Relativistic wavefunctions on the Poincare group

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大学物理下册常用英文单词

大学物理下册常用英文单词大学物理下册常用英文单词及短语Mechanical Oscillation（ vibration）机械振动Electromagnetic Oscillation 电磁振荡Simple Harmonic Motion (SHM) simple harmonic oscillator 简谐振动Superposition 合成、叠加Spring 弹簧Period 周期frequency Angular frequency 频率、角频率Phase initial phase 相位、初相rotating vector 旋转矢量vertical directions 垂直方向Mechanical Wave 机械波Plane Harmonic Waves 平面简谐波Propagation 传播Huygen’s Principle惠更斯原理Interference of Waves 波的干涉Standing Wave (Stationary Waves) 驻波 traveling waveDoppler Effect 多普勒效应Medium 介质Transverse waves 横波 Longitudinal waves 纵波:Wave Front：(波前)Wavelength 波长Crest 波峰 trough 波谷Wave speed 波速Randomly 随机Position 位置Displacement 位移wave equation 波动方程oscillating curve 振动曲线kinetic, potential and total energy 动能、势能和总能量energy flow density 能流密度spherical waves 球面波Principle of Independent Propagation 波的独立传播原理Principle of Superposition of Waves 波的叠加原理Coherent wave 相干波phase difference 相位差wave path difference 波程差constructive 干涉相长destructive 干涉相消interface draft(干涉图样)Node 波节 Antinode 波幅important features:重要性质Half-Wave Loss 半波损失wavy thinner medium 波疏介质wavy denser medium 波密介质toward 向着far away 背离Optics 光学Visible light可见光 light vector 光矢量Luminescence 发光Interference of Light 光的干涉 Coherence Light 相干光incoherent superposition 非相干叠加 coherent superposition 相干叠加Optical Distance 光程 Optical Path Difference 光程差Film Interference 薄膜干涉Reflection反射 Refraction折射 Diffraction 衍射Slit 缝 Fraunhofer 夫朗和费Grating 光栅Polarized Light偏振光stable interference pattern 稳定杆射花样light intensity 光强Young’s Interference杨氏干涉central bright fringe 中央明纹first-order bright fringes 第一级明纹adjacent bright (or dark) fringes 相邻明暗纹Lloyd’s mirror experiment络唉镜实验the refractive index折射率Thin Lens 薄透镜parallel light 平行光in-phase 同相位primary optical axis 主光轴secondary optical axis 副光轴primary focus 主焦点 focal plane 焦平面optical center 光心Equal-inclination interference, isoclinal interference 等倾干涉Film 薄膜Equal-thickness interference(等厚干涉)Wedge Interference 劈尖干涉bright or dark fringes 明暗纹distance 距离Newton’s ring 牛顿环Michelson’s Interferometer迈克尔逊干涉仪Huygens-Fresnel’s Principle惠更斯菲涅尔原理incident lights 入射光Fresnel Diffraction 菲涅尔衍射Vertical 垂直的center bright fringe 中央明纹Circular Hole Diffraction圆孔衍射Ariy disk艾丽斑Rayleigh criterion 瑞丽判据Ruling 刻痕glass plate 平板玻璃grating constant 光栅常数diffraction angle 衍射角grating equation 光栅方程Primary maximum fringe 主机大条纹spectrum line 谱线visible frings 可见条纹incident normally 垂直入射missing order 缺级Dark fringe 暗纹equal to 等于Grating spectrum 光栅光谱Oblique incidence 斜入射Nature Light 自然光Polarized Light 偏振光Linearly Polarized Light 线偏振光plane polarized light 平面偏振光Partially Polarized Light 部分偏振光Polarizer 起偏、偏振片Analyser 检偏The polarized direction 偏振方向Malus Law 马吕斯定律Extinction消光Snell’s Law (refraction law)死聂耳折射定律Brewster’s Law 布如斯特定律full polarization angle全偏振角or Brewster angle布儒斯特角the Kinetic Theory of Gases气体动理论Equilibrium State平衡态Ideal Gas 理想气体Equipartition Theory of Energy 能量均分理论Internal energy 内能Maxwell Speed Distribution麦克斯韦速率分布Boltzmann Distribution 波尔滋蔓速率分布The Mean Free Path 平均自由程vast number of molecular 大量分子Thermodynamics 热力学 Thermology 热学macroscopic quantities 宏观态microcosmic quantities 微观态Avogadro’s constant 阿伏伽德罗常量Brownian Motion 布朗运动Attractive 吸引Repulsive 排斥isolated System 孤立系统State Parameters 状态参量Ideal Gas Equation of State 理想气体状态方程geometric parameter 几何参数mechanical parameter 力学参数thermal parameter 热学参数fixed 固定的The root-mean-square(RMS) speed 方均根速率The degrees of freedom of molecule 分子自由度monatomic gas 单原子分子diatomic gas双原子分子polyatomic gas 多原子分子Energy Equipartition theorem 能量均分定理degree of translational freedom 平动自由度translational kinetic energy 平动动能average kinetic energy 平均动能 rotational kinetic energy 转动动能thermodynamic temperature 热力学温度speed distribution function 速率分布函数the mean value 平均值collisions 碰撞Normalization归一化Curve 曲线the most probable speed 最可几速率average speed平均速度The mean free path 平均自由程The First Law of Thermodynamics热力学第一定律Adiabatic Process绝热过程Carnot Cycle卡诺循环Cyclical Processes 循环过程The Second Law of Thermodynamics热力学第二定律Reversible可逆的 & Irreversible不可逆的 ProcessStatistical Meaning 统计意义Entropy 熵Quasistataic process 准静态过程Work功、Heat热、Internal Energy内能Piston 活塞Integration 积分Isochoric process等容过程Isobaric process等压过程Isothermal Processes等温过程Heat Capacities热容Specific Heat capacity比热容molar heat capacity 摩尔热熔Mayer’s formular 迈耶公式Adiabatic equation 绝热过程adiabatic curve 绝热曲线isotherm curve 等温曲线heat engine 热机p-V diagram p-V图original state 初态Clockwise 正循环Anticlockwise 逆循环Refrigerator 电冰箱The efficiency of heat engines 热机效率refrigeration coefficient 制冷系数Carnot cycle 卡诺循环isothermal expansion 等温膨胀adiabatic expansion 绝热膨胀compression 压缩Kelvin statement开尔文说法Clausius statement克劳修斯说法work transfers into heat 功热转换Carnot Theorem卡诺定理Statistical Meaning 统计平均。

The Physics of Ghost Imaging

obj
dρo f (ρo ) δ (ρo +
ρi ) m
(2)
where I (ρi ) is the intensity in the image plane, ρo and ρi are 2-D vectors of the transverse coordinates in the object and image planes, respectively, and m = si /so is the magniﬁcation factor. In reality, limited by the ﬁnite size of the imaging system, we may never have a perfect pointto-point correspondence. The incomplete constructive-destructive interference turns the pointto-point correspondence into a point-to-“spot” relationship. The δ -function in the convolution of Eq. (2) will be replaced by a point-spread function: I (ρ i ) =
obj
dρo f (ρo ) somb
R ω ρi ρo + so c m
(3)
where the sombrero-like function, or the Airy disk, is deﬁned as somb(x) = 2J1 (x) , x
and J1 (x) is the ﬁrst-order Bessel function, and R the radius of the imaging lens, and R/so is known as the numerical aperture of an imaging system. The ﬁnite size of the spot, which is deﬁned by the point-spread function or the Airy disk, determines the spatial resolution of the imaging setup. It is clear from Eq. (3) that a larger imaging lens and shorter wavelength will result in a narrower point-spread function, and thus a higher spatial resolution of the image. Ghost imaging, in certain aspects, has the same basic feature of classical imaging, such as the unique point-to-point image-forming relationship between the object plane and the image plane. Diﬀerent from classical imaging, the radiation stopped on the imaging plane does not “come” from the object plane, instead it comes directly from the light source. More importantly, the image is observed in the joint detection between two independent photodetectors, one measures the scattered or transmitted light from the object, another measures light directly coming from the source at each point in the image plane. The point-to-point correlation is determined by the nonlocal behavior of a pair of photons: neither photon-one nor photon-two “knows” where to arrive. However, if one of them is observed at a point on the object plane, its twin must arrive at a unique corresponding point on the image plane. The ﬁrst ghost imaging experiment was demonstrated by Pittman et al. in 1995 [1] [2]. The schematic setup of the experiment is shown in Fig. 2. A continuous wave (CW) laser is used to pump a nonlinear crystal to produce an entangled pair of orthogonally polarized signal (e-ray of the crystal) and idler (o-ray of the crystal) photons in the nonlinear optical process of spontaneous parametric down-conversion (SPDC). The pair emerges from the crystal collinearly with ωs ∼ = ωi ∼ = ωp /2 (degenerate SPDC). The pump is then separated from the signal-idler pair by a dispersion prism, and the signal and idler are sent in diﬀerent directions by a polarization beam splitting Thompson prism. The signal photon passes through a convex lens of 400mm focal 2

On the negative effective mass density in acoustic metamaterials

There is no actual negative mass/mass density in real materials indeed. The negative effective mass/mass density is the result of inaccurate modeling of acoustic metamaterials. In the present study, we consider a one-dimensional lattice consisting of lattice masses each of which contains an internal mass. This simple lattice system is employed to illustrate how the
On the negative effective mass density in acoustic metamaterials
H.H. Huang a, C.T. Sun a,*,G.L. Huang b
a School of Aeronautics and Astronautics, Purdue University, W. Lafayette, IN 47907, USA b Department of Systems Engineering, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA
* Corresponding author. Tel.: +1 765 494 5130; fax: +1 765 494 0307. E-mail address: sun@ (C.T. Sun).

Dislocations in wave trains

[ 165
I
11-2
166
J. F. Eye aiad 31. V. Berry
within a s~noothlyvarying (approximately Gaussian) envelope (figure l a ) . The echo from the rough surface was of longer duration (figme I b), coiic;istiirg of 50 or more approximately sinr~soidaawaves of fluctuating amplitude. By moving the microphone along s line it was possible to find points in space where the envelope of the echo had zero strength for some particular time deiay. Figure 2 c shows a
Dislocations in wave trains
167
symmetrical example of the time variation of the signal a t such a point; the broken line is the envelope. By moving the microphone first to one side (figures 2u, b) and then the other (figure 2d) one could follow the crests A and C continuously through the transition and see that a new crest B was created between them. Figures 3 a,b, c: d show an antisymmetrical example of the same thing; again a new crest B appears between crests A and C. I n general, the phase of the carrier wave relative to the envelope is such that one observes an unsymmetrical curve which is a lineartisymmetrical cases.

OpticsExpress

#189766 - $15.00 USD (C) 20132013; revised 3 Jun 2013; accepted 3 Jun 2013; published 18 Jul 2013 29 July 2013 | Vol. 21, No. 15 | DOI:10.1364/OE.21.017769 | OPTICS EXPRESS 17769
©2013 Optical Society of America
OCIS codes: (190.4370) Nonlinear optics, fibers; (320.7110) Ultrafast nonlinear optics; (320.5540) Pulse shaping; (060.5295) Photonic crystal fibers.
Abstract: Formation of parabolic pulses at femtosecond time scale by means of passive nonlinear reshaping in normally dispersive optical fibers is analyzed. Two approaches are examined and compared: the parabolic waveform formation in transient propagation regime and parabolic waveform formation in the steady-state propagation regime. It is found that both approaches could produce parabolic pulses as short as few hundred femtoseconds applying commercially available fibers, specially designed all-normal dispersion photonic crystal fiber and modern femtosecond lasers for pumping. The ranges of parameters providing parabolic pulse formation at the femtosecond time scale are found depending on the initial pulse duration, chirp and energy. Applicability of different fibers for femtosecond pulse shaping is analyzed. Recommendation for shortest parabolic pulse formation is made based on the analysis presented.

J. Phys. Condens. Matter 15 (2003) R841–R896

I NSTITUTE OF P HYSICS P UBLISHING J OURNAL OF P HYSICS:C ONDENSED M ATTER J.Phys.:Condens.Matter15(2003)R841–R896PII:S0953-8984(03)35571-7TOPICAL REVIEWNanomagneticsR SkomskiDepartment of Physics and Astronomy and Center for Materials Research and Analysis,University of Nebraska,Lincoln,NE68588,USAReceived17January2003Published12May2003Online at /JPhysCM/15/R841AbstractMagnetic nanostructures,such as dots and dot arrays,nanowires,multilayersand nanojunctions,are reviewed and compared with bulk magnets.Theemphasis is on the involved physics,but some applications are also outlined,including permanent magnets,soft magnets,magnetic recording media,sensors,and structures and materials for spin electronics.The consideredstructural length scales range from a few interatomic distances to aboutone micrometre,bridging the gap between atomic-scale magnetism and themacroscopic magnetism of extended bulk and thin-ﬁlm magnets.This leads toa rich variety of physical phenomena,differently affecting intrinsic and extrinsicmagnetic properties.Some speciﬁc phenomena discussed in this revieware exchange-spring magnetism,random-anisotropy scaling,narrow-wall andconstricted-wall phenomena,Curie temperature changes due to nanostructuringand nanoscale magnetization dynamics.Contents1.Introduction8422.Magnetic nanostructures8442.1.Particles and clusters8442.2.Thinﬁlms and multilayers8452.3.Particle arrays and functional components8462.4.Nanowires8472.5.Nanocomposites and other bulk materials8483.Atomic-scale effects8483.1.Magnetic moment8493.2.Magnetization and magnetic order8503.3.Anisotropy8534.Mesoscopic magnetism8554.1.Phenomenology of hysteresis8564.2.Micromagnetic background8574.3.Fundamental magnetization processes860 0953-8984/03/200841+56$30.00©2003IOP Publishing Ltd Printed in the UK R841R842Topical Review4.4.Nucleation in nanocomposites and multilayers8634.5.Grain boundaries and nanojunctions8664.6.Textured magnets and random-anisotropy behaviour8694.7.Magnetic localization and cooperativity of magnetization reversal8715.Magnetization dynamics8745.1.Fundamental equations8755.2.Spin waves8765.3.Magnetic viscosity and sweep rate dependence of coercivity8785.4.Freezing behaviour8825.5.Conduction phenomena and spin electronics8836.Summary and conclusions884Acknowledgments886 Appendix.Magnetic materials886A.1.Permanent magnets886A.2.Magnetic recording media887A.3.Soft magnetic materials889References889 1.IntroductionThousands of years of human curiosity have led to the discovery of magnetism,and for many centuries magnetism has stimulated progress in science and technology.For a long time, the focus had been on macroscopic magnetism,as exempliﬁed by the compass needle,by the geomagneticﬁeld and by the ability of electromagnets and permanent magnets to do mechanical work.Atomic-scale magnetic phenomena,such as quantum-mechanical exchange[1–4], crystal–ﬁeld interaction[5]and relativistic spin–orbit coupling[6,7],were discovered in theﬁrst half of the last century and are now exploited,for example,in advanced permanent–magnet intermetallics such as SmCo5and Nd2Fe14B[8].However,only in recent decades it became clear that solid-state magnetism is,to a large extent,a nanostructural phenomenon. The scientiﬁc and technological importance of magnetic nanostructures has three main reasons: (i)there is an overwhelming variety of structures with interesting physical properties,ranging from naturally occurring nanomagnets and comparatively easy-to-produce bulk nanocomposites to demanding artiﬁcial nanostructures,(ii)the involvement of nanoscale effects in the explanation and improvement of the properties of advanced magnetic materials,and(iii)nanomagnetism has opened the door for completely new technologies.A naturally occurring biomagnetic phenomenon is magnetite(Fe3O4)nanoparticles precipitated in bacteria,molluscs,insects and higher animals.Magnetostatic bacteria live in dark environments and contain chains of40–100nm magnetite particles used for vertical orientation[9].Similar magnetite particles have been found in the brains of bees,pigeons and tuna,and it is being investigated whether and how the particles serve asﬁeld sensors for migration[10].Magnetite and other oxide particles are also responsible for rock magnetism, exploited for example in archaeomagnetic dating and for monitoring changes in the Earth’s magneticﬁeld[11,12].Due to dilution and incomplete saturation,the thermoremanent magnetism of oxide particles in volcanic rocks is between0.0001and1µT,as compared to the geomagneticﬁeld of the order of100µT.Typical particle sizes,varying between less than1and 100µm,are at the upper end of the structures of interest here,but the magnetization dynamics in these particles is a nanoscale phenomenon.Smaller oxide particle sizes,less than10nm,Topical Review R843Figure1.Two schematic bulk nanostructures:(a)sintered Sm–Co and(b)magnetic clusters(white)embedded in a matrix.The two structures are very different from the point of view ofsize,geometry,origin and functionality.The Sm–Co magnets,consisting of a rhombohedralSm2Co17-type main phase(grey),a Cu-rich SmCo5-type grain-boundary phase(black)and a Zr-rich hexagonal Sm2Co17-type platelet phase(white),are produced by a complicated annealingprocess and widely used in permanent magnets[8,26].Nanostructures such as that shown in(b)can be produced,for example,by mechanical alloying and are used as permanent magnets[27],soft magnets[24]and magnetoresistive materials[28,29].are observed in gels having the nominal composition FeO(OH)·n H2O[13].Fine particles are also encountered in superparamagnetic systems[14],ferroﬂuids[15]and meteorites[16].The further improvement of current magnetic materials heavily relies on nanostructuring. This refers not only to materials such as permanent magnets,soft magnets and recording media but also to emerging areas such as spin electronics.An example of improving the performance of magnetic materials by nanostructuring is hard–soft permanent-magnet composites[8,17–22].As analysed in[19],atomic-scale magnetism does not support substantial improvements of permanent magnets beyond existing intermetallics such as SmCo5,Sm2Co17and Nd2Fe14B, but adding a soft phase to a hard phase in a suitable nanostructure improves the permanent-magnet performance beyond that of the hard phase.This‘metamaterials’approach exempliﬁes the materials-by-design strategy and makes it possible to produce materials not encountered in nature.Other nanoscale effects are exploited in soft magnetic nanostructures,for example in Fe73.5Si13.5B9Cu1Nb3[23–25],where soft magnetic Fe3Si grains are embedded in an amorphous matrix.Figure1shows two structures of interest in this context.A fascinating approach is artiﬁcial nanostructuring to create completely new materials and technologies.One area is the ever-progressing miniaturization in computer technology,as epitomized by the use of nanostructured media for ultra-high density magnetic recording[30–39].A related area is spin electronics[40,41],and various types of nanostructures, such as multilayers and nanojunctions,are being investigated in this context.One problem of current interest is spin injection into nonferromagnetic materials[42,43] and magnetic semiconductors[44,45],whereas the use of spin degrees of freedom in quantum computing[46,47]remains a challenge to future research.Another area is magnetoresistive sensors exploiting magnetoresistance effects in metallic thinﬁlms[48–51],granular systems[28,29,52]and magnetic oxides[41,53–55].Some other present or future applications are nanoparticle ferroﬂuids for cancer treatment,guided by a magnet and delivering high local doses of drugs or radiation[56],micro-electromechanical systems(MEMS)and other nanodevices,and nanoscale magnetic-force nanotips made from PtCo[57,58].From a theoretical point of view,nanostructural phenomena are often described by differential equations of the type∇2φ−κ2φ=f(r),whereκ−1is an interaction length.ThisR844Topical Review must be contrasted to the inhomogeneous Laplace(or Poisson)equation∇2φ=f(r)which implies long-range interactions and describes,for example,electrostatic and magnetostatic phenomena.The interaction length reﬂects competition between different atomic energy contributions.When the competition is between the electrons’kinetic energy(hopping) and electrostatic energies(Coulomb interaction and exchange),thenκ−1scales as k F or a0. However,when the main competition involves relativistic interactions,then the interaction length increases to l0=a0/α,whereα=4πε0e2/¯h c≈1/137is Sommerfeld’sﬁne-structure constant[8,59].An example is the competition between magnetocrystalline anisotropy and exchange,because the spin–orbit coupling necessary to create anisotropy is a higher-order relativistic correction to the leading electrostatic contributions.Length scales of the order of l0=7.52nm are indeed encountered in many nanomagnetic problems[8,21,59],indicating that nanomagnetism goes beyond a‘mixture’of atomic and macroscopic physics.A related question concerns the transition from nanoscale to macroscopic behaviour.How many atoms are necessary to make a nanostructure indistinguishable from a bulk magnet?As we will analyse below,the answer to this reduced-dimensionality problem depends not only on the geometry of the structure but also on whether one considers intrinsic or extrinsic magnetic properties.Intrinsic properties,such as the spontaneous magnetization M s,theﬁrst uniaxial anisotropy constant K1and the exchange stiffness A,refer to the atomic origin of magnetism. As a rule,intrinsic properties are realized on length scales of at most a few interatomic distances and tend to approach their bulk values on a length scale of less than1nm,although there are exceptions to this rule[8,60].Extrinsic properties,such as the remanence M r and the coercivity H c,are nonequilibrium properties—related to magnetic hysteresis—and exhibit a pronounced real-structure dependence[8,61–63].For example,the coercivity of technical iron doubles by adding0.01wt%nitrogen[63].Such small concentrations have little effect on the intrinsic properties but lead to inhomogeneous lattice strains on a scale of many interatomic distances,affecting the propagation of magnetic domain walls and explaining the observed coercivity increase.Magnetic nanostructures exhibit a particularly rich extrinsic behaviour, including phenomena such as random-anisotropy scaling[64],remanence enhancement[17], micromagnetic localization[65],bulging-type nucleation modes[66]and a variety of grain-boundary[67]and exchange-coupling effects[68,69].This review deals with the physics of magnetic nanostructures.Section2is devoted to the geometrical aspect of nanomagnetism,introducing various types of nanostructures,section3 focuses on the relation between atomic physics and nanomagnetism and section4investigates nanoscale phenomena in a narrower sense.Section5discusses zero-andﬁnite-temperature dynamic effects,section6summarizes this work and draws some tentative conclusions.Finally, the appendix summarizes some information on materials of interest in nanomagnetism.2.Magnetic nanostructuresAdvanced magnetic nanostructures are characterized by a fascinating diversity of geometries, ranging from complex bulk structures(ﬁgure1)to a broad variety of low-dimensional systems. Figure2shows some examples.This section introduces typical geometries of interest in nanomagnetism and outlines their key features;the division into subsections is somewhat arbitrary,because many structuresﬁt into two or more categories.2.1.Particles and clustersVarious types of small magnetic particles exist in nature(section1)or are produced artiﬁcially. Fine-particle systems,such as Fe in Al2O3with particle diameters of the order of5nm, have been investigated over many decades[70].So-called‘elongated single-domain(ESD)Topical Review R845Figure2.Typical nanostructure geometries:(a)chain ofﬁne particles,(b)striped nanowire,(c)cylindrical nanowire,(d)nanojunction,(e)vicinal surface step,(f)nanodots,(g)antidots and(h)particulate medium.particles’[71]are used,for example,in magnetic recording.The properties of particles are also of interest for the investigation of nanowires(section2.4),dot arrays(section2.3)and thin-ﬁlm(section2.2)and bulk(section2.5)composites.A crude criterion for the survival of the individuality of dots,particles and clusters in complex nanostructures is the strength of the exchange and magnetostatic interparticle interactions(section4.7).Interesting applications of small particles are stable colloidal suspensions known as ferroﬂuids[15,72].A variety of materials can be used,such as Fe3O4,BaFe12O19,Fe,Co and Ni,and a typical particle size is10nm.Most ferroﬂuids are based on hydrocarbons or other organic liquids,whereas water-based ferroﬂuids are more difﬁcult to produce.They are used as liquids in bearings and to monitor magneticﬁelds and domain conﬁgurations.Very small nanoparticles are also known as clusters.Their production by various techniques and typical structural properties have been reviewed by Sellmyer et al[39].In both free and embedded clusters,nanoparticle effects are particularly important.First,the large surface-to-volume ratio of clusters leads to a comparatively strong diameter dependence of the intrinsic properties such as anisotropy[73]and magnetization[74].Second,clusters tend to be superparamagnetic[14,75],particularly at high temperatures(section5.3).The ground-state domain conﬁguration and the mechanism of magnetization reversal in small magnetic particles[75–80]depend on the particle size.At the macroscopic end of the range there are,e.g.,arrays of(110)Fe dots on sapphire,having a thickness of about 50nm and lateral dimensions of the order of1µm[78].Such dots are characterized byﬂux closure[78,81].In contrast,clusters are single-domain magnets(section4.2)and their reversal starts by coherent rotation(section4.3).2.2.Thinﬁlms and multilayersMany magnetic thinﬁlms and multilayers[51,82–87]can be considered as nanostructures, but since thin-ﬁlm magnetism has developed into a separate branch of condensed matterR846Topical ReviewFigure3.An example of high resolution TEM from5.6nm Co clusters produced in our system.(Courtesy D J Sellmyer.)physics,a comprehensive introduction to these structures goes beyond the scope of this work. Nanostructured thinﬁlms with intermediate or high coercivities[20,21,88,89]have been studied in the context of permanent magnetism and magnetic recording.Thin-ﬁlm structures exhibit a number of interesting properties.Examples are anisotropies of ideal and vicinal surfaces and of interfaces[84,90,91],moment modiﬁcations at surfaces and interfaces[92,93],thickness-dependentdomain-wall and coercive phenomena[59,82,86,94], interlayer exchange coupling[48–50]andﬁnite-temperature magnetic ordering[95].Two speciﬁc examples are the nanoscale exchange-coupling or‘exchange-spring’effects in multilayers[18,19,88,96–99]and the pinning of domain walls in sesquilayer iron–tungsten thinﬁlms[86].2.3.Particle arrays and functional componentsTwo-dimensional arrays of nanoparticles are interesting scientiﬁc model systems with many present or future applications.In particular,advanced magnetic recording media can be characterized as a complex array of magnetic particles,and interest in dot arrays[30,75,100–104]has been sparked by the search for ever-increasing storage densities in magnetic recording. In very small dots,quantum-mechanical effects are no longer negligible and there are phenomena such as quantum-well states.Quantum-dot effects are of interest in quantum computing and spin electronics[51,105].There are many methods for producing nanoparticle arrays[38,51,106].A traditional, though somewhat cumbersome,method to produce periodic arrays of nanoscale magnetic particles,dots and wires is nanolithography[107,108].Other examples are molecular-beam epitaxy[109],the use of STMs[110],chemical vapour deposition[101]and e-beam nanolithography[107,108].The call for well-characterized large-area arrays of nanoparticles has stimulated the search for advanced production methods such as laser-interference lithography(LIL),where laser-intensity maxima effect a local decomposition of a nonferromagnetic material into ferromagnetic islands[103].Another development is the use of ion beams[34,111],for example focused ion-beam(FIB)milling[111],to create small particles and particle arrays with well-deﬁned properties.Most easily produced and investigated are submicron dots made from iron-series transition metals,such as Ni[101],but it is also possible to use intermetallics,such as permalloy[81,112], and to reduce the dot size to less than100nm.The arrays may be square or hexagonal,orTopical Review R847Figure4.Advanced MFM tips made by(a)ion milling,(b)electron beam deposition,(c)FIBmilling.(Courtesy S-H Liou.)the dots may form other structures such as corrals.Among the investigated phenomena are the properties of individual dots and interdot interactions[46,112,113].A related class of nanostructures is antidots,that is,holes in aﬁlm rather than dots on aﬁlm[109,114,115]. Antidots exhibit interesting resistive and magnetoresistive properties[114],but magnetic domains in antidots have been studied too[115].Potential applications include magnetic recording,sensors,magnetic and quantum computing,micron-and submicron-size mechanical devices,short-wavelength optics and spin electronics.In section4we will discuss some magnetic properties of dots and dot arrays.Other functional structures are,for example,nanojunctions[40,116],spin valves (section5.5)and tips for magnetic-force microscopy(MFM tips).Figure4shows three MFM tips made by various techniques[57].Some properties of nanojunctions and spin valves will be discussed in sections4.4and5.5.2.4.NanowiresThere is a smooth transition from elongated dots and thin-ﬁlm patches[117,118]to nanowires[38,119–121].Magnetic nanowires are scientiﬁcally interesting and have potential applications in many areas of advanced nanotechnology,including patterned magnetic media, magnetic devices and materials for microwave applications.Thin-ﬁlm nanowires,such as inﬁgure2(b),are comparatively easily obtained by depositing magnetic materials on vicinal surfaces[51,117]and by exploiting structural anisotropies of the substrate[86].They can be produced with thicknesses down to one or two monolayers.Electrodeposition of magnetic materials into porous alumina may be used to produce regular wire arrays[38,119,121].Other ways of fabricating cylindrical nanowires include the deposition into molecular sieves[38,122–125],track-etched polymer membranes[126,127]and mica templates[128].By electrodeposition into porous anodic alumina[124,129,130]it is now possible to produce Fe,Co and Ni wires with diameters ranging from4to200nm,depending on the anodization conditions,and lengths of up to about1µm[38,106,119,121,131–135]. Typically,the nanowires form nearly hexagonal columnar arrays with variable centre-to-centre spacings of the order of50nm[38,121,131,135].The resulting materials are of interest as magnetic recording media[132,136],for optical and microwave applications[137,138]and as electroluminescent display devices[139].Aside from the above-mentioned iron-series transition-metal elements,there is interest in depositing alloys and multilayers,such as Fe/Pt, into porous templates[38,140,141].On the other hand,magnetoresistive effects have been investigated in electrodeposited Co–Cu alloy nanowires[142]and Co–Ni–Cu/Cu multilayered nanowires[143].R848Topical Review Much of the early work on magnetic nanowire arrays was concerned with exploratory issues,such as establishing an easy axis for typical preparation conditions,the essential involvement of shape anisotropy,as opposed to magnetocrystalline anisotropy,and the description of magnetostatic interactions between wires(see,e.g.,[38,127,135,144]and references therein).More recently,attention has shifted towards the understanding of magnetization processes[145–147].On a nanometre scale,interatomic exchange is no longer negligible compared to magnetostatic interactions.This leads to a transition from curling-type to quasi-coherent nucleation(section4.3).For Fe,Co and Ni,the corresponding diameters are about11,15and25nm,respectively,irrespective of the critical single-domain radius[8]. Furthermore,in section4.6we will see that the reversal behaviour is affected by the deposition-dependent polycrystallinity[38]of typical transition-metal nanowires[148].Some other interesting phenomena are magnetic-mode localization(section4.7),as evident,e.g.,from experimental activation volumes(section5.3),spin waves(section5.2)and current-induced magnetization reversal[149].2.5.Nanocomposites and other bulk materialsEmbedded clusters,granular materials and other bulk nanostructures are of great importance in nanoscience.The structural correlation lengths of typical nanocomposite materials range from about1nm in x-ray amorphous structures to several100nm in submicron structures and can be probed,for example,by small-angle neutron scattering(SANS)[150]and electron microscopy[21].Magnetic glasses[13,151]and atomic-scale defect structures are beyond the scope of nanomagnetics,but they are of indirect interest as limiting cases and because nanomagnetic phenomena have their quantum-mechanical origin in atomic-scale magnetism.Structures similar toﬁgure1(b)can be produced by methods such as mechanical alloying[152]and chemical reactions[27,153].Depending on grain size and microchemistry, they are used,for example,as permanent magnets(Nd–Fe–B),soft magnets(Fe–Cu–Nb–Si–B)and magnetoresistive materials(Co–Ag).There are two types of exchange-coupled permanent magnets:isotropic magnets[17,154–158],which exhibit random anisotropy and remanence enhancement(section4.6),and oriented hard–soft composites[19,21,88],which utilize exchange coupling of a soft phase with a high magnetization to a hard skeleton.Closely related systems with many potential applications are magnetic clusters deposited in a matrix. For example,the narrow size distribution of10–20%makes this material interesting as a granular media for magnetic recording[39].A well-known soft magnetic nanocomposite is the‘Yoshizawa’alloy Fe73.5Si13.5B9Cu1Nb3[23,159],which consists of iron–silicon grains embedded in an amorphous matrix.The Fe–Si nanocrystallites,which provide most of the magnetization,crystallize in the cubic DO3structure and have a composition close to Fe3Si.Nanoscale composites must be distinguished from amorphous metals(magnetic glasses) and spin glasses,whose exchange and anisotropy disorder is on an atomic scale[13,151,160–162].However,the boundary is smooth and spin glasses and amorphous materials exhibit various nanostructural phenomena.On the other hand,spin-glass-like phenomena are observed in some nanostructures.For example,interacting particles give rise to spin-glass-like(cluster-glass)dynamics[70,163]and isotropic nanostructures can be considered as random-anisotropy magnets[164].3.Atomic-scale effectsIntrinsic magnetic properties,such as magnetization and anisotropy,are determined on an atomic scale.For example,the magnetization ofα-Fe,µ0M s=2.15T,is associated with the body-centred cubic structure of elemental iron.However,some intrinsic effects are realizedTopical Review R849on a length scale of several interatomic distances.Examples are Ruderman–Kittel–Kasuya–Yosida(RKKY)interactions between localized moments embedded in a Pauli paramagnetic matrix and the disproportionally strong contribution of surface and interface atoms to the magnetic anisotropy of nanostructures.3.1.Magnetic momentThe magnetic moment m of solids nearly exclusively originates from the electrons in partly ﬁlled inner electron shells of transition-metal atoms.Of particular importance are the iron-series transition-metal or3d elements Fe,Co and Ni and the rare-earth or4f elements,such as Nd,Sm,Gd and Dy.Palladium series(4d),platinum series(5d)and actinide(5f)atoms have a magnetic moment in suitable crystalline environments.There are two sources of the atomic magnetic moment:currents associated with the orbital motion of the electrons and the electron spin.The magnetic moment of iron-series transition-metal atoms in metals(Fe,Co,Ni,YCo5) and nonmetals(Fe3O4,NiO)is largely given by the spin and the moment,measured inµB,is equal to the number of unpaired spins.For example,Fe2+(ferrous iron)has four unoccupied 3d↓orbitals,so that the moment per ion is4µB.The orbital moment is very small,typically of the order of0.1µB,because the orbital motion of the electrons is quenched by the crystal ﬁeld[8,165,166].In contrast,rare-earth moments are given by Hund’s rules,which predict the spin and orbital moment as a function of the number of inner-shell electrons[165].The moment per atom is largely determined by intra-atomic exchange.Exchange is an electrostatic many-body effect,caused by1/|r−r |Coulomb interactions between electrons located at r and r .Physically,↓↑electron pairs in an atomic orbital are allowed by the Pauli principle but are unfavourable from the point of view of Coulomb repulsion.In the case of parallel spin alignment,↑↑,the two electrons are in different orbitals,which is electrostatically favourable,but the corresponding gain in Coulomb energy competes against an increase in one-electron energies.(Only one electron beneﬁts from the low ground-state energy—the second electron must occupy an excited one-electron level.)The magnetic moments of insulating transition-metal oxides and rare-earth metals are located on well-deﬁned atomic sites.However,in Fe,Co and Ni,as well as in many alloys,the moment is delocalized or itinerant.Itinerant ferromagnetism is characterized by non-integer moments and explained in terms of the metallic band structure[167–170].Nonmagnetic metals(Pauli paramagnets) have two equally populated↑and↓subbands;an applied magneticﬁeld may transfer a few electrons from the↓band to the↑band,but the corresponding spin polarization is very small, of the order of0.1%.Itinerant ferromagnetism is realized by narrow bands,where the intra-atomic exchange is stronger than the bandwidth-related gain in single-electron hybridization (Stoner criterion).Atomic magnetic moments are affected by several nanoscale mechanisms.First, nonmagnetic atoms may become spin-polarized by neighbouring ferromagnetic atoms.A semiquantitative description of these effects is provided by the Landau–Ginzburg type[171] expression−A2∇2M+A0M=H ex(r).(3.1) Here M(r)is the induced magnetization(moment per unit volume),H ex is the intra-atomic exchangeﬁeld and A0and A2describe the electronic properties of the system.Essentially,χ(k,T)=1/(A0+k2A2)is the wavevector-dependent exchange-enhanced spin susceptibility, which is known for a variety of systems[172].Equation(3.1)predicts an exponential decay of the magnetization with a decay length of1/κ=(A2/A0)1/2.In simple metals,κscales as the Fermi wavevector(κ∼k F)and ferromagnetism is difﬁcult to induce.However,exchange enhanced Pauli paramagnets,such as Pd and Pt,are very close to satisfying the Stoner criterion,R850Topical ReviewFigure5.Intrinsic properties of multilayered Pt–Fe structures(after[169]).so that A0andκare small[8,172].A similar A0reduction is encountered in semiconductors [44]and in semimetals such as Sb,where the decay length is of the order of1nm[173].Nanoscale moment modiﬁcations are important at surfaces and interfaces[173,174],but they do not extend very far into the bulk.By deﬁnition,Bloch wavefunctions extend to inﬁnity, but nanoscaleﬁnite-size effects yield only small corrections to the metallic moment.This can be seen,for example,from real-space approaches based on the moment’s theorem[8,175–178].When only nearest neighbours are taken into account,these methods yield the correct bandwidth but ignore details of the band structure,such as peaks in the density of states. Increasing the number of neighbours improves the resolution of the density of states and makes it possible to distinguish between bulk sites and sites close to surfaces.Figure5shows the modiﬁcation of the moment and of the effective interatomic exchange in multilayered Fe–Pt magnets,as obtained fromﬁrst-principle electronic-structure calculations[169].As a rule,nanoscale intrinsic phenomena are caused by small differences between atomic interaction energies.In terms of(3.1),this occurs when A0≈0due to competing hopping and intra-atomic exchange energies.A loosely related phenomenon,observed for example in rare-earth elements and alloys,is noncollinear spin structures[13].Helimagnetic rare-earth noncollinearity is characterized by k vectors depending on the ratio of the nearest-and next-nearest-neighbour exchange(section3.2),and k may be,in principle,a very small fraction of k F.Even more complicated spin arrangements are possible in disordered magnets with competing interatomic exchange interactions(spin glasses)and at surfaces and interfaces. Furthermore,surface states[51,178]modify the magnetic moment of surface atoms[51,174]. Another type of noncollinearity is caused by spin–orbit coupling.The orbit of an electron, and therefore its crystal-ﬁeld interaction,depend on the spin direction,so that electrons on sites without inversion symmetry can minimize the crystal-ﬁeld energy by forming a slightly noncollinear spin structure.In spin glasses,this is known as Dzyaloshinskii–Moriya interaction[151],but the same effect occurs in other low-symmetry structures[179].Noncollinear states must not be confused with micromagnetic structures,such as domains and domain walls(section4.2).For example,small particles may exhibit some noncollinearity due to competing exchange,particularly at the surface,but an applied magneticﬁeld merely changes the direction of the net magnetization,leaving the atomic-scale noncollinear correlations M(r i)·M(r j) unchanged.By contrast,micromagnetic magnetization processes,such as domain-wall motion,change the relative magnetization directions of well-separated spins in comparatively small magneticﬁelds.3.2.Magnetization and magnetic orderIn a strict sense,ferromagnetism is limited to inﬁnite magnets,because thermal excitations inﬁnite magnets cause the net moment toﬂuctuate between opposite directions.In。

Is Minisuperspace Quantum Gravity Reliable

a r X i v :g r -q c /0101108v 2 13 F eb 2001Is Minisuperspace Quantum Gravity Reliable?O.Saremi ∗Feb.12,2001Department of PhysicsSharif University of Technology,Azadi St.Tehran,Iran P.O.Box:11365-9161AbstractWe study a minisuperspace quantum cosmology for a 2+1dimensional de Sitter universe and ﬁnd the wave function both exactly and in WKB approxi-mation.Then we extend the model to a canonically quantized ﬁeld theory for quantum gravity,i.e.,a midisuperspace,and obtain the wave functional of the resulting ﬁeld theory in the saddle point approximation.It is shown that these two approaches yield diﬀerent results.1IntroductionQuantum cosmology (QC)was initiated by B.S.Dewitt in his seminal paper on Hamil-tonian quantization of gravity [1].Up to advent of reasonable boundary conditions to pick out a unique solution to the Wheeler-Dewitt equation (WDW),QC was out of attention.In the early eighties,proposals on possible boundary conditions revitalized QC.[2,3,4].Equivalent to Hamiltonian quantization of 4-dimensional gravity,one can path integrate over all metric conﬁgurations and sum over all possible 4-topologies which have a given ∂M as their boundary [5].The space of all conﬁgurations for 4-metric is called superspace (SS)where diﬀeomorphically equivalent conﬁgurations are factored out.Mathematically,dealing with the full SS is too diﬃcult,if not im-possible.Inevitably,we must invoke to some approximate schemes.A commonly used approximate model is called ”minisuperspace”model [1].In this approximation,one conﬁnes his attention to a restricted region of SS.In practice,one freezes or sus-pends many inﬁnite degrees of freedom of gravitational ﬁeld on a time constant slice of 3-geometry and retains a few of them alive.Despite the problems surrounding the interpretation of the wave function of the universe,by using some generally accepted interpretational rules [4],many attempts have been made to present explanations for some observed features of the universe in the context of ”minisuperspace”models,e.g.,cosmological constant problem [7]or lacking the existence of primordial black holes at the present state of the universe [8].There are some other and more realistic approaches.Inhomogenous perturbations and midisuperspace models are examples of more realistic approximations to the full quantized theory.Inhomogeneous perturbations approach to quantum cosmology is a perturbative way to pass beyond the minisuperspace models[9].In a typical midis-uperspace model,one retains some dependence of metric conﬁgurations on spatial coordinates.This means that the less symmetric conﬁgurations are considered as the space of path integration.The spherically symmetric midisuperspaces,the BCMN models[10],have been used as models for quantized Schwarzchild black holes.In the light of these models some of expected features of quantum black holes have been derived successfully[11].Also a Hamiltonian quantization of false-vacuum bubbles within the framework of a spherically symmetric midisuperspace has been presented [12].There is no support in favor of this claim that the minisuperspace conﬁgura-tions have the dominant contribution to the partition function of quantum gravity. Moreover minisuperspace is not known to be a part of a systematic approximation to the full theory.Therefore,it is possible that some of derived predictions from these minisuperspace calculations to be artifact of this approximation.In this way, a comparison between amplitudes derived from a minisuperspace calculation and of midi one,will be useful.Through this paper,it is shown that a minisuperspace wave function can diﬀer from midi one.The organization of the paper is as follows:In section2,we take a minisuperspace for QC in2+1dimensions and obtain the”wave function”in saddle point approximation.Through Section3,we will give the exact solutions to the Wheeler-Dewitt equation.Section4has been devoted to extension of this minisuperspace model to a midi one.We solve the resultingﬁeld theory in the saddle point approximation and obtain the wave functional of the theory.Finally,we present a comparison between the obtained wave functions and wave functionals.Paper will end up with a conclusion.2Minisuperspace Saddle Point Wave FunctionsIn this section,we review the quantum cosmology of a2+1dimensional de Sitter universe.Take the following2+1dimensional FRW line element as a minisuperspace:ds2=−N(t)2+a(t)2(dr24π (.a2dt ( .aand the Hamiltonian constraint:H=1√Λt+β),(5) whereβis an integration constant.This solution describes classical dynamics of a dS3universe.Euclideanized version of the action,resulting equation of motion and the Hamil-tonian constraint read:S E=−m2pN2+N−ΛNa2)dτ,(6) dN)=−ΛNa,(7)H E=1√Λτ).(9) Note that the condition da√√2√√2√√ΛτIm).(10)This part of the instanton describes half of a Lorentzian dS3universe.There is also another instanton which satisﬁes NBP conditions and contributes to the saddle point approximation[6].Resulting NB wave function,for a<1Λwill be:ψNB∼exp(−2a √Λa)(11),and for a>1Λ:ψNB∼e−πΛcos(2a √Λa−πVilenkin wave function in WKB approximation [3,4]will have the following form for a <1Λ:ψV ∼exp (2a√Λa )(13),and for a >1Λ:ψV ∼exp (πΛ)exp [−i (2a√Λa )].(14)3Exact SolutionsAccording to Dirac prescription for quantization [13],the wave function of a constraint system should be annihilated by the operator version of classical constraints.Replac-ing πa by −i ¯h ∂4a P∂∂a)+(1−Λa 2)ψ=0,(15)where P carries some part of factor ordering ambiguity due to indeﬁniteness of measure of path integral or equivalently quadratic form of Hamiltonian in πa .We will set it to 0.There are exact solutions to the diﬀerential equation(15)in terms of Whittaker functions of type W and M [14]:ψ(a,Λ)=ζmaW M (−1−Λ,1−Λa 2)+ζwaW W (−1−Λ,1−Λa 2).(16)Constant coeﬃcients ζm and ζw should be determined by considering appropriate boundary conditions.Asymptotic expansion of ψ(a,Λ)can be obtained easily [9]:ψ(a,Λ)∼1a Λi Λ((i −1)√4Γ(34√4√2√2√√8√π4−iΛ))ζm a−i ΛeiΛa 2.(17)a →∞Vilenkin wave function,has only outgoing sector at very large values of a ,therefore:ζw =−(i −1)√4Γ(34√4√√4√4,√√2π4+i Λ)e1ΛW W (−1−Λ,1−Λa 2)}.(19)4Toﬁnd the Hartle-Hawking state,we note that the ingoing and outgoing sectors of ψ(a,Λ)should have the same amplitudes for a→∞.This implies:ζw=(√2Γ(34√2π(i−1)4+iΛ)e1Λ)ζm.(20)Therefore,the resulting Hartle-Hawking state will be:ψH−H(a,Λ)=ζmaW M(−1−Λ,1−Λa2)+1a(√2Γ(34√2π(i−1)4+iΛ)e1Λ)ζm W W(−1−Λ,1−Λa2).(21)4Midisuperspace Saddle Point Wave FunctionalsIs there any dramatic discrepancy between using a minisuperspace instead of full theory or even a midisuperspace?A way to answer to this question is to compare a minisuperspace amplitude with a midi one.Here we present and,approximately,solve a”midi”superspace with axial symmetry for quantum cosmology of dS3universes which is described by the following line element:ds2=−N(t,r)2dt2+φ(t,r)dr2+ψ(t,r)dθ2.(22) By axial symmetry we mean that the metric components are considered to beθindependent.Substituting the above metric ansatz,Lagrangian density reads:£=N(ψφ)12N2φψ(−.φ+2N′r−φ′φN r)+2R+2Λ](23),where2R is the scalar curvature of spatial sector of three geometry.2R is given by:2R=2ψ′′ψφ−ψ′2φ−φ′ψψ′∂.φ=12(.ψ−ψ′∂.ψ=12(.φ+φ′2N(φψ)1φN r)+.ψ(.φ+φ′H=12(.ψ.φ+N2rψ′φ′φψ′)−N2ψ′′ψφ−ψ′2φ−φ′ψψ′2φ32Λ.(28)Now,we should express the time derivatives of theﬁled variables in terms of momenta,ﬁelds and their spatial derivatives:.φ=2N(φψ)1φN r,(29).ψ=2N(φψ)1φN r.(30) Integration over a spacelike hypersurfaceΣleads to the Hamiltonian constraint: H=2π Σ{N[2(φψ)122R−2Λ(φψ)1φπψ−φ′2πφπψ−2Λ(φψ)12ψ32=0,(32)H r=ψ′φπφ−2π′φ=0.(33)Hamiltonian constraint carries time reparametrization invariance of the classical the-ory and momentum constraint is the generator of inﬁnitesimal coordinate transfor-mations within spacelike hypersurfaces.The number ofﬁeld variables and constraints are equal,then it is tempting to solve πφandπψin terms ofﬁeld variables and their spatial derivatives by constructing a superposition of constraints to eliminateπψ.Note that2R can be rewritten as follows:2R=1ψφ).(34) Now,consider the below superposition:2(φψ)1ψ′πφH r−H0=(φψ)1ψ′(−2φ′π2φ−4φπφπ′φ+2Rψ′+2Λψ′)=0.(35)By considering the eqs.(34)and(35),we obtain:∂r(−2φπ2φ+ψ′212ψφ+2Λψ−2).(37)6Regarding eq.(32),πψwill become:πψ=1ψφ)′+2Λ12ψφ+2Λψ−2).(38)The leading term in a semiclassical expansion of the”wave functional”of thisﬁeld theory will be e±iS cl,where S cl is the action evaluated for classical path:δS= dr(πφδφ+πψδψ).(39)We can choose a speciﬁc contour of integration to simplify integration process.First of all,we holdψconstant and then integrate overφ,to a conﬁguration ofφsuch that:ψ′22ψ′(ψ′2 2 2φ(ψ′2ψ′2ψ′ 2φ,(41)then this part will not contribute to the action.Therefore,the action reads:S= 2φ(ψ′2ψ′2ψ′2ψ′24ψ))dr.(43)Wave functional for classically forbidden region will be a superposition of exponen-tially decaying and growing forms and in the allowed region is a linear superposition of oscillating exponentials of the action.Now,we are in the position to calculate the wave functional of a2+1dimensional FRW-like universe.It will be suﬃcient to consider a homogeneous and isotropic universe on theﬁnal hyprsuface of simultaneity.By substituting”FRW”form for the metric on theﬁnal hypersurface and regarding to eq.(43),we will reach the following saddle point”wave functionals”:7Vilenkin wave functional in tunneling region will be:ΨV ∼e2a√(1−r 2)11−Λa 2]dr(44),and in classical region:ΨV ∼e−i (2a√(1−r 2)1−1+Λa 2]dr.(45)Hartle-Hawking wave functional in tunneling region will be:ΨH −H ∼e−2a√(1−r 2)11−Λa 2]dr(46),and in classical region:ΨH −H ∼cos(2a(1−r 2)1−1+Λa 2]dr −π5.G.W.Gibboins,S.W.Hawking,Euclidean Quantum Gravity,World Scientiﬁc(1993)6.P.D.D’eath,Supersymmetric Quantum Cosmology,Cambridge University Press(1995).7.A.Strominger,Nucl.Phys.B319(1989)722-7328.R.Bousso and S.W.Hawking,Phys.Rev.D52,No.10,5659(1995)9.J.J.Halliwell,S.W.Hawking,Phys.Rev.D31,No.8,1777(1985).10.B.K.Berger,D.M.Chitre,V.E.Moncrief,and Y.Nutku,Phys.Rev.D5,2467(1972).11.C.Vaz,Phys.Rev.D61(2000),064017.12.W.Fischler,D.Morgan,and J.Polchinski,Phys.Rev.D42,No.12(1990).13.P.A.M.Dirac,Lectures on quantum mechanics(New York,Belfer Graduate School of Science,Yeshiva University,1964).14.M.Abramowitz and I.Stegun,Hand Book of Mathematical Functions,Dover Publication,INC.,New York,1965.9。

A Birdeye View of DFT

Preface
This text is the outgrowth of lectures the author gave at the Chemistry Institute of the University of S˜ ao Paulo at S˜ ao Carlos, Brazil, and at the VIII’th Summer School on Electronic Structure of the Brazilian Physical Society [1]. It is an attempt to introduce density-functional theory (DFT) in a language accessible for students entering the ﬁeld or researchers from other ﬁelds. It is not meant to be a scholarly review of DFT, but rather an informal guide to its conceptual basis and some recent developments and advances. The Hohenberg-Kohn theorem and the Kohn-Sham equations are discussed in some detail, including comparisons with the equations of Thomas-Fermi, Hartree-Fock, and Dyson. Approximate density functionals, selected aspects of applications of DFT, and a variety of extensions of standard DFT are also discussed, albeit in less detail. Throughout it is attempted to provide a balanced treatment of aspects that are relevant for chemistry and aspects relevant for physics, but with a strong bias towards conceptual foundations. The text is intended to be read before (or in parallel with) one of the many excellent more technical reviews available in the literature. The author apologizes to all researchers whose work has not received proper consideration. The limits of the author’s knowledge, as well as the limits of the available space and the nature of the intended audience, have from the outset prohibited any attempt at comprehensiveness.1

Nonlinear Rashba Model and Spin Relaxation in Quantum Wells

the interfaces. According to the Ehrenfest’s theorem, the external electric ﬁeld contribution ∆En(2) is approxi-
and HSO are the diagonal elements of the light-hole and spin-orbit split-oﬀ bands in the eight-band Hamiltonian.
From Eq. (1), we ﬁnd that the dominant contribution to the RSS consists of the interface term ∆En(1) in the valence band and the external electric ﬁeld term ∆En(2),
We ﬁnd that the Rashba spin splitting is intrinsically a nonlinear function of the momentum, and the linear Rasha model may overestimate it signiﬁcantly, especially in narrow-gap materials. A nonlinear Rashba model is proposed, which is in good agreement with the numerical results from the eight-band k·p theory. Using this model, we ﬁnd pronounced suppression of the D’yakonov-Perel’ spin relaxation rate at large electron densities, and a non-monotonic dependence of the resonance peak position of electron spin lifetime on the electron density in [111]-oriented quantum wells, both in qualitative disagreement with the predictions of the linear Rashba model.

体外冲击波治疗脑卒中后功能障碍的研究进展

- 167 -*基金项目：江西省中医优势病种（中风）防治中心项目（赣中医药医政字〔2021〕8号）①江西中医药大学附属医院康复科　江西　南昌　330006通信作者：张衍辉体外冲击波治疗脑卒中后功能障碍的研究进展*曾宝瑶①　张衍辉①　施凤飞①　洪恩四①　王鹏① 【摘要】　体外冲击波疗法（ESWT）由于其安全、无创、方便及疗效好等特点，一直以来被广泛应用于康复医学中，特别是肌肉骨骼康复的骨科术后或外伤炎症康复当中，而相比之下，ESWT 在神经康复中的应用较少。

神经系统疾患中的脑卒中，已成为我国居民的第一位死亡原因，具有高发病率、高死亡率和高致残率等特点。

脑卒中后遗留的功能障碍，给无数患者及其家庭造成了极大的困扰。

近些年来，越来越多的研究者开始尝试把体外冲击波应用在脑卒中后功能障碍的康复治疗中，本文对其近十年的相关临床应用进行统合、对比及分析，希望能为未来的脑卒中康复提供新的思路与应用依据。

【关键词】　体外冲击波疗法　脑卒中　功能障碍　综述 Research Progress of Extracorporeal Shock Wave Therapy for Post-stroke Dysfunction/ZENG Baoyao, ZHANG Yanhui, SHI Fengfei, HONG Ensi, WANG Peng. //Medical Innovation of China, 2024, 21(11): 167-175 [Abstract]　Extracorporeal shock wave therapy (ESWT) has been widely used in rehabilitation therapy, especially in orthopedic surgery or traumatic inflammation rehabilitation for musculoskeletal rehabilitation, due to its characteristics of safety, non-invasive, convenience and good curative effect. In contrast, ESWT has less application in neurological rehabilitation. Stroke, one of the neurological diseases, has become the first cause of death in China, with high incidence rate, high mortality and high disability rate. The post-stroke dysfunction has caused great distress to countless patients and their families. In recent years, more and more researchers have begun to apply extracorporeal shock wave to the rehabilitation treatment of post-stroke dysfunction. This article reviews its clinical application in the past decade to provide new ideas and evidence for future clinical rehabilitation treatment. [Key words]　Extracorporeal shock wave therapy　Stroke　Dysfunction　Review First-author's address: Department of Rehabilitation, Affiliated Hospital of Jiangxi University of Chinese Medicine, Nanchang 330006, China doi：10.3969/j.issn.1674-4985.2024.11.037 脑卒中又称中风，是我国常见的一种中枢神经系统疾病，分为缺血性脑卒中和出血性脑卒中，其高致残率和高死亡率是危害国民生命和健康的两大特征。

hyperbolic functions

Hyperbolic functionsThe hyperbolic functions have similar names to the trigonmetric functions,but they are deﬁned in terms of the exponential function.In this unit we deﬁne the three main hyperbolic functions, and sketch their graphs.We also discuss some identities relating these functions,and mention their inverse functions and reciprocal functions.In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.After reading this text,and/or viewing the video tutorial on this topic,you should be able to:•deﬁne the functions f(x)=cosh x and f(x)=sinh x in terms of the exponential function, and deﬁne the function f(x)=tanh x in terms of cosh x and sinh x,•sketch the graphs of cosh x,sinh x and tanh x,•recognize the identities cosh2x−sinh2x=1and sinh2x=2sinh x cosh x,•understand the meaning of the inverse functions sinh−1x,cosh−1x and tanh−1x and spec-ify their domains,•deﬁne the reprocal functions sech x,csch x and coth x.Contents1.Introduction22.Deﬁning f(x)=cosh x23.Deﬁning f(x)=sinh x44.Deﬁning f(x)=tanh x75.Identities for hyperbolic functions86.Other related functions91.IntroductionIn this video we shall deﬁne the three hyperbolic functions f(x)=sinh x,f(x)=cosh x and f(x)=tanh x.We shall look at the graphs of these functions,and investigate some of their properties.2.Deﬁning f(x)=cosh xThe hyperbolic functions cosh x and sinh x are deﬁned using the exponential function e x.We shall start with cosh x.This is deﬁned by the formulacosh x=e x+e−x2=1+1 2+e−x2for large x.But the graph of cosh x will always stay above the graph of e x/2.This is because,even though e−x/2(the second part of the sum)gets very small,it is always greater than zero.As x gets larger and larger the diﬀerence between the two graphs gets smaller and smaller.Now suppose that x<0.As x becomes more negative,e−x increases quickly,but e x decreases quickly,so theﬁrst part of the sum e x/2+e−x/2gets very small.As x gets more and more negative,cosh x gets closer and closer to e−x/2.We write this ase−xcosh x≈.2The function satisﬁes the conditions cosh0=1and cosh x=cosh(−x).The graph of cosh x is always above the graphs of e x/2and e−x/2.3.Deﬁning f(x)=sinh xWe shall now look at the hyperbolic function sinh x.In speech,this function is pronounced as ‘shine’,or sometimes as‘sinch’.The function is deﬁned by the formulasinh x=e x−e−x2=1−1 2−e−x2for large x.But the graph of sinh x will always stay below the graph e x/2.This is because,even though −e−x/2(the second part of the diﬀerence)gets very small,it is always less than zero.As x gets larger and larger the diﬀerence between the two graphs gets smaller and smaller.Next,suppose that x is negative.As becomes more negative,−e−x becomes large and negative very quickly,but e x decreases very quickly.So as x becomes more negative,theﬁrst part of the diﬀerence e x/2−e−x/2gets very small.So sinh x gets closer and closer to−e−x/2.We write this as−e−xsinh x≈.2The function satisﬁes the conditions sinh0=0and sinh(−x)=−sinh x.The graph of sinh x is always between the graphs of e x/2and e−x/2.We have seen that sinh x gets close to e x/2as x gets large,and we have also seen that cosh x gets close to e x/2as x gets large.Therefore,sinh x and cosh x must get close together as x gets large.Sosinh x≈cosh x for large x.Similarly,we have seen that sinh x gets close to−e−x/2as x gets large and negative,and we have seen that cosh x gets close to e−x/2as x gets large and negative.Therefore,sinh x and −cosh x must get close together as x gets large and negative.Sosinh x≈−cosh x for large negative x.We can see this by sketching the graphs of sinh x and cosh x on the same axes.Key PointFor large values of x the graphs of sinh x and cosh x are close together.For large negative values of x the graphs of sinh x and−cosh x are close together.4.Deﬁning f(x)=tanh xWe shall now look at the hyperbolic function tanh x.In speech,this function is pronounced as ‘tansh’,or sometimes as‘than’.The function is deﬁned by the formulatanh x=sinh x2÷e x+e−xe x+e−x.We can use what we know about sinh x and cosh x to sketch the graph of tanh x.Weﬁrst take x=0.We know that sinh0=0and cosh0=1,sotanh0=sinh01=0.As x gets large,sinh x≈cosh x,so tanh x gets close to1:tanh x≈1for large x.But sinh x is always less than cosh x,so tanh x is always slightly less than1.It gets close to1 as x gets very large,but never reaches it.As x gets large and negative,sinh x≈−cosh x,so tanh x gets close to−1:tanh x≈−1for large negative x.But sinh x is always greater than−cosh x,so tanh x is always slightly greater than−1.It gets close to−1as x gets very large and negative,but never reaches it.We can now sketch the graph of tanh x.Notice that tanh(−x)=−tanh x.5.Identities for hyperbolic functionsHyperbolic functions have identities which are similar to,but not the same as,the identities for trigonometric functions.In this section we shall prove two of these identities,and list some others.Theﬁrst identity iscosh2x−sinh2x=1.To prove this,we start by substituting the deﬁnitions for sinh x and cosh x:cosh2x−sinh2x= e x+e−x2 2.If we expand the two squares in the numerators,we obtain(e x+e−x)2=e2x+2(e x)(e−x)+e−2x=e2x+2+e−2xand(e x−e−x)2=e2x−2(e x)(e−x)+e−2x=e2x−2+e−2x,where in each case we use the fact that(e x)(e−x)=e x+(−x)=e0=ing these expansions in our formula,we obtaincosh2x−sinh2x=e2x+2+e−2x4.Now we can move the factor of14 (e2x+2+e−2x)−(e2x−2+e−2x) . If,ﬁnally,we remove the inner brackets and simplify,we obtaincosh2x−sinh2x=14×4=1,which is what we wanted to prove.2.We want to manipulate the right-hand side to achieve this.So we shall start by substituting the deﬁnitions of sinh x and cosh x into the right-hand side:2sinh x cosh x=2 e x−e−x2 .We can cancel the2at the start with one of the2’s in the denominator,and then we can take the remaining factor of12(e x−e−x)(e x+e−x).Now we can multiply the two brackets together.This gives us2sinh x cosh x=12(e2x−e−2x)=sinh2x,which is what we wanted to achieve.2=1+cosh x2=cosh x−1sinh x.Instead,sinh−1x means the‘inverse function’.This means that f−1(x)=y whenever f(y)=x. So,for instance,sinh−1x=y whenever sinh y=x.This inverse function is deﬁned for all values of x.We can also deﬁne the inverse functions for cosh x and tanh x.We deﬁnecosh−1x=y whenever cosh y=x,and this function is valid for x≥1.We also deﬁnetanh−1x=y whenever tanh y=x,and this function is valid for−1<x<1.We have also mentioned the reciprocal functions,and these have special names related to the names of the trigonometric reciprocal functions.They aresech x=1sinh x,coth x=1。

Global Fluctuation Spectra in Big CrunchBig Bang String Vacua

2
Department of Physics, University of Pennsylvania, Philadelphia, PA 19104–6396
ABSTRACT We study Big Crunch/Big Bang cosmologies that correspond to exact worldsheet superconformal ﬁeld theories of type II strings. The string theory spacetime contains a Big Crunch and a Big Bang cosmology, as well as additional “whisker” asymptotic and intermediate regions. Within the context of free string theory, we compute, unambiguously, the scalar ﬂuctuation spectrum in all regions of spacetime. Generically, the Big Crunch ﬂuctuation spectrum is altered while passing through the bounce singularity. The change in the spectrum is characterized by a function ∆, which is momentum and timedependent. We compute ∆ explicitly and demonstrate that it arises from the whisker regions. The whiskers are also shown to lead to “entanglement” entropy in the Big Bang region. Finally, in the Milne orbifold limit of our superconformal vacua, we show that ∆ → 1 and, hence, the ﬂuctuation spectrum is unaltered by the Big Crunch/Big Bang singularity. We comment on, but do not attempt to resolve, subtleties related to gravitational backreaction and light winding modes when interactions are taken into account.

不同波段滤光片完美脉冲技术治疗面部毛细血管扩张症的效果及安全性对比

ʌ文章编号ɔ1006-6233(2024)01-0061-05不同波段滤光片完美脉冲技术治疗面部毛细血管扩张症的效果及安全性对比李㊀微,㊀朱李霞,㊀党㊀辉,㊀陈小燕(空军军医大学第一附属医院整形外科,㊀陕西㊀西安㊀710032)ʌ摘㊀要ɔ目的:探讨应用改良完美脉冲技术(AOPT )的治疗模式下不同波段滤光片治疗面部毛细血管扩张症(FT )的效果及安全性㊂方法:选取128例FT 患者的资料,并随机分为对照组和观察组,每组64例,两组均给予AOPT 治疗,其中对照组采用波长560~1200nm 血管滤光片;观察组采用波长530~650nm 和波长900~1200nm 血管滤光片,评估两组治疗5个月后及术后随访3个月的临床疗效,比较治疗前㊁治疗5个月后及术后3个月的面部美学分区评分及毛细血管形态评分,观察两组术后不良反应发生情况以及调查术后随访3个月的面容修复满意度㊂结果:观察组治疗后及术后随访3个月的临床总有效率分别为93.75%及100.00%,均显著高于对照组的75.00%及90.63%(P <0.05);面部美学分区评分的时间效应㊁组间效应有统计学意义(P <0.05),毛细血管形态评分的时间效应有统计学意义(P <0.05),两组面部美学分区评分㊁毛细血管形态评分均较术前有降低,对照组上述评分高于观察组,差异均有统计学意义(P <0.05);观察组的术后不良反应总发生率为1.56%,显著低于对照组的12.50%(P <0.05);观察组术后随访3个月后的面容修复满意度为100.00%,显著高于对照组的93.75%(P <0.05)㊂结论:应用AOPT 技术并采用波长530~650nm 和波长900~1200nm 血管滤光片治疗FT 患者的近㊁远期疗效明确,能明显改善面部美观度和毛细血管形态,减少术后不良反应,提升患者面容修复满意度㊂ʌ关键词ɔ㊀血管滤光片;㊀完美脉冲技术;㊀术后不良反应;㊀面容修复满意度ʌ文献标识码ɔ㊀A㊀㊀㊀㊀㊀ʌdoi ɔ10.3969/j.issn.1006-6233.2024.01.011Comparison of Effect and Safety of AOPT with Different BandFilters in the Treatment of Facial TelangiectasiaLI Wei ,ZHU Lixia ,DANG Hui ,et al(The First Affiliated Hospital of Air Force Medical University ,Shaanxi Xi 'an 710032,China )ʌAbstract ɔObjective :To investigate the effectiveness and safety of using Advanced Optimal Pulse Tech-nology (AOPT )with different wavelength band filters in the treatment of facial telangiectasia (FT ).Methods :Clinical data from 128FT patients were collected and they were randomly divided into a control group and an observation group ,each consisting of 64patients.Both groups received AOPT treatment ,with the control groupusing a vascular filter with a wavelength of 560~1200nm ,and the observation group using vascular filters withwavelengths of 530~650nm and 900~1200nm.The clinical efficacy of the two groups was assessed after 5months of treatment and at a 3-month post -surgery follow -up.The facial aesthetic zone score and capillarymorphology score were compared before treatment ,after 5months of treatment ,and at 3months after surgery.Postoperative adverse reactions in both groups were observed ,and the satisfaction with facial repair at the 3-month post -surgery follow -up was investigated.Results :The total clinical effectiveness rates after treatment and at the 3-month post -surgery follow -up in the observation group were 93.75%and 100.00%,significantly higher than the 75.00%and 90.63%in the control group (P <0.05).There were statistically significant effects of time and between -group differences in facial aesthetic zone scores (P<0.05),and the capillary mor-㊃16㊃ʌ基金项目ɔ陕西省重点研发计划项目,(编号:2022SF -493)ʌ通讯作者ɔ陈小燕phology score showed a significant effect over time(P<0.05).Both groups had reduced facial aesthetic zone scores and capillary morphology scores compared to before surgery,with the scores in the control group being higher than those in the observation group(P<0.05).The total incidence of postoperative adverse reactions in the observation group was1.56%,significantly lower than the12.50%in the control group(P<0.05).The facial repair satisfaction in the observation group was100.00%at the3-month post-surgery follow-up,signifi-cantly higher than the93.75%in the control group(P<0.05).Conclusion:The use of AOPT technology with vascular filters having wavelengths of530-650nm and900-1200nm in the treatment of FT patients provides clear short-term and long-term efficacy.This treatment significantly improves facial aesthetics and capillary morphology,reduces postoperative adverse reactions,and enhances patient satisfaction with facial repair.ʌKey wordsɔ㊀Vascular filter with wavelength;㊀Optimal pulse technology;㊀Postoperative adverse re-actions;㊀Facial repair satisfaction㊀㊀面部毛细血管扩张症(facial telangiectasia,FT)是指好发于面部㊁鼻翼两侧颧部的损容性皮肤病,其发病原因与血液循环障碍㊁风吹日晒㊁温差变化㊁情绪波动及滥用外用药物等因素有关[1]㊂目前,临床治疗FT 的方法包含激光治疗㊁外敷用药㊁电外科手术等,其中激光技术趋于成熟,具有效果佳㊁不良反应少㊁操作简便及安全性高等特点,深受医患青睐[2]㊂新型强脉冲光(intense pulsed light,IPL)治疗仪应用改良完美脉冲技术(ameliorate optimal pulse technology,AOPT)治疗FT具备良好治疗效果,能通过发射出一般在515~ 1200nm的IPL波长,利用选择性光热分解原理将能量脉冲释放至皮损处,从而达到良好治疗效果[3]㊂本文研究探讨在用AOPT治疗模式下分别采取560~ 1200nm㊁590~1200nm的双段波长滤光片和560~ 1200nm波长滤光片治疗我院FT的效果及安全性,现将结果进行论述㊂1㊀资料与方法1.1㊀一般资料:选取2022年8月至2022年12月期间在本院收治的FT患者128例作为本次试验的受试对象㊂纳入标准:符合FT的诊断标准,可见面部两侧及鼻翼旁的毛细血管呈持续扩张发展,毛细血管为红色或紫色,形态呈细丝状㊁蛛网状或星状缓慢扩张或增多者;符合IPF的治疗适应症,包括光子嫩肤㊁血管性疾病㊁色素性疾病㊁多毛症㊁炎症性皮肤㊁寻常型痤疮㊁酒糟鼻等;临床资料完整,能配合本试验完成治疗和随访者㊂排除标准:伴有面部炎症性皮肤疾病者;经询问遗传史或家族史,既往瘢痕疙瘩以及腿或手臂划痕等方法判定为瘢痕体质者;伴有过敏性皮肤者;面部患有皮肤癌或癌前病变者;既往3个月内存在皮肤感染或严重溃烂者;面部患区皮肤出现开放性伤口者;既往2周内存在日光性暴晒史者;服用维生素A酸药物者;免疫功能低下者;伴有糖尿病或血液系统疾病者;既往2周内及受试期间存在糖皮质激素药物治疗史者;精神系统疾病或认知功能障碍者;合并其他严重系统性疾病者;处于妊娠阶段或哺乳阶段女性者;医嘱依从性差;受试中途擅自中止试验,或放弃治疗,或中途死亡者㊂128例受试病例入组后,无脱组病例,依照随机数字表的分组方式,以1ʒ1分组比例随机生成对照组和观察组各64例,其中观察组男14例㊁女50例,年龄在18~59(35.56ʃ8.12)岁,病程在2~40(15.59ʃ3.89)年;对照组男18例㊁女46例,年龄在20~58(36.82ʃ8.25)岁,病程在4~38(15.99ʃ4.02)年㊂两组受试病例的基线资料导入SPSS2.0软件进行描述性分析得出无显著性差异(P>0.05),组间资料具备可比性㊂1.2㊀方法:治疗前,清洁受试者的面部并拍照记录,使用一次性手术帽将受试者眼部进行遮盖,使用医用耦合剂均匀涂抹于患区皮肤㊂对照组采用波长560~ 1200nm血管滤光片;观察组采用波长530~650nm和波长900~1200nm血管滤光片㊂两组所使用的滤光片手具光斑大小均为15mmˑ35mm,应用AOPT模式,将子脉冲能量调整为9~17J/cm2,脉宽调整为4.0~6. 0ms,脉冲延迟调整为20~40ms,选择接触式冷却方式㊂首次治疗先采取用低能量进行光斑试验,并依据受试者的毛细血管扩张程度及皮肤种类调整合适的治疗参数,将治疗头竖直紧贴于面部患区皮肤处作均匀扫描,确保光斑无重叠现象,针对皮肤损伤较为严重区域则反复扫描,针对额部㊁颞部及下眼睑部位可将能量略微调低,针对鼻部可适当增加能量,男性患者可适当增加能量进行扫描㊂治疗全程3~5次无需麻醉,治疗结束后将面部用冷水洗净,再外敷舒缓面膜30min,治疗后均嘱受试者使用防晒霜(SPF>50,PA+++)作防晒处理㊂两组治疗5次,每次治疗间隔28d,治疗时长在5个月,随访3个月左右㊂1.3㊀观察指标㊃26㊃1.3.1㊀临床疗效判定:于治疗后(治疗5个月后)及术后随访3个月后,对两组受试病例的临床疗效标准,根据受测者的毛细血管扩张区域占面颊解剖区域面积的百分比进行量化评分,毛细血管扩张区域/面颊解剖区域面积ˑ100%在<5%,6%~11%㊁12%~21%㊁22%~31%㊁32%~41%㊁42%~51%㊁52%~61%㊁62%~71%㊁72%~81%㊁82%~91%㊁92%~100%分别记为1~10分,并依据毛细血管扩张严重程度积分减少率进行疗效评价,将疗效分为治愈㊁显效㊁有效及无效共4个等级,治愈:毛细血管扩张严重程度积分减少率在90%以上,可见局部皮肤潮红,毛细血管管径缩小或消失;显效:积分减少率在60%~90%,可见扩张的毛细血管基本消退;有效:积分减少率在30%~60%,可见扩张的毛细血管有一定消退;无效:积分减少率<30%,未见扩张的毛细血管出现消退甚至加重㊂毛细血管扩张严重程度积分减少率=(治疗前积分-治疗前后积分)/治疗前积分ˑ100%;总有效率为治愈率㊁显效率和有效率之和㊂1.3.2㊀面部毛细血管美容效果和面部毛细血管形态评估:①面部毛细血管美容效果:对整个面部细分为49个美学亚单位标准,此次评估主要观察受测者面颊及鼻部共8个美学亚单位,单个美学亚单位出现扩张的毛细血管记为1分,总分为8分,得分越高,提示受测者的面部毛细血管美容效果越差㊂②面部毛细血管形态:依据面部毛细血管程度分级的评分标准评估受测者的面部毛细血管形态,涉及形态(1~2分)㊁数量(1~2分)㊁血管直径(1~3分)及皮肤颜色(0~1分)共4项内容及评分细则标准,得分越高,提示受测者的面部毛细血管形态越差㊂于治疗前㊁治疗5个月后及术后随访3个月后,评估两组面部美学分区评分和面部毛细血管形态评分㊂1.3.3㊀术后不良反应:记录两组受试病例的术后不良反应,如水泡㊁红肿㊁皮肤灼伤感㊁瘢痕㊁色素沉淀等㊂1.3.4㊀面容修复满意度:于术后随访3个月,对两组患者的面容修复满意度进行调查,调查内容涉及面部光滑度㊁色素㊁毛细血管改善程度及皮损消退情况,总分为1~10分,将得分换算百分制后,当分值在85分以上记为满意,70~85分记为比较满意,不足70分记为不满意,记录两组术后随访3个月的面容修复满意度㊂满意度为满意率与比较满意率之和㊂1.4㊀统计学方法:所有数据导入SPSS20.0软件分析,对计量资料进行验算得出符合正态分布和具备方差齐性特征,选择( xʃs)进行数据描述,组间选择独立t检验,组内不同时间点采用重复测量设计资料的方差分析;计数资料以n(%)表示,选择卡方检验㊂采用双侧检验,α=0.05㊂2㊀结㊀果2.1㊀两组治疗5个月后和术后随访3个月后的临床疗效对比:观察组治疗5个月后及术后随访3个月的临床总有效率均显著高于对照组(P<0.05)㊂见表1㊂表1㊀两组治疗后和术后随访3个月后的临床疗效对比n(%)组别n治疗5个月后治愈㊀㊀㊀㊀显效㊀㊀㊀㊀有效㊀㊀㊀㊀总有效术后随访3个月后治愈㊀㊀㊀㊀显效㊀㊀㊀㊀有效㊀㊀㊀㊀总有效观察组6432(50.00)18(28.13)10(15.63)60(93.75)40(62.50)15(23.44)9(14.06)64(100.00)对照组6421(32.81)15(23.44)12(18.75)48(75.00)32(50.00)16(25.00)10(15.63)58(90.63)χ28.533 6.295P0.0030.0122.2㊀两组治疗前及治疗后的面部美学分区评分及毛细血管形态评分对比:面部美学分区评分的时间效应㊁组间效应有统计学意义(P<0.05),毛细血管形态评分的时间效应有统计学意义(P<0.05),两组面部美学分区评分㊁毛细血管形态评分均较术前有降低,对照组上述评分高于观察组,差异均有统计学意义(P<0.05)㊂见表2㊂2.3㊀两组术后不良反应发生率对比:观察组的术后不良反应总发生率显著低于对照组(P<0.05)㊂见表3㊂㊃36㊃表2㊀两组治疗前及治疗后的面部美学分区评分及毛细血管形态评分对比( xʃs)组别n面部美学分区评分治疗前㊀㊀㊀治疗5个月后㊀㊀术后3个月毛细血管形态评分治疗前㊀㊀㊀治疗5个月后㊀㊀术后3个月观察组64 6.05ʃ1.38 2.52ʃ0.54a 1.21ʃ0.21a 5.56ʃ1.12 1.89ʃ0.31a 1.18ʃ0.18a 对照组64 6.18ʃ1.52 3.21ʃ0.68a 1.69ʃ0.35a 5.61ʃ1.09 2.68ʃ0.51a 1.56ʃ0.22a t,P0.507,0.613 6.357,<0.0019.408,<0.0010.256,0.79810.589,<0.00110.695,<0.001F时间点,P855.621,<0.0011553.252,<0.001F组间,P21.063,<0.0010.058,0.810F时间点ˑ组间,P 2.992,0.05140.58,0.944㊀㊀注:与治疗前对比,aP<0.05表3㊀两组不良反应发生率对比n(%)组别n水泡红肿皮肤灼伤感不良反应合计观察组640(0.00)0(0.00)1(1.56)1(1.56)对照组642(3.13)3(4.69)3(4.69)8(12.50)χ2 5.856P0.0162.4㊀两组术后随访3个月的面容满意度对比:观察组术后随访3个月后的面容修复满意度显著高于对照组(P<0.05)㊂见表4㊂表4㊀两组术后随访3个月的面容满意度对比n(%)组别n满意比较满意满意度观察组6446(71.88)18(28.13)64(100.00)对照组6432(50.00)28(43.75)60(93.75)χ2 4.129 P0.042 3㊀讨㊀论FT又称红血丝,与激素治疗㊁缺氧及寒冷刺激等因素所致有关,可见扩张后管径直径会从6~9μm增至0.1~1.0mm,且血管通透性变低,可见局部皮肤表面有点状㊁红丝网状㊁树枝状㊁星芒状或丘疹状结构,按压皮肤可出现褪色,偶见刺痛或灼热感,且好发于女性群体,严重影响患者的面容美观度[4]㊂目前,FT治疗手段包括二氧化碳激光㊁高频电刀㊁微波㊁同位素辐射治疗等㊂但极易损伤皮肤,造成皮肤出现溃疡㊁疤痕及放射性坏死等严重不良反应,严重影响患者颜面部修复效果[5]㊂近年来,IPL应用于临床皮肤科及整形科领域中越发广泛,其治疗FT具有光斑大㊁疗效均匀㊁术后并发症轻㊁无术后停工期㊁耐受性良好等优势㊂IPL治疗FT其治疗原理是通过APOT技术将IPL产生的选择性光热作用,作用于组织中的氧合血红蛋白,促使其转换为去氧血红蛋白,再进一步化学反应转变为高铁血红蛋白,同时将所产生的热能促使血管壁完整性被破坏和血红蛋白变性,从而可对血管起到封闭作用,达到改善红血丝的治疗目的[6]㊂此外,IPL治疗仪的APOT 技术属于非侵袭性操作,不仅能有效强化FT患者的皮肤屏障功能,促使皮肤在应对外界刺激时表现出更强的抵御能力,而且还能解决皮肤敏感问题,有效刺激面部胶原蛋白新生,利于血管支撑作用提升,有效保证血管壁完整性,进而提升受损肌肤及毛细血管的修复能力,改善微循环,并提高血管收缩扩张功能,从而有效改善FT带来的皮肤问题[7]㊂然后,选择不同波长及不同波段的血管滤光片所产生的强光,也会对皮肤㊁血管及局部血液流速等产生不同影响㊂㊃46㊃本文研究结果显示,采取IPL 中AOPT 治疗模式下,并选择双波段(波长530~650nm 和波长900~1200nm 血管滤光片)治疗FT 患者,能提高治疗后㊁术后随访3个月的临床总有效率至93.75%及100.00%,明显高于采取单段波长(波长在560~1200nm 血管滤光片)的75.00%及90.63%,患者的面部美学分区及毛细血管形态评分明显降低,且低于采取单段波长治疗,提示双波段的血管滤光片治疗FT 患者的近远期效果明确,能明显改善FT 患者的面部美学分区及毛细血管形态㊂屈欢欢等[8]报道也证实了,应用AOPT 技术选择双波段(波长530~650m 和波长900~1200nm )血管滤光片治疗50例FT 患者,其临床总有效率可提升至98.00%,明显高于选择其他单波段血管滤光片(波长560~1200nm ,或590~1200nm ),可支持本文论点㊂分析以上改善原因在于,由于血红蛋白可优先特定波长(如418nm ㊁542和577nm 波长)的强光,且血红蛋白二次吸收峰谱在800~1100nm ,也便于治疗靶向皮肤较为深处的血管;将IPL 从宽谱改为530~650nm 的窄谱波段,穿透力较浅,能针对面部浅表血管的黑素进行有效吸收;再选择波长900~1200nm 的窄谱波段进行皮肤深层穿透,能将强光有效集中于血红蛋白进行病理性破坏治疗,达到封闭血管的效果;此外,结合IPL 和AOPT 治疗模式更适合亚洲人的肤质治疗,能针对其不同肤色㊁血管血液流速及管径进行每个子脉冲的能量与脉宽的调节,并利用血红蛋白的双峰吸收,提高临床治疗效果,能最大程度避免无效光谱对皮肤的损伤,降低术后不良反应发生风险,提高光谱治疗有效性和安全性㊂因此,本文笔者对比2种不同血管滤光片治疗的安全性发现,采取双波段(波长530~650nm 和波长900~1200nm )的血管滤光片能有效降低术后不良反应,治疗安全性较高㊂最后,由于FT 患者经双波段血管滤光片治疗后达到满意的面容修复效果,且术后不良反应少,能有效提升患者对面容修复的满意度㊂综上所述,在AOPT 治疗模式下采取双段波长(波长530~650nm 和波长900~1200nm )血管滤光片治疗FT 患者能明显提升近㊁远期疗效,显著改善患者的面部美容效果和毛细血管形态,降低术后不良反应发生风险,利于提升患者对自身颜面部修复的满意度,临床应用价值明确,可积极推广于整形科㊂ʌ参考文献ɔ[1]㊀Turkmen D ,Altunisik N ,Sener S.The efficacy of a single -session pro -yellow laser in the treatment of facial telangiec-tasia [J ].Lasers in Medical Science ,2021,36(2):401-406.[2]㊀郑方,祝露露,李美红,等.火针治疗面部毛细血管扩张症的临床效果[J ].中国医药导报,2021,18(36):99-102.[3]㊀陆星宇,张爱华,杨雪,等.自拟清肺消痤汤联合强脉冲光治疗冲任失调型寻常痤疮疗效观察[J ].海南医学,2022,33(20):2657-2660.[4]㊀高妮,高琳.MaxG 强脉冲光治疗面部毛细血管扩张症疗效观察[J ].中国美容医学,2020,29(3):53-55.[5]㊀金清美,李周娜,南美兰,等.超脉冲CO 2点阵激光治疗鼻部毛细血管扩张症的临床疗效[J ].中国激光医学杂志,2021,30(3):177-178.[6]㊀廖康宁.强脉冲光治疗面部血管性疾患的疗效观察[D ].重庆:重庆医科大学,2020.[7]㊀张荣利,高妮,高琳,等.不同波段强脉冲光治疗面颈部毛囊红斑黑变病临床疗效分析[J ].实用皮肤病学杂志,2021,14(5):279-282.[8]㊀屈欢欢,高妮,李凯,等.强脉冲光不同波段滤光片治疗面部毛细血管扩张症的临床观察[J ].临床皮肤科杂志,2020,49(10):603-605.ʌ文章编号ɔ1006-6233(2024)01-0065-06多因子降维法分析MTHFR C677T 基因多态性-Hcy 交互作用对高血压合并冠心病的影响张明超,㊀汪㊀娟,㊀鲁大胜,㊀杨凌飞(皖南医学院第二附属医院心血管内科,㊀安徽㊀芜湖㊀241000)ʌ摘㊀要ɔ目的:基于多因子降维法(GMDR )分析亚甲基四氢叶酸还原酶(MTHFR )C677T 基因多态性-Hcy 交互作用对高血压合并冠心病的影响㊂方法:选取2021年3月至2022年10月我院收治的㊃56㊃ʌ基金项目ɔ国家自然科学基金,(编号:国青81800445)ʌ通讯作者ɔ杨凌飞。

kink效应

100IEEE ELECTRON DEVICE LETTERS,VOL.30,NO.2,FEBRUARY2009 Anomalous Kink Effect in GaN HighElectron Mobility TransistorsGaudenzio Meneghesso,Senior Member,IEEE,Franco Zanon,Michael J.Uren,Member,IEEE,and Enrico Zanoni,Fellow,IEEEAbstract—An anomalous kink effect has been observed in the room-temperature drain current I D versus drain voltage V DS characteristics of GaN high electron mobility transistors.The kink is originated by a buildup(at low V DS)and subsequent release (at high V DS)of negative charge,resulting in a shift of pinch-off voltage V P toward more negative voltages and in a sudden in-crease in I D.The kink is characterized by extremely long neg-ative charge buildup times and by a nonmonotonic behavior as a function of photon energy under illumination.The presence of traps in the GaN buffer may explain both spectrally resolved photostimulation data and the slow negative charge buildup.Index Terms—AlGaN/GaN,GaN,high electron mobility tran-sistor(HEMT),kink effect.I.I NTRODUCTIONK INK effects have been observed in various kinds of FETs, like Si MOSFETs,GaAs MESFETs,and GaAs-and InP-based high electron mobility transistors(HEMTs).Depending on the device structure and layout,on the gate length and on the properties of epitaxial materials,the kink effect may occur at different drain voltage values V DSkink and may have a different bias dependence[1],light sensitivity[2],relationship with im-pact ionization[1],[3],dynamic behavior[3],and dependence on aging and surface treatments[4],[5].Three explanations suggested for the kink effect have been clearly summarized in [2].These are the following:1)channel impact ionization and subsequent hole accumulation leading to changes of surface or channel/substrate interface potential(as in silicon-on-insulator MOSFETs[1]);2)ﬁeld-dependent trapping/detrapping in deep levels;and3)a combined effect of impact ionization and deep levels whereby generated holes modify the occupation state of surface,bulk,or channel/substrate interface deep levels [3].Kink is a detrimental phenomenon for the performance of FETs,possibly resulting in output-conductance increase, transconductance compression,and dispersion between dc and RF characteristics.Recently,kink effects have been described in GaN HEMTs[6]–[10]and were attributed to impact ioniza-tion coupled either withﬂoating-body effects[6],[7]or withManuscript received October9,2008;revised November7,2008.First published December22,2008;current version published January28,2009. This work was supported in part by EDA under the KORRIGAN Project and in part by the Italian Ministry for University and Research under the PRIN2007 Project“GaN wideband microwave and millimeter-wave integrated circuits for low-noise and high-power subsystems,”coordinated by Prof.Giovanni Ghione. The review of this letter was arranged by Editor J.A.del Alamo.G.Meneghesso, F.Zanon,and E.Zanoni are with the Department of Information Engineering,University of Padova,35131Padova,Italy (e-mail:gauss@dei.unipd.it).M.J.Uren is with QinetiQ Ltd.,WR143PS Malvern,U.K.Digital Object Identiﬁer10.1109/LED.2008.2010067traps[9],[10].In this letter,we describe an anomalous kink effect in GaN HEMTs,with a remarkable light sensitivity which is ascribed to the presence of slow traps in the epitaxial layers under the gate,possibly in the GaN buffer.II.S AMPLES AND E XPERIMENTAL R ESULTS AlGaN/GaN HEMTs were grown on4H vanadium-dopedsilicon carbide by MOVPE.The undoped multilayer structureconsisted of a1.2-μm GaN buffer layer,followed by a30-nmAl0.28Ga0.72N barrier layer.The source and drain ohmic con-tacts and gate Schottky contacts were Ti/Al/Pt/Au and Ni/Au,respectively.The devices were passivated using SiN depositedby PECVD.The device gate length and width were0.6and100μm,respectively,the gate-to-drain spacing was2.4μm,andthe drain-to-source distance was4μm.The dc electrical characterization of the devices was carriedout by means of an Agilent HP4142semiconductor parame-ter analyzer;a kink in the dc I D–V DS characteristics wasobserved only when long time constants and slow voltagesweeps were adopted,see Fig.1.For increasing V GS fromthe pinch-off voltage value of≈−6.5–0V,the kink voltageV DSkink(identiﬁed as the point where the output conductanceis maximum)has a nonmonotonic behavior as a function ofV GS.It initially decreases from≈9V(at V GS=−6V)toless than5V(at V GS=−5V)and then increases again up toV DSkink≈10V for V GS=0V.By comparing the I D versus V GS characteristics taken at V DS<V DSkink(hereafter identi-ﬁed as“prekink”region)with those taken at V DS>V DSkink(“postkink”),a shift in the pinch-off voltage V P is evident(see Fig.2).The two threshold voltage values are V p-prekink=−6V and V p-postkink=−6.3V,respectively,the most nega-tive pinch-off voltage value corresponding to the higher drain current measured when V DS>V DSkink.For these devices,the presence of the kink is inﬂuenced only by the properties of the epitaxial layers and was found not to depend on the device fabrication processing(lithography,metallization,annealing processes,plasma treatments,and passivation).In particular, both unpassivated and SiN-passivated devices show the kink. Memory effects can hinder kink characterization during re-peated measurements with a fast sweep rate(provided that V DS exceeds V DSkink during the sweep).Only theﬁrst curve will show a“kink,”i.e.,the kink is temporarily removed as a result of theﬁrst measurement(see the inset of Fig.1).The effect is recovered by keeping the device at V GS=V DS=0V at RT for at least5min.Pulsed measurements(not shown)demonstrate this“memory”effect:If V DS is pulsed from V DS>V DSkink0741-3106/$25.00©2009IEEEMENEGHESSO et al.:ANOMALOUS KINK EFFECT IN GaN HIGH ELECTRON MOBILITY TRANSISTORS101Fig. 1.Long integration time I D versus V DS with a small V GS step (−0.2V from 0to −7V).Gray diamonds correspond to the maximum of the output conductance in the kink region (V KINK ).(Inset)Repeated short integration time I DS versus V DS measurements at V GS =0V.Fig.2.Long integration time I D versus V GS characteristics measured (solid)in the dark and (dashed)under illumination using an incandescent lamp.to V DS <V DSkink ,a very slow decreasing transient of I D is observed (from milliseconds to several seconds depending on the chosen ﬁnal prekink V DS value).When V DS is pulsed from the prekink to the postkink value,I D can reach its high steady-state postkink value in 100μs to 1ms,depending on the ﬁnal bias point.Conventional wisdom suggests that traps may be associated with the charge buildup and transient phenomena related to the kink.In order to investigate a possible role of traps,we carried out dc measurements under continuous illumination and compared them with those obtained in the dark.Rather surprisingly,if the device is illuminated using an incandescent lamp,the kink amplitude increases due to an enhanced prekink shift of pinch-off voltage toward positive values,see Fig.2.We have repeated the characterization using monochromatic light in the 0.8–3.32-eV energy range and measured the amplitude of the kink.Illumination affects the behavior of drain current in the prekink region (see Fig.3).For 1eV <E <2eV,where E is the photon energy,the prekink I D value is lowered,for E >2eV,the prekink I D recovers,and for E >2.5eV,the prekink drain current value is higher than that in the dark.III.D ISCUSSION AND C ONCLUSIONThe long time constants associated with the kink effect suggest the involvement of deep levels,a hypothesis which is conﬁrmed by the peculiar spectral dependence of the kink—the kink amplitude can be modulated using photons withenergiesFig.3.I D values measured at V GS =−3V in postkink region (V DSkink +2V )and in prekink region (V DSkink −2V ))as a function of photon energy from 0.80to 3.32eV .An I D –V DS curve (V GS =−3V )is also presented in the inset to clarify the measurement conditions.Intensity of light sources was kept constant.lower than the GaN energy gap.Deep levels must be located within the epitaxial layers below the gate,since the kink is caused by a shift in V P .This cannot be explained by surface traps;the fact that the kink is not affected by passivation or surface treatments is in agreement with this interpretation.When deep levels are negatively charged,V P is shifted toward more positive values,thus decreasing the drain current:This occurs for V DS <V DSkink or,more slowly,when the device is unbiased.When V DS >V DSkink ,electron detrapping or com-pensation by holes occurs,thus V P returns to more negative values,see Fig.2.The actual mechanism responsible for the modulation of charge on deep levels cannot be due to ﬁeld-assisted detrapping,which should depend monotonically on the electric ﬁeld and,as a consequence,on V GD .Detrapping would be most efﬁcient close to pinch-off,where the V GD and electric ﬁeld are at their maximum.In fact,V DSkink has a nonmonotonic dependence on V GS ,see Fig.1,which is typical of impact-ionization effects [11].When the device is in the ON -state,energetic channel elec-trons could possibly impact ionize traps [11],thus releasing electrons,or can create electron-hole pairs.However,if hole creation by impact ionization does take place,it should be a very rare event,considering the large energy gap of the GaN channel and the low V DS values (≈5–10V)at which the kink occurs,see Fig.1.Moreover,in the electrical conditions corresponding to the occurrence of the kink,we did not observe any of the signs of electron-hole pair creation,like increase in gate current,or photon emission with energy corresponding to the GaN energy gap.A possible mechanism could be the following:When deep levels are negatively charged,electron detrapping can occur due to the impact ionization of traps by hot electrons [12].This requires substantially lower energies than electron-hole pair creation by band-to-band ionization and,consequently,can take place at substantially lower drain voltages.The onset of the kink appears at roughly a constant voltage above the knee volt-age indicating that the hot-electron-assisted detrapping process102IEEE ELECTRON DEVICE LETTERS,VOL.30,NO.2,FEBRUARY2009requires a threshold electricﬁeld and,hence,electron energy.The simultaneous requirement to have a supply of electronsresults in the increase in kink voltage near pinch-off.Illumination with photons with energy E lower than≈1eVdoes not affect the kink.When the device is biased in theprekink region,electron trapping is enhanced by illuminatingthe device with photons having energies between≈1and ≈2eV.A possible explanation could involve the presence of deep states with energies at≈E V+1eV.In this case,photo-stimulation may promote the transfer of an electron from thevalence band to the deep level,resulting in a negatively chargedtrap and reducing|V P|and I D,see Fig.3.When the photon energy exceeds≈2eV,a transition from the deep level to shallow donor states close to E C becomes possible,thus turning on a competitive mechanism removing the negative charge.As a consequence,|V P|and I D recover,and the kink amplitude is reduced,see Fig.3.Similar transitions between deep levels with energies in the2.0–2.5-eV range have been invoked in order to explain the yellow luminescence in GaN[12].Therefore, although traps could be located either within the Al0.28Ga0.72N barrier layer under the gate or in the GaN buffer,spectral data would suggest that they are located in the GaN epitaxial layer. Another consideration which tends to suggest that the traps are located in the GaN is that hot electrons must be present for the detrapping mechanism to occur.In the GaN under the gate(and including the depletion region below the channel), the density of electrons is roughly proportional to V GS−V P, whereas in the AlGaN,the density changes exponentially with gate bias as a result of the rapidly changing electricﬁeld in this region.Hence,the electron density changes much quicker in the AlGaN region,and we would not expect to see the observed V DSkink−V knee behavior.Assuming the presence of deep levels in the GaN buffer re-mote from the2-D electron gas(2DEG),three regimes need tobe considered for prekink trapping dynamics at V DS<V DSkink.1)In unbiased conditions without illumination,only a low con-centration of thermally activated electrons can reach the traps,resulting in an extremely slow trapping and charge buildup rate(hundreds of seconds).2)When the device is unilluminatedand biased in the prekink region,hot electrons from the2DEGcan be injected into the GaN where they can be capturedby traps(the electron concentration is higher than in process1)so that trapping is faster and can take place during typicalvoltage ramp measurements which use a“long”time constantof approximately a second).3)When the device biased in theprekink state is illuminated with monochromatic light withphoton energy E,trapping is promoted for1eV<E<2eV,while detrapping prevails for E>2eV.Although the quotedmechanisms could potentially be applied to trapping/detrappingin the AlGaN,spectroscopic data correspond to those reportedfor deep levels connected to yellow luminescence in GaN[12].In all cases,at V DS>V DSkink,hot electrons may reach sufﬁcient energies to impact ionize traps or possibly generate electron-hole pairs by impact ionization.However,since we did not observe any sign of impact ionization up to V DS=20V, we think that the former is the dominant mechanism in these devices.In conclusion,we have identiﬁed an“anomalous”kink effect in GaN HEMTs,with extremely long negative charge buildup times and a nonmonotonic behavior as a function of photon energy under illumination.Since kink is correlated to pinch-off voltage shifts,traps should be located under the gate,either in the AlGaN barrier or in the GaN buffer.The presence of deep traps in the GaN buffer would explain both the spectrally resolved photostimulation experiments and the slow negative charge buildup.R EFERENCES[1]M.H.Somerville,A.Ernst,and J.A.del Alamo,“A physical model for thekink effect in InAlAs/InGaAs HEMTs,”IEEE Trans.Electron Devices, vol.547,no.5,pp.922–930,May2000.[2]J.Haruyama,H.Negishi,Y.Nishimura,and Y.Nashimoto,“Substrate-related kink effects with a strong light-sensitivity in AlGaAs/InGaAs PHEMTs,”IEEE Trans.Electron Devices,vol.44,no.1,pp.25–33, Jan.1997.[3]A.Mazzanti,G.Verzellesi, C.Canali,G.Meneghesso,andE.Zanoni,“Physics-based explanation of kink dynamics in AlGaAs/GaAsHFETs,”IEEE Electron Device Lett.,vol.23,no.7,pp.383–385, Jul.2002.[4]T.Suemitsu,H.Yokoyama,T.Ishii,T.Enoki,G.Meneghesso,andE.Zanoni,“30-nm two-step recess gate InP-based InAlAs/InGaAsHEMTs,”IEEE Trans.Electron Devices,vol.49,no.10,pp.1694–1700, Oct.2002.[5]G.Meneghesso,B.Cogliati,G.Donzelli,D.Sala,and E.Zanoni,“De-velopment of‘kink’in the output I–V characteristics of pseudomor-phic HEMTs after hot-electron accelerated aging,”Microelectron.Reliab., vol.37,no.10/11,pp.1679–1682,Oct./Nov.1997.[6]B.Brar,K.Boutros,R. E.DeWames,V.Tilak,R.Shealy,andL.Eastman,“Impact ionization in high performance AlGaN/GaN HEMTs,”in Proc.IEEE Lester Eastman Conf.High Performance Devices,2002,pp.487–491.[7]S.Nuttinck,E.Gebara,skar,and M.Harris,“Floating-body effect inAlGaN/GaN power HEMT,”in Proc.GaAs Symp.,2002,pp.33–36. [8]K.Kunihiro,K.Kasahara,Y.Takahashi,and Y.Ohno,“Experimental eval-uation of impact ionization coefﬁcients in GaN,”IEEE Electron Device Lett.,vol.20,no.12,pp.608–610,Dec.1999.[9]C.-H.Lin,W.-K.Wang,P.-C.Lin,Y-J.Chang,and Y.-J.Chen,“Tran-sient pulsed analysis on GaN HEMTs at cryogenic temperatures,”IEEE Electron Device Lett.,vol.26,no.10,pp.710–712,Oct.2005.[10]H.F.Sun and C.R.Bolognesi,“Anomalous behavior of AlGaN/GaNheterostructureﬁeld-effect transistors at cryogenic temperatures:From current collapse to current enhancement with cooling,”Appl.Phys.Lett., vol.90,no.12,p.123505,Mar.2007.[11]G.Meneghesso,E.De Bortoli,A.Paccagnella,E.Zanoni,and C.Canali,“Recovery of low temperature electron trapping in AlGaAs/InGaAs PM-HEMTs due to impact-ionization,”IEEE Electron Device Lett., vol.16,no.7,pp.336–338,Jul.1995.[12]E.Calleja,F.J.Sanchez,D.Basak,M.A.Sanchez-Garcia,E.Muñoz,I.Izpura,F.Calle,J.M.G.Tijero,J.L.Sanchez-Rojas,B.Beaumont,P.Lorenzini,and P.Gibart,“Yellow luminescence and related deep states in undoped GaN,”Phys.Rev.B,Condens.Matter,vol.55,no.7, pp.4689–4694,Feb.1997.。

Nuclear Parity-Violation in Effective Field Theory

Department of Physics, Peking University Beijing 100871, China b Kellogg Radiation Laboratory, California Institute of Technology Pasadena, CA 91125, USA Departamento de F´ ısica, Funda¸ c˜ ao Universidade Federal do Rio Grande Campus Carreiros, PO Box 474 96201, Rio Grande, RS, Brazil d Department of Physics-LGRT, University of Massachusetts Amherst, MA 01003, USA e Theory Group, Thomas Jeﬀerson National Accelerator Facility Newport News, VA 23606, USA f Department of Physics, University of Connecticut Storrs, CT 06269, USA g Institute for Nuclear Theory, University of Washington Seattle, WA 98195, USA h Department of Physics, University of Arizona Tucson, AZ 85721, USA i RIKEN-BNL Research Center, Brookhaven National Laboratory Upton, NY 11973, USA
1
Introduction

wave effects on multiple bodies

3M.Kashiwagi (ed.),Hydrodynamics in Ship and Ocean Engineering,RIAM,Kyushu University,April 2001(pp.3–26)Wave Eﬀects on Multiple BodiesJ.N.NEWMANMassachusetts Institute of TechnologyCambridge,MA 02139,USAAbstractSigniﬁcant hydrodynamic interactions occur when bodies are located in closeproximity on the ocean surface.This situation exists in many applications ofpractical importance,which require rational engineering analyses.Solutionsbased on panel methods and other direct numerical methods can be usedfor conﬁgurations involving relatively small numbers of bodies.When thenumber of bodies is very large,asymptotic approximations are required.Thispaper reviews the extensive analytical and numerical accomplishments in thisﬁeld.N ew computations are included to illustrate ﬁrst-and second-orderinteraction eﬀects.Special consideration is given to conﬁgurations where theinteractions are singular at certain frequencies.1.IntroductionMany applications occur in the ﬁeld of marine hydrodynamics where two or more vessels are in suﬃciently close proximity to experience signiﬁcant interactions.Catamarans and other multi-hull ships,oﬀshore platforms supported by multiple columns,ﬂoating bridges,and arrays of wave-power devices are all examples where the proximity is a permanent feature of the design.In other cases,such as marine operations involving multiple vessels and platforms or replenishment operations of two ships,the proximity is temporary but nevertheless important.Hydrodynamic interactions related to wave eﬀects are particularly signiﬁcant,due to the oscilla-tory phase of the waves in relation to the spacing,and the large horizontal scale of the wave inﬂuence.Multiple bodies can be studied with the same experimental and theoretical methods that are applied to wave eﬀects on a single body.Typically,the analysis of two or three interacting bodies is a straightforward extension,but the analysis of very large conﬁgurations is fundamentally more diﬃcult.On the experimental side,the physical size of the model may exceed the practical limits of the wave basin,and the sensitivity of the response to the wave period and direction may dictate an extensive series of tests.Thus there is a great need for reliable theories and associated computational tools suitable for analyzing these problems.Moreover,the variety of interesting interactions that occur for multiple bodies provides a rich source of stimulus for fundamental research.4J.N.NEWMANIn most cases of practical importance,the eﬀects of ocean waves onﬂoating and submerged bodies can be analyzed by the linear potential theory.This theory is well established forﬁxed structures,and for vessels which have no substantial forward velocity.Classical solutions exist for relatively simple body shapes such as circular cylinders.In some cases it is necessary to account for second-order eﬀects,including mean drift forces and more complex time-varying nonlinearities. The same fundamental theory can be extended to the analysis of wave eﬀects on multiple bodies.In some of the examples cited above the diﬀerent bodies are connected structurally,and in others they are dynamically independent.The distinction between structurally connected or independent bodies is not important from the hydrodynamic standpoint,except insofar as the total number of modes of body motion is reduced if the connections are rigid.Much recent attention has been devoted to‘Very Large Floating Structures’(VLFS)suitable for use asﬂoating airports and‘Mobile Oﬀshore Bases’(MOB). On the order of104separate buoyancy elements are used in some of the proposed designs forﬂoating airports.Typical MOB designs involve joining several large semi-submersibles withﬂexible connectors which permit rotational deﬂections be-tween the vessels.Among the challenging technical issues for this type of structure is the coupling and decoupling of the vessels in a seaway.A similar issue exists for the pontoon-type of VLFS,an array of rectangular shallow barges which are joined together in the water.The complexity of multi-body solutions increase rapidly with the number of elements.To meet this challenge there has been steady progress in the development of numerical methods which extend the capabilities of direct-solution techniques, including panel methods suitable for the analysis of arbitrary structures and special multipole methods suitable for arrays of vertical cylinders and other axisymmetric elements.To complement these direct solutions,and oﬀer computationally feasible alternatives for very large arrays,approximations have been developed which are based in varying degrees on the hydrodynamic properties of inﬁnite arrays.A brief historical review of this subject is presented in Section2,with special attention given to the seminal works of Professor Makoto Ohkusu.The theoretical formulation is outlined in Section3,with emphasis on techniques for represent-ing the various radiation modes in a computationally eﬃcient and general form. Section4describes two examples of drift forces on multiple bodies,one the slow oscillations of two independent bodies and the other the drift force on individ-ual elements of a large array.Periodic arrays of bodies which extend to inﬁnity, and the analogous problems of bodies in channels,are considered in Section5. Closely related work onﬁnite arrays is described in Section6,with emphasis on the phenomenon of near-trapping.In Section7the extension to second-order second-harmonic eﬀects is reviewed,with a comparison made between diﬀerent models for a tension-leg platform(TLP).Section8describes recent work on doubly-inﬁnite periodic arrays,and their relation to large rectangularﬁnite arrays such as the column-supported VLFS.Conclusions are summarized in Section9.Wave Eﬀects on Multiple Bodies5 2.HistoryOne of theﬁrst papers on this topic was published in1969by Ohkusu[44].He extended the classical solution for a single heaving circular cylinder,ﬁrst devel-oped by Ursell[55],to the case of two cylinders in a catamaran conﬁguration.This type of two-dimensional analysis is most relevant to the interactions between ad-jacent shiphulls in beam seas.Such interactions are p articularly strong,indeed nearly singular,when the spacing between the hulls is an integer multiple of half a wavelength.More extensive computations and experiments were published in a subsequent paper[45]and a subset of these results is included in a more recent survey[50].Motivated by practical designs for oﬀshore platforms with multiple columns, Ohkusu[46]also developed a three-dimensional technique using the eigenfunction expansions for single axisymmetric cylinders to account for their mutual wave interactions.Hisﬁrst paper presented at an international symposium[47]summa-rized and extended both of these pioneering contributions,and brought his work to the attention of the international community.Coincidentally,the Proceedings of that symposium include on adjoining pages one of theﬁrst papers devoted to the direct numerical solution of wave eﬀects on three-dimensionalﬂoating bodies, by Faltinsen and Michelsen[15].That panel method was extended subsequently to analyze two independent bodies by van Oortmerssen[52,53]and Løken[31], but the computational resources of that time restricted their work to simple body shapes represented by rather large panels.Second-order slowly-varying drift forces are particularly important for vessels in close proximity.Over many cycles of theﬁrst-order oscillatory motions,the drift forces may cause substantial changes in the relative positions,possibly leading to collisions.Ohkusu[48]demonstrated the importance of hydrodynamic interactions on the drift force.His presentation of that paper included a memorable cinematic record of experiments,showing the behavior in beam seas of a small vessel on the weather side of a largerﬁxed structure.The sign of the drift force was shown to change depending on the spacing relative to the wavelength.As a result,the drift motion of the smaller vessel oscillated slowly in time.Another illustration of the same phenomenon is presented in Section4.Over the years,as computational power has increased and software has been reﬁned,the panel method has been applied to more complex arrays and multi-body conﬁgurations.Lee&Newman[26]show results for a proposed MOB consisting ofﬁve large semi-subs joined by hinges,with ten columns on each semi-sub.A more extensive hydroelastic analysis of a similar MOB conﬁguration is reported by Kim et al[23].Maniar and Newman[33]present results for the diﬀraction past an array of100vertical cylinders,with emphasis on the phenomenon of near-trapping which is described in Section6.Accelerated methods,such as those based on the pre-corrected FFT algorithm[24,40,42],can extend the computational limits substantially.6J.N.NEWMANThe multiple-scattering method is an alternative approach which is particularly eﬃcient for arrays of axisymmetric bodies.It is based on(1)expanding the poten-tial due to each body in cylindrical harmonics,(2)using Graf’s addition theorem to transform to the local coordinates of the other cylinders,and(3)solving the gen-eralized scattering problems for each of the other cylinders in the presence of the ﬁrst.This method,which was introduced to water-wave problems by Ohkusu[47], has been developed and extended by Kagemoto&Yue[17],Linton&Evans[28], Mavrakos[34],and in other works cited by these authors.Accelerated methods also can be used here,as demonstrated by Murai et al[39],Kashiwagi[20],and Utsunomiya et al[58].Murai et al[39]analyze VLFS structures of the column-supported and pontoon types,as well as hybrid combinations of the two types. For the pontoon type they compute the relative motions between one sub-element barge and the remaining very large structure,with application to the operation of joining the sub-elements in the ocean.Williams&Li[59]use the multiple-scattering method to analyze arrays of cylin-ders with porous walls.Chakrabarti[3,4]has used the multiple-scattering method to represent the interactions between diﬀerent bodies,in combination with local solutions at each body based on the panel method.Budal[1]and Falnes[14]have analyzed arrays of small wave-power absorbers, motivated by the economy of building large numbers of small devices,and also by the favorable outputs that could be achieved with optimum spacing of the elements in the array relative to the wavelength.Simpliﬁed theories based on long-wavelength and large-spacing approximations were used initially.A summary and comparison of diﬀerent approximations is given by Mavrakos&McIver[35], together with accurate results based on the multiple-scattering method.McIver& McIver[38]analyze a line of submerged Bristol cylinders with intermediate gaps.In cases where a very large number of identical buoyancy elements are arranged in a periodic array,it is logical to develop approximations based on the limiting case where the array extends to inﬁnity.Kagemoto et al[18]describe approximations for analyzing wave diﬀraction by long rectangular pontoon-type bodies,and also by large arrays of small columns.Ohkusu and Namba[51]have developed local approximations valid near the edges of a rectangular mat-type structure.A similar approach may be required to analyze the edges of a large periodic array.This topic is discussed more extensively in§8.3.Theoretical formulationFollowing the usual conventions,a Cartesian coordinate system x=(x,y,z)is used,with z=0the equilibrium plane of the free surface and z positive upwards. Except where otherwise noted we shall simplify the discussion by assuming that the ﬂuid depth is inﬁnite,and that plane incident waves propagate in the+x-direction with radian frequencyωand wavenumber K=ω2/g.Theﬂuid velocity is deﬁned as the gradient of the velocity potential,whichWave Eﬀects on Multiple Bodies7 is expressed in the time-harmonic form as the real part of the complex product φ(x)e iωt.The potentialφis governed by Laplace’s equation in theﬂuid domain. The linearized free-surface condition Kφ−∂φ/∂z=0is applied on z=0.One or more bodies are present in theﬂuid,eitherﬂoating on the free surface or submerged.Each body is identiﬁed by an index k,where k=1,2,...,N and N is the total number of bodies.The submerged surface of each body is denoted by S k and the global body surface S is deﬁned by the union of all body surfaces.Except in the special case of porous bodies,the normal components of the body andﬂuid velocity vectors are equal on S.Theﬂuid motion vanishes at large depths,and in the farﬁeld the radiated waves due to the presence of the bodies must be outgoing.It is convenient to decompose the potential in the alternative formsφ=φD+φR=φI+φS+φR.(1) HereφD=φI+φS is the solution of the diﬀraction problem where∂φD/∂n=0 on S,φI is the potential of the incident-wave system,andφS is the scatteredﬁeld due to the presence of the bodies.The radiation potentialφR represents theﬂuid disturbance due to the motions of the bodies.In the simplest case of a single rigid body with six degrees of freedom,it is logical to express the radiation potential in the formφR=iω6j=1ξjφj,(2)whereξj is the complex amplitude of the body motion in each degree of free-dom(surge,sway,heave,roll,pitch,yaw).The boundary condition on S is then expressed in the form∂φj/∂n=n j,(3) where(n1,n2,n3)=n,(n4,n5,n6)=x×n.(4) The hydrodynamic pressure force(and moment)acting on the body are repre-sented by the exciting-force coeﬃcientsX i=−iωρSφD n i dS,(5)and by the added-mass and damping coeﬃcientsA ij−(i/ω)B ij=ρSφj n i dS.(6)One way to extend this notation if N>1bodies are present is to use anadditional index to identify each body.Thus the exciting force X(k)i acting onbody k is the contribution to the global integral(5)from the surface S k,and the radiation potentials due to body k are deﬁned similarly in the formφ(k)j.Withand rotational motions of two bodies,e.g.a catamaran with independentheave and roll motions of each hull.The physical modes are deﬁned sep-arately for each body,with the other bodyﬁxed.The generalized modesare deﬁned to be symmetric(j odd)or antisymmetric(j even)about themidpoint between the two hulls.this notation the added-mass and damping coeﬃcients require double-superscripts to distinguish the eﬀects of each body’s motions on the others[25].This facilitates the interpretation of each body’s role,and it is most logical in when each body is physically separate from the others.An alternative and more compact notation follows if the original indices(i,j) are extended to include all of the relevant separate eﬀects of each body.For two bodies,each having six conventional rigid-body modes,the extended indices j=7,8,...,12are used to deﬁne the six modes and force components of the second body.This simpliﬁes the changes required both in the theoretical formulation and in programming.This notation is more logical if the separate bodies are viewed as elements of a single global body with the submerged surface S,and the various modes j represent the appropriate modes of normal velocity on this surface.This notation is also useful in its extension to represent generalized modes,as described below.Generalized modes,also known as generalized coordinates,can be used in vari-ous ways[41]to represent the diﬀerent rigid-body modes of multiple bodies,struc-tural deﬂections of bodies,and even the motions of interior free surfaces such as moonpools and oscillating water columns.One or more of these physically diﬀerent extensions can be analyzed simultaneously with a logical framework.For exam-ple,Lee&Newman[26]combine hinge deﬂection modes and bending modes in a single set of generalized-mode shapes suitable for the hydroelastic analysis ofﬁve semi-submersibles connected by hinges.If a set of generalized modes j=1,2,...,JWave Eﬀects on Multiple Bodies9symmetric or antisymmetric about x=0.Theﬁrst two modes correspondto global heave and pitch without hinge deﬂections.The remaining modesrepresent the hinge deﬂections with zero displacement at the ends.are deﬁned by a corresponding set of generalized normal-velocity components n j, no changes are required to extend the deﬁnitions of the hydrodynamic force coef-ﬁcients(5-6).Fundamental properties including symmetry of the added-mass and damping coeﬃcients,and the Haskind relations for the exciting forces,also are unchanged.It is straightforward to express the conventional rigid-body modes of motion in terms of a diﬀerent set of generalized modes,or vice versa.The use of generalized modes has two computational advantages.Firstly,if the global body surface S is symmetric about one or more planes of symmetry,the symmetric/antisymmetric components of the radiation potentials can be deﬁned separately in terms of corre-sponding generalized modes.A simple example with two identical bodies is illus-trated in Figure1.Secondly,if the bodies are connected by structural constraints such as hinges,the number of modes can be reduced and the constraints imposed without special programming changes.Appropriate modes for representing up to four hinged bodies are shown in Figure2.4.Drift forces on multiple bodiesAs noted in§2,the second-order slowly-varying drift forces are particularly im-portant for vessels in close proximity.In most cases the mean drift force acting on aﬂoating body is in the downwave direction.Thusﬂoating objects,like the wa-10J.N.NEWMANter particles themselves,drift slowly in the same direction as the incident waves. It is easy to verify this statement for a single body,ﬂoating freely on an un-bounded free surface.Ohkusu[48]demonstrated a remarkable exception when he analyzed a small shiplying in beam seas near the weather side of a largeﬁxed vessel.The physical explanation for this phenomenon was clear from the cinematic observations of experiments:resonant standing waves occur at critical values of the separation distance between the two vessels,and the resulting radiation stress eﬀectively pushes the two vessels apart.This phenomenon is obvious in two dimensions,and it can be expected to play an important role in the interactions of ship-like vessels in beam seas,but the ex-istence of a negative drift force is less obvious for three-dimensional conﬁgurations which are not elongated parallel to the wave crests.To study this further we con-sider the same example used by Danmeier[7,8],where a freelyﬂoating hemisphere is upwave of a circular cylinder with vertical axis.The hemisphere and cylinder have the same radius a and the cylinder draft is2a.Figure3shows how the drift force on the hemisphere depends on the spacing for four diﬀerent wavenumbers. In long waves,corresponding to the wavelength ratioλ/a=10,the drift force is relatively small but oscillatory with amplitude much greater than the limiting value when the two bodies are far apart.Since the oscillations diminish slowly with increasing separation distance,there are many intervals of the spacing where the drift force is negative.In shorter waves,e.g.λ/a=5where the maximum drift force occurs,the sign is always positive.For the intermediate case shown in Figure3,λ/a=6.67,both the oscillatory and limiting values are substantial,but only two negative regimes exist.If the wavelength is not too short,it follows from a quasi-steady analysis that the drift motion of the hemisphere will oscillate slowly in time,moving back and forth between zones of positive and negative drift force in precisely the same manner as in the experiments of Ohkusu[48].This has been conﬁrmed by Danmeier[7,8], using a time-domain analysis,but the computational cost of that method limited the duration to slightly less than one oscillatory drift cycle.For a single body the mean drift force can be evaluated directly,from integration of the second-order pressure on the body surface,or alternatively from momentum conservation.The direct approach is generally more diﬃcult since it requires a higher degree of numerical precision.The diﬃculties are especially signiﬁcant for bodies with sharpcorners such as truncated cylinders.For multip le bodies it is generally not possible to evaluate the drift force separately for each body from the momentum analysis in the farﬁeld.A local momentum analysis has been developed by Ferreira&Lee[16]to overcome this problem,but this approach requires the computation of theﬂuid velocity and pressure at a large number of control points in the nearﬁeld.An interesting example of recent numerical progress in thisﬁeld is presented by Kashiwagi[22],who computes the local drift force on various elements of an array of64truncated circular cylinders.The array includes four rows with16Fig.3Mean drift force acting on a freelyﬂoating hemisphere upwave of aﬁxed vertical cylinder,plotted vs.the separation distance s.The two bodies havethe same radius a,the cylinder draft is2a,and s is the distance betweentheir centers.The drift force is normalized by the productρgaA2whereρis theﬂuid density,g is gravity,and A is the incident-wave amplitude.Thenormalized wavenumber Ka and wavelengthλ/a are shown on the right.The dashed lines show the values of the drift force on the hemisphere alone,or in the limit s/a→∞.elements in each row.For cylinders at the upwave end,the drift force oscillates rapidly as a function of the wavenumber below the critical wavenumber where near-trapping occurs(as explained below).Above this wavenumber the upwave elements experience a drift force similar to a single isolated cylinder,but with larger magnitude.Elements in the middle of the array experience a mean drift force which oscillates less rapidly below the critical wavenumber,with a sharp peak at the critical wavenumber,followed by small magnitude above this critical point due to sheltering.The elements near the downwave end of the array experience drift forces which are smaller,or vary more slowly,except near the critical wavenumber where oscillatory features occur.These computations are substantially conﬁrmed by parallel experimental measurements.12J.N.NEWMAN5.Bodies in channels and inﬁnite periodic arraysA direct analogy exists between two diﬀerent problems,(A)a single body in a channel with parallel vertical walls,and(B)a periodic array consisting of the same body and an inﬁnite set of images reﬂected about the walls.Both problems have practical applications,including the estimation of wall reﬂection in experiments carried out in narrow wave tanks,and the use of large periodic arrays for the support of piers or bridges and very largeﬂoating structures(VLFS).Only(B) can be considered strictly as involving multiple bodies,but the analogy justiﬁes considering the two problems together here.Ohkusu[49]analyzed the reﬂection and transmission of waves by a periodic array of vertical circular cylinders,using the eigenfunction expansion for a single cylinder and summing the inﬁnite series for each parison was made with experimental results for a cylinder in a channel,and also for two cylinders situated with their axes on the channel centerline.The strong dependence on the wavenumber was generally conﬁrmed,with good agreement between the theory and experiments.Singular results were found at the frequency corresponding to theﬁrst symmetric transverse standing wave in the channel.No singular results are evident at the lower frequency where an antisymmetric trapped mode might be observed,perhaps because the forcing incident wave system is symmetric.The existence of trapped modes was established for this class of problems by Callan et al[2].For the case of a single vertical cylinder of radius a on the centerplane y=0of a channel with vertical walls at y=±d,they showed that a nontrivial motion of theﬂuid and free surface exists with no forcing from incident waves or motions of the cylinder.This solution exists at a particular wavenumber K in the range1.32<Kd<π/2,with the precise value dependent on a/d. Following earlier work on trapped modes along the axis of horizontal cylinders[56], this type of motion is referred to as a‘trapped mode’since it is conﬁned to the domain near the(vertical)cylinder and no energy radiation occurs in the far ﬁeld.The trapped mode discovered by Callan et al[2]is antisymmetric about y=0.Physically it resembles a localized‘sloshing mode’in the channel.However it diﬀers,both in wavenumber and in localization,from the simple transverse standing wave that exists at Kd=π/2in an unobstructed channel.Other cases of trapping by one or more bodies in a channel have been discovered subsequently by Evans et al[9],Evans&Porter[10],McIver et al[37],Utsunomiya &Eatock Taylor[57],and Linton&McIver[30].Closely related works which are relevant toﬁnite arrays are discussed in the following section.In the references discussed above the bodies are cylindrical throughout the ver-tical water column,andﬁxed.In these circumstances the boundary-value prob-lem for the velocity potential can be reduced to a solution of the modiﬁed wave equation in the horizontal plane,and direct analogies exist with two-dimensional acoustic problems.The full three-dimensional problem must be considered if the body geometry changes with depth,or if radiation problems are considered.Twoapproaches have been used in such cases:(1)superposing a largeﬁnite number of images and summing their eﬀects,or(2)using special multipoles or Green functions which satisfy the boundary conditions on the channel walls.The method based on superposing aﬁnite number of images has the advantage that it can be used with computer codes intended for use with a single compact body,simply by extending the deﬁnition of the body to include the images.This method has been criticized,on the basis of slow convergence of the corresponding series for the Green function and incorrect far-ﬁeld asymptotics,but the conver-gence of computed local quantities such as the hydrodynamic force coeﬃcients is usually faster.Newman[41]gives illustrative results for the hydrodynamic force coeﬃcients showing the convergence with increasing numbers of images.The method based on special multipoles which satisfy the boundary condi-tions on the channel walls has been applied to inﬁnite arrays of truncated circular cylinders by Yeung&Sphaier[60]and Linton&Evans[29].Linton&Evans de-vote special eﬀorts to the derivation of integral representations for the multipoles which satisfy exactly the conditions on the channel walls and in the farﬁeld.They show the existence of trapped modes for cylinders which are either submerged,ex-tending from the bottom part-way up to the free surface,or surface-piercing and extending part-way down to the bottom.Linton[27]has derived a more eﬃcient representation for the Green function in a channel ofﬁnite depth.6.Finite arrays and near-trappingIn work that was intended to test the limits of a higher-order panel method, Maniar&Newman[33]analyzed long arrays with upto100vertical cylinders. They discovered that,as the number of cylinders is increased,the local solution in the interior of the array increases without apparent limit in narrow bands of wavenumbers.The locations of these bands are slightly below the values Kd= nπ/2,where2d is the spacing between the axes of adjacent cylinders and n= 1,2,3,....The odd modes(n=1,3,5,...)have a phase diﬀerence of180◦at adjacent elements and zero normal velocity(∂φ/∂y=0)on the vertical planes y=±d,±3d,...,corresponding to the channel walls and their images(in the analagous case of a cylinder on the center of a channel).Conversely,the even modes(n=2,4,6,...)are in phase at adjacent elements andφ=0on the vertical planes y=±d,±3d,....In view of these boundary conditions,the odd/even modes are called Neumann or Dirichlet modes,respectively.For theﬁrst Neumann and Dirichlet modes,Maniar&Newman[33]showed that very large local exciting forces act on the cylinders near the middle of the array(up to35times the force on a single cylinder,in the case N=100).The connection with trapped modes was suggested by the coincidence of theﬁrst wavenumber (n=1)with the trapped-mode wavenumber reported by Callan et al(1991). Since the array isﬁnite,the analogy with trapped modes on inﬁnite arrays(or bodies in channels)is approximate,but with increasing correspondence as the length of the array increases toward inﬁnity.When a distinction is appropriate,。

微扰动在气流中的传播及马赫锥膨胀波的形成及普朗特迈耶

一、膨胀波的形成
超声速直匀流沿如图6-3所示的外凸壁流动，在壁面转折处，产生一道马赫波，其马赫角 sin1(1/Ma) 。气流通过马赫波之后，流动方向将沿波后壁面折转一个dθ ，称为气流折转角。通常规定相对于来流方向逆时针方向折转为正，而顺气流方向折转角为负。
动画演示PLAY
Ma>1
O
dθ
除了超声速气流沿外凸壁流动外，在其它一些情况下，如扰动源
为压强差，也可能会产生膨胀波。
图6-3 气流经膨胀波后的折转
二、普朗特—迈耶流动的形成
对于定常均匀平面超声速流绕外凸壁面的流动，通常称为普朗特迈耶流动，如图6-4所示。
Ma1
μ1
μ2ห้องสมุดไป่ตู้
Ma2
δ
图6-4 普朗特——迈耶流动
当气流流过图6-4所示的外凸壁时，可以
四、微弱压缩波
• 微弱压缩波相交形成强压缩波
强压缩波
Ma 1
弱压缩
d 3
波
d 2
d1
微弱压缩波相交
第二种扰动源
p* Ma<1
t Mat=1
t
Mae<1
Pe=Pb=P3
e
§6.3 波的相互作用
膨胀波在固体壁面上的反射与消波膨胀波的相交膨胀波在自由边界上的反射膨胀波与压缩波的相交
（波的方向要改变）。
从上述几种典型的膨胀波的相交与反射问题中，可以看出在流场中的同一个区内，气流的方向一致和静压相等。不同方向，不同压强的超声速气流
M a1
1
A′
δ
2
B
3
C′
4
M a4
2δ E
C

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↑ special Lorentz group [26]). As is known, a universal covering P↑ + of the group P+ is deﬁned
by a semidirect product P↑ + = SL(2, C) ⊙ T4 ≃ Spin+ (1, 3) ⊙ T4 , where T4 is a subgroup of
PACS numbers: 03.65.Pm, 03.65.Ge, 02.30.Gp
1
Introduction
In 1988, Biedenharn et al. [5] introduced the Poincar´ e group representations of wavefunctions on the space of complex spinors. The construction presented in [5] is an extension of the Wigner’s group theoretical method [37]. On the other hand, in accordance with basic principles of quantum ﬁeld theory, a wavefunction of the particle is a solution of some relativistic wave equation. Relativistic wavefunctions of the work [5] were introduced without explicit reference to wave equations and for this reason they represent purely group theoretical constructions. However, if we further develop the Poincar´ e group representations of wavefunctions with reference to such basic notions of QFT as a Lagrangian and wave equations, then we come to a quantum ﬁeld theory on the Poincar´ e group1 (QFTPG) introduced by Lur¸ cat in 1964 [20] (see also [3, 18, 7, 2, 19, 31, 21, 8, 11, 12] and references therein). In contrast to the standard QFT (QFT in the Minkowski spacetime) case, for QFTPG all the notions and quantities are constructed on a ten-dimensional group manifold F of the Poincar´ e group. It should be noted here that a construction of relativistic wave equations theory on the group manifold F is one of the primary problems in this area, which remains incompletely solved. Wavefunctions and wave equations on a six-dimensional submanifold L ⊂ F, which is a group manifold of the Lorentz group, have been studied in the recent work [33].
arXiv:math-ph/0308038v2 18 May 2004
Abstract The Biedenharn type relativistic wavefunctions are considered on the group manifold of the Poincar´ e group. It is shown that the wavefunctions can be factorized on the group manifold into translation group and Lorentz group parts. A Lagrangian formalism and ﬁeld equations for such factorizations are given. Parametrizations of the functions obtained are studied in terms of a ten-parameter set of the Poincar´ e group. An explicit construction of the wavefunction for the spin 1/2 is given. A relation of the proposed description with the quantum ﬁeld theory and harmonic analysis on the Poincar´ e group is discussed.
1
1
ቤተ መጻሕፍቲ ባይዱ
Ù
Ù
Z2
Z2
Ù
Ù
1
Û
T4
Û
P↑ +
Ù
Û
SL(2, C)
Ù
Û
1
Ù
Ù
1
Û
T4
Û
P↑ +
Ù
Û
L↑ +
Ù
Û
1
Ù
Ù
1
1
↑ ↑ The diagram shows that P↑ + (P+ ) is the semidirect product of SL(2, C) (L+ ) and T4 . The each transformation Tα ∈ P↑ + is deﬁned by a parameter set α(α1 , . . . , α10 ), which can be represented by a point of the space F10 . The space F10 possesses locally euclidean properties; therefore, it is a manifold, called a group manifold of the Poincare group. It is easy to see that the set α can be divided into two subsets, α(x1 , x2 , x3 , x4 , g1 , g2 , g3 , g4 , g5 , g6 ), where xi ∈ T4 are parameters of the translation subgroup, gj are parameters of the group
2
Matrix elements of both spinor and principal series representations of the Lorentz group are expressed via the hyperspherical functions [32, 34].
↑ ↑ four-dimensional translations. The relations between the groups P↑ + , P+ , SL(2, C) and L+ are deﬁned by the following diagram of exact sequences:
2
Preliminaries
Let us consider some basic facts concerning the Poincar´ e group P. First of all, the group P has the same number of connected components as the Lorentz group. Later on we will ↑ consider only the component P↑ + corresponding the connected component L+ (the so-called
In 1955, Finkelstein [9] showed that elementary particle models with internal degrees of freedom can be described on manifolds larger then Minkowski spacetime (homogeneous spaces of the Poincar´ e group). A consideration of the ﬁeld models on the homogeneous spaces leads to a generalization of the concept of wavefunction. One of the ﬁrst examples of such generalized wavefunctions was studied by Nilsson and Beskow [23].
Relativistic wavefunctions on the Poincar´ e group
V. V. Varlamov
Department of Mathematics, Siberia State University of Industry, Kirova 42, Novokuznetsk 654007, Russia