239 resultados para Legendre Formula


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The rare-earth trichloroacetates having the formula Ln(Cl3CCOO)3. 3H2O, where Ln = La, Pr, Nd, Sm, Gd or Y, were investigated by conductimetric, cryoscopic and infrared methods. The results suggest a structure in which the acetate anion coordinates to the metal through both the oxygen and the water molecules remain outside the coordination sphere.

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Dimethyl sulphoxide complexes of lanthanide and yttrium nitrates of the general formula M(DMSO)n(NO3)3 where M = La, Ce, Pr, Nd, Sm or Gd; n = 4 and M = Y, Ho or Yb; n = 3 have been isolated and characterized. The i.r. data besides excluding the presence of D3h nitrate, reveal co-ordination through the oxygen atom of the dimethyl sulphoxide. The complexes are monomeric in acetonitrile. Molecular conductance data in acetone, acetonitrile, dimethyl formamide and dimethyl sulphoxide suggest a co-ordination number of eight for the lighter lanthanides and seven for yttrium and the heavier lanthanides.

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Rare earth perchlorate-antipyrine (ap) complexes of the formula Ln (ClO4)3.6 ap have been prepared and characterised. Infrared and electronic spectra showed the co-ordination through carbonyl oxygen. Conductivity and molecular weight data indicated a co-ordination number of six for these complexes.

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Dimethylsulphoxide (DMSO) complexes of rare-earth perchlorates of the formula M(ClO4)3·n DMSO (M = La, Ce, Pr and Nd, n = 8; M = Sm, Gd and Y, n = 7) have been prepared. I.r. studies indicate co-ordination through oxygen. Cryoscopic and conductivity data show co-ordination number of 7 and 8.

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Raman spectra of single crystals of adipic and sebacic acids have been photographed for the first time using λ 2537 excitation. The spectra have been divided into four regions: (a) internal frequencies; (b) summations and overtones; (c) external vibrations; and (d) low-frequency hydrogen bond oscillations. Tentative correlations have been given for all the internal frequencies and summations and overtones. A series of diffuse weak bands observed in the spectra of both these acids in the not, vert, similar2400–2800 cm−1 have been explained as a superposition of O---H frequencies lowered due to hydrogen bond formation over the summations and overtones of fundamentals mainly in the not, vert, similar1000–1500 cm−1 region. Rotatory type of external oscillations of the two formula units of these molecules in their unit cells have been identified at 76, 99, 118 and 165 cm−1 in adipic acid and 66, 95, 117 and 177 cm−1 in the spectrum of sebacic acid. A brief discussion of the low frequency hydrogen bond vibrations in these acids has been made. Making use of the Lippincott—Schroeder potential and assuming a highly anharmonic potential curve for the hydrogen bond, the vibrational frequencies of the bond have been theoretically evaluated. There is very good agreement between these and the experimental values. The results for adipic acid in cm−1 are: 304 (0 → 1), 270 (1 → 2), 241 (2 → 3), 222 (3 → 4) 201 (4 → 5), 183 (5 → 6). In the case of sebacic acid some of the intermediate and higher transitions are absent in the spectrum recorded by the author. From the above data for adipic acid the dissociation energy of the hydrogen bond was evaluated as 5·9 kcal/mole in fair agreement with the values derived from conventional methods.

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Masonry strength is dependent upon characteristics of the masonry unit,the mortar and the bond between them. Empirical formulae as well as analytical and finite element (FE) models have been developed to predict structural behaviour of masonry. This paper is focused on developing a three dimensional non-linear FE model based on micro-modelling approach to predict masonry prism compressive strength and crack pattern. The proposed FE model uses multi-linear stress-strain relationships to model the non-linear behaviour of solid masonry unit and the mortar. Willam-Warnke's five parameter failure theory developed for modelling the tri-axial behaviour of concrete has been adopted to model the failure of masonry materials. The post failure regime has been modelled by applying orthotropic constitutive equations based on the smeared crack approach. Compressive strength of the masonry prism predicted by the proposed FE model has been compared with experimental values as well as the values predicted by other failure theories and Eurocode formula. The crack pattern predicted by the FE model shows vertical splitting cracks in the prism. The FE model predicts the ultimate failure compressive stress close to 85 of the mean experimental compressive strength value.

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The role of cobalt centers in promoting the recombination and trapping processes in n-type germanium has been investigated. Data on lifetime measurements carried out by the steadystate photoconductivity and photo-magneto-electric methods in the temperature range 145 to 300°K on n-type germanium samples containing cobalt in the concentration range 1·1013 to 5.·014/cm3 are presented. The results are analysed on the basis of Sah-Shockley's multi-level formula to yield the capture cross-sections Sp= (hole capture cross-section at doubly negatively charged center) and Sn-(electron capture cross-section at singly negatively charged center) and temperature dependence. It is found that Sp= is (22 ± 6). 10-16 cm2 and Sn- is ∼ 0·1. 10-16 cm2 at 145°K. Sp= varies (n = 3·5 to 4·5) in the range 145-220°K; above 225°K the index 'n' tends to a smaller value. Sn- is practically temperature independent below 180°K and increases with increase of temperature above 180°K. The value of Sp= and its temperature variation lead one to the conclusion that during capture at attractive centers, the phonon cascade mechanism is responsible for the dissipation of the recombination energy.

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Copper(II) complexes of 1-benzyl-2-phenylbenzimidazole (BPBI) of the general formula Cu(BPBI)2X2, nH2O [X= Cl-, Br-, NO3 or OAc- (n = O) and X = NO3- or 1 2SO42-(n = 2H2O)] have been prepared. The complexes are found to be nonelectrolytes in nitrobenzene. Conductivity in nonaqueous media, magnetic susceptibilities and i.r. and electronic spectra of the complexes are reported. A tetragonally distorted octahedral structure has been suggested for these complexes.

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ORANGE red and amorphous peroxy-titanium complexes of oxalic, malonic and maleic acids1-3, when vacuum-dried, have co-ordinated water molecules firmly bonded to the central titanium atom as shown in formula (I). The peroxy-oxygen from these compounds is slowly lost even at room temperature because of the strained peroxy-group3,4. The compounds, when kept at 95°-100°C. for about three days, give deperoxygenated compounds of the type (II). However, a sample of peroxy-titanium oxalate sealed in a glass tube lost all its peroxy-oxygen in about four years and gave a white crystalline basic oxalate (II). The amorphous nature of the compounds may be due to random hydrogen bonding in the complexes. The crystallinity observed in one of the deperoxygenated titanyl oxalates may be due to the rearrangement of the molecules during ageing for more than four years. The infra-red absorption of these compounds was studied to find out the effect of co-ordination and hydrogen bonding on the infra-red bands of the free water.

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The recombination properties of cobalt centers in p-type germanium containing cobalt in the concentration range 1014 to 1016 atoms/cm3 have been investigated. The measurement of lifetime has been carried out by steady-state photoconductivity and photo-magneto-electric methods in the temperature range 145 to 300°K. The cross-sections Sno (electron capture cross-section at neutral centers). Sn- (electron capture cross-section at singly negatively charged centers) and their temperature variations have been estimated by the analysis of the lifetime data on the basis of Sah-Shockley's multi-level formula. The value of Sno is (15±5).10-16 cm2 and is temperature independent. The value of Sn- is ≈4·10-16 cm2 around 225°K and it increases with increase of temperature. The possible mechanisms for capture at neutral and repulsive centers are discussed and a summary of the capture cross-sections for cobalt centers is given. A comparison of the cross-section values of cobalt and their temperature variations with those of the related impurities-manganese, iron and nickel-in germanium has been made.

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A careful comparison of the distribution in the (R, θ)-plane of all NH ... O hydrogen bonds with that for bonds between neutral NH and neutral C=O groups indicated that the latter has a larger mean R and a wider range of θ and that the distribution was also broader than for the average case. Therefore, the potential function developed earlier for an average NH ... O hydrogen bond was modified to suit the peptide case. A three-parameter expression of the form {Mathematical expression}, with △ = R - Rmin, was found to be satisfactory. By comparing the theoretically expected distribution in R and θ with observed data (although limited), the best values were found to be p1 = 25, p3 = - 2 and q1 = 1 × 10-3, with Rmin = 2·95 Å and Vmin = - 4·5 kcal/mole. The procedure for obtaining a smooth transition from Vhb to the non-bonded potential Vnb for large R and θ is described, along with a flow chart useful for programming the formulae. Calculated values of ΔH, the enthalpy of formation of the hydrogen bond, using this function are in reasonable agreement with observation. When the atoms involved in the hydrogen bond occur in a five-membered ring as in the sequence[Figure not available: see fulltext.] a different formula for the potential function is needed, which is of the form Vhb = Vmin +p1△2 +q1x2 where x = θ - 50° for θ ≥ 50°, with p1 = 15, q1 = 0·002, Rmin = 2· Å and Vmin = - 2·5 kcal/mole. © 1971 Indian Academy of Sciences.

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Nickel(II) complexes of 1-benzyl-2-phenylbenzimidazole (BPBI) of the general formula [Ni(BPBI)2X2](X=Cl-, Br-, NCS- or NO3-) have been prepared and their magnetic moments, i.r. and electronic spectra studied. [Ni(BPBI)2Cl2] has a pseudotetrahedral structure while [Ni(BPBI)2 Br2] exists as square planar and speudotetrahedral isomers. [Ni(BPBI)2I2] and [NI(BPBI)2(NCS)2] have square planar stereochemistry. The nitrato complex [Ni(BPBI)2(NO)3)2] exists in two different octahedral modifications in the solid state.

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The crystal structure of the complex La(NO3)3.4(CH3)2SO has been solved by the heavy-atom method. The complex crystallizes in the monoclinic space group C2/e with four formula units in a unit cell of dimensions a= 14.94, b= 11.04, c= 15.54 A and fl= 109 ° 10'. The parameters have been refined by threedimensional least-squares procedures with anisotropic thermal parameters for all atoms except hydrogen. The final R index for 1257 observed reflexions is 0.094. The La 3 + ion is coordinated by ten oxygen atoms with La-O distances varying from 2.47 to 2.71 A. The geometry of the coordination polyhedron is described.

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The complex crystallizes in the space group P21/c with four formula units in a unit cell of dimensionsa= 12.747, b= 7.416, c= 17.894 A and/3= 90.2 °. The structure has been solved by the symbolic addition procedure using three dimensional photographic data and refined to an R value of 0.079 for 2019 observed reflexions. The pyramidal nature of the two hetero nitrogen atoms in the antipyrine molecule is inter:nediate between that observed in free antipyrine and in some of its metal complexes. The molecule is more polar than that in crystals of free antipyrine but less so compared with that in metal complexes. In the salicylic acid molecule, the hydroxyl group forms an internal hydrogen bond with one of the oxygen atoms in the carboxyl group. The association between the salicylic acid and the antipyrine molecules is achieved through an intermolecular hydrogen bond with the other carboxyl oxygen atom in the salicylic acid molecule as the proton donor and the carboxyl oxygen atom of the antipyrine molecule as the acceptor

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We examine the stability of hadron resonance gas models by extending them to include undiscovered resonances through the Hagedorn formula. We find that the influence of unknown resonances on thermodynamics is large but bounded. We model the decays of resonances and investigate the ratios of particle yields in heavy-ion collisions. We find that observables such as hydrodynamics and hadron yield ratios change little upon extending the model. As a result, heavy-ion collisions at the RHIC and LHC are insensitive to a possible exponential rise in the hadronic density of states, thus increasing the stability of the predictions of hadron resonance gas models in this context. Hadron resonance gases are internally consistent up to a temperature higher than the crossover temperature in QCD, but by examining quark number susceptibilities we find that their region of applicability ends below the QCD crossover.