Dynamic Form Factor Research Articles

The `Frozen-Lattice' Approach for Incoherent Phonon Excitation in Electron Scattering. How Accurate Is It?

The `frozen-lattice' model is a semi-classical approach for calculating electron diffuse scattering in crystals that has arisen from thermal vibration of crystal atoms. This quasi-elastic scattering approach is, however, unproven since its equivalence with the incoherent phonon-excitation model is not yet established. As quantitative electron microscopy is becoming a realistic method, it is necessary to examine the accuracy of the model. In this paper, based on a rigorous quantum-mechanical phonon excitation theory, it is proved that an identical result would be obtained using the frozen-lattice model and the formal phonon-excitation model if (i) the incoherence between different orders of thermal diffuse scattering is considered in the frozen-lattice-model calculation and (ii) the specimen thickness and the mean-free-path length for phonon excitation are both smaller than the distance travelled by the electron within the lifetime of the phonon (τ0v, which is 5 µm for 100 kV electrons). Condition (ii) is usually absolutely satisfied and condition (i) can be precisely accounted for in the calculation with the introduction of the mixed dynamic form factor S(Q, Q′). The conclusion holds for each and all orders of diffuse scattering, thus, the quantum-mechanical basis of the frozen-lattice model is established, confirming the validity, reliability and accuracy of using this model in quantitative dynamical electron diffraction and imaging calculations. It has also been shown that the frozen-lattice model is suitable for low-energy electrons.

An optical potential approach to incoherent multiple thermal diffuse scattering in quantitative HRTEM

The theory for the absorption potential (or optical potential) in electron scattering was first proposed by Yoshioka in 1957 based on an approximation that the Green's function is replaced by its form in free-space. This approximation has dramatically simplified the calculation, but the function of the optical potential has been partially lost. In this paper, a rigorous theoretical proof is given based on the quantum inelastic excitation theory to show that the inclusion of the optical potential in the dynamic calculation automatically recovers the contributions made by the high-order diffuse scattering although the calculation is done using the equation derived for single diffuse scattering. This conclusion generalizes the existing first-order diffuse scattering theories to cases in which the incoherent multiple diffuse scattering are important. It is suggested that multiple thermal diffuse scattering (or phonon excitations) is likely to be the dominant source for affecting image contrast in quantitative electron microscopy because of the channeling effect and the localized scattering nature of phonon excitation. Surprisingly, phonon scattering is anticipated to contribute fine atomic-scale contrast in the image, which is believed to be more pronounced than the “white noise” effect of conventional understanding. The optical potential is no longer a simple potential function, rather it is a non-local function strongly dependent on the dynamic diffraction in the crystal because of the involvement of Green's function. These characteristics can be properly taken into the computation using a proposed Bloch wave–multislice approach, in which the non-local effect is resolved in the Bloch wave matrix diagonalization and the dynamical effect is taken care of using Green's function calculated by the multislice theory under the small angle (or high-energy) scattering approximation. In the ground-state approximation, by which we mean that the crystal is in its ground state before each and every inelastic excitation, an introduction of the mixed dynamic form factor and the density matrix automatically produces the incoherence between different order and different phonon excitation processes, resolving a big problem encountered by many other theories. In the conventional dynamical calculation for elastic wave, the inclusion of the Debye–Waller factor and the optical potential accounts only for the effects of the phonon excitation on the elastic wave rather than the contribution made by the diffusely scattered electrons to the image/diffraction pattern.

Thomson scattering from ion acoustic waves in laser plasmas

This work is concerned with the description of ion acoustic fluctuations in electron-ion plasmas relevant to laser-plasma interaction experiments. A nonlocal closure to the linearized moment equations for the fluctuating hydrodynamic quantities is introduced. These equations are used to construct practical expressions for the dynamical form factor and Thomson scattering cross section, which are valid in the entire region of particle collisionality in plasmas with high $Z$ and large ${\mathrm{ZT}}_{e}{/T}_{i}.$

Role of temperature in the physics of metallic clusters

The finite temperature of the ionic degrees of freedom plays either a determining or an important role in a large variety of cluster properties. We briefly review here the role played by the temperature in different cluster phenomena, such as: occurrence of phase transitions, shape fluctuations, detailed properties of the static and dynamic structure formfactors, of the electron single-particle spectra and of the optical response, and the emergence of the 1/f-noise in ionic dynamics.

Inhomogeneous eigenmode localization, chaos, and correlations in large disordered clusters

Statistical and localization properties of dipole eigenmodes (plasmons) of fractal and random nonfractal clusters are investigated. The problem is mathematically equivalent to the quantum-mechanical eigenproblem for vector (spin-1) particles with a dipolar hopping amplitude in the same cluster. In fractal clusters, individual eigenmodes are singular on the small scale and their intensity strongly fluctuates in space. They possess neither strong nor weak localization properties. Instead, an inhomogeneous localization pattern takes place, where eigenmodes of very different coherence radii coexist at the same frequency. Chaotic behavior of the eigenmodes is found for fractal clusters in the region of small eigenvalues, i.e., in the vicinity of the plasmon resonance. The observed chaos is ``stronger'' than for quantum-mechanical problems on regular sets in the sense that the present problem is characterized by (deterministically) chaotic behavior of the amplitude correlation function (dynamic form factor). This chaotic behavior consists of rapid changes of the phase of the amplitude correlation in spatial and frequency domains, while its magnitude is a very smooth function. A transition between the chaotic and scaling behavior with increase of eigenvalue is observed. In contrast to fractal clusters, random clusters with nonfractal geometry do not exhibit chaotic behavior, but rather a mesoscopic delocalization transition of the eigenmodes with decrease of eigenvalue.

Neutron study of mesoscopic magnetic clusters:Mn12O12

Neutron diffraction and inelastic neutron scattering in zero field have been performed on mesoscopic magnetic Mn{sub 12}O{sub 12} clusters, well known for their macroscopic quantum effects observed at low temperature. In addition to the static spin correlations of the cluster in its ground state, we have observed some energy levels related both to the anisotropy and to the exchange energies of the cluster, with their respective dynamical form factor. The temperature and Q dependences of the anisotropy energy levels can be qualitatively explained using a quantum model. Besides these expected modes, the most striking result is the observation of extended energy modes at energy values below those related to anisotropy. Their temperature and Q dependences differ from those expected for the energy levels of usual magnetic clusters. They indicate an additional spin coupling which must play a role in tunneling properties. {copyright} {ital 1997} {ital The American Physical Society}

Effects of non-Maxwellian (super-Gaussian) electron velocity distribution on the spectrum of Thomson scattering

The dynamic form factor for Thomson scattering with a non-Maxwellian (super-Gaussian) electron velocity distribution is analytically and numerically studied for the first time. Both the ion and electron features of the spectra for α>1 are found to be affected strongly. For the ion feature, the use of the usual theoretical spectra with the assumption of a Maxwellian distribution for fitting the measured Thomson scattering spectra may have resulted in some errors in calculating the plasma parameters in high-Z plasmas irradiated with high intensity light. However, it is impossible to use the ion feature of the measured spectra alone to deduce the form of the electron distribution. For the electron feature, the spectral shape (in particular the peak amplitude) changes sensitively with the distribution index. It would be possible to detect the super-Gaussian distribution by simultaneously measuring the ion and electron features of the spectra.

High-Order Harmonic Generation in Dense Plasmas with a High Intensity Light Field

High-order harmonic generation by an oscillating current due to intense laser electric fields in dense plasmas is investigated. In solid target plasmas produced by an intense ultra-short pulse laser, the intense laser radiation directly interacts with overdense plasmas. In such a case, plasma screening effects are important in the plasma response to the laser field. For such plasmas, the higher harmonics are evaluated by using a dynamical form factor for plasma electrons and ions. There is a peak in the harmonic spectrum at twice the plasma frequency.

Statistical multiple diffuse scattering from rough surfaces in RHEED — beyond the distorted-wave Born approximation

In reflection high-energy electron diffraction (RHEED) of growing surfaces in molecular beam epitaxy (MBE), diffuse scattering is generated by atom vibrations, point vacancies and growth islands (or surface roughness). Most of the existing RHEED theories have been developed under the first-order diffuse scattering approximation, and thus they are restricted for surfaces whose roughness is relatively low. In fact, crystal surfaces grown by MBE are usually rough; the change of surface coverage from 0 to 1 monolayer accounts for the observed RHEED oscillation. In this paper, a formal dynamical theory of RHEED has been developed to calculate the diffuse scattering produced by both atom vibrations and point vacancies at surfaces. The theory is aimed at recovering the multiple diffuse scattering that has been dropped by the distorted-wave Born approximation (DWBA). With the inclusion of a complex potential in the dynamical calculation, a rigorous proof is given to show that the high-order diffuse scattering terms are recovered in the calculation using the equation originally derived under the DWBA. This conclusion establishes the basis for expanding the RHEED theories developed under the first-order diffuse scattering to cases where the degree of surface roughness is high, allowing dynamical calculation of RHEED rocking curves for any growing surface. The statistical time and structure averages over the distorted crystal potential are evaluated analytically before numerical calculation. The dynamic form factor is calculated with consideration of anisotropic surface atom vibration and point vacancies at a growing surface.

Electron Statistical Dynamical Diffuse Scattering in Crystals Containing Short-Range-Order Point Defects

A formal statistical dynamical theory is developed to calculate diffuse scattering produced by short-range order (SRO) in a distorted crystal structure with consideration of atomic thermal vibrations. Diffuse scattering not only produces fine details in diffraction patterns but also introduces a non-local imaginary potential function that reduces the intensities of the Bragg reflected beams. The distribution of the diffusely scattered electrons and the Fourier coefficients of the absorption potential are directly related to a dynamic form factor S(Q,Q'), which has been calculated with consideration of SRO in the distorted lattices. The statistical structure average on imperfections is performed analytically and the final result is correlated to Cowley's short-range-order parameters. The theory is formulated in the Bloch-wave scheme (Bethe theory) for the convenience of numerical calculation in transmission electron diffraction. A rigorous theoretical proof is given to show that the inclusion of a complex potential in the dynamical calculation automatically recovers the contributions made by the high-order diffuse scattering, although the calculation is done using the equation derived for single diffuse scattering. This simply expands the capability of conventional single diffuse scattering theories. Therefore, the complex potential has a much richer meaning than the conventional interpretation of absorption effect.

Dynamical diffraction in electron-energy-loss spectrometry: The independent Bloch-wave model.

The double-differential cross section for inner-shell ionization in a crystal by fast electrons is calculated within the theory of dynamical electron diffraction. Bloch waves are used for the incident and the emerging electrons. We show that interference terms are still present in the independent Bloch-wave model. In the two-beam case, the cross section contains but six terms, three of them caused by interference. From the six terms, three are identified as the Kikuchi excess and defect lines and the Kikuchi band. The remaining three terms have not yet been described explicitly. As an example, the Si K-shell ionization cross section is calculated for a (220) two-beam case, in a dipole approximation for the mixed dynamic form factor. It is shown that in electron spectroscopic diffraction the maximum of the fundamental Lorentzian can be significantly shifted with respect to the direction of incidence, and that the interpeak minimum between the fundamental and the Bragg-excited Lorentzians is depressed relative to the kinematical case. \textcopyright{} 1996 The American Physical Society.

Interference in electron Compton scattering for silicon

Starting from an expression for the double-differential cross section for electron Compton scattering in a crystal for a two-beam case, valid for a small Ewald sphere, the contributions of M-shell and L-shell electrons in silicon to the mixed dynamic form factors are calculated. The shape and intensity of the interference effects that superimpose on the Compton profile are determined. It is shown that the appearance of interference terms renders the cross section asymmetric and dependent on the momentum transfer also in the usual momentum representation. A procedure for an experimental investigation of the interference terms is proposed.

Progress in electron Compton scattering

Use of electrons as a probe instead of photons for Compton scattering experiments has been suggested in 1981 by B.G. Williams et al. The advantages, compared to conventional Compton scattering, are evident: a cross section higher by five orders of magnitude, extraordinary spatial resolution, on-line specimen manipulation, and all the additional information that is available in an analytical electron microscope. With PEELS nowadays available, and with the solution of the background problem, electron Compton profiles (CPs) can now be studied. Directional electron CPs of Si could be obtained by use of a two-beam geometry. In the intermediate momentum region, theory fits our results better than the results from conventional γ-ray and X-ray experiments. Electron CPs of amorphous allotropes of carbon could be interpreted in terms of a charge transfer between s and p orbitals. In a crystal, the CP contains interference terms from scattering with different momentum transfer. They are caused by momentum correlation (off-diagonal elements of the momentum density in the reciprocal lattice), and can be described by the mixed dynamical form factor. In the two-beam case these terms are proportional to the excitation error. An experiment is suggested in order to obtain the off-diagonal elements of the momentum density.

Induced electrostatic fields in dense plasmas with an intense ultrashort pulse laser.

The electrostatic fields induced by an intense laser field around an ion in a solid density plasma are investigated. In a plasma produced by irradiating an intense ultrashort pulse laser on a solid target, the electron and ion temperatures are not in thermal equilibrium, and the electron temperature is higher than the ion temperature. For such a plasma, the dynamical form factor of a plasma electron is derived. By using the dynamical form factor, the screened potential and the induced oscillating electrostatic field around an ion are obtained. As a result, the plasma screening effects on the direct ionization by the laser are clarified.

Exact calculations of the dynamic form factor of hydrogen chemisorbed on a transition metal with one electron or hole in the energy band

The dynamic form factor S a(ω) of hydrogen chemisorbed on a transition metal with one electron or hole in the energy band is calculated exactly for the Anderson-Newns model. The influence of the electron correlation effects on S a(ω) is studied in different regions of the parameter space and the nature of the hydrogen-induced electron-energy losses is discussed. The functions S a(ω) are calculated for the sets of parameters, imitating hydrogen chemisorption on Ni and Pd, and the results are compared with measured electron-energy-loss spectra of the H/Ni adsorption system.

Correlations in space and time and dynamical diffraction of high-energy electrons by crystals.

The problem of scattering of high-energy electrons by solids is formulated on the basis of a kinetic equation (KE) for the one-particle density matrix. This equation provides a general treatment of spatial and temporal coherence of electrons and takes account of both elastic and inelastic scattering for arbitrary geometry of diffraction. Formulating the KE, we prove that if the energy of an electorn is sufficiently high (E\ensuremath{\sim}100 keV), the problem of multiple elastic and inelastic scattering by a solid is entirely determined by two universal functions, namely, the Coulomb potential averaged over the motion of the crystal particles and the mixed dynamic form factor of inelastic excitations, which is related to the time-dependent correlation function of the positions of electrons and nuclei in a solid. We show that in all the diffraction experiments the scattering cross section contains information about both these functions and discuss the possibility of their separate determination. The KE method generalizes previous theoretical approaches to the description of multiple inelastic scattering of high-energy electrons, and in the case of single inelastic interactions the solution of the KE reduces to the distorted-wave approximation. As an illustration, we consider an application of the KE to the problem of multiple scattering of high-energy electrons by collective electronic excitations of a crystal. Numerical solution of the KE is shown to be consistent with experimental observations, and evidence is found for the existence of a mechanism of damping of coherence by small-angle inelastic scattering of high-energy electrons by a crystal.

Magnetic effects, dynamical form factors, and electronic instabilities in the Hubbard chain.

We identify and classify the possible divergences of the charge and spin response functions of the Hubbard chain in a constant external magnetic field H=Hz^. When 0H${\mathit{H}}_{\mathit{c}}$, where ${\mathit{H}}_{\mathit{c}}$ is the critical field for full spin polarization, these divergences occur at the momenta \ifmmode\pm\else\textpm\fi{}2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$ for all values of U and density n=${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$+${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$; here ${\mathit{k}}_{\mathit{F}}$=\ensuremath{\pi}n/2 and ${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\sigma}}}$=\ensuremath{\pi}${\mathit{n}}_{\mathrm{\ensuremath{\sigma}}}$, so that 2${\mathit{k}}_{\mathit{F}}$=${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$+${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\downarrow}}}$. Divergences at \ifmmode\pm\else\textpm\fi{}4${\mathit{k}}_{\mathit{F}}$ show up in the charge function when H=0 and U\ensuremath{\rightarrow}\ensuremath{\infty}. Finally, divergences at \ifmmode\pm\else\textpm\fi{}2${\mathit{k}}_{\mathit{F}\mathrm{\ensuremath{\uparrow}}}$ appear only for certain regions of these parameters. We discuss the possible connection of our results to the physics of the quasi-one-dimensional metals.

Monte Carlo simulation of the dynamics and quasielastic scattering in many-chain polymer systems

A dynamic Monte Carlo algorithm has been employed to stimulate different dynamic properties for dilute and nondilute polymer systems, mainly in the good solvent regime (whith excluded volume conditions). The results for translational diffusion coefficients, relaxation times of the end-to-end vector and the internal Rouse corrdinates, the dynamic form factor of individual chains, and the dynamic collective scattering function of the systems are analyzed. The variations of these properties whith chain length and concentration are discussed and component with previous simulation results, theoretical predictions, or experimental behaviors

Narrow central peak of the magnetization dynamics at and below Tc in the singlet-singlet ferromagnet LiTbF 4

Approaching the Curie temperature T c from above, a narrow relaxation peak appears in the dynamical form factor of the homogeneous magnetization on top of the broad, noncritical central intensity detected previously by dynamical susceptibility and neutron scattering work. Below T c, this central peak substantially narrows and the resulting shape is described by an empirical model which couples the damping to two relaxation dynamics of unknown origin.

Dynamic light scattering from flexible ring macromolecules

The molecular dynamic form factor for the Rouse-Zimm free-deaining Gaussian ring is calculated in a form which facilitates comparison with experimental scattering data. The relative contributions of the pure translation mode and various interanl modes are given. At zero time, the dynamic factor form reduces to the well known Casassa result for the static form factor. The differences in the dynamic form factors of linear and circular free-draining gaussian chain are discussed