On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion

00Greek that of the eastern half.
00230Both Latin and Greek belong to Indo-European language.
00240The Roman writer Horace(:e0lQCQMR65-8t^ WlN0) said captive Greece took her rude conqueror captive 0
00The world s first vast interior space.NLu
N,{ Colosseum('Yt珐) is an enormous.2)Yvsb_q_gRb
00410Sculpture(QX) She-wolf(
00; Father of History ! Herodotus ! war(between Greeks and Persians)
00This war is called Peleponicion wars. ZSWTY嬒d 3
00fH[ v^l g_洺0
00; The greatest historian that ever lived. ( geggO'YvS[) ! Thucydides ! war (Sparta Athens and Syracuse)
00200The burning of Corinth in 146 B.C. Marked Roman conquest of Greece.
00210The melting between Roman Culture and Greek Culture. (Wl_ g ^Jvh)
00220From 146 B.C. Latin was the language of the western half of the Roman Empire.

小学上册英语第1单元测验卷英语试题一、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.What is the largest breed of dog?A. ChihuahuaB. BeagleC. Great DaneD. PoodleC2.What do we call a person who travels to different countries for pleasure?A. TouristB. TravelerC. ExplorerD. NomadA3.The process of making biodiesel involves _______ oils.4.What is the sound a cow makes?A. BarkB. MeowC. MooD. RoarC5.The girl has a kind ________.6.The giraffe is known for its long ________________ (脖子).7.I thin k it’s fun to go ________ (参加舞会).8.What do you call the action of making something less dirty?A. CleaningB. TidyingC. OrganizingD. DustingA9.My friend is a great __________ (听众) and always supports me.10.The _____ (sun/cloud) is shining.11.The horse gallops across the ________________ (田野).12.The ________ (香味) of flowers can be pleasant.13.The __________ (铁路) connects different cities.14.What do you drink from?A. PlateB. CupC. TableD. ChairB15.He is _____ (tall/short) like his father.16.The chemical symbol for xenon is ______.17.What is the name of the planet known as the "Red Planet"?A. VenusB. MarsC. JupiterD. SaturnB18.I want to be a ___ (scientist/artist).19.The __________ is often unpredictable in spring. (天气)20.What is the capital city of Burkina Faso?A. OuagadougouB. Bobo DioulassoC. BanforaD. Koudougou21.I like to draw _____ in my notebook.22.What is the primary color of a lemon?A. GreenB. YellowC. RedD. BlueB23.What is the primary color of a lemon?A. GreenB. YellowC. RedD. Orange24.We can grow __________ (蔬菜) in our backyard.25.I see a ___ in the garden. (flower)26.The _____ (土壤改良) can enhance plant growth.27.The baby is _______ (笑着).28. A __________ is a characteristic of a substance that can be observed without changing it.29.My sister enjoys __________ (参加) local festivals.30.I can ________ my bike.31.What is the term for a baby cow?A. CalfB. FoalC. KidD. Lamb32.We had a _________ (玩具交换) at school, and I got a new _________ (玩具).33. A __________ is a natural elevation of the Earth's surface.34.The chemical formula for cobalt(II) nitrate is _____.35.He is _______ at playing soccer.36. A _______ can help illustrate how energy is transferred in a circuit.37.The _______ (鲸鱼) breaches the surface.38.I enjoy taking my camera to capture beautiful ______ (瞬间).39.I enjoy making ______ (手工艺品) from recycled materials. It’s a fun way to be creative and eco-friendly.40.What do you call the time before noon?A. AfternoonB. EveningC. MorningD. Midnight答案:C41.I like to bake ______ (美味) treats for my friends.42.I think being a __________ (志愿者) is very important.43.The ________ (植物资源管理) is crucial.44.The hamster runs in its _______ (仓鼠在它的_______里跑).45.What is the capital of Malta?A. VallettaB. MdinaC. RabatD. BirkirkaraA46.The dog is _____. (barking/sleeping/jumping)47.What is the capital city of Slovakia?A. BratislavaB. KošiceC. PrešovD. Nitra48.My favorite _____ is a friendly little puppy.49.I love to ______ (与家人一起) explore different cuisines.50. A _____ is a large body of gas and dust in space.51. A ________ (蝎子) has a stinger and can be dangerous.52.We use ______ (草) to make lawns green.53.What is the name of the first person to walk on the moon?A. Yuri GagarinB. Neil ArmstrongC. Buzz AldrinD. John Glenn54.What is the name of the famous scientist who formulated the laws of gravity?A. Albert EinsteinB. Isaac NewtonC. Galileo GalileiD. James Clerk Maxwell55.Rabbits are known for their strong ______ (后腿).56.Which season is the hottest?A. WinterB. SpringC. SummerD. AutumnC57.My brother collects ____ (stamps) from different countries.58.This boy, ______ (这个男孩), enjoys fishing.59.What do you call the distance between two points?A. LengthB. WidthC. HeightD. Measurement60.What do we call the process of a liquid turning into a gas?A. FreezingB. MeltingC. EvaporationD. CondensationC61.I enjoy _______ (listening) to classical music.62.The ______ helps with the communication between cells.63.My ________ (姐姐) loves to bake cookies and cakes.64. A ______ (刺猬) curls up when it feels threatened.65.The chemical formula for calcium chloride is ______.66.I love camping in the mountains. My favorite part is __________.67.The _______ (The Age of Enlightenment) emphasized reason and scientific thought in society.68.Many plants are ______ (适应性强) to their surroundings.69.I can see a ___ (bird) in the tree.70.The main component of cell membranes is ______.71.My favorite game is ______ (国际象棋).72.Plant cells have ______ that capture sunlight.73.I love to watch the __________ dance in the wind. (树叶)74.What do we call the act of traveling to different countries?A. ExploringB. AdventuringC. TouringD. VacationingC75.ration of Independence was signed in ________ (1776). The Decl76.What is the capital of Nepal?A. KathmanduB. PokharaC. BhaktapurD. LalitpurA77.The parrot repeats everything it _________. (听到)78.Which food is made from milk?A. BreadB. CheeseC. RiceD. PastaB79.My teacher is very __________ (耐心).80.The __________ is a famous city known for its beaches and nightlife. (迈阿密)81.The ant carries food back to its ______ (巢).82.My uncle is a skilled ____ (blacksmith).83.小龙虾) scuttles across the riverbed. The ____84.What is the capital of Tunisia?A. TunisB. SfaxC. KairouanD. Bizerte85.His favorite food is ________.86.What do you call a place where animals are kept for public viewing?A. FarmB. ZooC. AquariumD. ParkB87.The process of breaking down food is a type of _____ reaction.munity gardens promote ______ (邻里关系).89.What do you call a collection of stories or articles published together?A. AnthologyB. NovelC. MagazineD. JournalA90.The capital city of Vietnam is __________.91. f Enlightenment emphasized reason and __________. (科学) The Age92.island) is surrounded by water on all sides. The ____93.My friend’s _________ (玩具飞机) can fly high in the sky!94.The Stone Age is known for the use of ________ tools.95.What is the main ingredient in pizza?A. RiceB. DoughC. MeatD. SaladB96.The ice cream is ________ cold.97.What is the fastest land animal?A. ElephantB. CheetahC. HorseD. LionB98.Solar systems can contain a variety of celestial _______.99.My favorite thing about school is ________ (友谊).100.An ecosystem is a community of living organisms interacting with their ______ environment.。

a r X iv:mat h /2818v1[mat h.AG ]23Aug202TORELLI’S THEOREM FOR HIGH DEGREE SYMMETRIC PRODUCTS OF CUR VES NAJMUDDIN F AKHRUDDIN Abstract.We show that two smooth projective curves C 1and C 2of genus g which have isomorphic symmetric products are isomorphic unless g =2.This extends a theorem of Martens.Let k be an algebraically closed field and C 1and C 2two smooth projective curves of genus g >1over k .It is a consequence of Torelli’s theorem that if Sym g −1C 1∼=Sym g −1C 2,then C 1∼=C 2.The same holds for the d ’th symmetric products,for 1≤d <g −1as a consequence of a theorem of Martens [2].In this note we shall show that with one exception the same result continues to hold for all d ≥1,i.e.we have the following Theorem 1.Let C 1and C 2be smooth projective curves of genus g ≥2over an algebraically closed field k .If Sym d C 1∼=Sym d C 2for some d ≥1,then C 1∼=C 2unless g =d =2.It is well known that there exist non-isomorphic curves of genus 2over C with isomorphic Jacobians.Since the second symmetric power of a genus 2curve is isomorphic to the blow up of the Jacobian in a point,it follows that our result is the best possible.Proof of Theorem.Let C i ,i =1,2be two curves of genus g >1with Sym d C 1∼=Sym d C 2for some d ≥1.Since the Albanese variety of Sym d C i ,d ≥1,is iso-morphic to the Jacobian J (C i ),it follows that J (C 1)∼=J (C 2).If d ≤g −1,the theorem follows immediately from [2],since the image of Sym d C i in J (C i )(after choosing a basepoint)is W d (C i ).Note that in this case it sufices to have a birational isomorphism from Sym d C 1to Sym d C 2.Suppose g ≤d ≤2g −3.Then the Albanese map from Sym d C i to J (C i )is surjective with general fibre of dimension d −g .Interpreting the fibres as complete linear systems of degree d on C i ,it follows by Serre duality that the subvariety ofJ (C i )over which the fibres are of dimension >d −g is isomorphic to W 2g −2−d (C i ).Therefore if Sym d C 1∼=Sym d C 2,then W 2g −2−d (C 1)∼=W 2g −2−d (C 2),so Martens’theorem implies that C 1∼=C 2.Now suppose that d >2g −2and g >2.By choosing some isomorphism we identify J (C 1)and J (C 2)with a fixed abellian variety A .If φ:Sym d C 1→Sym d C 2is our given isomorphism,from the universal property of the Albanese morphism we obtain a commutative diagramSym d C 1φSym d C 2π2A12NAJMUDDIN FAKHRUDDINwhere theπi’s are the Albanese morphisms corresponding to some basepoints and f is an automorphism of A(not necessarily preserving the origin).By replacing C2 with f−1(C2)we may then assume that f is the identity.Since d>2g−2,the mapsπi,i=1,2make Sym d C i into projective bundles over A.By a theorem of Schwarzenberger[5],Sym d C i∼=P roj(E i),where E i is a vector bundle on A of rank d−g+1with c j(E i)=[W g−j(C i)],i=1,2,0≤j≤g−1,in the group of cycles on A modulo numerical equivalence.Sinceφis an isomorphism of projective bundles,it follows that there exists a line bundle L on A such that E1∼=E2⊗L.Letθi=[W g−1(C i)],so by Poincar´e’s formula[W g−j(C i)]=θj i/j!,i=1,2,i≤j≤g−1.The lemma below implies thatθg−1i =θg−12in the group of cyclesmodulo numerical equivalence on A.Sinceθg i=g!,this implies thatθ1·[C2]=g. By Matsusaka’s criterion[3],it follows that W g−1(C1)is a theta divisor for C2, which by Torelli’s theorem implies that C1∼=C2.If d=2g−2and g>2,then we can still apply the previous argument.In this case we also have that Sym d(C i)∼=P roj(E i),i=1,2but E i is a coherent sheaf which is not locally free.However on the complement of some point of A it does become locally free and the previous formula for the Chern classes remains valid.The above argument clearly does not suffice if g=2.To handle this case we shall use some properties of Picard bundles for which we refer the reader to[4]. Suppose that d>2and C i,i=1,2are two non-isomorphic curves of genus2with Sym d C1∼=Sym d ing the same argument(and notation)as the g>2case, it follows that there exist embeddings of C i,i=1,2,in A and a line bundle L on A such that E1∼=E2⊗L and L⊗d−1∼=O(C1−C2)(we identify C i,i=1,2with their images).For i≥1,let G i denote the i-th Picard sheaf associated to C2,so that P roj(G i)∼= Sym i(C2).(G i is the sheaf denoted by F2−i in[4]and G d∼=E2).There is an exact sequence([4,p.172]):0→O A→G i→G i−1→0(1)for all i>1.We will use this exact sequence and induction on i to compute the cohomology of sheaves of the form E1⊗P∼=E2⊗L⊗P,where P∈P ic0(A).Considerfirst the cohomology of G1,which is the pushforward of a line bundle of degree1on a translate of C2.Since we have assumed that C1≇C2,it followsthat C1·C2>2.Since C21=C22=2,deg(L|C2)=(C1−C2)·C2/(d−1)>0.By Riemann-Roch it follows that h j(A,G1⊗L⊗P),j=1,2is independent of P,except possibly for one P if deg(L|C2)=1,and h2(A,G1⊗L⊗P)=0since G1issupported on a curve.Now C1·C2>2also implies that c1(L)2<0.By the index theorem,it follows that h0(A,L⊗P)=h2(A,L⊗P)=0and h1(A,L⊗P)is independent of P. Therefore by tensoring the exact sequence(1)with L⊗P and considering the long exact sequence of cohomology,we obtain an exact sequence(2)0→H0(A,G i⊗L⊗P)→H0(A,G i−1⊗L⊗P)→H1(A,L⊗P)→H1(A,G i⊗L⊗P)→H1(A,G i−1⊗L⊗P)→0 and isomorphisms H2(A,G i⊗L⊗P)→H2(A,G i−1⊗L⊗P)for all i>1.By induction,it follows that H2(A,G i⊗L⊗P)=0for all i>0.Since the Euler characteristic of G i⊗L⊗P is independent of P,the above exact sequence(2)TORELLI’S THEOREM FOR HIGH DEGREE SYMMETRIC PRODUCTS OF CURVES3 along with induction shows that for all i>0and j=0,1,2,h j(G i⊗L⊗P)is independent of P,except for possibly one P.In particular,this holds for i=d hence h j(A,E1⊗P)is independent of P except again for possibly one P.We obtain a contradiction by using the computation of the cohomology of Picard sheaves in Proposition4.4of[4]:This implies that h1(A,E1⊗P)is one or zero depending on whether the point in A corresponding to P does or does not lie on a certain curve (which is a translate of C1).Lemma1.Let X be an algebraic variety of dimension g≥3and let E i,i=1,2 be vector bundles on X of rank r.Suppose c1(E i)=θi,c j(E i)=θj i/j!for i=1,2 and j=2,3(j=2if g=3),and E1∼=E2⊗L for some line bundle L on X.Then θj1=θj2for all j>1(j=2if g=3).Proof.Since E1∼=E2⊗L,c1(E1)=c1(E2)+rc1(L),hence c1(L)=(θ1−θ2)/r.For a vector bundle E of rank r and a line bundle L on any variety,we have the following formula for the Chern polynomial([1],page55):c t(E⊗L)=rk=0t k c t(L)r−k c i(E).Letting E=E1,E⊗L=E2,and expanding out the terms of degree2and3,one easily sees thatθj1=θj2for j=2and also for j=3if g>3.(Note that this only requires knowledge of c j(E i)for j=1,2,3).Since any integer n>1can be written as n=2a+3b with a,b∈N,the lemma follows.We do not know whether the theorem holds if k is not algebraically closed.Remark.S.Ramanan has suggested it may also be possible to prove the theorem when g>2by computing the Chern classes of the push forward of the tangent bundle of Sym d C to J(C).Acknowledgements.We thank A.Collino for some helpful correspondence,in par-ticular for informing us about the paper[2],and S.Ramanan for the above remark.References[1]W.Fulton,Intersection theory,Springer-Verlag,Berlin,second ed.,1998.[2]H.H.Martens,An extended Torelli theorem,Amer.J.Math.,87(1965),pp.257–261.[3]T.Matsusaka,On a characterization of a Jacobian variety,Memo.Coll.Sci.Univ.Kyoto.Ser.A.Math.,32(1959),pp.1–19.[4]S.Mukai,Duality between D(X)and D(ˆX)with its application to Picard sheaves,NagoyaMath.J.,81(1981),pp.153–175.[5]R.L.E.Schwarzenberger,Jacobians and symmetric products,Illinois J.Math.,7(1963),pp.257–268.School of Mathematics,Tata Institute of Fundamental Research,Homi Bhabha Road,Mumbai400005,IndiaE-mail address:naf@math.tifr.res.in。

—— Range Instrumentation prism, right.
BM 60 90 right
Ray Tracing
Reflection from Flat Surface
—— Range Instrumentation prism, left, roof.
BM 100 90 left roof
CR 180 roof
Ray Tracing
Reflection from Flat Surface
—— Rhomb prism. It has two reflective faces 斜方棱镜，又名菱形棱镜
BC 0 (Rhomb)
Ray Tracing
Reflection from Flat Surface
Reflection from Flat Surface
—— Isosceles prism, three reflective faces, roof
CR 45 roof (Schmidt)
Ray Tracing
Reflection from Flat Surface
—— Isosceles prism, three reflective faces, roof

Reflection Prism
—— Isosceles prism (Classification code: R), single reflective face. 等腰棱镜(代号:D), 一次反射型
The Dove Prism (AR45)
Ray Tracing
Reflection from Flat Surface
• In fact, if the UV and IR are included, most any substance will sow some absorption. So anomalous dispersion exist somewhere throughout the spectrum

麦克斯韦方程组洛伦兹协变性的两种证明方法朱永乐（天水师范学院物理系甘肃天水 741000）摘要：麦克斯韦方程组的证明一般有电磁场张量分析法和洛伦兹微分变换法，电磁场张量分析法数学上是简洁的，洛伦兹微分变换法则具有明显的物理意义，其结论都显示了电磁场的统一性，本文通过两种方法来证明麦克斯韦方程组具有相对论不变性。

关键词:伽利略变换洛伦兹变换麦克斯韦方程组协变性相对性原理Lorentz covariance of Maxwell's equations that the two methodsZhu yong le(The department of Physics Tianshui normal university ,Gansu Tianshui 741000)Abstract: Maxwell's equations that are generally electromagnetic field tensor analysis methods and Lorenz differential transform method, electromagnetic field tensor analysis method is simple math, Lorenz differential transform method has obvious physical meaning, its conclusions are shows the unity of the electromagnetic field, this paper two methods to prove the relativistic invariance of Maxwell's equations with.Key words: Galilean transformation ;Lorentz transformation; the covariance of Maxwell's equations of relativity theory1．引言相对性原理要求任何物理规律在不同的惯性系中形式相同。

a r X i v :g r -q c /0607043v 2 2 O c t 2006DF/IST-4.2006Lorentz Symmetry Derived from Lorentz Violation in the BulkOrfeu Bertolami ∗and Carla Carvalho †Departamento de F´ısica,Instituto Superior T´e cnico Avenida Rovisco Pais 1,1049-001Lisboa,PortugalAbstractWe consider bulk fields coupled to the graviton in a Lorentz violating fashion.We expect that the overly tested Lorentz symmetry might set constraints on the induced Lorentz violation in the brane,and hence on the dynamics of the interaction of bulk fields on the brane.We also use the requirement for Lorentz symmetry to constrain the cosmological constant observed on the braneI.INTRODUCTIONLorentz invariance is one of the most well tested symmetries of physics.Nevertheless, the possibility of violation of this invariance has been widely discussed in the recent liter-ature(see e.g.[1]).Indeed,the spontaneous breaking of Lorentz symmetry may arise in the context of string/M-theory due to the existence of non-trivial solutions in stringfield theory[2],in loop quantum gravity[3],in noncommutativefield theories[4,5]or via the spacetime variation of fundamental couplings[6].This putative breaking has also implica-tions in ultra-high energy cosmic ray physics[7,8].Lorentz violation modifications of the dispersion relations viafive dimensional operators for fermions have also been considered and constrained[9].It has also been speculated that Lorentz symmetry is connected with the cosmological constant problem[10].However,the main conclusion of these studies is that Lorentz symmetry holds up to about2×10−25[1,8].Efforts to examine a putative breaking of Lorentz invariance have been concerned mainly with the phenomenological aspects of the spontaneous breaking of Lorentz symmetry in particle physics and only recently have the implications for gravity been more closely studied [11,12].The idea is to consider a vectorfield coupled to gravity that undergoes spontaneous Lorentz symmetry breaking by acquiring a vacuum expectation value in a potential.Moreover,recent developments in string theory suggest that we may live in a braneworld embedded in a higher dimensional universe.In the context of the Randall-Sundrum cosmo-logical models,the warped geometry of the bulk along the extra spacial dimension suggests an anisotropy which could be associated with the breaking of the bulk Lorentz symmetry.In this paper we study how spontaneous Lorentz violation in the bulk repercusses on the brane and how it can be constrained.We consider a vectorfield in the bulk which acquires a non-vanishing expectation value in the vacuum and introduces spacetime anisotropies in the gravitationalfield equations through the coupling with the graviton.For this purpose, we derive thefield equations and project them parallel and orthogonally to the brane. We then establish how to derive brane quantities from bulk quantities by adopting Fermi normal coordinates with respect to the directions on the brane and continuing into the bulk using the Gauss normal prescription.We parameterize the worldsheet in terms of coordinates x A=(t b,x b)intrinsic to the ing the chain rule,we may express the brane tangent and normal unit vectors interms of the bulk basis as follows:ˆe A=∂∂xµ=XµAˆeµ,ˆe N=∂∂xµ=Nµˆeµ,(1)withgµνNµNν=1,gµνNµXνA=0,(2) where g is the bulk metricg=gµνˆeµ⊗ˆeν=g ABˆe A⊗ˆe B+g ANˆe A⊗ˆe N+g NBˆe N⊗ˆe B+g NNˆe N⊗ˆe N(3)To obtain the metric induced on the brane we expand the bulk basis vectors in terms of the coordinates intrinsic to the brane and keep the doubly brane tangent components only. It follows thatg AB=XµA XνB gµν(4)is the(3+1)-dimensional metric induced on the brane by the(4+1)-dimensional bulk metric gµν.The induced metric with upper indices is defined by the relationg AB g BC=δA C.(5)It follows that we can write any bulk tensorfield as a linear combination of mutually orthogonal vectors on the brane,ˆe A,and a vector normal to the brane,ˆe N.We illustrate the example of a vector Bµand a tensor Tµνbulkfields as followsB=B Aˆe A+B Nˆe N,(6)T=T ABˆe A⊗ˆe B+T ANˆe A⊗ˆe N+T NBˆe N⊗ˆe B+T NNˆe N⊗ˆe N.(7) Derivative operators decompose similarly.We write the derivative operator∇as∇=(XµA+Nµ)∇µ=∇A+∇N.(8)Fixing a point on the boundary,we introduce coordinates for the neighborhood choosing them to be Fermi normal.All the Christoffel symbols of the metric on the boundary are thus set to zero,although the partial derivatives do not in general vanish.The non-vanishing connection coefficients are∇Aˆe B=−K ABˆe N,∇Aˆe N=+K ABˆe B,∇Nˆe A=+K ABˆe B,∇Nˆe N=0,(9)as determined by the Gaussian normal prescription for the continuation of the coordinates offthe boundary.For the derivative operator∇∇wefind that∇∇=gµν∇µ∇ν=g AB[(XµA∇µ)(XνB∇ν)−XµA(∇µXνB)∇ν]+g NN[(Nµ∇µ)(Nν∇ν)−Nµ(∇µNν)∇ν] =g AB[∇A∇B+K AB∇N]+∇N∇N.(10)We can now decompose the Riemann tensor,Rµνρσ,along the tangent and the normal directions to the surface of the brane as followsR ABCD=R(ind)ABCD+K AD K BD−K AC K BD,(11) R NBCD=K BC;D−K BD;C,(12)R NBND=K BC K DC−K BC,N,(13) from which wefind the decomposition of the Einstein tensor,Gµν,obtaining the Gauss-Codacci relationsG AB=G(ind)AB +2K AC K CB−K AB K−K AB,N−12 −R(ind)−K CD K DC+K2 .(16) II.BULK VECTOR FIELD COUPLED TO GRA VITYWe consider a bulk vectorfield B with a non-trivial coupling to the graviton in afive-dimensional anti-de Sitter space.The Lagrangian density consists of the Hilbert term, the cosmological constant term,the kinetic and potential terms for B and the B–graviton interaction term,as followsL=14BµνBµν−V(BµBµ±b2),(17)where Bµν=∇µBν−∇νBµis the tensorfield associated with Bµand V is the potencial which induces the breaking of Lorentz symmetry once the Bfield is driven to the minimumat BµBµ±b2=0,b2being a real positive constant.As discussed in the introduction,this model has been proposed in order to analyse the impact on the gravitational sector of the breaking of Lorentz symmetry[11,12].Furthermore,in the present modelκ2(5)=8πG N= M3P l,M P l is thefive-dimensional Planck mass andλis a dimensionless coupling constant that we have inserted to track the effect of the interaction.In the cosmological constant termΛ=Λ(5)+Λ(4)we have included both the bulk vacuum valueΛ(5)and that of the braneΛ(4),described by a brane tensionσlocalized on the locus of the brane,Λ(4)=σδ(N).By varying the action with respect to the metric,we obtain the Einstein equation1Tµν,(18)2where1Lµν=[∇µ∇ρ(BνBρ)+∇ν∇ρ(BµBρ)−∇2(BµBν)−gµν∇ρ∇σ(BρBσ)](20)2are the contributions from the interaction term andTµν=BµρBνρ+4V′BµBν+gµν −1+2λ[B C(K CD;D−K;C)+B N(K CD K CD−K;N)]=0,(24)parallelly and orthogonally to the brane respectively,which we include here for the purpose of illustration.Next we proceed to derive the induced equations of motion for both the metric and the vectorfield in terms of quantities measured on the brane.The induced equations on the brane are the(AB)projected components after the singular terms across the brane are subtracted out by the substitution of the matching conditions.Considering the brane as a Z2-symmetric shell of thickness2δin the limitδ→0,derivatives of quantities discontinuous across the brane generate singular distributions on the brane.Integration of these terms in the coordinate normal to the brane relates the induced geometry with the localization of the induced stress-energy in the form of matching conditions.First we consider the Einstein equations and then the equations of motion for B which,due to the coupling of B to gravity,will also be used as complementary conditions for the dynamics of the metric on the brane.Combining the Gauss-Codacci relations with the projections of the stress-energy tensor and the interaction source terms,we integrate the(AB)component of the Einstein equation in the coordinate normal to the brane to obtain the matching conditions for the extrinsic curvature across the brane,i.e.the Israel matching conditions.From the Z2symmetry it follows that B A(−δ)=+B A(+δ)but that B N(−δ)=−B N(+δ),and consequently that (∇N B A)(−δ)=−(∇N B A)(+δ)and(∇N B N)(−δ)=+(∇N B N)(+δ).Moreover,g AB(N=−δ)=+g AB(N=+δ)implies that K AB(N=−δ)=−K AB(N=+δ).Hence,wefind for the(AB)matching conditions that12 +δ−δdN −g ABΛ(4)λ+component,we note thatG AN =K AB ;B−K ;A=−∇B+δ−δdN G AB=−κ2(5)∇B T AB =0(26)which vanishes due to conservation of the induced stress-energy tensor T AB on the brane.From integration of the (NN )component in the normal direction to the brane,we find the following junction condition∇C (B C B N )+3KB N B N −K CD B C B D =σ,(27)which we substitute back in,obtaining12−R(ind )−K CD K CD +K2=14B CD B CD −V+1which becomes∇C B C +λKB N =0.(32)The junction conditions Eq.(30)and Eq.(32)offer the required (4+1)boundary conditions respectively for B A and B N on the brane.Substituting the junction condition for B A back in Eq.(23)and using the result from G AN =0,we find for the induced equation of motion for B A on the brane that∇C (∇C B A −∇A B C )+2K AC (∇C B N )−2V ′B A +2λB CR (ind )AC+2K AD K DC=0.(33)Similarly,substituting the junction condition for B N back in Eq.(24)we obtain∇C ∇C B N −2V ′B N +λ[K (∇N B N )+B N K CD K CD ]=0.(34)Thus,Eq.(30)provides the value at the boundary for ∇N B A and Eq.(34)provides that for ∇N B N .Using the results derived above in the Israel matching conditions we find that12(∇A B B )B N +(∇B B A )B N+B A B C K CB +K AC B C B B −K AB B N B N+g AB−∇C (B C B N )+12KB N B N +1κ2(5)G (ind )AB +2K AC K BC −K AB K +12−B AC B BC −4V ′B A B B +12g AB−2∇C ∇D (B C B D )−∇C ∇C (B N B N )+12B N ∇C ∇C B N −20V ′B N B N+4(K CD −g CD K )∇D (B C B N )+6K CD B D (∇C B N )+KB C (∇C B N )+B C B D R (ind )CD+9K CD K CD B N B N +14K CE K DE B C B D −Kσ+1+KB N(∇A B B+∇B B A)−2K AB B N(∇C B C) 8−∇C B C =0(38)−K CD B C B D =σ(39)1κ2(5)12−1κ2(5)G (ind )AB+2K AC K BC −12g ABR(ind )−K CD K CD −K2+g AB Λ(5)−1214B B ∇C (5∇C B A −9∇A B C )+∇A ∇C ( B B B C )+∇B ∇C ( B A B C )−2(∇C B A )(∇C B B )+52B B BC R (ind )AC −6K AC K BD B C B D +2K AB σ+1κ2(5)G (ind )AB+2K AC K BC −12g ABR (ind )−2K CD K CD −K 2+g AB Λ(5)=12 B A B C R (ind )CB +52g ABB C B D R (ind )CD+2Kσ.(44)Hence,in order to obtain a vanishing cosmological constant and ensure that Lorentz in-variance holds on the brane,we must impose respectively thatΛ(5)=Kσ(45)and that2K AC K BC−12g AB R(ind)−2K CD K CD−K2=κ2(5) 54 B B B C R(ind)AC+1the matching of the observed cosmological constant in four dimensions.This tuning does not follow from a dynamical mechanism but is imposed by phenomenological reasons only. From this point of view,both the value of the cosmological constant and the Lorentz symmetry seem to be a consequence of a complexfine-tuning.We aim to further study the implications of our mechanism by considering also the inclusion of a scalarfield in a forthcoming publication[14].AcknowledgmentsC.C.thanks Funda¸c˜a o para a Ciˆe ncia e a Tecnologia(Portuguese Agency)forfinancial support under the fellowship/BPD/18236/2004. C.C.thanks Martin Bucher,Georgios Kofinas and Rodrigo Olea for useful discussions,and the National and Kapodistrian Uni-versity of Athens for its hospitality.[1]CPT and Lorentz Symmetry III,Alan Kosteleck´y,ed.(World Scientific,Singapore,2005);O.Bertolami,Gen.Rel.Gravitation34(2002)707;O.Bertolami,Lect.Notes Phys.633 (2003)96,hep-ph/0301191;R.Lehnert,hep-ph/0312093.[2]V.A.Kosteleck´y and S.Samuel,Phys.Rev.D39(1989)683;Phys.Rev.Lett.63(1989)224;V.A.Kosteleck´y and R.Potting,Phys.Rev.D51(1995)3923;Phys.Lett.B381(1996)89.[3]R.Gambini and J.Pullin,Phys.Rev.D59(1999)124021;J.Alfaro,H.A.Morales-Tecotland L.F.Urrutia,Phys.Rev.Lett.84(2000)2318.[4]S.M.Carroll,J.A.Harvey,V.A.Kosteleck´y,ne and T.Okamoto,Phys.Rev.Lett.87(2001)141601.[5]O.Bertolami and L.Guisado,Phys.Rev.D67(2003)025001;JHEP0312(2003)013;O.Bertolami,Mod.Phys.Lett.A20(2005)1359.[6]V.A.Kosteleck´y,R.Lehnert and M.J.Perry,Phys.Rev.D68(2003)123511;O.Bertolami,R.Lehnert,R.Potting and A.Ribeiro,Phys.Rev.D69(2004)083513.[7]H.Sato,T.Tati,Prog.Theor.Phys.47(1972)1788;S.Coleman and S.L.Glashow,Phys.Lett.B405(1997)249;Phys.Rev.D59(1999)116008;L.Gonzales-Mestres, hep-ph/9905430.[8]O.Bertolami and C.Carvalho,Phys.Rev.D61(2000)103002.[9]O.Bertolami and J.G.Rosa,Phys.Rev.D71(2005)097901.[10]O.Bertolami,Class.Quantum Gravity14(1997)2785.[11]V.A.Kosteleck´y,Phys.Rev.D69(2004)105009;R.Bluhm and V.A.Kosteleck´y,Phys.Rev.D71(2005)065008.[12]O.Bertolami and J.P´a ramos,Phys.Rev.D72(2005)044001.[13]M.Bucher and C.Carvalho,Phys.Rev.D71(2005)083511.[14]O.Bertolami and C.Carvalho,in preparation.。

NIM （负折射率材料）专题研究严 杰一、有关折射的基本概念1、基本定义与关系式电磁学的早期即由实验发现了以下规律：各向同性介电物质中电位移矢量与电场强度矢量方向一致，大小成正比，故有 E ε=D ，式中ε是比例系数，称为介电率或介电常数．另外，实验还证明，对各向同性非铁磁性物质，磁感应强度矢量与磁场强度矢量方向一致，大小成正比，故有H B μ=，式中μ比例系数称为导磁率．ε和μ被看成表征物质电磁性质的宏观参数．在自由空间(无电荷源及传导电流)，由麦克斯韦方程组导出的电磁波波方程为由此得无色散电磁波传播速度rr cv μεεμ==1式中，0/εεε=r 是相对介电常数；,/0μμμ=r 是相对磁导率00με，则为ε，μ在真空中的值；而c 为自由空间(真空中)光速，001με=c 。

实际上，按照麦克斯韦场理论，电磁作用过程是经过场(波)而完成的，在真空条件下，这个作用传递的速度就是c ．可见，麦克斯韦由于提出电磁场方程组而被后人认为是伟大的科学家这点没错；但由于时代的局限(经典场论产生于距今136年前)，他的理论不可能解释近年来以量子力学、量子光学为基础而完成的超光速、超慢光速实验．2、折射折射是自然界最基本的电磁现象之一。

当电磁波以任意角度入射到两种不同折射率的介质交界面处时，波传播的方向会发生变化。

那么，介质的折射率是如何定义的？图一表示介质1中的入射波在介质2中折射，虚线AC ，BE 为波前，由于,sin ,sin 2211t v CB CE t v CB AB ====θθ故有此式即为Snell 定律，由它可以计算折射波前进的方向，式中1v ，2v 均为相速。

0,0222222=∂∂-∇=∂∂-∇t H H t E E εμεμ1211222121sin sin n n v v ===μεμεθθ这个比值被称为折射率，用n 表示，1122μεμε=n ，如0101,μμεε==，（介质1为真空），μμεε==22,，，则有r r vcn με==。

洛伦茨吸引子维基百科，自由的百科全书跳转到：导航，搜索ρ=28、σ=10、β=8/3时的洛伦兹系统轨迹洛伦茨吸引子是洛伦茨振子（Lorenz oscillator）的长期行为对应的分形结构，以爱德华·诺顿·洛伦茨的姓氏命名。

洛伦茨振子是能产生混沌流的三维动力系统，以其双纽线形状而著称。

映射展示出动力系统（三维系统的三个变量）的状态是如何以一种复杂且不重复的模式，随时间的推移而演变的。

目录[隐藏]1 简述2 洛伦茨方程3 瑞利数4 源代码 4.1 GNU Octave4.2 Borland C4.3 Borland Pascal4.4 Fortran4.5QBASIC/FreeBASIC("fbc -lang qb")5 参见 6 参考文献7 外部链接简述洛伦茨方程的一条轨迹被描绘成金属线，以展现方向以及三维结构洛伦茨吸引子及其导出的方程组是由爱德华·诺顿·洛伦茨于1963年发表，最初是发表在《大气科学杂志》（Journal of the Atmospheric Sciences）杂志的论文《Deterministic Nonperiodic Flow》中提出的，是由大气方程中出现的对流卷方程简化得到的。

这一洛伦茨模型不只对非线性数学有重要性，对于气候和天气预报来说也有着重要的含义。

行星和恒星大气可能会表现出多种不同的准周期状态，这些准周期状态虽然是完全确定的，但却容易发生突变，看起来似乎是随机变化的，而模型对此现象有明确的表述。

从技术角度看来，洛伦茨振子具有非线性、三维性和确定性。

2001年，沃里克·塔克尔（Warwick Tucker）证明出在一组确定的参数下，系统会表现出混沌行为，显示出人们今天所知的奇异吸引子。

这样的奇异吸引子是豪斯多夫维数在2与3之间的分形。

彼得·格拉斯伯格（Peter Grassberger）已于1983年估算出豪斯多夫维数为2.06 ±0.01，而关联维数为2.05 ±0.01。

表 面粗 糙 度 小 于 ３．２５ ｎｍ。并 且 ＰＶＢ加 入 后 增 加 了膜 层 胶 粒 间 的 黏 附性 ，使 得 膜 层 强 度 增 大 。ＰＶＢ加 入 使 膜
层 的激 光 损 伤 阈 值 有 所 增 加 。 当 ＰＶＢ 的 添 加 量 为 １ 时 ，膜 层 的 激 光 损 失 阈 值 从 ３０．０ Ｊ／ｃｍ。增 加 到
Ｖ ｏ１．２４，Ｎ ｏ．７ Ｊｕ１．，２０１２
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晏 良宏 ， 吕海兵， 严鸿维 ， 赵松楠， 王 韬， 李熙斌 ， 王海军， 袁晓东
（中 国 工 程 物 理 研 究 院 激 光 聚 变研 究 中 心 ，四川 绵 阳 ６２１９００）
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摘 要 ： 采 用 溶 胶 一凝 胶 技 术 制 备 了二 氧 化 硅 增 透 膜 ，通 过 向溶 胶 中 添加 高 分 子 聚 乙烯 醇 缩 丁醛 （ＰＶＢ），
调 控 胶 体 的粒 径 ，进 而 控 制 膜 层 微 观 结 构 ，研 究 膜 层 微 观 结 构 与 激 光 损 伤 阈值 的关 系 。纳 米 粒 度仪 和 扫 描 探 针
２ 结 果 与 讨 论
２．１ 膜层 折射 率 与孔 隙率 的调控 膜 层 的孔 隙率 影响膜 层 的折 射率 ，孔 隙率 越大 ，折 射率 越低 ［１ ；在 溶胶 中加 入 ＰＶＢ后 ，可 以调节 膜层 的孔

Derivation of the Lorentz Force Law and the Magnetic Field Concept using an Invariant Formulation of the LorentzTransformationJ.H.FieldD´e partement de Physique Nucl´e aire et Corpusculaire Universit´e de Gen`e ve.24,quaiErnest-Ansermet CH-1211Gen`e ve4.e-mail;john.field@cern.chAbstractIt is demonstrated how the right hand sides of the Lorentz Transformation equa-tions may be written,in a Lorentz invariant manner,as4–vector scalar products.The formalism is shown to provide a short derivation,in which the4–vector elec-tromagnetic potential plays a crucial role,of the Lorentz force law of classical elec-trodynamics,and the conventional definition of the magneticfield in terms spatialderivatives of the4–vector potential.The time component of the relativistic gen-eralisation of the Lorentz force law is discussed.An important physical distinctionbetween the space-time and energy-momentum4–vectors is also pointed out.Keywords;Special Relativity,Classical Electrodynamics.PACS03.30+p03.50.De1IntroductionNumerous examples exist in the literature of the derivation of electrodynamical equa-tions from simpler physical hypotheses.In Einstein’s original paper on Special Relativ-ity[1],the Lorentz force law was derived by performing a Lorentz transformation of the electromagneticfields and the space-time coordinates from the rest frame of an electron (where only electrostatic forces act)to the laboratory system where the electron is in motion and so also subjected to magnetic forces.A similar demonstration was given by Schwartz[2]who also showed how the electrodynamical Maxwell equations can be derived from the Gauss laws of electrostatics and magnetostatics by exploiting the4-vector char-acter of the electromagnetic current and the symmetry properties of the electromagnetic field tensor.The same type of derivation of electrodynamic Maxwell equations from the electrostatic and magnetostatic ones has recently been performed by the present author on the basis of‘space-time exchange symmetry’[3].Frisch and Wilets[4]discussed the derivation of Maxwell’s equations and the Lorentz force law by application of relativistic transforms to the electrostatic Gauss law.Dyson[5]published a proof,due originally to Feynman,of the Faraday-Lenz law of induction,based on Newton’s Second Law and the quantum commutation relations of position and momentum,that excited considerable interest and aflurry of comments and publications[6,7,8,9,10,11]about a decade ndau and Lifshitz[12]presented a derivation of Amp`e re’s Law from the electro-dynamic Lagrangian,using the Principle of Least Action.By relativistic transformation of the Coulomb force from the rest frame of a charge to another inertial system in rela-tive motion,Lorrain,Corson and Lorrain[13]derived both the Biot-Savart law,for the magneticfield generated by a moving charge,and the Lorentz force law.In many text books on classical electrodynamics the question of what are the funda-mental physical hypotheses underlying the subject,as distinct from purely mathematical developments of these hypotheses,used to derive predictions,is not discussed in any de-tail.Indeed,it may even be stated that it is futile to address the question at all.For example,Jackson[14]states:At present it is popular in undergraduate texts and elsewhere to attempt to derive magneticfields and even Maxwell equations from Coulomb’s law of electrostatics and the theory of Special Relativity.It should immediately obvious that,without additional assumptions,this is impossible.’This is,perhaps,a true statement.However,if the additional assumptions are weak ones,the derivation may still be a worthwhile exercise.In fact,in the case of Maxwell’s equations,as shown in References[2,3],the‘additional assumptions’are merely the formal definitions of the electric and magneticfields in terms of the space–time derivatives of the 4–vector potential[15].In the case of the derivation of the Lorentz force equation given below,not even the latter assumption is required,as the magneticfield definition appears naturally in the course of the derivation.In the chapter on‘The Electromagnetic Field’in Misner Thorne and Wheeler’s book ‘Gravitation’[16]can be found the following statement:Here and elsewhere in science,as stressed not least by Henri Poincar´e,that view isout of date which used to say,“Define your terms before you proceed”.All the laws and theories of physics,including the Lorentz force law,have this deep and subtle chracter, that they both define the concepts they use(here B and E)and make statements about these concepts.Contrariwise,the absence of some body of theory,law and principle deprives one of the means properly to define or even use concepts.Any forward step in human knowlege is truly creative in this sense:that theory concept,law,and measurement —forever inseperable—are born into the world in union.I do not agree that the electric and magneticfields are the fundamental concepts of electromagnetism,or that the Lorentz force law cannot be derived from simpler and more fundamental concepts,but must be‘swallowed whole’,as this passage suggests. As demonstrated in References[2,3]where the electrodynamic and magnetodynamic Maxwell equations are derived from those of electrostatics and magnetostatics,a more economical description of classical electromagentism is provided by the4–vector potential. Another example of this is provided by the derivation of the Lorentz force law presented in the present paper.The discussion of electrodynamics in Reference[16]is couched entirely in terms of the electromagneticfield tensor,Fµν,and the electric and magnetic fields which,like the Lorentz force law and Maxwell’s equations,are‘parachuted’into the exposition without any proof or any discussion of their interrelatedness.The4–vector potential is introduced only in the next-but-last exercise at the end of the chapter.After the derivation of the Lorentz force law in Section3below,a comparison will be made with the treatment of the law in References[2,14,16].The present paper introduces,in the following Section,the idea of an‘invariant for-mulation’of the Lorentz Transformation(LT)[17].It will be shown that the RHS of the LT equations of space and time can be written as4-vector scalar products,so that the transformed4-vector components are themselves Lorentz invariant quantities.Consid-eration of particular length and time interval measurements demonstrates that this is a physically meaningful concept.It is pointed out that,whereas space and time intervals are,in general,physically independent physical quantities,this is not the case for the space and time components of the energy-momentum4-vector.In Section3,a derivation of the Lorentz force law,and the associated magneticfield concept,is given,based on the invariant formulation of the LT.The derivation is very short,the only initial hypothesis being the usual definition of the electricfield in terms of the4-vector potential,which,in fact,is also uniquely specified by requiring the definition to be a covariant one.In Section 4the time component of Newton’s Second Law in electrodynamics,obtained by applying space-time exchange symmetry[3]to the Lorentz force law,is discussed.Throughout this paper it is assumed that the electromagneticfield constitutes,to-gether with the moving charge,a conservative system;i.e.effects of radiation,due to the acceleration of the charge,are neglected2Invariant Formulation of the Lorentz Transforma-tionThe space-time LT equations between two inertial frames S and S’,written in a space-time symmetric manner,are:x =γ(x−βx0)(2.1)y =y(2.2)z =z(2.3)x 0=γ(x0−βx)(2.4) The frame S’moves with velocity,v,relative to S,along the common x-axis of S and S’.βandγare the usual relativistic parameters:vβ≡√−(∆x0)2+∆x2=∆x(2.12) since,for the measurement procedure just described,∆x0=0.Notice that∆x is not necessarily defined in terms of such a measurement.If,following Einstein[1],the interval ∆x is associated with the length, ,of a measuring rod at rest in S and lying parallel to the x-axis,measurements of the ends of the rod can be made at arbitarily different times in S.The same result =∆x will be found for the length of the rod,but the corresponding invariant interval,S x,as defined by Eqn(2.12)will be different in each case.Similarly,∆x0may be identified with the time-like invariant interval corresponding to successive observations of a clock at afixed position(i.e.∆x=0)in S:S0≡the same value,∆x0,for the time difference between two events in S,but with different values of the invariant interval defined by Eqn(2.13).In virtue of Eqns(2.12)and(2.13)the LT equations(2.8)and(2.11)may be written the following invariant form:S x=−¯U(β)·S(2.14)S 0=U(β)·S(2.15) where the following4–vectors have been introduced:S≡(S0;S x,0,0)=(∆x0;∆x,0,0)(2.16)U(β)≡(γ;γβ,0,0)(2.17)¯U(β)≡(γβ;γ,0,0)(2.18)The time-like4-vector,U,is equal to V/c,where V is the usual4–vector velocity,whereas the space-like4–vector,¯U,is‘orthogonal to U in four dimensions’:U(β)·¯U(β)=0(2.19) Since the RHS of(2.14)and(2.15)are4–vector scalar products,S x and S 0are manifestly Lorentz invariant quantites.These4–vector components may be defined,in terms of specific space-time measurements,by equations similar to(2.12)and(2.13)in the frame S’.Note that the4–vectors S and S are‘doubly covariant’in the sense that S·S and S ·S are‘doubly invariant’quantities whose spatial and temporal terms are,individually, Lorentz invariant:S·S=S20−S2x=S ·S =(S 0)2−(S x)2(2.20) Every term in Eqn(2.20)remains invariant if the spatial and temporal intervals described above are observed from a third inertial frame S”moving along the x-axis relative to both S and S’.This follows from the manifest Lorentz invariance of the RHS of Eqn(2.14)and (2.15)and their inverses:S x=−¯U(−β)·S (2.21)S0=U(−β)·S (2.22) Since the LT Eqns(2.1)and(2.4)are valid for any4–vector,W,it follows that:W x=−¯U(β)·W(2.23)W 0=U(β)·W(2.24) Again,W x and W 0are manifestly Lorentz invariant.An interesting special case is the energy-momentum4–vector,P,of a physical object of mass,m.Here the‘doubly in-variant’quantity analagous to S·S in Eqn(2.20)is equal to m2c2.Choosing the x-axis parallel to p andβto correspond to the object’s velocity,so that S’is the object’s proper frame,and since P≡mcU(β),Eqns(2.23)and(2.24)yield,for this special case:P x=−mc¯U(β)·U(β)=0(2.25)P 0=mcU(β)·U(β)=mc(2.26)Since the Lorentz transformation is determined by the single parameter,β,then it follows from Eqns(2.25)and(2.26)that,unlike in the case of the space and time intervals in Eqns(2.8)and(2.11),the spatial and temporal components of the energy momentum 4–vector,in an arbitary inertial frame,are not independent.In fact,P0is determined in terms of P x and m by the relation,that follows from the inverse of Eqns(2.25)and(2.26):P0=Thus,from rotational invariance,the general covariant definition of the electricfield is:E i=∂i A0−∂0A i(3.4) This is the‘additional assumption’,mentioned by Jackson in the passage quoted above, that is necessary,in the present case,to derive the Lorentz force law.However,as written, it concerns only the physical properties of the electricfield:the magneticfield concept has not yet been introduced.A further a posteriori justification of Eqn(3.4)will be given after derivation of the Lorentz force law.Here it is simply noted that,if the spatial part of the4–vector potential is time-independent,Eqn(3.4)reduces to the usual electrostatic definition of the electricfield.The force F on an electric charge q at rest in the frame S’is given by the definition of the electricfield,and Eqn(3.4)as:F i=q(∂ i A 0−∂ 0A i)(3.5) Equations analagous to(2.24)may be written relating A and∂ to the corresponding quantities in the frame S moving along the x’axis with velocity−v relative to S’:∂ 0=U(β)·∂(3.6)A 0=U(β)·A(3.7) Substituting(3.6)and(3.7)in(3.5)gives:F i=q∂ i(U(β)·A)−(U(β)·∂)A i(3.8)This equation expresses a linear relationship between F i,∂ i and A i.Since the coefficients of the relation are Lorentz invariant,the same formula is valid in any inertial frame,in particular,in the frame S.Hence:F i=q∂i(U(β)·A)−(U(β)·∂)A i(3.9)This equation gives,in4–vector notation,a spatial component of the Lorentz force on the charge q in the frame S,and so completes the derivation.To express the Lorentz force formula in the more familiar3-vector notation,it is convenient to introduce the relativistic generalisation of Newton’s Second Law[19]:dPdτ=γdP iIntroducing now the magneticfield according to the definition[20]:B k≡− ijk(∂i A j−∂j A i)=( ∇× A)k(3.12) enables Eqn(3.11)to be written in the compact form:dP idγ βt=mc(3.15)∂twhere Eqn(3.12)has been used.Eqn(3.15)is just the Faraday-Lenz induction law,i.e.the magnetodynamic Maxwell equation.This is only apparent,however,once the‘magnetic field’concept of Eqn(3.12)has been introduced.Thus the initial hypothesis,Eqn(3.4),is actually a Maxwell equation.This is the a posteriori justification,mentioned above,for this covariant definition of the electricfield.It is common in discussions of electromagnetism to introduce the second rank electro-magneticfield tensor,Fµνaccording to the definition:Fµν≡∂µAν−∂νAµ(3.16) in terms of which,the electric and magneticfields are defined as:E i≡F i0(3.17)B k≡− ijk F ij(3.18) From the point of view adopted in the present paper both the electromagneticfield tensor and the electric and magneticfields themselves are auxiliary quantities introduced only for mathematical convenience,in order to write the equations of electromagnetism in a compact way.Since all these quantities are completly defined by the4–vector potential, it is the latter quantity that encodes all the relevant physical information on any electro-dynamic problem[21].This position is contrary to that commonly taken in the literature and texbooks where it is often claimed that only the electric and magneticfields have physical significance,while the4–vector potential is only a convenient mathematical tool. For example R¨o hrlich[22]makes the statement:These functions(φand A)known as potentialsmanner!In other cases(e.g.Maxwell’s equations)simpler expessions may be written interms of the4–vector potential.The quantum theory,quantum electrodynamics,thatunderlies classical electromagnetism,requires the introduction the4–vector photonfield Aµin order to specify the minimal interaction that provides the dynamical basis of the theory.Similarly,the introduction of Aµis necessary for the Lagrangian formulation ofclassical electromagnetism.It makes no sense,therefore,to argue that a physical conceptof such fundamental importance has‘no physical meaning’.The initial postulate used here to derive the Lorentz force law is Eqn(3.4),whichcontains,explicitly,the electrostatic force law and,implicitly,the Faraday-Lenz inductionlaw.The actual form of the electrostatic force law(Coulomb’s inverse square law)is notinvoked,suggesting that the Lorentz force law may be of greater generality.On theassumption of Eqn(3.4)(which has been demonstrated to be the only possible covariantdefinition of the electricfield),the existence of the‘magneticfield’,the‘electromagneticfield tensor’,andfinally the Lorentz force law itself have all been derived,without furtherassumptions,by use of the invariant formulation of the Lorentz transformation.It is instructive to compare the derivation of the Lorentz force law given in the presentpaper with that of Reference[13]based on the relativistic transformation properties of theCoulomb force3–vector.Coulomb’s law is not used in the present paper.On the otherhand,Reference[13]makes no use of the4–vector potential concept,which is essential forthe derivation presented here.This demonstrates an interesting redundancy among thefundamental physical concepts of classical electromagnetism.In Reference[2],Eqns(3.4),(3.12)and(3.16)were all introduced as a priori initialpostulates without further justification.In fact,Schwartz gave the following explanationfor his introduction of Eqn(3.16)[23]:So far everything we have done has been entirely deductive,making use only ofCoulomb’s law,conservation of charge under Lorentz transformation and Lorentz in-variance for our physical laws.We have now come to the end of this deductive path.Atthis point when the laws were being written,God had to make a decision.In generalthere are16components of a second-rank tensor in four dimensions.However,in anal-ogy to three dimensions we can make a major simplification by choosing the completelyantisymmetric tensor to represent ourfield quantities.Then we would have only6inde-pendent components instead of the possible16.Under Lorentz transformation the tensorwould remain antisymmetric and we would never have need for more than six independentcomponents.Appreciating this,and having a deep aversion to useless complication,Godnaturally chose the antsymmetric tensor as His medium of expression.Actually it is possible that God may have previously invented the4–vector potentialand special relativity,which lead,as shown above,to Eqn(3.4)as the only possible co-variant definition of the electricfield.As also shown in the present paper,the existence ofthe remaining elements of the antisymmetricfield tensor,containing the magneticfield,then follow from special relativity alone.Schwartz derived the Lorentz force law,as inEinstein’s original Special Relativity paper[1],by Lorentz transformation of the electricfield,from the rest frame of the test charge,to one in which it is in motion.This requiresthat the magneticfield concept has previously been introduced as well as knowledge ofthe Lorentz transformation laws of the electric and magneticfields.In the chapter devoted to special relativity in Jackson’s book[24]the Lorentz forcelaw is simply stated,without any derivation,as are also the defining equations of theelectric and magneticfields and the electromagneticfield tensor just mentioned.Noemphasis is therefore placed on the fundamental importance of the4–vector potential inthe relativistic description of electromagnetism.In order to treat,in a similar manner,the electromagnetic and gravitationalfields,thediscussion in Misner Thorne and Wheeler[16]is largely centered on the properties of thetensor Fµν.Again the Lorentz force equation is introduced,in the spirit of the passagequoted above,without any derivation or discussion of its meaning.The defining equationsof the electric and magneticfields and Fµν,in terms of Aµ,appear only in the eighteenthexercise of the relevant chapter.The main contents of the chapter on the electromagneticfield are an extended discussion of purely mathematical tensor manipulations that obscurethe essential simplicity of electromagnetism when formulated in terms of the4–vectorpotential.In contrast to References[2,24,16],in the derivation of the Lorentz force law andthe magneticfield presented here,the only initial assumption,apart from the validityof special relativity,is the chosen definition,Eqn(3.4),of the electricfield in terms ofthe4–vector potential Aµ,which is the only covariant one.Thus,a more fundamentaldescription of electromagnetism than that provided by the electric and magneticfieldconcepts is indeed possible,contrary to the opinion expressed in the passage from MisnerThorne and Wheeler quoted above.4The time component of Newton’s Second Law in ElectrodynamicsSpace-time exchange symmetry[3]states that physical laws inflat space are invariantwith respect to the exchange of the space and time components of4-vectors.For example,the LT of time,Eqn(2.4),is obtained from that for space,Eqn(2.1),by applying the space-time exchange(STE)operations:x0↔x,x 0↔x .In the present case,application of the STE operation to the spatial component of the Lorentz force equation in the secondline of Eqn(3.11)leads to the relation:dP00(4.1)where Eqns(2.5)and(3.4)and the following properties of the STE operation[3]have been used:∂0↔−∂i(4.2)A0↔−A i(4.3)C·D↔−C·D(4.4)Eqn(4.1)yields an expression for the time derivative of the relativistic energy,E=P0:d E(4.5)Integration of Eqn(4.5)gives the equation of energy conservation for a particle moving from an initial position, x I,to afinal position, x F,under the influence of electromagnetic forces:E F E I d E=qxFx IE·d x(4.6)Thus work is done on the moving charge only by the electricfield.This is also evident from the Lorentz force equation,(3.14),since the magnetic force β× B is perpendicular to the velocity vector,so that no work is performed by the magneticfield.A corollary is that the relativistic energy(and hence the magnitude of the velocity)of a charged particle moving in a constant magneticfield is a constant of the motion.Of course,Eqn(4.5) may also be derived directly from the Lorentz force law,so that the time component of the relativistic generalisation of Newton’s Second Law,Eqn(4.1),contains no physical information not already contained in the spatial components.This is related to the fact that,as demonstrated in Eqns(2.25)and(2.26),the spatial and temporal components of the energy-momentum4–vector are not independent physical quantities.AcknowledgementsI should like to thank O.L.de Lange for asking the question whose answer,presented in Section4,was the original motivation for the writing of this paper,and an anonymous referee of an earlier version of this paper for informing me of related material,in the books of Jackson and Misner,Thorne and Wheeler,which is discussed in some detail in this version.References[1]A.Einstein,17891(1905).[2]M.Schwartz,‘Principles of Electrodynamics’,(McGraw-Hill,New York,1972)Ch3.[3]J.H.Field,Am.J.Phys.69569(2001).[4]D.H.Frisch and L.Wilets,Am.J.Phys.24574(1956).[5]F.J.Dyson,Am.J.Phys.58209(1990).[6]N.Dombey,Am.J.Phys.5985(1991).[7]R.W.Breheme,Am.J.Phys.5985(1991).[8]J.L.Anderson,Am.J.Phys.5986(1991).[9]I.E.Farquhar,Am.J.Phys.5987(1991).[10]S.Tanimura,Ann.Phys.(N.Y.)220229(1992).[11]A.Vaidya and C.Farina,Phys.Lett.153A265(1991).[12]ndau and E.M.Lifshitz,‘The Classical Theory of Fields’,(Pergamon Press,Oxford,1975)Section30,P93.[13]P.Lorrain, D.R.Corson and F.Lorrain,‘Electromagnetic Fields and Waves’,(W.H.Freeman,New York,Third Edition,1988)Section16.5,P291.[14]J.D.Jackson,‘Classical Electrodynamics’,(John Wiley and Sons,New York,1975)Section12.2,P578.[15]Actually,a careful examination of the derivation of Amp`e re’s from the Gauss lawof electrostatics in Reference[3]shows that,although Eqn(3.4)of the present paper is a necessary initial assumption,the definition of the magneticfield in terms of the spatial derivatives of the4–vector potential occurs naturally in the course of the derivation(see Eqns(5.16)and(5.17)of Reference[3])so it is not necessary to assume, at the outset,the expression for the spatial components of the electromagneticfield tensor as given by Eqn(5.1)of Reference[3].[16]C.W.Misner,K.S.Thorne and J.A.Wheeler,‘Gravitation’,(W.H.Freeman,San Fran-cisco,1973)Ch3,P71.[17]This should not be confused with a manifestly covariant expression for the LT,whereit is written as a linear4-vector relation with Lorentz-invariant coefficients,as in:D.E.Fahnline,Am.J.Phys.50818(1982).[18]A time-like metric is used for4-vector products with the components of a4–vector,W,defined as:W t=W0=W0,W x,y,z=W1,2,3=−W1,2,3and an implied summation over repeated contravariant(upper)and covariant(lower) indices.Repeated Greek indices are summed from0to3,repeated Roman ones from1to3.Also∂µ≡(∂∂x1,−∂∂x3)=(∂0;− ∇)[19]H.Goldstein,‘Classical Mechanics’,(Addison-Wesley,Reading Massachusetts,1959)P200,Eqn(6-30).[20]The alternating tensor, ijk,equals1(−1)for even(odd)permutations of ijk.[21]The explicit form of Aµ,as derived from Coulomb’s law,is given in standard text-books on classical electrodynamics.For example,in Reference[13],it is to be found in Eqns(17-51)and(17-52).Aµis actually proportional to the4-vector velocity,V, of the charged particle that is the source of the electromagneticfield.[22]F.R¨o hrlich,‘Classical Charged Particles’,(Addison-Wesley,Reading,MA,1990)P65.[23]Reference[2]above,Ch3,P127.[24]Reference[14]above,Section11.9,P547.。

《负折射研究综述》负折射现象是俄国科学家Veselago 在1968 年提出的：当光波从具有正折射率的材料入射到具有负折射率材料的界面时，光波的折射与常规折射相反，入射波和折射波处在于界面法线方向同一侧。

直到本世纪初这种具有负折射率的材料才被制备出来。

这种材料由金属线和非闭合金属环周期排列构成，也被称为metmaterial 。

在这种材料中，电场、磁场和波矢方向遵守“左手”法则，而非常规材料中的“右手”法则。

因此，这种具有负折射率的材料也被称为左手材料，光波在其中传播时，能流方向与波矢方向相反。

英国科学家Pendry 提出折射率为-1的一个平板材料可以作为透镜实现完美成像，可以放大衰势波,使成像的大小突破光学衍射极限。

负折射现象实验和超透镜提出时引起极大的争议，因为这些概念违反人们的直觉。

通过查询相关的论文，我找到了两种理论来解释负折射是存在的。

第一种是根据法拉第、洛伦兹等人提出的电极化方程，经过对比后得到折射率的表达式，然后说明其为负的可能性。

1837年，法拉第最先提出电介质在电场中极化的概念．1850年，0.F.Mosotti 提出了电介质极化理论方程。

1880年，H.F.Lorenntz 和L ．V ．Lorenz 用光学方法导出了一个包含折射率的公式，称为Lorentz-Lorenz 方程。

由这两个方程对比可知道r n ε=2。

r r n με=2。

因而，r r n με±=。

这里的负号不能随便丢掉．在某种材料同时具有0,0<<r r με时，上式右端可能取负值。

这就是负折射材料。

第二种则是由麦克斯韦方程组出发，推导出折射率的表达式，同样也可以证明折射率是可以为负的。

根据麦克斯韦电磁场理论，对于无损耗、各向同性、均匀的介质得到正弦时变光波的亥姆霍兹方程为： 022=+∇E K E 022=+∇B K B 其中：002222002221,,μεεμεεμμωμεω=====c n n c w k r r r r 式中n 代表折射率，c 是真空中的光速。

a r X i v :p h y s i c s /9910034v 1 [p h y s i c s .c l a s s -p h ] 22 O c t 1999On the form of Lorentz-Stern-Gerlach forceSameen Ahmed KHAN ∗Modesto PUSTERLA †Dipartimento di Fisica Galileo Galilei Universit`a di Padova Istituto Nazionale di Fisica Nucleare (INFN)Sezione di PadovaVia Marzolo 8Padova 35131ITALYAbstractIn recent times there has been a renewed interest in the force experienced by a charged-particle with anomalous magnetic moment in the presence of external fields.In this paper we address the basic question of the force experienced by a spin-12point-like charged-particlewith anomalous magnetic and anomalous electric moments in the presence of space-and time-dependent external electromagnetic fields,based ab initio on the Dirac equation via the Foldy-Wouthuysen (FW)transformation technique.In the present derivation we neglect the radiation reaction and the electromagnetic fields are treated as classical.In absence of spin the force experienced by a point-like charged-particle is completely described by the Lorentz force law (F L =q (E +v ×B )).In the regime where the spin and the magnetic moment are to be taken into account the question of the form of the force obtained from the relativistic quantum theory is still unresolved to this day,though extensive studies,using diverse approaches have been done since the discovery of quantum mechanics.This is evident from the numerous approaches/prescriptions which have been tried to address this basic question and are still being tried.Before proceeding further we note that the expression quoted above constitutes the Lorentz force.The total force whichwe call as the Lorentz-Stern-Gerlach(LSG)force includes the Lorentz force as the basic constituent and all the other contributions coming from the spin,anomalous magnetic and electric moments etc,.The reason for this nomenclature will be clear as we proceed.Here we quote a few approaches which have been used to address the question of the force and acceleration experienced by a charged-particle.A Lagrangian formalism based on the action principle has been suggested[1]-[3].A Hamiltonian formalism is considered in[4]–[5].In the case of slowly varying electromagneticfields an approach based on the Dirac equation via the WKB approximation scheme has been presented[6].In the context of the Aharonov-Bohm and Aharonov-Casher effects[7]-[8],the question as to whether neutron acceleration can occur in uniform electromagneticfields is also raised[9]-[11].In the very recent work of Chaichian[4]it has been rightly pointed out that in the nonrelativistic limit the results of the above approaches do not coincide.This motivates us to examine the form of the force derived from the Dirac equation using the FW-transformation[13]-[14] scheme;note that the FW-transformation technique is the only one in which we can take the meaningful nonrelativistic limit of the Dirac equation[15].The FW-approach gives the expression for the force in the presence of external time-dependentfields,the nonrelativistic limit and a systematic procedure to obtain the relativistic corrections to a desired degree of accuracy.In such a derivation we also take into account the anomalous electric moment.We compare the results of our derivation with other approaches mentioned above.One should also note that a novel approach for producing polarized beams has been suggested using the Stern-Gerlach forces[16]-[17].II.SECTIONLet us consider the Dirac particle of rest mass m0,charge q,anomalous magnetic moment µa and anomalous electric momentǫa.In presence of the external electromagneticfields, the Dirac equation is∂i¯hthe Dirac Hamiltonian.Such a transformation is available in the case of the free-particle. In the very general case of time-dependentfields such a transformation is not known to exist.Therefore,one has to be content with an approximation procedure which reduces the strength of the odd-part to a desired degree of accuracy in powers of1m0c2.The result to the leading order,that is to order1∂t|ψ =ˆH(2)|ψ ,ˆH(2)=m0c2β+ˆE+1(m0c2)3is given byi¯h∂2m0c2βˆO2−1∂tˆO −1 2m0 ˆπ2−q¯hσ·B+18m30c2 ˆπ4+¯h2q2B2−¯h q ˆπ2(σ·B)+(σ·B)ˆπ2 (7) A detailed formula including theµa contibutions is given by(A1)in the appendix.III.SECTIONNow we use the Hamiltonian derived in(7)to compute the acceleration,a experienced by the particle using the Heisenberg representation,d¯h ˆH,ˆO +∂m 0˙r=m 0d¯hˆH,r=ˆπ−14m 0c 2(σ×E )+¯h qdt˙r=m 0¨r =q E −q2m 0(ˆπ×B −B ׈π)−q4m 20c 2{(ˆπ·E +E ·ˆπ)ˆπ+ˆπ(ˆπ·E +E ·ˆπ)}+¯h q8m 30c2ˆπ2∇(σ·B )+∇(σ·B )ˆπ2+¯h q 28m 20c2∇(σ·(ˆπ×E −E ׈π))+¯h q 28m 30c2∇B 2+¯h q 24m 0c 2∂4m 20c2ˆπ∂∂t(σ·B )ˆπ+R(10)where the r k -th component of R is(R )r k =−¯h qm 0−→v where v is the velocity of the particle.With such asubstitution and with β=|v |2β2 qm 0v2β2¯h q4m 0c 2∇(σ·(v ×E ))12m 0(σ·B )+···(12)The above derivation is consistent with the result[19]of classical electrodynamicsa=q1−β2 E+v×B−v2m0c2 c2 ˆπ2−q¯hσ·B +(µa E+ǫa B)2+µa cσ·(ˆπ×E−E׈π)+ǫa cσ·(ˆπ×B−B׈π) (14)Next,to leading order Hamiltonian is given in(A1)of the appendix.Now we use the above derived Hamiltonians in(14)to compute the Lorentz-Stern-Gerlach force and we get˙r=dm0 ˆπ−µa c(σ×B)=1dtˆ =i∂tˆ=q E+12m0 ∇(σ·B)−µa∂t(σ×E)−µa¯h c µa+¯h q2m0c2∇ µ2a E2+iµ2a2m0c2¯h{(σ×(ˆπ×E−E׈π))×E−E×(σ×(ˆπ×E−E׈π))} +R(16) where the r k-th component of R is(R)r k =−µac∂2m0c∇(σ·(ˆp×E−E׈p))+2µ2a2m0c2∇ µ2a E2+iµ2a2m0c2¯h {(σ×(ˆp×E−E׈p))×E−E×(σ×(ˆp×E−E׈p))}+···+O µ3a (18) In the above expression in(18)the leading terms have been retained and the“···”indicates the higher order terms.The complete expression is given in(A2)in the appendix.The detailed formulae shall be given in an appendix at the end.This is the case where ever the“···”appear in the expressions.From the expression(18)we conclude that the leading order(linear inµa)contributions to the neutron acceleration come through the gradients and the time derivatives of the electromagneticfields.Such contributions disappear in the case of uniform and constant fields respectively.The next-to-leading order contributions come from the terms of the type µ2a E×(σ×B).Such contributions do not vanish and hence we have neutron acceleration even in the presense of uniformfields.Such accelerations are quadratic(and higher powers) inµa and hence are very small.In the expression(16)for the Lorentz-Stern-Gerlach force if we substituteµa=g¯h|q|2m0(ˆπ×B−B׈π)+ µa+q¯hc∂2m0c ∇(σ·(ˆπ×E−E׈π))−ǫaIV.CONCLUSIONS AND SUMMARYAs can be seen above we get a variety of terms contributing to the Lorentz-Stern-Gerlach force.The nonrelativistic static limit coincides with the usual“classical”formula if B is time-independent,inhomogeneous and E is absent in the lab system.Otherwise there are differ-ences even at low non-relativistic velocities.In particular one may consider the force terms µa∇ µ2a E2 that are present whenever a spin-12m0c2ˆπ22m0− ¯h q2m0cσ·(ˆπ×E−E׈π)+µ2a8m20c4 +¯h qc2σ·(ˆπ×E−E׈π)+2µa¯h qcE2−µ2a((σ·ˆπ)(ˆπ·B+B·ˆπ)+(ˆπ·B+B·ˆπ)(σ·ˆπ))−µ2a¯h cσ·(∇(E·B+B·E))+4µ3a(σ·E)(E·B)1−+µ3a c E2σ·(ˆπ×E−E׈π)+σ·(ˆπ×E−E׈π)E2+µ4a E4 .(A1) The total acceleration(or equivalently the force)experienced by a neutron when bothµa andǫa are taken into account is:F=µa∇(σ·B)−ǫa∇(σ·E)−1∂t(µa(σ×E)+ǫa(σ×B))−µa2m0c∇(σ·(ˆp×B−B׈p))−1¯h c (σ×B)×E−2ǫ2a¯h c((σ×E)×E−(σ×B)×B)−iµ2a2m0c2¯h((ˆp×B)×B−B×(B׈p))+µ2a2m0c2¯h{(σ×(ˆp×B−B׈p))×B−B×(σ×(ˆp×B−B׈p))}−iµaǫa2m0c2¯h (σ×(ˆp×B−B׈p))×E−E×(σ×(ˆp×B−B׈p))+(σ×(ˆp×E−E׈p))×B−B×(σ×(ˆp×E−E׈p)) +R(A2) where the r k-th component of R is(R)r k =−µa2m0c ˆp·∇ (σ×B)r k +∇ (σ×B)r k ·ˆpr k=x,y,z,k=1,2,3.(A3)In the presence of the anomalous electric momentǫa the Lorentz-Stern-Gerlach force is:F =qE +12m∇(σ·B )−ǫa ∇(σ·E )−1∂t (µa (σ×E )+ǫa (σ×B ))−µa 2m 0c∇(σ·(ˆπ×B −B ׈π))−1m 0c ((σ×E )×B −(σ×B )×E )−2µ2a¯h c(σ×E )×B +2µa ǫa 2m 0c 2¯h ((ˆπ×E )×E −E ×(E ׈π))−iǫ2a2m 0c 2¯h {(σ×(ˆπ×E −E ׈π))×E −E ×(σ×(ˆπ×E −E ׈π))}+ǫ2a2m 0c 2¯h(ˆπ×B −B ׈π)×E +E ×(ˆπ×B −B ׈π)+(ˆπ×E −E ׈π)×B +B ×(ˆπ×E −E ׈π)+µa ǫa2m 0cˆπ·∇ (σ×E )r k +∇ (σ×E )r k ·ˆπ−ǫaREFERENCES[1]Patrick L.Nash,A Lagrangian theory of the classical spinning electron,J.Math.Phys.25(6)(1984)2104-2108.[2]Patrick L.Nash,Order¯h corrections to the classical dynamics of a particle with intrinsicspin moving in a constant magneticfield,acc-phys/9411002(19November1994)pp.15.[3]J.P.Costella and Bruce H.J.McKellar,Electromagnetic deflection of spinning particle,Int.J.Mod.Phys.A9(1994)461-473.Also in:hep-ph/9312256(10December1993) pp.18.[4]M.Chaichian,R.Gonz´a lez Felipe,D.Lois Martinez,Spinning relativistic particle inan external electromagneticfield,Phys.Lett.A236(1997)188-192.Also in:hep-th/9601119(23January1996)pp.10.[5]K.Heinemann,On Stern-Gerlach forces allowed by special relativity and the special caseof the classical spinning particle of Derbenev-Kondratenko,e-print:physics/9611001.Barber,D.P.,Heinemann,K.and Ripken,G.Z.Phys.C,64,117(1994).Barber,D.P., Heinemann,K.and Ripken,G.(1994).Z.Phys.C,64,143(1994).[6]J.Anandan,Electromagnetic effects in the quantum interference of dipole,Phys.Lett.A138(8)(1989)347-352;ERRATA Phys.Lett.A152(9)(1991)504.[7]Timothy H.Boyer,Proposed Aharonov-Casher effect:Another example of an Aharonov-Bohm effect arising from a classical lag,Phys.Rev.A36(10)(1987)5083-5086. [8]Y.Aharonov,P.Pearle and L.Vaidman,Comments on“Proposed Aharonov-Cashereffect:Another example of an Aharonov-Bohm effect arising from a classical lag”,Phys.Rev.A37(10)(1988)4052-4055.[9]Russell C.Casella and Samuel A.Werner,Electromagnetic acceleration of neutronsPhys.Rev Lett.69(11)(1992)1625-1628.[10]Y.Aharonov and A Casher,Topological quantum effects for neutral particles,Phys.Rev.Lett.53(4)(1984)319-321.[11]J.Anandan and C.R.Hagen,Neutron acceleration in uniform electromagneticfields,Phys.Rev.A50(4)(1994)2860-2864.Also in:hep-th/9301110(26January1993)pp.11.[12]Jeeva S.Anandan,The secret life of the dipole,Nature387(1997)558-559.[13]Leslie L.Foldy and S.A.Wouthuysen,On the Dirac theory of spin1/2particles andits non-relativistic limit,Phys.Rev.78(1950)29-36.[14]J.D.Bjorken and S.D.Drell,Relativistic Quantum Mechanics(McGraw-Hill,NewYork,San Francisco,1964).[15]John P.Costella and Bruce H.J.McKellar,The Foldy-Wouthuysen transformation,AmJ.Phys.63(12)(1995)1119-1121.Also in:hep-ph/9503416.[16]M.Conte,A.Penzo and M.Pusterla,Spin splitting due to longitudinal Stern-Gerlachkicks,Il Nuovo Cimento A108(1995)127-136.[17]M.Conte,R.Jagannathan,S.A.Khan and M.Pusterla,Beam optics of the Diracparticle with anomalous magnetic moment,Particle Accelerators56(1996)99-126.[18]B.Thaller,The Dirac Equation(Springer Berlin1992).[19]Section17in,ndau and E.M.Lifshitz,The Classical theory of Fields(PergamonPress1962).。

诺贝尔演讲稿1.12.23.3奥巴马获诺贝尔和平奖的获奖感言演讲稿全文，我知道最近几十天来有关我的获奖引起多方的质疑和争论，甚至有人认为这不过是给我下的一个圈套而已，在我获奖的翌日有一位来自中国的道长送了一本书给我道德经。

诺贝尔演讲稿2017-08-06 20:46:15 | #1楼UniversityofCaliforniaatBerkeleygraduationspeechThomas J. Sargent.May 16, 2016I rememberhowhappy Ifelt whenIgraduatedfromBerkeley manyyearsago. But I thought the graduation speeches were long. I will economize onwords.Economics is organized common sense. Here is a short list of valuablelessons that our beautiful subject teaches.1. Many things that are desirable are not feasible.2. Individuals and communities face trade-offs.3. Other people have more information about their abilities,their efforts,and their preferences than you do.4. Everyone respondstoincentives,includingpeopleyou wanttohelp. Thatis why social safety nets don’t always end up working as intended.5. There are tradeoffs between equality and efficiency.6. In an equilibrium ofagame or an economy,people are satisfied withtheirchoices. That is why it is difficult for well meaning outsiders to changethings for better or worse.7. In the future, you too will respond to incentives. That is why there aresomepromisesthatyou’dliketo makebut can’t. No one willbelievethosepromises because they know that later it will not be in your interest todeliver. The lesson here is this: before you make a promise, think aboutwhetheryou will want tokeepitif and whenyour circumstances change.Thisishowyou earn a reputation.8. Governments and voters respond to incentives too.That is why governments sometimes default on loans and other promises that they havemade.. NewYorkUniversityandHooverInstitution.Email: .19. Itis feasiblefor onegeneration to shift costs to subsequent ones. Thatiswhat national government debts and the U.S. social security system do(but not the socialsecurity system of Singapore).10. When a government spends, its citizens eventually pay, either today ortomorrow, either through explicit taxes or implicit ones like inflation.11. Most people want other people to pay for public goods and governmenttransfers(especially transfers tothemselves).12. Because market prices aggregate traders’ information, it is difficult toforecast stock prices and interest rates and exchange rates.21902诺贝尔奖获得者洛仑兹演讲稿2017-08-06 20:48:02 | #2楼Hendrik A. Lorentz – Nobel LectureNobel Lecture, December 11, 1902The Theory of Electrons and the Propagation of LightWhen Professor Zeeman and I received the news of the great honour of the high distinction awarded to us, we immediately began to consider how we could best divide our roles with respect to our addresses. Professor Zeeman was first to have described the phenomenon discovered by him, given the explanation of it, and given an outline of his later experimental work. My task should have been to consider rather more deeply our present-day knowledge of electricity, in particular the so-called electron theory. I am more sorry than I can say that Professor Zeeman has been prevented by illnefrom undertaking the journey to Stockholm, and that therefore you will now only be able to hear the second half of our programme. I hope you will excuse me if under these circumstances I say only a little about the main theme, Zeeman's fine discovery. A short description of it, however, might well precede my further thoughts.As is well known to you, Faraday even in his day discovered that magnetic forces can have an effect on the propagation of light. He showed in fact that in suitable conditions the vibrations of a beam of polarized light can be made to rotate by such forces. Many years later Kerr found that such a beam of light also undergoes similar changes when it is simply reflected from the polished pole of a magnet. However, it remained for Zeeman's talent to show that a magnetic field affects not only the propagation andreflection of light but also the processes in which the beamof light originates, that is to say that the rays emitted by a light source assume different properties if this source is placed in the gap between a magnetic north and south pole. The difference is shown in the spectral resolution of the light, when one is working with the type of light source whose spectrum consists of single bright lines - that is, with a coloured flame, an electrical spark, or a Geissler tube. To have a specific case before your eyes, imagine that my hands are the two poles, only much closer together than I am holding them now, and that the light source is between these poles, that is to say in the space immediately in front of me. Now if the spectrum of the light which shines on a point directly opposite me is investigated, there can be observed, instead of a single spectral line such as can be seen under normal circumstances, a three-fold line, or triplet, whose components admittedly are separated from each other by a very small distance. Since each position in the spectrum corresponds to a specific frequency of light, we can also say that instead of light of one frequency the source is, under the influence of the magnetic field, emitting light of three different frequencies. If the spectrum consists of more than one line, then you can imagine that each line is resolved into a triplet. I must, however, add that the situation is not always as simple as this, and many spectral lines resolve into more than three components.Before turning to the theory, I should like to remark that thanks to the speedy publication of research and the consequent lively exchange of views between scientists much progremust be considered as the result of a great deal of joint effort. Since it is expected of me, I am going to talk principally of my own ideas and the way in which I have come to them. I do beg of you, however, not to lose sight of the fact that many other physicists,not all of whom I can name in this short space of time, have arrived at the same or very similar conclusions.The theory of which I am going to give an account represents the physical world as consisting of three separate things, composed of three types of building material: first ordinary tangible or ponderable matter, second electrons, and third ether.I shall have very little to say about ponderable matter, but so much the more about ether and electrons. I hope it will not be too much for your patience.As far as the ether - that bearer of light which fills the whole universe - is concerned, after Faraday's discovery which I have already mentioned and also independently of it, many attempts were made to exploit the ether in the theory of electricity also. Edlund went so far as to identify the electric fluid with the ether, ascribing to a positively charged body an exceof ether and toa negatively charged one a deficiency of ether. He considered this medium as a liquid, subject to the Archimedean principle, and in this way succeeded in imputing all electrostatic effects to the mutual repulsion of particles of ether.There was also a place in his theory for the electrodynamic attraction and repulsion between two metallic wires with electrical current flowing in them. Indeed, he formed a most remarkable conception of these effects. He explained them by the condition that the mutual repulsion of two particles of ether needs a certain time to be propagated from the one to the other; it was in fact an axiom with him that everything which occurs in Nature takes a certain length of time, however short this may be. This idea, which has been fully developed in our present-day views, is found also in the work of other older physicists. I need only mention Gauss, of whom we know that he did not followthis up only because he lacked a clear picture of the propagation. Such a picture, he wrote to Wilhelm Weber, would be the virtual keystone of a theory of electrodynamics.The way pioneered by Edlund, in which the distinction between ether and electricity was completely swept aside, was incapable of leading to a satisfactory synthesis of optical and electrical phenomena. Lorenz at Copenhagen came nearer the goal. You know, however, that the true founders of our present views on this subject were Clerk Maxwell and Hertz. In that Maxwell developed further and constructed a basis for the ideas put forward by Faraday, he was the creator of the electromagnetic theory of light, which is undoubtedly well known to you in its broad outline. He taught us that light vibrations are changes of state of the same nature as electric currents. We can also say that electrical forces which change direction extremely rapidly - many billions of times a second - are present in every beam of light. If you imagine a tiny particle in the path of a sunbeam, something like the familiar dust motes in the air, only considerably smaller, and if you also imagine that this particle is electrically charged, then you must also suppose that it is set into a rapid quivering movement by the light vibrations.Immediately after Maxwell I named Hertz, that great German physicist, who, if he had not been snatched from us too soon, would certainly have been among the very first of those whom your Academy would have considered in fulfilling your annual task. Who does not know the brilliant experiments by which he confirmed the conclusions that Maxwell had drawn from his equations ? Whoever has once seen these and learnt to understand and admire them can no longer be in any doubt that the features of the electromagnetic waves to be observed inthem differ from light beams only in their greater wavelength.The result of these and other investigations into the waves propagated in the ether culminate in the realization that there exists in Nature a whole range of electromagnetic waves, which, however different their wavelengths may be, are basically all of the same nature. Beginning with Hertz's "rays of electrical force", we next come to the shortest waves caused by electromagnetic apparatus and then, after jumping a gap, to the dark thermal rays. We traverse the spectrum far into the ultraviolet range, come acroanother gap, and may then put X-rays, as extremely short violent electromagnetic disturbances of the ether, at the end of the range. At the beginning of the range, even before the Hertzian waves, belong the waves used in wireletelegraphy, whose propagation was established last summer from the southwest tip of England to as far as the Gulf of Finland.Although it was principally Hertz's experiments that turned the basic idea of Maxwell's theory into the common property of all scientists, it had been possible to start earlier with some optimism on the task of applying this theory to special problems in optics. We will begin with the simple phenomenon of the refraction of light. It has been known since the time of Huygens that this is connected with the unequal rate of propagation of the beams of light in different substances. How does it come about, however, that the speed of light in solid, liquid, and gaseous substances differs from its speed in the ether of empty space, so that it has its own value for each of these ponderable substances; and how can it be explained that these values, and hence also the refractive index, vary from one colour to another?In dealing with these questions it appeared once more, as in many other cases, that much can be retained even from a theorywhich has had to be abandoned. In the older theory of undulation, which considered the ether as an elastic medium, there was already talk of tiny particles contained in ponderable substances which could be set in motion by light vibrations. The explanation of the chemical and heating action of light was sought in this transmission of motion, and a theory of colour dispersion had been based on the hypothesis that transparent substances, such as glaand water, also contained particles which were set intoco-vibration under the influence of a beam of light. A successor to Maxwell now has merely to translate this conception of co-vibrating particles into the language of the electromagnetic theory of light.Now what must these particles be like if they can be moved by the pulsating electrical forces of a beam of light? The simplest and most obvious answer was: they must be electrically charged. Then they will behave in exactly the same way as the tiny charged dust motes that we spoke of before, except that the particles in glaand water must be represented, not as floating freely, but as being bound to certain equilibrium positions, about which they can vibrate.This idea of small charged particles was otherwise by no means new; as long as 25 years ago the phenomena of electrolysis were being explained by ascribing positive charges to the metallic atoms in a solution of a salt, and negative charges to the other components of the salt molecule. This laid the foundation of modern electrochemistry, which was to develop so rapidly once Prof. Arrhenius had expressed the bold idea of progressive dissociation of the electrolyte with increasing dilution.We will return to the propagation of light in ponderable matter. The covibrating particles must, we concluded, be electrically charged; so we can conveniently call them "electrons", the name that was introduced later by Johnstone Stoney. The exact manner in which this co-vibration takes place, and what reaction it has on the processes in the ether, could be investigated with the aid of the well-known laws of electromagnetism. The result consisted of formulae for the velocity of propagation and the refractive indices, in their dependence on the one hand on the vibration period - i.e. on the colour of the light - and on the other hand on the nature and number of the electrons.You will forgive me if I do not quote the rather complicated equations, and only give some account of their significance. In the first place, as regards the dependence of the refractive index on vibration period - that is, colour dispersion: in the prismatic spectrum and in rainbows we see a demonstration of the fact that the electrons in glaand water possea certain mass; consequently they do not follow the vibrations of light of different colours with the same readiness.Secondly, if attention is focussed on the influence of the greater or smaller number of particles in a certain space an equation can be found which puts us in a position to give the approximate change in the refractive index with increasing or decreasing density of the body - thus, for example, it is possible to calculate the refractive index of water vapour from that of water. This equation agrees fairly well with the results of experiments.When I drew up these formulae I did not know that Lorenz at Copenhagen had already arrived at exactly the same result,even though he started from different viewpoints, independent of the electromagnetic theory of light. The equation has therefore often been referred to as the formula of Lorenz and Lorentz.This formula is accompanied by another which makes it possible to deduce the refractive index of a chemical compound from its composition, admittedly only in rough approximation as was possible earlier with the aid of certain empirical formulae. The fact that such a connection between the refraction of light and the chemical composition does exist at all is of great importance in the electromagnetic theory of light. It shows us that the power of refraction is not one of those properties of matter which are completely transformed by the action of chemical combination. The relative positions of, and type of bond between, the atoms are not of primary importance as concerns the speed of propagation in a compound. Only the number of atoms of carbon, hydrogen, etc. is of importance; each atom plays its part in the refraction of light, unaffected by the behaviour of the others. In the face of these results we find it hard to imagine that the forces which bind an electron to its equilibrium position and on the intensity of which depends the velocity of light are generated by a certain number of neighbouring atoms. We conclude rather that the electron, together with whatever it is bound to, has its place within a single atom; hence, electrons are smaller than atoms.Permit me now to draw your attention to the ether. Since we learnt to consider this as the transmitter not only of optical but also of electromagnetic phenomena, the problem of its nature became more pressing than ever. Must we imagine the ether as an elastic medium of very low density, composed of atoms whichare very small compared with ordinary ones? Is it perhaps an incompressible, frictionlefluid, which moves in accordance with the equations of hydrodynamics, and in which therefore there may be various turbulent motions? Or must we think of it as a kind of jelly, half liquid, half solid?Clearly, we should be nearer the answers to these questions if it were possible to experiment on the ether in the same way as on liquid or gaseous matter. If we could enclose a certain quantity of this medium in a vessel and compreit by the action of a piston, or let it flow into another vessel, we should already have achieved a great deal. That would mean displacing the ether by means of a body set in motion. Unfortunately, all the experiments undertaken on these lines have been unsuccessful; the ether always slips through our fingers. Imagine an ordinary barometer, which we tilt so that the mercury rises to the top, filling the tube completely. The ether which was originally above the mercury must be somewhere; it must have either passed through the glaor been absorbed into the metal, and that without any force that we can measure having acted upon it. Experiments of this type show that bodies of normal dimensions, as far as we can tell, are completely permeable to the ether. Does this apply equally to much larger bodies, or could we hope to displace the ether by means of some sort of very-large, very-fast moving piston? Fortunately, Nature performs this experiment on a large scale. After all, in its annual journey round the sun the earth travels through space at a speed more than a thousand times greater than that of an expretrain. We might expect that in these circumstances there would be an end to the immobility of the ether; the earth would push it away in front of itself, and the ether would flow to the rear of the planet, either along its surface or ata certain distance from it, so as to occupy the space which the earth has just vacated. Astronomical observation of the positions of the heavenly bodies gives a sharp means of determining whether this is in fact the case; movements of the ether would assuredly influence the course of the beams of light in some way. Once again we get a negative answer to our question whether the ether moves. The direction in which we observe a star certainly differs from the true direction as a result of the movement of the earth - this is the so-called aberration of light. However, by far the simplest explanation of this phenomenon is to assume that the whole earth is completely permeable to the ether and can move through it without dragging it at all. This hypothesis was first expressed by Fresnel and can hardly be contested at present. If we wish to give an account of the significance of this result, we have one more thing to consider. Thanks to the investigations of Van der Waals and other physicists, we know fairly accurately how great a part of the space occupied by a body is in fact filled by its molecules. In fairly dense substances this fraction is so large that we have difficulty in imagining the earth to be of such loose molecular structure that the ether can flow almost completely freely through the spaces between the molecules. Rather are we constrained to take the view that each individual molecule is permeable. The simplest thing is to suggest further that the same is true of each atom, and this leads us to the idea that an atom is in the last resort some sort of local modification of the omnipresent ether, a modification which can shift from place to place without the medium itself altering its position. Having reached this point, we can consider the ether as a substance of a completely distinctive nature, completely different from all ponderable matter. Withregard to its inner constitution, in the present state of our knowledge it is very difficult for us to give an adequate picture of it.I hardly need to mention that, quite apart from this question of constitution, it will always be important to come to a closer understanding of the transmission of apparent distant actions through the ether by demonstrating how a liquid, for example, can produce similar effects. Here I am thinking in particular of the experiments of Prof. Bjerknes in Christiania* on transmitted hydrodynamic forces and of his imitation of electrical phenomena with pulsating spheres.I come now to an important question which is very closely connected with the immobility of the ether. You know that in the determination of the velocity of sound in the open air, the effect of the wind makes itself felt. If this is blowing towards the observer, the required quantity will increase with the wind speed, and with the wind in the opposite direction the figure will be reduced by the same amount. If, then, a moving transparent body, such as flowing water, carries along with it in its entirety the ether it contains, then optical phenomena should behave in much the same way as the acoustical phenomena in theseexperiments. Consider for example the case in which water is flowing along a tube and a beam of light is propagated within this water in the direction of flow. If everything that is involved in the light vibrations is subject to the flowing movement, then the propagation of light in the flowing water will in relation to the latter behave in exactly the same way as in still water. The velocity of propagation relative to the wall of the tube can be found by adding the velocity of propagation in the water to the rate of flow of the water, just as, if a ball is rolling along the deck of aship in the direction in which it is travelling, the ball moves relative to an observer on the shore at the sum of two speeds - the speed of the ship and the speed at which the ball is rolling on it. According to this hypothesis the water would drag the light waves at the full rate of its own flow.We come to a quite different conclusion if we assume, as we now must, that the ether contained in the flowing water is itself stationary. As the light is partly propagated through this ether, it is easy to see that the propagation of the light beams, for example to the right, must take place more slowly than it would if the ether itself were moving to the right. The waves are certainly carried along by the water, but only at a certain fraction of its rate of flow. Fresnel has already demonstrated the size of this fraction; it depends on the refractive index of the substance - the value for water, for example, being 0.44. By accepting this figure it is possible to explain various phenomena connected with aberration. Moreover, Fresnel deduced it from a theoretical standpoint which, however ingenious it may be, we can now no longer accept as valid.In 1851 Fizeau settled the question by his famous experiment in which he compared the propagation of light in water flowing in the direction of the beam of light with its propagation in water flowing in the opposite direction. The result of these experiments, afterwards repeated with the same result by Michelson and Morley, was in complete agreement with the values assumed by Fresnel for the drag coefficient.There now arose the question of whether it is possible to deduce this value from the new theory of light. To this end it was necessary first of all to develop a theory of electromagnetic phenomena in moving substances, with the assumption that theether does not partake of their motion. To find a starting-point for such a theory, I once again had recourse to electrons. I was of the opinion that these must be permeable to the ether and that each must be the centre of an electric and also, when in motion, of a magnetic field. For conditions in the ether I introduced the equations which have been generally accepted since the work of Hertz and Heaviside. Finally I added certain assumptions about the force acting on an electron, as follows: this force is always due to the ether in the immediate vicinity of the electron and is therefore affected directly by the state of this ether and indirectly by the charge and velocity of the other electrons which have brought about this state. Furthermore, the force depends on the charge and speed of the particle which is being acted upon; these values determine as it were the sensitivity of the electron to the action due to the ether. In working out these ideas I used methods deriving from Maxwell and partly also relied on the work of Hertz. Thus I arrived at the drag coefficient accepted by Fresnel, and was able to explain in a fairly simple way most of the optical phenomena in moving bodies.At the same time, a start was made on a general theory which ascribed all electromagnetic processes taking place in ponderable substances to electrons. In this theory an electrical charge is conceived as being a surplus of positive or negative electrons, but a current in a metallic wire is considered to be a genuine progression of these particles, to which is ascribed a certain mobility in conductors, whereas in non-conductors they are bound to certain equilibrium positions, about which, as has already been said, they can vibrate. In a certain sense this theory represents a return to the earlier idea that we were dealing with two electrical substances, except that now, in accordance withMaxwell's ideas, we have to do with actions which are transmitted through ether and are propagated from point to point at the velocity of light. Since the nature and manner of this transmission can be followed up in all its details, the demand that Gaumade for a theory of electrodynamics is fulfilled. I cannot spend any more time on these matters, but would like to mention that Wiechert at Gttingen and Larmor at Cambridge have produced very similar results, and that Prof. Poincaré has also contributed much to the development and evaluation of the theory.I must also paover many phenomena investigated in recent years, in which the concept of electrons has proved a useful guide, in order not to stray too far from the theory of the Zeeman effect.When Prof. Zeeman made his discovery, the electron theory was complete in its main features and in a position to interpret the new phenomenon. A man who has peopled the whole world with electrons and made them covibrate with light will not scruple to assume that it is also electrons which vibrate within the particles of an incandescent substance and bring about the emission oflight. An oscillating electron constitutes, as it were, a minute Hertzian vibrator; its effect on the surrounding ether is much the same as the effect we have when we take hold of the end of a stretched cord and set up the familiar motion waves in the rope by moving it to and fro. As for the force which causes a change in the vibrations in a magnetic field, this is basically the force, the manifestations of which were first observed by Oersted, when he discovered the effect of a current on a companeedle.I will leave the explanation of triplets to Prof. Zeeman. I will confine myself to remarking that it is the negative electrons which oscillate, and that from the distance between the。

基于三角函数组合的洛伦兹曲线模型周递芝【摘要】构建一个基于三角函数与幂函数乘积的凹凸组合的洛伦兹曲线模型,并利用1977年美国收入分配分组数据检验该模型的合理性.%A Lorentz curve model based on the product of trigonometric function and power function is constructed,and the rationality of the model is tested by the income distribution grouping data in 1977.【期刊名称】《经济研究导刊》【年(卷),期】2018(000)005【总页数】2页(P13-14)【关键词】洛伦兹曲线模型;凹凸组合;三角函数;幂函数【作者】周递芝【作者单位】贵州民族大学教务处,贵阳550025【正文语种】中文【中图分类】F014.4;F016引言作为刻画社会收入分配的有效工具，洛伦兹曲线模型已得到广泛而深入的研究。

一般情况下，可通过两种途径获取收入分配数据的洛伦兹曲线，一种是通过已知数据拟合收入分配的概率密度函数，再导出洛伦兹曲线；另一种是由收入分配数据直接构造洛伦兹曲线。

由于收入分配的统计分布不易确定，导致很难拟合出合适的概率密度函数。

因此，数理经济理论界学者更倾向使用第二种途径，即直接构造洛伦兹曲线。

设收入分配的概率密度函数为（fx），对应的概率分布函数为F（x），则p=F（x）表示收入低于或等于x的人口比例。

记收入低于或等x于的人口群体拥有收入占总收入的比例为L（p），则；记F（x）的反函数为F-（1p），μ为平均收入，则也被称为收入分配的洛伦兹曲线。

实际应用中，可通过入户调查获得家庭收入与消费等数据其中pi与Li分别为低收入群体的累计比例和该群体的总收入比例。

利用最小二乘拟合的方法，先确定L（p，τ）参数向量τ的估计值，再用作为洛伦兹曲线对收入分配进行近似分析。

MgF 2antireflective coatings by sol–gel processing:film preparation and thermal densificationJohannes Noack,a Kerstin Scheurell,a Erhard Kemnitz,*a Pl a cido Garcia-Juan,b Helge Rau,b Marc Lacroix,b Johannes Eicher,b Birgit Lintner,c Thomas Sontheimer,d Thomas Hofmann,d Jan Hegmann,e Rainer Jahn e and Peer L €o bmann *eReceived 24th May 2012,Accepted 17th July 2012DOI:10.1039/c2jm33324dMagnesium fluoride sols for the wet chemical processing of porous MgF 2antireflective coatings were prepared by the reaction of MgCl 2with HF.The formation and crystallisation of MgF 2nanoparticles were followed by 19F NMR spectroscopy,X-ray diffraction (XRD)and dynamic light scattering (DLS)in the liquid phase.The crystallization of the resulting films was monitored by XRD experiments.At temperatures exceeding 550 C the film material and glass substrates undergo a chemical reaction,MgO is formed and SiF 4evaporates as a volatile product.Microstructure and optical properties werecharacterized as a function of the annealing temperature.The mechanical stability of MgF 2films was evaluated by the Crockmeter test using both felt and steel wool.It is shown that porous MgF 2films prepared by this synthesis have a vast potential for the large-area processing of antireflective coatings.1.IntroductionPorous films on optical surfaces such as windows or solar panels can facilitate antireflective properties and thus an enhanced transmission of radiation.Since the best performance can be achieved when the film thickness equals a quarter of the incident light,such systems are denoted as l /4layers.If silica (n ¼1.5)is applied to common glasses (n $1.46–1.65)a porosity of 50%is required to achieve the optimum refractive index of 1.22for the film.1The mechanical stability of these layers may allow applications on e.g.photovoltaic cells where no abrasive stress is expected due to mechanical cleaning of the surfaces and only weathering resistance is required.For archi-tectural glazing a higher mechanical stability is essential which may be achieved by the application of films with a lower porosity.To maintain the desired optical performance,materials with a lower refractive index than silica are required for the film back-bone.MgF 2(n ¼1.38)is a promising candidate for this purpose.Sputtering may be used for film deposition,2,3but it is difficult to adjust the correct metal/fluorine ratio.A better control of film composition is achieved by evaporation techniques,4but the point-shape of vaporization sites prohibits an effective large-areadeposition.In any case neither of the methods allows for an accurate adjustment of film porosity.Porous metal fluoride can be produced by applying a sol–gel solution of metal trifluoroacetates as a thin film followed by calcination at temperatures between 200 C and 300 C (TFA-method).5The thermal decomposition of a mixture of magnesium acetate and trifluoroacetic acid yields magnesium oxo-fluoride films with extraordinarily low refractive indices between 1.08and 1.2.6,7Although optical data seem to be promising,large scale coating of glass is limited due to corrosive and toxic reaction products such as CF 3COF,COF 2and HF.The fluorolytic sol–gel synthesis route,which circumvents this drawback,has been developed for the preparation of transparent metal fluoride sols.8,9The reaction of magnesium precursors,usually Mg(OMe)2dissolved in methanol,leads to the formation of nanocrystalline MgF 2particles with a crystallite size below 5nm.10Thin film coatings,prepared from these sols,exhibit a low refractive index of $1.32and high film homogeneity.11Unfor-tunately,the synthesis of these sols and large-scale coating under industrial conditions is difficult because of the release of hydrogen gas during the preparation of Mg(OMe)2and the use of methanol as solvent.Other alkoxides or non-toxic solvents do not yield transparent sols.In recent years,the crystallisation of magnesium fluoride of different morphologies has been reported for the reaction of MgCl 2and NaF or NH 4F.12,13Depending on the F to Mg ratio,either platelets,cubic or spherical particles of crystalline magne-sium fluoride were formed.All of these experiments were con-ducted with a large excess of MgCl 2and contained large amounts of NH 4+or Na +.None of these factors allow the coating of thin films with optical quality.In this context,the stoichiometricaDepartment of Chemistry,Humboldt-Universit €at zu Berlin,Brook-Taylor-Str.2,12489Berlin,Germany.E-mail:erhard.kemnitz@chemie.hu-berlin.de;Tel:+4903020937555bSolvay Fluor,Hans-B €ockler-Allee 20,30173Hannover,Germany cPrinz Optics GmbH,Simmerner Strasse 7,55442Stromberg,Germany dCentrosolar Glas GmbH &Co.KG,Siemensstr.3,90766F €urth,Germany eFraunhofer-Institut f €ur Silicatforschung ISC,Neunerplatz 2,97082W €urzburg,Germany.E-mail:peer.loebmann@isc.fraunhofer.de;Tel:+4909314100-404Dynamic Article Links CJournal ofMaterials ChemistryCite this:J.Mater.Chem.,2012,22,/materialsPAPERD o w n l o a d e d b y B e i j i n g U n i v e r s i t y o n 31 O c t o b e r 2012P u b l i s h e d o n 18 J u l y 2012 o n h t t p ://p u b s .r s c .o r g | d o i :10.1039/C 2J M 33324DView Online / Journal Homepage / Table of Contents for this issuereaction of magnesium chloride with HF to form nanoscopic magnesium fluoride according to eqn (1),and its utilisation for thin flim coatings is investigated in the present publication.MgCl 2+2HF /nano-MgF 2+2HCl(1)Despite these promising reports on the synthesis of MgF 2precursors for the sol–gel processing of MgF 2thin films there have not been,to our knowledge,any attempts of scaling up the respective processes to an industrial level so far.This is not surprising since many side-conditions regarding upscaling of the synthesis,shelf-life of the coating solution,large-area processing and environmental stability of the final product have to be considered.In the case of SiO 2-based AR coatings it took over 50years from the initial reports on the concept 14to its commercialization.15,16In this paper we report as members of a joint academic and industrial consortium about the cost-efficient synthesis of magnesium fluoride sols from a commercially available magne-sium chloride precursor in a stoichiometric reaction with 2equivalents of HF and the related preparation of porous AR coatings with special focus on practical issues.The vast potential of the overall process for commercialization is highlighted.2.Experimental procedurePrecursor synthesisAll chemicals for the synthesis of magnesium fluoride were used as received without drying or further processing.For better handling and dosing the HF solution used for these experiments was prepared by condensation of HF (Solvay Fluor GmbH)in ethanol with a 19.2M concentration.A series of understoichiometric sols (Mg :F <1:2)were prepared by dissolution of 2.4g magnesium chloride (anhydrous,Aldrich,25mmol)in 50ml ethanol (99.8%,Roth,0.5M solu-tion)and dropwise addition of the required amount of HF solution under vigorous stirring at ambient conditions.The stoichiometric sols (Mg :F ¼1:2)for coating of glass slides were prepared by the reaction of 35.7g of MgCl 2(0.375mol)with 39.0ml (0.75mmol;2eq.)of HF in 700ml ethanol.The sols for the coating experiments were prepared by the stoichiometric reaction of 35.7g of MgCl 2(0.375mol)with 39.0ml (0.75mmol;2eq.)of HF in 700ml ethanol.In order to investigate the influence of prolonged heat treatment on the sol properties,such samples were refluxed for 24h under inert gas atmosphere.In order to investigate the influence of prolonged heat treatment on the sol properties,samples were refluxed for 24h in ambient atmosphere.Film deposition and thermal treatmentThe MgF 2thin films were prepared by dip coating on borosilicate glass (Schott Borofloat Ò)at the size of 3.3Â150Â100mm.Additionally soda-lime glass (Pilkington Optiwhite Float),display glass (Eagle XG,Corning)and fused silica (Quarzglas-technik GmbH &Co.KG,Germany)were used.Before the coating experiment the substrates were cleaned in a laboratory dishwasher by an alkaline cleaning procedure with a final neutralization step.After each coating,the samples werepre-dried for 10min at 80 C in a vented furnace (Model D-6450,Heraeus Instruments,Hanau,Germany),followed by sintering in a pre-heated vented air oven (Model Thermicon P,Heraeus Instruments,Hanau,Germany)for 10min at 300–650 C and slow cooling in the furnace after switching off the power supply.Material characterizationDynamic light scattering (DLS)measurements were performed using a Zetasizer Nano ZS (Malvern Instruments,Worcester-shire,UK)using disposable PMMA cuvettes.Hydrodynamic diameters were calculated from the correlation functions by the Malvern Nanosizer Software.The viscosity was determined simultaneously to DLS measurements with a microviscometer from Anton Paar (AMVn,Graz,Austria)at 25 C.The XRD investigation of the sols was performed at the m Spot beamline at BESSY II (Helmholtz Zentrum,Berlin,Germany)employing a wavelength of 1.00257 A.17The data obtained were corrected for background scattering of the pure solvent.For direct measurement of the sol particles without drying and to avoid any surface effects,the XRD experiments were carried out using an ultrasonic trap (Tec5,Oberusel,Germany)as a sample holder.The 19F NMR spectra of the sols with varying fluorine content were carried out using a Bruker AVANCE II 300(Larmor frequency of 282.4MHz).The 19F isotropic chemical shifts are given with respect to the CFCl 3standard.Fractured surfaces of the films were examined by scanning electron microscopy (SEM),using a Zeiss Ultra 25(Carl Zeiss SMT,Oberkochen,Germany),Pt was applied prior to the investigation by sputtering.Grazing incidence X-ray diffractometry (GIXRD)was per-formed with a Siemens D-5005diffractometer (Bruker AXS GmbH,Karlsruhe,Germany)at the angle of incidence of 1 and at an angular range of 20–70 .Reflection curves were measured with an UV-Vis-spectrometer (Shimadzu UV-3100,Kyoto,Japan),in the range of 300–1400nm.The refractive index n (550nm)and thickness t were determined by spectroscopic ellipsometry.The ellipsometric parameters tan psi and cos delta were measured with a GES-5E instrument (Sopra,Paris,France)and evaluated with the software ‘‘WinElli II’’.The measured functions were fitted by the Levenberg–Marquard-algorithm in the range of 1.38–4.5eV on the suppo-sition of Sellmeier’s law of dispersion.Open porosity was determined with atmospheric ellipsometric porosimetry (EPA),which uses the change of the refractive index (n (633nm))during water-vapor adsorption and desorption.Before measurement,the samples were cleaned by rinsing with de-ionized water,immersing 5min in ethanol and drying for 5min at 180 C in a vented furnace.To calculate open porosity the ellipsometric results of completely empty pores (n 1)and fully water-filled pores (n 2)can be evaluated with the Lorentz–Lorenz equation,without requirement of any information on the back-bone material:P L ¼n 22À1n 22þ2Àn 12À1n 12þ2n water 2À1n water 2þ2(2)D o w n l o a d e d b y B e i j i n g U n i v e r s i t y o n 31 O c t o b e r 2012P u b l i s h e d o n 18 J u l y 2012 o n h t t p ://p u b s .r s c .o r g | d o i :10.1039/C 2J M 33324DThe mechanical stability of the MgF2thinfilms was tested by a custom made Crockmeter-test using felt and steel wool of the fineness0000.The stamp(contact area4.5cm2)was pressed on the sample with a force of4N.The kinematic viscosity of the sols was determined with an Ubbelohde viscometer(Schott-Ger€a te Hofheim,Germany)with an instrument constant of K¼0.02908.3.Results and discussionThe reaction of MgCl2with anhydrous hydrogenfluoride,both dissolved in ethanol,under the conditions used results directly inthe formation of nanosized particles.This reaction was outlined in eqn(1),clear sols are formed that are suitable for coating operations.The formation of magnesiumfluoride nanoparticles from MgCl2and HF as afluorination agent was investigated by 19F NMR spectroscopy.Fig.1shows the NMR spectra of the sols with different ratios of HF to magnesium ranging from0.5 to2.All spectra exhibit a very broad signal aroundÀ198ppm, which is attributed to thefluorine in the rutile structure of magnesiumfluoride.18In previous investigations the reaction of MgOMe2with HF proceeded via the formation of intermediate dicubane magnesium methoxidefluoride species19that were identified by NMR and crystal structure analysis.10In the present study even at very lowfluorine to magnesium ratios MgCl2 directly reacts to form MgF2.There is no indication of any intermediate phases which might act as seeds for crystallization in this reaction.As small sol particles in the lower nm-range possess a high mobility in the solvent,the spectra are comparable to those of MAS-NMR of the dried xerogels.20Agglomerates of the particles would inevitably lead to a broad background signal. Hence,liquid NMR spectra are also a good measure for very small sol particles.Even though MgCl2readily reacts with HF to form MgF2,in samples with understoichiometric amounts of fluorine,a second broad signal betweenÀ172andÀ180ppm (line width200–500Hz)is present in all spectra,which corre-sponds to unreacted HF adsorbed to the particles’surface. During ageing of the sol,this signal vanishes and no free HF can be determined.No otherfluorine-containing soluble species are identified by NMR.Mechanistically,we assume a protonation of the molecularly dissolved MgCl2in afirst step followed by an attack by the HF-molecule resulting in the formation of nanosized MgF2.These particles obviously are stabilised–as in the classical oxide-based sol gel approach–by the donating solvent molecules ethanol. All the as-prepared sols are highly transparent and possess a low viscosity,indicating small sol particles.The mean hydrody-namic diameter of the as-synthesised MgF2particles was deter-mined to be8nm(Fig.2a)by dynamic light scattering(DLS).In order to investigate the changes in particle and crystallite size induced by refluxation of the sol,the hydrodynamic diameter of as-prepared sol particles is compared to the values obtained for sols refluxed for24h(Fig.2b).It is obvious,that the maximum of the size distribution increases from8nm to60–70nm by thermal treatment of the sol.It has to be noted,though,that the hydrodynamic diameter as measured by dynamic light scattering(DLS)not necessarily equals the crystallite sizes but may be correlated to aggregates consisting of smaller primary particles.To prove the crystallinity of the magnesiumfluoride sol particles,XRD patterns(Fig.3)of levitated sol droplets have been measured using an ultrasonic trap.The as-synthesised sol(a)shows broad reflections which correspond to magnesiumfluoride(c;PDF:41-1443)with a crystallite size well below5nm.Since these values are farbelow Fig.119F NMR spectra of sols with0.5(a),1.0(b),1.5(c),1.75(d)and2.0eq.(e)HF(normalized to equalfluorinecontent).Fig.2Hydrodynamic diameter of particles in as-prepared MgF2sols(a)and in samples after boiling for24hours(b)as measured by dynamic lightscattering.Fig.3X-ray diffraction pattern of levitated droplets of as-preparedMgF2sols(a)and of samples after boiling for24hours(b).The PDFpattern(41–1443)of MgF2(sellaite)is given as a reference.DownloadedbyBeijingUniversityon31October212Publishedon18July212onhttp://pubs.rsc.org|doi:1.139/C2JM33324Dthe hydrodynamic diameters in Fig.2,the species in solution governing the viscosity obviously consist of aggregated crystal-lites.MgF 2sol particles prepared from Mg(OMe)2show signif-icant changes in particle size and crystallinity during ageing.10In the case of MgCl 2as the precursor of the sol–gel reaction no such changes are observed by DLS and X-ray diffraction.Yet,when the sol is refluxed for 24h the reflections of MgF 2sharpen due to crystal growth via Ostwald-ripening or by coalescence.At the same time,the 19F NMR signal of MgF 2broadens as a conse-quence of agglomeration and the peak of adsorbed HF vanishes (data not shown).The viscosity of MgF 2sols that had been stored at ambient conditions was measured as a function of the ageing time.Fig.4reveals that only a minute increase in viscosity of 0.01mm 2s À1is observed within 4months even though the coating solution had been repeatedly used for film deposition experiments.This minor raise can mainly be attributed to solvent evaporation during pro-cessing and thus does not represent any difficulty relating to the manufacturing process.Unlike many alkoxide-based solutions for the preparation of oxide films such as SiO 2,TiO 2or ZrO 2the MgF 2system is obviously stable against traces of air moisture introduced by the ambient atmosphere.Even though the uptake of moisture by the hygroscopic HCl during processing cannot fully be ruled out,only minor viscosity increase is observed.The viscosity of the solution significantly increases upon refluxation under ambient atmosphere.In Fig.4the viscosity of a solution boiled for 24hours under reflux conditions is given.As these samples are then stored at ambient temperature their viscosity gradually decreases again.This observation suggests that even though the system may be altered by the introduction of thermal energy it subsequently re-approaches a thermody-namically more stable state again.It is noteworthy that these changes occur over a period of days and weeks.Films were deposited on glasses by dip-coating with subse-quent thermal treatment.In Fig.5a XRD patterns of samples prepared on borosilicate glass are given.In the temperature range between 300 C and 550 C all reflections can be assigned to MgF 2(Sellaite,PDF 00-041-1443).XRD measurements without external standards do not provide reliable information about the level of crystallization,but the width of the signals slightly decreases.The respective increase of crystallite sizes as derived from the Scherrer equation is shown in Fig.5b.After annealing at 550 C traces of a second phase become apparent.The intensity for MgO (Periclase,PDF 00-045-0946)increases at 600 C at the expense of the MgF 2signals until at 650 C it becomes the dominant phase.This observation is equally made for soda lime glass,borosil-icate glass,alkali-free display glass (Eagle 2000,Corning)and fused silica (data not shown).As the onset of appearance of MgO is slightly shifted to higher temperatures in this order of the substrates it clearly correlates to the respective glass transition temperatures T g .Obviously the film material undergoes a reac-tion with the respective glass.Since no other crystalline inter-mediates such as hexafluorosilicates are detected the reaction may proceed as indicated in eqn (3):2MgF 2+SiO 2/2MgO +SiF 4(g)(3)The presumed volatile product SiF 4is removed from the equilibrium.As the substrates do provide a surplus of Si this reaction goes to completion until the MgF 2materialisFig.4Viscosity of MgF 2coating solutions as a function of the ageingtime.Fig.5X-ray diffraction pattern of MgF 2films prepared on borosilicate glass and annealed at the temperatures indicated (a)the MgF 2crystallite size as calculated by the Scherrer equation (b).The single data marker denotes the value of a film prepared from a refluxed coating solution after heat treatment at 500 C.D o w n l o a d e d b y B e i j i n g U n i v e r s i t y o n 31 O c t o b e r 2012P u b l i s h e d o n 18 J u l y 2012 o n h t t p ://p u b s .r s c .o r g | d o i :10.1039/C 2J M 33324Dconsumed.It has to be noted,though,that no volatile products are observed when nano-MgF 2is mixed with SiO 2powder (Aerosil)and heated up to 900 C.This observation suggests that the close contact between MgF 2and the respective glass substrates as established by the wet chemical coating is a requirement for the proposed reaction.The increase in crystallite size induced by boiling the sol (Fig.3)is also reflected in the grain size of the corresponding films.In Fig.5b a single data point for such a sample treated at 500 C is given.Since the high viscosity requires low withdrawal rates in dip-coating experiments and the visual homogeneity of the resulting films falls behind the samples from as-prepared solutions,no complete series were prepared from the boiled precursor.In order to avoid glass deformation during industrial film processing,the maximum treatment temperature has to be limited to 500 C in any case.Therefore the reaction of MgF 2with the glass substrates at higher temperatures is of no practical importance and does not limit the applicability of the method.Therefore all subsequent investigations are mainly focused on annealing temperatures below 500 C.In Fig.6a cross-sectional SEM view of a MgF 2film annealed at 500 C is given.The film has a thickness of $118nm and consists of globular grains with diameters between 10nm and 20nm.This range corresponds well with the crystallite sizes as determined by XRD (Fig.5b),the grains visualized by SEM obviously represent crystalline subunits.As SEM investigations hardly provide quantitative results regarding film microstructures,ellipsometric porosimetry (EP)measurements were performed on films that had been annealed at different temperatures.In Fig.7a adsorption–desorption-isotherms of several films are compiled and Fig.7b shows the pore radius distributions as calculated from the respective desorption branch.It can be seen that in the temperature range from 300 C to 500 C the overall film porosity of approximately 35%remains constant whereas the average pore radius increases.This obser-vation can be attributed to coarsening of the porous network during the growth of the crystalline grains.In addition to film porosity and pore radius distribution EP allows determining the film thickness and differentiating between the indices of refraction of the overall film (n film )and the solid backbone material (n backbone );21these data are given in Fig.8.In the temperature range of 300 C to 600 C the refractive index of the film backbone material remains constant.The value of approximately 1.41is higher than the theoretical value of MgF 2(1.38)which may be attributed to e.g.oxygen impurities within the solid.Applying the effective medium approximation a mixture of 91vol%MgF 2(n ¼1.38)and 9vol%MgO (n ¼1.74)would account for the deviation observed.As the oxide contaminations are presumably evenly distributed in the material and no separation of MgO crystallites is expected,this level of oxide contamination is below the detection limit of XRD experiments.Together with the XRD pattern shown in Fig.5it can be concluded that the film crystallinity is not significantly increased above 300 C,changes within the film are mainly due to the growth of the MgF 2crystallites.As already suggested by Fig.7,this process does not go along with a significant loss ofporosityFig.6Cross-sectional SEM image of MgF 2thin film annealed at 500C.Fig.7Adsorption (filled symbols)and desorption (open symbols)isotherms of MgF 2thin films treated at 300 C,400 C and 500 C measured by ellipsometric porosimetry (a).Derivative pore radius distributions as calculated from the respective desorption branch (b).D o w n l o a d e d b y B e i j i n g U n i v e r s i t y o n 31 O c t o b e r 2012P u b l i s h e d o n 18 J u l y 2012 o n h t t p ://p u b s .r s c .o r g | d o i :10.1039/C 2J M 33324Dand film shrinkage.It has to be noted that the determination of n backbone becomes unreliable at treatment temperatures above 600 C due to the inhomogeneous film microstructure as MgF 2is successively converted to MgO.Regarding the thermal processing,the film material can be expected to be largely fault-tolerant.This assumption is confirmed by the UV-Vis spectra (Fig.9)of films that have been treated at different temperatures:With a minimum reflectance of 0.2%at 600nm the films treated at 500 C show excellent antireflective properties.The minimum of the reflectance curves as it is governed by thickness and n film can easily be modified by the withdrawal rate during the dip-coating procedure.As already outlined the stability of porous antireflective coat-ings under environmental conditions is of crucial importance for any practical application.MgF 2films annealed at 500 C were subjected to the Crockmeter test (Fig.10).No scratches were observed even after 500abrasion cycles with standard felt testing probes.Therefore in addition to felt,steel wool was also applied.Whereas even under these harsh conditions the MgF 2films prepared on borosilicate substrates remain visually unaltered,scratches can be found in films deposited on soda-lime glass.The specific stability of the MgF 2coatings on the different surfaces will be a subject of future investigations.Taken as a whole,however,the porous AR coatings exhibit an extraordi-nary stability under mechanical stress.4.ConclusionsMgF 2precursor synthesis is based on the inexpensive raw material MgCl 2,the resulting coating solution is non-toxic and can be handled under environmental conditions over months without significant changes in its properties.Porous antireflective films can be applied on large areas by dip-coating and subsequent thermal treatment.Even though crystallite growth is induced as the annealing temperature is increased,no major changes in optical properties are observed in the temperature range between 300 C and 500 C.The resulting films exhibit excellent antire-flective properties along with a high mechanical stability.These results offer good perspectives for a commercialization of the system in the near future.AcknowledgementsThis project was funded by the German Federal Ministry of Economics and Technology (grant 0329800).The authors wish to thank Tatjana Shinkar and Angelika Schmitt forperformingFig.10Photographs of MgF 2thin films on (a)soda-lime and (b)borosilicate glass substrates (10Â15cm 2)that had undergone the Crockmeter test with felt and steel wool.The numbers at the right side of the images correspond to the respective number of abrasion cycles.All samples have been treated at 500 C prior toanalysis.Fig.8Refractive indices of the solid film backbone (n backbone )and the overall film (n film )(left axis)and film thickness (right axis)as determined by ellipsometric porosimetry(EP).Fig.9Reflectance of MgF 2films annealed at 300 C,400 C and 500 C.D o w n l o a d e d b y B e i j i n g U n i v e r s i t y o n 31 O c t o b e r 2012P u b l i s h e d o n 18 J u l y 2012 o n h t t p ://p u b s .r s c .o r g | d o i :10.1039/C 2J M 33324Dthe coating experiments and ellipsometric porosimetry measurements.Franziska Emmerling’s contribution of XRD investigation of the sols is gratefully acknowledged.References1W.Glaubitt and P.L €obmann,J.Eur.Ceram.Soc.,2012,32,2995.2L.Martinu and H.Biederman,Vacuum ,1985,35,531.3T.H.Allen,J.P.Lehan and L.C.McIntyre,Jr,Proc.SPIE–Int.Soc.Opt.Eng.,1990,1323,277.4D.Jacoba,F.Peiro,E.Quesnel and D.Ristau,Thin Solid Films ,2000,360,133.5S.Fujihara,M.Tada and T.Kimura,Thin Solid Films ,1997,304,252.6P.Joosten,P.Heller,M.Nabben,H.van Hal,T.Popma and J.Haisma,Appl.Opt.,1985,24,2674.7J.Bass,C.Boissiere,L.Nicole,D.Grosso and C.Sanchez,Chem.Mater.,2008,20,5550.8S.R €udiger and E.Kemnitz,Dalton Trans.,2008,1117.9E.P.1666411,EP 1732853.10J.Noack,F.Emmerling,H.Kirmse and E.Kemnitz,J.Mater.Chem.,2011,21,15015.11H.Kr €uger,E.Kemnitz,A.Hertwig and U.Beck,Phys.Status Solidi A ,2008,205,821.12I.Sevonkaev and E.Matijevic,Langmuir ,2009,25,10534.13A.Nandiyanto,F.Iskandar,T.Ogi and K.Okuyama,Langmuir ,2010,26,12260.14H.Moulton,US Pat.,2474061,1949.15W.Glaubitt,M.Kursawe,A.Gombert and T.Hofmann,PCT/EP 2002/010*******.16A.Gombert and W.Glaubitt,Sol.Energy ,2000,68,357.17O.Paris,C.Li,S.Siegel,G.Weseloh,F.Emmerling,H.Riesemeier,A.Erko and P.Fratzl,J.Appl.Crystallogr.,2007,40,466.18G.Scholz,C.Stosiek,J.Noack and E.Kemnitz,J.Fluorine Chem.,2011,132,1079.19S.Wuttke, A.Lehmann,G.Scholz,M.Feist, A.Dimitrov,S.Troyanov and E.Kemnitz,Dalton Trans.,2009,4729.20M.Karg,G.Scholz,R.K €onig and E.Kemnitz,Dalton Trans.,2012,41,2360.21A.Bittner,A.Schmitt,R.Jahn and P.L €obmann,Thin Solid Films ,2012,520,1880.D o w n l o a d e d b y B e i j i n g U n i v e r s i t y o n 31 O c t o b e r 2012P u b l i s h e d o n 18 J u l y 2012 o n h t t p ://p u b s .r s c .o r g | d o i :10.1039/C 2J M 33324D。

辛督 强 。 ，刘 薇
（ ．西京学 院 基础部 ，陕西 西安 １ ７ ０ ２ ；２ １１ ３ ．北京 师范大学 核科学与技术学 院 ，北京 １０７ ） ０ ８ ５
摘要 ： 出了一种计算不 同温度 、 提 压强下溶 液密度的新方 法 ： 先用 液体状 态方程 外推 法计 算各种 温度 、 压强 下溶液 各 纯组分的密度 ， 然后再利用 Ｌ ｒｎｚ ｏｅｚ ｏｅｔ Ｌ ｒｎ 公式对各 纯组 分密度进行 合成 ， — 从而 获得相 同条件 下溶液 的密度 。以 多种二元溶液 为例 进行 了计 算 ， 结果 表明 ： 在多种摩 尔分 数 、 温度和 压强下 ， 计算 值与实 验测 量值 吻合得非 常好 ， 平 均绝对偏差仅 为 ０ ０ ５３ｇ ｃ 。最 大绝 对偏 差也不超过 ０ ０ １８ｇ ｃ ．０ ／ ｍ ， ． １ ／ ｍ 。表 明该方 法 的计 算值 准确 、 可靠 ， 为科 这 研和生产 中 ， 特别是高 温 、 高压下 ， 强腐蚀性 、 毒性 溶液密度 的获得提供 了方便 ， 大大 降低 了实 验费用 和测试 的 有 并
Ａｐ ｌｃ ｔｏ ｆｅ ｔ ａ ｏ ａ ｉ ｎ ａ ｄ Ｌｏ ｅ ｔ — ｒ ｎ ｐ ｉ ａ ｉ ｎ ｏ ｘ ｒ ｐ ｌ ｔｏ ｎ ｒ ｎ ｚ Ｌｏ ｅ ｚ ｆ ｒ ｕ ａ ｔ ｏ ｕ ｉ ｎ ｄ ｎ ｉｙ ｃ ｌ ｕ ａ ｉ ｎ ｏ ｍ ｌ ｏ ｓ ｌ ｔｏ ｅ ｓｔ ａ ｃ ｌ ｔｏ
劳动强度 。
关键 词 ： 外推法 ；ｏ ｎ ．ｏｅｚ Ｌ ｒ ｔ Ｌ ｒｎ 公式 ； ｅｚ 密度计算 ； 溶液
中 图 分 类 号 ： ４ ．２ Ｏ６ ２４ 文 献 标 识 码 ： Ａ 文章 编 号 ：０ ５９ ５ （ ０２ ０ － ４ －３ １０ － ４ ２ １ ） ６０ ００ ９ ０

lorentz-lorenz公式的简单推导Lorentz-Lorenz公式是物理学中一个特殊的现象，它是由Hendrik Lorentz和ε-Lorenz发现的，用来描述分子和原子的吸收现象。

Lorentz-Lorenz公式可以精确地计算几何体内某种物质的折射率，其形式如下：n²=ε-2ρ/ε-ρ，其中，n是空气内折射率，ε是介质内折射率，ρ是上下文内介电强度。

该公式有三种优点：它可以精确估算折射率，而且公式非常简单；它可以提供很多相关信息，可以准确描述介质内微小变化；它可以用于表示空气中不同波长和折射率的光学分布。

简单来说，Lorentz-Lorenz公式被用来确定某种物质的折射率，以及其在介质内的变化过程。

它的基本思想是由Hendrik Lorentz等人发现的，利用这种特殊公式，可以精确计算几何体内某种物质的折射率，它关系到介质内光线的向下传播，以及某种物质吸收情况，用它可以准确预测几何体内的光线透射以及反射现象。

Lorentz-Lorenz公式的应用非常广泛，几乎涉及到物理学的所有领域，最为运用的范围是传播技术方面，比如电路设计以及超导技术。

用它可以预测几何体内的折射率，推测出不同波长的光线在介质内的分布和变化，因此可以有效地满足电路设计的需求，用它制作的线路投射出的信号稳定有效，还可以减少电缆买卖时的维修成本。

另外，Lorentz-Lorenz公式还被运用在原子能课堂，有助于学生理解原子吸收现象，以及分子在介质内的变化，同时也可以在气候学中提供重要信息。

总的来说，Lorentz-Lorenz公式是一种特殊的公式，它可以准确预测空气内物质折射率、介质有效变化过程，以及在设计电路以及超导技术中的应用，它为各种物理学领域的进步提供了重要的贡献。