Abstract
The present study deals with how stochastic stackings of tetrahedral/octahedral phengitic sheets bearing diverse cation distributions affect diffraction signals and the structural inferences therefrom derived. The interest for such minerals is dictated by that the stability of phengite polytypes, their cation distributions and P/T conditions of crystallization are related to each other. We focus our attention on layers’ probabilistic sequences that preserve the topology of the polytypes 2M 1(SG: C2/c) and 3T(SG: P3112). Neutron diffraction intensities are modelled by a Monte Carlo approach and then used as artificial experimental data for conventional structure refinements that yield the occupancy factors in the fourfold (Si, Al) and sixfold (Al, Mg) coordination sites of 2M 1 and 3T. The cation ordering from structure refinement tallies with the one of the “average structure” of a stochastic stacking, but it can significantly differ from those of the individual tetrahedral/octahedral sheets. For instance, sheets having ordered cation arrangements can lead to a stochastic structure which is supposed to bear a fully disordered cation partitioning according to structure refinement. This affects the configuration entropy contributions: the values obtained by conventional refinements can deviate from the correct ones up to 30 %. The analysis of the equivalent reflection intensities brings to light the anomalies hinting at the occurrence of such stacking disorder (using modelled reflections, the mean ratio between standard deviation and average intensity of symmetry equivalent reflections is ideally 0 for perfect crystal structures, but it can amount up to 6 in stochastically disordered phengites). However, taking into account the instrumental uncertainties and the deviations from ideality of actual crystals, such phenomena are very difficult to be detected experimentally.
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Acknowledgments
Italian Ministry for University and Research (M.U.R.S.T) and Italian National Research Council (C·N.R) are kindly acknowledged for contributing to fund the investigations in question. The authors are grateful to the two anonymous referees who really contribute to improve the quality of the manuscript.
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Appendices
Appendix 1: Structure factor
The intensity of a scattering process with momentum transfer H, from a stack of (N + 1) layers, each one having a structure factor F j (H), is described by
where
T j−1,j is the vector connecting the origin of the j−1th layer to that of the jth layer (Treacy et al. 1991). Once the transition probability array, p(i → j), is known, one can stack layer on layer, choosing them by random numbers extracted from a uniform probability distribution. The Monte Carlo stacking-generation cycle converges to I(H) when
where Δ and 〈〉 are the e.s.d and average calculated over the set of the last K terms of the I stack,N (H)-series; K and ε are fixed on the basis of preliminary numerical trials so as to guarantee convergence to the I stack,N (H)-series. Setting ε ~ 10−5 and K ~ 150–200, one attains a precision on I(H) much better than 1 % for more than 99 % reflections.
In the present study, we have modelled neutron diffraction intensities from a single crystal, using isotropic thermal parameters (Biso = 1.0 Å2), T–O bond length of 1.64, octahedral-layer thickness of 2.10 and interlayer thickness of 3.41 Å; ditrigonal distortion angle of 9.5°. Note that such figures are irrelevant to the aim of the work but Biso that has been set so as to simulate low-temperature diffraction processes. The choice of neutron radiation is dictated by the need of distinguishing between quasi-isoelectronic species, that is, Mg–Al–Si.
Diffuse scattering, I(H)diff, due to a disordered distribution of two chemical species over L-sites, has been modelled (Warren 1969) as
where Δb is the coherent neutron scattering length difference, and x j,1 is the occupancy factor of the species “1” at the jth site. The equation above has been generalized in order to account for tetrahedral (Si/Al) and octahedral (Al/Mg) sites, in phengites. Taking into account that the topology is conserved, such correction significantly affects low-intensity reflections only, for the minerals under study.
Appendix 2: Symmetry breach quantification
Let us define LSB and PSB in the following way:
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1.
$$\begin{aligned} \quad 2M_{ 1} \left( {\left\langle {^{(0,a,b)} {\mathbf{v}}_{\alpha ,j - 1} } \right\rangle \left\langle {^{(c,d,e,f)} Tet_{j} }\right\rangle \left\langle {^{(0,g,h)} {\mathbf{v}}_{\alpha ,j + 1} } \right\rangle \left\langle {^{{({\it{i}},{\it{l}},{\it{m}},{\it{n}})}}Tet_{j + 2} } \right\rangle } \right) \\ \quad T - LSB = \, \left( {\left| {\left( {c - d} \right) - \left( {e - f} \right)} \right| + |\left( {i -{\it{l}}} \right) - \left( {m - n} \right)|} \right)/ 2\\ \quad O - LSB = \left( {\left| {a - b} \right| + |g - h|} \right)/ 2\\ \quad sLSB = \left({|\left( {a - b} \right) - \left( {g - h} \right)|} + \right|\left( {c - d} \right) + \left( {i - {\it{l}}} \right)\left| + \right|\left( {e - f} \right) + \left({m - n} \right)|)/ 3\\ \end{aligned}$$
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2.
$$\begin{aligned} 3T\left( {\left\langle {^{(a,b,c,d)} Tet_{j - 2} } \right\rangle \left\langle {^{(0,e,f)} {\mathbf{v}}_{\alpha ,j - 1} } \right\rangle \left\langle {^{(g,h,i,l)} Tet_{j} } \right\rangle \left\langle {^{(0,m,n)} {\mathbf{v}}_{\alpha ,j + 1} } \right\rangle \left\langle {^{(o,p,q,r)} Tet_{j + 2} } \right\rangle \left\langle {^{(0,s,t)} {\mathbf{v}}_{\alpha ,j + 3} } \right\rangle } \right) \\ \quad a_{T} = \, \left( {\left( {a - b} \right) + \left( {c - d} \right) + \left( {g - h} \right) + \left( {i - l} \right) + \left( {o - p} \right) + \left( {q - r} \right)} \right)/ 6\\ \quad a_{O} = \left( {\left( {e - f} \right) + \left( {m - n} \right) + \left( {s - t} \right)} \right)/ 3\\ \quad T - LSB = \, \left( {\left| {\left( {a - b} \right) - a_{T} } \right| + \left| {\left( {c - d} \right) - a_{T} } \right| + \left| {\left( {g - h} \right) - a_{T} } \right| + \left| {\left( {i - l} \right) - a_{T} } \right| + \left| {\left( {o - p} \right) - a_{T} } \right| + |\left( {q - r} \right) - a_{T} |} \right)/ 6\\ \quad sLSB = \left( {\left| {\left( {e - f} \right) - a_{O} } \right| + \left| {\left( {m - n} \right) - a_{O} } \right| + |\left( {s - t} \right) - a_{O} |} \right)/ 3 \\ \end{aligned}$$
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Pavese, A., Diella, V. How stacking disorder can conceal the actual structure of micas: the case of phengites. Phys Chem Minerals 40, 375–386 (2013). https://doi.org/10.1007/s00269-013-0568-6
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DOI: https://doi.org/10.1007/s00269-013-0568-6