Analysis of some combination-overtone infrared bands of SO3

Several new infrared absorption bands for ³²S¹⁶O₃ have been measured and analyzed. The principal bands observed were ν₁+ν₂ (at 1561 cm⁻¹), ν₁+ν₄ (at 1594 cm⁻¹), ν₃+ν₄ (at 1918 cm⁻¹), and 3ν₃ (at 4136 cm⁻¹). Except for 3ν₃, these bands are very complicated because of (a) the Coriolis coupling between ν₂ and ν₄, (b) the Fermi resonance between ν₁ and 2ν₄, (c) the Fermi resonance between ν₁ and 2ν₂, (d) ordinary l-type resonance that couples levels that differ by 2 in both the k and l quantum numbers, and (e) the vibrational l-type resonance between the A₁^′ and A₂^′ levels of ν₃+ν₄. The unraveling of the complex pattern of these bands was facilitated by a systematic approach to the understanding of the various interactions. Fortunately, previous work on the fundamentals permitted good estimates of many constants necessary to begin the assignments and the fit of the measurements. In addition, the use of hot band transitions accompanying the ν₃ band was an essential aid in fitting the ν₃+ν₄ transitions since these could be directly observed for only one of four interacting states. From the hot band analysis we find that the A₁^′ vibrational level is 3.50 cm⁻¹ above the A₂^′ level, i.e., r₃₄=1.75236(7) cm⁻¹. In the case of the 3ν₃ band, the spectral analysis is straightforward and a weak Δk=±2, Δl₃=±2 interaction between the l₃=1 and l₃=3 substates locates the latter A₁^′ and A₂^′ “ghost” states 22.55(4) cm⁻¹ higher than the infrared accessible l₃=1 E^′ state.     

ON THE CONVERGENCE OF THE PARABOLIC APPROXIMATION OF A CONSERVATION LAW IN SEVERAL SPACE DIMENSIONS

1.IntrodnctionWegiveaproofofthestrongconvergenceinofthesolutionoftheparabolicapproximationtowardstheentropicsolutiontothescalarconservationlawwhereuo(RN),udenotessomeapproximationofuosuchthatandthefluxfsatisfiesTheconvergenceoftheapproximatesolutions...     

Coherent Raman and Infrared Studies of Sulfur Trioxide

High-resolution (0.001 cm⁻¹) coherent anti-Stokes Raman scattering (CARS) was used to observe the Q-branch structure of the IR-inactive ν₁ symmetric stretching mode of ³²S¹⁶O₃ and its various ¹⁸O isotopomers. The ν₁ spectrum of ³²S¹⁶O₃ reveals two intense Q-branches in the region 1065–1067 cm⁻¹, with surprisingly complex vibrational–rotational structure not resolved in earlier studies. Efforts to simulate this with a simple Fermi-resonance model involving ν₁ and 2ν₄ states do not reproduce the spectral detail, nor do they yield reasonable spectroscopic parameters. A more subtle combination of Fermi resonance and indirect Coriolis interactions with nearby states, 2ν₄(1=0, ±2), ν₂+ν₄(1=±1), 2ν₂(1=0), is suspected and a determination of the location of these coupled states by high-resolution infrared measurements is under way. At medium resolution (0.125 cm⁻¹), the infrared spectra reveal Q-branch features from which approximate band origins are estimated for the ν₂, ν₃, and ν₄ fundamental modes of ³²S¹⁸O₃, ³²S¹⁸O₂¹⁶O, and ³²S¹⁸O¹⁶O₂. These and literature data for ³²S¹⁶O₃ are used to calculate force constants for SO₃ and a comparison is made with similar values for SO₂ and SO. The frequencies and force constants are in excellent agreement with those obtained by Martin in a recent ab initio calculation.     

High-Resolution Infrared Spectra of the ν2, ν3, ν4, and 2ν3 Bands of SO3

New measurements are reported for the infrared spectrum of sulfur trioxide, ³²S¹⁶O₃, with resolutions ranging from 0.0015 cm⁻¹ to 0.0025 cm⁻¹. Rovibrational constants have been measured for the fundamentals ν₂, ν₃, and ν₄ and the overtone band 2ν₃. Comparisons are made with the earlier high-resolution measurements on SO₃, and the high correlation among some of the constants related to the Coriolis coupling of the ν₂ and ν₄ levels is discussed in order to understand the areas of disagreement with the earlier work. Splittings of some of the levels are observed and the splitting constant for K=3 of the ground state is determined for the first time. Other observed splittings include the K=1 levels of 2ν₃ (l=2), the K=2 levels of ν₃ and ν₄, and the K=3 levels of ν₂. The analysis shows that there are level crossings between the l=0 and l=2 states of 2ν₃ that allow one to determine the separation of the subband centers for these two states even though access to the l=0 state from the ground state is electric-dipole forbidden. This is a generalized phenomenon that should be found for many other molecules with the same symmetry. The l-type resonance constant, q₃, that causes the splitting of the l₃=±1, k=±1 levels of ν₃ also couples the l₃=0 and 2 states of 2ν₃.     

ON LACUNARY STATISTICAL CONVERGENCE OF ORDER α

In this article, we introduce the concept of lacunary statistical convergence of order α of real number sequences and give some inclusion relations between the sets of lacunary statistical convergence of order α and strong Nαθ(p)-summability. Furthermore, some relations between the spaces Nαθ(p) and Sαθare examined.     

Measurement of CrO in flames by cavity ringdown spectroscopy

CrO is an important intermediate in the high temperature oxidation chemistry of chromium containing species. This work reports the first detection of CrO in a flame. The B⁵-X⁵ electronic transition was probed by cavity ringdown spectroscopy (CRDS) in a lean (=0.38), low-pressure, flat, laminar hydrogen-oxygen-argon flame seeded with Cr(CO)₆. The previous B⁵-X⁵ CrO spectrum of Hocking et al. (605.0 nm-606.5 nm) is extended from the band head located at 605.6 nm to 614.4 nm. The temperature profiles of the doped and undoped flames were obtained from measurements of OH laser- induced fluorescence. Seeding the flame with Cr(CO)₆ increased the flame temperature by approximately 150 K. The concentration profile of CrO was measured as a function of height above the burner. CrO absorption signals were converted to concentration using the measured temperature profile and absorption cross-section calculated from lifetimes by Hedgecock et al. A lower limit peak CrO concentration of 1.6 ppb was found in the flame. Some uncertainty in the cross-section remains. Comparisons to calculations using STANJAN indicate that CrO is present in flames at super equilibrium concentrations. PACS 82.33.Vx; 42.62.Fi     

SOME GEOMETRIC PROPERTIES OF A NEW DIFFERENCE SEQUENCE SPACE INVOLVING LACUNARY SEQUENCES

In this paper, we define a new generalized difference sequence space involving lacunary sequence. Then, we examine k-NUC property and property （β） for this space and also show that it is not rotund where p = （pr） is a bounded sequence of positive real numbers with pr ≥ 1 for all r ∈ N.     

Analysis of high-resolution infrared and CARS spectra of SO3

Three fundamental modes and several hot bands of ³⁴S¹⁸O₃ have been investigated using both infrared spectroscopy and coherent anti-Stokes Raman scattering spectroscopy (CARS). Coriolis coupling effects are particularly noticeable in ³⁴S¹⁸O₃ due to the close proximity of the ν₂ and ν₄ fundamental vibrations, whose wavenumber values are 477.50864(5) and 502.05565(4) cm⁻¹. The uncertainties in the last digits are shown in parentheses and are two standard deviations. Hot band transitions from ν₂, ν₄ levels give access to infrared inactive ν₂, ν₄ combination/overtone levels which interact strongly with levels of the Raman-active ν₁ symmetric stretching mode due to indirect Coriolis couplings, l-resonances, and Fermi resonances. The result is a complex ν₁ CARS Q-branch spectrum that is the most perturbed of the four SO₃ isotopomers we have studied. The relative importance of these interaction terms on the ν₁ CARS spectrum is examined in some detail and accurate rovibrational constants are determined for all of the mixed states, leading to deperturbed values of 1004.662(24), 0.0003503(9), and 0.0007066(12) cm⁻¹ for ν₁, α₁^B, and α₁^C, respectively. The B_e value is found to be 0.310875(12) cm⁻¹, which gives an equilibrium bond length r_e of 141.7339(3) pm, in excellent agreement with values of 141.7340(1) and 141.7347(7) pm reported earlier for ³²S¹⁶O₃ and ³⁴S¹⁶O₃.