Measurement of Compton scattering on bound electrons by the coincidence method

A review of results of the new method for measuring the Compton scattering on bound electrons in germanium, introduced by the presented authors, is given. It is based on the application of two detectors that operate in the coincidence mode. One detector is used as the scatterer and the other as the detector of scattered radiation. Two conditions, simultaneity of pulses from the two detectors and constant energy sum, result in very clean spectra in broad energy regions. Normalization of the Compton spectra to the Ge K X-ray escape peaks, which are measured simultaneously with the Compton spectrum, gives reliable double-differential Compton-scattering cross sections on an absolute scale. Several versions of the impulse approximation are compared to the cross sections obtained by the present method for incident photon energies in the range from 60 to 105 keV. The non-relativistic impulse approximation gives the best agreement to the experimental data. We point out the suitability of the new method for an investigation of the incoherent scattering function at small photon momentum transfer.     

Monte Carlo calculation of angular transmission functions for small angle scattering

We define an angular transmission function η in the center of mass system. The convolution of the differential cross section σ with η yields the signal in the laboratory system. For the case of elastic small angle scattering by spherically symmetric potentials we calculate η by a Monte Carlo method. Random positions are taken in the beam defining collimators, resulting in a trajectory with a deflection angle at the scattering centre. These deflection angles are transformed to the c.m. system with the small angle tranformation formulae. From the distribution we calculate η as a histogram and the central moments of η. The function η depends on the velocity ratio and on the mass ratio of the scattering partners. We store the results in such a way that the central moments can be calculated afterwards for all mass and velocity ratios. By using the central moments the convolution integral can be reduced to a simple weighted sum of σ-values at different scattering angles. The r.m.s. deviations of the central moments scale with N^{$\text{1}\text{2}$}, with N the number of Monte Carlo trajectories. A typical deviation is 1% in the second order moment for N = 2 × 10⁴, increasing slightly with increasing order of the moments. This method of calculation gives a large degree of freedom for optimisation of the collimation geometry. The use of an angular transmission function defined in the center of mass system gives a good insight in the experimental reflection of the physical events. As an example we apply the method to the case of small angle scattering of Ar as a primary beam by Kr as a secondary beam and the inverse configuration of Kr by Ar.     

Positron-hydrogen scattering at intermediate energies using CCA

Positron-hydrogen scattering has been studied using close coupling approximation (CCA) in the energy range 19.584–200 eV. The coupling scheme: H(1s), H(2s), H(2p), H(3d), Ps(1s) is used to study elastic, 1s–2s excitation and ground state capture cross sections. The effect of capture channel is found to influence the scattering parameters.     

A phase-corrected coupled-states approximation for inelastic scattering

A simple phase correction is derived for the coupled-states (CS) approximation based on WKB phase shifts for the electric part of the potential. The resulting phase-corrected CS (PCCS) scattering matrix agrees well in both phase and magnitude with the exact one. The PCCS approximation should then give accurate m-state information for any quantization axis.     

Multiple collisions in molecular beam scattering experiments

Due to the forward peaked differential cross section for elastic atom—atom scattering the effect of multiple collisions has to be considered in the analysis of crossed beam measurements of the total cross section and especially of the small angle differential cross section at large values of the beam attenuation. At angles θ ≈ θ₀, with θ₀ the quantum mechanical scaling angle of the elastic differential cross section, the correction for the latter case amounts to 20% at beam attenuations I/I₀ = exp(?1). Firstly, a careful analysis of the probabilities for single and multiple scattering is given, resulting in an expression for the measured beam signals which is correct for all values of the beam attenuation. The probability for multiple scattering is then calculated for an inverse power potential V(r) = ?C_sr^?s, with s = 4 through s = 7, which include both the case of ion—atom scattering (s = 4) and atom—atom scattering (s = 6). The results are given as effective differential cross sections σ_n(θ) for n-fold scattering. They are described by a single, simple analytical function with four free parameters that have been determined for n = 2, 3 and 4 by a least squares method. The σ_n(θ) are normalised to the total cross section Q.     

Inelastic and superelastic scattering of electrons from sodium

Recent experimental results of the NIST, Flinders and Münster groups on superelastic scattering of intermediate-energy polarized electrons from laser-excited polarized and unpolarized sodium atoms have been analyzed theoretically together with those on inelastic scattering data. Based on the Persival-Seaton hypothesis specific model-independent relations between correlation and polarization parameters measured in superelastic scattering experiments of different types have been suggested. The differential cross section as well as the parametersL _⊥,L ^S(T) _⊥, γ,r, and \(\bar S\) _A (J) for the 3²S?3² P transition in a broad range of scattering angels have been calculated within the distorted-wave approximation. Problem of constructing the e + Na optical potential is discussed.     

Vibrational force constants from generalized X-ray scattering factors

Generalized X-ray scattering factors have been extracted from an accurate one-electron density function of molecular hydrogen. With the approximation that the atomic densities perfectly follow the nuclei, vibrational force constants are given in terms of sum rules for the generalized X-ray scattering factors. Calculated (experimental) results for H₂ are k_e = 5.76 (5.73) mdyne Å^?1, l_e = -38.7 (~36.9) mdyne Å^?2 and m_e = 246 (235) mdyne Å^?3.     

Electron scattering with H2S and PH3 molecules

Calculated total, differential and momentum transfer cross sections are reported for the vibrationally elastic scattering of electrons from H₂S and PH₃ molecules in the range of energy 0.1–50 eV. The scattering process is approximated by two incoherent scatterings caused, separately, by a central field and a long-range electric dipole interaction. The central field is calculated with a spherical approximate molecular wave function, in which the exchange interaction is treated in two ways: (i) exactly within the accuracy of the molecular wave function; (ii) approximately by a local model potential. The scattering by the central field is calculated with partial wave expansion technique, while the scattering by the electric dipole potential is calculated by using the first Born approximation for a rotating dipole model with experimental values of the dipole moments of H₂S and PH₃. The total cross sections are approximated by the incoherent sum of the cross section due to the central potential and the cross section of 00→10 rotational transition caused by the electric dipole potential. The effects of the polarization interaction are also tested. The contribution of small-angle scattering to the integral cross section is analyzed for these weakly polar molecules with some quantitative comparison.     

At-edge minima in elastic photon scattering amplitudes for dilute aqueous ions

Elastic photon scattering and absorption in the vicinity of core atomic orbital energies give rise to resonances in the elastic photon scattering cross-section. Of interest is whether a dilute-ion aqueous system provides an environment suitable for testing independent particle approximation (IPA) predictions. Predictions of the energy of these resonances have been determined for a Dirac–Slater exchange potential with a Latter tail. At BM28 (ESRF), tuneable X-rays were obtained at eV resolution using a 1 1 1 Si monochromator. From target systems including Cu²⁺ and Zn²⁺, the X-rays were scattered through high angle from an aqueous medium contained in a thin Perspex cell provided with 8 μm kaplan windows. An energy resolution of ∼500 eV from the HPGe detector was adequate to separate the elastic scattering signal from K_α radiation but not from Compton or K_β contributions. The Compton contribution from the medium was removed assuming validity of the relativistic impulse approximation. The contribution due to K_β fluorescence and the resonant X-ray Raman scattering process were handled by assuming the branching ratio for K_α and K_β contributions to be constant and to be accurately described by fluorescent yields measured above edge. At ionic concentrations ranging from 0.01 to 0.1 mol/l, resonance structures accord with predictions of elastic scattering cross-sections calculated within IPA. Amplitudes calculated using modified form-factors and anomalous scatter factors computed from a Dirac–Slater exchange potential were convolved with a Lorentzian of several eV (FWHM).     

On the validity of using the TKR sum rule to place zero angle electron impact spectra on an absolute scale

The use of the Thomas—Kuhn—Reiche (TKR) sum rule to place electron impact spectra on an absolute scale is only rigorously correct if the observed intensities can be extrapolated to their first Born zero momentum transfer limit. This note investigates the error involved in using the TKR sum rule at zero scattering angle instead of zero momentum transfer. By considering an expansion of the generalized oscillator strength at zero angle it is shown that the first order correction to the TKR sum rule can be written as ($\text{1}\text{3}\text{k}{}^2$)[|E_o|?($\text{4}\text{5}$)$\text{V}$_ee] for target systems randomly oriented in space where k² is the incident electron energy and |E_o| and $\text{V}$_ee are the magnitude of the total electronic energy and the electron—electron repulsive energy of the ground state of the target system respectively. Estimates based on Hartree—Fock calculations indicate a 0.27% error in the use of the TKR sum rule for F and 0.79% for Ar at an incident electron energy of 25 keV.     

A rotational thermalization model for the calculation of collisionally narrowed isotropic raman scattering spectra - application to the SRS N2 Q-branch

A model for the calculation of collisionally narrowed isotropic. Raman scattering spectra is proposed. In this model, the rotational transition probabilities are calculated within the strong collision approximation, allowing the rotational energy transfer rates to be expressed in terms of the sole individual Q(J) line broadening coefficients. These transfer rates satisfy both detailed balance principle and unitarity of the scattering matrix in contrast with most of the previous approaches. Under further approximation concerning the rotational distribution of the collisional frequency, simpler expressions for transfer rates are deduced, which do not satisfy necessarily both unitarily and detailed balance. A simple analytical expression for the Q-branch profile is then obtained. An experimental study of the isotropic Q-branch for N₂ as a function of pressure has been conducted at room temperature by stimulated Raman spectroscopy (SRS). The Q-branch profiles calculated from the present model show a good agreement with SRS experiments, in particular when the lines overlap and when collisional narrowing takes place. This agreement is quite similar to that obtained by using a polynomial inverse energy gap law to describe the rotational energy transfer rates, and the results of these two models are closer for higher temperatures. The simple analytical expression mentioned above for the Q-branch profile, which is inaccurate at room temperature, becomes reliable at high temperature.     

Dimensionality control of coupled scattering equations using partitioning techniques

A method of variable reduction of the dimensionality of the coupled equations for inelastic scattering is presented, based upon a projection operator P with a restricted range of orbital angular momentum states. For rotational states in the range O?j ?j^* and total angular momentum large, the coupled equations have dimensionality (j^* + 1) ? N ?(j^* + 1)², where the value of N is controlled by the choice of P. This is in contrast to conventional partitioning techniques which utilize further restrictions on the important molecular rotational states. The equations for the P subspace and its complementary Q subspace are decoupled by an approximation on the equation of motion of Qψ_scat. Information about scattering into the Q subspace is retained, within this degree of approximation, and is reintroduced at the end of the computation with little additional labor. The theory is developed in terms of atom-rigid-rotor scattering, although addition of vibrational modes would not in any way interfere with the basic techniques used.     

Direct determination of scattering time delays using the S-matrix version of the Kohn variation method

A direct method for determining time delays for scattering processes is developed using the S-matrix version of the Kohn variation method recently reported by Miller. In this paper we develop a procedure for simultaneously determining both the S matrix and its energy derivative. Since only a small number of additional terms are needed to generate the energy derivative of the S matrix, this yields a direct method for determining scattering time delays that should be more attractive than the traditional methods based on numerical differentiation of #S#-matrix elements calculated at many closely spaced energies.     

Atom-molecule scattering with the average wavefunction method

The average wavefunction method (AWM) is applied to atom-molecule scattering. In its simplest form the labor involved in solving the AWM equations is equivalent to that involved for elastic scattering in the same formulation. As an initial illustration, explicit expressions for the T-matrix are derived for the scattering of an atom and a rigid rotor. Results are presented for low-energy scattering and corrections to the Born approximation are clearly evident. In general, the AWM is particularly suited to polyatom scattering due to its reduction of the potential in terms of a separable atom-atom potential.     

Energy distribution of charge carriers in solids in highly non-equilibrium states considering cascade transitions and inelastic scattering

A study is made of the cascade process, which describes the energy loss and multiplication of highly non-equilibrium secondary electrons and holes in crystalline platinum irradiated by low-energy electrons. The pair-creation scattering rates are evaluated in the framework of statistical model that takes into account the electron band structure of platinum. Kinetic equations for the excited electron and hole energy distributions are solved numerically in the isotropic scattering approximation for some primary (excitation) energies E_p that do not exceed the plasma energy E_F+ℏω_pl.     

Anomalous small-angle x-ray scattering from mesoporous noble metal catalysts

We present the analysis of a catalyst containing platinum nanoparticles supported on mesoporous MCM-41 silica by anomalous small-angle x-ray scattering (ASAXS). The analysis of this composite system by ASAXS is first studied by use of model calculation. Here, it is shown that the full analysis must proceed by decomposing the scattering data measured at different energies of the incident beam into three partial intensities. This evaluation is compared to a simplified method in which scattering curves measured at two different energies are subtracted from each other. The different methods are applied to experimental data obtained from platinum nanoparticles on an MCM-41 support material. The model calculations show that the simplified method leads to large deviations especially at low q in ordered systems. In the semi-ordered material MCM-41, these deviations are less pronounced, and the method of simple subtraction proves to be a good approximation for q values higher than 0.1 nm^?1.     

The elastic scattering on jellium clusters

The following calculations are based on the local density approximation potential (LDA) of W. Ekardt for the spherical jellium-background model (SJBM). Taking into account the smooth shape of the potential, the WKB approximation was used to calculate the energy and angular dependence of the electron scattering cross-sections fo rsmall Na clusters. The number of phase shifts needed to describe the scattering in the range of energies <4.5 eV increases with the size of cluster. The calculated elastic electron scattering cross-sections for the Na clusters, corresponding to the shell closings (8, 20, 40), are exhibiting a pronounced peak structure, correlated with resonance states. The computed peaks of the angular dependences of the cross-sections on energy are shifted to small angles with increasing the cluster size. The absence of fragmentation at these small electron energies presents a challenge for the experimentalists.     

Urbach rule in the optical spectra of crystalline and amorphous organic-solids

In this paper we explore the effects of microscopic structural disorder on the low energy optical absorption edges in some crystalline and amorphous organic solids. We advance a theoretical model which incorporates the simultaneous effects of static disorder and of exciton—phonon coupling on Frenkel exciton states. Explicit theoretical expressions were derived for the self energy and for the absorption lineshapes by the introduction of a two-step effective-field approximation, defining an effective exciton—phonon hamiltonian which was subsequently utilized within the framework of a single site average t-matrix approximation for disorder scattering. Numerical results for the low energy lineshapes over a broad temperature range demonstrate that exciton scattering by a static disorder field results in Urbach type low energy absorption tails, where the Urbach slope is temperature independent.