United Atom Limit Research Articles

Study of high energetic electrons emitted during heavy ion collisions

Double differential cross sectionsd 2 σ/dΩdE were measured for high energetic electrons emitted in heavy ion collisions. Electrons were detected in the energy range of 60 keV-500 keV for various target projectile combinations 66≦Z u =Z t +Z p ≦145 and projectile velocities between 7 % and 10 % of the speed of light. Clear evidence was found that these electrons stem from the united atom formed during the collision. Slope and height of the spectra are discussed with respect to the momentum distribution of strongly bound states (i.e. theL-shell) at momenta far above the mean value. In addition for the systems S, Ni, Br→Pb electrons were detected in coincidence withK x-rays of Pb. By this method the contribution of theK-shell of the combined system to the total spectrum could be separated. Binding energies of theK-shell were estimated by a slope comparison between the coincident and single spectrum. The resulting values are close to the united atom limit.

A scaling law for the 1sσ molecular orbital direct ionization

In the united atom limit of the molecular orbital ionization theory a scaling rule for the 1sσ molecular orbital direct ionization cross sections is proposed, and it is shown that a large amount of published slow ion-atom collision data can be organized on a universal curve. This scaling is tentatively extended also to very heavy collision systems.

A simple theoretical model for the van der Waals potential at intermediate distances. IV. The bond distance dependence of the potential hypersurfaces for He–H2 and Ne–H2 also for the repulsive region

A simple theory for the van der Waals potential in the region of the well minimum, which previously has been successfully applied to the prediction of the isotropic atom–atom [J. Chem. Phys. 66, 1496 (1977)] and the anisotropic atom–diatom potentials [J. Chem. Phys. 68, 5501 (1978); 74, 1148 (1981)], has been extended to calculate the full potential hypersurface including the H2 bond distance dependence for He–H2 and Ne–H2. By taking advantage of the known potential parameters in the united atom limits He–He and Ne–He, respectively, the potential hypersurface is predicted over a wide range of bond distances. The model is modified to also provide a good estimate of the true potential in the repulsive region (V≃1 eV). The results for He–H2 are compared with a recently calculated CI type hypersurface [Meyer, Hariharan, and Kutzelnigg, J. Chem. Phys. 73, 1880 (1980)] and found to be in good agreement in the region of intermediate distances of the center of masses (R≈3.0 Å), and for H2 bond distances r in the range (0.50–1.00 Å). The R dependence of the vibrational coupling matrix elements are presented for both systems and the implications for the vibrational relaxation rates of both systems are discussed.

Charge-state dependence of inner-shell vacancy production in heavy ion-atom collisions with atomic number larger than 10

X-ray production cross sections have been measured for Ne K X-rays in Ne-Ar, Kr and Ar-Ne collisions, Kr L X-rays in Ar-Kr and Kr-Kr, Xe and Ar K X-rays in Ar-Ar, Xe. Projectile energies used were 60 to 1150 keV for neon ions, 300 to 2000 keV for argon ions and 175 to 1900 keV for krypton ions. The range of charge states used was 1+ to 4+ for neon and 2+ to 7+ for argon and krypton ions. The vacancy production cross sections extracted from the X-ray data show no charge-state dependence and indicate that two-step processes are not important. It is proposed that near the united-atom limit a direct coupling from molecular orbitals (MO) to the continuum is important. A large difference in the excitation function was seen depending on whether 2p sigma , 3d sigma or 4f sigma MO were involved. It is proposed that this strong dependence is caused by the relative difference in the energy gap and the shape of the MO involved. The data are compared with estimates from the modified BEA scaling law and the statistical model. For the latter model systematic deviations with experiment are found.

Vibration-rotation intensities of SiO

Dipole moment matrix elements have been computed for a large number of transitions of astrophysical interest for the more abundant isotopes of SiO. The wave functions utilized were obtained from a direct solution of the Schrödinger equation with an accurate RKR potential. The dipole moment function, in the form of a Padé approximant, was chosen to reproduce the experimental measurements near equilibrium, to have the proper united and separated atom limits, and to have the correct long-range asymptotic dependence on internuclear separation. Because of the large number of transitions involved, and to facilitate applications, the squares of the dipole moment matrix elements were fitted by a least-squares procedure to polynomials in v and J. In addition, Einstein A coefficients are given for observed maser transitions and for selected vibrational bands. These latter are compared with previous calculations, and it is concluded that for the higher Δ v transitions, the present results represent a significant improvement.

Production of x rays beyond theK-series limit in gaseous and solid targets

X rays of energy beyond the K-series limit have been detected during the bombardment of various low-Z, solid and gaseous targets by C/sup +/, N/sup +/, and O/sup +/ ions of energy 0.75 to 1.9 MeV. K..cap alpha cap alpha.. x rays, most likely from the transition (1s/sup -2/2p/sup -1/)..-->..(2s/sup -1/2p/sup -2/), are emitted by both collision partners in gaseous targets. The low-resolution energy measurements are in good agreement with published calculations. The relative yield from the heavier collision partner is larger than simple vacancy-sharing models would indicate, but electron screening may account for the excess. Molecular-orbital (MO) continuum radiation completely obscures the Kaa lines from thin solid carbon targets. The exponential shape of the MO spectrum beyond the united-atom limit is in reasonable agreement with collision-broadened spontaneous emission.

MCSCF–CI calculations of the ground state potential curves of LiH, Li2, and F2

The potential curves for LiH(X 1Σ+), Li2(X 1Σ+g), and F2(X 1Σ+g) have been calculated using a wave function comprising all single and double replacements from an MCSCF reference state. It is shown that in order to obtain satisfactory results, the reference function should describe properly not only the dissociation of the molecule, but also the united atom limit. This implies that for Li2, 1πu, and for F2,2πu orbitals should be included in the reference state. Calculated dissociation energies are 2.48 (2.52) eV for LiH, and 1.00 (1.05) eV for Li2 (experimental values within parentheses). For F2 only 1.31 (1.66) eV is obtained with 2πu excluded from the reference orbital space. A calculation including this orbital yields 1.40 eV even at the MCSCF level.

Non-Adiabatic Transitions Induced by Rotational Coupling

. The Landau-Zener-Stueckelberg formula based on the “dynamical” adiabatic representation is shown to be useful for the study of the title transitions, The following two cases are investigated: (a) curve crossing at the finite internuclear distance ( R = R x ), (b) degeneracy at the united atom limit ( R =0).

MO X-ray angular distributions from 1 GeV208Pb+208Pb and209Bi+209Bi collisions

The Doppler shift and the anisotropy of the MO X-rays above the separated-atomK-lines have been measured in208Pb+208Pb and209Bi+209Bi collisions at beam energies between 4.2 and 4.8 MeV/u. The Doppler shift is found to be consistent with a source moving with 90% of the center of mass velocity of the colliding system. One peak is observed in the anisotropy spectrum. This peak might be attributed to transitions from the continuum into the minimum of the 2p 1/2 σ MO or from the 3p 1/2 σ and 2p 3/2 σ MO's into the 2 sσ MO near the united atom limit. Radiative electron capture and secondary electron bremsstrahlung could be excluded as major sources of the observed radiation. Contrary to lighter systems no anisotropy corresponding to transitions into the 1 sσ orbital could be observed.

A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach

A density matrix formulation of the super-CI MCSCF method is presented. The MC expansion is assumed to be complete in an active subset of the orbital space, and the corresponding CI secular problem is solved by a direct scheme using the unitary group approach. With a density matrix formulation the orbital optimization step becomes independent of the size of the CI expansion. It is possible to formulate the super-CI in terms of density matrices defined only in the small active subspace; the doubly occupied orbitals (the inactive subspace) do not enter. Further, in the unitary group formalism it is straightforward and simple to obtain the necessary density matrices from the symbolic formula list. It then becomes possible to treat very long MC expansions, the largest so far comprising 726 configurations. The method is demonstrated in a calculation of the potential curves for the three lowest states (1Σ+g, 3Σ+u and 3Πg) of the N2 molecule, using a medium-sized gaussian basis set. Seven active orbitals were used yielding the following results: De: 8.76 (9.90), 2.43 (3.68) and 3.39 (4.90) eV; re: 1.108 (1.098), 1.309 (1.287) and 1.230 (1.213) Å; ωe: 2333 (2359), 1385 (1461) and 1680 (1733) cm−1, for the three states (experimental values within parentheses). The results of these calculations indicate that it is important to consider not only the dissociation limit but also the united atom limit in partitioning the occupied orbital space into an active and an inactive part.

Collision broadening of continuum radiation from Ni+Ni collisions

The projectile energy dependence of characteristic K X-ray and high-frequency quasimolecular X-ray production cross sections in Ni+Ni collisions has been measured. The extracted half widths for the frequency distribution of the continuum radiation above the united-atom limit increase as the projectile velocity to the 0.59 power. Theoretical descriptions of the collision-broadened quasimolecular radiation are inadequate to explain these results.

Electron scattering from diatomic polar molecules. II. Treatment by frame transformations

The theory of Chang and Fano is applied to electron scattering by polar molecules. For collision energy $Eg{E}_{min}$ the problem can be treated accurately without numerical radial integration; the solution is presented in a form suited to an $R$-matrix treatment of the molecular core. The value of ${E}_{min}$ (in a.u.) is determined through $D\frac{\mathrm{Bj}}{{E}_{min}}l\ensuremath{\delta}\ensuremath{\ll}1$; when $\frac{E}{{E}_{min}}\ensuremath{\approx}\mathrm{a}\mathrm{few}$ eV, more extensive calculation is required beyond the bodyframe Born-Oppenheimer region. For this purpose a generalization of the Chang-Fano theory is developed. It consists of an adiabatic continuous transformation of frame, the frame used at each electron-molecule distance $r$ being determined directly by the dynamics of the problem. This procedure generates a coordination diagram for the known solutions to limiting cases, a diagram analogous to that used in quantum chemistry for connecting the united-atom and separated-atom limits of molecular-orbital theory. A new type of symmetry in the problem is made apparent. Preliminary discussion is made of the applicability of this method to more general problems of close coupling in the asymptotic region.

The Radial and Rotational Coupling Terms between the Quasi-Molecular States of He2++

The radial and rotational coupling matrix elements between the 1 Σ g + (2sσ 2 ,2pσ 2 ,2pπ 2 ), \(^{1}\varPi_{\text{g}}(2\text{p}\sigma 2\text{p}\pi)\) and 1 Δ g (2pπ 2 ) quasi-molecular states of He 2 ++ have been calculated using an elaborate CI wave function in the range of internuclear separation from 0.0 to 0.5 atomic units. It is pointed out that, because of the configuration mixing, the rotational coupling matrix elements which are zero at the united atom limit or in the single configuration approximation have finite values at small R . The radial coupling matrix element between the 1 Σ g + (2sσ 2 ) and 1 Σ g + (2pσ 2 ) states is about 10 atomic units at the pseudo-crossing near R =0.2. The accuracy of the calculation of the radial coupling matrix elements is discussed.

Representation of a b i n i t i o energy surfaces by analytic functions

A method for the analytical representation of ab initio energy surface results is described. For a three-atom surface, the potential function is written as a sum of atomic, two-body, and three-body terms. This allows an easy fitting of the whole surface, including the dissociated and united atom limits. The theory is applied to an H3 surface of limited accuracy calculated for these purposes. Comparisons are made, where possible, with more accurate calculations. Extensions to surfaces involving four or more atoms are also discussed.

Radial matrix elements and dipole moment function for the ground state of CO

Radial matrix elements 〈 vJ| x k | v′ J′〉 for k = 0–5, v = 0–12, | v′- v| = 0–4, and J up to 150 have been calculated for CO using accurate wavefunctions obtained from the numerical solution of the Schrödinger equation with a second-order RKR potential curve. These are used in conjunction with a model dipole moment function (a Padé approximant which has the correct united and separated atom limits and R −4 long-range behavior) to analyze the experimental intensity data. For all the levels considered, we conclude that an adequate representation of the dipole moment function is provided by a five-term power series expansion. This simplifies the computation of dipole moment matrix elements, typical results of which are presented to illustrate the dependence on the rotational and vibrational quantum numbers.

Molecular orbital correlation diagrams for He2, He2+, N2, N2+, CO, and CO+

After a preliminary check with He2 and He2+, self-consistent field calculations have been carried out for the nitrogen and carbon monoxide molecules and some of their positive ions for the range of internuclear distances from about 1.5 times equilibrium down to 0.01 bohr. To adequately describe the passage from the separated to the united atom limits, basis sets comprised of even-tempered expansions of Slater type s and p functions were centered at each nucleus and at a point midway between them. This three center procedure is compared with that of reoptimizing basis functions at each internuclear distance. It is found that the three-center approach almost completely obviates the need of the polarization functions which must be added if a two-center approach is used. The correlation diagrams for nitrogen and carbon monoxide and the first positive ions of each symmetry type are all similar in shape, especially for the innermost orbitals. Ionization potential curves plotted on the diagrams for the neutral systems are nearly parallel to the corresponding orbital energy curves. This indicates that Koopmans’ theorem is consistent at all distances and implies that correlation diagrams for neutral molecules should be reasonable approximations to the ionization potential curves. For the most easily ionized orbital of both N2 and especially CO, the ionization potential has a remarkably low value at about one-third the equilibrium separation. To adequately describe this orbital an extra s function having a smaller zeta than the smallest one in the united atom basis set was required. This implies that merely using the optimized basis sets for the separated and united atoms in a three-center procedure is scarcely acceptable in the region where the outermost orbital energy has the lowest value.

United-atom limits for molecular-orbital continua and the characteristics of broadening beyond the limit

In collisions of 0.2-3.0-MeV ${\mathrm{Na}}^{+}$ with thick targets of Ti through Zn, relatively flat molecular-orbital continua are observed that extend from the characteristic separated-atom lines to the united-atom $L$-shell limits. The broadening of these continua beyond their respective united-atom limits is essentially target independent and can, to a good approximation, be fitted with half-Gaussians characterized by widths proportional to ${E}^{\frac{1}{4}}$, where $E$ is the energy of the incident ion. Similar continua are observed for ${\mathrm{Na}}^{+}$-target combinations that have as their limit the united-atom $M$ shell.

Noncharacteristic Radiations in Atomic and Molecular Collisions

Noncharacteristic radiations in atomic and molecular collisions are examined. A new mechanism is considered in which electrons are induced to radiate collectively. The collective radiations are present even in the absence of inner-shell vacancies. The importance of this mechanism increases with collision energy and the number of electrons. The emitted photon energy increases with collision energy and can exceed the united-atom limit. The asymmetries of the noncharacteristic radiation are examined.

Variable-screening model for the quasimolecular treatment of ion-atom collisions

An effective single-electron potential is proposed which is appropriate for a realistic description of the electronic motion in quasimolecular systems formed in near-adiabatic ion-atom collisions. The potential is represented as a sum of two spherical potentials, each centered about one of the colliding nuclei. For finite internuclear distances, these potentials are obtained by smoothly interpolating the screening parameters of suitable atomic potentials (Thomas-Fermi or phenomenological potentials) between the united-atom and separated-atom limits (variable-screening model). The use of phenomenological potentials entails a great accuracy in describing the atomic states and opens up the possibility of treating ionized quasimolecular systems which were hitherto inaccessible to molecular-orbital Hartree-Fock calculations. Molecular correlation diagrams calculated from the variable-screening model for various neutral and ionized systems are presented. For the neutral cases the agreement with the results of Hartree-Fock calculations is surprisingly good. The scaling properties of the variable-screening model are discussed.

Semiempirical model for angular distribution of electrons emitted in metastable atom-atom collisions

Electron angular distributions from ionization in collisions of $\mathrm{He}(2^{1,3}S)$ and $\mathrm{Ar}(^{1}S)$ have been studied in terms of molecular-orbital correlation diagrams and a semiclassical and probabilistic model for the collision process. Consideration of the united-atom limit for this system indicates that the electronic angular momenta $\ensuremath{\lambda}=0, 1, \mathrm{and} 2$ play an important role. The interaction potential responsible for electron emission was assumed to decrease exponentially with increasing distance, and was parametrized by fitting experimental results at a fixed collision energy. The values of the parameters show that $\ensuremath{\lambda}=0$ is the dominant partial wave and lead to backward-peaked distributions. Calculations performed at other energies show an increasing anisotropy with increasing collision energies.