Well-orderIn mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element.
AdatomAn adatom is an atom that lies on a crystal surface, and can be thought of as the opposite of a surface vacancy. This term is used in surface chemistry and epitaxy, when describing single atoms lying on surfaces and surface roughness. The word is a portmanteau of "adsorbed atom". A single atom, a cluster of atoms, or a molecule or cluster of molecules may all be referred to by the general term "adparticle". This is often a thermodynamically unfavorable state. However, cases such as graphene may provide counter-examples.
Crystal growthA crystal is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. Crystal growth is a major stage of a crystallization process, and consists of the addition of new atoms, ions, or polymer strings into the characteristic arrangement of the crystalline lattice. The growth typically follows an initial stage of either homogeneous or heterogeneous (surface catalyzed) nucleation, unless a "seed" crystal, purposely added to start the growth, was already present.
Well-ordering theoremIn mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also ). Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem.
NucleationIn thermodynamics, nucleation is the first step in the formation of either a new thermodynamic phase or structure via self-assembly or self-organization within a substance or mixture. Nucleation is typically defined to be the process that determines how long an observer has to wait before the new phase or self-organized structure appears. For example, if a volume of water is cooled (at atmospheric pressure) below 0 °C, it will tend to freeze into ice, but volumes of water cooled only a few degrees below 0 °C often stay completely free of ice for long periods (supercooling).
Partially ordered setIn mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, transitive and antisymmetric.
Self-assemblySelf-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the constitutive components are molecules, the process is termed molecular self-assembly. Self-assembly can be classified as either static or dynamic. In static self-assembly, the ordered state forms as a system approaches equilibrium, reducing its free energy.
Self-organizationSelf-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when sufficient energy is available, not needing control by any external agent. It is often triggered by seemingly random fluctuations, amplified by positive feedback. The resulting organization is wholly decentralized, distributed over all the components of the system.
CrystalA crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations. The scientific study of crystals and crystal formation is known as crystallography.
Crystal structureIn crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure.
Transfinite inductionTransfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Let be a property defined for all ordinals . Suppose that whenever is true for all , then is also true. Then transfinite induction tells us that is true for all ordinals. Usually the proof is broken down into three cases: Zero case: Prove that is true. Successor case: Prove that for any successor ordinal , follows from (and, if necessary, for all ).
Surface diffusionSurface diffusion is a general process involving the motion of adatoms, molecules, and atomic clusters (adparticles) at solid material surfaces. The process can generally be thought of in terms of particles jumping between adjacent adsorption sites on a surface, as in figure 1. Just as in bulk diffusion, this motion is typically a thermally promoted process with rates increasing with increasing temperature. Many systems display diffusion behavior that deviates from the conventional model of nearest-neighbor jumps.
Self-assembly of nanoparticlesNanoparticles are classified as having at least one of three dimensions be in the range of 1-100 nm. The small size of nanoparticles allows them to have unique characteristics which may not be possible on the macro-scale. Self-assembly is the spontaneous organization of smaller subunits to form larger, well-organized patterns. For nanoparticles, this spontaneous assembly is a consequence of interactions between the particles aimed at achieving a thermodynamic equilibrium and reducing the system’s free energy.
Colloidal crystalA colloidal crystal is an ordered array of colloidal particles and fine grained materials analogous to a standard crystal whose repeating subunits are atoms or molecules. A natural example of this phenomenon can be found in the gem opal, where spheres of silica assume a close-packed locally periodic structure under moderate compression. Bulk properties of a colloidal crystal depend on composition, particle size, packing arrangement, and degree of regularity.
Quantitative geographyQuantitative geography is a subfield and methodological approach to geography that develops, tests, and uses mathematical and statistical methods to analyze and model geographic phenomena and patterns. It aims to explain and predict the distribution and dynamics of human and physical geography through the collection and analysis of quantifiable data. The approach quantitative geographers take is generally in line with the scientific method, where a falsifiable hypothesis is generated, and then tested through observational studies.
BoilingBoiling is the rapid phase transition from liquid to gas or vapor; the reverse of boiling is condensation. Boiling occurs when a liquid is heated to its boiling point, so that the vapour pressure of the liquid is equal to the pressure exerted on the liquid by the surrounding atmosphere. Boiling and evaporation are the two main forms of liquid vapourization. There are two main types of boiling: nucleate boiling where small bubbles of vapour form at discrete points, and critical heat flux boiling where the boiling surface is heated above a certain critical temperature and a film of vapour forms on the surface.
SupersaturationIn physical chemistry, supersaturation occurs with a solution when the concentration of a solute exceeds the concentration specified by the value of solubility at equilibrium. Most commonly the term is applied to a solution of a solid in a liquid. A supersaturated solution is in a metastable state; it may be brought to equilibrium by forcing the excess of solute to separate from the solution. The term can also be applied to a mixture of gases.
Thin filmA thin film is a layer of material ranging from fractions of a nanometer (monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many applications. A familiar example is the household mirror, which typically has a thin metal coating on the back of a sheet of glass to form a reflective interface. The process of silvering was once commonly used to produce mirrors, while more recently the metal layer is deposited using techniques such as sputtering.
Quantitative revolutionThe quantitative revolution (QR) was a paradigm shift that sought to develop a more rigorous and systematic methodology for the discipline of geography. It came as a response to the inadequacy of regional geography to explain general spatial dynamics. The main claim for the quantitative revolution is that it led to a shift from a descriptive (idiographic) geography to an empirical law-making (nomothetic) geography. The quantitative revolution occurred during the 1950s and 1960s and marked a rapid change in the method behind geographical research, from regional geography into a spatial science.
NanowireA nanowire is a nanostructure in the form of a wire with the diameter of the order of a nanometre (10−9 metres). More generally, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less and an unconstrained length. At these scales, quantum mechanical effects are important—which coined the term "quantum wires". Many different types of nanowires exist, including superconducting (e.g. YBCO), metallic (e.g. Ni, Pt, Au, Ag), semiconducting (e.g.
