A novel numerical approximation for the space fractional advection-dispersion equation

In this paper, we consider a space fractional advection–dispersion equation. The equation is obtained from the standard advection–diffusion equation by replacing the first- and second-order space derivatives by the Riesz fractional derivatives of order β1 ∈ (0, 1) and β2 ∈ (1, 2], respectively. The fractional advection and dispersion terms are approximated by using two fractional centred difference schemes. A new weighted Riesz fractional finite-difference approximation scheme is proposed. When the weighting factor θ equals 12, a second-order accuracy scheme is obtained. The stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.

The molecular structure of the phosphate mineral väyrynenite : a vibrational spectroscopic study

We have studied the mineral väyrynenite from the Viitaniemi pegmatite, located in the Eräjärvi area, Finland using a combination of electron microscopy electron microprobe and vibrational spectroscopic techniques. Chemical analysis shows the formula of the mineral to be (Mn0.88,Fe0.08,Mg0.01)∑0.97Be1.02(PO4)1.00(OH)1.02. Vibrational spectroscopy enables an assessment of the molecular structure of väyrynenite to be assessed. An intense Raman band at 1004 cm−1 is to the ν1 symmetric stretching mode. The observation of multiple bands in the phosphate stretching region, offers support for the concept of different phosphate units in the väyrynenite structure. Infrared spectroscopy confirms this multiplicity of vibrational bands. Multiple bands are observed in the phosphate bending region.

Vibrational spectroscopy of the mineral meyerhofferite CaB3O3(OH)5·H2O – An assessment of the molecular structure

Meyerhofferite is a calcium hydrated borate mineral with ideal formula: CaB3O3(OH)5�H2O and occurs as white complex acicular to crude crystals with length up to �4 cm, in fibrous divergent, radiating aggregates or reticulated and is often found in sedimentary or lake-bed borate deposits. The Raman spectrum of meyerhofferite is dominated by intense sharp band at 880 cm�1 assigned to the symmetric stretching mode of trigonal boron. Broad Raman bands at 1046, 1110, 1135 and 1201 cm�1 are attributed to BOH in-plane bending modes. Raman bands in the 900–1000 cm�1 spectral region are assigned to the antisymmetric stretching of tetrahedral boron. Distinct OH stretching Raman bands are observed at 3400, 3483 and 3608 cm�1. The mineral meyerhofferite has a distinct Raman spectrum which is different from the spectrum of other borate minerals, making Raman spectroscopy a very useful tool for the detection of meyerhofferite in sedimentary and lake bed deposits.

Vibrational spectroscopy of the phosphate mineral kovdorskite – Mg2PO4(OH)⋅3H2O

The mineral kovdorskite Mg2PO4(OH)�3H2O was studied by electron microscopy, thermal analysis and vibrational spectroscopy. A comparison of the vibrational spectroscopy of kovdorskite is made with other magnesium bearing phosphate minerals and compounds. Electron probe analysis proves the mineral is very pure. The Raman spectrum is characterized by a band at 965 cm�1 attributed to the PO3� 4 m1 symmetric stretching mode. Raman bands at 1057 and 1089 cm�1 are attributed to the PO3�4 m3 antisymmetric stretching modes. Raman bands at 412, 454 and 485 cm�1 are assigned to the PO3�4 m2 bending modes. Raman bands at 536, 546 and 574 cm�1 are assigned to the PO3�4 m4 bending modes. The Raman spectrum in the OH stretching region is dominated by a very sharp intense band at 3681 cm�1 assigned to the stretching vibration of OH units. Infrared bands observed at 2762, 2977, 3204, 3275 and 3394 cm�1 are attributed to water stretching bands. Vibrational spectroscopy shows that no carbonate bands are observed in the spectra; thus confirming the formula of the mineral as Mg2PO4(OH)�3H2O.

Vibrational spectroscopic characterization of the phosphate mineral series eosphorite–childrenite–(Mn,Fe)Al(PO4)(OH)2·(H2O)

The phosphate mineral series eosphorite–childrenite–(Mn,Fe)Al(PO4)(OH)2·(H2O) has been studied using a combination of electron probe analysis and vibrational spectroscopy. Eosphorite is the manganese rich mineral with lower iron content in comparison with the childrenite which has higher iron and lower manganese content. The determined formulae of the two studied minerals are: (Mn0.72,Fe0.13,Ca0.01)(Al)1.04(PO4, OHPO3)1.07(OH1.89,F0.02)·0.94(H2O) for SAA-090 and (Fe0.49,Mn0.35,Mg0.06,Ca0.04)(Al)1.03(PO4, OHPO3)1.05(OH)1.90·0.95(H2O) for SAA-072. Raman spectroscopy enabled the observation of bands at 970 cm−1 and 1011 cm−1 assigned to monohydrogen phosphate, phosphate and dihydrogen phosphate units. Differences are observed in the area of the peaks between the two eosphorite minerals. Raman bands at 562 cm−1, 595 cm−1, and 608 cm−1 are assigned to the �4 bending modes of the PO4, HPO4 and H2PO4 units; Raman bands at 405 cm−1, 427 cm−1 and 466 cm−1 are attributed to the �2 modes of these units. Raman bands of the hydroxyl and water stretching modes are observed. Vibrational spectroscopy enabled details of the molecular structure of the eosphorite mineral series to be determined.

The molecular structure of the phosphate mineral senegalite Al2(PO4)(OH)3⋅3H2O – A vibrational spectroscopic study

We have studied the mineral senagalite, a hydrated hydroxy phosphate of aluminium with formula Al2(PO4)(OH)3⋅3H2O using a combination of electron microscopy and vibrational spectroscopy. Senegalite crystal aggregates shows tabular to prismatic habitus and orthorhombic form. The Raman spectrum is dominated by an intense band at 1029 cm−1 assigned to the PO43- ν1 symmetric stretching mode. Intense Raman bands are found at 1071 and 1154 cm−1 with bands of lesser intensity at 1110, 1179 and 1206 cm−1 and are attributed to the PO43- ν3 antisymmetric stretching vibrations. The infrared spectrum shows complexity with a series overlapping bands. A comparison is made with spectra of other aluminium containing phosphate minerals such as augelite and turquoise. Multiple bands are observed for the phosphate bending modes giving support for the reduction of symmetry of the phosphate anion. Vibrational spectroscopy offers a means for the assessment of the structure of senagalite.

Vibrational spectroscopic characterization of the phosphate mineral kulanite Ba(Fe2+,Mn2+,Mg)2(Al,Fe3+)2(PO4)3(OH)3

The mineral kulanite BaFe2Al2(PO4)3(OH)3, a barium iron aluminum phosphate, has been studied by using a combination of electron microscopy and vibrational spectroscopy. Scanning electron microscopy with EDX shows the mineral is homogenous with no other phases present. The Raman spectrum is dominated by an intense band at 1022 cm−1 assigned to the PO43-ν1 symmetric stretching mode. Low intensity Raman bands at 1076, 1110, 1146, 1182 cm−1 are attributed to the PO43-ν3 antisymmetric stretching vibrations. The infrared spectrum shows a complex spectral profile with overlapping bands. Multiple phosphate bending vibrations supports the concept of a reduction in symmetry of the phosphate anion. Raman spectrum at 3211, 3513 and 3533 cm−1 are assigned to the stretching vibrations of the OH units. Vibrational spectroscopy enables aspects on the molecular structure of kulanite to be assessed.

Vibrational spectroscopy of the phosphate mineral lazulite – (Mg, Fe)Al2(PO4)2·(OH)2 found in the Minas Gerais, Brazil

This research was done on lazulite samples from the Gentil mine, a lithium bearing pegmatite located in the municipality of Mendes Pimentel, Minas Gerais, Brazil. Chemical analysis was carried out by electron microprobe analysis and indicated a magnesium rich phase with partial substitution of iron. Traces of Ca and Mn, (which partially replaced Mg) were found. The calculated chemical formula of the studied sample is: (Mg0.88, Fe0.11)Al1.87(PO4)2.08(OH)2.02. The Raman spectrum of lazulite is dominated by an intense sharp band at 1060 cm-1 assigned to PO stretching vibrations of of tetrahedral [PO4] clusters presents into the HPO2/4- units. Two Raman bands at 1102 and 1137 cm-1 are attributed to both the HOP and PO antisymmetric stretching vibrations. The two infrared bands at 997 and 1007 cm-1 are attributed to the m1 PO3/4- symmetric stretching modes. The intense bands at 1035, 1054, 1081, 1118 and 1154 cm-1 are assigned to the v3PO3/4- antisymmetric stretching modes from both the HOP and tetrahedral [PO4] clusters. A set of Raman bands at 605, 613, 633 and 648 cm-1 are assigned to the m4 out of plane bending modes of the PO4, HPO4 and H2PO4 units. Raman bands observed at 414, 425, 460, and 479 cm-1 are attributed to the m2 tetrahedral PO4 clusters, HPO4 and H2PO4 bending modes. The intense Raman band at 3402 and the infrared band at 3403 cm-1 are assigned to the stretching vibration of the OH units. A combination of Raman and infrared spectroscopy enabled aspects of the molecular structure of the mineral lazulite to be understood.

Vibrational spectroscopic characterization of the phosphate mineral bermanite–Mn2+Mn23+(PO4)2(OH)2.4(H2O)

Bermanite Mn2þMn3þ2 ðPO4Þ2ðOHÞ2 � 4ðH2OÞ is a mixed valent hydrated hydroxy phosphate mineral. The mineral is reddish-brown and occurs in crystal aggregates and as lamellar masses. Bermanite is a common mineral in granitic pegmatites. The chemical composition of bermanite was obtained using EDS techniques. We have studied the molecular structure of bermanite using vibrational spectroscopy. The mineral is characterized by a Raman doublet at 991 and 999 cm-1 attributed to the phosphate stretching mode of two non-equivalent phosphate units. Raman bands at 1071, 1117 and 1142 cm-1 are assigned to the phosphate antisymmetric stretching modes. The hydroxyl stretching spectral region is complex with overlapping bands attributed to water and hydroxyl stretching vibrations. Vibrational spectroscopy proves most useful for the study of the mineral bermanite.

Thermal analysis and vibrational spectroscopic characterization of the boro silicate mineral datolite – CaBSiO4(OH)

The objective of this work is to determine the thermal stability and vibrational spectra of datolite CaBSiO4(OH) and relate these properties to the structure of the mineral. The thermal analysis of datolite shows a mass loss of 5.83% over a 700–775 °C temperature range. This mass loss corresponds to 1 water (H2O) molecules pfu. A quantitative chemical analysis using electron probe was undertaken. The Raman spectrum of datolite is characterized by bands at 917 and 1077 cm−1 assigned to the symmetric stretching modes of BO and SiO tetrahedra. A very intense Raman band is observed at 3498 cm−1 assigned to the stretching vibration of the OH units in the structure of datolite. BOH out-of-plane vibrations are characterized by the infrared band at 782 cm−1. The vibrational spectra are based upon the structure of datolite based on sheets of four- and eight-membered rings of alternating SiO4 and BO3(OH) tetrahedra with the sheets bonded together by calcium atoms.

Vibrational spectroscopy of the borate mineral olshanskyite Ca3[B(OH)4]4(OH)2

The mineral olshanskyite is one of many calcium borate minerals which has never been studied using vibrational spectroscopy. The mineral is unstable and decomposes upon exposure to an electron beam. This makes the elemental analysis using EDX techniques difficult. Both the Raman and infrared spectra show complexity due to the complexity of the structure. Intense Raman bands are found at 989, 1,003, 1,025 and 1,069 cm-1 with a shoulder at 961 cm-1 and are assigned to trigonal borate units. The Raman bands at 1,141, 1,206 and 1,365 cm-1 are assigned to OH in-plane bending of BOH units. A series of Raman bands are observed in the 2,900–3,621cm-1 spectral range and are assigned to the stretching vibrations of OH and water. This complexity is also reflected in the infrared spectra. Vibrational spectroscopy enables aspects of the structure of olshanskyite to be elucidated.

Vibrational spectroscopic characterization of the sulphate mineral khademite Al(SO4)F⋅5(H2O)

Vibrational spectroscopy has been used to characterize the sulphate mineral khademite Al(SO4)F∙5(H2O). Raman band at 991 cm-1 with a shoulder at 975 cm-1 is assigned to the ν1 (SO4)2- symmetric stretching mode. The observation of two symmetric stretching modes suggests that the sulphate units are not equivalent. Two low intensity Raman bands at 1104 and 1132 cm-1 are assigned to the ν3 (SO4)2- antisymmetric stretching mode. The broad Raman band at 618 cm-1 is assigned to the v4 (SO4)2- bending modes. Raman bands at 455, 505 and 534 cm-1 are attributable to the doubly degenerate v2 (SO4)2- bending modes. Raman bands at 2991, 3146 and 3380 cm-1 are assigned to the OH stretching bands of water. Five infrared bands are noted at 2458, 2896, 3203, 3348 and 3489 cm-1 are also due to water stretching bands. The observation of multiple water stretching vibrations gives credence to the non-equivalence of water units in the khademite structure. Vibrational spectroscopy enables an assessment of the structure of khademite.

Vibrational spectroscopy of the borate mineral pinnoite MgB2O(OH)6

There is a large number of boron containing minerals with water and/or hydroxyl units of which pinnoite MgB2O(OH)6 is one. Some discussion about the molecular structure of pinnoite exists in the literature. Whether water is involved in the structure is ill-determined. The molecular structure of pinnoite has been assessed by the combination of Raman and infrared spectroscopy. The Raman spectrum is characterized by an intense band at 900 cm−1 assigned to the BO stretching vibrational mode. A series of bands in the 1000–1320 cm−1 spectral range are attributed to BO antisymmetric stretching modes and in-plane bending modes. The infrared spectrum shows complexity in this spectral range. Multiple Raman OH stretching vibrations are found at 3179, 3399, 3554 and 3579 cm−1. The infrared spectrum shows a series of overlapping bands with bands identified at 3123, 3202, 3299, 3414, 3513 and 3594 cm−1. By using a Libowitzky type function, hydrogen bond distances were calculated. Two types of hydrogen bonds were identified based upon the hydrogen bond distance. It is important to understand the structure of pinnoite in order to form nanomaterials based upon the pinnoite structure.

Vibrational spectroscopic study of the uranyl selenite mineral derriksite Cu4UO2(SeO3)2(OH)6⋅H2O

Raman spectrum of the mineral derriksite Cu4UO2(SeO3)2(OH)6⋅H2O was studied and complemented by the infrared spectrum of this mineral. Both spectra were interpreted and partly compared with the spectra of demesmaekerite, marthozite, larisaite, haynesite and piretite. Observed Raman and infrared bands were attributed to the (UO2)2+, (SeO3)2−, (OH)− and H2O vibrations. The presence of symmetrically distinct hydrogen bonded molecule of water of crystallization and hydrogen bonded symmetrically distinct hydroxyl ions was inferred from the spectra in the derriksite unit cell. Approximate U–O bond lengths in uranyl and O–H⋯O hydrogen bond lengths were calculated from the Raman and infrared spectra of derriksite.