Partial Wave Sum Research Articles

Differential cross sections for chemically reactive systems

A quantum mechanical theory is developed for the differential and total elastic and reactive scattering cross sections for an electronically adiabatic bimolecular exchange reaction, with the restriction that the three atoms are constrained to move on a straight line, but with the whole system free to rotate in three dimensions. The introduction of a set of natural collision coordinates, together with a vibrationally adiabatic approximation, is used to reduce the scattering problem to the solution of one-dimensional Schrodinger equations. Semi-classical expressions for the elastic and reactive phase shifts are derived. The partial wave summations that occur in the theory are evaluated by semi-classical techniques and elastic and reactive differential cross sections are calculated for three different kinds of potential surface. A feature of the calculations is the appearance of a new kind of rainbow, which is named a ‘cubic’ rainbow since it arises when the deflection function varies cubically with impact p...

Born cross sections for the ionization of He and Li+ by electron impact

First Born approximation cross sections are calculated for the ionization of the ground states of He and Li+ by electron impact, in the energy range from threshold to 9 kev. An open-shell two-parameter wave function is used for the ground state of the target. A partial wave expansion is used for the ejected electron, the radial functions being continuum Hartree-Fock or Sloan polarized orbitals. The partial wave sums converge to within 01% of the total by l = 4. The calculations were carried out using both the length and velocity formulations of the Born approximation. The length results are approximately 9% higher than the velocity results for He and 6% higher for Li+. The cross sections obtained are in excellent agreement with experiment at energies in excess of twenty-five times threshold and approach the Bethe limit (within 3%) at energies in excess of fifty times threshold.

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.

Partial Wave Theory of Diatomic Molecules. II

The method of expanding the wavefunction of diatomic molecules in a series of orbital angular momentum eigenfunctions (partial waves) is extended to two-electron homonuclear molecules. The angular momentum is a function of the Euler angles only; a symmetric choice of these angles is used which greatly facilitates the description of the exchange character of the wavefunction. As a result, explicit equations for the three-dimensional ``radial'' functions can be derived for all the different magnetic, parity, and exchange states. For Σ states the partial wave sums only go over alternate values of the angular momentum giving rise to the expectation of an even more rapidly convergent series in these cases.

Summation of Partial Waves

For high-energy potential scattering, when the partial wave expansion converges too slowly to be directly useful, a technique is proposed for extracting the differential cross section more effectively from the phase shifts by means of a weight function and orthogonal polynomial approach.

π−πScattering with Unitarity and Crossing

A model of pion-pion scattering is presented in which each partial-wave amplitude in the s channel satisfies unitarity including inelastic states. This is accomplished by the use of the N/D technique. The full amplitude, i.e., the explicit, continued, partial-wave sum does not satisfy unitarity in the t and u channels, but it does have the correct branch points in these channels. These characteristics are guaranteed by construction and do not depend on arbitrary parameters occurring in the N functions. The parameters are fixed by imposing the crossing relations and by demanding the existence of a Pomeranchuk trajectory in the T = 0 channel. The existence of the p is not assumed. Since crossing is not satisfied exactly, the parameters are determined as those that yield the best fit to the crossing relations. Once the parameters are fixed, the behavior of each partial wave in each isospin channel is determined. Phase shifts are presented for the S, P, and D waves. The P wave is repulsive at low energies but becomes quite attractive at higher energies due to inelastic effects. A resonance occurs at s = 33.7 mu 2 with a width about three times that experimentally observed; this width ismore » extremely sensitive to the parameters, the position is not. An S-wave ghost occurs at s = --57.1 mu 2 (whose residue is zero) as well as the Pomeranchuk trajectory at s - 0. The S waves are strongly repulsive at low energies. In fact, they are so repulsive that rather broad peaks are produced in their cross sections near 400 Mev when the phase shifts pass through -- pi /2. The peak in the T = 0 channel may very well correspond to the Abashian, Booth, and Crowe (ABC) anomaly. There are no resonances in the D waves. In panticular, the existence of an f/sub 0/ is incompatible with any choice of the parameters unless it is accompanied by another strong D-wave resonance at low energy, and even this possibility violates crossing badly. This may be due to a poor choice for the trial function. (auth)« less

Higher Born Approximations for the Coulomb Scattering of a Spinless Particle

The Coulomb scattering amplitude of a Klein-Gordon particle is computed exactly to the third order in $\ensuremath{\beta}=\frac{Z{e}^{2}}{\ensuremath{\hbar}v}$ by perturbation theory using the potential $\frac{\mathrm{Ze}{e}^{\ensuremath{-}\ensuremath{\lambda}r}}{r}$ in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. It contains only finite terms except for terms containing powers of $\mathrm{ln}(\frac{|\mathrm{Q}|}{\ensuremath{\lambda}})$ which can be shown to arise from the expansion of a common phase factor. Thus, first- and second-order corrections to the relativistic Rutherford cross section are obtained.Our results are compared to the exact scattering amplitude in the form of the partial-wave summation, expanded in powers of $\ensuremath{\beta}$, which we have succeeded in summing to third order.

Addition theorems for arbitrary plane and spherical waves

is a representation for u as a sum of partial waves; i.e., as an infinite series each of whose terms solves the wave equation. The purpose of this note is to establish addition theorems for plane waves, u(x, y, z, t) = f(t-z), and spherical waves,