h pattern with those calculated for models similar to the optimal one from which one of the contributions is systematically modified or subtracted. As a result of these structural modifications, the quality of fit is systematically deteriorated in specific parts of the calculated diagram, and more specifically in the high intensity "background" regions between the main reflections, supporting the existence of each of these elementary phases.

The best fit to the experimental KBi 7 pattern (Figure 10a -R WP = 9.1%) was obtained for a mixture in a 7:29:41:23 ratio of defect-free 2H polytype with interstratified 2O/2H and 2H/3R and 3R/2H MLSs containing respectively 50%, 70%, and 10% of hexagonal layer pairs. The respective contributions of each of these elementary phases to the diffracted intensity are shown on Figures 10b-d which compare the experimental KBi 7 XRD pattern with those calculated for models similar to the optimal one from which one of the contributions is systematically modified or subtracted. As for sample KBi 8h , the quality of fit is then systematically deteriorated in specific parts of the experimental diagram, and more specifically in the high intensity "background" regions between the main reflections, supporting the existence of each of these elementary phases.

Discussion

Phase and Structural Heterogeneity of KBi. The results obtained in the present article and in the companion papers [START_REF] Gaillot | [END_REF]25 show that phase and structural heterogeneity of KBi samples synthesized at 800°C and 1000°C strongly depends on the heterogeneity of physico-chemical conditions (temperature, PO 2 , …) within the KMnO 4 powder during the synthesis at these elevated temperatures. Homogeneous KBi samples are obtained when flat crucibles covered by a very thin layer of KMnO 4 powder are used. For example by heating a flat crucible covered with a thin layer of KMnO 4 powder (~180 mg.cm -2 ) to 1000°C a unique defect-free 2O phase is obtained (sample KBi 10o ). 25 If the thickness of the layer of KMnO 4 powder is increased as for the synthesis of KBi 10h (~475 mg.cm -2 ), the resulting product contains various 2O varieties with slightly different unit-cell parameters (Table 9).

Similarly, at 800°C phase heterogeneity increases with the thickness of the layer of KMnO 4 powder. In particular, the decomposition of a thin layer of KMnO 4 powder (100 mg.cm -2 ) covering a flat crucible leads to the formation of the defect-free 2H phase in sample KBi 8o which contains in addition a defective interstratified 3R/2H phase. Increasing the layer thickness of KMnO 4 powder up to 955 mg.cm -2 leads to a dramatic increase of phase heterogeneity as observed for KBi 8h sample (Figure 9, Table 10), most likely reflecting temperature or PO 2 heterogeneity within the KMnO 4 powder during the synthesis.

Structure of KBi as a Function of Synthesis Temperature. Synthetic analogues of birnessites are usually obtained at low-to-medium temperatures, that is between 5°C 1,2,11,[START_REF] Bach | [END_REF] and 200-500°C. 19,20,34,35 Kim et al. 21 assumed that a significant increase of temperature, up to 1000°C, would promote the formation of birnessite-like compounds having a high structural perfection and a high density, which are essential parameters for the phase stability required in their sought electrochemical application. These authors were the first to obtain hightemperature KBi by using K + as interlayer cations to prevent the formation of Mn oxides with tunnel or anion close-packed structures (pillaring effect), and highly oxidizing conditions to prevent the transition towards mixed-valence oxides. As a result, a variety of synthetic birnessite in which successive layers are rotated with respect to each other by 180° around the c axis was obtained. 21,22,[START_REF] Gaillot | [END_REF]25 However, the structural characterization of KBi samples obtained at 800°C and 1000°C temperatures was confined to the determination of idealized models, 21,22 and additional insights in the KBi structure will be provided below.

Layer Stacking and Symmetry. Our experimental work and results show that temperature plays a key decisive role in the occurrence of the different layer stacking modes, as well as for layer and interlayer composition. A temperature increase from 200°C to 1000°C results in a complex sequence of structural transitions from one metastable phase to the other through intermediate mixed-layered varieties. At 400°C, the turbostratic stacking of KBi 2 is replaced by a highly disordered 3R polytype. The structural order of this 3R phase increases significantly by increasing temperatures to 600-700°C as attested by the presence of sharp and intense reflections (112 R , 115 R and 118 R ) which are diagnostic for this phase (Figure 1). In addition, these temperatures (600-700°C) correspond to the onset of the 2H polytype formation, the proportion of which increases at the expense of the 3R phase with increasing temperature.

The detailed phase characterization of KBi samples obtained at temperatures higher than 600°C reveals the main trends of phase transitions affecting KBi structures. With this respect, the composition of KBi 7 and KBi 8 samples is especially remarkable as both samples contain the same four main phases. However the temperature increase from 700°C to 800°C leads to significant modifications between the two samples. The overall tendency is the decreasing proportion of the 3R layer pairs with increasing temperature and the related increase of 2H and 2O layer pair proportions (Figure 11 -Table 10). This tendency is systematically observed in the four constituting phases as the proportion of 2H layer pairs increases from 70 to 75% in the 2H/3R MLS and from 10 to 15% in the 3R/2H MLS when increasing the pyrolysis temperature from 700 to 800°C. In addition, the proportion of 2O layer pairs in the 2O/2H MLS increases from 50% in KBi 7 to 60% in KBi 8 , the relative proportion of this 2O/2H MLS increasing slightly from KBi 7 to KBi 8h at the expense of the 3R/2H MLS (Table 10). At higher temperature (1000°C) 3R layer pairs are absent and the sample is overwhelmingly dominated by the 2O polytype (Table 10). In KBi 10h phase heterogeneity is minimum as the defect-free 2O phase makes up most of the sample together with a minor amount (24%) of a defect-free 2H phase.

Layer Structure. Difference in KBi formation temperature modifies not only the layer stacking mode but also the crystal chemistry of KBi layers and interlayers. In particular, a temperature increase from 800°C to 1000°C leads to a much higher proportion of Mn 3+ cations in the structure (0.14 in KBi 8h as compared to 0.24 in KBi 10h ). The contrasting amount of Mn 3+ cations is combined with distinct distributions of these cations between layer and interlayer sites in the two varieties, as described earlier for the ordered KBi 8o and KBi 10o samples. [START_REF] Gaillot | [END_REF]25 In KBi 8o interlayer Mn 3+ cations are located above or below vacant layer octahedra in VI TC sites, whereas they are present in the octahedral sites of vacancy-free layers in KBi 10o . In this latter variety, the unique azimuthal orientation of Mn 3+ -octahedra, which are elongated because of Jahn-Teller distortion, leads to the departure from the hexagonal layer symmetry. However, by heating this variety up to 350°C it is possible to obtain a random orientation of elongated Mn 3+ octahedra as described by Gaillot et al. 25 Mn 3+ -containing layers of edge-sharing MnO 6 octahedra may have hexagonal symmetry such as in lithiophorite as a result of this random orientation with respect to the a axis (n60° rotations) of Mn 3+ octahedra. In addition to considerable lattice strains, such azimuthal distribution of Mn 3+ octahedra leads to increased b parameters (2.925 Å in lithiophorite 36 ) as compared to those determined for birnessites whose layers are devoid of Mn 3+ cations or for birnessites with an ordered distribution of heterovalent Mn octahedra. For these two birnessite families the experimentally determined b parameters range 2.844 -2.854 Å. 6,[13][14][15][16][START_REF] Gaillot | [END_REF]25 Accordingly, XRD patterns recorded for samples KBi 8h and KBi 10h heated in situ at 350°C (350-KBi 8h and 350-KBi 10h , respectively) may be indexed with a unique 2H phase with a b parameter equal to 2.875 Å and 2.894 Å, respectively. It should be noted that the unit-cell parameters obtained for sample KBi 10o heated in situ at 350°C (b = 2.895 Å and c = 12.848 Å) 25 almost coincide with those determined for KBi 10h -350 sample (b = 2.894 Å and c = 12.920 Å -Table 9) pleading for a similar content of Mn 3+ layer in the two samples. The different b parameters determined for 350-KBi 8h and 350-KBi 10h are in agreement with the contrasting contents of Mn 3+ layer in the two samples. From the comparison of Equations 2 and 5, an even stronger contrast is expected as hexagonal layers of KBi 8h are presumably devoid of Mn 3+ . However 2O layer pairs, which contain a significant amount of Mn 3+ layer (Table 8), represent 19% of KBi 8h sample and will contribute to a significant increase of the average b parameter in the 2O/2H phase (Figure 10 -Table 10), leading in turn to the increase of the b parameter observed when heating sample KBi 8h to 350°C.

In both KBi 8h and KBi 10h , the 2O polytype and A structural fragments consist of vacancy-free layers, whereas layers of the 2H and 3R polytypes, as well as B and C structural fragments, contain a significant proportion of vacant layer sites (up to 0.15 per octahedron -Tables 6,8). These vacant octahedra originate mostly from the migration of Mn 3+ cations from the layer to the interlayer. As a result, 2O, 2H and 3R structures have contrasting interlayer compositions and different distributions of these interlayer species. The interlayers of the 2O layer pairs are homogeneous and contain only K + cations and associated H 2 O molecules, whereas the interlayers of the 2H and 3R fragments are heterogeneous as they contain octahedrally coordinated Mn 3+ cations along with K + cations and H 2 O molecules. Mn 3+ octahedra are elongated along the a axis, whereas in crystals with the minimum a/b ratio the long Mn 3+ -O bonds of some Mn 3+ octahedra are likely oriented at ±60° with respect to the a axis. Such different azimuthal orientations of elongated Mn 3+ octahedra should result in an increased b parameter whereas the a parameter should decrease, as observed experimentally.

KBi Heterogeneity and FWHM of Reflections in

After heating to 350°C the FWHM of 11l,20l reflections recorded for 350-KBi 10o and 350-KBi 10h are almost independent of l (Figure 13) pleading for a similar content of Mn 4+ and Mn 3+ in all KBi 10h crystals, as in KBi 10o . The similar unit-cell parameters obtained for 350-KBi 10o and 350-KBi 10h (2.894 Å vs. 2.895 Å, respectively) also support this second hypothesis of a partial disorientation of Mn 3+ octahedra with respect to the a axis. It should be noted that the presence of the 2H polytype, even though minor, does not affect the FWHM of 11l,20l reflections recorded for 350-KBi 10h . This is likely due to the presence of a relatively high amount of Mn 3+ layer in the layers building up this 2H polytype (Table 6). This hypothesis is supported by the high value (2.873 Å -Table 9) determined for the b parameter of this 2H polytype. As discussed above, such a high b parameter most likely results from the random orientation of elongated Mn 3+ octahedra within the octahedral layer of birnessite.

Origin of the Structural Heterogeneity for KBi 8h . KBi 8h is a complex physical mixture of a periodic 2H phase and of various 2O/2H and 3R/2H MLSs. In these different phases, individual layers have different layer symmetry most likely resulting from contrasting contents of Mn 3+ layer . As described above, A fragments likely consist of vacancy-free layers containing a significant amount of Mn 3+ layer (~25%) whereas B and C fragments are likely devoid of Mn 3+ layer cations as a result of their layer-to-interlayer migration. As can be seen in Figure 13, the FWHM values measured for 350-KBi 8h sample decrease systematically with increasing l values to indicate the coexistence of crystals with close but different unit-cell parameters as described above. The difference of unit-cell parameters after heating to 350°C

and the induced random orientation of elongated Mn 3+ octahedra likely originates from the coexistence of crystals with contrasting contents of Mn 3+ layer . By combining chemical data (Equations 3 and 5) and structural details derived from XRD simulations ( [START_REF] Moore | Jr X-ray Diffraction and the Identification and Analysis of Clay Minerals[END_REF] Well defined stacking faults, which make the other type of structural disorder, are conveniently described as resulting from the interstratification of layers having similar thickness but different interlayer displacements. [START_REF] Drits | [END_REF] The random interstratification of 2O (A) and 2H (B) structural fragments in the 2O/2H phase described in both KBi 7 and KBi 8h represents a new type of structural disorder in lamellar structures. In this specific case, interstratified layers have the same thickness and the same interlayer stacking mode, but their layer dimensions in the ab plane are incommensurate. Such crystal structures consisting of alternating layers having different unitcell parameters have been previously reported in the literature. For example, alternation of incommensurate layers has been described in the structures of asbolanes, 39 valleriite and tochilinite. [40][START_REF] Drits | Electron diffraction and high-resolution elctron microscopy of mineral structures[END_REF][START_REF] Organova | [END_REF] However, in all these "hybrid structures" 40 incommensurate layers are regularly alternating along the c axis and their structures may be described by a set of unit cells, which have not only different sizes but also different shapes.

In contrast, the 2O and 2H structural fragments having different a parameters are randomly interstratified in the 2O/2H phase, and the description of the diffraction effects by such a structure should be analogous to that developed for mixed-layered structures [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF][START_REF] Drits | X-Ray structure analysis of mixed-layer minerals[END_REF]44 and for the structures containing stacking faults. [START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF]29,45 In the reciprocal space, 11 rods of the interstratified fragments completely overlap each other because of their identical d( 110) 3 and4). 5). Stars indicate the 11l reflections of a 3R-type phase as listed in Table 5. 6d). -0.263 ±0.210 3.570 0.12 Note: Parameters defining the layer cation composition. Optimal values were determined by trial-and-error fitting of KBi 8h and KBi 7 experimental pattern (Figures 9a,10a). All notations as in Table 6. Debye-Waller factors are 0.5, 1.0, 1.5, 2.0, and 2.0 for Mn layer , O layer , Mn interlayer , K and H 2 O respectively.

values
Occupancies are given for the sum of all symmetrical sites. For all elementary contributions, the radius of the coherent scattering domains in the ab plane is 350 Å (300 Å for the 2H/3R MLS with a high 2H content), whereas the mean coherent scattering domain along the c* axis is 15 layers (20 layers for the 2O/2H MLS). All elementary contributions are devoid of random stacking faults. Please print in one column format

KBi 8h .

 8h The three sharp peaks present at 1.448 Å, 1.426 Å and 1.398 Å (Figure 2), can be indexed with two different unit cells. The first one has an orthogonal layer symmetry and a two-layer periodicity along the c axis with a = 5.060 Å, b = 2.855 Å, c = 14.200 Å, whereas the second one considers a hexagonal layer symmetry with a ort = a hex √3 = 4.945 Å, b ort = b hex = 2.855 Å, c = 14.320 Å. However, d(20l,11l) values calculated for these two unit cells are systematically shifted with respect to the experimental ones (Table

KBi 7 .

 7 As for KBi 8h , reflections were indexed using both 2H and 2O unit cells with a = 4.945 Å, b = 2.855 Å, c = 14.320 Å, α = β = γ = 90° (a = b√3), and a = 5.060 Å, b = 2.855 Å, c = 14.200 Å, α = β = γ = 90°, respectively. As for the KBi 8h sample, the d-values calculated for (20l,11l) reflections of the 2H phase are systematically shifted with respect to experimental ones towards those calculated for the 2O unit cell (Table 4). Similarly, the three maxima at 2.410 Å, 2.143 Å and 1.818 Å, were indexed with a 3R unit cell with a = 4.945 Å, b = 2.855 Å, c = 21.420 Å, α = β = γ = 90° (112, 115 and 118 reflections, respectively). Powder XRD Patterns of Heated KBi Samples. Additional XRD patterns were recorded in situ from KBi 10h and KBi 8h samples heated to 350°C under air conditions (Figure 3). For both patterns, reflections are indexed with a two-layer hexagonal (2H) unit cell with a = 4.980 Å, b = 2.875 Å, c = 12.995 Å (350-KBi 8h ), and a = 5.013 Å, b = 2.894 Å, c = 12.920 Å (350-KBi 10h ), α = β = γ = 90°, and a = b√3 (Table

  compares the experimental XRD pattern obtained for KBi 10h with those calculated for three defect-free 2O modifications having slightly different layer unit-cell parameters. The XRD pattern calculated for a mixture of these three modifications exhibits 11l reflections similar to those in each elementary XRD pattern, whereas 20l reflections are significantly broadened, as observed in KBi 10h experimental pattern. In addition, intensity and positions of the 11l reflections in KBi 10h experimental pattern are similar to those of 20l,11l reflections calculated for a 2H modification with a = 4.976 Å, b = 2.873 Å and c = 14.240 Å (Figure 6c). Finally, the best possible fit to the experimental KBi 10h XRD pattern (Figure 6d -R WP = 11.1%) was obtained for a physical mixture of three 2O modifications (a 1 = 5.124 Å, b 1 = 2.846 Å, a 2 = 5.101 Å, b 2 = 2.850 Å, and a 3 = 5.084 Å, b 3 = 2.853 Å -c = 14.240 Å) and of a 2H phase with a = 4.976 Å, b = 2.873 Å, and c = 14.240 Å. Relative proportions of these different phases are 32, 31, 13, and 24%, respectively. Powder XRD Pattern Simulation for KBi 8h and KBi 7 . From the indexation of experimental XRD patterns, the description of KBi 8h and KBi 7 samples as a physical mixture of various KBi polytypes devoid of well-defined stacking faults may be rejected.

  occupancies of the 2H (a = 4.945 Å, b = 2.855 Å, c = 14.320 Å) and 2O (a = 5.043 Å, b = 2.850 Å, c = 14.240 Å) cells providing the best fit to the experimental XRD patterns are given in Table

  Experimental Powder XRD Patterns. FWHM values measured for 20l and 11l reflections of KBi 10o and for 11l maxima of sample KBi 10h are almost independent of l (Figure 4). In contrast, FWHM values measured for 20l reflections of KBi 10h dramatically decrease with increasing l values. To account for such a contrasting behavior between 11l and 20l reflections of KBi 10h , one has to consider that KBi 10h consists of several 2O varieties with slightly different a and b parameters but identical d(110) values. As a consequence, 11 rods corresponding to these different varieties coincide in the reciprocal lattice, whereas their 20 rods are located close to each other along the a* axis. These latter rods are not strictly superimposed but partially overlap each other.Diffraction effects arising from this particular case are described below. Powder X-ray diffraction effects are conventionally derived in the reciprocal space from the rotation of the reciprocal lattice with respect to a fixed Ewald sphere. Brindley and Méring proposed an alternative and effective view in which the reciprocal lattice is fixed and the radius of Ewald sphere is continuously increased.37 In this alternative model, the integrated intensity at 1/d sums up intensities of the reciprocal lattice nodes located on the surface of the Ewald sphere having a 1/d radius. This alternative approach may be used to predict reflection profiles for a physical mixture of two phases having closely related a and b unit-cell parameters. The reciprocal lattice of a layered structure may be represented as a set of hk rods parallel to the c* axis along which hkl nodes with different l values are located (Figure12a).[START_REF] Drits | X-ray diffraction by disordered lamellar structures: Theory and applications to microdivided silicates and carbons[END_REF] Two partially overlapping reciprocal rods having the same hk indices and containing reflections with the same l values are shown on Figure12btogether with the Ewald spheres defining hkl reflections for these two reciprocal rods. With increasing l values, the two rods contribute simultaneously to the diffracted intensity over an increased range of 1/d values. This leads to a shorter distance between the respective maxima corresponding to the two overlapping reflections (dashed lines -Figure12b) and to a lower FWHM for the resulting combined diffraction maximum (Figure12b) as observed experimentally after correction for crystal-size broadening. The observed decrease of the 20l reflection FWHM with increasing l values (Figure4) can thus be considered as an independent evidence of such type of phase heterogeneity. Because all 2O varieties have identical d(110) values, 11 reciprocal rods completely overlap in the reciprocal space and, as a consequence, the FWHM of 11l reflections is independent of l despite the differences in a and b parameters. One may note (Figure4) that for the homogeneous KBi 10o sample the correlations between FWHM and l for 20l and 11l reflections lead to slightly different slopes. This difference most likely reflects the presence in this sample of a small proportion of particles having 2O structure and unit-cell parameters slightly different from those of the main 2O phase. The relation between the FWHM of 11l,20l reflection and the l indice may be used as an independent criterion to estimate the degree of phase heterogeneity in KBi samples.Origin of theStructural Heterogeneity for KBi 10h . Heterogeneous heating to ~1000°C of individual KMnO 4 particles leads essentially to the formation of KBi crystals having different a and b parameters. The a/b ratios obtained for the three identified subpopulations of KBi crystals range 1.784-1.800 (Table9) as a consequence of lattice distortion induced by the Jahn-Teller distortion of Mn 3+ octahedra. Two hypotheses may account for the observed scatter of the a/b ratio. The first one is the local fluctuation of the redox conditions in which individual KBi crystals are formed resulting in the presence of contrasting amounts of Mn 3+ cations from one crystal to the other. According to the other hypothesis, all KBi crystals have almost the same content of Mn 3+ cations but differ from each other by their respective mean orientation of the long Mn 3+ -O bonds with respect to the a axis. In this second hypothesis, the maximum a/b ratio likely correspond to crystals in which most or all

  leading to the coherence of the diffracted waves for 11l reflections. A similar effect exists for the 20l and 11l reflections of the hexagonal fragments as these reflections coincide with 11l reflections of the 2O fragments. Such a high coherence gives rise to the sharp and intense maxima observed for the combined 20l,11l reflections of the 2H fragments and 11l reflections of the 2O fragments (Figures 2, 9, 10). In contrast, a significant phase misfit arises for the incomplete overlap of 20 rods from 2O fragments with those of the 2H fragments leading to a significant loss of coherence of the diffracted waves when 2O and 2H fragments are interstratified. As a result, 20l reflections of the 2O fragments are broadened and shifted towards the partially overlapped 20l,11l and 11l reflections scattered by the 2H and 2O layer pairs, respectively, and are only observed as diffuse shoulders (Figures 9, 10). (44) Reynolds, R. C., Jr In Crystal structures of clay minerals and their X-ray identification; Brindley, G. W., Brown, G., Eds.; The Mineralogical Society: London, 1980, pp 249-359. (45) Plançon, A. J. Appl. Cryst. 1981, 14, 300-304.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Experimental XRD patterns of KBi samples obtained from thermal decomposition

Figure 3 .

 3 Figure 3. Experimental XRD patterns of KBi 10h (top) and KBi 8h (bottom) samples recorded in

Figure 4 .

 4 Figure 4. Evolution of the full width at half maximum intensity (FWHM) for 20l and 11l

Figure 5 .

 5 Figure 5. Structure model for KBi 10h . a) Projection on the ab plane. The upper surface of the

Figure 6 .

 6 Figure 6. Comparison between experimental and calculated XRD patterns for KBi 10h .

Figure 7 .

 7 Figure 7. Schematic view along the b axis of the interlayer structure in mixed-layered

Figure 8 .

 8 Figure 8. Comparison between experimental and calculated XRD patterns for KBi 8h . Patterns

Figure 9 .

 9 Figure 9. Comparison between experimental and calculated XRD patterns for KBi 8h . Patterns

Figure 10 .

 10 Figure 10. Comparison between experimental and calculated XRD patterns for KBi 7 . Patterns

Figure 11 .

 11 Figure 11. Relative proportions of 3R, 2H, and 2O layer pairs (solid bars, shaded bars, and

Figure 12 .

 12 Figure 12. (a) Schematic description of the intensity diffracted at a hkl node (shaded

Figure 13 .

 13 Figure 13. Evolution of the FWHM for 20l, 11l reflections as a function of the Miller index

= 4 .

 4 976 Å, b = 2.873 Å, c = 14.240 Å, α = β = γ = 90° (2H polytype), and a = 5.101 Å, b = 2.850 Å, c = 14.240 Å, α = β = γ = 90° (2O polytype). Calculated d cal (hkl) reflection positions (in Å) are compared to experimental d exp (hkl) ones. a indicates additional lines related to the supercell of the 2O polytype.25

  = 4.945 Å, b = 2.855 Å, c = 14.320 Å (2H polytype), a = 5.060 Å, b = 2.855 Å, c = 14.200 Å (2O polytype), and a = 4.945 Å, b = 2.855 Å, c = 21.420 Å (3R polytype). α = β = γ = 90° for all three polytypes. Calculated d cal (hkl) reflection positions (in Å) are compared to experimental d exp (hkl) ones.

  : a = 5.013 Å, b = 2.894 Å, c = 12.920 Å (350-KBi 10h sample), and a = 4.980 Å, b = 2.875 Å, c = 12.995 Å (350-KBi 8h sample). α = β = γ = 90° for the two samples. Calculated d cal (hkl) reflection positions (in Å) are compared to experimental d exp (hkl) ones. a indicates unindexed additional weak lines. b indicates 11l reflections of a 3R-type additional phase

  x and y coordinates are expressed as fractions of the orthogonal a and b parameters, respectively. Coordinates along the c* axis, ζ, are expressed in Å to emphasize the thickness of layer and interlayer polyhedra. a = b√3 = 4.976 Å, b = 2.873 Å (2H polytype -P6 3 /mmc space group), a 1 = 5.124 Å, b 1 = 2.846 Å, a 2 = 5.101 Å, b 2 = 2.850 Å, and a 3 = 5.084 Å, b 3 = 2.853 Å for the three 2O polytypes (Cmcm space group). c = 14.240 Å, α = β = γ = 90° for all 4 phases. Debye-Waller factors were 0.5, 1.0, 2.0, 2.0 for Mn layer , O layer , K and H 2 O, respectively. Occupancies are given for the sum of all symmetrical sites. For all elementary contributions, the radius of the coherent scattering domains in the ab plane is 350 Å, whereas the mean coherent scattering domain along the c* axis is 17 layers. All elementary contributions are devoid of random stacking faults.

25 350 25 CM035236R

 2525 All unit-cell parameters are given in Å. α = β = γ = 90° for all phases. a Sample described by Gaillot et al. 24 b Sample described by Gaillot et al. All unit-cell parameters are given in Å. α = β = γ = 90° for all phases. c Sample described by Gaillot et al.

  et al., Fig. 07 Please print in one column format CM035236R -Gaillot et al., Fig. 08

  Please print in two column format CM035236R -Gaillot et al., Fig.10

  Please print in two column format CM035236R -Gaillot et al., Fig.

Main Features of the Experimental Powder XRD Patterns. XRD

	27 (Mn 4+ 0.77 Mn 3+ 0.21 0.02 )O 2 •0.53 H 2 O	KBi 10h , (2)
	K + 0.265 (Mn 4+ 0.825 Mn 3+ 0.145 0.03 )O 2 •0.68 H 2 O	KBi 7 , KBi 8h , (3)
		patterns collected
	for KBi samples synthesized at temperatures ranging 200-1000°C all exhibit basal reflections
	at ~7.13 Å and ~3.56 Å characteristic of the layered structure of birnessite (Figure

  the corresponding reflections of the defect-free 2H and 3R polytypes in agreement with the conceptual model proposed by Drits et al.[START_REF] Drits | [END_REF] These authors showed that for a random interstratification of layers fragments having similar thickness but different interlayer displacements observed hkl reflections are located between neighboring hkl reflections of phases whose layer fragments are interstratified, the exact position depending on the relative proportion of the elementary components.

	Description of KBi 8h and KBi 7 Samples as a Mixture of Defective KBi Phases. The best
	fit to the experimental KBi 8h XRD pattern (Figure 9a -R WP = 8.2%) was obtained for a
	mixture of the defect-free 2H polytype with interstratified 2O/2H, 2H/3R and 3R/2H MLSs
	containing respectively 40%, 75% and 15% of hexagonal layer pairs in a 9:34:48:8 ratio. The
	quality of the fit shown on Figure

Similarly, when looking at XRD patterns calculated for 3R/2H MLSs containing 75% and 15% of 2H layer pairs (Figures 8d, f, respectively) 20l,11l reflections are located between

Table 8

 8

	), it is

New Type of Structural Disorder in Birnessite. Except for the frequent presence of random stacking faults, two types structural defects are extremely common in layered structures. The random interstratification of layers having different thicknesses, which is the first type of such structural disorder, is especially widespread in mixed-layered clay minerals.

Table 1 .

 1 Experimental conditions (shape and size of crucible and density of KMnO 4 powder) used for the synthesis of the KBi samples. Sample described by Gaillot et al.25 b Sample described by Gaillot et al.[START_REF] Gaillot | [END_REF]

	Sample	Amount of KMnO 4 powder	Surface area (crucible)	Density of powder (mg/cm 2 )
	KBi 10o (homogeneous) a	2g	4.5 x 2.5 cm 2	178
	KBi 10h	1.5g	π x 1 2 cm 2	477
	KBi 8o (homogeneous) b	3g	6.8 x 4.45 cm 2	100
	KBi 8h	3g	π x 1 2 cm 2	955
	KBi 7	1g	4.5 x 2.5 cm 2	89
	KBi 6	2g	π x 2.25 2 cm 2	126
	KBi 4	2g	π x 2.25 2 cm 2	126
	KBi 2	2g	π x 2.25 2 cm 2	126
	Note: a

Table 2 .

 2 Main chemical parameters of KBi samples

		KBi 7	KBi 8h	KBi 10h
	Weight loss due to interlayer H 2 O	6.3%	6.5%	8.9%
	Weight loss due to hydroxyl groups	2.6%	2.3 %	-
	H 2 O/Mn ratio	0.70	0.68	0.53
	K/Mn ratio	0.270	0.274	0.271
	Mn mean oxidation state	3.87	3.85	3.78
	Note: K/Mn ratio is determined by ICP-AES. Weight losses
	correspond to the first two endotherms (~150°C and 370°C,
	respectively) observed on the DTA-TG curves. H 2 O/Mn ratio includes
	both interlayer H 2 O and hydroxyl groups.

Table 3 .

 3 Indexing of the experimental XRD pattern of KBi 10h with 2O and 2H cells.

		2H polytype	2O polytype
	d exp (hkl)	hkl	d cal (hkl)	hkl	d cal (hkl)
	7.705 a
	7.131	002	7.120	002	7.120
	4.404 a
	3.700 a
	3.562	004	3.560	004	3.560
	2.750 a
	2.552			200	2.551
	2.485	200, 110	2.488	110	2.488
	2.451	201, 111	2.451	111	2.451
	2.404			202	2.401
	2.347	202, 112	2.349	112	2.349
	2.249			203	2.247
	2.204	203, 113	2.204	113	2.204
	2.074			204	2.073
	2.040	204, 114	2.039	114	2.039
	1.898			205	1.900
	1.874	205, 115	1.874	115	1.874
	1.781	008	1.780	008	1.780
	1.738			206	1.737
	1.718	206, 116	1.717	116	1.717
	1.460			208 / 310	1.460
	1.448	208, 118	1.448	118	1.448
	1.431	310, 020	1.436	312	1.430
	1.425			020	1.425
	1.397	312, 022	1.408	022	1.397
	1.352			314	1.351
	1.344			209	1.345
	1.334	209, 119 314, 024	1.335 1.332	119	1.335
	1.323			024	1.323
	1.243	220, 400	1.244	220 / 316 20.10	1.244 1.243
	1.236	221, 401	1.239	221 / 11.10	1.239 / 1.236
	1.225	316, 026 222, 402	1.229 1.225	222 / 026	1.225 / 1.222
	1.204	223, 403	1.203	223	1.203
	1.174	224, 404	1.174	224	1.174
	1.128			318 / 406	1.129 / 1.123
	1.102	226, 406	1.102	226	1.102
	Note: a

Table 4 .

 4 Indexing of the experimental XRD patterns of KBi 8h and KBi 7 with 2O, 2H, and 3R cells.

	KBi 7	KBi 8h	2 H polytype	2O polytype	3R polytype
	d exp (hkl) d exp (hkl)	hkl	d cal (hkl)	hkl	d cal (hkl)	hkl	d cal (hkl)
	7.140	7.136	002	7.160	002	7.110	003	7.140
	3.574	3.567	004	3.580	004	3.555	006	3.570
	2.514	2.517			200	2.530
	2.479				110	2.487
	2.471	2.472	200, 110	2.473
	2.456				111	2.449	201	2.456
		2.428	201, 111	2.436
	2.410	2.412					112	2.409
	2.378	2.378	006	2.387	202, 006 2.384, 2.370	009	2.380
	2.341	2.342	202, 112	2.337	112	2.347
	2.223	2.218			203	2.232	204	2.245
	2.195	2.196	203, 113	2.195	113	2.202
	2.143	2.144					115	2.141
		2.058			204	2.061
	2.036	2.036	204, 114	2.034	114	2.038
	1.929						207	1.923
	1.871	1.871	205, 115	1.872	115	1.872
	1.818	1.817					118	1.816
	1.785	1.785	008	1.790	008	1.778	00.12	1.785
		1.728			206	1.730
	1.717	1.717	206, 116	1.717	116	1.716
	1.626						20.10	1.619
	1.531						11.11	1.530
	1.448	1.448	208, 118	1.450	208, 310, 118	1.454, 1.452, 1.446
	1.426	1.426	310, 020	1.427	020, 312 1.428, 1.423	310, 020	1.427
	1.416				00.10	1.422
	1.399	1.398	312, 022	1.400	022	1.400	313, 023	1.400
	1.340	1.338	209, 119	1.338	314, 119 1.344, 1.332
	1.325	1.325	314, 024	1.326	024	1.325	316, 026	1.325
	1.235	1.237	20.10, 11.10 400, 220	1.238 1.236	316 11.10	1.238 1.234	221	1.234
	1.223	1.222	316, 026 402, 222	1.225 1.218	222 026	1.225 1.223	402 319, 029	1.228 1.224
	1.191	1.190	403, 223	1.197	404	1.192	224	1.205
	1.169	1.170	404, 224	1.169	224	1.174	405	1.188
		1.114	318, 028	1.116	406, 028 1.116, 1.113 31.12, 02.12	1.115
	Note: a

Table 5 .

 5 Indexing of the experimental XRD patterns of KBi 10h and KBi 8h recorded in situ at 350°C with a 2H cell.

		350-KBi 10h	350-KBi 8h
	hkl	d exp (hkl)	d cal (hkl)	d exp (hkl)	d cal (hkl)
	002	6.471	6.460	6.506	6.498
		4.818 a
		4.332 a
		4.124 a
	004	3.231	3.230	3.248	3.249
	200,	2.504	2.506	2.490	2.490
	201,	2.462	2.460	2.423 b
	202,	2.336	2.337	2.326	2.325
	203,	2.166	2.166	2.159	2.159
				2.096 b
	204,	1.980	1.980	1.976	1.976
	205,	1.801	1.799	1.785 b	1.798
	206,	1.634	1.633	1.634	1.634
	008	1.618	1.615	1.618	1.624
	310,	1.447	1.447	1.437	1.438
	312,	1.412	1.412	1.403	1.404
	208,			1.360	1.360
	314,			1.313	1.315
	400,			1.242	1.245
	402,			1.221	1.223
	316,			1.193	1.198
	404,			1.152	1.162
	Note

Table 6 .

 6 Optimum structural parameters (atomic positions and occupancies) used for the simulation of sample KBi 10h with 2H and 2O polytypes. Parameters defining the layer cation composition. Optimal values were determined by trial-and-error fitting of KBi 10h experimental pattern (Figure

			2 H polytype			2O polytype
		x	y	ζ (Å) Occ.	x	y	ζ (Å)	Occ.
	Mn layer	0	0	0	0.98	0	0	0	0.98
	O layer	±0.333	0	±1.000 2.00	±0.341	0	±1.000 2.00
	K	-0.220	0	3.560 0.09	-0.250	0	3.560	0.09
	K	0.110 ±0.330 3.560 0.18	0.125	±0.375 3.560	0.18
	H 2 O	0.150	0	3.560 0.17	0.150	0	3.560	0.53
	H 2 O	0.075 ±0.225 3.560 0.34	-	-	-	-
	Note:

Table 7 .

 7 Selected inter-atomic distances calculated from the optimal atomic coordinates for KBi samples.

			KBi 10h		KBi 8h and KBi 7
		2 O-1	2O-2	2O-3	2H	2 O	2H	3R
	Height of Mn layer	2.00	2.00	2.000	2.000	2.000	2.000	2.000
	Mn layer -Mn ayer	2.846 x2 2.931 x4	2.850 x2 2.922 x4	2.853 x2 2.915 x4	2.873 x6	2.855 x2 2.905 x4	2.855 x6	2.855 x6
	Average Mn layer -Mn layer	2.903	2.898	2.894	2.873	2.888	2.855	2.855
	Mn layer -O layer	1.929 x4 2.023 x2	1.929 x4 2.016 x2	1.929 x4 2.011 x2	1.946 x6	1.927 x4 2.003 x2	1.940 x6	1.938 x6
	Average Mn layer -O layer Mn 3+ interlayer -Mn layer Mn 3+ interlayer -O layer Mn 3+ interlayer -H 2 O interlayer	1.960 ---	1.958 ---	1.956 ---	1.946 ---	1.952 ---	1.940 3.512 1.940 2.251	1.938 3.509 1.938 2.248
	O layer -H 2 O interlayer	-	-	-	-	-	3.043	3.037
	K interlayer -O layer	2.951 x2 3.295 x1	2.952 x2 3.289 x1	2.952 x2 3.284 x1	2.933 x2 3.377 x1	2.929 x2 3.373 x1	2.933 x2 3.444 x1	2.927 x2 3.439 x1
	Average K interlayer -O layer	3.066	3.064	3.063	3.081	3.077	3.103	3.098
	K interlayer -H 2 O interlayer	3.074 x1 3.507 x2	3.061 x1 3.505 x2	3.050 x1 3.504 x2	2.731 x2 3.135 x1 3.305 x1 3.412 x2	3.188 x1 3.414 x2	2.731 x2 3.223 x2 3.440 x2	2.731 x2 3.223 x2 3.440 x2
	O layer -H 2 O interlayer	2.725	2.723	2.722	2.702	-	2.730	2.723
	Note: All distances are given in Å.

Table 8 .

 8 Optimum structural parameters (atomic positions and occupancies) used for the simulation of sample KBi 8h and KBi 7 with 2H, 2O, and 3R

	polytypes.
			2H polytype			2O polytype			3R polytype
		x	y	ζ(Å)	Occ.	x	y	ζ(Å)	Occ.	x	y	ζ(Å)	Occ.
	Mn layer	0	0	0	0.85	0	0	0	0.98	0	0	0	0.85
	O layer	±0.333	0	±1.000	2.00	±0.341	0	±1.000	2.00	±0.333	0	±1.000	2.00
	Mn interlayer	0	0	±2.100	0.11	-	-	-	-	0	0	±2.100	0.11
	H 2 O inter.	-0.333	0	3.580	0.33	-	-	-	-	-0.333	0	3.570	0.33
	K inter.	-0.200	0	3.580	0.09	-0.220	0	3.550	0.09	-0.200	0	3.570	0.045
	K inter.	0.100	±0.300	3.580	0.18	0.110	±0.330	3.550	0.18	0.100	±0.300	3.570	0.09
	K inter.									-0.533	0	3.570	0.045
	K inter.									-0.233	±0.300	3.570	0.09
	H 2 O inter.	0.140	0	3.580	0.12	0.150	0	3.550	0.54	0.140	0	3.570	0.06
	H 2 O inter.	-0.070	±0.210	3.580	0.24					-0.070	±0.210	3.570	0.12
	H 2 O inter.									-0.193	0	3.570	0.06
	H 2 O inter.

Table 9 .

 9 Unit-cell parameters of the elementary 2O, 2H, and 3R polytypes present in KBi samples.

			KBi 8h KBi 7	KBi 8o	a	KBi 10h		KBi 10o	b
		a	5.060	-	5.124	5.101	5.084	5.155
		b	2.855	-	2.846	2.850	2.853	2.846
	2O	a/b	1.772	-	1.800	1.790	1.782	1.811
		c	14.220	-		14.240		14.088
		a	4.945	4.928	4.976		-
		b	2.855	2.845	2.873		-
	2H	a/b	1.732	1.732	1.732		-
		c	14.320	14.235	14.240		-

Acknowledgements

VAD is grateful to the Environmental Geochemistry Group of the LGIT (Grenoble, France) and to the Russian Science Foundation for financial support. BL acknowledges financial support from INSU/Géomatériaux, and CNRS/PICS709 programs. Céline Boissard (Hydr'ASA -Poitiers), Martine Musso and Delphine Tisserand (LGIT -Grenoble) are thanked for their technical support (DT-TG analyses and chemical analyses, respectively).