of free space. Inside a medium, the velocity of light is reduced by the index of refraction of the medium v light = c/n. velocityovershoot When a high electricfield is applied to a solid, the drift velocity of elec- trons or holes rapidly rises, reaches a peak, and then drops to the steady-state value. This is known as velocity overshoot, whereby the ve- locity can temporarily exceed the steady-state value. This happens because the scattering rate increases when the electrons or holes become hot (their energy increases). The time taken for the energy to increase is roughly the so-called energy-relaxation time, whereas the time taken for the velocity to respond to the electric field is the momentum relaxation time. The former can be much larger than the latter. Hence the velocity responds much faster than the energy, causing the overshoot. Temporal response of the drift velocity of electrons to a suddenly applied strong electric field. The velocity overshoots the steady-state velocity momentarily and then settles down to the steady-state value gradually. velocity potential Scalar function φ which satisfies both u ≡ ∂φ ∂x The drift velocity of chargecarriersinasolid vs.applied electric field. The velocity at first rises linearly with the field and then saturates to a fixed value. and v ≡ ∂φ ∂y whichexistsfor allirrotationalflows. Theveloc- ity potential also satisfies the Laplace equation ∇ 2 φ = ∂ 2 φ ∂x 2 + ∂ 2 φ ∂y 2 = 0 exactly. velocity saturation When an electric field is applied to a solid, an ordered drift motion of electrons and holes is superimposed on the random motion of these entities. Whereas the random motion results in no resultant drift ve- locity, the ordered motion gives riseto anet drift velocity and a current. When an electric field is applied to a solid, the electrons and holes in the solid are acceler- ated. However, the scattering of the electrons and holes due to static scatterers such as im- purities and dynamic scatterers such as phonons (latticevibrations) retards the electrons. Finally, a steady-state velocity is reached where the ac- celerating force due to the electric field just bal- ances the decelerating force due to scattering. In the Drude model, scattering is viewed as a frictional force which is proportional to the © 2001 by CRC Press LLC velocity. Hence, Newton’s law predicts m dv dt + v τ =qE where v is the velocity, t is the time, τ is a char- acteristic scattering time, q is the charge of the electron or hole, and E is the applied electric field. The second term on the left side is the frictional force due to scattering. In a steady-state (time-derivative = 0), the velocity is found to be given by v= qτ m E which predicts that the velocity is linearly pro- portional to the electric field. Indeed, the drift velocity is found to be proportional to the elec- tric field (the proportionality constant is called the mobility, which can be written down from the above equation) if the electric field is small. At high electric fields, the dependence is non- linear because the characteristic scattering time τ becomes a function of the electric field E.In fact, in many materials like silicon, the veloc- ity saturates to a constant value at high electric fields. This phenomenon is known as velocity saturation. It must be mentioned that in some materi- als like GaAs, the velocity never saturates but instead exhibits non-monotonic behavior as a function of the electric field. The velocity first rises with the applied electric field, reaches a peak, and then drops. This non-monotonic be- havior can arise from various sources. In GaAs, it is caused by the Ridley–Hilsum–Gunn effect associated with the transfer of electrons from one conduction band valley to another. The neg- ative differential mobility associated with such non-monotonic behavior has found applications in high frequency oscillators. vena contracta The region just downstream of the discharge of a liquid jet emanating from an orifice. Thejet slightly contracts in the area after leaving the orifice due to momentum effects. venturi A nozzle consisting of a converging– diverging duct. Often used in gases to accel- erate a flow from subsonic to superonic. See converging–diverging nozzle. Possible flow states in a venturi. venturi meter A flow-rate meter utilizing a venturi. Measurement ofthepressuredifference upstream of the venturi and at the venturi throat can be used to determine the flow rate using em- pirical relations. vertex detector Detector designed to mea- sure particle traces as precisely as possible near the vertex or site of collision. vertical cavity surface emitting lasers (VCSEL) A laser is a device that emits co- herent light based on amplification via stimu- lated emission of photons. There are two condi- tions that must be satisfied for a laser to operate: the medium comprising the laser must exhibit optical gain or amplification (meaning it emits more photons than it absorbs; alternately, one can viewtheabsorptioncoefficientasbeingneg- ative), and there has to be acavitywhichactslike a feedback loop so that the closed-loop optical gain can be infinite (an infinite gain amplifier is an oscillator that produces an output without an input). The above two conditions are referred to as the Bernard–Durrefourg conditions. The cavity is the structure within which the laser light is repeatedly reflected and amplified. The walls of the cavity are partial mirrors that allow some of the light to escape (most of it is reflected). The vertical cavity surface light emitting laser (VCSEL) is a laser to which the cavity is vertically placed and light is emitted from the top surface which is one of the walls. It is often realized by a quantum well laser which consists © 2001 by CRC Press LLC of a narrow bandgap semiconductor (with a high refractive index) sandwiched between two semi- conductor layers with a wider gap and smaller refractive index. The narrow gap layer is called a quantum well which traps both electrons and holes as well as photons. The quantum well thus acts as a cavity. Cross-sectional view of a quantum well based vertical cavity surface emitting laser. very large-scale integrated circuits Elec- tronic circuits where more than 10,000 func- tional devices (e.g., transistors) are integrated on a single chip. V-groove wire A V-shaped groove is etched into a quantum well. Electrons accumulate near theedgeofthegrooveandconstitutetwoparallel one-dimensional conductors (quantum wires). vibrationalenergy Theenergycontentofthe vibrational degrees of freedom of a molecular state. Because of the interaction with rotational and electronic degrees of freedom, it is not a directly measurable quantity except in certain simple circumstances. vibrational level An energy level of a mole- cule which is a member of a vibrational pro- gression and is characterized by a vibrational quantum number. V-groove quantum wires. vibrational model of a nucleus This model describes a nucleus as a drop of fluid. Proper- ties of a nucleus can be described as phenomena of the surface tension of the drop and the vol- ume energy of the drop. The spherical shape of the nucleus is the state of equilibrium (potential energy is minimum). The spherical model is a simple one; spherical nuclei have no rotational degrees of freedom. Many nuclei are deformed and rotational degrees of freedom have to be in- cluded. The vibrational quantum of energy is called a phonon. See also shape vibrations of nuclei. vibrational quantum number A quantum number ν indicating the vibrational motion of nuclei in a molecule neglecting rotational and electronic excitation so that the vibrational en- ergycanbeapproximatelygivenas ¯ hω(ν+1/2), where ¯ h is Planck’s constant and ω is the vibra- tional frequency (multiplied by 2π). vibrational spectrum Also called vibra- tional progression. The part of a sequence of molecular spectral lineswhichresults from tran- sitions between vibrational levels of a molecule andwhichresemblesthespectrumofaharmonic quantum oscillator. © 2001 by CRC Press LLC vibration of strings In string theory, parti- cles (quanta) have extensions and they can vi- brate (analogous to ordinary strings). That is different from standard theories where particles (quanta) are point like. The harmonics (nor- mal vibrations) are determined by the tension of the strings. Each vibrational mode of strings corresponds to some particle. The vibrational frequency of the mode of the string determines the energy of that particle and, hence, its mass. The familiar particles are understood as differ- ent modes of a single string. Superstring the- ory combines string theory with supersymmet- ric mathematical structures. In such a way, the problem of combining gravity and quantum me- chanic is overcome. This allows the considera- tion of all four forces as a manifestation of one underlying principle. The vibrational frequen- cies of a string are determined by its tension. This energy is extremely high 10 19 GeV . violet cell A solar cell with a shallow p–n junction which has a high spectral response in the violet region of the solar spectrum. virtual mass See added mass. virtual process A process which has the po- tential to interfere with a real physical process although it is not observable by itself. The inter- ference may be constructive or destructive and is usually expressed in the framework of pertur- bation theory. virtual quantum Also called virtual particle. A particle or photon which, in an intermediate state, acts as the agent of an interaction (e.g., the Coulomb interaction) and does not satisfy the energy–momentum relation of a free particle. It cannot be directly observed. virtual state An unstable state of an excited atom, molecule, or nucleus with a lifetime that far exceeds typical single particle time scales, e.g., the time it takes an electron to traverse the linear dimension of a molecule. viscoelastic fluid Non-Newtonian fluid in which the fluid partially or completely returns to its original state once the deforming stress is removed. viscosity A measure of a fluid’s resistance to motion due primarily to friction of the fluid molecules. Seeabsoluteviscosity and kinematic viscosity. visibility of fringes A measure of the depth of a fringe. It is defined as V = (I max −I min )/ (I max +I min ). It is equal to 1 for perfect fringes and 0 for no fringes. vitreous state The state of a supercooled liq- uid appearing in the form of glass, the viscosity being very high. Voigt profile The line shape of a transi- tion that is simultaneously homogeneously and Dopplerbroadened,S(ω)= (4ln2/π) 1/2 (e b 2 /δ ω D erf c(b)). The parameter b = (4ln2) 1/2 δω 0 /δω D where δω 0 is the homogeneous width and δω D is the Doppler width. Volterra dislocation A dislocation affected by cutting a material in the form of a ring and putting it back together after the cut surfaces are dislocated. Voronoy polyhedron A generalized Wigner–Seitz cell chosen about a lattice point where the set of lattice points do not necessarily form a Bravais lattice. vortex A structure that has a circulatory or rotational motion. A vortex can be either ro- tational or irrotational depending upon the lo- cal value of vorticity. An irrotational vortex of strength can be most easily represented by the tangential (circumferential) velocity field u θ = 2πr where r is the distance from the center of the vortex and = C u ·dl or using Stoke’s theorem = S ω · dS where S is the area of integration inside of C. Thus, the fluid is irrotational everywhere except © 2001 by CRC Press LLC at the center of the vortex. Common vortical representations include the Rankine and Lamb– Oseen vortices. Common types of vortices. vortex line (1) A curve such that its tangent at any point gives the direction of the local vorticity vector. Vortex lines obey the Helmholtz vortex theorems such that a vortex line can only end at a solid boundary or form a closed loop (vortex ring). (2) When a type II superconductor is sub- jected to a magnetic field whose strength is in- termediate between the lower and upper criti- cal fields, the superconductor exists in a mixed state which is neither completely superconduct- ing nor completely normal. Rather, the sam- ple consists of a complicated structure of nor- mal and superconducting regions. The mag- netic field partially penetrates the sample in the form of thin filaments of flux. Within each fila- ment, the field is high and the material is normal (not superconducting). Outside the filament, the material remains superconducting and the field decays exponentially with distance, with a de- cay constant equal to the London penetration depth. Circulating around each film is a vortex of screening current which is called a vortex line. vortex pair A pair of vortices, either two- dimensional or three-dimensional, separated by a distance b, which move under mutually in- duced motion. For a case of same-signed (co- rotating) vortices, the motion is circular about a common center of vorticity (similar to plane- tary motion about a center of gravity). For the case of opposite-signed (counter-rotating) vor- tices, the direction of motion is perpendicular to the line connecting the centers of the vortex pair. If the vortices are of equal strength (cir- culation) , then the motion is a straight line. In either case, the induced velocity of the vor- tices is U=/2πb. Three-dimensional vortex pairs may experience long-wavelength (Crowe) or short-wavelength instabilities. vortex ring A line vortex whose ends link to form a ring. Due to the velocity induction from one part of the vortex on every other part, the vortex ring translates much like an opposite- signed vortex pair. Due to the three-dimensional nature, it experiences a short-wavelength insta- bility. vortex sheet An infinite number of vortex fil- aments generated by a discontinuity in velocity. The junction between the velocity jump forms the sheet. Though the sheet may be idealized as infinitesimal, in reality the sheet or veloc- ity change has a finite thickness. Vortex sheets result from Kelvin–Helmholtz formations and flow over wings. vortex street See Kármán vortex street. vortex wake The wake behind a body consisting of vortices created at the three- dimensional boundaries. For a rectangular wing, vortices are created at the wing-tips. For a delta wing, vortices are created at the lead- ing edge. Corners also generate vortical wake structures. See trailing vortex wake. vorticity Kinematic definition relating the amount of rotation in a flow field given by the curl of the velocity vector ω =∇×u . In an irrotational or potential flow, ω = 0. © 2001 by CRC Press LLC W Wafer scale integration The concept of us- ing every area — no matter how small — on a chip to perform some useful circuit function (e.g., computation or signal processing). The entire surface of the chip is therefore utilized for a giant circuit. waist For a Gaussian beam inside an optical cavity, thatis, onewhosetransverseintensityhas a Gaussian distribution of I∝e −2(x 2 +y 2 )/w 2 (z) , one refers to the minimal value of the spot size w(z) as the beam waist, where the radius of cur- vature is infinite. waiting time distribution(W(τ)) Gives the probability of a photon emission at time τ given that aproton emission happened at t= 0 and no other emission occurred in the intervening time. wake Region behind a body in a viscous flow where the flow field has a velocity deficit due to momentum loss in the boundary layer. In an ir- rotational upstream flow, vorticity generation in the boundary layer creates a wake which is ro- tational (nonzero vorticity), resulting in a flow field downstream of the body with irrotational and rotational portions. Wakes are generally classified as laminar or turbulent, but can also be related to a wave phenomenon as well (see Kelvin wedge). In surface flow (such as a ship), both turbulent and wave wakes are present, each with a distinct shape. Boundary layer forma- tion and separation have a large impact on the characteristics of the subsequent wake. Wakefields Producedinacceleratorsbyelec- tromagnetic interaction of charged beam parti- cles and metallic surfaces of the beam cham- ber. These fields can change trajectory of beam particles. Wake fields depend on geometry and material of the chamber. wake vortex Any vortex in the wake of a flow whose generation is linked to the existence of the wake itself. Prevelant in lift-generating and juncture flows. wall energy Energy of the boundary between domains in any ferromagnetic substance that are oppositely directed, measured per unit area. wall layer The region in a boundary layer immediatelyadjacenttothewallcontainingboth the viscous sublayer and the overlap region. Wannier functions The wave function of an electron possessing a momentum ¯ hk in a crystal can be written as ψ k ( r ) =e i k·r u k(r) where the function u k(r) is the Bloch function that is periodic in space and has the same period as that of the crystal lattice. The above equation is the statement of Bloch theorem. Since the statement of theBloch theorem im- plies that u k(r+n R) = u k(r) where n is an integer and R is the lattice vector whosemagnitudeisthelatticeconstant, itiseasy to see that the wave function of an electron in a crystal obeys the relation ψ k r + n R = e i k·n R ψ k ( r ) . The Bloch function can be written as u k(r) = n e i k·n R φ r − n R where the functions φ(r − n R) are called Wan- nier functions. They are orthonormal in that φ r − n R φ r − m R dr = δ mn where the δ is a Krönicker delta. wave Any of a number of information and energy transmitting motions which do not trans- mit mass. Different types of fluid waves include sound waves and shock waves which are longi- tudinal compressive waves and surface waves. In fluid dynamics, waves are either dispersive or non-dispersive. © 2001 by CRC Press LLC Transverse, longitudinal, and surface waves. wave equation The classical wave equation, or Helmholtz equation, is one that relates the second timederivative of a variable to its second spatial derivative via ∂ 2 E∂x 2 − (1/v 2 )∂ 2 E∂t 2 = 0, where v is the wave velocity. The solu- tion to this equation is any function E(kx −vt), where k = 2π/λ is the wave number. This is also known as D’Alembert’s equation. This term is also used for other equations that have wavelikesolutions, for example the Schrödinger equation. wave function The function (r,t) that sat- isfies the Schrödinger equation in the position representation. It can also be defined as the pro- jection of the state vector onto a position eigen- state, (r,t) ≡x|(r,t). wavelength The distance from peak to peak of a wave disturbance. wave mechanics There are two popular representations in non-relativistic quantum me- chanics: the matrix representation attributed to Heisenberg and the wave representation at- tributed to Schrödinger. The backbone of the latter is the Schrödinger equation which has the mathematical form of a wave equation. The wave function can be viewed as the amplitude of a scalar wave in time and space as described by the Schrödinger equation. wave mixing If n beams are incident on a non-linear medium producing a new beam, the process is referred to as n + 1 wave mixing. wave number The wave number is desig- nated by k, and is equal to 2π divided by the wavelength λ. wave packet A wave that is spatially lo- calized. This wave packet can be formed by a superposition of monochromatic waves using Fourier’s theorem. wave–particle duality The observation that, depending on the experimental setup, quantum particles can behave sometimes as waves and sometimes as particles. Likewise, electromag- netic radiation can exhibit particle properties as well as the expected wave nature. The dual aspect of matter waves is expressed by the de Broglie relations and quantified in Heisenberg’s uncertainty relations. wave vector A vector whose magnitude is the wave number, pointing in the direction of propagation of a plane electromagnetic wave. wave vector space The momentum space for the wave vector, the latter acting normal to the wave front. W-boson (gauge bosons of weak interaction) The charged intermediate bosons (weak inter- action) discovered in January 1983, and several months later Z neutral. The discovery was made in CERN using an antiproton-proton collider. W-bosons have a mass of 82 Gev. The mass of Z is 92 GeV. These particles were predicted by the Glashow– Salam–Weinberg (GSW) electroweak theory. weak interactions This kind of interaction is mediated by the W-mesons. These bosons change the flavor of quarks, but not color. The range of weak interaction is extremely short — only 10 −3 fm, which is three orders of mag- nitude less than the long-range part of nuclear force. In nuclear physics, this interaction can be considered a zero-range or contact interac- tion. W-bosons carry charges and they change the charge state of a particle. Z-bosons are a © 2001 by CRC Press LLC source of neutral weak current and are respon- sible for the neutrino-electron scattering type of reaction (ν +e − → ν +e − ). weak link A tunneling barrier between two conductors. This is a highly resistive connection between the two conductors and a charge carrier can tunnel through this region from one conduc- tor to another. A Josephson junction consists of twosuperconductorswithaweaklink interposed between them. weak localization This is a quantum me- chanical correction to the conductivity of two- dimensional electron gases. The conductivity of a two-dimensional solid can be viewed in the transmission framework that was established by Rolf Landauer (Landauer’s formalismapplies to one- and three-dimensional solids as well). The more a solid transmits electrons, the more cur- rent it passes at a constant voltage and the more conductive it is. Similarly, more reflection (due to scattering of electrons within the solid) leads to higher resistance. There is a special set of reflected trajectories that can be grouped pair- wise into time-reversed pairs which correspond to two paths that trace out exactly the same re- gion of space inside the solid but in opposite directions. These paths always interfere con- structively (since they are exactly in phase) and, hence reinforce the resistance. Thus, the resis- tance is always a little more than it would have been otherwise. Since electrons can maintain phase coherence only if they suffer no inelastic collisions, low temperatures are a pre-requisite for observing this additional quantum mechani- cal contribution to the resistance. A manifesta- tion of weak localization is seen at low tempera- tures when a sample is subjected to a transverse magnetic field. The resistance of the sample de- creases as the quantum mechanical correction gradually goes to zero with increasing magnetic field (negative magnetoresistance). The mag- netic field introduces a phase shift between the time-reversedtrajectories(calledtheAharonov– Bohm phase shift), which depends on the mag- netic field and the area enclosed by the time- reversed pair. Since different pairs enclose dif- ferent trajectories, different pairs interfere dif- ferently and the net interference gradually av- erages to zero. Thus, the resistance gradually drops to the classical value as the magnetic field is increased. weakly ionized plasma A plasma in which only a small fraction of the atoms are ionized, as opposed to a highly ionized plasma, in which nearly all atoms are ionized, or a fully ionized plasma, in which all atoms are stripped of all electrons nearly all the time. Weber number Ratio of inertial forces to surface tension important in free-surface flow We ≡ ρU 2 L σ where σ is the surface tension. Webstereffect When abipolarjunction tran- sistor is operated athighcurrent levels (high col- lector and emitter currents), the carriers (elec- trons in the case of npn transistors and holes in the case of pnp transistors) that enter the base from the emitter raise the majority carrier con- centration in the base to maintain charge neu- trality. This effectively decreases the emitter injection efficiency, which is the ratio of current injected from the emitter to the base to the cur- rent injected from the base to the emitter. As a result, the current gain of the transistor de- creases. Weiner–Khintchine theorem This theorem defines the spectral density of a stationary ran- dom process (t) via S(ω) = (1/2π) ∞ −∞ (τ )e iωτ dτ. Weiner process A stochastic process that is Gaussian distributed. In numerical simulations of stochastic differential equations, the Weiner increment is given by dW = B √ dt where B is the standard deviation of the Gaussian distribu- tion and is physically related to a damping rate involved in the problem being modeled. Wein’s displacement law This law states that λ max T = 0.2898 × 10 −2 , where T is the temperature of a blackbody radiator, and λ max is thewavelengthatwhichthe blackbody spectrum is maximized. © 2001 by CRC Press LLC weir A dam used in an open channel over which water flows which is used for flow mea- surement by measuring the height of the fluid flowing over the dam. For low upstream veloc- ities, the flow rate for a sharp-crested weir is given by Q = 2 3 C d · width 2g ·(height) 1.5 where C d is an empirical discharge coeffi- cient. Various types of weirs are sharp-crested, broad-crested, triangular, trapezoidal, propor- tional (Suttro wier), and ogee spillways. Weissenberg method A photographic method of studying the crystal structure by X- rays. The single crystal is rotated and the X-ray beam is allowed to fall on it at right angles to the axis of rotationand the photographicfilm moves parallel to the axis. The crystal is screened in such a way that only one layer line is exposed at one time. Weisskopf–Wignerapproximation Intreat- ing spontaneous emission using perturbation theory, an approximation that leads to exponen- tial decay of probability of being in the excited state. Weiss law The inverse dependence of sus- ceptibility on absolute temperature χ ∝ 1 T while the susceptibility of ferromagnets empir- ically follows the dependence χ ∝ 1 T −θ c where θ c is the Curie temperature. Weiss oscillations The electrical conductiv- ity of a periodic two-dimensional array of po- tential barriers (called an antidot lateral surface superlattice) oscillates in a magnetic field. The peaks or troughs occur whenever the cyclotron radius associated with the motion of an electron in a magnetic field is commensurate with the period of the lattice. Weizsäcker–Williams method The method allows a collision between two particles by al- lowingone particle at rest while the otherpasses by, and thereby generates bremstrahlung radia- tion. This is measured. Wentzel–Kramers–Brillouin method (WKB method) Semiclassical approximation of quantumwavefunctionsandenergylevelsbased on an expansion of the wave function in powers of Planck’s constant. Werner–Wheeler method An approximate method to compute parameters of cylindrically symmetric small deformation of nuclei using irrotational-flow model. Weyl ordering Also known as symmetric ordering. whistler A plasma wave which propagates parallel to the magnetic field produced by cur- rents outside the plasma at a frequency less than that of the electron cyclotron frequency, and which is circularly polarized, rotating about the magnetic field in the same sense as the elec- tron gyromotion. The whistler is also known as the electron cyclotron wave. The whistler was discovered accidentally during World War I by large ground-loop antennas intended for spy- ing on enemy telephone signals. Ionospheric whistlers are produced by distant lightning and get their name because of a characteristic de- scending audio-frequency tone, which is a result of the plasma dispersion relation for the wave, lower frequencies travel somewhat slower and therefore arrive at the detector later. white noise This is a stochastic process that has a constant spectral density, that is all fre- quencies are equally represented in terms of in- tensity. Wiedemann–Franz law An empirical law of 1853 that postulates that the ratio of thermal to electrical conductivity of a metal is propor- tional to the absolute temperature T with a pro- portionality constant that is about the same for all metals. © 2001 by CRC Press LLC Wiggler magnets Specific combinations of short bending magnets with alternating field used in electron accelerators to produce coher- ent and incoherent photon beams or to manipu- late electron beam properties. They are used to produce very intense beams of synchrotron ra- diation, or to pump a free electron laser. There are two designs of Wiggler magnets: flat design with planar magnetic field components, and he- lical design in which transverse component ro- tates along the magnetic axle. Wigner distribution function (1) A quasi- probability function used in quantum optics. It isdefinedastheFouriertransformationofasym- metrically ordered characteristic function by W(α)= 1 π 2 exp(η ∗ α−ηα ∗ ) Tr[ρe ηa † −η ∗ a ]d 2 η. Here, ρ is the density operator of some open quantum system, and alpha is a complex var- iable. This function always exists, but is not always positive. (2) A quantum mechanical function which is a quantum mechanical equivalent of the Boltz- mann distribution function. The latter describes the classical probability of finding a particle at a given region of space with a given momentum at a given instant of time. It is difficult, however, to interpret the Wigner distribution function as a probability since it can be complex and even negative. There is a Wigner equation that describes the evolution of the Wigner function in time and (real and momentum) space. The Wigner distri- butionfunction canbeusedtocalculatetransport variables such as current density, carrier density, energy density, etc. within a quantum mechan- ical formalism. Therein lies its utility. Wigner–Eckart theorem (1) Describes cou- pling of the angular momentum. The matrix el- ement of an operator rank of k between states with angular momentum J and J is JM T kq J M =(−1) J−M JkJ −MqM J T k J <JT k J > is the reduced matrix element, its invariant under rotation of the coordinate sys- tem. (2) A theorem in the quantum theory of an- gular momentum which states that the matrix elements of a spherical tensor operator can be factored into two parts, one which expresses the geometry and another which contains the rele- vant information about the physical properties of the states involved. The first factor is a vec- tor coupling coefficient and the second is a re- duced matrix element independent of the mag- netic quantum numbers. Wigner–Seitz cell (1) The smallest volume of space in a crystal, which when repeated in all directions without overlapping, reproduces the complete crystal without leaving any void is called the primitive unit cell. Integral multiples of the primitive cell are also unit cells, since by repeating them in space one can reproduce the crystal. However, they are not primitive because they are not the smallest such unit. The Wigner– Seitz cell is a primitive cell chosen about a lattice point in a crystal such that any region within the cell is closer to the chosen lattice point than to any other lattice point in the crystal. (2) When all lines, each of which connects a lattice point to its nearest lattice points, are bi- sected, the cell enclosed by all bisects is defined as the Wigner-Seitz cell. After a translation op- eration, the cell can also fill in all the crystal space. Wigner–Seitzmethod Themethodestimates the band structure by evaluating the energy lev- els of electrons, based on the assumption of spherical symmetry for electronsaroundtheion. Wignertheorem Itpredicts theconservation of electron-spin angular momentum. Wigner three-j symbol See three-j coeffi- cients. Woods–Saxon potential form Represents the radial distribution of nuclear density with a diffused edge in the form: ρ(r) = ρ 0 1 + exp {(r − c)/z} , where ρ 0 is the nuclear matter density (roughly 310 14 mass of water), z is a parameter that mea- sures the diffuseness of the nuclear surface with © 2001 by CRC Press LLC [...]... to the n-type material and a large negative voltage to the p-type material) , the edge of the valence band in the p-type material can rise above the edge of the conduction band in the ntype material Electrons from the mostly filled valence band of the p-type material can then tunnel into the mostly empty conduction band of the n-type material leading to a large reverse-biased current The voltage over... CRC Press LLC yrast band A rotational band consisting of the lowest energy member of each spin that is formed in composite systems created by collision of heavy ions Yukawa, Hideki Postulated that the existence of meson exchange creates a strong force between the proton and neutron The first discovered particle of this kind was the pion (1947) In 1935, Yukawa suggested the existence of a vector boson as... example of a compound that exhibits the zinc-blende crystal type is GaAs zone folding In a superlattice, the reduced Brillouin zone breaks up into smaller zones The number of smaller zones is the ratio of the period of the superlattice to the lattice constant of the constituent materials Each of these smaller zones is called a minizone, and the energy bands within the minizones are called minibands It... represents the plane of the airfoil and ζ (ξ, η) represents the plane of the circle; b is a constant To transform a circle into a cambered airfoil, the center of the circle is offset from the origin of the ζ -plane by some finite amount From this transformation, it can be shown that the circulation about an airfoil is given by = π U∞ c sin(α) and the lift coefficient is given by Energy band diagram of a reverse... only in directions and at wavelengths for which the rays radiated from all lattice points interfere constructively Y Yang–Mills particles The particles that intermediate in gauge interaction are named after C.N Yang of the State University of New York at Stony-Brook and Robert L Mills of Ohio State University Photons in the theory of electromagnetism are an example of Yang–Mills particles yin-yang... being made out of parallel planes of ions spaced a distance d apart Bright interference fringes (corresponding to constructive interference of X-rays reflected off two different planes of ions) occur if the following condition is met nλ = 2d sin θ where n is an integer, λ is the wavelength of the incident X-ray, d is the distance between two successive lattice planes, and θ is the angle of incidence X...a typical value of 0.5 fm (related to the thickness of the surface region, 1fm= 10−15 m), and c is the distance from the center to the point in which density drops on half value work function The energy difference between the Fermi energy and vacuum energy of electrons in a metal It is the minimum energy that must be supplied to the metal to release an electron... widely used in voltage regulators zero-coupled pair and approximation Pairs of identical nucleons in nuclei in the ground state of a nucleus prefer to couple to angular momentum zero For those nucleons the dipole contribution of magnetic moments vanishes zero-point energy The ground state of the harmonic oscillator has a nonzero ground state energy of hω/2 according to quantum mechan¯ ics This is due... the distance between the nucleons and V0 and µ are constants, is in use to parameterize relevant features of the nuclear force, such as strength and range of the force respectively Z Z (neutral current) Neutral intermediate boson discovered in proton-antiproton collisions Zeeman effect The splitting of spectral lines of atomic or molecular radiation due to the presence of a static magnetic field One distinguishes... boson as an intermediate particle in weak interaction Yukawa meson A particle postulated by Yukawa as the agent of the strong, short-range forces between nucleons This particle, now identified as a pion, has to have a finite rest mass to account for the short range of the nuclear force Pions account for only part of the nuclear force Yukawa potential A simple potential function of the form V (r) = V0 exp(−r/µ)/r, . the tension of the strings. Each vibrational mode of strings corresponds to some particle. The vibrational frequency of the mode of the string determines the energy of that particle and, hence,. interaction of charged beam parti- cles and metallic surfaces of the beam cham- ber. These fields can change trajectory of beam particles. Wake fields depend on geometry and material of the chamber. wake. n-type material and a large nega- tive voltage to the p-type material) , the edge of the valence band in the p-type material can rise above the edge of the conduction band in the n- type material.