Marco Di Renzo
email: marco.direnzo@centralesupelec.fr

Habibi Fadil

Danufane

Xiaojun Xi

Julien De Rosny

Sergei Tretyakov

Analytical Modeling of the Path-Loss for Reconfigurable Intelligent Surfaces -Anomalous Mirror or Scatterer ?

Reconfigurable intelligent surfaces (RISs) are an emerging field of research in wireless communications. A fundamental component for analyzing and optimizing RIS-empowered wireless networks is the development of simple but sufficiently accurate models for the power scattered by an RIS. By leveraging the general scalar theory of diffraction and the Huygens-Fresnel principle, we introduce simple formulas for the electric field scattered by an RIS that is modeled as a sheet of electromagnetic material of negligible thickness. The proposed approach allows us to identify the conditions under which an RIS of finite size can or cannot be approximated as an anomalous mirror. Numerical results are illustrated to confirm the proposed approach.

I. INTRODUCTION

In contemporary wireless networks, transmitters and receivers can be programmed and controlled for optimizing the system performance. The environmental objects (buildings, walls, ceilings, etc.) that constitute the wireless environment cannot, on the other hand, be customized based on the network conditions. This status quo has recently been challenged by the emerging technology of reconfigurable intelligent surfaces (RISs) -Thin sheets of electromagnetic materials that are capable of shaping the radio waves in arbitrary ways [START_REF] Tretyakov | Smart radio environments empowered by reconfigurable AI meta-surfaces: An idea whose time has come[END_REF], [START_REF] Zappone | Wireless networks design in the era of deep learning: Model-based, AI-based, or both?[END_REF]. The overarching vision consists of coating the environmental objects with RISs and optimizing their properties, in order to, e.g., reflect an impinging radio wave towards a desired direction with the objective of capitalizing from multipath propagation rather than being negatively affected by it [START_REF] Liaskos | A new wireless communication paradigm through software-controlled metasurfaces[END_REF]- [START_REF] Huang | Holographic MIMO surfaces for 6G wireless networks: Opportunities, challenges, trends[END_REF].

In simple terms, an RIS is the two-dimensional equivalent of a reconfigurable meta-material, and is made of elementary elements called scattering particles or meta-atoms [START_REF] Yu | Light propagation with phase discontinuities: Generalized laws of reflection and refraction[END_REF]. Depending on the arrangement and configuration of the scattering particles, an RIS is capable of altering the wavefront of the radio waves impinging upon it. For example, RISs can modify the direction of the reflected or refracted waves and their polarization, or can encode data onto the shape of the scattered waves [START_REF] Asadchy | Perfect control of reflection and refraction using spatially dispersive metasurfaces[END_REF], [START_REF] Estakhri | Wave-front transformation with gradient metasurfaces[END_REF], [START_REF] Liu | Intelligent metasurfaces with continuously tunable local surface impedance for multiple reconfigurable functions[END_REF]. The two-dimensional nature of RISs make them easier to design, less lossy, less expensive, and easier to deploy than their three-dimensional counterpart. Broadly speaking, RISs are special surfaces that are engineered to possess properties that cannot be found in surfaces made of naturally occurring materials [START_REF] Liu | Intelligent metasurfaces with continuously tunable local surface impedance for multiple reconfigurable functions[END_REF]. Thanks to these properties, RISs are receiving major attention from the wireless community, and are considered to be the key enabler of the emerging concept of smart radio environments [START_REF] Tretyakov | Smart radio environments empowered by reconfigurable AI meta-surfaces: An idea whose time has come[END_REF].

A major open research issue for analyzing the ultimate performance limits, optimizing the operation, and assessing the advantages and limitations of RIS-empowered wireless networks is the development of simple but sufficiently accurate models for the power received at a given location in space when a transmitter emits radio waves that illuminate an RIS. Recently, a few research works have tackled this research issue. In [START_REF] Tang | Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimental measurement[END_REF], the authors have performed a measurement campaign in an anechoic chamber and have shown that the power reflected from an RIS follows a scaling law that depends on many parameters, including the size of the RIS, the mutual distances between the transmitter/receiver and the RIS (i.e., near-field vs. far-field conditions), and whether the RIS is used for beamforming or broadcasting applications. In [START_REF] Bucheli | Reconfigurable intelligent surfaces: Bridging the gap between scattering/reflection[END_REF], the authors have employed antenna theory to compute the electric field in the near-field and far-field of a finite-size RIS, and have proved that an RIS is capable of acting as an anomalous mirror in the near-field of the array. The results are obtained numerically and no explicit analytical formulation of the received power as a function of the distance is given. Similar results have been obtained in [START_REF] Ellingson | Path loss in reconfigurable intelligent surfaceenabled channels[END_REF]. In [START_REF] Khawaja | Coverage enhancement for NLOS mmWave links using passive reflectors[END_REF], the power measured from passive reflectors in the millimeter-wave frequency band is compared against ray tracing simulations. By optimizing the area of the surface that is illuminated, it is shown that a finite-size passive reflector can act as an anomalous mirror. The study in [START_REF] Ozdogan | Intelligent reflecting surfaces: Physics, propagation, and pathloss modeling[END_REF] relies on the assumption of plane waves and is valid only in the far-field of the RIS (i.e., long distances).

In this paper, we leverage the general scalar theory of diffraction and the Huygens-Fresnel principle, and introduce simple closed-form expressions to compute the power reflected from an RIS as a function of the distance between the transmitter/receiver and the RIS, the size of the RIS, and the phase transformation applied by the RIS. With the aid of the stationary phase method, we identify sufficient conditions under which an RIS acts as an anomalous mirror, and, therefore, the received power decays as a function of the reciprocal of the sum of the distances between the transmitter and the RIS, and the RIS and the receiver. For simplicity, the analytical formulas are reported without proof and for a one-dimensional RIS. The proofs, discussions on the boundary conditions to solve Maxwell's equations, the impact of the direct link, the analysis of refraction, and two-dimensional RISs can be found in the companion journal paper [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF].

The rest of this paper is organized as follows. In Section II, the system model is introduced. In Section III, the analytical formulation of the electric field emitted by a point source and scattered by a finite-size RIS is reported. Explicit expressions of the electric field in the near-field and far-field are given. In Section IV, numerical results are provided to illustrate the scaling laws of the received power as a function of the transmission distances. Finally, Section V concludes this paper.

II. SYSTEM MODEL

In a two-dimensional space, we consider a system that consists of a transmitter (Tx), a receiver (Rx), and a flat surface (S) of zero-thickness. Without loss of generality, we assume that S is located such that its center coincides with the origin. Furthermore, S lies in the x-axis and spans along [-L, L], i.e., S = {(x, 0) : -L ≤ x ≤ L}. In other words, S is a straight line. The locations of Tx and Rx are denoted by (x T , y T ) and (x R , y R ), respectively. We consider only the scenario where Tx and Rx are on the same side of the surface S, i.e., we focus our attention on modeling reflections from the surface S. Therefore, y T and y R take positive values, while there is no restriction on the values taken by x T and x R .

Tx is modeled as a point source that emits cylindrical electromagnetic (EM) waves through the vacuum. The EM waves emitted by Tx travel at the speed of light c. The frequency of the EM waves is denoted by f , and the wavelength and wavenumber are λ = c/f and k = 2π/λ, respectively. We are interested in computing the intensity of the electric field emitted by Tx and observed at an arbitrary point, i.e. Rx, on the positive y-axis, with the exception of the location of the point source. In vacuum, the x and y components of the electric field are not coupled, and we assume that S does not change the polarization of the EM waves. Under these assumptions, we can analyze any components of the electric field. We consider the tangential (to the surface S) component of the electric field, which is denoted by

E x (x R , y R ).
For every point (x, 0) ∈ S, the Tx-to-S and S-to-

Tx distances are denoted by d T (x) = (x -x T ) 2 + y 2 T and d R (x) = (x R -x) 2 + y 2 R
, respectively. In particular, d T (x) is the radius of the wavefront of the EM wave that is emitted by Tx and intersects S at (x, 0), and d R (x) is the radius of the wavefront of the EM wave that originates from S at (x, 0) and is observed at Rx. With a similar terminology, the angle of incidence of the EM wave at (x, 0) ∈ S is denoted by θ T (x). It represents the angle formed by the y-axis and the wavefront of the EM wave that originates from Tx and intersects S at (x, 0). The angle of reflection of the EM wave at (x, 0) ∈ S is denoted by θ R (x), and it represents the angle formed by the y-axis and the wavefront of the EM wave that is emitted by S at (x, 0) and is observed at Rx.

For simplicity, we assume d T (x) ≫ λ and d R (x) ≫ λ, which usually hold true in practical setups [START_REF] Tang | Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimental measurement[END_REF]. The complete analysis is available in [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF]. Under these assumptions, the electric field emitted by the point source (Tx) and observed at Rx in the absence of S corresponds to the Green function in the plane, which is well approximated as follows [START_REF] Economou | Green's functions in quantum physics[END_REF]:

E x (x R , y R ) ≈ E 0 exp (-jkd T R (x R , y R )) kd T R (x R , y R ) (1)
where E 0 = -j 1/(8π) exp (-jπ/4), j is the imaginary unit, and

d T R (x R , y R ) = (x R -x T ) 2 + (y R -y T )
2 is the distance between Tx and Rx. The surface S is modeled as a spatially-inhomogeneous reflector that is capable of modifying the phase of the incident field. We assume that the electromagnetic properties of the surface S vary slowly, as compared with the wavelength, along the surface itself. Under this approximation, the surface S can be well modeled as a local structure: the reflected field at (x, 0) ∈ S depends, approximately, only on the incident field at (x, 0) ∈ S [START_REF] Vahabzadeh | Computational analysis of metasurfaces[END_REF]. More precisely, the reflection coefficient at (x, 0) ∈ S can be written as follows:

Γ r (x) = C(x) exp (jΦ(x)) (2)
where C(x) ∈ R + and Φ(x) ∈ [0, 2π) denote the amplitude and phase of the reflection coefficient, respectively. In this paper, we are interested in analyzing only reflections. Furthermore, we assume that the surface S operates in the regime of a phase-gradient reflector and, therefore, assume C(x) = 1.

III. ELECTRIC FIELD REFLECTED FROM S

Based on the assumptions in Section II, the intensity of the electric field emitted by Tx, reflected by the surface S, and observed at (x R , y R ) for y R > 0 is given as follows.

Theorem 1. Let us assume d T (x) ≫ λ and d R (x) ≫ λ. The electric field E x (x R , y R ) can be formulated as follows:

E x (x R , y R ) = I 0 +L -L I (x) exp (-jkP (x)) dx (3)
where I 0 = 1/(8π) and:

P (x) = d T (x) + d R (x) -Φ (x) (4)
I (x) = 1 d T (x) d R (x) y T d T (x) + y R d R (x) (5)
Proof. It follows by formulating in mathematical terms the Huygens-Fresnel principle by using the general scalar theory of diffraction and by applying appropriate boundary conditions at the surface S. The details can be found in [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF].

The electric field in (3) is formulated in a simple integral form, which, however, does not explicitly unveil the dependency of the electric field as a function of the transmission distances. Also, the electric field depends on the specific phase shift Φ(x) applied by the surface S. In the following three sub-sections, we consider three case studies for the choice of Φ(x). Due to space limitations, only the first two case studies are analyzed in detail. For both cases, we introduce explicit approximate closed-form expressions for the electric field in (3) for short and long transmission distances. The definition of long transmission distance is given as follows.

Definition 1. Let us define d Q (x) = (x Q -x) 2 + y 2 Q and sin (θ Q0 ) = sin (θ Q (0)) = -qx Q /d Q0 for Q = {T, R}, where q = 1 if Q = T and q = -1 if Q = R, as well as d Q0 = x 2 Q + y 2 Q for Q = {T, R}.
The system is said to operate in the long distance regime if the approximation

d Q (x) ≈ d Q0 + qx sin (θ Q0 ) holds true for Q = {T, R}.
Otherwise, it is said to operate in the short distance regime.

Loosely speaking (with a slightly abuse of terminology), the long and short distance operating regimes can be identified with the far-field and near-field regimes, respectively.

A. S Acting as a Uniform Reflecting Surface

In this section, we analyze the case study in which the surface S operates as a mirror reflector. This operation is obtained by setting Φ (x) = φ 0 for x ∈ S in (3), where φ 0 ∈ [0, 2π) is a fixed phase shift. The following two propositions report approximate closed-form expressions of the intensity of the electric field in (3) under the assumption of short and long distance regime, respectively. Proposition 1. In the short distance regime, the intensity of the electric field in (3) can be approximated as follows:

|E x (x R , y R )| ≈ 1 √ 8πk 1 d T (x s ) + d R (x s ) (6)
where

x s ∈ [-L, L]
is the unique solution of the equation:

x s -x T d T (x s ) - x R -x s d R (x s ) = 0 (7)
Proof. It follows from (3) by applying the stationary phase method. The details can be found in [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF].

Remark 1. Proposition 1 holds true only if [START_REF] Karasik | Beyond max-SNR: Joint encoding for reconfigurable intelligent surfaces[END_REF] has at least one solution x s ∈ [-L, L]. The case study when this does not hold true can be found in [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF]. Similar comments hold for similar case studies analyzed in the following sub-sections.

From Proposition 1, the following conclusions follow.

• In the short distance regime, the surface S behaves as a specular mirror. In particular, the (end-to-end) intensity of the electric field reflected from the surface decays as a function of the reciprocal of the square root of the sum of the Tx-to-S and S-to-Rx distances. The presence of the square root originates from the assumption of twodimensional space (see the emitted field in (1)).

• Equation ( 6) can be regarded as an approximation of (3) under the condition of geometric optics propagation. More precisely, [START_REF] Ntotnin | Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison[END_REF] unveils that the intensity of the electric field is approximately the same as that obtained from a single ray (i.e., the direction of propagation of the wavefront of the EM wave) that is obtained from the two line segments that connect Tx with the point x s ∈ [-L, L] that fulfills [START_REF] Karasik | Beyond max-SNR: Joint encoding for reconfigurable intelligent surfaces[END_REF], and the latter point with Rx. Therefore, the point x s can be referred to as reflection point.

• From the definition of angles of incidence and reflection, we have

(x s -x T )/d T (x s ) = sin (θ T (x s )) and (x R -x s )/d R (x s ) = sin (θ R (x s ))
, respectively. From [START_REF] Karasik | Beyond max-SNR: Joint encoding for reconfigurable intelligent surfaces[END_REF], this implies that the angles of incidence and reflection coincide at the reflection point x s ∈ [-L, L]. In other words, (6) allows us to retrieve the law of reflection.

Proposition 2. In the long distance regime, the intensity of the electric field in (3) can be approximated as follows:

|E x (x R , y R )| ≈ 1 4π cos (θ T 0 ) + cos (θ R0 ) √ d T 0 d R0 × sin (kL (sin (θ T 0 ) -sin (θ R0 ))) k (sin (θ T 0 ) -sin (θ R0 )) (8)
where cos (θ

Q0 ) = y Q /d Q0 for Q = {T, R}.
Proof. It follows by using the approximation d Q (x) ≈ d Q0 + qx sin (θ Q0 ) for Q = {T, R}. The details are in [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF].

From Proposition 2, we evince that the surface S does not behave as a specular mirror for long transmission distances. The scaling law that governs the intensity of the electric field as a function of the distances is, in addition, no straightforward to be identified. To shed light on the impact of the transmission distances, i.e., d T 0 and d R0 in [START_REF] Huang | Holographic MIMO surfaces for 6G wireless networks: Opportunities, challenges, trends[END_REF], we consider the case study in which Tx and Rx move along two straight lines such that the angles θ T 0 and θ R0 are kept constant (the two angles do not have to be necessarily the same), but d T 0 and d R0 are different. In this case, we observe from (8) that the intensity of the electric field decays as a function of the square root of the product of the distances between Tx and the (center of the) surface S, and the (center of the) surface S and Rx. In this regime, therefore, the surface S is better modeled as a scatterer, since its size is relatively small in comparison with the transmission distances involved (i.e., d T 0 and d R0 ).

The findings in Propositions 1 and 2 provide us with evidence that justifies the validity of Theorem 1, and, therefore, substantiate the approach embraced in this paper for modeling and analyzing RISs in wireless networks. This constitutes the departing point of the following two sub-sections.

B. S Acting as a Reconfigurable Intelligent Surface

In this section, we analyze the case study in which the surface S operates as an RIS whose phase Φ (x) can be appropriately optimized. In particular, we assume that S acts as an anomalous reflector that is configured for reflecting the EM waves emitted by Tx towards a given direction. Due to the assumption C(x) = 1, we implicitly ignore parasitic scattering. To this end, Φ (x) in ( 3) is chosen as follows:

Φ (x) = φT -φR x + φ 0 /k (9)
where

φ 0 ∈ [0, 2π) is a fixed phase shift, φT = -x T x2 T + ȳ2 T , φR = xR x2 R + ȳ2
R , and (x T , ȳT ) and (x R , ȳR ) are parameters that are optimized for obtaining the desired reflection capability, as detailed in further text.

In contrast with Section III-A, this case study needs more elaboration to unveil the scaling law of the intensity of the electric field as a function of the distances. To this end, we consider the specific setup in which φT = sin (θ T (0)) and φR = sin (θ R (0)), which corresponds to a surface S that is configured by taking into account the angles of incidence and reflection of the EM with respect to (0, 0). Other case studies are analyzed in [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF]. It is worth mentioning that this setup does not necessarily imply (x T , ȳT ) = (x T , y T ) and (x R , ȳR ) = (x R , y R ), which would imply that the locations of Tx (the point source) and Rx (the observation point) need to be exactly known. Setups corresponding to different locations of Tx and Rx, but yielding the same angles of incidence and reflection, are included in the considered case study. For example, Tx and Rx move along two straight lines in which the angles with the y-axis at (0, 0) are kept constant but the distances are not.

The following two propositions report approximate expressions of the intensity of the electric field in (3) under the assumption of short and long distance regime, respectively. Proposition 3. Assume φT = sin (θ T (0)) and φR = sin (θ R (0)). In the short distance regime, the intensity of the electric field in (3) can be approximated as follows:

|E x (x R , y R )| ≈ 1 4 √ 2πk 1 -φ2 T + 1 -φ2 R 1 -φ2 R d T 0 + 1 -φ2 T d R0 ( 10
)
Proof. The proof is similar to Proposition 1 [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF].

Proposition 4. Assume φT = sin (θ T (0)) and φR = sin (θ R (0)). In the long distance regime, the intensity of the electric field in (3) can be approximated as follows:

|E x (x R , y R )| ≈ L 4π 1 -φ2 T + 1 -φ2 R √ d T 0 d R0 (11)
Proof. The proof is similar to Proposition 2 [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF].

Since φT and φR in Propositions 3 and 4 do not depend on the distances, the following conclusions can be drawn.

• From (10), RISs behave as anomalous mirrors in the short distance regime: the intensity of the electric field decays with the square root of a weighted sum of the distances, but the angles of incidence and reflection can be different.

• From [START_REF] Estakhri | Wave-front transformation with gradient metasurfaces[END_REF], RISs behave as scatterers in the long distance regime: the intensity of the electric field decays as a function of the square root of the product of the distances.

• In ( 10) and [START_REF] Estakhri | Wave-front transformation with gradient metasurfaces[END_REF], φT is configured based on the direction of incidence (at (0, 0) and with the y-axis) of the EM wave emitted by Tx, and φR is configured based on the desired direction of reflection (at (0, 0) and with the yaxis) of the EM wave reflected by the RIS. By optimizing φR , RISs can be configured to reflect EM waves towards, predominantly, any directions. The limitations are discussed in, e.g., [START_REF] Asadchy | Perfect control of reflection and refraction using spatially dispersive metasurfaces[END_REF]. This is different from [START_REF] Ntotnin | Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison[END_REF], in which the direction of reflection and incidence coincide. This is the difference between specular and anomalous mirrors.

C. S Acting as a Passive Reflecting Beamformer

In this section, we analyze the case study in which the surface S operates as an RIS whose phase Φ (x) is appropriately optimized in order for S to act as a beamformer. The difference with the previous sub-section can be summarized as follows.

• In Section III-B, the desired functionality of the RIS consists of reflecting (or steering) the incident EM wave towards a predetermined direction. All the receivers located in the direction of reflection benefit from the RIS. This setup is, therefore, more suitable for RISs that are employed for broadcasting applications [START_REF] Tang | Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimental measurement[END_REF].

• In this sub-section, on the other hand, the desired functionality of the RIS is to focus the EM wave towards a predetermined location. In this case, a single or a few receivers at specific locations benefit from the RIS. In particular, we consider that S acts as a beamformer (or a reflecting lens) that focuses the signal towards a single location (x R , ȳR ). To this end, Φ (x) in ( 3) is chosen as follows:

Φ (x) = (x -x T ) 2 + y 2 T + (x -xR ) 2 + ȳ2 R ( 12
)
From a mathematical point of view, Φ (x) in Section III-B is optimized such that the first-order derivative of P (x) is equal to zero at x s ∈ [-L, L] (if it exists). The phase Φ (x) in ( 12) is, by contrast, optimized such that P (x) is equal to zero when evaluated at the location of interest, i.e., (x R , ȳR ). The design criterion for optimizing the surface S is, thus, different.

The intensity of the electric field can be computed, in the long distance regime, by using analytical steps similar to those reported in Sections III-A and III-B [START_REF] Renzo | On the path-loss of reconfigurable intelligent surfaces: An approach based on the theory of diffraction[END_REF]. Due to space limitations, the details are omitted. Numerical illustrations are, however, reported in the next section in order to showcase the difference among the three configurations for the surface S.

IV. NUMERICAL RESULTS

In this section, we provide some illustrative numerical results in order to showcase the difference among the three configurations for S that are elaborated in Section III, and in order to numerically evaluate the intensity of the electric field as a function of the transmission distances. For ease of writing, the directions of incidence and reflection are identified by the angles θ T 0 and θ R0 , respectively. In all simulation results, we consider the following setup: (i) L = 0.75 m; (ii) f = 28 GHz; (iii) x T = -d T 0 sin (θ T 0 ) and y T = d T 0 cos (θ T 0 ) with θ T 0 = π/4; (iv) (x T , ȳT ) = (x T , y T ); and (v) φ 0 = 0. This setup corresponds to a scenario in which S is employed in the millimeter-wave frequency band, and its size, 2L, corresponds to, approximately, the diagonal of a two-dimensional surface of size 1 m 2 . This is compatible and in agreement with other recent papers and experimental activities [START_REF] Tang | Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimental measurement[END_REF], [START_REF] Khawaja | Coverage enhancement for NLOS mmWave links using passive reflectors[END_REF], [21].

More precisely, we consider two case studies.

• In the first case study, we are interested in illustrating the difference among the three different configurations for S analyzed in Section III. To this end, we plot the intensity of the electric field emitted by a fixed location (Tx) and observed at different locations (x R , y R ). The following setup is considered: The results corresponding to the first case study are reported in Fig. 1, which shows the intensity of the electric field obtained from (3). The figure substantiates the findings in Section III: (i) the angles of incidence and reflections of a uniform reflecting surface are the same (Section III-A); (ii) an RIS configured as described in Section III-B is capable of steering the reflected signal towards desired (anomalous) directions; and (iii) an RIS configured as described in Section III-C is capable of focusing the signal towards desired locations.

The results corresponding to the second case study are reported in Fig. 2, which compares (3) with the approximated closed-form expressions obtained in Sections III-A and III-B. The figure substantiates the findings in Section III. In particular: (i) the closed-form approximations for the short distance regime are accurate for end-to-end distances (2d 0 ) up to 100-150 m; and (ii) the closed-form approximations for the long distance regime are accurate for end-to-end distances (2d 0 ) greater than 200-250 m. For the considered setup, we conclude that RISs are capable of acting as anomalous mirrors for distances of the order of tens of meters. The range of distances for which the approximation holds depend, among other parameters, on the size of the surface and the operating frequency. In general, the larger the size of the surface is and the higher the operating frequency is, the more accurate the approximation as an anomalous mirror becomes, i.e., it can be used for longer transmission distances.

V. CONCLUSION

In this paper, we have leveraged the general scalar theory of diffraction in order to obtain approximate closed-form expressions of the intensity for the electric field reflected by RISs in the short and long transmission distance regimes. We have observed different scaling laws in the two considered operating regimes. The proposed approach and results constitute a first attempt to identify appropriate path-loss models for analyzing the achievable performance of RISs in wireless networks.

•Fig. 1 :Fig. 2 :

 12 Fig. 1: Intensity of the electric field from Theorem 1.

ACKNOWLEDGMENT

This work was supported in part by the European Commission through the H2020 5GstepFWD project under grant 722429 and the H2020 ARIADNE project under grant 871464.


Justine Touja

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Effect of the electrolyte on K-metal batteries

teaching and research institutions in France or abroad, or from public or private research centers.

The comparison of different electrolytes showed that both salt concentration and anion are key parameters for controlling the performance of K-metal batteries. Among the different tested electrolytes, 5 M KTFSI in DME exhibits the best stability at high potential and good performance in K|Prussian blue cells.

To meet the demand for high energy density batteries, alternative systems beyond Li-ion batteries (LIBs) have to be developed. Several reasons make potassium-based batteries a perfect alternative to LIBs for large-scale energy storage applications. First, the widespread availability of potassium makes it approximately six times cheaper than lithium, and leads to cheaper precursors and salts. The electrochemical properties of the K + /K redox couple are attractive, especially because the redox potential of K + /K in organic solvents is lower than that of Li + /Li, which can lead to a large potential window when associated with a high voltage cathode and thus high energy densities. 1 Moreover, in carbonate-based electrolytes, K + has the highest ion mobility, ion conductivity and ion transport number among alkaline cations (Li + , Na + , K + ). Finally, a recent study on existing commercial cell architectures and upto-date manufacturing know-how concluded that potassium batteries are capable of achieving 78 % of the energy density of LIBs. 2 Renewed design of electrochemical cells using metal negative electrodes is presently among the most promising paths to reach higher energy density batteries. In the case of lithium metal, by far the most studied system nowadays, the main issue is the uneven deposition of Li during charge that results in the detrimental growth of dendrites. 3 Their formation promotes the increase of the exposed metal surface favouring the continuous decomposition of the electrolyte and the growth of the solid-electrolyte interphase (SEI), and thus a progressive loss of active metal during cycling. The same issues are observed and even enhanced when using potassium metal electrodes. 4 In fact, the reactivity of potassium with the electrolyte is higher, and often influences the performance of the positive electrode when the latter is tested in half cells. 5,6 To counter this high reactivity and stabilize the metallic electrode, some solutions have been proposed. [7][8][9] One of the strategies to limit dendrite growth in metal anodes is using highly concentrated electrolytes. These contain fewer free solvent molecules and thus form thinner SEI, improving the cycling stability, as it has been observed for both Li 10,11 and Na. 12 Highly concentrated potassium electrolytes have also been investigated, for instance with potassium bis(fluorosulfonyl)imide (KFSI) dissolved in ether solvents, for graphite electrodes 13,14 and positive electrodes such as Prussian blue analogues (PBA). 15,16 Concentrated KFSI in 1,2dimethoxyethane (DME) electrolytes, as for Li and Na, seem to prevent the corrosion of the aluminium current collector, increasing the oxidation stability up to 5 V vs. K + /K and allowing the stable and reversible plating/stripping of potassium metal with an uniform and stable SEI. 16,15 Herein a pioneer study is presented on the effect of electrolyte concentration and of the salt anion on the performance of a Kmetal cell using four electrolytes with KFSI or KTFSI at 1 or 5 moles per litre of solvent (named 1M or 5M) dissolved in DME. These electrolytes will be referred to as KFSI-1M, KFSI-5M, KTFSI-1M and KTFSI-5M. Their respective impacts on the stability at high potential in the presence of Al current collectors, on the potassium plating/stripping efficiency and on the performance of a cell built with the PBA K1.73Mn2/3Fe1/3[Fe(CN)6] nH2O as the positive electrode is evaluated. This type of PBA is similar to that proposed as cathode material in aqueous batteries, 17 and is used here as a a. ICGM, Univ. Montpellier, CNRS, ENSCM, Montpellier, France. b. Réseau sur le Stockage Electrochimique de l'Energie (RS2E), FR CNRS 3459, Hub de l'Energie, Amiens, France. c. ALISTORE-European Research Institute, FR CNRS 3104, Hub de l'Energie, Amiens, France.

benchmark cathode. A full characterisation of this material will be the object of a forthcoming paper. The stability at high voltage for each electrolyte was studied by linear sweep voltammetry (LSV) in coin cells built with a potassium negative electrode and an aluminium foil as the positive electrode to compare the influence of both anions and electrolytes concentrations on the aluminium corrosion (Figure 1). It is important to note that, in lithium batteries, where the electrolytes are usually based on lithium hexafluorophosphate (LiPF6), PF6 -is known to protect the aluminium current collector of the positive electrode by forming an insoluble passivation layer, preserving the electrode from excessive corrosion during cycling, whereas FSI -does not lead to such a favourable feature. 18 In potassium batteries, the use of KFSI-based electrolytes has the advantage of reducing the large first cycle irreversible capacities obtained with KPF6-based ones, but brings as a drawback the loss of the interesting passivation ability of PF6 -. 19 The LSV measurements were performed with the four electrolytes at a scan rate of 0.1 mV s -1 up to 6 V vs. K + /K. The anodic current increases suddenly for KFSI-1M from 4.2 V, which confirms its limited stability at high potential in the presence of aluminium (Figure 1a). With the highly concentrated electrolyte (KFSI-5M), the electrolyte decomposition or the aluminium corrosion is slightly delayed as the anodic current increases only starting from 4.4 V. Changing from KFSI to KTFSI has a more important impact on the LSV. The voltage stability window is enhanced, since the current increase starts at 4.6 V for KTFSI-1M and at 4.7 V for KTFSI-5M, in agreement with previous observations. 20 Note that the first current raise occurring during LSV with both KTFSI-based electrolytes could be also due to the oxidation of TFSI -anion, as previously observed in lithium systems. 21 The surface of the Al current collectors after the LSV (Fig. 1b-e) shows several pits when low concentrated electrolytes are used, and very few or no pits when employing highly concentrated electrolytes. This result indicates that by increasing the concentration, the corrosion at the aluminium current collector is reduced. In summary, the stability of highly concentrated electrolytes, and especially of that based on the KTFSI salt, observed high potential might enable high energy batteries obtained by coupling new cathode materials working at high potential with K metal anodes. The plating/stripping stability of potassium metal in these electrolytes was compared in symmetric K|K cells at different current densities, namely 0.025 mA cm -2 and 1.00 mA cm -2 . Figures 2a and2b show the K plating/stripping voltage profiles obtained at 0.025 mA cm -2 with KFSI-based electrolytes and KTFSI ones, respectively. At low current density, the cell using KFSI-1M has a stable overpotential near 0.1 V for 300 h whereas with KTFSI-1M, the polarization increases rapidly after 50 h, reaching 0.5 V. A sharp increase of potential like this one has been proven to be due to the growth of dendrites which can short circuit the batteries. 22 In these conditions, the cells with the highly concentrated electrolytes present a lower polarization than the ones with the low concentrated electrolytes. However, the overpotential measured for KFSI-5M decreases from 800 mV to 300 mV within the ten first cycles, whereas the one for KTFSI-5M remains constant for all the experiment. The reaction between the K-metal and the electrolytes leading to different nature of SEI can explain this difference. 5 The electrochemical impedance spectroscopy (EIS) measured on the symmetrical cells before cycling, show that the global resistance of the system is lower with KTFSI-5M than with KFSI-5M (Fig. S2). This is in agreement with the higher polarization of the K-metal plating/stripping with the latter electrolyte for the first cycle. The chemically formed SEI changes with the electrolyte nature and is then more resistive with the KFSI-based electrolyte than with the KTFSI-based one. After 10 cycles, the impedance decreases in both cases but remains nearly twice higher for KFSI-5M. Both highly concentrated KFSI and KTFSI-based electrolytes are, however, very stable for 300 h revealing that no, or at least fewer, dendrites are formed with these electrolytes. No colour change of the separator was observed after cycling for both 5M electrolytes (Fig. S3a). It means that neither the degradation of the salt nor the dendrites growth through the separators can be observed for low current density, as suggested by the K metal plating/stripping electrochemical measurements (Fig. 2a,b). At a higher current density, i.e., 1 mA cm -2 , (Fig. 2c,d) the plating/stripping curves with both 1 M-electrolytes show poor stability, since the overpotential is higher than 1 V after few hours. However, although KFSI-5M does not improve the overpotential stability (Fig. 2c), the overpotential of the cell using KTFSI-5M remains stable for 300 h around 400 mV (Fig. 2d). The use of concentrated KTFSI instead of KFSI seems to improve the stability of the plating/stripping of potassium metal, especially at high current densities. After opening the batteries, the separator of the KFSI-5M cell was partially covered with K-metal whereas the one of KTFSI-5M cell was fully covered at the location of the metallic electrode (Fig. S3b). This also suggests that the plating of K-metal is more homogeneous with KTFSI-5M than with KFSI-5M. The reactions between the potassium and the electrolyte may have a lower impact on the galvanostatic measurement of a half-cell with KTFSI-5M. The electrolytes were finally benchmarked using PBA cathodes (K1.73Mn2/3Fe1/3[Fe(CN)6]•nH2O) between 4.3 and 3.0 V vs. K + /K at a current rate of 25 mA g -1 . The galvanostatic curves for K|PBA cells using either low or highly concentrated electrolytes are presented in Fig. S4 and Fig. 3a,b, respectively. The average shape of the galvanostatic curves is similar for the four electrolytes, and is consistent with that observed by Jiang et al. in different electrolyte systems. 17 The initial coulombic efficiency (CE) is clearly improved for the highly concentrated electrolytes. For KFSI-1M, the first charge capacity is surprisingly high (316 mA h g -1 ) compared to the first charge of KFSI-5M (145 mA h g -1 ) but the CE is stable only for five cycles before dropping (Fig. S5a). Even though the initial CE is high for KFSI-5M, with a value of 85%, it drastically drops after a few cycles (Fig. 3c) due to the steady increase of the charge capacity after four cycles and the irreversible cell failure after ten cycles. Such a behaviour, already reported for KFSI (1 M) in EC/DEC, results from the formation of a non-conductive passivation layer for K + ions on the potassium metal, as well as from the growth of potassium dendrites leading to short-circuits. 23 This behaviour is consistent with the plating/stripping results reported above. Moreover, the instability above 4 V of the KFSI-based electrolytes observed by LSV, due to parasitic oxidation reactions such as electrolyte decomposition and/or Al corrosion, could also explain the observed coulombic efficiency drops.

For the KTFSI-based electrolytes, the increase in salt concentration allows higher specific capacities, and the longterm cycling highlights the better stability and reversibility of the reactions, in comparison with KTFSI-1M (Fig. 3d, Fig. S5b). This is probably due to the thinner and more inorganic SEI formed during the galvanostatic experiment with concentrated electrolytes. 24 On the other hand the higher polarization observed in the first cycle with KFSI-5M electrolyte could due to the SEI formed on the K-metal. Indeed, the plating/stripping and the EIS measurements showed that the SEI is more resistive with the highly concentrated KFSI-based electrolyte than with the KTFSIbased one, especially for the first cycles. The reactivity of the Kmetal with the electrolyte has an impact on the electrochemical performance of the half-cells as suggested. The anion used in the electrolyte plays also an important role for the cycling stability of the cathode. Indeed, for both 5 M KFSI and KTFSI in DME, the capacities are quite the same for the first cycles but the use of TFSI instead of FSI enhances the capacity retention, which remains at 110 mA h g -1 after 100 cycles (i.e. 72 % of the initial capacity), as well as the CE (96 %). The electrochemical performance of PBA was shown to be controlled by different parameters such as crystallinity, defects or lattice water content, 25,26 which can explain the slight differences between the results shown here and those previously obtained with highly concentrated KFSI electrolytes. 15 In conclusion, both the salt and its concentration in the electrolyte have a strong impact on the electrochemical stability of potassium metal batteries. By increasing the concentration from 1 M to 5 M in both KFSI and KTFSI in DME electrolytes, the Al current collector corrosion is delayed in potential, allowing the use of high voltage cathodes. Spectroscopic analyses of the Al current collector surface are required to get deeper insight on its behaviour with respect to electrolyte salt and concentration. Moreover, concentrated electrolytes improve the stability of potassium metal plating/stripping, especially for the KTFSI-5M system, demonstrating the crucial role of the salt anion on the SEI formation. The SEI formed with KTFSI is likely more favourable for the plating/stripping of potassium, reducing parasitic reactions and dendrite growth. These results suggest that potassium metal electrode is more stable with KTFSI-5M, less impacting the electrochemical performance of the positive electrode. This assumption is confirmed here in K|PBA cells, showing enhanced capacity as well as CE performance after 100 cycles with the highly concentrated KTFSI electrolyte. Further studies are ongoing to analyse and compare the SEI formed with these different electrolytes, and to better understand their impact on potassium metal plating/stripping as well as on the K|PBA battery performance.

Figure 1 .Figure 2 .

 12 Figure 1. Linear sweep voltammetry of K-metal vs Al with the four electrolytes (a). Optical microscopy images (40X) of the Al current collector after the LSV of KFSI-1M (b), KFSI-5M (c), KTFSI-1M (d) and KTFSI-5M (e) Figure 2. K-metal plating/stripping curves of K|K symmetric cells with KFSI-1M and KFSI-5M and KTFSI-1M and KTFSI-5M at a current density of 25 µA/cm² (a,b) and 1 mA/cm² (c,d)

Figure 3 .

 3 Figure 3. Galvanostatic profiles and long-term cycling evolution of PBA with KFSI-5M (a,c) and KTFSI-5M (b,d) recorded at 25 mA/g between 3.0 and 4.3 V

Acknowledgements

Jérôme Long is gratefully acknowledge for providing the PBA K1.73Mn2/3Fe1/3[Fe(CN)6]•nH2O used for preparing the benchmark cathodes. The authors thank the Alistore-European Research Institute for their financial support through the PhD grant to P. N. Le Pham. Authors also thank A. Mohammadi for the graphical abstract. The authors acknowledge also the French RS2E Network (RS2E) (STORE-EX Labex ProjectANR-10-LABX-76-01).

Conflicts of interest

There are no conflicts to declare.

Notes and references


Anne-Claire Gaillot
email: claire.gaillot@cnrs-imn.fr

Victor A Drits

Bruno Lanson

Buseck Peter

Kenneth Livi
email: klivi@jhu.edu

POLYMORPH AND POLYTYPE IDENTIFICATION FROM INDIVIDUAL MICA PARTICLES USING SELECTED AREA ELECTRON DIFFRACTION

Keywords: Cis-vacant, Dioctahedral mica, Electron diffraction, Illite, Muscovite, Polymorph, Polytype, Trans-vacant, Zone-axis orientation

whether they are published or not. The documents may come

INTRODUCTION

K-bearing, aluminous, dioctahedral micas such as illite, muscovite, or phengite, are most abundant in nature and occur, for example, in diagenetically altered sediments, soils, low-grade metamorphic rocks, and igneous and hydrothermal systems [START_REF] Bailey | Classification and structure of the micas[END_REF]Mottana et al. 2004 and references therein). Ideally, their 2:1 layers are composed of a sheet of edge-sharing Al 3+ octahedra set between two sheets of Si 4+ tetrahedra. In dioctahedral micas, one out of three octahedral sites is vacant. Isomorphic substitutions can occur both in octahedral and tetrahedral sheets to produce a layer charge deficit which is compensated by interlayer K + cations. Interactions between adjacent layers are relatively weak and produce energetic similarity of polytypes with different layer stacking. Two or more polytypes can coexist even within a crystal. Whereas polytypes differ mainly by their stacking sequences, polymorphs result from different locations of the octahedral vacancy within the 2:1 layer. The average polytypic/polymorphic components of a sample can be determined using X-ray diffraction or thermo-gravimetric analysis [START_REF] Drits | Diffraction effects calculated for structural models of K-saturated montmorillonite containing different types of defects[END_REF][START_REF] Drits | X-ray identification of one-layer illite varieties: Application to the study of illites around uranium deposits of Canada[END_REF][START_REF] Drits | Semiquantitative determination of trans-vacant and cis-vacant 2:1 layers in illites and illite-smectites by thermal analysis and X-ray diffraction[END_REF][START_REF] Bailey | Hydrous phyllosilicates (exclusive of micas)[END_REF][START_REF] Emmerich | Dehydroxylation behavior of heat-treated and steam-treated homoionic cis-vacant montmorillonites[END_REF][START_REF] Drits | Potential problems in the interpretation of powder X-ray diffraction patterns from fine-dispersed 2M1 and 3T dioctahedral micas[END_REF][START_REF] Zviagina | X-ray diffraction criteria for the identification of trans-and cis-vacant varieties of dioctahedral micas[END_REF]). Determining the structure of individual particles requires a local probe and electron microscopy techniques are ideal candidates. Periodicity normal to 2:1 layers and occurrence and nature of stacking faults can be determined from electron diffraction (ED) patterns or high-resolution images collected with the incident electron beam parallel to the layers [START_REF] Kogure | Direct identification of the six polytypes of chlorite characterized by semi-random stacking[END_REF][START_REF] Kogure | First occurrence of a stacking sequence including (+60°, 180°) rotations in Mg-rich annite[END_REF][START_REF] Kameda | XRD and HRTEM analyses of stacking structures in sudoite, di-trioctahedral chlorite[END_REF]Kogure & Kameda 2008;Kogure et al. 2008). As a possible alternative to the latter approach, [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF] showed that ED patterns collected along the [001] zone axis can also be used to determine the periodicity along the stacking direction from hk0 intensities and to obtain structural information such as location of the octahedral vacancy. This method is more limited than when crystals are oriented normal to [001]. A minimal sample preparation (a drop of a diluted suspension dried on the support) produces many well-oriented crystals quickly and both morphological and structural information is then available.

Morphology of illite, muscovite, or phengite and possibly their (qualitative) chemistry, were shown to be correlated to crystal structure [START_REF] Lanson | Illitization of diagenetic kaolinite-to-dickite conversion series: Late-stage diagenesis of the lower Permian Rotliegend sandstone reservoir, offshore of The Netherlands[END_REF][START_REF] Lanson | Authigenic kaolin and illitic minerals during burial diagenesis of sandstones: A review[END_REF][START_REF] Patrier | High-grade diagenetic dickite and 2M1 illite from the middle Proterozoic Kombolgie formation (Northern Territory, Australia)[END_REF][START_REF] Laverret | Mineralogy and geochemistry of the host-rock alterations associated with the Shea creek unconformity-type uranium deposits (Athabasca basin, Saskatchewan, Canada). Part 1. Spatial variation of illite properties[END_REF]). This link could not be unambiguously established using global methods such as X-ray diffraction (XRD), thermogravimetric analysis (TGA), or infrared (IR) spectroscopy, however, and the method proposed here thus provides a unique opportunity to (in)validate the established qualitative relationship from the characterization of individual finely divided crystals.

If valid, this relationship may be important in determining structure characteristics of mica crystals coating quartz grains in relation to their impact on oil reservoir quality, especially on permeability (Morris & Shepperd 1982;Pallatt et al. 1984;Kantorowicz 1990;Pevear 1999;Wilson et al. 2014).

Crystal morphologies ranging from one-dimensional (1D) "hairy" illite to more isometric pseudohexagonal 2D plates were reported for illite as a function of maximum burial depth and size fraction in sandstone reservoirs (Lanson et al. 2002 and references therein;Wilson et al. 2014). A structural control on crystal morphology could determine the origin of the morphological variability. The possibility of determining the morphology, dimensions, and crystal structure of individual plates may have implications for crystal growth mechanisms of the different polytype/polymorphs. Convergent-beam electron diffraction (CBED) patterns can also be used to determine periodicity along the c*-axis from both hk0 and hk1 reflections of the zero-and first-order Laue zones (ZOLZ and FOLZ) using minimal sample preparation and crystal orientation [START_REF] Beermann | Structure and analysis of montmorillonite crystallites by convergent-beam electron diffraction[END_REF]. CBED application to clay minerals has been limited. First, diffraction intensity is very weak beyond hk0 spots and hk1 spots from the FOLZ because clay-size micas occur as thin crystals composed mainly of weakly scattering atoms (Si, Al, Mg, O). Second, stacking defects are common even in well-ordered dioctahedral aluminous micas, and the defects induce elongated reciprocal rods along the c*-axis. Finally, CBED patterns allows collection of diffraction patterns from a limited crystal surface area (a few nm 2 ), thus limiting variation in crystal orientation.

However, the high electron dose required for CBED pattern collection, with the electron beam fully condensed on mica crystals, often results in sample damage, ranging from amorphization to crystal "drilling". Beam damage also occurs when cis-vacant (see below) layers of illite and smectite and trans-vacant layers of nontronite are imaged at high resolution. In these cases, layers dehydroxylate owing to the high electron dose which results in cation migration from one octahedral site to another [START_REF] Kogure | Structural change in celadonite and cis-vacant illite by electron radiation in TEM[END_REF]. In contrast, conventional parallel-illumination ED (selected-area electron diffraction, SAED) requires very low electron dose compared to CBED, while producing averaged intensities over a larger ZOLZ area [START_REF] Vincent | Double conical beam-rocking system for measurement of integrated electron diffraction intensities[END_REF]. SAED allows retrieving layer stacking information in the direction parallel to the electron beam [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF].

Information on the vacancy location (polymorphismcis-or trans-vacant configuration) can also be obtained for the 1M polytype.

Charge-coupled devices (CCD) cameras on microscopes allow recording quantitative intensities for ED spots. Structure identification remains challenging, however, owing to the camera dynamics needed to record ED patterns equal to the high quality achievable with films or recording plates. A small beam stop commonly used to avoid saturating the CCD camera is essential also to avoid masking part of the ED pattern. A small beam stop allows the optimization of orientation during data collection and the analysis of the diffraction intensity distribution.

However, intrinsic properties of mica crystals further impair quantitative analysis. The coexistence and interstratification of polytypic and polymorphic varieties within a crystal was addressed by [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF], together with the impact of variable crystal thickness on mica ED patterns. In addition, mica plates may be bent, or have growth steps, or variable chemical compositions, which result in inhomogeneous local diffraction conditions. The influence of these additional parameters on the intensity diffracted by dioctahedral 2:1 micas is investigated in the present study.

Identification criteria of mica polytypes are proposed using qualitative intensity distributions within [001] ED patterns, together with a database of ED patterns for a selected set of crystal orientations.

STRUCTURE MODELS AND METHODS

Structure models

The structural formula for K-bearing aluminous dioctahedral micas [START_REF] Bailey | Classification and structure of the micas[END_REF]) is:

2 10 3 x 4 x - 4 [IV] 2 y 3 y - 2 [VI] y x (OH) )O Al (Si ) R (Al K       , 0.9  x + y  1 (1)
where octahedral R 2+ cations are Mg 2+ and Fe 2+ . To simplify structure models, the interlayer site was assumed to host 1.0 K + cation per O10(OH)2. Al 3+ -for-Si 4+ , and Mg 2+ -for-Al 3+

substitutions and H were ignored because they have negligible effects on the structure factors of hk reflections, leading to the following simplified composition:

12 4 [IV] 2 [VI] O Si Al K (to show idealized occupancy). (2)
A complete Fe 2+,3+ -for-Al 3+ substitution in the octahedra, as in celadonite [ideally

[VI] 3 2 [IV] 4 10 2 K (Fe Fe ) Si O (OH)  ]
, was investigated to show the effect of the greater scattering.

Unit-cell parameters and atomic coordinates were kept constant despite the ionic radii difference between Fe and Al. Similarly, Na + -for-K + replacement, as in paragonite [ideally Trans-and cis-vacant 1M polymorphs. In ideal one-layer monoclinic (1M) mica polytypes, all 2:1 layers have the same azimuthal orientation and the layer displacement Tx is about -a/3 along the a-axis, a being defined from the orthogonal C-centered cell described by [START_REF] Bailey | Hydrous phyllosilicates (exclusive of micas)[END_REF] and given in Table 1 (b = a3, γ = 90°). Where the octahedral M1 site is empty and M2 and M2' are filled, the 2:1 layer is trans-vacant (tv) with a C2/m symmetry. When the M2 (or M2') site is vacant and M1 and M2' (or M2) sites are occupied, the 2:1 layer is cis-vacant (cv) with a C2 symmetry [START_REF] Bailey | Classification and structure of the micas[END_REF]. The two polymorphs possess identical layer plane unit-cell dimensions (a, b, γ) and layer-to-layer distance (Tz = c.sinβ) for identical compositions [START_REF] Drits | Diffraction effects calculated for structural models of K-saturated montmorillonite containing different types of defects[END_REF]. They differ however by their layer displacement along the a-axis with Tx = c.cosβ values of -0.383a and -0.308a, for tv-and cv-1M, respectively (Table 1; [START_REF] Drits | Diffraction effects calculated for structural models of K-saturated montmorillonite containing different types of defects[END_REF]. Coordinates of the octahedral sites also differ slightly between the two polymorphs. Because tv and cv layers can coexist within 1M mica crystals [START_REF] Mccarty | Rotationally disordered illite/smectite in paleozoic Kbentonites[END_REF][START_REF] Drits | The nature of diffraction effects from illite and illite-smectite consisting of interstratified trans-vacant and cis-vacant 2:1 layers: A semiquantitative technique for determination of layer-type content[END_REF][START_REF] Ylagan | Reaction mechanisms of smectite illitization associated with hydrothermal alteration from ponza island[END_REF], random interstratifications are averaged by diffraction to produce a simplified cell (M-cell).

In the average M-cell, the octahedral sheet is not distorted, octahedral sites are partially occupied, and the interlayer displacement Tx is -0.333a [START_REF] Drits | Diffraction effects calculated for structural models of K-saturated montmorillonite containing different types of defects[END_REF]. Crystals with interstratifications having contrasting proportions of tv and cv layers may be described by varying the occupancy of M1, M2, and M2' sites (w1, w2, and w2', respectively). For example, w1 = 0 and w2 = w2' = 1 [Mcell with (0:1:1)] correspond to tv-1M micas, whereas w1 = w2 = 0.5 and w2' = 1 [M-cell with (0.5:0.5:1)] correspond to the interstratification of tv and cv layers in a 1:1 ratio.

2M1, 2M2, and 3T polytypes. In two-layer monoclinic polytypes, successive layers are alternately rotated by ±120° or ±60° (2M1 and 2M2 polytypes, respectively). Each c-axis forms an oblique angle β with [100] and their projection on the a-axis corresponds to the overall layer displacement of ~-a/3 (for 2 layers). In contrast to all other mica polytypes, the a vector of the 2M2 polytype corresponds to the long dimension of the orthogonal cell in the ab-plane (with a ~ b3), thus producing major differences for the orientation of ED patterns and subsequent analysis (see below). In the three-layer trigonal 3T polytype, successive layers are rotated by 120° with respect to the preceding layer and the overall layer displacement is null. Structure models of 2M1 and 3T polytypes were refined from muscovite and phengite X-ray and neutron diffraction data (#86622 and #75952 of the ICSD database -Amisano- [START_REF] Amisano-Canesi | Muscovite-3T and phengite-3T -crystal-structure and conditions of formation[END_REF][START_REF] Liang | Triclinic muscovite: X-ray diffraction, neutron diffraction and photo-acoustic FTIR spectroscopy[END_REF], whereas the 2M2 model corresponds to the refinement of oblique texture ED data from a natural mica [START_REF] Zhoukhlistov | Structure of a dioctahedral mica 2M2 according to high-voltage electron diffraction data (in Russian)[END_REF]. Layer chemistry was simplified for each model to match Eq. 2. Cv structures were modeled from those of tv polymorphs by changing occupancy of octahedral sites, while unit-cell parameters and atomic positions were kept unchanged. Unit-cell parameters are listed in Table 1 for all polytypes considered.

Theoretical ED calculations

Dynamical effects are minor for most fined-grained mica crystals of thickness 10-25 nm if the structure consists essentially of light elements [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF]). In addition, natural micas occur as mosaic crystals, thus reducing the likelihood of multiple scattering and further minimizing dynamical effects on ED patterns [START_REF] Drits | Electron diffraction and high-resolution electron microscopy of mineral structures[END_REF][START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF]). To assess this assumption, SAED patterns were collected using 1° and 2° precession angles of the incident electron beam to minimize dynamical diffraction effects [START_REF] Vincent | Double conical beam-rocking system for measurement of integrated electron diffraction intensities[END_REF], and compared with ED patterns obtained with no beam precession (Fig. 1). The similarity of ED patterns confirms the quasi absence of dynamical diffraction effects. Consequently, qualitative interpretation of the ED patterns used the kinematical approximation (single scattering event) only with the JEMS simulation software package [START_REF] Stadelmann | Electron Microscopy Suite, Java version[END_REF]. See [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF] for simulation parameters. Intensities are proportional to the square of the structure factor (F 2 ), modulated by the excitation error s. A s value of 0.2 nm -1 allows matching the extension of the experimental ZOLZ region while excluding contribution from the FOLZ. Calculated intensities are displayed with a spot radius proportional to F to mimic intensity differences. Crystal thickness was not considered.

Experimental ED patterns

Origin and chemical compositions of the two illite specimens investigated are described by [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF]. Briefly, the sample from the Athabasca basin, Canada, contains both tvand cv-1M crystals, whereas the sample from Kombolgie, Australia, is essentially a 2M1 illite.

Samples were ultrasonically dispersed in de-ionized water and deposited on Cu-mesh grids coated with a holey carbon membrane. SAED patterns were collected on Kodak negative films (Eastman Kodak company, Rochester NY, USA) or with a GATAN CCD camera (Gatan, Pleasanton CA, USA) using a Hitachi H9000 NAR transmission electron microscope (TEM, Hitachi, Okinawa, Japan) equipped with a LaB6 source and operated at 300 kV and a JEOL 2000fx LaB6 TEM operated at 200kV (Jeol, Tokyo, Japan). Complementary Precession ED patterns were acquired on the Nant'Themis, a S/TEM Themis Z G3 (Thermo Fisher Scientific, Whaltman MA, USA) Csprobe corrected, monochromated, operated at 300 kV and equipped with a Complementary Metal Oxide Semiconductor (CMOS) GATAN OneView camera and the 'Digistar' precession system from NanoMEGAS SPRL (Bruxelles, Belgium).

RESULTS AND DISCUSSION

Finely dispersed mica crystals tend to settle on basal surfaces and lay on grids with the c*-direction quasi parallel to the electron beam. Intensity distributions of various polytypes oriented along the c*-axis are not sufficiently contrasting to allow for polytype/polymorph identification, however (see below). Contrast is enhanced when microcrystals are oriented with their [001], i.e., c-axis, parallel to the electron beam [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF]. Qualitative criteria based on the intensity distribution over hk0 spots that are used for identification of the different mica polymorphs and polytypes from their [001] ED patterns are given below. unique cis site (Fig. 2b), except for a limited weakening of 11 reflections and thus a lower contrast with 02 reflections. Similar differences are observed for X-ray diffraction intensity distributions calculated for tv and cv polymorphs (See for example [START_REF] Drits | Diffraction effects calculated for structural models of K-saturated montmorillonite containing different types of defects[END_REF][START_REF] Drits | X-ray identification of one-layer illite varieties: Application to the study of illites around uranium deposits of Canada[END_REF][START_REF] Drits | The nature of diffraction effects from illite and illite-smectite consisting of interstratified trans-vacant and cis-vacant 2:1 layers: A semiquantitative technique for determination of layer-type content[END_REF]. Where tv and cv layers coexist with equal probability [M-cell (0.5:0.5:1), Fig. 2k], the resulting ED pattern resembles that calculated for a pure tv layer (Fig. 2a). For a pure tv layer, weaker 02 reflections occur, and a lower intensity contrast exists between 02 and 11 reflections.

ED along [001]: geometry, symmetry and indexing

Intensities of 11 and 02 reflections are identical for tv and cv layers occurring in a 1:2 ratio within crystals. For a 1:2 tv:cv ratio, the intensity of 22 and 04 reflections are increased when vacancies occur equally over the three octahedral sites [M-cell (0.67:0.67:0.67), 3T polymorphs. In contrast to previous cases, the [001] direction is parallel to both c-and c*-axes, and the ED pattern for tv-3T polytype (Fig. 2e) is hexagonal and exhibits six-fold rotational symmetry, with very strong and of equal intensity 06 and 33 reflections, whereas 11, 02, 22, 04, 26, and 40 spots are weak (Fig. 2j). Vacancy location has a minor influence on the intensity distribution over the ED pattern: 02 and 11 reflections are slightly more intense for the tv-3T polymorph whereas 04, 22, 40, and 26 reflections are more intense for the cv-3T polymorph (Fig. 2o). These differences are not sufficient to allow experimental differentiation of tv-3T and cv-3T polymorphs.

Validity of identification criteria

Precession ED criteria. Precession ED (PED) is increasingly used for structure identification or refinement owing to the minimal influence of dynamical effects, with limited electron beams being excited simultaneously [START_REF] Vincent | Double conical beam-rocking system for measurement of integrated electron diffraction intensities[END_REF][START_REF] Gjonnes | Structure model for the phase AlmFe derived from three-dimensional electron diffraction intensity data collected by a precession technique. Comparison with convergentbeam diffraction[END_REF]Gemmi & Nicolopoulos 2007;[START_REF] Nicolopoulos | From powder diffraction to structure resolution of nanocrystals by precession electron diffraction[END_REF][START_REF] Moeck | Precession electron diffraction and its advantages for structural fingerprinting in the transmission electron microscope[END_REF]. These authors showed that intensities are proportional to the structure factor amplitude rather than to its squared value as in conventional ED. Although PED intensities are not equal to the present calculations, qualitative distribution of intensity among bright and weak beams are similar, and identification criteria established above for SAED remain valid (Fig. 1).

Qualitative vs. quantitative criteria. Qualitative, rather than quantitative, identification criteria are proposed in the present work (Figs. 2f-j) because of the limited availability of cameras with sensor dynamics to allow the recording of high-quality ED patterns. In addition, qualitative criteria allow a quick assessment of the polytypic form during observation and a real-time decision for data collection strategy. More importantly, and as discussed hereafter, numerous parameters influence actual ED intensities beside the crystal structure: i) crystal thickness, which is possible to assess in the microscope by using electron-energy loss spectroscopy (EELS) and analyzing the plasmon peak, or by using beforehand shadowing treatment of the sample; ii) chemical composition, with the implicit requirement of systematic energy dispersive X-ray spectroscopy;

and iii) nature and number of structural defects (i.e., random or well-defined translational or rotational stacking faults, interstratification of cv-and tv-layers or of polytypic fragments, etc.).

Consequently, systematic quantitative processing of diffracted intensities is unrealistic whereas qualitative identification criteria are useful for polytypic and polymorphic varieties of dioctahedral K-bearing aluminous mica, if only few defects are present and if the mica occurs as thin flat crystals. Intensity ratios may be used to determine structure details, such as the relative proportion of cv and tv layers in 1M crystals from the intensity ratio between 02 and 11 reflections (Figs. 2k,l).

Influence of crystal thickness. ED intensities were calculated here with the kinematical approximation, thus neglecting multiple scattering events that may occur for thick crystals as described by [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF]. In the absence of beam precession capability to minimize multiple scattering effects, crystal thickness can be estimated in situ EELS. Alternatively, the low intensity of diffracted beams compared to the direct beam will ensure that crystals are sufficiently thin to interpret intensity distribution among hk spots (h and k  6). In any case, reported intensity distributions based on kinematical approximation, are probably sufficient to qualitatively interpret SAED patterns and to identify polytypes and polymorphs for most crystals of microcrystalline micas, containing mostly light elements [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF].

Influence of local unit-cell distortions. Except for the 1M polytype, structure models [START_REF] Zhoukhlistov | The crystal structure of the dioctahedral mica 2M2 determined by high voltage electron diffraction[END_REF][START_REF] Zhoukhlistov | Structure of a dioctahedral mica 2M2 according to high-voltage electron diffraction data (in Russian)[END_REF][START_REF] Drits | Diffraction effects calculated for structural models of K-saturated montmorillonite containing different types of defects[END_REF]Drits et al. , 2010aDrits et al. , 2010b;;[START_REF] Amisano-Canesi | Muscovite-3T and phengite-3T -crystal-structure and conditions of formation[END_REF][START_REF] Liang | Triclinic muscovite: X-ray diffraction, neutron diffraction and photo-acoustic FTIR spectroscopy[END_REF]) used for the calculations reported here were systematically refined for tv layer configuration. As noted above, diffraction effects from cv polymorphs and structures where tv and cv layers coexist were calculated by varying the wi occupancy of M1, M2, and M2' sites without adjustment of atomic positions. Comparison of ED patterns calculated for tv-1M and cv-1M models with adjustment of atomic positions and for the average M-layer with equivalent octahedral site occupancies (0:1:1 and 1:0:1, respectively) indicates the sensitivity of ED patterns to local distortions within the unit cell (Figs. 3a,b). This influence remains limited, however, and qualitative identification criteria remain valid for cv polymorphs of 2M and 3T polytypes. Similarly, local distortions induced by cationic substitutions can be ignored as a first approximation. However, as the occupancy of M2 and M2' sites within cv-1M crystals influences the intensity distribution, the coexistence of vacant M2 and M2' sites within cv-1M crystals possibly accounts for the experimental absence of expected dynamical effects, such as center of symmetry loss for cv layers [START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF]).

Influence of crystal bending. Owing to the size of the selected-area aperture, ED intensities are averaged over a large surface area of the crystal (typically ~200 nm in diameter, although a smaller aperture can be used, if available). Variation in surface orientation with respect to the electron beam, often revealed by bending contours in the image, results in diffracted intensities being averaged, similar to mosaicity effects. Moving a small aperture over the crystal allows assessing the effect of bending. For investigated crystals, diffraction mapping at the nanoscale showed that the orientation fluctuation remains <1-2° (not shown). As long as the crystal surface bending is small (<1-2°), a distinctive pattern of a given polytype should be recognizable.

Influence of chemical composition on dioctahedral mica ED patterns

Mg 2+ -for-Al 3+ or Al 3+ -for-Si 4+ substitutions in octahedral and tetrahedral sheets have marginal influence on diffracted intensity distributions owing to the similar scattering factors of involved cations. Other cation substitutions are frequent in natural dioctahedral micas, the most common being Na + -for-K + in paragonite and the partial substitution of Fe 2+ /Fe 3+ for Al 3+ in illite and phengite, and in celadonite and glauconite [START_REF] Bailey | Classification and structure of the micas[END_REF][START_REF] Mottana | Micas: Crystal chemistry and metamorphic petrology[END_REF]). The influence of Na + -for-K + and Fe 2+ /Fe 3+ -for-Al 3+ substitutions on ED intensity is described below for the mica polytypes. Expected lattice distortions induced by cationic radii variations or vacant octahedral sites are not considered owing to their limited effect on diffraction intensity.

Na

ED patterns along other crystal orientations

Except for the 3T polytype, whose c-axis is perpendicular to the layer plane, all mica crystals must be rotated to be oriented along the [001] axis, the direction of which is unknown a priori. To reach the [001] axis, which forms an oblique angle with [100] (Fig. 5a), crystals that lay on the ab-plane must be rotated by a θth angle around the [010] axis. Two directions of rotation Experimental zone axis angle. Ideally, the b*-axis is aligned with the sample holder primary rotation axis (the "Tilt" axis) and the tilt angle to reach the desired [u0w] orientation can be compared directly to the theoretical th value (indicated in the figures). In practice, this is achieved with a "Tilt-Rotation" holder that allows rotating the sample within the observation plane.

With commonly available "Double Tilt" holders, obtaining the correct orientation requires combining two rotation axes (Tilt and Azimuth angles, Fig. 5b), both normal to the electron beam.

In cylindrical coordinates, the inclination of the ED pattern zone axis with respect to the crystal c*axis is defined by two angles:  and , the angles with the z-direction and between the projection of the zone axis on the ab-plane and the sample holder primary rotation (tilt) x-axis, respectively.

Experimental  and  angles can be computed from sample holder tilt (Tt) and azimuth (Az) rotation angles using the following equations: tan tan tan

y x d Tt d Az    ,  = 90° if Az = 0° (3) 22 2 2 2 2 tan tan tan xy dd Az Tt h      (4)
where dx, dy, h are the coordinates (in Å) of a virtual point (M) on the zone axis, along the two rotation axes (x, y) and the electron beam (z) directions, respectively (Fig. 5c). Conversely, it is possible to calculate Tt and Az values to be used experimentally from the  and  angles of the sought zone axis: bearing aluminous mica crystals (illite, muscovite, or phengite), including morphology and possibly chemistry [START_REF] Lanson | Illitization of diagenetic kaolinite-to-dickite conversion series: Late-stage diagenesis of the lower Permian Rotliegend sandstone reservoir, offshore of The Netherlands[END_REF][START_REF] Lanson | Authigenic kaolin and illitic minerals during burial diagenesis of sandstones: A review[END_REF][START_REF] Patrier | High-grade diagenetic dickite and 2M1 illite from the middle Proterozoic Kombolgie formation (Northern Territory, Australia)[END_REF][START_REF] Laverret | Mineralogy and geochemistry of the host-rock alterations associated with the Shea creek unconformity-type uranium deposits (Athabasca basin, Saskatchewan, Canada). Part 1. Spatial variation of illite properties[END_REF]).

  Al)O (OH) ], on ED intensity distribution was assessed.

  Geometry of [001] ED patterns is similar and all models have the same pseudo-hexagonal arrangement of hk0 reflections. Departure from hexagonal symmetry is marginal for 1M, 2M1, and 2M2 polytypes. For 1M and 2M1 polytypes, indexing was performed with an orthogonal a*b* unit cell with a C-centered lattice. For the 2M2 polytype (a* < b*), h'k' indices are obtained from from h'k' indices of the 2M1 by h' = k and k' = -h. The orthogonal cell was used for the 3T polytype despite its hexagonal symmetry. All ED patterns present mirror planes perpendicular to a*-and b*-axes, and hk0, ℎ ̅ k0, h𝑘 ̅ 0, and ℎ ̅ 𝑘 ̅ 0 reflections are equivalent and are hereafter referred to as hk reflections. Intensity distributions for hk reflections differ significantly for each polytype[START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF].ED patterns calculated along [001] for 1M polymorphsPure tv-and cv-1M polymorphs. ED patterns calculated along [001] for pure tv-and cv-1M micas were described by[START_REF] Gaillot | Polytype and polymorph identification of finely divided aluminous dioctahedral mica individual crystals with SAED. Kinematical and dynamical electron diffraction[END_REF]. The intensity distribution depends on the layer structure and the occupancy of M1 and M2 sites. For both polymorphs, 06 and 20 spots are strong and 13 and 26 reflections have similar intermediate intensities. The main difference is the intensity of 02 spots, which are much stronger than the 11 reflections for tv-1M (Figs.2a,f), whereas the opposite is observed for cv-1M layers (M2 or M2' site vacant, Figs.2b,g). Similar intensity distributions are obtained when considering M-layers and site occupancies matching those of pure tv-and cv-layers [(0:1:1) and (1:0:1) occupancy probabilities, respectively, Figs.3a,b].Partial occupation of octahedral sites. Diffraction patterns for 1M crystals in which octahedral vacancies are distributed with equal probability over the two symmetrically related cis sites [M-cell with (1:0.5:0.5), Fig.3c] are similar to that obtained where all vacancies occur in a

  Fig.2i). Differentiation of cv and tv with SAED is difficult although 31 reflections dominate the

  -bearing aluminous dioctahedral mica. Intensity distributions among hk reflections calculated for tv-1M, cv-1M, tv-2M1, tv-2M2, and tv-3T structure models and an idealized Figs.2a-e). However, 02 and/or 11 reflections are stronger for Na vs. K micas, whereas 06 reflections are weaker. Furthermore, because 02 reflections dominate the tv-1M ED pattern, and 11 reflections dominate the cv-1M polymorph, the difference between tv-vs. cv-1M is enhanced for sodic micas.K-bearing ferruginous dioctahedral mica. Differences in intensity distributions calculated for tv-1M, cv-1M, tv-2M1, tv-2M2 and tv-3T models and either an idealized celadonite composition ( 4f-j) or a muscovite composition (Figs.2a-e) are marginal despite the scattering contrast between Fe and Al/Mg. For example, [001] ED patterns are similar for ferruginous and aluminous varieties of tv-3T and tv-2M2 micas. For tv-2M1 micas, all reflections but 06 and 40 are weaker for the ferruginous variety compared to the aluminous mica, whereas for tv-1M and cv-1M polymorphs, 13 reflections are weaker and 26 spots stronger for the ferruginous mica compared to the aluminous mica. In all cases, the overall intensity distribution, and the sets of bright hk spots are similar for both ferruginous and aluminous varieties.

(

  ±θ) are possible to reach the [001] zone axis. In practice, the closest zone axis is easily reached, but may differ from the [001] axis. Indeed[001] and[101] axis are tilted at about the same angle (~10°) in opposite directions for 1M polytypes. To more rapidly allow for a correct identification of SAED patterns oriented along [001], additional calculations were performed along a selection of[hk] zone axes near the c-axis (Fig.5a). Theoretical θth angles between the c*-axis and[hk] directions are provided in Figures6-7, although experimentally determined values may deviate slightly if crystals are not normal to the electron beam.ED patterns calculated along the c*-axis. The c*-axis is close to [103] for 1M, 2M1, and 2M2 polytypes (θth  1.5°) and perpendicular to the 2:1 layers, thus producing hexagonal ED patterns, similar to the 3T polytype along[001]. For all polytypes, hk reflections from the third hexagon (06 and 33 reflections) are of equal intensity and dominate the ED pattern (Figs.6a-e).The ED pattern calculated for the 3T polytype is the only ED showing a six-fold axis, whereas those calculated for monoclinic polytypes (1M, 2M1, and 2M2) exhibit two mirror planes normal to a//* (projection of a* on the ZOLZ plane) and b* vectors, hkl reflections with h = 3n being the only ones observed. For well-ordered crystals, extinction rules can be used as an identification criterion.Extinction rules may be violated, owing to reciprocal lattice node elongation for defective or very thin crystals. Tilt around [010]. The geometry and intensity distributions of calculated ED patterns depend on the [u0w] zone axis considered. As a rule, relative intensities of 0k0 spots along the common b*-axis are preserved. For crystals oriented along the [102] zone axis, the ZOLZ also contains -2l k l reflections (with the h + k = -w × l + k = 2n extinction rule from C-centering). hk reflections with h ≠ 2l do not theoretically appear (unless the reciprocal rods are sufficiently elongated) leading to rectangular patterns of the brightest spots (Figs. 6f-j). For increasing tilt values (Fig. 5a), the geometry of [101] and [1 ̅ 01] ED patterns (Figs. 6k-t) remains similar to that of patterns calculated along [001] (Figs. 2a-e) with only a faint stretching along the a//* direction. This elongation becomes significant for ED patterns calculated along [h01] with |h|  2, although the intensity distribution may be still useful for polytype identification (Figs. 7a-o). ED patterns calculated along other [h0] axes are less informative. ED patterns for tv-2M1 along [101] and for cv-1M along [001] are alike (Figs. 6m and 4b, respectively), as are those for tv-3T along [101] and for 1M with the vacancies equally distributed over the three octahedral sites (Figs. 6o and 4l). Similarly, ED patterns for tv-2M1 along [1 ̅ 01] and for cv-1M along [101] are alike (Figs. 6r and 6l), as are those for tv-3T along the c-or c*-axis and for tv-1M, cv-1M, and tv-2M1 along [103], at least for the intensity distribution among most intense reflections (Figs. 6e and 6a-c, respectively). These similarities may require recording an additional ED pattern along another [10w] ZA to determine the [001] axis. Tilt around [100]. ED patterns calculated along [011] for tv-1M, cv-1M, tv-2M1, tv-2M2, and tv-3T polytypes are shown in Figure 7p-t. For monoclinic polytypes, [001] and [100] form an oblique angle. The angle between a//* and b*-axis thus differs from 90° and ED patterns calculated along [011] appear significantly distorted. In addition, ED patterns calculated along [0k1], with k  0, possess lower symmetry than those calculated along [001] and the former no longer present mirror planes (even for the 3T polytype) because intensities of hk, h; _ k, and equivalent reflections are no longer equal. The center of symmetry is preserved however with I(hk) = I(ℎ ̅ 𝑘 ̅  ̅ ) for kinematical diffraction (Friedel's law). Symmetry lowering makes ED patterns recorded along [0k1] recognizable. Advanced identification method Different polytypes can have similar ED patterns along different zone axes, thus impairing polytype identification if the [001] zone axis is not correctly determined. Although complementary powder X-ray diffraction data indicate the major polytypes present in a sample, a suitable orientation along [001] is key to avoid polytype misidentification using the proposed criteria (Figs. 2f-j). Orientation along [001] relies on the correct identification of the [010] rotation axis and on the inclination θ of the zone axis relative to the c*-axis. ED patterns along several crystallographic directions can be useful to further constrain polytype/polymorph identification if the crystal orientation is uncertain.Experimental considerations. In practice, determining the appropriate rotation direction to reach the [001] axis can be challenging. A large number of stacking faults present may negate any effort, however, as the [001] direction is not unique, and thus thin crystals are usually preferred.Crystal morphology is also used to determine ED pattern orientation in the a*-b*-plane because unit-cell a or b vectors are often parallel to a crystal edge. In addition, illite lath-shaped crystals grow parallel to the [100][START_REF] Rex | Authigenic kaolinite and mica as evidence for phase equilibria at low temperature[END_REF], and the [001] axis can be reached by rotating the crystal around the direction parallel to the short dimension (Figs.8a-b). There is no absolute method to determine a priori how the rotation should be performed and a trial-and-error approach is necessary. However, the [001] ED patterns must possess two perpendicular mirror planes common to all patterns oriented along [u0w] axes. If this symmetry is fulfilled, experimental SAED pattern can be compared with calculated patterns (Figs.2 and 6) for identification. Additional tilt in the same or opposite directions allows reaching other [u0w] orientations to confirm polytype identification.

  aid in finding the orientation of the first [u0w] zone axis from the initial crystal orientation with the c*-axis close to the beam direction. The equations can also be used for multiple zone-axis data collection. In the latter case, the crystal shift upon tilting must be compensated if the crystal is not flat. To illustrate, SAED patterns were obtained for two orientations from isomorphic plates of illite from Kombolgie (Figs.8c-d). For both patterns, the |exp| angle calculated from the TEM holder angles (~12° and 3°, respectively) is consistent with the theoretical |th| value calculated for the [101] and [001] ZA of the 2M1 polytype (~9° and 6°, respectively), thus obtaining a determination of the 2M1 polytype, despite the minor discrepancy between experimental and calculated tilt angles.SUMMARY AND PERSPECTIVESQualitative criteria are provided to determine the polymorph/polytype of individual crystals of K-bearing aluminous dioctahedral micas from the intensity distribution of hk reflections from the zero-order Laue zone. Differences are enhanced for ED patterns calculated along the [001] zone-axis. ED patterns calculated along adjacent zone-axes are provided to avoid potential difficulties in obtaining unambiguously the optimum [001] orientation. The method allows retrieving morphological and structural information from individual mica crystals. Matching the observed data to calculated patterns allows the determination of structural information from K-

Figure 1 .

 1 Figure 1. (a) Transmission electron micrograph of an isomorphic illite particle from Athabasca

Figure 2 .

 2 Figure 2. ED patterns calculated using the kinematical approximation along the [001] zone axis for

Figure 3 .

 3 Figure 3. ED patterns calculated using the kinematical approximation along the [001] zone axis for

Figure 4 .

 4 Figure 4. ED patterns calculated along the [001] zone axis for different polytypes and polymorphs

Figure 5 .

 5 Figure 5. (a) Orientation of different zone axes with respect to the a (or a') = [100] and c = [001]

Figure 6 .

 6 Figure 6. ED patterns calculated for various mica polymorphs and polytypes along different zone

Figure 7 .

 7 Figure 7. ED patterns calculated for various mica polymorphs and polytypes along different zone

Figure 8 .

 8 Figure 8. (a) Transmission electron micrographs of Athabasca Basin illite isomorphic and lath-

  Gaillot et al. -Figure1

Figure

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Table 1 .

 1 Unit-cell parameters for tv-1M, cv-1M, tv-2M1and tv-3T polytypes (α = β = 90°) 613

	Polytype a (Å) b (Å)	c (Å)	γ (°)
	tv-1M	5.199 9.005 10.164 101.3
	cv-1M	5.199 9.005 10.090 99.13
	M-layer	5.199 9.005	10.12	99.86
	tv-2M1	5.177 8.987 20.072 95.76
	tv-2M2 *	9.023 5.197 20.171 99.48
	tv-3T	5.212 9.027 29.804	90

* Unit-cell parameters listed in the Russian version of the article

[START_REF] Zhoukhlistov | The crystal structure of the dioctahedral mica 2M2 determined by high voltage electron diffraction[END_REF]

slightly differ from those reported in an earlier English version and found in the ICSD database

(#89823)

ACKNOWLEDGMENTS

Daniel Beaufort (IC2MP, Poitiers -France) is thanked for providing the 1M illite and 2M1 muscovite samples. Funded by the French Contrat Plan État-Région and the European Regional Development Fund of Pays de la Loire, the CIMEN Electron Microscopy Center in Nantes is greatly acknowledged. ISTerre is part of Labex OSUG@2020 (ANR10 LABX56). Comments by two anonymous reviewers improved and clarified the initial manuscript. On behalf of all authors, the corresponding author states that there is no conflict of interest.