ated 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 ω 1 and ω 2 . 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- tainsystemsthataresimplefractionsofthenum- 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 atomic mass number 223 has the longest half-life of 22 minutes. Because francium is the heaviest known element which chemically acts as a one- electron atom, interest has developed in using it to increase the sensitivity of atomic parity ex- periments. Franck–Condon factors The overlap inte- grals of vibrational wave functions of differ- ent electronic states. According to the Born– Oppenheimer approximation, a molecular wave functioncanbewrittenastheproductoftheelec- tronic wave function e , the vibrational wave function v , the rotational wave function r , and the nuclear wave function N . In the case ofanelectronictransition,thetransitionmoment is given by the integral M = |q ˆr| 2 = e |q ˆr| e 2 v | v 2 r | r 2 N | N 2 , where | v | v | 2 is called the Franck–Condon factor. It constitutes the overlap integral of the vibrational wave functions in the initial and fi- nal states. Since the dipole transition moment is carried by the electronic transition, no se- lection rules apply between the initial and fi- nal vibrational states. However, only electronic transitions from one particular vibrational state of an electronic state to another electronic state with a different vibrational state can be excited, for these, the Franck–Condon factor is large. Classically, this reflects the fact that the nu- clei move much slower than the electrons, and, consequently, electronic transitions are favored where the kinetic energy of the nuclei does not change. Quantum mechanically, this results in large overlap functions | v | v | 2 . The factors | r | r | 2 are called Hoenl- London factors. Franck–Condon principle A classical prin- ciple which reflects the fact that electronic tran- sitions in molecules occur at points on the molecular energy surface where the kinetic en- © 2001 by CRC Press LLC 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- tenceof sharp atomicenergylevelsandprovided 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 iscalled a Franck–Readsourceand 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. Bydetectingcoincidencecounts 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 is equivalent 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 = I 0 sin 2 δ δ 2 with δ = πd sin α λ , where α is the angle from the optical axis and λ is the wavelength of the light. free electrons Electrons detached from an atom. free energy An energy quantity assigned to each substance, such that a reaction in a system held 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 con- tainer to fill the entire container by opening a valve separating the two compartments. In this process, no heat flows into the system since it is thermally isolated, and no work is done. Thus, conservation of energy requires that the inter- nal energy of the system remain unchanged. If the gas is ideal, there will be no temperature change; however, for a real gas, the temperature decreases in a free expansion. free induction decay Term originally coined in the area of nuclear magnetic resonance to de- scribe the decay of the induction signal of a macroscopic sample of matter containing nu- clear spins which are initially tipped over from their equilibrium orientation and then undergo free precession. The term reflects the decay of the signalin asmall fractionof thespins’ natural lifetimes, which occurs due to inhomogeneities in the magnetic field in the sample. The term is now used in all other types of magnetic reso- nance, as well as in the area of resonance optics, i.e., the interaction of atoms with coherent light tuned to an atomic transition. © 2001 by CRC Press LLC 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 range FSR, FSR = c L ring resonator FSR = c 2L linear resonator FSR = c 4L confocal resonator . The value for the free spectral range is a conse- 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 tan- 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 not enclosing the origin is zero. The flow is thus irrotational. 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 ad≈λ. friction coefficient The ratio of the force re- quired to move one solid surface over another surface to the total force pressing two surfaces together. friction drag Also called viscous drag since it is generated by the shear stresses. Thefriction 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 parameter that is related to the pressure required to move a fluid at a certain rate. It is generally a func- tion ofthe Reynolds number,surface roughness, and body geometry. In a pipe or duct, the rela- tion between the friction factor, flow velocity, pressure drop, and geometry is given by p = ρV 2 2 f L D . This relation can be written also as h L = f L D V 2 2g where f is the Darcy–Weisbach factor. For a Mach number, Ma, less than one, the friction © 2001 by CRC Press LLC factor 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 ∗ = ( τ w /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 Froudenumber isimportantinflowswherethere 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 plas- mas, the magnetic field lines are tied to the plas- masothattheymoveandoscillatetogether. 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 or excitations 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 the squares of the Fermi and Gammow–Teller ma- trix elements. In forbidden transitions, it does not give a measure of these matrix elements, but does indicate the order of forbiddenness of the decay. Since ft-value has a large range, it is usually quoted in terms of a log 10 . fully developed flow Beyond the entrance region of the flow into a pipe or a duct, the mean flow properties do not change with downstream distance. The velocity profile is fully developed andtheflowiscalledafullydevelopedflow. The entrance length, beyond which the flow is fully developed, varies between 40 to 100 diameters along the pipe and is dependent on the Reynolds number. See also entrance region. fundamental vectors Vectors thatcan define the atomic arrangement in an entire lattice by translation. fusion Whentwonuclei coalesce intoalarger 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 re- leaseofa neutron andenergy. Thelatterreaction is the fundamental reaction in a hydrogen bomb. © 2001 by CRC Press LLC G g (2) (τ) See intensity correlation function. g a The weak interaction can be described in terms of a leptonic current interacting with a hadronic current. In general, these currents couldconsistoffiveforms—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 Anelement with atomic number (nuclear charge) 64 and atomic weight 157.25. Theelement hassevenstableisotopes. 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 P gage = P abs − P atm . The gage pressure is negative whenever the ab- solute pressure isless thanthe atmosphericpres- sure; it is then called a vacuum. gain Growth rate of the number of photonsin alasercavity. Gaininamediumoccurswhenthe 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 g(ν) = λ 2 A 8π N 2 − g 2 g 1 N 1 S(ν) , where S(ν) is the Lorentzian line shape S(ν) = 1 π δν o ( ν −ν o ) 2 + ( δν o ) 2 . Here, A is Einstein’sAcoefficient, λ is the wave- length, g 1 and g 2 are degeneracies of lower and excited states, N 1 and N 2 are the number of atoms in the lower and excited states, ν and ν o are field and atomic frequency, and δν o is the Lorentzian line width. gainfactor, photoconductivity The increase oftheelectricalconductivityduetoillumination. gain saturation Gain of a lasing medium decreases with an increase in photon flux in the cavity, resulting in gain saturation. For very large photon flux, gain approaches zero. Gain saturation restricts the maximum output powerofalaser. In ahomogeneouslybroadened medium, gainsaturation causes powerbroaden- ing which is given by g(ν) = g o (ν) 1 + I (ν)/I sat (ν) , where g o (ν) is unsaturatedgain, I(ν)isthe pho- ton flux, and I sat (ν) is the saturated photon flux at frequency ν. In a homogeneously broad- ened medium, gain saturation causes spatial holeburning, and in aninhomogeneouslybroad- ened medium, gain saturation causes spectral hole burning. gain switching A technique used for gener- ating high power laser pulses of very short dura- tion. Using a fast pumping pulse, the inversion is raised 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 gain results in a short pulse of high power. Depend- ing on the duration of the pump pulse, a laser pulse 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 (nu- clear charge) 31 and atomic weight 69.72. The element has three stable isotopes. In the form © 2001 by CRC Press LLC 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 γ 0 = 0 1 1 0 ,γ i = 0 −σ i σ i 0 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 a Couloumb 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 Intheprocessofbetadecayofanucleusinwhich the nucleus emits a beta particle and a neutrino with 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: I =±1, or 0, no change of parity . Gamow–Teller transition See Fermi transi- tion. Gamow theory (alpha decay) Atheory pro- posed in 1928 by G. Gamow, and independently by R.W. Gurney and E.U. Condon, to describe the decay of a nucleus into an alpha particle (an He nucleus with charge 2e) plus a daughter nu- cleus. Gamow assumed that alpha particles ex- ist for a short time before emission inside the nucleus. He further assumed that the potential 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)= 2Z 1 e 2 ( 4π 0 ) r ,r>R. Z 1 e isthechargecarriedby the daughter nucleus and 0 is the permittivity of free space. © 2001 by CRC Press LLC gas 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, PV=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 234 U at .0055%, 2 35U at 0.72 %, and 2 38U at 99.27%. Nuclear fuel composed of 3.2% 235 U 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 q is a parameter. For the theory to be gauge in- variantwith respectto theabovetransformation, , 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 gauge invariance may be generalized to include a the- ory built up by requiring invariance under a set of local phase transformations. These transfor- mations can be based on non-Abelian groups. Yang and Mills studied the generalized theory of these fields in 1954. gauge invariance (of the electromagnetic field) In describing the quantum interaction of an electron with an electromagnetic field, one often chooses a specific gauge to perform cal- culations. For an example of a gauge condi- 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 unal- teredbythesubstitutionsA −→ A+∇χ ,φ −→ 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 normallychosen iscalled theCoulomb 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 © 2001 by CRC Press LLC that 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 interaction 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: B= µ 0 4π I × r r 2 dl. 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 is proportional to exp −ik z+ x 2 +y 2 2q(z) 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 TEM lm mode. The electric field of a TEM lm mode is proportional to H l √ 2 x w(z) H m √ 2 y w(z) exp − (x 2 +y 2 ) w(z) 2 exp −i k(x 2 +y 2 ) 2R(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 w 2 (z)=w 2 0 [ 1 + ( z/z s ) ] R(z)=z 1 + ( z s /z ) 2 , φ(z)= tan −1 ( z/z s ) , z s =πw 2 0 /λ. The beam waistw 0 is spot-size atz= 0, λ is the wavelength, and z s is the Rayleigh range. The lowest order Gaussian mode TEM 00 is used in many applications because of its circular cross- section and Gaussian intensity profile exp − 2 x 2 +y 2 w(z) 2 . The intensity profile of a higher order gaussian mode is obtained by squaring the electric field. 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 a Gaussian form represented by P(x)= (2/π)e −(x−x 0 ) 2 /2σ 2 . where x 0 is the mean value of the distribution andσ representsthe1/e width. Thisdistribution 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 Dopplerbroadening. Seeinhomogeneousbroad- ening. Gaussian (probability) distribution Also called normal distribution. It is a probability distribution of a continuous variable x(t) of the form 1 σ √ 2π exp −(x −x o ) 2 2σ 2 , where x o is the mean and σ is the standard de- viation. σ 2 is called variance. Gaussian random processes Involves the Gaussian probability distribution, which is de- termined by two parameters, meanand variance. © 2001 by CRC Press LLC For a Gaussian random process, all higher order correlations can be expressed in terms of second order correlations x ( t 1 ) x ( t 2 ) ···x(t n ) = all possible pairs x ( t 1 ) x ( t 2 ) ··· x ( t n−1 ) x(t n ) . 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 Gaussianrandomprocesswithmeanzero(η(t) = 0) and varianceη(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•dA = Enclosed Charge /. Here, is the dielectric constant and E is the 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 simple integers. This law helped to confirm the atomic nature 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 + r 0 ,t)=e ip·r 0 / ¯ h (r,t) where r 0 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+t 0 ) =e −iHt 0 / ¯ h (r,t) where H is the Hamiltonian, t 0 is any constant time, 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 spin- dependent parton distribution function, g 1 ,at Q 2 = 0. It relates the spin-dependent scattering cross-section of circularly polarized photons on longitudinally polarized nucleons to the anoma- lous magnetic moment of the nucleon. lim Q 2 →0 M 2 Q 2 1 = lim Q 2 →0 2M 2 Q 2 1 0 g 1 (x, Q 2 )dx =−κ 2 N /4 . Here, κ istheanomalousmomentofthenucleon, i.e., foraproton, the magneticmomentisdefined as µ p = (1 + κ p )µ B , where µ B is the nuclear magneton. See gyromagnetic ratio. Geiger counter A particle detector which is sensitive to the passage of ionizing radiation. A Geiger counter is constructedby inserting a thin wire along the axis of a cylindrical tube filled with a mixture of a noble gas (He, Ar, etc.) with a small amount of a quenching gas. When a © 2001 by CRC Press LLC voltage 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). Theexplanationofthisrulewasanearlyachieve- 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 lambda particles, 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 the third component of isospin by Q/e=B/s+S/2 +I 3 . Gell-Mann–Okubo mass formula Using the static quark model, a relation between the masses of the pseudoscalar mesons can be ob- tained. This relation is: 4M k −π= 3η 8 . The prediction for the mass of η 8 is 613 MeV compared to the known η(550) and η (960).If one assumes that the physical mass eigenstates are admixtures of the singlet and octet represen- tations of the pseudoscalar mesons, the mixing angle can be calculated to be tan 2 (θ)= 0.2 ; from the Gell-Mann–Okubo mass formula. generalized Ohm’s law One of four basic equationsof magnetohydrodynamics,which de- scribes therelationship between the time deriva- tive of a current in an MHD fluid and various forces acting on the current. In the limit of a stationary, inhomogeneous, non-magnetized plasma, this law reduces to the usual Ohm’s law. See also magnetohydrodynamics. generalized oscillator strength In discus- sing inelastic electron scattering by a one elec- tron atom, one defines a generalized oscillator strength, F qq ,as F qq () = E q − E q 2 2 S qq () 2 where E q and E q are the energies of the atom before and after the scattering process. is the magnitude of the wave vector difference −→ = (k q −k q ) between the initial and final states of the one-electron atom. S qq () is the inelastic form factor defined as S qq () = ∗ q e i −→ ·r q dr © 2001 by CRC Press LLC [...]... polycrystalline solids, limits of crystalline structure graphite An amorphous form of carbon Because of the low neutron cross-section of carbon and the abundance and ease of production, it was originally used as a moderator in nuclear reactors grand canonical distribution Gives the probability of finding a system with Ns particles in a state of energy Es in equilibrium with a temperature and particle reservoir... system is constant and energy is conserved, the system is called a grand canonical ensemble grand partition function For a system at constant temperature T and chemical potential µ, all thermodynamic properties can be obtained from the grand partition function defined as Z(T , µ) = exp ((µNs − Es ) /kB T ) s where Ns is the number of particles in a state of energy Es grand unification Physics attempts... first measurements of the second order intensity correlation function and the results were published in Nature, 177, 27, 1 957 , Proc Roy Soc., 142, 300, 1 957 , and Proc Roy Soc., 143, 241, 1 958 In the original experiment, light with in frequency of 4 35. 8 Hz from a mercury lamp was split into two parts at a beam splitter, and the intensity of each beam was detected by separate detectors One of the detectors... Principle of a Hanbury–Brown–Twiss experiment to measure the photon statistics of a stream of photons and qualitative plot of the second order correlation function g (2) for thermal and non-classical light The characteristic time τc is called the coherence time and is a measure for the temporal coherence of the light source handedness A property of a particle associated with the direction of its spin... manyelectron atom consists of a nucleus of charge Ze (Z being the atomic number of the atom), and N electrons each of charge −e For an infinitely heavy nucleus and considering only the attractive Coulomb interactions between the electrons and the nucleus and the coulomb repulsions between the electrons, we write the Hamiltonian of the N -electron atom (ion) in the absence of external fields as N H =... that this phenomenon has been known for many years, because of the complexities of this phenomenon and difficulties involved in its measurements, many of the details of glow discharge remain unresolved glueball A quantum of energy composed of gluons in a colorless arrangement, representing an excitation of the gluonic field gluon (1) A quantum of the strong interaction There are eight gluons which couple... with the speed of light from the point © 2001 by CRC Press LLC of source and represent a time-dependent distortion of the local space and time coordinates They follow an inverse square law similar to electromagnetic waves The effects of gravitational waves are very small and very difficult to measure Possible detectable events include the collision of astronomical objects and the collapse of a large astronomical... Iliopoulos, and Maiani (GIM) proposed that there must be a charmed quark that decays into a strange and © 2001 by CRC Press LLC for the first generation and c s cos(θc ) − d sin(θc ) for the second Ginsburg–Landau theory of superconductivity Seven years before the BCS (BardeenCooper-Schrieffer) theory of superconductivity was developed, Ginsburg and Landau proposed a phenomenological quantum theory of superconductivity... atomic number 32 and atomic weight 72 .59 Germanium has five stable isotopes It is extensively used in the electronic industry since, doped with other elements, it is one component of semiconductor devices It is also used in infrared and gamma photon spectroscopy as a detector Germer (experiment of Davisson and Germer) In this famous experiment, a beam of 54 eV electrons irradiated a crystal of nickel normally... interfaces beneath the ocean surface and they are referred to as internal gravity waves gray (See dose.) The unit of exposure to ionizing radiation (dose) and equal to the deposition of 6.24 × 1012 Mev/kg of matter (1 joule/kg) scribing the wave function, , of a particle moving in a potential V can be written as =Q ∇ 2 + k2 √ where k ≡ 2mE and Q ≡ 2m V The inh ¯ h2 ¯ tegral form of the Schrödinger equation . the number of particles in a state of energy E s . grand unification Physics attempts to ex- plain natural phenomenon in terms of a set of fundamental axioms. It is the general goal of physics to. wave- length, g 1 and g 2 are degeneracies of lower and excited states, N 1 and N 2 are the number of atoms in the lower and excited states, ν and ν o are field and atomic frequency, and δν o is the Lorentzian. isotope 2 35 to about 3% from the natu- ral abundance of the uranium isotopes, which in- clude 234 U at .0 055 %, 2 35U at 0.72 %, and 2 38U at 99.27%. Nuclear fuel composed of 3.2% 2 35 U can