Bloch wave - Citizendia

Bloch wave in silicon

A Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle (usually, an electron) placed in a periodic potential. This page addresses only the Swiss physicist for the man accused of espionage see Felix Bloch (diplomatic officer Felix Bloch ( October 23 A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. It consists of the product of a plane wave envelope function and a periodic function (periodic Bloch function) $\, u_{nk}(r)$ which has the same periodicity as the potential:

$\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}).$

The result that the eigenfunctions can be written in this form for a periodic system is called Bloch's theorem. In the Physics of Wave propagation (especially Electromagnetic waves, a plane wave (also spelled planewave) is a constant-frequency wave whose In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in The corresponding energy eigenvalue is Є_n(k)= Є_n(k + K), periodic with periodicity K of a reciprocal lattice vector. In Crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that e^{i\mathbf{K}\cdot\mathbf{R}}=1 Because the energies associated with the index n vary continuously with wavevector k we speak of an energy band with band index n. Because the eigenvalues for given n are periodic in k, all distinct values of Є_n(k) occur for k-values within the first Brillouin zone of the reciprocal lattice. In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the

More generally, a Bloch-wave description applies to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. A dielectric is a nonconducting substance ie an insulator. The term was coined by William Whewell in response to a request from Michael Faraday. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of Photonic crystals are periodic Optical (nanostructures that are designed to affect the motion of Photons in a similar way that periodicity of a Semiconductor A phononic crystal is a material which exhibits Stop bands for Phonons preventing phonons of selected ranges of frequencies from being transmitted through the material It is generally treated in the different forms of the dynamical theory of diffraction. The dynamical theory of diffraction describes the interaction of Waves with a regular lattice

The plane wave wavevector (or Bloch wavevector) k (multiplied by the reduced Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. In solid state physics crystal momentum is a momentum -like vector associated with Electrons in a crystal lattice. In Crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that e^{i\mathbf{K}\cdot\mathbf{R}}=1 In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the For a given wavevector and potential, there are a number of solutions, indexed by n, to Schrödinger's equation for a Bloch electron. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wavevectors, it is called a (complete) band gap. In Solid state physics and related applied fields a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states The band structure is the collection of energy eigenstates within the first Brillouin zone. In Solid-state physics, the electronic band structure (or simply band structure) of a Solid describes ranges of Energy that an Electron In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within the independent electron approximation. Independent electron approximation both in the case of free electron theory and Nearly-free electron approximation we use independent electron approximation

A corollary of this result is that the Bloch wavevector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the group velocity of the wave is conserved. The group velocity of a Wave is the Velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that electrical resistance in a crystalline conductor only results from things like imperfections that break the periodicity. Electrical resistance is a ratio of the degree to which an object opposes an Electric current through it measured in Ohms Its reciprocal quantity is

It can be shown that the eigenfunctions of a particle in a periodic potential can always be chosen in this form by proving that translation operators (by lattice vectors) commute with the Hamiltonian. In Physics, translation is movement that changes the position of an object as opposed to Rotation. In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system More generally, the consequences of symmetry on the eigenfunctions are described by representation theory. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

The concept of the Bloch state was developed by Felix Bloch in 1928, to describe the conduction of electrons in crystalline solids. This page addresses only the Swiss physicist for the man accused of espionage see Felix Bloch (diplomatic officer Felix Bloch ( October 23 Year 1928 ( MCMXXVIII) was a Leap year starting on Sunday (link will display full calendar of the Gregorian calendar. The same underlying mathematics, however, was also discovered independently several times: by George William Hill (1877), Gaston Floquet (1883), and Alexander Lyapunov (1892). George William Hill ( March 3, 1838 &ndash April 16, 1914) was a U Year 1877 ( MDCCCLXXVII) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Common Achille Marie Gaston Floquet ( December 15 1847 &ndash October 7 1920) was a French Mathematician, best known for his work Year 1883 ( MDCCCLXXXIII) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Common Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов ( June 6 1857 &ndash November 3 1918, all Year 1892 ( MDCCCXCII) was a Leap year starting on Friday (link will display the full calendar of the Gregorian Calendar (or a Leap year As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory (or occasionally the Lyapunov-Floquet theorem). In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its Floquet theory is a branch of the theory of Ordinary differential equations relating to the class of solutions to Linear differential equations of the form Various one-dimensional periodic potential equations have special names, for example, Hill's equation:^[1]

$\frac {d^2y}{dx^2}+\left(\theta_0+2\sum_{n=1}^\infty \theta_n \cos(2nx) \right ) y=0$ ,

where the θ's are constants. In mathematics the Hill's equation or Hill differential equation ( is the second-order Ordinary differential equation \frac{d^2y}{dx^2}+\left(\theta_0+2\sum_{n=1}^\infty Hill's equation is very general, as the θ-related terms may viewed as a Fourier series expansion of a periodic potential. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions Other much studied periodic one-dimensional equations are the Kronig-Penney model and Mathieu's equation. In Quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. In Mathematics, the Mathieu functions are certain Special functions useful for treating a variety of interesting problems in applied mathematics including

References

^ W Magnus and S Winkler (2004). Hill's Equation. Courier Dover, p. 11. ISBN 0-0486495655.

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