Diffraction of light by adjacent fundamental and second or fourth harmonic ultrasound beams: Comparison of exact and simplified formulae with experiment

Theoretical expressions and their approximate formulae for the intensities of light diffracted by two adjacent ultrasonic beams are discussed and compared with experimental data for the case when the second beam is an even harmonic of the first (fundamental). In the special case of the fundamental and its second or fourth harmonic, relatively simple formulae are found for the first and second diffraction orders describing the variation of light intensity with phase shift and intensity ratio.The approximate formulae describe the experiments satisfactorily only for small values of ζ₁ and α_n. For higher values the exact formulae give better agreement. Only the Raman-Nath region is considered.     

Model studies on the systematics and interpolation of Franck-Condon factor arrays

A simple harmonic oscillator approximation is used to derive approximate algebraic expressions for low vibrational quantum number Franck-Condon factors for band systems for which molecular data are sparse. These simple expressions involve a transition parameter u which is characteristic of the electronic transition involved. The expressions provide Franck-Condon factors which agree quite well with RKR values when these are available. The formulae are used to explain quantitatively the dependence on u of the systematic properties of Franck-Condon factor arrays and to provide a method for their interpolation. The formulae are also used to describe quantitatively the dependence on u of the gross appearance of molecular band systems in emission and absorption.     

The intensity distribution in the CH(A2Δ − X2Π) spectrum produced by electron impact on acetylene

The intensity distribution in the rotational and vibrational structure of the CH(A²Δ ? X²Π) spectrum, formed by dissociative excitation of acetylene by electron impact, is analysed. This spectrum is built-up from three overlapping bands, namely, the 0-0, 1-1 and 2-2 bands. The intensity distribution in the rotational structure of the 0-0 band is determined by an analysis of the free rotational lines of the R-branch. The intensity distribution in the other bands is analysed by a comparison of the spectrum with a spectrum simulated on the computer. In this way, the overlap of the various rotational and vibrational transitions is taken into account. It turns out that the distribution of molecules over the rotational levels can be described by assuming one Boltzmann distribution for each vibrational level.     

On the accuracy of the approximate formulae for the photon counting statistics of mixed coherent and chaotic radiation

We compare the photon counting statistics for mixed coherent and chaotic radiation calculated with the help of approximate formulae and exact recurrence formulae recently obtained. We investigate Lorentzian, Gaussian and rectangular spectrum of chaotic radiation and show that the accuracy of the approximate formulae, which are relatively simple, is better than 1% if the signal-to-noise ratio is greater than 4 for arbitrary values of the other parameters. If the signal-to-noise ratio is less than one the agreement of the approximate and exact values is only fair. The simple closed-form approximate formulae may be useful particularly for systems employing laser radiation.     

Approximate,non-relativistic scattering phase shifts,bound state energies,and wave function normalization factors for a screened Coulomb potential of the Hulthén type

Non-relativistic scattering phase shifts, bound state energies, and wave function normalization factors for a screened Coulomb potential of the Hulthén type are presented in the form of relatively simple analytic expressions. These formulae have been obtained by a suitable renormalization procedure applied to the quantities derived from an approximate Schrödinger equation which contains the exact Hulthén potential together with an approximate angular momentum term. When the screening exponent vanishes, our formulae reduce to the exact Coulomb expressions. The interrelation between our formulae and Pratt's ‘analytic perturbation theory for screened Coulomb potentials’ is discussed.     

The dipole moment derivatives and charge flux parameters from vibrational absorption intensities of ethylene oxide

The signs of dipole moment derivatives for ethylene oxide obtained from ab initio calculations using the 6-31G¹ basis set are presented. From these signs and the experimental vibrational absorption intensity data, the atomic polar tensor elements and bond charge flux parameters were evaluated. These quantities were compared with those in the literature calculations employing approximate signs and force constants.     

Topological cross-section in hadron-nucleus collisions and multiple scattering theory

The multiple scattering theory supplemented with AGK cutting rules is applied to calculation of the hadron-nucleus interaction cross-sections. In contrast to standard Glauber approach neither smallness of the interaction radius compared to the nuclear radii nor Gaussian form of thehN-interaction profile function are assumed. We consistently use the theory of the supercritical Pomeron. However all the results are more general and do not depend on the specific parametrization of the Pomeron pole ampletude. The region of validity of the widely used approximate formulae for topological and totalhA-interaction cross sections are discussed. It is discussed the novel features of particle-nucleus interactions at superhigh energies.     

The Use of a hierarchy scheme for the evaluation of normal modes of vibration

In vibrational analysis it is the common practice to initially approximate a set of force constants rather than to initially approximate a set of normal modes. This does not allow a flexible and easily analyzed computational system. A hierarchy scheme of approximation has been developed. Systematic testing of alternative hierarchies and geometries is essential since a force field may always be evaluated to match exactly observed and calculated frequencies for any set of approximate normal modes. Other physical requirements must therefore be considered to establish the correctness of an answer. A set of N linearly independent internal coordinates of a particular symmetry type are placed in a hierarchy and a set of N orthormal modes are developed in such a way that the nth normal mode does not change the internal coordinates of the n + 1 to Nth members of the hierarchy. Constraints on the form of the force field require iteration from these well-defined initial approximations. The hierarchy scheme offers a much more satisfying physical picture of characteristic group frequencies and lends itself readily to the use of perturbation theory.     

Efficient and accurate method for radiation transfer problems

An efficient method of analysis, which utilizes trial functions based on Case's eigenvalues, is developed for solving radiation transfer in an absorbing and scattering homogeneous semi-infinite plane-parallel medium subjected to externally incident radiation. Expressions for the forward and backward intensities, reflectivity and total radiation intensity are included. Numerical results are given and compared involving different forms of the externally incident radiation on the boundary surface. It is shown that the solution converges rapidly to the exact results and that lower-order solutions predict values of the physical parameters that are accurate to five figures in all values of the single-scattering albedos in the range 0.1 ≤ ω ≤ 1. The method has been also used to get approximate formulae for calculating Chandrasekhar's characteristic H-functions and their moments.     

Vibrational anomalies and marginal stability of glasses

The experimentally measured vibrational spectrum of glasses strongly deviates from that expected in Debye’s elasticity theory: The density of states deviates from Debye’s ω² law (“boson peak”), the sound velocity shows a negative dispersion in the boson-peak frequency regime, and there is a strong increase in the sound attenuation near the boson-peak frequency. A generalized elasticity theory is presented, based on the model assumption that the shear modulus of the disordered medium fluctuates randomly in space. The fluctuations are assumed to be uncorrelated and have a certain distribution (Gaussian or otherwise). Using field-theoretical techniques one is able to derive mean-field theories for the vibrational spectrum of a disordered system. The theory based on a Gaussian distribution uses a self-consistent Born approximation (SCBA),while the theory for non-Gaussian distributions is based on a coherent-potential approximation (CPA). Both approximate theories appear to be saddle-point approximations of effective replica field theories. The theory gives a satisfactory explanation of the vibrational anomalies in glasses. Excellent agreement of the SCBA theory with simulation data on a soft-sphere glass is reached. Since the SCBA is based on a Gaussian distribution of local shear moduli, including negative values, this theory describes a shear instability as a function of the variance of shear fluctuations. In the vicinity of this instability, a fractal frequency dependence of the density of states and the sound attenuation ∝ ω^1+a is predicted with a ? 1/2. Such a frequency dependence is indeed observed both in simulations and in experimental data. We argue that the observed frequency dependence stems from marginally stable regions in a glass and discuss these findings in terms of rigidity percolation.     

双原子分子体系的振动结构研究

最近通过微扰理论得到了双原子分子体系的能量表达式、振动光谱常数和转动光谱常数 ,并建议用代数方法和势能变分法由有限的振动能级求得收敛的振动能级的完全谱和高阶的振动力常数 .用该方法对一些双原子分子电子态的计算结果表明 :(1)代数方法得到的最高振动能收敛于正确的分子离解能 ;(2 )代数方法产生的振动能级不但能重复已知的精确能级 ,而且还能得到实验上和其他理论方法难以得到的高振动激发态的能级 ;(3)可用获得的各阶振动力常数fn比较同一分子的不同电子态的化学键的相对强度. Alternative expressions for vibrational and rotational spectrum constants and energies of diatomic molecular electronic states are suggested based on the perturbation theory. An algebraic method (AM) is proposed to generate converged full vibrational spectrum from limited energy data, and a potential variational method (PVM) is suggested to produce the vibrational force constants f n s and rotational spectrum constants using the perturbation formulae and the AM vibrational constants. Applying this method...     

Electronic recoil effects in high-energy photoelectron spectroscopy

It is shown that the recoil energy imparted to the residual ion by the outgoing fast photoelectron leads to noticeable modifications of X-ray excited photoelectron spectra of molecules containing light atoms. The vibrational band envelopes may differ considerably from those predicted by the Franck-Condon principle. Al Kα excited valence electron bands are shown to exhibit, in addition to the translational recoil energy of the molecule, energy shifts of up to several tenth of an eV due to recoil-induced vibrational and rotational excitation. This effect has to be kept in mind when ionization potentials are determined from ESCA spectra. The recoil-induced vibrational and rotational excitation depends on the orbital quantum numbers of the ionized electron. Simple formulae for the shift of the centroid and the broadening of the band due to recoil effects are given for the special case of diatomic molecules.     

Inductive-resonant theory of nonradiative transitions in lanthanide and transition metal ions (review)

The review is devoted to the theory of nonradiative transitions in tricharged ions of lanthanides and transition metals in the condensed phase, which was proposed in 1971. The theory is based on the phenomenon of nonradiative energy transfer from an electronically excited ion to surrounding molecular groups with excitation of resonant vibrational states and makes it possible to calculate the nonradiative transition rate constant (k _nr) by a formula that is similar to the Förster formula. The primary emphasis is placed on recent experimental works that directly confirm the proposed theory. It is shown that the theory satisfactorily quantitatively accounts for (i) the effect of deuteration of molecular groups surrounding ions on k _nr, (ii) the energy gap law, and (iii) the dependence of k _nr on the distance between the ion and deactivating groups. Furthermore, it is shown that (iv) the theory makes it possible to satisfactorily quantitatively calculate in the dipole-dipole approximation the constant k _nr of the electronic transition based on the knowledge of the radiative rate constant and the vibrational absorption spectra of molecular groups in the range of overlap with the luminescence spectrum of the ion; (v) the temperature dependence of k _nr; and (vi) the anomalously low k _nr in the case where the corresponding radiative transition is caused by the magnetic rather than the electric dipole. Literature data are presented that directly experimentally support the proposed theory of nonradiative transitions. In addition, works where this approach is used to calculate k _nr of transitions in laser media are described.     

光谱仪谱线和谱带弯曲现象的精确表述方式

在矢量衍射理论基础上给出了任意倾斜入射下的光栅方程一般形式以及衍射极角和衍射方位角的完整解析表达式,由此首次导出了平面光栅光谱仪谱线和谱带弯曲量的精确计算公式和对应于衍射极角和衍射方位角的两类角色散公式,比较了谱线弯曲精确公式和近似公式的计算结果,进而指出和弥补了近似公式的不足,并数值考察了入射狭缝高度对光谱仪色散能力的影响程度。由于在以上各式的推导过程中未作任何近似,且涉及到了导致谱线和谱带弯曲的所有可能因素,因而它要比以往所用近似公式更为全面和可靠,可作为实际光谱仪器设计、测试、装调和使用的理论依据。     

混响强度及其衰减规律与脉宽的关系

本文由短脉宽下混响强度的普遍表达式I_r(t,τ)=Kτt^-me^-βt出发,导出对应的可用于长脉宽的混响强度精确表达式,并给出了忽略脉宽尺度范围的指数衰减损失后相应的近似表达式。当指数衰减系数β足够小和脉宽大于有效脉宽τ_m=a_mt(a_m随m增大而减小,m是大于2的实数)时,近程混响强度趋向饱和且随时间t的(m-1)次幂衰减。实验结果与理论符合较好。     

A theoretical study of the 21Ag ← 11Ag two-photon transition and its vibronic band in trans-stilbene

The two-photon spectrum of the 2¹A_g ← 1¹A_g transition in trans-stilbene has been calculated at the complete active space self-consistent field (CASSCF) level of theory. Energies were obtained at the complete active space second-order perturbation (CASPT2) level of theory, while the geometries of both the initial and final states were optimized at the CASSCF level. The energy and the geometry optimizations were performed using an active space of 14 electrons in 14 active π orbitals. The vibrational frequencies of both states and the two-photon transition (TPT) cross-section were calculated with a smaller active space where the two lowest π orbitals were kept inactive. A newly implemented algorithm, in the quantum chemical package Molcas was used to determine the two-photon transition intensity. This method requires only the linear response of the CASSCF wavefunction. Furthermore, the vibronic structure of this TPT was studied. The Franck-Condon factors were obtained by calculating the overlap between the vibrational states involved, which were determined from the force fields of both the initial and final states, at the CASSCF level of theory. The results are in agreement with experiment.     

NO分子宏观气体热力学性质的理论研究

采用量子统计系综理论,研究了基态NO分子宏观气体摩尔熵、摩尔内能、摩尔热容等热力学性质.首先应用课题组前期建立的变分代数法(variational algebraic method, VAM)计算获得了基态NO分子的完全振动能级,得到的VAM振动能级作为振动部分,结合欧拉-麦克劳林渐进展开公式的转动贡献,应用于经典的热力学与统计物理公式中,从而计算得到了1000-5000 K温度范围内NO宏观气体的摩尔内能、摩尔熵和摩尔热容.将不同方法计算得到的摩尔热容结果分别与实验值进行比较,结果表明基于VAM完全振动能级获得的结果优于其他方法获得的理论结果.振动部分采用谐振子模型对无限能级求和计算热力学性质的方法有一定的局限性,应当使用有限的完全振动能级进行统计求和.     

Polarization labeling spectroscopy of NO2

Polarization-labeling spectroscopy is used to simplify the visible spectrum of nitrogen dioxide. Nonlinear polarization effects are used to label and probe molecules in selected rotational-vibrational levels. Those labeled molecules can be detected by their depolarizing effect on a probe laser. The resulting spectra are considerably less complicated than normal absorption spectra. P- and R-branch transitions can be readily identified, and approximate excited-state rotational constants are determined. With more powerful lasers, it may be possible to use polarization labeling to unravel the very complicated vibrational structure of the NO₂ excited states.     

Improvement of the parametric method of the theory of vibronic spectra of polyatomic molecules: Absorption spectra and the structure of butadiene, hexatriene, and octatetraene in the excited state

The structure of the excited states and absorption spectra of butadiene, hexatriene, and octatetraene are calculated by the parametric method of the theory of vibronic spectra using models of the first-and second-order approximations. It is shown that these molecular models adequately reflect the molecular structure and allow one to predict quantitatively the shape and fine vibrational structure of the absorption spectra. When passing to the second-order approximation, only two additional (angular) parameters are used. These parameters are transferable in the series of polyenes. Compared to the first-order approximation model, the second-order approximation model more accurately takes into account the angular deformations of polyenes upon their excitation and describes the intensity distribution in the vibrational spectrum, including weak lines. In addition, the calculations also quantitatively predict spectral variations in the molecular series. The parametric method is more efficient for modeling polyatomic molecules in the excited states and their vibrational spectra compared to other semiempirical and ab initio methods.     

Universal functions for estimating total vibration-rotation band absorptance—I. Linear molecules and spherical tops

We estimate total band absorptances and their derivatives for nonoverlapping lines of vibration-rotation bands for linear molecules and spherical tops. We use universal functions obtained by replacing the sums of line contributions by integrals over the rotational quantum numbers. An optical path is introduced for the total band. Only general information is utilized on vibrational transitions and line shapes. Power and asymptotic series have been obtained for Doppler and Lorentz line shapes. For a linear molecule and the Lorentz shape, approximate formulae have been derived for the universal functions.