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This chapter presents the properties of thermal neutrons. Their wavelength (from the de Broglie equation) is well suited to the investigation of condensed matter, i.e., to the study of liquids, glasses (amorphous materials), and crystalline materials with varying degrees of order. That the neutrons carry magnetic moment also makes them well suited to the study of magnetic ordering. The theory of nuclear and magnetic scattering from individual atoms and from assemblies of atoms is presented, this leading to the definition of neutron scattering length and to the concepts of coherent and incoherent scattering. The focus then shifts to the direction and intensity of diffraction from crystalline materials (Bragg's law, structure factors), and to the description of this scattering when samples are presented in polycrystalline or powder form (Debye-Scherrer cones).

This chapter covers the rudiments of quantum theory, including the dual nature of light and matter, the Uncertainty Principle and probability concept, the electronic wavefunction, and the probability density function. Numerical examples are given to show that given the electronic wavefunction of a system, the probability of finding an electron in a volume element around a certain point in space can be readily calculated. Finally, the electronic wave equation, the Schrödinger equation, is introduced. This discussion is followed by the solution of a few particle-in-a-box problems, with the ‘box’ having the shape of a wire (one-dimensional), a cube, a ring, or a triangle. Where possible, the solutions of these problems are then applied to a chemical system.

This chapter discusses the electron wavelength, relativistic effects, ray diagram for a typical electron microscope, simple electron optics theory and approximations for ideal lenses. The constant field approximation, projector lenses, objective lenses, lens design, aberrations and the pre-field are also covered.

This discussion introduces the student to the reality, in quantum technology, that analysis of any problem necessarily begins with the Hamiltonian representing the system. The quantum Hamiltonian represents the total energy of the system, the sum of kinetic energy plus potential energy, written in canonical coordinates and conjugate momenta, and where these variables become time independent quantum operators. The nature of the potential energy for the nano-vibrator, following Hooke’s law, serves to localize the particle. The relevance of the nano-vibrator Hamiltonian—sometimes called the harmonic oscillator Hamiltonian—is perhaps one of the most important Hamiltonians in quantum systems. Not only can it be extended to cover things like phonons in solids, vibrations in molecules, and the behavior of bosons, but it is also the basis for leading to the concept of a photon, the quantum radiation field, and the quantum vacuum. This chapter provides the basic introduction for vibration of a particle or a nano-rod and looks at the wave-like behavior that emerges from the solution to the time independent Schrödinger equation. When we include the time evolution, we can observe dynamical behavior and begin to examine the meaning of quantum measurement.

The most widely used analytic models for representing thermodynamic behavior in supercritical ßuids are of the mean-Þeld variety. In addition to the practical interest in studying this topic, this class of models is also the conceptual starting point for any microscopic discourse on critical phenomena. In this chapter we take up the basic ideas behind this approach, studying different physical models, showing how their mean-Þeld approximations can be constructed as well as investigating their critical behavior. A useful conceptual model for understanding mean-Þeld ideas is the Ising model whose properties we consider in some detail, especially its mean-Þeld approximation. The Ising model has the advantage of belonging to the same critical universality class as so-called simple fluids, deÞned as ßuids with short-range intermolecular potentials. Most supercritical ßuid solvent systems of practical interest fall within this class; hence results developed using the Ising model have important implications for understanding the critical behavior of this entire universality class. While we discuss universality and related ideas in more detail in subsequent chapters, sufÞce it to say here that the Ising system belongs to arguably the most important critical universality class from a process engineering standpoint. In its simplest form, the Ising model considers N spins arranged on a lattice structure (of 1, 2, or 3 dimensions) with each spin able to adopt one of two (up or down) orientations in its lattice position. A speciÞc state of the system is determined by a given conÞguration of all the spins. The model can be made more complex by considering additional degrees of freedom to the spin orientations. For example, the Heisenberg model considers a 3-dimensional lattice with the spin orientation at each lattice site described by a 3-dimensional vector quantity. All that is required to facilitate the use of statistical mechanics with this model is the deÞnition of the Hamiltonian (the systemÕs energy function) associated with a particular lattice state υ. This Hamiltonian usually consists of spinÐspin interaction terms, as well as a term representing the presence of a magnetic Þeld, which serves to orient the spins in its direction.