Radial Differential Equations Research Articles

Complex form of the equations of motion in the hyperspherical-expansion approach

Inspired with Surkov's modification of interaction matrix elements, radial differential equations of the hyperspherical-expansion approach are transformed by means of the quasi-Laplace transformation into a system of integro-differential equations. Using the method of the steepest descent they are reduced to eigenvalue equations of the usual shell models; also transition amplitudes of physical quantities can be approximated in a similar manner. The obtained results are independent of the contigent introduction of the auxiliary fields, the characteristic quantity being the nuclear radius.

The hyperspherical-expansion approach and shell models

By means of Talmi integrals the matrix elements of nuclear interaction in the set of radial differential equations of the hyperspherical-expansion approach are related to the elements of usual shell models. The introduced average field equated to the diagonal element of the core reproduces the form of the effective (two-body) cut-off oscillator. The corresponding eigenvalue problem with a flat (heavy nuclei) and rounded (light nuclei) alternative for the well is solved and compared with the phenomenological relations: the asymptotic saturation can be achieved for the flat well. Using these results the starting set is transformed into the set of integral equations by means of the Green's functions. Then it is, successively, reduced to the eigenvalue equations of usual shell models identical for both the well alternatives. Compared with the usual models some improvements apply and the oscillator length is given automatically.

Two-centre wavefunctions in the theory of electron scattering by hydrogen atoms

The scattering wavefunctions of the H – ion are expanded in a basis of two-centre wavefunctions with radial components. Two expansions are considered characterized by the charge ratio q being either – 1 or + 1. These expansions may be regarded as an extension of the polarized-orbital method. The q = + 1 expansion has some novel features. The radial components are determined as the solutions of a set of coupled radial differential equations. Imposition of exchange antisymmetry transforms these equations into integro-differential equations for the q = - 1 expansion, but for the q = + 1 expansion the equations remain simply differential. The cross-sections are determined by partial wave analysis, utilizing the Mathieu function form of the radial wavefunctions at moderately large values of R. Numerical work is reported based upon truncation of the q = + 1 expansion after two terms. Phase shifts derived from static and adiabatic q = + 1 potentials for the l = 0 - 7 partial waves are presented for 59 values of the energy between 0 and 13.6eV. The corresponding differential and total cross-sections are compared with experimental, and other theoretical results.

A new recurrence relation for hydrogenic radial matrix elements

A recurrence relation is presented which connects the general (n' not=n) hydrogenic radial matrix element of rbeta , where beta is a complex number to matrix elements involving exponents beta +2, beta -1, and beta -2 but the same sets of quantum numbers. This identity is found solely from the boundary conditions on the wavefunctions and the radial differential equation and should be of use in simplifying rk forms given elsewhere. For n'=n, the recurrence relation reveals a proportionality between the - beta -2, beta -1 matrix elements that includes the Pasternack-Sternheimer identity as a special case. Further results may be obtained by beta -differentiation, and new identities for matrix elements of rk log (r) are given. The n'=n matrix elements of rk are studied for k>or=-1; closed-product expressions are found for k-values near the sum and difference of the angular momenta, so that simple prescriptions can be given for the iterative calculation of n-dependent explicit forms. It is suggested that complex-variable theory be applied to a study of these results.

Trapped ion instability in plasmas with magnetic shear

The radial structure of the dissipative trapped ion mode is studied in order to determine the effect of shear and the possibility of finding ranges of tokamak operation which are stable to this mode even in the presence of a temperature gradient. It is found that owing to the presence of increasing numbers of resonant untrapped ions near the mode rational surfaces, where lq is an integer, the radial wavelengths are reduced and the modes are strongly Landau damped. These effects increase with increasing shear and with increasing values of l, the toroidal mode number. Numerical estimates of the growth rates and marginal stability conditions are obtained from eigenmode solutions of the radial differential equation and from algebraic dispersion relations which simulate the effects of the radial structure. It is concluded that if (r /q )d q / d r exceeds one, then the temperature threshold for instability is raised by approximately a factor of two from that of the no shear case. It is also proposed that even in unstable regimes, the anomalous diffusion should be less than indicated by previous estimates, since both the growth rates and wavelengths are reduced.

Resonant scattering from inhomogeneous nonspherical targets

A method is proposed for solving problems in which either scalar or vector waves impinge at an arbitrary angle on an inhomogeneous nonspherical target whose size is comparable to the wavelength of the incident radiation. This method reduces a partial differential equation, the Helmholtz wave equation, to an ordinary differential equation through selection of angular trial functions composed of weighted sums of spherical harmonics. The wave equation then becomes a coupled set of radial differential equations which are discretized and solved by matrix methods, enforcing boundary conditions on the surface of the smallest sphere which completely encloses the target. The method is an extension of partial wave expansion and reduces to it exactly when the target is spherically symmetric.

Nuclear beta decay

New formulae for nuclear beta decay are derived by using a modified expansion of the electron radial wave functions. This expansion is obtained by means of the integral equations which correspond to the Dirac radial differential equations. As a consequence new types of form factor coefficients or nuclear matrix elements occur which depend on the shape of the nuclear charge distribution. The implications of the new formulae for the analysis of experiments are discussed. The sensitivity to the detailed shape of the charge distribution turns out to be weak usually, even in the case of non-unique forbidden transitions. An elementary treatment of the induced interactions leads in a simple way to the formulae which express the form factor coefficients in terms of coupling constants and nuclear matrix elements. Expressions are given for the single-particle values of all the nuclear matrix elements in terms of radial integrals. References

Scattering by Spherically Symmetric Inhomogeneities

The Jost function formulation of quantum scattering theory is applied to classical problems involving the scattering of a scalar plane wave by a medium in which the velocity is a function only of the spherical radial coordinate. This technique is used to solve the radial differential equation for scattering from a constant spherical inhomogeneity. The radial equation can be converted into an integral equation incorporating the Jost boundary conditions. The l = 0 partial wave integral equation for a constant inhomogeneity is solved using an iteration procedure (the first two iterations are considered). The Jost function and l = 0 cross section σ0 are plotted as a function of kR1, where k is the wavenumber in the surrounding medium and R1 is the sphere radius. The iterative technique is good for long wavelengths (kR1≪1) and any ratio of wavenumbers in the scattering and surrounding media. For shorter wavelengths and small ratio of wavenumbers (e.g., k1/k = 1.1, where k1 is the wavenumber in the scattering medium), it gives a good approximation to σ0 for the entire range of kR1 considered (0⩽kR1⩽2π). For shorter wavelengths and larger ratio of wavenumbers (e.g., k1/k = 1.5, 2.0), it gives a good approximation to σ0 out to approximately kR1 = 3π/4. More general problems using this method are also discussed.

Scattering by Spherically Symmetric Inhomogeneities

The Jost-function formulation of quantum scattering theory is applied to classical problems involving the scattering of a scalar plane wave by a medium in which the velocity is a function only of the spherical radial coordinate. This technique is used to solve the radial differential equation for scattering from a constant spherical inhomogeneity. The radial equation can be converted into an integral equation incorporating the Jost boundary conditions. The l−0 partial-wave integral equation for a constant inhomogeneity is solved using an iteration procedure (the first two iterations are considered). The Jost function and l−0 cross section σ0 are plotted as a function of kR1, where k is the wavenumber in the surrounding medium and R1 is the sphere radius. The iterative technique is good for long wavelengths (kR1≪1) and any ratio of wavenumbers in the scattering and surrounding media. For shorter wavelengths and small ratio of wavenumbers (e.g., k1/k = 1.1, where k1 is the wavenumber in the scattering medium), it gives a good approximation to σ0 for the entire range of kR1 considered (0⩽kR1⩽2π). For shorter wavelengths and larger ratios of wavenumbers (e.g., k1/k = 1.5, 2.0), it gives a good approximation to σ0 out to approximately kR1 = 3π/4. More general problems using this method are also discussed.

Quantal corrections and the WKB approximation of multiple Coulomb excitation

For an exact quantal calculation of multiple Coulomb excitation a set of coupled radial differential equations must be solved. The WKB method can be used to facilitate such computations. For large values of the angular momenta the set of coupled radial equations can be approximated by the simpler semiclassical equations. Also for large distances of the projectile from the target nucleus the WKB method simplifies the computation. The earlier suggested symmetrization of the relevant parameters of Coulomb excitation has been proved. For the case of the reorientation effect quantal corrections have been computed numerically for the excitation probability and the angular distribution of the de-excitation γ-rays. The slow convergence of the sum over the orbital momenta in the reaction amplitude can be improved by a method similar to the one used in electron scattering calculations. The numerical computations allow an estimate of the quantal correction for the reorientation effect for all experimental conditions up to a few percent.

Nuclear Photodisintegration in the1sShell: A Perturbative Approach to the Dipole Sum Rules

The integrated and bremsstrahlung-weighted $E1$ photoabsorption cross sections, ${\ensuremath{\sigma}}_{\mathrm{int}}$ and ${\ensuremath{\sigma}}_{\mathrm{b}}$, have been calculated for the lightest nuclei, ${\mathrm{H}}^{2}$, ${\mathrm{H}}^{3}$, ${\mathrm{He}}^{3}$, and ${\mathrm{He}}^{4}$, within the framework of a second-order perturbation procedure. For the purpose of comparison, two Gaussian nucleon-nucleon potentials were employed: one containing a repulsive core and central attractive and tensor components, the other only central attractive and tensor components. The results for ${\ensuremath{\sigma}}_{\mathrm{int}}$ indicate that while reasonable over-all agreement with experiment may be achieved with either potential, the component contributions differ widely because of the admixture of the repulsive core. Furthermore, the results for ${\ensuremath{\sigma}}_{\mathrm{b}}$ for the deuteron seem to indicate a deficiency in the present choice of oscillator basis functions when applied to the loosely bound system. Considerable improvement for ${\ensuremath{\sigma}}_{\mathrm{b}}$ is noted for ${\mathrm{H}}^{3}$, ${\mathrm{He}}^{3}$, and ${\mathrm{He}}^{4}$, where the nuclei are more tightly bound and less extended structures. It is found by analyzing exact and approximate third-order contributions to both the deuteron binding energy and integrated cross section that the use of a wave function containing parameters determined by minimization of the perturbation expansion through second order is probably not acceptable, at least for this nucleus. This is further substantiated by comparison with the binding-energy results obtained from an exact numerical solution of the coupled $S$ and $D$ radial differential equations.

2sand2pElectron Impact Excitation in Atomic Hydrogen

A numerical calculation has been carried out to evaluate the 3\ifmmode\times\else\texttimes\fi{}3 cross-section matrix involved in the electron impact excitation of the ground state of H atom to the $2s$ and $2p$ levels. The method of solution is that of atomic eigenstates expansion. In this paper, instead of the iterative technique used by other authors, the definite integral terms in the coupled radial differential equations are eliminated through some linear transformation of the radial functions, thus avoiding iteration of these equations. The accuracy of the numerical integration is tested by satisfying the equation of reciprocity and the equation of continuity of currents with an error-to-value ratio less than 1 per 1000 on the average; and the maximum of this ratio, except for a few cases, has been kept below 5%. The results are in agreement with the results of an interative technique. To evaluate the effect of the long range and the centrifugal potential, a simple perturbation theory is developed. The six cross sections $1s\ensuremath{\rightarrow}2s$, $1s\ensuremath{\rightarrow}2p$, $1s\ensuremath{\rightarrow}1s$, $2s\ensuremath{\rightarrow}2s$, $2s\ensuremath{\rightarrow}2p$, and $2p\ensuremath{\rightarrow}2p$ are tabulated elsewhere, only the $2s\ensuremath{\rightarrow}2p$ and the $2p\ensuremath{\rightarrow}2p$ cross sections are reported here. The $2p\ensuremath{\rightarrow}2p$ cross section requires the solution of the sets of differential equations with different parities. Assuming the validity of the eigenstates expansion, it is found by comparison with the eigenstates expansion calculation that the Born approximation, despite its simplicity, gives meaningful results for low and close-to-the-threshold energies of the bombarding electrons. The effect of the exchange potentials on the cross sections is also investigated. Finally, an interesting structure of the $1s\ensuremath{\rightarrow}2s$ excitation cross section above threshold is found.

Perturbation treatment of the molecular helium ion

The binding energy of the /sup 1/Eu state of the molecular calculated by the method of distorted waves, the resonance approximation and of Seaton's version of the modified Born 3 x 3 cross-section matrix involved in the electron impact excitation of the ground state of H atom to the 2s and 2p levels. The method of solution is that of atomic eigenstates expansion. Instead of the iterative technique used by others, the definite integral terms in the coupled radial differential equations are eliminated through some linear transformation of the radial functions, thus avoiding iteration of these equations. The accuracy of the numerical integration is tested by satisfying the equation of reciprocity and the equation of continuity of currents with an error-to-value ratio less than 1 per 1000 on the average; and the maximum of this ratio, except for a few cases, was kept below 5%. The results are in agreement with the results of an iterative technique. To evaluate the effect of the long range and the centrifugal potential, a simple perturbation theory is developed. Only the 2s yields 2p and the 2p yields 2p cross more » sections are reported. The 2p yields 2p cross section requires the solution of the sets of differential equations with different parities. Assuming the validity of the eigenstates expansion, it is found by comparison with the eigenstates expansion calculation that the Born approximation, despite its simplicity, gives meaningful results for low and closeto-the-threshold energies of the bombarding electrons. The effect of the exchange potentials on the cross sections is also investigated. Finally, an interesting structure of the 1s yields 2s excitation cross section above threshold is found. (auth) « less

A method of computing phase shifts and normalizations of continuum-state dirac wave functions

A new method is proposed for computing the phase shifts and normalizations of the spherical wave functions of an unbound Dirac electron in the electric field of a point or finite-sized nucleus. The method uses the phase-amplitude formsG(r)=A cosϕ andF(r)=A sinϕ for the radial factors. The first step is to computeϕ(r 1), wherer 1 is larger than the nuclear radius but is still reasonable for machine computations in many cases of interest. This first step requires a numerical solution of radial differential equations fromr=0 tor=r 1. The second step is to compute the phase shift and a normalization correction factor directly from a new asymptotic series, whosen-th term is of the formP(ϕ)r 1 − , whereP(ϕ) is a finite Fourier series inϕ(r 1). Formulae for upper bounds on the error in the series are given and are used to estimate suitable values ofr 1 for several cases. The series is especially suitable for problems, like the decay of a μ-mesic atom, where the wave functions must be computed in any case up to distances ≳r 1. No comparison is made with other methods.

Analytic Properties of Radial Wave Functions

This is a review article about the properties of radial wave functions and other quantities relevant to the partial wave analysis of scattering theory, as functions of the energy or wave number. The treatment is restricted to the nonrelativistic Schrödinger equation for two particles with a local potential. In addition to regular and irregular solutions of the radial differential equations, the Jost function, S matrix, and Green's functions are analyzed and completeness is proved. The examples investigated in detail include the Bargmann potentials and their generalizations.

Energy Bands in Solids—The Quantum Defect Method

This paper describes a procedure for the calculation of electron energy bands in certain solids from spectroscopic data for the corresponding free atom. This method is an improved version of one used by Kuhn and Van Vleck to calculate the energy bands of sodium, potassium, and rubidium. It avoids explicit construction of a one-electron potential to represent the interaction between the valence and core electrons.We assume that the interaction between a valence electron and an ion in the crystal is approximately the same as in the isolated atom. If the interaction is accurately represented by a Coulomb potential outside the ion core, we may express solutions of the radial differential equation in this region as linear combinations of standard Coulomb functions. The combination corresponding to the solution which is well-behaved at the nucleus involves a coupling constant which depends upon the ion potential through a parameter that is a slowly varying function of the energy. At an eigenvalue this parameter can be evaluated from the quantum defect. Hence if the eigenvalue spectrum is known, we may obtain this parameter by extrapolation for arbitrary nearby energies, and the regular solution of the radial equation is consequently determined explicitly, outside the core. This is sufficient information for the calculation of energy bands with available techniques.We establish an approximate formula for the ratio of the amplitude of the wave function near the nucleus to its value at a point outside the core. For an $s$ function this relation involves only the nuclear charge, standard Coulomb functions, and the aforementioned parameter derived from spectroscopic data. It therefore provides a convenient means of calculating ${P}_{F}$ and ${P}_{A}$, the squared amplitudes at the nucleus appropriate to the Knight shift and the atomic hyperfine splitting, respectively. In the latter case our result is identical with a formula given by Fermi and Segr\`e, which gives reasonable agreement with experiment.Arguments are presented in support of the thesis that the quantum defect method takes very general account of exchange and correlation interactions between the valence electron and core electrons. Relativistic effects, including spin-orbit coupling, are also included naturally. We also discuss modifications in the method to take into account deviations of the ion or crystal potential outside the core from Coulomb form. Tables of the essential data, including improved polarization corrections, are given for the alkali metals.

Gravitational and thermal oscillations in the Earth's upper atmosphere

The recent work of Sen and White [1] on the excitation of large-scale oscillations in an atmosphere on a rotating globe by gravitational and thermally induced forces is extended to include heating in any portion of the atmosphere. The region where the heating occurs is unrestricted regarding temperature profile. The solution χ(z) of the radial differential equation of motion consists of (1) a complementary function yex/2, and (2) a particular solution q(z)/γH(z) + ex/2 I(x) where I is obtained by integrating the heat source function (with appropriate weight factor) from the top of the atmosphere to the level in question, γ is the ratio of specific heats, H(z) is the scale height, x = ∫dz/H where z is the altitude, q is the heat source function, and χ(z) is the divergence of the velocities. With this representation of the wave function, one recovers the expressions for the velocity components and pressure variations of Pekeris [2] and of Weekes and Wilkes [3] for the purely gravitational case, provided the wave function y is replaced by [y + I]. General expressions are also obtained for the vertical and horizontal displacements and for the amplification over equilibrium tide at any level. The latter expression is useful for establishing the so-called solar control of the dynamics of a given layer by radiation absorbed in any other layer. The boundary conditions of Weekes and Wilkes [3] are shown valid for a "top" with constant positive temperature gradient, believed to be the case above 100 km. Finally, the solar and lunar semi-diurnal winds as collected by Briggs and Spencer [4] for the E region when tested by present theory are found to have the same azimuthal dependence as exists at the ground. This is consistent with the original assumption of “variables separable.”