Perturbative spectrum of Trapped Weakly Interacting Bosons in Two Dimensions

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CCP3SURF ACESCIENCENEWSLETTERCollaborative Computational Project3on Surface ScienceNumber26-January2000ISSN1367-370XDaresbury LaboratoryContents1Editorial1 2Scientiﬁc Articles2 3SRRTNet-a new global network124High performance computing164.1Cluster-Computing Developments in the UK (16)4.2New HPC Support Mechanisms (22)5Reports on visits256Meetings,Workshops,Conferences296.1Reports on Bursaries (29)6.2Reports from Meetings (32)6.3Upcoming meetings (34)7Abstracts of forthcoming papers37 8Surface Science Related Jobs40 9Members of the working group43 Contributions to the newsletter from all CCP3members are welcome and should be sent to ccp3@eful Links:CCP3Home Page /Activity/CCP3CCP3Program Library /Activity/CCP3+896 SRRTNet /Activity/SRRTNetDLV /Activity/DLVCRYSTAL /Activity/CRYSTALCASTEP /Activity/UKCPMany useful items of software are available from the UK Distributed Computing Support web site,DISCO /Activity/DISCOEditors:Dr.Klaus Doll and Dr.Adrian Wander,Daresbury Laboratory,Daresbury, Warrington,WA44AD,UK1EditorialThe renewal of CCP3,which is due in the summer,is on all our minds,and this edition of the newsletter reminds us of our aims and achievements.Theﬂagship project supported by the post-doc is at the heart of the CCP–new programs can be developed which would not get oﬀthe ground otherwise.Over the last three years CCP3has been very lucky to have had Klaus Doll working on the development of analytic gradient methods for the CRYSTAL electronic structure package.This will lead to much more eﬃcient structure optimization,of particular beneﬁt to surface scientists where surface relaxation and reconstructions are so important.Klaus’achievements so far are described in theﬁrst article,and it is most satisfactory that tests on the CO molecule and bulk MgO have proved successful.The next step is to build in symmetry,to achieve greater computational eﬃciency,and then the new code can be released.As theﬂagship in the renewal proposal,the working group has chosen the development of methods to study the electronic structure and physical properties of large clusters.These clusters themselves possess surface-like properties, but at the same time it is proposed to study their interaction with surfaces.Such systems are a topic of active research for several of the members of the working group,both theoretical and experimental,and it is expected that their expertise will contribute greatly to the success of the project.Having Adrian Wander as a permanent member of staﬀat Daresbury supporting CCP3will lead to very welcome support for the synchrotron radiation community in the development of new surface program packages.In this newsletter he describes developments in SRRTNet,originally an American collaboration for providing sup-port for surface scientists using synchrotron radiation,which is likely to develop into an international collaboration based on the CCP model.Adrian is also in discussion with the Daresbury-based X-ray community with a view to developing new codes for the analysis of near-edge spectra in a wider range of systems than can be tackled at the moment,using the improved self-consistent electron potentials available for complex materials.This work would be based at Daresbury,and will form part of the CCP3collaboration.This issue contains short articles on our visitor programme,and by Ally Chan (Nottingham)and Yu Chen(Birmingham)who received student bursaries for participating in ECOSS-18.Please continue to apply for CCP3support!It is interesting to read in the pieces by Ally and Yu what most impressed them at ECOSS–I was struck by Ally’s comment that surface science has broadened to include nanoparticles and nanowires.Just what we thought in our choice ofﬂagship project next time round.John Inglesﬁeld12Scientiﬁc ArticlesAnalytical Hartree-Fock gradients for periodic systemsK.Doll,V.R.Saunders,and N.M.HarrisonCLRC,Daresbury Laboratory,Daresbury,Warrington,WA44AD,UKWe report on the progress of the implementation of analytic gradients in the program package CRYSTAL.The algorithm is brieﬂy summarised and tests illustrate that highly accurate analytic gradients of the Hartree-Fock energy can be obtained for molecules and periodic systems.IntroductionComputational materials science has been a fast growingﬁeld in the last years. This is mainly because methods which were developed earlier(density functional theory,molecular dynamics,Hartree-Fock and correlated quantum chemical meth-ods,Monte Carlo schemes,the GW method,etc)can now be applied to demanding realistic systems due to the increase in computational resources(faster CPUs,par-allelisation,cheaper memory and diskspace).CCP3is a collaboration in the area of surfaces and interfaces where progress de-pends on an interaction between experimental and theoretical approaches.There-fore,codes which provide a better theoretical understanding are important.One of the key issues in surface science is the determination of surface structure and adsorption energetics.From the computational point of view,a fast structural optimisation must be possible.Availability of numerical or analytical gradients facil-itatesﬁnding a minimum energy structure,and availability of analytical gradients can make optimisation algorithms more eﬃcient.As a rule of thumb,analyticalgradients are about N3times more eﬃcient than numerical gradients(with N beingthe number of variables).Also,for future developments such asﬁnding transition states,gradients are essential.Analytical gradients in quantum chemistry were pioneered by Pulay who did the ﬁrst implementation for multicentre basis sets[1].In many molecular codes based on quantum chemistry methods,analytical gradients are now implemented and gradient development has become an important task in quantum chemistry[2,3,4,5].Simi-larly,in solid-state codes such as CASTEP,WIEN,or LMTO,analytic gradients are available.Analytic Hartree-Fock gradients have already been implemented in a code for systems periodic in one dimension[6].CRYSTAL[7,8]was born in Turin and is now jointly developed in Turin and Daresbury.CRYSTAL was initially designed to deal with the exact exchange in and to solve the Hartree-Fock equations for real systems.With the modern versions of the code,density-functional2calculations or calculations using Hybrid functionals such as B3LYP with the ad-mixture of exact exchange are also possible.The target of this project,which began in October1997,was the implementation of analytical gradients in CRYSTAL and in autumn1999,theﬁrst test calculations on periodic systems were performed.In this article,we try to outline the theory and implementation of analytical gra-dients.We try to keep the mathematics at a minimum;a more formal publication is intended in the near future[9].A very comprehensive summary of the theory underpinning CRYSTAL will appear in the future[10].Total energyFirst,we want to brieﬂy summarise how the total energy is obtained.The total energy consists of•kinetic energy of the electrons•nuclear-electron attraction•electron-electron repulsion•nuclear-nuclear repulsionCRYSTAL,similar to molecular codes such as GAMESS-UK,MOLPRO(Stuttgart and Birmingham),GAUSSIAN,TURBOMOLE,etc,solves the single particle Schr¨o dinger equation and a wavefunction is calculated.The wavefunction is based on crystalline orbitalsΨi( r, k)which are linear combinations of Bloch functionsΨi( r, k)= µaµ,i( k)ψµ( r, k)(1) with the Bloch functions constructed fromψµ( r, k)= gφµ( r− Aµ− g)exp(i k g)(2) g are direct lattice vectors, Aµdenotes the coordinate of the nuclei.φµare the basis functions which are Gaussian type orbitals.For example,an s-type function centred at R a=(X a,Y a,Z a)is expressed asφ(α, r− R a,n=0,l=0,m=0)=φµ( r− R a)= Nexp(−α( r− R a)2).In molecular calculations,no mathematical problem arises from any of the interac-tions.In periodic systems,however,there are several divergent terms which have to3be dealt with:for example,in a one dimensional periodic system with lattice con-stant a and n being an index numerating the cells,the electron-electon interaction per unit cell would have contributions like:∞n=1e2na(3)This sum is divergent(similarly in two and three dimensions).Therefore,an indi-vidual treatment of this term is not possible.Instead,all the charges(nuclei and electrons)are partitioned and a scheme based on the Ewald method is used to sum the interactions[11].The Hartree-Fock equations are solved in terms of Bloch functions because the Hamiltonian becomes block-diagonal(i.e.at each k-point the equations are solved independently).The wavefunction coeﬃcients aµ,i are optimised due to this procedure and the total energy can be evaluated.For the computation of gradients,the dependence of the total energy on the nuclear coordinates must be analysed.There are three dependencies of the total energy on the nuclear coordinates:•nuclear-nuclear repulsion and nuclear-electron attraction:obviously,the coor-dinates of the nuclei enter•wavefunction coeﬃcients(or density matrix):we will obtain a diﬀerent solution with diﬀerent density matrix when moving the nuclei•basis functions:the basis functions are centred at the position of the nu-clei and therefore moving the nuclei will change integrals over the basis func-tions.These additional terms are called Pulay forces.They are missing when the Hellmann-Feynman theorem is applied and therefore Hellmann-Feynman forces often diﬀer substantially from energy derivative forces in the case of a local basis set(see[1]and references therein).Density matrix derivatives are diﬃcult to evaluate.However,for the solution of the Hartree-Fock equations,this problem can be circumvented and instead a new term is introduced,the so-called energy-weighted density matrix which is easily evaluated [12].However,this is only strictly correct for the exact Hartree-Fock solution. In practice,convergence is achieved up to a certain numerical threshold(e.g.a convergence of10−6E h of the total energy corresponding to27.2114×10−6eV).For very accurate gradient calculations,it may be necessary to make this threshold even lower.The remaining main problem is to generate all the derivatives of the integrals. In a second step,these derivatives have to be mixed with the density matrix.4Evaluation of integralsIn this section we summarise the types of integral which occur.The simplest type is the overlap integral between two basis functions at two centres:Sµν R k R l= φµ( r− R k)φν( r− R l)d3r(4) Obviously we can shift R k to the origin,and suppressing 0in the notation,we obtain:Sµν R i= φµ( r)φν( r− R i)d3r(5) with R i= R l− R k.A kinetic energy integral has the form:Tµν R i= φµ( r)(−12∆ r)φν( r− R i)d3r(6) the nuclear attraction integral has the form:Nµν R i= φµ( r)Z c| r− A c|φν( r− R i)d3r(7) and the electron-electron interaction has the form:Bµν R iτσ R j = φµ( r)φν( r− R i)φτ( r′)φσ( r′− R j)| r− r′|d3rd3r′(8)These integrals are in principle suﬃcient to deal with molecules.In the case of periodic systems,new types of integrals appear(e.g.multipolar integrals,integrals over the Ewald potential and its derivatives)[11,13,14].The fast evaluation of integrals is one of the main issues in the development of quan-tum chemistry codes.CRYSTAL uses a McMurchie-Davidson algorithm[15].Its idea is to map a product of two Gaussian type orbitals at two centres in an expan-sion of Hermite polynomials at an intermediate centre.This algorithm has proven to eﬃciently evaluate integrals,although in recent years progress in this specialised ﬁeld of quantum chemistry has been made(see for example the introduction in[16] or two recent reviews[17,18]).The expansion[15,14]looks like:5φ(α, r− A,n,l,m)φ(β, r− B,n′,l′,m′)= t,u,v E(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)(9)withγ=α+βand P=α A+β Bα+β.Λis a so-called Hermite Gaussian type function Λ(γ, r− P,t,u,v)= ∂∂P x t ∂∂P y u ∂∂P z v exp(−γ( r− P)2)(10)The start value E(0,0,0,0,0,0,0,0,0)=exp(−αβ( B− A)2)can be veriﬁed by inserting it in equation9.It can be derived from the Gaussian product rule[19,20]:exp(−α( r− A)2)exp(−β( r− B)2)=exp −αβα+β( B− A)2 exp −(α+β) r−α A+β Bα+β 2(11) General values E(n,l,m,n′,l′,m′,t,u,v)are obtained from recursion relations[15, 14].The E-coeﬃcients depend on the distance( B− A),but not on P or r.All the integrals can be expressed in terms of E-coeﬃcients[15,14,11,13].Evaluation of gradients of the integralsOne of the issues of the gradient project is to generalise the algorithms used to generate the energy integrals to obtain the gradients of the integrals.This madea new implementation of recursion relations necessary which are used to obtainthe coeﬃcients G in the expansion of the gradients of the integrals in Hermite polynomials.∂Φ(α, r− A,n,l,m)Φ(β, r− B,n′,l′,m′)∂A x= t,u,v G A x(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)(12)Once the coeﬃcients are known,the integration can be performed.The integrationfor the case of gradients of integrals is similar to the case of integrals for the total energy.The only diﬀerence is that,instead of the coeﬃcientsE(n,l,m,n′,l′,m′,t,u,v)which enter the energy expression,the gradient coeﬃcientsG A x(n,l,m,n′,l′,m′,t,u,v),G A y,G A z,G B x,G B y,and G B z6are used.The coeﬃcients G B x can eﬃciently be obtained together with the coeﬃ-cients G A x[21].For example,the evaluation of the overlap integral is done as follows:Sµν R i = φµ( r)φν( r− R i)d3r=t,u,v E(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)d3r=E(n,l,m,n′,l′,m′,0,0,0)Λ(γ, r− P,0,0,0)d3r=ME(n,l,m,n′,l′,m′,0,0,0)From theﬁrst line to the second,we have used the McMurchie-Davidson scheme, from the second to the third we exploited a property of the Hermite Gaussian type functions:all the integrals of the type Λ(γ, r− P,t,u,v)d3r with t=0or u=0or v=0vanish because of the orthogonality of these functions.The integration(fromthe third to the fourth line)is trivial.M is a normalisation constant. Calculating the gradient is easy once we know the new expansion:∂Sµν R i ∂A x =∂∂A xφµ( r)φν( r− R i)d3r=∂ t,u,v E(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)∂A x d3r=t,u,v G A x(n,l,m,n′,l′,m′,t,u,v)Λ(γ, r− P,t,u,v)d3r=G A x(n,l,m,n′,l′,m′,0,0,0)Λ(γ, r− P,0,0,0)d3r=MG A x(n,l,m,n′,l′,m′,0,0,0)This way,all the derivatives can be calculated!There are some integrals which involve three centres(for example nuclear attraction)where we exploit translational invariance:∂∂C x =−∂∂A x−∂∂B x(13)because the value of the integral is invariant to a simultaneous uniform translation of the three centres.Four centre integrals can be reduced to a product of two integrals over two centres which makes the calculation of gradients straightforward.As a whole,the calculation of gradients of the integrals is closely related to calcu-lating the integrals itself.This means that most of the subroutines can be used for7the gradient code.One of the main diﬀerences is that array dimensions need to be changed-dealing with gradients is similar to increasing the quantum number(a derivative of an s-function is a p-function,and so on).However,the task of adjusting the subroutines should not be underestimated.After obtaining the derivatives of the integrals,we mix them with the density ma-trix just like in the energy calculation.We have to take into account the new term which arose because we did not calculate a density matrix derivative—the energy weighted density matrix.Again,coding this additional term can be done by modi-fying existing subroutines.After this,weﬁnally obtain the forces on the individual atoms.Results from test calculationsIn this section,we summarise results from test calculations.We have considered the CO molecule which was arranged as a single molecule,as a molecule which is peri-odically reproduced with a periodicity of4˚A in one spatial direction(”polymer”), periodically reproduced with a periodicity of4˚A in two spatial directions(”slab”), and periodically reproduced with a periodicity of4˚A in three spatial directions (”solid”).Because of the large distance of4˚A,the molecules can be considered as nearly independent and the forces are quite similar.Still,the calculation of energy and gradient is completely diﬀerent and therefore this is an important test of Ewald technique and multipolar expansion.The results are given in table1.The results agree in the best case to at least6digits which is the numerical noise and in the worst case up to4digits.The diﬀerence between analytic and numerical gradi-ents in periodic systems mainly originates from an approximation made within the evaluation of the integrals[22]and from the number of k-points which aﬀects the accuracy of the energy-weighted density matrix.In table2,we display results from a MgO solid with one oxygen atom slightly distorted from the symmetrical position.Again,the forces agree well up to5digits with numerical derivatives.Future developments and ConclusionThe present version of the code is able to calculate Hartree-Fock forces for periodic systems up to a precision of4and more digits.There is no extra diskspace needed and the additional memory usage is moderate.This code will certainly be useful for structural optimisation and for future program development towards molecular dynamics or the calculation of response functions.The present version,however,is not yet ready for a release.Instead,the following steps are necessary:Firstly,the usage of symmetry must be implemented.This is of highest importance to make the code fast enough so that it can be used for practical optimisations.We expect8Table1:Force on a CO molecule with a carbon atom located at(0˚A,0˚A,0˚A)and an oxygen atom located at(0.8˚A,0.5˚A,0.4˚A).In the periodic case,the molecule is generated with a periodicity of4˚A.This means,that in one dimension,for example,there would be other molecules with a carbon atom at(n×4˚A,0˚A,0˚A)and an oxygen atom at((n×4+0.8)˚A,0.5˚A,0.4˚A),with n running overall positive and negative integers.Forces are given in E h,with E h=27.2114eVand a0=0.529177˚A.Higher ITOLs means a lower level of approximation in the evaluation of the integrals[22].ITOLs) k-points) numerical force0.3769140.37660(0.37664)0.376310.37566(0.37566) analytical force0.3769130.37663(0.37665)0.376330.37588(0.37578))on the atoms of an MgO solid.The MgO solid was chosen Table2:Forces(in E ha0to have an artiﬁcially high lattice constant of6.21˚A to make the calculation faster. Coordinates are given in fractional units,e.g.the second Mg is at0˚A,0.5×6.21˚A,0.5×6.21˚A.A normal fcc lattice would be obtained if the sixth atom(Oxygenat0.53,0,0)was at(0.5,0,0).Moving this atom from its normal position has ledto the nonvanishing forces.Mg(0.00.00.0)-0.03018-0.03019Mg(0.00.50.5)-0.00314-0.00314Mg(0.50.00.5)0.008950.00895Mg(0.50.50.0)0.008950.00895O(0.50.50.5)-0.00379-0.00379O(0.530.00.0)0.004290.00430O(0.00.50.0)0.007460.00746O(0.00.00.5)0.007460.00746that a version of the present code with symmetry will already be fast enough to compete with numerical derivatives.Further developments will be the coding of the bipolar expansion(a method to evaluate the electron-electron repulsion integrals faster),and sp-shells(s and p shells are often chosen to have the same exponentsto make the evaluation of integrals faster).Also,the newly written subroutines arenot yet optimal and they will certainly go through a technical optimisation(moreeﬃcient coding).In later stages,the code should be made applicable to metals (there is an extra term coming from the shape of the Fermi surface[23]which is notyet coded)and to magnetic systems(unrestricted Hartree-Fock gradients).Finally, pseudopotential gradients and density functional gradients should be included. References[1]P.Pulay,Mol.Phys.17,197(1969).[2]P.Pulay,Adv.Chem.Phys.69,241(1987).[3]P.Pulay,in Applications of Electronic Structure Theory,edited by H.F.Schae-fer III,153(Plenum,New York,1977).[4]H.B.Schlegel,Adv.Chem.Phys.67,249(1987).[5]T.Helgaker and P.Jørgensen,Adv.in Quantum Chem.19,183(1988)[6]H.Teramae,T.Yamabe,C.Satoko,A.Imamura,Chem.Phys.Lett.101,149(1983).[7]C.Pisani,R.Dovesi,and C.Roetti,Hartree-Fock Ab Initio Treatment of Crys-talline Systems,edited by G.Berthier et al,Lecture Notes in Chemistry Vol.48(Springer,Berlin,1988).[8]V.R.Saunders,R.Dovesi,C.Roetti,M.Caus`a,N.M.Harrison,R.Orlando,C.M.Zicovich-Wilson crystal98User’s Manual,Theoretical Chemistry Group, University of Torino(1998).[9]K.Doll,V.R.Saunders,N.M.Harrison(in preparation)[10]V.R.Saunders,N.M.Harrison,R.Dovesi,C.Roetti,Electronic StructureTheory:From Molecules to Crystals(in preparation)[11]V.R.Saunders,C.Freyria-Fava,R.Dovesi,L.Salasco,and C.Roetti,Mol.Phys.77,629(1992).[12]S.Bratoˇz,in Calcul des fonctions d’onde mol´e culaire,Colloq.Int.C.N.R.S.82,287(1958).[13]R.Dovesi,C.Pisani,C.Roetti,and V.R.Saunders,Phys.Rev.B28,5781(1983).[14]V.R.Saunders,in Methods in Computational Molecular Physics,edited by G.H.F.Diercksen and S.Wilson,1(Reidel,Dordrecht,Netherlands,1984).[15]L.E.McMurchie and E.R.Davidson,put.Phys.26,218(1978).[16]R.Lindh,Theor.Chim.Acta85,423(1993).[17]T.Helgaker and P.R.Taylor,in Modern Electronic Structure Theory.Part II,World Scientiﬁc,Singapore,725(1995)[18]P.M.W.Gill,in Advances in Quantum Chemistry,edited by P.-O.L¨o wdin,141(Academic Press,New York,1994)[19]S.F.Boys,Proc.Roy.Soc.A200,542(1950).[20]R.McWeeny,Nature166,21(1950).[21]T.Helgaker and P.R.Taylor,Theor.Chim.Acta83,177(1992).[22]The integrals Bµν R iτσ R j =Bτσ R jµν R iwhich should have the same value,are notnecessarily evaluated within the same level of approximation—this is nearly inevitable for periodic systems,as enforcing this symmetry would require a much higher computational eﬀort and much more data storage.The derivation of the equations for the analytic gradients,however,relies on these integrals be-ing equivalent.Therefore,the introduced asymmetry will lead to inaccuracies in the gradients.This can be controlled with the ITOL-parameters(tolerances as described in the CRYSTAL manual[8])which control the level of approx-imation.Higher ITOLs lead to a higher accuracy in the forces.However,the defaults appear to give forces with an accuracy up to4digits which should be good enough for most purposes.[23]M.Kertesz,Chem.Phys.Lett.106,443(1984).3SRRTNet-a new global networkFrascati’99-Birth of a NetworkScientiﬁc MeetingFrom the23rd to the25th September1999,a workshop on Theory and Computation for Synchrotron Radiation was held at the laboratory in Frascati just outside Rome, Italy.This was the third in an ongoing series of meetings on various aspects of synchrotron radiation,and follows meetings on Theory and Computation for Syn-chrotron Applications held at the Advanced Light Source in Berkeley in October 1997and Needs for a Photon Spectroscopy Theory Center held at the Argonne National Laboratory in August1998.This was an excellent meeting,featuring a variety of high quality scientiﬁc pre-sentations from both experimental and theoretical participants.Theﬁrst day was devoted to presentations concentrating on resonant x-ray processes and orbital or-dering eﬀects,particularly in V2O5.The second day then moved on to discussions of photoemission,photoelectron diﬀraction and holography,and studies of high T c superconductors.This day was concluded with an excellent conference dinner which ﬁnished rather late!Theﬁnal day then concluded with discussions of EXAFS,and x-ray spectroscopies.The overheads used in all the presentations can be viewed on line at http://wwwsis.lnf.infn.it/talkshow/srrtnet99.htmSRRTNet DiscussionsThe Friday programme also included a two hour session devoted to the idea of forming a global network concentrating on theory for synchrotron radiation re-search based research.The session began with a talk from Michel Van Hove of the Lawrence Berkeley National Laboratory who outlined the purposes and function of the proposed network.This was then followed by presentations by John Rehr of the University of Washington who highlighted moves to extend the synchrotron radia-tion research theory network(SRRTNet)in North America,by Maurizio Benfatto of the INFN Frascati,who presented the European perspective,by Kenji Makoshi of Himeji Institute of technology who discussed the Japanese eﬀorts and by Adrian Wander of the Daresbury Laboratory who presented CCP3as a possible model of how the network could be run.The concept of establishing a global network was received with enthusiasm from all present.OutcomeGiven the support of the meeting for the concept of global network of this sort,it was decided to extend SRRTNet into the global arena.The aims of the network are:•To provide a central repository for information of relevance to synchrotron radiation research•To develop theoretical methods pertaining to the experiments performed on synchrotron facilities•To provide state of the art and user friendly software for the analysis and interpretation of experiments•To provide training in the use of relevant software through workshops and site visits•To host visiting scientists•To hold periodic workshops for the dissemination of new results and method-ologiesThe directors of the network are Michel Van Hove and John Rehr.As aﬁrst step in the development of the network,Daresbury has agreed to host the web pages, and theoretical groups have been contacted and ask to provide input to this central web hub of what will grow into a globe encompassing network.If you are interested in contributing to the network and missed our e-mail announcement,the invitation letter follows;Dear Colleague,You may know of the recently established Synchrotron Radiation Research Theory Network(SRRTNet).We are contacting you to invite you,and all theorists inter-ested in this topic,to actively participate in the next phase of the network. SRRTNet aims to provide theory for experiments that use synchrotron radiation,by means of a global,web-based network linking theoretical and experimental research groups.The driving philosophy is to promote interactions between theory and exper-iment for mutual beneﬁt,by means of web-based information,workshops,exchange of theoretical methods and computer codes,as well as establishing visiting scientist programs.At the last SRRTNet workshop,conducted at Frascati near Rome in September1999, it was decided to strengthen the global character of this network by establishing a cen-tral,web-based source of information.Daresbury Laboratory is hosting this web site with Dr.Adrian Wander acting as editor.It is anticipated that all synchrotron facilities will provide direct links for their users to this web site,and consequently we expect this site to grow into an essential resource for synchrotron radiation re-searchers.An importantﬁrst function of the web site will be to provide information about theorists’research interests and links to relevant web pages.The network will be all the more valuable as this coverage becomes complete:it will thus allow theorists and experimentalists alike toﬁnd the best sources of information about the various methods for solving speciﬁc scientiﬁc problems.The purpose of this message is to ask you to provide such information and links about your group.You may visit the new web site/Activity/SRRTNetand see not only an overview of the network in general,but also the beginnings of such information about speciﬁc theoretical groups.The idea is to put a list of your research topics on the SRRTNet web site,while providing links to your own web site for more detailed and up-to-date information. If you prefer,the SRRTNet site can itself host a more complete web page covering your activities.The information we wish to present(or link to)includes as many as possible of the following items:•your topics of scientiﬁc activity related to synchrotron radiation(directly or by methodology);•your computer codes,with their capabilities and availability;•your publications,such as abstracts,papers,databases and web-presentations;•how to contact you or your group.。

Power-spectrum condition for energy-efficient watermarking

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Hale Waihona Puke ABSTRACTattacks. However, others have proposed placing the watermark in the middle or high frequencies to make it easier to separate from the original image 7, 8 or making it white, as in conventional spread spectrum. This paper elaborates on a simple theoretical watermarking and attack model from 9 . Analysis leads to a meaningful way to evaluate robustness. It is shown that watermarks that resist the attack should satisfy a powerspectrum condition. Finally, experiments with theoretical signal models and natural images verify and reinforce the importance of this condition.

气科院大气物理面试英语专业词汇[1]

大气科学系微机应用基础Primer of microcomputer applicationFORTRAN77程序设计FORTRAN77 Program Design大气科学概论An Introduction to Atmospheric Science大气探测学基础Atmospheric Sounding流体力学Fluid Dynamics天气学Synoptic Meteorology天气分析预报实验Forecast and Synoptic analysis生产实习Daily weather forecasting现代气候学基础An introduction to modern climatology卫星气象学Satellite meteorologyC语言程序设计 C Programming大气探测实验Experiment on Atmospheric Detective Technique云雾物理学Physics of Clouds and fogs动力气象学Dynamic Meteorology计算方法Calculation Method诊断分析Diagnostic Analysis中尺度气象学Meso-Microscale Synoptic Meteorology边界层气象学Boundary Layer Meteorology雷达气象学Radar Meteorology数值天气预报Numerical Weather Prediction气象统计预报Meteorological Statical Prediction大气科学中的数学方法Mathematical Methods in Atmospheric Sciences专题讲座Seminar专业英语English for Meteorological Field of Study计算机图形基础Basic of computer graphics气象业务自动化Automatic Weather Service空气污染预测与防治Prediction and Control for Air Pollution现代大气探测Advanced Atmospheric Sounding数字电子技术基础Basic of Digital Electronic Techniqul大气遥感Remote Sensing of Atmosphere模拟电子技术基础Analog Electron Technical Base大气化学Atmospheric Chemistry航空气象学Areameteorology计算机程序设计Computer Program Design数值预报模式与数值模拟Numerical Model and Numerical Simulation接口技术在大气科学中的应用Technology of Interface in Atmosphere Sciences Application海洋气象学Oceanic Meteorology现代实时天气预报技术（MICAPS系统）Advanced Short-range Weather Forecasting Technique(MICAPS system)1) atmospheric precipitation大气降水2) atmosphere science大气科学3) atmosphere大气1.The monitoring and study of atmosphere characteristics in near space as an environment forspace weapon equipments and system have been regarded more important for battle support.随着临近空间飞行器的不断发展和运用,作为武器装备和系统环境的临近空间大气特性成为作战保障的重要条件。

Large-Scale Mass Power Spectrum from Peculiar Velocities

a rXiv:as tr o-ph/98792v19J ul1998LARGE-SCALE MASS POWER SPECTRUM FROM PECULIAR VELOCITIES I.ZEHAVI Racah Institute of Physics,The Hebrew University,Jerusalem 91904,Israel This is a brief progress report on a long-term collaborative project to measure the power spectrum (PS)of mass density ﬂuctuations from the Mark III and the SFI catalogs of peculiar velocities.1,2The PS is estimated by applying maximum likelihood analysis,using generalized CDM models with and without COBE normalization.The applica-tion to both catalogs yields fairly similar results for the PS,and the robust results are presented.1Introduction In the standard picture of cosmology,structure evolved from small density ﬂuctua-tions that grew by gravitational instability.These initial ﬂuctuations are assumed to have a Gaussian distribution characterized by the PS.On large scales,the ﬂuc-tuations are linear even at late times and still governed by the initial PS.The PS is thus a useful statistic for large-scale structure,providing constraints on cosmol-ogy and theories of structure formation.In recent years,the galaxy PS has been estimated from several redshift surveys.3In this work,we develop and apply like-lihood analysis 4in order to estimate the mass PS from peculiar velocity catalogs.Two such catalogs are used.One is the Mark III catalog of peculiar velocities,5a compilation of several data sets,consisting of roughly 3000spiral and elliptical galaxies within a volume of ∼80h −1Mpc around the local group,grouped into ∼1200objects.The other is the recently completed SFI catalog,6a homogeneously selected sample of ∼1300spiral ﬁeld galaxies,which complies with well-deﬁned criteria.It is interesting to compare the results of the two catalogs,especially in view of apparent discrepancies in the appearance of the velocity ﬁelds.7,82MethodGiven a data set d ,the goal is to estimate the most likely model m .Invoking a Bayesian approach,this can be turned to maximizing the likelihood function L ≡P (d |m ),the probability of the data given the model,as a function of the model parameters.Under the assumption that both the underlying velocities and the observational errors are Gaussian random ﬁelds,the likelihood function can be written as L =[(2π)N det(R )]−1/2exp −1Figure1:Likelihood analysis results for theﬂatΛCDM model with h=0.6.ln L contours in theΩ−n plane are shown for SFI(left panel)and Mark III(middle).The best-ﬁt parameters are marked by‘s’and‘m’on both,for SFI and Mark III respectively.The right panel shows the corresponding PS for the SFI case(solid line)and for Mark III(dashed).The shaded region is the SFI90%conﬁdence region.The three dots are the PS calculated from Mark III by Kolatt andDekel(1997),10together with their1σerror-bar.maximum likelihood.Conﬁdence levels are estimated by approximating−2ln L as a χ2distribution with respect to the model parameters.Note that this method,based on peculiar velocities,essentially measures f(Ω)2P(k)and not the mass density PS by itself.Careful testing of the method was done using realistic mock catalogs,9 designed to mimic in detail the real catalogs.We use several models for the PS.One of these is the so-calledΓmodel,where we vary the amplitude and the shape-parameterΓ.The main analysis is done with a suit of generalized CDM models,normalized by the COBE4-yr data.These include open models,ﬂat models with a cosmological constant and tilted models with or without a tensor component.The free parameters are then the density parameter Ω,the Hubble parameter h and the power index n.The recovered PS is sensitive to the assumed observational errors,that go as well into R.We extend the method such that also the magnitude of these errors is determined by the likelihood analysis, by adding free parameters that govern a global change of the assumed errors,in addition to modeling the PS.Weﬁnd,for both catalogs,a good agreement with the original error estimates,thus allowing for a more reliable recovery of the PS.3ResultsFigure1shows,as a typical example,the results for theﬂatΛCDM family of models, with a tensor component in the initialﬂuctuations,when setting h=0.6and varying Ωand n.The left panel shows the ln L contours for the SFI catalog and the middle panel the results for Mark III.As can be seen from the elongated contours,what is determined well is not a speciﬁc point but a high likelihood ridge,constraining a degenerate combination of the parameters of the formΩn3.7=0.59±0.08,in this case.The right panel shows the corresponding maximum-likelihood PS for the two catalogs,where the shaded region represents the90%conﬁdence region obtained from the SFI high-likelihood ridge.These results are representative for all other PS models we tried.For each2catalog,the diﬀerent models yield similar best-ﬁt PS,falling well within each oth-ers formal uncertainties and agreeing especially well on intermediate scales(k∼0.1h Mpc−1).The similarity,seen in theﬁgure,of the PS obtained from SFI to that of Mark III is illustrative for the other models as well.This indicates that the peculiar velocities measured by the two data sets,with their respective error estimates,are consistent with arising from the same underlying mass density PS. Note also the agreement with an independent measure of the PS from the Mark III catalog,using the smoothed densityﬁeld recovered by POTENT(the three dots).10 The robust result,for both catalogs and all models,is a relatively high PS,with P(k)Ω1.2=(4.5±2.0)×103(h−1Mpc)3at k=0.1h Mpc−1.An extrapolation to smaller scales using the diﬀerent CDM models givesσ8Ω0.6=0.85±0.2.The error-bars are crude,reﬂecting the90%formal likelihood uncertainty for each model,the variance among diﬀerent models and between catalogs.The general constraint of the high likelihood ridges is of the sortΩh50µnν=0.75±0.25,whereµ=1.3and ν=3.7,2.0forΛCDM models with and without tensorﬂuctuations respectively. For open CDM,without tensorﬂuctuations,the powers areµ=0.9andν=1.4.For the span of models checked,the PS peak is in the range0.02≤k≤0.06h Mpc−1. The shape parameter of theΓmodel is only weakly constrained toΓ=0.4±0.2. We caution,however,that these results are as yet preliminary,and might depend on the accuracy of the error estimates and on the exact impact of non-linearities.2 AcknowledgmentsI thank my close collaborators in this work A.Dekel,W.Freudling,Y.Hoﬀman and S.Zaroubi.In particular,I thank my collaborators from the SFI collaboration, L.N.da Costa,W.Freudling,R.Giovanelli,M.Haynes,S.Salzer and G.Wegner, for the permission to present these preliminary results in advance of publication. References1.S.Zaroubi,I.Zehavi,A.Dekel,Y.Hoﬀman and T.Kolatt,ApJ486,21(1997).2.W.Freudling,I.Zehavi,L.N.da Costa,A.Dekel,A.Eldar,R.Giovanelli,M.P.Haynes,J.J.Salzer,G.Wegner,and S.Zaroubi,ApJ submitted(1998).3.M.A.Strauss and J.A.Willick,Phys.Rep.261,271(1995).4.N.Kaiser,MNRAS231,149(1988).5.J.A.Willick,S.Courteau,S.M.Faber,D.Burstein and A.Dekel,ApJ446,12(1995);J.A.Willick,S.Courteau,S.M.Faber,D.Burstein,A.Dekel and T.Kolatt,ApJ457,460(1996);J.A.Willick,S.Courteau,S.M.Faber,D.Burstein,A.Dekel and M.A.Strauss,ApJS109,333(1997).6.R.Giovanelli,M.P.Haynes,L.N.da Costa,W.Freudling,J.J.Salzer and G.Wegner,in preparation.7.L.N.da Costa,W.Freudling,G.Wegner,R.Giovanelli,M.P.Haynes and J.J.Salzer,ApJ468,L5(1996).8.L.N.da Costa,A.Nusser,W.Freudling,R.Giovanelli,M.P.Haynes,J.J.Salzer and G.Wegner,MNRAS submitted(1997).39.T.Kolatt,A.Dekel,G.Ganon and J.Willick,ApJ458,419(1996).10.T.Kolatt and A.Dekel,ApJ479,592(1997).4。

MHD Turbulence Revisited

MHD Turbulence Revisited
P. Goldreich 1 and S. Sridhar 2 1 California Institute of Technology Pasadena, CA 91125, USA Inter-University Centre for Astronomy and Astrophysics Ganeshkhind, Pune 411 007, INDIA ABSTRACT Kraichnan (1965) proposed that MHD turbulence occurs as a result of collisions between oppositely directed Alfv´ en wave packets. Recent work has generated some controversy over the nature of non linear couplings between colliding Alfv´ en waves. We ﬁnd that the resolution to much of the confusion lies in the existence of a new type of turbulence, intermediate turbulence, in which the cascade of energy in the inertial range exhibits properties intermediate between those of weak and strong turbulent cascades. Some properties of intermediate MHD turbulence are: (i) in common with weak turbulent cascades, wave packets belonging to the inertial range are long lived; (ii) however, components of the strain tensor are so large that, similar to the situation in strong turbulence, perturbation theory is not applicable; (iii) the breakdown of perturbation theory results from the divergence of neighboring ﬁeld lines due to wave packets whose perturbations in velocity and magnetic ﬁelds are localized, but whose perturbations in displacement are not; (iv) 3–wave interactions dominate individual collisions between wave packets, but interactions of all orders n ≥ 3 make comparable contributions to the intermediate turbulent energy cascade; (v) successive collisions are correlated since wave packets are distorted as they follow diverging ﬁeld lines; (vi) in common with the weak MHD cascade, there is no parallel cascade of energy, and the cascade to small perpendicular scales strengthens as it reaches higher wave numbers; (vii) For an appropriate weak excitation, there is a natural progression from a weak, through an intermediate, to a strong cascade.

Collective dynamics of the fireball

a r X i v :n u c l -t h /0409075v 1 30 S e p 20041Collective dynamics of the ﬁreballBoris Tom´a ˇs ik ∗The Niels Bohr Institute,Blegdamsvej 17,2100Copenhagen Ø,DenmarkI analyse the identiﬁed single-particle p t spectra and two-pion Bose-Einstein correlations from RHIC.They indicate a massive transverse expansion and rather short lifetime of the system.The quantitative analysis in framework of the blast-wave model yields,however,unphysical results and suggests that the model may not be applicable in description of two-particle correlations.I then discuss generalisations of the blast-wave model to non-central collisions and the question how spatial asymmetry can be disentangled from ﬂow asymmetry in measurements of v 2and azimuthally sensitive HBT radii.1.INTRODUCTIONWhen studying nuclear collisions at highest energies we are interested in properties of strongly interacting matter.A clear signal of existence of an extended piece of matter is its collective expansion which is not present in simple nucleon-nucleon collisions.This can be deduced from the slopes of identiﬁed hadronic single-particle p t spectra:while the slope is universal for all particle species in proton-proton collisions,the spectra become ﬂatter with increasing particle mass in collisions of heavy ions.Hadrons interact strongly an so they can only decouple from the system when it becomes dilute enough.Their spectra are ﬁxed at the moment of freeze-out and carry information about the phase-space distribution in the ﬁnal state of the ﬁreball.We want to reconstruct this information and get an idea about the collective evolution of the ﬁreball by comparing its ﬁnal state to its initial state which is known from the energy and the collision geometry.The process of freeze-out is very complicated [1].Here I will simplify it and assume that all particles decouple along a speciﬁed three-dimensional freeze-out hypersurface [2].The phase-space distribution at the freeze-out hypersurface will be parametrised by the so-called blast-wave model .I will ﬁrst introduce the model and analyse single-particle p t spectra and two-pion correlations from RHIC.Then I show how the model is generalised to non-central collisions and focus on calculation of v 2and azimuthal dependence of HBT radii.2.THE BLAST-WAVE MODEL FOR CENTRAL COLLISIONSI will assume that in the end of its evolution the ﬁreball is in a state of local thermal equilibrium given by temperature T and chemical potentials µi for every species i .The2freeze-out hypersurface will stretch along the τ=√(2π)3expp ·u −µ√2∆τ2,(3)where the term m t cosh(y −η)comes from the Cooper-Frye pre-factor dσµp µ[2]and the exponential in τrepresents smearing in the freeze-out time.The energy in the statistical distribution is taken in the rest-frame of the ﬂuid:E =p ·u .This introduces “coupling”between momentum of the emitted particle and the ﬂow velocity of the piece of ﬁreball where it comes from.The strength of this coupling is controlled by 1/T .2.1.Calculation of spectraThe single-particle spectrum is obtained from the correlation function as E d 3N1As space-time coordinates I will be using longitudinal proper time τ=√m 2+p 2t ,and the azimuthal angle φsuch that p µ=(m t cosh y,p t cos φ,p t sin φ,m t sinh y ).3The inverse slope T ∗appears as a parameter of a ﬁt to the m t spectrum with the function N exp(−m t /T ∗).3 The full expression for the single-particle spectrum reads[5]E d3N2π2 ∞0dr r G(r)τ0(r)m t K1 m t cosh(ρ(r))T−p t dτ0T I1 p t sinh(ρ(r))const+z2+r2and the second term on the r.h.s.of eq.(6)gives acontribution.Thus the temperature obtained fromﬁts with that model cannot be directly compared to the one which will obtained below[8].2.2.Calculation of HBT radiiI will also calculate the HBT4radii in Bertsch-Pratt parametrisation(see e.g.[9]for more details on HBT interferometry).They appear as width parameters of a Gaussian parametrisation of the correlation functionC(q,K)−1=λexp −R2o(K)q2o−R2s(K)q2s−R2l(K)q2l ,(7) where q and K are momentum diﬀerence and the average momentum of the pair,respec-tively,and the R’s are the HBT radii.They measure the lengths of homogeneity[10],i.e., sizes of the part of the source which produces pions with speciﬁed momentum.The sizes are measured in three directions:longitudinal is parallel to the beam,outward parallel to transverse component of K,and sideward is the remaining Cartesian direction.The parametrisation(7)is valid in the CMS frame at mid-rapidity of central collisions;other-wise terms mixing two components of q may appear(like R2ol q o q l,for example;see[9]for4more details).The HBT radii will be determined asR2o(K)= (˜x−βt˜t)2 ,(8a) R2s(K)= ˜y2 ,(8b) R2l(K)= ˜z2 .(8c) In eqs.(8)x,y,z,t stand for the Cartesian space-time coordinates,z is the longitudinal coordinate and x the outward coordinate.Averaging and the tilde are deﬁned as f(x) (K)= d4x f(x)S(x,K)K0.(9) Because the lengths of homogeneity depend on momentum,the HBT radii show depen-dence on K t.For R2s(K t),an approximate formula5says[4]R2s=R2Gm2+K2t.(10)Thus the temperature T and transverse expansion measured byρ0are coupled together here.However,recalling eq.(5),they can be disentangled by analysing both the HBT radii and the single particle m t spectrum.3.THE FREEZE-OUT STATE IN CENTRAL COLLISIONSI willﬁt the model to single-particle spectra and HBT radii measured at RHIC.First, each of the identiﬁed spectra of pions,kaons,and protons of both charges will beﬁtted individually.This provides a consistency check for the assumption that all these species freeze-out simultaneously:if the results of theﬁts are incompatible,this assumption is invalid.After that,I willﬁt the HBT radii and look again whether they can be accommodated with the same model as the single-particle spectra.Resonance production of pions will be taken into account for spectra but not for HBT radii.A study of the inﬂuence of resonance production on HBT radii indicated that in the presence of transverseﬂow they are changed only marginally[11].Chemical potentials of the resonance species are determined in accord with the model of partially chemically frozen gas[12].It assumes that after the chemical freeze-out[13] the eﬀective numbers of particles decaying weakly(and thus slowly)stayﬁxed while the strong(and therefore fast)interactions stay in equilibrium.Eﬀective number of any given species includes this species plus particles which can be produced on average from decays of all present resonances.Adjectives“fast”and“slow”refer to comparison with the typical time scale of theﬁreball evolution.As an example:eﬀective number of pions ˜Nπ=Nπ+2Nρ+N∆+...,and chemical potential of∆will beµ∆=µπ+µN because it is in equilibrium with pions and nucleons.Au+Au at√5Actually,this formula was derived in a slightly diﬀerent model,where R G was the transverse size. Nevertheless,it can be used for the qualitative arguments here.5123456780.20.30.40.50.6R o u t [f m ]M t [GeV/c]STAR π-π-STAR π+π+PHENIX π-π-PHENIX π+π+123456780.20.30.40.50.6R s i d e [f m ]M t [GeV/c]fits also spectra best fit to HBT2468100.20.30.40.50.6R l o n g [f m ]M t [GeV/c]Figure 1.Two ﬁts to the HBT radii measured in Au+Au collisions at√s =200A GeV.Results from ﬁts to identiﬁed single-particle spectra ofpions,kaons and protons[17,18]overlap at 1σlevel at T ≈115MeV and v t ≈0.57.The ﬁt to HBT radii (Fig.2)with these parameters is marginally good.The best ﬁt is again obtained with an unphysically low freeze-out temperature.The distinguishing power between the two models will be improved when ﬁnal data will be ﬁtted.Note that a ﬁt to ﬁnal results from PHENIX seem to be in better agreement with the conventional parameter values [21].A very low freeze-out temperature is also obtained when the complete collection of HBT radii form Pb+Pb collisions at projectile energy of 158A GeV is ﬁtted.That ﬁt is not shown here due to lack of space.Conclusions from central collisions.The parameters obtained in the ﬁts to K t dependence of HBT radii have no direct physics interpretation.The low temperature just indicates the need for very strong coupling between momentum and ﬂow within the612345678R o u t [f m ]M t [GeV/c]12345678R s i d e [f m ]M t [GeV/c]0246810R l o n g [f m ]M t [GeV/c]Figure 2.Fits to the HBT radii measured in Au+Au collisions at√x 2R 2y,Rx =a R ,R y =R/a.(11)Here I deﬁne the direction of the x-coordinate to be parallel to the impact parame-ter,and y-coordinate as perpendicular to the reaction plane.The spatial anisotropy isparametrised by the parameter a .The transverse ﬂow rapidity—given by eq.(2)for7 central collisions—will also vary with the angleψwith respect to the reaction planeρ(r,ψ)=ra−2cos2ψ+a2sin2ψρ0(1+ρ2cos(2ψﬂow)).(12)I will discuss two models which diﬀer in the azimuthal dependence ofρ(r,ψ)[22]. Model1:Transverse velocity is perpendicular to the surface andψﬂow=Arctan y8R o,22/R o,02, K t = 0.3 GeV/c >0<01.4aR o,22/R o,02, K t = 0.9 GeV/c >0<00.70.80.911.11.21.3 R s,22/R s,02, K t = 0.3 GeV/c<0>0-0.4-0.200.20.4ρ21.4a R s,22/R s,02, K t = 0.9 GeV/c<0>0-0.4-0.200.20.4ρ20.70.80.911.11.21.31.4a R o,22/R o,02, K t = 0.9 GeV/c>0<00.70.80.911.11.21.322 1.4a R s,22/R s,02, K t = 0.9 GeV/c >0<0-0.4-0.200.20.4ρ20.70.80.911.11.21.3Figure 3.The second order Fourier terms normalised by the average HBT radii R 2i,2/R 2i,0(see eq.(13))as a function of a and ρ2.Upper row:outward radius;lower row:sideward radius.Thick lines show where there is no second-order oscillation of the radius as a func-tion of φ.Consecutive lines correspond to increments by 0.1.Left panel:calculation with Model 1at p t =300MeV /c (left)and p t =900MeV /c (right).Right panel:calculation with Model 2at p t =900MeV /c ;results at p t =300MeV /c are similar to Model 1.parison with dataBy comparing to data one can try to distinguish whether we observe an in-plane or out-of-plane elongated ﬁreball and which of the two introduced models is better.In Figure 4I show v 2(p t )compared to curves calculated with Model 1.Within this model we obtain an out-of-plane elongated source (a <1),i.e.,a source which remembers its original deformation.Recall,however,that the same theoretical curves can be obtained in Model 2if we change a into 1/a ,so we would have an in-plane elongated source.The key to resolve this ambiguity is a ﬁt to the data on azimuthal dependence of HBT radii.I can ﬁt their oscillation well with Model 1if I assume parameters from the ﬁt to v 2;this is shown in Figure 5.I checked that in Model 2the amplitude of oscillations is opposite to what the data show [22].The conclusion is that the observed ﬁreball is out-of-plane elongated and that Model 1reproduces data better than Model 2.With the used “conventional”values of temperature and average radial ﬂow gradient ρ0I could not reproduce the absolute size and K t dependence of the sideward HBT radius,but note that this is the same kind of problems as was observed in central collisions.90.020.040.060.080.100.51 1.52v 2p t [GeV]Figure 4.Elliptic ﬂow coeﬃcient v 2measured by the PHENIX col-laboration in Au+Au collisions at√10R o 2[f m 2]R s 2 [f m 2]R o s 2 [f m 2]φR l 2[f m 2]φFigure 5.Azimuthal dependence of the HBT radii at midrapidity from Au+Au collisions at 200A GeV and centrality class 20–30%measured by the STAR Collaboration [25].From highest till lowest the data are taken for K t values:0.2,0.3,0.4,and 0.52GeV/c .Theoretical curves are calculated with Model 1with the parameter values:T =120MeV,ρ0=0.99,ρ2=0.035,a =0.94646,R =9.4fm,τ0=5.0fm /c ,∆τ=2.9fm /c .8.U.Heinz,talk at the RIKEN BNL Workshop,Nov.17-19,2003,/ﬂow03/.9. B.Tom´a ˇs ik and U.A.Wiedemann,in Quark-Gluon Plasma 3,R.C.Hwa and X.-N.Wang eds.,World Scientiﬁc,2003,hep-ph/0210250.10.A.N.Makhlin and Y.M.Sinyukov,Z.Phys.C 39(1988)69.11.U.A.Wiedemann and U.W.Heinz,Phys.Rev.C 56(1997)3265.12.H.Bebie,P.Gerber,J.L.Goity and H.Leutwyler,Nucl.Phys.B 378(1992)95.13.P.Braun-Munzinger,D.Magestro,K.Redlich and J.Stachel,Phys.Lett.B 518(2001)41.14.K.Adcox et al.[PHENIX Collaboration],Phys.Rev.Lett.88(2002)242301.15.C.Adler et al.[STAR Collaboration],Phys.Rev.Lett.87(2001)082301.16.K.Adcox et al.[PHENIX Collaboration],Phys.Rev.Lett.88(2002)192302.17.J.Adams et al [STAR Collaboration],Phys.Rev.Lett.92(2004)11230.18.S.S.Adler et al.[PHENIX Collaboration],Phys.Rev.C 69(2004)034909.19.M.L´o pez Noriega for the STAR Collaboration,Nucl.Phys.A715(2003)623c.20.A.Enokizono for the PHENIX Collaboration,Nucl.Phys.A715(2003)595c.21.F.Reti`e re,J.Phys.G 30(2004)S827.22.B.Tom´a ˇs ik,arXiv:nucl-th/0409074.23.F.Reti`e re and M.A.Lisa,arXiv:nucl-th/0312024.24.S.S.Adler et al.[PHENIX Collaboration],Phys.Rev.Lett.91(2003)182301.25.J.Adams et al [STAR Collaboration],Phys.Rev.Lett.93(2004)012301.。

The Forgotten Process the emission stimulated by matter waves

a r X i v :q u a n t -p h /0407187v 2 17 N o v 2004THE ”FORGOTTEN”PROCESS :the emission stimulated by matter waves.via the introduction of a new coeﬃcient C in addition to the Einstein coeﬃcients A and B.Weﬁrst summarize the usual approach describing atom-photon interactions using Maxwell-Boltzmann statistics and the Einstein A and B coeﬃcients.This is followed by a short summary of matter waves and their properties that are relevant to our purpose.In section4we restore the symmetry between electro-magnetic and matter waves,which we then apply to the thermal equilibrium (section5)leading to the derivation of transition probabilities for all processes. We obtain the usual Einstein coeﬃcients and a”forgotten”coeﬃcient from these probabilities in section6and conclude in section7.2The usual approachWe consider two level atoms where E1and E2are the energy of the atom in each level(with E2−E1=¯hω0>0).The atoms are in an ideal electromagnetic cavity with perfectly reﬂecting walls.The thermal equilibrium,at temperature T,is achieved.The case of atoms in free space is obtained when the volume V of the cavity becomes inﬁnite.The electromagnetic energy is distributed among the diﬀerent modes of the cavity.It is described by Planck’s energy spectral density.ρϕ(ω,T)=dϕ×4πν2e¯hω/k B T−1(1) where k B is Boltzmann’s constant.Each term inρϕ(ω,T)has a precise meaning.•The electromagnetic wave is characterized by a wave vector−→k.We de-ﬁne2πν=c −→k .The electromagnetic waves in an ideal cavity must satisfy boundary conditions.As a consequence,νcan take only resonant values:the number of possible values betweenνandν+dνis d Nν=4πν2dν×V(2)c3•Finally,using Bose-Einstein statistics,the mean number of photons in a mode at thermal equilibrium is1˜ρϕ(ω,T)=ρϕ(ω,T)dνN1=d2d1B(em)and A=8πν20Therefore the existence of spontaneous emission(i.e.A=0),implies that absorption and stimulated emission also exist(i.e.A=0⇒B(em)=0and B(abs)=0).These results are well known and can be found in many textbooks[6,7,8]. 3Matter wavesMatter waves were introduced in de Broglie’s thesis in1924.At this time the photoelectric eﬀect had already been interpreted(Einstein1905)and the Comp-ton eﬀect observed(Compton1923).Thus,it was already known that light could behave as aﬂow of particles called photons[9].The merit of Louis de Broglie idea was to reverse the proposition and claim that particles such as electrons could behave like waves.Since these early days,the similarity between light waves and matter waves never ceased to be emphasized,from theﬁrst observation of electronic interfer-ences by Davisson and Germer in1927[10]to atom interferometry,and recently Bose-Einstein condensation of atoms,ﬁrst achieved in1995[11].Several examples can be given of matter waves and their applications:the electronic microscope and the Collela-Overhauser-Werner experiment with neu-trons[12]are well known.More recently laser cooling of atoms has lead to the development of atom interferometry used,for example,for ultra precise inertial sensors such as gravimeters[13],gradiometers[14]and gyroscopes[15].Ulti-mately the use of the forgotten process to produce Bose-Einstein condensates, i.e.coherent beams of matter waves with all atoms in the same mode,should result in atom lasers based on ampliﬁcation of the matterﬁelds via the forgotten process.These are expected to lead to signiﬁcant improvements in matter wave interferometry and to new applications.Presently,what is often called an atom laser is a Bose-Einstein condensate which has been produced by other means.Despite its complex structure,an atom can be considered as a particle with a mass M depending on its internal energy:M=M0+E/c2where E is the internal energy and M0the mass of the atom in its fundamental level.The total energy of the atom is then¯hω= 2M(7)whereωis its angular frequency,¯h−→k its momentum and the non-relativistic approximation holds for¯h −→k /M≪c.An atom is generally characterized by its angular momentum,−→J,which plays the role of an intrinsic spin.The value of−→J2is¯h2j(j+1).General results in quantum mechanics lead to the conclusion that2j is an integer which depends on the internal state of the atom,and that the atom is a boson when j itself is an integer.This is the only case that we consider[16].The energy can be degenerate.Therefore,for a given mass,the internal state is a multi-component vector which belongs to a d a dimensional space.4Keeping the deﬁnition2πν=c −→k ,the fundamental diﬀerence with re-spect to photons is that now the relation betweenωandνis the dispersion relation obtained from Eq.(7)¯hω=c3dν×V(9) and the number of modes d N m=d a×d Nν.As we consider bosons only,the mean number of particles per mode at thermal equilibrium is given by Bose-Einstein statisticse(¯hω−µ)/(k B T)−1whereµis the chemical potential(see ref.[17]for instance).In section2,we assumed that the atoms are not able to be created or an-nihilated,only their internal energy could change by emission or absorption of photons.However,considering what we know about the origin of the Universe, we have to admit that atoms can be created too.Of course,the correspond-ing mechanism can be complicated and slow but annihilation and creation of atoms via exchange of energy with the thermal reservoir are possible as well as annihilation and creation of photons;this is a matter of principle not of order of magnitude.Thus the number of atoms in the cavity must be determined from the condition of thermal equilibrium.The same happens to the photons with the same consequence,i.e.µ=0(see the black body radiation in[17]). Therefore:e(¯hω)/(k B T)−1.(10) Therefore,the spectral energy density isρa(ω,T)=d a×4πν2e(¯hω)/(k B T)−1.(11)Analogously to the electromagnetic case one can introduce a corresponding energy density per interval of angular frequency˜ρa(ω,T)deﬁned by Eq.(3). But now dν/dωis obtained from the dispersion relation(8)and therefore˜ρa(ω,T)=ρa(ω,T)2πω4Restoring the symmetry between electromag-netic and matter wavesIn section2,the atoms and the photons have been considered from two very diﬀerent points of view.For instance,the decay of an atom could be stimulated by the presence of the photons but the presence of the atoms did not produce any similar eﬀect.First,in order to restore the similarity between the atoms and the photons, let us modify the notation.Now,a is an atom in its fundamental state of energy E a.This atom can absorb a photonϕ,the corresponding excited state is aϕwhose internal energy is E aϕ.We assume that thermal equilibrium is driven by the chemical-like equationaϕ⇀↽a+ϕ(13) whereϕ,a and aϕare bosons described by waves(i.e.electromagnetic waves and matter waves)which are trapped in an ideal cavity at temperature T.An atom is considered as the quantum associated to a matter wave,therefore N aϕand N a are now occupation numbers of matter-wave-modes,similar to the occupation number of the electromagnetic modes(i.e.the number of photons).The mass of the atom depends on its internal energy.Thus we can interpret the change of the internal energy as the annihilation of an atom with the initial value of the mass and the creation of an atom with theﬁnal value of the mass.The atoms and the photons are assumed to be trapped in an ideal cavity without losses.The cavity is coupled to a thermal reservoir at temperature T. We assume the following properties:B1The photons and the atoms occupy respectively electromagnetic-modes and matter-wave-modes.The equilibrium is achieved when the mean numberof quanta per mode is given by the Bose-Einstein statisticse¯hω/k B T−1i.e.we substitute Eq.(10)into Eq.(4).B2Second we assume that photons and atoms can be either absorbed or emitted by the reservoir,in order to achieve the equilibrium spectral densities (1)and(11).B3Finally we assume that once thermal equilibrium is achieved,it remains without the help of the reservoir although it is in statistical equilibrium where annihilation and creation of photons and atoms,continue to happen.From the old point of view the dissymmetry between matter and light lies in the diﬀerence between the assumptions A1and A2of section2.Here,this dissymmetry has disappeared but the Maxwell-Boltzmann statistics used in as-sumption A1has been changed into Bose-Einstein relativistic statistics.The various modes of the excited atoms aϕare labelled by a set of indexes called m;the modes of the atoms a are labelled by n and the modes of thephotonsϕby a set of indexes k.We use the notation(aϕ)m for an excited atomaϕin the mode m,and the similar notations(a)n and(ϕ)k.With this notation6Eq.(13)becomesR(aϕ)m ⇀↽(a )n +(ϕ)k .L (14)Following assumption B3above,we assume that the equilibrium is achieved when,during a given arbitrary time,the number of reactions to the right (reac-tion R is the same as the number of reactions to the left (reaction L ).Moreover we accept the usual assumption that the number of reactions per unit time is proportional to the number of quanta in the modes involved.We introduce the concentrations [aϕ]m ,[a ]n and [ϕ]k i.e.the number ofatoms or photons per unit volume respectively in mode m ,n and k .The volume of the cavity is V .We deﬁne α,β(abs ),β(em )and γ:(i)The number of spontaneous reactions R per unit time is α×[aϕ]m V .(ii)The number of reactions L per unit time is β(abs )×[ϕ]k V ×[a ]n V .(iii)Given an excited atom,aϕ,in the mode m ,we consider its decay (reactionL )stimulated by the presence of the photons in the mode k .The number of such reactions per unit time is β(em )×[ϕ]k V ×[aϕ]m V .With Maxwell-Boltzmann statistics where M 0c 2≫¯h ω0and Mc 2≫k B T ,and within the ”broad band”approximation [18],these mechanisms result in Eq.(5).(iv)Finally,we assume that the reaction R can also be stimulated by the pres-ence of the atoms a in the mode n ,which restores the symmetry between elec-tromagnetic waves and matter waves.We call this mechanism the ”forgotten”process [19].The number of such reactions per unit time is γ×[a ]n V ×[aϕ]m V .5Thermal equilibrium and its consequencesCompared to the year 1917the conception of matter has changed dramatically.We will now explain why this conceptual change leaves the description of thermal equilibrium practically unchanged.However,we will emphasize the importance of the ”forgotten”process.Now we assume that equilibrium is achieved when the mean number of quanta per mode is given by Bose-Einstein statistics and when the number of reactions R per unit time is equal to the number of reactions L .Therefore[a ]n V =1e (¯h ωϕ)/(k B T )−1,(15)[aϕ]m V =1The energy of the atom a in the mode n is ¯h ωa ,similarly ¯h ωϕis the energy of the photon in the mode k and ¯h ωaϕis the energy of the excited atom aϕin the mode m .We assume that the energy (¯h ωa ,¯h ωϕor ¯h ωaϕ)deﬁnes the mode except for the polarization of the light and the degeneracy of the internal energy of the atoms [20].One can check that the equality (16)holds true at any temperature if and only ifωaϕ=ωa +ωϕand β(abs )=α=β(em )=γ(17)The ﬁrst relation in (17)expresses the conservation of energy while the other relations imply that the ”forgotten”process does exist (i.e.γ=0)because spontaneous decay is observed (because α=0).Let us now verify that at thermal equilibrium the results above are prac-tically the usual ones when the ”forgotten”process is neglected.We consider Eq.(16)with β(abs )=α=β(em )=γ.Then the sum of the last two terms is γ×[ϕ]k V ×[aϕ]m V 1+[a ]n [ϕ]k ≪1even with extreme (impossible!)values of ¯h ωϕand k B T .Therefore the contribu-tion γ×[a ]n V ×[aϕ]m V in Eq.(16)is completely negligible and the ”forgotten”process does not play any signiﬁcant role at thermal equilibrium.Moreover one can easily calculate N aϕ/N a where N aϕ(respectively N a )is the mean number of atoms with energy E aϕ=¯h ωaϕ(respectively E a =¯h ωa ).It is the number of atoms in one mode with energy ¯h ωaϕ(respectively ¯h ωa )times the number of modes with such an energy.From the preceeding assumptions we obtainN aϕ=d aϕe (¯h ωa )/(k B T )−1.(18)Finally,using the relation ¯h ωaϕ∼¯h ωa ∼M 0c 2≫k B T we obtainN aϕd ae (¯h ωa )/(k B T )d a e −(¯h ωaϕ−¯h ωa )/(k B T )(19)which is identical to Eq.(4)under the simple change of notation (aϕ→2,a →1and ¯h ωaϕ−¯h ωa =¯h ω0).We notice that at thermal equilibrium nothing is signiﬁcantly modiﬁed if we use Maxwell-Boltzmann statistics instead of Bose-Einstein statistics and if we neglect the ”forgotten”process.However,far from equilibrium this is not necessarily the case.Now we can use standard methods to give an estimation of γin the simplest case of a homogeneous line width.Let αbe the probability per unit time that an excited atom aϕ,in a given mode m with given energy ¯h ωaϕ,decays spontaneously into a +ϕwhere the pho-ton ϕis in the mode k with energy ¯h ωϕand the atom a in the mode n with energy ¯h ωa .Let us deﬁne the probability per unit time,dp ,that an excited atom de-cays spontaneously into a photon with angular frequency ωϕ∈[ωϕk ,ωϕk +dω]8and a ground state atom with angular frequencyωa∈[ωan−dω,ωan]where ωaϕ=ωa+ωϕ.We then havedp=α×dϕd a×d N a+ϕ(20) where d N a+ϕis the number of pairs of resonance values(νa,νϕ)that satisfy ωϕ∈[ωϕk,ωϕk+dω],ωa∈[ωan−dω,ωan]andωaϕ=ωa+ωϕ.For the photons,the number of resonant values ofνϕ,d Nϕ,over the band-width dωare given(c.f.Eq.(1))by the number of possible values ofν,i.e. 4πν2πc3V.(21) Similarly for the ground state atom,the number of resonant values ofνa over the bandwidth dωare given by4πν2πc3V=ωa 2π2c3¯h V≃ M0c2 √2π2c3¯h2V(22)where E k=¯hωa−M0c2is the kinetic energy of a,and where we have used ¯hωa+M0c2≃2M0c2.Comparing Eq.(21)to Eq.(22)we note that even in extreme conditions d N a is much bigger than d Nϕ.For example,with E k≈k B T≈10−13eV, M0c2≈1GeV and¯hωϕ≈20eV we have d N a≈104d Nϕ.As a result the number of possible energy pairs d N a+ϕis entirely determined by d Nϕbecause for eachωϕthere exists aωa such thatωaϕ=ωa+ωϕ(but not vice-versa),so we have d N a+ϕ≃d Nϕ.Under these conditions Eq.(20)becomesdp=α×dϕd a×ωϕνϕdωt spfϕ(ωϕ−ωϕ0)(24) where t sp is the time constant which characterizes the spontaneous emission(i.e. 1/t sp=A,the Einstein coeﬃcient).It is then straightforward to obtainαby eliminating dp between Eqs.(23) and(24)9α=βabs=βem=γ=πc3t spfϕ(ωϕ−ωϕ0)×1dϕωϕνϕ1¯hωϕ(26)where we have introduced the electromagnetic energy density of the mode k deﬁned as u k(ωϕ)=[ϕ]k×¯hωϕ.Similarly the probability per unit time,W(abs),of absorption of a photonϕin mode k by a ground state atom a in mode n to form an excited atom aϕin any mode isW(abs)=d aϕ×β(em)×[ϕ]k V=d aϕdϕωϕνϕ1¯hωϕ.(27)Expressions(26)and(27)are well known in the usual laser theory and can be found in many textbooks(e.g.[6]).On the other hand,the”forgotten”process leads to new results.We cal-culate the probability per unit time,W f,that an excited atom aϕin mode m undertakes the forgotten process i.e.that it decays into a ground state atom a in mode n and a photonϕin any mode.The decay due to the atoms a in moden,towards a special given mode ofϕ,has a probability per unit timeγ×[a]n V(see the property(iv)of section4above).Therefore,the probability of a decay towards the various modes k ofϕwhich have the same energy isW f=dϕ×γ×[a]n V=πc3tspfϕ(ωϕ−ωϕ0)×u n(ωa)6The Einstein and the”forgotten”coeﬃcients Now,on one hand we consider that the frequency width of the electromagnetic spectrum of the radiation in the cavity is much larger than the width of the natural decay spectrum of the atom(broad band approximation).On the other hand we model the spectrum of the radiation in the cavity by monochromatic radiation with angular frequenciesωϕℓwhereℓis an integer andωϕℓ+1−ωϕℓ=δωϕ.The function fϕ(ωϕℓ−ωϕ0)has a maximum forωϕℓ≃ω0;it is negligible for|ωϕℓ−ωϕ0|>∆ωwith∆ω≪ωϕ0,and it fulﬁlls the condition ∞0fϕ(ωϕℓ−ωϕ0)dω=1.Assumingδωϕ≪∆ωwe can write:ℓfϕ(ωϕℓ−ωϕ0)δωϕ≃1.(29)We consider a given initial state characterized by the number N aϕof atoms aϕwith angular frequencyωaϕ,and the number N a of atoms a with angular fre-quencyωa.We use the preceding results to calculate the number dN of photons which are emitted or absorbed during a time interval dt,through the various processes.Let us give theﬁnal results before we outline the derivation.Oneﬁnds(i)for spontaneous emission:dN sp=A N aϕdt=1d a dϕ×c3t sp(30)(iii)for emission stimulated by the photons: dN em=B(em)ρϕ(ωϕ0)N aϕdt withB(em)=14π¯hωϕ0ν2ϕ01d a×c3t sp(32)whereωa0=ωaϕ−ωϕ0.B(abs)and B(em)are the well known Einstein coeﬃcients but C is a new one.We now detail the derivation of the above expressions.Let us for instance calculate B(em).We assume that,at a given angular frequencyωϕ,the two electromagnetic polarizations display the same energy density u k(ωϕ).There-fore the number of photons produced during dt by the stimulated emission due to the electromagnetic radiation at angular frequencyωϕis dϕW(em)N aϕdt where W(em)is given by Eq.(26).The stimulated emission due to the radia-tions at the various frequenciesωϕℓresults in the number of emitted photons11dN (em )= ℓd ϕW (em )N aϕdt.The function f ϕ(ωϕ−ωϕ0)is a quickly vary-ing function of ωϕwhile u k (ωϕℓ)is slowly varying within the framework of the broad band approximation.More precisely,we assume u k (ωϕℓ)≃u k (ωϕ0),ωϕℓ≃ωϕ0and νϕℓ≃νϕ0,for ωϕ0−∆ω<ωϕℓ<ωϕ0+∆ω.Therefore we can use these expressions for u k (ωϕℓ),ωϕℓand νϕℓto calculate dN (em ).We obtaindN (em )= ℓd ϕ×W (em )N aϕdt= ℓd ϕ×1¯h ω2ϕℓνϕℓ1d ϕπc 3t sp d ϕu k (ωϕ0) ℓf ϕ(ωϕℓ−ωϕ0)Naϕdt(33)The spectral energy density is d ϕu k (ωϕ0)d ϕπc 3t sp ˜ρϕ(ωϕ0) ℓf ϕ(ωϕℓ−ωϕ0)δωϕN aϕdt.(34)Finally,using Eqs.(29)and (3)we ﬁnddN (em )=B (em )ρϕ(ωϕ0)N aϕdt(35)withB (em )=14π¯h ωϕ0ν2ϕ01d a πc 3t sp f ϕ(ωϕℓ−ωϕ0)×d a u n (ωa )d a πc 3t sp ×d a u n (ωa 0)=1ωϕ0νϕ01¯hωa0d a u n(ωa0)πc3δωϕ×V.In the samebandwidth there is only one resonant frequency for the photons(the consequence of the deﬁnition ofδωϕ)soδNϕ=ωϕ0νϕ0ωϕ0νϕ0.The spectral energy density is˜ρa(ωa0,T)=d a u n(ωa0)δN aδωϕωa0νa0d a×c3t sp.(40)One can notice the similarities between the expressions for C and B(em).A permanent state is achieved when dN sp+dN em+dN f=dN abs.Therefore, at thermal equilibrium,Eq.(5)becomesA N aϕ+B(em)ρϕ(ωϕ0,T)N aϕ+Cρa(ωa0,T)N aϕ=B(abs)ρϕ(ωϕ0,T)N a.(41) One can easily check that the expressions obtained above guarantee the validity of Eq.(41)at any temperature when N aϕand N a satisfy Bose-Einstein statistics.It is also possible to derive the conservation of energy and the coeﬃcients B(em),B(abs)and C directly from Eq.(41)where N aϕand N a fulﬁll Bose-Einstein rather than the Maxwell-Boltzmann statistics.However,in this paper, we chose to emphasize the elementary underlying processes providing,for ex-ample,the transition probabilities W(em),W(abs)and W f(Eqs.(26),(27)and (28))as functions of the observed electromagnetic line-shape fϕ(ωϕ−ωϕ0). 7ConclusionWe have derived a general description of atom-photon interactions of the form aϕ⇀↽a+ϕthat includes the”forgotten”process of decay stimulated by the matter waves a,additionally to the well known spontaneous decay and the decay stimulated by the electromagnetic wavesϕ.Our aim was to provide a description based on fundamental elementary principles,and to emphasize throughout this work the symmetry between atomic and matter waves.To do so,we have followed standard textbook descriptions of the involved known processes(in terms of transition probabilities,energy densities,observed line shapes,and Einstein coeﬃcients)but applied them to also derive analogous expressions for the”forgotten”process that cannot yet be found in textbooks.13Throughout this work the various mechanisms have been considered for the case of thermal equilibrium.But,similarly to the well known photonic pro-cesses,our results(in particular the transition probability W f and coeﬃcient C of the”forgotten”process)are quite general and remain valid far away from equilibrium.Non-equilibrium conditions(population inversions etc.)are the fundamental ingredients of lasers,and one expects the same to hold true for atom lasers i.e.sources of coherent matter waves.Atom lasers based on matter wave ampliﬁcation have not yet been built,but several proposals for a practi-cal realization based on the”forgotten”process can be found in the literature [2,4,5].Indeed,any practical realization will necessarily involve conditions far from thermal equilibrium,as can easily be seen from Eq.(15).For example,a single atom a inside a cavity corresponds to[a]n V=1and therefore the thermal equilibrium condition(15)is only satisﬁed for temperatures T≃1013K which are impossible to attain in practice.It is easy to generalize the results that we have obtained to the general bosonic case ab⇀↽a+b where ab,a and b are massive or massless bosons of any kind(photons,atoms,molecules,etc.).If we assume that the same concepts are valid for the decay of an atom and for the chemical reaction ab⇀↽a+b, i.e.the concept of line shape where fϕ(ω−ωϕ0)becomes f b(ω−ωb0)and the concept of stimulated and spontaneous reactions,then all our considerations apply to the general bosonic case.Moreover if we assume that in the same elementary angular frequency interval dω,the number of resonance frequencies is much higher for b than for a,the developments of the preceeding sections (sections3to6)remain unchanged,except for the substitutionϕ→b and νϕ=ωϕ/2π→νb=Scientiﬁc,7-12,(1996).Spreeuw R.J.C.,et al.,Scheme for a bosonic atom laser,Proc.XII Conf.on Laser Spectroscopy,Inguscio M.,Allegrini M., Sasso A.,ed.,World Scientiﬁc,301-302,(1996).Bord´e.Ch.J.,Ampliﬁcation of atomic waves by stimulated emission of atoms,Proc.XII Conf.on Laser Spectroscopy,Inguscio M.,Allegrini M.,Sasso A.,ed.,World Scientiﬁc,303-307,(1996).[3]Wiseman H.M.,Collett M.J.,An atom laser based on dark-state cooling,Phys.Lett.A202,246-252,(1995).[4]Spreeuw R.J.C.,et al.,Laser-like scheme for atomic-matter waves,Euro-phys.Lett.32,6,469-474,(1995).[5]Bord´e,Ch.J.,Ampliﬁcation of atomicﬁeld by stimulated emission of atoms,Phys.Lett.A204,217-222,(1995).[6]Yariv A.,Introduction to optical electronics,Holt,Rinehart and Wilson,(1976).[7]Sargent M.III,Scully M.O.and Lamb W.E.Jr,Laser Physics,Addison-Wesley,(1974).[8]Hilborn R.C.,Einstein coeﬃcients,cross sections,f values,dipole moments,and all that.,Am.J.Phys.50,11,982-986,(1982)and references therein, (erratum in Am.J.Phys.51,5,471,(1983)).[9]The word”photon”was invented several years later but the concept ofphoton was already there.[10]Davisson C.J.and Germer L.H.,Diﬀraction of electrons by a crystal ofnickel,Phys.Rev.30,705-740,(1927).[11]Anderson M.H.,Enscher J.R.,Matthews M.R.,Wieman C.E.and Cor-nell E.A.,Observation of Bose-Einstein condensation in dilute atomic va-por,Science269,198-201,(1995).[12]Colella R.,Overhauser A.W.,Werner S.A.,Observation of Gravitationallyinduced Quantum Interferences,Phys.Rev.Lett.34,1472-1474,(1975).[13]Peters A.,Chung K.Y.,Chu S.,A measurement ofgravitational accelerationby dropping atoms,Nature400,849-852,(1999).[14]Snadden M.J.,et al.,Measurement of the Earth’s gravity gradient with anatom interferometer-based gtravity gradiometer,Phys.Rev.Lett.81,971-974,(1998).[15]L.Gustavson,ndragin,and M.A.Kasevich.Rotation Sensing withaDual Atom-Interferometer Sagnac Gyroscope.Class.Quantum Gravity, 17:2385-98,(2000)15[16]The sensitivity of atom interferometers is directly related to the number ofatoms(presently of order of105).Therefore,because of the Pauli exclusion principle,the use of fermions is of less interest and not considered here. [17]Landau L.D.and Lifschitz E.M.,Statistical Physics,Pergamon Press,NewYork,(1969).[18]The”broad band”approximation is valid when the radiation in the cavitydisplays a line shape whose width is large compared to the width of the natural lineshape of the atomic decay.[19]The short chronology that we gave shows that this process could not beconsidered before1924.It was actually forgotten between1924and the 1990s when itﬁrst appeared in the context of Bose-Einstein condensation [2,3,4,5].[20]This is the case,for example,for an ideal rectangular cavity of suitableproportions.16。

Blue Perturbation Spectra from Hybrid Inflation with Canonical Supergravity

∆T T
and an appropriate potential for the inﬂaton has to be found [4]. In
particular, a variation of hybrid inﬂation, smooth hybrid inﬂation [5], in which the phase transition takes place gradually during inﬂation, has been invented to successfully address this issue. As soon as one replaces global by local supersymmetry it is well-known that the potential becomes very steep and inﬂation becomes, in general, impossible. This is, to a large extent, due to the generation of a mass for the inﬂaton which is larger than the Hubble constant H . For the simple superpotentials used so far in SUSY bybrid inﬂation such a 2
Abstract
We construct a hybrid inﬂationary model associated with the superheavy scale MX ∼ 1016 GeV of supersymmetric grand uniﬁed theories in which the inﬂaton potential is provided entirely by canonical supergravity. We ﬁnd that the spectrum of adiabatic density perturbations is characterized by a strongly varying spectral index which is considerably larger than unity. Moreover, the total number of e-foldings is very limited. Implications of our analysis for other hybrid inﬂationary scenarios are brieﬂy discussed.

The spectral energy distribution of self-gravitating protostellar disks

arXiv:astro-ph/01063
Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy Universit` a degli Studi di Milano, Dipartimento di Fisica, Via Celoria 16, I-20133 Milano, Italy
Received / Accepted Abstract The long wavelength emission of protostellar objects is commonly attributed to a disk of gas and dust around the central protostar. In the ﬁrst stages of disk accretion or in the case of high mass protostars, the disk mass is likely to be suﬃciently large, so that the disk self-gravity may have an impact on the dynamics and the emission properties of the disk. In this paper we describe the spectral energy distribution (SED) produced by a simple, non-ﬂaring, self-gravitating accretion disk model. Self-gravity is included in the calculation of the rotation curve of the disk and in the energy balance equation, as a term of eﬀective heating related to Jeans instability. In order to quantify in detail the requirements on the mass of the disk and on the accretion rate posed on the models by realistic situations, we compare the SEDs produced by these models with the observed SEDs of a small sample of well-studied protostellar objects. We ﬁnd that relatively modest disks - even lighter than the central star - can lead to an interesting ﬁt to the infrared SED of the FU Orionis objects considered, while in the case of T Tauri stars the required parameters fall outside the range suggested as acceptable by the general theoretical and observational scenario. On the basis of the present results, we may conclude that the contribution of a selfgravitating disk is plausible in several cases (in particular, for FU Orionis objects) and that, in the standard irradiation dominated disk scenario, it would help softening the requirements encountered by Keplerian accretion models. Key words. Accretion, accretion disks – Gravitation – Stars: pre-main sequence

Comparison of models for predicting band emissivity of carbon dioxide and water vapour

Comparison of models for predicting band emissivity of carbon dioxide and water vapour at hightemperaturesM.Alberti a ,⇑,R.Weber a ,M.Mancini a ,M.F.Modest ba Institute for Energy Process Engineering and Fuel Technology,Agricolastr.4,38678Clausthal-Zellerfeld,Germany bSchool of Engineering,University of California,Merced 95343,USAa r t i c l e i n f o Article history:Received 13November 2012Received in revised form 30April 2013Accepted 5May 2013Available online 7June 2013Keywords:Spectral transmissivity Band emissivity Carbon dioxide Water vapour High temperaturea b s t r a c tA comparison of different models for predicting band emissivity of CO 2and H 2O at temperatures up to 1550K is presented.The calculations do not contain line-by-line databases only;a narrow-band as well as a wide-band model are also included.The main objective of this work is in comparing the spectral transmissivity and band emissivity values obtained from line-by-line calculations using the most recent HITEMP-2010as well as CDSD-1000spectral database with measured data.Differences between the pre-vious HITEMP-2004and the HITEMP-2010spectral databases are depicted.The measurements are also compared with a narrow-band model as well as a wide-band model because both have frequently been used in heat transfer calculations.It is demonstrated that line-by-line calculations show high accuracy when computing CO 2transmissivities but inaccuracies still remain in case of H 2O at temperatures above 1000K.The narrow-band model as well as the wide-band model show larger discrepancies if compared to the line-by-line predictions.Ó2013Elsevier Ltd.All rights reserved.1.Introduction and objectivesRadiative heat transfer is due to the emission and absorption of photons and its energy ﬂux is maximum in vacuum.If a medium is present,it may or may not participate in the radiative energy ex-change.Generally,molecules with a deﬁned dipole moment take part in radiative heat transfer only.Carbon dioxide,water vapour,and carbon monoxide are relevant in combustion.Molecules can absorb and emit photons in certain wavenumber intervals only and in order to predict radiative ﬂuxes knowledge of radiative properties is required.It is known from quantum mechanics that energy –that is mainly the sum of electronic,rotational,and vibra-tional energies –of atoms and molecules is quantized.If an atom or molecule absorbs or emits a photon,its energy is either increased or decreased by the amount of the photon’s energy.Due to the fact that the energy states are quantized,only photons with a certain energy and,therefore,wavenumber can be absorbed or emitted.There are numerous possible transitions leading to several thou-sand individual absorption lines over an entire absorption band,which makes the calculation of radiative heat transfer in case of participating media cumbersome [1–6].Considering spectral intensity i 0g incident on an elemental gas volume leads to the radiative transfer equation (RTE).If both in-and out-scattering can be omitted,the radiative transfer equa-tion at local thermal equilibrium reads [3–6]d i 0g ðg ;s Þa g Ád s¼Ài 0g ðg ;s Þþi g ;b ðg ;T Þð1ÞIn the above equation a g is the linear absorption coefﬁcient in unitsof reciprocal length (cm À1),s is the pathlength,and i 0g ð;b Þis the spec-tral directional (blackbody)intensity in units of W cm sr À1m À2while g stands for wavenumber expressed typically in cm À1.The linear absorption coefﬁcient can be seen as the reciprocal mean free path that a photon can travel through a column of gas before being absorbed.Eq.(1)is valid for one direction and one wavenumber only and if the medium is homogeneous,a g =const.,its solution isi 0g ðg ;s Þ¼i 0g ðg ;0ÞÁexp ðÀa g Ás Þþi g ;b ðg ;T ÞÁ½1Àexp ðÀa g Ás Þð2ÞThen,the spectral directional transmissivity can be deﬁned ass g ¼exp ðÀa g Ás Þð3Þand the spectral absorptivity asa g ¼1Àexp ðÀa g Ás Þð4Þwhich equals the spectral emissivity ( g )in the case of thermal equilibrium [3–6].Now,the RTE can be written in terms of spectral transmissivity as well as spectral emissivityi 0g ðg ;s Þ¼s g Ái 0g ðg ;0Þþ g Ái g ;b ðg ;T Þð5Þ0017-9310/$-see front matter Ó2013Elsevier Ltd.All rights reserved./10.1016/j.ijheatmasstransfer.2013.05.011Corresponding author.Tel.:+495323722476.E-mail address:alberti@ievb.tu-clausthal.de (M.Alberti).URL:http://www.ievb.tu-clausthal.de/(M.Alberti).Theﬁrst term in the above equation accounts for the extinction of incoming radiation intensity due to absorption whereas the second term accounts for the increase of intensity due to spontaneous emissions of the radiatively participating species.Because the absorption coefﬁcient and,therefore,spectral transmissivity varies rapidly with wavenumber,it is common to use spectrally averaged –so-called–total values.For example,total emissivity,that is the emissive power divided by the blackbody emissive power is calcu-lated as followsðsÞ¼R1gÁi g;bðg;TÞÁd gR1i g;bðg;TÞÁd gð6ÞThe spectral intensity can be replaced by spectral emissive power divided by p number,(the factor p is due to the conversion of pro-jected area to real area,see e.g.[4,5])so thatðsÞ¼pÁR1gÁ_e g;bðg;TÞÁd gpÁR1_e g;bðg;TÞÁd gð7ÞIntroducting Planck’s Radiation Law as well as Stefan–Boltzmann-Law,the total emissivityﬁnally becomesðsÞ¼1ÁT ÁZ1gÁC1Ág3exp C2ÁgÀ1Ád gð8Þwhere r is the Stefan–Boltzmann constant,C1and C2are Planck’s First and Second constants.With properly averaged values of trans-missivity and emissivity,it is possible to solve the RTE without con-sidering the rapidly varying absorption coefﬁcient.If the integration in Eq.(7)is performed over a spectral interval instead of the whole spectrum,the value is called band emissivity[3–6].Radiative transfer equation is important not only because it provides the basis for spectroscopic measurements of monochro-matic transmissivity but it is also the starting point in the develop-ment of methods for calculating radiative transfer in absorbing-emitting media.In the latter case methods of averaging over both the wavenumber and direction are needed.Averaging over wave-number results in a spectral model for calculating either the absorption coefﬁcient or transmissivity(or both)in a certain wave-number band.Averaging over wavenumber can be carried out either experimentally or theoretically.In the past,the measure-ments of band transmissivity,carried our for several temperatures and pressures of the absorbing gas,provided the basis for the development of narrow-band-models(NBM)and wide-band-mod-els(WBM).More recently,line-by-line data are used for the devel-opment of narrow-band-models or emissivity correlations for speciﬁc spectral bands.Obviously,such developed NBMs or corre-lations are as good as the database which was used for their devel-opment.Recently,HITEMP-2010[7]database has been released as an update to its previous version called here as HITEMP-2004, which is just HITRAN-2004extended by the hot lines listed in Refs. [8,9].The main difference between HITEMP-2010and HITEMP-2004is in the inclusion of a number of CO2lines of the Russian CDSD-1000[10]database(for details see Ref.[7]).Furthermore, in case of H2O the number of lines has been increased roughly by a factor of ten.The main objective of our work is in comparing the emissivity/ transmissivity values obtained by performing line-by-line calcula-tions using HITEMP-2010with the measured transmissivity data of Bharadwaj et al.[11–14](see below).As a matter of fact,publica-tions of Bharadwaj et al.[11–13]contain not only the measured transmissivity but also HITEMP-2004based line-by-line calcula-tions.One may argue,that their work has identiﬁed the shortcom-ings of the HITEMP-2004and has perhaps prompted the update. Since the release of HITEMP-2010,a number of projects have been initiated to asses its value for combustion applications.Recently,Becher et al.[15]have validated HITEMP-2010(CO2and H2O mol-ecules)in the temperatures up to1773K using the transmissivity data measured by Fateev and Clausen[16,17].It will be demon-strated later that theﬁndings of our work are in agreement with those of Becher et al.[15].Over the last two decades or so,RADCAL narrow band model [18]has been used not only in numerous heat transfer calculations but also in interpreting in-ﬂame data after performing a narrow-angle radiometry acrossﬂames,as exempliﬁed by Refs.[19,20]. Since the model became widely used,we also include this model in our work to demonstrate the departure from HITEMP-2010. Similarly,the Exponential Wide Band Model of Edwards and Balakrishnan[21,22]has been included since it has been used in many works concerning combustion[23–25].2.Measured data[11–14]Medium resolution transmission measurements(resolution of approximately4cmÀ1)for the most important bands of both CO2 and H2O at ambient pressure,temperatures up to1550K,and pathlengths up to50cm were recently published in[11–14].In the case of CO2,the following bands were measured:2.0,2.7,4.3, and15.0l m,and for H2O:1.8,2.7,and6.3l m bands.Fig.1shows the location of the bands together with Planck’s function for three different temperatures.As expected,the importance of lower wavelengths bands increases at elevated temperatures,as shown in Fig.1.The measurements that have been performed by Bharadwaj and Modest[11–14]are used in this paper as a basis for the comparison between the two spectral databases CDSD-1000[10]and HITEMP-2010[7]as well as NBM and EWBM.It will be shown in Section5 that the current HITEMP-2010update brought the most improve-ments to the carbon dioxide database.Therefore,the focus in this paper is on CO2and fewer calculations are presented for H2O.3.Spectral models usedA software written in FORTRAN named ABS-EMI[26]is used to carry out line-by-line(LBL)calculations,see also Ref.[27].This software uses purely Lorentzian line shapes to account for line broadening.A short discussion whether the Lorentzian line shape is appropriate is presented below.A spectral averaging of4cmÀ1 is chosen for the spectra of Section4.It should be noted that the CDSD-1000version used in this paper includes pure12C16O2only and no isotoplogues are enclosed while HITEMP-2010includes se-ven isotopologues weighted by their standard atmospherical abun-dances.Therefore,the HITEMP-2010database used in this paper contains approximatelyﬁve times more lines than CDSD-1000. Nevertheless,it will be shown in Section4that there are no sub-stantial differences between transmissivities calculated using the two databases.Generally,absorption lines in spectral databases are mainly characterised by line position and intensity.Broadening mechanisms are accounted for by multiplying the intensity with a normalised lineshape.Each shape has its own maximum and half-width at half maximum.The most important broadening mechanisms are due to molecular collisions and due to the Doppler Effect;the correspponding lineshapes are named Lorentzian or Doppler.When both mechanisms are important it is convient to calculate the convolution of both shapes resulting in the Voigt line-shape.A measure of whether the Voigt function has a similar dis-tribution to Lorentz or Doppler is given by the ratio of Lorentz to Doppler half-width at half maximum.If the ratio is large,there are small differences between Voigt and Lorentz;if the ratio is small,the Voigt function approaches the Doppler lineshape.ItM.Alberti et al./International Journal of Heat and Mass Transfer64(2013)910–925911can be shown that this ratio is proportional to total pressure di-vided by temperature.Therefore,the Doppler contribution should become more important with increasing temperature[4,5].In the LBL calculations presented in this paper the pressure-shift of the line centre and the line mixing effects have been neglected[28].In order to visualise the difference between Lorentz and Voigt lineshapes,the absorption coefﬁcient of pure CO2at atmospheric pressure and1550K temperature is plotted in Fig.2and the corre-sponding transmissivity is shown in Fig.3.Bothﬁgures also show the difference between the values using Lorentz and Voigt line-shape.As it is shown in Fig.2,the absorption coefﬁcient differs by a small amount of maximum±0.0002cmÀ1only.In case of spec-tral transmissivity a value ofÀ0.0015is applicable,where the LBL calculation using Lorentz lineshape typically results in a somewhat larger transmissivity,see Fig.3.Nevertheless,the deviations are very small.Therefore,for the temperature range used in this paper, the Lorentzian lineshape is applicable which is consistent with observations of Wang and Modest[29].Instead of calculating each single line separately,NBMs average the spectral absorptivity over a narrow spectral bandwidth in a typical range of25cmÀ1leading to a smoother absorption coefﬁ-cient[4].RADCAL[18]is a FORTRAN-language software which cal-culates a NBM transmissivity with a spectral averaging between5 to50cmÀ1in the wavelength range from200to1l m.Some of the parameters are tabulated from measured data while others are modelled.In the calculations presented in this paper a C-language version[30]of the original Grosshandler’s program[18]is used.Generally,gases emit in no more thanﬁve strong bands.The ba-sic idea of EWBM is then to calculate an averaged transmissivity for each band.The model was origianally developed by Edwards and Balakrishnan[21,22].The EWBM used in this paper is a part of RADIANT[31]software written in1993at the International Flame Research Foundation and is identical to the model used by Lallemant et.al.[24,25,32,33].The band emissivities in the spectral region from g1to g2(see Section4)are calculated as the ratio of the emitted intensity inte-grated over the entire absorption band divided by the blackbody ing Eq.(4)leads toD g¼R g2g1i g;bðg;TÞÁð1Às gÞÁd gR g2g1i g;bðg;TÞÁd g¼R g2g1_e g;bðg;TÞÁð1Às gÞÁd grÁT4Áj fðg1ÁTÞÀfðg2ÁTÞjð9ÞThe denominator of Eq.(9)results from the Stefan–Boltzmann law and the conversion of intensity to emissive power is performed using_e g;b=pÁi g,b which is valid for isotropic radiation.The spectral blackbody emissive power_e g;b can be calculated using Planck’s radi-ation law;f(g iÁT)is the blackbody fractional function at a given wavenumber g i and temperature T[4].In case of EWBM the calcu-lation can be simpliﬁed further.Because of the constant spectral transmissivity,the factor(1Às g)can be taken out of the integral leadingto912M.Alberti et al./International Journal of Heat and Mass Transfer64(2013)910–925D g ;EWBM ¼ð1Às g ÞÁj f ðg L ÁT ÞÀf ðg U ÁT Þj j f ðg 1ÁT ÞÀf ðg 2ÁT Þjð10Þwhere g L and g U denote the lower and upper band wavenumber.In order to account for the inﬂuence of the whole vibrational–rotational band the spectral interval at which Eq.(9)is evaluated should be wide enough,so that the transmissivity reaches approx-imately unity at the boundaries of the band.Care must be taken in the case of LBL calculations.Due to the spectral line broadening,lines that are just outside the spectral range can inﬂuence the spec-tral behaviour inside this interval.Therefore,LBL calculations pre-sented here are evaluated over a spectral region which extends about 100cm À1to the left and the right hand side of the consid-ered regions.4.Results4.1.The 2.0l m-band of CO 2The 2.0l m-band of CO 2is an overtone/combination band and it is one of the weakest.Table 1shows the conditions at which the measurements [12]were taken.Both the measured and calculated transmissivities are shown in Figs.4–6.For all the conditions,the CDSD-calculated and the HITEMP-2010-calculated transmissivities represent very well the measured values.Minor differences occur between the CDSD and HITEMP-2010calculations.As shown in Fig.7,the overall band emissivity calculated using the CDSD data-base is only by 4.5%different from the measured value (on aver-age).For HITEMP-2010the value of 8.3%is applicable.Generally,the predicted band emissivity calculated using HITEMP-2010is al-ways a little bit larger than the one calculated using CDSD-1000.RADCAL substantially underestimates the spectral transmissiv-ities in the wavnumber range from 4750to 5150cm À1.The RAD-CAL calculated overall band emissivity is typically by a factor 2to 2.5larger than the measured value.As shown in Figs.4–6,the EWBM not only displaces the band location but also underesti-mates the overall band emissivity by as much as 33%(average va-lue is 19%).4.2.The 2.7l m-band of CO 2The 2.7l m-band of CO 2is a combination band.Table 2shows the conditions at which the measurements were taken [12].The measured and calculated spectral transmissivities are shown in Figs.8–10.At 1000K,see Fig.8,the LBL calculations using both databases show spectral transmissivities which are a bit lower than the measurments in the region 3450to 3750cm À1.The agree-ment becomes better with increasing temperature,see Fig.9.For condition 1,CDSD-1000as well as HITEMP-2010underestimates the band emissivity by around 30%while for the other conditions the deviations are between À3.8%and +7.9%.For all conditions with pure CO 2,the spectral transmissivity cal-culated using the NBM is somewhat smaller than the measured values.For 1000K temperature,the RADCAL calculated emissivites are 17%too small and around 11%too large for conditions 1and 2,respectively.At higher temperatures RADCAL values are typically 2.5%up to 10.6%too large if compared to the measured values,as can be seen in Fig.11.Altough the EWBM shifts the band center to higher wavenumbers by a value of approximately 100cm À1,see Figs.8–10,the calculated band emissivity is comparatively well predicted,see Fig.11.The deviations from measured values are be-tween À11.4%and 5.0%.4.3.The 4.3l m-band of CO 2The 4.3l m-band of CO 2is a fundamental band and,therefore,one of the strongest.It is a so-called headforming band (for details see [1–3]).Table 3shows the conditions at which themeasure-Table 1Description of the conditions for the 2.0l m-band of CO 2measurements [12,14].The remaining gas,in case of x CO 2<1,is N 2.Condition Temperature in K Pathlength in cm x CO 2p CO 2in atm q CO 2in kg m À3Figure 1100020 1.00 1.000.53642100050 1.00 1.000.536443130020 1.00 1.000.41264130050 1.00 1.000.4126551550400.500.500.173061550501.001.000.34606M.Alberti et al./International Journal of Heat and Mass Transfer 64(2013)910–925913ments were taken [12,14].The spectral transmissivity is shown in Figs.12–14.The band is so strong that the spectral transmissivity is almost zero for a relatively large region extending from 2400cm À1to 2600cm À1.For all the conditions,the CDSD-calculated and the HITEMP-2010-calculated spectral transmissivities represent well the mea-sured values.Minor differences occur between the CDSD and HI-TEMP-2010calculations in the wavenumber range 2000to 2150cm À1and the HITEMP-2010-calculated transmissivities are somewhat smaller than the CDSD-calculated.For conditions shown in Fig.12,HITEMP-2010seems to be less accurate than CDSD how-ever for Fig.13the opposite is observed.Similar statements are true for the spectral range on the right hand side of the band head around 2400cm À1.Nevertheless,the difference between the mea-sured and calculated band emissivities is from À5.0%to 6.8%in case of CDSD and À2.6%to À9%for HITEMP-2010(Fig.15).At lower temperatures,RADCAL overestimates the transmissiv-ity in the region from 2000to 2050cm À1,as shown in Figs.12and 13.At 1550K,the agreement becomes better,see Fig.14.On aver-age,the error between RADCAL-calculated band emissivity and the measurements is around À2.5%to 12.5%,see Fig.15.Because of the 4.3l m-band beeing a headforming one,the EWBM predicts a properly placed rectangle with an upper band head at a ﬁxed wavenumber of 2410cm À1[3,31].The band emissivity differs typ-ically by ±12%from the measurements,see Fig.15.4.4.The 15.0l m-band of CO 2The 15.0l m-band of CO 2is relatively strong since it is a funda-mental band.Table 4shows the conditions at which the measure-ments were taken [12,14].The spectral transmissivity is shown in Figs.16–18.As can be seen from the ﬁgures,the measuredtrans-914M.Alberti et al./International Journal of Heat and Mass Transfer 64(2013)910–925missivity reaches 1not until 1100cm À1and in addition to the 15l m-band,the 10.4l m-band (difference)as well as the 9.4l m-band (overtone/difference)occur.While calculating the band emissivity using the EWBM or RADCAL the band overlap-ping has to be accounted for.With decreasing CO 2partial pressure the overlapping disappears and the 10.4l m as well as 9.4l m bands can be neglected,as shown in Fig.18.At lower wavenum-bers,the measurement noise becomes signiﬁcant.Again,the CDSD-calculated and the HITEMP-2010-calculated spectral transmissivities represent well the measured values for all conditions,as shown in Fig.19.The largest differences occurwhen CO 2partial pressure is small,it means in condition 1and 5(see Table 4);the CDSD calculated and HITEMP-2010calculated emissivities are by 32%(condition 1)and 54%too small,if com-pared with the measured values.RADCAL shows a strange behaviour at wavenumbers above 800cm À1but the alternating over-and underestimations lead to a relatively good prediction of the band emissivity,see Fig.19.Appar-ently,the band emissivity for small amounts of CO 2is predicted with lower accuracy,see conditions 1and 5of Fig.19.In case of con-dition 5,the band emissivity is underestimated by 50%.In case of the EWBM,the three different bands can be well identiﬁed,see Figs.described Table 2Description of the conditions for the 2.7l m-band of CO 2measurements [12,14].The remaining gas,in case of x CO 2<1,is N 2.Condition Temperature in K Pathlength in cm x CO 2p CO 2in atm q CO 2in kg m À3Figure 11000400.050.050.02682100050 1.00 1.000.536483130040 1.00 1.000.41264130050 1.00 1.000.412695155020 1.00 1.000,346061550501.001.000.346010M.Alberti et al./International Journal of Heat and Mass Transfer 64(2013)910–92591516–18.It is worth noting that the EWBM shows no band overlap-ping between the fundamental and its adjacent bands while the other models show it (which is consistent with the measurements).The EWBM-calculated band emissivities agree well with the mea-sured values.A maximum error of À27.7%is applicable at condition 5,which is indeed better than LBL and RADCAL emissivities.4.5.The 1.8l m-band of H 2OThe 1.8l m-band of H 2O is a combination band and is,therefore,relatively weak.Table 5shows the conditions at which the mea-surements were taken [13,14].The spectral transmissivity is shown in Figs.20and 21.916M.Alberti et al./International Journal of Heat and Mass Transfer 64(2013)910–925The LBL calculations show good agreement with the measured transmissivities for both conditions.Minor differences occur at 1000K at the band wing around5600cmÀ1(Fig.20)and at 1550K at the around4900cmÀ1(Fig.21),where the spectral trans-missivity is a bit underestimated for both temperatures.At1000K, the band emissivity calculated using HITEMP-2010is around3.8% too large,if compared to the measurements,see Fig.22.At1550K, the departure is as large as7.2%.RADCAL generally shows correct dependence of the transmis-sivity with wavenumber but underestimates the transmissivity at wavenumbers lower than5100cmÀ1,as shown in Figs.20and 21.Therefore,the band emissivity is overestimated by as much as9.8%and24.3%at1000K and1550K,respectively,see Fig.22. With increasing temperature,the width of the rectangle predicted by the EWBM increases while the height decreases.But the mea-sured width of this band seems to be approximately unaffectedTable3Description of the conditions for the4.3l m-band of CO2measurements[12,14].The remaining gas,in case of x CO2<1,is N2.Condition Temperature in K Pathlength in cm x CO2p CO2in atm q CO2in kg mÀ3Figure11000400.050.050.02682100050 1.00 1.000.536412 3130040 1.00 1.000.41264130050 1.00 1.000.412613 51550500.010.010.00356155050 1.00 1.000.346014M.Alberti et al./International Journal of Heat and Mass Transfer64(2013)910–925917by the temperature increase of 550K,see Figs.20and 21.The EWBM underestimates the band emissivity by 8.8%at 1000K but overestimates by 16.7%at 1550K temperature.4.6.The 2.7l m-band of H 2OThe 2.7l m-band of H 2O is a strong overtone/fundamental band.Table 6shows the conditions at which the measurements were taken [13,14].The spectral transmissivity is shown in Figs.23and 24.The LBL calculations ﬁt well the data measured at 1000K,see Fig.23;small differences occur at the far wings below 3200cm À1and above 4100cm À1only.The LBL calculated band emissivity is underestimated by 3.4%only if compared to the measured value.At 1550K larger errors occur in the band wing from 3100to 3500cm À1(see Fig.24)resulting in an overestimation of the band emissivity by 17.5%,see Fig.25.RADCAL transmissivities are close to the LBL calculations and,therefore,show similar errors,especially at the higher tempera-ture,as can be seen in Figs.23and 24.The error for the band emis-sivity (Fig.25)is somewhat higher than for LBL;values of À4.7%and 19.8%are applicable at 1000K and 1550K,respectively.The EWBM places the band center at the proper position but underes-timates the transmissivity at 1550K.The band emissivitycalcu-918M.Alberti et al./International Journal of Heat and Mass Transfer 64(2013)910–925lated using the EWBM is by4.7%too small at1000K while it is as much as23.2%too large at1550K temperature,see Fig.25.4.7.The6.3l m-band of H2OThe6.3l m-band of H2O is a strong fundamental band.Table7 shows the conditions at which the measurements were taken [13,14].The spectral transmissivity is shown in Figs.26and27.Similarly to the1.8l m and2.7l m-band of H2O,the LBL calcu-lations using HITEMP-2010represent well the measurments at 1000K,see Fig.26;the band emissivity is overestimated by4.2%. As shown in Fig.27,large errors occur in the band wing at wave-numbers from1100to1500cmÀ1,so the band emissivity at 1550K temperature is overestimated by18.2%,see Fig.28.RADCAL also represents well the measured values at1000K (Fig.26),but substantially underestimates the transmissivity inin TableCondition Temperature in K Pathlength in cm x H2Op H2Oin atm q H2Oin kg mÀ3Figure1100040 1.00 1.000.219620 2155040 1.00 1.000.141721the left band wing(1100to1500cmÀ1)at1550K,see Fig.26.At 1000K the departure from the measured band emissivity is À0.3%only while at1550K it increases to21.5%,as shown in Fig.28.At1000K temperature,the EWBM predicts a band emissiv-ity that is very close to the measured value(an error ofÀ0.4%is applicable),but an error of29.4%is applicable at1550K,see Fig.28,which is the largest error for H2O.parison of different spectral databasesThe update brought a large number of new lines into the HI-TEMP-2010[7]database.In case of water vapour,the number of transitions have been augmented by a factor of around ten and, in case of carbon dioxide,by a factor ofﬁve[7–9].In this section a comparison between spectral transmissivities calculated using the HITEMP-2004,HITEMP-2010,CDSD-1000,and the measured [12,13]values is presented.It is worth mentioning that there are two HITEMP-2004versions for water vapour,which differ in the intensity cutoff value at a reference temperature leading to a dif-ferent number of lines,see Ref.[8]for details.The so-called 1500K version is used here.Calculations of the spectral transmis-sivities are carried out up to30000cmÀ1,although only HITEMP-2010provides lines for wavenumber larger than12000cmÀ1.The spectral region from0to6000cmÀ1for CO2and0to8000cmÀ1 for H2O is presented here only,because the transmissivity above this region is almost1and no measured values are available.Fig.29shows the LBL calculations using HITEMP-2010,HI-TEMP-2004,and CDSD-1000as well as the measured data [12,14]for pure CO2at0.5m pathlength and1550K temperature. The CDSD-1000and HITEMP-2010databases provide almost iden-tical transmissivities and both databases represent well the mea-surements.Such a good agreement between HITEMP-2010and CDSD-1000databases is to be expected since CDSD-1000lines have been included into HITEMP-2010(see introduction).Large differences can be observed in case of HITEMP-2004,whose trans-missivities are substantially underestimated in some spectral ranges,especially in the wings of the2.7and4.3l m-bands.This is consistent with the observations of Bharadwaj and Modest [12,14].Some minor differences occur in the wavenumber range from600to1000cmÀ1only.If one calculates the total emissivity of CO2using CDSD-1000and HITEMP-2010,the calculated values are0.141and0.144,respectively.For HITEMP-2004a value of 0.195is applicable.The calculated band emissivity using the mea-sured transmissivities is0.143.Fig.30shows the LBL calculations using HITEMP-2010and HI-TEMP-2004as well as the measured data[13,14]for pure H2O at 0.4m pathlength and1550K temperature.In addition to SectionCondition Temperature in K Pathlength in cm x H2Op H2Oin atm q H2Oin kg mÀ3Figure1100040 1.00 1.000.219623 2155040 1.00 1.000.141724。

Fluorescence spectroscopy and multi-way techniques PARAFA

Received 12th July 2013 Accepted 9th September 2013 DOI: 10.1039/c3ay41160e /methods
Introduction
a
University of New South Wales, Water Research Centre, Sydney, Australia. E-mail: krm@.au; Fax: +61 2 9313 8624; Tel: +61 2 9385 4601 Technical University of Denmark, National Institute for Aquatic Resources, Charlottenlund, Denmark. E-mail: cost@aqua.dtu.dk
PARAllel FACtor analysis (PARAFAC) is used in the chemical sciences to decompose trilinear multi-way data arrays and facilitate the identication and quantication of independent underlying signals, termed ‘components’. In 2011–2012, 334 Scopus-indexed journal and conference papers were published with keywords “PARAFAC” or “parallel factor analysis”. In the subset of papers where PARAFAC was used primarily as a tool for data interpretation (n ¼ 238, thus excluding 96 papers concerned primarily with developing or comparing algorithms, tools or statistical methodologies), PARAFAC was applied across research elds (medical, pharmaceutical, food, environmental, social, and information science) and to a wide range of data

Turbulence characteristics over complex terrain in West China

TURBULENCE CHARACTERISTICS OVER COMPLEX TERRAIN INWEST CHINAM.H.AL-JIBOORI ,XU YUMAO and QIAN YONGFU Department of Atmospheric Sciences,Nanjing University,Nanjing210093,P.R.China(Received inﬁnal form18January2001)Abstract.Meteorological data of velocity components and temperature have been measured on a mast of height4.9m at one site in the Heihe River Basin Field Experiment(HEIFE)conducted in west China.Mean and individual turbulence parameters,power spectra/cospectra,phase angles and their changes with fetch downwind of a change in surface roughness were analyzed.The tur-bulence characteristics depend strongly on the prevailing wind direction,which in turn is associated with changes in the upwind surface roughness pattern.The results show that values of horizontal velocity standard deviationsσu,v scaled with local friction velocity u under different conditions are larger than those overﬂat terrain,while the values ofσw/u have the same values as overﬂat terrain.The differences between variance values of the horizontal velocity components,u and v, over inhomogeneous terrain were found to be signiﬁcantly smaller than those overﬂat terrain.Since energy densities of the w spectra,uw and wT cospectra at low frequencies are relatively lower than those of longitudinal velocity spectra,total energies of w spectra,uw and wT cospectra tend to be in equilibrium with the local terrain.The values of phase angles at the low frequency end of the spectra showed obvious differences associated with changes of roughness.Keywords:Complex terrain,Phase angles,Power spectra and cospectra,Thermal internal boundary layer,Turbulence parameters.1.IntroductionAnalyses of spectra and cospectra provide a good basis to improve our knowledge of the characteristics of turbulence structure.During the last three decades spectral and cospectral characteristics in surface layers overﬂat,homogeneous surfaces and under various atmospheric stratiﬁcations have become both complete and reliable (Kaimal et al.,1972;Roth and Oke,1993).The study of turbulence parameters and spectral analyses over complex surfaces (varying topography and roughness)have special features in dealing with problems of wind energy conversion systems,pollutant transfer,etc.Several models have been developed by Panofsky and Townsend(1964),Peterson(1969),Peterson et al.(1976)and Højstrup(1981)to describe the development of wind proﬁles and surface stress proﬁles downstream of a change in surface roughness from a smooth to a rough surface.Power spectra of velocity components over various types of Supported by the National Sciences Foundation of China under Grant No.49735170.Current afﬁliation:Institute of Atmospheric Physics(JAP)in Beijing,P.R.China.Boundary-Layer Meteorology101:109–126,2001.©2001Kluwer Academic Publishers.Printed in the Netherlands.110M.H.AL-JIBOORI ET AL.complex terrain under neutral and unstable conditions have been measured by Højstrup(1981),Panofsky et al.(1982)and Founda et al.(1997).The most obvious physical factors that can inﬂuence the windﬁeld in areas on or near water(i.e.,oases,lakes,etc.)are the substantial and abrupt changes in sur-face roughness in going from the water to land or vice versa,and a corresponding change in the surface heatﬂux.Near the shoreline when the air crosses from a rough to a smooth surface or vice versa,the modiﬁed lower part of the boundary layer is called the internal boundary layer.If the airﬂows from a cooler to a warmer surface,the air will be thermally modiﬁed by the new surface properties,and the air layer modiﬁed over the new surface is called a thermal internal boundary layer (TIBL).The depth of this layer will increase with downwind distance.Above the modiﬁed layer,the air does not‘feel’the new surface.For an extensive review of the development and characteristics of the TIBL the reader can refer to the books of Stull(1988)and Garratt(1994).As will be demonstrated in Section2,thermal features as well as topograghic effects are important in generating TIBLs.This paper focuses on the study of characteristics of atmospheric turbulence over complex terrain as experienced for various fetch conditions arising under vari-ous wind directions,and different atmospheric stabilities.Dimensionless velocity standard deviations,σi/u ,power spectra of velocity components(u and w)and temperature T,power cospectra of uw and wT,and phase angles of uw and wT were computed.The main purpose of this study is to measure the inﬂuence of thermal and roughness changes on the properties of air turbulence,when the air crosses from oasis to desert,(similar to from water to land),based on the compar-ison with well-deﬁned results over homogeneous surfaces(no roughness change). The differences between different fetch conditions were discussed and compared to results reported by Miyake et al.(1970),Kaimal et al.(1972),Bradley(1980), Panofsky et al.(1977)and Xu et al.(1993).2.Observation Site,Data Analysis and Computation Procedures2.1.S ITE DESCRIPTIONThe data reported here were obtained from Huayin station in HEIFE,a Sino-Japanese co-operative program of atmosphere-land surface process experiments at the Heihe River Basin in the Hexi Corridor,Gansu Province,Western China. The experimental area(99◦30 –101◦00 E,38◦40 –39◦40 N)is70km×90km. Huayin station is located in the Gobi Desert about5km southwest of Linze County, surrounded by different topography.The Qilian Mountain is about40km south, 10km southwest and15km west from Huayin,while the oases lie about15km The main peak of the Qilian Mountain is at5062m,which lies about65km west-south-west of Huayin.Oases are natural water areas for producing crops(wheat and maize)and are cold and moisture sources not only during the day but also at night on clear or partly cloudy days.TURBULENCE CHARACTERISTICS OVER COMPLEX TERRAIN IN WEST CHINA111Figure1.Map of the Huayin Station.The numbers represent distances in km. northwest,2km north,3km east and30km southeast away as shown in a de-tailed map of the site(Figure1).Thus,under south to northwest wind directions, conditions can be considered as being locallyﬂat terrain,while for the other wind directions it should be regarded as complex.The data have been analyzed according to these different conditions and are presented in Sections3.1,3.2and3.3.The elevation of the station is about1410m with upslopes of1◦from east to west and 0.1◦from north to south.There are a few scattered plants around the station but the land surface is mostly composed of sand and gravel.Hu et al.(1994)estimated the roughness length,z◦,by the logarithmic wind proﬁle method from data under neutral conditions and the diagram method explained in detail by Stull(1988).The roughness length was obtained as0.25mm for both methods.This result is close to the experimental value of z◦=0.3mm on desert quoted by Oke(1987).112M.H.AL-JIBOORI ET AL.TABLE IClassiﬁcation data according to wind direction.Sector Wind direction(◦)Terrain No.of runsI180–230Flat5II280–310Oasis-to-ﬂat3III340–360Oasis-to-ﬂat72.2.D ATA ANALYSIS AND COMPUTATION PROCEDUREA three-dimensional sonic anemometer-thermometer instrument(Kaijo-Denki Dat-300,path length0.2m),installed on a mast of height4.9m above the ground, was used to measure theﬂuctuations of velocity components and temperature,on August16,1992.There were many hourly runs,the observation duration for each run was half an hour,with sampling16times per second(16Hz).In order to study the inﬂuence of inhomogeneity of terrain on turbulence characteristics,all the available data analyzed were divided into three categories according to different wind directions as indicated in Figure1and Table I.Although the data are only from one station,its geographic location has inter-esting features to study the characteristics and structures of atmospheric turbulence over complex terrain under various wind direction conditions.Power spectra,cospectra and phase angles of the three dimensional wind ve-locityﬂuctuations(u ,v ,w )and temperatureﬂuctuation(T )were computed by using the fast Fourier transform technique.A second-order polynomialﬁt was used for trend removal.The Hanning function was used to reduce leakage in the time series.Aliasing was minimized by choosing appropriate cut-off frequencies for these spectra.Finally,the raw spectral densities were block-averaged to provide smoothed estimates over frequency bands.3.Results and Discussions3.1.T HE INFLUENCE OF SURFACE INHOMOGENEITY ON TURBULENCEINTENSITYMean values of wind speed,U,friction velocity,u ,and the standard deviation componentsσi(i=u,v,w)were evaluated for all runs,which are listed in Table II.The friction velocity is deﬁned by,u =τ/ρ=−u w ,(1)where u and w are longitudinal and vertical velocityﬂuctuations,respectively. The corresponding results from other studies reported by Miyake et al.(1970)TURBULENCE CHARACTERISTICS OVER COMPLEX TERRAIN IN WEST CHINA113 over water asﬂat terrain and by Bradley(1980)on the crest of Black Mountain as complex terrain are presented in Table II for the purpose of comparisons.We can deduce several important results from Table II.When the airﬂows from a rough surface to a smooth surface the value of u for the sector II(280-310◦) is relatively smaller than those of other sectors.The large spread of this value in Table II is not unusual,and has been observed in many experimental studies over ﬂat and complex terrain(e.g.,Kaimal and Haugen,1969;Founda et al.,1997).The orderσu>σv>σw observed overﬂat terrain for sector I(180–230◦), is the same as that reported by Miyake et al.(1970),but the magnitudes of these parameters are signiﬁcantly large.The difference,of course,is expected because the surface roughness of the water is smaller than that of desert(Stull,1988). However,the value ofσv is almost larger than that ofσu over complex terrain. In this study the differences betweenσu andσv values in the sector II(280–310◦) and sector III(340–360◦)over inhomogeneous terrain are smaller than over the ﬂat terrain(sector I).The same behaviour was also observed over complex terrain (e.g.,Bradley,1980and Founda et al.,1997).The ratioσv/σu,0.92,in sector I forﬂat terrain is very similar to the value obtained by Miyake et al.(1970).This ratio value is1.14and0.99in sectors II and III respectively for complex terrain,indicating an increased transverse com-ponent of the turbulence.The relatively largeσv/σu ratio was also observed over inhomogeneous terrain by Bradley(1980)and Founda et al.(1997).The values ofσw are of the same order overﬂat and complex terrain.It may be relevant to the factor(as will be shown in Section3.3)that vertical velocity ﬂuctuations are produced by small eddies,which rapidly adjust to terrain change (Panofsky and Dutton,1984).But the value ofσw/u for sector II(280–310◦)has roughly twice the value of the others.This could be a stability effect,as will be discussed in a following section.Finally,the turbulence intensity values,σi/U are found to be increased in both sectors II and III.3.2.T URBULENCE UNDER DIFFERENT STRATIFICATIONThe standard deviations of velocity components normalized by u ,σi/u (i= u,v,w),could be analyzed according to atmospheric stability,although data for one day only were used in this paper.There were seven near-neutral runs,ﬁve unstable runs and three stable.According to Monin–Obukhov similarity,σi/u should be functions of only the dimensionless length scale z/L,where L is the Obukhov length,L=−u3 T/kgw T ,with k the von Kármán constant(here taken to be0.4);g the acceleration of gravity;and T the mean temperature.Unfortunately,there are no clear similarity relationships for the standard deviations of horizontal velocityσu,v/u for the sur-face layer that describe their behaviour under unstable and stable stratiﬁcation,but114M.H.AL-JIBOORI ET AL.T A B L E I IM e a n v a l u e s o f t u r b u l e n c e p a r a m e t e r s f o r a l l r u n s .W i n d d i r e c t i o n U u σu σv σw σu U σv U σw U σu u σv u σw us e c t o r (◦)m s −1m s −1m s −1m s −1m s −1I (180–230)5.100.210.730.670.350.140.130.073.573.191.66M i y a k e e t a l .(1970)4.440.160.380.350.220.090.080.052.382.191.38I I (280–310)2.770.100.590.670.300.210.240.115.906.703.00I I I (340–360)2.120.210.670.660.290.320.310.143.193.141.32B r a d l e y (1980)8.150.301.101.240.820.130.150.103.674.132.73TURBULENCE CHARACTERISTICS OVER COMPLEX TERRAIN IN WEST CHINA115 note the results for the u-component by Xu et al.(1993).Most authors however have found thatσw/u is a function of z/L in the surface layer overﬂat terrain (e.g.,Merry and Panofsky,1976,Panofsky et al.,1977and Xu et al.,1993).In Figures2a,b,c,the dimensionless turbulence parametersσi/u are plotted as functions of z/L.Weﬁtted the observational data for all wind direction sectors under unstable conditions with solid lines represented by a general function,σi u =a i1+b izL1/3,(2)where a i and b i are empirical constants.In this study they are found to be2.55,2.62 and1.20,and1.71,2.26and4.05for u,v and w components respectively,while a u,w=2.35and1.4,and b u,w=1.4and2.0were obtained by Xu et al.(1993);a w =1.3and b w=3.0are given by Panofsky et al.(1977).It is of interest to show that the data points were separated for different values ofﬂat and complex terrain. The lower data denoted by‘×’are from sector I(180–230◦)overﬂat terrain,while upper points symbolized as‘ ’and‘ ’are from sectors II(280–310◦)and III (340–360◦)over complex terrain.The horizontal dotted and dash-double dotted lines represent the mean value statistics ofσi/u under neutral conditions derived from a number of studies over ﬂat and complex terrain(Panofsky and Dutton,1984).It can be seen that both σu/u andσv/u parameters in this study exhibit larger values than those reported elsewhere forﬂat terrain near the neutral limit,with a slight increase ofσv/u values,while reported values are less than our observations over complex terrain. The vertical componentσw/u is in good agreement with the results over bothﬂat and complex terrain.Under unstable conditions,Figure2a shows that the values ofσu/u increase with increasing−z/L with the same behaviour observed by Xu et al.(1993). However,the results in this study are signiﬁcantly larger than those of Xu.The vertical componentσw/u also shows an increase with increasing instability(see Figure2c)which is in agreement with results overﬂat(e.g.,Panofsky et al.,1977; Xu et al.,1993),and complex terrain(Founda et al.,1997).We can see that the change of surface roughness does not seem to inﬂuence the properties ofσw/u . For stable conditions,we did not makeﬁtting curves because of too few data points. However,the data seem to show thatσi/u also increase with stability,andσw/u has slight increase which agrees with Founda s results.3.3.N ORMALIZED POWER SPECTRA/COSPECTRABecause of the special geographical environment for the site,especially from north-west to north winds,it is of particular importance to study theﬂow properties when air is transported from the oasis to the local desert terrain surrounding the observation site.In this section we examine spectral/cospectral characteristics of the turbulence for various wind directions.According to the second Kolmogorov116M.H.AL-JIBOORI ET AL.Figure2.Dimensionless standard deviations of velocity componentsσi/u (i=u,v,w)as a function of z/L.The solid lines represent similarity relations given by(2).Dashed and dash-dotted lines denote the relations given by Xu et al.(1993)and Panofsky et al.(1977),respectively.The horizontal dotted and dash-double dotted lines denote neutralﬂat and complex terrain intercepts for σi/u ,respectively.The symbols(×),( )and( )denote various values for sectors I(180–230◦),II (280–310◦)and III(340–360◦),respectively.TURBULENCE CHARACTERISTICS OVER COMPLEX TERRAIN IN WEST CHINA117 hypothesis for the inertial subrange,and using Taylor’s frozen turbulence hypo-thesis to convert to wavenumber k=2πn/U,the spectral density of longitudinal and vertical velocity components,S u,w(n),is given bynS u,w(n)=αu,w zε2π2/3f−2/3,(3)where n is the natural frequency,f=nz/U is non-dimensional frequency,εis the rate of dissipation of turbulent kinetic energy andαu,w are universal constants. Similarly,the temperature spectrum in the inertial subrange can be expressed bynS T(n)=βNε−1/3 z2π2/3f−2/3,(4)whereβis a constant analogous toαin(3)and N is the dissipation rate for temperature variance T 2/2.As for the power spectra above,a model for the cospectra of the shear stress uw and vertical heatﬂux wT in the inertial subrange,as proposed by Wyngaard and Coté(1972)isnCo uw(n)=−γ∂U∂zε1/3z2π4/3f−4/3(5)nCo wT(n)=−δ∂T∂zε1/3z2π4/3f−4/3,(6)whereγandδare the non-dimensional cospectral constants and the overbars de-note time averages.Generally,Equations(5)and(6)show that the cospectra fall off with f more rapidly than the power spectra in the inertial subrange.The phase angle, ij(f),represents the relative contribution of the quadrature spectrum,Q ij(f),to the cospectrum,Co ij(f).It is deﬁned asij(f)=tan−1Q ij(f)Co ij(f).(7)The phase spectrum can be expressed as the phase difference between two variables that yields the greatest correlation for any frequency(Stull,1988).Power spectra/cospectra have been derived from measurements for all runs.In this paper all u,w spectra and uw cospectra were normalized by local momentum ﬂux u2 ,while T spectra and wT cospectra are normalized by T2 and heatﬂux u T respectively,and the frequency was normalized as f=zn/U,where U is the mean wind speed.In order to isolate the inﬂuence of different geographic features on turbulence spectra/cospectra,near-neutral data have been picked from the available runs,except for sector II(280–310◦).It is interesting to compare these spectra/cospectra with corresponding representations of those resulting from the118M.H.AL-JIBOORI ET AL.well-known neutral Kaimal formulae(Kaimal et al.,1972)as an‘ideal’reference forﬂat terrain.These formulae were chosen because this is one of the few studies which includes the complete set of similarity relations of power spectra and cospec-tra at neutral stratiﬁcation.Power spectra/cospectra will be analyzed and discussed according to the wind directions in the following subsections.3.3.1.Sector I(180–230◦)In this sector the wind is blowing from the south and southwest to the station overﬂat terrain.The characteristics of u and w spectra,T spectra,and uw and wT cospectra for two runs in directions180.5◦and209.4◦are discussed.Figure3 shows power spectra of the longitudinal and vertical velocity components together with the Kansas spectrum for a uniform site represented by the solid lines.The u spectra results shown in Figures3a and3b seem to have the same behaviour for most of the frequency range and are in agreement with Kaimal et al.(1972) at high frequency with a‘−2/3’slope in the inertial subrange as predicted by the Kolmogorov hypothesis.Spectral curves for longitudinal velocity components have well-deﬁned peaks at about f=0.01and0.006.The vertical velocity spectra shown in Figures3c and3d are displaced toward high frequency compared with u spectra, and they are similar to Kaimal s curve,except for lower spectral densities in the high frequency region as shown in Figure3c.Furthermore,the spectral densities of vertical velocity are usually less than those of u,except at high frequencies.The peak of the w spectrum is located at about f=0.13and0.07for the wind direction 180.5◦and209.4◦,respectively.Temperature spectra are also plotted in Figure4with the corresponding curve of Kaimal et al.(1972).Lumley and Panofsky(1964)suggest that the components of u and w make contributions to theﬂuctuations in T.Thus,one might expect that power spectra of T are inﬂuenced by u and w.T spectra in sector I(180–230◦)seem to match the u spectra,and the values of their spectral densities are larger than those of w.In addition,the inertial subrange seems not signiﬁcant, because they have the characteristics of noise in the high frequency end.The same results were also found by Miyake et al.(1970)overﬂat terrain.In Figure4a the spectral densities are similar to the curve of Kaimal et al.(1972)with a little high power levels,while these densities do not seem to be similar to Kaimal’s curve in Figure4b.Spectral curves of T overﬂat terrain have their peaks roughly at f= 0.02and0.01in Figures4a and4b,which conform to the‘−2/3’power law over the ranges of f=0.06–0.16and f=0.014–0.5,respectively.Figures5a,b and5c,d give the normalized uw and wT cospectra overﬂat terrain,respectively.Although there are some differences in the magnitudes of cospectral densities,the shapes of the cospectra seem similar and coincide with each other over the entire frequency range.This cospectral behaviour was also observed by earlier authors(e.g.,Miyake et al.,1970;Kaimal et al.,1972;Roth and Oke,1993).There is a good agreement with Kaimal s expressions for uw and wT cospectra for wind direction209.4◦as shown in Figures5b and5d,while inFigure3.Normalized logarithmic spectra of longitudinal velocity component(a,b),and vertical ve-locity component(c,d)from sector I(180–230◦).The solid line represents the theoretical curve forneutral conditions of Kaimal et al.(1972).Figure4.Same as in Figure3,but for temperature spectra.Figure5.Same as in Figure3,but for normalized logarithmic cospectra of momentum uw(a,b),and heatﬂux wT(c,d).Figures5a and5c they also agree with Kaimal’s curves for most of frequency range with different power levels.Figures5a,b and5c,d show that uw and wT cospectra seem to follow the‘−4/3’power law at the high frequency end.The location of cospectral peaks for uw cospectra is about f=0.017and0.03for180.5◦and 209.4◦wind directions respectively,while for wT cospectra these peaks seem to be ratherﬂat in Figures5c,d for the same directions.The graphical representation of phase spectra is presented in Figure6.It shows that phase angles as a function of f for uw and wT for the wind directions180.5◦and209.4◦in a semi-log plot,which exhibits both positive and negative values. The phase spectra almost correspond to results of Caughey and Readings(1975), where in general,these spectra clearly seem to exhibit peaks at the low frequency end,and then gradually approach zero values at high frequencies.When the phase angles are large at the low frequency the coherence will be very small.However, they are relatively large at low frequency,where the cospectral peaks are usually found.This behaviour is clear in sector I(180–230◦)overﬂat terrain as shown inFigures6a and6b.Figure6.Phase angles of uw and wT represented by the solid and dashed lines respectively,with wind direction of(a)180.5◦,and(b)209.4◦in sector I(180–230◦).3.3.2.Sector II(280–310◦)and Sector III(340–360◦)In these sectors the wind is blowing from northwest and north,(i.e.,from oasis to desert)and different fetches over land arise.A thermal internal boundary layer is formed with distance downwind of the border.The TIBL depth,h,was found to be a function of distance or fetch,x,downwind from the border by several authors(e.g.,SethuRaman and Raynor,1980;Venkatram,1977).Venkatram(1977) proposed the following equation for the height of the TIBLh=uU2(T land−T sea)x(1−2F)1/2,(8)where T land and T sea are the air temperature over land and sea,respectively, is the vertical temperature gradient above the TIBL,F is an entrainment coefﬁcient, which ranges from0to0.22.Su et al.(1987)have studied the microclimate characteristics at both the Gobi and the oasis.They showed that the temperature over Gobi is higher than that over oasis farmland during the whole day,especially close to the surface.According to (8)the depth of the TIBL at x=2.0km is about20m for the sector III(340–360◦), and is deeper for sector II(280–310◦)since the distance,x,is greater.Sector II(280–310◦)and sector III(340–360◦)involve complex terrain.Thus, power spectra/cospectra for both sectors are discussed below.Unfortunately,there are no turbulence runs in near neutral conditions in sector II and a very unstable run(z/L=−5.5)is presented without comparisons,because of the lack of sim-ilarity relations and measurements spectra/cospectra in the literature for the same frequency range.Two runs of wind directions310.2◦and350.7◦are presented to represent the sectors II and III,respectively.The longitudinal and vertical velocity spectra of these runs are plotted in Figure7.Figures7a and7b show a large excess of u spectral energy at low frequencies, especially for wind direction310.2◦compared to w spectra in Figures7c and7dFigure7.Normalized spectra for longitudinal wind component(a,b),and vertical velocity component (c,d),for sector II(280–310◦)and sector III(340–360◦),respectively.Theoretical curves are from Kaimal et al.(1972).for near-neutral data.This excess is observed in many studies ofﬂat(Kaimal et al., 1972;Roth and Oke,1993)and complex terrain(Højstrup,1981).The roughness change has produced relatively large spectral densities at low frequencies as well as changes of spectral shape,which make spectral peaks broad.From comparisons with the line obtained from Kaimal et al.(1972)the spectra of the longitudinal ve-locity components at low frequency are affected by changes in roughness features, while they adjust more rapidly at high frequency,as shown in Figure7a.The w spectra in Figures7c and7d seem to have the same spectral properties as those over homogeneous terrain,because they have most of their energy at re-latively high frequencies,which respond rapidly to changes of surface roughness. Figure7d shows that w spectra for wind direction350.7◦agree well with the curve of Kaimal et al.(1972)over the entire frequency range.The peak of the w spectrum is at about f=0.03for sector II(280–310◦),and f=0.2in sector III(340–360◦). Moreover,these spectra follow the‘−2/3’power law at the inertial subrange at the high frequency end.Normalized spectra of temperature in both sectors II(280–310◦)and III(340–360◦)are presented in Figures8a and8b,respectively.The shape of the T spectrumFigure8.Normalized temperature spectra,for(a)sector II(280–310◦),and(b)sector III(340–360◦), with curve of Kaimal et al.(1972).for wind direction310.2◦is also not affected by the surface roughness for most of the frequency range(Figure8a),and is quite similar to the w spectrum with the same spectral peak location.This behaviour is in accordance with turbulence measurements for unstable conditions.As previously mentioned in3.3.1,with in-creasing−z/L the inﬂuence of w grows steadily,while that of u declines(Kaimal et al.,1972).This can be inferred from determining the trend of the correlation coefﬁcients of this run:r uw=−0.048,r wT=+0.40,and r uT=−0.046.There-fore,the u component becomes relatively unimportant in determining the spectral shape.The T spectrum for the350.7◦direction in Figure8b is affected by surface roughness,with higher power levels at low frequency,which respond slowly to changes of surface conditions,while at high frequencies they adjust rapidly.The power cospectra of uw and wT for wind directions310.2◦and350.7◦from sectors II and III are shown in Figures9a,b and9c,d,respectively.The inﬂuence of w-component onﬂuctuation T is still obvious in Figures9a and9c,and the slope ‘−4/3’appears clearly at the high frequencies,while cospectral peaks locate at approximately the same frequency at about f=0.03.In near neutral conditions, uw and wT cospectra have broad peaks and the spectral data follow a‘−4/3’slope at the high frequency end.At low frequency the power levels of uw-cospectrum agree with the line given by Kaimal et al.(1972),while the wT cospectra have relatively high energies for the whole frequency region.According to the above results,uw and wT cospectra do not seem to be affected by the roughness change.The phase spectra of uw and wT for the two runs in sectors II(280–310◦)and III(340–360◦)are presented in Figures10a and10b.As can be seen,they havevalues smaller than those in Figure6.。

神经细胞中五种常见物质的太赫兹波谱的推定

第17卷第1期太赫兹科学与电子信息学报Vo1．17，No．1 2019年2月 Journal of Terahertz Science and Electronic Information Technology Feb．，2019 文章编号：2095-4980(2019)01-0035-05神经细胞中五种常见物质的太赫兹波谱的推定罗洁，寇天一，施辰君，吴旭*(上海理工大学上海市现代光学系统重点实验室，上海 200093)摘要：利用太赫兹时域光谱技术(THz-TDS)对神经细胞中常见的5种物质(L-谷氨酸、γ-氨基丁酸、肌酸、盐酸多巴胺和肌醇)的纯品进行特征谱检测。

实验结果显示，5种物质在太赫兹波段有明显且独特的吸收谱线，并且同一种物质不同浓度的吸收系数符合朗伯比尔定律。

此外，利用CASTEP软件包对5种物质的光谱进行仿真，结果显示太赫兹波段光谱特征吸收峰的成因主要由分子基团的振动而引起。

之后，对肌醇、L-谷氨酸、γ-氨基丁酸和肌酸的1:1:1:1混合物的谱线进行检测，结合最小二乘法等算法倒推，可以准确推断每一种物质的含量及比例。

结果表明，当4种物质混合时，物质含量推断的准确率为94%以上，这一研究结果对于癌组织的早期检测有重大意义。

关键词：太赫兹时域光谱技术；混合物成分分析；吸收系数；光谱检测中图分类号：TN29文献标志码：A doi：10.11805/TKYDA201901.0035Estimation of five common substances in nerve cells by terahertz spectroscopyLUO Jie，KOU Tianyi，SHI Chengjun，WU Xu*(Shanghai Key Laboratory of Modern Optical System，University of Shanghai for Science and Technology，Shanghai 200093，China)Abstract：The spectra of five substances commonly in nerve cells including, L-glutamic acid, Gamma Amino Butyric Acid(GABA), creatine, dopamine hydrochloride and myo-inositol are investigated byTerahertz Time-Domain Spectroscopy(THz-TDS). The experimental results show that the five substanceshave obvious and unique absorption spectra in terahertz region. For the same material, absorbancecoefficient of different concentrations agrees with Lambert-Beer Law. In addition, the spectra of fivesubstances are simulated by using CASTEP software package. The results show that the absorption peaksof terahertz spectrum are mainly due to the vibration of molecular groups. Furthermore, the spectra of1:1:1:1 mixture of GABA, creatine, L-glutamic acid, myo-inositol are tested. By using the Least SquareMethod(LSM), the proportion and percentage of each substance are deduced exactly. The results show thatthe accuracy of substance content prediction is more than 94% when the four substances are mixed, whichis of great significance for the early detection of cancer tissue.Keywords：Terahertz Time-Domain Spectroscopy(THz-TDS)；mixture component analysis；absorption coefficient；spectroscopic inspection太赫兹(THz)波是指频率在0.1~10 THz范围的电磁波，介于微波与红外之间，是电磁波谱上电子学向光子学过渡的特殊区域。

新型无机闪烁体的能量分辨率(中英文对照)

成都理工大学学生毕业设计（论文）外文译文极，（b）光电子是后来ηNph，（c）这些∝ηNph电子在第一倍增极和到达（d）倍增极的k（k = 1，2…）放大后为δk 并且我们假设δ1=δ2=δ3=δk=δ的，并且δ/δ1≈1的。

我们可以得出：R2=Rlid2=5.56δ/[∝ηNph（δ-1）] ≈5.56/Nel （3）Nel表示第一次到达光电倍增管的数目。

在试验中，δ1≈10＞δ2=δ3=δk，因此，在实际情况下，我们可以通过（3）看出R2的值比实际测得大。

请注意，对于一个半导体二极管（不倍增极结构）（3）也适用。

那么Nel就是是在二极管产生电子空穴对的数目。

在物质不均匀，光收集不完整，不相称和偏差的影响从光电子生产过程中的二项式分布及电子收集在第一倍增极不理想的情况下，例如由于阴极不均匀性和不完善的重点，我们有：R2=Rsci2+Rlid2≈5.56[(νN-1/Nel)+1/Nel] (4)νN光子的产生包括所有非理想情况下的收集和1/Nel的理想情况。

为了说明，我们在图上显示，如图1所示。

ΔE/E的作为伽玛射线能量E的函数，为碘化钠：铊闪烁耦合到光电倍增管图。

1。

对ΔE/E的示意图（全曲线）作为伽玛射线能量E功能的碘化钠：铊晶体耦合到光电倍增管。

虚线/虚线代表了主要贡献。

例如见[9,10]。

对于Rsci除了1/（Nel）1/2的组成部分，我们看到有两个组成部分，代表在0-4％的不均匀性，不完整的光收集水平线，等等，并与在0-400代表非相称keV的最大曲线。

表1给出了E=662Kev时的数值（137Cs）在传统的闪烁体资料可见。

从图一我们可以清楚的看到在低能量E＜100Kev，如果Nel，也就是Nph增大的话，是可以提高能量分辨率的。

这是很难达到的，因为光额产量已经很高了（见表1）在能量E＞300Kev时，Rsci主要由能量支配其能量分辨率，这是没办法减小Rsci 的。

然而，在下一节我们将会讲到，可以用闪烁体在高能量一样有高的分辨率。

BA-TH05-521 CERN-PH-TH2005-216

BA-TH/05-521CERN-PH-TH/2005-216hep-th/0511039 Observable(?)cosmological signaturesof superstrings in pre-big bang models of inﬂationM.Gasperini1,2,3and S.Nicotri21CERN,Theory Unit,Physics Department,CH-1211Geneva23,Switzerland2Dipartimento di Fisica,Universit`a di Bari,Via G.Amendola173,70126Bari,Italy3Istituto Nazionale di Fisica Nucleare,Sezione di Bari,Bari,ItalyAbstractThe diﬀerent couplings of the dilaton to the U(1)gaugeﬁeld of heterotic and Type Isuperstrings may leave an imprint on the relics of the very early cosmological evolu-tion.Working in the context of the pre-big bang scenario,we discuss the possibility ofdiscriminating between the two models through cross-correlated observations of cosmicmagneticﬁelds and primordial gravitational-wave backgrounds.To appear in Phys.Lett.BCERN-PH-TH/2005-216November2005It is well known that theﬁve superstring models diﬀer,at the level of the lowest-order bosonic eﬀective action,for theirﬁeld content and for the way in which the variousﬁelds are coupled to the dilaton[1].Even in models with the same(boson)particle content,such as Type I and heterotic superstrings,there are diﬀerences due to the dilaton couplings,most notably for the Yang–Mills one form containing the U(1)gaugeﬁeld associated to standard electromagnetic(e.m.)interactions.The eﬀective tree-level action for the photon–dilaton interactions,after appropriate dimensional reduction from ten to four dimensions,can be parametrized(in the string frame)as follows:S=−1−g f(b i)e−ǫφFµνFµν,Fµν=∂µAν−∂νAµ,(1)whereφis the ten-dimensional dilaton,(normalized in such a way that the string coupling parameter g corresponds to g2=expφ),and where the“form factor”f(b i)takes into account the possibe dynamical contribution of the internal moduliﬁelds b i,i=1, (6)The dilaton parameterǫhas values1,1/2,respectively,for heterotic and Type I superstrings, which are probably the best candidates for representing the low energy limit of standard model interactions.The value ofǫcould be diﬀerent,however,were the e.m.ﬁeld identiﬁed not as a component of the higher-dimensional gauge group,but as a one form present in the Ramond–Ramond or Neveu-Schwarz–Neveu-Schwarz sector of other string theory models. In that case also the coupling to the internal moduli would be diﬀerent(see for instance [2]).The discussion of this paper will be concentrated on the case in which f(b i)simply represents the volume factor of the six-dimensional internal manifold,f(b i)=V6≡ 6i=1b i, and diﬀerent string models are characterized by diﬀerent values ofǫ.When the internal moduli are stabilized,indeed,the coupling function f(b i)becomes trivial,and the only relevant parameter isǫ.In any case,diﬀerent values ofǫamount to diﬀerent rescaling of the photon–dilaton coupling,and may have signiﬁcant impact on cosmological processes where large variations of the dilatonﬁeld,for long periods of time,may occur.The eﬀect we shall consider here will concern the ampliﬁcation of the quantumﬂuc-tuations of the e.m.ﬁeld,and the possible production of“seeds”for the magneticﬁelds currently observed on large scales[3].Working in the context of the so-called pre-big bang scenario of superstring cosmology[4,5]it will be shown in particular that,depending on the value ofǫ,the region of parameter space compatible with an eﬃcient production of seeds may or may not overlap with the diﬀerent regions of parameter space compatible with an observable production of cosmic gravitons.The cross-correlated analysis of cosmic magnetic ﬁelds and of the outputs of gravitational detectors may thus provide unique information, in principle,not only on the allowed region of parameter space of the considered class of string cosmology models,but also on the most appropriate superstring model–i.e.on the most appropriate M-theory limit–to be chosen for a realistic description of fundamental interactions during the primordial stages of our Universe.1We start by recalling that the e.m.ﬂuctuations of the vacuum,according to the action (1),are represented by the canonical variableψµ,ψµ=zγAµ,zγ= 6 i=1b i 1/2e−ǫφ/2,(2)which diagonalizes the four-dimensional action.In a conformallyﬂat four-dimensional ge-ometry,gµν=a2ηµν,and in the radiation gauge A0=0=∂i A i,ψµsatisﬁes the canonical equation[6]:ψ′′i−∇2ψi−z−1γz′′γψi=0,(3) where the primes denote diﬀerentiation with respect to the conformal timeη.Notice that, because of the conformal invariance of the four-dimensional Maxwell action,theﬂuctuations ψi are decoupled from the four-dimensional geometry;the canonical equation only contains a coupling to theﬁrst and second derivatives of the dilaton background and of the internal moduli.A time-dependent dilaton(with the possible contribution of a time-dependent internal geometry)may thus act as an external“pumpﬁeld”,and sustain the ampliﬁcation of the e.m.ﬂuctuations during the primordial phases of accelerated(inﬂationary)evolution.For the case of the heterotic couplingǫ=1the inﬂationary spectrum ofﬂuctuations produced by the dilaton has already been computed in[6,7].Here we shall derive the corresponding spectrum for a generic value ofǫ,using the same example of inﬂationary string-cosmology background as was adopted in[6],based on a“minimal”pre-big bang scenario consisting of two phases.During theﬁrst phase,ranging in conformal time from−∞toηs<0,the Universe is assumed to evolve from the string perturbative vacuum according to the exact(Kasner-like) solutions of the lowest order gravi-dilaton eﬀective action[5]:a(η)∼(−η)β0(1−β0),φ(η)∼ iβi+3β0−13/2,ǫ′= iβi−ǫ( iβi+3β0−1)3(1−β0)(6)(in the special case of frozen internal dimensions,βi=0,β0=−1/√the higher-dimensional perturbative vacuum towards the four-dimensional strong coupling regime–that the internal dimensions are shrinking,and that the four-dimensional coupling, controlled byφ4=φ−ln V6=φ−6ln b i,is growing.The intersection with the Kasner condition then deﬁnes,forβ0andβi,the following allowed ranges of values:−1/√6.(7) During the second,high-curvature phase,ranging fromηs toη1<0,the(string-frame) curvature stays frozen at a constant value H1controlled by the string scale,H1∼M s;the internal moduli are also frozen,while the dilaton(and the string coupling g)keeps growing from the initial value g2s=expφ(ηs)≪1to aﬁnal value g21=expφ(η1)∼1.The background evolution during this second“stringy”phase can be represented bya(η)∼(−η)−1,φ(η)∼−2βln(−η),zγ(η)∼(−η)ǫβ,β>0,ηs≤η≤η1<0,(8) whereβis a phenomenological parameter that possibly takes into account the eﬀects of the higher orderα′corrections to the eﬀective action.Eventually,forη>η1,the Universe is expected to enter the radiation-dominated,frozen-dilaton(decelerated)regime a∼η,φ= const,and to follow all subsequent steps of the standard cosmological evolution.For such a model of background,each polarization component of the canonical variable can be expanded in Fourier modesψk such that∇2ψk=−k2ψk,and Eq.(3)becomes a Bessel equation whose solution can be expressed in terms of theﬁrst-and second-kindHankel functions[5].By imposing the asymptotic normalization to a spectrum of quantum√vacuumﬂuctuations,ψk=exp(−ikη)/=Ωr(t0) H1ω1 3−|2ǫβ−1|,ωs<ω<ω1,d lnω=Ωr(t0) H1ω1 3−|2ǫβ−1| ω3−1|,ω<ωs,(9)where the parametersβ,βi,β0may vary in the ranges previously deﬁned,and where we have assumedǫ>0(forǫ=0there is no ampliﬁcation of e.m.ﬂuctuations during the3string phase,and the above spectrum cannot be applied).Hereρc=3H0M2P is the criticalenergy density,M P=(8πG)−1is the Planck mass,Ωr(t0)≃4×10−5h−20is the present fraction of critical energy density in radiation,with h0=H0/(100km sec−1Mpc−1)the present(dimensionless)value of the Hubble parameter.Finally,ω1=(a|η1|)−1≃H1(a1/a) andωs=(a|ηs|)−1≃H s(a s/a),where H1∼M s is the value of the curvature scale at the inﬂation→radiation transition,whileωs is a free parameter of the given inﬂationaryscenario.Previous detailed computations of the above spectrum for the pure heterotic caseǫ=1can be found in[5,9](see[10]for a recent computation of the spectrum for a genericvalue ofǫ,but for the special case of frozen internal dimensions,i.e.withǫ′=ǫ).This primordial spectrum of e.m.ﬂuctuations is distributed over a wide range of fre-quencies and can provide,in principle,the“seeds”required to trigger the galactic dynamoand to produce the cosmic magneticﬁelds B G(of microgauss strength)currently observedon the galactic scale[3,6].The identiﬁcation of the seedﬁelds with the vacuumﬂuctua-tions,ampliﬁed by inﬂation,requires however that suchﬂuctuations be coherent and largeenough over a proper length scale that today roughly corresponds to the megaparsec scale,asﬁrst pointed out in[11].For a conservative estimate of the requiredﬁeld strength[11]one can assume the exis-tence of the standard galactic dynamo mechanism,operating since the epoch of structureformation,characterized by an ampliﬁcation factor∼1013,and can also take into accountthe additional ampliﬁcation(∼104)due to magnetic-ﬂux conservation in the collapse ofthe galactic structure from the Mpc to the10kpc scale.One obtains,in this way,the lowerbound B s>∼10−23gauss on the present amplitude of the magnetic seeds at the Mpc scale; the identiﬁcation of the seeds with the inﬂationary spectrum of e.m.ﬂuctuations leads then to the condition[11]:B2s(ωG,t0)ρr(t0)=Ωγ(ωG,t0)spanned by the variables{ω1,ωs,β,β0,H1},and the purpose of this paper is to compare such regions with analogous allowed regions associated to the ampliﬁcation of tensor metric perturbations,and to the detectable production of cosmic gravitons,in the same given class of pre-big bang backgrounds.To this purpose we must consider the evolution equation of the canonical variable u ij, associated to the tensor perturbations h ij of the three-dimensional spatial sections of the metric background:u ij=z g h ij,z g=a 6 i=1b i 1/2e−φ/2.(12)In the transverse,traceless gauge∂i h i j=0=h i i,and in the string frame,such a canonical equation reads[12]u′′j i−∇2u i j−z−1g z′′g u i j=0,(13) for each spin-two polarization mode u i j.We will use the same model of background speciﬁed by Eqs.(4),(8),and we can notice that the graviton“pumpﬁeld”z g,at low energies, is completely insensitive to the dynamics of the internal dimensions[13,14],unlike e.m.ﬂuctuations:according to Eq.(4),indeed,z g∼(−η)1/2during the initial low-curvature phase,quite independently of the values ofβ0andβi(and even in the anisotropic case).In the second,high-curvature phase,the pumpﬁeld is parametrized instead by z g∼(−η)β−1, withβ>0,according to Eq.(8).We can now follow the same procedure as before,expanding theﬂuctuations in Fourier modes,imposing the canonical normalization atη→−∞,and matching the solutions atηs andη1.The computation of the spectral energy density stored in the ampliﬁed gravitational radiation leads to theﬁnal(already known[14,15,5])result:Ωg(ω,t0)=Ωr(t0) H1ω1 3−|2β−3|,ωs<ω<ω1,=Ωr(t0) H1ω1 3−|2β−3| ωthe gravitational instability of the background[16].We shall thus restrict our discussion to the case0≤β≤3.We should warn the reader,however,that the considered graviton spectrum is typical of a stringy phase at constant curvature and linearly growing dilaton [17],which takes into account higher-derivative corrections but remains at the tree-level in the string coupling:the spectrum could be signiﬁcantly diﬀerent if higher-loop corrections were included.There are two main upper bounds on the amplitude of such a growing spectrum:one is obtained from the observations of millisecond pulsars(in particular,from the absence of a detectable distortion of pulsar timing),which imposes the bound[18]h20Ωg(ωp,t0)<∼10−8,ωp≃10−8Hz;(15) the other is obtained from the standard nucleosynthesis analysis,which imposes a bound on the total integrated energy density of the relic graviton spectrum at the nucleosynthesis epoch[19].Such a bound can also be expressed as an upper limit on the peak intensity of the spectrum[20]:Ωg(ω,t0)<∼10−1Ωr(t0),(16) which is valid at all frequency scales.These two upper bounds are in competition with the lower bounds to be imposed on the spectrum for its possible detection by a gravitational antenna of given sensitivity.Here we will use,as a typical reference value,the planned sensi-tivity of the Advanced LIGO interferometers,which are expected to detect a background of relic gravitons through cross-correlated observations around the frequency bandωL∼102 Hz,provided[21]:h20Ωg(ωL,t0)>∼10−11,ωL/2π≃102Hz.(17) See[22,5]for other sensitivities associated to diﬀerent types of gravitational detectors.The set of constraints(15)–(17),imposed on the graviton spectrum(14),deﬁnes allowed regions in the parameter space of our class of background,which we shall now compare with the e.m.regions speciﬁed by the constraints(10),(11).For a clearer discussion we will ﬁrst reduce the dimensions of the parameter space by exploiting the direct relationship existing between H1andω1,determined by the details of the post-inﬂationary evolution: considering the standard radiation-dominated→matter-dominated evolution weﬁnd[5], in particular,ω1≃H1a1/a≃(H1/M P)1/21011Hz.Next,we willﬁx the transition scale H1 precisely at the present value of the string scale,H1=M s,assuming for M s the typical value M s=0.1M P,consistently with string models of uniﬁed gravitational and gauge interactions [23].This completely speciﬁes the position and the“end point”of the graviton and photon spectrum in terms ofω1andΩ(ω1),and leaves us with four parameters,ωs,β,β0andǫ. The graviton spectrum,however,only depends onωs andβ:we will thus plot the allowed regions in the two-dimensional space spanned by the variables{ωs,β},at diﬀerent,constant values of the parametersǫandβ0,chosen appropriately for the e.m.spectrum.6For the physical interpretation of theﬁnal result,however,it will be convenient to replaceωs andβwith an equivalent set of parameters,more directly related to the inﬂationarykinematics.This new set is obtained by noting that,according to Eq.(8),ωs is related tothe extension in time of the string phase through the e-folding factor z s=a1/a s=ηs/η1=k1/k s=ω1/ωs;the parameterβis related instead to the growth of the string coupling duringthe string phase,since g s/g1=(η1/ηs)β=z−βs.On the other hand,once we have imposed H1=M s=0.1M P,not onlyω1but also theﬁnal string coupling g1=exp(φ1/2)isﬁxed[20],at a value quite close to the currently expected value g1≃M s/M P=0.1(since the dilaton is assumed to be frozen during the subsequent standard evolution).The two variables{z s,g s} thus represent a complete and independent set of variables,equivalent to the set{ωs,β},for a convenient parametrization of the chosen class of string cosmology backgrounds[6].For practical purposes we shall work,in particular,with a decimal logarithmic scale,andwe shall plot theﬁgures in the plane parametrized by the coordinatesx=log z s=log(ω1/ωs)>0,y=log(g s/g1)=−βlog z s=−βx<0.(18) We are now in the position of discussing the possible overlap of the allowed regions for the photon and graviton spectra.We rewrite these in terms of the new variables{x,y} and we note,ﬁrst of all,that the condition(16)is automatically satisﬁed because of the choice M s/M P=0.1and of the monotonic growth of the graviton spectrum,which imposes 0≤β≤3,and which translates into the stringent constraint−3x<y<0(19) on the allowed gravitational region.A similar argument can be applied to the e.m.spectrum(9)by noting that its dilaton(low-frequency)branch,controlled by the parameterǫ′,is always growing for all values ofβ0,βi included in the allowed range(7),and forǫvarying from0to1.The high-frequency branch of the spectrum might be decreasing,in principle,if y<−(2x/ǫ):a decreasing spectrum,however,has a peak atω=ωs,and is only marginally compatible with the homogeneity condition(11),which imposes,when applied to the peak value of the spectrum,y>−(2x/ǫ)−1/ǫ.In addition,a decreasing spectrum is ruled out if one would take into account the slightly more stringent(but model-dependent)boundΩγ<0.1Ωr, which follows from the presence of strong magneticﬁelds at the nucleosynthesis epoch [24],which would impose y>−(2x/ǫ).As in the graviton case,we will thus restrict our discussion to a growing spectrum of e.m.ﬂuctuations,which automatically satisﬁes Eq.(11)and the nucleosynthesis bound,and which imposes the constraint−2x/ǫ<y<0(20)on the allowed e.m.region.The allowed regions of the plane{x,y}determined by the set of conditions(10)(eﬃcientseed production),(15)(pulsar bound),(17)(graviton detectability),(19)and(20)(growing7spectra)are illustrated in Fig.1,which presents in a compact form the main results of this paper.The regions marked by the thin border refer to the graviton spectrum(14),those marked by the bold border refer to the e.m.spectrum(9).We have plotted,for the e.m. spectrum,the two casesǫ=1andǫ=1/2,and we have explicitly indicated the application of the constraints to the high-frequency(string)or low-frequency(dilaton)branches of the spectrum,both for tensor and e.m.ﬂuctuations.It should be stressed that the existence of two graviton regions is due to the absolute value appearing in the power of the graviton spectrum,and to the diﬀerent kinematic behaviour of tensor perturbations outside the horizon[14,9],depending on the sign of 2β−3:the upper part of the allowed region corresponds to the class of backgrounds with pumpﬁeld characterized by the powerβ−1<1/2,in which the amplitude of tensor perturbations stays constant outside the horizon;the lower part corresponds to backgrounds withβ−1>1/2,in which the amplitude grows outside the horizon,and the inﬂationary ampliﬁcation is even more eﬀective[5].No such distinction exists,instead,for the allowed regions of the photon spectrum,once the condition(20)has been imposed.Finally,the dashed lines present in the dilaton sector of the photon regions illustrate the possible eﬀects of a non-trivial(primordial)dynamics of the internal dimensions,possibly producing diﬀerent values of the eﬀective parameterβ0according to Eqs.(4),(5).The bold line represents the seed condition imposed on the photon spectrum computed in the√case of frozen internal dimensions,and corresponding toβ0=−1/6),(1/3,1/3)for the caseǫ=1,and(−0.568,0.072),(−1/3,1/3), (0.063,0.406)for the caseǫ=1/2(the numerical values have been selected so as to include the maximal and minimal extension of the allowed region compatible with the range of variations of the parametersβ0andβi,given in Eq.(7)).Note that we have also included the possibility of a“bouncing”scenario with positiveβ0,in which the initial growth of the dilaton,at low energy,is associated with the shrinking of the four-dimensional geometry. The eﬀects of the internal dynamics may enhance or reduce the photon allowed regions but, as clearly illustrated in the picture,it cannot lead to the overlap of the photon and graviton regions in the caseǫ=1.The main message of Fig.1is that an eﬃcient production of seeds for the cosmic magneticﬁelds(ampliﬁed by the dilaton in a string cosmology context)is in principle compatible with the associated production of a cosmic graviton background detectable by Advanced LIGO ifǫ=1/2(as in the case of Type I superstrings),and incompatible ifǫ=1 (as in the case of heterotic superstrings).This may give us direct experimental information on the possible primordial strength of the photon–dilaton coupling,and on the choice of the superstring model most appropriate to the early cosmological evolution.Let us suppose,indeed,that further studies of cosmic magneticﬁelds will provide direct8x = log(ω1/ωs )y = l o g (g s /g 1) Figure 1:Allowed regions determined by the conditions (10),(15),(17),(19),(20),imposed on the spectra (9)and (14)with (H 1/M P )=0.1.The photon regions are plotted for the two cases ǫ=1and ǫ=1/2,and for appropriate values of β0and βi illustrating the maximal and minimal extension of the allowed region in the case of a non-trivial evolution of the moduli ﬁelds (dashed lines).independent conﬁrmation of the expected seed production,exactly as predicted in the class of pre-big bang models considered here:for instance,by measuring the primordial spec-trum through its eﬀects on the polarization and the anisotropy of the CMB radiation [3](primordial seeds can be produced,more or less naturally,with various mechanisms [10],but a monotonically growing spectrum seems to be peculiar only of the ampliﬁcation of the vacuum ﬂuctuations in pre-big bang cosmology).Then,a future detection of cosmic gravitons by the next generation of gravitational antennas will provide support in favour of Type I models,while the absence of detection (at the same sensitivity level)should be interpreted more in favour of the heterotic model of coupling (see [25]for a very recent,direct experimental bound on the energy density of a cosmic graviton background,and [26]for a discussion of the near-future sensitivities of LIGO in various regions of the parameter space of pre-big bang models).Such an argument cannot be directly applied to cosmological9models based on Type II superstrings,which are expected to describe the strong coupling limit of standard interactions.We may notice,as a conclusive remark,that the extension of the low-energy(dilaton) part of the allowed regions illustrated in Fig.1is rather strongly dependent on the values ofβ0andβi.A precise experimental determination of the spectrum of e.m.ﬂuctuations may thus,in principle,open a direct window on the primordial dynamics of the extra dimensions.Work along these lines is currently in progress(see also[2,27]for previous studies on higher-dimensional modiﬁcations of the photon spectrum). 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