Higher-order Energy Corrections Research Articles

The theory of equilibrium isotope fractionations for gaseous molecules under super-cold conditions

It is necessary to build a proper theoretical method that can precisely describe isotope fractionation processes under super-cold (<200 K) conditions, because there have been many isotopic data obtained in our solar system that are related to such processes. However, current methods of isotope fractionation calculation, i.e., the Bigeleisen-Mayer equation and its higher-order energy corrections, may not be applicable to super-cold conditions. Here, we have checked important assumptions and higher-order corrections that can affect isotope fractionations of gas-phase molecules under super-cold conditions and developed a new theoretical method for calculating equilibrium isotope fractionation factors. Compared with previous works, we have added three new corrections into our calculation, i.e., nuclear-spin weights for quantum mechanical rotation, correction for Born-Oppenheimer approximation (BOA), and inversion splitting effect for non-planar molecules such as NH3. We further examined gaseous molecules of geochemistry and cosmochemistry relevance, e.g., H2, HF, HCl, H2O, H2S, HCHO, NH3, CH4 and their deuterated isotopologues. We found that the correction for BOA, which was rarely considered in previous studies, is important for those gaseous molecules under super-cold conditions. In case of D/H, 13C/12C and 18O/16O exchanges among organic molecules, BOA correction cannot be ignored even at ambient or higher temperature conditions. Most isotope fractionation trends at super-cold conditions reported here are quite different from their counterparts at ambient or higher temperature conditions. The method proposed here will extend our capability in interpreting equilibrium isotope fractionations to super-cold conditions in the solar system.

High-order perturbed corrections with scaling transformation

The performance of the so-called superconvergent quantum perturbation theory (Wenhua Hai et al 2000 Phys. Rev. A 61 052105) is investigated for the case of the ground-state energy of the helium-like ions. The scaling transformation r → r/Z applied to the Hamiltonian of a two-electron atomic ion with a nuclear charge Z (in atomic units). Using the improved Rayleigh–Schrödinger perturbation theory based on the integral equation to helium-like ions in the ground states and treating the electron correlations as perturbations, we have performed a third-order perturbation calculation and obtained the second-order corrected wavefunctions consisting of a few terms and third-order energy corrections. We find that third-order and higher-order energy corrections are improved with decreasing nuclear charge. This result means that the former is quadratically integrable and the latter is physically meaningful. The improved quantum perturbation theory fits the higher-order perturbation case. This work shows that it is a development on the quantum perturbation problem of helium-like systems.

Undulator radiation in a parallel plate waveguide: Inclusion of off-axis motion and higher-order energy corrections

The theory of undulator radiation in a parallel plate waveguide is reconsidered with the inclusion of energy corrections upto K 3/ γ 3 and the relevant physical consequences. We also discuss the role of betatron oscillations and the modifications induced in the spectral brightness which arise when the electrons are injected off the undulator-axis.

Linear undulator brightness: Inclusion of sextupolar magnetic-field contributions and of higher-order energy corrections.

In this paper we discuss the theory of emission by relativistic electrons in linearly polarized undulators. We show that the analytical computations are greatly simplified by the introduction of the so-called generalized Bessel functions, originally introduced to treat problems for which the dipolar approximation does not hold. We also discuss the modification induced in the brightness by the inclusion of the sextupolar terms, which arise when electrons are injected off the undulator axis. We show that the phenomonology inherent to this problem is very rich, as additional harmonics come into play and may cause sizable effects. The analysis we develop may be useful to obtain further insight in the physics underlying the so-called inhomogeneously broadened regime. Finally, we also discuss the inclusion of energy corrections up to 1/${\ensuremath{\gamma}}^{3}$ and the relevant physical consequences.

Calculations of infinite series representing higher-order energy corrections for nonrelativistic hydrogen atoms in external fields

A new method is given for the evaluation of higher-order perturbation corrections to energy of the nonrelativistic hydrogen atom in an external field. The choice of a suitable basis in the Hilbert space of bound states for nonrelativistic hydrogen atom is exploited in the method. The infinite series which represents the higher-order energy correction in the Rayleigh-Schr\odinger perturbation theory may be expressed in such a basis in terms of finite series which are easily summed. As an illustrative example the summation of a few series, which are of importance for higher-order corrections to energy of the H atom in an external homogeneous magnetic field, is carried out.

Impurity-Band Tails in the High-Density Limit. II. Higher Order Corrections

In an earlier paper, I, the authors presented a theory of the low-energy tail of an electron band in a semiconductor in the presence of a high density of impurities. This theory was based on the approximation that all the local potential fluctuations binding states of a given energy have the same shape. We now discuss corrections to the density of states $\ensuremath{\rho}(E)$ and the spectral density $A(\mathbf{k},E)$ by treating the difference between the actual random potential and the average well shape as a small perturbation. An average higher-order energy correction to the density of states is proposed, which displaces all energies by the average difference between the perturbed and unperturbed energies, and the calculation of this correction is discussed for the limiting case of a random potential obeying Gaussian statistics. We also discuss the complete second-order correction proposed by Zittartz and Langer, which leads to the exact asymptotic form of $\ensuremath{\rho}(E)$ in a Gaussian potential, and we show how the formulas can be extended to impurity potentials of nonzero range. For three-dimensional Gaussian models, the approximate density of states of our earlier paper had the form $\ensuremath{\rho}(E)\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}2}A(E) \mathrm{exp}[\frac{\ensuremath{-}B(E)}{(2\ensuremath{\xi})}]$, where $\ensuremath{\xi}$ is a parameter proportional to the density of impurities and to the square of the strength of the individual impurity potentials. The corrections described in the present paper modify the function $A(E)$, but not $B(E)$. We also discuss briefly corrections to the momentum dependence of our earlier approximation to the spectral density $A(\mathbf{k},E)$.

Energy levels and transition probabilities for deformed nuclei

Higher-order energy corrections and intensity rules for K-allowed and K-forbidden transitions are discussed on the basis of a generalized Peierls-Yoccoz theory, which includes K mixing. The results put a number of predictions of the rotational model on a firmer basis, but show in addition some marked differences. A comparison with experimental data is given.