relative distances of less than approximately 0.5 fm (1 fm = 10 −15 m). hardness Property of a solid determined by its ability to abrade or indent another solid. hard sphere interaction The interaction of particles of finite size modeled with an inter- particle interaction that is infinitely repulsive if the separation between the centers of the two particles becomes less than the diameter of the particle, i.e., the spheres representing the parti- cles are completely impenetrable. This is also known as the excluded volume interaction. hard superconductor A superconductor that requires a strong magnetic field to destroy super- conductivity. harmonic generation A monochromatic light of frequency ω passing through a non- linear crystal generates non-linear polarization which is proportional to higher powers of the electric field. This effectively can generate higher order harmonics such as 2ω,3ω, etc. Second harmonic generation is most commonly used in nonlinear optics. Harmonic generation requires phase matching and energy conserva- tion within uncertainty limits. harmonic oscillator A particle acted on by a linear restoring force that is, a force proportional to the distance of the particle from its equilib- rium position and opposite the direction of the displacement. In the presence of such a force, a particle performs harmonic oscillations around its equilibrium position. A particle attached to a spring is one example. harmonic oscillator (linear) A prototype for systems exhibiting small vibrations about an equilibrium point. The Hamiltonian operator  H for such a system is  H=− ¯ h 2 2m d 2 dx 2 + 1 2 kx 2 where k is a force constant. The eigenvalues of the Hamiltonian, E n , consist of an infinite sequence of non-degenerate discrete levels as E n =  n+ 1 2  ¯ hω,n= 0, 1, 2, where ¯ h= h 2π and h is Planck’s constant, and ω is the angular frequency. harmonic oscillator potential The potential energy function associated with a linear restor- ing force; see also harmonic oscillator. For a simple harmonic oscillator, the potential energy function has the quadratic form V(x)= 1 2 kx 2 with k denoting the force constant and x denot- ing the displacement from the equilibrium posi- tion. harmonic oscillator wave function The so- lution of the Schrödinger equation in the pres- ence of a harmonic oscillator potential. See har- monic oscillator potential, harmonic oscillator. Harris instability A type of microinstability in plasmas driven by temperature anisotropes. Its basic energy source lies in the thermal en- ergy of the particle gyration, and thus it occurs in a magnetized anisotropic plasma in which the temperature perpendicular to the magnetic field is higher than the temperature parallel to the field. Excited electrostatic waves have fre- quencies around the electron (or ion) cyclotron frequency or its harmonics and travel obliquely to the magnetic field. They are observed mainly in laboratory plasmas. Hartree equation A single-particle SchrödingerequationiscalledtheHartreeequa- tion. Hartree–Fock method The Hartree method extended by properly antisymmetrizing the many-fermion wave function. See Hartree method theory. Hartree method theory A method to self- consistently derive the best single-particle po- tentials and wavefunctionsforasystemofmany interacting fermions. The method is based upon the assumption that the mutual interactions among the particles lead to an average poten- tial felt by each particle. The Hartree method and the closely related Hartree–Fock method al- low accurate predictions of atomic energy lev- els and wave functions. This method also works © 2001 by CRC Press LLC reasonably well for the shell model of the atomic nucleus. harvard classification A classification aris- ing out of the study of the absorption spectra of stars. Stars were classified according to the strength of the hydrogen lines in their spectra. Letters of the alphabet were used to identify the classes, with class A corresponding to the stars having the strongest hydrogen lines, class B the next strongest, etc. H center A lattice defect (hole center) in alkali crystal. head A term used to express different quanti- ties with the dimension of length. For instance, it is sometimes convenient to express the pres- sure in terms of a height of a column of fluid rather than in terms of a force per unit area. This height is then referred to as pressure head. In a similar manner, one can define a velocity head  V 2 2g  . The sum of the elevation or geodetic head, velocity head, and pressure head is then referred to as the total head. In a similar man- ner, it is also common to refer to the energy term,  ˙ W ˙mg  , associated with a pump or a turbine as a pump head or a turbine head, and to the energy loss per unit weight of a fluid as a head loss. head loss See head. heat The energy transferred to or from a system due to a thermal interaction with an- other system. Heat is more rigorously defined in terms of the first law of thermodynamics as Q=U−W, where U is the change in the internal energy and −W is the net work done by the system. heat capacity The amount of energy trans- ferred to a system that raises its temperature by one degree. heat conduction The process by which ther- mal energy is transported directly through a ma- terial across a temperature gradient from one place to another without a bulk transport of par- ticles. heat engine A device for the conversion of heat into work. heat exchanger A device used to remove heatfromahotobjectbytransferringthethermal energy to a large reservoir. heat of fusion The latent heat that is removed per unit of mass (or per unit mole) from a sub- stance undergoing a phase transformation from a liquid to a solid. heat of reaction The change in the enthalpy per unit of chemical reaction in the vicinity of the equilibrium state. heat of vaporization The latent heat that is addedperunitofmass(orperunitmole)toasub- stance undergoing a phase transformation from a liquid to a gas. heat pump A device that extracts heat from a cold reservoir and pumps it to an enclosure at a lower temperature with the input of work. heat reservoir A large system whose temper- ature remains essentially unchanged when heat flows into or out of it. heavy bosons The W ± (mass of 80 GeV/c 2 ) and the Z 0 (mass of 91 GeV/c 2 ) bosons. These bosons are understood to be the carriers of the weak interaction. heavy ions Charged particles resulting from adding charges to or removing charges from (heavier) atoms, a process known as ionization. See also ion, ionization chamber. heavy meson A strongly interacting parti- cle (see also hadron) with zero or integer spin. Heavy typically refers to the mass of a me- son relative to the lightest meson, the pion (π), which has a mass of approximately 140 MeV/c 2 . heavy-water reactor A nuclear reactor us- ing heavy water (D 2 O) as a moderator instead of ordinary (light) water (H 2 O). Light water re- actors have difficulties reaching critical condi- tions because of the large probability of neutron © 2001 by CRC Press LLC absorption by hydrogen (or a large neutron ab- sorption cross-section). Reaching critical con- ditions is therefore facilitated when hydrogen is replaced with deuterium, which has a smaller neutron capture cross-section. Compare with light-water reactor. Heisenberg–Langevin equations In quan- tum treatment of an atom-field system, damp- ing is introduced by coupling the system of in- terest with a large reservoir. It is assumed that the reservoir has a large degree of freedom, and therefore the system does not affect the reser- voir significantly. Generally, the evolution of the reservoir is not of much interest. The evolu- tion of the system is obtained by adding together the reservoir operators, which results in equa- tions of motion for the system operators. These equations for the system operators have a form similar to the classical Lengevin equation with damping and noise operator terms. These equa- tionsarecalledHeisenberg–Langevinequations. For example, a Heisenberg–Langevin equation for an atomic system correctly describes spon- taneous emission. Heisenberg uncertainty principle This principle was formulated by W. Heisenberg in 1927. The essence of the principle is that cer- tain pairs of variables describing for example, a particle, cannot be determined with arbitrary precision. These pairs of variables are often called complementary variables and some ex- amples are energy (E), time (t), position (r), and momentum (p). The following are the mathe- matical forms of the commutation relations: Et ¯ h xp x  ¯ h, yp y  ¯ h,zp z  ¯ h where r = ix+ jy+ kz is the particle posi- tion in terms of Cartesian components (x,y,z); i, j, and k are unit vectors along thex-,y-, andz- axes. x=  ( x−  x  ) 2  1 2 and p x = [  ( p x −  p x  ) 2  1 2 and  x  ,  p x  are expectation values of the position and momentum, respec- tively. Similar definitions apply to the y and z components. Heitler–London method In this method for obtaining the molecular wave functions, the or- bitals of the separated atoms are used as the trial wave functions in any variational method used to obtain the approximate molecular wave func- tions. helicity A property of a particle associated with the component of its spin (see intrinsic an- gular momentum) along the direction of the par- ticle motion. See also handedness. helicon modes Also called the whistler mode, the helicon mode is an electromagnetic right-hand circularly polarized mode present in a magnetic field; the mode propagates predom- inantly parallel to the magnetic field, and has a frequency somewhere between the proton (or ion) cyclotron frequency and the electron cy- clotronfrequency. Atfrequencieslowerthanthe proton (or ion) cyclotron frequency, the helicon mode branch of the dispersion relation is con- nected tothemagnetosonicmodebranch. These modes are of extreme importance in the study of ionospheric phenomena and condensed matter. Theyarealsoheavilyusedinplasma processing. helium (1) The second lightest chemical ele- ment. The helium atom contains two electrons. The nucleus of helium consists of two protons andtwoneutrons,andisknownasthe α-particle. Due to its closed-shell atomic structure, helium does not form chemical bonds with any other element and is therefore known as one of the noble gases. (2) A two-electron atom with atomic number Z = 2. Assuming that the nucleus of this atom is at rest, its Hamiltonian can be written as  H =  − ¯ h 2 2m ∇ 2 1 − 1 4πε 0 2e 2 r 1  +  − ¯ h 2 2m ∇ 2 2 − 1 4πε 0 2e 2 r 2  + 1 4πε 0 e 2 | r 1 − r 2 | which consists of the sum of two hydrogenic (with nuclear charge 2e) Hamiltonians, one for electron 1 and one for electron 2, with the sym- bols representing their usual meanings. The fi- nal term describes the repulsion energy of the two electrons. The spatial eigenfunctions of he- © 2001 by CRC Press LLC lium can be either space-symmetric (para states) or space-antisymmetric (ortho states). If the ef- fect of the total spin of the two electrons is in- cluded in the total wave function, one finds that the para state is coupled to the spin singlet state (S = 0 and M S = 0) and the ortho state is coupled to one of three spin states (spin triplet, S = 1,M S =−1, 0, 1). Helmholtz equation For a function ψ ( r ) , the inhomogeneous Helmholtz equation takes the form ∇ 2 ψ ( r ) + k 2 ψ ( r ) =−ρ ( r ) where k is a constant and ρ ( r ) is a scalar func- tion of the spatial coordinate r. The homoge- neous form of the equation is ∇ 2 ψ ( r ) + k 2 ψ ( r ) = 0 . In quantum mechanics, we often write the time-independent Schrödinger equation as an inhomogeneous Helmholtz equation, ∇ 2 ψ ( r ) + k 2 ψ ( r ) = Q where k = √ 2mE ¯ h and Q = 2m ¯ h 2 Vψ ( r ) ; m is the particle mass, V is the potential energy, E is the total energy, and ¯ h = h 2π where h is Planck’s constant. Helmholtz free energy (F ) Defined as F = U −TS, where U and S denote the internal en- ergy and entropy, respectively, at temperature T and volume V . The physical significance of F is that it is a minimum for a system of con- stant volume in equilibrium with a temperature reservoir. Helmholtz theorem A theorem that de- scribes the rate of change of vorticity and is stated as follows. If there exists a potential for all forces acting on a non-viscous fluid, no fluid particle can have a rotation if it did not origi- nally rotate, fluid particles always belong to the same vortex line, and vortex filaments must be either closed tubes or end on the boundaries of the fluid. The Helmholtz theorem applies for incompressible and homogeneous flow and is proven by taking the curl of Euler’s equation ∇x  Dv Dt  =∇x   f − ∇p ρ  . Using vector identities and manipulating the equations, one arrives at the following equation: D ω Dt =ω ·∇v −ω∇·v where ω is the vorticity vector. For incompress- ible flow, ∇·v = 0, and one is left with D ω Dt =ω ·  ∇v. The term on the right represents the action of velocity variations on the vorticity. Using this equation, one can show that the changes in length and direction of a line joining two ele- ments on a vortex line (line drawn in the direc- tion of local vorticity) are exactly equal to the changes of the corresponding vorticity vector. Therefore, fluid elementsonacertainvortex line will always remain there. The above equation also shows that if the vorticity is zero, then D ω Dt = 0 i.e., if a fluid element has no vorticity at some instant, it can never gain any vorticity. In other words, under the action of potential forces, all motions of an inviscid incompressible fluid set up from a state of rest or uniform motion are permanently irrotational. He-Ne lasers One of the most commonly used gas lasers. The first He-Ne laser was con- structedin1960 andoperatedatwavelength1.15 µm by A. Javan, W.R. Bennett, Jr., and D.R. Harriott. It was the first gas laser and first con- tinuous wave (cw) laser constructed. Laser ac- tion is achieved from the transition of Ne atoms. Ne atoms are excited by collision with He atoms which, being nearly resonant, facilitate the pumping process. Typically, the ratio of He to Ne varies from 5:1 to 10:1, and total gas pres- sure in the tube is about 1 torr. It can operate at various wavelengths such as 632.8 nm (red), 543 nm (green), 1.15 µm (infrared), and 3.39 µm (infrared). In an He-Ne laser, lasing of a red line is achieved by using mirrors which are © 2001 by CRC Press LLC highly reflecting for a 632.8 nm wavelength but not for wavelengths corresponding to other tran- sition lines, thus suppressing other modes. Hermite-Gaussian modes See Gaussian beam and TEM modes. Hermite polynomials The one-dimensional Schrödinger equation for a particle of mass m in a linear harmonic oscillator potential, V = 1 2 kx 2 , admits solutions of the form  n (ξ), such that  n (ξ) = e − ξ 2 2 H n (ξ), n = 1, 2, 3, where ξ =  mk ¯ h 2  1 4 x foraparticleofmassm and force constant k. H(ξ)are Hermite polynomials defined as H n (ξ) = (−1) n e ξ 2 d n e −ξ 2 dξ n . Hermitian operators In quantum mechan- ics, observable quantities are represented by Hermitian operators. A Hermitian operator  T satisfies  T † =  T , where  T † is the Hermitian con- jugate of  T . Hermitian operators havevery use- ful properties, including real eigenvalues, their eigenvectors belonging to distinct eigenvalues are orthogonal, and their eigenvectors span the space. heterodyne detection A technique used for reducing background noise. In heterodyne de- tection, a signal beam is superposed at a beam splitter with a coherent beam of a local oscilla- tor of different frequency and constant relative phase. It is used for demodulating FM and AM signals. heteronuclear/heteropolar molecule A molecule whose atoms are not identical. Com- pare with homonuclear/homopolar molecule. heterostructure lasers Semiconductor la- sers which are made of heterojunctions. A het- erojunction is made of layers of two different types of materials which are doped by p and n types of atoms. For example, layers of n and p type GaAs and AlGaAs can form a hetero- junction. The layered structure gives rise to a larger energy band gap than a homojunction, and the active region has greater confinement of electrons and holes. A larger refractive index of GaAs also helps in laser action by confining the electromagnetic radiation and current in the active region. These lasers have a low thresh- old currentandcanoperateatroomtemperature. Also, because of their small size, these lasers are suitable for many practical applications such as fiber-optics communications. hidden variables The theory of hidden vari- ables is based on the premise that the wave func- tionofasystemdoes not providealltheinforma- tion about a quantum system. Additional infor- mation is needed through variables called hid- den variables to describethesystemcompletely. It is believed that the indeterminism arising in quantum mechanics is due to our lack of knowl- edge of such variables. Higgs particle A boson whose rest mass is expected to be of the order of 1 TeV (10 12 eV). The Higgs boson has not been observed. The- oretical calculations show that the Higgs bo- son should be produced in head-on collisions of protons with energies of approximately 20 TeV. The existence of the Higgs boson is pre- dicted by the electroweak theory, which is part of the standard model. According to the elec- troweak theory, the electromagnetic and weak interactions shouldbeconsidereddifferentman- ifestations of the fundamental electroweak in- teraction. The fact that the electromagnetic in- teraction is mediated by the massless photon, whereas the weak interaction is mediated by the W ± and the Z 0 bosons, which have masses of about 100 GeV/c 2 , is explained in terms of a spontaneously broken symmetry. The sym- metry-breaking mechanism is provided by the Higgs particle. Thus, its discovery would be of enormous relevance as it would give the stron- gest support to the electroweak theory. higher order correlation Generalization of the second order correlation function, which can © 2001 by CRC Press LLC be defined as g (m,n) (r 1 , r 2 , ,r m+n ;t 1 ,t 2 ···,t m+n )=  E (−) ( r 1 ,t 1 ) ···E (−) ( r m ,t m ) E (+) ( r m+1 ,t m+1 ) ···E (+) ( r m+n ,t m+n )   E (−) ( r 1 ,t 1 ) E (+) ( r 1 ,t 1 )  ···  E (−) ( r n+m ,t n+m ) E (+) ( r n+m ,t n+m )  1/2 . Here, E (−) (r 1 ,t 1 ) and E (+) (r 1 ,t 1 ) are negative and positive frequency parts of the electric field. For a quantum system, E (−) (r 1 ,t 1 ) and E (+) (r 1 , t 1 ) become operators. higher order Gaussian modes See Gaussian beam and TEM modes. high powered unstable lasers Unstable res- onators are sometimes used to generate high power laser beams. These lasers must achieve very high gain in a short distance to compen- sate for all the losses. The advantages of un- stable resonators are larger mode volume, effi- cient power extraction, collimated beam output with low diffraction, and better far-field patters. These lasers are easier to align than lasers with stable resonators. Mirrors of these lasers must be cooled to avoid damage due to high power. high pressured gas laser Has a gaseous gain medium at very high pressures (> 50 torr). These lasers are also called transversely excited atmospheric pressure laser (TEA lasers). High pressure increases the gain but may lead to cur- rent in the gas, resulting in non-uniform excita- tion. Special design of the laser is required to eliminate this problem. One example of such a laser is the CO 2 laser. Hilbert space The wave function describ- ing a particle belongs to a function of space or set of functions called Hilbert space. The space has special properties such as completeness, and a well-defined inner product exists. Complete- ness means that there is a special set of vec- tors called a basis, such that every vector in the Hilbert space can be written as a linear com- bination of the members of this set. An inner product  φ | ψ  is defined between any two vec- tors or wave functions of the Hilbert space, φ and ψ,as  φ | ψ  =  ∞ −∞ φ ∗ (x)ψ(x) dx < ∞ . hohlraum A small metallic (typically gold) chamber used for converting high power laser beams into (soft) X-rays with efficiencies up to 50%. In addition to X-rays, interaction between the laser beams and a hohlraum produces an en- ergetic, rapidly expanding plasma. This mech- anism has also been applied to inertial confine- ment fusion as well as astrophysics. hole A mobile vacancy in the electronic va- lence structure of a material. hole burning (spatial) Multimode oscilla- tions observed in a homogeneously broadened media are due to spatial hole burning. A laser in a cavity forms a standing wave. At the nodes of the standing wave of a laser, with an intensity smaller than of other points, the inversion keeps growing, and gain saturation is much lower than in other regions. Thus, the spatial variation in inversion gives rise to spatial hole burning in the gain curve. Due to the spatial variation of the inversion, another mode may oscillate in the cavity resulting in multimode oscillation. hole burning (spectral) With an increase in intensity, gain in a medium saturates. In a homogeneously broadened medium, line shape cannot change. Therefore, the shape of the gain curve is restricted by the gain of the cen- tral mode. In an inhomogeneously broadened medium, the gain can saturate differently at dif- ferent frequencies. Due to the selective satu- ration, the frequencies resonant with the cavity modes can saturate more than it the nonreso- nant frequencies. This results in a spectral hole in the gain curve. This gives rise to multimode oscillation in a laser. hole state A vacancy left by a particle which has undergone a transition to a different energy level. For instance, an electron removed from its site leaves a vacancy at that site, namely a hole. The hole left by the electron behaves like a positive charge carrier. © 2001 by CRC Press LLC hologram Recording which contains holo- graphic images. See also holography. holography A three-dimensional imaging technique based on interferometeric techniques. In this technique, an interference pattern due to coherent superposition of the object wave and a reference wave is recorded in a plate or film medium. The recording, which is called a holo- gram, contains both phase and amplitude infor- mation of the object wave. The image is recon- structed by illuminating the recorded plate/film by a reference beam similar to the one used for recording and sending the reference beam from the same direction as the original refer- ence beam. The wave diffracted from the plate reconstructs the image which looks like the orig- inal object. Holography was invented by Dannis Gabor in 1948. homenergic flow A flow where the enthalpy is constant. See enthalpy. homentropic flow A flow where every fluid particle has the same value of entropy. See en- tropy. homodyne detection Superposing a signal beam with a coherent beam of a local oscillator of the same frequency and its constant relative phase at a beam splitter. The homodyne detec- tion technique has been used to detect squeezed light and enhance antibunching. homogeneous broadening Broadening mechanism which is identical for all absorbing or emitting atoms. Examples of homogeneous broadening are radiative broadening and colli- sional broadening, which have a Lorentzian line shape of the form δν o /π  ( ω − ω o ) 2 + ( δν o ) 2  . Here, δν o is the homogeneous line width. homogeneousturbulence Situationinwhich the average properties of the turbulent fluctua- tions are independent of the position in the fluid. In general, all turbulent flows are inhomoge- neous. Yet, the assumption of homogeneous turbulence in the theoretical treatment of tur- bulent flows gives a better understanding of cer- tain details that are the same in both homoge- neous and inhomogeneous flows, e.g., the tur- bulent energy transfer processes. Experimen- tally initiating homogeneous turbulent motion is extremely difficult. Even if this problem is overcome, maintaining the energy in such flows is difficult. As such, homogeneous turbulence is usually produced by placing grids in a flow, which renders the flow inhomogeneous, yet sta- tionary, in one direction. homojunction laser Simplest form of a semiconductor laser in which the active region is a p-n junction depletion region. Doping p and n type atoms in the same type of material creates the p-n junction in this laser. The thresh- old current density of this laser is very high at room temperature, therefore it is operated at a low temperature such as liquid nitrogen temper- ature. homonuclear/homopolar molecule A mol- ecule whose atoms are identical, such as the hy- drogen molecule, H 2 . Compare with heteronu- clear/heteropolar molecule. Hooke’s law For small displacements, the size of the deformation is proportional to the deforming force. hopping Microscopic motion of electrons in the presence of both lattice potential and the ex- ternal field. This motion consists of individual steps in which the electron hops from one local- ized state to the next. horseshoe vortex When a boundary layer encounters a surface-mounted obstacle, the ad- verse-pressure gradient causes a separation in which the near-wall vorticity of the boundary layer is reorganized into a vortex that has the shape of a horseshoe. This vortex is composed oftwostreamwiselegsofvorticity, eachleg hav- ing a vorticity of opposite sense. Horseshoe vortices are observed in many flows, including the flowsaround wing-fuselage junctures on air- craft, ship, and submarine appendages, bridge- piers, andturbomachineryblade-rotorjunctures. Horseshoe vortices are undesirable in many of © 2001 by CRC Press LLC these flows. For instance, they increase flow losses, they are a source of noise generation, and they cause scouring of stream beds around piers. The characteristics of the horseshoe vor- tex and its effects are dependent on the nature of the incoming boundary layer. hot wire (hot film) anemometry A tech- nique to measure fluid velocity. The principle of operation is based on the fact that the rate of cooling of a heated wire by a flow is depen- dent on the velocity. Hotwireprobesareusually made out of a thin (5µm in diameter) short plat- inum or tungsten wire through which a current of electricity is passed. The current causes the wire to heat up. In one method of operation, the current through the wire is kept constant. The velocity is then obtained by measuring the voltage across the wire, which depends on the resistance, and thus the temperature, of the wire. In a more common method of operation, a feed- back circuit is used to maintain the wire at a constant temperature. The current and voltage needed to do this are then related to fluid veloc- ity. Hot wiresareusuallyused in gasflowswhile hot films are used in liquids. In hot films, the heatedelementconsistsofathinmetallic film on the surface of a wedge-shaped probe. Single hot wires are used to measure a velocity component. To measure more than one component, differ- ent configurations are used. Because hot wires and hot films are not absolute instruments, they always require calibration of the voltage with the fluid velocity. Hot wires and hot films have several advantages over other velocity measure- ment techniques. For instance, in comparison with other techniques, they have a short time re- sponse and can thus pick up rapid fluctuations in velocity, which is necessary for measurements in turbulent flows. Moreover, they are small enough to give local measurements instead of averagevaluesovercomparativelylargeregions, as in the case of Pitot tube. On the other hand, their inability to fulfill the requirement of cal- ibration over a certain velocity range and their intrusive character are two main shortcomings of hot wires and hot films in comparison with other techniques such as laser doppler anemom- etry. Hubble’s law This term is encountered in the context of nuclear astrophysics. Hubble’s law states that the velocity of the recession of an object with respect to the earth is given by v = Hd with d representing the distance from the earth. H is the Hubble constant, which has a value of approximately 2×10 −18 s −1 . Hund–Mulliken (molecular orbital) method Electronic wave functions for molecular sys- tems containing several electrons are construc- ted from one-electron molecular orbitals. This is the molecular-orbital method. For example, in determining the wave functions of the hydro- gen molecule H 2 , we use the wave functions of H + 2 as a starting point. Hund’s rules A set of rules that have been established empirically to determine the ground state configuration of an atom. Use the Russell– Saunders notation to denote a particular state of the electrons in an atom, i.e., 2S+1 L J , where J is the total angular momentum quantum number andS represents thetotalspinof theelectrons. L takes on the code letters S, P, D, for values of L = 0, 1, 2, For example, the ground state for hydrogen is 2 S 1 2 . Hund’s rules are as follows: (1) the state with the largest possible value of S has the lowest energy; the energy of the other states increases with decreasing S, and (2) for a given value of S, the state having the maximum possible value of L has the lowest energy. hybrid frequency In a magnetized plasma, there are two types of hybrid frequencies that characterize electrostatic plasma waves prop- agating perpendicularly to the magnetic field. One is the lower-hybrid frequency and the other is the upper-hybrid frequency, both of which contain in their expressions either plasma or cy- clotron frequency of protons (or ions) and elec- trons. hybridization Phenomenon which occurs when atomic orbitals are combined to produce a molecular orbital of lower energy than the en- ergy of the individual orbitals. © 2001 by CRC Press LLC hybrid modes Modes of cylindrical dielec- tric wave guides such as optical fibers, in which both axial electric and magnetic field compo- nents are finite. Hybrid modes are classified into two groups: HE modes in which the axial elec- tric field is significant compared to transverse electric components and EH modes in which the axial magnetic field is significant compared to transverse magnetic field component. These modes are further characterized by two inte- gers corresponding to radial and azimuthal vari- ations. hydraulic diameter A term used to account for the shape as well as the size of a conduit. It is defined as four times the ratio of the cross- sectional area to the wetted perimeter of the cross-section. Wetted perimeter means the por- tion of the perimeter where there is contact be- tween the fluid and the solid boundary. hydraulic jump A phenomenon that takes place when a supercritical flow (Froude num- ber FR> 1) undergoes a transition to a sub- critical flow (Froude number FR< 1). Hy- draulic jumps occur downstream of overflow structures such as spillways or underflow struc- tures such as sluice gates where the velocities are very high. They are also observed in sinks or bathtubs when the tap water comes down at certain rates. Hydraulic jumps can be used as dissipaters of energy to prevent problems aris- ing from high speeds, such as the scouring of channel bottoms. They can be used in water and sewage treatment designs to enhance chemicals mixing with the flow. Depending on the up- stream Froude number, the hydraulic jump can assume an undular water surface (FR<2.5) or a rough water surface with intermittent jets from the bottom. The hydraulic bore, also known as surge, formed by rapidly releasing water into a channel or by abruptly lowering a downstream gate is one form of a hydraulic jump usually re- ferred to as a translating hydraulic jump. hydraulic radius A term used to take into consideration the shape, as well as the size, of a conduit. It is defined as the ratio of the cross- sectional area to the wetted perimeter of the cross-section. Wetted perimeter means the por- tion of the perimeter where there is contact be- tween the fluid and the solid boundary. For a circular pipe flowing full, the hydraulic radius is equal to one-fourth the diameter and is, there- fore, not equal to the radius of the pipe. For a circular pipe flowing half-full, the hydraulic radius is equal to one-half the diameter of the pipe. hydraulics Atermoriginallyusedtodescribe applied and experimental aspects of fluid behav- ior (mainly water) and develop empirical formu- las for practical problems. It stands in contrast to hydrodynamics, a term that was used to de- scribe theoretical and mathematical aspects of idealized or frictionless fluid behavior. The in- troduction of the concept of boundary layers, and interest in new fields such as aerodynam- ics at the beginning of the twentieth century led to the synthesis of both approaches to what is known today as the science of fluid mechanics. hydrodynamics See hydraulics. hydrodynamic stability Afield of study that deals with the prediction of whether a flow pat- ternisstable andwithits transitiontoturbulence. It involves the linear stability theory, which ex- amines the amplification rates of small distur- bances, the subsequent non-linear stages of the transition where the growing instabilities inter- act with each other, and the different mecha- nisms for breakdown to turbulence. hydrogen Chemicalelementwiththelightest atomic weight. The nucleus of hydrogen con- sists of one proton. Thus, the atom contains one proton and one electron. Hydrogenic iso- topes are deuterium and tritium, consisting of one proton and one neutron and one proton and two neutrons, respectively. hydrogen atom Considered the most impor- tant two-particle system in quantum physics. It consists of a relatively heavy nucleus of mass M containing one proton of charge e together with an electron orbiting around it with charge −e and mass m. The spherically symmetric poten- tial energy V ( r ) of the electron in the electric field of the nucleus is obtained from Coulomb’s © 2001 by CRC Press LLC law and is given as V ( r ) = −e 2 4πε 0 1 | r | (i) ground state of: The state of lowest en- ergy (ground state) is described by the spatial wave function (for an infinitely heavy nucleus), ψ 100 ( r ) ,as ψ 100 ( r ) = 1 √ πa 3 exp ( − | r | /a ) where a is the Bohr radius defined as a = 4πε 0 ¯ h 2 me 2 ≈ 0.529 × 10 −10 m ε 0 is the permittivity of free space, ¯ h = h 2π where h is Planck’s constant, and m is the mass of theelectron. The subscripts on thefunctionψ denote the values of the principal, orbital angu- lar momentum, and magnetic quantum numbers n, l,andm l , respectively, with n = 1,l = 0, and m l = 0. The energy of the ground state E 1 is −13.6 electron-volts. (ii) Schrödinger equation for: The time- independent Schrödinger equation for the hy- drogen atom in the center-of-mass system of co- ordinates in which the effect of the finite mass of the nucleus is taken into account is given as:  − ¯ h 2 2µ ∇ 2 − e 2 4πε 0 r  ψ ( r ) = Eψ ( r ) where µ is the reduced mass defined by µ = Mm M+m , r = r 1 − r 2 is the relative displacement between the proton position r 1 and the electron r 2 , and r = | r 1 − r 2 | . (iii) allowed energies of: The allowed en- ergies, E n , of the bare (unperturbed) hydrogen atom in the non-relativistic approximation and for a nucleus with mass taken as infinite form a discrete spectrum which depends only on the principal quantum number n. E n is defined by the following formula E n =−  m 2 ¯ h 2  e 2 4πε 0  2  1 n 2 where m is the mass of the electron, e is the magnitude of the electronic charge, ε 0 is the per- mittivity of free space, and ¯ h = h 2π where h is Planck’s constant. (iv) wave functions for: The spatial wave functions, ψ nlm (r), for hydrogen are labeled by three quantum numbers n, l, and m l . They are defined (up to some normalization constant) by ψ nlm l (r) = R nl (r)Y m l l (θ, φ) where ( r, θ , φ ) are spherical polar coordinates. R nl (r) is the radial part of the wave function defined as R nl (r) = 1 r ρ l+1 e −ρ v(ρ) where ρ = r an and a is the Bohr radius of the hydrogen atom. v(ρ) is a polynomial of degree i max = n − l − 1inρ such that v(ρ) = i max  i=0 a i ρ i . The coefficients in the expansion for v(ρ) are determined from the following recursion rela- tion a i+1 =  2 ( i +l +1 ) − A ( i +1 )( i +2l +2 )  a i where A is a constant given by A = me 2 2πε 0 ¯ h 2  √ −2mE n ¯ h  . The angularpartsY m l (θ, φ) are called the spherical harmonics and are defined as Y m l (θ, φ) = γ  ( 2l +1 )( l − | m | ) ! 4π ( l + | m | ) ! e imφ P m l ( cos θ ) where P m l ( cos θ ) is an associated Legendre polynomial and γ = ( −1 ) m for m ≥ 0 and γ = 1 for m ≤ 0. (v) spectrum of: When an excited hydrogen atom with energy E i decays to a lower energy levelE f , a photonisemittedwithenergy E ν and frequency ν. In terms of the principal quantum numbers n i and n f of the initial and final energy levels, respectively, the energy of the emitted photon is determined from the following E ν = E i − E f =−13.6eV  1 n 2 i − 1 n 2 f  = hν © 2001 by CRC Press LLC [...]... index of the crystal and its surrounding medium in nonlinear optics to minimize Fresnel losses incident flux The number of particles incident upon the target area per unit of time © 2001 by CRC Press LLC index of refraction The ratio of the velocity of a wave in a vacuum to that in a specified material indirect band gap semiconductor In an indirect band gap semiconductor, the conduction band edge and valence... inelastic cross-section For collisions, the ratio of the number of events of this type per unit of time and per of unit scatterer to the flux of the incident particles is defined as the inelastic cross-section where B is the span of the airfoil and C is the mean chord length The ratio B/C is the aspect ratio Induced drag is critical during takeoff and landing when the lift coefficient is especially large... The impulse is the integral of the left-hand side The right-hand side represents the change in momentum when integrated The equality of these two quantities is the impulse momentum principle impurity An atom that is foreign to the material in which it exists impurity band An energy level caused by the presence of impurity atoms incident energy The energy of the incoming particle in a collision process... effects between the fluid particles, and thus there is no boundary layer The motion of an ideal fluid is analogous to the motion of a solid body on a frictionless surface ideal gas ticles A gas of non-interacting point par- ideal gas law The equation of state for an ideal gas, P V = nRT Here, P , V , and T denote the pressure, volume, and temperature of n moles of an ideal gas, and R is the universal... symmetric (gerade) and is called the bonding orbital It represents the state of lowest energy u (R; r) is antisymmetric (ungerade) and is called the antibonding orbital The binding energy of H+ is 2.79 eV 2 (xi) fine structure of: In the study of the hydrogen atom, the calculation of the energy levels using the Bohr theory does not take into account the correction due to relativistic effects of the electron... acting on the particle: F = q (E + v × B) where E and B are electric and magnetic field vectors and v is the velocity of the particle It should be noted that this force cannot be written as the gradient of a scalar potential energy function The classical Hamiltonian describing this situation is H = 1 (p − qA)2 + qφ 2m where A is the vector potential and p is the generalized momentum of the particle; φ... and leptonic currents is mediated by some combination of four distinct quanta: W+ , W0 , W− , and B0 The coupling of the electromagnetic currents of quarks and leptons selects a particular combination of the above four particles Some combinations, like B0 and W0 , can produce a massless particle called a photon Other combinations produce massive particles through a process called spontaneous symmetry... by emission of γ -rays (electromagnetic radiation) In the process of internal conversion, however, an electron is ejected from the atom instead of a photon internal energy The total energy content of a system A thermodynamic equilibrium state of a system is characterized by a function of state called the internal energy, which is constant for an isolated system The state of equilibrium of a system... where n is the particle density and τ is the average confinement time of the plasma To achieve such a goal, extremely large compression factors of the order of 10,000 are required In addition, the confinement time τ in the inertial fusion is set by the time taken for the plasma to expand freely Therefore, extremely high pulsed powers of hundreds of terawatts [T W ] must be focused down and delivered to... ωo is applied with a secant pulse of frequency ω then each atom completes a whole cycle of excitation and returns back to the ground state, resulting in no absorption of the energy This gives rise to the phenomenon of self-induced transparency See soliton A property of a particle defined Y =A+S where A is the baryon number (equal to 1 for baryons and 0 for mesons), and S is the strangeness The strangeness . are made of heterojunctions. A het- erojunction is made of layers of two different types of materials which are doped by p and n types of atoms. For example, layers of n and p type GaAs and AlGaAs. exists. impurity band An energy level caused by the presence of impurity atoms. incident energy The energy of the incoming particle in a collision process. incident flux The number of particles inci- dent. terms of the principal quantum numbers n i and n f of the initial and final energy levels, respectively, the energy of the emitted photon is determined from the following E ν = E i − E f =−13.6eV  1 n 2 i − 1 n 2 f  =