Classical Origin of the Spin of Relativistic Pointlike Particles and Geometric interpretati

The electronic properties of grapheneA.H.Castro NetoDepartment of Physics,Boston University,590Commonwealth Avenue,Boston,Massachusetts02215,USAF.GuineaInstituto de Ciencia de Materiales de Madrid,CSIC,Cantoblanco,E-28049Madrid,SpainN.M.R.PeresCenter of Physics and Department of Physics,Universidade do Minho,P-4710-057,Braga,PortugalK.S.Novoselov and A.K.GeimDepartment of Physics and Astronomy,University of Manchester,Manchester,M139PL,United Kingdom͑Published14January2009͒This article reviews the basic theoretical aspects of graphene,a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations.The Dirac electrons can be controlled by application of external electric and magneticfields,or by altering sample geometry and/or topology.The Dirac electrons behave in unusual ways in tunneling,confinement,and the integer quantum Hall effect.The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers.Edge͑surface͒states in graphene depend on the edge termination͑zigzag or armchair͒and affect the physical properties of nanoribbons.Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties.The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.DOI:10.1103/RevModPhys.81.109PACS number͑s͒:81.05.Uw,73.20.Ϫr,03.65.Pm,82.45.MpCONTENTSI.Introduction110II.Elementary Electronic Properties of Graphene112A.Single layer:Tight-binding approach1121.Cyclotron mass1132.Density of states114B.Dirac fermions1141.Chiral tunneling and Klein paradox1152.Confinement and Zitterbewegung117C.Bilayer graphene:Tight-binding approach118D.Epitaxial graphene119E.Graphene stacks1201.Electronic structure of bulk graphite121F.Surface states in graphene122G.Surface states in graphene stacks124H.The spectrum of graphene nanoribbons1241.Zigzag nanoribbons1252.Armchair nanoribbons126I.Dirac fermions in a magneticfield126J.The anomalous integer quantum Hall effect128 K.Tight-binding model in a magneticfield128 ndau levels in graphene stacks130 M.Diamagnetism130 N.Spin-orbit coupling131 III.Flexural Phonons,Elasticity,and Crumpling132 IV.Disorder in Graphene134A.Ripples135B.Topological lattice defects136C.Impurity states137D.Localized states near edges,cracks,and voids137E.Self-doping138F.Vector potential and gaugefield disorder1391.Gaugefield induced by curvature1402.Elastic strain1403.Random gaugefields141G.Coupling to magnetic impurities141H.Weak and strong localization142I.Transport near the Dirac point143J.Boltzmann equation description of dc transport indoped graphene144 K.Magnetotransport and universal conductivity1451.The full self-consistent Born approximation͑FSBA͒146 V.Many-Body Effects148A.Electron-phonon interactions148B.Electron-electron interactions1501.Screening in graphene stacks152C.Short-range interactions1521.Bilayer graphene:Exchange1532.Bilayer graphene:Short-range interactions154D.Interactions in high magneticfields154VI.Conclusions154 Acknowledgments155 References155REVIEWS OF MODERN PHYSICS,VOLUME81,JANUARY–MARCH20090034-6861/2009/81͑1͒/109͑54͒©2009The American Physical Society109I.INTRODUCTIONCarbon is the materia prima for life and the basis of all organic chemistry.Because of the flexibility of its bond-ing,carbon-based systems show an unlimited number of different structures with an equally large variety of physical properties.These physical properties are,in great part,the result of the dimensionality of these structures.Among systems with only carbon atoms,graphene—a two-dimensional ͑2D ͒allotrope of carbon—plays an important role since it is the basis for the understanding of the electronic properties in other allotropes.Graphene is made out of carbon atoms ar-ranged on a honeycomb structure made out of hexagons ͑see Fig.1͒,and can be thought of as composed of ben-zene rings stripped out from their hydrogen atoms ͑Pauling,1972͒.Fullerenes ͑Andreoni,2000͒are mol-ecules where carbon atoms are arranged spherically,and hence,from the physical point of view,are zero-dimensional objects with discrete energy states.Fullerenes can be obtained from graphene with the in-troduction of pentagons ͑that create positive curvature defects ͒,and hence,fullerenes can be thought as wrapped-up graphene.Carbon nanotubes ͑Saito et al.,1998;Charlier et al.,2007͒are obtained by rolling graphene along a given direction and reconnecting the carbon bonds.Hence carbon nanotubes have only hexa-gons and can be thought of as one-dimensional ͑1D ͒ob-jects.Graphite,a three dimensional ͑3D ͒allotrope of carbon,became widely known after the invention of the pencil in 1564͑Petroski,1989͒,and its usefulness as an instrument for writing comes from the fact that graphite is made out of stacks of graphene layers that are weakly coupled by van der Waals forces.Hence,when one presses a pencil against a sheet of paper,one is actually producing graphene stacks and,somewhere among them,there could be individual graphene layers.Al-though graphene is the mother for all these different allotropes and has been presumably produced every time someone writes with a pencil,it was only isolated 440years after its invention ͑Novoselov et al.,2004͒.The reason is that,first,no one actually expected graphene to exist in the free state and,second,even with the ben-efit of hindsight,no experimental tools existed to search for one-atom-thick flakes among the pencil debris cov-ering macroscopic areas ͑Geim and MacDonald,2007͒.Graphene was eventually spotted due to the subtle op-tical effect it creates on top of a chosen SiO 2substrate ͑Novoselov et al.,2004͒that allows its observation with an ordinary optical microscope ͑Abergel et al.,2007;Blake et al.,2007;Casiraghi et al.,2007͒.Hence,graphene is relatively straightforward to make,but not so easy to find.The structural flexibility of graphene is reflected in its electronic properties.The sp 2hybridization between one s orbital and two p orbitals leads to a trigonal planar structure with a formation of a ␴bond between carbon atoms that are separated by 1.42Å.The ␴band is re-sponsible for the robustness of the lattice structure in all allotropes.Due to the Pauli principle,these bands have a filled shell and,hence,form a deep valence band.The unaffected p orbital,which is perpendicular to the pla-nar structure,can bind covalently with neighboring car-bon atoms,leading to the formation of a ␲band.Since each p orbital has one extra electron,the ␲band is half filled.Half-filled bands in transition elements have played an important role in the physics of strongly correlated systems since,due to their strong tight-binding charac-ter,the Coulomb energies are large,leading to strong collective effects,magnetism,and insulating behavior due to correlation gaps or Mottness ͑Phillips,2006͒.In fact,Linus Pauling proposed in the 1950s that,on the basis of the electronic properties of benzene,graphene should be a resonant valence bond ͑RVB ͒structure ͑Pauling,1972͒.RVB states have become popular in the literature of transition-metal oxides,and particularly in studies of cuprate-oxide superconductors ͑Maple,1998͒.This point of view should be contrasted with contempo-raneous band-structure studies of graphene ͑Wallace,1947͒that found it to be a semimetal with unusual lin-early dispersing electronic excitations called Dirac elec-trons.While most current experimental data in graphene support the band structure point of view,the role of electron-electron interactions in graphene is a subject of intense research.It was P .R.Wallace in 1946who first wrote on the band structure of graphene and showed the unusual semimetallic behavior in this material ͑Wallace,1947͒.At that time,the thought of a purely 2D structure was not reality and Wallace’s studies of graphene served him as a starting point to study graphite,an important mate-rial for nuclear reactors in the post–World War II era.During the following years,the study of graphite culmi-nated with the Slonczewski-Weiss-McClure ͑SWM ͒band structure of graphite,which provided a description of the electronic properties in this material ͑McClure,1957;Slonczewski and Weiss,1958͒and was successful in de-scribing the experimental data ͑Boyle and Nozières 1958;McClure,1958;Spry and Scherer,1960;Soule et al.,1964;Williamson et al.,1965;Dillon et al.,1977͒.From 1957to 1968,the assignment of the electron and hole states within the SWM model were oppositetoFIG.1.͑Color online ͒Graphene ͑top left ͒is a honeycomb lattice of carbon atoms.Graphite ͑top right ͒can be viewed as a stack of graphene layers.Carbon nanotubes are rolled-up cylinders of graphene ͑bottom left ͒.Fullerenes ͑C 60͒are mol-ecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice.From Castro Neto et al.,2006a .110Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009what is accepted today.In1968,Schroeder et al.͑Schroeder et al.,1968͒established the currently ac-cepted location of electron and hole pockets͑McClure, 1971͒.The SWM model has been revisited in recent years because of its inability to describe the van der Waals–like interactions between graphene planes,a problem that requires the understanding of many-body effects that go beyond the band-structure description ͑Rydberg et al.,2003͒.These issues,however,do not arise in the context of a single graphene crystal but they show up when graphene layers are stacked on top of each other,as in the case,for instance,of the bilayer graphene.Stacking can change the electronic properties considerably and the layering structure can be used in order to control the electronic properties.One of the most interesting aspects of the graphene problem is that its low-energy excitations are massless, chiral,Dirac fermions.In neutral graphene,the chemical potential crosses exactly the Dirac point.This particular dispersion,that is only valid at low energies,mimics the physics of quantum electrodynamics͑QED͒for massless fermions except for the fact that in graphene the Dirac fermions move with a speed v F,which is300times smaller than the speed of light c.Hence,many of the unusual properties of QED can show up in graphene but at much smaller speeds͑Castro Neto et al.,2006a; Katsnelson et al.,2006;Katsnelson and Novoselov, 2007͒.Dirac fermions behave in unusual ways when compared to ordinary electrons if subjected to magnetic fields,leading to new physical phenomena͑Gusynin and Sharapov,2005;Peres,Guinea,and Castro Neto,2006a͒such as the anomalous integer quantum Hall effect ͑IQHE͒measured experimentally͑Novoselov,Geim, Morozov,et al.,2005a;Zhang et al.,2005͒.Besides being qualitatively different from the IQHE observed in Si and GaAlAs͑heterostructures͒devices͑Stone,1992͒, the IQHE in graphene can be observed at room tem-perature because of the large cyclotron energies for “relativistic”electrons͑Novoselov et al.,2007͒.In fact, the anomalous IQHE is the trademark of Dirac fermion behavior.Another interesting feature of Dirac fermions is their insensitivity to external electrostatic potentials due to the so-called Klein paradox,that is,the fact that Dirac fermions can be transmitted with probability1through a classically forbidden region͑Calogeracos and Dombey, 1999;Itzykson and Zuber,2006͒.In fact,Dirac fermions behave in an unusual way in the presence of confining potentials,leading to the phenomenon of Zitter-bewegung,or jittery motion of the wave function͑Itzyk-son and Zuber,2006͒.In graphene,these electrostatic potentials can be easily generated by disorder.Since dis-order is unavoidable in any material,there has been a great deal of interest in trying to understand how disor-der affects the physics of electrons in graphene and its transport properties.In fact,under certain conditions, Dirac fermions are immune to localization effects ob-served in ordinary electrons͑Lee and Ramakrishnan, 1985͒and it has been established experimentally that electrons can propagate without scattering over large distances of the order of micrometers in graphene͑No-voselov et al.,2004͒.The sources of disorder in graphene are many and can vary from ordinary effects commonly found in semiconductors,such as ionized impurities in the Si substrate,to adatoms and various molecules ad-sorbed in the graphene surface,to more unusual defects such as ripples associated with the soft structure of graphene͑Meyer,Geim,Katsnelson,Novoselov,Booth, et al.,2007a͒.In fact,graphene is unique in the sense that it shares properties of soft membranes͑Nelson et al.,2004͒and at the same time it behaves in a metallic way,so that the Dirac fermions propagate on a locally curved space.Here analogies with problems of quantum gravity become apparent͑Fauser et al.,2007͒.The soft-ness of graphene is related with the fact that it has out-of-plane vibrational modes͑phonons͒that cannot be found in3D solids.Theseflexural modes,responsible for the bending properties of graphene,also account for the lack of long range structural order in soft mem-branes leading to the phenomenon of crumpling͑Nelson et al.,2004͒.Nevertheless,the presence of a substrate or scaffolds that hold graphene in place can stabilize a cer-tain degree of order in graphene but leaves behind the so-called ripples͑which can be viewed as frozenflexural modes͒.It was realized early on that graphene should also present unusual mesoscopic effects͑Peres,Castro Neto, and Guinea,2006a;Katsnelson,2007a͒.These effects have their origin in the boundary conditions required for the wave functions in mesoscopic samples with various types of edges graphene can have͑Nakada et al.,1996; Wakabayashi et al.,1999;Peres,Guinea,and Castro Neto,2006a;Akhmerov and Beenakker,2008͒.The most studied edges,zigzag and armchair,have drastically different electronic properties.Zigzag edges can sustain edge͑surface͒states and resonances that are not present in the armchair case.Moreover,when coupled to con-ducting leads,the boundary conditions for a graphene ribbon strongly affect its conductance,and the chiral Dirac nature of fermions in graphene can be used for applications where one can control the valleyflavor of the electrons besides its charge,the so-called valleytron-ics͑Rycerz et al.,2007͒.Furthermore,when supercon-ducting contacts are attached to graphene,they lead to the development of supercurrentflow and Andreev pro-cesses characteristic of the superconducting proximity effect͑Heersche et al.,2007͒.The fact that Cooper pairs can propagate so well in graphene attests to the robust electronic coherence in this material.In fact,quantum interference phenomena such as weak localization,uni-versal conductancefluctuations͑Morozov et al.,2006͒, and the Aharonov-Bohm effect in graphene rings have already been observed experimentally͑Recher et al., 2007;Russo,2007͒.The ballistic electronic propagation in graphene can be used forfield-effect devices such as p-n͑Cheianov and Fal’ko,2006;Cheianov,Fal’ko,and Altshuler,2007;Huard et al.,2007;Lemme et al.,2007; Tworzydlo et al.,2007;Williams et al.,2007;Fogler, Glazman,Novikov,et al.,2008;Zhang and Fogler,2008͒and p-n-p͑Ossipov et al.,2007͒junctions,and as“neu-111Castro Neto et al.:The electronic properties of graphene Rev.Mod.Phys.,V ol.81,No.1,January–March2009trino”billiards ͑Berry and Modragon,1987;Miao et al.,2007͒.It has also been suggested that Coulomb interac-tions are considerably enhanced in smaller geometries,such as graphene quantum dots ͑Milton Pereira et al.,2007͒,leading to unusual Coulomb blockade effects ͑Geim and Novoselov,2007͒and perhaps to magnetic phenomena such as the Kondo effect.The transport properties of graphene allow for their use in a plethora of applications ranging from single molecule detection ͑Schedin et al.,2007;Wehling et al.,2008͒to spin injec-tion ͑Cho et al.,2007;Hill et al.,2007;Ohishi et al.,2007;Tombros et al.,2007͒.Because of its unusual structural and electronic flex-ibility,graphene can be tailored chemically and/or struc-turally in many different ways:deposition of metal at-oms ͑Calandra and Mauri,2007;Uchoa et al.,2008͒or molecules ͑Schedin et al.,2007;Leenaerts et al.,2008;Wehling et al.,2008͒on top;intercalation ͓as done in graphite intercalated compounds ͑Dresselhaus et al.,1983;Tanuma and Kamimura,1985;Dresselhaus and Dresselhaus,2002͔͒;incorporation of nitrogen and/or boron in its structure ͑Martins et al.,2007;Peres,Klironomos,Tsai,et al.,2007͓͒in analogy with what has been done in nanotubes ͑Stephan et al.,1994͔͒;and using different substrates that modify the electronic structure ͑Calizo et al.,2007;Giovannetti et al.,2007;Varchon et al.,2007;Zhou et al.,2007;Das et al.,2008;Faugeras et al.,2008͒.The control of graphene properties can be extended in new directions allowing for the creation of graphene-based systems with magnetic and supercon-ducting properties ͑Uchoa and Castro Neto,2007͒that are unique in their 2D properties.Although the graphene field is still in its infancy,the scientific and technological possibilities of this new material seem to be unlimited.The understanding and control of this ma-terial’s properties can open doors for a new frontier in electronics.As the current status of the experiment and potential applications have recently been reviewed ͑Geim and Novoselov,2007͒,in this paper we concen-trate on the theory and more technical aspects of elec-tronic properties with this exciting new material.II.ELEMENTARY ELECTRONIC PROPERTIES OF GRAPHENEA.Single layer:Tight-binding approachGraphene is made out of carbon atoms arranged in hexagonal structure,as shown in Fig.2.The structure can be seen as a triangular lattice with a basis of two atoms per unit cell.The lattice vectors can be written asa 1=a 2͑3,ͱ3͒,a 2=a2͑3,−ͱ3͒,͑1͒where a Ϸ1.42Åis the carbon-carbon distance.Thereciprocal-lattice vectors are given byb 1=2␲3a͑1,ͱ3͒,b 2=2␲3a͑1,−ͱ3͒.͑2͒Of particular importance for the physics of graphene are the two points K and K Јat the corners of the graphene Brillouin zone ͑BZ ͒.These are named Dirac points for reasons that will become clear later.Their positions in momentum space are given byK =ͩ2␲3a ,2␲3ͱ3aͪ,K Ј=ͩ2␲3a ,−2␲3ͱ3aͪ.͑3͒The three nearest-neighbor vectors in real space are given by␦1=a 2͑1,ͱ3͒␦2=a 2͑1,−ͱ3͒␦3=−a ͑1,0͒͑4͒while the six second-nearest neighbors are located at ␦1Ј=±a 1,␦2Ј=±a 2,␦3Ј=±͑a 2−a 1͒.The tight-binding Hamiltonian for electrons in graphene considering that electrons can hop to both nearest-and next-nearest-neighbor atoms has the form ͑we use units such that ប=1͒H =−t͚͗i ,j ͘,␴͑a ␴,i †b ␴,j +H.c.͒−t Ј͚͗͗i ,j ͘͘,␴͑a ␴,i †a ␴,j +b ␴,i †b ␴,j +H.c.͒,͑5͒where a i ,␴͑a i ,␴†͒annihilates ͑creates ͒an electron with spin ␴͑␴=↑,↓͒on site R i on sublattice A ͑an equiva-lent definition is used for sublattice B ͒,t ͑Ϸ2.8eV ͒is the nearest-neighbor hopping energy ͑hopping between dif-ferent sublattices ͒,and t Јis the next nearest-neighbor hopping energy 1͑hopping in the same sublattice ͒.The energy bands derived from this Hamiltonian have the form ͑Wallace,1947͒E ±͑k ͒=±t ͱ3+f ͑k ͒−t Јf ͑k ͒,1The value of t Јis not well known but ab initio calculations ͑Reich et al.,2002͒find 0.02t Շt ЈՇ0.2t depending on the tight-binding parametrization.These calculations also include the effect of a third-nearest-neighbors hopping,which has a value of around 0.07eV.A tight-binding fit to cyclotron resonance experiments ͑Deacon et al.,2007͒finds t ЈϷ0.1eV.FIG.2.͑Color online ͒Honeycomb lattice and its Brillouin zone.Left:lattice structure of graphene,made out of two in-terpenetrating triangular lattices ͑a 1and a 2are the lattice unit vectors,and ␦i ,i =1,2,3are the nearest-neighbor vectors ͒.Right:corresponding Brillouin zone.The Dirac cones are lo-cated at the K and K Јpoints.112Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009f ͑k ͒=2cos ͑ͱ3k y a ͒+4cosͩͱ32k y a ͪcosͩ32k x a ͪ,͑6͒where the plus sign applies to the upper ͑␲*͒and the minus sign the lower ͑␲͒band.It is clear from Eq.͑6͒that the spectrum is symmetric around zero energy if t Ј=0.For finite values of t Ј,the electron-hole symmetry is broken and the ␲and ␲*bands become asymmetric.In Fig.3,we show the full band structure of graphene with both t and t Ј.In the same figure,we also show a zoom in of the band structure close to one of the Dirac points ͑at the K or K Јpoint in the BZ ͒.This dispersion can be obtained by expanding the full band structure,Eq.͑6͒,close to the K ͑or K Ј͒vector,Eq.͑3͒,as k =K +q ,with ͉q ͉Ӷ͉K ͉͑Wallace,1947͒,E ±͑q ͒Ϸ±vF ͉q ͉+O ͓͑q /K ͒2͔,͑7͒where q is the momentum measured relatively to the Dirac points and v F is the Fermi velocity,given by v F =3ta /2,with a value v F Ӎ1ϫ106m/s.This result was first obtained by Wallace ͑1947͒.The most striking difference between this result and the usual case,⑀͑q ͒=q 2/͑2m ͒,where m is the electron mass,is that the Fermi velocity in Eq.͑7͒does not de-pend on the energy or momentum:in the usual case we have v =k /m =ͱ2E /m and hence the velocity changes substantially with energy.The expansion of the spectrum around the Dirac point including t Јup to second order in q /K is given byE ±͑q ͒Ӎ3t Ј±vF ͉q ͉−ͩ9t Јa 24±3ta 28sin ͑3␪q ͉͒ͪq ͉2,͑8͒where␪q =arctanͩq x q yͪ͑9͒is the angle in momentum space.Hence,the presence of t Јshifts in energy the position of the Dirac point and breaks electron-hole symmetry.Note that up to order ͑q /K ͒2the dispersion depends on the direction in mo-mentum space and has a threefold symmetry.This is the so-called trigonal warping of the electronic spectrum ͑Ando et al.,1998,Dresselhaus and Dresselhaus,2002͒.1.Cyclotron massThe energy dispersion ͑7͒resembles the energy of ul-trarelativistic particles;these particles are quantum me-chanically described by the massless Dirac equation ͑see Sec.II.B for more on this analogy ͒.An immediate con-sequence of this massless Dirac-like dispersion is a cy-clotron mass that depends on the electronic density as its square root ͑Novoselov,Geim,Morozov,et al.,2005;Zhang et al.,2005͒.The cyclotron mass is defined,within the semiclassical approximation ͑Ashcroft and Mermin,1976͒,asm *=12␲ͫץA ͑E ͒ץEͬE =E F,͑10͒with A ͑E ͒the area in k space enclosed by the orbit andgiven byA ͑E ͒=␲q ͑E ͒2=␲E 2v F2.͑11͒Using Eq.͑11͒in Eq.͑10͒,one obtainsm *=E Fv F2=k Fv F.͑12͒The electronic density n is related to the Fermi momen-tum k F as k F2/␲=n ͑with contributions from the two Dirac points K and K Јand spin included ͒,which leads tom *=ͱ␲v Fͱn .͑13͒Fitting Eq.͑13͒to the experimental data ͑see Fig.4͒provides an estimation for the Fermi velocity andtheFIG.3.͑Color online ͒Electronic dispersion in the honeycomb lattice.Left:energy spectrum ͑in units of t ͒for finite values of t and t Ј,with t =2.7eV and t Ј=−0.2t .Right:zoom in of the energy bands close to one of the Diracpoints.FIG.4.͑Color online ͒Cyclotron mass of charge carriers in graphene as a function of their concentration n .Positive and negative n correspond to electrons and holes,respectively.Symbols are the experimental data extracted from the tem-perature dependence of the SdH oscillations;solid curves are the best fit by Eq.͑13͒.m 0is the free-electron mass.Adapted from Novoselov,Geim,Morozov,et al.,2005.113Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009hopping parameter as v F Ϸ106ms −1and t Ϸ3eV,respec-tively.Experimental observation of the ͱn dependence on the cyclotron mass provides evidence for the exis-tence of massless Dirac quasiparticles in graphene ͑No-voselov,Geim,Morozov,et al.,2005;Zhang et al.,2005;Deacon et al.,2007;Jiang,Henriksen,Tung,et al.,2007͒—the usual parabolic ͑Schrödinger ͒dispersion im-plies a constant cyclotron mass.2.Density of statesThe density of states per unit cell,derived from Eq.͑5͒,is given in Fig.5for both t Ј=0and t Ј 0,showing in both cases semimetallic behavior ͑Wallace,1947;Bena and Kivelson,2005͒.For t Ј=0,it is possible to derive an analytical expression for the density of states per unit cell,which has the form ͑Hobson and Nierenberg,1953͒␳͑E ͒=4␲2͉E ͉t 21ͱZ 0F ͩ␲2,ͱZ 1Z 0ͪ,Z 0=Άͩ1+ͯE t ͯͪ2−͓͑E /t ͒2−1͔24,−t ഛE ഛt4ͯE t ͯ,−3t ഛE ഛ−t ∨t ഛE ഛ3t ,Z 1=Ά4ͯE t ͯ,−t ഛE ഛtͩ1+ͯE tͯͪ2−͓͑E /t ͒2−1͔24,−3t ഛE ഛ−t ∨t ഛE ഛ3t ,͑14͒where F ͑␲/2,x ͒is the complete elliptic integral of thefirst kind.Close to the Dirac point,the dispersion is ap-proximated by Eq.͑7͒and the density of states per unit cell is given by ͑with a degeneracy of 4included ͒␳͑E ͒=2A c ␲͉E ͉v F2,͑15͒where A c is the unit cell area given by A c =3ͱ3a 2/2.It is worth noting that the density of states for graphene is different from the density of states of carbon nanotubes ͑Saito et al.,1992a ,1992b ͒.The latter shows 1/ͱE singu-larities due to the 1D nature of their electronic spec-trum,which occurs due to the quantization of the mo-mentum in the direction perpendicular to the tube axis.From this perspective,graphene nanoribbons,which also have momentum quantization perpendicular to the ribbon length,have properties similar to carbon nano-tubes.B.Dirac fermionsWe consider the Hamiltonian ͑5͒with t Ј=0and theFourier transform of the electron operators,a n =1ͱN c͚ke −i k ·R na ͑k ͒,͑16͒where N c is the number of unit ing this transfor-mation,we write the field a n as a sum of two terms,coming from expanding the Fourier sum around K Јand K .This produces an approximation for the representa-tion of the field a n as a sum of two new fields,written asa n Ӎe −i K ·R n a 1,n +e −i K Ј·R n a 2,n ,b n Ӎe −i K ·R n b 1,n +e −i K Ј·R n b 2,n ,͑17͒ρ(ε)ε/tρ(ε)ε/tFIG.5.Density of states per unit cell as a function of energy ͑in units of t ͒computed from the energy dispersion ͑5͒,t Ј=0.2t ͑top ͒and t Ј=0͑bottom ͒.Also shown is a zoom-in of the density of states close to the neutrality point of one electron per site.For the case t Ј=0,the electron-hole nature of the spectrum is apparent and the density of states close to the neutrality point can be approximated by ␳͑⑀͒ϰ͉⑀͉.114Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009where the index i =1͑i =2͒refers to the K ͑K Ј͒point.These new fields,a i ,n and b i ,n ,are assumed to vary slowly over the unit cell.The procedure for deriving a theory that is valid close to the Dirac point con-sists in using this representation in the tight-binding Hamiltonian and expanding the opera-tors up to a linear order in ␦.In the derivation,one uses the fact that ͚␦e ±i K ·␦=͚␦e ±i K Ј·␦=0.After some straightforward algebra,we arrive at ͑Semenoff,1984͒H Ӎ−t͵dxdy ⌿ˆ1†͑r ͒ͫͩ3a ͑1−i ͱ3͒/4−3a ͑1+i ͱ3͒/4ͪץx +ͩ3a ͑−i −ͱ3͒/4−3a ͑i −ͱ3͒/4ͪץy ͬ⌿ˆ1͑r ͒+⌿ˆ2†͑r ͒ͫͩ3a ͑1+i ͱ3͒/4−3a ͑1−i ͱ3͒/4ͪץx +ͩ3a ͑i −ͱ3͒/4−3a ͑−i −ͱ3͒/4ͪץy ͬ⌿ˆ2͑r ͒=−i v F͵dxdy ͓⌿ˆ1†͑r ͒␴·ٌ⌿ˆ1͑r ͒+⌿ˆ2†͑r ͒␴*·ٌ⌿ˆ2͑r ͔͒,͑18͒with Pauli matrices ␴=͑␴x ,␴y ͒,␴*=͑␴x ,−␴y ͒,and ⌿ˆi†=͑a i †,b i †͒͑i =1,2͒.It is clear that the effective Hamil-tonian ͑18͒is made of two copies of the massless Dirac-like Hamiltonian,one holding for p around K and the other for p around K Ј.Note that,in first quantized lan-guage,the two-component electron wave function ␺͑r ͒,close to the K point,obeys the 2D Dirac equation,−i v F ␴·ٌ␺͑r ͒=E ␺͑r ͒.͑19͒The wave function,in momentum space,for the mo-mentum around K has the form␺±,K ͑k ͒=1ͱ2ͩe −i ␪k /2±e i ␪k /2ͪ͑20͒for H K =v F ␴·k ,where the Ϯsigns correspond to the eigenenergies E =±v F k ,that is,for the ␲*and ␲bands,respectively,and ␪k is given by Eq.͑9͒.The wave func-tion for the momentum around K Јhas the form␺±,K Ј͑k ͒=1ͱ2ͩe i ␪k /2±e −i ␪k /2ͪ͑21͒for H K Ј=v F ␴*·k .Note that the wave functions at K and K Јare related by time-reversal symmetry:if we set the origin of coordinates in momentum space in the M point of the BZ ͑see Fig.2͒,time reversal becomes equivalent to a reflection along the k x axis,that is,͑k x ,k y ͒→͑k x ,−k y ͒.Also note that if the phase ␪is rotated by 2␲,the wave function changes sign indicating a phase of ␲͑in the literature this is commonly called a Berry’s phase ͒.This change of phase by ␲under rotation is char-acteristic of spinors.In fact,the wave function is a two-component spinor.A relevant quantity used to characterize the eigen-functions is their helicity defined as the projection of the momentum operator along the ͑pseudo ͒spin direction.The quantum-mechanical operator for the helicity has the formhˆ=12␴·p ͉p ͉.͑22͒It is clear from the definition of h ˆthat the states ␺K͑r ͒and ␺K Ј͑r ͒are also eigenstates of h ˆ,h ˆ␺K ͑r ͒=±12␺K͑r ͒,͑23͒and an equivalent equation for ␺K Ј͑r ͒with inverted sign.Therefore,electrons ͑holes ͒have a positive ͑negative ͒helicity.Equation ͑23͒implies that ␴has its two eigen-values either in the direction of ͑⇑͒or against ͑⇓͒the momentum p .This property says that the states of the system close to the Dirac point have well defined chiral-ity or helicity.Note that chirality is not defined in regard to the real spin of the electron ͑that has not yet ap-peared in the problem ͒but to a pseudospin variable as-sociated with the two components of the wave function.The helicity values are good quantum numbers as long as the Hamiltonian ͑18͒is valid.Therefore,the existence of helicity quantum numbers holds only as an asymptotic property,which is well defined close to the Dirac points K and K Ј.Either at larger energies or due to the presence of a finite t Ј,the helicity stops being a good quantum number.1.Chiral tunneling and Klein paradoxIn this section,we address the scattering of chiral elec-trons in two dimensions by a square barrier ͑Katsnelson et al.,2006;Katsnelson,2007b ͒.The one-dimensional scattering of chiral electrons was discussed earlier in the context on nanotubes ͑Ando et al.,1998;McEuen et al.,1999͒.We start by noting that by a gauge transformation the wave function ͑20͒can be written as115Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009。

Physics8.421Center for Ultracold Atoms Massachusetts Institute of TechnologySpring2006Lecture NotesFebruary13,2006ContentsPreface iii 1The Two-State System:Resonance11.1Introduction (1)1.2Resonance Studies and Q.E.D (2)1.2.1The language of resonance:a classical damped system (3)1.3Magnetic Resonance:Classical Spin in Time-varying B-Field (5)1.3.1The classical motion of spins in a static magneticfield (5)1.3.2Rotating coordinate transformation (6)1.3.3Larmor’s theorem (6)1.4Motion in a Rotating Magnetic Field (7)1.4.1Exact resonance (7)1.4.2Off-resonance behavior (8)1.5Adiabatic Rapid Passage:Landau-Zener Crossing (9)1.5.1Rotating frame argument (10)1.5.2Quantum treatment-Landau Zener (11)1.6Resonance Of Quantized Spin (12)1.6.1Expected projection of quantized spin (12)1.6.2Resonance of quantized spin12 (13)1.6.2.1The Rabi transition probability (13)1.6.2.2The Hamiltonian of a quantized spin12 (14)1.7Quantum Mechanical Solution for Resonance in a Two-State System (15)1.7.1Interaction representation (15)1.7.2Two-state problem (16)1.8Density Matrix (18)1.8.1General results (18)1.8.2Density matrix for two level system (20)1.8.3Phenomological treatment of relaxation:Bloch equations (21)1.8.4Introduction:Electrons,Protons,and Nuclei (23)iii CONTENTSPrefaceThe original incarnation of these notes was developed to accompany the lectures in the MIT graduate courses in atomic physics.AMO I was created in the late1960s as a one-term in-troductory course to prepare graduate students for research in atomic physics in the Physics Department.Over the years Dan Kleppner and David Pritchard changed the contents of the course to reflect new directions of research,though the basic concepts remained as a constant thread.With the growth of interest in atom cooling and quantum gases,a second one-term course,AMO II,was designed by me a few years ago and presented with AMO I in alternating years.AMO I is still taught in the traditional way.These lecture notes were created by”culling the best of Dan and Dave”and putting them into Latex form.As part of the Joint Harvard/MIT Center for Ultracold Atoms summer school in Atomic Physics, John Doyle got involved and improved the notes.However,they are still work in progress.Wolfgang Ketterleiiiiv CONTENTSChapter1The Two-State System:Resonance1.1IntroductionThe cornerstone of major areas of contemporary Atomic,Molecular and Optical Physics (AMO Physics)is the study of atomic and molecular systemsthrough their resonant in-teraction with applied oscillating electromagneticfields.The thrust of these studies has evolved continuously since Rabi performed thefirst resonance experiments in1938.In the decade following World War II the edifice of quantum electrodynamics was constructed largely in response to resonance measurements of unprecedented accuracy on the properties of the electron and thefine and hyperfine structure of simple atoms.At the same time,nu-clear magnetic resonance and electron paramagnetic resonance were developed and quickly became essential research tools for chemists and solid state physicists.Molecular beam magnetic and electric resonance studies yielded a wealth of information on the properties of nuclei and molecules,and provided invaluable data for the nuclear physicist and physical chemist.With the advent of lasers and laser spectroscopy these studies evolved into the creation of new species,such as Rydberg atoms,to studies of matter in ultra intensefields, to fundamental studies in the symmetries of physics,to new types of metrology,and to the manipulation of matter with laser light,notably the creation of atomic quantumfluids.Resonance techniques may be used not only to learn about the structure of a system, but also to prepare it in a precisely controlled way.Because of these two facets,resonance studies have lead physicists through a fundamental change in attitude-from the passive study of atoms to the active control of their internal quantum state and their interactions with the radiationfield.The chief technical legacy of the early work on resonance spectroscopy is the family of lasers which have sprung up like the brooms of the sorcerer’s apprentice.The scientific applications of these devices have been prodigious.They caused the resurrection of physical optics-now freshly christened quantum optics-and turned it into one of the liveliestfields in physics.They have had a similar impact on atomic and molecular spectroscopy.In ad-dition they have led to new families of physical studies such as single particle spectroscopy, multiphoton excitation,cavity quantum electrodynamics,and laser cooling and trapping.This chapter is about the interactions of a two-state system with a sinusoidally oscil-latingfield whose frequency is close to the natural resonance frequency of the system.The term“two-level”system is sometimes used,but this is less accurate than the term two-state, because the levels could be degenerate,comprising several states.)However,its misusage is so widespread that we adopt it anyway-The oscillatingfield will be treated classically,and12MIT8.421Notes,Spring2006Figure1.1.Spectral profile of the H a line of atomic hydrogen by conventional absorption ponents1)and2)arise from thefine structure splitting.The possibility that a third line lines at position3)was suggested to indicate that the Dirac theory might need to be revised. (From“The Spectrum of Atomic Hydrogen”-Advances.G.W.Series ed.,World Scientific,1988). the linewidth of both states will be taken as zero until near the end of the chapter where relaxation will be treated phenomenologically.The organization of the material is historical because this happens to be also a logical order of presentation.The classical driven oscillator is discussedfirst,then the magnetic resonance of a classical spin,and then a quantized spin.The density matrix is introduced last and used to treat systems with damping-this is a useful prelude to the application of resonance ideas at optical frequencies and to the many real systems which have damping.1.2Resonance Studies and Q.E.D.One characteristic of atomic resonance is that the results,if you can obtain them at all,are generally of very high accuracy,so high that the information is qualitatively different from other types.The hydrogenfine structure is a good example.In the late1930s there was extensive investigation of the Balmer series of hydrogen, (n>2→n=2).The Dirac Theory was thought to be in good shape,but some doubts were arising.Careful study of the Balmer-alpha line(n=3→n=2)showed that the line shape might not be given accurately by the Dirac Theory.Pasternack,in1939,suggested that the2s2S1/2and2p2P1/2states were not degenerate, but that the energy of the2s state was greater than the Dirac value by∼.04cm−1(or, in frequency,∼1,200MHz).However,there was no rush to throw out the Dirac theory on suchflimsy evidence.In1947,Lamb found a splitting between the2S1/2and2P1/2levels using a resonance method.The experiment is one of the classics of physics.Although his veryfirst observa-tion was relatively crude,it was nevertheless accurate to one percent.He foundS H=1h E(2S1/2)−E(2P1/2) =1050(10)MHz(1.1)The inadequacy of the Dirac theory was inescapably demonstrated.1.2.Resonance Studies and Q.E.D.3The magnetic moment of the electron offers another example.In 1925,Uhlenbeck and Goudsmit suggested that the electron has intrinsic spin angular momentum S =1/2(in units of ¯h )and magnetic momentµe =e ¯h 2m=µB (1.2)where µB is the Bohr magneton.The evidence was based on studies of the multiplicity of atomic lines (in particular,the Zeeman structure).The proposal was revolutionary,but the accuracy of the prediction that µe =µB was poor,essentially one significant figure.According to the Dirac theory,µe =µB ,exactly.However,our present understanding is µe µB−1=1.1596521884(43)×10−3(experiment,U.of Washington)(1.3)This result is in good agreement with theory,the limiting factor in the comparison being possible doubts about the value of the fine structure constant.The Lamb shift and the departure of µe from µB resulted in the award of the 1955Nobel prize to Lamb and Kusch,and provided the experimental basis for the theory of quantum electrodynamics for which Feynman,Schwinger and Tomonaga received the Nobel Prize in 1965.1.2.1The language of resonance:a classical damped systemBecause the terminology of classical resonance,as well as many of its features,are car-ried over into quantum mechanics,we start by reviewing an elementary resonant system.Consider a harmonic oscillator composed of a series RLC circuit.The charge obeys¨q +γ˙q +ω20q =0(1.4)where γ=R/L,ω20=1/LC .Assuming that the system is underdamped (i.e.γ2<4ω20),the solution for q is a linear combination of exp −γ2 exp ±iω t (1.5)where ω =ω0 1−γ2/4ω20.If ω0 γ,which is often the case,we have ω ≡ω0.The energy in the circuit is W =12C q 2+12L ˙q 2=W 0e −γt (1.6)where W 0=W (t =0).The decay time of the stored energy is τ=1γ.If the circuit is driven by a voltage E 0e iωt ,the steady state solution is q o e iωt whereq 0=E 02ω0L 1(ω0−ω+iγ/2).(1.7)(We have made the usual resonance approximation:ω2o −ω2≈2ω0(ω0−ω).)The average power delivered to the circuit isP =12E 20R 11+ ω−ω0γ/2 2(1.8)4MIT8.421Notes,Spring2006Figure1.2.Sketch of a Lorentzian curve,the universal response curve for damped oscillators and for many atomic systems.The width of the curve(full width at half maximum)is∆ω=γ,where γis the decay constant.The time constant for decay isτ=γ.In the presence of noise(right),the frequency precision with which the center can be located,δω,depends on the signal-to-noise ratio, S/N:δω=∆ω/(S/N).The plot of P vsω(Fig.1.2)is a universal resonance curve often called a“Lorentzian curve”.The full width at half maximum(“FWHM”)is∆ω=γ.The quality factor of the oscillator isQ=ω0∆ω(1.9)Note that the decay time of the free oscillator and the linewidth of the driven oscillator obeyτ∆ω=1(1.10) This can be regarded as an uncertainty relation.Assuming that energy and frequency are related by E=¯hωthen the uncertainty in energy is∆E=¯h∆ωandτ∆E=¯h(1.11) It is important to realize that the Uncertainty Principle merely characterizes the spread of individual measurements.Ultimate precision depends on the experimenter’s skill:the Uncertainty Principle essentially sets the scale of difficulty for his or her efforts.The precision of a resonance measurement is determined by how well one can“split”the resonance line.This depends on the signal to noise ratio(S/N).(see Fig.1.2)As a rule of thumb,the uncertaintyδωin the location of the center of the line isδω=∆ωS/N(1.12)In principle,one can makeδωarbitrarily small by acquiring enough data to achieve the required statistical accuracy.In practice,systematic errors eventually limit the precision. Splitting a line by a factor of104is a formidable task which has only been achieved a few1.3.Magnetic Resonance:Classical Spin in Time-varying B-Field5 times,most notably in the measurement of the Lamb shift.A factor of103,however,is not uncommon,and102is child’s play.1.3Magnetic Resonance:Classical Spin in Time-varying B-Field1.3.1The classical motion of spins in a static magneticfieldNote:angular momentum will always be expressed in a form such as¯h J,where the vector J is dimensionless.The interaction energy and equation of motion of a classical spin in a static magnetic field are given byW=− µ·B,(1.13)F=−∇W=∇( µ·B),(1.14)torque= µ×B.(1.15) In a uniformfield,F=0.The torque equation(d¯h/dt=torque)givesd¯h J= µ×B.(1.16)dtSince µ=γ¯h J,we haved J=γJ×B=−γB×J.(1.17)dtTo see that the motion of J is pure precession about B,imagine that B is alongˆz and that the spin,J,is tipped at an angleθfrom this axis,and then rotated at an angleφ(t) from theˆx axis(ie.,θandφare the conventionally chosen angles in spherical coordinates). The torque,−γB×J,has no component along J(that is,alongˆr),nor alongθ(because the J−B plane containsθ),hence−γB×J=−γB|J|sinθˆφ.This implies that J maintains constant magnitude and constant tipping angleθ.Since theφ-component of J is d J/dt =|J|sinθ(dφ/dt)it is clear thatφ(t)=−γBt.This solution shows that the moment precesses with angular velocityΩL=−γB(1.18) whereΩL is called the Larmor Frequency.For electrons,γe/2π=2.8MHz/gauss,for protonsγp/2π=4.2kHz/gauss.Note that Planck’s constant does not appear in the equation of motion:the motion is classical.6MIT8.421Notes,Spring20061.3.2Rotating coordinate transformationA second way tofind the motion is to look at the problem in a rotating coordinate system. If some vector A rotates with angular velocityΩ,thend A=Ω×A.(1.19)dtIf the rate of change of the vector in a system rotating atΩis(d A/dt)rot,then the rate of change in an inertial system is the motion in plus the motion of the rotating coordinate system.d A dt= d A dt rot+Ω×A.(1.20)inertThe operator prescription for transforming from an inertial to a rotating system is thusd dt= d dt inert−Ω×.(1.21)rotApplying this to Eq.1.17givesd J dt=γJ×B−Ω×J=γJ×(B+Ω/γ).(1.22)rotIf we letB eff=B+Ω/γ,(1.23) Eq.1.22becomesd J dt=γJ×B eff.(1.24)rotIf B eff=0,J is constant in the rotating system.The condition for this isΩ=−γB(1.25) as we have previously found in Eq.1.18.1.3.3Larmor’s theoremTreating the effects of a magneticfield on a magnetic moment by transforming to a rotat-ing co-ordinate system is closely related to Larmor’s theorem,which asserts that the effect of a magneticfield on a free charge can be eliminated by a suitable rotating co-ordinate transformation.1.4.Motion in a Rotating Magnetic Field7Consider the motion of a particle of mass m,charge q,under the influence of an applied force F0and the Lorentz force due to a staticfield B:F=F0+q v×B.(1.26) Now consider the motion in a rotating coordinate system.By applying Eq.1.20twice to r,we have¨r rot=¨r inert−2Ω×v rot−Ω×(Ω×r).(1.27)F rot=F inert−2m(Ω×v rot)−mΩ×(Ω×r),(1.28) where F rot is the apparent force in the rotating system,and F inert is the true or inertial force.Substituting Eq.1.26givesF rot=F inert+q v×B+2m v×Ω−mΩ×(Ω×r).(1.29)If we chooseΩ=−(q/2m)B,and take B=ˆz B,we haveF rot=F inert−mΩ2B2ˆz×(ˆz×r).(1.30) The last term is usually small.If we drop it we haveF rot=F inert.(1.31)The effect of the magneticfield is removed by going into a system rotating at the Larmor frequency qB/2m.Although Larmor’s theorem is suggestive of the rotating co-ordinate transformation, Eq.1.22,it is important to realize that the two transformations,though identical in form, apply to fundamentally different systems.A magnetic moment is not necessarily charged-for example a neutral atom can have a net magnetic moment,and the neutron possesses a magnetic moment in spite of being neutral-and it experiences no net force in a uniform magneticfield.Furthermore,the rotating co-ordinate transformation is exact for a mag-netic moment,whereas Larmor’s theorem for the motion of a charged particle is only valid when the B2term is neglected.1.4Motion in a Rotating Magnetic Field1.4.1Exact resonanceConsider a moment µprecessing about a staticfield B0,which we take to lie along the z axis.Its motion might be described byµz=µcosθ,µx=µsinθcosω0t,µy=−µsinθsinω0t(1.32) whereω0is the Larmor frequency,andθis the angle the moment makes with B o.8MIT8.421Notes,Spring2006 Now suppose we introduce a magneticfield B1which rotates in the x-y plane at the Larmor frequencyω0=−γB0.The magneticfield isB(t)=B1(ˆx cosω0t−ˆy sinω0t)+B0ˆz.(1.33) The problem is tofind the motion of µ.The solution is simple in a rotating coordinate system.Let systemˆx ,ˆy ,ˆz =ˆz)precess around the z-axis at rate−ω0.In this system the field B1is stationary(andˆx is chosen to lie along B1),and we haveB(t)eff=B(t)−(ω0/γ)ˆz=B1ˆx +(B0−ω0/γ)ˆz=B1ˆx .(1.34) The effectivefield is static and has the value of B1.The moment precesses about the field at rateωR=γB1,(1.35) often called the Rabi frequency.This equation contains a lot of history:the RF magnetic resonance community con-ventionally calls this frequencyω1,but the laser resonance community calls it the Rabi FrequencyωR in honor of Rabi’s invention of the resonance technique.If the moment initially lies along the z axis,then its tip traces a circle in theˆy −ˆz plane. At time t it has precessed through an angleφ=ωR t.The moment’s z-component is given byµz(t)=µcosωR t.(1.36) At time T=π/ωR,the moment points along the negative z-axis:it has“turned over”.1.4.2Off-resonance behaviorNow suppose that thefield B1rotates at frequencyω=ω0.In a coordinate frame rotating with B1the effectivefield isB eff=B1ˆx +(B0−ω/γ)ˆz.(1.37)The effectivefield lies at angleθwith the z-axis,as shown in Fig.1.3Thefield is static, and the moment precesses about it at rate(called the effective Rabi frequency)ω R=γB eff=γ (B0−ω/γ)2+B21= (ω0−ω)2+ω2R(1.38) whereω0=γB0,ωR=γB1,as before.Assume that µpoints initially along the+z-axis.Findingµz(t)is a straightforward problem in geometry.The moment precesses about B effat rateω R,sweeping a circle1.5.Adiabatic Rapid Passage:Landau-Zener Crossing9 as shown.The radius of the circle isµsinθ,where sinθ=B1/ (B0−ω/γ)2+B21=ωR/ (ω−ω0)2+ω2R.In time t the tip sweeps through angleφ=ω R t.The z-component of the moment isµz(t)=µcosαwhereαis the angle between the moment and the z-axis after it has precessed through angleφ.As the drawing shows,cosαis found from A2=2µ2(1−cosα).Since A=2µsinθsin(ω R t/2),we have4µ2sin2θsin2(ω R t/2)= 2µ2(1−cosα)andµz(t)=µcosα=µ(1−2sin2θsin2ω R t/2)=µ 1−2ω2R(ω−ω0)+ω2R sin212 (ω−ω0)2+ω R t(1.39)=µ 1−2(ωR/ω R)2sin2(ω R t/2) (1.40)The z-component of µoscillates in time,but unlessω=ω0,the moment never completely inverts.The rate of oscillation depends on the magnitude of the rotatingfield;the amplitude of oscillation depends on the frequency difference,ω−ω0,relative toωR.The quantum mechanical result will turn out to be identical.1.5Adiabatic Rapid Passage:Landau-Zener CrossingAdiabatic rapid passage is a technique for inverting a spin population by sweeping the system through resonance.Either the frequency of the oscillatingfield or the transition frequency(e.g.,by changing the applied magneticfield)is slowly varied.The principle is qualitatively simple in the rotating coordinate system.The problem can also be solved analytically.In this section we give the qualitative argument and then present the analytic quantum result.The solution is of quite general interest because this physical situation arises frequently,for example in inelastic scattering,where it is called a curvecrossing.10MIT 8.421Notes,Spring 2006B 1dB eff(t )dt =1B 11γdωdtγB 1,(1.41)or using ωR =γB 1,dωdtω2R ,(1.42)In this example we have shown that a slow change from ω γB 0to ω γB 0will flip the spin;the same argument shows that the reverse direction of slow change will also flip1.5.Adiabatic Rapid Passage:Landau-Zener Crossing11 the spin.For a two-state system the problem can be solved rigorously.Consider a spin1/2system in a magneticfield B effwith energiesW±=±12¯hγB eff.(1.43)For a uniformfield B0(with B1=0),the effectivefield in the rotating frame is B0−ω/γ, andW±=±12¯h(ω0−ω),(1.44)whereω0=γB0.Asωis swept through resonance,the two states move along their charging eigenenergies.The energies change,but the states do not.There is no coupling between the states,so a spin initially in one or the other will remain so indefinitely no matter how ωchanges relative toω0.In the presence of a rotatingfield,B1,however,the energy levels look quite different:in-stead of intersecting lines they form non-intersecting hyperbolas separated by energy±¯hωR. If the system moves along these hyperbolas,then↑→↓and↓→↑.2ωRFigure1.5.An avoided crossing.A system on one level can jump to the other if the parameter q that governs the energy levels is swept sufficiently rapidly.1.5.2Quantum treatment-Landau ZenerWhether or not the system follows an energy level adiabatically depends on how rapidly the energy is changed,compared to the minimum energy separation.To cast the problem in quantum mechanical terms,imagine two non-interacting states whose energy separation,ω,depends on some parameter x which varies linearly in time,and vanishes for some value x0.Now add a perturbation having an off-diagonal matrix element V which is independent12MIT8.421Notes,Spring2006 of x,so that the energies at x0are±V,as shown in Fig.1.11e.The probability that the system will“jump”from one adiabatic level to the other after passing through the“avoided crossing”(i.e.,the probability of non-adiabatic behavior)isP na=e−2πΓ(1.45) whereΓ=|V|2¯h2dωdt −1(1.46)This result was originally obtained by Landau and Zener.The jumping of a system as it travels across an avoided crossing is called the Landau-Zener effect.Further description, and reference to the initial papers,can be found in[5].Inserting the parameters for our magneticfield problem,we haveP na=exp −π2ω2R dω/dt (1.47) Note that the factor in the negative exponential is related to the inequality in Eq.1.42. When Eq.1.42is satisfied,the exponent is large and the probability of non-adiabatic be-havior is exponentially small.Incidentally the“rapid”in adiabatic rapid passage is something of a misnomer.The technique was originally developed in nuclear magnetic resonance in which thermal relax-ation effects destroy the spin polarization if one does not invert the population sufficiently rapidly.In the absence of such relaxation processes one can take as long as one pleases to traverse the anticrossings,and the slower the crossing the less the probability of jumping.1.6Resonance Of Quantized Spin1.6.1Expected projection of quantized spinBefore solving the quantum mechanical problem of a magnetic moment in a time varying field,it is worthwhile demonstrating that its motion is classical.By“its motion is classical”we mean the time evolution of the expectation value of the magnetic moment operation obeys the classical equation of motion.Specifically,we shall show thatddtµop =γ µop ×B.(1.48) Proof:Recall that,for any operator Od dt O =i¯h[H,O] +∂O∂t.(1.49)If the operator is not explicitly time dependent the last term vanishes.The interaction of µwith a staticfield B0z is1.6.Resonance Of Quantized Spin13H=− µop·B0=−γJ·B0=−γB0J z,(1.50) Note that J has dimensions of angular momentum.Thusd µopdt=−iγB0[J z, µop]/¯h.(1.51) Using µop=γJ,we can rewrite this asd Jdt=−iγB0[J z,J]/¯h.(1.52) The commutation rules for J are[J x,J y]=i¯h J z,etc.,or J×J=i¯h J.(This is a shorthand way of writing[J i,J j]= ij,J k.)Hence˙Jx=γB0J y(1.53)˙Jy=−γB0J x(1.54)˙Jz=0(1.55) These describe the uniform precession of J about the z axis at a rate−γB0.ThusddtJ =γ J ×B(1.56) and since µop=γJ,this directly yields Eq.1.48:ddtµop =γ µop ×B.(1.57) Thus the quantum mechanical and classical equation of motion are identical.This fact underlies the great utility of classical magnetic resonance in providing intuition about res-onance in quantum spin systems.1.6.2Resonance of quantized spin11.6.2.1The Rabi transition probabilityFor a spin1/2particle we can push the classical solution further and obtain the amplitudes and probabilities for each state.Consider µz /¯h=γ J z =γm,where m is the usual “magnetic”quantum number.For a spin1/2particle m has the value+1/2or−1/2.Let the probabilities for having these values be P+and P−respectively.ThenJ z =12P+−12P−,(1.58)or,since P++P−=1,J z =12(1−2P−),(1.59)µz =12γ¯h(1−2P−).(1.60)14MIT8.421Notes,Spring2006 If µlies along the z axis at t=0,thenµz(0)=γ¯h/2,and we haveµz(t)=µz(0)(1−2P−).(1.61) In this case,P−is the probability that a spin in state m=+1/2at t=0has made a transition to m=−1/2at time t,P↑→↓(t).Comparing Eq.1.61with1.39,we seeP↑→↓(t)=ω2Rω2R+(ω−ω0)2sin212 ω2R+(ω−ω0)2t(1.62)P↑→↓(t)=(ωR/ω R)2sin2(ω R t/2)(1.63) This result is known as the Rabi transition probability.It is important enough to memo-rize.We have derived it from a classical correspondence argument,but it can also be derived quantum mechanically.In fact,such a treatment is essential for a complete understanding of the system.1.6.2.2The Hamiltonian of a quantized spin12Now we investigate the time dependence of the wave functions for a quantized spin12systemwith moment µ=γ¯h S is placed in a uniform magneticfield B=ω0k/γand,starting at t =0,subject to afield B(t)which rotates in the x−y plane with frequencyω.Thesefields are the same as thefields discussed in the preceding section on the motion of classical spin and a time-varyingfield.The only difference is that now we are discussing their effect on a quantized system,so we must use Schr¨o dingerr’s equation rather than the laws of classical Electricity and Magnetism to discuss the dynamics of the system.The basis states are(using the standard column vector representation):|1 = 10 (1.64)|2 = 01 (1.65) with state|1 lower in energy.The unperturbed Hamiltonian is(remember,ω0=γB0,and µis an operator)H0=− µ·B0=−¯h S zω0=−12¯hω0 100−1 =−12¯hω0σz(1.66)whereσz is a Pauli spin matrix.The energies areE1=−¯hω0/2ω1=−ω0/21.7.Quantum Mechanical Solution for Resonance in a Two-State System15E2=+¯hω0/2ω2=+ω0/2(1.68) The perturbation Hamiltonian is simplified by usingωR=γB R where B R is the magnetic field which rotates in the x−y plane.(In the magnetic resonance community the subscript R is often replaced by1.)H (t)=− µ·B R(t)=− µ·(ωR/γ)[ˆx cosωt−ˆy sinωt]=−ωR[S x cosωt−S y sinωt]=−¯hωR2 0110cosωt− 0−i i0 sinωt=−¯hωR2 0e iωte−iωt0 (1.69)ˆx andˆy are unit vectors,and µ·ˆx=γS x.In the penultimate line we have replaced the operators S x and S y with(¯h/2)σx and(¯h/2)σy whereσx andσy are Pauli spin matrices. The perturbation matrix element is just the entry H 21(t)in row2and column1:2|H |1 =−¯hωR2e−iωt(1.70)We have thus derived the following Hamiltonian for the spin12problem:H=¯h2 −ω0−ωR e+iωt−ωR e−iωtω0 (1.71)This Hamiltonian is of the famous form for the“dressed atom”which we will discuss elsewhere.1.7Quantum Mechanical Solution for Resonance in a Two-State SystemAs has been emphasized,a two-state system coupled by a periodic interaction is an archetype for large areas of atomic/optical physics.The quantum mechanical solution can be achieved by a variety of approaches,the most elegant of which is the dressed atom picture in which the atom and radiationfield constitute a single quantum system and onefinds its eigenstates. That approach will be introduced later.Here we follow a rather different approach,less elegant,but capable of being generalized to a variety or problems including multi-level resonance and radiative decay in the presence of oscillatingfields.The starting point is the interaction representation.1.7.1Interaction representationWe consider a complete set of eigenstates to a Hamiltonian H0,ψ=|1>,|2>...,such thatH0|j>=E j|j>.(1.72) The problem is tofind the behavior of the system under an interaction V(t),i.e.tofind solutions to。

小学上册英语第3单元测验卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.His dream is to be a ________.2.The pelican dives into the _________ (水) to catch fish.3.They like to watch ________ movies.4. A chemical reaction can lead to a change in ______ state.5.Snakes can be ______ or harmless.6.The _____ (露水) on the grass is refreshing in the morning.7.What is the name of the famous scientist known for his work on electromagnetism?A. James Clerk MaxwellB. Nikola TeslaC. Thomas EdisonD. Michael Faraday答案: A8.The _____ (fauna) interacts with plant life.9.The ________ (chick) is cute and fluffy.10.The ancient Egyptians used ________ for religious texts.11. A ______ is a positively charged particle in the nucleus of an atom.12.My grandma is a wonderful __________ (导师).13. A sea turtle can live for _______ (很多年).14. A ____(participatory approach) involves residents in decision-making.15.I like to eat ______.16. A rabbit can jump really ______ (高).17.The __________ helps to circulate blood in the body.18.I like to help my mom ________ (整理) my room.19.We have a ______ (大) backyard to play in.20.The library is _____ (quiet/loud) and peaceful.21.What is the term for a young dolphin?A. CalfB. PupC. KitD. Chick答案:a22.The ________ (initiative) encourages innovation.23.What do we call a baby deer?A. FawnB. CalfC. KidD. Lamb答案: A. Fawn24.My cousin is a ______. She enjoys storytelling.25.What is the capital of Panama?A. Panama CityB. ColonC. DavidD. Chiriquí答案: A26.I love watching the ________ (星星) at night.27.The bat can fly in the _____.28.I enjoy _____ (聊天) with friends.29.My mom is very _______ (形容词) when it comes to gardening. 她的花园很_______ (形容词).30.The ancient Mayans were known for their _____ calendar.31.The engineer, ______ (工程师), solves complex problems.32.My dad works as a _______ (工程师).33.Rabbits like to dig _________. (洞)34.What do we call a group of lions?A. PackB. HerdC. PrideD. Flock答案:C35.Which of these is a type of fruit?A. CarrotB. BananaC. PotatoD. Cucumber答案:B Banana36.She is ___ (laughing/sobbing) at the joke.37. A turtle moves slowly but has a hard _______.38.The ________ was a notable treaty that settled numerous disputes.39.Which insect can make honey?A. AntB. ButterflyC. BeeD. Fly答案:C40.My little sister has a _________ (跳跳球) that she loves to bounce around the house.41.My mom gives me __________ (建议) when I need help.42.She enjoys ________.43.What is the opposite of "short"?A. TallB. BigC. ThinD. Heavy答案:A Tall44.His favorite animal is a ________.45.Chemical reactions can be represented using ________ equations.46.The cake is ________ for my birthday.47.The ______ (小鱼) swims gracefully in the tank.48.The kitten loves to cuddle with its _________. (主人)49.The __________ is a popular tourist destination.50.Electric circuits need a ______ (complete) path to work.51.My favorite _____ is a bright blue kite.52.I play pretend with my ________ (玩具名称).53.The _____ (温暖) climate is suitable for tropical plants.54.She is __________ her friend a secret.55.The ______ (植物的交互作用) with animals is complex.56.The __________ is a large area of land that is covered with grass.57.My favorite hobby is ______ (绘画).58. A kitten purrs when it feels ______ (舒适).59.She is _______ (running) in the race.60.What do we use to read?A. BookB. KnifeC. PlateD. Spoon答案:A Book61.Gases fill the entire _____ they are in.62.The chemical formula for sulfuric acid is __________.63.The capital city of Latvia is __________.64._____ (绿化) projects improve city landscapes.65.The __________ is a famous mountain range in South America.66.Which of these animals is a mammal?A. FishB. BirdC. DogD. Reptile答案:C Dog67.The __________ (历史的内容) engages different audiences.68.The Industrial Revolution started in the ________.69. A __________ is a geological feature that can be shaped by erosion.70.The _____ (waterfall) is beautiful.71.mation led to the rise of __________ (新教). The Refo72.What is the name of the famous scientist known for his theory of relativity?A. Isaac NewtonB. Albert EinsteinC. Galileo GalileiD. Nikola Tesla 答案: B73.The element with the symbol Te is __________.74.My aunt loves to __________. (看书)75. A _______ is a good choice for beginning gardeners.76.What do you call the main part of a plant that supports it?A. RootB. StemC. LeafD. Flower答案:B77.What do you call a young female goat?A. KidB. CalfC. LambD. Pup答案: A78.My cousin is a talented ____ (photographer).79._____ (多肉植物) store water in their leaves.80.The ancient Romans used ________ for building strong structures.81. A __________ is a piece of land that projects into a body of water.82.I want to buy a new __________.83.The ______ (植物生长) cycle is fascinating.84.The _____ (penguin) is waddling.85.The ancient Egyptians used ________ for their writing system.86.The ____ has a fluffy tail and likes to chase after insects.87.The finch is a small _______ (鸟) that sings beautifully.88.The __________ is a major river in South America. (亚马逊河)89.I have a pet ________ that loves to play.90.She has a ___ (nice) dress.91.What is the name of the famous rock band from Liverpool?A. The Rolling StonesB. The BeatlesC. Led ZeppelinD. Pink Floyd 答案:B92.How many sides does a triangle have?A. 3B. 4C. 5D. 6答案:A 393.The snapping turtle can bite very _________ (痛).94.The __________ is the part of a plant that holds leaves and flowers.95.continent) of Australia is both a country and a landmass. The ____96.The chemical formula for rust is _______.97.The first man-made satellite was ________ (斯普特尼克).98.What do you call a place where you can buy books?A. LibraryB. StoreC. SchoolD. Park答案:B99.The formula for carbon dioxide is _______.100.An electrolyte conducts ______ when dissolved in water.。

对飞向太空的看法英语作文The Allure of Space Exploration.Humanity's quest to explore the unknown has always been a defining characteristic of our species. From the earliest cave drawings depicting animals and celestial bodies to the modern-day telescopes scanning the vastness of the universe, our curiosity has pushed the boundaries of knowledge and understanding. Among these endeavors, the concept of space exploration stands out as a testament to our resolve to conquer the final frontier.The allure of space exploration is multifaceted. At its core, it represents a quest for knowledge and understanding. Space is a vast and mysterious domain, filled withmysteries that have fascinated people for centuries. The idea of understanding the origin of the universe, thenature of black holes, and the existence ofextraterrestrial life is incredibly compelling. Space exploration offers a window to the past, present, andfuture of our existence, allowing us to gain insights into our place in the cosmos.Moreover, space exploration serves as a barometer for technological advancement. The development of spacecraft, rockets, and satellites requires cutting-edge technology. Each mission to space pushes the limits of engineering, physics, and materials science, leading to advancements that trickle down to improve our daily lives. The spin-off technologies from space exploration, such as GPS systems, satellite communications, and water purification systems, have revolutionized the way we live.Apart from technological benefits, space exploration also holds the promise of addressing some of the most pressing issues facing humanity. Climate change, resource depletion, and population growth are among the many challenges that require global cooperation and innovative solutions. Space exploration offers a potential avenue for addressing these issues by enabling us to access resources beyond Earth's limits, such as water and rare metals, and by providing a platform for conducting research in areaslike agriculture and energy efficiency.Additionally, space exploration is a powerful symbol of human unity and aspiration. It transcends political and cultural divides, unifying people under a common goal of scientific discovery and exploration. The International Space Station (ISS) is a prime example of this, with multiple countries collaborating to conduct research and experiments in space. Such collaborations foster international cooperation and understanding, promoting peace and mutual respect.However, space exploration is not without its challenges and controversies. The cost of space missions is often immense, both financially and in terms of human resources. The risks associated with space travel are also significant, with the potential for disasters like the Challenger and Columbia shuttle accidents. Furthermore, the ethics and implications of extraterrestrial life discovery and colonization are complex and controversial topics that require careful consideration.Despite these challenges, the benefits of space exploration far outweigh the costs. It serves as a powerful reminder of our potential as a species, pushing us to reach beyond our current limitations and embrace the unknown. Space exploration is not just about science and technology; it is about our inherent desire to explore, to learn, and to grow. As long as we approach it with humility, respect, and a sense of wonder, the final frontier will continue to inspire and challenge us, driving us forward into a future filled with possibilities.。

第一单元1.Condensed matter physics 凝聚态物理2.Atomic, molecular and optical physics 原子、分子、光学物理3.Particle and nuclear physics 粒子与原子核物理4.Astrophysics and physical cosmology 天体物理学和物理宇宙学5.Current research frontiers 当前研究前沿6.natural philosophy 哲学7.natural science 自然科学8.matter 物质9.motion 运动10.space and time 时空11.energy 能量12.force 力13.the universe 宇宙14.academic disciplines 学科15.astronomy 天文学16.chemistry 化学17.mathematics 数学18.biology 生物19.Scientific Revolution 科学革命20.interdisciplinary各学科间的21.biophysics 生物物理22.quantum chemistry 量子化学23.mechanism 机制24.avenues 渠道；大街25.advances 前进26.electromagnetism电磁学27.nuclear physics原子核物理28.domestic appliances家用电器29.nuclear weapons核武器30.thermodynamics热力学31.industrialization工业化32.mechanics力学33.calculus微积分34.the theory of classical mechanics经典力学35.the speed of light 光速36.remarkable卓越的37.chaos混沌38.quantum mechanics量子力学39.statistical mechanics 统计力学40.special relativity狭义相对论41.acoustics声学42.statics静力学43.at rest静止44.kinematics运动学45.causes原因46.dynamics动力学47.solid mechanics 固体力学48.fluid mechanics 流体力学49.continuum mechanics 连续介质力学50.hydrostatics流体静力学51.hydrodynamics流体动力学52.aerodynamics气体动力学53.pneumatics气体力学54.sound 声音55.ultrasonics超声学56.sound waves 声波57.frequency 频率58.bioacoustics生物声学59.electroacoustics电声学60.manipulation操作61.audible听得见的62.electronics电子63.visible light 可见光64.infrared红外线65.ultraviolet radiation 紫外线辐射66.reflection 反射67.refraction折射68.interference干涉69.diffraction衍射70.dispersion色散71.polarization偏振72.Heat 热度73.the internal energy内能74.Electricity 电力75.magnetism磁学76.electric current电流77.magnetic field磁场78.Electrostatics静电学79.electric charges电荷80.electrodynamics电动力学81.magnetostatics静磁学82.poles磁极83.matter and energy 物质和能量84.on the very large or very small scale 非常大或非常小的规模85.atomic and nuclear physics 原子与核物理学86.chemical elements化学元素87.The physics of elementary particles基本粒子88.high-energy physics 高能物理学89.particle accelerators 粒子加速器90.Quantum theory 量子论91.discrete离散92.subatomic原子内plementary互补94.The theory of relativity 相对论95.a frame of reference参考系96.the special theory of relativity 狭义相对论97.general theory of relativity 广义相对论98.gravitation万有引力99.universal law 普遍规律100.absolute time and space 绝对的时间和空间101.space-time 时空ponents组成103.Max Planck 普朗克104.quantum mechanics 量子力学105.probabilistic概率性106.quantum field theory量子场107.dynamical动态的108.curved弯曲的109.massive巨大的110.candidate候选111.quantum gravity 量子重力112.macroscopic宏观113.properties属性114.solids 固体115.liquids 液体116.electromagnetic force电磁力117.atom 原子118.superconducting超导119.conduction electrons 传导电子120.ferromagnetic 铁磁体121.the ferromagnetic and antiferromagnetic phases of spins铁磁和反铁磁的阶段的旋转122.atomic lattices原子晶格123.solid-state physics 固体物理124.subfields分区；子域125.nanotechnology纳米技术126.engineering工程学127.quantum treatments 量子治疗128.Atomic physics 原子物理129.electron shells电子壳层130.trap捕获131.ions离子132.collision碰撞133.nucleus原子核134.hyperfine splitting超精细分裂135.fission and fusion 分裂与融合136.Molecular physics 分子物理137.optical fields 光场138.realm范围139.properties属性140.distinct区别141.Particle physics 粒子物理142.elementary constituents基本成分143.interactions 相互作用144.detectors探测器puter programs程序146.Standard Model 标准模型147.quarks and leptons轻子-夸克148.gauge bosons规范波色子149.gluons胶子150.photons光子151.nuclear power generation核发电152.nuclear weaponsh核武器153.nuclear medicine 核医学154.magnetic resonance imaging磁共振成像155.ion implantation离子注入156.materials engineering 材料工程157.radiocarbon dating放射性碳测定年代158.geology 地质学159.archaeology考古学.160.Astrophysics天体物理学161.astronomy天文学162.stellar structure恒星结构163.stellar evolution恒星演化164.solar system太阳系165.cosmology宇宙学166.disciplines学科167.emitted射出168.celestial bodies天体169.Perturbations扰动170.interference干扰171.Physical cosmology 宇宙物理学172.Hubble diagram哈勃图173.steady state 定态，稳恒态174.Big Bang nucleo-synthesis核合成175.cosmic microwave background宇宙微波背景176.cosmological principle 宇宙论原理；宇宙论原则177.cosmic inflation宇宙膨胀178.dark energy 暗能量179.dark matter暗物质of high-temperature superconductivity 高温超导180.spintronics自旋电子学181.quantum computers 量子电脑182.the Standard Model 标准模型183.neutrinos中微子184.solar太阳185.the TeV万亿电子伏186.the super-symmetric particles 超对称粒子187.quantum gravity 量子重力188.superstring超弦189.theory and loop圈190.ultra-high energy cosmic rays高能宇宙射线,191.the baryon asymmetry重子不对称,192.the acceleration of the universe and the anomalous宇宙的加速和异常193.rotation旋转194.galaxies星系.195.turbulence动荡196.water droplets 水滴197.mechanisms of surface tension catastrophes表面紧张灾难198.heterogeneous多相的199.aerodynamics 气体力学第二单元所有的红色单词，重要的我标有星号1.classical mechanics 经典力学*2.physical laws 物理定律3.forces 力4.macroscopic 宏观的5.Projectiles 抛射体6.Spacecraft 太空飞船7.Planets 行星8.Stars 恒星9.Galaxies 星系,银河系10.gases, liquids, solids 气体,液体固体11.the speed of light 光速12.quantum mechanics 量子力学*13.the atomic nature of matter 物质的原子性质14.wave–particle duality 波粒二象性*15.special relativity 狭义相对论*16.General relativity 广义相对论*17.Newton's law of universal gravitation 牛顿万有引力*18.Newtonian mechanics 牛顿力学*grangian mechanics 拉格朗日力学*20.Hamiltonian mechanics 哈密顿力学*21.analytical mechanics 分析力学*22.as point particles 质点*23.Negligible 微不足道的可忽略的24.position, mass 位置,质量25.Forces 力26.non-zero size 不计形状27.the electron 电子*28.quantum mechanics 量子力学*29.degrees of freedom 自由度*30.Spin 旋转posite 组合的32.center of mass 质心33.the principle of locality 局部性原理34.Position 位置35.reference point 参照点(参照物)*36.in space 在空间37.Origin 原点*38.the vector 矢量39.Particle 质点*40.Function 函数41.Galilean relativity 伽利略相对性原理*42.Absolute 绝对43.time interval 时间间隔44.Euclidean geometry 欧几里得几何学45.Velocity 速度46.rate of change 变化率47.Derivative 倒数*48.Vector 矢量49.Speed 速度50.Acceleration 加速度*51.second derivative 二阶导*52.Magnitude 大小(量级)53.the direction 方向54.or both55.Deceleration 加速度56.Observer 观察者57.reference frames 参考系*58.inertial frames 惯性系*59.at rest60.in a state of uniform motion 运动状态一致61.Straight 直的62.physical laws 物理学定理63.non-inertial 非惯性系64.accelerating 加速65.fictitious forces 虚拟力(达朗贝尔力)*66.equations of motion 运动学方程*67.the distant stars 遥远的恒星68.Newton 牛顿69.force and momentum 力和动量70.Newton's second law of motion 牛顿第二定律*71.(canonical) momentum 动量* force 净力73.ordinary differential equation 常微分方程*74.the equation of motion 运动学方程*75.gravitational force 重力*76.Lorentz force 洛伦兹力*77.Electromagnetism 电磁学*78.Newton's third law 牛顿第三定律*79.opposite reaction force 反作用力80.along the line 沿直线81.displacement 位移*82.work done 做功83.scalar product 标极*84.the line integral 线积分*85.path 路径86.conservative. 守恒*87.Gravity 重力88.Hooke's law 胡克定律*89.Friction 摩擦力*90.kinetic energy 动能*91.work–energy theorem 功能关系(动能定理)*92.the change in kinetic energy 动能改变量93.gradient 梯度*94.potential energy 势能*95.Conservative 保守的,守恒的96.potential energy 势能97.total energy 总能量(机械能)*98.conservation of energy 能量守恒**99.linear momentum 线动量100.translational momentum 平移动量101.closed system 封闭系统*102.external forces 外力*103.total linear momentum 总(线)动量线动量就是动量区别于角动量104.center of mass 质心*105.Euler's first law 欧拉第一定律106.elastic collision 弹性碰撞*107.inelastic collision 非弹性碰撞*108.slingshot maneuver 弹弓机动109.Rigidity 硬度(刚性)*110.Dissipation 损耗**111.inelastic collision 非弹性碰撞112.heat or sound 热或声113.new particles 新粒子114.angular momentum 角动量*115.moment of momentum 瞬时动量*116.rotational inertia 转动惯量*117.rotational velocity 转速*118.rigid body 刚体**119.moment of inertia 惯性力矩*120.angular velocity 角速度*121.linear momentum 线动量122.Crossed 叉乘*123.Position 位置124.angular momentum 角动量125.pseudo-vector 赝矢量*126.right-hand rule 右手规则 external torque 净外力转矩128.neutron stars 中子星129.angular momentum 角动量*130.Conservation 守恒131.Gyrocompass 陀螺罗盘132.no external torque 无外力炬133.Isotropy 各向同性*134.Torque 转矩135.central force motion 中心力移动136.white dwarfs, neutron stars and black holes 白矮星,中子星,黑洞第三单元ThermodynamicsThermodynamics: 热力学；热力的Heat ：热；热力；热度Work：功macroscopic variables：肉眼可见的；宏观的，粗观的，粗显的。

小学上册英语第四单元测验卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.An electromagnet uses electric current to create a ______ field.2.How many fingers do we have on one hand?A. 4B. 5C. 6D. 73.My mom is very __________ (支持的) of my goals.4.The jellyfish floats gracefully in the ______ (海洋).5.My best friend is very __________. (善良)6.The ancient Greeks held their athletic competitions in honor of ______ (宙斯).7.I often ________ (动词) my toys after playing. It helps keep my room ________ (形容词). My favorite place to play is in my ________ (名词).8.The ________ (农业创新) drives progress.9.What is the capital of the United States?A. New YorkB. Los AngelesC. Washington,D.C. D. Chicago答案:C10.I wear _____ (运动鞋) to school.11.The __________ is a popular destination for tourists in Europe.12._____ (草原) are home to many wildflowers.13.The eagle is a symbol of _________ (力量).14.What do you call a book of maps?A. DictionaryB. AtlasC. NovelD. Journal答案:B15.How do you say "water" in French?A. L'eauB. AquaC. WasserD. H2O16.The __________ is a significant natural resource found in forests. (木材)17.The __________ (历史的多元性) enrich discussions.18.What is the term for the slow movement of continents caused by tectonic forces?A. Continental DriftB. Plate TectonicsC. Geological ShiftD. Earth Movement19. A magnet can attract _______ objects.20.The elephant is very ___ (large).21.What do you call the time after noon?A. MorningB. AfternoonC. EveningD. Night答案:B22.The ______ is a symbol of peace.23.I like to draw with my _____ (水彩笔).24.The ______ of the plant world is vast and varied. (植物界的多样性是广泛而多样的。