Sectorial Mindlin Plates from Kirchhoff Plates

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地震的原因英语作文

Earthquakes are one of the most powerful and destructive natural phenomena on Earth.They occur due to the sudden release of energy in the Earths crust,which creates seismic waves.Here are the primary causes of earthquakes:1.Tectonic Plate Movement:The Earths crust is divided into several tectonic plates that are constantly moving.The boundaries where these plates meet are zones of high seismic activity.There are three types of plate boundaries:Convergent Boundaries:Where plates move towards each other,often resulting in the formation of mountain ranges or subduction zones where one plate slides under another. Divergent Boundaries:Where plates move away from each other,typically creating new crust as magma rises from the mantle.Transform Boundaries:Where plates slide past each other horizontally,often causing earthquakes when the plates get stuck and then suddenly slip.2.Fault Lines:Earthquakes often occur along fault lines,which are fractures in the Earths crust where rocks on either side have moved past each other.There are three main types of faults:Normal Faults:Where the hanging wall moves downward relative to the footwall. Reverse Faults:Where the hanging wall moves upward relative to the footwall. StrikeSlip Faults:Where the movement is horizontal.3.Volcanic Activity:Some earthquakes are triggered by volcanic activity.As magma rises towards the Earths surface,it can cause the overlying rocks to fracture and move, resulting in earthquakes.4.Human Activities:Although less common,human activities can also induce earthquakes.This can occur through processes such as:ReservoirInduced Seismicity:The filling of large reservoirs behind dams can increase pressure on fault lines,potentially triggering earthquakes.Oil and Gas Extraction:The injection of fluids into the ground or the removal of fluids like oil and gas can alter the stress on fault lines.Underground Mining:The removal of large amounts of rock can change the stress distribution in the Earths crust.5.Geological Processes:Other geological processes,such as the cooling and solidification of magma chambers,can also lead to earthquakes.As the magma cools,the volume of the chamber decreases,which can cause the overlying rock to collapse and move.6.Aftershocks:Earthquakes are often followed by a series of smaller earthquakes knownas aftershocks.These occur as the Earths crust adjusts to the new stress distribution caused by the main shock.Understanding the causes of earthquakes is crucial for developing strategies to mitigate their impacts.Seismologists use a variety of tools and techniques,including seismographs to measure seismic waves and GPS technology to monitor the movement of tectonic plates,to study earthquakes and predict where they are most likely to occur. Despite these efforts,predicting the exact time and location of an earthquake remains a challenge.。

tectonic plates英译 (2)

tectonic plates英译Tectonic Plates: Understanding the Earth's Dynamic CrustIntroduction:Tectonic plates, also known as lithospheric plates, are large sections of the Earth's crust that move and interact with each other. These plates are responsible for shaping the Earth's surface and are crucial in understanding various geological phenomena. In this article, we will delve into the concept of tectonic plates, exploring their movement, boundaries, and the impact they have on our planet.Body:1. Movement of Tectonic Plates:1.1 Plate Tectonics Theory:- Plate tectonics theory explains the movement of tectonic plates.- The theory suggests that the Earth's lithosphere is divided into several plates that float on the semi-fluid asthenosphere.- These plates move due to convection currents in the mantle, driven by heat from the Earth's core.1.2 Types of Plate Boundaries:- Convergent Boundaries: Plates collide, leading to the formation of mountains, volcanic activity, and earthquakes.- Divergent Boundaries: Plates move apart, creating new crust through seafloor spreading and rift valleys.- Transform Boundaries: Plates slide past each other, causing lateral movement and resulting in earthquakes.1.3 Plate Interactions:- Subduction Zones: Occur when one plate is forced beneath another, leading to the formation of trenches and volcanic arcs.- Mid-Ocean Ridges: Underwater mountain ranges where new crust is formed through seafloor spreading.- Transform Faults: Zones where plates slide horizontally past each other, causing earthquakes.2. Major Tectonic Plates:2.1 Pacific Plate:- The largest tectonic plate, covering the Pacific Ocean and parts of the western United States, Japan, and Australia.- Known for its high seismic activity, including the infamous Ring of Fire, where several tectonic plates meet.2.2 North American Plate:- Covers most of North America, including the United States and Canada.- Interacts with the Pacific Plate along the San Andreas Fault, resulting in frequent earthquakes in California.2.3 Eurasian Plate:- Encompasses Europe, Russia, and parts of Asia.- Collides with the Indian Plate, leading to the formation of the Himalayas, the world's highest mountain range.2.4 African Plate:- Covers the African continent and parts of the Atlantic Ocean.- Interacts with the Eurasian Plate along the Mid-Atlantic Ridge, contributing to the formation of the Great Rift Valley in East Africa.2.5 South American Plate:- Includes South America and parts of the Atlantic Ocean.- Converges with the Nazca Plate, giving rise to the Andes Mountains and the Pacific Ring of Fire.3. Geological Phenomena:3.1 Earthquakes:- Tectonic plate interactions often result in earthquakes.- Earthquakes occur when stress builds up along plate boundaries and is released in the form of seismic waves.3.2 Volcanic Activity:- Volcanoes are commonly found at convergent and divergent plate boundaries.- The subduction of one plate beneath another triggers volcanic eruptions, releasing magma from the Earth's mantle.3.3 Mountain Formation:- Collisions between tectonic plates lead to the formation of mountain ranges.- The uplift and folding of rock layers occur as plates converge, resulting in the creation of large-scale mountain systems.4. Plate Tectonics and Plate Tectonics and the Continents:4.1 Continental Drift:- Plate tectonics theory explains the movement of continents.- The theory suggests that continents were once part of a single landmass called Pangaea, which later broke apart and drifted to their current positions.4.2 Plate Tectonics and Fossil Evidence:- Fossil evidence supports the theory of continental drift and plate tectonics.- Fossils of similar species found on different continents provide evidence of past connections when the continents were joined.4.3 Plate Tectonics and Climate Change:- Plate tectonics influence climate patterns.- The movement of tectonic plates can alter ocean currents, leading to changes in global climate over long periods.5. Conclusion:In conclusion, tectonic plates play a vital role in shaping the Earth's surface and are responsible for a range of geological phenomena. Understanding the movement, boundaries, and interactions of these plates provides valuable insights into earthquakes, volcanic activity, mountain formation, and even climate change. Through ongoing research and study, scientists continue to unravel the complexities of tectonic plates, deepening our knowledge of the dynamic nature of our planet.。

semiconductor optics1

Maxwell’s Equations, Photons and the Densityof StatesIn this chapter we consider Maxwell’s equations and what they reveal about the propagation oflight in vacuum and in matter. We introduce the concept of photons and present their density ofstates. Since the density of states is a rather important property in general and not only for photons,we approach this quantity in a rather general way. We will use the density of states later also forother (quasi-) particles including systems of reduced dimensionality. In addition, we introduce theoccupation probability of these states for variousgroups of particles.It should be noted, that we shall approach the concept of photons on an elementary level only,in correspondence with the concept of this book. We do not delve into present research topics onphoton physics itself like photoncorrelation and -statistics, squeezed light, photon anti-bunching,entangled photon states, etc., but give some introductory references for those interested in thesefields [89S1, 92M1, 94A1, 01M1, 01T1, 01T2, 02B1, 02D1, 02G1, 02L1, 02Y1]. Einstein, whoobtained the Nobel prize for physics in 1921 for the explanation of the photo-electric effect (notfor the theory of relativity!), once stated: “Was das Licht sei, das wei ic h nicht” (What the lightmight be, I do not know). So there still seems to be ample place for research in these fields.2.1 Maxwell’s EquationsMaxwell’s equations can be written in different ways. We use here the macroscopic Maxwell’sequations in their differential form. Throughout this book the internationally recommended systemof units known as SI (systeme international) is used. These equations are given in their generalform in (2.1a –f), where bold characters symbolize vectors and normal characters scalar quantities.D ρ∇= 0B ∇= (2.1a,b)E B∇⨯=- H j D ∇⨯=+ (2.1c,d ) 0D E P ε=+ 0B H M μ=+ (2.1e,f )The various symbols have the following meanings and units:E = electric field strength; 1V /m = 1mkg s -3 A -1D = electric displacement; 1As /m 2 = 1C /m 2H = magnetic field strength; 1A /mB = magnetic induction or magnetic flux density ; 1V s /m 2= 1T =1Wb /m 2ρ = charge density; 1As /m 3 = 1C /m 3j = electrical current density; 1A /m22P = polarization density of a medium, i.e., electric dipole moment per unit volume; 1 As /m 2M = magnetization density of the medium, i.e., magnetic dipole moment per unit volume1;1Vs /m 2ε0 = 8.859 × 10-12 As /Vm is the permittivity of vacuumμ0 = 4π × 10-7 Vs /Am is the permeability of vacuum∇ = Nabla-operator, in Cartesian coordinates ∇ = (∂/∂x, ∂/∂y, ∂/∂z )˙ = ∂/∂t i.e., a dot means differentiation with respect to time.The applications of ∇ to scalar or vector fields are usually denoted by∇ · f (r ) = grad f,∇ ·A (R ) = div A ,∇×A (r ) = curl A ,and the Laplace operator Δ is defined asΔ ≡ ∇2If Δ is applied to a scalar field ρ we obtain222222x y z ρρρρ∂∂∂∆=++∂∂∂ (2.2)Application to a vector field E results in(2.3)Further rules for the use of ∇ and of Δ and their representations in other than Cartesiancoordinates (polar or cylindrical coordinates) are found in compilations of mathematical formulae[84A1, 91B1, 92S1].Equations (2.1a,b) show that free electric charges ρ are the sources of theelectric displacement and that the magnetic induction is source-free. Equations (2.1c,d)demonstrate how temporally varying magnetic and electric fields generate each other. In addition, the H field can be created by a macroscopic current density j . Equations (2.1e,f) are the materialequations in their general form. From them we learn that the electric displacement is given by the sum of electric field and polarization, while the magnetic flux density is given by the sum ofmagnetic field and magnetization. Some authors prefer not to differentiate between H and B . This leads to difficulties, as can be easily seen from the fact that B is source-free (2.1b) but H isnot, as follows from the inspection of the fields of every simple permanent magnet.By applying ∇· to (2.1d) we obtain the continuity equation for the electric charges(2 .4)which corresponds to the conservation law of the electric charge in a closed system.The integral forms of 2.1 can be obtained from the differential forms by integration and the use ofthe laws of Gauss or Stokes resulting inwhere d V , d f and d s give infinitesimal elements of volume, surface or area and line, respectively.In their microscopic form, Maxwells equations contain all charges as sources of the electric fieldE micro including all electrons, protons bound in atoms as ρ bound and not only the free spacecharges ρ. By analogy, not only the microscopic current density j has to be used as a source ofH micro but all spins and 0l ≠orbits of charged particles have to be included as “bound” currentdensity j bound . The transition to macroscopic quantities can then be performed by averaging oversmall volumes (larger than an atom but smaller than the wavelength of light) and replacing ρboundby P -∇and j bound by P + curl M /μ0. For more details see [98B1,98D1] or Chap. 27.Concerning the units, some theoreticians still prefer the so-called c g s(cm, g, second) system.Though it has only marginal differences in mechanics to the SI system, which is based on the units1m, 1 kg, 1 s, 1A, 1K, 1 mol and 1 cd, the c g s system produces strange units in electrodynamicslike the electrostatic units (esu), which contain square roots of mass and are therefore unphysicaland even ill-defined. For conversion tables see [96L1].2.2 Electromagnetic Radiation in VacuumIn vacuum the following conditions are fulfilled(2 .6)With the help of (2.1e,f) this simplifies (2.1c,d) to(2.7a,b)Applying ∇× to (2.7a) and ∂/∂t to (2.7b) yields(2 .8)From (2.8) we find with the help of the properties of the ∇ operator(2.9) With (2.6), (2.3) and (2.1a) we see that0E ∇= (2.10)and (2.9) reduces to the usual wave equation, written here for the electric field2000E E με∇-= (2.11) An analogous equation can be obtained for the magnetic field strength. Solutions of this equationare all waves of the form0(,)()E r t E f kr t ω=- (2.12)E 0 is the amplitude, f is an arbitrary function whose second derivate exists. As can be shown byinserting the ansatz (2.12) into (2.11) the wave vector k and the angular frequency ω obey therelation(2.13)In the following we use for simplicity only the term “frequency” for ω = 2π/T where T is thetemporal period of the oscillation.In (2.13), c is the vacuum speed of light and λv is the wavelength in vacuum. From all possiblesolutions of the form (2.12) we shall concentrate in the following on the most simple ones, namelyon plane harmonic waves, which can be written as0(,)exp[()]E r t E i kr t ω=- (2.14) For all waves (not only those in vacuum), the phase and group velocities v ph and v g are given by(2.15)where v ph gives the velocity with which a certain phase propagates, (e.g.,a maximum of amonochromatic wave) while v g gives the speed of the center of mass of a wave packet with middlefrequency ω and covering a small frequency interval d ω as shown schematically in Figs. 2.1a,b,respectively. The formulas (2.15) are of general validity. The grad k on the r.h.s. of (2.15) means adifferentiation with respect to k ; in the sense of ∇k = (∂/∂k x , ∂/∂k y, ∂/∂k z ) and has to be usedinstead of the more simple expression ∂ω/∂k in anisotropic media. For the special case ofelectromagnetic radiation in vacuum we find from (2.13), (2.15)(2.16)Fig.2.1. A harmonic wave (a) and a wave packet (b) shown at two different times t and t+▽t toillustrate the concepts of phase and group velocity, respectivelyNow we want to see what constraints are imposed by Maxwell’s equations on the variousquantities such as E 0and k . Inserting (2.12) or (2.14) into (2.10) gives0exp[()]0E iE k i kr t ω∇=-= (2.17)This means that0E k ⊥ (2.18)or, in other words, the electromagnetic wave is transverse in E . What can we learn fromMaxwell’s equations for the other fields? From (2.7) we have for plane waves( 2.19a)With(2.19b) Furthermore we have with (2.1e,f) and (2.6)The electromagnetic wave is, according to (2.19b), also transverse in B and the electric and magnetic fields are perpendicular to each other, that is, we have in general⊥⊥⊥(2.19e)D k B DIn vacuum and isotropic media one has in addition(2.19f)As we shall see later in connection with (2.17) and (2.43),(2.44) one has in matter usually transverse waves, which obey (2.19e) but additionally, longitudinal waves exist under certain conditions.The momentum density Π of the electromagnetic field is given by(2.20) and the energy flux density by the Poynting vector S=⨯(2.21)S E Hwith S // Π in vacuum and isotropic materials.S is a rapidly oscillating function of space and time. The average value <S>is usually called the intensity I or the energy flux density The intensity is proportional to the amplitude squared for all harmonic waves. For the plane monochromatic waves treated here, we obtain(2.22)Equations (2.20) and (2.21) are also valid in matter.2.3 Electromagnetic Radiation in Matter; Linear OpticsNow we treat Maxwell’s equations in matter. Doing so we have in principle to use the equations in their general from 2.1. However we will still make in several steps some assumptions which are reasonable for semiconductors: we assume first that there are no macroscopic free space charges i.e. ρ = 0. Then a treatment of 2.1 in analogy with (2.7)–(2.11) results in(2.23) Actually, there also occurs in the derivation of (2.23) a term ∇(∇P) or ∇(∇E). This term vanishes for transverse waves and is therefore neglected if not mentioned otherwise. This equation is the inhomogeneous analogue of (2.11) telling us that the sources of an electro-magnetic radiation field can be– A dipole moment p or a polarization P with a non-vanishing second time derivative–– A temporally varying current density–The curl of a temporally varying magnetizationAgain a similar equation can be obtained for the magnetic field. We continue now with the application of further simplifications and assume that we have a nonmagnetic material, i.e., that the third term on the r.h.s. of (2.23) vanishes. Actually, all matter has some diamagnetism. But this is a rather small effect of the order of 10-6 so it can be neglected for our purposes. Paramagneticand especially ferromagnetic contributions can be significantly larger for low frequencies.However, even these contributions diminish rapidly for higher frequencies. Consequently theassumption of a nonmagnetic material is a good approximation over a wide range of theelectromagnetic spectrum starting in the IR even for ferromagnetic materials. Furthermore, themore common semiconductors are not ferro-, ferri- or antiferromagnetic and have only a smallconcentration of paramagnetic centres which may be seen in electron paramagnetic resonance(EPR), but which have negligible influence on the optical properties. The only exceptions aresemiconductors which contain a considerable amount of e.g., Mn or Fe ions as does Zn 1-y Mn y Se.We refer the reader to [88D1,91O1,92G1,92Y1,94D1,94G1,96H1,03D1] or to Sect. 16.1 andreferences therein for this class of materials.The current term j in (2.1d) deserves some more consideration. The current is driven by theelectric fieldj E σ= (2.24)where σ is the conductivity. For intrinsic or weakly doped semiconductors, the carrier density issmall and consequently σ is as well. Then the following inequality holds(2.25)In the following we will consider this case and neglect j in (2.1d). For heavily dopedsemiconductors (2.25) is no longer valid and ζ will have some influence on the optical propertiesat least in the infrared (IR). We come back to this situation in connection with plasmons in Chaps.10 and 12.The basic material equation still left in comparison with the vacuum case is now (2.1e)D =ε0E + P .If we proceed with this equation again in the manner of (2.7)–(2.11) the result is(2.26)Equation (2.26) states the well-known fact that every dipole p and every polarization P with anon vanishing second derivative in time radiates an electromagnetic wave.As long as we have no detailed knowledge about the relationships between D , E and P wecannot go beyond (2.26). Now we make a very important assumption. We assume a linearrelationship between P and E :01P E χε= (2.27a)or 00(1)D E E εχεε=+= (2.27b)with1εχ=+ (2.27c)This linear relation is the reason why everything that is treated in the following Chaps. 3 to 18is called linear optics. A linear relation is what one usually assumes between two physicalquantities as long as one does not have more precise information. In principle we can alsoconsider (2.27a) as an expansion of P (E ) in a power series in E which is truncated after the linearterm. We come back to this aspect in Chap. 19.2 The quantities ε and χ are called the dielectricfunction and the susceptibility, respectively. They can be considered as linear response functions[93S1, 98B1,98D1].Both quantities depend on the frequency ω and on the wave-vector k , and they both have a realand an imaginary part as shown for ε.(,);(,)(,)1k k k εεωχχωεω===- (2.28)12(,)(,)(,)k k i k εωεωεω=+ (2.29)The frequency dependence is dominant and will be treated first in Chaps. 3 to 4. We drop the kdependence for the moment but come back to it in connection with spatial dispersion in Chap. 5.In Chap. 6 we discuss the properties of ε as a function of frequency and wave vector or as afunction of time and space.The value of ε(ω) for 0ω is usually called the dielectric constant.In general ε and χ are tensors. For simplicity we shall consider them to be scalar quantities ifnot stated otherwise, e.g., in connection with birefringence in Sect. 3.1.7.Using the linear relations of (2.27) we can transform (2.23) into200()0E E μεεω∇-=(2.30a) where we assumed also that ε(ω) is spatially constant on a length scale of the order of thewavelength of light. Deviations of this assumption are treated in Sects. 17.2–4.If magnetic properties are to be included, a corresponding linear approach would lead to200()()0E E μμωεεω∇-= (2.30b)where μ(ω) is the magnetic permeability. As outlined above we have in the visible for mostsemiconductors μ(ω) - 1.As for (2.12) the solutions of (2.30) are again all functions of the type0()E E f kr t ω=- (2.31)or for our present purposes, i.e. again for the case of plane harmonic waves0exp[()]E E i kr t ω=- (2.32)The relationship between k and ω is however now significantly differentfrom (2.13). It followsagain from inserting the ansatz (2.31 or 32) into (2.30)and now reads:222()c k εωω= (2.33)This relation appears in Chap. 5 again under the name “polariton equation”. It can also bewritten in other forms:(2.34)where λv and k v v refer to the vacuum values of the light wave. For the square root of ε we introduce for simplicity a new quantity ()nω which we call the complex index of refraction1/2()()()()nn ik ωωωεω=+= .3 (2.35) The equations (2.13) and (2.33–35) can be interpreted in the following way. In vacuum anelectromagnetic wave propagates with a wave vector k v which is real and given by (2.13). Inmatter, light propagates with a wave vector k which can be a complex quantity given by (2.34), or, with the help of (2.35), by(2.36)We should notice that k is for complex nnot simply |k | since |k | is always a positive, real quantity. Here k means just neglecting the vector character of k but k can still be a real, imaginaryor complex quantity according to (2.36). The direction of the real part of k , which describes theoscillatory part of the wave, is still parallel to D × B as in (2.20). Writing the plane wave explicitly we have:(2.37)Where ˆkis the unit vector in the direction of k , i.e., in the direction of propagation. Obviously n (ω) describes the oscillatory spatial propagation of light in matter; it is often calledthe refractive index in connection with Snells’ law of refraction. This means that the wavelength λin a medium is connected with the wavelength λv in vacuum by1()v n λλω-= (2.38)In (2.37) κ(ω) describes a damping of the wave in the direction of propagation. This effect isusually called absorption or, more precisely, extinction. We give the precise meaning of these twoquantities in Sect. 3.1.5. Here we compare (2.37) with the well-known law of absorption for thelight intensity I of a parallel beam propagating in z -direction(0)z I I z e α-== (2.39a)With I following from (2.22) to be now(2.39b) where α(ω) is usually called the absorption coefficient, especially in Anglo Saxon literature. InGerman literature α(ω) is also known as “Absorptionskonstante”(absorption constant) anddimensionless quantities proportional to κ(ω) are calle d “Absorptionskoeffizient” or “Absorptionsindex” (absorption coefficient or absorption index). So some care has to be taken regarding what is meant by one or the other of the above terms.Bearing in mind that the intensity is still proportional to the amplitude squared (2.24), a comparison between (2.37) and (2.39) yields(2.40) The phase velocity of light in a medium is now given by (2.15)(2.41) For the group velocity we can get rather complicated dependencies originating fromg v kω∂=∂ (2.42) We return to this aspect later.2.4 Transverse, Longitudinal and Surface WavesThe only solution of (2.9) for light in vacuum is a transverse electromagnetic wave (2.19). This solution exists for light in matter as well. However (2.9) has now with the use of (2.27) the form 0()0D E εεω∇=∇= (2.43) Apart from the above-mentioned transverse solution with E ⊥ k there is a new solution which does not exist in vacuum (εvac ≡ 1), namely()0εω= (2.44a) This means that we can find longitudinal solutions at the frequencies at which ε(ω) vanishes. We call these frequencies correspondingly ωL and note that for(2.44b)In matter, the Maxwell’s equation E B∇⨯=- is still valid. This leads for plane waves in nonmagnetic material to1000()H k E ωμ-=⨯ (2.46) For the longitudinal wave it follows from (2.44) that0H = and 00B H μ== (2.47) The longitudinal waves which we found in matter are not electromagnetic waves but pure polarization waves with E and P opposed to each other with vanishing D , B and H .Until now we were considering the properties of light in the bulk of a medium. The boundary of this medium will need some extra consideration e.g., the interface between vacuum (air) and a semiconductor. This interface is crucial for reflection of light and we examine this problem in Sects. 3.1.1–4; 5.4.2 and 5.6. Here we only want to state that the boundary conditions allow asurface mode, that is, a wave which propagates along the interface and has field amplitudes which decay exponentially on both sides. These waves are also known as surface polaritons for reasons discussed in more detail in Sect. 5.6.2.5 Photons and Some Aspects of Quantum Mechanicsand of Dispersion elationsMaxwell’s equations are the basis of the classical theory of light. They describe problems like light propagation and the diffraction at a slit or a grating e.g., in the frame of Huygen’s principle or of Fourier optics [93S1].In the interaction of light with matter, its quantum nature becomes apparent, e.g., in the photoelectric effect which shows that a light field of frequency ω can exchange energy with matter only in quanta ω . Therefore, the proper description of light is in terms of quantum mechanics or of quantum electrodynamics. However, we shall not go through these theories here in detail nor do we want to address the aspects of quantum statistics of coherent and incoherent light sources, but we present in the following some of their well-known results and refer the reader to the corresponding literature [85G1,92M1, 94A1, 94B1,01M1, 01T1, 02B1, 02D1, 02G1, 02L1, 02Y1] for a comprehensive discussion.The electromagnetic fields can be described by their potentials A and φ byE grad Aφ=-- ; B A =∇⨯ (2.48) where A is the so-called vector potential. Since ∇ · (∇×A ) ≡ 0 the notation of (2.48) fulfills automatically ∇ · B = 0 and reduces the six components of E and B to four.The vector potential A is not exactly defined by (2.48). A gradient of a scalar field can be added. We can choose the so-called Coulomb gauge0A ∇= (2.49) In this case φ is the usual electrostatic potential obeying the Poisson equation:(2.50) In vacuum we still have ρ = 0 and we assume the same for the description of the optical properties of matter.Now we should carry out the procedure of second quantization, for simplicity again for plane waves. A detailed description of how one begins with Maxwell’s equations and arrives at photons within the framework of second quantization is beyond the scope of this book see [55S1, 71F1, 73H1, 76H1, 80H1,85G1,92M1,94A1,94B1]. On the other hand we want to avoid that thecreation and annihilation operators appear like a “deus ex machina”. Therefore we try at least to outline the procedure.First we have to write down the classical Hamilton function H which is the total energy of the electromagnetic field using A and φ. Then we must find some new, suitable quantities p k ,s and q k ,s which are linear in A and which fulfill the canonic equations of motion(2.51) and are thus canonically conjugate variables. Here k is the wave vector of our plane electromagnetic or A -wave and s the two possible transverse polarizations. The Hamilton function reads in these variables:(2.52) This is the usual form of the harmonic oscillator. The quantization condition(2.53) for all k and s = 1, 2 gives then the well-known result for the harmonic oscillator: The electromagnetic radiation field has for every k and polarization s energy steps1()2k k k E n ω=+ with 0,1,2...k n = (2.54) It can exchange energy with other systems only in units of ω . These energy units or quanta are called photons. The term ω /2 in (2.54) is the zero-point energy of every mode of the electromagnetic field.The so-called particle-wave dualism, that is, the fact that light propagates like a wave showing, e.g., diffraction or interference and interacts with matter via particle-like quanta, can be solved by the simple picture that light is an electromagnetic wave, the amplitude of which can have only discrete values so that the energy in the waves just fulfills (2.54).From the above introduced, or better, postulated quantities p k,s and q k,s we can derive by linear combinations operators a k,s and a k,s with the following properties: If a k,s acts on a state which contains n k,s quanta of momentum k and polarization s it produces a new state with n k,s −1 quanta. Correspondingly, a k,s increases n k,s by one. We call therefore a k,s and a k,s annihilation and creation operators, respectively. Since the operators a k,s and a k,s describe bosons (see below), their permutation relation is(2.55a) This holds for equal k and s . The commutator is zero otherwise. The operator a k,s a k,s acting on a photon state gives the number of photons n k,s times the photon state and is therefore called the number operator. Summing over all possible k -values and polarizations s gives finally the Hamilton operator(2.55a) It is clear to the author that the short outline given here is not sufficient to explain the procedure to a reader who is not familiar with it. However, since the intent is not to write a textbook on quantum electrodynamics, we want to stress here only that the electromagnetic radiation field in vacuum can be brought into a mathematical form analogous to that of the harmonic oscillator, and that quantum mechanics gives for every harmonic oscillator theenergetically equidistant terms of (2.54).The harmonic oscillator is one of the fundamental systems, which has been investigated in physics and is understood in great detail. In theoretical physics a problem can be considered as “solved” if it can be rewritten in the form of the harmonic oscillator. Apart from the electromagnetic radiation field in vacuum, we will come across some other systems which are treated in this way. For those readers who are not familiar with the concept of quantization and who wish to study the procedure in a quiet hour by themselves, we recommend the above given references.Here are some more results: The two basic polarizations of single quanta of the electromagnetic field, – of the photons – are left and right circular ζ-and ζ+, respectively. A linearly polarized wave can be considered as a coherent superposition of a left and right circularly polarized one with equal frequencies and wave vector k . The term coherent means that two light beams have a fixed-phase relation relative to each other. The component of the angularmomentum s in the direction of the quantization axis which is parallel to k is for photons thuss =± (2.56) This means that photons have integer spin and are bosons. The third possibility s // = 0 expected for spin one particles is forbidden, because longitudinal electromagnetic waves do not exist at least in vacuum.Photons in thermodynamic equilibrium are described by Bose-statistics. The occupation probability f BE of a state with frequency ω is given by(2.57) where T is the absolute temperature and k B is Boltzmann’s constant.The chemical potential μ which could appear in (2.57) is zero in thermal equilibrium, since the number of photons is not conserved.An approach to describe non-thermal photon fields e.g. luminescence by a non-vanishing μ and Kirchhoff’s law in the sense of a generalized Planck’s law is found in [82W1, 92S2,95D1]. The momentum p of a photon with wave vector k is given, as for all quanta of harmonic waves, byp k = (2.58) where k is the real part of the wave vector, which describes as already mentioned the oscillatory, propagating aspect of the plane wave.To summarize, we can state that photons are bosons with spin ± , energy ω and momentum k which propagate according to the wave equations.A very important property of particles in quantum mechanics is their dispersion relation. By this we mean the dependence of energy E or frequency ω on the wave vector k i.e., the E (k ) or ω(k ) relation. For photons in vacuumwe find the classical relation given already in (2.13)E ck ω== (2.59) The dispersion relation for photons in vacuum is thus a linear function with slope c as shown in Fig. 2.2. Correspondingly we find again both for phase and group velocity with (2.15) ph g v v c == (2.60)We conclude this subsection with an explanation of energy units. In the SI system the energy unit is 1Nm = 1 kgm2/s2 with the following identity relations1Nm = 1mkg s-2= 1VAs = 1Ws = 1J. (2.61a)Since the energies of the quanta in optical spectroscopy are much smaller, we frequently use the unit 1 eV. This is the energy that an electron gains if it passes, in vacuum, through a potential difference of one volt, resulting in1 eV = 1.60217733× 10-19 J ≈ 1.6 × 10-19 J (2.61b)In spectroscopy another measure of energy is frequently used the wave number. The definition is as follows. One expresses the energy of a (quasi-) particle by the number of wavelengths per cm of a photon with the same energy. So(2.61c)Another quantity that is sometimes confused with the wave number, which gives the energy and is therefore a scalar quantity, is the wave vector, since it has also the dimension 1/length.The amount of the (real part of the) wave vector is given by k = 2π/λ, where λ is the wavelength of the corresponding quantum or particle (electron, phonon, photon, etc.). The direction of k is the direction of propagation, i.e., k is normal to the wave-front, in the case of light in vacuum or in matter normal to D×B. The quantity k is very closely related to the (quasi-)momentum of the (quasi-)particle p throughp k(2.58)For the discussion of the concept of quasi-momentum see, e.g., Sects. 5.2, 3 and 5 or [98B2]. The dispersion relation of (quasi-) particles is thus given by E(k). The wave vector of light is in the visible in vacuum, falling in the range of a few times 104 cm-1 while the border of the first Brillouin zone (see Sect. 7.2) is of the order of 108cm-1.It is obvious, that a quantity like a wave-number vector (Wellenzahlvektor) is ill-defined and does not exist!2.6 Density of States and Occupation ProbabilitiesA quantity which is crucial in quantum mechanics for the properties of particles is their density of states. It enters, e.g., in Fermi’s golden rule which allows one to calculate transition probabilities.。

GRE(VERBAL)阅读模拟试卷4(题后含答案及解析)

GRE（VERBAL）阅读模拟试卷4(题后含答案及解析) 题型有：1. PART ONEPART ONE （Time：30 minutes 38 Questions）SECTION 3Directions: Each passage in this group is followed by questions based on its content. After reading a passage, choose the best answer to each question. Answer all questions following a passage on the basis of what is stated or implied in the passage.Late-eighteenth-century English cultural authorities seemingly concurred that women readers should favor history, seen as edifying, than fiction, which was regarded as frivolous and reductive. Readers of Marry Ann Hanway’s novel Andrew Stewart, or the Northern Wanderer, learning that its heroine delights in David Hume’s and Edward Gibbon’s histories, could conclude that she was more virtuous and intelligent than her sister, who disdains such reading. Likewise, while the na?ve, novel-addicted protagonist of Jane Austen’s Northanger Abbey, Catherine Morland, finds history a chore, the sophisticated, sensible character Eleanor Tilney enjoys it more than she does the Gothic fiction Catherine prefers. Yet in both cases, the praise of history is more double-edged than it might actually appear. Many readers have detected a protofeminist critique of history in Catherine’s protest that she dislikes reading books filled with men “and hardly any women at all.”Hanway, meanwhile, brings a controversial political edge to her heroine’s reading, listing the era’s two most famous religious skeptics among her preferred authors. While Hume’s history was generally seen as being less objectionable as his philosophy, there were widespread doubts about his moral soundness even as a historian by the time that Hanway was writing, and Gibbon’s perceived tendency to celebrate classical paganism sparked controversy from the first appearance of his history of Rome.1．The author’s primary purpose is thatA．the evidence used in support of a particular argument is questionableB．a distinction between two genres of writing has been overlookedC．a particular issue is more complex than it might appearD．two apparently different works share common featuresE．two eighteenth-century authors held significantly different attitudes toward a particular正确答案：A解析：A选项中的a particular argument指的是文化权威们认为“女人应该多读历史”的观点，evidence指的是第二、三句。

小学上册A卷英语第六单元真题

小学上册英语第六单元真题英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The __________ is beautiful with all the colors of fall. (树木)2.Which month has Thanksgiving in the United States?A. OctoberB. NovemberC. DecemberD. JanuaryB3.My uncle takes me fishing ____.4.Which animal is known for being very intelligent?A. GoldfishB. DogC. SlothD. TurtleB5.What do we call a young male horse?A. ColtB. FillyC. FoalD. Pony6.What do you call the hard outer covering of an egg?A. ShellB. MembraneC. YolkD. AlbumenA7. A _______ (马) can be very strong.8.My grandma has a wealth of __________ (知识) about history.9.What do you call the person who drives a bus?A. TeacherB. DriverC. PilotD. Engineer10.What is 5 plus 7?A. 11B. 12C. 13D. 14C11.The _______ (小果子狸) has a long tail and is very agile.12.The ________ is a busy animal that collects food.13.I like to go ________ (购物) for new books.14. A _____ (54) is a large group of islands.15.What do you call a type of large mammal that swims?A. WhaleB. SharkC. DolphinD. All of the above16.The dog is ___ the kids. (watching)17.What is the primary color of the sky on a clear day?A. RedB. BlueC. GreenD. YellowB18. A __________ (温室) is ideal for growing delicate plants.19.In a chemical reaction, products are formed from ______.20.The nucleus of an atom contains protons and ______.21.What is the name of the famous ancient city in Israel?A. JerusalemB. Tel AvivC. HaifaD. NazarethA Jerusalem22.What do we call a young female goat?A. KidB. CalfC. LambD. FoalA Kid23.The _______ (Radio) became popular in the early 20th century.24.Plants are vital for our _____ (生存).25.We eat dinner _____ the dining room. (in)26.I have a pet _______ (小鸟) named Tweety.27.I have a _____ (卡片游戏) with my friends.28.The invention of the airplane revolutionized ________.29. A _____ (shrub) can provide privacy in a yard.30.What do you call a baby polar bear?A. CubB. KitC. PupD. Calf31.The __________ (工厂) produces goods for the city.32.An element's properties are determined by its ______ structure.33.The term "catalyst" refers to a substance that speeds up a chemical _______.34.Which is a primary color?A. GreenB. BlueC. OrangeD. Purple35.What do you call the first month of the year?A. DecemberB. JanuaryC. FebruaryD. MarchB January36.I want to ___ a friend. (make)37._____ (dish) garnished with fresh herbs is delightful.38.Which of these is a type of pasta?A. BreadB. SpaghettiC. RiceD. ChickenB39.What is the opposite of "hot"?A. WarmB. CoolC. ColdD. TemperatureC40.The ancient Egyptians built their pyramids using ______ (石块).41.What do we call a large area of flat land?A. MountainB. PlainC. PlateauD. ValleyB Plain42. A __________ is a measurement of how much matter is in an object.43.What is 3 + 5?A. 6B. 7C. 8D. 944.The __________ (历史的影响) shapes our experiences.45.What do you call a small, soft fruit?A. GrapeB. CherryC. RaspberryD. All of the above46.My sister loves her _______ (我妹妹爱她的_______).47.My brother is _____ (young/old).48.What is the capital of Hungary?A. BudapestB. PragueC. ViennaD. BucharestA49.The chemical symbol for potassium is _______.50.The Earth's core is primarily made up of ______ and nickel.51.How many days are in a week?A. FiveB. SixC. SevenD. Eight52. A __________ is a small creature that often hides under rocks.53.Sodium bicarbonate is another name for _____ (baking soda).54.The dolphin swims _______ (迅速) in the ocean.55.What do we call the person who studies space?A. AstronomerB. AstrologerC. MeteorologistD. PhysicistA Astronomer56.What is the capital city of the Maldives?A. MaléB. Addu CityC. FuvahmulahD. Kulhudhuffushi57.How many teeth does a typical adult human have?A. 28B. 30C. 32D. 34C58.What is the name of the famous American holiday celebrated on the last Thursday of November?A. ThanksgivingB. Labor DayC. Independence DayD. Veterans DayA59.What do we call the study of living things?A. ChemistryB. BiologyC. PhysicsD. Astronomy60.What do we call a person who studies prehistoric life?A. PaleontologistB. ArchaeologistC. HistorianD. AnthropologistA61.What is the name of the boundary around a black hole?A. Event HorizonB. SingularityC. Photon SphereD. Gravity Well62.This girl, ______ (这个女孩), is very creative.63.What is the main ingredient in pancakes?A. EggsB. FlourC. SugarD. Water64.__________ (实验数据) must be recorded carefully for analysis.65.What is the capital of Saint Lucia?A. CastriesB. Vieux FortC. SoufrièreD. Gros IsletA66.What do you call a vehicle with two wheels?A. CarB. TruckC. BicycleD. Bus67.The lizard can lose its _______ (尾巴) to escape.68.The first successful blood transfusion was performed in ______ (17世纪).69.What is the tallest animal in the world?A. ElephantB. GiraffeC. HorseD. BearB70.ers bloom in ______ (夜间). Some flo71.Gold is a _______ metal that does not tarnish easily. (贵重)72.crater) is formed by volcanic activity. The ____73.The ________ was a famous explorer who sailed around the world.74.I love to ride my ______ (bike) in the park.75.The skin of a snake is ______.76.What is the currency used in the United States?A. EuroB. YenC. DollarD. Peso77.What is the name of the longest river in the world?A. AmazonB. NileC. MississippiD. Yangtze78.My dream job is to _______ a scientist.79.I found a _____ (shell/stone) on the beach.80.I have _____ (one/two) brother(s).81.My grandma is very creative at ____.82.I love going to the _______ (图书馆) after school.83.I enjoy making new friends with my toy ________ (玩具名称).84.What do you call the study of animals?A. ZoologyB. BotanyC. BiologyD. Ecology85.What is the value of Pi (π.approximately?A. 2.14B. 3.14C. 4.14D. 5.14B86.Liquid water freezes at _____ ( degrees Celsius).87. A chemical change produces new ______.88.What is the first letter of the English alphabet?A. AB. BC. CD. DA89.The arrangement of electrons around an atom's nucleus determines its ______ properties.90.I see a squirrel in the ___. (park)91.I feed my dog _______ (狗粮) twice a day.92.I enjoy listening to ________ (民谣音乐) at home.93.I love to bake ______ with my grandma.94. A wallaby is smaller than a _______ (袋鼠).95.My brother loves to read ____.96.I want to _____ (learn/forget) new things.97.What is the name of the famous clock in London?A. Big BenB. Eiffel TowerC. Leaning Tower of PisaD. ColosseumA98.The _______ (蟑螂) is a creepy insect.99.My sister is very _______ (形容词) at dancing. 她的舞蹈很 _______ (形容词). 100.The __________ (历史的遗产) shapes our collective memory.。

地球介绍英语作文

Earth,the third planet from the Sun in our solar system,is a unique celestial body teeming with life and diverse ecosystems.It is the only known planet to support life,and its characteristics make it a fascinating subject for study.Geographical Composition:The Earth is composed of several layers,including the crust,mantle,outer core,and inner core.The crust,which is the outermost layer,is divided into tectonic plates that float on the semifluid asthenosphere.These plates movement causes geological phenomena such as earthquakes and volcanic eruptions.Atmosphere:The atmosphere of Earth is a mixture of gases,primarily nitrogen78%and oxygen21%, with trace amounts of other gases like argon,carbon dioxide,and water vapor.This atmosphere protects life by absorbing harmful solar radiation and regulating the planets temperature.Hydrosphere:Water covers approximately71%of the Earths surface,mostly in the form of oceans.The remaining fresh water is found in rivers,lakes,and ice caps.Water is essential for life, and its distribution has a significant impact on the planets climate and ecosystems.Biosphere:The biosphere is the global sum of all ecosystems,including all living organisms and their interactions with the environment.It extends from the bottom of the oceans to the top of the atmosphere and includes a wide variety of habitats,from deserts to rainforests and from the poles to the equator.Climate and Weather:Earths climate is influenced by various factors,including the tilt of its axis,the distance from the Sun,and the composition of its atmosphere.Weather patterns,driven by the movement of air and water,result in a wide range of conditions from extreme cold in the polar regions to intense heat in the tropics.Human Impact:Human activities have significantly altered the Earths environment.Deforestation, pollution,and climate change are some of the challenges that threaten the planets ecosystems and the survival of many species.Efforts to mitigate these impacts,such as conservation and sustainable development,are crucial for the future of our planet.Exploration and Future:The exploration of Earth,both its surface and its depths,continues to reveal new insights into its history and the processes that shape it.As we look to the future,understanding and preserving Earths natural systems will be essential for the continued wellbeing of all its inhabitants.In conclusion,Earth is a complex and dynamic planet with a rich tapestry of life and natural phenomena.Its study offers a wealth of knowledge about our universe and our place within it,emphasizing the importance of environmental stewardship and the pursuit of scientific discovery.。

断层解剖学测试题

In neuroanatomy, a neuropil is a region between neuronal cell bodies in the gray matter of the brain and spinal cord (i.e. the central nervous system). It consists of a dense tangle of axon terminals, dendrites and glial cell processes. It is where synaptic connections are formed between branches of axons and dendrites.[1]
断层解剖学测试题NeuroilFrom Wikipedia, the free encyclopedia
Jump to: navigation, search
In neuroanatomy, a neuropil is a region between neuronal cell bodies in the gray matter of the brain and spinal cord (i.e. the central nervous system). It consists of a dense tangle of axon terminals, dendrites and glial cell processes. It is where synaptic connections are formed between branches of axons and dendrites.[1]
第三章
胸部
1、胸骨角平面的标志性结构有那些？
White matter, which is mostly composed of axons and glial cells, is generally not considered to be a part of the neuropil.[citation needed]

当代研究生英语第七单元 B课文翻译

价格的利润生物公司正在吞噬可改变动物DNA序列的所有专利。

这是对阻碍医学研究发展的一种冲击。

木匠认为他们的贸易工具是理所当然的。

他们买木材和锤子后，他们可以使用木材和锤子去制作任何他们所选择的东西。

多年之后来自木材厂和工具储藏室的人并没有任何进展，也没有索要利润份额。

对于那些打造明日药物的科学家们来说，这种独立性是一种罕见的奢侈品。

发展或是发现这些生物技术贸易中的工具和稀有材料的公司，对那些其他也用这些工具和材料的人进行了严格的监控。

这些工具包括关键基因的DNA序列，人类、动物植物和一些病毒的基因的部分片段，例如，HIV,克隆细胞，酶，删除基因和用于快速扫描DNA样品的DNA 芯片。

为了将他们这些关键的资源得到手，医学研究人员进场不得不签署协议，这些协议可以制约他们如何使用这些资源或是保证发现这些的公司可以得到最终结果中的部分利益。

许多学者称这抑制了了解和治愈疾病的进程。

这些建议使Harold得到了警示，Harold是华盛顿附近的美国国家卫生研究院的院长，在同年早期，他建立了一个工作小组去调查此事。

由于他的提早的调查，下个月出就能发布初步的报告。

来自安阿伯密歇根大学的法律教授，该工作组的主席Rebecea Eisenberg说，她们的工作组已经听到了好多研究者的抱怨，在它们中有一份由美国联合大学技术管理组提交的重量级的卷宗。

为了帮助收集证据，NIH建立了一个网站，在这个网站上研究者们可以匿名举报一些案件，这些案件他们相信他们的工作已经被这些限制性许可证严重阻碍了。

迫使研究人员在出版之前需要将他们的手稿展示给公司的这一保密条款和协议是投诉中最常见的原因之一。

另一个问题是一些公司坚持保有自动许可证的权利，该许可证是有关利用他们物质所生产的任何未来将被发现的产品，并且这些赋予他们对任何利用他们的工具所赚取的利润的支配权利的条款也有保有的权利。

Eisenberg说：“如果你不得不签署了许多这样的条款的话，那真的是一个大麻烦”。

《学术英语》预习材料Unit 1昆工

Unit One EconomicsWhat to Learn1.There is talk of raising the admission requirements to the number of students on campus.有传闻称将通过提高录取条件来限制在校学生的数目。

restricternment officials visited the earthquake zone on Thursday morning to the relief effort.星期四上午，政府官员视察地震灾区，以协调救援工作。

coordinate3.The movement towards democracy in Latin America and the foreign debt problems that haveit have gone out of focus.拉丁美洲的民主运动和困扰该地区的外债问题已经淡出人们的视线。

plagued4. Supporters of the death penalty argue that it would deter criminals from carrying guns.死刑的支持者认为它能阻吓罪犯携带枪支。

deter5.Integrating the pound with other European currencies could cause difficulties.把英镑与欧洲其他货币合并会引发许多难题。

Integrating6. The party vowed to incorporate environmental considerations into all its policies.该党宣誓要把环境因素纳入它所有的政策当中。

incorporate7. For some inexplicable reason, the investors decided to pull out.由于某些无法说明的原因, 投资商们决定撤出投资。

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is deﬁned by
͸ϱ
q(r, ␪) = q¯m(r)sin ␮␪
ห้องสมุดไป่ตู้
(1)
m =1
where ␮ = m␲/␣.
Kirchhoff Plate Theory
First consider the bending problem of the aforementioned sectorial plate under the loading deﬁned in (1), based on the Kirchhoff plate theory. The equilibrium equations of the plate problem at hand are given by Timoshenko and WoinowskyKrieger (1959):
INTRODUCTION
A plate is typically considered to be thin when the ratio of its thickness to a representative lateral dimension is less than 1/20. Using the Kirchhoff (or classical thin) plate theory, the bending solutions may be determined with good accuracy. However, if the aforementioned ratio is greater than 1/20, the bending solutions will not be accurate. For example, the deﬂections will be underpredicted from neglecting the effect of transverse shear deformation. These inaccuracies may be largely eliminated by using the Mindlin plate theory, which takes into consideration the ﬂexibility of the plate due to the transverse shear deformation.
clamped or free. The ﬂexural rigidity of the isotropic plate is given by D = Eh3/[12(1 Ϫ ␯2)], where E is the Young’s modulus and ␯ Poisson’s ratio. The shear modulus of the plate is G = E/[2(1 ϩ ␯)]. For analysis, it is expedient to adopt the polar coordinates system for such a plate shape, with the origin
BENDING SOLUTIONS OF SECTORIAL MINDLIN PLATES FROM KIRCHHOFF PLATES
By C. M. Wang,1 Member, ASCE, and G. T. Lim2
ABSTRACT: This study presents exact relationships between the bending solutions of sectorial plates based on the Kirchhoff (or classical thin) plate theory and the Mindlin plate theory. While the former plate theory neglects the effect of transverse shear deformation, the latter theory allows for this effect, which becomes signiﬁcant when dealing with thick plates and sandwich plates. The considered sectorial plates have simply supported radial edges, while the circular curved edge may be either simply supported, or clamped or free. The availability of such relationships allow easy conversion of the existing Kirchhoff sectorial plate solutions into the corresponding Mindlin solutions, thus bypassing the need to solve the more complicated bending equations of the Mindlin plates. The use of the relationships is illustrated using some sectorial plate examples, and sample solutions obtained were checked with previous researchers’ results and those computed from the software ABAQUS.
1Assoc. Prof., Dept. of Civ. Engrg., National Univ. of Singapore Kent Ridge, Singapore 119260. E-mail: cvewcm@.sg
2Grad. Student, Dept. of Civ. Engrg., National Univ. of Singapore, Kent Ridge, Singapore.
Wang and his coworkers embarked on a research project to develop relationships between the solutions of classical and higher-order plate theories. So far, the research has yielded very interesting bending relationships that are exact and yet simple to use, but they can only be applied to certain classes of plate shapes, boundary, and loading conditions. For instance, exact deﬂection relationships were found for (1) simply supported polygonal plates under arbitrary loading conditions (Wang and Alwis 1995), (2) axisymmetric bending of circular and annular plates (Wang and Lee 1996; Wang 1997), and (3) the bending of Levy plates (Wang et al. 1999). As a continuation of this project, this paper focuses on the elastic bending problem of sectorial plates with simply supported radial edges while the circular edge may be either simply supported, or clamped or free. The relationships between the bending solutions for such plates are derived herein. On the basis of existing exact Kirchhoff solutions for these plates [see books by Timoshenko and Woinowsky-Krieger (1959) and Mansﬁeld (1989)], the relationships enable the easy deduction of the corresponding exact Mindlin plate solutions. These new sectorial Mindlin plate solutions (and thus the relationships) are veriﬁed by comparison studies with previous researchers’ results [e.g., those given by Cheung and Chan (1981)] wherever possible. For some cases, the writers used the software ABAQUS to provide independent checks to ensure the correctness of the results.
Note. Associate Editor: Roberto Ballarini. Discussion open until September 1, 2000. To extend the closing date one month, a written request must be ﬁled with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 26, 1999. This paper is part of the Journal of Engineering Mechanics, Vol. 126, No. 4, April, 2000. ᭧ASCE, ISSN 0733-9399/00/0004-0367– 0372/$8.00 ϩ $.50 per page. Paper No. 20139.