Showing papers in "Physical Review A in 1980"
TL;DR: In this article, a general basis for the formalism is developed, and it is then applied to find transition probabilities for any order of interaction for both linearly and circularly polarized plane-wave fields.
Abstract: The approximation method introduced by Keldysh is revised and extended. The technique is applicable to the photodetachment by a plane-wave field of an electron bound by a short-range potential. The approximation is to neglect the effect of the binding potential as compared to the field effects on the final state of the detached electron. By choice of a different gauge than that used by Keldysh, the formalism becomes very simple and tractable. A general basis for the formalism is developed, and it is then applied to find transition probabilities for any order of interaction for both linearly and circularly polarized plane-wave fields. The low-intensity, first-order limit and the high-intensity, high-order limit yield the correct results. Two intensity parameters are identified. The fundamental one is $z=\frac{{e}^{2}{a}^{2}}{4m\ensuremath{\omega}}$, where $a$ is the magnitude of the vector potential (in radiation gauge) of the field of circular frequency $\ensuremath{\omega}$. The second parameter is ${z}_{1}=\frac{2z\ensuremath{\omega}}{{E}_{B}}$, where ${E}_{B}$ is binding energy, and it becomes important only in the asymptotic case. With the assumption that the field leaves the neutral atomic core relatively unaffected, the formalism is applied to the example of the negative hydrogen ion irradiated by circularly or linearly polarized 10.6-\ensuremath{\mu}m radiation. Photodetachment angular distributions and total transition probabilities are examined for explicit intensity effects. It is found that total transition probability $W$ is not sensitive to intensity since $\frac{d(logW)}{d}$ ($logz$) retains low-intensity straight-line behavior up to quite high values of $z$. An important intensity effect is the major significance of higher-than-lowest-order terms when $z$ is large, especially for circular polarization. A sensitive indicator of intensity is the ratio of photodetachment probabilities in circularly and linearly polarized fields, which increases sharply with intensity. An investigation of the convergence of perturbation expansions gives the upper limit $zl[\frac{{E}_{B}}{\ensuremath{\omega}}]\ensuremath{-}\frac{{E}_{B}}{\ensuremath{\omega}}$, where the square bracket means "smallest integer containing" the quantity in brackets. This limit is $zl0.59$ for ${\mathrm{H}}^{\ensuremath{-}}$ in 10.6-\ensuremath{\mu}m radiation. The failure of perturbation theory is not necessarily manifest in qualitative ways. For example, it is not apparent in total photoelectron yield as a function of intensity.
1,125 citations
TL;DR: In this article, a density-functional theory for describing polarization-type many-body effects influencing the photoresponse of small electronic systems is presented. But it does not take account of the time-dependent fields induced by an external radiation field.
Abstract: We present a formalism based on density-functional theory capable of describing polarization-type many-body effects influencing the photoresponse of small electronic systems. The self-consistent field approach we describe incorporates correlations into a local, effective single-particle potential which takes account of the time-dependent fields induced by an external radiation field. In this paper we present calculations of the static polarizabilities, total photoabsorption cross sections, and selected partial photoabsorption cross sections of the rare gases which yield results in good agreement with experiment. A study of the energy and spatial dependence of the local field leads to a clear physical picture of the dielectric properties of these systems.
718 citations
TL;DR: In this paper, the authors generalize the response given long ago by a number theorist to a student who asked “Why is number theory useful?” His reply: “It is useful because one can get a Ph.D with it!”
Abstract: Why is the study of the relativistic theory of many-electron bound systems interesting and useful? As regards utility, a minimal answer can be given by generalizing the response given long ago by a number theorist to a student who asked “Why is number theory useful?” His reply: “It is useful because one can get a Ph.D. with it!”
623 citations
TL;DR: In this paper, the authors derived from quantum theory the optical force, its first-order velocity dependence, and its fluctuations for arbitrary light intensity, and applied the results to the problem of creating a stable optical trap for sodium atoms.
Abstract: The force exerted by optical-frequency radiation on neutral atoms can be quite substantial, particularly in the neighborhood of an atomic resonance line. In this paper we derive from quantum theory the optical force, its first-order velocity dependence, and its fluctuations for arbitrary light intensity, and apply the results to the problem of creating a stable optical trap for sodium atoms. New results include the position dependence of the velocity-dependent force, a complete expression for the momentum diffusion constant including the substantial contribution from fluctuations of the dipole force, and an estimate of trapping times in excess of 1 sec even in the absence of effective damping. The paper concludes with a discussion of the prospects and difficulties in providing sufficient damping to stabilize such a trap.
503 citations
TL;DR: In this paper, the Coulomb free energy of the classical one-component plasma was derived by using converged Monte Carlo chains and a more accurate potential approximation than that used by Hansen.
Abstract: We compute the internal energy of the classical one-component plasma, for values of the Coulomb coupling parameter $\ensuremath{\Gamma}$ between 1 and 300, by using converged Monte Carlo chains and a more accurate potential approximation than that used by Hansen. The liquid data are fitted to a simple, very accurate four-parameter formula from which the Helmholtz free energy is derived. The solid data are likewise fitted to a one-parameter formula. The intersection of the two free-energy curves gives an estimate of the fluid-solid transition at $\ensuremath{\Gamma}=168\ifmmode\pm\else\textpm\fi{}4$.
428 citations
TL;DR: In this article, a set of rate equations is proposed to describe nucleation and growth of droplets in metastable, near-critical fluids, and these equations are used in conjunction with steady-state nucleation theory to compute completion times for phase separation in binary mixtures.
Abstract: A simple set of rate equations is proposed to describe nucleation and growth of droplets in metastable, near-critical fluids. These equations are used in conjunction with steady-state nucleation theory to compute completion times for phase separation in binary mixtures. Reexamination of available experimental data provides little if any evidence for the major failure of conventional nucleation theory that has been postulated on the basis of these data.
357 citations
TL;DR: In this paper, an individual barium ion, continuously observed by laser fluorescence, has been isolated in a Paul rf quadrupole trap at room temperature and its image has been reduced in thickness to 2 εm in the object plane, the diffraction limit.
Abstract: An individual barium ion, continuously observed by laser fluorescence, has been isolated in a Paul rf quadrupole trap at room temperature. By optical sideband cooling its microscopically measured image has been reduced in thickness to \ensuremath{\sim} 2 \ensuremath{\mu}m in the object plane, the diffraction limit. Estimated ion temperatures reached are ${T}_{i}\ensuremath{\simeq}10 \mathrm{to} l36$ mK. With cooling, the ion could be held indefinitely, without cooling \ensuremath{\sim} 30 s. In the future the technique seems capable of attaining kinetic temperatures \ensuremath{\sim} ${10}^{\ensuremath{-}8}$ K, much lower than realized so far by other means, with corresponding far-reaching implications.
344 citations
TL;DR: In this article, a detailed analysis of the dependence of the cross sections and rate coefficients on electron energy and temperature, and on atomic parameters is presented, including sudden as well as adiabatic collisions, for electron-induced ionization, excitation, and de-excitation.
Abstract: For electron-induced ionization, excitation, and de-excitation, mainly from excited atomic states, a detailed analysis is presented of the dependence of the cross sections and rate coefficients on electron energy and temperature, and on atomic parameters. A wide energy range is covered, including sudden as well as adiabatic collisions. By combining the available experimental and theoretical information, a set of simple analytical formulas is constructed for the cross sections and rate coefficients of the processes mentioned, for the total depopulation, and for three-body recombination. The formulas account for large deviations from classical and semiclassical scaling, as found for excitation. They agree with experimental data and with the theories in their respective ranges of validity, but have a wider range of validity than the separate theories. The simple analytical form further facilitates the application in plasma modeling.
285 citations
TL;DR: In this article, the authors examined experimental cross sections for charge transfer in collisions of partially stripped heavy ions with atomic hydrogen at low collision energies, using a model in which the projectiles are replaced by bare nuclei of a given effective charge, and showed the presence of a strong oscillatory dependence of the cross sections on effective charge due to the crossings of diabatic potential curves in the low energy region below 10 keV/amu.
Abstract: To examine experimental cross sections for charge transfer in collisions of partially stripped heavy ions with atomic hydrogen at low collision energies, unitarized-distorted-wave-approximation calculations are performed using a model in which the projectiles are replaced by bare nuclei of a given effective charge. The results show the presence of a strong oscillatory dependence of the cross sections on effective charge due to the crossings of diabatic potential curves in the low-energy region below 10 keV/amu. The considerable differences in the measured cross sections for impacts of ions of different elements (B, C, N, and O) observed by Bayfield et al. and Crandall et al. at low impact energies are attributed to this oscillatory behavior.
277 citations
TL;DR: In this article, a new Hamiltonian method for deformation simulations is related to the Green-Kubo fluctuation theory through perturbation theory and linear response theory, in a way consistent with enhanced "long-time tails".
Abstract: A new Hamiltonian method for deformation simulations is related to the Green-Kubo fluctuation theory through perturbation theory and linear-response theory. Numerical results for the bulk and shear viscosity coefficients are compared to corresponding Green-Kubo calculations. Both viscosity coefficients depend similarly on frequency, in a way consistent with enhanced "long-time tails."
256 citations
TL;DR: An electronic Hamiltonian appropriate for describing core-hole processes is derived and its properties are discussed in this article, where a many-body theory is presented, and a useful linked-cluster theorem is proven.
Abstract: An electronic Hamiltonian appropriate for describing core-hole processes is derived and its properties are discussed. Based on this Hamiltonian a many-body theory is presented, and a useful linked-cluster theorem is proven. A distinction between relaxation and correlation terms can be made in each order of the perturbation expansion. Under certain well-defined conditions the core-ionization process can be visualized as a Franck-Condon transition between two multidimensional potential-energy surfaces in an abstract bosonic space. The corresponding approximation, the boson approximation, is discussed and applied to the water molecule.
TL;DR: In this article, an integral equation determining the density induced by an external perturbation is derived and the independent-particle response function for an inhomogeneous system appearing in this equation is obtained in a form convenient for numerical calculations, applied to the calculation of dipole polarizabilities of spherically symmetric atoms and ions.
Abstract: The linear response of an arbitrary electronic system is considered within the framework of the density-functional theory. An integral equation determining the density induced by an external perturbation is derived. The independent-particle response function for an inhomogeneous system appearing in this equation is obtained in a form convenient for numerical calculations. The equation is applied to the calculation of dipole polarizabilities of spherically symmetric atoms and ions. Results for the rare-gas atoms obtained by means of the local-density approximation are in good agreement with experimental values, except for the case of He. Polarizabilities are also evaluated for the alkali-metal ions, the alkaline-earth and other closed-shell metallic atoms.
TL;DR: In this article, a strong steady dense-fluid shock wave is simulated with 4800-atom nonequilibrium molecular dynamics, and the resulting density, stress, energy, and temperature profiles are compared with corresponding macroscopic profiles derived from Navier-Stokes continuum mechanics.
Abstract: A strong steady dense-fluid shock wave is simulated with 4800-atom nonequilibrium molecular dynamics. The resulting density, stress, energy, and temperature profiles are compared with corresponding macroscopic profiles we derive from Navier-Stokes continuum mechanics. The differences found are relatively small.
TL;DR: In this paper, the authors considered the problem of minimum entropy production in a heat engine subject only to thermal-resistance losses and derived an expression for the minimum total entropy production and used this to give a bound on the maximum work which can be produced by such engines.
Abstract: We consider the problem of minimum entropy production in a heat engine subject only to thermal-resistance losses. For such engines, minimizing the total entropy production is equivalent to minimizing the loss of availability. We show for any engine operating with a given cycle time that minimum total entropy production is achieved in a heat engine by operating it so as to keep the entropy production rate constant along each branch. For the limit of slow engine operation, the entropy production rate should be the same constant for all branches of the cycle. We obtain an expression for the minimum total entropy production and use this to give a bound on the maximum work which can be produced by such engines. This bound is significantly more realistic than the reversible one. Analogous results are derived for a working fluid which carries arbitrary flows from one potential to another.
TL;DR: In this article, the angular distributions for elastically scattered electrons were measured in a crossed-beam geometry using a collimated, differentially pumped atomic-beam source which requires no effective-path-length correction.
Abstract: Absolute differential, integral, and momentum-transfer cross sections for electrons elastically scattered from helium are reported for the impact energy range of 5 to 200 eV. Angular distributions for elastically scattered electrons are measured in a crossed-beam geometry using a collimated, differentially pumped atomic-beam source which requires no effective-path-length correction. Below the first inelastic threshold the angular distributions were placed on an absolute scale by use of a phase-shift analysis. Above this threshold, the angular distributions from 10 to 140 deg were fitted using the phase-shift technique, and the resulting integral cross sections were normalized to a semiempirically derived integral elastic cross section. Depending on the impact energy, the data are estimated to be accurate to within 5 to 9%.
TL;DR: In this paper, the authors measured the energy and angular dependence of the cross-sections for scattering of helium ions at four different energies, using self-supporting vacuum-deposited polycrystalline carbon and gold foils.
Abstract: Total differential cross-section ratios for scattering of ${\mathrm{H}}^{+}$, ${\mathrm{He}}^{+}$, and ${\mathrm{Li}}^{+}$ ions incident on bismuth-zinc and gold-carbon systems have been measured. The energy dependence of the cross sections was measured for each species at a fixed laboratory backscattering angle (${\ensuremath{\phi}}_{\mathrm{lab}}=170\ifmmode^\circ\else\textdegree\fi{}$), using an amorphous carbon target implanted with 10-keV ${\mathrm{Zn}}^{+}$ and ${\mathrm{Bi}}^{+}$ at a depth of \ensuremath{\sim}2.7 \ensuremath{\mu}g/${\mathrm{cm}}^{2}$. Angular distributions (${\ensuremath{\phi}}_{\mathrm{lab}}=15\ifmmode^\circ\else\textdegree\fi{}, \dots{}, 170\ifmmode^\circ\else\textdegree\fi{}$) were measured for helium ions at four different energies, using selfsupporting vacuum-deposited polycrystalline carbon and gold foils. For the case of backscattering, the cross-section ratios $\frac{{(d\ensuremath{\sigma})}_{\mathrm{Bi}}}{{(d\ensuremath{\sigma})}_{\mathrm{Zn}}}$ deviate significantly from both the Rutherford-scattering law and the Lindhard, Nielsen, and Scharff differential-scattering cross section. The deviations of the absolute cross sections from the Rutherford cross sections amount to 3.5% for 1-MeV and 16% for 0.2-MeV helium on bismuth. The experimental results are in good agreement with exact classical differential-scattering cross sections based on the Lenz-Jensen and Dirac-Hartree-Fock-Slater atomic potentials. Simple analytical formulas describing the energy and angular dependence of the cross sections are presented.
TL;DR: In this paper, the equivalence of the two-body interactions between the final particles of the ion-atom ionization process is considered in the $T$-matrix formalism.
Abstract: The equivalence of the two-body interactions between the final particles of the ion-atom ionization process is considered in the $T$-matrix formalism. This defines a final wave function which is product of Coulomb waves and describes simultaneously the capture to the continuum and the direct ionization. The relation of that approach with the wave equation for the three-particle system and the asymptotic behavior are discussed. The doubly differential cross section for the ionization of hydrogen atoms by proton impact is calculated, in first order, and compared with experimental values.
TL;DR: In this article, the effects of gravity, inertia, and quantum mechanics on the motion of the neutron were investigated using a neutron interferometer of the type developed by Bonse and Hart for x rays, and the importance of these experiments with regard to the role of the principle of equivalence in quantum mechanics is discussed.
Abstract: The experiments described in this paper probe the simultaneous effects of gravity, inertia, and quantum mechanics on the motion of the neutron. Using a neutron interferometer of the type developed by Bonse and Hart for x rays, we have observed quantum-mechanical interference phenomena induced by the gravitational field of the Earth and by the Earth's rotation relative to the fixed stars. The importance of these experiments with regard to the role of the principle of equivalence in quantum mechanics is discussed.
TL;DR: In this paper, it was shown that any cross section, or process rate, involving charged particles, exhibits noise as an infrared phenomenon, and the observed noise is the sum of these contributions, and can be used to detect new infraquanta.
Abstract: On the basis of the known experimental properties of $\frac{1}{f}$ noise, some previous models are analyzed. The presence of $\frac{1}{f}$ noise in the simplest systems such as beams of charged particles in vacuum, the existence of $\frac{1}{f}$ noise in currents limited by the surface recombination rate, bulk recombination rate, or by the finite mobility determined by interaction with the phonons in solids, suggests a fundamental fluctuation of the corresponding elementary cross sections. This leads to fluctuations of the kinetic transport coefficients such as mobility $\ensuremath{\mu}$ or recombination speed, observable both in equilibrium and nonequilibrium. In the first case the available Johnson noise power $\mathrm{kT}$, determined by the Nyquist theorem, is free of this type of $\frac{1}{f}$ fluctuation. An elementary calculation is presented which shows that any cross section, or process rate, involving charged particles, exhibits $\frac{1}{f}$ noise as an infrared phenomenon. For single-particle processes, the experimental value of Hooge's constant is obtained as an upper limit, corresponding to very large velocity changes of the current carriers, close to the speed of light. The obtained ${sin}^{2}(\frac{\ensuremath{\theta}}{2})$ dependence on the mean scattering angle predicts much lower $\frac{1}{f}$ noise for (low-angle) impurity scattering, showing a strong ($\ensuremath{\sim}\frac{{\ensuremath{\mu}}^{2}}{{\ensuremath{\mu}}_{\mathrm{latt}}^{2}}$) noise increase with temperature at the transition to lattice scattering. This is in qualitative agreement with measurements on thin films and on heavily doped semiconductors, or on manganin. The theory is based on the infrared quasidivergence present in all cross sections (and in some autocorrelation functions) due to interaction of the current carriers with massless infraquanta: photons, electron-hole pair excitations at metallic Fermi surfaces, generalized spin waves, transverse phonons, hydrodynamic excitations of other quanta, very low-energy excitations of quasidegenerate states observed, e.g., in disordered materials, at surfaces, or at lattice imperfections, etc. The observed $\frac{1}{f}$ noise is the sum of these contributions, and can be used to detect and study new infraquanta.
TL;DR: In this paper, the authors used X-ray photoelectron spectroscopy (XPS) to measure lifetime broadening of core levels with binding energies less than 1500 eV in approximately 25 elements.
Abstract: X-ray photoelectron spectroscopy (XPS) has been used to measure lifetime broadening of $L$, $M$, $N$, $O$, and $P$ core levels with binding energies less than 1500 eV in approximately 25 elements. The results show that the framework provided by previously accepted theoretical estimates of lifetime broadening is sometimes misleading. Lifetimes derived from theory and experiment often differ by a factor of 2 or more. In the transition metals of the first period plots of measured widths of the ${L}_{1}(2s),{L}_{2}(2{p}_{\frac{1}{2}})$, and ${L}_{3}(2{p}_{\frac{3}{2}})$ levels as a function of atomic number show broad maxima which are not adequately described by theory. The origin of this broadening is extensively discussed, as is the overestimation of lifetimes by theory in other areas of the periodic table.
TL;DR: In this article, the magnitude and sign of laser saturation resonances induced in atomic transitions with level degeneracy can be dramatically altered by the presence of optical pumping, and the physical principles underlying the laser optical-pumping saturation effect are discussed.
Abstract: The magnitude and sign of laser saturation resonances induced in atomic transitions with level degeneracy can be dramatically altered by the presence of optical pumping. This paper presents a detailed theoretical treatment of optical-pumping Lamb dips and crossover resonances, and their experimental observation in the 553-nm barium resonance transition. The physical principles underlying the laser optical-pumping saturation effect are also discussed.
TL;DR: In this paper, the contribution of elastic photon scattering from an atom due to scattering off the bound atomic electrons (Rayleigh scattering) is calculated and compared with other theories, particularly the form-factor approximation.
Abstract: We calculate the contribution to elastic photon scattering from an atom due to scattering off the bound atomic electrons (Rayleigh scattering). We compare predictions resulting from our numerical evaluation of the relativistic second-order $S$ matrix in a screened central potential with other theories, particularly the form-factor approximation. We give a prescription for accurate $O(1%)$ evaluation of total-atom Rayleigh amplitudes (summed over electrons) and present sample tabulations for lead ($Z=82$) for energies of experimental interest in the range from 22.1 to 2750 keV. Based on our prescription we compare elastic-scattering cross sections using Rayleigh amplitudes with selected experiments and are able to remove the large factor-of-two discrepancies previously reported.
TL;DR: In this paper, the Sternheimer equation for atomic polarizability is modified to account for the self-consistent field, and the equations are quite simple for closed-shell ions, and numerical results are presented.
Abstract: The Sternheimer equation for atomic polarizability is modified to account for the self-consistent field. The equations are quite simple for closed-shell ions. Numerical results are presented.
TL;DR: The positions of the /sup 2/Pi/sub g/ resonance peaks were determined with much greater accuracy ( +- 15 meV) than in previous transmission measurements, indicating that these features are not a property of the N/sub 2/ ground vibronic state.
Abstract: Absolute total electron scattering cross sections for N/sub 2/ from 0.5 to 50 eV have been measured with an estimated uncertainty of +- 3% using a transmission time-of-flight method previously described. The results are compared to previous experimental results and to recent calculations. The positions of the /sup 2/Pi/sub g/ resonance peaks were determined with much greater accuracy ( +- 15 meV) than in previous transmission measurements. The structure reported by Golden (1966) below the /sup 2/Pi/sub g/ resonance was clearly not present, indicating that, if real, these features are not a property of the N/sub 2/ ground vibronic state. The shape resonance predicted at 11 eV by Dill and Dehmer (1977) was not seen, perhaps because it was too weakly manifested in the total cross section. A weak broad band centered at 25 eV may be interpreted as being due to a sigma/sub u/ shape resonance as predicted by Dehmer, Siegel, Welch, and Dill.
TL;DR: In this paper, a general analysis of free electron lasers in which a static periodic magnetic pump field is scattered from a relativistic electron beam is presented, and scaling laws for the growth rates and efficiencies at a fixed radiation frequency as a function of the magnetic pump amplitude are obtained.
Abstract: Abstract : A general analysis is presented of free electron lasers in which a static periodic magnetic pump field is scattered from a relativistic electron beam. The steady state formulation of the problem is fully relativistic and contains beam thermal effects. Growth rates associated with the radiation field, efficiencies, and saturated field amplitudes are derived for various modes of operation. Effects of space charge on the scattering process are included and shown to play a dominant role in certain situations. Scaling laws for the growth rates and efficiencies at a fixed radiation frequency as a function of the magnetic pump amplitude are obtained. The shear in beam axial velocity due to self fields is discussed and various methods of reducing it are suggested. Finally, a detailed illustration of a far infrared (lambda = 2mu m) two-stage free electron laser using a 3 MeV electron beam and a 2 cm wavelength magnetic pump field is presented.
TL;DR: In this paper, relativistic calculations of atomic K-shell Auger transition probabilities are reported for 25 elements with Z between 18 and 96, inclusive, and the theoretical relative intensities of K-LX Auger transitions in heavy elements, calculated in j-j coupling, agree well with measurements; for low Z, configuration interaction and intermediate coupling must apparently be included.
Abstract: Systematic relativistic (Dirac-Hartree-Slater) calculations of atomic K-shell Auger transition probabilities are reported for 25 elements with Z between 18 and 96, inclusive. K-level Auger widths are found to be enhanced by relativistic effects, while total K-level widths are reduced. Relativistic theoretical K-shell fluorescence yields are in excellent agreement with experiment. Theoretical relative intensities of K-LX Auger transitions in heavy elements, calculated in j-j coupling, agree well with measurements; for low Z, configuration interaction and intermediate coupling must apparently be included.
TL;DR: In this paper, the Thomas-Fermi computation of the total binding energy of neutral atoms was adapted to strongly bounded electrons, and the correct treatment of strongly bound electrons was grafted smoothly onto the Thomas Fermi computations.
Abstract: The correct treatment of strongly bound electrons is grafted smoothly onto the Thomas-Fermi computation of the total binding energy of neutral atoms. This provides a clearcut demonstration of the leading correction of relative order ${Z}^{\ensuremath{-}\frac{1}{3}}$ which, with effects of relative order ${Z}^{\ensuremath{-}\frac{2}{3}}$, gives an accurate account of the binding energy over a wide range of $Z$ values. There is a brief discussion of relativistic corrections, with results that are somewhat at variance with previous numerical estimates.
TL;DR: In this article, the Coulomb-Born and distorted-wave approximations of the electron-impact ionization cross sections for highly ionized atoms in the hydrogen and lithium isoelectronic sequences have been computed in several variants of the CBE and distorted wave approximation.
Abstract: Electron-impact ionization cross sections for highly ionized atoms in the hydrogen and lithium isoelectronic sequences have been computed in several variants of the Coulomb-Born and distorted-wave approximations Electron exchange in the transition matrix element and Coulomb distortion of the partial waves were found to be important The results are compared to recent crossed-beam experimental data and to other theoretical predictions