Amplitude-phase Method Research Articles

Evaluation of integrals involving products of oscillatory wavefunctions and inverse powers of r

The simple contour and high-order phase amplitude methods are described for the numerical evaluation of integrals involving products of inverse powers of r and oscillatory wavefunctions of particles in short-range potentials. The merits of these methods are assessed.

A numerically accurate investigation of black-hole normal modes

The oscillations of a Schwarzschild black hole, describing the late time ringing expected after, for example, a gravitational collapse, are discussed in terms of the characteristic normal-mode frequencies. A condition determining these frequencies is derived within the phase-amplitude method. The numerical results obtained using this condition are of very high accuracy, and the phase-amplitude analysis seems to provide a powerful alternative to the previous investigations of the normal-mode problem.

Amplitude-phase method of measuring acoustic system impedance in the low- and infralow frequency bands

Amplitude-phase method of measuring acoustic system impedance in the low- and infralow frequency bands

An amplitude-phase method of measuring the radiation patterns of linear antennas

An amplitude-phase method of measuring the radiation patterns of linear antennas

Molecular Rydberg states. XI. Quantum defect analogies between molecules and rare gases

The s and d Rydberg series of hydrogen iodide and methyl iodide are reported. These series are assigned using analogies to the corresponding series in xenon. It is shown that a comparison between molecular spectra and pertinent atomic data provides a facile assignment technique for molecular Rydberg spectra. Using the phase amplitude method, the Rydberg data are analyzed in an attempt to provide information on the residual atomic and molecular potentials and to categorize them qualitatively as attractive/repulsive, long range/short range. It is shown that, for l?2, the centrifugal barrier inhibits ingress of the Rydberg electron into the core and that the molecule is effectively spherically symmetric.

A generalized approach to the phase-amplitude method

A generalized formulation of the phase-amplitude method is given for scattering by a complex nonlocal potential. It is shown that the constraint on the derivative of the wavefunction introduced by earlier workers is redundant to the development of the approach. The cases of real local and nonlocal potentials as well as a complex local potential can be obtained in a straightforward manner.

Measurement of angular displacements by a photoelectric amplitude-phase method

Measurement of angular displacements by a photoelectric amplitude-phase method

Electron-optical properties of atomic fields

We present a unified discussion and illustrations of the electron-optical aspects of electron penetration into, or escape from, the inner region of atoms. Both processes may focus or defocus the amplitudes of wavefunctions and shift their phases, as manifested in countless phenomena ranging from level shifts to $\ensuremath{\beta}$-decay rates. A background survey begins by discussing the Fermi-Segr\`e formula for hyperfine splittings and emphasizes the interplay of hydrogenic and WKB approximations. The Phase-Amplitude Method, which determines amplitude ratios and phase shifts directly, proves useful for interpreting the systematics of these parameters along the Periodic System. We present results of survey calculations, carried through the Periodic System using Hartree-Slater potential fields, of: (a) $\frac{{\ensuremath{\alpha}}_{l}(0)}{{\ensuremath{\alpha}}_{l}(\ensuremath{\infty})}$, the ratio of the wavefunction's amplitude at $r=0$ to that outside the atom; (b) ${\ensuremath{\delta}}_{l}(E=0)$ and ${\frac{d{\ensuremath{\delta}}_{l}}{\mathrm{dE}}|}_{E=0}$, the phase shift and its energy derivative at $E=0$, and (c) the changes in ${\ensuremath{\delta}}_{l}(E=0)$ and $\frac{{\ensuremath{\alpha}}_{l}(0)}{{\ensuremath{\alpha}}_{l}(\ensuremath{\infty})}$ induced by either a unit perturbation localized near $r=0$ or a relativistic correction. Thus we provide a broad mapping of certain fundamental parameters based on rather crude but realistic calculations. These results are meant to serve as a bench mark in surveying problems and in checking new results, while standard methods are preferable for working out specific applications accurately.

Comment on Rapid Numerical Solution of the One Dimensional Schrodinger Equation

In a recent article Wills alleges that the method for rapid numerical solution of the one dimensional Schrodinger equation devised by Newman and Thorson "is equivalent to the well known phase-amplitude method". A critique of this opinion is given.

Scattering cross sections for 40 eV to 1 keV electrons colliding elastically with nitrogen and oxygen

A phase-amplitude method has been used in deriving differential and total elastic scattering cross sections for electron collisions in atomic and diatomic molecular nitrogen and oxygen. The energy range considered has been from 1 keV to below 50 eV. Available theoretical and experimental cross sections for both atoms and molecules have been compared with the present calculations.

On the Rapid Numerical Solution of the One-Dimensional Schrödinger Equation

It is shown that the "new" method of rapid solution of the one-dimensional Schrödinger equation proposed by Newman and Thorson is equivalent to the well known phase–amplitude method.

Clinical equipment for measurement of middle-ear muscle reflexes and tympanometry.

The range of applications of middle-ear muscle reflex measurements is reviewed. Different methods for measuring the relative impedance change, including extratympanic manometry and a method of tympanometry, are discussed in relation to the authors' equipment, which is of four main kinds: the intra-aural reflex indicator, a pressure transducer system for extratympanic manometry, a system for changing and monitoring the air pressure in the ear canal for tympanometry, and a four-channel recorder. The authors have found that intra-aural reflex thresholds are highly similar, whether determined by changes in amplitude or amplitude-phase. Furthermore, extratympanic manometry performed by the authors' method does not show reflex thresholds to be as sensitive as do the other methods. In studies of reflex growth and preliminary measurements of the decay of the stapedial reflex, amplitude-phase method seems superior because of larger responses. The method of tympanometry for measuring ear-drum mobility and the status of the ossicular chain and middle-ear cavity will identify certain pathological conditions of the middle ear. We have found 800 Hz to be the preferred working frequency for this measurement.

Phase-Amplitude Method in Atomic Physics. I. Basic Formulas for an Electron in an Ionic Potential

The radial Schr\odinger equation for a modified Coulomb field is separated into two first-order equations for a function and an function which vary only in the non-Coulomb region $rl{r}_{0}$. At $r={r}_{0}$, these functions are related to the mixing parameter and normalization constant of quantum-defect theory. Integrals over the range [$0,{r}_{0}$] give the scattering phase shift and the wave function's amplitude near the nucleus. Phase shifts for negative energies serve to calculate the discrete eigenvalues of an atom or positive ion. A numerical procedure for this approach is outlined.

The measurement of permittivity of high-loss liquids at microwave frequencies using an unmatched phase changer

The behaviour of an adjustable length of coaxial line containing a standing wave is considered theoretically. It is shown that a relationship can be obtained between the change in the line length and the resulting change of phase. This is then followed by a description of how the complex permittivity of a liquid such as water can be measured using the phase changer in conjunction with a phase-amplitude balance method.

Very low frequency measurements in northern Sweden

As an extension of the experiments, made in 1963–1964, regarding the use of very low frequency (V.L.F.) radio signals for ore prospecting, the Geological Survey of Sweden carried out further field studies of the V.L.F. technics for this purpose in 1965–1966. The measurements included the determination of amplitude and dip of very low frequency signals or the determination of three mutually perpendicular field components over known ores. The observations were based on signals from two specific transmitters: G.B.R. in England ( ƒ= 16 kc/sec ) and N.A.A. in U.S.A. ( ƒ= 17.8 kc/sec ). Investigations have been made on ores having different properties with respect to electrical conductivity and magnetic susceptibility. Measurements along profiles and over an area have been carried out. The utility of the so called amplitude-phase method was tested. The results obtained show the usefulness of the V.L.F. technique in some arrangements which offer different facilities with regard to simplicity, speed and abundance of details.

Dynamic Mechanical Properties of Chlorinated Biphenyls and Polystyrene—Chlorinated Biphenyl Solutions at 40 MHz

Measurements of dynamic shear impedance at 40 MHz and 25°C were carried out on solutions of monodisperse polystyrenes of molecular weight 2.67×105 and 2.39×105 in chlorinated biphenyl solvents (Aroclors). Measurements were also obtained on solutions of monodisperse polystyrenes of molecular weight 2.03×103 and 1.8×106 in di-n-butyl phthalate (DBP). In addition, measurements were obtained on one binary mixture of Aroclors and on five pure Aroclor solvents. These covered a range of 106 in steady-flow viscosity. The dynamic measurements were made by an ultrasonic reflectance technique employing pulse super-position or by a phase-amplitude balancing method. Results for the polystyrene—Aroclor system indicate that the effect of a solvent slightly better thermodynamically than DBP is observed as an increase in the in-phase modulus, whereas little effect is observed in the magnitude of the polymer contribution to the out-of-phase modulus (G″—ωηs), where ω is the angular frequency, ηs is the solvent viscosity, and G″ is the modulus. Measurements on polystyrene dissolved in an Aroclor solvent for which ηs is about 3 P indicate a reduced dynamic viscosity lower by a factor of 2.5 than that obtained at 73 kHz by other workers. Results for the low-molecular-weight polymer (2.3×103) indicate a behavior more nearly Newtonian than that predicted by the Zimm theory, while results for the high-molecular-weight polymer (1.8×106) conform closely to it. Measurements for the Aroclors show that viscoelastic behavior is already significant for an Aroclor with a steady-flow viscosity of 3 P. Logarithmic plots of reduced values of G′ and G″ cross at a reduced frequency of about 109.90 and both G″/ρ and η″/η exhibit a maximum at a reduced frequency of about 1010 Hz. The series is analyzed in terms of several theories of viscoelastic behavior.

Elastic Electron Scattering Amplitudes for Neutral Atoms Calculated Using the Partial Wave Method at 10, 40, 70, and 100 kV for Z = 1 to Z = 54

Elastic electron scattering amplitudes for neutral atoms were calculated using the partial wave expansion method for 10-, 40-, 70-, and 100-kV incident electrons. The partial wave phase shifts were calculated by numerical intergration using the phase amplitude method until the results converged to values obtained using the WKBJ and first Born approxmations, which were then used in the remainder of the partial wave sum. The static potential field of the target atoms was represented by an analytical expression involving a sum of Yukawa terms. Potential field parameters for the expression were obtained by a least-squares fit of the radial electron distribution function, D (r), using Hartree—Fock and relativistic wavefunctions for all the neutral atoms from Z=1 to Z=54. For the smaller atoms from Z=1 to Z=21, Clementi Hartree—Fock wavefunctions were used and for the atoms from Z=22 to 54 Liberman—Waber—Cromer D (r) curves calculated using the Dirac Hamiltonian incorporating a j—j coupling scheme and the Slater ⅔ approximation in the exchange potential were used.

Rapid Calculation of Electron Scattering Factors

The phase amplitude method is used to reduce the radial Schrödinger equation to two separate differential equations, one for the phase and one for the amplitude. These functions are both smooth as opposed to the rapidly oscillating solution of the radial Schrödinger equation for electron energies in the kilovolt range. The partial-wave phase shifts were obtained rapidly by integrating the differential equations for the phase and amplitude numerically. Hartree—Fock and Thomas—Fermi—Dirac fields were used in the calculation. Results for argon and uranium are given in order to compare with previous results. It was found that the WKBJ approximation to the partial-wave phase shift is a good approximation for the energies used in electron diffraction. This rapid method of computing electron-scattering factors will make routine analysis of electron diffraction data more rapid as well as more exact.

Bessel functions for large arguments

Calculations of Bessel Functions of real order and argument for large values of the argument can be greatly facilitated by the use of the so called phase-amplitude method [1]. In this method two auxiliary functions, the amplitude and phase functions, are defined in terms of the regular and irregular solution of Bessel's equation. These auxiliary functions have the great advantage that they are monotonic functions of the argument; moreover, for large arguments these functions are slowly varying and, hence, easily amenable to computation and interpolation. The Bessel Functions of the third kind, so-called Hankel Functions, are defined as follows: (1.1) H,(')(x) = J,(x) + iY,(x) and (1.2) H,(2)(x) = J,(x) iY,(x). These definitions suggest the alternative definition of the Bessel Function in terms of an amplitude and phase function, viz., (2.1) AV ,1(x) I--II(2x(x) I or (2.2) Hp(') = Av+ifv (2.3) = A so that (2.4) J,(x) =A, cos 4, and (2.5) Y,(x) = A, sin 4, The phase function P,(x) obeys the first order differential equation

A note on the calculation of Coulomb penetration and Coulomb wave functions

The phase-amplitude method is applied to the calculation of Coulomb penetration and Coulomb wave functions. The methods discussed apply for η = Z 1Z 2e 2/ h ̵ v>1 .