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This chapter considers the interactions of X-rays with matter. It opens by developing X-ray wave equations in the presence of scatterers, taking the Maxwell equations as a starting point. The projection approximation is then discussed. The concept of a Green function, which is of immense importance in the formalism of X-ray scattering, is introduced. Equipped with this, an integral from of the X-ray wave equation is developed, approximate solutions to which are furnished by the famous first Born approximation. Second and higher-order Born approximations are also considered, heralding the transition from so-called kinematical diffraction to dynamical diffraction. Other subjects treated in the chapter include the Ewald sphere, the multislice approximation, the eikonal approximation, the link between refractive index and electron density, Compton scattering, photoelectric absorption, fluorescence, and the information content of scattered fields.

This chapter examines the key approximations in crystal lattice dynamics in a way that is particularly helpful for defect studies. It discusses the Born-Oppenheimer approximation, the adiabatic approximations, the harmonic approximations, and the dipole approximation. It also details electron-phonon interaction, the Hellman-Feynman theorem, and electron-lattice coupling.

This chapter deals with the theory of the double differential photon scattering cross-section in both non-relativistic and relativistic treatments. It shows how the Impulse Approximation leads to a cross-section interpretable in terms the ground state electron momentum density distribution, either directly through the Compton profile or in coordinate space though the reciprocal form factor. The nature of these derived quantities for atoms, molecules, and solids, including the treatment of electron-electron correlations is explained. The chapter also deals with spin-dependent scattering theory, and provides an introduction to the theory of the photon inelastic scattering cross section for all x-ray physicists.

This chapter looks at problems that deal with the Born approximation; scattering theory (partial-wave analysis); low-energy scattering (resonant scattering); scattering of fast particles (Eikonal approximation); scattering of particles with spin; analytic properties of the scattering amplitude; and scattering of composite quantum particles (inelastic collisions).

This chapter explains how to incorporate scattering by random impurity atoms into the general Green function formalism of the theory of superconductivity. The cross-diagram technique based on the averaging over random impurity positions is derived using the Born approximation for the scattering amplitude. Impurity self-energy is derived. Homogeneous state of an s-wave superconductor is considered.

This chapter examines the case of a penetrable scatterer with an index of refraction that can be space-dependent and is assumed to be different from the constant background index. The inverse scattering problem is to determine the support D of the contrast from far field measurements. The chapter begins with a simple scattering model where the scatterers consists of a finite number of point scatterers. The inverse problem is to determine the locations of these point scatterers from the multistatic response matrix F, which is the discrete analog of the far field operator. In this situation, the Factorization Method is nothing else but the MUSIC-algorithm which is well known in signal processing. The chapter then formulates direct and inverse scattering problem for the scattering by an inhomogeneous medium, reformulates the direct problem as the Lippmann-Schwinger integral equation, and justifies the popular Born approximation. The chapter formulizes the far field operator and proves a characterization of D by the convergence of a Picard series which involves only known data derived from the far field operator. This characterization holds only if the frequency is not an eigenvalue of an unconventional eigenvalue problem of transmission type. The last section shows that there exist at most a quantifiable number of these values.

This chapter begins by calculating the Wigner transform for the von Neumann equation for the one-body density operator. It shows how the Liouville equation follows in leading order in an expansion to ℏ. Properties of this expansion and of the resulting equation are discussed with respect to their physical and practical importance. Semi-classical approximations to the collision term are described and interpreted in terms of relevant transition rates. In Born approximation, equations of Boltzmann-Uehling-Uhlenbeck (BUU)- or Landau-Vlasov-type are obtained. The relevance of the conservation laws for particle number, energy, and momentum is discussed. For relaxation processes to equilibrium, self-energies are introduced and the relaxation-time approximation to the collision term is presented. The physical meaning of self-energies is discussed, together with the formula for the leading-order dependence of their imaginary part on energy, chemical potential, and temperature.

This chapter reviews G. N. Lewis' seminal paper of 1916 that introduced the concept of the electron-pair bond. The potential energy curves for the two lowest electronic states of the hydrogen molecule ion (H₂ ⁺) are described. The molecular orbital (MO) concept is introduced and a set of approximate molecular orbitals formed by linear combination of the 1s atomic orbitals of the two atoms (LCAO MOs). The potential energy curve for a neutral hydrogen molecule in its ground state calculated from a wavefunction consisting of the product of one LCAO MO for each electron is shown to be much higher than the experimental curve for all values of the internuclear distance R. The electron correlation energy is defined. The non-zero experimental dipole moment of hydrogen deuteride (HD) shows that the Born-Oppenheimer approximation is not completely valid.

Furnished with basic ideas about the scattering on a single impurity, the motion of a particle scattered by many randomly distributed impurities is approached. In spite of having a single particle only, this system already belongs to many-body physics as it combines randomising effects of high-angle collisions with mean-field effects due to low-angle collisions. The averaged wave function leads to the Dyson equation. Various approximations are systematically introduced and discussed ranging from Born, averaged T-matrix to coherent potential approximation. The effective medium and the effective mass as wave function renormalisations are discussed and the various approximations are accurately compared.

Semi-empirical π‐electron theories of conjugated polymers are introduced, starting from the fundamentals of the Born-Oppenheimer approximation and sp-hybridization. The models described include the simplest non-interacting (Hückel) theory, models of electron-nuclear coupling (i.e., the Su-Schrieffer-Heeger and Peierls models), and models of interacting electrons (i.e., the Pariser-Parr-Pople model). All of them are introduced in the second quantized notation.There is also a brief discussion of the role of spatial and electron-hole symmetries, and quantum numbers in classifying the electronic states of π‐ conjugated polymers.

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born-Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

Chapter 1 introduces several of the common principles, techniques and approximations that will be employed throughout the text. Classically, the Hamiltonian is the sum of kinetic energy and potential energy. In quantum mechanics, it is an operator that, by operating on the statefunction, leads to the energy observable. The chapter begins with a preliminary description of the crystal’s Hamiltonian and then introduces approximation techniques that permit useful solutions. Beginning with the simple jellium model, Hartree and Hartree-Fock approaches are developed, exchange interactions and correlation effects are explored, and both time-independent perturbation and time-dependent perturbation techniques discussed. Examples illustrate scattering by perturbation as well as adiabatic evolution. The centrality of fast-and-slow interactions is stressed, the Born-Oppenheimer approximation is illustrated through the configuration coordinate diagram, and interacting electron systems are analyzed. The multi-electron aspects are stressed by discussing static screening, dynamic screening and the meaning of permittivity therein.

This chapter spells out the description of molecular structure and dynamics in terms of the adiabatic or Born-Oppenheimer approximation. Adiabatic electronic eigenstates are defined using a unitary transformation parametrized by nuclear coordinate operators. Conditions are specified under which the Born-Oppenheimer approximation applies.

This chapter discusses scattering in three spatial dimensions. It introduces the concepts of the scattering amplitude and cross-section. The chapter discusses the Born approximation and how to systematically go beyond this approximation. It also discusses the concepts of phase shifts and partial wave expansion.

This chapter discusses some elementary molecular physics. Its goal is to understand the reason why neutral atoms form aggregates, that is, molecules, solids, and the matter which we see around us. The answer that it is simply because of the dipole–dipole attraction is not correct. In the case of the hydrogen molecule, for example, the size of its binding energy is orders of magnitude larger than the size of the dipole–dipole attraction between two isolated hydrogen atoms. In fact, in the hydrogen molecule, the two electrons are shared by both nuclei, so the integrity of each atom is completely lost in the molecule. The chapter analyzes within the Born–Oppenheimer approximation the H2+ ion. The chapter introduces the idea of hybridization, the tight-binding approximation, and the meaning of electron correlations in the hydrogen molecule which we treat within the Hubbard model.

This chapter derives the Hamiltonian for the electronic Schrödinger equation of a molecule in the presence of external and internal static electric or magnetic fields. It starts off with the Born-Oppenheimer approximation and discusses in detail the minimal coupling approach to the interaction of molecules with fields from a non-relativistic as well as relativistic point of view. In this context, the Dirac equation of an electron is derived and reduced to the Schrödinger-Pauli equation via the elimination of the small component approach. It also considers the relation between electric and magnetic fields and their scalar and vector potentials through Maxwell's equations, and introduces the problems related to transformations of the gauge of these potentials.

This chapter focuses on the coupling of electron dynamics and nuclear motion. In principle, electronic and nuclear degrees of freedom both have to be treated quantum mechanically and on an equal footing. This can be formally achieved using multicomponent time-dependent density-functional theory (TDDFT), but this approach has so far been of limited use. In practice one starts from the Born-Oppenheimer approximation and discusses the nuclear dynamics in terms of potential-energy surfaces. It is shown how TDDFT performs in the calculation of potential-energy surfaces. A particular challenge is the so-called conical intersections. Then, various schemes of ab initio molecular dynamics are discussed: Born-Oppenheimer dynamics, the TDDFT-Ehrenfest approach, and surface hopping schemes. It is shown how TDDFT is used for calculating nonadiabatic couplings between potential-energy surfaces. Finally, the Car-Parrinello approach is briefly discussed.

The atomic and electronic degrees of freedom for a crystalline solid are separated using the Born–Oppenheimer approximation. The Hartree–Fock approximation is then applied to the electronic problem, which is solved for a gas of free electrons, culminating in the Slater approximation for the exchange energy. This sets the stage for the density-functional theory that is used to give the Kohn–Sham–Schrödinger equation to be solved for an effective single-particle problem. The electron spin is treated by Dirac’s theory, which is reduced to a two-component theory encompassing the spin–orbit interaction, finally leading to the spin-density-functional theory in the local density-functional approximation (LDA) that is formulated for an arbitrary spin configuration. To close this chapter, some important formal properties of density-functionals are described and used in the general gradient approximation (GGA) for exchange and correlation devised by Perdew and others.

Scattering experiments provide the main source of information on the properties of elementary particles. Here we present the theory of scattering in both classical and non-relativistic quantum physics. We introduce the basic notions of cross section and of range and strength of interactions. We work out some illustrative examples. The concept of resonant scattering, central to almost all applications in particle physics, is explained in the simple case of potential scattering, where we derive the Breit–Wigner formula. This framework of non-relativistic potential scattering turns out to be very convenient for introducing several other important concepts and results, such as the optical theorem, the partial wave amplitudes and the corresponding phase shifts and scattering lengths. The special cases of scattering at low energies, and that in the Born approximation, are studied. We also offer a first glance at the problem of the infrared divergences for the case of Coulomb scattering.

The formulation of generalize electromagnetic scattering is given. Previously derived multipole expansions using the language of scattering are presented and applied to resonant and plasmon resonances. Formal scattering theory is introduced, and the integral scattering equation is derived and used to find the Born expansion and to prove the optical theorem. Partial wave analysis for the scaler scattering problem is discussed with connections between quantum (wave theory) and classical views. Vector spherical harmonics and the extension of partial wave analysis to the scattering of vector fields of electromagnetic waves are presented. Finally, Mie scattering is considered in detail with applications including glory scattering and whisper gallery mode resonances.