From a viewpoint of a quantized field, one photon out of each of the pump beams is annihilated, a photon in the phase conjugate wave is generated, and the signal beam gets am-plified.. I
Trang 1ated field The signal and generated fields are
therefore subject to a phase matching condition,
which does not apply to the propagation of the
input fields ω1 and ω2
As opposed to other schemes of frequency
generation by the interaction of coherent beams,
such as lasing without inversion, no population
transfer is occurring within the atomic or
molec-ular system
A special case of four wave mixing is the
de-generate four wave mixing, which leads to phase
conjugation In this case, the two incoming
fields with the same frequency are
copropagat-ing through the medium, which causes a
station-ary grating to build up It can be shown that the
generated field is the phase conjugate of the
sig-nal beam and, consequently, co-propagates with
respect to it From a viewpoint of a quantized
field, one photon out of each of the pump beams
is annihilated, a photon in the phase conjugate
wave is generated, and the signal beam gets
am-plified
fractional charge An electric charge less
than that of the electron, generally by a
fac-tor expressible as a rational fraction made of
small integers The quasiparticles in the
frac-tional quantum Hall effect, e.g., are believed to
possess fractional charges.
fractional quantum numbers Quantum
numbers associated with quasiparticles in
cer-tain systems that are simple fractions of the
num-bers for elementary particles See fractional
charge
fractional statistics Term used to describe
certain field theories in which the wave function
of the many-particle system does not get
multi-plied by +1 or−1 during the exchange of any
two particles, as would be the case for Bose or
Fermi statistics respectively Instead, the wave
function is multiplied by a phase factor with a
phase angle that is a fraction of π
fragmentation function In a high
momen-tum transfer reaction, a recoiling quark-parton
will eventually hadronize The fragmentation
function represents the probability that a
quark-parton of a specific type will produce a hadron
in an interval dz about z, where z is the spatial
direction of the recoiling hadron
francium An element with atomic number
87 The element has 39 known isotopes, none
of which are stable The isotope with atomicmass number 223 has the longest half-life of
22 minutes Because francium is the heaviest
known element which chemically acts as a electron atom, interest has developed in using it
one-to increase the sensitivity of aone-tomic parity periments
ex-Franck–Condon factors The overlap grals of vibrational wave functions of differ-ent electronic states According to the Born–Oppenheimer approximation, a molecular wavefunction can be written as the product of the elec-
inte-tronic wave function e, the vibrational wave
function v , the rotational wave function r,
and the nuclear wave function N In the case
of an electronic transition, the transition moment
is given by the integral
se-of an electronic state to another electronic statewith a different vibrational state can be excited,
for these, the Franck–Condon factor is large.
Classically, this reflects the fact that the clei move much slower than the electrons, and,consequently, electronic transitions are favoredwhere the kinetic energy of the nuclei does notchange Quantum mechanically, this results inlarge overlap functions| v|v | 2
Trang 2prin-ergy of the nuclei remains constant or at least
very similar Quantum mechanically, at these
locations, the Franck–Condon factors are large.
Franck–Hertz experiment Experiment by
J Franck and G Hertz in 1914, in which atoms
were bombarded by low energy electrons
Franck and Hertz discovered that the electron
beam current decreased sharply whenever
cer-tain thresholds in the electron energy were
ex-ceeded, The experiment demonstrated the
exis-tence of sharp atomic energy levels and provided
strong support for N Bohr’s model of the atom
Franck–Read source In a closed circular
dislocation, a dislocation segment pinned at
each end is called a Franck–Read source and can
lead to the generation of a large number of
con-centric dislocation loops on a single slip plane
Frank, Ilya M. Nobel Prize winner in 1958
who, with Igor Tamm, explained the Cerenkov
effect
Franson interferometer A special type of
interferometer used in photon correlation
mea-surements A correlated pair of photons is sent
through an interferometric setup as depicted in
the figure One photon is sent one way and the
other photon is sent down the other path in the
interferometer By detecting coincidence counts
between the two detectors on each side, the
in-terferences between the two cases when either
both photons have taken the long path or both
photons have taken the short path are detected
These are second order interference effects
Setup of a Franson interferometer.
Fraunhofer diffraction The diffraction
pat-tern when observed in the farfield, i.e., a large
distance a from the diffracting object with a
di-mension d The size of the object must be in the
same order of magnitude as the wavelength λ of
the light The diffraction pattern of an object isequivalent to the spatial Fourier transformation
of the diffracting object
For instance, the diffraction pattern from a
single slit with width d is given by
I = I0sin2δ
δ2with δ=π d sin α
where α is the angle from the optical axis and λ
is the wavelength of the light
free electrons Electrons detached from anatom
free energy An energy quantity assigned toeach substance, such that a reaction in a systemheld at constant temperature tends to proceed if
it is accompanied by a decrease in free energy The free energy is the sum of the enthalpy and
entropy
free expansion The process of expansion of
a gas contained in one part of an isolated tainer to fill the entire container by opening avalve separating the two compartments In thisprocess, no heat flows into the system since it isthermally isolated, and no work is done Thus,conservation of energy requires that the inter-nal energy of the system remain unchanged Ifthe gas is ideal, there will be no temperaturechange; however, for a real gas, the temperature
con-decreases in a free expansion.
free induction decay Term originally coined
in the area of nuclear magnetic resonance to scribe the decay of the induction signal of amacroscopic sample of matter containing nu-clear spins which are initially tipped over fromtheir equilibrium orientation and then undergofree precession The term reflects the decay ofthe signal in a small fraction of the spins’ naturallifetimes, which occurs due to inhomogeneities
de-in the magnetic field de-in the sample The term
is now used in all other types of magnetic nance, as well as in the area of resonance optics,i.e., the interaction of atoms with coherent lighttuned to an atomic transition
Trang 3reso-free particle A particle not under the
influ-ence of any external forces or fields
free precession In magnetic resonance, the
precession of the magnetic moment in a uniform
static magnetic field
free shear flows Shear flows are flows where
the velocity varies principally in a direction at a
right angle to the flow direction In free shear
flows, this variation is caused by some upstream
variation or disturbance Downstream of the
disturbance, the free shear flow decelerates,
en-trains ambient fluids, and spreads Examples of
free shear flows include jets, wakes, mixing
lay-ers, and separated boundary layers Viscosity
has the effect of smoothing the velocity field,
which causes it to become self-similar Free
shear flows are unstable and are characterized
by large-scale structures
free spectral range The frequency
separa-tion between adjacent transmission maxima for
an optical cavity This is an important
consider-ation in determining the mode spacing in laser
resonators or the resolution of Fabry–Perot
in-terferometers For a resonator with an optical
path length L, one finds for the free spectral
quence of the boundary conditions of the
elec-tric field amplitudes in linear resonators and the
condition for constructive interference of
con-secutive passes in ring resonators, respectively
free surface A surface that consists of the
same fluid particles and along which the
pres-sure is constant In studying free surface flows,
the shape of the free surface is not known
ini-tially Rather, it is a part of the solution
free vortex A flow field with purely
gential motion (circular streamlines) The
tan-gential velocity is inversely proportional to the
radius Consequently, the origin is a singular
point The circulation around any contour notenclosing the origin is zero The flow is thusirrotational
Frenkel defect A point defect in a lattice in
which an atom is transferred from the lattice site
to an interstitial position
Frenkel exciton An exciton in which the
ex-citation is localized on or near a single atom
Fresnel diffraction The diffraction in close
proximity to the diffracting object with size d, i.e., when the wavelength of the light λ, d, and the distance from the object a are in the same
order of magnitude:
a ≈ d ≈ λ
Fresnel diffraction is in contrast to Fraunhofer
diffraction, which is observed when
a d ≈ λ
friction coefficient The ratio of the force
re-quired to move one solid surface over anothersurface to the total force pressing two surfacestogether
friction drag Also called viscous drag since
it is generated by the shear stresses The friction drag scales with the Reynolds number It is im-
portant in flows with no separation and depends
on the amount of surface area of the object that
is in contact with the fluid See also form drag
friction factor A dimensionless parameterthat is related to the pressure required to move
a fluid at a certain rate It is generally a tion of the Reynolds number, surface roughness,and body geometry In a pipe or duct, the rela-
func-tion between the fricfunc-tion factor, flow velocity,
pressure drop, and geometry is given by
Trang 4factor is considered to be independent of the
Ma.
frictionless flow A flow where viscous
ef-fects are neglected See also inviscid flow
friction velocity Defined for boundary layers
as
u∗= (τw /ρ) 1/2 where τ w is the shear stress at the wall and ρ is
the density
Froude number A dimensionless parameter
that represents the relative importance of inertial
forces acting on a fluid element to the weight
of the element It is given by V /√
g, where
V is the fluid velocity and is a characteristic
length such as body length or water depth The
Froude number is important in flows where there
is a free surface such as open channel flows,
rivers, surface waves, flows around floating
ob-jects, and resulting wave generation
frozen field lines In non-resistive MHD
mas, the magnetic field lines are tied to the
plas-ma so that they move and oscillate together This
state is called field lines frozen into the plasma
or, simply, frozen field lines.
f-sum rule Relates the total amount of
scat-tering of light, neutrons, or any other probe from
any physical system, when integrated over all
energies, to the number of scatterers The rule
is closely related to the dipole sum rule, and
can be used in the same way to estimate the
contribution of various physical mechanisms orexcitations to scattering
ft-value In allowed transitions in beta decay,
the ft-value is a measure of the probability of the
decay rate It is proportional to the sum of thesquares of the Fermi and Gammow–Teller ma-trix elements In forbidden transitions, it doesnot give a measure of these matrix elements, butdoes indicate the order of forbiddenness of thedecay Since ft-value has a large range, it isusually quoted in terms of a log10
fully developed flow Beyond the entrance
region of the flow into a pipe or a duct, the meanflow properties do not change with downstreamdistance The velocity profile is fully developedand the flow is called a fully developed flow Theentrance length, beyond which the flow is fullydeveloped, varies between 40 to 100 diametersalong the pipe and is dependent on the Reynoldsnumber See also entrance region
fundamental vectors Vectors that can definethe atomic arrangement in an entire lattice bytranslation
fusion When two nuclei coalesce into a larger
nucleus, nuclear fusion has occurred Nuclear fusion is usually associated with the combina-
tion of two deuterium atoms to form a helium
atom or the fusion of a deuterium atom with a
tritium atom to form a helium atom with the lease of a neutron and energy The latter reaction
re-is the fundamental reaction in a hydrogen bomb
Trang 5g(2) (τ) See intensity correlation function
ga The weak interaction can be described
in terms of a leptonic current interacting with
a hadronic current In general, these currents
could consist of five forms — scalar, pseudoscalar,
vector, pseudovector, and tensor— but the weak
interaction may be described only in terms of
vector and pseudovector terms The charge
in-volved in this interaction is called the coupling
constant, and the pseudovector coupling
con-stant is g a
gadolinium An element with atomic number
(nuclear charge) 64 and atomic weight 157.25
The element has seven stable isotopes
Gadolin-ium has the highest thermal neutron
cross-section of the known elements
gage pressure The pressure relative to
atmo-spheric pressure The gage pressure is related
to the absolute pressure by
Pgage = Pabs− Patm The gage pressure is negative whenever the ab-
solute pressure is less than the atmospheric
pres-sure; it is then called a vacuum
gain Growth rate of the number of photons in
a laser cavity Gain in a medium occurs when the
rate of stimulated emission of radiation is larger
than the rate of absorption, which requires that
population inversion must be achieved For a
laser action, gain must be larger than the losses
in the cavity
gain coefficient Provides a measure of the
growth rate of intensity as a function of distance
in a laser gain medium It is proportional to the
population inversion and given by
.
Here, A is Einstein’s A coefficient, λ is the length, g1and g2are degeneracies of lower and
wave-excited states, N1 and N2 are the number of
atoms in the lower and excited states, ν and ν o are field and atomic frequency, and δν o is theLorentzian line width
gain factor, photoconductivity The increase
of the electrical conductivity due to illumination
gain saturation Gain of a lasing mediumdecreases with an increase in photon flux in
the cavity, resulting in gain saturation. Forvery large photon flux, gain approaches zero
Gain saturation restricts the maximum output
power of a laser In a homogeneously broadened
medium, gain saturation causes power
broaden-ing which is given by
g(ν)= g o (ν)
1+ I (ν)/Isat(ν) ,
where g o (ν) is unsaturated gain, I (ν) is the ton flux, and Isat(ν)is the saturated photon flux
pho-at frequency ν. In a homogeneously
broad-ened medium, gain saturation causes spatial
hole burning, and in an inhomogeneously
broad-ened medium, gain saturation causes spectral
hole burning
gain switching A technique used for ating high power laser pulses of very short dura-tion Using a fast pumping pulse, the inversion
gener-is ragener-ised rapidly to a value high above threshold.The rapid increase in gain does not allow pho-tons to build up inside the cavity and, therefore,depletion is negligible Build-up of large gainresults in a short pulse of high power Depend-ing on the duration of the pump pulse, a laserpulse of about a nanosecond can be achieved
Gain switching can be achieved in any laser.
Typical gain-switched lasers are diode lasers,and dye lasers
gallium Element with atomic number clear charge) 31 and atomic weight 69.72 Theelement has three stable isotopes In the form
Trang 6(nu-of gallium arsenide it is used in solid state lasers
and fast switching diodes
galvanometric effects Transformation of
electrical current into mechanical motion
gamma decay An excited nucleus can lose
energy through the radiation of electromagnetic
energy, or gamma rays The energy of these
photons is the difference between the initial and
final energy levels in the nucleus
gamma-matrices The Dirac equation for
massive spin half-particles is usually written in
terms of the four γ -matrices, γ µ (µ = 0, ,
3), as
γ µ p µ − m (p)= 0
where p µ is the particle four-momentum, m is
the mass, and (p) is the momentum-space
wave function The γ -matrices are defined as
where 1 is the unit 2 × 2 matrix, and σ i is the
well-known Pauli spin matrix
gamma ray A quantum of electromagnetic
energy emitted from an excited nucleus as it
de-cays electromagnetically A gamma ray is a
photon, but is differentiated from photons in that
the source of gamma rays is the atomic nucleus.
gamma ray microscope A gedanken
micro-scope first proposed by Heisenberg to measure
the position of a particle Consider the following
schematic diagram:
M is the microscope, L is a lens, and P is the
particle positioned along the x-axis The
parti-cle is irradiated with gamma rays of wavelength
λ The microscope can only resolve the particle
position x to precision x given by x= λ
sin a ,where a is the half-angle subtended by the lens
A particle entering the microscope imparts a
re-coil momentum to the particle with uncertainty
in the x-direction p x , given by p x = h
λ sin
(a), where h is Planck’s constant When
com-bining, we obtain p x x ≈ h, which is
con-sistent with Heisenberg’s uncertainty relation
Schematic of gamma ray microscope.
Gamow factor In the alpha decay of heavy
elements, the alpha particle must penetrate aCouloumb barrier The approximate transpar-ency for s-waves through a very high (or thick)
barrier is called the Gamow factor, G It is
writ-ten as:
G ≈ e −π(2Zz/137β)
Gamow–Teller selection rules for beta decay
In the process of beta decay of a nucleus in whichthe nucleus emits a beta particle and a neutrinowith their spins parallel, the selection rules for
the emission process are known as the Gamow– Teller selection rules In terms of changes in the
angular momentum quantum numbers in units
of¯h (I) of the nuclear state due to beta decay,
for allowed transitions:
He nucleus with charge 2e) plus a daughter
nu-cleus Gamow assumed that alpha particles ist for a short time before emission inside thenucleus He further assumed that the potential
ex-energy V (r) of the alpha particle is such that it
is negative and constant in the nucleus of radius
R (r < R) and falls off as
V (r)= 2Z1e2
( 0) r, r > R
Z1eis the charge carried by the daughter nucleus
0is the permittivity of free space
Trang 7gas State (or phase) of matter in which the
molecules are relatively far apart (spacing is of
an order of magnitude larger than the
molecu-lar diameter) and are practically unrestricted by
intermolecular forces Consequently, a gas can
easily change its volume and shape This is in
contrast to solids, where both volume and shape
are maintained In the solid state, the molecules
are relatively close (spacing is of the same order
of magnitude as a molecular diameter) and are
subject to large intermolecular forces
gas constant (R) The constant of
propor-tionality R, in the ideal gas law, P V = nRT ,
where P , V , and T denote the pressure,
vol-ume, and absolute temperature of n moles of an
ideal gas The value of R = 8.31 J/(mol.K) is
a universal constant, and is equal to the product
of the Boltzmann constant k B and the Avogadro
number See ideal gas law
gas dynamics The study of compressible
flows, since compressible effects are more
im-portant in gas flow
gaseous diffusion The name given to a
pro-cess that is used to increase the percentage of the
uranium isotope 235 to about 3% from the
natu-ral abundance of the uranium isotopes, which
in-clude 234U at 0055%, 235U at 0.72 %, and 238U
at 99.27% Nuclear fuel composed of 3.2% 235U
can be used in the power producing reactors
(light-water reactors) in the United States The
separation occurs based on the very small mass
difference between the isotopes, which results
in a slight difference in the diffusion rates
gas lasers Lasers with gaseous gain medium.
Most gas lasers are excited by electron
colli-sions in various types of gas discharge which
have narrow absorption bands Common gas
lasers are He-Ne, argon, carbon dioxide See
He-Ne lasers
gauge bosons A quantum of a gauge field
gauge field A field which has to be
intro-duced into a theory so that gauge invariance is
preserved at all points in space and time
(lo-cally) For example, consider a charged particle
with wave function which transforms to
under a local gauge transformation
(r,t) = e iq ( r,t) (r, t)
where (r, t) is an arbitrary scalar function and
qis a parameter For the theory to be gauge variant with respect to the above transformation,
in- , and must describe the same physics If
one takes (r, t) as an arbitrary scalar function
such that a transformation of the scalar (φ) and
vector (A) electromagnetic potentials φ −→
φ− ∂
∂t and A−→ A + ∇ leaves the
elec-tric and magnetic fields invariant, then φ and A
are the gauge fields that have to be introduced into the theory for to preserve the above local
gauge transformation
gauge field theories The concept of gaugeinvariance may be generalized to include a the-ory built up by requiring invariance under a set
of local phase transformations These mations can be based on non-Abelian groups.Yang and Mills studied the generalized theory
tion, consider the electric field E(r,t) and the magnetic field B(r,t) They can both be ob- tained from scalar and vector potentials φ(r,t) and A(r,t), respectively, by
E(r,t) = −∇φ(r,t) − ∂
∂t A(r,t)
B(r,t) = ∇ × A(r,t)
The potentials are not completely defined by
the above equations because E and B are tered by the substitutions A−→ A+∇χ, φ −→
unal-φ− ∂
∂t χ , where χ is any scalar This property
of the invariance of E and B under such
trans-formations is known as gauge invariance of the electromagnetic field A particular gauge which
is normally chosen is called the Coulomb gauge,
in which∇ · A =0 It should be noted that in
this gauge,∇ and A commute.
gauge transformation In classical
electro-magnetic theory, a gauge transformation is one
Trang 8that changes the vector and scalar potentials,
leaving the electric and magnetic fields
un-changed This transformation is associated with
conservation of electric charge The
symme-try is introduced in quantum mechanics by
in-troducing a phase change in the wave function,
where the phase change can be global or local
(dependent on position) A gauge
transforma-tion is dependent on the interactransforma-tion of a long
range field and the conservation of a quantity
such as electric charge
gauss A unit of magnetic field It is equal to
10−4 tesla (MKS unit), which results from the
Biot-Savart law given below:
Gaussian beam A very important class of
beam-like solutions of Maxwell’s equation for
an electromagnetic field It retains its functional
form as it propagates in free space The field of
a Gaussian beam propagating in the z-direction
where q(z) is a complex beam parameter given
by the ABCD law for Gaussian beams In the
paraxial approximation, the electric and
mag-netic fields of a Gaussian beam are transverse
to the direction of propagation, and are therefore
denoted by TEMlm mode The electric field of
a TEMlm mode is proportional to
H l
√
2 x w(z)
H m
√
2 y w(z)
+ i(1 + l + m)φ
where H l (x) stands for the Hermite polynomial
of order l with argument x Here, the spot size
w(z), radius of curvature of the spherical
wave-front of the Gaussian beam R(z), and
longitudi-nal phase factor φ are
wavelength, and z s is the Rayleigh range The
lowest order Gaussian mode TEM00 is used inmany applications because of its circular cross-section and Gaussian intensity profile
Gaussian beams are very directional Laser is
an example of a Gaussian beam.
Gaussian error In the limit of large
num-bers, a binomial probability distribution has aGaussian form represented by
P (x)=(2/π )e −(x−x0)2/ 2σ2
where x0 is the mean value of the distribution
and σ represents the 1/e width This distribution
is called the normal distribution, and σ is the Gaussian error.
Gaussian line shape The absorption
spec-trum of light with a Gaussian line shape is seen
in inhomogeneous broadening One of the
mechanisms resulting in Gaussian line shape is
Doppler broadening See inhomogeneous
Trang 9For a Gaussian random process, all higher order
correlations can be expressed in terms of second
Gaussian statistics Statistics of random
vari-ables which can be described by Gaussian
ran-dom processes See Gaussian random
pro-cesses
Gaussian white noise A delta correlated
Gaussian random process with mean zero (η(t)
= 0) and variance η(t)η(t) = δ(t −t), where
δ(t − t) is the Dirac delta function.
Gauss–Markov process A random process
x(t ) which is Gaussian and Makovian It
satis-fies the linear differential equation
dx(t )
dt = A(t)x(t) + B(t)q(t) ,
where q(t ) is Gaussian white noise with zero
mean (q(t) = 0) and delta correlated
vari-ance (q(t)q(t) = δ(t − t); A(t ) and B(t) are
functions of time For time independent
coeffi-cients A(t ) and B(t ), the mean and variance of
the random process decay exponentially, and the
power spectrum is Lorentzian This special case
is known as the Orenstein–Uhlenbeck process
See also Markov process; Gaussian random
pro-cesses
Gauss’s Law Gauss’s law is a combination of
Couloumb’s law giving the force between
elec-tostatic charges, and the law of superposition,
which states that the force law is linearly
addi-tive, so the total force is obtained by adding all
the charges In integral form in MKS units, the
law is
surface
E
electric field The enclosed charge is that
con-tained within the surface integral
Gay-Lussac’s law In 1808, J.L Gay-Lussac
discovered that when two gases combine to form
a third, the volumes are in the ratio of simpleintegers This law helped to confirm the atomicnature of matter
1 Generator of translations in space:
For a wave function (r, t) that satisfies
Schrödinger’s equation and that can be
ex-panded in a Taylor series in r, it demonstrated
that
(r + r0, t ) = e ip·r0/ ¯h (r, t)
where r0 is any constant displacement, ¯h is
Planck’s constant, andp= −i ¯h∇ is the
mo-mentum operator.p/ ¯h is called the generator of
translations in space (r, t) is a point in
space-time
2 Generator of translations in time:
For a wave function (r, t) that satisfies
Schrödinger’s equation, it is shown that
(r, t + t0) = e −iHt0/ ¯h (r, t)
where H is the Hamiltonian, t0 is any constanttime, and ¯h is Planck’s constant - H
¯h is called
the generator of translations in time
GDH (Gerasimov–Drell–Hearn) sum rule
A prediction of the first moment, 1, of the dependent parton distribution function, g1, at
spin-Q2 = 0 It relates the spin-dependent scattering
cross-section of circularly polarized photons onlongitudinally polarized nucleons to the anoma-lous magnetic moment of the nucleon
i.e., for a proton, the magnetic moment is defined
as µ p = (1 + κp )µ B , where µ B is the nuclearmagneton See gyromagnetic ratio
Geiger counter A particle detector which issensitive to the passage of ionizing radiation A
Geiger counter is constructed by inserting a thin
wire along the axis of a cylindrical tube filledwith a mixture of a noble gas (He, Ar, etc.) with
a small amount of a quenching gas When a
Trang 10voltage is applied between the cylinder and the
wire, electrons from the primary ionization of a
passing charged particle are accelerated in the
high electric field near the wire surface These
electrons knock-out other atomic electrons from
the gas causing an avalanche and creating an
electronic signal
Geiger counter.
Geiger, H (the experiment of H Geiger, E.
Marsden, and E Rutherford) In 1906, H.
Geiger, E Marsden, and E Rutherford carried
out a series of experiments on the scattering of
alpha particles by metallic foils of various
thick-nesses They found that most of the alpha
par-ticles are deflected through very small angles
(< 1◦), but some are deflected through large
an-gles These measurements helped to establish
that all the positive charge of an atom is
concen-trated at the center of the atom in the nucleus of
very small dimensions
Geiger–Nuttall law In 1911, Geiger and
Nuttall noticed that the higher the released
en-ergy in α decay, the shorter the half-life
Al-though variations occur, smooth curves can at
least be drawn for nuclei having the same (Z).
The explanation of this rule was an early
achieve-ment of quantum mechanics and nuclear
struc-ture
Gell-Mann, Murry Nobel Prize winner in
1969 who exploited the symmetries of the
known elementary particles to classify them in
a proposed scheme, the eightfold way
Gell-Mann–Nishijima relation Gell-Mann
and Nishijima proposed that in order to account
for the weak decay of the kaon and the lambdaparticles, a quantum number called strangeness,
S, which was conserved in the strong
interac-tions, could be defined This quantum number
is related to the charge, baryon number, and thethird component of isospin by
Q/e = B/s + S/2 + I3 .
Gell-Mann–Okubo mass formula Using
the static quark model, a relation between themasses of the pseudoscalar mesons can be ob-tained This relation is:
4Mk − π = 3η8 The prediction for the mass of η8 is 613 MeV
compared to the known η(550) and η(960) If
one assumes that the physical mass eigenstatesare admixtures of the singlet and octet represen-tations of the pseudoscalar mesons, the mixingangle can be calculated to be
tan2(θ ) = 0.2 ;
from the Gell-Mann–Okubo mass formula
generalized Ohm’s law One of four basicequations of magnetohydrodynamics, which de-scribes the relationship between the time deriva-tive of a current in an MHD fluid and variousforces acting on the current In the limit of
a stationary, inhomogeneous, non-magnetizedplasma, this law reduces to the usual Ohm’s law
See also magnetohydrodynamics
generalized oscillator strength In sing inelastic electron scattering by a one elec-
discus-tron atom, one defines a generalized oscillator strength, F qq, as
F qq()=E q − Eq 2
2Sqq()2
where E q and E qare the energies of the atom
before and after the scattering process is the
magnitude of the wave vector difference−→ =
(k q− kq)between the initial and final states ofthe one-electron atom S qq()is the inelasticform factor defined as
Trang 11where q and q are the electron wave
func-tions before and after the scattering, respectively
The generalized oscillator strength is often
used to obtain the differential scattering
cross-section
generalized Rabi frequency Consider the
interaction of an intense optical field E(t) =
Ee −iωt+ (complex conjugate) with a two-level
atom The field is assumed to be nearly resonant
with the allowed transition between the ground
state|a and the upper state |b of the atom The
solution for the atomic wave function (r, t ) in
the presence of the applied field is given as
(r,t ) = Ca (t )u a (r)e −iωat
+ Cb (t )u b (r)e −iωb t
where u a (r)e −iωa trepresents the wave function
of the atomic ground state,|a, and ub (r)e −iωbt
is the wave function of the excited state,|b The
solutions for C a (t ) and C b (t )are of the form
C a (t ) ∝ e −iλt
C b (t ) ∝ λe −i(λ+)t
where λ is the characteristic frequency λ can
have two possible values λ±given by
where = 2µba E/ ¯h denotes the complex Rabi
frequency, and = ω − ωba represents the
detuning, ω babeing the transition frequency of
the atom; E is the magnitude of the electric field
vector E.
general relativity Special relativity requires
that the velocity of light in a vacuum is constant
in inertial reference (non-accelerating) frames
The extension of special relativity to non-inertial
frames is called general relativity and is based
on the equivalence principle which states that
gravitational mass and inertial mass are
equiva-lent Thus, gravitating bodies change the
struc-ture of space-time so that a gravitational
attrac-tion is explained through the curvature of space
For example, general relativity predicts the vature of light in a gravitational field
cur-generating function Provides a way of culating probability distribution and moments of
cal-a distribution The genercal-ating function for ton counting distribution P (m, T ) is defined as
photons in the counting interval[0 − T ] The
photon counting distribution and the factorial
moments can be obtained from the generating function as
The generating function G(1, T ) gives the
prob-ability of detecting zero photons in the countinginterval[0, T ].
generation It is possible to order quarks andleptons into sets of mutually corresponding par-ticles in a symmetry based on SU(2)× U(1)
These are called generations, and there are three generations now accepted in the so-called stan- dard model The figure below shows the gener- ations, which are coupled by weak decays be-
tween the quarks Experiments counting trino flavors strongly suggest that there are only
neu-three possible generations.
Trang 12In this figure, the family doublets are all
left-handed (left helicity) In principle, there are
also corresponding right-handed singlets which
are sterile (have no interactions) but are usually
ignored The diagram indicates that, moving left
to right, the generations become more massive.
geometric phase For a time-dependent
Hamiltonian, H (t ), the eigenfunctions, n (t ),
and eigenvalues, E n (t ), of the Schrödinger
equation satisfy:
H (t ) n (t ) = En (t ) n (t )
If H (t ) changes gradually, then according to
the adiabatic theorem, a particle starting in an
initial nth eigenstate remains in this state apart
from an additional phase factor which appears
in the state vector,
γ n (t ) is called the geometric phase.
geometric phase of light Due to its
vec-tor nature, an electromagnetic wave, during its
propagation through material medium, may
ac-quire a phase in addition to the dynamic phase
This additional contribution to the phase may
be reflected in the polarization state of the field
This phase, which depends only on the
geom-etry of the path followed is called the
geomet-ric phase This is also called the Panchratnam
phase, as it was first discussed by Panchratnam
(see Panchratnam, S., Generalized theory of
in-terference, and its applications, Proc Ind Acad.
Sci., A44, 247, 1956) This is a special case
of topological phase discussed by Berry (Berry,
M.V., Proc R Soc London, A392, 45, 1984).
geometric probability distribution Also
known as the Bose–Einstein distribution Has
a form
P (m)= n m
(1 + n) m+1,
wheren is the mean of the distribution The
generating function for this distribution is given
pro-of similarities exist namely, geometric,
kine-matic, and dynamic similarities The geometric similarity requires that the shapes of both model
and prototype be the same This requirement issatisfied by making sure that all lengths of themodel and prototype have the same ratio, andthat all corresponding angles are equal
gerade wave function Molecular wave
func-tions (r, t) that are even under a parity change,
i.e.,
(r,t) = (−r,t)
are said to be gerade Note that those that are
odd are ungerade
germanium Element with atomic number 32
and atomic weight 72.59 Germanium has five
stable isotopes It is extensively used in the tronic industry since, doped with other elements,
elec-it is one component of semiconductor devices
It is also used in infrared and gamma photonspectroscopy as a detector
Germer (experiment of Davisson and mer) In this famous experiment, a beam of 54
Ger-eV electrons irradiated a crystal of nickel
nor-mally (θ = 0◦) The angular distribution of the
number of scattered electrons from the crystalwas measured Davisson and Germer found that
the distribution falls from a maximum at θ = 0◦
to a minimum near 35◦, then rises to a peak
near 50◦ The peak could only be explained by
constructive interference of electron waves tered by the regular lattice of the crystal Thiswas one of the most important experiments toconfirm de Broglie’s hypothesis
scat-g-factor The ratio of the number of Bohr
magnetons to the units of h of angular
momen-tum
g-factor, Landé In the interaction of a electron atom with a weak uniform external mag-