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Register a new userAnalysis of the $tilde{A}-tilde{X}$ bands of the Ethynyl Radical near 1.48$mu$m and Re-evaluation of $tilde{X}$ State Energies
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We report the observation and analysis of spectra in part of the near-infrared spectrum of C$_2$H, originating in rotational levels in the ground and lowest two excited bending vibrational levels of the ground $tilde{X},^2Sigma^+$ state. In the analysis, we have combined present and previously reported high resolution spectroscopic data for the lower levels involved in the transitions to determine significantly improved molecular constants to describe the fine and hyperfine split rotational levels of the radical in the zero point, $v_2=1$ and the $^2Sigma^+$ component of $v_2=2$. Two of the upper state vibronic levels involved had not been observed previously. The data and analysis indicate the electronic wavefunction character changes with bending vibrational excitation in the ground state and provide avenues for future measurements of reactivity of the radical as a function of vibrational excitation.
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Frequency-modulated diode laser transient absorption spectra of the ethynyl radical have been recorded at wavelengths close to 1.66 $mu$m. The observed spectrum includes strong, regular, line patterns. The two main bands observed originate in the ground $tilde{X},^2Sigma^+$ state and its first excited bending vibrational level of $^2Pi$ symmetry. The upper states, of $^2Sigma^+$ symmetry at 6055.6 cm$^{-1}$ and $^2Pi$ symmetry at 6413.5 cm$^{-1}$, respectively, had not previously been observed and the data were analyzed in terms of an effective Hamiltonian representing their rotational and fine structure levels to derive parameters which can be used to calculate rotational levels up to J = 37/2 for the $^2Pi$ level and J = 29/2 for the $^2Sigma$ one. Additionally, a weaker series of lines have been assigned to absorption from the second excited bending, (020), level of $^2Sigma$ symmetry, to a previously observed state of $^2Pi$ symmetry near 6819 cm$^{-1}$. These strong absorption bands at convenient near-IR laser wavelengths will be useful for monitoring CCH radicals in chemical systems.
Transient diode laser absorption spectroscopy has been used to measure three strong vibronic bands in the near infrared spectrum of the C$_2$H, ethynyl, radical not previously observed in the gas phase. The radical was produced by ultraviolet excimer laser photolysis of either acetylene or (1,1,1)-trifluoropropyne in a slowly flowing sample of the precursor diluted in inert gas, and the spectral resolution was Doppler-limited. The character of the upper states was determined from the rotational and fine structure in the observed spectra and assigned by measurement of ground state rotational combination differences. The upper states include a $^2Sigma ^+$ state at 6696 cm$^{-1}$, a second $^2Sigma ^+$ state at 7088 cm$^{-1}$, and a $^2Pi$ state at 7110 cm$^{-1}$. By comparison with published calculations (R. Tarroni and S. Carter, textit{J. Chem. Phys} textbf{119}, 12878 (2003) and textit{Mol. Phys}. textbf{102}, 2167 (2004)), the vibronic character of these levels was also assigned. The observed states contain both $X^2Sigma^+$ and $A^2Pi$ electronic character. Several local rotational level perturbations were observed in the excited states. Kinetic measurements of the time-evolution of the ground state populations following collisional relaxation and reactive loss of the radicals formed in a hot, non-thermal, population distribution were made using some of the strong rotational lines observed. The case of C$ _{2} $H may be a good place to investigate the behavior at intermediate pressures of inert colliders, where the competition between relaxation and reaction can be tuned and observed to compare with master equation models, rather than deliberately suppressed to measure thermal rate constants.
We show that every smooth Schubert variety of affine type $tilde{A}$ is an iterated fibre bundle of Grassmannians, extending an analogous result by Ryan and Wolper for Schubert varieties of finite type $A$. As a consequence, we finish a conjecture of Billey-Crites that a Schubert variety in affine type $tilde{A}$ is smooth if and only if the corresponding affine permutation avoids the patterns $4231$ and $3412$. Using this iterated fibre bundle structure, we compute the generating function for the number of smooth Schubert varieties of affine type $tilde{A}$.
We explore the signatures of the $tilde{R}_2$ class of leptoquark (LQ) models at the proposed $e^- p$ and $e^+p$ colliders. We carry out an analysis for the proposed colliders LHeC and FCC-eh with center of mass (c.m.) energy 1.3 TeV and 3.46 TeV, respectively. For $tilde{R}_2$ class of LQ models, there are a number of final states that can arise from LQ production and its subsequent decay. In this report we do a detailed cut-based analysis for the $l^{pm}j$ final state. We also discuss the effect of polarized electron and positron beams on LQ production and in turn on $l^{pm}j$ production. At LHeC, the final state $l^+j$ has very good discovery prospect. We find that, only 100 $text{fb}^{-1}$ of data can probe LQ mass upto 1.2 TeV with $5sigma$ significance, even with a generic set of cuts. On the contrary, at FCC-eh, one can probe LQ masses upto 2.2 TeV (for $e^-$ beam) and 3 TeV (for $e^+$ beam), at more than $5sigma$ significance with luminosity $1000,text{fb}^{-1}$ and $500,text{fb}^{-1}$, respectively.
In this paper, we study the alternating Euler $T$-sums and related sums by using the method of contour integration. We establish the explicit formulas for all linear and quadratic Euler $T$-sums and related sums. Some interesting new consequences and illustrative examples are considered.
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