Advances in Imaging and Electron Physics

Time Resolved Electron Diffraction: For Chemistry, Biology And Material Science
1. Auflage 2014
ISBN: 978-0-12-800309-1
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, Band Volume 184, 310 Seiten

Reihe: Advances in Imaging and Electron Physics

Advances in Imaging & Electron Physics merges two long-running serials-Advances in Electronics & Electron Physics and Advances in Optical & Electron Microscopy. The series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains. - Contributions from leading authorities - Informs and updates on all the latest developments in the field

Autoren/Hrsg.

Leseproben

Chapter One

Theory and Data Analysis in Time-Resolved Electron Diffraction

Anatoli A.IschenkoSergei A.Aseyev

Abstract

In contrast to the traditional approach to electron diffraction and X-ray structural analysis of equilibrium systems, the data analysis process for time-resolved pump-probe electron and X-ray diffraction requires the inclusion of the interaction between the molecular ensemble and the laser field explicitly. The interference term that arises in the molecular scattering intensity of the electrons upon the coherent excitation of a molecular system under study and its Fourier transform makes it fundamentally possible to determine the density matrix and carry out the tomographic reconstruction of the molecular quantum state of this system. Thus, the time sequence of measurements of the scattering intensity and the use of the Fourier transform, which implements a transition from the space of scattering variables to the space of internuclear distances, gives necessary information for the tomographic reconstruction of the Wigner function. This chapter describes the theory and data analysis for time-resolved electron diffraction and presents the basic assumptions and the approximations, the illustration of the diffraction signatures of the excited molecules, simplified and complete cumulant analysis of the time-resolved electron diffraction (TRED) data, and the manifestation of chaotic nuclear dynamics in TRED studies of highly excited nonequilibrium ensembles. Here, the results of several internationally renowned research groups are included and cited.

Keywords

Electron scattering intensitycoherent excitationpump-probe experimentsdensity matrixWigner functionpotential energy surfacetomographic reconstructionexcited molecules

Basic Assumptions and Approximations

A plane wave electron that is elastically scattered by an atom emerges as a spherical wave with an amplitude as given by Bonham and Fink (1974):

(R,?)={exp(ikR)/R}f(?),

(1.1)

where R is the distance between the scattering center and the detector plane and the absolute value of the wave vector k is given by k = |k| = 2?/?, with ? being the wavelength of the electron. For an isolated atom, the atomic electron scattering amplitude f(?) determines the amplitude of the electron beam scattered into the angle ? (Figure 1.1). As the electron traverses the atom, it experiences a phase delay, which makes the scattering factor complex. Meanwhile, in scattering from a single atom, this phase shift is inconsequential, and scattering from multiple atoms may involve different phase shifts from each individual atom.

The amplitude of the wave scattered by atom i within a molecule is written as given in Bonham and Fink (1974):

i(R,?)={exp(ik|R-ri|)/|R-ri|}exp(ik0zi)fi(?),

(1.2)

where zi is the projection of the atomic position vector ri onto the z-axis (Figure 1.1) and R is the scattering distance. Since R is a macroscopic parameter (i.e., ri << R), Eq. (1.2) can be expressed as

i (R,?)={exp(ikR)/R}exp(i(k0-ks)ri)fi(?),

(1.3)

where k0 and ks are the wave vectors of the incident and scattered electrons, respectively, and |k0|=|ks| for elastic scattering.

Figure 1.1Definition of scattering coordinates used for the development of intensity equations in electron diffraction. ? is the scattering angle and ? the azimuthal angle in the detector plane; k0 and ks are the wave vectors of the incident and scattered electrons, respectively; s is the momentum transfer vector; rij is the internuclear distance vector between the nuclei of atoms i and j, which are positioned at ri and rj, respectively; and ? and ? give the orientation of the molecular framework with respect to the XYZ laboratory frame.

Introducing the momentum transfer vector s with a magnitude of |s|=|k0-ks|= (4?/?)sin(?/2), and invoking the superposition principle, one obtains the amplitude of the electron wave scattered by the molecular system of N atoms as

=?i=1,N?i={exp(ikR)/R}?i=1,Nfi(s)exp(isri).

(1.4)

The intensity of the scattered electrons can be expressed in terms of the electron current density j (Ewbank, Schäfer, & Ischenko, 2000):

(s)=(he/4?mei)(?i???i??i??i?),

(1.5)

where e and me are the electron charge and mass, ? the gradient operator, and ?? the complex conjugate wave function.

Using Eqs. (1.4) and (1.5), one obtains for the intensity:

(s)=I0(jsc/j0)=I0Re{(1/2ik0)?i=1,N(?i???i??i??i?)}=(I0/R2)Re{?i=1,Nfi(s)exp(isri)?j=1,Nfj?(s)exp(-isrj)}=(I0/R2){?i=1,N|fi(s)|2+Re?i?j=1,Nfi(s)fj?(s)exp(is(ri?rj))}.

(1.6)

In Eq. (1.6), I0 is the intensity of the incident electron beam; j0 (=hk0e/2?me) and jsc are the current densities for the incident and the scattered electrons, respectively; and Re denotes the real part of the function. Higher-order terms, corresponding to multiple scattering, are disregarded for the current purpose.

Eq. (1.6) is often written as I(s) = Ia(s) + Imol(s), where the first term is associated with incoherent “atomic scattering” because it does not depend on the internuclear distances. The second term, which does depend on the internuclear distances, is associated with coherent “molecular scattering.” For each pair of atoms (i,j), separated by the instantaneous internuclear distance, rij = ri-rj, Eq. (1.6) yields the following molecular intensity function:

mol(s)=(I0/R2)Re{?i?j=1,N|fi(s)||fj(s)|exp(i??ij(s))exp(isrij)},

(1.7)

where ??ij(s) is the difference in the phase shifts incurred by the electrons while scattering from atoms i and j, respectively (Hargittai, 1988).

Inherent in Eq. (1.7) is an approximation known as the Independent Atom Model (IAM), which assumes that the electronic wave function of each atom in a molecule is just that of the isolated atom (Bonham and Fink, 1974). This implies that the effects of chemical bonding on the electron density distribution of the atoms are ignored. Within the IAM approximation, the molecular surrounding of an atom does not affect its scattering, so that tabulated atomic scattering factors can be used for each atom in a molecule.

Assuming single scattering processes for fast electrons (> 10 keV) with a short (attosecond) coherence time, the electrons encounter molecules that are essentially “frozen” in their rotational and vibrational states. Thus, the latter can be accounted for by using a probability density function (p.d.f) that characterizes the ensemble under investigation. If the molecular systems investigated are not at equilibrium, as is the case in studies of laser-excited molecules, a time-dependent p.d.f must be used to describe the structural evolution of the system. In addition, rotational and vibrational motions can be separated adiabatically, since the latter involves much faster processes. The time-dependent molecular intensities then can be represented by averaging Eq. (1.7) with the p.d.f that represents the spatial and vibrational distributions of the scattering ensemble (Ischenko, Schäfer, & Ewbank, 1996, 1997):

mol(s,t)=??Imol(s)?vib?sp=(I0/R2)?i?j=1,N|fi(s)||fj(s)|Re{exp[i??ij(s)]??exp(isrij)?vib?sp}=(I0/R2)?i?j=1,N|fi(s)||fj(s)|cos(??ij(s))?0,?Pvib(rij,t)[?0,??0,2?Psp(?ij,?ij,t)exp(isrij)sin?ijd?ijd?ij]drij.

(1.8a)

In Eq. (1.8a), ?...? denotes the vibrational and spatial (orientational) averaging over the scattering ensemble, Pvib(rij,t) and Psp(?ij,?ij,t) are the vibrational and spatial p.d.fs, respectively, and ?ij and ?ij are the angles of the spherical polar coordinate system (Figure 1.1) that define the orientation of the internuclear distance vector rij in the scattering coordinate frame.

For spatially isotropic, randomly oriented molecules, Psp(?ij,?ij) = 1/4?, and Eq. (1.8a) simplifies to the following expression for the time-dependent molecular intensity function:

mol(s,t)=(I0/R2)?i?j=1,N|fi(s)||fj(s)|cos(??ij(s))×?Pvib(rij,t)[sin(srij)/srij]drij.

(1.8b)

The time-dependent p.d.fs, Psp(?ij,?ij,t) and...