Continuum State Wave Functions Research Articles

Triple-differential cross sections for the fast-electron impact ionization of helium in a hybrid second-order model.

A hybrid second-order model, in which the first-Born-approximation amplitude is evaluated by using an improved choice of the target continuum-state wave function and the second-Born-approximation amplitude in the usual way, is used to obtain the triple-differential cross sections for the ionization of helium. The results are compared with the recent data of Jung et al. in the coplanar asymmetric geometry at an incident electron energy of 600 eV and are found to be in very good agreement.

Atomic inner-shell ionization.

Cross sections for the K-shell ionization of medium-heavy atoms by relativistic electrons have been calculated in a relativistic framework in the Coulomb gauge to order ${\ensuremath{\alpha}}_{0}$ (=${e}^{2}$/\ensuremath{\Elzxh}c) in the interaction Hamiltonian. Here exchange is neglected. The incident- and scattered-electron wave functions are described by Dirac plane waves. Only the bound- and ejected-electron wave functions are described nonrelativistically for a screened Coulomb potential, used earlier by the present authors. Thus the screening effect is taken into account in this calculation in a satisfactory manner. The calculation is a repetition of what has been published recently by the present authors with one important exception: Here, the ejected-electron continuum state wave function that is used is determined variationally for the above screened Coulomb potential. Thereby some perturbation approximation of ad hoc nature could be avoided. As a consequence, considerable improvement is noticed in the total cross-section results. Comparison with experimental results for $^{29}\mathrm{Cu}$, $^{47}\mathrm{Ag}$, and $^{79}\mathrm{Au}$ shows a good agreement. The calculation may easily be extended to ionization from other shells.

Photoionization of sodium atoms and electron scattering from ionized sodium.

The polarized-orbital method [originally developed by A. Temkin, Phys. Rev. 107, 1004 (1957)] is used to calculate the photoionization cross sections of sodium atoms, from threshold to about 60 eV. The polarized orbitals are calculated from Sternheimer's equation. The polarizability of ${\mathrm{Na}}^{+}$ is found to be 1.0914${a}_{0}^{3}$, which is very close to the experimental value. The scattering equation is solved in the exchange, the exchange-adiabatic, and the polarized-orbital approximations. For consistency, both the bound-state and the continuum-state wave functions are obtained in the same approximation. In the calculation of these wave functions, all perturbed orbitals are taken into account in the direct polarization potential and only the perturbation of the very tightly bound 1s orbital is neglected in the exchange-polarization terms. The length form for the photoionization-cross-section formula is used and all terms are included in the photoionization matrix element. Our results are in good agreement with the many-body calculations of Chang and Kelly. However, no calculation to date, including this one, agrees well with experiment except at low energies. The phase shifts thus obtained for s, p, and d waves for the e+${\mathrm{Na}}^{+}$ system are used to calculate the differential cross sections; the phase shifts, as well as these cross sections, agree very well with other available results.

Caution: Approximation of the nonrelativistic but high-energy continuum state of a photoelectron by a plane wave can be dangerous in the evaluation of the matrix element

Most quantum mechanics textbooks approximate the nonrelativistic but high-energy continuum-state wave function by a plane wave in the evaluation of photoelectric effect matrix element. This approximation leads to very wrong results if the gauge is chosen so that the EM interaction Hamiltonian is −(e/m)A⋅p and the matrix element is evaluated directly for the momentum operator. On the other hand, the same approximation is a fair one if the interaction Hamiltonian is written in the −eE⋅r gauge, or when p is replaced by the commutator −im[H,r], where H is the atomic Hamiltonian. This is explicitly demonstrated for the photoelectric effect for the hydrogen atom in the ground state.

Resonance and threshold effects in atomic and molecular photoionization

Photoionization processes depend critically on the behavior of the bound and continuum state wave functions involved in the ionization process. Resonance and threshold effects, as they show up in the photoemission cross section, are discussed within this context.

AtomicL-shell Compton profiles and incoherent scattering factors: Theory

Compton profiles are calculated for $L$-shell electrons within the Born approximation by use of nonrelativistic "exact" hydrogenic bound- and continuum-state wave functions. The wave functions are expressed in parabolic coordinates and the resulting matrix elements are evaluated following the method of F. Bloch (1934). Impulse-approximation profiles are calculated, and comparisons with the "exact" profiles show that, as expected, the two profiles lie very close to one another for weak binding and high incident photon energies. However, even when these conditions are not fulfilled, the two curves have a tendency to cross one another in the neighborhood of the profile center. This tendency has been observed in previous studies of $K$-shell profiles. However, unlike $K$-shell profiles, the $2S$ Compton profiles exhibit a secondary maximum. The secondary maxima occur at approximately the same region in $q$, where the impulse-approximation profiles exhibit a plateau. The location of this plateau is shown to be related to a node in the bound-state wave function and is around $q=\frac{Z}{2}$ for the $2S$ profiles. The impulse approximation, being a monotonic decreasing function in $|q|$, cannot exhibit the secondary-maximum structure appearing in the "exact" hydrogenic profiles. The intensity of the secondary maximum in the $2S$ profiles is reduced by over an order of magnitude from the central peak. The $2{P}^{(0)}$ Compton profiles also exhibit structure, however, unlike the relatively small secondary maximum in the $2S$, the two maxima in the $2{P}^{(0)}$ profiles are of the same order of magnitude. Integrated profiles (incoherent-scattering factors) are calculated and the impulse-approximation results agree with the "exact" results over a wide range of binding energies owing to profile crossover near the center. Waller-Hartree incoherent-scattering factors give a closer agreement with the "exact" results than observed for $K$-shell electrons; however, for low-momentum transfer the Waller-Hartree results can differ from "exact" results by more than 50%. In such regions, impulse-scattering factors represent a considerable improvement over the Waller-Hartree factors.

Compton Scattering byK-Shell Electrons. I. Nonrelativistic Theory with Retardation

An exact analytic calculation is presented for the cross section of the nonrelativistic Compton-type inelastic scattering of photons by electrons bound in the ground state of a hydrogenlike atom. The matrix element of the process was taken to be of the Kramers-Heisenberg-Waller form. It was integrated by the Coulomb-Green's-function method, developed previously by the author in connection with Rayleigh scattering. This is based on replacing the sum over intermediate states by the Green's function which is expressed in terms of the integral representation in momentum space given by Schwinger. The final continuum-state wave function was also described by an integral representation. Thus the evaluation of the matrix element requires the carrying out of momentum-space integrations, followed by two contour integrations. Its final form was expressed in terms of hypergeometric functions of four variables of Lauricella's type ${F}_{D}$. At small photon energies, where the dipole approximation is valid, the result simplifies considerably. The low-energy end of the scattered-photon spectrum is considered in detail in connection with the infrared divergence problem. The shape of the spectrum is discussed and a comparison with previous results is given.

Photoionization and Autoionization of H2

The cross sections are calculated for direct photoionization and photoexcited autoionization of H2 for light in the region 600–800 Å. Angular distributions of directly produced photoelectrons, lifetimes for vibrationally induced autoionization, some spectral assignments, and relative intensities of photoelectrons of various energies (from autoionizing states) are also given. The method by which the Rydberg and continuum state wavefunctions are derived is similar to the correlated pseudopotential method of Tully. The results indicate that correlation has only a small effect on the differential and total cross sections for photoionization. Agreement of theoretical and experimental lifetimes for autoionization is improved by the inclusion of correlation in the wavefunction. Critical tests of the calculations could be made by experimental study of the photoelectron spectra produced when H2 is excited to certain specific autoionizing states; for these states, the theory predicts anomalously efficient autoionization associated with transfer of two vibrational quanta into electronic excitation.

Charge asymmetry of nucleon widths of the lowest [formula omitted] states of 13N and 13C

The isospin-forbidden widths of the lowest T = 3 2 states of 13N and 13C are calculated by including isospin mixing due to Coulomb interaction and charge dependent nuclear forces in the first-order perturbation approximation. The widths are obtained by using the Feshbach formula with the continuum-state wave function distorted by an optical potential. It is found that the widths are sensitive to the exchange character of the charge-independent nuclear potential. The absolute magnitude of the widths, and the charge asymmetry in the branching ratio to the ground and first excited states of 12C are found to be compatible with the theoretical results, and the presence of a charge-dependent nuclear force is not essential for this agreement.

A zero-range treatment of the three-body photodisintegration cross section of 3H and 3He

A zero-range approach is used to construct bound and continuum state wave functions. The total and differential electric dipole photodisintegration cross sections for three-body breakup of 3He and 3H are calculated using these wave functions.

RelativisticL-Shell Photoeffect

The expressions of the relativistic $L$ shell photoeffect differential and total cross sections, for the case of light elements, are established. Coulomb wave functions are used for the description of the electron states in the calculation of the matrix elements, but it is shown how the results should be corrected so as to take screening effects into consideration. For the final continuum state wave function of the ejected electron, whose exact analytic form is not known, the Born approximation is used. In the case of ${L}_{\mathrm{I}}$ subshell the calculation runs similarly to the one performed previously by the author for the $K$ shell. For the other two subshells new tedious trace evaluations have to be carried out. Thus, for the ${L}_{\mathrm{I}}$ subshell, cross sections correct to first order in $\ensuremath{\alpha}Z$ (inclusive) are determined, whereas in the case of the ${L}_{\mathrm{II}}$ and ${L}_{\mathrm{III}}$ subshells, only their zero-order approximation is calculated. The cross sections are discussed and compared. It is shown that they reduce in the nonrelativistic and extreme relativistic limits to results established by other means.