Randbemerkungen zur historischen Entwicklung des Tangentenbegriffs

Zusammenfassung Die griechische Tangentendefinition verlangte, da? die Kurve ganz auf einer Seite der berührenden Geraden liegt. Eine allgemeine Methode, die Tangenten an eine beliebige Kurve zu finden, gab es nicht. Es wurden spezielle Konstruktionen (erraten und) angegeben und nachtr?glich bewiesen, da? sie Tangenten lieferten. Roberval und Torricelli leiteten auf Grund der Erzeugung mancher Kurven durch zwei Bewegungen Tangentenkonstruktionen aus dem Parallelogramm der Geschwindigkeiten ab. Barrow bemerkte, da? man jede Kurve durch zwei Bewegungen erzeugen kann: die Koordinaten werden als Funktionen der Zeit aufgefa?t —x(t), y(t). Fermat und Descartes fanden eine neue Methode zur Berechnung der Tangenten an algebraische Kurven. Sie beruht darauf, da? eine Gerade zwei zusammenfallende Punkte mit der Kurve gemeinsam hat, und da? dies einen quadratischen Faktor [(y−y ₀)² odere ²] in der Gleichung für die Schnittpunkte verlangt. Diese Methode enth?lt eine neue Definition der Tangente, die auch in Wendepunkten gilt. Die Methode wurde auch — etwas leichtfertig — auf transzendente Kurven übertragen, und lieferte auch richtige Resultate. Die Rechtfertigung dafür erbrachte erst die Differentialrechnung, von der hier nicht mehr gesprochen wird.

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Herrn Professor Dr. Dr.h.c.mult. Otto Haupt mit den besten Wünschen zum 100. Geburtstag gewidmet     

Collective cooling and self-organization of atoms in a cavity

We theoretically investigate the correlated dynamics of N coherently driven atoms coupled to a standing-wave cavity mode. For red detuning between the driving field and the cavity as well as the atomic resonance frequencies, we predict a light force induced self-organization of the atoms into one of two possible regular patterns, which maximize the cooperative scattering of light into the cavity field. Kinetic energy is extracted from the atoms by superradiant light scattering to reach a final kinetic energy related to the cavity linewidth. The self-organization starts only above a threshold of the pump strength and atom number. We find a quadratic dependence of the cavity mode intensity on the atom number, which demonstrates the cooperative effect.     

Doping‐Induced Absorption Bands in P3HT: Polarons and Bipolarons

In this work, we focus on the formation of different kinds of charge carriers such as polarons and bipolarons upon p‐type doping (oxidation) of the organic semiconductor poly(3‐ hexylthiophene‐2,5‐diyl) (P3HT). We elucidate the cyclic voltammogram during oxidation of this polymer and present spectroscopic changes upon doping in the UV/Vis/near‐IR range as well as in the mid‐IR range. In the low‐oxidation regime, two absorption bands related to sub‐gap transitions appear, one in the UV/Vis range and another one in the mid‐IR range. The UV/Vis absorption gradually decreases upon further doping while the mid‐IR absorption shifts to lower energy. Additionally, electron paramagnetic resonance (EPR) measurements are performed, showing an increase of the EPR signal up to a certain doping level, which significantly decreases upon further doping. Furthermore, the absorption spectra in the UV/Vis range are analyzed in relation to the morphology (crystalline vs. amorphous) by using theoretical models. Finally, the calculated charge carriers from cyclic voltammogram are linked together with optical transitions as well as with the EPR signals upon p‐type doping. We stress that our results indicate the formation of polarons at low doping levels and the existence of bipolarons at high doping levels. The presented spectroscopic data are an experimental evidence of the formation of bipolarons in P3HT.     

Formaldehyde Formation on Vanadium Oxide Surfaces V2O3(0001) and V2O5(001): How does the Stable Methoxy Intermediate Form?

Hydroxy‐mediated methoxy formation or stabilization is probably an important process in many methanol adsorption systems. Hydrogen atoms originating from the scission of the methanol O? H bond react with the substrate and form water. This process may result 1) in the production of additional surface defects as reactive centers for methoxy formation and 2) in the stabilization of methoxy groups by suppression of methanol formation.

     

Quantitative Depth Profiling Using Online-Laser Ablation of Solid Samples in Liquid (LASIL) to Investigate the Oxidation Behavior of Transition Metal Borides

The increased demand for sustainability requires, among others, the development of new materials with enhanced corrosion resistance. Transition metal diborides are exceptional candidates, as they exhibit fascinating mechanical and thermal properties. However, at elevated temperatures and oxidizing atmospheres, their use is limited due to the fact of their inadequate oxidation resistance. Recently, it was found that chromium diboride doped with silicon can overcome this limitation. Further improvement of this protective coating requires detailed knowledge regarding the composition of the forming oxide layer and the change in the composition of the remaining thin film. In this work, an analytical method for the quantitative measurement of depth profiles without using matrix-matched reference materials was developed. Using this approach, based on the recently introduced online-LASIL technique, it was possible to achieve a depth resolution of 240 nm. A further decrease in the ablation rate is possible but demands a more sensitive detection of silicon. Two chromium diboride samples with different Si contents suffering an oxidation treatment were used to demonstrate the capabilities of this technique. The concentration profiles resembled the pathway of the formed oxidation layers as monitored with transmission electron microscopy. The stoichiometry of the oxidation layers differed strongly between the samples, suggesting different processes were taking place. The validity of the LASIL results was cross-checked with several other analytical techniques.     

Correlation between thermal and mass spectral techniques for the characterization of allenylidene and carbene complexes

The fragmentation pathways of allenylidene and carbene complexes have been studied using FAB mass spectrometry in comparison with thermal analyses (TGA, DrTG and DTA). Both the decomposition modes are investigated and the possible fragmentation pathways are suggested. The use of mass and thermal analyses (TGA and DTA) in the analyses of allenylidene and carbene complexes allowed the characterization of the fragmentation pathways in MS. The major pathway includes successive loss of carbon monoxide followed by fragmentation of the organic part of the allenylidene or carbene molecules. This is also confirmed by thermogravimetric analysis (TGA) where the first step involves the loss of carbon monoxide followed by the organic ligand. The nature of each step; exothermic or endothermic, is also studied using DTA technique. The kinetic parameters of the thermal decomposition are also studied using the Coates-Redfern method.     

Intramolecular azide-alkyne cycloaddition for the fast assembly of structurally diverse, tricyclic 1,2,3-triazoles

The synthesis of novel tricyclic 1,2,3-triazoles starting from cyclic epoxides via the sequential azidolysis, propargylation and 1,3-dipolar cycloaddition is described. Derivatization by N-arylation reaction and the synthesis of enantiomerically pure compounds is also reported. Some of these compounds exhibit significant affinity for the sigma-1 receptor.     

On the Role of the Electronic Structure of the Heteronuclear Oxide Cluster [Ga2Mg2O5].+ in the Thermal Activation of Methane and Ethane: An Unusual Doping Effect

The reactivity of the heteronuclear oxide cluster [Ga₂Mg₂O₅]^.+, bearing an unpaired electron at a bridging oxygen atom (O_b^.?), towards methane and ethane has been studied using Fourier transform ion cyclotron resonance mass spectrometry (FT‐ICR‐MS). Hydrogen‐atom transfer (HAT) from both methane and ethane to the cluster ion is identified experimentally. The reaction mechanisms of these reactions are elucidated by state‐of‐the‐art quantum chemical calculations. The roles of spin density and charge distributions in HAT processes, as revealed by theory, not only deepen our mechanistic understanding of C? H bond activation but also provide important guidance for the rational design of catalysts by pointing to the particular role of doping effects.