Fibonacci Numbers and an Area Puzzle: Connecting Geometry and Algebra in the Mathematics Classroom

A mathematical puzzle that asks about “missing” area leads to an exploration of the Fibonacci sequence as well as genuine inquiry in plane geometry connected to algebra. This article discusses the inquiry, the concepts, the solution, and an extension that deepens all students’ understanding of connections between algebra and geometry.     

Electrical transport and magnetic ordering in R 2Ti3Ge4 (R=Dy, Ho and Er) compounds

New R ₂Ti₃Ge₄ (R=Dy, Ho and Er) intermetallic compounds have been synthesized and characterized by X-ray diffraction and low temperature ac magnetic susceptibility, electrical resistivity and thermoelectric power measurements were carried out. The compounds crystallize in the parent, Sm₅Ge₄-type orthorhombic structure (space group Pnma) and lanthanide contraction is observed as one moves along the rare-earth series. The changeover from paramagnetic to antiferromagnetic phase happens at low temperatures and the ordering temperature scales with the de Gennes factor. The electrical resistivity is metallic with a negative curvature above 100 K. Thermopower displays a weak maximum at temperatures less than 50 K signifying the possible phonon and magnon drag effects.     

Effect of surface photoreactions on the photocoloration of a wide band gap metal oxide: probing whether surface reactions are photocatalytic

A nonphotocatalytic reaction occurring on the surface of an irradiated wide band gap metal oxide, such as ZrO2, can affect the process of photoinduced formation of Zr3+, F- and V-type color centers. The effect of such reactions is seen as the influence of photostimulated adsorption on the photocoloration of the metal oxide specimen. In particular, photoadsorption of electron donor molecules leads to an increase of electron color centers, whereas photoadsorption of electron acceptor molecules leads to an increase of hole color centers. Monitoring the photocoloration of a metal oxide during a surface photochemical reaction probes whether the reaction is photocatalytic: accordingly, the influence of simple photoreactions on the photocoloration of ZrO2, reactions that involved the photoreduction of molecular oxygen, the photooxidation of molecular hydrogen, the photooxidation of hydrogen by adsorbed oxygen, and the photoinduced transformation of ammonia and carbon dioxide. Kinetics of the photoprocesses are reported, as well as the photoinduced chesorluminscence (PhICL effect) of ammonia. Thermoprogrammed desorption and mass spectral monitoring of the photoreaction involving NH3 identified hydrazine as an intermediate and molecular nitrogen as the final product. The photoreactions involving NH3 and CO2 are nonphotocatalytic processes, in contrast to the photooxidation of hydrogen which is photocatalytic. Carbon dioxide and carbonate radical anions are formed by interaction of CO2 with Zr3+ centers and hole states (OS-*), respectively. Mechanistic implications are discussed.     

Mechanistic studies of the formation of different states of oxygen on irradiated ZrO2 and the photocatalytic nature of photoprocesses from determination of turnover numbers

The photoadsorption of oxygen (photoreduction on electron surface-active centers) and the photoadsorption of hydrogen (photooxidation on hole surface-active centers) as well as the photooxidation of hydrogen in the presence of oxygen were examined over irradiated zirconia (ZrO2) specimens by thermoprogrammed desorption spectroscopy (TPD) and kinetically to assess the states (forms) of oxygen species formed on the surface of zirconia. The three TPD spectral maxima observed inferred three oxygen species of varying activity in the photooxidation of hydrogen. The number of surface-active sites on the zirconia surface were quantitatively estimated (ca. 1 x 10(16) centers), thereby permitting an estimate of the turnover numbers (TON) for the photooxidation of hydrogen (TON > 14.5) and for the photoreduction of oxygen (TON > 6.6). These demonstrate for the first time that a photoreaction occurring on the surface of a metal-oxide photocatalyst is truly photocatalytic.     

Elastic properties of the Ta–V system: bcc Ta0.33V0.67 and C15 TaV2

Resonant ultrasound spectroscopy was used to measure the elastic constants of bcc Ta_0.33V_0.67 over the temperature range 3.5–300?K; the results were compared to earlier measurements on C15 TaV₂. The temperature dependence of the polycrystalline shear modulus is completely different in the two phases; that of the bcc phase decreases with temperature whereas that of the C15 phases increases in an anomalous fashion. This difference is consistent with a model involving doubly-degenerate levels at the X point of the Brillouin zone in the C15 phase with the Fermi level lying near the doubly degenerate level. This model accounted for the unusual behaviour of the C15 phase. Debye temperatures were determined from the ultrasonic measurements: 295?K for the C15 phase and 315?K for the bcc phase.