leading role in the development of modern nu- clear physics. See Copenhagen interpretation. Bohr quantization Rule that determines the allowed electron orbits in Bohr’s theory of the hydrogen atom. In an early atomic theory, Bohr suggested that electrons orbit parent nuclei much like planets orbit the sun. Because elec- trons are electrically charged, classical physics predicts that such a system is unstable due to radiative energy loss. Bohr postulated that elec- trons radiate only if they “jump” between al- lowed prescribed orbits. These orbits are called Bohr orbits. The conditions required for the al- lowed angular momenta, hence orbits, is called Bohr quantization and is given by the formula L=n ¯ h, where L is the allowed value of the an- gular momentum of a circular orbit, n is called the principal quantum number, and ¯ h is the Planck constant divided by 2π. Bohr radius (a 0 ) (1) The radius of the elec- tron in the hydrogen atom in its ground state, as described by the Bohr theory. In Bohr’s early atomic theory, electrons orbit the nucleus on well defined radii, the smallest of which is called the first Bohr radius. Its value is 0.0529 nm. (2) According to the Bohr theory of the atom (see Bohr atom), the radius of the circle in which the electron moves in the ground state of the hydrogen atom, a 0 ≡ ¯ h 2 /m 2 e = 0.5292 Å. A full quantum mechanical treatment of hydrogen gives a 0 as the most probable distance between electrons and the nucleus. Boltzmann constant ( k B ) A fundamental constant which relates the energy scale to the Kelvin scale of temperature, k B = 1.3807 × 10 −23 joules/kelvin. Boltzmann distribution A law of statistical mechanics that states that the probability of find- ing a system at temperature T with an energy E is proportional to e −E/KT , where K is Boltz- mann’s constant. When applied to photons in a cavity with walls at a constant temperature T , the Boltzmann distribution gives Planck’s dis- tribution law of E k = ¯ hck/(e ¯ hck/KT − 1). Boltzmann factor The term, exp(−ε/k B T), that is proportional to the probability of finding a system in a state of energy ε at absolute tem- perature T . Boltzmann’s constant A constant equal to theuniversalgasconstantdividedbyAvogadro’s number. It is approximately equal to 1.38 × 10 −23 J/K and is commonly expressed by the symbol k. Boltzmann statistics Statistics that lead to the Boltzmann distribution. Boltzmann statis- tics assume that particles are distinguishable. Boltzmann transport equation An integro- differential equation used in the classical theory of transport processes to describe the equation of motion of the distribution function f(r, v,t). The number of particles in the infinitesimal vol- ume drdv of the 6-dimensional phase space of Cartesian coordinates r and velocity v is given by f(r, v,t)drdv and obeys the equation ∂f ∂t +α·∇ v f+v·∇ r f= ∂f ∂t coll. . Here,α denotestheacceleration, and(∂f/∂t) coll. is the change in the distribution function due to collisions. The integral character of the equa- tion arises in writing the collision term in terms of two particle collisions. bonding orbital See anti-bonding orbital. bootstrap current Currents driven in tor- oidal devices by neo-classical processes. Bornapproximation Anapproximationuse- ful for calculation of the cross-section in colli- sions of atomic and fundamental particles. The Born approximation is particularly well-suited for estimates of cross-sections at sufficiently large relative collision partner velocities. In po- tential scattering, the Born approximation for the scattering amplitude is given by the ex- pression f(θ)=− 2µ ¯ h 2 q ∞ 0 rsin(qr)V(r)dr, where θ is the observation angle, ¯ h is Planck’s constant divided by 2π, µ is the reduced mass, V(r)is the spherically symmetric potential,q≡ 2ksin(θ/2), and k is the wave number for the collision. See cross-section. Born-Fock theorem See adiabatic theorem. © 2001 by CRC Press LLC Born, Max (1882–1970) German physi- cist. A founding father of the modern quan- tum theory. His name is associated with many applications of the modern quantum theory, such as the Born approximation, the Born- Oppenheimer approximation, etc. Professor Born was awarded the Nobel Prize in physics in 1954. Born-Oppenheimer approximation An ap- proximation scheme for solving the many- few- atom Schrödinger equation. The utility of the approximation follows from the fact that the nu- clei of atoms are much heavier than electrons, and their motion can be decoupled from the elec- tronicmotion. TheBorn-Oppenheimerapproxi- mation is the cornerstone of theoretical quantum chemistry and molecular physics. Born postulate The expression |ψ(x,y,z, t)| 2 dxdydz gives the probability at time t of finding the particle within the infinitesimal re- gion of space lying betweenx anddx, y anddy, and z and dz. |ψ(x,y,z,t)| 2 is then the prob- ability density for finding a particle in various positions in space. Born-Von Karman boundary condition Also called the periodic boundary condition. To one dimensional crystal, it can be expressed as U 1 =U N + 1, where N is the number of parti- cles in the crystal with length L. Bose-Einstein condensation A quantum phenomenon, first predicted and described by Einstein, in which a non-interacting gas of bosons undergoes a phase transformation at crit- ical values of density and temperature. A Bose- Einstein condensate can be considered a macro- scopic system described by a quantum state. Bose-Einstein condensates have recently been observed, about 70 years following Einstein’s prediction, in dilute atomic gases that have been cooled to temperatures only about 10 −9 Kelvin above absolute zero. Bose-Einstein statistics Statistical treatment of an assembled collection of bosons. The dis- tinction between particles whose wave functions are symmetric or antisymmetric leads to differ- ent behavior under a collection of particles (i.e., different statistics). Particles with integral spin are characterized by symmetric wave functions and therefore are not subject to the Pauli exclu- sion principal and obey Bose-Einstein statistics. Bose, S.N. (1894–1974) Indian physicist and mathematician noted for fundamental contribu- tions to statistical quantum physics. His name is associated with the term Bose statistics which describes the statistics obeyed by indistinguish- able particles of integer spin. Such particles are also called bosons. His name is also as- sociated with Bose-Einstein condensation. See Bose-Einstein condensation. Bose statistics Quantum statistics obeyed by a collection of bosons. Bose statistics lead to the Bose-Einstein distribution function and, for crit- ical values of density and temperature, predict the novel quantum phenomenon of the Bose- Einstein condensation. See Bose-Einstein con- densation. boson (1) A particle that has integer spin. A boson can be a fundamental particle, such as a photon, or a composite of other fundamental particles. Atoms are composites of electrons and nuclei; if the nucleus has half integer spin and the total electron spin is also half integral, then the atom as a whole must possess integer spin and can be considered a composite boson. (2) Particles can be divided into two kinds, boson and fermion. The fundamental difference between the two is that the spin quanta number of bosons is integer and that of fermions is half integer. Unlikefermions,whichcan onlybe cre- ated or destroyed in particle-antiparticle pairs, bosons can be created and destroyed singly. bounce frequency The average frequency of oscillation of a particle trapped in a magnetic mirror as it bounces back and forth between its turning points in regions of high magnetic field. boundary layer (1) A thin layer of fluid, ex- isting next to a solid surface beyond which the liquid is moving. Within the layer, the effects of viscosity are significant. The effects of viscos- ity often can be neglected beyond the boundary layer. © 2001 by CRC Press LLC (2) The transition layer between the solid boundary of a body and a moving viscous fluid as required by the no-slip condition. The thick- ness of the boundary layer is usually taken to be the point at which the velocity is equal to 99% of the free-stream velocity. Other measures of boundary layer thickness include the displace- ment thickness and momentum thickness. The boundary layer gives rise to friction drag from viscous forces and can also lead to separation. It also is responsible for the creation of vortic- ity and the diffusion thereof due to viscous ef- fects. Thus, a previously irrotational region will remain so unless it interacts with a boundary layer. This leads to the separation of flows into irrotational portions outside the boundary layer and viscous regions inside the boundary layer. The thickness of a boundary decreases with an increasing Reynolds number, resulting in the ap- proximation of high speed flows as irrotational. A boundary layer can be laminar, but will even- tually transition to turbulence given time. The boundary layer conceptwas introduced by Lud- wig Prandtl in 1904 and led to the development of modern fluid dynamics. See boundary layer approximation. Boundary layer. boundary layer approximation Simplifica- tion of thegoverning equationsof motion within a thin boundary layer. If the boundary layer thicknessis assumedto besmall comparedtothe length of the body, then the variation along the direction of the boundary layer (x) is assumed to be much less than that across the boundary layer (y)or ∂ ∂x ∂ ∂y and ∂ 2 ∂x 2 ∂ 2 ∂y 2 where the velocity in the y-direction (v) is also assumed to be much smaller than the velocity in the x-direction (u), v u. The continuity equation remains ∂u ∂x + ∂v ∂y = 0 while the x- and y-momentum equations reduce to u ∂u ∂x + v ∂u ∂y =− 1 ρ ∂p ∂x + ν ∂ 2 u ∂y 2 and 0 =− ∂p ∂y respectively. This results in a tractable solution with three equations and three unknowns. bound state An eigenstate of distinct energy that a particle occupies when its energy E<V of a potential well that confines it near the force center creating the potential. The discrete en- ergy values are forced on the system by the re- quirement of continuity of the wave function at the boundaries of the potential well, beyond which the wave function must diminish (or van- ish). When E>Veverywhere, the particle is not bound but instead is free to occupy any of an infinite continuum of states. Bourdon tube Classical mechanical device used for measuring pressure utilizing a curved tube with a flattened cross-section. When pres- surized, the tube deflects outward and can be calibrated to a gauge using a mechanical link- age. Bourdon tubes are notable for their high accuracy. Boussinesq approximation Simplification oftheequations ofmotion byassuming thatden- sity changes can be neglected in certain flows due to the compressibility. While the density may vary in the flow, the variation is not due to fluid motion such as occurs in high speed flows. Thus, the continuity equation is simplified to ∇·u = 0 from its normal form. box normalization A common wave func- tion normalization convention. If a particle is contained in a box of unit length L, the wave function isconstrainedto vanishat the boundary © 2001 by CRC Press LLC and requires quantization of momentum. An in- tegral of the probability density|ψ| 2 throughout the box is required to sum to unity and typically leads to a normalization pre-factor for the wave function given by 1/ √ V , whereV is the volume of the box. In most applications, the volume of the box is taken to have the limit as L→∞. Boyle’slaw Anempiricallawforgaseswhich states that at a fixed temperature, the pressure of a gas is inversely proportional to its volume, i.e., pV= constant. This law is strictly valid for a classical ideal gas; real gases obey this to a good approximation at high temperatures and low pressures. bracket, or bra-ket An expression repre- senting the inner (or dot) product of two state vectors, ψ † α ≡<α|β>which yields a simple scalar value. The first and last three letters of the bracket name the notational expression in- volving triangular brackets for the two kinds of state vectors that form the inner product. Bragg diffraction A laboratory method that takes advantage of the wave nature of electro- magneticradiationinordertoprobethestructure of crystalline solids. Also called X-ray diffrac- tion, the method was developed and applied by W.L. Bragg and his father, W.H. Bragg. The pair received the Nobel Prize for physics in 1915. bra vector Defined by the bra-ket formal- ism of Dirac, which allows a concise and easy- to-use terminology for performing operations in Hilbert space. According to the bra-ket formal- ism, aquantumstate, oravectorinHilbertspace, can be described by the ket symbol. For any ket |a>there exists a bra<a|. This is also called a dual correspondence. If <b| is a bra and |a>a ket, then one can define a complex number rep- resented by the symbol <b|a>, whose value is given by an inner product of the vectors |a> and |b>. breakeven (commercial, engineering, scien- tific, and extrapolated) Several definitions existforfusionplasmas: Commercialbreakeven is when sufficient fusion power can be con- verted into electric power to cover the costs of the fusion power plant at economically compet- itive rates; engineering breakeven is when suf- ficient electrical power can be generated from the fusion power output to supply power for the plasma reactor plus a net surplus without the economic considerations; scientific breakeven is when the fusion power is equal to the input power; i.e., Q= 1. (See also Lawson crite- rion); extrapolated breakeven is when scientific breakeven is projected for actual reactor fuel (e.g., deuterium and tritium) from experimental results using an alternative fuel (e.g., deuterium only) by scaling the reaction rates for the two fuels. Breit–Wigner curve The natural line shape of the probability density of finding a decaying stateat energyE. Ratherthanexistingata single well-defined energy, the state is broadened to a full width at half max, , which is related to its lifetimeby τ = ¯ h. Thecurveof theprobability density is given by P(E)= 2π 1 ( E −E 0 ) 2 + (/2) 2 . Breit–Wigner curve. Breit–Wignerform Functionalformof cross sections in the vicinity of a resonance. In res- onance scattering, the presence of a metastable state, or a significant time delay, is signaled by the behavior of the scattering cross-section ac- cording to this particular functional form. Near the resonance energy E r , the Breit–Wigner form for the cross-section σ(E), is given by σ(E) ∼ (E−E r ) 2 +(/2) 2 , where E is the collision energy, and is the lifetime of the resonance state. Bremsstrahlung Electromagnetic radiation that is emitted by an electron as it is accelerated or deceleratedwhile moving throughthe electric field of an ion. © 2001 by CRC Press LLC Bremsstrahlung radiation Occurs in plas- ma when electrons interact (“collide”) with the Coulomb fields of ions; the resulting deflection of the electrons causes them to radiate. Brillouin–Wigner perturbation Perturba- tion treatment that expresses a state as a se- ries expansion in powers of λ (the scale of the perturbation from an unperturbed Hamiltonian, H=H 0 +λV) with coefficients that depend on the perturbed energy values E n (rather than the unperturbed energies εn of the Rayleigh- Schrödinger perturbation method). An initial unperturbed eigenstate, ϕ n , becomes, n =ϕ n + m=n ϕ m 1 E n −E m λ ϕ m |V|ϕ n . Brillouin zone Similar to the first Brillouin zone, bisect all lines, among which each con- nects a reciprocal lattice point to one of its sec- ondly nearest points. The region composed of all the bisections is defined as the second Bril- louin zone. Keeping on it, we can get all Bril- louin zones of the considered reciprocal lattice point. Each Brillouin zone is center symmetric to the point. broken symmetry Property of a system whose ground state is not invariant under sym- metry operations. Suppose L is the generator of some symmetry of a system described by Hamil- tonian H. Then [L,H]=0, and if |a>is a non-degenerate eigenstate of H, it must also be an eigenstate of L. If there exists a degeneracy, L|a>is generally a linear combination of states in the degenerate sub-manifold. If the ground state |g>of the system has the property that L|g>=c|g>, where c is a complex number, then the symmetry corresponding to the gener- ator of that symmetry, L, is said to be broken. Brownian motion The disordered motion of microscopic solid particles suspended in a fluid or gas, first observed by botanist Robert Brown in 1827 as a continuous random motion and attributed to the frequent collisions the particles undergo with the surrounding molecules. The motionwasqualitativelyexplainedbyEinstein’s (1905) statistical treatment of the laws of mo- tions of the molecules. Brunt–Väisälä frequency Natural frequen- cy, N, of vertical fluid motion in stratified flow as given by the linearized equations of motion: N 2 ≡− g ρ o d¯ρ dz where ¯ρ(z)=ρ−ρ . Also called buoyancy frequency. bubble chamber A large tank filled with liq- uid hydrogen, with a flat window at one end and complex optical devices for observing and photographing the rows of fine bubbles formed when a high-energy particle traverses the hydro- gen. Buckingham’s Pi theorem For r number of required dimensions (such as mass, length, time, and temperature), n number of dimen- sional variables can always be combined to form exactly n−r independent dimensionless vari- ables. Thus, for a problem whose solution re- quires seven variables with three total dimen- sions, the problem can be reduced to four dimen- sionless parameters. See dimensional analysis, Reynolds number for an example. bulk viscosity Viscous term from the consti- tutive relations for a Newtonian fluid, λ+ 2 3 µ, whereλ andµ are measures of the viscous prop- erties of the fluid. This is reduced to a more usable form using the Stokes assumption. buoyancy The vertical force on a body im- mersed in a fluid equal to the weight of fluid displaced. A floating body displaces its own weight in the fluid in which it is floating. See Archimede’s law. © 2001 by CRC Press LLC C calorie (Cal) A unit of heat defined as the amount of heat required to raise the temperature of 1 gm. of water at 1 atmosphere pressure from 14.5 to 15.5 C. It is related to the unit of energy in the standard international system of units, the Joule, by 1 calorie = 4.184 joules. Note that the calorie used in food energy values is 1 kilo- calorie ≡ 1000 calories, and is denoted by the capital symbol Cal. camber Curvature of an airfoil as defined by the line equidistant between the upper and lower surfaces. Important geometric property in the generation of lift. canonical ensemble Ensemble that de- scribes the thermodynamic properties of a sys- tem maintained at a constant temperature T ,by keeping itin contact with aheatreservoir attem- perature T . The canonical distribution function gives the probability of finding the system in a non-degenerate state of energy E i as P ( E i ) = exp ( −E i /k B T ) / i exp ( −E i /k B T ) , where k B is the Boltzmann constant, and the summation is over all possible microstates of the system, denoted by the index i. canonical partition function For a system of N particles at constant temperature T and volume V , all thermodynamic properties can be obtained from the canonical partition function defined as Z(T ,V, N) = i exp(−E i /k B T), where E i is the energy of the system of N par- ticles in the ith microstate. canonical variables In the Hamiltonian for- mulation of classical physics, conjugate vari- ables are defined as the pair, q, p = ∂L ∂ ˙q , where L is the Lagrangian and q is a coordinate, or variable of the system. capacitively coupled discharge plasma Plasmacreatedby applyinganoscillating, radio- frequency potential between two electrodes. Energy is coupled into the plasma by collisions between the electrons and theoscillatingplasma sheaths. If the oscillation frequency is reduced, the discharge converts to a glow discharge. capillarity Effect of surface tension on the shape of the free surface of a fluid, causing cur- vature, particularly when in contact with a solid boundary. The effect is primarily important at small length scales. capillary waves Free surface waves due to the effect of surface tension σ which are present at very small wavelengths. The phase speed, c, of capillary waves decreases as wavelength increases, c = kσ ρ as opposed to surface gravity waves, whose phase speed increases with increasing wave- length. Carnot cycle A cyclical process in which a system, for example, a gas, is expanded and compressedin foursteps: (i)an isothermal(con- stant temperature) expansion at temperature T h , until its entropy changes from S c to S h , (ii) an adiabatic (constant entropy) expansion during which the system cools to temperature T c , fol- lowedby (iii) an isothermalcompression at tem- perature T c , and (iv) an adiabatic compression until the substance returns to its initial state of entropy, S c . The Carnot cycle can be repre- sented by a rectangle in an entropy–temperature diagram, as shown in the figure, and it is the same regardless of the working substance. carrier A charge carrier in a conduction pro- cess: either an electron or a positive hole. cascade Arowof bladesin aturbine orpump. cascade, turbulent energy Transfer of en- ergyinaturbulentflowfromlargescales tosmall scalesthrough variousmeanssuchas dissipation and vortex stretching. Energy fed into the tur- bulent flow field is primarily distributed among © 2001 by CRC Press LLC Carnot cycle. large scale eddies. These large eddies generate smaller and smaller eddies until the eddy length scale is small enough for viscous forces to dis- sipate the energy. Dimensional analysis shows that the relation between the energy E, the en- ergy dissipation ε, and wavenumber k is E∝ε 2/3 k −5/3 which is known as Kolmogorov’s -5/3 law. See turbulence. Casimir operator Named after physicist H.A. Casimir, these operators are bi-linear com- binations of the group generators for a Lie group that commute with all group generators. For the covering group of rotations in three-dimensional space, there exists one Casimir operator, usually labeled J 2 , where J are the angular momentum operators. See angular momentum. cation A positively charged ion, formed as a resultoftheremovalofelectronsfromatomsand molecules. In an electrolysis process, cations will move toward negative electrodes. Cauchy–Riemann conditions Relations be- tween velocity potential and streamfunction in a potential flow where ∂φ ∂x = ∂ ∂y ∂φ ∂y =− ∂ ∂x such that either φ or can be determined if the other is known. causality The causal relationship between a wavefunction at an initial time ψ(t o ) and a wavefunction at any later timeψ(t)as expressed through Schrödinger’s equation. This applies only to isolated systems and assumes that the dynamical state of such a system can be repre- sented completely by its wave function at that instant. See complementarity. cavitation Spontaneous vaporization of a liquid when the pressure drops below the va- por pressure. Cavitation commonly occurs in pumps or marine propellers where high fluid speeds arepresent. Excessivespeed ofthe pump or propeller and high liquid temperatures are standard causes of cavitation. Cavitation de- grades pump performance and can cause noise, vibration, and even structural damage to the de- vice. cavitation number Dimensionless parame- ter used to express the degree of cavitation (va- por formation) in a liquid: Ca ≡ ( p a − p v ) /ρU 2 where p a is the atmospheric pressure and p v is the vapor pressure of the liquid. cellular method Onemethod of energy band calculation in crystal. It was addressed by Wigner and Seitz. They divided a crystal into atomic cells. For a given potential, because of symmetry, the calculation reduced into a single cell. The assumption for the cellular method is that the normal component of the gradient of wave function will vanish at the single cell sur- face or at the Wigner–Seitz sphere. Celsius temperature scale (C) Defined by setting the temperature at which water at 1 at- mospheric pressure freezes at 0 ◦ C and boils at 100 ◦ C. Alternatively, the Celsius scale can be defined in terms of the Kelvin temperature T as temperature in Celsius = T − 273.16K. center-of-momentum (c.o.m.) coordinates A coordinate system in which the centers of mass of interacting particles are at rest. The particles are located by position vectors ρ r i de- fined by the center of mass of the rest frame of the system, which, in general, moves with re- spect to the particles themselves. © 2001 by CRC Press LLC c.o.m. coordinates In the center-of-momentum system, a pair of colliding particles both approach the c.o.m. head on, and then recede from the center with equal but opposite momenta: ρ p 1 + ρ p 2 = ρ p 1 + ρ p 2 = 0 even if, in the laboratory frame, the target particle is at rest (as depicted above). The velocity of the c.o.m. for such a collision is ρ ν cm = m 1 ρ ν 1 m 1 +m 2 . While in the laboratory frame two angles measured with respect to the line of motion of the incident parti- cle are necessary to describe the final directions of the particles, φ 1 and φ 2 , a single common angle θ suffices in the c.o.m.: tanφ 1 = sinθ γ+ cosθ where γ= ν cm ν cm 1 = m 1 ν 1 ν cm 1 ( m 1 +m 2 ) . central force A force always directed toward or away from a fixed center whose magnitude is a function only of the distance from that center. In terms of spherical coordinates with an origin at the force’s center, ρ F =ˆrF(r), where r= x 2 +y 2 +z 2 or in cartesian coordinates, F x = x r F(r),F y = y r F(r), and F z = z r F(r). centrifugal barrier A centrifugal force-like term that appears in Schrödinger’s equations for central potentials that prevents particles with non-zero angular momentum from getting too close to the potential’s center. The symmetry of Hamiltonians with central potentials allows the state function to be separated into radial and an- gular parts: ψ(r)=f λ (r)Y λm (θ,φ). If the ra- dial part is written in the formf λ (r)=u λ (r)/r, the function u λ (r) can satisfy − η 2 2m d 2 dr 2 + η 2 2m λ(λ+ 1) r 2 +V(r)−E u λ (r)= 0 a one-dimensional Schrödinger equation carry- ing an additional potential-like term η 2 λ(λ+ 1)/2mr 2 which grows large as r→ 0. centrigual instability Present in a circular Couette flow driven by the adverse gradient of angular momentum which results in counter- rotating toroidal vortices. Also known as the Taylor or Taylor-Couette instability. cesium chloride structure In cesium chlo- ride, the bravais lattice is a simple cube with primitive vectors ax,ay, and az and a basis composed of a cesium positive ion and a chlo- ride negative ion. CFD Computational fluid dynamics. change of state Refers to a change from one state of matter to another (i.e., solid to liquid, liquid to gas, or solid to gas). chaos The effect of a solution on a system which is extremely sensitive to initial condi- tions, resulting in different outcomes from small changes in the initial conditions. Deterministic chaos is often used to describe the behavior of turbulent flow. characteristic Mach number AMach num- ber such that M = u/a where a is the speed of sound for M = 1. Thus, M is not a sonic Mach number, but the Mach number of any velocity based on the sonic Mach number speed of sound. This merely serves as a useful reference condition and helps to simplify the governing equations. See Prandtl relation. character of group representation The trace of a matrix at a representation in group theory. © 2001 by CRC Press LLC charge conjugation (1) The symmetry op- eration associated with the interchange of the role of a particle with its antiparticle. Equiva- lent to reversing the sign on all electric charge and the direction of electromagnetic fields (and, therefore, magnetic moments). (2) A unitary operator ζ:j µ (x)→−j µ (x) which reverses the electromagnetic current and changes particles into antiparticles and vice versa. chemical bond Term used to describe the na- ture of quantum mechanical forces that allows neutral atoms to bind and form stable molecules. The details of the bond, such as the bind- ing energy, can be calculated using the meth- ods of quantum chemistry to solve the Born- Oppenheimer problem. See Born-Oppenheimer approximation. chemical equilibrium For a reaction at con- stant temperature and pressure, the condition of chemical equilibrium is defined in terms of the minimum Gibbs free energy with respect to changes in the proportions of the reactants and the products. This leads to the condition, j v j µ j = 0, where v j is the stoichiometric coefficientofthejthspeciesinthereaction(neg- ative for reactants and positive for products), and µj is the chemical potential of the jth species. chemical potential (1) At absolute zero tem- perature, the chemical potential is equal to the Fermi energy. If the number of particles is not conserved, the chemical potential is zero. (2) The chemical potential (µ) represents the change in the free energy of a system when the number of particles changes. It is defined as the derivative of the Gibbs free energy with re- spect to particle number of thejth species in the system at constant temperature and pressure, or, equivalently, as the derivative of the Helmholtz free energy at constant temperature and volume: µ j (T,P)= ∂G ∂N j T,P ; µ j (T,V)= ∂F ∂N j T,V . Chézy relations For flow in an open chan- nel with a constant slope and constant channel width, the velocity U and flow rate Q can be shown to obey the relations U=C R h tanθQ=CA R h tanθ where C= √ 8g/f and is known as the Chézy coefficient; f is the friction factor and R h is the hydraulic radius. Child–Langmuir law Description of elec- tron current flow in a vacuum tube when plasma conditions exist that result in the electron cur- rent scaling with the cathode–anode potential to the 3/2 power. choked flow Condition encountered in a throat in which the mass flow rate cannot be increased any further without a change in the upstream conditions. Often encountered in high speed flows where the speed at a throat cannot exceed a Mach number of 1 (speed of sound) regardless of changes in the upstream or down- stream flow field. circularly polarized light A light beam whose electric vectors can be broken into two perpendicularelements havingequalamplitudes but differing in phase by l/4 wavelength. circulation The total amount of vorticity within a given region defined by ≡ C u ·ds . Circulationis ameasureof theoverallrotationin a flow field and is used to determine the strength of a vortex. See Stokes theorem. classical confinement Plasma confinement in which particle and energy transport occur via classical diffusion. classical diffusion In plasma physics, dif- fusion due solely to the scattering of charged particles by Coulomb collisions stemming from the electric fields of the particles. In classical transport (i.e., diffusion), the characteristic step size is one gyroradius (Larmor orbit) and the characteristic time is one collision time. © 2001 by CRC Press LLC classical limit Used to describe the limit- ing behavior of a quantum system as the Planck constant approaches the limit ¯ h→ 0. classical mechanics The study of physical systems that states that each can be completely specified by well-defined values of all dynamic variables (such as position and its derivatives: velocity and acceleration) at any instant of time. The system’s evolution in time is then entirely determined by a set of first order differential equations, and, as a consequence, the energy of a classical system is a continuous quantity. Under classical mechanics, phenomena are classified as involving matter (subject to Newton’s laws) or radiation (obeying Maxwell’s equations). Clausius–Clapeyron equation The change of the boiling temperature T , with a change in the pressure at which a liquid boils, is given by the Clausius–Clapeyron equation: dP dT = L T v g −v l . Here, L denotes the molar latent heat of vapor- ization, and v g and v l are the molar volumes in the gas and liquid phase, respectively. This equation is also referred to as the vapor pressure equation. Clebsch–Gordon coefficients Coefficients that relate total angular momentum eigenstates with product states that are eigenstates of in- dividual angular momentum. For example, let |j 1 m 1 > be angular momentum eigenstates for operators J 1 (i.e., its square, and z-component), and let |j 2 m 2 > be the eigenstates of angu- lar momentum J 2 . We require the components of J 1 to commute with those of J 2 . We de- fine J = J 1 + J 2 , and if states |JM> are angular momentum eigenstates of J 2 and J z , then |JM>= <j 2 m 2 j 1 m 1 |JM>|j 1 m 1 j 2 m 2 >, where the sum extends over all al- lowed values j 1 j 2 m 1 m 2 . The complex num- bers <j 2 m 2 j 1 m 1 |JM> are called Clebsch– Gordon coefficients. See angular momentum states. Clebsch–Gordon series Identity involving Wigner rotation matrices, given the Wigner ma- tricesD j a m a m a (R)andD j b m b m b (R), wherethefirst matrix is a representation, with respect to an angular momentum basis, of rotation R. The second rotation is a representation of the same rotation R but is defined with respect to an- other angular momentum basis. The matrices act on direct product states of angular momen- tum. For example, the first Wigner matrix op- erates on spin states for particle 1, whereas the second operates on the spin states for particle 2. The Clebsch-Gordon series relates products of these matrices with a third Wigner rotation matrix D j mm (R), which is a representation of the rotation R with respect to a basis given by the eigenstates of the total angular momentum (for the above example, the total spin angular momentum of particle 1 and 2). closed system A thermodynamic system of fixed volume that does not exchange particles or energy with its environment is referred to as a closed system. Such a system is also called an isolated system. All other external parameters, such as electric or magnetic fields, that might affectthesystemalsoremainconstantinaclosed system. closure See completeness. closure relation Satisfied by any complete orthonormal set of vectors |n>, the relation n |n><n|=1, valid when the spectrum of eigenvalues is entirely discrete, allows the ex- pansion of any vector |u>as a series of the basis kets of any observable. When the spec- trum includes a continuum of eigenvalues, the relation is sometimes expressed in terms of a delta function identity: ρ δ ρ r − ρ r = n φ ∗ n ρ r φ n ρ r + φ ∗ n ρ p , ρ r φ ρ p , ρ r d 3 p where n enumerates the discrete eigenfunctions (the vectors above) and φ ρ p , ρ r general- izes the expression to the continuous case. © 2001 by CRC Press LLC [...]... function of scattering angle θ and is h given by Compton’s formula λ = 2 mc sin2 θ , 2 where m is the rest mass of the electron Compton wavelength The ratio λ = h/mc, ¯ where h is the Planck constant divided by 2 , ¯ m is the mass of the electron, and c is the speed of light Its value is λ = 2. 4 × 10−10 cm and provides the scale of length which is important for describing the scattering of radiation... arises in any system of particles with a total half-integral spin in the absence of external influences, such as a magnetic field, that break time-reversal symmetry The theorem is of great value in understanding magnetic ions in crystals degeneracy, lifting of, or removal of Slight inequality of energy of a group of quantum states due to the presence of a perturbation, usually © 20 01 by CRC Press LLC... exp (−ı 2 t) exp −ık2 z ac |c a| , where the are the Rabi frequencies of the fields Solution of the problem shows that it is possible to prepare the system in a superposition of the two lower states |b and |c so that no absorption to state |a occurs, even in the presence of the fields This is possible because of appropriate choice of the Rabi frequencies and phases of the fields The superposition of the... data as a function of these parameters usually leads to poor statistical accuracy and a complicated presentation in which systematic trends are difficult to extract Thus, the spectra © 20 01 by CRC Press LLC are integrated over the relative angles of the outgoing particles, which leaves two independent variables These may be taken as the energies of two of the particles, E1, E2, and E3 Energy is conserved... coherence Property of the density matrix Coherences of the off-diagonal elements of the density matrix say something about the statistical properties of a quantum system coherent Refers to waves or sources of radiation that are always in phase The laser is an example of a single source of coherent radiation Coanda effect coefficient of linear expansion The fractional change in length per unit of change in... same order of magnitude, new phenomena can be observed such as Bose-Einstein condensation and the formation of degenerate Fermi gases de Broglie waves Profound and far reaching concept attributed to L de Broglie (1 924 ), stating that matter, which until then had been conceived of as particulate, would sometimes behave as a wave Specifically, a particle of momentum p also behaves as a wave of the socalled... emission of radiation or other particles The term is especially applied to atoms, often as spontaneous emission, where it stands in sharp distinction to induced decay or emission, and alpha decay of nuclei decay width ( ) For any decaying state, system, particle, etc., with a lifetime τ , a quantity with the dimensions of energy, given by = h /2 τ , where h is Planck’s constant Through the energy time... concentration (number of particles per unit of volume) from the average concentration in a system capable of exchanging particles with a reservoir condensation Compression region in an acoustic wave where the density is higher than the ambient density conduction A process in which there is net energy transfer through a material without movement of the material itself For example, energy transfer could... gas A system of noninteracting fermions at a temperature much lower than the Fermi temperature degenerate gas (1) A gas of quantum mechanical particles (fermions or bosons) at temperatures low enough, or densities high enough, that the low-lying single particle energy levels are multiply occupied in equilibrium (2) A gas at temperatures and densities such that the thermal energy of a particle is comparable... Method developed by A Dalgarno and J.T Lewis (1955) that occasionally enables the second order perturbative correction to the energy of a state to be evaluated exactly Dalitz pair A high energy gamma can convert into an electron positron pair in the electric field of a nucleus In this situation, energy and momentum are conserved by the three particles (electron, positron, and recoil nucleus) in the final . radial and an- gular parts: ψ(r)=f λ (r)Y λm (θ,φ). If the ra- dial part is written in the formf λ (r)=u λ (r)/r, the function u λ (r) can satisfy − η 2 2m d 2 dr 2 + η 2 2m λ(λ+ 1) r 2 +V(r)−E u λ (r)=. eigenstates of angu- lar momentum J 2 . We require the components of J 1 to commute with those of J 2 . We de- fine J = J 1 + J 2 , and if states |JM> are angular momentum eigenstates of J 2 and. angles of the out- going particles, which leaves two independent variables. These may be taken as the energies of two of the particles, E1,E2, and E3. Energy is conserved so that E1 + E2 + E3