medium (mostly rare earth atoms like Nd, Yb, Er, etc.), will also gain importance due to their high power capabilities, compactness, and reli- ability. In the Free-electron laser, in which the radiation given off by accelerated electrons is used, the wavelength range extends further into the VUV as well as the longer wavelengths. laser cooling Is the reduction of the tem- perature of atoms in the gas or bulk phase by means of laser radiation. Most often the cool- ing is associated with a reduction in the speed of the atoms and a narrowing of their velocity distribution. Laser cooling can be performed by irradi- ating the atoms with light red-detuned from the atomic resonance. Each absorption process transfers a momentum kick to the atoms. This is followed by spontaneous emission. The latter has no net-effect since it occurs randomly in a 4π radian. Due to the red-detuning of the laser beam, the atom is more likely to absorb from a laser beam which is counterpropagating with the atom leading to a slowing of the atom. In or- der to keep the decelerating atom on resonance with the laser, the atom or the laser frequencies must be tuned. The former can be achieved with a spatially varying magnetic field ( see Zeeman slower), the latter by sweeping the frequency of the lasers in synchronous with the loss in veloc- ity. See also magneto-optical trap. laser fluctuations Are fluctuations in phase and amplitude of a laser. Intensity and phase fluctuations stem from spontaneous emission. The photons in a laser follow the Poisson statis- tics and scale with the square root of the photon number. The phase undergoes a random walk which is also termed the phase diffusion. Phase locking allows the locking of the phases of two lasers with respect to each other. laserfusion Aprocessinwhichintenselasers are used to implode a pellet containing ther- monuclear fuel. The power delivered by the lasers causes the surface material of the pellet to ablate, which compresses and then heats the material in the center of the pellet to produce nuclear fusion reactions. laser induced fluorescence (LIF) Is an important tool in spectroscopy of atoms and molecules. Afterexcitationofa singletransition from state |a>→|b>with a narrow linewidth laser, the system decays spontaneously to lower levels. The emitted fluorescence is spectrally analyzed. The selective emission of single lev- els facilitates a high degree of simplification in the spectra, which enables us to draw conclu- sions about the transition strengths. The re- quirementof theselective excitations isanarrow linewidthlaser and that the Dopplerlinewidth of the different transitions is smaller than the sep- aration between lines. laser wakefield accelerator Particle accel- erator that uses an intense short pulse of laser light to excite plasma oscillations that are used to accelerate charged particles to high energy. latent heat (L) The heat absorbed or given off from a system undergoing a first order phase transition. It is related to the molar change in entropy of the two phases, s = s I − s II ,by L = Ts. lattice conductivity The contribution of lat- tice vibrations to the thermal conductivity of the crystal. The thermal conductivity K is given by K ∼ 1 3 Cvl, by a simple argument attributed to Debye, where C is the specific heat, v is an av- erage velocity for lattice waves, and l is a mean free path which is proportional to 1/T at high temperature T . Peierls pointed out the impor- tance of three lattice wave processes for which the conservation of k brings in a reciprocal lat- tice vector G (umklapp processes). lattice, crystal lattices Perfect crystals are periodic structures, and it is this periodicity which makes their study easier. A lattice is a mathematical set of points defined by the vec- tors r = n 1 a 1 + n 2 a 2 + n 3 a 3 , where n 1 ,n 2 , and n 3 are integers, and the vectors a 1 , a 2 , and a 3 are linearly independent, but their choice is not unique. A crystal structure results when the atoms are assigned positions in this lattice (such an assignment is denoted by a basis). When one atom is assigned per lattice site, the crystal has a Bravais lattice. The three cubic lattices, simple cubic, body-centered cubic, and face-centered © 2001 by CRC Press LLC cubic, are all Bravais lattices. The physical properties of a crystal, such as the electron den- sity and the potential V(r ) which an electron sees, are periodic functions with the periodicity of its lattice, and it is convenient to describe such properties in terms of a Fourier series. For this purpose, we introduce reciprocal lattice which is spanned by the vectors, G =m 1 b 1 +b 2 +m 3 b 3 , where m 1 ,m 2 , and m 3 are integers, and b 1 = 2πa 2 ×a 3 /v c , b 2 = 2πa 3 ×a 1 /v c , and b 3 = 2πa 1 ×a 2 /v c , where v c is the volume of the unit cell in the direct lattice, namely,a 1 ·a 2 ×a 3 . The volume of the unit cell in reciprocal space is 8π3/v c . Thus, the potentialV(r)can be written as V r = G V G exp iG ·r , V G = 1 v c V r exp −iG ·r d 3 r. where the integration is carried out over the unit cell. lattice gauge theory Gauge field theories performed in discrete space-time intervals, i.e., on a lattice, by means of numerical techniques. See also lattice QCD. lattice QCD Quantum chromodynamics (QCD) is the accepted theory of strong inter- actions. To facilitate theoretical studies within QCD (which is a highly non-linear theory), nu- merical calculations are performed in a discrete space-time, namely on a lattice. lattice vibrations An application of the the- ory of small oscillations in classical mechanics. The potential energy of a crystal is developed as a quadratic function of the atomic displace- ments from their equilibrium positions in the lattice (this is often called the harmonic approx- imation). The kinetic energy is also a quadratic function of the velocities. The periodicity of the crystal requires the atomic displacements to have the wave form exp i k · n − ωt where k is a propagation vector of the wave, ω is its frequency, and n is an abbreviation for the lattice vector n 1 a 1 + n 2 a 2 + n 3 a 3 . This reduces the number of equations from 3Ns (ac- tually 3Ns − 6)to3s equations, where N is the number of unit cells in the crystal and s is the number of atoms in a unit cell; s is one for silver and gold, for example, and two for dia- mond. For a given k , we obtain 3s values of ω, which, when k is varied, give 3s surfaces or branches. To illustrate, consider a linear chain of atoms of mass m at x = 0, ±a,±2a, , and atoms of mass M at x =±a, ±3/2a; coupled withspringsofspring constantsα, weobtaintwo branches: the lower branch is called an acousti- cal branch since ω = ck for small k as in sound waves, and the upper branch is called an optical branch by convention. Notethat k is determined to within 2π/a,orω as a function of k is peri- odic with the period 2π/a, which is a reciprocal lattice vector for this one-dimensional crystal. The interval −π a ≤ k ≤ π a is the Brillouin zone for this crystal. In general for a crystal, we ob- tain three acoustical branches, 3(s − 1) optical branches, and 3Ns harmonic oscillators, which are uncoupled and can be quantized. The spe- cific heat of the crystal is the sum of the spe- cific heats of these oscillators. If we include the potential energy of the crystal cubic terms and atomic displacements, the oscillators will be coupled and lattice waves will scatter each other or break and form other waves. The terms are important in explaining the thermal conduc- tivity of the crystal and the thermal expansion of solids. 2α / m 2α / M 2α (m+M)/mM -π_ a π_ a k ω Lattice vibrations. © 2001 by CRC Press LLC Laue’s condition method In1912, Max von Laue recognized that a crystal can serve as a three-dimensional diffraction grating for X-rays of wavelengths λ of about 1 Å. Electrons in the atoms of the crystal are excited by the electric field of the incident X-rays and radiate X-rays with the same frequency. The wavelets from different atoms combine (interfere) to form the scattered (diffracted) wave. Constructive inter- ference will result if the phase difference be- tween two wavelets from any two atoms is 2πn, where n is an integer. For atoms A and B sep- arated by a vector r in the crystal and with an incident wave vector k and scattered wave vec- tor k (here, 2π/λ = k = k, elastic scattering), we see that the phase difference between B and Aisr ·(k −k ), corresponding to a shorter path by CA + AD. If we assume, for simplicity, a Bravais lattice, where any vector r joining two atoms is given by n 1 a 1 + n 2 a 2 + n 3 a 3 where n 1 ,n 2 , and n 3 are integers and a 1 ,a 2 , and a 3 are three primitive translation vectors, we ob- tain Laue’s conditions: (k − k ) · a 1 = 2π (integer) (k −k ) · a 2 = 2π (integer) (k −k ) · a 3 = 2π (integer) which are equivalent to the statement k − k is a reciprocal lattice vector G . From the triangle, we see that G is perpendicular to the bisector of the angle between k and k , and 2k sin θ = G. k l k l G k k θ θ B C A D Laue’s condition method. The diffractionappearsasreflectionfromthe atomic planes perpendicular to G whose spac- ing d = 2π m/G , which, when substituted for G gives the Bragg condition 2d sinθ = mλ, where m is an integer denoting the order of the reflection. In Laue’s method, a well-collimated X-ray beam containing a range of wavelengths (polychromatic) is incident on a single crystal whose orientation has been chosen. A flat film can receiveeither the reflectedor thetransmitted beam. lawofcorrespondingstates Hypothesispro- posed by Van der Waal that the equation of state expressed in terms of the reduced pressure, tem- perature, and volume (reduced variables defined as the ratio to the value of the variable at the crit- ical point) becomes the same for all substances. This holds true for Van der Waal’s equation of state; real gases do not obey this rule to a high accuracy. law of mass action In a chemical reaction with ideal gases, the condition of equilibrium can be expressed in terms of the law of mass action. Denoting the chemical reaction of the species A j in terms of the stoichiometric coef- ficients ν j as j ν j A j = 0, one can write the equilibrium constant K(T ) as k(T ) = j A j ν j , where [A j ]denotes the concentration of the j th species in the reaction. Note that the stoichio- metric coefficients for reactants and products have opposite signs. law of the wall Variation of velocity in a turbulent boundary layer as given by U/u ∗ = f y + where u ∗ = √ τ o /ρ is the friction velocity and y + = yu ∗ /ν is the dimensionless distance from the wall. The velocity profiles are divided into tworegions, aviscous sublayernear thewall and an outer layer near the free-stream. An overlap layer connects the two. The regions are given by U/u ∗ = y + (viscous sublayer) U/u ∗ = 2.5lny + + 5 (logarithmic layer) . Lawson criterion Attributed to the British physicist J.D. Lawson, this criterion establishes a condition under which a net energy output © 2001 by CRC Press LLC would be possible in fusion. If n is the ion den- sity and τ is the confinement time (namely, the time during which the ions are maintained at a temperature at least equal to the critical ignition temperature), then the Lawson criterion states that nτ> 10 16 s/cm 3 for deuterium–deuterium reactions, and nτ> 10 14 s/cm 3 for deuterium– tritium reactions. LDV Laser-Doppler velocimetry. Optical method of measuring flow velocity at a point through use of a crossed laser beam which forms fringes due to interference. Scattered light from particles passing through the laser intersection is measured by a photodetector and processed to determine the velocity. Le Chatelier’s principle States that the crite- rionforthermodynamicstabilityisthatthespon- taneous processes induced by a deviation from equilibrium must be in a direction to restore the system to equilibrium. left-handed particle A particle whose spin is antiparallel to the direction of its momentum. See handedness. Lehmann representation In the quantum many-particle problem, a standard technique is to use the one-particle Green’s function. The space-time Fourier transform of Green’s func- tion is useful. The related object is the spectral function defined as follows. Consider a large system of interacting particles. Insert a particle with a fixed momentum in this system. The en- ergy spectrum of the obtained system defines the spectral function. The Lehman representation is the expression for the space-time Fourier trans- formation of the one-particle Green’s function in the integral form of the spectral function. Lennard–Jones potential The interaction energy between two atoms, such as inert gas atoms, as a function of r , the distance between them, is given by, U r = 4ε (σ/r) 12 −(σ/r) 6 , where ε and σ are energy and distance param- eters. This potential is used in calculating the cohesive energy of inert gas crystals. lepton A particle which does not interact via the strong interaction. Leptons interact via the weak or electromagnetic interaction. For in- stance, electrons are leptons. leptonic interactions Interactions among leptons. See lepton. lepton number A lepton number equal to +1(−1) is assigned to leptons (antileptons), while a lepton number equal to zero is assigned to all nonleptons. The lepton number, L, is al- ways conserved. That is, reactions or decays that would violate conservation of the lepton number have never been observed. level In the context of nuclear or atomic physics, it usually denotes an energy level, namely, one of the allowed (quantized) values of the energy a quantum system can have. level width The energy of a small quantum system is quantized and is represented as an en- ergy level. In many cases, the system is dynam- ically coupled with a large degree of freedom. Then the energy of the small system spreads. The distribution functionof thisenergy spreadis observed, for example, through an intensity dis- tribution of the emission or absorption of pho- tons. In many cases, the width is defined as the difference between the energies at which the value of the distribution function is one-half its maximum value. lever rule In a first order phase transition such as in a liquid–gas system, the ratio of the mole fraction in the coexisting liquid vs. the gas phase, x l /x g , for a liquid–gas mixture with total volume is v T , is inversely related to the ratio of the difference of the volume v T from the molar volumes of the liquid and gas phases, v l and v g , respectively. Mathematically stated, this gives x l /x g = (v g − v T )/(v T − v l ). Levinson’s theorem In the S-matrix theory of scattering, the angular momentum represen- tation is the most interesting. For the elastic scattering by a potential, the S-matrix is diag- onal in this representation. The eigenvalues of S, the S-matrix, are closely related to the phase © 2001 by CRC Press LLC shifts; S l (k)= exp[2iδ l (k)], where k is the mo- mentum of the incoming particle,l is the angular momentum of its partial wave, and δ l (k) is the phase shift. The Levinson theorem is that δ l (0)−δ l (∞)=[number of the bound states with angular momentum l]π. levitron Toroidal plasma experimental de- vice that includes a current-carrying coil levi- tated within the plasma. lifetime A characteristic time associated with the decay of an unstable system. The law of radioactive decay is N(t)=N(0)e −λt with N(t) symbolizing the number of nuclei present at any time t, N(0) denoting the initial number of nuclei, and λ representing the disin- tegration constant. τ= 1 λ is the lifetime or mean life of the sample. Com- pare with half-life. lift Force perpendicular to the direction of motion generated by pressure differences. The lift can be generated by a symmetric body in- clined at an angle to the flow, from flow about an asymmetric body, or a combination of both. lift coefficient Lift non-dimensionalized by dynamic pressure: C L = L 1 2 ρU 2 A . A lift coefficient is primarily used to determine the lifting capability of a wing and is plotted vs. the attack angle or drag coefficient (drag polar). Lift coefficients for an arbitrary symmetric and cambered wing are shown. lifting line theory Theory for determining the lift of a wing by assuming the lift is created by a number of discrete line vortices. lift-to-drag ratio Measure of the efficiency of a airfoil: L/D= C L C D . Lift coefficient vs. angle of attack. The greater the lift-to-drag ratio(L/D), the bet- ter a wing is at producing lift with minimal drag. light emitting diode (LED) A p–n junction made from a direct gap semiconductor such as GaAs, where the electron gas (in the n region) and the hole gas (in the p region) are degener- ate. When biased in the forward direction (p is connected to the positive terminal and n to the negative terminal), electrons travel to the p side and holes travel to the n side where they re- combine with opposite charge carriers emitting radiation. The transition which occurs is that of an electron from the conduction band filling a hole in the valence band. Such a device is a candidate for a laser. light ion A charged particle obtained from stripping charges from or adding charges to the neutral atom. As opposed to heavy ions, light ions are obtained from lighter atoms. See ion. light quantum See photon. light-water reactor A reactor which uses ordinary water as a moderator, unlike a heavy- water reactor. Compare with heavy-water reac- tor. limiter Material structure used to define the edge of the plasma and to protect the first wall in a magnetic confinement device. See also di- vertor, plasma divertor. Lindemann melting formula Assumes that at the melting temperature of a solid, the root- mean-square of the atomic displacement due to vibrationis afraction ofthe distancebetween the atoms. For the melting temperature T m ,itgives © 2001 by CRC Press LLC the formula T m =Mx 2 m r 2 s kθ 2 /(9 ¯ h 2 ), where M is the mass of the atom, x m is a fraction 0.2 – 0.25, r s is the radius of a sphere assigned to an atom in a crystal,k is Boltzmann’s constant, and θ is the Debye temperature. linear accelerator An accelerator which (through electric fields) accelerates particles (typically protons, electrons, or ions) in a straight line, as opposed to a cyclotron or syn- crotron, where particle trajectories are bent by magnetic fields into circular shapes. linear combination of atomic orbitals (LCAO) For example, let φ(r ) be an s wave function for an atomic level of a single Na atom. For a sodium crystal, we might qualitatively construct from thisφ a trial Bloch wave function k (r) of an energy band corresponding to this atomic level. Let k (r)= n φ r −n exp ik ·n , where k is the wave vector of the Bloch func- tion and n is a direct lattice vector, and calcu- late the energy E(k ) as the expectation value of the single electron Hamiltonian(p 2 /2m), the kinetic energy, plus V(r ) the crystal potential. This LCAO is known as the tight binding ap- proximation in energy band calculations. See pseudopotential. linear response theory (1) Most transport problems and other phenomena such as electric and magnetic properties deal with currents pro- duced by forces, or responses toexcitations: We assume four things. First, we assume a linear system: if R(t) is a response to excitation E(t), thenc 1 R 1 +c 2 R 2 istheresponsetoc 1 E 1 +c 2 E 2 . Second, we assume a stationary medium whose properties are independent of time. If R(t) is the response to E(t), R(t − t 0 ) is the response to E(t − t 0 ).IfG(t) is the response to δ(t), then G(t − t ) is the response to δ(t − t ).If E(t) = exp(−iωt), then R(t) = ∞ −∞ G(t) exp(iωt) dt exp(−iωt) . Third, we assume causality which means that G(t) = 0, for t<0, and R(t) = ∞ 0 G(t) exp(iωt) dt exp(−iωt) = χ(ω)E(ω) where the susceptibility function (χ) is given by χ 1 +iχ 2 =χ(ω) = ∞ 0 G(t) exp(iωt) dt . Finally, we assume that the total response to a finite excitation can be shown to be finite. The above equation shows that χ(ω) is an analytic function of ω in the upper half of the ω plane and leads to the dispersion relations, χ 1 (ω) − χ 1 (∞) = 1 π P ∞ −∞ χ 2 (ω ) ω 2 − ω 2 dω χ 2 (ω) =− 2ω π P ∞ −∞ χ 1 (ω ) − χ 1 (∞) ω 2 − ω 2 dω of which the dielectric constant and the index of refraction are examples. (2) Kubo developed a quantum mechanical linear response theory for transport problems without writing a transport equation. The trans- port coefficients can be obtained from calculat- ing appropriatecorrelation functions forthe sys- tem at thermal equilibrium. For example, the electrical conductivity σ µν (ω) (relating the cur- rent density in the µ-direction due to an electric field in the ν-direction) is given by σ µν (ω) = ∞ 0 e −iωt dt β 0 J ν ( −i ¯ hλ ) J µ (t) dλ, where β is the reciprocal of the thermal energy kT , and the angular brackets denote an average at thermal equilibrium, namely <A>= trace (A exp −βH )/Z, where Z is the trace of the density matrix exp(−βH). line spectrum A spectrum is obtained by analyzing the intensity of the radiation emitted by a source as a function of its wavelength. A line spectrum is observed when a source emits radiation only at specific (discrete) frequencies (or wavelengths). line tying Boundary conditions for perturba- tions of magnetically confinedplasmas in which © 2001 by CRC Press LLC the background magnetic field intersects a con- ducting material wall or a dense gravitation- ally confined plasma (as in the case of solar prominences). Line tying tends to stabilize in- terchange instabilities in plasmas. line vortex See vortex line. Lippmann–Schwinger equation (1)In quantum mechanical problems of potentialscat- tering or interparticle collisions, we start from a very simple system given by the Hamiltonian H o for which all eigenvalues and eigenvectors are known. In most cases H o is the Hamiltonian for all free particlesbut does not includeinterac- tions responsible for collisions. Its eigenvector n is related to eigenvalue E n . The real Hamil- tonian H is taken to be a sum of H o and H I .For largecontinuous systemswhere theenergyspec- trum is continuous, we may safely assume that n is related to n , which has the same energy E n . Then the Lippmann–Schwinger equation gives a formal solution for n as + n = n + ( E n − H o + iε ) −1 + n where + n represents the state of an incoming wave and ε is a positive infinitesimal. A simi- lar equation holds for − n , the state of outgoing wave, by substituting −iε in place of +iε. (2) Equation encountered in the context of quantum scattering theory. Inoperator notation, it reads T = V + VGT where T is the T -matrix (to be solved for) and V is thepotential actingbetween thetwo scattering particles. G is the Green’s function, defined as G = lim →0 1 E −H 0 + i with E representing the energy and H 0 denoting the free-particle Hamiltonian, i.e., the kinetic energy operator. liquid crystals Some organic crystals, when heated, go through one or more phases before they melt into the pure liquid phase. These intermediate phases, known as mesophases or mesomorphic phases are called liquid crystals. Their structure is less regular than a crystal but more regular than a liquid. Their physical and mechanical properties are intermediate between those of crystals and liquids. There are many types of liquid crystals. Nematics have rod-like molecules. They are uniaxial, and the optical axis can be rotated by the walls of a container or an external agent such as an electric field. They canbeswitched electricallyfrom cleartoopaque and are used in image display devices. Smec- tic liquid crystals have many phases. They are soap-like and have a layered structure. Smectic B is almost a crystal, and smectic D is a cubic gel. Hexactic smectic is uniaxial. Cholestics are made from thin layers (one molecule thick). The orientation of the molecules in a layer can change gradually from layer to layer, leading to a helical structure with intriguing optical prop- erties. liquid drop model The simplest kind of col- lective modelfor the nucleus. Typically,nuclear models can be subdivided in two groups: the independent particle models, and the collective models. The former assume that the nucleons move essentially independently of one another in an average potential. In the collective mod- els, the nucleons are strongly coupled to one an- other. The nucleons are treated like molecules in a drop of fluid. They interact strongly and have frequent collisions with one another. The resulting motion canbe compared to the thermal motion of molecules in a liquid drop. liquid metals A fluid of randomly dis- tributed ions with an electron gas glue between them. The thermal and electrical conductivities, though a few times lower than those of the crys- tals, are still high. The electron screening of the interactions is still as effective as in regular crystals. L-mode(lowmode) Plasma confinementob- tained in tokamak experiments with significant auxiliary heating power (such as neutral beam injection or radio frequency heating) and high recyclingor gas puffingof neutrals at theplasma edge. local gauge transformation The transfor- mation ψ = e iQ ψ © 2001 by CRC Press LLC appliedto thewavefunction ψ of aquantum me- chanical system, where is an arbitrary real pa- rameter and Q is an operator associated with the physical observable q, is called a global gauge transformation. Invariance under such a trans- formation implies conservation of the quantity q.If is an arbitrary function of space and time coordinates, (r,t), the transformation above becomes a local gauge transformation. locality The property of depending upon the location in space. localization Local orlocalizedmodeor wave, refers to a damped wave such as a localized lattice vibrational wave which is damped away fromanatom,whichis heavierorlighterthanthe other atoms, an electron wave around a donor or an acceptor in a semiconductor, or an electron wave localized by disorder (Anderson localiza- tion). local thermal equilibrium (LTE) model Model for computing radiation from dense plas- mas in which it is assumed that the population of electrons in bound levels (such as the elec- trons still attached to impurity ions) follows the Boltzmann distribution. Londons’ equations Hans and Fritz London obtained the following two equations for super- conductivity: ∂J s ∂t = E A =−cJ , where J s is the supercurrent density, E is the electric field, = m/(n s e 2 ), c is the speed of light, e is the charge, m is the mass of the carrier of supercurrent, n s is the density of the carriers, and A is the vector potential with ∇· A = 0. The above equations, together with the Maxwell equations, show that the magnetic fields and currents penetrate a superconductor only to distances of around λ L , where λ 2 L = c 2 /4π. longitudinal polarization A particle is said to be longitudinally polarized when the direc- tion ofits spin isparallel to thedirection of prop- agation. longitudinalwave Waveinaplasmainwhich the oscillating electric field is partially or totally parallel to the wave number (the direction of wave propagation). Examples include electron plasma oscillations and sound waves. long wavelength limit This term describes the situation where the wavelength of the elec- tromagnetic radiation is much larger than the nuclear dimensions. This is a valid assumption up to several MeV and therefore applies to most nuclear γ -rays. Lorentz force Force acting on a charged particle moving through a magnetic field. The Lorentz force is given by q v×B, where q is the particle charge, v is the particle velocity, and B is the magnetic field. Lorentz invariance The property of being invariant upon a Lorentz transformation be- tween reference frames. Lorentz ionization The process of ionizing neutral atoms by using the electric field asso- ciated with their motion through a background magnetic field. Lorentz–Lorenz formula The formula 4π Nα/3 = (n 2 −1)/(n 2 +2), where N isthe num- ber of molecules (atoms) per unit of volume, α is the molecularpolarizability, and n is the index of refraction, was discovered independently by H.A. Lorentz and L. Lorenz in 1880. A formula which replaces n 2 by ε, the dielectric constant, is known as the Clausius–Mossotti relation. For the field polarizing, the formula uses a molecule of the local fieldwhich is E , the external applied field, plus 4πP /3, where P is the polarization which is the electric dipole moment per unit of volume. Lorentz model (Lorentz gas approximation) Kinetic theory model for the collisions of charged particles off cold charged particles with infinite masses. Lorentz scalar Term used in the context of special relativity. It is the scalar product be- tween two four-dimensional vectors. Namely, A · B = A µ B µ © 2001 by CRC Press LLC with µ= 1, ,4. A Lorentz scalar is invari- ant under Lorentz transformations. See Lorentz transformations. Lorentz transformations Relativistically valid transformations between inertial obser- vers. They reduce to the Galilean transforma- tions in the non-relativistic limit. For instance, if a primed system moves with speedv along the xx axes, the Lorentz transformations between the space-time coordinates of a point are x =γ(x−vt) y =y z =z t =γ t− v c 2 x with γ= 1 1 −(v 2 /c 2 ) . Lorenz number (L) The ratio K/(σT), where K is the electron thermal conductivity, σ is the electrical conductivity, and T is the abso- lutetemperature. Formetals, thisnumberisL= K/(σT)=(π 2 /3)(k/e) 2 = 2.7×10 −13 e. s. u, which is an expression of the Wiedemann–Franz law of 1853. For semiconductors and nondegen- erate electron gases, where the relaxation time varies as v p where v is the speed of the elec- tron, L= 1/2(p+ 5)(k/e) 2 . This remarkable result depends on the existence of an isotropic relaxation time. loss coefficient Dimensionless coefficient of the head or pressure loss in a piping system, K= h U 2 /2g = p 1 2 ρU 2 where h and p are the measured head loss and pressure drop, respectively. Values of K are generally determined experimentally for turbu- lent flow conditions for various pipe types and sizes. loss cone Region in velocity space of the plasma in a magnetic mirror device in which the charged particles have so much velocity parallel to the magnetic field that they pass through the magnetic mirror. loss, minor Any loss in a pipe or piping sys- tem not due to purely frictional effects of the wall, including pipe entrances and exits, sud- den and gradual expansions and contractions, valves, and bends and tees. Values of the loss coefficient K for each loss must be determined experimentally. low energy electron diffraction (LEED) A slow electron whose kinetic energy isV electron voltshasadeBrogliewavelengthλwhichequals (12.26/ √ V)Å. Thus, electrons in the energy range 5–500eV have wavelengths in the range of 6 to 1/2 Å which is comparable to the dis- tances between the atoms in crystals. However, such electrons, unlike X-rays and slow neutrons, penetrate only a few angstroms in a crystal, and therefore are not suited for obtaining diffrac- tion patterns from crystals. They are, however, highlysuited tostudy crystalsurfaces bydiffrac- tion methods. Assume that the atoms on the surface have a two-dimensional lattice whose primitive translation vectors are a 1 and a 2 (see lattice, crystal lattices). If the incident electron wave vector is k (usually normal to the surface) and k is the scattered wave vector, then we have only two Laue conditions for constructive inter- ference: (k −k ) ·a 1 = 2π (integer) and (k −k) ·a 2 = 2π (integer). This means that k − k is an integral combination of the reciprocal lattice vectors b 1 and b 2 , but has an arbitrary compo- nent in the b 3 direction, and the diffraction pat- tern consistsof linesor rods. Notethat |k |=|k| aswedealwith elasticscattering. LEEDisnowa highly developed technique which reveals many unusual features of surfaces. Electron diffraction is also carried out using medium energy electrons (500 eV –5keV ), MEED, and high energy electrons (5 keV – 500 keV ), HEED. lower hybrid frequency Frequency of a lon- gitudinalplasmaionoscillation propagatingper- pendicular to the background magnetic field. The lower hybrid frequency is intermediate to the high frequency of the electron extraordinary waveand thelowfrequencyof themagnetosonic wave. At a sufficiently high plasma density, the lower hybrid frequency is approximately the square root of the ion cyclotron frequency times the electron cyclotron frequency. © 2001 by CRC Press LLC lowerhybridresonantheating Plasma heat- ing by lower hybrid waves in which power is absorbed at the lowerhybridresonantfrequency for sufficiently high density plasmas or by Lan- dau damping at lower densities. LS coupling A possible coupling scheme for spins and angular momenta of the individ- ual nucleons in a nucleus, alternative to the j –j scheme. In the LS scheme, the orbital angular momenta of all nucleons are added together to provide the total orbital angular momentum L. The same is done with the intrinsic spins, which yields the total nuclear spin S. L and S are then coupled with each other to give the total angular momentum of the nucleus. lubricationtheory Hydrodynamictheory re- lating the motion oftwo solid surfaces separated by a liquid interface, where the relative motion of the solid surfaces generates an excess pres- sure in the fluid layer. This pressure allows the fluid to support a load force. In hydrostatic lu- brication, the excesspressure is maintained with an external pressure source. luminescence An excitation of a system resulting in light emission which does not include black body radiation. The excitation can be due to photons, cathode rays (electrons), electric field, chemical reactions, heat, or sound waves, for example, and the luminescence which results is called photoluminescence, cathodoluminescence, electroluminescence, chemiluminescence, thermoluminescence, and sonoluminescence respectively. Luminescence occurs in gases, liquids, and solids. The radia- tive transitions causing luminescence are simple in gases and are given by atomic spectroscopy, but they are more complex in liquids and solids dueto thestrong interactionsbetween theatoms. Luminescent solids, such as ZnS and CdS, are known as phosphors. They contain impurity ac- tivators such as Ag and Cu which act as lumi- nescent centers. They usually absorb ultraviolet light and emit light in the visible range with an efficiency of about 1/2 (one photon emitted for two absorbed). Fluorescence and phosphorescence are two forms of luminescence. After the excitation ceases, fluorescence decays exponentially with a time constant independent of temperature, but phosphorescence (afterglow) persists, and the decay is temperature-dependent. luminosity Term used within the context of accelerator physics in conjunction with the op- erational costs of the accelerator. The rate at which a reaction takes place is written as R = lσ where l is the luminosity and σ is the cross- section. The luminosity is a characteristic of the particular accelerator and its working condi- tions. It can be determined by calibration using a known cross-section. Lundquist number A dimensionless plasma parameter equal to the Alfvén speed times a characteristic scalelengthdividedbythe plasma resistivity. Lyddane–Sachs–Teller relation For a cu- bic polar crystal with two atoms/unit cell, the relation ω 2 L /ω 2 T = ε(0)/ε(∞) is known as the Lyddane–Sachs–Teller relation. Here, ω L and ω T are the longitudinal and transverse optical (angular) frequencies, ε(0) is the static dielec- tric constant, and ε(∞) is the dielectric constant at optical frequencies. © 2001 by CRC Press LLC [...]... Method of heating plasmas by oscillating the magnetic field strength During this process, there is a net gain of plasma energy because the changing magnetic field causes parallel and perpendicular components of charged particle kinetic energy to change at different rates, while particle collisions transfer energy between these components these energy levels The resonance frequency depends on the strength of. .. the square of it energy minus the square of its three-momentum, which is also equal to the square of the rest mass of the particle: 2 pµ = E 2 −p2 c2 = m2 c4 A particle which satisfies this condition is said to be on mass shell Virtual particles do not necessarily satisfy this condition mass tensor Every energy band has critical points where ∇E(k) = 0 At these points, such as k o , we can expand E(k)... however, will decrease due to the overall broadening of the distribution (2) A description of the distribution of speeds present in a collection of gas molecules A combination of the statistical arguments proposed by James Clerk Maxwell ( 183 1– 187 9) and Ludwig Boltzmann ( 184 4–1906) The probability that the speed of any given gas molecule lies in the range of c to c + dc is given by fc (c) dc = m 2πkB T 3/2... (T), and volume (V) The macrostate is defined by the probabilities of its constituent microstates Madelung constant A constant which is introduced in calculating the electrostatic energy of ionic crystals Consider a crystal of unit cells containing two ions of opposite charge per cell Denote the positions of the ions by the vectors r i and their charges by qi , where qi = ±q The electrostatic energy of. .. anti-bunching Mandelstam variables In a two-body reaction where the incoming particles have four momenta, p1 and p2 , and the outgoing particles have four momenta, p3 and p4 , the three Lorentz-invariant Mandelstam variables are defined by s = ( p1 + p2 )2 t = ( p1 − p3 )2 , and u = ( p1 − p4 )2 These variables satisfy the condition s + t + u = m2 +m2 + m2 + m2 , 1 2 3 4 where m1 , and m2 are the rest masses of. .. principle of conservation of energy must not be violated, the intensity in the other exit of the interferometer (i.e., the path back to the light source) is always 180 ◦ out of phase with the other arm The Michelson interferometer was first set up by the American physicist A.A Michelson in order to measure the diameter of distant stars In the history of physics, the Michelson interferometer is of utmost... electric and magnetic quantities The consequences for quantum mechanics of the existence of magnetic charges were studied by P.A.M Dirac The existence of magnetic charges arising from non-Abelian gauge theories was investigated by A.M Polyakov and G.t’Hooft magnetic multipole radiation Radiation due to higher terms in the series expansion of the interaction operator between an atomic system and electromagnetic... borderline between stability and instability Markov process A statistical process without any memory, i.e., each instant is independent of the past evolution of the process An example of such a process is the random walk maser Acronym that stands for microwave amplification by stimulated emission of radiation The principle of operation is very similar to the laser except that instead of light, microwave radiation... magnetic energy is usually required to be a maximum for the amount of magnetic material used magnetic energy density Contribution to the local energy density that is proportional to the magnetic field strength squared magnetic field energy The work done in establishing the presence of a magnetic field If we consider that the magnetic field is a consequence of circulating currents, then the magnetic field energy. .. light wave consisting of matter According to quantum mechanics, particles can also exhibit wave characteristics The wavelength is given by the deBroglie wavelength Matter waves are the basis of atom optics, in which the physics of matter waves analogous to the laws of optics are explored maximum and minimum thermometer A device that is designed to measure and record both the minimum and maximum temperatures . reduction of the tem- perature of atoms in the gas or bulk phase by means of laser radiation. Most often the cool- ing is associated with a reduction in the speed of the atoms and a narrowing of their. equilibrium. left-handed particle A particle whose spin is antiparallel to the direction of its momentum. See handedness. Lehmann representation In the quantum many-particle problem, a standard technique. conservation of the lepton number have never been observed. level In the context of nuclear or atomic physics, it usually denotes an energy level, namely, one of the allowed (quantized) values of the energy