Method Of Stationary States Research Articles

Charge Transfer Processes Involving Highly Excited Hydrogen Atoms. III. The Perturbed Stationary State Method

The perturbed stationary state (PSS) method is applied to the resonant charge transfer processes involving highly excited hydrogen atoms at low energies. The momentum transfer effect of the electron is included. It plays an important role in the energy dependence of the cross sections. The energy dependence of the calculated cross sections agree with that of the experimental results but the magnitude is about five times smaller than the experimental value.

Modified Perturbed Stationary State Method Applied to Heavy Ion Scattering. I

Modified Perturbed Stationary State Method Applied to Heavy Ion Scattering. I

Large angle scattering in slow H+-H(1s) collisions

Collisions between H+ ions and H(1s) atoms are studied theoretically, allowance being made for the fact that the relative motion of the nuclei is not rectilinear. A simple physical argument is given for supposing that if the encounter is sufficiently close and fast and if the axis of quantization is taken to be parallel to the final direction of the velocity of relative motion vector then the probability of excitation to the 2p+or-1 state around either nucleus is 1/2 sin2 Theta where Theta is the scattering angle in the centre of mass system. This prediction is verified by computations in which the electronic motion is treated by the perturbed stationary state method (in a 4-state approximation) and the motion of the nuclei is treated semi-classically with a forced common turning point. Other aspects of H+-H(1s) collisions are covered by the computations. The results are in accord with the results of the full quantal investigation recently carried out by Knudson and Thorson (abstr. A30138 of 1970). Special attention is given to the case of 6 degrees scattering. Satisfactory agreement with the measurements of Lockwood and Everhart (abstr. A3715 of 1962) is achieved.

Collisional Transfer of Triplet Excitations between Helium Atoms

The cross sections for 2 3 S and 2 3 P excitation transfer processes between He atoms are calculated by the perturbed stationary state method. The potential energy curves for He(1 1 S)+He * (2 3 P) and He(1 1 S)+He * (2 3 S) systems are obtained by the Heitler-London method taking the configuration interaction between them into account. The calculation involves the transfer, the diffusion and the total elastic cross sections. The results show that the transfer cross section for 2 3 P is 15∼23π a 0 2 at thermal energy and larger than that for 2 3 S. The transfer cross section for 2 3 S dose not so much differ from the calculation using rigorous potentials by Kolker and Michels [J. chem. Phys. 50 (1969) 1762].

Ionization Cross-Section of an Atom by De-Excitation of Another Metastable Atom through Collision

A method for calculation of the ionization cross-section of an atom by de-excitation of another metastable atom through collision (Penning ionization cross-section) is proposed. The perturbed stationary state method is applied and Jeffreys' approximation is used of the determination of the phase shift. Fano's formula is applied to the coupling between the discrete state and the continuum and a parameter β is introduced by which the wave function is made to satisfy the initial condition. The method is applied to a long-range interaction V =- C n / R n . The velocity dependence obtained is the same as that of the previous work. The cross-section in the case of n =3 is about half the value obtained by the impact parameter method.

Single-electron capture by slow protons in helium

The perturbed stationary state method is applied to He(1s2)+H+ -> He+(1s)+H(1s) but with neglect of momentum transfer. The calculations of Green et al. and of Sin Fai Lam show that the approximation is certainly untenable at energies above 16 keV. The experimental results of Hasted and Stedeford and of Stier and Barnett for the sum total of all processes involving charge transfer from helium to protons are exceeded from 3 to 10 keV. The differential cross section at small angles of deflection compares reasonably well with the experiments of Helbig and Everhart for energies below 5 keV.

Nonadiabatic Transition in the Slow Collision between Heavy Particles

Inelastic atom-atom collision is investigated under semi-classical condition by the use of perturbed stationary state method. Wave functions of the perturbed states of the system are expanded in terms of products of wave functions of isolated atoms. The expression for cross-section is derived, using these wave functions. Dependence of the cross-section on velocity is discussed. Further it is shown that the cross-section is expressed only in terms of the exact adiabatic potentials concerned. The obtained formula is applied to electron capture by α particles in hydrogen. Results of calculation are discussed and compared with the experimental data by Fite et al.

Collisional Excitation Transfer between Hydrogen Atoms II. H(2S)+H(1S) → H(1S)+H(2S)

The cross-section for the exact resonance excitation transfer process H(2S)+H(1S)→H(1S)+H(2S), of which the interaction is a short range one, is calculated at energies of relative motion of 0.03, 0.1, 0.3, 1.0, 3.0, 5.0 and 10.0 eV by the perturbed stationary state method (PSS method). The interaction energies between hydrogen atoms are calculated by the Heitler-London method. The results are in good agreement with the results of the impact parameter method for energies of relative motion over 1.0 eV. The disagreement below 1.0 eV is due to the presence of the potential barrier in the 3 Σ g adiabatic potential of the excited hydrogen molecule. The cross-section calculated by PSS method is 33.1π au (atomic unit) at 0.03 eV.

Theory of Slow Atomic Collisions. I. H2+

A theory of zero-order elastic and first-order inelastic scattering for slow atomic collisions in the system H2+ is presented. The method of stationary states and partial wave analysis is used and is the first such treatment which meets the formal requirement of scattering theory that the zero-order eigenfunctions become correct eigenfunctions at R→∞; the ``perturbed-stationary-states'' method, for example, fails to meet this criterion. The method is applicable only to electronic states which are ``tightly bound'' (≳0.25 eV below the ionization limit); no treatment of scattering to ``continuum'' states of the electron is given. The use of partial wave analysis makes possible the elimination of practical difficulties which appear in the time-dependent treatments due to nonorthogonality of the electronic basis functions. The zero-order eigenfunctions may properly be termed ``adiabatic'' solutions, since they are physically the logical extension of the Born—Oppenheimer concept to unbound states, and for many problems zero-order elastic scattering is indeed just potential scattering by the Born—Oppenheimer potential. First-order inelastic scattering cross sections can be computed from electronic transition matrix elements which differ in significant respects from those of the ``perturbed stationary states'' method.

Destruction of Metastable Helium Atoms in Binary Collisions

A calculation of the cross section for the reaction He′+He′=He+He++e (destruction of the metastable 23S state by ionization) has been performed by means of the perturbed stationary state method. The cross section obtained is of the order of magnitude 10—18 cm2 at room temperature and increases slowly with decreasing kinetic energy of the colliding particles. The magnitude of the cross section fits well with a tentative scheme which considers processes of the type A′+B=A+B++e as being intermediate between true resonance collisions and nonresonance collisions. It disagrees by 4 orders of magnitude with experimental findings [A. V. Phelps and J. P. Molnar, Phys. Rev. 89, 1202 (1953)]. The reason for this discrepancy is not known.

Inhibited thermal polymerization of styrene. Kinetics of copolymerization of quinones with styrene

AbstractIn research on the kinetics of the inhibition of the thermal polymerization of styrene, we came to the conclusion that quinones participate not only in the inhibition reaction but also in the copolymerization. We carried out the derivation of the basic kinetic equations of the process in two ways: (1) Using the usual simplifying assumptions of the theory of copolymerization (Melville‐Watson method) and (2) using the stationary state method. The kinetic equations derived by the second method transform under certain conditions to the corresponding equations of simple inhibition, while the equations from the first method lead to contradictions. On the basis of the kinetic analysis it was found that the inhibition period during copolymerization of the inhibitor does not increase linearly within increasing concentration of the inhibitor, and that on the curve of conversion versus time there is an inflection point if the concentration of inhibitor is higher than a certain value. The curve of initial polymerization velocity against concentration of inhibitor has a minimum at a certain concentration (“inhibition optimum”). Experiments carried out with p‐benzoquinone at 90, 105, 120, and 135°C. completely agree with what has been stated above. At the lower temperatures, except for a slight shortening of the inhibition period, no significant indications of copolymerization are apparent; the course of these experiments is well described by the equation of simple inhibition. At 120 and 135°C. copolymerization occurs to a marked extent and is well characterized by the equations derived by the second method. Finally, from the temperature dependence of the appropriate velocity constants, we determined the activation energy of the inhibition reaction (4.3 kcal./mol.) and the activation energy of copolymerization of quinone (9.4 kcal./mole).

Charge Transfer and the Mobility of H-Ions in Atomic Hydrogen

The energies of interaction of H- and H are calculated over a wide range of nuclear separations using molecular wave functions based upon various forms of atomic orbitals and these energies used for the calculation of the charge transfer and diffusion cross sections of H- ions in atomic hydrogen by the perturbed stationary state method. The charge transfer cross sections decrease from 2 × 10-14 cm2 at an impact energy of 1 eV to 0.3 × 10-14 cm2 at 1000 eV and the diffusion cross sections lead to values of the mobility of H- ions in atomic hydrogen decreasing from 3.5 cm2 volt-1 sec-1 at 100°K to 1.8 cm2 volt-1 sec-1 at 600°K.

Inelastic collisions between atoms I. General theoretical considerations

The probability of slow inelastic collisions between two atomic systems is investigated theoretically using the perturbed stationary state method (p.s.s. method) in which the kinetic energy of the relative motion is treated as the perturbation responsible for the transitions. The p.s.s. method has been extended to include cases in which resonance degeneracy occurs and when the transition concerned involves a change of the axial component of angular momentum. The scope and limitations of the method are examined in detail and compared with those of Born’s approximation. It is shown that two modifications of the latter approximation are of comparable importance in most cases, one involving a change in the effective transition potential, the other the distortion of the plane waves representing the relative motion of the colliding systems. The first of these is the only one included in previous applications. It is shown that, under semi-classical conditions, the formulae of the p.s.s. method reduce to Mott’s impact parameter formulae provided the important distortion effect is neglected. New semi-classical formulae, including this effect as well as the effect of resonance degeneracy, are derived in a form suitable for numerical application.

Electron Capture I: Resonance Capture from Hydrogen Atoms by Fast Protons

The cross sections for resonance electron capture from hydrogen atoms by protons of incident energy up to 100 keV are calculated theoretically using the Perturbed Stationary State method and a comparison is made with the results of the Born approximation both with and without neglect of momentum transfer.