Long-range Tail Research Articles

Elastic Scattering of Bloch Waves in Absorbing Crystals Containing Lattice Defects

The differential equations of Bloch wave type which are used in transmission electron microscopy for the calculation of the diffraction contrast from lattice defects are discussed with special reference to crystals with anomalous absorption (i. e. with a complex crystal potential). anomalous absorption is included, the eigenvalue equation by which the Bloch waves are defined in the perfect reference lattice, becomes non-Hermitian. It is shown that for some particular contrast effects not only the eigenvalues but also the eigenvectors must be corrected for the imaginary part of the crystal potential. As a consequence the orthogonality of the Bloch waves is violated. The correction of the eigenvectors may result in long-range contrast tails of the images of lattice defects (intraband scattering). If, however, only the peak positions of the contrast profiles are considered, it suffices in practical cases to correct only the eigenvalues as is usually assumed for the diffraction in perfect crystals.

Accurate Analytic Static and Dynamic Polarizabilities forH−via the Asymptotic Approximation

Accurate closed analytic expressions for certain properties of ${\mathrm{H}}^{\ensuremath{-}}$ that strongly depend on the long-range tail of its wave function are given. These properties are $〈{{\mathcal{r}}_{1}}^{n}+{{\mathcal{r}}_{2}}^{n}〉$, the static electric multipole polarizabilities, and the dynamic dipole polarizability. From the last, the ${\mathrm{H}}^{\ensuremath{-}}$ photodetachment cross section is calculated and the simple but accurate result obtained by Armstrong is recovered. The only assumption made is that the exact wave function can be replaced by its rigorous asymptotic form, for the purpose of calculating these properties. No model potentials or arbitrary cutoffs are introduced. The results for the dipole polarizabilities agree to better than 5% with the extensive recent calculations of Chung. For $〈{\mathcal{r}}_{1}^{n}+{\mathcal{r}}_{2}^{n}〉$, $n\ensuremath{\ge}2$, the formula obtained agrees to within 0.5% with values obtained from large calculations. For all these quantities, the Hartree-Fock results are far inferior.

Critical Correlations of Certain Lattice Systems with Long-Range Forces

We show the way perturbation procedures originated by Brout and Hemmer and further developed by Lebowitz and co-workers can be used to study the critical behavior of lattice systems with a potential $v(\mathrm{r})$ having a weak, long-ranged tail that approaches ${\ensuremath{\gamma}}^{d}\ensuremath{\phi}(\ensuremath{\gamma}\mathrm{r})$ as $\ensuremath{\gamma}$ approaches zero, where $d$ is dimensionality. We consider both the Ising and the spherical models, and note that the results for the latter model are also those that follow from the Ornstein-Zernike assumption that the direct correlation function behaves like $\ensuremath{-}\frac{v(\mathrm{r})}{\mathrm{kT}}$, when $v(\mathrm{r})\ensuremath{\ll}\mathrm{kT}$. We recover by a graph technique the previously known result that in one dimension, quantities of interest such as the inverse correlation length $\ensuremath{\kappa}$ cannot be expanded in $\ensuremath{\gamma}$ at the mean-field critical temperature ${T}_{0}$ and density ${\ensuremath{\rho}}_{0}$ that characterize the critical point in the $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ limit. Instead, at (${T}_{0}, {\ensuremath{\rho}}_{0}$), $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. This is true for both the Ising and the spherical models. In the two-dimensional spherical model, we find $\ensuremath{\kappa}$ to vary as ${\ensuremath{\gamma}}^{2}{[\mathrm{ln}(\frac{1}{\ensuremath{\gamma}})]}^{\frac{1}{2}}$ when $\ensuremath{\gamma}\ensuremath{\rightarrow}0$, while in three dimensions $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{5}{2}}$. In the Ising case for $d=1$, we characterize topologically the infinite sum of graphs that contribute to the $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}^{\frac{4}{3}}$ term in the expansion of the pair correlation function. (In the spherical model, a chain graph gives the entire contribution to the pair correlation function for all $d$). For $d=3$ we do not attempt to deal with the correlation length in the Ising case, but instead consider the shift in the critical temperature as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$. We find that the spherical-model result that gives a shift of order ${\ensuremath{\gamma}}^{3}$ is exact to that order in $\ensuremath{\gamma}$ for the Ising model. We also find that the lowest-order correction to the spherical-model result is of the form ($\mathrm{const}{\ensuremath{\gamma}}^{6}\mathrm{ln}(\mathrm{const}\ensuremath{\gamma})$, in agreement with recent work by Thouless; but in general we expect to find terms of order ${\ensuremath{\gamma}}^{4}$ and ${\ensuremath{\gamma}}^{5}$ from the spherical-model result dominating this correction.

Uniform approximations for glory scattering and diffraction peaks

Semiclassical approximations are derived for the angular dependence of the scattering cross section σ(θ) for two cases, involving the forward and backward directions, where the classical scattering is infinite. The results are approximations uniform in angle and valid from the ordinary semiclassical region where σ(θ) is O(0) right round to θ = 0 or π for a glory σ(θ) is then O(-1) and the forward diffraction peak σ(θ) is O(-2) or larger, depending on the form of the long-range tail of the scattering potential. The formulae are all expressed in terms of the action functions along the paths of the classical problems.

Coulombic Modified Effective-Range Theory for Long-Range Effective Potentials

A modified effective-range theory (MERT) was introduced previously to describe the scattering of a charged particle by a neutral polarizable system. The long-range components of the resultant effective one-body (generalized optical-model) potential, which cause the phase shift to have a rapidly varying energy dependence, are taken account of exactly by solving a one-body problem numerically. The short-range potential components generate only a slowly varying energy-dependent effect on the phase shift, and this effect can be accounted for by a few terms in a power series in ${k}^{2}$. The procedure is here extended to the case for which the polarizable system is itself charged. An MERT expansion is derived for the difference $\ensuremath{\delta}(k)$ between the total phase shift $\ensuremath{\eta}(k)$ and the phase shift $\ensuremath{\rho}(k)$ due to the long-range tail alone; both $\ensuremath{\eta}(k)$ and $\ensuremath{\rho}(k)$ are defined relative to pure Coulomb scattering. With the strongly energy-dependent Coulombic and other long-range effects accounted for exactly by the numerical solution of a one-body scattering problem, low-energy scattering data can be matched by the proper choice of the coefficients of just the first few terms in the MERT expansion; these terms will then determine the scattering in the (experimentally inaccessible) energy domain extending down to zero energy. For a repulsive Coulomb field, the leading term in $\ensuremath{\eta}(k)$ is determined exactly for all $L$ by the Born approximation; for $V(r)=(\frac{2\ensuremath{\mu}}{{\ensuremath{\hbar}}^{2}})(\ensuremath{-}\frac{{\ensuremath{\beta}}^{2}}{{r}^{4}})$, where $\ensuremath{\alpha}=\frac{{\ensuremath{\beta}}^{2}{\ensuremath{\hbar}}^{2}}{|\ensuremath{\mu}{e}^{2}{Z}_{1}{Z}_{2}}|={\ensuremath{\beta}}^{2}{a}_{0}$ is the electric-dipole polarizability of the target, $tan\ensuremath{\eta}(k)=\frac{{\ensuremath{\beta}}^{2}{{a}_{0}}^{3}{k}^{5}}{15}$ for $k{a}_{0}L\ensuremath{\ll}1$.

Extrapolation of Cross-Section Data to Zero Energy for Long-Range Effective Potentials

The effective one-body potential representing a particle-compound-system interaction usually has a very complicated short-range behavior, but a relatively simple, and more or less known, long-range tail $U$ which can be nonlocal and energy-dependent. With $U$ present, the usual effective-range-theory (ERT) expansion for a single-channel scattering phase shift $\ensuremath{\eta}$ is inapplicable. Recent extensions of ERT include the effect of long-range potentials, but since $\ensuremath{\eta}$ then becomes very energy-dependent, one would need many terms to obtain the desired accuracy, so the results are of restricted use. We show that one can first solve, numerically, a one-body problem which includes only $U$, and then use a modified ERT expansion for the difference $\ensuremath{\delta}$ between $\ensuremath{\eta}$ and the phase shift due to $U$ alone. Since the rapid energy dependence due to $U$ is thus extracted out (exactly), terms in the expansion which depend upon the short-range potential in the presence of $U$ are more slowly varying in energy; they involve a number of adjustable parameters to be determined experimentally, but only a few are usually needed. The procedure thus makes possible the extrapolation of low-energy scattering data down to zero energy, even when effective long-range potentials are present.