On cubature formulae with a minimal number of knots

Summary In this paper an approach is outlined to the two-dimensional analogon of the Gaussian quadrature problem. The main results are necessary and sufficient conditions for the existence of cubature formulae which are exact for all polynomials of degree m and which have a minimal number of 1/2k(k+1) knots,k=[m/2]+1. Ifm is odd, similar results are due to I.P. Mysovskikh ([5, 6]) which will be derived in a new way as a special case of the general characterization given here. Furthermore, it will be shown how this characterization can be used to construct minimal formulae of even degree.     

Anharmonicity in rotational and vibrational excitation of H2 by Li+ collisions

Integral cross sections for pure rotational and vibrational-rotational excitation of H₂(X¹Σ⁺_g) by Li⁺(¹S) impact are computed by close-coupling methods at 0.2, 0.6, and 1.2 eV in the c.m. system using vibrational functions that are numerical solutions of the one-dimensional radial Schrödinger equation for harmonic, Morse, and adiabatically corrected Kolos-Wolniewicz (KW) potential functions. Comparison of results employing KW and Morse functions shows excellent agreement for all transitions studied. Findings using harmonic oscillator functions, however, differ noticeably from KW and Morse values for vibrational (0 → 1) and very large rotational (Δj = 10) transitions, but are satisfactory for lower order (0 → 2, 4, 6, 8) rotational transitions.     

Poly(ethyleneimine) as reducing and stabilizing agent for the formation of gold nanoparticles in w/o microemulsions

This paper is focused on the use of branched poly(ethyleneimine) (PEI) as reducing as well as stabilizing agent for the formation of gold nanoparticles in different media. The process of nanoparticle formation was investigated, in the absence of any other reducing agents, in microemulsion template phase in comparison to the nucleation process in aqueous polymer solution.

On the one hand, it was shown that the polyelectrolyte can be used for the controlled single-step synthesis and stabilization of gold nanoparticles via a nucleation reaction and particles with an average diameter of 7.1 nm can be produced.

On the other hand, it was demonstrated that the polymer can also act as reducing and stabilizing agent in much more complex systems, i.e. in water-in-oil (w/o) microemulsion droplets. The reverse microemulsion droplets of the quaternary system sodium dodecylsulfate (SDS)/toluene–pentanol (1:1)/water were successfully used for the synthesis of gold nanoparticles. The polymer, incorporated in the droplets, exhibits reducing properties, adsorbs on the surface of the nanoparticles and prevents their aggregation. Consequently, nanoparticles of 8.6 nm can be redispersed after solvent evaporation without a change of their size.

Nevertheless, the polymer acts already as a “template” during the formation of the nanoparticles in water and in microemulsion, so that an additional template effect of the microemulsion is not observed.

The particle formation for both methods is checked by means of UV–vis spectroscopy and the particle size and size distribution are investigated via dynamic light scattering and transmission electron microscopy (TEM).     

Gerüstumlagerung eines α,β -ungesättigten γ,δ-Epoxiketons bei der Birch-Reduktion: Strukturaufklärung mittels 13C-INADEQUATE-NMR-Spektroskopie

Skeleton Rearrangement of an α-β-Unsaturated γ,δ-Epoxyketone during Birch Reduction: Structure Elucidation by Means of ¹³C-INADEQUATE-NMR Spectroscopy When the γ-epoxide 2 of β-ionone is treated under standard Birch-reduction conditions, unexpectedly a 70% combined yield of regioisomeric octalones 4 and 5 is isolated. These products unquestionably result form cleavage of the central epoxide C?C bond. The structure of compounds 4 and 5 could be determined by means of ¹³C-INADEQUATE-NMR spectroscopy.     

Beiträge zur Chemie der Silicium—Stickstoff-Verbindungen, 16. Mitt.: N-Silyl-carbonamide und N-Silyl-carbamidsäureester

Zusammenfassung Die durch direkte Silylierung von Carbonamidgruppen mit Halogensilanen nicht zugänglichen Verbindungen der Gruppierung R-CONHSiR₃ konnten durch Umsetzung von Säureamiden oder Säurechloriden mit Hexamethyldisilazan gewonnen werden. So wurden Derivate mit R=CH₃, C₂H₅, C₆H₅, CONHSiR₃ , OCH₃, OC₂H₅, OCH(CH₃)₂; R=CH₃ (vgl. Tab. 1) dargestellt und in ihren physikalischen und chemischen Eigenschaften näher studiert.Herrn Prof. Dr.A. Zinke zum 70. Geburtstag gewidmet.15. Mitt.:U. Wannagat undH. Kuckertz, Angew. Chem.74, 117 (1962).Auszug aus der DissertationJ. Pump, Techn. Hochsch. Aachen 1962.     

Topologisch und elektronisch bemerkenswerte „reduzierte”︁ Cluster des Typs [V18O42(X)]n− (X = SO4, VO4) mit Td-Symmetrie und davon abgeleitete Cluster [V(18–p)As2pO42(X)]m− (X = SO3, SO4, H2O; p = 3, 4)

Reduced Clusters with Remarkable Topological and Electronic Properties of the Type of [V₁₈O₄₂(X)]^n? (X = SO₄, VO₄) with T_d-Symmetry and Related Clusters [V_(18—p)As_2pO₄₂(X)]^m? (X = SO₃, SO₄, H₂O; p = 3, 4) The novel cluster-compounds Na₆[V₁₈O₄₂H₉(VO₄)] · 21 H₂O, (NH₄)₈[V₁₈O₄₂(SO₄)] · 25 H₂O, K₆[V₁₅As₆O₄₂(H₂O)] · 8 H₂O, (NH₄)₆[V₁₄As₈O₄₂(SO₃)], (NH₄)₆[V₁₄As₈O₄₂(SO₄)] and [N(CH3)₃]₄[₄V₁₄As₈0₄₂(H₂0)] were prepared and characterized by IR- and UV/Vis/NIR-spectroscopy, magnetic measurements and complete crystal structure analysis. For structural data see Inhaltsübersicht. Topological relations to the rhombicuboctahedron spanned by 24 0-atoms of the genuine hypothetical a-Keggin ion, at which the square planes are capped by V?O or As₂O groups, are discussed. Of particular interest are the ?extended”? Keggin ions [V₁₈O₄₂(X)]ⁿ- (X = SO₄ VO₄), (formaly derived from the hypothetical genuine a-Keggin ion by addition of six V?O groups) which have quite different electron populations in spite of the same structure of their cluster shells.     

In Situ Study of the Surface Oxidation of FeCr Alloys Using Grazing Incidence X‐ray Absorption Spectroscopy (GIXAS)

The surface oxidation of FeCr alloys with 18, 28, and 43 mass‐% Cr was investigated in situ using grazing‐incidence X‐ray absorption spectroscopy (GIXAS) at the chromium and iron K‐edges. Oxidation in air was monitored in situ in the temperature range from 290 K to 680 K. The standard GIXAS data analysis is extended for the treatment of a single layer model in order to estimate the chromium concentrations of the oxide layer and of the near‐interface substrate as well as the oxide layer thickness. XANES analysis shows transitions from b.c.c. Fe to corundum type Fe₂O₃ and from b.c.c. Cr to corundum type Cr₂O₃. The initial oxide layers are 1.1‐1.4 nm thick and contain 60‐90 mass‐% chromium, while the near‐interface substrate is depleted in Cr. During heating, iron oxide growth dominates up to 560‐600 K. Then the chromium oxide layer loses its passivation effect and Cr oxidation sets in.