A critique of the term value approach to determining molecular constants from the spectra of diatomic molecules

Two alternative methods for the reduction of observed line positions to spectroscopic constants are compared. In the direct approach, the line positions are fit directly to expressions for the energy level differences in which the upper and lower state molecular constants enter as unknowns. In the traditional term value approach, a two-step procedure is followed where first a set of intermediate unknowns, the upper and lower state term values are calculated from the observed line positions and secondly these term values for each vibrational state, are fit separately to energy expressions in which the molecular constants of that state enter as unknowns. It is shown that there is correlation between the term values of all vibrational states included in the reduction. Thus, the separate fitting of term values to energy level expressions for each vibrational state does not retain this correlation and the molecular constants found in this manner cannot be the minimum-variance, linear, unbiased (MVLU) estimates and the accompanying standard deviations cannot be unbiased estimates. However, if the molecular constants of all the vibrational levels are determined simultaneously from all the term values in a correlated least-squares fit, then the molecular constants will be equal to those determined from the direct approach, i.e., the MVLU set. Furthermore, missing lines and the frequent occurrence of two independent sets of lines within a band both cause the molecular constants obtained from the traditional term value approach to differ even more from the MVLU set. Finally, the limitations of reducing groups of bands simultaneously are discussed. Certain singlet systems are given as examples of the above conclusions.     

Influence of the angle-dependence of mode coupling on optical power distribution in step-index plastic optical fibers

Determination of the power distribution in step index plastic optical fibers by the power flow equation has been reported in the literature both with and without the simplifying assumption of constant coupling that is independent of the angle of light propagation. The need for this assumption is evaluated in this paper. Results with the angle-dependent coupling coefficient are compared to those derived under the simplifying assumption of constant coupling. Benchmarked to values measured experimentally, this comparison covered the coupling lengths L_c (denoting where the equilibrium mode distribution is achieved in the fiber) and lengths z_s (for achieving the steady-state distribution). Results differ slightly but only for longer fiber lengths, thus largely vindicating the simplifying assumption of constant coupling.     

Determination of rovibronic term values of a diatomic molecule from experimental data on the wavenumbers of spectral lines

It has been shown that, if experimental data are available on the lines of the rovibronic spectrum of a diatomic molecule that pairwise couple three or more vibronic states, then there is a relation between the wavenumbers measured for the lines and the absolute values of all corresponding rovibronic term values. A method is proposed for determining the optimal set of rovibronic term values with error estimations by varying the desired values of the rovibronic term values so as to minimize the weighted standard deviation of the wavenumbers that are calculated according to the Rydberg-Ritz combination principle from the wavenumbers measured for a sufficiently large number of rovibronic spectral lines of various systems of bands. It is substantial that this method does not require any assumptions on the molecular structure. The new method is applied to determine the rovibronic term values of 12 singlet electronic states of the isotopomer ¹¹B¹H of the boron hydride molecule, which have already been studied experimentally, with the use of all available experimental data for 15 systems of bands, which are obtained in 10 works. Thus, the energy spectrum of the diatomic molecule has been determined from the experimental data on the wavenumbers of rovibronic spectral lines without any assumption on its structure.     

Rotational analysis of the ultraviolet bands of the antimony monofluoride molecule

A rotational analysis is performed on several bands of the ultraviolet systems of the SbF molecule obtained at high resolution in emission and in absorption. The results confirm the electronic term value diagram proposed earlier [J. Mol. Spectrosc.49, 377 (1974)] and yield molecular constants for the first excited state a2 and for the highly excited C₃1 state of the molecule.     

Fundamental electrical standards and the quantum metrological triangle

The advent of the fundamental constants R_K (the von Klitzing constant) and K_J (the Josephson constant) in electrical metrology and the growing development of nanotechnologies have totally changed the vision and the practice of the National Metrology Institutes (NMIs), opening a modern era of metrology and arousing a growing interest in a possible re-definition of the international system of units (SI). The Josephson effect (JE) and the Quantum Hall effect (QHE), at the origin of these fundamental constants, constitute the keystone of a new approach to electrical units, when one considers the very high level of reproducibility of these units, never seen before. On the other hand, the Watt balance experiment in which these constants play a part could be the origin of a new SI definition, replacing the mass unit ‘the kilogram’ as a fundamental unit by the Planck constant h. It thus seems that the implementation of experiments aimed at demonstrating the coherency between the theoretical and phenomenological values of these constants is a major objective. In this framework the metrological triangle experiment associating QHE, JE and single electron tunnelling effect would play a major role in checking the consistency of these fundamental constants in terms of the Planck and electron charge constants. This article gives briefly an outline of these quantum phenomena and their metrological applications in NMIs for the realisation of electrical units and the determination of the fundamental constants. To cite this article: F. Piquemal et al., C. R. Physique 5 (2004).     

Use of calculated centrifugal distortion constants (Dν, Hν, Lν and Mν) in the analysis of the B ← X system of I2

Fourier spectroscopy measurements of 14 703 lines from the B-X band system of I₂ are analyzed using centrifugal distortion constants calculated from RKR curves by the perturbation theory method of Hutson [J. Phys. B14, 851–857 (1981)]. Only vibrational and rotational constants are varied in the least-squares fits, with the centrifugal distortion constants fixed at the calculated values. The fit obtained in the present work is as good as when the centrifugal distortion constants are treated as independent parameters, but the molecular constants obtained are physically more meaningful because the statistical correlation between parameters is reduced. The analysis includes B-state vibrational levels up to v′ = 72, which is bound by only 0.5% of the well depth; the method used for calculating centrifugal distortion constants remains reliable even this close to dissociation. It is found that centrifugal distortion constants up to fifth order make significant contributions to the observed line positions.     

Revised molecular constants and term values for the XΣ and AΠ states of NH

The vibration–rotation spectra of NH have been reinvestigated using laboratory spectra and infrared solar spectra recorded from orbit by the ACE and ATMOS instruments. In addition to identifying the previously unobserved 6–5 vibration–rotation band in the laboratory spectra, many additional high N rotational lines have been observed. By combining the new observations with the previously published data and recent far infrared data, an improved set of molecular constants and term values have been derived for the X³Σ⁻ and A³Π states.     

Accurate Measurement of Small Spin–Spin Couplings in Partially Aligned Molecules Using a Novel J-Mismatch Compensated Spin-State-Selection Filter

The accurate measurement of small spin–spin coupling constants in macromolecules dissolved in a liquid crystalline phase is important in the context of molecular structure investigation by modern liquid state NMR. A new spin-state-selection filter, DIPSAP, is presented with significantly reduced sensitivity to J-mismatch of the filter delays compared to previously proposed pulse sequences. DIPSAP presents an attractive new approach for the accurate measurement of small spin–spin coupling constants in molecules dissolved in anisotropic solution. Application to the measurement of ¹⁵N–¹³C′ and ¹H^N–¹³C′ coupling constants in the peptide planes of ¹³C, ¹⁵N labeled proteins demonstrates the high accuracy obtained by a DIPSAP-based experiment.     

The collisional energy transfer between the D 1Π and d 3Π states of the NaK dimer induced by argon buffer gas H. Zhiwei et al.: Appl. Phys. B 66, 471 (1998)

It is shown in this Comment that the method proposed by Zhiwei et al. to extract cross sections for collisional energy transfer between molecular states mediated by argon atoms, is neither correct nor suitable, as it fails to reproduce the molecular behaviour they observe. An alternative model based on a consecutive scheme is put forward, and it is shown that a minimal assumption of one intermediate collisionally activated step is enough to satisfactorily reproduce the observations by Zhiwei et al. However, the extracted parameters are functions of the involved rate constants that are too complex to leave any hope of attaining information on the desired cross sections from them by just varying the argon buffer gas pressure, as done by Zhiwei et al. Received: 30 November 1998 / Published online: 14 July 1999     

Cercignani's Conjecture is Sometimes True and Always Almost True

 We establish several new functional inequalities comparing Boltzmann's entropy production functional with the relative H functional. First we prove a longstanding conjecture by Cercignani under the nonphysical assumption that the Boltzmann collision kernel is superquadratic at infinity. The proof rests on the method introduced in [39] combined with a novel use of the Blachman-Stam inequality. If the superquadraticity assumption is not satisfied, then it is known that Cercignani's conjecture is not true; however we establish a slightly weakened version of it for all physically relevant collision kernels, thus extending previous results from [39]. Finally, we consider the entropy-entropy production version of Kac's spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension. The first two results are sharp in some sense, and the third one is likely to be, too; they contain all previously known entropy estimates as particular cases. This gives a first coherent picture of the study of entropy production, according to a program started by Carlen and Carvalho [12] ten years ago. These entropy inequalities are one step in our study of the trend to equilibrium for the Boltzmann equation, in both its spatially homogeneous and spatially inhomogeneous versions. Received: 25 July 2002 / Accepted: 13 September 2002 Published online: 10 February 2003 Communicated by J. L. Lebowitz     

改进的双曲等效法用于双站合成孔径雷达前视成像

双站合成孔径雷达(SAR)前视成像有着很多的潜在应用,比如在能见度很低的天气实现飞机的导航和盲降,提高飞机飞行的安全度等.然而实现双站SAR前视成像的难点在于其距离历程的双根式性,直接采用驻定相位原理对其分解将得到复杂的解析式,从而后续算法的推导变得困难.现有的双曲等效法可以将双根式距离历程等效为单根式,但由于该方法为距离历程的二阶近似,当用于双站SAR前视成像时,由于其较大的前视角,未完全等效的距离历程高次项造成的误差已经超出可以接受的范围.针对以上问题,本文首先提出了一种改进的双曲等效法,该方法通过引入新的补偿变量来提高距离历程的等效精度;然后在此基础上推导回波信号的二维频谱解析式,并给出了一种适用于双站SAR前视成像的距离多普勒算法;最后通过仿真实验,将原双曲等效法与改进方法的等效误差以及成像结果做对比,验证了本文算法的有效性和优越性.     

Confined diffusion

Summary An equation for the unidimensional confined diffusion is proposed. The equation coincides with the well-known homogeneous equation except the presence of a source term. This term which has the form of a dipole distribution is located on a moving front which sharply separates two distinct regions. In the first region (from the boundary up to the front) the confined solution coincides with a suitable solution of the homogeneous equation; in the second region (besides the front) it vanishes. The source term, moreover, switches off the diffusing flux at the front. The sharp confinement allows to relax the original boundary conditions of the homogeneous equation. Precisely, to the function depending on the time at the boundary, another arbitrary function depending on the space at the initial time is added. This new function (provided not vanishing) allows to obtain in general an acceptable evolution of the front and does not prevent the validity of the conservation law: flux at the boundary is equal to the time variation of the diffusing quantity contained between the boundary and the front. By a suitable choice of this new function, so that it results to be connected to the other boundary condition (that depending on time) it is possible to arrive at an evolution of the front such as: , where λ,K, corresponding, respectively, to a dimensionless parameter and diffusivity, depend on the medium. Under such simplifying assumption, it is possible to obtain an analytical expression for the confined solution. This solution, evaluated in a point of the space, arrives asymptotically at the same value reached by the solution of the homogeneous equation.     

Correcting reaction rates measured by saturation-transfer magnetic resonance spectroscopy

Off-resonance or spillover irradiation and incomplete saturation can introduce significant errors in the estimates of chemical rate constants measured by saturation-transfer magnetic resonance spectroscopy (MRS). Existing methods of correction are effective only over a limited parameter range. Here, a general approach of numerically solving the Bloch-McConnell equations to calculate exchange rates, relaxation times and concentrations for the saturation-transfer experiment is investigated, but found to require more measurements and higher signal-to-noise ratios than in vivo studies can practically afford. As an alternative, correction formulae for the reaction rate are provided which account for the expected parameter ranges and limited measurements available in vivo. The correction term is a quadratic function of experimental measurements. In computer simulations, the new formulae showed negligible bias and reduced the maximum error in the rate constants by about 3-fold compared to traditional formulae, and the error scatter by about 4-fold, over a wide range of parameters for conventional saturation transfer employing progressive saturation, and for the four-angle saturation-transfer method applied to the creatine kinase (CK) reaction in the human heart at 1.5 T. In normal in vivo spectra affected by spillover, the correction increases the mean calculated forward CK reaction rate by 6-16% over traditional and prior correction formulae.     

The uv spectrum of 18OD

The combined high-resolution ultraviolet (uv) absorption spectrum of ¹⁶OD and ¹⁸OD was obtained. State selective measurements of the transitions from the electronic ground state to the first excited electronic state, $\text{A}\text{?}{}^2\text{Σ}{}^{\mathrm{+}}\text{←}\text{X}\text{?}{}^2\text{Π}$, in the 0-0 vibronic band were performed by means of a narrow bandwidth dye laser system. Evaluation of these transition frequencies in wave-numbers yielded molecular constants as well as rotational term values for each of the isotopic species. A computer program based on a linearized least-squares procedure was used to determine the molecular constants and term values. The term value formulas which were employed for this purpose, take into account the Λ splitting and the centrifugal distortion of the diatomic species. The transitions, recalculated from the semiempirically determined term values agree with the measured absorption lines to better than 0.1 cm^?1. The following molecular constants are reported: B, D, H, the rotational constants of the ²Π and ²Σ⁺ states; O₀, P₀, Q₀, the constants of the Λ splitting of the ²Π state; A and A₁; the constants of the spin-orbit coupling of the ²Π state; and γ₀, the constant of the p doubling of the ²Σ⁺ state. Futhermore, term values up to J″ and J′ of 25.5 and the corresponding uv transitions are given.     

A Metropolis Monte Carlo Implementation of Bayesian Time-Domain Parameter Estimation: Application to Coupling Constant Estimation from Antiphase Multiplets

The Bayesian perspective on statistics asserts that it makes sense to speak of a probability of an unknown parameter having a particular value. Given a model for an observed, noise-corrupted signal, we may use Bayesian methods to estimate not only the most probable value for each parameter but also their distributions. We present an implementation of the Bayesian parameter estimation formalism developed by G. L. Bretthorst (1990,J. Magn. Reson.88, 533) using the Metropolis Monte Carlo sampling algorithm to perform the parameter and error estimation. This allows us to make very few assumptions about the shape of the posterior distribution, and allows the easy introduction of prior knowledge about constraints among the model parameters. We present evidence that the error estimates obtained in this manner are realistic, and that the Monte Carlo approach can be used to accurately estimate coupling constants from antiphase doublets in synthetic and experimental data.     

A planarity relation for sextic centrifugal distortion constants

The relations between the molecular parameters of a planar molecule are used to derive a planarity relation for the sextic centrifugal distortion constants from the recently published expressions [Aliev and Watson, J. Mol. Spectrosc.61, 29 (1976)]. This relation, which contains only the rotational constants and the quartic and sextic distortion constants, is strictly valid only for the equilibrium constants, but will be useful as a constraint in the fitting of rotational energies if the experimental error is larger than the vibrational defect. The planarity relation is compared with the observed constants for the SO₂, O₃, NO₂, H₂O, and H₂S molecules. There appears to be a significant defect for SO₂ and for H₂O and its isotopes. The defect for SO₂ may be too large to be genuine, and possibly results from truncation of the Hamiltonian at the sextic terms.     

Nuclear magnetic relaxation by spin-rotation interaction.

Spin-rotation interaction constants, c, are calculated from the paramagnetic term of the nuclear magnetic shielding constant, σ _p, using a relationship originally due to Ramsey. Calculated values show excellent agreement with experimental determinations from molecular beam measurements. Since σ _p can be easily estimated from the chemical shift of the nuclear resonance this provides a general method for estimating spin-rotation interaction constants.

Chemical shift anisotropies allow the components c _⊥ and c _‖ of the spin-rotation interaction tensor to be determined. Generally both components have the same sign and are of similar magnitude. The r.m.s. value of the spin-rotation interaction constant, required to calculate nuclear spin-lattice relaxation times, is not expected to differ appreciably from average values obtained from molecular beam measurements or magnetic shielding.

Values for the molecular angular velocity correlation time, τ _sr, calculated directly from nuclear spin-lattice relaxation times are about a factor of four longer than those predicted by the relationship between τ _sr and the re-orientation correlation time, τ _d, as given by Hubbard.     

Revised molecular constants and term values for the XΠ state of CH

An improved set of molecular constants and term values are given for the X²Π (v = 0-5) state of the CH radical. They are derived from a fit of previously published data and additional lines taken from infrared solar spectra recorded on orbit and from new laboratory IR emission data.     

由实验数据计算氢气的维里系数问题

我们对计算气体物态方程中维里系数的方法作了全面的探讨,提出了如何判定物态方程中项数的四个原则。这四个原则是:(1)用∑v²的大小的变化来判断项的数目应如何选取,其中v为pV的计算值减去其实验值。(2)多项式要满足逐项在数值上减少的级数性质。(3)多项式的末一项大小要与实验误差大小相同。(4)误差v的正负号数目要大致相等。我们建议在计算中要用A₀+B₀+C₀+…=1条件,但不要用A=RT条件。此外还讨论了用压强展开和用密度展开的问题,级数展开式中省去奇数幂次项问题,维里系数的有效位数问题,若干实验数据的修正问题等。我们根据上述原则和建议,重新计算了氢气的维里系数,结果与各原作者所得的有相当的不同。在同一温度用不同实验窒的数据所得的维里系数比各原作者所得的要互相接近些,一般在两倍标准误差内一致。     

Direct Digital Frequency Control Based on the Phase Step Change Characteristic between Signals

We present a new digital phase lock technology to achieve the frequency control and transformation through high precision multi-cycle group synchronization between signals without the frequency transformation circuit.In the case of digital sampling, the passing zero point of the phase of the controlled signal has the phase step characteristic, the phase step of the passing zero point is monotonic continuous with high resolution in the phase lock process, and using the border effect of digital fuzzy area, the gate can synchronize with the two signals,the quantization error is reduced. This technique is quite different from the existing methods of frequency transformation and frequency synthesis, the phase change characteristic between the periodic signals with different nominal is used. The phase change has the periodic phenomenon, and it has the high resolution step value. With the application of the physical law, the noise is reduced because of simplifying frequency transformation circuits,and the phase is locked with high precision. The regular phase change between frequency signals is only used for frequency measurement, and the change has evident randomness, but this randomness is greatly reduced in frequency control, and the certainty of the process result is clear. The experiment shows that the short term frequency stability can reach 10~(-12)/s orders of magnitude.