Ein v. Staudt-Clausenscher Satz für periodische Bernoullizahlen

Letk, n be relatively prime integers, 1k<n. The numbersB _k ^m =m(i/2) ^m cot^(m–1)(k/n),m2, belong to then-th cyclotomic field. They occur as values of certain simple Dirichlet series at integral arguments and have a natural interpretation as generalized Bernoulli numbers. Therefore the arithmetic properties of these numbers arouse some interest. Ifn is not a prime power,B _k ^m /m is an algebraic integer. In the other case, i.e.n=p ^e ,p prime, the fractional part ofB _k ^m /m resp.B _k ^m is determined in this paper. It is periodic modulop–1 and exhibits a connection between these Bernoulli numbers and the Stirling numbers of the second kind.     

Dynamic percolation transition induced by phase separation: A Monte Carlo analysis

The percolation transition of geometric clusters in the three-dimensional, simple cubic, nearest neighbor Ising lattice gas model is investigated in the temperature and concentration region inside the coexistence curve. We consider quenching experiments, where the system starts from an initially completely random configuration (corresponding to equilibrium at infinite temperature), letting the system evolve at the considered temperature according to the Kawasaki spinexchange dynamics. Analyzing the distributionn _l(t) of clusters of sizel at timet, we find that after a time of the order of about 100 Monte Carlo steps per site a percolation transition occurs at a concentration distinctly lower than the percolation concentration of the initial random state. This dynamic percolation transition is analyzed with finite-size scaling methods. While at zero temperature, where the system settles down at a frozen-in cluster distribution and further phase separation stops, the critical exponents associated with this percolation transition are consistent with the universality class of random percolation, the critical behavior of the transient time-dependent percolation occurring at nonzero temperature possibly belongs to a different, new universality class.     

Influence of a continuous quenching procedure on the initial stages of spinodal decomposition

Instead of the standard assumption in the theory of phase separation where an instantaneous quench from an initial equilibrium state to the final state in the two-phase region is assumed, we consider the more realistic situation that the change of the external control parameter (e.g. temperature) can only be performed with finite rates. During the initial stages of spinodal decomposition the system then has some memory of the states intermediate between the initial and the final one. This influence of the finite quench rate in continuous quenching procedures is studied within the linearized theory of spinodal decomposition, with the Langer-Baron-Miller decoupling, and with Monte Carlo simulations. Both the case of thermally activated mobilities (applicable to solid metallic alloys) and the case of nearly temperature-independent mobilities (applicable to fluid polymer mixtures) are treated, and possible experimental applications are discussed. We find drastic deviations from the standard instantaneous quench situations in all cases of experimental interest.     

Coordination compounds of disulphur dinitride

Summary Coordination compounds of the S₂N₂ molecule including methods for their preparation, reactivities, i.r. data, structures, and aspects of chemical bonding are reviewed. Methods of synthesis include reactions of S₂N₂, S₄N₄ or (NSCl)₃ with metal halides, metal complexes such as carbonyls, or even metals themselves. In all cases, the planar S₂N₂ ring is coordinated, usuallyvia both, of its nitrogen atoms so that S₂N₂ acts as a bridging ligand between two metal centres; short contact distances imply that halogen atoms linked to the metal atoms show some interaction with the sulphur atoms. The stability of S₂N₂ is greatly enhanced by coordination. In the i.r. spectra, two characteristic S₂N₂ vibrations assist identification of the S₂N₂ species, a ring stretching mode being observable atca. 850 cm^–1 and the out-of-plane deformation at 450–490 cm^–1.     

π-Bindungsbeitrag in Boranen, 2. Mitt.:Ab initio- und Kraftfeld-Rechnungen von Vinyldifluorboran

Ab initio calculations on vinyldifluoroborane yield a -contribution of 23 kJ/mol (5.5 kcal/mol) mainly due to the C=C bond and not to the lone pairs of the fluorine atoms. The rotational barrier was also determined. The force field calculations favour the interpretation of the bond structure derived from the theab initio results.

     

Artificial multiple criticality and phase equilibria: an investigation of the PC-SAFT approach

The perturbed-chain statistical associating fluid theory (PC-SAFT) is studied for a wide range of temperature, T, pressure, p, and (effective) chain length, m, to establish the generic phase diagram of polymers according to this theory. In addition to the expected gas-liquid coexistence, two additional phase separations are found, termed "gas-gas" equilibrium (at very low densities) and "liquid-liquid" equilibrium (at densities where the system is expected to be solid already). These phase separations imply that in one-component polymer systems three critical points occur, as well as equilibria of three fluid phases at triple points. However, Monte Carlo simulations of the corresponding system yield no trace of the gas-gas and liquid-liquid equilibria, and we conclude that the latter are just artefacts of the PC-SAFT approach. Using PC-SAFT to correlate data for polybutadiene melts, we suggest that discrepancies in modelling the polymer density at ambient temperature and high pressure can be related to the presumably artificial liquid-liquid phase separation at lower temperatures. Thus, particular care is needed in engineering applications of the PC-SAFT theory that aims at predicting properties of macromolecular materials.     

Synthese und Kristallstrukturen der Phosphanimin-Komplexe MCl2(Me3SiNPMe3)2 mit M = Zn und Co,und CoCl2(HNPMe3)2

Syntheses and Crystal Structures of the Phosphaneimine Complexes MCl₂(Me₃SiNPMe₃)₂ with M = Zn and Co, and CoCl₂(HNPMe₃)₂ The molecular complexes MCl₂(Me₃SiNPMe₃)₂ (M = Zn, Co) have been prepared by the reaction of the dichlorides of zinc and cobalt with Me₃SiNPMe₃ in CH₃CN and CH₂Cl₂, respectively, whereas the complex CoCl₂(HNPMe₃)₂ has been prepared by the reaction of CoCl₂ with NaF in boiling acetonitrile in the presence of Me₃SiNPMe₃. All complexes were characterized by IR spectroscopy and by crystal structure determinations. The complexes MCl₂(Me₃SiNPMe₃)₂ crystallize isotypically. ZnCl₂(Me₃SiNPMe₃)₂: Space group P2₁2₁2₁, Z = 4, 2677 observed unique reflections, R = 0.024. Lattice dimensions at ?70°C: a = 1243.6; b = 1319.0; c = 1464.7 pm. CoCl₂(Me₃SiNPMe₃)₂: Space group P2₁2₁2₁, Z = 4, 3963 observed unique reflections, R = 0,071. Lattice dimensions at ?80°C: a = 1236.3; b = 1317.4; c = 1457.6 pm. CoCl₂(HNPMe₃)₂ · CH₂Cl₂: Space group Pbca, Z = 8, 1354 observed unique reflections, R = 0.055. Lattice dimensions at ?80°C: a = 1247.3; b = 998.4; c = 2882.4 pm. All complexes have monomeric molecular structures, in which the metal atoms are coordinated in a distorted tetrahedral fashion by the two chlorine atoms and by the nitrogen atoms of the phosphaneimine molecules.     

Chlorthionitrenkomplexe des Wolframs Die Kristallstruktur von [WCl4(NSCl)]2

Chlorothionitrene Complexes of Tungsten. Crystal Structure of [WCl₄(NSCl)]₂ Tungsten hexachloride reacts with trithiazyl chloride, (NSCl)₃, yielding the chlorothionitrene complex [WCl₄(NSCl)]₂, from which AsPh₄[WCl₅(NSCl)] can be obtained by reaction with AsPh₄Cl. Both complexes are characterized by their i.r. spectra. The crystal structure of [WCl₄(NSCl)]₂ was determined and refined with X-ray diffraction data (1059 reflexes, R = 0.055). It crystallizes in the monoclinic space group P2₁/n with the lattice constants a = 1523, b = 904, c = 583 pm and β = 91.35°. In the unit cell there are two centrosymmetric [WCl₄(NSCl)]₂ molecules in which the W atoms are linked via two chloro bridges; short and long W? Cl distances (244 and 265 pm) alternate in the W₂Cl₂ ring, the NSCl groups are found in the trans positions to the longer W? Cl bonds. The WNS bond angle (175°) and short bond distances correspond to a formulation .     

On the phase CrTe3

A new phase CrTe₃ was discovered and its existence was confirmed by differential thermal analysis and X-ray investigations. Symmetry and lattice parameters as well as the temperature of its peritectic decomposition were determined.

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Über die PhaseCrTe ₃

Zusammenfassung Eine neue Phase CrTe₃ wurde entdeckt und ihre Existenz wurde mittels differential-thermoanalytischer und röntgenographischer Untersuchungen sichergestellt. Es wurden Symmetrie und Gitterparameter sowie auch die Temperatur des peritektischen Zerfalls bestimmt.