SU(3) Einstein-Yang-Mills Sphalerons and Black Holes
Running Coupling in SU(3) Yang-Mills Theory
cal running coupling constant at small separation r. This requires control of the force from short to long distance in one simulation with a r L holding for the whole range of physical r involved. With L=a always limited to feasible lattice sizes like 32 or 48, compromises on the above conditions have to be accepted, and it is hardly possible to vary all scale ratios signi cantly to check for the stability of the results. State of the art calculations along these lines are reported in 1,2]. It has to be noted that the highest physical energies r 1 that can be reached here are below about 2 GeV, if one only stays a factor 2 : : : 3 away from the cuto energy. Cuto e ects are corrected semi-empirically using the lattice Coulomb propagator. While these are di cult and careful simulations, we nd it somewhat hard to assess the systematic errors in a completely convincing fashion. An alternative attempt to derive the coupling in QCD has been pioneered by the Fermilab group 3]. Here, in a quenched simulation, the spin averaged 1P-1S charmonium splitting is determined on a physically large lattice. Although this is a nice experimentally known scale with little sensitivity to the quark masses, also other masses could in principle be used here to set the scale. The point relevant in the present context is, that they extract from such a simulation the bare lattice coupling g0 together with the corresponding lattice spacing a in GeV. A perturbative method is then used to relate g0 to a physical coupling at a scale of the order of the cuto . The scale problem is clearly alleviated in comparison to the quark force method, as e ectively the cuto a 1 is identi ed with the high energy physical scale.
杨-米斯尔方程
杨-米斯尔方程
杨一米尔斯方程(Yang-Millsequation)是一个重要的微分方程，指杨一米尔斯作用量所确定的欧拉一拉格朗日方程。
杨一米尔斯方程也叫做杨—米尔斯理论。
杨氏理论是基于SU(N)组的一种量规理论，或者更普遍地说，是一个紧凑、半简单的李群。
杨振宁米尔斯理论旨在描述基本粒子的行为使用这些非阿贝尔李群和统一的核心的电磁和弱力(即U(1)×SU(2))以及量子色动力学理论的强力(基于SU(3))。
从而形成了对粒子物理标准模型理解的基础。
在一份私人信件中，沃尔夫冈，泡利在1953年提出了爱因斯坦的广义相对论的六维理论，将Kaluza、Klein、Fock等五维理论扩展到高维的内部空间。
然而，没有证据表明泡利发展了一个量子场的拉格朗日或它的量子化。
因为泡利发现他的理论“导致了一些非物质的阴影粒子”，他没有正式公布结果。
虽然保利没有发表他的六维理论，但他在苏黎世发表了两份关于它的演讲。
最近的研究表明，扩展的kaluza-klein理论一般不等同于杨斯-米尔斯理论，因为前者包含了额外的术语。
冀教版九年级Lesson 8 课件
The lifetime of Einstein
Lead-in
Albert Einstein
The greatest scientist of the 20th century and one of the greatest of all time.Among the several important discoveries Einstein made in his life, the greatest is the creation of his famous Theory of Relativity.
世纪
相对论
一生；终生
宇宙；万象
解决
New words
①可数名词 “聪明人”
e.g. She was one of the greatest minds of her generation.
②名词 “头脑；主意”
e.g. Are you quite clear in your own mind what you should do?
Unit 2 Great PeopleA Universe of Thought
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Particle-like solutions to the Yang--Mills-dilaton system in d=4+1 dimensions
a r X i v :h e p -t h /0611270v 3 20 F eb 2007Particle-like solutions to the Yang–Mills-dilatonsystem in d =4+1dimensionsEugen Radu †,Ya.Shnir ‡and D.H.Tchrakian †⋆†Department of Mathematical Physics,National University of Ireland Maynooth,Maynooth,Ireland ‡Institut f¨u r Physik,Universit¨a t Oldenburg,Postfach 2503D-26111Oldenburg,Germany ⋆School of Theoretical Physics –DIAS,10Burlington Road,Dublin 4,Ireland February 7,2008Abstract We construct static solutions to a SU (2)Yang–Mills (YM)dilaton model in 4+1dimensions subject to bi-azimuthal symmetry.The YM sector of the model consists of the usual YM term and the next higher order term of the YM hierarchy,which is required by the scaling condition for the existence of finite energy solutions.The basic features of two different types of configurations are studied,corresponding to (multi)solitons with topological charge n 2,and soliton–antisoliton pairs with zero topological charge.1IntroductionMulti-instantons and composite instanton-antiinstanton bound states subject to bi-azimuthal symmetry were reported in a recent paper [1].These were constructed numerically,for the usual (p =1)SU (2)Yang-Mills (YM)system in 4Euclidean dimensions,the spherically symmetric special case being the usual BPST [2]instanton.In a work [3]unrelated to [1],regular and black hole static and spherically symmetric solutions to a Einstein–YM (EYM)system in 4+1dimensional spacetime were constructed numerically.The YM system in that model had gauge group SO (4),with the connection taking its values in (one of the two)chiral spinor representations of SO (4),namely in SU (2).Given that the solutions in [3]were static,i.e.that the YM field is defined on a 4dimensional Euclidean space,the SU (2)YM field in [3]is the same one as that in [1].This is the relation between the two works [1]and [3],and our intention here is to exploit this relation.The present work serves two distinct purposes.Thefirst and main purpose is to pave the way for the construction of more general,non-spherically symmetric solutions to EYM systems in afive dimensional spacetime.To our knowledge,no such results in EYM theory have appeared in the literature to date.Although considerable progress has been made in constructing asymptoticallyflat higher dimensional EYM solutions1all known configurations were subject to spherical symmetry.Our choice of a YM-dilaton(YMd) model is made because it has been shown that,at least in d=3+1dimensions,the classical solutions of this system mimic the corresponding EYM solutions[13],so the dilaton–YM exercise serves as a warmup for the considerably more complex gravitational problem. Our choice of a YMd model is made as an expedient in attempting the analysis of the corresponding EYM model,the last being of physical interest low energy effective actions of string theory,descended from11dimensional supergravity.It is also a coincidence here,that these supergravity descended low energy effective actions include the dilaton in addition to gravities and non Abelian matter.But here,the dilaton appears only as a substitute for gravity.A particular feature of the model to be introduced in the next section is that it features a term that is higher order in the YM curvature.As will be explained in section2,such terms are necessary to ensure that the solution yields afinite mass.Such terms were employed in previous works[4,3,5,6,7,8]with precisely the same purpose.The physical justification for introducing higher order YM terms,which goes hand in hand with the inclusion of higher order gravitational terms,is that these occur in the low energy effective action of string theory[9].Thus in principle the choice of higher dimensional EYM models involves the selection of higher order terms in the gravitational and non Abelian curvatures,namely the Riemann and the YM curvatures,which are reparametrisation and gauge invariant. Because we are concerned withfinding classical solutions,we impose a pragmatic but important further restriction,namely that we consider only those Lagrange densities that are constructed from antisymmetrised2p curvature forms,and exclude all other powers of both Riemann and YM curvature2-forms.(In the gravitational case this results in the familiar Gauss–Bonnet type Lagrangians,while in the case of non Abelian matter to the YM hierarchy pointed out in footnote3below.)As a result,only velocity–squared fields appear in the Lagrangian,which is what is needed both for physical reasons and for solving the classicalfield equations.In practice we add only the minimal number of such higher order terms that are necessitated by the requirements offinite mass.This criterion makes the inclusion of higher order gravitational terms unnecessary since we know from the(numerical)results of[4]that the qualitative properties of the classical solutions are insensitive to them.In addition to this argument based on numerical results there is an independent argument advanced at the end of section2of[6],based on the symmetries of the(higher order)gravitational terms,which in the absence of a dilaton dispenses with theeffectiveness of employing such terms.(Note here that the present YMd model is being used as a prototype for a EYM model,without an additional dilatonfield.)This leaves one with higher order YM curvature terms only,whose status in the context of the string theory effective action is complex and as yet not fully resolved.While YM terms up to F4 arise from(the non Abelian version of)the Born–Infeld action[10],it appears that this approach does not yield all the F6terms[11].Terms of order F6and higher can also be obtained by employing the constraints of(maximal)supersymmetry[12].The results of the various approaches are not identical.In this background,we restrict our considerations to terms in the YM hierarchy(see footnote3)only,in particular to thefirst two terms.Concerning our particular choice of4+1spacetime dimensions here,our reasons are: When imposing axial symmetry on a YMfield in d=D+1dimensions the simplest way is,following[14],to impose spherical symmetry in the D−1dimensional subspace of the d spacelike dimensions.In this case the Chern-Pontryagin topological charge is fixed by the boundary conditions imposed on thefirst polar angle,and no analogue of the vortex number appearing in the axially symmetric Ansatz for d=3[15]is featured[14]. Technically,the absence of a vortex number makes the numerical integration much harder. Imposing axial symmetry in turns in the x−y and z−u planes of D=4Euclidean space as in[1]on the other hand,features two(equal)vortex numbers,making the numerical work technically more accessible.It is our intention to use the particular bi-azimuthally symmetric Ansatz of[1]in D=4that has led us to restrict ourselves to d=4+1 dimensional spacetime.(Numerical work on implementing axial symmetry like in[14]is at present in active progress.)Of course,the exploitation of this type of symmetry is not restricted to4+1spacetime,but can be extended to any odd2q+1spacetime where q distinct azimuthal symmetries are imposed,but this in practice results in residual PDE’s of order three and higher for q≥3.The second and subsidiary aim of this work is to break the scale invariance of the usual YM system in D=4studied in[1],and the introduction of the dilatonfield does just that.The question of instanton–antiinstanton bound states in a scale breaking model is an interesing enough matter in itself,presenting a second important motivation for this work.In Section2we present the model,impose the symmetry and state the boundary con-ditions,in successive subsections.The numerical results are presented in Section3,pre-senting both solutions with spherical and bi-azimuthal symmetry.We give our conclusions and remarks in thefinal section.2The modelThe model in5spacetime dimensions with coordinates x M=(x0,xµ)that we study here is described by the LagrangianL m=12·2!e2aφTr F2MN+τ2whereφis the dilatonfield,F MN=∂M A N−∂N A M+[A M,A N]is the2-form YM cur-vature and F MNRS={F M[N,F RS]}is the4-form YM curvature consisting of the totally antisymmetrised product of two YM2-form YM curvatures.(The bracket[νρσ]implies cyclic symmetry.)τ1andτ2are dimensionful coupling strengths which will eventually be scaled out against the constant a in the exponent,which has the inverse dimension of the dilatonfieldφ.Similar to the d=3+1case,the form we choose for the coupling of the dilatonfield to the nonabelian matter was found by requiring that a shiftφ→φ+φ0of the dilatonfield to be compensated by a suitable rescaling of the coordinates.Let us give a brief justification for the choice of the model(1).At the most basic level it is a YM–dilaton(YMd)model designed to simulate qualitatively a EYM model in d=5.Repacing the dilaton in(1)by a gravitational term is a physically relevant model, representing part of a low energy effective action in d=5.Adding gravitational terms to (1)as it stands is a EYMd model,which is just as physically relevant.The YM system,which scales as L−4,in d=4+1supports static solitons,namely the BPST instantons in d=4+0dimensions.When the usual Einstein–Hilbert gravity, which scales as L−2,is added to the YM term,the soliton collapses because of the(Derrick) scaling mismatch.To compensate for this scaling mismatch,a term scaling as L−ν,with ν≥5must be added.If one is to restrict to positive definite terms2,νwill be even,and the most economical choice isν=6.A typical such term would be Tr(F∧DX)2,where X is a scalarfield,e.g.a Higgs or sigma–modelfield.This necessitates the introduction of a completely new(scalar)field which unlike the dilaton is not directly recognised as a constituent of a low energy effective action.For this reason we eschew this choice,and restrict our attention instead to systems featuring only YM(and eventually YMd)fields. The most economical choice then is to compensate with the YM term Tr(F∧F)2,scaling withν=4.We note,finally,that adding a(positive or negative)cosmological constant does not remedy the scaling mismatch since these terms do not scale at all.Indeed,in all higher dimensional EYM cases studied,withΛ=0[4,3],Λ<0[6]andΛ>0[7],the mass turns out to be infinite when higher order YM terms are not employed.The YM sector of the action density in4+1dimensions employed here,is that one used in[3],namely the superposed p=1and p=2members of the YM hierarchy3.The dilaton breaks the scale invariance of the usual p=1YM system,and a simple Derrick-type scaling argument shows that nofinite mass/energy solution can exist ifτ2=0,i.e. the p=2term in(1)is necessary.The YM and dilatonfield equations readτ1Dµ e2aφFµν +12π2 2e2aφˆL1+6e6aφˆL2 .(3)5AµAνAρAσ scaling as L−5isa possibility,albeit a considerably harder problem technically,and is at present under active consideration.3The YM hierarchy labeled by the integer p was introduced in[16]in the context of self-dual solutions in4p Euclidean dimensions,but superpositions of various p members were employed ubiquitously since.In(3)we have used the notationˆL 1=τ12·4!Tr F2MNRS.(4)2.1Imposition of symmetry and residual actionIn the YM connection A M=(A0,Aµ),we choose the temporal component A0=0to vanish and the spacelike components Aµis subjected to two successive axial symmetries,described in[1].We denote the Euclidean four dimensional coordinates as xµ=(x,y;z,u)≡(xα;x i), withα=1,2and i=3,4,and use the following parametrisationxα=r sinθˆxα≡ρˆxα,x i=r cosθˆx i≡σˆx i,(5) where r2=|xµ|2=|xα|2+|x i|2,with the unit vectors appearing in(5)parametrised as ˆxα=(cosϕ1,sinϕ1),ˆx i=(cosϕ2,sinϕ2),with0≤θ≤πρ Σαβˆxβ+φm2A ab iΣab,Aρ=−1index a in(9)-(10)runs only over three values,and reassigning the values of the index i=1,2,the analogues of(7)and(8)contract to giveA i= χ4+n2ρ (εˆx)i(εn(2))jΣj3+A34σˆx i n(2)jΣj3,(11)Aρ=A34σn(2)jΣjm,(12) exhibiting the Abelian connection A34σanalogous to the Abelian connection A34ρappearing in(7)and the isodoublet function(χ3,χ4).The corresponding axially symmetric decomposition ofΦin(10)isΦ=ξ1n(2)jΣj4+ξ2Σ34.(13)In(11)-(12)and(13)we have used the unit vector n(2)i=(cos n2ϕ2,sin n2ϕ2),with vorticity integer n2.Thefinal stage of symmetry imposition is to treat the two azimuthal symmetries imposed in the x−y and the z−u planes on the same footing,leading to the equality of the two vortex numbers,n1=n2≡n.Denoting the residual functions(A34ρ,A34σ)=(aρaσ,),(χ3,χ4)=χA,(ξ1,ξ2)=ξA, and regarding(aρ,aσ)as an Abelian connection on the quater plane defined by(ρ,σ),the residual action densities can be expressed exclusively in terms of the SO(2)curvaturefρσ=∂ρaσ−∂σaρ(14) and the covariant derivativesDρχA=∂ρχA+aρ(εχ)A,DσχA=∂σχA+aσ(εχ)A,(15)DρξA=∂ρξA+aρ(εξ)A,DσξA=∂σξA+aσ(εξ)A.The residual two dimensional YM action densities descending from the p=1and the p=2 termsˆL1andˆL2defined by(4)are,respectively,L1=τ1σ |DρχA|2+|DσχA|2 +σρσ(εABχAξB)2,L2=τ2Since our numerical constructions will be carried out using the coordinates(r,θ)we display (16)also asL1=τ1cosθ |D rχA|2+1sinθ |D rξA|2+1r sinθcosθ(εABχAξB)2(18)L2=τ2g L m= ∞0dr π/20dθ 1r2φ2,θ)+(e2aφL1+e6aφL2) ,(20) and equals the total action of solutions,viewed as solitons in a d=4Euclidean space.2.2Boundary conditionsTo obtain regular solutions withfinite energy density we impose at the origin(r=0)the boundary conditionsa r=0,aθ=0,χA= 0−n2 ,ξA= 0−n1 ,(21)which are requested by the analyticity of the YM ansatz,and∂rφ|r=0=0for the dilaton field.In order tofindfinite mass solutions,we impose at infinitya r=0,aθ=−2m,χA=(−1)m+1n2 sin2mθcos2mθ ,ξA=−n1 sin2mθcos2mθ ,φ=0,(22)m being a positive integer.Similar considerations lead to the following boundary conditions on theρandσaxes:a r=1n1∂θξ1,χ1=0,ξ1=0,∂θχ2=0,ξ2=−n1,∂θφ=0,(23)forθ=0,anda r=1n2∂θχ1,χ1=0,ξ1=0,χ2=−n2,∂θξ2=0,∂θφ=0,(24)forθ=π/2,respectively.2.3Topological chargeIn our normalisation,the topological charge is defined as q =12εµν 14 εµν∂µ(χA D νξA −ξA D νχA )d 2x.(27)The integration in (26)iscarriedoutover the 2dimensional space x µ=(x ρ,x σ).As expected this is a total divergence expressed by (27).Using Stokes’theorem,the two dimensional integral of (27)reduces to the one dimen-sional line integralq =12[1−(−1)m ]n 1n 2.(29)3Numerical resultsApart from the coupling constants τ1and τ2the model contains also the dilaton constant a .Dimensionless quantities are obtained by rescalingφ→φ/a,r →r (τ2/τ1)1/4,(30)This reveals the existence of one fundamental parameter which gives the strength of the dilaton-nonabelian interactionα2=a 2τ3/21/τ1/22,(31)which is a feature present also in the EYM case [3].We use this rescaling to set τ1=1,τ2=1/3in the numerical computation,without any loss of generality.One can see that the limit α→0can be approached in two ways and two different branches of solutions may exist.The first limit corresponds to a pure p =1YM theory with vanishing dilaton and p =2YM terms,the solutions here replicating the (multi-)instantons and composite instanton-antiinstanton bound states discussed in [1].The other possibility corresponds to a finite value of the dilaton coupling a as τ1→0.Thus,the second limitingconfiguration is a solution of the truncated p=2YM system interacting with the dilaton, with no p=1YM term.We have studied YMd solutions with m=1,2.From our knowledge of the tolopogical charges(29),the m=1solutions will describe(multi)solitons and the m=2solutions, soliton-antisoliton configurations.Also,to simplify the general picture we set n1=n2=n in the boundary conditions(21)-(24).The spherically symmetric solutions are found by using a standard differential equations solver.The numerical calculations in the bi-azimuthally symmetric case were performed with the software package CADSOL,based on the Newton-Raphson method[17].In this case,thefield equations arefirst discretized on a nonequidistant grid and the resulting system is solved iteratively until convergence is achieved.In this scheme,a new radial variable x=r/(1+r)is introduced which maps the semi-infinite region[0,∞)to the closed region[0,1].As will be described below,solutions exist for certain ranges of the parameterα.It turns out that m=1solutions with all n and m=2solutions with n=1exist for a range ofαstarting from aα→0limit,but do not persist all the way up to the second α→0limit.(However,the way the solutions approach the limitα→0depends on m.) By contrast wefind that m=2solutions with all n>1,exist for allαbetween the two limits.3.1m=1configurationsn=1spherically symmetric solutionsIn the spherically symmetric limit,which case we shall analyse numericallyfirst,the angular dependence of these functions isfixed and the only remaining independent function depends on the variable r.The independent function in this case is aθ=w(r)−1,with the remaining fuctions(a r,χA,ξA)given bya r=0,χ1=−ξ1=1r )+9e6φτ2(w2−1)2r4) ′=2e2φw(w2−1)r2 .(33)The asymptotic solutions to these functions can be systematically constructed in both regions,near the origin and for r≫1.The small r expansion isw(r)=1−br2+O(r4),φ=φ0+4α2(τ1with b,φ0two real parameters,while as r→∞wefindw(r)=±1−4φ127r6+O(1/r8),φ=φ127r6+O(1/r8).(35)We numerically integrate the Eqs.(33)with the above set of boundary conditions forτ1=1,τ2=1/3and varyingα.The picture we found is very similar to that found for the EYM system[3],the dilaton coupling constant playing the role of the Newton constant. First,for a givenα,solutions with the right asymptotics exist for a single value of the ”shooting”parameter b which enters the expansion(34).Forαsmall enough,a branch of solutions smoothly emerges from the BPST configuration[2].Whenαincreases,the mass M and the absolute value of the dilaton function at the origin increase,as indicated in Figure1.These solutions exist up to a maximal valueαmax≃0.36928of the parameterα.As in the corresponding gravitating case[3],we found another branch of solutions inthe intervalα∈[αcr(1),αmax]withα2cr(1)≃0.2653.On this second branch of solutions, bothφ(0)and M continue to increase but stayfinite.However,a third branch of solutions exists forα∈[0.2653,0.2652],on which the two quantities increase further.A fourth branch of solutions has also been found,with a correspondingαcr(3)≃0.2642.The mass M,the value of the dilatonfield at the originφ(0)and the initial(shooting)parameter b increase along these branches.Further branches of solutions,exhibiting more oscillations aroundα≃0.264are very likely to exist but their study is a difficult numerical problem. This critical behaviour is described as a conicalfixed point in the analytic analysis in[5]. Therefore we conclude that,as in the spherically symmetric gravitating case[3],the limit τ1=0is not approached for solutions with m=1,n=1.As a general feature,all solutions discussed here present only one node in the gauge function w(r).As in the higher dimensional EYM models discussed in[4,5],no multinode solutions were found.n>1Solutions with bi-azimuthal symmetry with nontrivial dependence on both r andθare found for(n1,n2)=1subject to the boundary conditions(21)-(24).We have studied solutions for m=1with2≤n≤5.The general features of the m=1solutions are the same for all n>1.Also,as seen in(29),the m=1configurations carry a topological charge q=n2.The corresponding solutions of the F2MN model are self-dual and have been considered already in[18],[19](for a different parametrization of the gaugefield,however).These solutions are constructed by starting with the known spherically symmetric con-figuration and increasing the winding number n in small steps.The iterations converge, and repeating the procedure one obtains in this way solutions for arbitrary n.The physical values of n are integers.The typical numerical error for the functions is estimated to be of the order of10−3or lower.Any spherically symmetric configuration appears to result in generalisations with higher winding numbers n.Moreover,the branch structure noticed for the m=1,n=1case seems to be retained by the higher winding number m=1solutions.Again,thefirst branchof solutions exists up to a maximal value ofα,where another branch emerges,extending backwards inα.We managed to construct higher winding number n counterparts of the first two branches of spherically symmetric solutions.The mass M and the absolute value of the dilaton function at the origin increase along these branches,as shown in Figure2. Note that the value of the dilaton function at the origin exhibited in thefigures is actually φ(r=0,θ=0),restricting toθ=0.This restriction is reasonable since for all solutions with bi-azimuthal symmetry discussed in this paper,the dilaton function at r=0presents almost no dependence on the angleθ.We expect that the oscillatory pattern ofφ(0)arising from the conicalfixed point observed above for the spherically symmetric n=1solutions,will also be discovered for the n>1solutions here,but their construction is a difficult numerical problem beyond the scope of the present work.In Figure3we present the gauge functions,the dilaton,and the topological charge density1̺=parametrised by the effective coupling constantα,while the latter has no such parameter. As will be described below,m=2n=1solutions exist for a certain range ofα,and thisrange excludes the limiting case where the contribution to the action of the dilaton term and the p=2YM term in(1)disappear,i.e.a F2MN model.Wefind that in the limitα→0resulting from a→0,cf.(31),no solutions of thistype exist.However in the limitα→0corresponding to afinite value of the dilaton coupling a asτ1→0,such solutions exist.This limiting configuration is then a solution of the truncated system consisting of the dilaton term and p=2YM term F2MNRS,whichdominate.Its characteristic feature is that for this configuration both nodes of the effective Higgsfields|χ|and|ξ|merge on theθ=π/4axis.A family of solutions of the model(1) emerges from this configuration.Asαincreases,the nodes move towards the symmetry axes,ρandσ,respectively,forming two identical vortex rings whose radii slowly decrease while the separation of both rings from the origin increase.At the critical valueαcr≃0.265 the node structure of the configuration changes,both vortex rings shrink to zero size and two isolated nodes appear on each symmetry axis.This structure is known for the usual YM system in d=4+0[1],indeed,increasing ofαalong this branch can be associated with increasing of the couplingτ1w.r.t.τ2as the dilaton coupling a remainsfixed;then the term F2MN becomes leading.The maximum of the action density however is still located onθ=π/4axis.Another similarity with the instanton-antiinstanton solution of the d=4+0p=1YM theory is that the gauge functions a r,aθas well the dilaton functionφof the n=1,m=2 solutions also are almostθ-independent.Along this branch the mass of the solutions grows with increasingαsince with increasing couplingτ1the contribution of the term F2MN also increases.As the effective coupling increases further beyondαcr the relative distance between the nodes increases,one lump moving towards the origin while the other one moves in the opposite direction.Along this branch both the value of the dilatonfield at the origin|φ(0)| and mass of configuration M increase asαincreases.This branch extends up to a maximal valueα(1)max≃0.311beyond which the dilaton coupling becomes too strong for the static configuration to persist.The second branch,whose energy is higher,extends backwards up toα(2)max≃0.279.Along this branch both|φ(0)|and the mass of the configuration continue to increase asαdecreases.Also the separation between the nodes decreases and both nodes invert direction of the motion,moving toward each other along this branch. In Figure5we present the values of the dilaton function at the originφ(0)and the total mass(rescaled byα2)of these configurations as functions ofα.n=2This configuration also resides in the topologically trivial sector and can be considered as consisting of two solitons of charges n=±2.Then the interaction between the nonabelian matterfields becomes stronger than in the case of unit charge constituents and the expected pattern of possible branches of solutions is different from the n=1case above.Indeed,the n=2,m=2solutions show a different dependence on the coupling con-stantα,with two branches of solutions.The lower branch emerges from the correspondingsolution in pure p=1YM theory with vanishing dilaton and p=2YM terms,replicating the corresponding solution in[1].The variation of the effective coupling along this branchis associated with the decrease ofτ1,atfixedτ2andfixed dilaton coupling a.The secondbranch emerges from a solution of the p=2YM-dilaton system,the unrescaled mass M diverging in this limit,with the rescaled mass Mα2vanishing as seen from Figure5a.Atthe maximal valueαmax≃0.2372this branch bifurcates with the lower YM branch.For larger values ofα,the dilaton coupling becomes too strong for the static configurations to persist.Thus for0≤α<αmax we notice the existence of(at least)two distinct solutionsfor the same value of coupling constant.For the same value ofα,the mass of the second branch solution is larger that that ofthe corresponding lower branch configuration(s).One should also notice the existence ofa curious backbending of the lower branch for0.193<α<0.218.Four distinct solutions exist in this case for the same value ofα(three of them located on the lower branch), distinguished by the value of the mass and the dilatonfield at the origin.This pattern is illustrated in Figure5.Again,observation of the positions and structure of the nodes of the effective scalarfields allows us to better understand the behaviour of the solutions.For lower branch solutions with small values ofαthere are two(double)nodes of thefields|χ|and|ξ|on the ρandσsymmetry axes respectively.The locations of nodes correspond to the locations of the two individual constituents and the action density distribution posesses two distinct maxima on theθ=π/4axis.The distance between these nodes changes only slightly along the lower mass branch.The backbending inαobserved in this case is reflected also for in the relative positions of the nodes.At the maximal value ofα,the inner node is located atρ(1)0=σ(1)0≃2.97and the outer node is located atρ(2)0=σ(2)0≃4.18.Along the upper branch,asαslightly decreases belowαmax,the inner node inverts direction of its movement toward the outer node which still moves inwards.Thus,both nodes on the symmetry axis rapidly approach each other and merge forming a two vortex ring solution asα≃0.2355.The action density then has a single maximum onθ=π/4 axis.Asαdecreases further both nodes move away from the symmetry axis and their positions do not coincide with the location of the maximum of the action density.Further decreasingαresults in the increase of the radii of the two rings around the symmetry axis, and in the limitα→0the rings touch each other on theθ=π/4axis.In Figure4we give three dimensional plots of the modulus of the effective Higgsfield ξfor the n=m=2upper branch vortex solution atα=0.20and the n=m=2lower branch double node solution at the same value ofα.The action density as given by(1)is also plotted atα=0.20both for the upper and for the lower branches.The numerical calculations indicate the possibility that the solutions of the fundamentalYM branch,namely the branch on which the p=1YM term dominates,are not unique. It is possible that higher linking number configurations with higher masses might exist. This possibility will be explored elsewhere.。
Unit3Albert Einstein爱因斯坦
Achievements
• • • • • • I. General relativity 广义相对论 II. Special relativity 狭义相对论 III. Brownlan effect 布朗运动 IV. Photoelectric effect 光电效应 V. E=mc^2 E=mc^2 VI. Einstein field equation 爱因斯坦场方 程 • VII. Bose-Einstein statistics 波色-爱因斯 坦统计
Albert Einstein died in Nineteen-FiftyFive. He was seventy-six years old.
THE END
Einstein’s famous sayings
1. Imagination is more important than knowledge. 想象力比知识更重要。 2. Try not to become a man of success, but rather try to become a man of value. 试着不去做一个成功的人，而去做一个有价值的人。 3.Anyone who has never made a mistake has never tried angthing new. 一个人从未犯错是因为他从不尝试新鲜事物。 4.Weakness of attitude becomes weakness of character. 态度上的弱点会变成性格上的弱点。 5. Life is like riding a bicycle. To keep your balance you must keep moving. 人生就像骑单车。想保持平衡就得往前走。 6.Logic will get you from A to B. Imagination will take you everywhere. 逻辑会把你从A带到B，想象力能带你去任何地方。
物理学家：托马斯·杨
生平简介科学成就趣闻轶事一、生平简介托马斯·杨（Thomax Young,1773—1829年）英国医生兼物理学家，光的波动说的奠基人之一。
1773年6月13日生于萨默塞特郡的米菲尔顿。
他从小就有神童之称，兴趣十分广泛。
后来进入伦敦的圣巴塞罗缪医学院学医，21岁时，即以他的第一篇医学论文成为英国皇家学会会员。
为了进一步深造，他到爱丁堡和剑桥继续学习，后来又到德国哥廷根去留学。
在那里，他受到一些德国自然哲学家的影响，开始怀疑起光的微粒说。
1801年进行了著名的杨氏干涉实验，为光的波动说的复兴奠定了基础。
1829年5月10日杨氏在伦敦逝世。
二、科学成就1．著名的杨氏干涉实验，为光的波动说奠定一基础。
杨氏干涉实验的巧妙之处在于，他让通过一个小针孔S0的一束光，再通过两个小针孔S1和S2，变成两束光。
这样的两束光因为来自同一光源，所以它们是相干的。
结果表明，在光屏上果然看见了明暗相间的干涉图样。
后来，又以狭缝代替针孔，进行了双缝干涉实验，得到了更明亮的干涉条纹。
在他之前，不少人曾进行过光的干涉实验。
由于他们是用两个独立的非相干光源发出的两束光迭加，因此，这些实验都失败了。
他用这个实验首先引入干涉概念论证了波动说，又利用波动说解释了牛顿环的成因和薄膜的彩色。
1801年他引入叠加原理，把惠更斯的波动理论和牛顿的色彩理论结合起来，成功地解释了规则光栅产生的色彩现象。
1803年，他又用波动理论解释了障碍物影子具有彩色毛边的现象。
1820年他用比较完善的波动理论对光的偏振作出了比较满意的解释，认为只要承认光波是横波，必然会产生偏振现象。
2．对人眼感知颜色的研究，建立三原色原理他还第一个测量了7种颜色光的波长。
他曾从生理角度说明了人眼的色盲现象；他还建立了三原色原理，指出一切色彩都可以从红、绿、蓝这三种原色的不同比例的混和而得到。
3．对弹性力学的研究托马斯·杨对弹性力学很有研究，特别是对胡克定律和弹性模量。
The string solution in SU(2) Yang-Mills-Higgs theory
a rXiv:h ep-th/966124v12J un1996The string solution in SU(2)Yang-Mills-Higgs theory V.D.Dzhunushaliev ∗and A.A.Fomin Theoretical physics department,the Kyrgyz State National University,720024,Bishkek,Kyrgyzstan Abstract The tube solutions in Yang -Mills -Higgs theory are received,in which the Higgs field has the negative energy density.This solutions make up the discrete spectrum numered by two integer and have the finite linear energy density.Ignoring its transverse size,such field configuration is the rest infinity straight string.PACS number:03.65.Pm;11.17.-w At the end of 50-th years W.Heisenberg has been investigate the non-linear spinor matter theory (see,for example,[1],[2]).It is supposed that on the basis one or another nonlinear spinor equation the basic parameters of the elementary particles existing at that time will be derived:masses,charges and so on.The mathematical essence of this theory lies in the fact that the nonlinear spinor Heisenberg equation (HE)(or in the simpler case the nonlinear boson equation like nonlinear Schr¨o dinger equation)has the discrete spectrum of the solutions having physical meaning (possesing,for example,the finite energy).This solutions give the mass spectrum in clas-sical region even.This gave hope that after quantization more or less likely mass spectrum and the charges of the elementary particles would be derive.Now the string can to arise in Dirac theory with the massive vector field A µby interaction 2magnetic charges with opposite sign [3].At present timethe investigations continue along this line and explore not1-dimensional ob-ject(string)stretched between quarks(see,for example,[4])but3-dimensional (tube)filled byfield(see,for example,[5],[6]).So,for example,a tube of the chromodynamicalfield and its properties in[5]is considered.But this consideration is phenomenological because a question on the reason of the field pinching isn’t affected,also a question on thefield distribution in the tube isn’t analyzed.In this article we shown that the Yang-Millsfield interacted with Higgs scalarfield is confined in tube.In this case the Higgsfield have the negative energy density.In[2]it is showed that the nonlinear Klein-Gordon and Heisenberg equations have the regular solutions.They are the spherical symmetric par-ticlelike solutions numered by integer,i.e.they form discrete spectrum with the corresponding energy value.One would expect(and this will be showed below)that we have in axial symmetric case as well as in spherical-sym-metric case the physical interesting(string)solutions withfinite energy per unith length.Finally,we present some qualitative argument in favour of the existence suchfield configurations(tube,string)according[4].In QCD vacuumfield taken external pressure on the gluon tube.Diameter of such tube will be defined from equilibrium condition between external pressure of the vacuum field and internal pressure of the gluonfield in tube.It can be evaluate by minimizing the energy density of such tube which is the difference between the positive energy density of the chromodynamicfield and negative energy density of vacuumfield in QCD.This diameter R0after corresponding cal-culations is equal:ΦR0=F aµνFµνa−14g2where a =1,2,3is SU (2)colour index;µ,ν=0,1,2,3are spacetime indexes;F aµν=∂µA aµ−∂νA aµ+ǫabc A bµA aνis the strength tensor of the SU (2)gauge field;F µν=F aµνt a ,t a are generators of the SU (2)gauge group;D µΦ=(∂µ+A µ)Φ;V (Φ)=λ(Φ+Φ−4η2)/32;g,η,λare constant;Φis an isodoublet of the Higgs scalar field;The Yang -Mills -Higgs equations system look by following form in this model:D µF µνa =(−γ)−1/2∂µ (−γ)1/2F µνa +ǫabc A bµF µνc =g 2∂Φ+,(4)where γis the metrical tensor determinant.We seek the string solution in the following form:the gauge potential A aµand the isodoublet of the scalar field Φwe chosen in cylindrical coordinate system (z,r,θ)as :A 1t =2ηf (r ),(5)A 2z =2ηv (r ),(6)A 3θ=2ηrw (r ),(7)Φ= 2ηϕ(r )0 (8)By substituting Eq’s(5-8)in Eq’s(3-4)we receive the following equations system:f ′′+f ′x =v 4 −f 2+w2 −g 2ϕ2 ,(10)w ′′+w ′x 2=w 4 −f 2+v 2 −g 2ϕ2,(11)ϕ′′+ϕ′λis introduced;(′)means thederivative with respect to x ;and the following renaming are made:g 2λ−1/2→3g2,f(x)λ−1/2→f(x),v(x)λ−1/2→v(x),w(x)λ−1/2→w(x).We will study this system by the numerical tools.In this article we investigate the easiest case v=f=0.Thus system(9-12)look as:w′′+w′x2=−g2wϕ2,(13)ϕ′′+ϕ′2+···,(15) w=w0x+w3x32ϕ0 1−ϕ20 ,(17) w3=−3w(x)≈1√x−Cg2x2,(22)where integers m and n enumerate the knot number ofϕ(x)and w(x)func-tions respectively.According to this we shall denote the boundary value ϕ(0)and parameter g in the following manner:ϕ∗mn and g∗mn.The result of numerical calculations on Fig.1,2are displayed(w1=0.1).The asymptotic behaviour of theϕmn(x)and w mn(x)functions as in(21)-(22)results in that the energy density of thisfields drop to zero as exponent on the infinity and this means that this tube has thefinite energy per unit. It is easy to show that aflux of colour”magnetic”field H z across the plane z=const isfinite.Thus we can to speak that the Yang-Mills-Higgs theory have the tube solution if the Higgsfield have the negative energy density.It is notice that this solutions are not topological nontrivial thread.Ignoring the transversal size of obtained tube we receive the rested boson string withfinite linear energy density.References5[1]Nonlinear quantumfield theory.Ed.D.D.Ivanenko,Moskow,IL,p.464,1959.[2]R.Finkelstein,R.LeLevier,M.Ruderman,Phys.Rev.,83,326(1951).[3]Nambu Y.,Phys.Rev.1974,D10,p.4262.[4]Bars I.,Hanson A.J.,Phys.Rev.,1976,D13,p.1744.[5]Nussinov S.,Phys.Rev.D.1994,v.50,N5,p.3167.[6]Olson C.,Olsson M.G.,Dan LaCourse,Phys.Rev.D.1994,v.49,N9,p.4675.[7]Barbashow B.M.,Nesterenko W.W.Relativistic string model inhadronic physics,Moskow,Energoatomizdat,p.179,1987.[8]V.D.Dzhunushaliev,Superconductivity:physics,chemistry,technique,v.7,N5,767,1994.6。
Exact solutions in the Yang-Mills-Wong theory
2
of gauge invariant operators obeys the so-called factorization relation, and quantum fluctuations disappear [8]. Thus QCD becomes a classical theory as N → ∞. We suggest that the large-N YMW theory is intimately related to the classical limit of QCD. Note, however, that the confinement problem is out of the question now. Indeed, it is conceivable that quarks constituting a hadron experience an attractive constant force originating from a term of the potential Aµ which linearly rises with distance between the quarks, and such a behavior of Aµ is to provide the area law for the Wilson loop functional [9]. Are we correct in interpreting the area law as the evidence of the constant attractive force? As will be shown, an exact classical solution Aµ with the linearly rising term actually exists. Although this term contributes to the field strength, it produces no force. The general reason for such a surprising result is the conformal invariance. The linearly rising term violates the scale invariance. While such a violation being allowable for the gauge quantities Aµ and Fµν , it cannot be tolerated for observables. One may expect a dimensional parameter, measuring a gap in the energy spectrum and violating the scale symmetry, to emerge only upon quantization leading to anomalies. Meanwhile exact classical solutions are crucial in learning the symmetry of the vacuum. One believes two phases of the strong interacting matter to exist, hot and cold, which must be distinguished by their symmetry. At high temperatures, the asymptotical freedom dominates, hence the conventional SU(3)c symmetry is inherent in the hot phase. On the ˇ cki [10] developed an exhaustive phenomenological classifiother hand, Ne’eman and Sijaˇ cation of hadrons on the basis of infinite-dimensional unitary representations of SL(4, R), which hints that SL(4, R) is the cold phase symmetry. Where does this SL(4, R) come from? Coleman [11] argued that the symmetry of the vacuum is the symmetry of world. Given the vacuum invariant under SL(4, R), excitations about it possess the same symmetry. Since the symmetry of the gluon vacuum is nothing but the symmetry of the background field, the responsibility for SL(4, R) rests with the background described by a certain solution of the QCD equations in the classical limit. It is the background generated by quarks in hadrons that provides the SL(4, R) relief for gluon excitations. We will find two classes of exact retarded solutions to the classical Yang-Mills (YM) equations. Solutions of the first class, invariant under SU(N ), appear to be related to the background in the hot phase. Solutions of the second class might be treated as the background generated by bound quarks in the cold phase. These solutions are complex valued with respect to the Lie algebra su(N ), but one can convert them to the real form to yield the invariance under SL(N, R) or its subgroups. In particular, the background generated by any three-quark cluster is invariant under SL(4, R), and that generated by any two-quark cluster is invariant under SL(3, R). ˇ cki operates in spacetime while the present Notice that SL(4, R) of Ne’eman and Sijaˇ SL(4, R) acts in the color space. However, we attempt to interweave two arenas by reference to that color degrees of freedom may be convertible into spin degrees of freedom, the fact discovered by Jackiw and Rebbi, and Hasenfratz and ’t Hooft [12]. The paper is organized as follows. Section II outlines the general formalism of the YMW theory. The next section is devoted to a justification of the Ansatz whereby we seek exact retarded solutions of the YM equations with the source composed of several arbitrarily moving quarks. Finding such solutions is traced by the simplest example of the single-quark source, Sec. IV. Properties of the background generated by two-quark sources 3
Was Einstein a Space Alien原文及翻译(精品文档)
Was Einstein a Space Alien?1 Albert Einstein was exhausted. For the third night in a row, his baby son Hans, crying, kept the household awake until dawn. When Albert finally dozed off ... it was time to get up and go to work. He couldn't skip a day. He needed the job to support his young family.1. 阿尔伯特.爱因斯坦精疲力竭。
他幼小的儿子汉斯连续三个晚上哭闹不停，弄得全家人直到天亮都无法入睡。
阿尔伯特总算可以打个瞌睡时，已是他起床上班的时候了。
他不能一天不上班，他需要这份工作来养活组建不久的家庭。
2 Walking briskly to the Patent Office, where he was a "Technical Expert, Third Class," Albert worried about his mother. She was getting older and frail, and she didn't approve of his marriage to Mileva. Relations were strained. Albert glanced at a passing shop window. His hair was a mess; he had forgotten to comb it again.2. 阿尔伯特是专利局三等技术专家。
在快步去专利局上班的路上，他为母亲忧心忡忡。
母亲年纪越来越大，身体虚弱。
她不同意儿子与迈尔娃的婚事，婆媳关系紧张。
Flavoring the gravity dual of N=1 Yang-Mills with probes
´ Carlos N´ un ˜ ez † 1 , Angel Paredes ∗
† 2
and Alfonso V. Ramallo ∗
3
Center for Theoretical Physics, Massachusetts Institute of Technology Cambridge, MA 02139, USA
US-FT-3/03 MIT-CPT/3441 hep-th/0311201
November 2003
1 2
nunez@ angel@c.es 3 alfonso@c.es
1
Introduction
The gauge/string correspondence, an old proposal due to ’t Hooft [1], is now well understood in the context of maximally supersymmetric super Yang-Mills (SYM) theories. Indeed, the so-called AdS/CFT correspondence is a conjectured equivalence between type IIB string theory on AdS5 × S 5 and N = 4 SYM theory [2]. In the large ’t Hooft coupling limit, the N = 4 SYM theory is dual to the type IIB supergravity background corresponding to the near-horizon geometry of a stack of parallel D3-branes, whose metric is precisely that of the AdS5 × S 5 space. There are nowadays a lot of non-trivial tests of this duality (for a review see [3]). The extension of the gauge/string correspondence to theories with less supersymmetries is obviously of great interest. A possible way to obtain supergravity duals of SYM theories with reduced supersymmetry is to consider branes wrapping supersymmetric cycles of CalabiYau manifolds [4]. In order to preserve some supersymmetry the normal bundle of the cycle within the Calabi-Yau space has to be twisted [5]. Gauged supergravities in lower dimensions provide the most natural framework to implement this twisting. In these theories the gauge field can be used to fiber the cycle in which the brane is wrapped in such a way that some supersymmetries are preserved. In this paper we will restrict ourselves to the case of the supergravity dual of N = 1 SYM. This background, which corresponds to a fivebrane wrapping a two-cycle, was obtained in ref. [6] from the solution found in ref. [7] representing non-abelian magnetic monopoles in four dimensions. The geometry of this background is smooth and leads to confinement and chiral symmetry breaking. Actually, if only the abelian part of the vector field of seven dimensional gauged supergravity is excited, one obtains a geometry which is singular at the origin and coincides with the smooth one at large distances, i.e. in the UV. Therefore, the singularity at the origin is resolved by making the gauge field non-abelian, in complete analogy to what happens with the resolution of the Dirac string by the ’t Hooft-Polyakov monopole. Moreover, as argued in ref. [8], the same mechanism that de-singularizes the supergravity solution also gives rise to gaugino condensation. Based on this observation, the NSVZ beta function can be reproduced at leading order [9, 10, 11]. Other aspects of this supergravity dual have been studied in ref. [12](for a review see [13]). Most of the analysis carried out with the background of [6] do not incorporate quarks in the fundamental representation which, in a string theory setup, correspond to open strings. In order to introduce an open string sector in a supergravity dual it is quite natural to add Dbrane probes and see whether one can extract some information about the quark dynamics. As usual, if the number of brane probes is much smaller than those of the background, one can assume that there is no backreaction of the probe in the bulk geometry. In this paper we follow this approach and we will probe with D5-branes the supergravity dual of N = 1 SYM. The main technique to determine the supersymmetric brane probe configurations is kappa symmetry [14], which tells us that, if ǫ is a Killing spinor of the background, only those embeddings for which a certain matrix Γκ satisfies Γκ ǫ = ǫ . (1.1)
Rigidity of Quasi-Einstein Metrics
1. Introduction Einstein metrics and their generalizations are important both in mathematics and physics. A particular example is from the study of smooth metric measure spaces. Recall a smooth metric measure space is a triple (M n , g, e−f dvolg ), where M is a complete n-dimensional Riemannian manifold with metric g , f is a smooth real valued function on M , and dvolg is the Riemannian volume density on M . A natural extension of the Ricci tensor to smooth metric measure spaces is the m-Bakry-Emery Ricci tensor 1 df ⊗ df for 0 < m ≤ ∞. m When f is constant, this is the usual Ricci tensor. We call a triple (M, g, f ) (a Riemannian manifold (M, g ) with a function f on M ) (m-)quasi-Einstein if it satisfies the equation (1.1) Ricm f = Ric + Hessf − (1.2) 1 df ⊗ df = λg m for some λ ∈ R. This equation is especially interesting in that when m = ∞ it is exactly the gradient Ricci soliton equation; when m is a positive integer, it corresponds to warped product Einstein metrics (see Section 2 for detail); when f is constant, it gives the Einstein equation. We call a quasi-Einstein metric trivial when f is constant (the rigid case). Many geometric and topological properties of manifolds with Ricci curvature bounded below can be extended to manifolds with m-Bakry-Emery Ricci tensor bounded from below when m is finite or m is infinite and f is bounded, see the survey article [18] and the references there for details. Quasi-Einstein metrics for finite m and for m = ∞ share some common properties. It is well-known now that compact solitons with λ ≤ 0 are trivial [8]. The same result is proven in [10] for quasi-Einstein metrics on compact manifolds with finite m. Compact shrinking Ricci solitons have positive scalar curvature [8, 5]. Here we show Ricm f = Ric + Hessf −
Nature Chemistry 3, 634-641(2011)--Single-atom catalysis of CO oxidation using Pt1_FeOx
Single-atom catalysis of CO oxidation using Pt 1/FeO xBotao Qiao 1,Aiqin Wang 1,Xiaofeng Yang 2,Lawrence F.Allard 3,Zheng Jiang 4,Yitao Cui 5,Jingyue Liu 1,6*,Jun Li 2*and T ao Zhang 1*Platinum-based heterogeneous catalysts are critical to many important commercial chemical processes,but their efficiency is extremely low on a per metal atom basis,because only the surface active-site atoms are used.Catalysts with single-atom dispersions are thus highly desirable to maximize atom efficiency,but making them is challenging.Here we report the synthesis of a single-atom catalyst that consists of only isolated single Pt atoms anchored to the surfaces of iron oxide nanocrystallites.This single-atom catalyst has extremely high atom efficiency and shows excellent stability and high activity for both CO oxidation and preferential oxidation of CO in H 2.Density functional theory calculations show that the high catalytic activity correlates with the partially vacant 5d orbitals of the positively charged,high-valent Pt atoms,which help to reduce both the CO adsorption energy and the activation barriers for CO oxidation.Supported noble-metal catalysts are the most widely used in industry because of their high activity and /or selectivity for a large number of important chemical reactions.Generally,in such systems noble metals are dispersed finely on a support with a high surface area for the efficient use of catalytically active com-ponents.The size of metal particles is therefore one of the most important factors that dictate the performance of a catalyst 1–4.Recent theoretical and experimental results demonstrated that sub-nanometre clusters have better catalytic activity and /or selectivity than nanometre-sized particles 3–7.Low-coordination,unsaturated atoms often function as active sites 8,so downsizing the particles or clusters to single atoms is highly desirable for catalytic reactions.However,fabrication of practical and stable single-atom catalysts remains a significant challenge because,typically,single atoms are too mobile and easy to sinter under realistic reaction conditions 9,10.Moreover,it remains unclear whether single atoms can be catalyti-cally active or have better performance than the corresponding (sub)nanometre-sized clusters or particles 3,9,11,12.Although mass-selected clusters (or even single atoms)soft-landed on oxide supports are used for fundamental studies 6,13,14,such an expensive and low-yield fabrication method is not suitable for practical applications.A more realistic approach is to tune con-ventional chemical methods to anchor single metal atoms on special sites of oxide supports.Many studies on oxide-supported metal clusters show that surface defects of the supports could serve as anchoring sites for metal clusters or even single atoms 2,15,16.Here we report the first practical fabrication of a Pt single-atom catalyst that consists only of isolated single atoms anchored onto iron oxide (FeO x )nanocrystallites.This Pt single-atom catalyst exhibits very high activity and stability for both CO oxidation and preferen-tial oxidation (PROX)of CO in H 2,attributed to the partially vacant 5d orbitals of positively charged,high-valent Pt atoms.Results and discussionPreparation.The single-atom Pt 1/FeO x catalyst (denoted as sample A,with a Pt /Fe atomic ratio of 1/1430and a Pt loading of 0.17weight percent (wt%))was prepared by a co-precipitation method reported previously for supported Au and Pd catalysts 17,18,but with a finely tuned co-precipitation temperature and pH value.Specifically,to anchor Pt atoms onto the defects of the FeO x surfaces and to isolate the Pt atoms sufficiently,we controlled the Pt loading to 0.17wt%and used a FeO x nanocrystallite support with a high surface area (approximately 290m 2g 21for the as-synthesized samples).To evaluate the effect of Pt loading on the structure and performance of the final catalysts,a similar catalyst with a Pt loading of 2.5wt%(denoted as sample B,with a Pt /Fe atomic ratio of 1/95)was also prepared,characterized and tested.Electron microscopy.Suba˚ngstro ¨m-resolution,aberration-corrected scanning transmission electron microscopy (STEM)was used to characterize the dispersion and configuration of the Pt clusters in the catalysts.Individual heavy atoms in practical catalysts can be discerned in the atomic resolution high-angle annular dark-field (HAADF)images 3,9,10,19–25.For sample A,Fig.1a clearly shows individual Pt atoms (marked by the white circles)uniformly dispersed on the surfaces of FeO x nanocrystals.Examination of different regions revealed that only Pt single atoms are present in sample A;additional representative imagesare shown in Supplementary Fig.S1.The suba˚ngstro ¨m-resolution HAADF image in Fig.1b reveals clearly that individual Pt atoms (white circles)occupy exactly the positions of the Fe atoms.As plan-view HAADF images represent the projection of atoms along the incident beam direction,surface atoms cannot be distinguished from the subsurface atoms in the HAADF images.However,whether the Pt atoms are inside the FeO x nanocrystallites can be determined by changing the focus of the electron beam (or ‘depth sectioning’)26,27.By analysing images obtained sequentially with varying beam focus settings,we concluded that the observed Pt atoms were not located inside the individual FeO x nanocrystals.In addition,by examining many HAADF images of sample A,we estimated that the density of Pt single atoms was about 0.07Pt atoms nm 22,which is very close to the actual Pt1State Key Laboratory of Catalysis,Dalian Institute of Chemical Physics,Chinese Academy of Sciences,Dalian 116023,China,2Department of Chemistry,T singhua University,Beijing 100084,China,3Materials Science and T echnology Division,Oak Ridge National Laboratory,Oak Ridge,T ennessee 37831,USA,4Shanghai Synchrotron Radiation Facility,Shanghai Institute of Applied Physics,Chinese Academy of Sciences,Shanghai 201204,China,5State Key Laboratory of Molecular Reaction Dynamics,Dalian Institute of Chemical Physics,Chinese Academy of Sciences,Dalian 116023,China,6Centerfor Nanoscience,Department of Physics &Astronomy,and Department of Chemistry &Biochemistry,University of Missouri-St Louis,Missouri 63121,USA.*e-mail:taozhang@;liuj@;junli@loading (about 0.09Pt atoms nm 22)of sample A.Therefore,all the observed individual Pt atoms are located either on the surfaces or in the near subsurfaces of the FeO x nanocrystallites.The Pt dispersion in sample B is quite different from that in sample A.As revealed clearly in Fig.1c,d,sample B contained individual Pt atoms (white circles),two-dimensional Pt rafts with fewer than ten Pt atoms (black circles)and three-dimensional Pt clusters (white squares)of size ≤1nm.On examining variousregions of sample B,we did not detect any Pt nanoparticles with sizes .2nm in diameter;Supplementary Fig.S2shows more examples of such images.Quantitative analysis of many images suggested that the individual Pt atoms and the Pt (sub)nanometre structures in sample B account for about 27%and 73%,respectively,in terms of their frequency of observation (Supplementary Fig.S3).However,a rough estimate revealed that the single atoms represent only about 1.8atom%of the total amount of Pt in sample B.R (Å)E (keV)11.5511.6011.65I n t e n s i t y (a .u .)I n t e n s i t y (a .u .)Sample A (used)Sample A Sample BPtO 2Pt foilabFigure 2|X-ray absorption studies.a ,The k 3-weighted Fourier transform spectra from EXAFS.D k ¼2.8–10.0Å21was used for samples A and B,but D k ¼2.8–13.8Å21was used for Pt foil and PtO 2.The peaks at 1.7Åin the first shell of samples A and B are fitted to the Pt–O contribution from the interaction between Pt and the FeO x support,and the very weak peaks at 2.5Åin the second shell are fitted to the Pt–Fe coordination in sample A and the Pt–Pt coordinationin sample B,respectively.b ,The normalized XANES spectra at the Pt L 3edge of sample A,sample B,PtO 2and Pt foil.The data show a decreasing trend in the white-line intensities:PtO 2.sample A (used).sample A .sample B .Pt foil,which indicates that the Pt single atoms in sample A carry positive charges and were oxidized further during the reaction.a.u.¼arbitrary units.badc2 nmFigure 1|HAADF-STEM images of samples A and B.a ,b ,In sample A,Pt single atoms (white circles)are seen to be uniformly dispersed on the FeO xsupport (a )and occupy exactly the positions of the Fe atoms (b ).Examination of different regions reveals that only Pt single atoms are present in sample A.c ,d ,In sample B,a mixture of single atoms (white circles),two-dimensional Pt rafts consisting of fewer than 10Pt atoms (black circles)and three-dimensional Pt clusters of size about 1nm or less (white squares)are observed clearly.DOI:10.1038/NCHEM.1095X-ray absorption fine structure studies.Consistent with the HAADF image analyses,X-ray diffraction patterns of samples A and B (Supplementary Fig.S4)did not show any Pt-containing crystal phases,primarily because of the insensitivity of X-ray diffraction to small clusters.To verify further that sample A contained only atomically dispersed individual Pt atoms throughout the whole catalyst,extended X-ray absorption fine structure (EXAFS)spectra were measured on freshly reduced catalysts.The EXAFS spectra of both sample A and sample B at the Pt L 3edge are characterized by the absence of oscillations at a high k region of k .8Å21(Supplementary Fig.S5)that indicates the dominance of low Z backscatters,which should be oxygen in our system.Correspondingly,in the Fourier transforms (r space,Fig.2a)of the EXAFS data,there is one prominent peak at 1.7Åfrom the Pt–O contribution and a very weak peak at 2.5Åfrom either the Pt–Pt or the Pt–Fe contributions.Owing to the high disordering in the higher shells,only the two main peaks in the r range of 1.0–3.2Åwere considered in the EXAFS curve-fitting (fitting parameters are given in Table 1).There were Pt–O contributions at a distance of 2.02Åfor sample A and of 1.98Åfor sample B;their coordination number was around 2.0.The samples were pre-reduced before the EXAFS measurements,so PtO x phases should not be in both samples.As a result,the Pt–O coordination originated most probably from the interaction between Pt and the FeO x support.The observation that the Pt–O bonding distance is close to that in PtO 2suggests the strong metal–support interaction.For sample B,there was a Pt–Pt contribution at a distance of 2.53Åwith an average coordination number of 3.8.Both the coordination number and the Pt–Pt bonding distance in sample B are much lower than those in bulk platinum (coordination number of 12at a distance of 2.78Å)(ref.28).The significantly shorter Pt–Pt bond distance in small Pt clusters was predicted bytheoretical studies 29.Our suba˚ngstro ¨m-resolution HAADF images of sample B also show that most of the clusters are two-dimensional rafts and that the Pt–Pt distances in these raft-like subnanometre clusters range from 2.1to 2.5Å,in good agreement with the EXAFS results discussed above.From these data,we conclude that sample B contains very small Pt x clusters.In contrast to sample B,the second shell in sample A does not fit with Pt–Pt parison of three different models (Supplementary Table S1)shows that the best fit for the second shell of sample A is with a contribution of Pt–Fe at a distance of 2.88Å,with an average coordination number of 0.9.As the nearest neighbours of Pt are O atoms,the Pt–Fe coordination orig-inates from the next-nearest neighbour Fe bridged by the nearestoxygen atom.The EXAFS data do not reveal any Pt–Pt contribution in sample A,in agreement with the HAADF result that sample A contains only single Pt atoms.Figure 2b shows the normalized X-ray absorption near-edge structure (XANES)spectra of samples A and B,and the reference spectra of Pt foil and PtO 2.The white line intensities in the spectra reflects the oxidation state of Pt in different samples 30,so the white-line intensity in sample A,which is between the intensities of Pt foil and PtO 2,suggests that the Pt single atoms in sample A carry positive charges.In contrast,the white-line intensity of sample B is clearly lower than that of sample A,and is similar to that of the Pt foil,which indicates the dominance of Pt 0clusters in sample B.Fourier transform infrared studies of CO adsorption.We investigated further the CO adsorption behaviour of the catalysts using Fourier-transform infrared (FTIR)spectroscoy to provide additional information about the dispersion and oxidation state of Pt.T able 1|EXAFS parameters of samples A and B.Sample Shell N R (Å)D s 23103(Å2)D E 0(eV)Pt foil Pt–Pt 12.0 2.81PtO 2Pt–O 6.0 2.07Pt–Pt 6.0 3.01Sample BPt–O 2.0 1.98 1.76 2.3Pt–Pt 3.8 2.537.22210.0Sample A (Model I*)Pt–O 1.9 2.02 4.199.8Pt–Fe 0.9 2.88 1.73210.0Sample A (used)Pt–O 3.6 2.01 4.938.5Pt–Fe1.42.5613.010.0N ,coordination number;R ,distance between absorber and backscatter atoms;Ds 2,change in the Debye–Waller factor value relative to the Debye–Waller factor of the reference compound;D E 0,inner potential correction to account for the difference in the inner potential between the sample and the reference compound.Error bounds (accuracies)that characterize the structural parameters obtained by EXAFS spectroscopy were estimated as N ,+20%;R ,+1%;Ds 2,+20%;D E 0,+20%.Pt foil parameter from data_41525-ICSD;PtO 2parameter from data_24923-ICSD;r space fit,D k ¼2.8–10.0Å21,D r ¼1.0–3.2Å,13statistically justified free parameters;S 02fitting from PtO 2foil defined as 0.95.*For details see the Supplementary Information.2,080 cm –12,030 cm –11,950 cm –11,860 cm –1a2,3002,2002,1002,0001,9001,800A b s o r b a n c e (a .u .)Evacuation for 30 min10.0 torr 5.00 torr 0.50 torr 0.05 torr0.005A b s o r b a n c e (a .u .)Wavenumber (cm –1)2,3002,2002,1002,0001,9001,800Wavenumber (cm –1)b0.1Evacuation for 30 min 8.1 torr 5.0 torr 1.0 torr2.9 × 10–1 torr 1.0 × 10–2 torr 7.0 × 10–3 torr3.0 × 10–3 torr 1.0 × 10–3 torrFigure 3|In situ FTIR spectra of CO adsorption for samples A and B.a ,For sample A,the band at 2,080cm 21is ascribed to CO,which is linearlyadsorbed on Pt d þ.The independence of frequency with CO pressure suggests a lack of interaction between the adsorbed CO molecules,and confirms that the single Pt atoms in sample A are well isolated.b ,For sample B,the bands at 2,030cm 21,1,860cm 21and 1,950cm 21are,respectively,ascribed to the linearly bonded CO on Pt 0sites,bridge-bonded CO on two Pt atoms and CO adsorbed on the interface between Pt clusters and the support.That bridge-bonded CO was detected and the frequency was blue shifted with CO pressure suggests the presence of Pt clusters in sample B.DOI:10.1038/NCHEM.1095Figure 3shows that,for sample B,the adsorption of CO produces a strong vibration band at 2,030cm 21and two other weak bands at 1,860cm 21and 1,950cm 21,respectively.The main band at 2,030cm 21can be ascribed to linearly bonded CO on Pt 0sites,and the bands at 1,860cm 21and 1,950cm 21are caused by bridged adsorption of CO on two Pt atoms,as well as CO adsorbed on the interface between Pt clusters and the support 31.The formation of bridge-bonded CO indicates the existence of dimer or Pt clusters,consistent with the STEM and EXAFS results that Pt clusters coexist with individual Pt atoms in sample B.In comparison to other supported Pt catalysts 32,33(for example,Pt /SiO 2and Pt /Al 2O 3),the terminal absorption band of CO on Pt 0of sample B had a large red shift,consistent with the data obtained from the very small size of Pt clusters on the FeO x support 33,34.With an increase of CO pressure,the absorption band was blue shifted because of the coupling of adsorbed CO molecules 35.After evacuation for 30minutes,the intensity of the terminal band at 2,030cm 21remained almost unchanged,which indicates that the adsorption of CO on Pt 0is irreversible.To verify the above assignment of the band at 2,030cm 21,sample B was exposed further to oxygen following CO adsorption and evacuation.The result (Supplementary Fig.S6)shows that,with an increase of the O 2pressure,the CO adsorption band at 2,030cm 21became weak as a new band at about 2,070cm 21evolved gradually;this suggests that Pt 0was oxidized gradually by O 2.Thus,the band at 2,030cm 21was,indeed,caused by the linearly adsorbed CO on Pt 0and the new band at 2,070cm 21can be assigned to CO adsorbed on Pt d þ(refs 36,37).For CO adsorption on sample A,only a weak band appeared at 2,080cm 21,which can be ascribed to CO adsorbed on Pt d þ.Unlike that for sample B,the band position of CO adsorbed on sample A was almost unchanged with increasing CO pressure,which suggests a lack of interaction between adsorbed CO molecules on Pt because of the isolation of single Pt atoms.Moreover,the adsorption of CO on single Pt d þatoms was rather weak and partially reversible,as indicated by the significant decrease of the peak intensity on evacuation.A more detailed investigation of CO adsorption on sample A revealed that the introduction of either H 2or O 2did not result in noticeable changes of the CO adsorption band (Supplementary Fig.S7),which indicates that the anchored single Pt d þatoms are stable and are not reduced in the presence of H 2at room temperature.The systematic FTIR spectroscopy study further confirmed that sample A contained only positively charged Pt single atoms and that sample B contained both single atoms and Pt clusters.In summary,all the characterization data discussed above provide strong evidence that sample A was composed of only iso-lated,positively charged single Pt atoms.The strong electrostatic and covalent interactions between the positively charged single Pt atoms and the FeO x support helped to stabilize the single Ptatoms on the FeO x support.As we show below,these FeO x -stabil-ized single Pt atoms exhibited remarkable catalytic performance for CO oxidation and PROX reactions.Catalytic performance.We chose CO oxidation and PROX as probe reactions to investigate the catalytic performance of FeO x -supported single Pt atoms,as such reactions are critical in cleaning air 38and in fuel-cell applications 39.Table 2lists the reaction rates and turnover frequencies (TOFs)of the two samples for CO oxidation and PROX reactions.As a reference,also the standard Au /Fe 2O 3catalyst (provided by the World Gold Council),well-known for its high activity for CO oxidation 40,was tested under the same reaction conditions.For CO oxidation,sample A gave a specific reaction rate of 0.435mol CO h 21g Pt 21at the reaction temperature of 278C,which is double that of Au /Fe 2O 3and almost triple that of sample B.The TOFs,calculated based on the dispersion of metals on the catalysts,also revealed that sample A was 2–3times more active than both sample B and Au /Fe 2O 3.For the PROX reaction,all three catalysts were more active than for CO oxidation,and sample A was again 2–3times as active as the other two samples.This result is surprising given that the commercial Pt /Al 2O 3catalyst is at least 1–2orders of magnitude less active than the standard gold nanocatalysts for low-temperature CO oxidation and PROX reactions 41.In fact,our single-atom Pt catalyst was the most active for CO oxidation and PROX reactions among a number of supported Pt catalysts (Supplementary Table S2).The high activity of sample A must have originated from the intrinsic nature of single Pt atoms dispersed onto FeO x surfaces,as our tests showed that the FeO x support itself was essentially inactive for the PROX reaction at 808C.Elemental analyses showed that,in addition to surface hydroxyl groups (Supplementary Fig.S8),some impurities (mainly Na þ)were present on the surface of the Pt 1/FeO x catalyst;however,they had little effect on the catalytic activity (Supplementary Fig.S9).The single-atom Pt catalyst was stable during the PROX reaction.As shown in Supplementary Fig.S10,at a space velocity of 2.1×106ml g Pt 21h 21,both the CO conversion and the CO 2selectivity over sample A remained at constant values over a 1,000minute run at 808C (the typical working temperature for polymer electrolyte membrane fuel cells)without any decay,which suggests that sintering of single Pt atoms did not occur during the catalytic reactions.To evaluate the deactivation trend of sample A,we increased the space velocity by about five times to 1.1×107ml g Pt 21h 21and found that the CO conversion began to decrease gradually only after a 300minute run.Concurrently,the selectivity of CO 2began to increase (Supplementary Fig.S10),which suggests that the catalyst deactiva-tion took place during the accelerated deactivation test and this deactivation was more prominent for the oxidation of H 2.When the catalyst bed was purged with He for 30minutes at 2008C,however,the deactivated catalyst regenerated completely.ThisT able 2|Comparison of reaction rates and TOFs.Metal loadings (wt%)Reaction type T emperature (8C)Specific rate 3102(mol CO h 21g metal 21)TOF 3102(s 21)†Sample A 0.17CO oxidation 2743.513.6Sample B 2.5CO oxidation 2717.78.01Au /Fe 2O 3* 4.4CO oxidation 2721.7 4.76Sample A 0.17PROX 2767.621.2Sample B 2.5PROX 2720.39.15Au /Fe 2O 3* 4.4PROX 2739.38.60Sample A 0.17PROX 8099.231.1Sample B 2.5PROX 8035.816.2Au /Fe 2O 3*4.4PROX8080.317.6*Provided by World Gold Council.†TOFs were calculated based on the metal dispersion.For samples A and B,the metal dispersion was measured by CO chemisorption conducted on a BT2.15heat-flux calorimeter and on an Auto Chem II 2920,respectively;for Au /Fe 2O 3,the dispersion was estimated by the metal particle size according to D ¼1/d Au .DOI:10.1038/NCHEM.1095result indicates that the deactivation of sample A during the acceler-ated deactivation test was not caused by the sintering of Pt atoms,but most probably by accumulation of carbonate species on the catalyst surface 31,42.Further studies revealed that the Pt 1/FeO x catalyst was stable even after two cycles of oxidation–reduction treatments (Supplementary Fig.S11).The STEM images of the used sample A shown in Supplementary Fig.S12reveal that the used catalyst still consists pri-marily of isolated single Pt atoms.The STEM images of the used cat-alyst provides direct evidence that the working Pt 1/FeO x catalyst was stable and that there was no observable agglomeration of the individual Pt atoms during the catalytic reactions.To study further the changes of the oxidation state of the Pt 1/FeO x catalyst during the reaction,we conducted in situ diffuse reflectance infrared Fourier transform spectroscopy (Supplementary Fig.S13)exper-iments under the PROX reaction conditions.The absorption band of CO adsorbed on sample A had a small blue shift with the time on stream,which suggested that the Pt single atoms were oxidized slightly during the PROX reaction.The XANES and EXAFS data of the used catalyst also confirmed that the oxidation state of the Pt single atoms was increased,as shown by the increased white-line intensity and the Pt–O coordination number (Fig.2and Table 1).There was still no evidence of the formation of Pt–Ptbonding in the used catalyst (Table 1),which corroborates the STEM results discussed above.Density functional theory studies.To elucidate the nature of the binding of Pt single atoms to the FeO x support and the exceptionally high catalytic activity of single Pt atoms,we carried out extensive theoretical investigations using relativistic density functional theory (DFT).a -Fe 2O 3has a hexagonal corundum structure with antiferromagnetism.The (001)surface is one of the predominant growth surfaces,and single Fe-terminated,double Fe-terminated and O 3-terminated surfaces can form when different surface atoms are exposed 43–45.However,previous experimental and theoretical results revealed that the Fe-terminated Fe 2O 3(001)surface is the inherently most stable surface 43,46–48,which indicates that most probably the Fe-terminated Fe 2O 3(001)surface was exposed in the supported catalysts.We compared the stabilities of Pt located at different sites on the three differently terminated Fe 2O 3(001)surfaces.As shown in Supplementary Fig.S14and Supplementary Table S3,our DFT calculations indicate that the most probable sites for single Pt atoms on the various possible Fe 2O 3surfaces are the three-fold hollow sites on the O 3-terminated surface,where each Pt atom is coordinated by three surface oxygen atoms,with thei. Oxygen vacancyPt–O: 1.93 ÅPt–Fe: 2.52 ÅO–O: 1.46 ÅO–O: 1.44 ÅC–O: 1.15 ÅPt–C: 1.88 ÅO–O: 2.06 ÅC–O: 1.15 ÅPt–C: 1.90 ÅPt–O: 1.92 ÅC–O: 1.16 ÅPt–C: 1.87 ÅPt–C: 1.98 ÅC–O cyan : 1.21 ÅC–O green : 1.35 ÅPt–C: 2.13 ÅC–O cyan : 1.19 ÅC–O green : 1.26 ÅTS-1ii. O 2 adsorptioniii. CO adsorptioniv. Oxygen fillTS-2v. CO adsorptionvi. CO migration bai. O ovii. O 2iii. CO adsorptionTS-1O 2O atom in CO C atom in CO O atom in O 2O atom in FeO x surface O atom in bulk FeO x Fe atom in bulk FeO xPt atomFigure 4|The proposed reaction pathways for CO oxidation on the Pt 1/FeO x catalyst (sample A).a ,b ,T op view (a )and side view (b ).After pre-treatment by H 2,the stoichiometric haematite surfaces near the Pt atoms were reduced partially to form an O vac (step i)that can adsorb the O 2reactants (step ii)as CO is adsorbed on the single Pt atoms (step iii).Through an activation barrier of 0.49eV (TS-1),the first CO 2molecule is released and the surface oxygen vacancy is healed by the remaining O ad atom of the O 2reactant (step iv).When the second CO molecule is adsorbed at the Pt atom (step v),it migrates to a neighbouring oxygen atom (step vi)to form a transition state with a barrier of 0.79eV (TS-2),which leads to a new CO oxidation.By releasing the second CO 2,the Pt-embedded stoichiometric surface is reduced again to form a new O vac (step i).The inset in the cycle (a )shows the calculated energy profile,with thepartially reduced sample A system as the reference for the energies (in eV).After one catalytic cycle,the catalyst is recovered and releases two CO 2molecules.DOI:10.1038/NCHEM.1095third-layer Fe atoms below the Pt atoms(Supplementary Fig.S15).This location of Pt can be viewed as the surface Fe atoms on thesurface terminated by single Fe atoms being replaced by single Ptatoms,consistent with the results from the HAADF images that Pt atoms occupy exactly the positions of the Fe atoms.The DFTcalculations show that the formation of an oxygen vacancy(O vac), defined as the energy needed to form one oxygen vacancy on thesurface and half an oxygen molecule in the gas phase,is much lower on the Pt-terminated surface( 1.06eV)than that on themost-stable Fe-terminated surface of the Fe2O3support( 2.99eV),in agreement with the temperature-programmed reductionresult(Supplementary Fig.S16).This result indicates that thepresence of individual Pt atoms can improve greatly the reducibility of the Fe2O3support.Moreover,the Bader charge analysis(Supplementary Table S4)shows that the Pt atoms carry considerablepositive charges(þ0.45|e|),which indicates significant electrontransfer from Pt atoms to the FeO x support.The stretchingvibration frequency of CO adsorbed on Pt single atoms was calculated as2,062cm21,a blue shift of44cm21relative to thefrequency of CO adsorbed on a free Pt10cluster.This frequencyshift is in excellent agreement with our observed infrared spectra.The calculated adsorption energy for CO is much lower(1.96eV)on the single Pt atoms embedded in the FeO x support than on the free Pt10clusters(3.05eV).All these results suggest that the Ptsingle atoms embedded into the Fe2O3surface had a highoxidation state because of coordination by the three surface Oatoms,significantly different from that of Pt atoms in clusters ormetal surfaces.Bonding analysis on model systems showed that the back-donating interaction between CO and Pt was much less forPt1/FeO x than for Pt x clusters,and so the more-vacant d orbitals of the individual Pt atoms in the Pt1/FeO x catalyst play a vital role inthe remarkable catalytic activity of this catalyst(Supplementary Fig.S17).These explain the low CO-adsorption energy,whichreduces the effect of CO poisoning and facilitates the adsorption ofoxygen,and thus enhances the activity of CO oxidation on the Ptsingle-atom sites.The catalytic cycle of CO oxidation over the single atom Pt catalyst from the DFT calculations is summarized in Fig.4.Afterpre-reduction by H2,the stoichiometric haematite surfaces near the Pt atoms are reduced partially to form an oxygen vacancy(step i),which can adsorb the O2reactants.In this model,an O vacnear the Pt atoms is used to model the reduced FeO x surfaces.In this case,the oxygen-coordination number of Pt changes fromthree to two,consistent with the EXAFS-fitting result on freshlyreduced Pt1/FeO x catalyst.The calculated adsorption energy of O2(1.05eV)and the O–O bond length(1.46Å)suggest that theadsorbed oxygen(O2,ad)is well activated over Pt single atoms (step ii).The adsorption of CO on the single Pt atoms has abinding energy of1.27eV(step iii),which is much lower thanthat we calculated for a Pt x cluster.The calculated activationbarrier for reaction of CO adþO–O ad CO2þO ad(TS-1)is 0.49eV.After the release of the CO2molecule the surface oxygenvacancy is healed by the remaining O ad atom of the O2reactant,which restores the Pt-substituted stoichiometric haematite surfaces(step iv).When the second CO molecule is adsorbed at Pt(stepv),a new reaction of CO adþO ad CO2þO vac occurs(step vi and TS-2).The calculated barrier of this reaction is0.79eV.Afterreleasing the second CO2,the Pt-embedded stoichiometric surfaceis reduced again to form a new oxygen vacancy(step i).This multi-step procedure thusfinishes the catalytic cycle.The two CO oxi-dation reactions of CO adþO–O ad and CO adþO ad both follow Langmuir–Hinshelwood mechanisms.The activation barriers on the single Pt atoms are remarkably lower than that( 1eV)of CO oxidation on Pt(111)(ref.49).All the elementary steps in the catalytic cycle are exothermic and the barriers are low enough for CO oxidation at low temperatures.Thus the Pt1/FeO x catalyst has well-balanced thermodynamics and kinetics for CO oxidation and the PROX reactions.ConclusionsIn summary,we synthesized and characterized a novel catalyst that consists of only single Pt atoms uniformly dispersed on a FeO x support of high surface area.This catalyst showed extremely high activity for both CO oxidation and PROX reactions.The chemical reactivity of our positively charged single Pt atoms is quite different from that of single Au atoms or cations on the same support;the commercial Au/FeO x catalyst contains polydispersed nanostructures from single atoms to clusters,with single Au atoms reported to be inactive for CO oxidation3,27,50.The more vacant d orbitals of the single Pt atoms,because of the electron transfer from Pt atoms to the FeO x surface,are responsible both for the strong binding and stabilization of single Pt atoms and for providing positively charged Pt atoms,which ultimately account for the excellent catalytic activity of the Pt1/FeO x catalyst.Although we used CO oxidation to demon-strate the high activity of the Pt1/FeO x catalyst,the observed behav-iour of single Pt atoms is,nonetheless,not limited to CO oxidation. Moreover,the stabilization of single atoms on practical oxide supports via the charge-transfer mechanism is not limited to the Pt1/FeO x system,but can be extended further and made applicable to other pre-cious-metal systems.The discovery of this single-atom Pt catalyst not only proves the concept of single-atom heterogeneous catalysis,but also has a great potential to reduce the high cost of commercial noble-metal catalysts in industry.MethodsSample preparation.Both sample A and sample B were prepared byco-precipitation of an aqueous solution of chloroplatinic acid(H2PtCl6.6H2O, 7.59×1022mol l21)and ferric nitrate(Fe(NO3)3.9H2O,1mol l21)with sodium carbonate(Na2CO3,1mol l21)at508C,with the pH value of the resulting solution controlled at about8.The recovered solid was dried at608C forfive hours and calcined at4008C forfive hours.Prior to being characterized and tested,the samples were reduced in10%H2/He at2008C for0.5hours.The actual Pt loadings in the two samples,determined by inductively coupled plasma spectroscopy on an IRIS Intrepid II XSP instrument(Thermo Electron Corporation),were0.17wt%for sample A and2.50wt%for sample B.Sample characterization.Suba˚ngstro¨m resolution HAADF STEM images were obtained on a JEOL JEM2200FS STEM/TEM,equipped with a CEOS(Heidelburg, Ger)probe corrector,and a nominal image resolution of0.07nm,and on a JEOL JEM-ARM200F STEM/TEM,also equipped with a CEOS probe corrector,with a guaranteed resolution of0.08nm.Before microscopy examination,the samples were dry dispersed onto a copper grid coated with a thin holey carbonfilm.Pt L3-edge absorption spectra(EXAFS)were performed on two beamlines.One was the BL14W1beamline at the Shanghai Synchrotron Radiation Facility(SSRF), Shanghai Institute of Applied Physics(SINAP),China,operated at3.5GeV with injection currents of140–210mA.The other was the1W1B beamline of the Beijing Synchrotron Radiation Facility(BSRF)operated at 200mA and 2.2GeV.On both beamlines,a Si(111)double-crystal monochromator was used to reduce the harmonic component of the monochrome beam.Pt foil and PtO2were used as reference samples and measured in the transmission mode,and both sample A and sample B were measured influorescence mode.Before measurement,the powder sample was pre-reduced at2008C for30minutes(the same pre-treatment condition as that used for the activity test)in a specially designed reactor equipped with shut-off valves at both ends.Prior to disconnection with theflowing hydrogen,the valves on the reactor were closed so that the freshly reduced sample was not exposed to air. After that,the reactor was transferred into an argon-filled glove box and the sample taken out and pressed into a self-supporting disk.The disk was then sealed with Kapton membrane and subjected to EXAFS measurement.For comparison,sample A after the PROX reaction(denoted as used catalyst)was also measured for EXAFS, but without sealing treatment.To obtain a much better signal-to-noise ratio,filters were used to reduce thefluorescence of Fe in addition to extending the integral time. We used IFEFFIT software to calibrate the energy scale,to correct the background signal and to normalize the intensity.A similar approach was used to analyse the EXAFS data at the Pt L3edge.Reliable parameters for the high Z(Pt,Fe)and low Z (O)contributions were determined by multiple-shellfitting in r space with application of k3and k1weightings in the Fourier transformations.FTIR spectra for CO adsorption on the two samples were collected at208C with a Bruker Equinox55spectrometer equipped with a mercury cadmium telluride detector at a resolution of4cm21using60scans for sample B and300scans for sample A.Before CO adsorption,the samples were reduced in situ at2008C withDOI:10.1038/NCHEM.1095。
Topological susceptibility for the SU(3) Yang--Mills theory
ar X i v:h e p -l a t /0409100v 1 16 S e p 2004Topological susceptibility for the SU(3)Yang–Mills theory Luigi Del Debbio a ,Leonardo Giusti b ∗and Claudio Pica c a CERN,Department of Physics,TH Division,CH-1211Geneva 23,Switzerland b Centre de Physique Th´e orique,Case 907,CNRS Luminy,F-13288Marseille Cedex 9,France c Dipartimento di Fisica dell’Universit`a di Pisa and INFN,Via Buonarroti 2,I-56127Pisa,Italy We present the results of a computation of the topological susceptibility in the SU(3)Yang–Mills theory performed by employing the expression of the topological charge density operator suggested by Neuberger’s fermions.In the continuum limit we find r 40χ=0.059(3),which corresponds to χ=(191±5MeV)4if F K is used to set the scale.Our result supports the Witten–Veneziano explanation for the large mass of the η′.1.Introduction The topological susceptibility in the Yang–Mills (YM)gauge theory can be formally defined in Euclidean space-time as χ= d 4x q (x )q (0) ,(1)where the topological charge density q (x )is q (x )=−12N f =χ,(3)where F πis the pion decay constant.The dis-covery of a fermion operator [5]that satisfies the Ginsparg–Wilson (GW)relation [6]triggered a breakthrough in the understanding of the topo-logical properties of the YM vacuum [7,8,4,9,10],and made it possible to give a precise and unam-biguous implementation of the WV formula [4].Indeed the naive lattice definition of the topo-logical susceptibility has a finite continuum limit,V (4)with Q ≡ x q (x )=n +−n −being the topo-logical charge,V the volume,and n +(n −)the number of zero modes of D with positive (nega-tive)chirality in a given ing new simulation algorithms [11],it is now possible to investigate the WV scenario from first principles for the first time.More precisely,the aim of the work presented here,and fully described in [12],is to achieve an accurate and reliable de-termination of χin the continuum limit,which in turn allows a verification of the WV mecha-nism for the η′mass.Several exploratory com-putations have already studied the susceptibility employing the GW definition of the topological charge [13,14,15,16,17,18,19]ttice computation The ensembles of gauge configurations are gen-erated with the standard Wilson action and peri-odic boundary conditions,using a combination of heat-bath and over-relaxation updates.More de-tails on the generation of the gauge configurations can be found in Refs.[18,19].Table 1shows the 1list of simulated lattices,where the bare coupling constantβ=6/g20,the linear size L/a in each direction and the number of independent config-urations are reported for each lattice.The topo-logical charge density is defined asq(x)=−¯aL/a N conf Q2 r40χ1213492.79(12)0.0543(24)F5.951212911.955(79)0.0551(24)G6.01235861.489(37)0.0596(18)H6.1169622.45(13)0.0599(33)J6.21817212.114(76)0.0591(24)3mensional product r40χ.The s=0.0and s=0.4data sets are represented by black circles andwhite squares respectively.The dashed lines rep-resent the results of the combinedfit describedin the text.Thefilled diamond at a=0is theextrapolated value in the continuum limit.earfit of the data.Thisfit gives a very goodvalue ofχ2dof when all sets are included,and isvery stable if some points at larger values of a2/r20are removed.In particular a combinedfit of allpoints with a2/r20<0.05gives c0=0.059(3)withχ2dof≈0.73.Since r0is not directly accessibleto experiments,we express our result in phys-ical units by using the lattice determination ofr0F K=0.4146(94)in the quenched theory[21]and we obtainχ=(191±5MeV)4,(6)which has to be compared with[2]F2π。
杨振宁的英文作文
杨振宁的英文作文Yang Zhenning, also known as Chen Ning Yang, is a renowned Chinese physicist who has made significant contributions to the field of theoretical physics. Born in 1922 in Hefei, China, Yang went on to study at theUniversity of Chicago, where he earned his Ph.D. in physics.Yang's research has focused on a wide range of topics, including particle physics, statistical mechanics, and condensed matter physics. He is perhaps best known for his work on the theory of non-conservation of parity, which he developed alongside Tsung-Dao Lee in 1956. This theory helped to explain certain phenomena in particle physics and earned the two scientists the Nobel Prize in Physics in 1957.In addition to his research, Yang has also been a dedicated teacher and mentor throughout his career. He has taught at several prestigious universities, including the University of Chicago, Princeton University, and the StateUniversity of New York at Stony Brook. He has also supervised numerous graduate students and postdoctoral researchers, many of whom have gone on to successfulcareers in physics.Despite his many accomplishments, Yang has remained humble and focused on his work throughout his career. Hehas said that he is motivated by a desire to understand the fundamental laws of the universe and to contribute to the advancement of human knowledge. His dedication and passion for physics have inspired countless others in the field and helped to shape our understanding of the universe.In conclusion, Yang Zhenning is a remarkable physicist whose contributions to the field of theoretical physicshave had a profound impact on our understanding of the universe. His work on the theory of non-conservation of parity, in particular, has helped to explain some of the most fundamental phenomena in particle physics. Furthermore, his dedication to teaching and mentoring has helped to inspire a new generation of physicists and ensured that his legacy will continue for many years to come.。
杨振宁获诺奖英文作文
杨振宁获诺奖英文作文英文：Yang Zhenning, a Chinese physicist, was awarded the Nobel Prize in Physics in 1957 for his work on parity non-conservation. This discovery fundamentally changed our understanding of the laws of physics and had a profound impact on the field of particle physics.I am particularly impressed by Yang's ability to think outside the box and approach problems in a unique way. For example, in his research on parity non-conservation, he used a method of statistical analysis that was not commonly used in physics at the time. This innovative approach allowed him to make groundbreaking discoveries and ultimately win the Nobel Prize.In addition to his scientific achievements, Yang's life story is also inspiring. He grew up during a time of political turmoil in China and faced many challenges inpursuing his education and career. Despite these obstacles, he persevered and became one of the most respectedphysicists of his time.中文：杨振宁是一位中国物理学家，因其在反演不守恒方面的工作而于1957年获得诺贝尔物理学奖。
爱因斯坦 波多尔斯基 罗森佯谬
在圆括弧内，称第一个项目为直积态 I，是两个量子态、的张量乘积，第二个项目为直积态 II，是两个量 子态、的张量乘积。在直积态 I里，量子态为的电子，其自旋的z轴分量为正值；量子态为的正电子，其为负值。 在直积态 II里，量子态为的电子，其为负值；量子态为的正电子，其为正值。但假若不做测量，则无法知道这 两个粒子中任何一个粒子的；根据哥本哈根诠释，这变量根本不存在。
EPR论文接着描述，先前相互作用的两个粒子，在分离之后的物理性质。假设两个粒子A、B在原点位置相互 作用之后，以相反方向移动分离。根据不确定性原理，由于位置算符与动量算符不对易，无法同时确定粒子B的位 置与动量；位置越确定则动量越不确定，反之亦然。假设准确测量出粒子A的位置，则由于粒子A与粒子B之间相 隔很远，测量粒子A不会搅扰到粒子B，粒子B的位置可以准确地预测为（概率为1），因此，按照实在性判据，对 于测量粒子B的位置，必定存在物理实在的要素。在这里，作者假设测量粒子A这动作遵守定域论，另外，由于存 在物理实在的要素，遵守实在论，粒子B的位置可以被预测。类似地，假设准确测量出粒子A的动量，则由于测量 粒子A不会搅扰到粒子B，粒子B的动量可以准确地预测为（概率为1），因此，按照实在性判据，对于测量粒子B 的动量，必定存在物理实在的要素。
EPR论文表明，假若定域实在论成立，则可以推导出量子力学的不完备性。
理论概述
EPR论文
玻姆版本
爱因斯坦003914(2)英语课件PPT
Born: 14 March 1879 in Ulm, Wü rttemberg, Germany
Died: 18 April 1955 in Princeton, New Jersey, USA
制作，宣讲 康乐
German-born American theoretical physicist whose special and general theories of relativity revolutionized modern thought on the nature of space and time and formed a theoretical base for the exploitation of atomic energy. He won a 1921 Nobel Prize for his explபைடு நூலகம்nation of the photoelectric effect.
16.付出真心，才会得到真心，却可能伤的彻底；保持距离，才能保护自己，却注定永远寂寞。 93.自信是走向成功的第一步，缺乏自信是失败的主要原因。——莎士比亚 17.成功的信念在人脑中的作用就如闹钟，会在你需要时将你唤醒。 54.真正决定一个人成就的，不是天分，也不是运气，而是严格的自律和高强度的付出，一直坚持做正确的事情，简单的事情重复做，重复的事情用心做，成功离你就不远了！ 65.没有退路时潜能就发挥出来了。 69.现在的你，应该明白，没有什么捷径能让你出类拔萃，没有哪些艰难困苦是白白煎熬。你的每一份经历，不管是顺境还是坎坷，都会增加你生命的厚度。世界很小，请带着梦想一起奔跑；世 界很大，请带着坚持努力向前！
47.如果要后退，上帝就会在我们的后脑长双眼睛了。
核糖体研究获2009年度诺贝尔化学奖
核糖体研究获2009年度诺贝尔化学奖李升伟/编译拉马克里希南（左）尤纳斯（中）施泰茨（右）瑞典皇家科学院10月7日宣布，3位科学家因揭示了DNA链状结构上的编码信息是如何翻译组成生命物质的蛋白质，获得了2009年度诺贝尔化学奖。
来自剑桥大学分子生物学实验室（LMB）的文卡特拉曼·拉马克里希南(Venkatraman Ramakrishnan)、耶鲁大学的托马斯·施泰茨(Thomas Steitz)和以色列魏茨曼科学研究所的阿达·尤纳斯(Ada Yonath)，将于12月10日在斯德哥尔摩分享总额为1000万瑞典克朗（约140万美元）的奖金。
核糖体研究获殊荣3位科学家和他们的合作者各自独立地利用由强场粒子加速器产生的X线照射，并运用巨型计算机进行计算，成功地绘出了细胞内称为核糖体的大分子复合体中数十万个原子的分布图。
在一次新闻发布会上，瑞典皇家科学院称，他们得奖是因为“揭示了核糖体的结构及其是如何在分子水平上发挥作用的”。
根据细菌核糖体设计的一些抗生素，他们的工作在医学上已经有了重要的应用，可以使一些细菌中止对它们的宿主的伤害。
瑞典科学院称，核糖体的研究正在被用于开发新的抗生素。
拉马克里希南博士于1952年生于印度奇丹巴拉姆邦，在俄亥俄大学获得博士学位，并加入了美国国籍。
施泰茨博士则于1940年出生于密尔沃基市，在哈佛大学获得博士学位。
尤纳斯博士于1939年生于巴勒斯坦耶路撒冷，在以色列魏茨曼科学研究所获得博士学位并一直在以色列工作。
当接受电话采访的时候，尤纳斯说，其实有些人早就对她说，她的研究项目会成为诺奖的赢家。
然而她补充说，其实“有很多很多的人有很杰出的工作，但是还没有得到这项大奖”。
尤纳斯说，当她收到获奖消息时正在工作，同时还在照顾着她的13岁的孙女。
最早向她道贺的人中，有以色列总统西蒙·佩雷斯(Shimon Peres)，他与已故总统伊扎克·拉宾(Yitzhak Rabin)和（巴勒斯坦）亚西尔·阿拉法特(Yasir Arafat)一起获得过1994年的诺贝尔和平奖。
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a rXiv:h ep-th/95453v11A pr1995SU(3)Einstein-Yang-Mills Sphalerons and Black Holes Burkhard Kleihaus 1,Jutta Kunz 1,2and Abha Sood 11Fachbereich Physik,Universit¨a t Oldenburg,Postfach 2503D-26111Oldenburg,Germany 2Instituut voor Theoretische Fysica,Rijksuniversiteit te Utrecht NL-3508TA Utrecht,The Netherlands February 1,2008Abstract In the SU(3)Einstein-Yang-Mills system sequences of static spherically sym-metric regular solutions and black hole solutions exist for both the SU(2)and the SO(3)embedding.We construct the lowest regular solutions of the SO(3)embedding,missed previously,and the corresponding black holes.The SO(3)solutions are classified according to their boundary conditions and the numberof nodes of the matter functions.Both,the regular and the black hole solutions are unstable.Utrecht-Preprint THU-95/81IntroductionThe SU(2)Einstein-Yang-Mills system possesses a sequence of static spherically sym-metric regular particle-like solutions[1],which are unstable[2,3].The n-th solution of the sequence has n nodes and2n unstable modes[4,5].The lowest solution has been interpreted in analogy to the electroweak sphaleron[6]as the top of a barrier between vacua[3].Beside the regular solutions the SU(2)Einstein-Yang-Mills system possesses static spherically symmetric black hole solutions.There are in fact two different types of black hole solutions having the same mass.These are the Schwarzschild black holes with vanishing gaugefields and the SU(2)coloured black holes[7,8,9].Like the regular solutions the coloured black hole solutions are unstable[10,11,5].Thus for a certain range of masses the system possesses two distinct types of black holes,providing a counterexample to the“no-hair conjecture”for black holes,unless the coloured black holes are discarded as a counterexample because of their instability.Here we consider static spherically symmetric regular solutions and black holes of the SU(3)Einstein-Yang-Mills system(with vanishing time component of the gauge field).Such solutions are obtained for both,the SU(2)embedding and the SO(3)em-bedding.The SU(2)embedding reproduces the known SU(2)solutions,while the SO(3) embedding leads to new interesting solutions.Missing the lowest regular solutions,sev-eral regular SO(3)solutions have been found previously by K¨u nzle[12],but he did not succeed in obtaining the corresponding black holes.We here construct thefirst few solutions of a new class of regular SO(3)solutions. These solutions include the lowest regular SO(3)solution,which we have obtainedfirst as a limiting solution of the SU(3)Einstein-Skyrme system[13](in analogy to the SU(2) case[14]).We classify the SO(3)solutions according to their boundary conditions and the number of nodes of the matter functions.Analogous to the SU(2)Einstein-Yang-Mills system asymptoticallyflat SO(3)black hole solutions emerge from the regular solutions by requiring regularity at afinite event horizon.We construct the black hole solutions corresponding to thefirst few solutions with the least nodes.The stability of regular and black hole solutions of arbitrary gauge groups has been studied recently[15].We apply the theorems of Ref.[15]to demonstrate the instability of both regular and black hole SO(3)solutions.2SU(3)Einstein-Yang-Mills Equations of Motion We consider the SU(3)Einstein-Yang-Mills actionS=S G+S M= L G√−gd4x(1)withL G=12Tr(FµνFµν),(3) whereFµν=∂µAν−∂νAµ−ie[Aµ,Aν],(4)Aµ=1r.(7) Generalized spherical symmetry for the gaugefield is realized by embedding the SU(2) or the SO(3)generators T i in SU(3).In the SU(2)-embedding T=12re( e r× τ)i,(8)with the SU(2)Pauli matrices τ=(τ1,τ2,τ3).In the SO(3)-embedding T=(λ7,−λ5,λ2), and the corresponding ansatz for the gaugefield with vanishing time component is A0=0,A i=2−K(r)2re ( e r× Λ)i, e r· Λ)+,(9)where[,]+denotes the anticommutator,and Λ=(λ7,−λ5,λ2).The SU(2)-embedding,eq.(8),leads to the well studied SU(2)Einstein-Yang-Mills equations[1,7,8,9].To obtain the SU(3)Einstein-Yang-Mills equations for the SO(3)-embedding,eq.(9),we also employ the tt and rr components of the Einstein equations, yielding for the metric functionsµ′=N(K′2+H′2)+12A′=√√4πG)r,and the prime indicates the deriva-tive with respect to x.For the matterfield functions we obtain the equations1(ANK′)′=AH H2+7K2−4 .(14)4x2With help of eq.(11)the metric function A can be eliminated from the matterfield equations.Note,that the equations are symmetric with respect to an interchange of the functions K(x)and H(x),and to the transformations K(x)→−K(x),and H(x)→−H(x),yielding degenerate solutions.Comparing the equations of the SO(3)embedding to those of the SU(2)embed-ding[1]shows that to each SU(2)solution there corresponds a scaled SO(3)solution. Defining x=2˜x,andµ=2˜µthe functions K(x)=2w(˜x),H(x)=0satisfy the SO(3) equations with coordinate x,when the function w satisfies the SU(2)equations with coordinate˜x.Thus these SO(3)solutions have precisely double the mass of their SU(2) counterparts.3Regular SolutionsLet usfirst consider the regular solutions of the SU(3)Einstein-Yang-Mills system. Requiring asymptoticallyflat solutions implies that the metric functions A andµboth approach a constant at infinity,and that the matter functions approach a vacuum configuration of the gaugefield.We here adoptA(∞)=1,(15) thusfixing the time coordinate,andK(∞)=±2,H(∞)=0,(16)K(∞)=0,H(∞)=±2.(17) At the origin regularity of the solutions requiresµ(0)=0,(18)and the gaugefield functions must satisfyK(0)=±2,H(0)=0,(19)K(0)=0,H(0)=±2.(20) Because of the symmetries of the SO(3)Einstein-Yang-Mills equations it is sufficient to study solutions with K(0)=2and H(0)=0.The other boundary conditions lead to degenerate solutions.In the following we present some numerical results for the regular solutions of the SO(3)embedding.In Table1we show the mass˜µ=µ/2of the lowest SO(3)solutions.Their ADM mass ism ADM=µ(∞)√e.(21)We observe,that the two lowest solutions,missed in the previous analysis by K¨u nzle [12],have a smaller mass than the lowest scaled SU(2)solution.To compare with the SO(3)solutions found by K¨u nzle[12]we note,that his func-tions u1and u2are related to the functions K and H as followsu1(x)=K(x)+H(x)2,(23) with the boundary conditions at the origin u1(0)=u2(0)=1,and at infinity u1(∞)=±1and u2(∞)=±1.Let us adopt the classification of the solutions with respect to their boundary con-ditions at infinity,the nodes(n1,n2)of the functions(u1,u2)[12],and the total number of nodes n=n1+n2.We see in Table1,that the lowest SO(3)solution has the node structure(0,1),i.e.n=1.In contrast,the lowest scaled SU(2)solution,being the lowest SO(3)solution found by Kuenzle[12],has the node structure(1,1),i.e.n=2. Naturally,the mass of the SO(3)solution with one node only is lower than the mass of the scaled SU(2)solution with n=2.But there is a second SO(3)solution with a lower mass.This solution has the node structure(0,2),i.e.a total of two nodes like the low-est scaled SU(2)solution.Evidently,the whole class of solutions with node structure (0,n)has been missed before[12].This class contains the lowest SO(3)solution,and for a given total number of nodes,these new solutions appear to be lowest.Table1further gives the coefficientsβ1andβ2for the numerical integration with the shooting method[12]u1(˜x)=1+β1˜x2+β2˜x3+...,(24)u2(˜x)=1+β1˜x2−β2˜x3+ (25)In Figs.1-4we show the lowest SO(3)solution.It is obtained independently in the limit of vanishing coupling constant on the unstable upper branch of the Einstein-Skyrme system[13].The excited solutions and further details will be given elsewhere [16].The instability of the regular solutions follows from Theorem1of Ref.[15].There the instability of the solutions of K¨u nzle[12]was demonstrated.Wefind that the theorem applies also to the new class of solutions with node structure(0,n),including the lowest mass solution(whereα=1[15]as well).4Black Hole SolutionsWe now turn to the black hole solutions of the SU(3)Einstein-Yang-Mills system. Imposing again the condition of asymptoticflatness,the black hole solutions satisfy the same boundary conditions at infinity as the regular solutions.The existence of a regular event horizon at x H requiresµ(x H)=x H4x2K K2+7H2−4 ,(27)N′H′=12+µout√e=µ(∞)√e,(29)as a function of the horizon x H.For x H→0the black hole solutions approach the regular solutions.With increasing horizon x H these black hole solutions keep their identity in terms of the boundary conditions and the node structure.Only the second solution with node structure(1,1)(#4of Table1)disappears.This solution has the same boundary conditions and node structure as the lowest scaled SU(2)solution(#3of Table1),but a slightly higher mass.It in fact merges into the scaled SU(2)solution at a horizon of x H=1.7146,leaving a unique black hole solution with node structure(1,1). (The same feature holds for the two solutions with node structure(2,2)of Ref.[12]. The solution with higher mass merges into the solution with lower mass,again a scaled SU(2)solution,at a horizon of x H=1.3745.)Note,that the order of the solutionschanges from the order of the regular solutions as the horizon increases.For instance, beyond x H=0.703solution#5has a lower mass than solution#3.In Table2we present some properties of the black hole solutions with a horizon x H=1,emerging from thefirstfive regular solutions of Table1,using again the notation of Ref.[12].The table contains the values of the functions u1and u2at the horizon,needed for a numerical shooting procedure.The families of SO(3)black hole solutions change continuously as a function of the horizon x H.As examples we show the radial functions for the lowest SO(3)black hole solutions for the horizons x H=1,2,3,4in Figs.(1)-(4).Further details of these solutions and the excited SO(3)Einstein-Yang-Mills black holes will be given elsewhere [16].The instability of the black hole solutions follows from Theorem2of Ref.[15].We find that the theorem applies to all black hole solutions constructed(whereα=1[15] as well).5ConclusionThe SU(3)Einstein-Yang-Mills system possesses a sequence of regular spherically sym-metric solutions based on the SO(3)embedding,besides the well-studied sequence based on the SU(2)embedding[1].The SO(3)solutions can be labelled according to their node structure with two integers(n1,n2)and the total number of nodes n.The low-est solution has node structure(0,1)and n=1.Thefirst excited solution has node structure(0,2),while the second excited solution,the lowest scaled SU(2)solution,has node structure(1,1).The third excited solution[12]has the same node structure as the lowest scaled SU(2)solution,being only slightly higher in mass.The next solution then has n=3,again with node structure(0,n),suggesting that this class of solutions has the lowest mass for a given total number of nodes.The regular SU(2)solutions are known to be unstable[2,3],the solution with n nodes has2n unstable modes[4,5].The SO(3)solutions are unstable as well,since Theorem1of Ref.[15]applies.It is an interesting open problem to study the number of unstable modes andfind a relation to the number of nodes.The lowest SU(2)solution has been interpreted in analogy to the electroweak sphaleron[6]as the top of a barrier between vacua[3].Furthermore,like the elec-troweak sphaleron[17,18],the gravitating sphaleron also possesses a fermion zero mode[19]and gives rise to level-crossing[19,20].It appears to be interesting to study fermions also in the background of the lowest SO(3)solution.Corresponding to each regular SO(3)solution there exist black hole solutions.These solutions keep their identity in terms of the node structure,for arbitrary horizon.If there are several solutions with the same structure of nodes,solutions may disappearby merging with the lowest solution of a given node structure,as is for instance the case for the lowest scaled SU(2)solution and its excitation with node structure(1,1).The SU(2)black holes are known to be unstable[10,11,5].The SO(3)black hole solutions are unstable as well,since Theorem2of Ref.[15]applies.It is an interesting open problem,especially with respect to the bifurcations,to study the number of unstable modes of the SO(3)black holes.Since the SU(3)Einstein-Yang-Mills system also contains Schwarzschild black holes, there are then many static,neutral black hole solutions(including the SU(2)black holes)for a given mass,enlarging the counterexample to the“no-hair conjecture”.But only the Schwarzschild solution is stable.The coloured black holes are all unstable.Charged SU(3)black hole solutions have been considered previously[21],and SU(2)×U(1) solutions have been constructed[21].Here a natural extension is to consider chargedSO(3)black hole solutions.AcknowledgementWe gratefully acknowledge discussions with M.Volkov.References[1]R.Bartnik,and J.McKinnon,Particlelike solutions of the Einstein-Yang-Millsequations,Phys.Rev.Lett.61(1988)141.[2]N.Straumann,and Z.H.Zhou,Instability of the Bartnik-McKinnon solutions ofthe Einstein-Yang-Mills equations,Phys.Lett.B237(1990)353.[3]D.V.Gal’tsov,and M.S.Volkov,Sphalerons in Einstein-Yang-Mills theory,Phys.Lett.B273(1991)255.[4]vrelashvili,and D.Maison,A remark on the instability of the Bartnik-McKinnon solutions,Phys.Lett.B343(1995)214.[5]M.S.Volkov,O.Brodbeck,vrelashvili,and N.Straumann,The numberof sphaleron instabilities of the Bartnik-McKinnon solitons and nonabelian blackholes,preprint ZU-TH-3-95,hep-th/9502045.[6]F.R.Klinkhamer,and N.S.Manton,A saddle-point solution in the Weinberg-Salam theory,Phys.Rev.D30(1984)2212.[7]M.S.Volkov,and D.V.Galt’sov,Black holes in Einstein-Yang-Mills theory,Sov.J.Nucl.Phys.51(1990)747.[8]on,Colored black holes,Phys.Rev.Lett.64(1990)2844.[9]H.P.K¨u nzle and A.K.M.Masoud-ul-Alam,Spherically symmetric static SU(2)Einstein-Yang-Millsfields,J.Math.Phys.31(1990)928[10]N.Straumann,and Z.H.Zhou,Instability of colored black hole solutions,Phys.Lett.B243(1990)33.[11]M.S.Volkov,and D.V.Gal’tsov,Odd-parity negative modes of Einstein-Yang-Mills black holes and sphalerons,Phys.Lett.B341(1995)279.[12]H.P.K¨u nzle,Analysis of the static spherically symmetric SU(n)-Einstein-Yang-Mills equations,Comm.Math.Phys.162(1994)371.[13]B.Kleihaus,J.Kunz,and A.Sood,SU(3)Einstein-Skyrme solitons and blackholes,Utrecht preprint THU-95/6,hep-th/9503087.[14]on,and T.Chmaj,Gravitating skyrmions,Phys.Lett.B297(1992)55.[15]O.Brodbeck,and N.Straumann,Instability proof for Einstein-Yang-Mills soli-tons and black holes with arbitrary gauge groups,ZU-TH-38-94,gr-qc/9411058.[16]B.Kleihaus,J.Kunz,and A.Sood,in preparation.[17]J.Boguta,and J.Kunz,Hadroids and sphalerons,Phys.Lett.B154(1985)407.[18]J.Kunz,and Y.Brihaye,Fermions in the background of the sphaleron barrier,Phys.Lett.B304(1993)141.[19]G.W.Gibbons,and A.R.Steif,Anomalous fermion production in gravitationalcollapse,Phys.Lett.B314(1993)13.[20]M.S.Volkov,Einstein-Yang-Mills sphalerons and level crossing,Phys.Lett.B334(1994)40.[21]D.V.Gal’tsov,and M.S.Volkov,Charged non-abelian SU(3)Einstein-Yang-Millsblack holes,Phys.Lett.B274(1992)173.˜µ(∞)u1u2β1β2u1(∞)u2(∞)123scaled SU(2)4K¨u nzle56K¨u nzleTable1:Properties of the lowest regular SO(3)solutions are given in the notation of Ref.[12].The ADM mass is obtained from the second column via Eq.(21)with µ(∞)=2˜µ(∞),the third and forth column give the number of nodes of the functions u1and u2defined in Eqs.(22)-(23),thefifth and sixth column provide the expansion coefficients of these functions as defined in Eqs.(24)-(25),and the seventh and eighth column give the values of the functions u1and u2at infinity.˜µ(∞)u1u2u1(˜x H)u2(˜x H)u1(∞)u2(∞)123scaled SU(2)45Table2:Properties of the lowest SO(3)black hole solutions are given for the horizon x H=1=2˜x H following the classification of Table1.The ADM mass is obtained from the second column via Eq.(21)withµ(∞)=2˜µ(∞),the third and forth column give the number of nodes of the functions u1and u2defined in Eqs.(22)-(23),thefifth and sixth column provide the value of these functions at the horizon,and the seventh and eighth column give the values of the functions u1and u2at infinity.Figure1:The function K(x)is shown for the regular solution and for the black hole solutions with horizons x H=1,2,3and4as a function of x.Figure2:The function H(x)is shown for the regular solution and for the black hole solutions with horizons x H=1,2,3and4as a function of x.Figure3:The functionµ(x)is shown for the regular solution and for the black hole solutions with horizons x H=1,2,3and4as a function of x.Figure4:The function A(x)is shown for the regular solution and for the black hole solutions with horizons x H=1,2,3and4as a function of x.Figure5:The mass fraction outside the horizon,µout,of the SO(3)black hole solutions of Table2is shown as a function of the horizon x H.When the mass fraction within the horizon,x H/2,is added,the ADM mass is obtained.。