An upper bound for the minimum diameter of integral point sets

For alln> d there existn points in the Euclidean spaceE ^d such that not all points are in a hyperplane and all mutual distances are integral. It is proved that the minimum diameter of such integral point sets has an upper bound of 2 ^{c logn log logn} .     

Lattice Defects and Oxygen Absorption/Migration in Ceria/Ceria-Zirconia Solid Solutions: Analysis by Semiempirical Interacting Bonds Method

Semiempirical Interacting Bonds Method was used to calculate the enthalpies of oxygen removal from the bulk of ceria and ceria-zirconia solid solutions. The energy of the bulk reduction was obtained to agree reasonably with the experimental data when the complex defect comprised of the oxygen vacancy and interstitial oxygen atom surrounded by six cations was suggested to emerge due to an oxygen removal from the bulk. The zirconium cations dissolved in ceria affect the energies of the bulk reduction and bulk oxygen mobility due to a decrease in the local Me-O coordination number from 8 to 7.     

Rotation alignment in 190Pt and 192Pt

Levels with spins up to 18 (20) in the ground-state bands of ¹⁹⁰Pt and ¹⁹²Pt have been observed. The very small energy spacing between the 12⁺ and 10⁺ levels in both nuclides is interpreted as a sudden decoupling of two i_{$\text{13}\text{2}$} neutrons from the collective core.     

Synthesis and Crystal Structure of RbKLiAlF6 – the First Al‐Elpasolite with Three Different Alkali Metals

The synthesis of RbKLiAlF₆ via a sol‐gel route is described. The crystal structure has been determined using Rietveld‐refinement. It is shown that the compound crystallizes in the hexagonal 12L‐structure of the Cs₂NaCrF₆‐type (space‐group ) with a = 5.7195Å, c = 27.8655(4) Å and therefore belongs to the family of hexagonal elpasolites being the first example for an Al‐elpasolite with three different alkali metals. Further insights regarding structural details could be gained using ⁸⁷Rb MAS NMR spectroscopy.     

Structure of (NH4)3GaF6 investigated by multinuclear magic-angle spinning NMR spectroscopy in comparison with rietveld refinement

The structure of ammonium gallium cryolite (NH(4))(3)GaF(6) was investigated by (19)F and (69,71)Ga magic-angle spinning (MAS) NMR in comparison with X-ray powder diffraction followed by Rietveld refinement. In agreement with previous thermodynamic measurements, NMR experiments on (NH(4))(3)GaF(6) support the model of rigid GaF(6) octahedra. At high spinning speeds (30 kHz), the scalar coupling between the six equivalent (19)F nuclei and (69,71)Ga can be directly observed in the powder spectra. The coupling constants are J(19)F(69)Ga = 197 Hz and J(19)F(71)Ga = 264 Hz. To explain the (71)Ga spectra recorded at 3 kHz a small distribution of quadrupolar frequencies has to be included. The spread of the spinning sidebands hints to a largest nu(Q) value of 28 kHz for (71)Ga. This can be explained by the occurrence of highly symmetric GaF(6) octahedra, which are tilted against the surrounding atoms. In addition, the incomplete motional excitation does not average out the quadrupolar effects. NMR findings are in discrepancy to those of Rietveld refinement. As result it appears that X-ray diffraction is not sensitive enough to deliver proper results.     

Six IPR isomers of C90 fullerene captured as chlorides: carbon cage connectivities and chlorination patterns

Three C(90) fractions were isolated by multi-step HPLC from fullerene soot obtained from direct current (DC) arc discharge of undoped graphite rods. C(90) of each fraction was chlorinated with VCl(4) or SbCl(5) in ampoules at 290-310 °C, affording a series of C(90)Cl(n) compounds. Single-crystal X-ray crystallography with the use of synchrotron radiation resulted in structure elucidation of seven C(90)Cl(n) compounds containing six different isolated pentagon rule (IPR) C(90) cages (the number of the C(90) isomer is given in the parentheses): C(90)(46)Cl(32) (I), C(90)(34)Cl(32) (II), C(90)(35)Cl(24) (III), and C(90)(35)Cl(28) (IV), C(90)(32)Cl(24) (V as co-crystals with III), co-crystals of C(90)(30)Cl(22) and C(90)(28)Cl(24) (VI), and C(90)(28)Cl(24) (VII). Cage connectivities of C(90) isomers 35 and 28 have been crystallographically confirmed for the first time. The chlorination patterns of the C(90)Cl(n) molecules are discussed in terms of the formation of isolated aromatic systems and isolated C=C double bonds on the fullerene cage. The distribution of six C(90) isomers in three HPLC fractions is compared with data from the literature.     

[1,1,t]-Colorings of Complete Graphs

Given non-negative integers $r, s,$ and $t,$ an $[r,s,t]$ -coloring of a graph $G = (V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the color set $\{1,\ldots ,k\}$ such that $\left|c(v_i) - c(v_j)\right| \ge r$ for every two adjacent vertices $v_i,v_j, \left|c({e_i}) - c(e_j)\right| \ge s$ for every two adjacent edges $e_i,e_j,$ and $\left|c(v_i) - c(e_j)\right| \ge t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$ -chromatic number $\chi _{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an $[r,s,t]$ -coloring. In this note we examine $\chi _{1,1,t}(K_p)$ for complete graphs $K_p.$ We prove, among others, that $\chi _{1,1,t}(K_p)$ is equal to $p+t-2+\min \{p,t\}$ whenever $t \ge \left\lfloor {\frac{p}{2}}\right\rfloor -1,$ but is strictly larger if $p$ is even and sufficiently large with respect to $t.$ Moreover, as $p \rightarrow \infty $ and $t=t(p),$ we asymptotically have $\chi _{1,1,t}(K_p)=p+o(p)$ if and only if $t=o(p).$      

Synthesis and molecular structures of heptafluoroisopropylated fullerenes: C60(i-C3F7)8, C60(i-C3F7)6, and C60(CF3)2(i-C3F7)2

One isomer of C₆₀(i-C₃F₇)₈, three isomers of C₆₀(i-C₃F₇)₆, and the first mixed perfluoroalkylated fullerene, C₆₀(CF₃)₂(i-C₃F₇)₂, have been isolated by HPLC from a mixture prepared by reaction of C₆₀ with heptafluoroisopropyl iodide in a glass ampoule at 260-290 °C. The molecular structures of the four new compounds have been determined by means of X-ray single crystal diffraction partially also by use of synchrotron radiation. Theoretical calculations at the DFT level of theory have been employed to rationalize the energetics of isomers and of C₆₀-R_f binding.