Even circuits in planar graphs

We prove that a planar graph can be partitioned into edge-disjoint circuits of even length, if and only if every vertex has even valency and every block has an even number of edges.     

Low-temperature phases of the solid electrolyte Ag26I18W4O16

Single crystal X-ray diffraction photographs taken with a Buerger precession camera, at temperatures 250, 214, and 122 K, corroborate the existence of three low-temperature phases of Ag₂₆I₁₈W₄O₁₆. These phases are labeled α′, β, and γ in order of decreasing temperature. The α′ phase is monoclinic, space group P2₁, Z = 2; the β phase is triclinic, space group $\text{P}\text{1}$ or P1, Z = 2; and the γ phase is triclinic, space group P1, Z = 1. Lattice constants at the aforementioned temperatures are given. Twins in the β and γ phases are related by the albite and pericline laws, as are twins in the feldspars. The highest symmetry known to be attained by the (W₄O₁₆)^8? entity is 2(C₂), which, strictly, it must lose at the transition to the α′ phase.     

Knight shifts in binary liquid alloys of equi-valent metals

The N.M.R. Knight shift depends upon the conduction electron density at the nucleus under observation. An interpretation of its absolute magnitude in liquid metals cannot be undertaken in terms of a pseudopotential formulation, but relative changes of shift with some parameter such as temperature, volume or alloy composition can be investigated. Knight shift changes with concentration have been measured for a series of binary liquid alloys. Since application of a pseudopotential treatment is simplest when the two constituents have the same valency, the alloys chosen have mostly been of this type, e.g. In-Ga, In-Tl, Sn-Pb, Sb-Bi. Although theory predicts the correct qualitative behaviour of the shift changes, the precise theoretical predictions are very sensitive to details of both the pseudopotentials and the liquid structure factor. In the case of In-Sn and In-Pb, where the shifts of both constituents decrease with increasing indium content for In-Sn, but increase for In-Pb, the difference between the pseudopotentials of tin and lead does not appear to be sufficient to explain the observed differences in changes of shift.     

Characterisation of Ba2In2 − x Sn x O5 + x/2 oxide ion conductors

The Ba₂In_2 − x Sn _x O_5 + x/2 solid solution was confirmed up to x = 1 by solid-state reaction. X-ray diffraction at room and at elevated temperatures, Raman scattering and impedance spectroscopy were used to characterise the samples. The structure refinement of the composition x = 0.1 from neutron diffraction data reveals that tin is preferentially located in the tetrahedral layers of the brownmillerite. Paper presented at the 11th EuroConference on the Science and Technology of Ionics, Batz-sur-Mer, Sept. 9–15, 2007     

Cycles in dense digraphs

Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let β(G) denote the size of the smallest subset X ? E(G) such that G?X has no directed cycles, and let γ(G) be the number of unordered pairs {u, v} of vertices such that u, v are nonadjacent in G. It is easy to see that if γ(G) = 0 then β(G) = 0; what can we say about β(G) if γ(G) is bounded?

We prove that in general β(G) ≤ γ(G). We conjecture that in fact β(G) ≤ ½γ(G) (this would be best possible if true), and prove this conjecture in two special cases:

• when V(G) is the union of two cliques
• when the vertices of G can be arranged in a circle such that if distinct u, v, w are in clockwise order and uw is a (directed) edge, then so are both uv, vw.

     

Electron super-rich radicals in the gas phase. A neutralization-reionization mass spectrometric and ab initio/RRKM study of diaminohydroxymethyl and triaminomethyl radicals

Diaminohydroxymethyl (1) and triaminomethyl (2) radicals were generated by femtosecond collisional electron transfer to their corresponding cations (1+ and 2+, respectively) and characterized by neutralization-reionization mass spectrometry and ab initio/RRKM calculations at correlated levels of theory up to CCSD(T)/aug-cc-pVTZ. Ion 1+ was generated by gas-phase protonation of urea which was predicted to occur preferentially at the carbonyl oxygen with the 298 K proton affinity that was calculated as PA = 875 kJ mol-1. Upon formation, radical 1 gains vibrational excitation through Franck-Condon effects and rapidly dissociates by loss of a hydrogen atom, so that no survivor ions are observed after reionization. Two conformers of 1, syn-1 and anti-1, were found computationally as local energy minima that interconverted rapidly by inversion at one of the amine groups with a <7 kJ mol-1 barrier. The lowest energy dissociation of radical 1 was loss of the hydroxyl hydrogen atom from anti-1 with ETS = 65 kJ mol-1. The other dissociation pathways of 1 were a hydroxyl hydrogen migration to an amine group followed by dissociation to H2N-C=O* and NH3. Ion 2+ was generated by protonation of gas-phase guanidine with a PA = 985 kJ mol-1. Electron transfer to 2+ was accompanied by large Franck-Condon effects that caused complete dissociation of radical 2 by loss of an H atom on the experimental time scale of 4 mus. Radicals 1 and 2 were calculated to have extremely low ionization energies, 4.75 and 4.29 eV, respectively, which belong to the lowest among organic molecules and bracket the ionization energy of atomic potassium (4.34 eV). The stabilities of amino group containing methyl radicals, *CH2NH2, *CH(NH2)2, and 2, were calculated from isodesmic hydrogen atom exchange with methane. The pi-donating NH2 groups were found to increase the stability of the substituted methyl radicals, but the stabilities did not correlate with the radical ionization energies.