Dictionary of Material Science and High Energy Physics Part 14 pdf

velocity overshoot When a high electric field 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.. T

of free space Inside a medium, the velocity of

light is reduced by the index of refraction of the

medium vlight= c/n.

velocity overshoot When a high electric field

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 charge carriers in a solid vs applied electric field The velocity at first rises linearly with the field and then saturates to a fixed value.

and

v≡∂φ

∂y

which exists for all irrotational flows The veloc-ity potential also satisfies the Laplace equation

∇2φ=∂2φ

∂x2 +∂2φ

∂y2 = 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 rise to a net 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 (lattice vibrations) 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

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 The jet 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 of the pressure difference 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 view the absorption coefficient as being neg-ative), and there has to be a cavity which acts like

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

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

the edge of the groove and constitute two parallel

one-dimensional conductors (quantum wires)

vibrational energy The energy content of the

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

number ν indicating the vibrational motion of

nuclei in a molecule neglecting rotational and electronic excitation so that the vibrational en-ergy can be approximately given as¯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 lines which results from tran-sitions between vibrational levels of a molecule and which resembles the spectrum of a harmonic quantum oscillator

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 thethe-ory 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 manifestaconsidera-tion of one

underlying principle The vibrational

frequen-cies of a string are determined by its tension

This energy is extremely high 1019 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 See absolute viscosity and kinematic viscosity

visibility of fringes A measure of the depth

of a fringe It is defined as V = (Imax −Imin )/ (Imax+Imin ) 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

Doppler broadened, S(ω) = (4ln2/π) 1/2 (e b2/δ

ω D erf c(b)) The parameter b = (4ln2) 1/2

δω0/δω D where δω0is the homogeneous width

and δω Dis 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

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 irroro-tational 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 = 

2π r 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

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-two-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.

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, that is, one whose transverse intensity has

a Gaussian distribution of I ∝ e −2(x2+y2)/w2(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 separaforma-tion have a large impact on the

characteristics of the subsequent wake.

Wake fields Produced in accelerators by

elec-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

immediately adjacent to the wall containing both 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 the Bloch theorem im-plies that

u k(r+n R) = u k(r) where n is an integer and  Ris the lattice vector whose magnitude is the lattice constant, it is easy

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



d r = δmn 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

Transverse, longitudinal, and surface waves.

wave equation The classical wave equation,

or Helmholtz equation, is one that relates the

second time derivative of a variable to its second

spatial derivative via ∂2E∂x2− (1/v2)∂2E∂t2

= 0, where v is the wave velocity The

solu-tion to this equasolu-tion is any funcsolu-tion 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

wavelike solutions, 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

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.

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

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

two superconductors with a weak link 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 formalism applies 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 localizamanifesta-tion 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-reversed trajectories (called the Aharonov–

Bohm phase shift), which depends on the

mag-netic field and the area enclosed by the

time-reversed pair Since different pairs enclose

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

W e≡ ρU2L

σ

where σ is the surface tension.

Webster effect When a bipolar junction tran-sistor is operated at high current 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 λmaxT = 0.2898 × 10−2, where T is the

temperature of a blackbody radiator, and λmaxis the wavelength at which the blackbody spectrum

is maximized

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

3C d· width2g · (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

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 rotation and the photographic film moves

parallel to the axis The crystal is screened in

such a way that only one layer line is exposed at

one time

Weisskopf–Wigner approximation In

treat-ing spontaneous emission ustreat-ing 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

T − θc where θcis 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

allows a collision between two particles by al-lowing one particle at rest while the other passes

by, and thereby generates bremstrahlung radia-tion This is measured

Wentzel–Kramers–Brillouin method (WKB method) Semiclassical approximation of quantum wave functions and energy levels based

on an expansion of the wave function in powers

of Planck’s constant

method to compute parameters of cylindrically symmetric small deformation of nuclei using irrotational-flow model

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

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

Wiggler magnets Specific combinations of

short bending magnets with alternating field

used in electron accelerators to produce

coher-ent and incohercoher-ent 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

is defined as the Fourier transformation of a

sym-metrically ordered characteristic function by

W (α)= 1

π2



exp(η∗α −ηα∗) Tr[ρe ηa†−η∗a ]d2

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-bution function can be used to calculate transport

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



J MT kqJ M

= (−1) J −M

−M q M



J Tk J

< J TkJ > 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–Seitz method The method estimates the band structure by evaluating the energy lev-els of electrons, based on the assumption of spherical symmetry for electrons around the ion

Wigner theorem It predicts the conservation

of electron-spin angular momentum

Wigner three-j symbol See three-j

coeffi-cients

the radial distribution of nuclear density with

a diffused edge in the form:

ρ(r)= ρ0

1+ exp{(r − c)/z} ,

where ρ0is the nuclear matter density (roughly

31014mass of water), z is a parameter that

mea-sures the diffuseness of the nuclear surface with

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. .. with the motion of an electron

in a magnetic field is commensurate with the

period of the lattice

allows a collision between two particles by al-lowing one particle at rest...

by, and thereby generates bremstrahlung radia-tion This is measured

Wentzel–Kramers–Brillouin method (WKB method) Semiclassical approximation of quantum wave functions and energy