Classical Deflection Function Research Articles

On the Approximation of the Classical Deflection Function

On the Approximation of the Classical Deflection Function

Classical theory of elastic scattering and vibro-rotational dissociation in N2+on He collisions at moderate energies (0.5 keV-3 keV)

The scattering of N2+ ions by He atoms is studied excluding any electronic excitation. A two centre-screened Coulomb model is taken for the scattering potential. In a first step the classical deflection function is determined for a specific molecular orientation, in assuming that the vibrational and rotational motions are negligible during the collision time. Then the corresponding differential cross section is calculated for an isotropic distribution of the molecular orientations. In a second step the amount of translational energy transferred into the internal degrees of freedom is evaluated and the vibro-rotational dissociation is taken into account. The 'pseudo-elastic' differential cross section is obtained and it is compared to experiment.

Numerical calculation of the classical deflection function

The singular point at the distance of closest approach r 0 in the integrand of the classical deflection function presents great difficulties. Without problems the integral can be calculated numerically on the interval r 0 + Δ → ∞. Analytical expressions are derived for upper and lower bounds of the integral on the interval r 0 → r 0 + Δ. With an adjustable value of Δ the difference between the bounds can be made arbitrarily small. By replacing the integral around r 0 by these upper and lower bounds the deflection angles can be calculated for any potential function with a high degree of accuracy.

Approximate Formula for Rainbow Scattering Angle

An integral expression of the classical deflection function by the use of Mellin transformation has been presented, which is convenient for the differentitation with respect to the impact parameter b to give the relation between the rainbow scattering angle θ R and the reduced kinetic energy K of the incident particle. A simple and useful approximate formula, -θ R⋅K=0.29 γ+1.75, valid for the value of K not less than five, has been derived for the three-parameter(12, 6, 4)potential.

Inversion Problem for Ion-Atom Differential Elastic Scattering

An inversion procedure to recover the intermolecular potentials from low-energy (4-14 eV) differential scattering data for the systems He${\mathrm{H}}^{+}$, Ne${\mathrm{H}}^{+}$, Ar${\mathrm{H}}^{+}$, and Kr${\mathrm{H}}^{+}$ is presented. The first step in this procedure is to construct the phase shifts {$\ensuremath{\eta}(l)$} from the experimental differential cross section $\ensuremath{\sigma}(\ensuremath{\theta})$ and, for reasons of intuition, the corresponding classical deflection function. This is accomplished by employing a new technique developed recently by Remler which involves singularities in the $S$ matrix. The Remler method is as accurate as, but less cumbersome than, the frequently employed standard partial-wave sum and in addition leads to an intuitive connection between $\ensuremath{\eta}(l)$ and $\ensuremath{\sigma}(\ensuremath{\theta})$. Having found {$\ensuremath{\eta}(l)$}, the intermolecular potential may be determined by means of the transformation method of Vollmer. To test the entire inversion procedure, the intermolecular potential obtained from the ${\mathrm{H}}^{+}$ + He scattering data is compared to the ab initio calculation by Wolniewicz of the ground-state potential for He${\mathrm{H}}^{+}$. This method should be applicable to any spherically symmetric scattering system.

Interpretation of Experimental Differential Elastic Scattering Cross Section forH++ Ne

The interatomic V(r) found by potential-model calculations is compared to an ab initio calculation of the intermolecular potential for NeH+ due to Peyerimhoff (1965). The reason for a rather significant difference in the two results is discussed, and a new method due to Remler (1971) involving Regge poles is applied. The method can be used to calculate the differential cross section, where the starting point for such calculations is based on intuition in light of the classical deflection function.

Determination of Intermolecular Potentials by the Inversion of Molecular Beam Scattering Data. I. The Inversion Procedure

A practical solution of the inversion problem within the realms of the WKB approximation is reported. The basic input information is only the measured relative differential cross section, from which the WKB phase function or the classical deflection function is constructed. The absolute potential curve is then determined by the application of a modified Firsov method. The problem of which part of the differential cross section is readily measurable and also suitable for the inversion procedure is studied. A practical procedure is proposed and tested by using a simulated example.

WKB Solution of Inversion Problems for Potential Scattering

The construction of a local, not necessarily monotonic, potential is given in the WKB approximation for the two following sets of “input” information: (1) the phase shift as a function of impact parameter (or orbital angular momentum) at one energy; (2) the phase shift as a function of energy for one value of l, plus the eigenvalues of the potential for this same l. It is also shown that in the high-energy WKB approximation (Eikonal approximation) the energy dependence of the total cross section can be inverted to obtain the potential. Procedures for constructing the classical deflection function (or, equivalently, the phase shift as a function of impact parameter) from the observed differential cross section are also discussed.

Derivatives of the Classical Deflection Function

Integral expressions have been derived for the second and third derivatives with respect to the impact parameter of the classical deflection function. These are then given in a useful form for numerical quadrature and their evaluation is discussed. The expressions are investigated for their behavior as various quantities become zero, and in particular, formulas valid at zero impact parameter are obtained. Tables of results are presented for (12,6) and (12,4,6) potentials, the former given by V(r) = 4ε[(σ / r)12 − (σ / r)6], and the latter by V(r) = (ε / 2) [(1 + γ) (rm / r)12 − 4γ(rm / r)6 − 3(1 − γ) (rm /r )4], in which ε, σ, rm, and γ are adjustable parameters.

Molekularstrahlmessungen von Sto�querschnitten f�r �berg�nge zwischen definierten Rotationszust�nden zwei-atomiger Molek�le

Previously reported experimental results for inelastic cross sections for rotational excitation of TlF molecules in low-lying, well defined rotational states are interpreted in terms of a time dependent perturbation theory formulation of the high energy approximation. In order to calculate inelastic cross sections for the observed small angle scattering the Born approximation and the classical deflection function are shown to be applicable. In this approximation the ΔJ selection rules are characteristic of the individual terms in the expansion of the potential, whereas the ΔM selection rules depend on the orientation of the molecule with respect to the scattering trajectory. An approximation for dealing with a scattering gas consisting of molecules is introduced and the appropriate orientation averaging is carried out for the case of a generalized electrostatic potential. The measured results for the transition TlF(2.0)→ (3.0) in collisions with the rare gases and CH4 and SF6 are more than a factor three larger than calculated results for the induced dipole-quadrupole (α, μ, Q) interaction. Rough argeement is found between calculated results for a dipole-quadrupole interaction and the experimental results for the above mentioned transition produced by the scattering gases O2, N2 (air), N2O, and H2O. Finally, the dipole-dipole potential appears to provide an explanation for the large inelastic cross sections observed with NH3 and CF2Cl2. Calculated inelastic and total cross sections are however considerably larger (about a factor 2) than the measured results with NH3. Possible explanations are discussed.

Comparison and Appraisal of Approximation Formulas for Total Elastic Molecular Scattering Cross Sections

The Massey—Mohr (MM), Schiff (S), and Landau—Lifshitz (LL) approximations for the total elastic cross section (Q) are intercompared. All can be shown to follow from the same assumption, (i.e., the classical small-angle deflection function, thence the Jeffreys—Born phases via the semiclassical equivalence relationship), sufficing to determine the velocity dependence of Q. Thus, for V=±C(s)/rs, Q(s)=p(s)[C(s)/hv ]2/(s—1) The coefficient p(s) is the same for the S and LL approximations; the ratio pSLL(s)/pMM(s)≥1, (<1.075); it is 1.0709 and 1.0458 for s=6 and 12, respectively. A numerical calculation for a repulsive (s=12) interaction shows that the SLL formula reproduces the partial-wave calculated Q to within ½%. A graphical presentation suggests the generality of this result; it also indicates the source of bias in the MM approximation. For a ``realistic'' intermolecular potential, (restricting consideration to collisions in the ``thermal'' energy range), the influence of the repulsion is only to produce undulations in Q(v); the correct value of C(6) may be obtained by velocity averaging the ``apparent'' CSLL(6).

Semiclassical description of scattering

The quantum-mechanical scattering amplitude can be simply related to the classical deflection function when the conditions for a semiclassical analysis of the quantum-mechanical scattering are met. Various interesting characteristic features of the scattering are related to special features of the classical deflection function. The characteristic types of scattering discussed are: interference, when the deflection function possesses more than one branch at a given angle; rainbow scattering, when the deflection function has a relative maximum or minimum; glory scattering, when the deflection function passes smoothly through 0° or through an integral multiple of ± π; and orbiting, when the deflection function possesses a singularity. The consideration of the characteristic features of semiclassical scattering makes possible the analysis of an observed differential cross section to yield the classical deflection function, which in turn may be used to construct the potential.