Analysis of the ν1 and ν2 + ν4 bands of 12CH4

Line assignments, positions, strengths, and experimentally determined upper states are reported for the ν₁ and ν₂ + ν₄ infrared bands of ¹²CH₄. The bands have been analyzed using infrared spectra recorded with different optical densities and temperatures at 0.02- and 0.01-cm^?1 resolution using a four-passed grating spectrometer at Florida State University and a Fourier transform spectrometer at Kitt Peak National Observatory. Both ν₁ and ν₂ + ν₄ lines are assigned through J′ = 14. For J′ ≥ 10, the upper state of the vibrationally infrared inactive ν₁ band interacts strongly with ν₂ + ν₄ and is observed in the infrared spectrum with line strengths on the order of 10^?3 cm^?2 atm^?1. The upper-state energies and transition intensities are calculated from the molecular constants and transition moment matrix elements obtained through a simultaneous analysis of ν₁, ν₃, ν₂ + ν₄, 2ν₄, and 2ν₂.     

Magnetic ordering in Fe<Subscript>1.087</Subscript>Te under pressure

The crystal and magnetic structures of Fe_1.087Te have been studied by neutron powder diffraction in the temperature range from 1.7 to 80 K at pressures of  ≈0.4 and ≈1.2 GPa. No symmetry change of the tetragonal paramagnetic ambient pressure phase (space group P4/nmm) was observed for temperatures above 60 K and pressures up to  ≈1.2 GPa. A novel pressure-induced phase of Fe_1.087Te having orthorhombic symmetry (space group Pmmn) and incommensurate antiferromagneticbicollinear order was observed in the temperature range from 50 to 60 K at  ≈1.2 GPa. The known monoclinic ambient pressure phase of Fe_1.087Te (space group P2 ₁/n) with commensurate antiferromagnetic order was found to be stable up to at least  ≈1.2 GPa at low temperature.     

Perturbation of the Laplacian supported by null sets,with applications to polymer measures and quantum fields

We discuss hamiltonians in L²(R^d, dx) of the form H = ?Δ + V, with V a potential supported by a zero measure set C. In particular if C is a path of a brownian motion b such that V(x) = ∫₀¹λ(x, ω)δ(x-b(s, ω)) ds, we show that H exists as a nontrivial, self-adjoint, lower bounded perturbation of ?Δ when d ?5. We must choose λ to be an infinitesimal, negative function for d = 4,5, but for d ? 3 any bounded real-valued function λ will do. The connection with Edward's model of polymers as well as with quantum fields of the ?_d⁴-type is also discussed. The proofs use methods of nonstandard analysis.     

A theoretical model for the interacting upper states of the ν1, ν3, 2ν2, ν2 + ν4, and 2ν4 bands in methane

The effective vibration-rotation Hamiltonians complete to fourth order in the Amat-Nielsen scheme for the upper states of the ν₁, ν₃, 2ν₂, ν₂ + ν₄, and 2ν₄ bands in methane are reviewed, and the major vibration-rotation interactions (H₃₀, $\text{H}\text{?}{}_{40}$, $\text{H}\text{?}{}_{21}$, $\text{H}{}_{31}$, $\text{H}\text{?}{}_{22}$) connecting the different vibrational states are discussed. Explicit matrix elements in a basis of harmonic oscillator-symmetric rotor basis functions are given for the purely vibrational terms and for the vibration-rotation interactions. Expressions for spectral intensities of infrared and Raman spectra are presented, and the selection rules and transition moment matrix elements are stated. A computer program is described which, incorporating all these features, can be used for prediction of infrared and Raman spectra and for determination of molecular constants from observed spectra by a least-squares routine. As an example the program is applied to the 2ν₄ isotropic Raman spectrum of ¹²CH₄, leading to a very good agreement between the experimental and calculated spectra.     

Measurement and analysis of the infrared absorption spectrum of the 2ν2 band of 12CH4

Approximately 500 infrared absorption lines with room-temperature strengths between 3 × 10^?5 and 1 × 10^?2 atm^?1 were assigned to the 2ν₂ band of ¹²CH₄ in the region from 2930 to 3250 cm^?1. These determine 207 of the 212 upper-state energy levels through J′ = 12 as well as a number of levels with J′ = 13 and 14. All but 17 of the levels with J′ ≤ 12 are calculated to 0.03 cm^?1 or better on the basis of a Hamiltonion that contains Coriolis and Fermi interaction terms coupling the upper states of the five bands, 2ν₄, ν₂ + ν₄, ν₁, ν₃, and 2ν₂.     

The ν2 + ν4 and 2ν2 isotropic Raman bands of 12CH4

The purely isotropic Raman spectrum of the ν₁ band, the ν₂ + ν₄ band (enhanced through interaction with ν₁), and the 2ν₂ band of ¹²CH₄ was obtained with a spectral resolution of 0.30–0.35 cm^?1 from exposures with different orientations of the linearly polarized exciting light. The ν₂ + ν₄ and 2ν₂ bands show partially resolved rotational structure. The spectra are interpreted in terms of a model which takes explicitly into account vibrational and rovibrational interactions with other vibrational states, using molecular constants determined primarily from infrared spectra. The computed contours are in excellent agreement with the experimental ones and the observed and calculated peak wavenumbers agree within one tenth of the spectral resolution limit, except for a small region near the ν₁ band. The good overall agreement represents an independent check on the overall correctness of the previously reported molecular constants. A detailed discussion is given of the contributions to the intensities of individual transitions from the three transition moment matrix elements, which in an isolated-band model are the intensity parameters of the ν₁, 2ν₄, and 2ν₂ isotropic bands, respectively.     

Some Raman bands of C3O2

The ν₁, ν₅, 2ν₅, and 2ν₆ Raman band accumulations of carbon suboxide, C₃O₂, have been photographed with a resolution of 0.2–0.3 cm^?1. Each band accumulation consists, in addition to the main band, of a large number of “hot” bands due to the extremely low, highly anharmonic ν₇ fundamental vibration. In the 2ν₆ band accumulation a few series of unresolved Q branches have been assigned. In the ν₁ and 2ν₅ band accumulations most Q branches almost coincide, forming a very intense peak, whereas the dominating feature of the ν₅ band accumulation is a minimum, in agreement with the expectation of an extremely weak Q branch for a Π_g fundamental band. Tentative values of ν₁ = 2196.9 ± 0.1 cm^?1 and ν₅ = 580.2 ± 0.5 cm^?1 as well as several energy values in the ν₇ manifold of the 2ν₆⁰ state are obtained. Further, improved exposures of the ν₂ + 2ν₇⁰ band accumulation yield some levels in the ν₇ manifold of the ν₂ state, in addition to those determined previously.     

The pentad rotation-vibrational states of 12CD4: Wavenumber analysis of infrared and Raman spectra of the ν1, ν3, 2ν2, ν2 + ν4, and 2ν4 bands

The so-called pentad of ¹²CD₄ consists of the vibrational states v₁ = 1(symmetry A₁), v₃ = 1(F₂), v₂ = 2(A₁ + E), v₂ = v₄ = 1(F₁ + F₂), and v₄ = 2(A₁ + E + F₂). All states are located in the 1950 to 2250-cm^?1 region and all are strongly interacting. In the present work we have assigned more than 5000 infrared rotation-vibrational transitions and 163 isotropic Raman transitions from the vibrational ground state to the pentad. We have used infrared and Raman spectra of a resolution better than 0.01 cm^?1. From the experimental wavenumbers 2567 pentad rotation-vibrational energy levels with J ≦ 20 have been determined. These levels are reported in the paper. The levels have been used for refinements of the spectroscopic constants of two physically different effective Hamiltonians for the pentad states. For all levels with J ≦ 12 an unweighted standard deviation of 0.004 cm^?1 is obtained for both Hamiltonians, whereas the standard deviation increases more or less rapidly with J above 12 due to the imperfections of the Hamiltonians. The values of the spectroscopic constants of both Hamiltonians (85 and 106, respectively) are reported and the effects of the approximations are discussed.     

Molecular constants for the interacting upper states of the ν1, ν3, 2ν2, ν2 + ν4, and 2ν4 bands in 12CH4

In a previous paper (J.-E. Lolck and A. G. Robiette, J. Mol. Spectrosc.88, 14 (1981)) a theoretical model for the interacting upper states of the ν₁, ν₃, 2ν₂, ν₂ + ν₄, and 2ν₄ bands in methane was described. The present paper summarizes the results obtained, using this model, in a comprehensive analysis of the five bands of ¹²CH₄ through J′ = 12. Values of 80 molecular constants, of which 17 correspond to vibrationally off-diagonal operators, are reported. In addition the computed energy levels of the v₃ = 1 state are compared to the experimental ones and to the result of the previous isolated band approach.     

Formation of γ-Fe2O3 nanoparticles and vacancy ordering: An in situ X-ray powder diffraction study

The formation of maghemite, γ-Fe₂O₃ nanoparticles has been studied by in situ X-ray powder diffraction. The maghemite was formed by thermal decomposition of an amorphous precursor compound made by reacting lauric acid, CH₃(CH₂)₁₀COOH with Fe(NO₃)₃·9H₂O. It has been shown that cubic γ-Fe₂O₃ was formed directly from the amorphous precursor and that vacancy ordering starts about 45 min later at 305 °C resulting in a tripled unit cell along the c-axis. The kinetics of grain growth was found to obey a power law with growth exponents n equal to 0.136(6) and 0.103(5) at 305 and 340 °C, respectively. Particles with average sizes of 12 and 13 nm were obtained in 86 and 76 min at 305 and 340 °C, respectively. The structure of cubic and vacancy ordered phases of γ-Fe₂O₃ was studied at 305 °C by Rietveld refinements.