Dictionary of Material Science and High Energy Physics Part 2 pptx

The conditions required for the al-lowed angular momenta, hence orbits, is called Bohr quantization and is given by the formula L = n¯h, where L is the allowed value of the an-gular mom

leading role in the development of modern

nu-clear physics See Copenhagen interpretation

Bohr quantization Rule that determines the

allowed electron orbits in Bohr’s theory of the

hydrogen atom In an early atomic theory,

Bohr suggested that electrons orbit parent nuclei

much like planets orbit the sun Because

elec-trons are electrically charged, classical physics

predicts that such a system is unstable due to

radiative energy loss Bohr postulated that

elec-trons radiate only if they “jump” between

al-lowed prescribed orbits These orbits are called

Bohr orbits The conditions required for the

al-lowed angular momenta, hence orbits, is called

Bohr quantization and is given by the formula

L = n¯h, where L is the allowed value of the

an-gular momentum of a circular orbit, n is called

the principal quantum number, and ¯h is the

Planck constant divided by 2π

Bohr radius (a0 ) (1) The radius of the

elec-tron in the hydrogen atom in its ground state, as

described by the Bohr theory In Bohr’s early

atomic theory, electrons orbit the nucleus on

well defined radii, the smallest of which is called

the first Bohr radius Its value is 0.0529 nm.

(2) According to the Bohr theory of the atom

(see Bohr atom), the radius of the circle in which

the electron moves in the ground state of the

hydrogen atom, a0 ≡ ¯h2/m2e = 0.5292 Å A

full quantum mechanical treatment of hydrogen

gives a0 as the most probable distance between

electrons and the nucleus

Boltzmann constant ( k B) A fundamental

constant which relates the energy scale to the

Kelvin scale of temperature, k B = 1.3807 ×

10−23 joules/kelvin.

Boltzmann distribution A law of statistical

mechanics that states that the probability of

find-ing a system at temperature T with an energy E

is proportional to e −E/KT , where K is

Boltz-mann’s constant When applied to photons in a

cavity with walls at a constant temperature T ,

the Boltzmann distribution gives Planck’s

dis-tribution law of E k = ¯hck/(e ¯hck/KT − 1).

Boltzmann factor The term, exp( −ε/kB T ),

that is proportional to the probability of finding

a system in a state of energy ε at absolute perature T

tem-Boltzmann’s constant A constant equal to

the universal gas constant divided by Avogadro’s

number It is approximately equal to 1.38 ×

10−23 J/K and is commonly expressed by the

symbol k.

Boltzmann statistics Statistics that lead to

the Boltzmann distribution Boltzmann tics assume that particles are distinguishable.

statis-Boltzmann transport equation An

integro-differential equation used in the classical theory

of transport processes to describe the equation

of motion of the distribution function f (r, v, t).

The number of particles in the infinitesimal

vol-ume dr dv of the 6-dimensional phase space of

Cartesian coordinates r and velocity v is given

by f (r, v, t)dr dv and obeys the equation

.

Here, α denotes the acceleration, and (∂f/∂t )coll.

is the change in the distribution function due tocollisions The integral character of the equa-tion arises in writing the collision term in terms

of two particle collisions

bonding orbital See anti-bonding orbital bootstrap current Currents driven in tor-

oidal devices by neo-classical processes

Born approximation An approximation

use-ful for calculation of the cross-section in sions of atomic and fundamental particles The

colli-Born approximation is particularly well-suited

for estimates of cross-sections at sufficientlylarge relative collision partner velocities In po-

tential scattering, the Born approximation for

the scattering amplitude is given by the

ex-pression f (θ ) = −2µ

¯h2q

∞

0 r sin(qr)V (r) dr,

where θ is the observation angle, ¯h is Planck’s

constant divided by 2π , µ is the reduced mass,

V (r) is the spherically symmetric potential, q ≡

2ksin(θ/2), and k is the wave number for the

collision See cross-section.

Born-Fock theorem See adiabatic theorem.

Born, Max (1882–1970) German

physi-cist A founding father of the modern

quan-tum theory His name is associated with many

applications of the modern quantum theory,

such as the Born approximation, the

Born-Oppenheimer approximation, etc Professor

Born was awarded the Nobel Prize in physics

in 1954

Born-Oppenheimer approximation An

ap-proximation scheme for solving the many-

few-atom Schrödinger equation The utility of the

approximation follows from the fact that the

nu-clei of atoms are much heavier than electrons,

and their motion can be decoupled from the

elec-tronic motion The Born-Oppenheimer

approxi-mation is the cornerstone of theoretical quantum

chemistry and molecular physics

Born postulate The expression |ψ(x, y, z,

t )|2 dx dy dz gives the probability at time t of

finding the particle within the infinitesimal

re-gion of space lying between x and dx, y and dy,

and z and dz |ψ(x, y, z, t)|2 is then the

prob-ability density for finding a particle in various

positions in space

Born-Von Karman boundary condition

Also called the periodic boundary condition To

one dimensional crystal, it can be expressed as

U1 = UN+ 1, where N is the number of

parti-cles in the crystal with length L.

Bose-Einstein condensation A quantum

phenomenon, first predicted and described by

Einstein, in which a non-interacting gas of

bosons undergoes a phase transformation at

crit-ical values of density and temperature A

Bose-Einstein condensate can be considered a

macro-scopic system described by a quantum state

Bose-Einstein condensates have recently been

observed, about 70 years following Einstein’s

prediction, in dilute atomic gases that have been

cooled to temperatures only about 10−9 Kelvin

above absolute zero

Bose-Einstein statistics Statistical treatment

of an assembled collection of bosons The

dis-tinction between particles whose wave functions

are symmetric or antisymmetric leads to

differ-ent behavior under a collection of particles (i.e.,

different statistics) Particles with integral spinare characterized by symmetric wave functionsand therefore are not subject to the Pauli exclu-sion principal and obey Bose-Einstein statistics

Bose, S.N (1894–1974) Indian physicist and

mathematician noted for fundamental tions to statistical quantum physics His name

contribu-is associated with the term Bose statcontribu-istics whichdescribes the statistics obeyed by indistinguish-able particles of integer spin Such particlesare also called bosons His name is also as-sociated with Bose-Einstein condensation SeeBose-Einstein condensation

Bose statistics Quantum statistics obeyed by

a collection of bosons Bose statistics lead to the

Bose-Einstein distribution function and, for ical values of density and temperature, predictthe novel quantum phenomenon of the Bose-Einstein condensation See Bose-Einstein con-densation

crit-boson (1) A particle that has integer spin.

A boson can be a fundamental particle, such as

a photon, or a composite of other fundamentalparticles Atoms are composites of electronsand nuclei; if the nucleus has half integer spinand the total electron spin is also half integral,then the atom as a whole must possess integer

spin and can be considered a composite boson.

(2) Particles can be divided into two kinds,

boson and fermion The fundamental differencebetween the two is that the spin quanta number

of bosons is integer and that of fermions is halfinteger Unlike fermions, which can only be cre-ated or destroyed in particle-antiparticle pairs,

bosons can be created and destroyed singly.

bounce frequency The average frequency

of oscillation of a particle trapped in a magneticmirror as it bounces back and forth between itsturning points in regions of high magnetic field

boundary layer (1) A thin layer of fluid,

ex-isting next to a solid surface beyond which theliquid is moving Within the layer, the effects ofviscosity are significant The effects of viscos-

ity often can be neglected beyond the boundary layer.

(2) The transition layer between the solid

boundary of a body and a moving viscous fluid

as required by the no-slip condition The

thick-ness of the boundary layer is usually taken to be

the point at which the velocity is equal to 99%

of the free-stream velocity Other measures of

boundary layer thickness include the

displace-ment thickness and modisplace-mentum thickness The

boundary layer gives rise to friction drag from

viscous forces and can also lead to separation

It also is responsible for the creation of

vortic-ity and the diffusion thereof due to viscous

ef-fects Thus, a previously irrotational region will

remain so unless it interacts with a boundary

layer This leads to the separation of flows into

irrotational portions outside the boundary layer

and viscous regions inside the boundary layer.

The thickness of a boundary decreases with an

increasing Reynolds number, resulting in the

ap-proximation of high speed flows as irrotational

A boundary layer can be laminar, but will

even-tually transition to turbulence given time The

boundary layer concept was introduced by

Lud-wig Prandtl in 1904 and led to the development

of modern fluid dynamics See boundary layer

approximation

Boundary layer.

boundary layer approximation

Simplifica-tion of the governing equaSimplifica-tions of moSimplifica-tion within

a thin boundary layer If the boundary layer

thickness is assumed to be small compared to the

length of the body, then the variation along the

direction of the boundary layer (x) is assumed

to be much less than that across the boundary

where the velocity in the y-direction (v) is also

assumed to be much smaller than the velocity

in the x-direction (u), v u The continuity

bound state An eigenstate of distinct energy

that a particle occupies when its energy E < V

of a potential well that confines it near the forcecenter creating the potential The discrete en-ergy values are forced on the system by the re-quirement of continuity of the wave function

at the boundaries of the potential well, beyondwhich the wave function must diminish (or van-

ish) When E > V everywhere, the particle is

not bound but instead is free to occupy any of

an infinite continuum of states

Bourdon tube Classical mechanical deviceused for measuring pressure utilizing a curvedtube with a flattened cross-section When pres-surized, the tube deflects outward and can becalibrated to a gauge using a mechanical link-

age Bourdon tubes are notable for their high

accuracy

Boussinesq approximation Simplification

of the equations of motion by assuming that sity changes can be neglected in certain flowsdue to the compressibility While the densitymay vary in the flow, the variation is not due tofluid motion such as occurs in high speed flows.Thus, the continuity equation is simplified to

den-∇ · u = 0

from its normal form

box normalization A common wave tion normalization convention If a particle is

func-contained in a box of unit length L, the wave

function is constrained to vanish at the boundary

and requires quantization of momentum An

in-tegral of the probability density|ψ|2 throughout

the box is required to sum to unity and typically

leads to a normalization pre-factor for the wave

function given by 1/√

V , where V is the volume

of the box In most applications, the volume of

the box is taken to have the limit as L→ ∞

Boyle’s law An empirical law for gases which

states that at a fixed temperature, the pressure

of a gas is inversely proportional to its volume,

i.e., pV = constant This law is strictly valid

for a classical ideal gas; real gases obey this to

a good approximation at high temperatures and

low pressures

bracket, or bra-ket An expression

repre-senting the inner (or dot) product of two state

vectors, ψ α† ≡< α|β > which yields a simple

scalar value The first and last three letters of

the bracket name the notational expression

in-volving triangular brackets for the two kinds of

state vectors that form the inner product

Bragg diffraction A laboratory method that

takes advantage of the wave nature of

electro-magnetic radiation in order to probe the structure

of crystalline solids Also called X-ray

diffrac-tion, the method was developed and applied by

W.L Bragg and his father, W.H Bragg The pair

received the Nobel Prize for physics in 1915

bra vector Defined by the bra-ket

formal-ism of Dirac, which allows a concise and

easy-to-use terminology for performing operations in

Hilbert space According to the bra-ket

formal-ism, a quantum state, or a vector in Hilbert space,

can be described by the ket symbol For any ket

|a > there exists a bra < a| This is also called a

dual correspondence If < b | is a bra and |a > a

ket, then one can define a complex number

rep-resented by the symbol < b |a >, whose value

is given by an inner product of the vectors |a >

and |b >.

breakeven (commercial, engineering,

scien-tific, and extrapolated) Several definitions

exist for fusion plasmas: Commercial breakeven

is when sufficient fusion power can be

con-verted into electric power to cover the costs of

the fusion power plant at economically

compet-itive rates; engineering breakeven is when

suf-ficient electrical power can be generated fromthe fusion power output to supply power for theplasma reactor plus a net surplus without the

economic considerations; scientific breakeven

is when the fusion power is equal to the input

power; i.e., Q = 1 (See also Lawson

crite-rion); extrapolated breakeven is when scientificbreakeven is projected for actual reactor fuel(e.g., deuterium and tritium) from experimentalresults using an alternative fuel (e.g., deuteriumonly) by scaling the reaction rates for the twofuels

Breit–Wigner curve The natural line shape

of the probability density of finding a decaying

state at energy E Rather than existing at a single

well-defined energy, the state is broadened to a

full width at half max, , which is related to its lifetime by τ  = ¯h The curve of the probability

the resonance energy E r , the Breit–Wigner form for the cross-section σ (E), is given by σ (E)∼

 (E −E r )2+(/2)2, where E is the collision energy, and  is the lifetime of the resonance state.

Bremsstrahlung Electromagnetic radiationthat is emitted by an electron as it is accelerated

or decelerated while moving through the electricfield of an ion

Bremsstrahlung radiation Occurs in

plas-ma when electrons interact (“collide”) with the

Coulomb fields of ions; the resulting deflection

of the electrons causes them to radiate

Brillouin–Wigner perturbation

Perturba-tion treatment that expresses a state as a

se-ries expansion in powers of λ (the scale of the

perturbation from an unperturbed Hamiltonian,

H = H0 + λV ) with coefficients that depend

on the perturbed energy values E n (rather than

the unperturbed energies εn of the

Rayleigh-Schrödinger perturbation method) An initial

unperturbed eigenstate, ϕ n, becomes,

 n = ϕn+

m =n

E n − Em λ ϕm|V |ϕn

Brillouin zone Similar to the first Brillouin

zone, bisect all lines, among which each

con-nects a reciprocal lattice point to one of its

sec-ondly nearest points The region composed of

all the bisections is defined as the second

louin zone Keeping on it, we can get all

Bril-louin zones of the considered reciprocal lattice

point Each Brillouin zone is center symmetric

to the point

broken symmetry Property of a system

whose ground state is not invariant under

sym-metry operations Suppose L is the generator of

some symmetry of a system described by

Hamil-tonian H Then [L, H] = 0, and if |a > is a

non-degenerate eigenstate of H , it must also be

an eigenstate of L If there exists a degeneracy,

L |a > is generally a linear combination of states

in the degenerate sub-manifold If the ground

state |g > of the system has the property that

L |g >= c|g >, where c is a complex number,

then the symmetry corresponding to the

gener-ator of that symmetry, L, is said to be broken.

Brownian motion The disordered motion of

microscopic solid particles suspended in a fluid

or gas, first observed by botanist Robert Brown

in 1827 as a continuous random motion and

attributed to the frequent collisions the particlesundergo with the surrounding molecules Themotion was qualitatively explained by Einstein’s(1905) statistical treatment of the laws of mo-tions of the molecules

Brunt–Väisälä frequency Natural

frequen-cy, N , of vertical fluid motion in stratified flow

as given by the linearized equations of motion:

Also called buoyancy frequency

bubble chamber A large tank filled with

liq-uid hydrogen, with a flat window at one endand complex optical devices for observing andphotographing the rows of fine bubbles formedwhen a high-energy particle traverses the hydro-gen

Buckingham’s Pi theorem For r number

of required dimensions (such as mass, length,

time, and temperature), n number of

dimen-sional variables can always be combined to form

exactly n − r independent dimensionless

vari-ables Thus, for a problem whose solution quires seven variables with three total dimen-sions, the problem can be reduced to four dimen-sionless parameters See dimensional analysis,

re-Reynolds number for an example

bulk viscosity Viscous term from the

consti-tutive relations for a Newtonian fluid, λ+2

3µ, where λ and µ are measures of the viscous prop-

erties of the fluid This is reduced to a moreusable form using the Stokes assumption

buoyancy The vertical force on a body

im-mersed in a fluid equal to the weight of fluiddisplaced A floating body displaces its ownweight in the fluid in which it is floating See

Archimede’s law

calorie (Cal) A unit of heat defined as the

amount of heat required to raise the temperature

of 1 gm of water at 1 atmosphere pressure from

14.5 to 15.5 C It is related to the unit of energy

in the standard international system of units, the

Joule, by 1 calorie= 4.184 joules Note that

the calorie used in food energy values is 1

kilo-calorie≡ 1000 calories, and is denoted by the

capital symbol Cal.

camber Curvature of an airfoil as defined

by the line equidistant between the upper and

lower surfaces Important geometric property

in the generation of lift

canonical ensemble Ensemble that

de-scribes the thermodynamic properties of a

sys-tem maintained at a constant sys-temperature T , by

keeping it in contact with a heat reservoir at

tem-perature T The canonical distribution function

gives the probability of finding the system in a

non-degenerate state of energy E ias

P (E i ) = exp (−Ei / k B T ) /

i

exp ( −Ei / k B T ) ,

where k B is the Boltzmann constant, and the

summation is over all possible microstates of

the system, denoted by the index i.

canonical partition function For a system

of N particles at constant temperature T and

volume V , all thermodynamic properties can be

obtained from the canonical partition function

defined as Z(T , V , N )= i exp( −Ei / k B T ),

where E i is the energy of the system of N

par-ticles in the ith microstate.

canonical variables In the Hamiltonian

for-mulation of classical physics, conjugate

vari-ables are defined as the pair, q, p=∂L

∂ ˙q, where

L is the Lagrangian and q is a coordinate, or

variable of the system

capacitively coupled discharge plasma

Plasma created by applying an oscillating, frequency potential between two electrodes.Energy is coupled into the plasma by collisionsbetween the electrons and the oscillating plasmasheaths If the oscillation frequency is reduced,the discharge converts to a glow discharge

radio-capillarity Effect of surface tension on theshape of the free surface of a fluid, causing cur-vature, particularly when in contact with a solidboundary The effect is primarily important atsmall length scales

capillary waves Free surface waves due to

the effect of surface tension σ which are present

at very small wavelengths The phase speed,

c, of capillary waves decreases as wavelength

increases,

c=



kσ ρ

as opposed to surface gravity waves, whosephase speed increases with increasing wave-length

Carnot cycle A cyclical process in which

a system, for example, a gas, is expanded andcompressed in four steps: (i) an isothermal (con-

stant temperature) expansion at temperature T h,

until its entropy changes from S c to S h, (ii) anadiabatic (constant entropy) expansion during

which the system cools to temperature T c, lowed by (iii) an isothermal compression at tem-

fol-perature T c, and (iv) an adiabatic compressionuntil the substance returns to its initial state of

entropy, S c The Carnot cycle can be sented by a rectangle in an entropy–temperaturediagram, as shown in the figure, and it is thesame regardless of the working substance

repre-carrier A charge carrier in a conduction cess: either an electron or a positive hole

pro-cascade A row of blades in a turbine or pump

cascade, turbulent energy Transfer of ergy in a turbulent flow from large scales to smallscales through various means such as dissipationand vortex stretching Energy fed into the tur-bulent flow field is primarily distributed among

en-Carnot cycle.

large scale eddies These large eddies generate

smaller and smaller eddies until the eddy length

scale is small enough for viscous forces to

dis-sipate the energy Dimensional analysis shows

that the relation between the energy E, the

en-ergy dissipation ε, and wavenumber k is

E ∝ ε 2/3 k −5/3

which is known as Kolmogorov’s -5/3 law See

turbulence

Casimir operator Named after physicist

H.A Casimir, these operators are bi-linear

com-binations of the group generators for a Lie group

that commute with all group generators For the

covering group of rotations in three-dimensional

space, there exists one Casimir operator, usually

labeled J2, where J are the angular momentum

operators See angular momentum

cation A positively charged ion, formed as a

result of the removal of electrons from atoms and

molecules In an electrolysis process, cations

will move toward negative electrodes

Cauchy–Riemann conditions Relations

be-tween velocity potential and streamfunction in

a potential flow where

other is known

causality The causal relationship between

a wavefunction at an initial time ψ(t o ) and a

wavefunction at any later time ψ (t) as expressed

through Schrödinger’s equation This appliesonly to isolated systems and assumes that thedynamical state of such a system can be repre-sented completely by its wave function at thatinstant See complementarity.

cavitation Spontaneous vaporization of aliquid when the pressure drops below the va-

por pressure Cavitation commonly occurs in

pumps or marine propellers where high fluidspeeds are present Excessive speed of the pump

or propeller and high liquid temperatures are

standard causes of cavitation Cavitation

de-grades pump performance and can cause noise,vibration, and even structural damage to the de-vice

cavitation number Dimensionless ter used to express the degree of cavitation (va-por formation) in a liquid:

parame-Ca ≡ (pa − pv ) /ρU2where p a is the atmospheric pressure and p visthe vapor pressure of the liquid

cellular method One method of energy bandcalculation in crystal It was addressed byWigner and Seitz They divided a crystal intoatomic cells For a given potential, because ofsymmetry, the calculation reduced into a single

cell The assumption for the cellular method

is that the normal component of the gradient ofwave function will vanish at the single cell sur-face or at the Wigner–Seitz sphere

Celsius temperature scale (C) Defined bysetting the temperature at which water at 1 at-mospheric pressure freezes at 0◦C and boils at

100◦C Alternatively, the Celsius scale can be

defined in terms of the Kelvin temperature T as

temperature in Celsius= T − 273.16K.

center-of-momentum (c.o.m.) coordinates

A coordinate system in which the centers ofmass of interacting particles are at rest Theparticles are located by position vectors ρ

r i

de-fined by the center of mass of the rest frame ofthe system, which, in general, moves with re-spect to the particles themselves

c.o.m coordinates

In the center-of-momentum system, a pair

of colliding particles both approach the c.o.m.

head on, and then recede from the center with

equal but opposite momenta:

ρ

p1 + p ρ

2 = p ρ

1 + p ρ

2 = 0 even if, in the

laboratory frame, the target particle is at rest (as

depicted above) The velocity of the c.o.m for

respect to the line of motion of the incident

parti-cle are necessary to describe the final directions

of the particles, φ1 and φ2, a single common

angle θ suffices in the c.o.m.:

central force A force always directed toward

or away from a fixed center whose magnitude is

a function only of the distance from that center

In terms of spherical coordinates with an origin

at the force’s center,

centrifugal barrier A centrifugal force-like

term that appears in Schrödinger’s equations for

central potentials that prevents particles with

non-zero angular momentum from getting too

close to the potential’s center The symmetry of

Hamiltonians with central potentials allows the

state function to be separated into radial and

an-gular parts: ψ (r) = fλ (r)Y λm (θ, φ) If the dial part is written in the form f λ (r) = uλ (r)/r, the function u λ (r) can satisfy

a one-dimensional Schrödinger equation

carry-ing an additional potential-like term η2λ(λ+

1)/2mr2 which grows large as r→ 0

centrigual instability Present in a circular

Couette flow driven by the adverse gradient ofangular momentum which results in counter-rotating toroidal vortices Also known as theTaylor or Taylor-Couette instability

cesium chloride structure In cesium

chlo-ride, the bravais lattice is a simple cube with

primitive vectors ax, ay, and az and a basis

composed of a cesium positive ion and a ride negative ion

chlo-CFD Computational fluid dynamics change of state Refers to a change from one

state of matter to another (i.e., solid to liquid,liquid to gas, or solid to gas)

chaos The effect of a solution on a system

which is extremely sensitive to initial tions, resulting in different outcomes from smallchanges in the initial conditions Deterministic

condi-chaos is often used to describe the behavior of

turbulent flow

characteristic Mach number A Mach ber such that

num-M= u/a

where ais the speed of sound for M= 1 Thus,

Mis not a sonic Mach number, but the Machnumber of any velocity based on the sonic Machnumber speed of sound This merely serves as auseful reference condition and helps to simplifythe governing equations See Prandtl relation.

character of group representation Thetrace of a matrix at a representation in grouptheory

charge conjugation (1) The symmetry

op-eration associated with the interchange of the

role of a particle with its antiparticle

Equiva-lent to reversing the sign on all electric charge

and the direction of electromagnetic fields (and,

therefore, magnetic moments)

(2) A unitary operator ζ : jµ (x) → −jµ (x)

which reverses the electromagnetic current and

changes particles into antiparticles and vice

versa

chemical bond Term used to describe the

na-ture of quantum mechanical forces that allows

neutral atoms to bind and form stable molecules

The details of the bond, such as the

bind-ing energy, can be calculated usbind-ing the

meth-ods of quantum chemistry to solve the

Born-Oppenheimer problem See Born-Oppenheimer

approximation

chemical equilibrium For a reaction at

con-stant temperature and pressure, the condition

of chemical equilibrium is defined in terms of

the minimum Gibbs free energy with respect

to changes in the proportions of the reactants

and the products This leads to the condition,



j v j µ j = 0, where vj is the stoichiometric

coefficient of the j th species in the reaction

(neg-ative for reactants and positive for products), and

µj is the chemical potential of the j th species.

chemical potential (1) At absolute zero

tem-perature, the chemical potential is equal to the

Fermi energy If the number of particles is not

conserved, the chemical potential is zero.

(2) The chemical potential (µ) represents the

change in the free energy of a system when the

number of particles changes It is defined as

the derivative of the Gibbs free energy with

re-spect to particle number of the j th species in the

system at constant temperature and pressure, or,

equivalently, as the derivative of the Helmholtz

free energy at constant temperature and volume:

Chézy relations For flow in an open

chan-nel with a constant slope and constant chanchan-nel

width, the velocity U and flow rate Q can be

shown to obey the relations

U = CR h tan θ Q = CAR h tan θ where C=√8g/f and is known as the Chézy coefficient; f is the friction factor and R h is thehydraulic radius

Child–Langmuir law Description of

elec-tron current flow in a vacuum tube when plasmaconditions exist that result in the electron cur-rent scaling with the cathode–anode potential tothe 3/2 power

choked flow Condition encountered in a

throat in which the mass flow rate cannot beincreased any further without a change in theupstream conditions Often encountered in highspeed flows where the speed at a throat cannotexceed a Mach number of 1 (speed of sound)regardless of changes in the upstream or down-stream flow field

circularly polarized light A light beam

whose electric vectors can be broken into twoperpendicular elements having equal amplitudesbut differing in phase by l/4 wavelength

circulation The total amount of vorticitywithin a given region defined by

≡



C

u· ds

Circulation is a measure of the overall rotation in

a flow field and is used to determine the strength

of a vortex See Stokes theorem.

classical confinement Plasma confinement

in which particle and energy transport occur viaclassical diffusion

classical diffusion In plasma physics, fusion due solely to the scattering of chargedparticles by Coulomb collisions stemming fromthe electric fields of the particles In classicaltransport (i.e., diffusion), the characteristic stepsize is one gyroradius (Larmor orbit) and thecharacteristic time is one collision time

dif-classical limit Used to describe the

limit-ing behavior of a quantum system as the Planck

constant approaches the limit ¯h → 0.

classical mechanics The study of physical

systems that states that each can be completely

specified by well-defined values of all dynamic

variables (such as position and its derivatives:

velocity and acceleration) at any instant of time

The system’s evolution in time is then entirely

determined by a set of first order differential

equations, and, as a consequence, the energy of a

classical system is a continuous quantity Under

classical mechanics, phenomena are classified

as involving matter (subject to Newton’s laws)

or radiation (obeying Maxwell’s equations)

Clausius–Clapeyron equation The change

of the boiling temperature T , with a change in

the pressure at which a liquid boils, is given by

the Clausius–Clapeyron equation:

dP

T

v g − vl

Here, L denotes the molar latent heat of

vapor-ization, and v g and v l are the molar volumes

in the gas and liquid phase, respectively This

equation is also referred to as the vapor pressure

equation

Clebsch–Gordon coefficients Coefficients

that relate total angular momentum eigenstates

with product states that are eigenstates of

in-dividual angular momentum For example, let

|j1 m1 > be angular momentum eigenstates for

operators J1 (i.e., its square, and z-component),

and let |j2 m2 > be the eigenstates of

angu-lar momentum J2 We require the components

of J1 to commute with those of J2 We

de-fine J = J1 + J2, and if states |J M > are

angular momentum eigenstates of J2 and J z,

then |J M >= < j2m2j1m1|J M > |j1m1

j2m2 >, where the sum extends over all

al-lowed values j1 j2 m1 m2 The complex

num-bers < j2m2j1m1|J M > are called Clebsch–

Gordon coefficients See angular momentum

states

Clebsch–Gordon series Identity involving

Wigner rotation matrices, given the Wigner

ma-trices D ma ja m

a (R) and D mbm jb

b (R), where the first

matrix is a representation, with respect to an

angular momentum basis, of rotation R The

second rotation is a representation of the same

rotation R but is defined with respect to

an-other angular momentum basis The matricesact on direct product states of angular momen-tum For example, the first Wigner matrix op-erates on spin states for particle 1, whereas thesecond operates on the spin states for particle

2 The Clebsch-Gordon series relates products

of these matrices with a third Wigner rotation

matrix D j mm(R), which is a representation of the rotation R with respect to a basis given by

the eigenstates of the total angular momentum(for the above example, the total spin angularmomentum of particle 1 and 2)

closed system A thermodynamic system of

fixed volume that does not exchange particles orenergy with its environment is referred to as a

closed system Such a system is also called an

isolated system All other external parameters,such as electric or magnetic fields, that might

affect the system also remain constant in a closed system.

closure See completeness.

closure relation Satisfied by any completeorthonormal set of vectors |n >, the relation



n |n >< n| = 1, valid when the spectrum of

eigenvalues is entirely discrete, allows the pansion of any vector |u > as a series of the

ex-basis kets of any observable When the trum includes a continuum of eigenvalues, therelation is sometimes expressed in terms of adelta function identity:

spec-ρ δ

+



izes the expression to the continuous case

number of particles changes It is defined as

the derivative of the Gibbs free energy with

re-spect to particle number of the j th... interchange of the

role of a particle with its antiparticle

Equiva-lent to reversing the sign on all electric charge

and the direction of electromagnetic fields (and,

therefore,... speed of the pump

or propeller and high liquid temperatures are

standard causes of cavitation Cavitation

de-grades pump performance and can cause noise,vibration, and