SICs and Algebraic Number Theory

We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert’s 12th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.     

High Carbon-Resistance Ni@CeO<Subscript>2</Subscript> Core–Shell Catalysts for Dry Reforming of Methane

Ni@CeO₂ core–shell catalysts were synthesized via a facile surfactant-assisted hydrothermal method and their catalytic performance in the dry reforming of methane (DRM) reaction was evaluated. A variety of techniques including XRD, N₂ adsorption–desorption, SEM, TEM, TPO, TGA were employed to characterize the prepared or spent catalysts. The encapsulation by the CeO₂ shell, on one side, can restrict the sintering and growth of Ni nanoparticles under harsh reaction conditions. On the other side, compared to the conventional shell material of SiO₂, CeO₂ can provide more lattice oxygens and vacancies, which is helpful to suppress coke deposition. Consequently, the Ni@CeO₂ core–shell catalysts exhibited better catalytic activity and stability in the DRM reaction with respect to the referenced Ni@SiO₂ core–shell catalysts and Ni/CeO₂ supported catalysts.     

Capping [H<Subscript>8−n</Subscript>Ni<Subscript>42</Subscript>C<Subscript>8</Subscript>(CO)<Subscript>44</Subscript>]<Superscript>n−</Superscript> (n = 6, 7, 8) Octa-carbide Carbonyl Nanoclusters with [Ni(CO)] and [CuCl] Fragments

The reactions of [Ni₁₆(C₂)₂(CO)₂₃]^4? and [Ni₃₈C₆(CO)₄₂]^6? with CuCl afforded mixtures of the previously reported [HNi₄₂C₈(CO)₄₄(CuCl)]^7? bimetallic octa-carbide cluster and the new [HNi₄₃C₈(CO)₄₅]^7? and [HNi₄₄C₈(CO)₄₆]^7? homo-metallic octa-carbides. The three species have very similar properties resulting always in co-crystals such as [NMe₄]₇[HNi_42+2xC₈(CO)_44+2x(CuCl)_1?x]·6.5MeCN (x = 0.14) (86% [HNi₄₂C₈(CO)₄₄(CuCl)]^7?, 14%[HNi₄₃C₈(CO)₄₅]^7?/[HNi₄₄C₈(CO)₄₆]^7?) and [NMe₄]₇[HNi_42+2xC₈(CO)_44+2x(CuCl)_1?x]·5.5MeCN (x = 0.30) (70% [HNi₄₂C₈(CO)₄₄(CuCl)]^7?, 30% [HNi₄₃C₈(CO)₄₅]^7?/[HNi₄₄C₈(CO)₄₆]^7?). The new homo-metallic octa-carbides can be obtained free from the Ni–Cu octa-carbido cluster by reacting [Ni₁₀(C₂)(CO)₁₆]^2? in thf with a stoichiometric amount of CuCl, and crystals of [NMe₄]₆[H₂Ni_43+xC₈(CO)_45+x]·6MeCN (x = 0.72), which contain [H₂Ni₄₄C₈(CO)₄₆]^6? (72%) and [H₂Ni₄₃C₈(CO)₄₅]^6? (28%), have been obtained. Despite the different charges and compositions, these anions display almost identical structures, which are also closely related to those previously reported for the bimetallic Ni–Cd octa-carbido clusters [Ni_42+xC₈(CO)_44+x(CdCl)]^7? and [HNi_42+xC₈(CO)_44+x(CdBr)]^6?. Indeed, all these clusters are based on the same Ni₄₂C₈ cage decorated by miscellaneous [CdX]⁺ (X = Cl, Br), [CuCl] and [Ni(CO)] fragments.