Triangles II: Complex triangle coordinates

Summary This paper is the second in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first paper of the series, we examined triangle shapes. In this paper, we coordinatize the Euclidean plane using cross ratios, and use these triangle coordinates to prove theorems about triangles. We develop a complex version of Ceva's theorem, and apply it to proofs of several new theorems. The remaining paper of this series will deal with complex triangle functions.     

A note on the stress singularity at a penny-shaped crack in three-dimensional electrostriction

Zusammenfassung Diese Mitteilung behandelt eine vollständige Bewertung der Spannungs-Singularität am Rande eines flachen kreisförmigen Spaltes unter der Einwirkung von Elektrostriktion. Die Ergebnisse bestätigen die Schlussfolgerungen, welche früher bei zweidimensionalen Problemen gezogen wurden.Die der Elektrostriktion zuzuschreibende Singularität zeitigt eine rein radiale Spannung, welche nicht dazu neigt, die schon bestehenden Spalten ihrer Ausdehnungs-Oberfläche entlang auszubreiten.

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Research supported by the National Science Foundation under grant NSF GP 22567.

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Research supported by the United States Atomic Energy Commission.     

Matrix reactions of alkaline earth metal atoms with ozone : Infrared spectra of the metal ozonide and metal oxide molecules

The 15°K deposition of alkaline earth metal atoms and ozone molecules at high dilution in argon yielded intense bands near 800 cm⁻¹ and in the region, 450–650 cm⁻¹. The bands near 800 cm⁻¹ showed the appropriate oxygen isotopic shifts for assignment to ν₃ of the ozonide ion; the use of scrambled isotopic ozones indicated that the metal cation is symmetrically bound to the ozonide anion which contains three oxygen atoms with two equivalent oxygens. For the case of Ca and Ba atoms and ozone, infrared absorptions appeared between 450–650 cm⁻¹ which showed appropriate oxygen isotopic shifts for vibrational assignment to several metal oxide species. In the calcium experiments, bands at 635.7 and 575.5 cm⁻¹ which showed diatomic oxygen-18 isotopic shifts were tentatively identified as (CaO)₂ species; a pair of bands at 593.0 and 592.2 cm⁻¹ were tentatively assigned to CaO₂. For the barium reactions, bands at 634.7, 571.3, and 460.0 showed appropriate oxygen-18 frequency shifts for assignment to BaO, BaO₂, and (BaO)₂, respectively. The BaO assignment was confirmed by the N₂O-nitrogen matrix reaction which yielded a nitrogen matrix counterpart for BaO at 612.4 cm⁻¹.     

Nonrelativistic dyon-dyon scattering

The nonrelativistic problem of the scattering of two dyons (including the case of electron scattering by magnetic monopoles) is systematically studied, both classically and quantum mechanically, with a view toward the discrimination between various combinations of electric and magnetic charges. We analyze the classical cross section with particular attention to the interesting phenomena which occur for large angle scattering, the “rainbows” and “glory,” where the cross section becomes infinite. Quantum mechanically, we find that these infinities do not occur and that, when the partial wave scattering amplitude is summed, a very elaborate structure emerges for the cross section, which depends sensitively upon the electric and magnetic charges of the particles, as well as on their relative speed. We further discuss a large modification, leading to spin flip and nonflip amplitudes, due to the dipole moments of the particles. Numerical results are presented for a variety of values of these parameters. In principle, these results could be used to distinguish the δ-ray distributions produced by the various species of electrically and magnetically charged particles. Quite apart from the experimental implications of our numerical results, we have made a number of theoretical improvements and extensions. Numbered among these are the consideration of dyons and particles having dipole moments, and the explicit demonstration, based on the methods of angular momentum, that the differential cross section is independent of the choice of singularity line.     

One-dimensional BeH2 polymers: infrared spectra and theoretical calculations

Laser-ablated beryllium atoms react with H2 upon co-condensation in excess hydrogen and neon to form BeH2 and (BeH2)2, which are identified through isotopic substitution and DFT calculations. Unreacted Be atoms isolated in solid neon or hydrogen are excited to the 1P0 state and react further with H2 to enhance the BeH2 and (BeH2)2 concentrations and produce (BeH2)n polymers. The series of strong infrared-active parallel Be-H-Be bridge-bond stretching modes observed for (BeH2)n polymers suggests one-dimensional structures, and this conclusion is supported by DFT calculations. The computed polymerization energy per BeH2 unit is about 33 kcal/mol.     

Matrix infrared spectra and density functional theory calculations of molybdenum hydrides

Laser-ablated Mo atoms react with H2 upon condensation in excess argon, neon, and hydrogen. The molybdenum hydrides MoH, MoH2, MoH4, and MoH6 are identified by isotopic substitution (H2, D2, HD, H2 + D2) and by comparison with vibrational frequencies calculated by density functional theory. The MoH2 molecule is bent, MoH4 is tetrahedral, and MoH6 appears to have the distorted trigonal prism structure.     

FTIR spectroscopic studies of the matrix photoionization and photolysis products of methylene halides

Matrix photoionization of methylene bromide produced absorptions at 1019, 897, and 788 cm^?1 identified previously as CBr₂⁺, CHBr₂⁺, and CHBr₂. High-resolution FTIR spectra revealed overlapping 1/2/1 triplets for natural bromine isotopes with individual linewidths near 0.2 cm^?1. New absorptions at 3121, 2897, and 1345 cm^?1 are assigned to the (CH₂Br⁺)Br cation complex which yields CHBr₂⁺ on photolysis. A substantially increased yield of the CHCl₂⁺ species made possible observation of the CH stretching mode at 3033 cm^?1 and the symmetric CCl₂ stretching mode at 845 cm^?1 along with the previously observed stronger 1291- and 1044-cm^?1 fundamentals. The high resolution and enhanced signal-to-noise capability of the FTIR are clearly demonstrated in this investigation.