Dictionary of Material Science and High Energy Physics Part 11 ppsx

quasi-classical distribution Representa-tions of the density operator for the electromag-netic field in terms of coherent rather than pho-ton number states.. Rabi oscillation When a two-

ference of π/2 with the aid of a quarter-wave

plate The doubly refracting transparent plates

transmit light with different propagation

veloc-ities in two perpendicular directions

quasi-Boltzmann distribution of fluctuations

Any variable, x, of a thermodynamic system that

is unconstrained will fluctuate about its mean

value The distribution of these fluctuations

may, under certain conditions, reduce to an

ex-pression in terms of the free energy, or other

such thermodynamic potentials, of the

thermo-dynamic system For example, the fluctuations

in x of an isolated system held at constant

tem-perature are given by the expression

f (x) ∼ e −F (x)/kT

where f (x) is the fluctuation distribution and

F (x)is the free energy, both as a function of

the system variable, x Under these conditions,

the fluctuation distribution is said to follow a

quasi-Boltzmann distribution.

quasi-classical distribution

Representa-tions of the density operator for the

electromag-netic field in terms of coherent rather than

pho-ton number states Two such distributions are

given by the Wigner function W (α) and the

Q-function Q(α) The Q-Q-function is defined by

Q(α) = 1

π < α |ρ|α >, where |α > is a herent state The Wigner function W (p, q) is

co-characterized by the position q and momentum

pof the electromagnetic oscillator and is defined

W (p, q)is quasi-classical owing to the lack of

positive definiteness for such distributions

quasi-continuum Used to describe quantum

mechanical states which do not form a

continu-ous band but are very closely spaced in energy

quasi-geostrophic flow Nearly geostrophic

flow in which the time-dependent forces are

much smaller than the pressure and Coriolis

forces in the horizontal plane

quasi-linear approximation A weaklynon-linear theory of plasma oscillations whichuses perturbation theory and the random phaseapproximation to find the time-evolution of theplasma state

quasi-neutrality The condition that the tron density is almost exactly equal to the sum ofall the ion charges times their densities at everypoint in a plasma

elec-quasi one-dimensional systems A systemthat is reasonably confined in one-dimension inorder to be considered onedimensional A typ-ical example would be a polymer chain which

is separated from neighboring chains by largesidegroups acting as spacers

quasi-particle (1) A conceptual particle-like

picture used in the description of a system of

many interacting particles The quasi-particles

are supposed to have particle-like propertiessuch as mass, energy, and momentum TheFermi liquid theory of L.D Landau, which ap-plies to a system of conduction electrons in met-als and also to a Fermi liquid of 3He, gives

rise to quasi-particle pictures similar to those

of constituent particles Landau’s theory of uid4He postulated quasi-particles of phonons

liq-and rotons, which carry energy liq-and momentum.Phonons of a lattice vibration could be regarded

as quasi-particles but they can not carry

momen-tum, though they have wave number vectors

(2) An excitation (not equivalent to the

ground state) that behaves as a particle and is

regarded as one A quasi-particle carries

prop-erties such as size, shape, energy, and tum Examples include the exciton, biexciton,phonon, magnon, polaron, bipolaron, and soli-ton

momen-quasi-static process The interaction of a

sys-tem A with some other syssys-tem in a process

(in-volving the performance of work or the change of heat or some combination of the two)

ex-which is carried out so slowly that A remains

arbitrarily close to thermodynamic equilibrium

at all stages of the process

quenching The rapid cooling of a material inorder to produce certain desired properties For

example, steels are typically quenched in a

liq-uid bath to improve their hardness, whereas

cop-per is quenched to make it softer Other methods

include splat quenching where droplets of

mate-rial are fired at rotating cooled discs to produce

extremely high cooling rates

q-value (magnetic q-value) In a toroidalmagnetic confinement device, the ratio of thenumber of times a magnetic field line winds thelong way around the toroid divided by the num-ber of times it winds the short way around, with

a limit of an infinite number of times

Rabi oscillation When a two-level atom

whose excited and ground states are denoted

re-spectively by a and b, interacts with radiation

of frequency ν (which is slightly detuned by δ

from the transition frequency ω = ω a − ω b,

i.e., δ = ω − ν), quantum mechanics of the

problem tells that the atom oscillates back and

forth between the ground and the excited state in

the absence of atomic damping This

phenom-enon, discovered by Rabi in describing spin 1/2

magnetic dipoles in a magnetic field, is known

as Rabi oscillation The frequency of the

os-cillation is given by  = √δ2+ R2, where

R = pE0/ ¯h, p is the dipole matrix element, and

E0is the amplitude of the electromagnetic field

If the radiation is treated quantum mechanically,

the Rabi oscillation frequency is given by =

δ2+ 4g2(n + 1), where g is the atom–field

coupling constant and n is the number of

pho-tons

radial distribution function The

probabil-ity, g(r), of finding a second particle at a

dis-tance r from the particle of interest

Particu-larly important for describing the liquid state

and amorphous structures

radial wave equation The Schrödinger

equation of a particle in a spherically symmetric

potential field of force is best described by polar

coordinates The equation can be separated into

ordinary differential equations The solution is

known for the angular variable dependence The

differential equation for the radial part is called

the radial wave equation.

radial wave function A wave function

de-pending only on radius, or distance from a

cen-ter It is most useful in problems with a central,

or spherically symmetric, potential, where the

Schrödinger equation can be separated into

fac-tors depending only on radius or angles; one

such case is the hydrogen atom, for which the

radial part R(r) obeys an equation of the form

1

system

radiation The transmission of energy from

one point to another in space The radiation

intensity decreases as the inverse square of the

distance between the two points The term diation is typically applied to electromagnetic

ra-and acoustic waves, as well as emitted particles,such as protons, neutrons, etc

radiation damping In electrodynamics, anelectron or a charged particle produces an elec-tromagnetic field which may, in turn, act on theparticle The self interaction is caused by virtualemissions and absorptions of photons The selfinteraction cannot disappear even in a vacuum,because of the zero-point fluctuation of the field.This results in damping of the electron motion in

the vacuum which is called the radiation ing.

damp-radiation pressure De Broglie wave–particle duality of implies that photons carrymomentum ¯hk, where k is the wave vector of

the radiation field When an atom absorbs aphoton of momentum ¯hk, it acquires the mo-

mentum in the direction of the beam of light Ifthe atom subsequently emits a photon by spon-taneous emission, the photon will be emitted in

an arbitrary direction The atom then obtains arecoil velocity in some arbitrary direction Thusthere is a transfer of momentum from photons

to the gas of atoms following spontaneous sion This transfer of momentum gives rise to

emis-radiation pressure.

radiation temperature The surface ature of a celestial body, assuming that it is aperfect blackbody The radiation temperature istypically obtained by measuring the emission ofthe star over a narrow portion of the electromag-netic spectrum (e.g., visible) and using Stefan’s

temper-law to calculate the equivalent surface

tempera-ture of the corresponding blackbody

radiative broadening An atom in an

ex-cited state would decay by spontaneous

emis-sion in the absence of photons, described by an

exponential decrease in the probability of being

found in that state In other words, the atomic

level would be populated for a finite amount of

time The finite lifetime can be represented by

γ−1, where γ is the decay rate The finite

life-time introduces a broadening of the level

Spon-taneous emission is usually described by treating

the radiation quantum mechanically, and since

it can happen in the absence of the field, the

process can be viewed as arising from the

fluc-tuations of the photon vacuum The

sponta-neous emission decay rate γ , for decay from

level two to level one of an atom, is given by γ

= e2r122ω3/(3π 0¯hc3), where r12 is the dipole

matrix element between the levels and ω is the

transition frequency γ is also related to the

Ein-stein A coefficient by γ = A/2.

radiative correction (1) The change

pro-duced in the value of some physical quantity,

such as the mass, charge, or g-factor of an

elec-tron (or a charged particle) as the result of its

interaction with the electromagnetic field

(2) A higher order correction of some process

(e.g., radiative corrections to Compton

scatter-ing) or particle property (e.g., radiative

correc-tions to the g-factor of the electron) For

ex-ample, an electron can radiate a virtual photon,

which is then reabsorbed by the electron In

terms of Feynman diagrams, radiative

correc-tions are represented by diagrams with closed

loops Radiative corrections can affect the

be-havior and properties of particles

radiative decay Decay of an excited state

which is accompanied by the emission of one or

more photons

radiative lifetime The lifetime of states if

their recombination was exclusively radiative

Usually the lifetime of states is determined by

the inverse of the sum of the reciprocal lifetimes,

both radiative and nonradiative

radiative transition Consider a microscopicsystem described by quantum mechanics Atransition from one energy eigenstate to another

in which electromagnetic radiation is emitted is

called the radiative transition.

radioactivity The process whereby heaviernuclei decay into lighter ones There are three

general types of radioactive decay: α-decay

(where the heavy nucleus decays by emitting

an helium nucleus), β-decay (where the heavy

nucleus decays by emitting an electron and

neu-trinos), and γ -decay (where the heavy nucleus

decays by emitting a gamma ray photon)

radius, covalent Half the distance betweennuclei of neighboring atoms of the same speciesbound by covalent bonds

radius, ionic Half the distance betweenneighboring ions of the same species

raising operator An operator that increasesthe quantum number of a state by one unit The

most common is the raising operator for the

eigenstates of the quantum harmonic

oscilla-tor a† Harmonic oscillator states have energy

Its Hermitian conjugate a has the opposite effect

and is known as the lowering or annihilation erator

op-Raman effect (active transitions) Light teracting with a medium can be scattered ine-leastically in a process which either increases ordecreases the quantum energy of the photons

in-Raman instability A three-wave interaction

in which electromagnetic waves drive electronplasma oscillations In laser fusion, this processproduces high energy electrons that can preheatthe pellet core

Raman scattering When light interacts with

molecules, part of the scattered light may

oc-cur with a frequency different from that of the

incident light This phenomenon is known as

Raman scattering The origin of this inelastic

scattering process lies in the interaction of light

with the internal degrees of freedom, such as the

vibrational degrees of freedom of the molecule

Suppose that an incident light of frequency ω i

produces a scattered light of frequency ω s, while

at the same time, the molecule absorbs a

vibra-tional quantum (phonon) of frequency ω v

mak-ing a transition to a higher vibrational level The

frequencies would be related by ω v + ω s = ω i

In this case, the frequency of the scattered light

is less than that of the incident light, a

phenom-enon known as the Stokes shift Alternately, a

molecule can give up a vibrational quanta in the

scattering process In this case the frequencies

are related by ω i + ω v = ω s, and the scattered

frequency is greater than that of the incident

light, an effect known as the anti-Stokes shift

Raman scattering also exists for rotational and

electronic transitions

Ramsey fringes In a Ramsey fringes

exper-iment, an atomic beam is made to traverse two

spatially separated electromagnetic fields, such

as two laser beams or two microcavities For

instance, if two-level atoms are prepared in the

excited state and made to go through two fields,

transition from the upper to the lower state can

take place in either field Consequently, the

tran-sition probability would demonstrate

interfer-ence The technique of Ramsey fringes is used

in high-resolution spectroscopy

random phases Consider a quantum system

whose state, represented by

a superposition of orthonormal states{|ϕ n >},

n a n |ϕ n > The elements of the density matrix are given by ρ nm = a n a∗

m Thedensity matrix has off-diagonal elements and

the state is said to be in a coherent

superposi-tion The expansion coefficients have phases,

i.e., a n = |a n |e iθ n, and if the phases are

un-correlated and random, an average would make

the off-diagonal elements of ρ vanish, as would

be the case if the system is in thermal

equilib-rium The nonzero off-diagonal elements of the

density matrix, therefore, imply the existence of

correlations in the phases of the members of theensemble representing the system

Rankine body Source and sink in potentialflow in a uniform stream that generates flow over

an oval shaped body

Rankine cycle A realistic heat engine cyclethat more accurately approximates the pressure-volume cycle of a real steam engine than the

Carnot cycle The Rankine cycle consists of

four stages: First, heat is added at constant sure p1 through the conversion of water to su-perheated steam in a boiler Second, steam ex-

pres-pands at constant entropy to a pressure p2 inthe engine cylinder Third, heat is rejected at

constant pressure p2in the condenser Finally,condensed water is compressed at constant en-

tropy to pressure p1by a feed pump

The Rankine cycle.

Rankine efficiency The efficiency of an idealengine working on the Rankine cycle undergiven conditions of steam pressure and temper-ature

Rankine–Hugoniot relation Jump tion across a shock wave relating the change in

condi-internal energy e from the upstream to

down-stream side

e2− e1=1

2(p1+ p2) (v1− v2) where v is the specific volume.

Rankine propeller theory A propeller erating in a uniform flow has a velocity at thepropeller disk half of that behind the propeller

op-in the slipstream Half of the velocity op-increase is

predicted to occur upstream of the propeller and

half downstream of the propeller, indicating that

the flow is accelerating through the propeller

Rankine temperature scale An absolute

temperature scale based upon the Fahrenheit

scale Absolute zero, 0◦ R, is equivalent to

−459.67◦ F, while the melting point of ice at

−32◦ F is defined as 491.67◦ R

Rankine vortex Vortex model where a

rota-tional core with finite vorticity is separated from

a irrotational surrounding flow field The

rota-tional core can be idealized with a velocity

pro-file

u θ = 1

2ω o r c

where r c is the radius of the core Matching

velocities at r = r c, this makes the irrotational

flow outside the core

u θ = 1

2ω o

r c2r

and the vortex circulation

 = πω o r c2.

This distribution has a region of constant

vor-ticity at r < r c and a discontinuity at r = r c,

beyond which the vorticity is zero See vortex.

RANS Reynolds Averaged Navier–Stokes.

See Reynolds averaging.

Raoult’s law The partial vapor pressure of a

solvent above a solution is directly proportional

to the mole fraction (number of moles of solvent

divided by the total number of moles present) of

the solvent in solution If p0 is the pressure

of the pure solvent and X is the solvent mole

fraction, then the partial vapor pressure of the

solvent, p, is given by:

p = p0X Any solution that obeys Raoult’s law is termed

an ideal solution In general, only dilute

solu-tions obey Raoult’s law, although a number of

liquid mixtures obey it over a range of

concen-trations These so-called perfect solutions occur

when the intermolecular forces of the pure stance are similar to those between molecules ofthe mixed liquids

sub-rapidity A quantity which characterizes aLorentz boost on some system such as a parti-cle If a particle is boosted into a Lorentz frame

where its energy is E and its momentum in the direction of the boost is p, then the rapidity is given by y = tanh−1p

E



rare-earth elements A group of elementswith atomic numbers from 58 to 71, also known

as the lanthanides Their chemical propertiesare very similar to those of Lanthanum; like it,

they have outer 6s2electrons, differing only in

the degree of filling of their inner 5d and 4f

shells

rare earth ions Ions of rare earth elements,viz lanthanides (elements having atomic num-bers 58 to 71) and actinides (elements havingatomic numbers 90 to 103)

rarefaction Expansion region in an acousticwave where the density is lower than the ambientdensity

Rarita–Schwinger equation (1) An

elemen-tary particle with spin 1/2 is described by theDirac equation:



γ µ ∂ µ + κψ = 0 , where γ1, γ4 are the Dirac’s γ -matrices, obeying the anti-commutation relations γ µ γ ν+

γ ν γ µ = 2δ µν , κ is the rest mass energy, and

ψis the four-component wave function A

par-ticle with spin 3/2 is described by the Rarita– Schwinger equation:



γ µ ∂ µ + κψ λ = 0, γ λ ψ λ = 0 Each of the wave functions ψ1, , ψ4 havefour components (two components represent thepositive energy states and the other two rep-resent the negative energy states), and hencethe particle is described by 16 component wavefunctions

(2) Equation which describes a spin 3/2

par-ticle The equation can be written as (i
γ α ∂ α−

m o µ (x) = 0 and the constraint equation

γ µ µ = 0 In these equations, γ α are Dirac

µ (x) is a vector-spinor,

equation

Rateau turbine A steam turbine that consists

of a number of single-stage impulse turbines

ar-ranged in series

rate constant The speed of a chemical

equa-tion in moles of change per cubic meter per

sec-ond, when the active masses of the reactants are

unity The rate constant is given by the

con-centration products of the reactants raised to the

power of the order of the reaction For example,

for the simple reaction

A → B

the rate is proportional to the concentration of

A, i.e., rate = k[A], where k is the rate constant.

rate equation In general, the rate equation

is complex and is often determined empirically

For example, the general form of the rate

equa-tion for the reacequa-tion A + B → products is given

by rate= k[A] x [B] y , where k is the rate constant

of the reaction, and x and y are partial orders of

the reaction

rational magnetic surface See mode rational

surface

ratio of specific heats The ratio of the

spe-cific heat at constant pressure and spespe-cific heat

at constant volume used in compressible flow

calculations

γ = C p

C v . For air, γ = 1.4.

Rayleigh–Bérnard instability See Bérnard

instability

Rayleigh criteria Relates, for just resolvable

images, the lens diameter, the wavelength, and

the limit of resolution

Rayleigh flow Compressible

one-dimension-al flow in a heated constant-area duct Assuming

the flow is steady and inviscid in behavior, the

governing equations simplify to the following:

total temperatureq = c p



T02− T0 1



The behavior varies depending upon whether

heat is being added (q > 0) or withdrawn (q < 0) and whether the flow is subsonic (M < 1) or supersonic (M > 1) Trends in the parameters

are shown in the table below as increasing ordecreasing in value along the duct Note that the

variation in temperature T is dependent upon the ratio of specific heats γ

Rayleigh flow Mollier Diagram.

A Mollier diagram shows the variation in tropy and enthalpy for heating and cooling sub-sonic and supersonic flows Heating a flow al-ways tends to choke the flow It is theoretically

en-possible to heat a flow and then cool it to

transi-tion from subsonic to supersonic flow and

vice-versa

Rayleigh inflection point criterion To

deter-mine flow instability in a viscous parallel flow,

a necessary but not sufficient criterion for

insta-bility is that the velocity profile U (y) has a point

of inflection See Fjortoft’s theorem.

Rayleigh-Jeans law Describes the energy

distribution from a perfect blackbody emitter

and is given by the expression

E ω dω= 8π ω2kT

c3 dω where E ω is the energy density radiated at a

temperature T into a narrow angular frequency

range from ω to ω +dω, c is the velocity of light,

and k is Boltzmann’s constant This expression

is only valid for the energy distribution at low

frequencies Indeed, attempting to apply this

law at high frequencies results in the so-called

UV catastrophe, which ultimately led to the

de-velopment of Planck’s quantized radiation law

and the birth of quantum mechanics

Rayleigh number Dimensionless quantity

relating buoyancy and thermal diffusivity effects

Re= gαT L3

νκ where α, ν, and κ are the expansion coefficient,

kinematic viscosity, and thermal diffusivity

re-spectively

Rayleigh scattering First described by Lord

Rayleigh in 1871, Rayleigh scattering is the

elas-tic scattering of light by atmospheric molecules

when the wavelength of the light is much larger

than the size of the molecules The wavelength

of the scattered light is the same as that of the

incident light The Rayleigh scattering

cross-section is inversely proportional to the fourth

power of the wavelength

Rayleigh–Schrödinger perturbation

expan-sion Rigorously solving the Schrödinger

equation of a system is difficult in almost all

cases In many cases we start from a simplified

system described by the Hamiltonian H0, whose

eigenvectors  n and eigenvalues E nare known,and take account of the rest of the Hamiltonian

H I as a weak action upon the exactly knownstates This is perturbation approximation The

Rayleigh–Schrödinger expansion is that in the

α , its energy E a, which issupposed to be non-degenerate, is expressed as

of the ray by normalizing the state to unity Even

so, a phase factor of a magnitude of one is leftunspecified Text books say that a transforma-tion from a set of eigenvectors as a basis forrepresentation to another set for another repre-sentation is unitary That statement is better ex-pressed in operator algebra, where symmetries

of our system are clarified in mathematical guage If a symmetry exists it will be described

lan-by a unitary or anti-unitary operator, connectingthe representations before and after the symme-try operation or transformation Furthermore,consider groups of symmetry transformations;i.e., a set of symmetry transformations forming

a group in the mathematical sense The set ofoperators representing the transformations form

a representation of the group This

representa-tion is called the ray representarepresenta-tion.

ray tracing Calculation of the trajectorytaken by a wave packet (or, equivalently, bywave energy) through a plasma Normally thiscalculation uses the geometrical optical approx-imation that gradient scale lengths are muchlonger than the wavelength of the wave

R-center One of many centers (e.g., F, M,

N, etc.) arising out of different types of

treat-ment to which a transparent crystal is subjected

to rectify some defects in the form of absorption

bands affecting its color Prolonged exposure

with light or X-rays producing bands between F

and M bands are responsible for R-centers

reabsorption Depending on the spectral

shape of photon emission and absorption

spec-tra in some media, one observes a strong

absorp-tion of emitted photons, i.e., reabsorpabsorp-tion This

process determines the line width of the

electro-luminescence of most inorganic light emitting

diodes

real gas See perfect gas.

Reaumur temperature scale A temperature

scale that defines the boiling point of water as

80◦ R and the melting point of ice as 0◦ R

reciprocal lattice A set of imaginary points

constructed in such a way that the direction of

a vector from one point to another coincides

with the direction of a normal to the real space

planes, and the separation of those points

(abso-lute value of the vector) is equal to the reciprocal

of the real interplanar distance

reciprocal relations See Onsager’s

recipro-cal relation

reciprocating engine An engine that uses the

pressure of a working fluid to actuate the cycling

of a piston located in a cylinder

recirculating heating system Typically used

in industrial ovens or furnaces to maintain the

atmosphere of the working chamber under

con-stant recirculation throughout the entire system

recoil energy The term can be illustrated by

the behavior of a system in which one particle

is emitted (e.g., hot gas in a jet-engine) The

re-coil energy is determined by the conservation of

momentum which governs the velocity of both

the gas and the jet Since the recoil energy is

equivalent to the kinetic energy of the jet

ob-tained by the emission of the gas, this energy

depends on the rifle If it is held loosely during

firing, its recoil, or kick, will be violent If it

is firmly held against the marksman’s shoulder,the recoil will be greatly reduced The differ-ence in the two situations results from the factthat momentum (the product of mass and veloc-ity) is conserved: the momentum of the systemthat fires a projectile must be opposite and equal

to that of the projectile By supporting the fle firmly, the marksman includes his body, withits much greater mass, as part of the firing sys-tem, and the backward velocity of the system

ri-is correspondingly reduced An atomic nucleus

is subject to the same law When radiation isemitted in the form of a gamma ray, the atomwith its nucleus experiences a recoil due to themomentum of the gamma ray A similar recoiloccurs during the absorption of radiation by anucleus

recombination The process of adding an

electron to an ion In the process of radiative combination, momentum is carried off by emit- ting a photon In the case of three-body recom- bination, momentum is carried off by a third

re-particle

recombination process The process bywhich positive and negative ions combine andneutralize each other

rectification The process of converting analternative signal into a unidirectional signal

recycling Processes that result in plasma ionsinteracting with a surface and returning to theplasma again, usually as a neutral atom

reduced density matrix For the ground state

of an identical particle system described by the

1, x2, , x n ), the cle reduced density matrix is

and so forth.

reduced density operator Many physical

systems consist of two interacting sub-systems

Denoting these by A, and B, the density

opera-tor of the total system can be denoted by ρ AB

Quite often, one is only interested in the

dynam-ics of the subsystem A, in which case a reduced

density operator ρ A is formally obtained from

the full density operator by averaging over the

degrees of freedom of the system B This can

be expressed by ρ A =TrB (ρ AB ) For

exam-ple, consider the interaction of an atom with the

modes of the electromagnetic field within a

cav-ity If the atom is the system A, the many modes

of an electromagnetic field could be considered

as the other system While the atom interacts

with the field modes, one might be interested in

pursuing the dynamics of the atom by

consider-ing the density operator ρ Aafter formally

aver-aging over the reservoir R of the field modes.

reduced mass A quantity replacing, together

with total mass, the individual masses in a

two-body system in the process of separation

of variables It is equal to

µ= m1m2

m1+ m2

.

reduced matrix element The part of a

spherical tensor matrix element between

angu-lar momentum eigenstates that is independent of

magnetic quantum numbers According to the

Wigner–Eckart theorem, the matrix element of

a spherical tensor operator of rank k with

mag-netic quantum number q between angular

mo-mentum eigenstates of the type|α, jm > has

The double-bar matrix element, which is

inde-pendent of m, m, and q, is also called the

re-duced matrix element.

reflectance The ratio of the flux reflected by

a body to the flux incident on it

reflection The reversal of direction of part

of a wave packet at the boundary between tworegions separated by a potential discontinuity.The fraction of the packet reflected is given by

the reflection coefficient which is equal to one

minus the transmission coefficient

reflection, Bragg The beam reinforced

by successive diffraction from several crystalplanes obeying the Bragg equation

reflection coefficient Ratio of reflected to

incident voltage for a transmission line (Z0−

Z R )/(Z0+ Z R ), where Z0and Z Rare teristic and load impedances, respectively

charac-refractive index When light travels from onemedium to another, refraction takes place The

refractive index for the two media (n12)is the

ratio of the speed of light in the first medium (c1)

to the speed of light in second medium (c2) The refractive index is thus defined by the equation

n12= c1/c2

refrigeration cycle Any thermodynamic cle that takes heat at a low temperature and re-jects it at a higher temperature From the sec-ond law of thermodynamics, any refrigerationcycle must receive power from an external en-ergy source

cy-refrigerator A machine designed to use chanical or heat energy to produce and maintain

me-a lower temperme-ature

regenerator A device that acts as a heatexchanger, transferring heat of exit or exhaustgases to the air entering a furnace or the waterfeeding a boiler Such a device tends to increasethe efficiency of the overall thermodynamic sys-tem

Regge poles A singularity which occurs inthe partial wave amplitude for some scatteringprocesses For some processes, the scattering

amplitude, f (E, cos θ ), where E is the energy and θ is the scattering angle, can be written as a

contour integral in the complex angular

momen-tum (J ) plane: f (E, cos θ )= 1

2π i



C dJ sin π J π (2J +1)a(E, J ) P J ( − cos θ), where a(E, J ) is

the partial wave amplitude A Regge pole is a

singularity in a(E, J ) for some value of J

Regge trajectory By plotting the angular

momentum (J ) and the mass square (m2) of

a given hadron and its rotational excitations, a

linear relationship is found to exist of the form

J = αm2 + J0, where α is a slope and J0

is an intercept This plotted lines form Regge

trajectories.

regularization A modification of a theory

that renders divergent integrals finite In a

quan-tum field theory, divergent momenquan-tum integrals

generally arise when radiative corrections are

calculated Some of the more common

regular-ization schemes are Pauli-Villars regularregular-ization,

dimensional regularization, and lattice

regular-ization.

relative density The density of a material

divided by the density of water It is also known

as specific gravity

relative permeability See permeability.

relative permittivity See permittivity

relativistic quantum mechanics A theory

that is compatible with both the special

the-ory of relativity and the quantum thethe-ory It

is based on the Dirac equation which replaces

the Schrödinger equation for spin-1/2 particles

with a four-component vector, or spinor, as the

wave function Developed in the 1930s, it forms

the basis of quantum electrodynamics, the

quan-tum theory of electromagnetism, as well as other

modern quantum field theories

relaxation time The characterisitic time

af-ter which a disequilibrium distribution decays

toward an equilibrium distribution The

elec-tron relaxation time in a metal, for example,

describes the time required for a disequilibrium

distribution of electron momenta (e.g., in a

flow-ing current) to decay toward equilibrium in the

absence of an ongoing driving force and can be

interpreted as the mean time between scattering

events for a given electron

relaxation time approximation mation to relaxation time, time by which thetime-measurable quantity of a physical phenom-enon changes exponentially to 1/eth of its orig-inal value

Approxi-renormalizability Interacting quantum fieldtheories contain technical difficulties, originat-ing from the basic notion of the infinite freedom

of field up to endlessly small region of space;

up to mathematical points This is, however,

an unphysical difficulty because in an extremelysmall region certain new field theories or physicswould be required Yet we hope that the knowntheory can give consistent descriptions and pre-dictions for the phenomena at desired energyrange and hence at necessary space dimensions.For some quantum fields, this is shown to betrue In fact, all infinite quantities can be ab-sorbed into a renormalization of physical param-

eters such as mass and charge This is the malizability of the quantum field theories The

renor-quantum electrodynamics is a typical examplefor providing such renormalizability

renormalization A rescaling or redefinition

of the original bare quantities of the Lagrangian

of a theory, such as mass or charge This ing gives a relationship between the original (of-ten infinite) parameters of the theory and the fi-nite real physical quantities

rescal-renormalization group In a particular

renor-malization scheme R, a renormalized quantity,

 R, is related to the unrenormalized quantity,

0, via  R = Z(R)0, where Z(R) is the malization constant associated with R Under a different scheme R, this relationship becomes

renor- R = Z(R)

0 A relationship can be obtained

between  R and  R, namely  R = Z(R, R)

quan-ical examples are representations in position or

momentum space Since position and

momen-tum operators do not commute, the

correspond-ing quantum numbers cannot be specified

simul-taneously and a choice of representation must be

made

reservoir A thermal reservoir is an

ideal-ized large thermodynamic system that can gain

or lose heat from the thermodynamic system of

interest without affecting its internal energy and

hence its temperature A particle reservoir is

the analogous case for particle exchange

residual resistivity The resistivity of a metal

that does not depend on temperature It is

pre-sented even at low temperature and is caused by

impurities

resistance, electrical The property of a

con-ducting substance determining the magnitude of

a current that would flow when a certain

poten-tial difference is applied across it

resistance, minimum The minimum

resis-tance is due to the scattering of conduction

elec-trons showing unexpected features if the

scat-tering center has a magnetic moment given by

Kondo theory

resistance thermometer A device that uses

the dependence of a material’s electrical

resis-tance upon temperature as a measure of

tem-perature For high precision measurements, a

platinum wire is typically used, whereas

semi-conductor materials are the material of choice

for high sensitivity (thermistor)

resistive ballooning mode Pressure-driven

mode in which instability is caused or

signifi-cantly enhanced by electrical resistivity, and the

perturbation is concentrated mostly on the

out-board edge of a toroidal magnetic confinement

device

resistive drift wave A magnetic drift mode

of plasma oscillation that is unstable because of

electrical resistivity

resistive instability Any plasma instability

that is significantly enhanced or made unstable

by electrical resistivity

resistive interchange mode Instability ven by plasma pressure gradient together withmagnetic reconnection in a magnetic confine-ment device

dri-resistivity The property of a material to pose the flow of electric current Resistivity (symbol is ρ) depends on temperature For a wire of length L, cross-sectional area A, and resistivity ρ, the resistance (R) is defined as:

op-R = ρL/A.

resolvent For the Schrödinger equation

R(E) = 1/(E − H )

resonance (1) The dramatic increase in a

transition probability or cross-section for a cess observed when an external applied periodicfield matches a characteristic frequency of thesystem In particle physics, the term is oftenused to describe a particle which has a lifetimetoo short to observe directly, but whose pres-ence can be deduced by an increase in a reactioncross-section when the center-of-mass energy is

pro-in the vicpro-inity of the particle’s mass

(2) A particle with a lifetime which is so short

that the particle is detected via its resonance

peak in the cross-section for some process For

example, in the process π++ p → π++ p,

a resonance peak in the cross-section occurs at some particular energy This resonance peak

is associated with the ++ particle which is

thought to occur between the initial and final

states (π++ p → ++→ π++ p).

resonance absorption The absorption ofelectromagnetic waves by a quantum mechan-ical system through its transition from one en-ergy level to another The frequency of thewave should satisfy the Bohr frequency condi-

tion hv = E2− E1, where E1and E2are, spectively, the energies of the levels before andafter the transition

re-resonance fluorescence The emission of anatom irradiated by a continuous monochromaticelectromagnetic radiation The situation is dif-ferent from that of spontaneous emission of an