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The influence of stereochemically active lone-pair electrons on crystal symmetry and twist angles in lead apatite-2H type structures
- T. Baikie, M. Schreyer, F. Wei, J. S. Herrin, C. Ferraris, F. Brink, J. Topolska, R. O. Piltz, J. Price, T. J. White
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- Journal:
- Mineralogical Magazine / Volume 78 / Issue 2 / April 2014
- Published online by Cambridge University Press:
- 05 July 2018, pp. 325-345
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- Article
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Lead-containing (Pb-B-X)-2H apatites encompass a number of [AF]4[AT]6[(BO4)6]X2 compounds used for waste stabilization, environmental catalysis and ion conduction, but the influence of the stereochemically active lone-pair electrons of Pb2+ on crystal chemistry and functionality is poorly understood. This article presents a compilation of existing structural data for Pb apatites that demonstrate paired electrons of Pb2+ at both the AF and AT results in substantial adjustments to the PbFO6 metaprism twist angle, φ. New structure refinements are presented for several natural varieties as a function of temperature by single-crystal X-ray diffraction (XRD) of vanadinite-2H (ideally Pb10(VO4)6Cl2), pyromorphite-2H (Pb10(PO4)6Cl2), mimetite-2H/M (Pb10(As5+O4)6Cl2) and finnemanite-2H (Pb10(As3+O3)6Cl2). A supercell for mimetite is confirmed using synchrotron single-crystal XRD. It is suggested the superstructure is necessary to accommodate displacement of the stereochemically active 6s2 lone-pair electrons on the Pb2+ that occupy a volume similar to an O2− anion. We propose that depending on the temperature and concentration of minor substitutional ions, the mimetite superstructure is a structural adaptation common to all Pb-containing apatites and by extension apatite electrolytes, where oxide ion interstitials are found at similar positions to the lonepair electrons. It is also shown that plumbous apatite framework flexes substantially through adjustments of the PbFO6 metaprism twist-angles (φ) as the temperature changes. Finally, crystalchemical [100] zoning observed at submicron scales will probably impact on the treatment of diffraction data and may account for certain inconsistencies in reported structures.
10 - Twenty points in ℙ3
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- By D. Eisenbud, University of California, Berkeley, R. Hartshorne, University of California, Berkeley, F.-O. Schreyer, Universität des Saarlandes
- Edited by Christopher D. Hacon, University of Utah, Mircea Mustaţă, University of Michigan, Ann Arbor, Mihnea Popa, University of Illinois, Chicago
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- Book:
- Recent Advances in Algebraic Geometry
- Published online:
- 05 January 2015
- Print publication:
- 15 January 2015, pp 180-199
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Summary
Abstract
Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in ℙ3 over an algebraically closed field of characteristic 0. As a consequence, we show that a general set of d points is glicci (that is, in the Gorenstein linkage class of a complete intersection) if d ≤ 33 or d = 37, 38. Computer algebra plays an essential role in the proof. The case of 20 points had been an outstanding problem in the area for a dozen years [8].
For Rob Lazarsfeld on the occasion of his 60th birthday
1 Introduction
The theory of liaison (linkage) is a powerful tool in the theory of curves in ℙ3 with applications, for example, to the question of the unirationality of the moduli spaces of curves (e.g., [3, 26, 29]). One says that two curves C, D ⊂ ℙ3 (say, reduced and without common components) are directly linked if their union is a complete intersection, and evenly linked if there is a chain of curves C = C0, C1, …, C2m = D such that Ci is directly linked to Ci+1 for all i. The first step in the theory is the result of Gaeta that any two arithmetically Cohen-Macaulay curves are evenly linked, and in particular are in the linkage class of a complete intersection, usually written licci. Much later Rao [23] showed that even linkage classes are in bijection with graded modules of finite length up to shift, leading to an avalanche of results (reported, e.g., in [19, 20]). However, in codimension > 2 linkage yields an equivalence relation that seems to be very fine, and thus not so useful; for example, the scheme consisting of the four coordinate points in ℙ3 is not licci.
A fundamental paper of Peskine and Szpiro [22] laid the modern foundation for the theory of linkage.