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Bloch Wave
A Bloch wave (also called Bloch state or Bloch function or Bloch wavefunction), named after Swiss physicist Felix Bloch, is a kind of wave function which can be written as a plane wave modulated by a periodic function. By definition, if a wave is a Bloch wave, its wavefunction can be written in the form: where [math]\displaystyle{ \mathbf{r} }[/math] is position, [math]\displaystyle{ \psi }[/math] is the Bloch wave, [math]\displaystyle{ u }[/math] is a periodic function with the same periodicity as the crystal, the wave vector [math]\displaystyle{ \mathbf{k} }[/math] is the crystal momentum vector, [math]\displaystyle{ \mathrm{e} }[/math] is Euler's number, and [math]\displaystyle{ \mathrm{i} }[/math] is the imaginary unit. Bloch waves are important in solid-state physics, where they are often used to describe an electron in a crystal. This application is motivated by Bloch's theorem, which states that the energy eigenstates for an electron in a crystal can be written as Bloch waves (more precisely, it states that the electron wave functions in a crystal have a basis consisting entirely of Bloch wave energy eigenstates). This fact underlies the concept of electronic band structures. These Bloch wave energy eigenstates are written with subscripts as [math]\displaystyle{ \psi_{n\mathbf{k}} }[/math], where [math]\displaystyle{ n }[/math] is a discrete index, called the band index, which is present because there are many different Bloch waves with the same [math]\displaystyle{ \mathbf{k} }[/math] (each has a different periodic component [math]\displaystyle{ u }[/math]). Within a band (i.e., for fixed [math]\displaystyle{ n }[/math]), [math]\displaystyle{ \psi_{n\mathbf{k}} }[/math] varies continuously with [math]\displaystyle{ \mathbf{k} }[/math], as does its energy. Also, for any reciprocal lattice vector [math]\displaystyle{ \mathbf{K} }[/math], [math]\displaystyle{ \psi_{n\mathbf{k}}=\psi_{n(\mathbf{k+K})} }[/math]. Therefore, all distinct Bloch waves occur for values of [math]\displaystyle{ \mathbf{k} }[/math] which fall within the first Brillouin zone of the reciprocal lattice.
  • 3.6K
  • 30 Nov 2022
Topic Review
Deformation (Engineering)
In engineering, deformation refers to the change in size or shape of an object. Displacements are the absolute change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain is the relative internal change in shape of an infinitesimally small cube of material and can be expressed as a non-dimensional change in length or angle of distortion of the cube. Strains are related to the forces acting on the cube, which are known as stress, by a stress-strain curve. The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic. The linear relationship for a material is known as Young's modulus. Above the yield point, some degree of permanent distortion remains after unloading and is termed plastic deformation. The determination of the stress and strain throughout a solid object is given by the field of strength of materials and for a structure by structural analysis. Engineering stress and engineering strain are approximations to the internal state that may be determined from the external forces and deformations of an object, provided that there is no significant change in size. When there is a significant change in size, the true stress and true strain can be derived from the instantaneous size of the object. In the figure it can be seen that the compressive loading (indicated by the arrow) has caused deformation in the cylinder so that the original shape (dashed lines) has changed (deformed) into one with bulging sides. The sides bulge because the material, although strong enough to not crack or otherwise fail, is not strong enough to support the load without change. As a result, the material is forced out laterally. Internal forces (in this case at right angles to the deformation) resist the applied load. The concept of a rigid body can be applied if the deformation is negligible.
  • 3.5K
  • 24 Oct 2022
Topic Review
Vacuum State
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. Zero-point field is sometimes used as a synonym for the vacuum state of an individual quantized field. According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space". According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of existence. The QED vacuum of quantum electrodynamics (or QED) was the first vacuum of quantum field theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by Feynman, Tomonaga and Schwinger, who jointly received the Nobel prize for this work in 1965. Today the electromagnetic interactions and the weak interactions are unified (at very high energies only) in the theory of the electroweak interaction. The Standard Model is a generalization of the QED work to include all the known elementary particles and their interactions (except gravity). Quantum chromodynamics (or QCD) is the portion of the Standard Model that deals with strong interactions, and QCD vacuum is the vacuum of quantum chromodynamics. It is the object of study in the Large Hadron Collider and the Relativistic Heavy Ion Collider, and is related to the so-called vacuum structure of strong interactions.
  • 3.5K
  • 17 Oct 2022
Topic Review
G-Factor
A g-factor (also called g value or dimensionless magnetic moment) is a dimensionless quantity that characterizes the magnetic moment and angular momentum of an atom, a particle or the nucleus. It is essentially a proportionality constant that relates the different observed magnetic moments μ of a particle to their angular momentum quantum numbers and a unit of magnetic moment (to make it dimensionless), usually the Bohr magneton or nuclear magneton.
  • 3.5K
  • 28 Oct 2022
Topic Review
Deformation
In physics, deformation is the continuum mechanics transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body. A deformation can occur because of external loads, intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc. Strain is related to deformation in terms of relative displacement of particles in the body that excludes rigid-body motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered. In a continuous body, a deformation field results from a stress field due to applied forces or because of some changes in the temperature field of the body. The relation between stress and strain is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deformations which cease to exist after the stress field is removed are termed as elastic deformation. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain. They exist even after stresses have been removed. One type of irreversible deformation is plastic deformation, which occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Another type of irreversible deformation is viscous deformation, which is the irreversible part of viscoelastic deformation. In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material.
  • 3.4K
  • 09 Oct 2022
Topic Review
Luminiferous Aether
Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space, something that waves should not be able to do. The assumption of a spatial plenum of luminiferous aether, rather than a spatial vacuum, provided the theoretical medium that was required by wave theories of light. The aether hypothesis was the topic of considerable debate throughout its history, as it required the existence of an invisible and infinite material with no interaction with physical objects. As the nature of light was explored, especially in the 19th century, the physical qualities required of an aether became increasingly contradictory. By the late 1800s, the existence of the aether was being questioned, although there was no physical theory to replace it. The negative outcome of the Michelson–Morley experiment (1887) suggested that the aether did not exist, a finding that was confirmed in subsequent experiments through the 1920s. This led to considerable theoretical work to explain the propagation of light without an aether. A major breakthrough was the theory of relativity, which could explain why the experiment failed to see aether, but was more broadly interpreted to suggest that it was not needed. The Michelson-Morley experiment, along with the blackbody radiator and photoelectric effect, was a key experiment in the development of modern physics, which includes both relativity and quantum theory, the latter of which explains the particle-like nature of light.
  • 3.4K
  • 17 Oct 2022
Topic Review
Degrees of Freedom (Physics and Chemistry)
In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the dimensions of the phase space. The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic (where the state at one instant uniquely determines its past and future position and velocity as a function of time) such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom. In classical mechanics, the state of a point particle at any given time is often described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism. In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space. In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer. It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system. Depending on what one is counting, there are several different ways that degrees of freedom can be defined, each with a different value.
  • 3.4K
  • 25 Nov 2022
Topic Review
Physical Information
Physical information is a form of information. In physics, it refers to the information of a physical system. Physical information is an important concept used in a number of fields of study in physics. For example, in quantum mechanics, the form of physical information known as quantum information is used in many descriptions of quantum phenomena, such as quantum observation, quantum entanglement and the causal relationship between quantum objects that carry out either or both close and long-range interactions with one another. In a general sense, information is that which resolves uncertainty, which is due to the fact that it describes the details of that which is associated with the uncertainty. The description itself is, however, divorced from any type of language. When clarifying the subject of information, care should be taken to distinguish between the following specific cases: As the above usages are all conceptually distinct from each other, overloading the word "information" (by itself) to denote (or connote) several of these concepts simultaneously can lead to confusion. Accordingly, this article uses more detailed phrases, such as those shown in bold above, whenever the intended meaning is not made clear by the context.
  • 3.3K
  • 09 Oct 2022
Topic Review
Distance Measures (Cosmology)
Distance measures are used in physical cosmology to give a natural notion of the distance between two objects or events in the universe. They are often used to tie some observable quantity (such as the luminosity of a distant quasar, the redshift of a distant galaxy, or the angular size of the acoustic peaks in the cosmic microwave background (CMB) power spectrum) to another quantity that is not directly observable, but is more convenient for calculations (such as the comoving coordinates of the quasar, galaxy, etc.). The distance measures discussed here all reduce to the common notion of Euclidean distance at low redshift. In accord with our present understanding of cosmology, these measures are calculated within the context of general relativity, where the Friedmann–Lemaître–Robertson–Walker solution is used to describe the universe.
  • 3.3K
  • 19 Oct 2022
Topic Review
Materials Science, Glasses
Glasses are solid amorphous materials which transform into liquids upon heating through the glass transition. The International Commission on Glass defines glass as a state of matter, usually produced when a viscous molten material is cooled rapidly to below its glass transition temperature, with insufficient time for a regular crystal lattice to form. The solid-like behaviour of glasses is separated from the liquid-like behaviour at higher temperatures by the glass transition temperature, Tg. The IUPAC Compendium on Chemical Terminology defines glass transition as a second order transition in which a supercooled melt yields, on cooling, a glassy structure. It states that below the glass-transition temperature the physical properties of glasses vary in a manner similar to those of the crystalline phase. Moreover, it is deemed that the bonding structure of glasses has the same symmetry signature in terms of Hausdorff-Besikovitch dimensionality of chemical bonds as for the crystalline materials. 
  • 3.2K
  • 09 May 2024
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