212 resultados para Legendre Formula


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Vibrational spectroscopy has been used to study the rare earth mineral churchite of formula (REE)(PO4)-⋅2H2O. The mineral contains a range of rare earth metals including yttrium depending on the locality. The Raman spectra of churchite-(REE) are characterized by an intense sharp band at 984 cm-1 assigned to the v1 (PO¾-) symmetric stretching mode. A lower intensity band observed at around 1067 cm-1 is attributed to the v3 (PO¾-) antisymmetric stretching mode. The (PO¾-) bending modes are observed at 497 cm-1 (v2) and 565 cm-1(v4). Raman bands at 649 and 681 cm-1 are assigned to water librational modes. Vibrational spectroscopy enables aspects of the structure of churchite to be ascertained.

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We have studied the hydrated hydroxyl silicate mineral inesite of formula Ca2(Mn,Fe)7Si10O28(OH)⋅5H2O using a combination of scanning electron microscopy with EDX and Raman and infrared spectroscopy. SEM analysis shows the mineral to be a pure monomineral with no impurities. Semiquantitative analysis shows a homogeneous phase, composed by Ca, Mn2+, Si and P, with minor amounts of Mg and Fe. Raman spectrum shows well resolved component bands at 997, 1031, 1051, and 1067 cm-1 attributed to a range of SiO symmetric stretching vibrations of [Si10O28] units. Infrared bands found at 896, 928, 959 and 985 cm-1 are attributed to the OSiO antisymmetric stretching vibrations. An intense broad band at 653 cm-1 with shoulder bands at 608, 631 and 684 cm-1 are associated with the bending modes of the OSiO units of the 6- and 8-membered rings of the [Si10O28] units. The sharp band at 3642 cm-1 with shoulder bands at 3612 and 3662 cm-1 are assigned to the OH stretching vibrations of the hydroxyl units. The broad Raman band at 3420 cm-1 with shoulder bands at 3362 and 3496 cm-1 are assigned to the water stretching vibrations. The application of vibrational spectroscopy has enabled an assessment of the molecular structure of inesite to be undertaken.

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The mineral beraunite from Boca Rica pegmatite in Minas Gerais with theoretical formula Fe2+Fe5 3+(PO4)4(OH)5⋅4H2O has been studied using a combination of electron microscopy with EDX and vibrational spectroscopic techniques. Raman spectroscopy identifies an intense band at 990 cm-1 and 1011 cm-1. These bands are attributed to the PO4 3- v, symmetric stretching mode. The m3 antisymmetric stretching modes are observed by a large number of Raman bands. The Raman bands at 1034, 1051, 1058, 1069 and 1084 together with the Raman bands at 1098, 1116, 1133, 1155 and 1174 cm-1 are assigned to the m3 antisymmetric stretching vibrations of PO4 3- and the HOPO3 2- units. The observation of these multiple Raman bands in the symmetric and antisymmetric stretching region gives credence to the concept that both phosphate and hydrogen phosphate units exist in the structure of beraunite. The series of Raman bands at 567, 582,601, 644, 661, 673, and 687 cm-1 are assigned to the PO4 3- v2 bending modes. The series of Raman bands at 437, 468, 478, 491, 503 cm-1 are attributed to the PO4 3- and OPO3 2- v4 bending modes. No Raman bands of beraunite which could be attributed to the hydroxyl stretching unit were observed. Infrared bands at 3511 and 3359 cm-1 are ascribed to the OH stretching vibration of the OH units. Very broad bands at 3022 and 3299 cm-1 are attributed to the OH stretching vibrations of water. Vibrational spectroscopy offers insights into the molecular structure of the phosphate mineral beraunite.

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The mineral leightonite, a rare sulphate mineral of formula K2Ca2Cu(SO4)4.2H2O, has been studied using a combination of electron probe and vibrational spectroscopy. The mineral is characterized by an intense Raman band at 991 cm-1 attributed to the SO2- 4 m1 symmetric stretching mode. A series of Raman bands at 1047, 1120, 1137, 1163 and 1177 cm-1 assigned to the SO2- 4 m3 antisymmetric stretching modes. The observation of multiple bands shows that the symmetry of the sulphate anion is reduced. Multiple Raman and infrared bands in the OH stretching region shows that water in the structure of leightonite is in a range of molecular environments.

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A natural single-crystal specimen of the kröhnkite from Chuquicamata, Chile, with the general formula Na2Cu(SO4)2 · 2H2O, was investigated by Raman and infrared spectroscopy. The mineral kröhnkite is found in many parts of the world's arid areas. Kröhnkite crystallizes in the monoclinic crystal system with point group 2/m and space group P21/c. It is an uncommon secondary mineral formed in the oxidized zone of copper deposits, typically in very arid climates. The Raman spectrum of kröhnkite dominated by a very sharp intense band at 992 cm−1 is assigned to the ν1 symmetric stretching mode and Raman bands at 1046, 1049, 1138, 1164, and 1177 cm−1 are assigned to the ν3 antisymmetric stretching vibrations. The infrared spectrum shows an intense band at 992 cm−1. The Raman bands at 569, 582, 612, 634, 642, 655, and 660 cm−1 are assigned to the ν4 bending modes. Three Raman bands observed at 429, 445, and 463 cm−1 are attributed to the ν2 bending modes. The observation that three or four bands are seen in the ν4 region of kröhnkite is attributed to the reduction of symmetry to C2v or less.

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The metal lithium is very important in industry, including lithium batteries. An important source of lithium besides continental brines is granitic pegmatites as in Australia. Lithiophilite is a lithium and manganese phosphate with chemical formula LiMnPO4 and forms a solid solution with triphylite, its Fe analog, and belongs to the triphylite group that includes karenwebberite, natrophilite, and sicklerite. The mineral lithiophilite was characterized by chemical analysis and spectroscopic techniques. The chemical is: Li1.01(Mn0.60, Fe0.41, Mg0.01, Ca0.01)(PO4)0.99 and corresponds to an intermediate member of the triphylite-lithiophilite series, with predominance of the lithiophilite member. The mineral lithiophilite is readily characterized by Raman and infrared spectroscopy.

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For transmedia to be acknowledged as worthy of investment by the business world, and even by those considering a career in development of transmedia creative products, there first needs to be established a business case for the concept. This chapter seeks to inform transmedia advocates about the concept of value and the ROI of transmedia more generally. While it is by no means a template or formula for measurement of value, it is a reminder to transmedia professionals and theorists, that intangible benefits are neither valueless nor unquantifiable. The chapter is divided into four sections: 1. Definitions of transmedia – concept and scope of transmedia, expressed in a manner that is intelligible for a business audience. 2. Value and cost – discussion of the terms from an economic perspective. 3. Audience interaction and collaborative content development – discussion of how feedback and engagement systems of transmedia have facilitated rich experiences which offer more than mere content and audience reach outputs. 4. ROI metrics for transmedia – measurable criteria for articulation of value to business investors.

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Three different radical anions of the empirical formula C5H2 have been generated by negative ion chemical ionization mass spectrometry in the gas phase. The isomers C4CH2 •-, and HC5H•- have been synthesized by unequivocal routes and their connectivities confirmed by deuterium labeling, charge reversal, and neutralization reionization experiments. The results also provided evidence for the existence of neutrals C4CH2, C2CHC2H, and HC5H as stable species; this is the first reported observation of C2CHC2H. Ab initio calculations confirm these structures to be minima on the anion and neutral potential energy surfaces.

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Three anion isomers of formula C7H have been synthesised in the mass spectrometer by unequivocal routes. The structures of the isomers are \[HCCC(C-2)(2)](-), C6CH- and C2CHC4-. One of these, \[HCCC(C-2)(2)](-), is formed in sufficient yield to allow it to be charge stripped to the corresponding neutral radical.

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The mineral sturmanite is a hydrated calcium iron aluminium manganese sulphate tetrahydroxoborate hydroxide of formula Ca6(Fe, Al, Mn)2(SO4)2(B(OH)4)(OH)12•26H2O. We have studied the mineral sturmanite using a number of techniques, including SEM with EPMA and vibrational spectroscopy. Chemical analysis shows a homogeneous phase, composed by Ca, Fe, Mn, S, Al and Si. B is not determined in this EPMA technique. An intense Raman band at 990 cm−1 is assigned to the SO42− symmetric stretching mode. Raman spectroscopy identifies multiple sulphate symmetric stretching modes in line with the three sulphate crystallographically different sites. Raman spectroscopy also identifies a band at 1069 cm−1 which may be attributed to a carbonate symmetric stretching mode, indicating the presence of thaumasite. Infrared spectra display two bands at 1080 and 1107 cm−1 assigned to the SO42− antisymmetric stretching modes. The observation of multiple bands in this ν4 spectral region offers evidence for the reduction in symmetry of the sulphate anion from Td to C2v or even lower symmetry. The Raman band at 3622 cm−1 is assigned to the OH unit stretching vibration and the broad feature at around 3479 cm−1 to water stretching bands. Infrared spectroscopy shows a set of broad overlapping bands in the OH stretching region. Vibrational spectroscopy enables an assessment of the molecular structure of sturmanite to be made.

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In this article we study the azimuthal shear deformations in a compressible Isotropic elastic material. This class of deformations involves an azimuthal displacement as a function of the radial and axial coordinates. The equilibrium equations are formulated in terms of the Cauchy-Green strain tensors, which form an overdetermined system of partial differential equations for which solutions do not exist in general. By means of a Legendre transformation, necessary and sufficient conditions for the material to support this deformation are obtained explicitly, in the sense that every solution to the azimuthal equilibrium equation will satisfy the remaining two equations. Additionally, we show how these conditions are sufficient to support all currently known deformations that locally reduce to simple shear. These conditions are then expressed both in terms of the invariants of the Cauchy-Green strain and stretch tensors. Several classes of strain energy functions for which this deformation can be supported are studied. For certain boundary conditions, exact solutions to the equilibrium equations are obtained. © 2005 Society for Industrial and Applied Mathematics.

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The mineral lomonosovite has been studied using a combination of scanning electron microscopy with energy dispersive X-ray analysis and vibrational spectroscopy. Qualitative chemical analysis gave Si, P, Na and Ti as the as major elements with small amounts of Mn, Ca, Fe and Al. The mineral lomonosovite has a formula Na5Ti2(Si2O7)(PO4)O2. Raman bands observed at 909, 925 and 939 cm−1 are associated with phosphate units. Raman bands found at 975, 999, 1070, 1080 and 1084 cm−1 are attributed to siloxane stretching vibrations. The observation of multiple bands in both the phosphate stretching and bending regions supports the concept that the symmetry of the phosphate anion in the structure of lomonosovite is significantly reduced. Infrared spectroscopy identifies bands in the water stretching and bending regions, thus suggesting that water is involved with the structure of lomonosovite either through adsorption on the surface or by bonding to the phosphate units.

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The mineral kidwellite, a hydrated hydroxy phosphate of ferric iron and sodium of approximate formula NaFe93+(PO4)6(OH)11⋅3H2O, has been studied using a combination of electron microscopy with EDX and vibrational spectroscopic techniques. Raman spectroscopy identifies an intense band at 978 cm−1 and 1014 cm−1. These bands are attributed to the PO43− ν1 symmetric stretching mode. The ν3 antisymmetric stretching modes are observed by a large number of Raman bands. The series of Raman bands at 1034, 1050, 1063, 1082, 1129, 1144 and 1188 cm−1 are attributed to the ν3 antisymmetric stretching bands of the PO43− and HOPO32− units. The observation of these multiple Raman bands in the symmetric and antisymmetric stretching region gives credence to the concept that both phosphate and hydrogen phosphate units exist in the structure of kidwellite. The series of Raman bands at 557, 570, 588, 602, 631, 644 and 653 cm−1are assigned to the PO43− ν2 bending modes. The series of Raman bands at 405, 444, 453, 467, 490 and 500 cm−1 are attributed to the PO43− and HOPO32− ν4 bending modes. The spectrum is quite broad but Raman bands may be resolved at 3122, 3231, 3356, 3466 and 3580 cm−1. These bands are assigned to water stretching vibrational modes. The number and position of these bands suggests that water is in different molecular environments with differing hydrogen bond distances. Infrared bands at 3511 and 3359 cm−1 are ascribed to the OH stretching vibration of the OH units. Very broad bands at 3022 and 3299 cm−1 are attributed to the OH stretching vibrations of water. Vibrational spectroscopy offers insights into the molecular structure of the phosphate mineral kidwellite.

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The mineral sulphohalite – Na6(SO4)2FCl is a rare sodium halogen sulphate and occurs associated with evaporitic deposits. Sulphohalite formation is important in saline evaporites and in pipe scales. Sulphohalite is an anhydrous sulphate–halide with an apparent variable anion ratio of formula Na6(SO4)2FCl. Such a formula with oxyanions lends itself to vibrational spectroscopy. The Raman band at 1003 cm−1 is assigned to the (SO4)2− ν1 symmetric stretching mode. Shoulders to this band are found at 997 and 1010 cm−1. The low intensity Raman bands at 1128, 1120 and even 1132 cm−1 are attributed to the (SO4)2− ν3 antisymmetric stretching vibrations. Two symmetric sulphate stretching modes are observed indicating at least at the molecular level the non-equivalence of the sulphate ions in the sulphohalite structure. The Raman bands at 635 and 624 cm−1 are assigned to the ν4 SO42− bending modes. The ν2 (SO4)2− bending modes are observed at 460 and 494 cm−1. The observation of multiple bands supports the concept of a reduction in symmetry of the sulphate anion from Td to C3v or even C2v. No evidence of bands attributable to the halide ions was found.

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Throughout Australia, regulation of the power of sale is highly inconsistent. In response to the uncertain nature of the mortgagee’s duty at common law, many legislatures have intervened. As a result, there has been a proliferation of statutory formula conferring varying degrees of protection on mortgagors. The differences in approach indicate a lack of consensus as to the best method of regulation. This article exposes the extent of the inconsistency and provides a comparative assessment of the various provisions with reference to the policy concerns that underpin legislative intervention. The article identifies a number of deficiencies associated with existing provisions and concludes that mortgagees and mortgagors alike would benefit from improved clarity and consistency. To that end, the article proposes a model provision that seeks to address the deficiencies identified.