Ch2 Thermodynamics of Superconductors
超导物理 第一讲 Overview of Superconductivity ppt课件
Overview of Superconductivity
浙江大学凝聚态物理研究所CGH
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课程介绍
参考书: 1) 张裕恒：《超导物理》，中国科技大学出版社 2) 章立源等：《超导物理学》，电子工业出版社 3) 张其瑞：《高温超导电性》，浙江大学出版社 4) M. Tinkham: “Introduction to Superconductivity”,
=(r)=Aei(r)
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BCS Theory of Superconductivity
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超导机制－Cooper电子对
动量空间
实空间
正常态
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BCS Theory of Superconductivity
1972 Nobel Prize for Physics
McGraw-Hill, Inc. 考试等：
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Outline
Discovery of Superconductivity Development of Superconducting Materials Basic Properties of Superconductors Phenomenological Theories Microscopic Theories New Challenges (HTSC)
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Superconducting Critical Magnetic of Tc, Hc and Jc
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Type I and Type II Superconductors
热力学专业英语作文
热力学专业英语作文Title: Thermodynamics in EnglishThermodynamics is the branch of physics that deals with the relationships between heat, work, energy, and temperature.It is one of the fundamental sciences that help us understand and predict the behavior of systems.In this essay, we will explore some key concepts and terms related to thermodynamics in English.Firstly, let"s talk about the laws of thermodynamics.There are four laws of thermodynamics, but the first and second laws are the most fundamental.The first law of thermodynamics, also known as the law of conservation of energy, states that energy cannot be created or destroyed, only transformed from one form to another.The second law of thermodynamics states that in a closed system, the total entropy always tends to increase over time, meaning that processes tend to become more disordered.ext, let"s discuss some common units of measurement in thermodynamics.The joule (J) is the unit of energy in the International System of Units (SI), while the calorie (cal) is a non-SI unit of energy commonly used in nutrition.The watt (W) is the unit of power, which is the rate at which work is done or energy is transferred.The kilowatt-hour (kWh) is a common unit of energy consumption, often used in the context of electricity usage.Thermodynamic properties are characteristics of a system that can be used to describe its state and predict its behavior.Some common thermodynamic properties include temperature, pressure, volume, and internal energy.Temperature is a measure of the average kinetic energy of the particles in a system, while pressure is a measure of the force exerted by the particles on the walls of the container.Volume is the amount of space occupied by the system, and internal energy is the total energy of the system, including both kinetic and potential energy.ow, let"s talk about some thermodynamic processes.An isothermal process is a process in which the temperature of the system remains constant.A reversible process is one that can be undone by a small change in the system"s state, while an irreversible process is not reversible and may involve a large change in the system"s state.An adiabatic process is one in which there is no heat transfer between the system and its surroundings, while a diabatic process involves heat transfer.In conclusion, thermodynamics is a fundamental science that helps us understand the behavior of systems.By studying the laws of thermodynamics, units of measurement, thermodynamic properties, and processes, we can gain a deeper understanding of how energy and heat are transformed and transferred.With this knowledge, we can apply thermodynamics to various fields, such as engineering, physics, andchemistry, to solve real-world problems and improve our lives.。
Advanced Thermodynamics
Advanced ThermodynamicsThermodynamics, a branch of physics that deals with heat and its conversion to and from other forms of energy, is a fascinating subject that has been a cornerstone of scientific exploration for centuries. It is a field that has not only shaped our understanding of the universe, but also has practical implications in various industries, from power generation to refrigeration and air conditioning.The beauty of thermodynamics lies in its ability to explain the behavior of systems in terms of energy transfer and transformations. It provides a framework for understanding how energy moves and changes from one form to another, and how this movement can be harnessed for practical applications. The first law of thermodynamics, also known as the law of conservation of energy, states that energy cannot be created or destroyed, only converted from one form to another. This principle is fundamental to our understanding of how energy is used and conserved in various processes.On the other hand, the second law of thermodynamics introduces the concept of entropy, which is a measure of the disorder or randomness in a system. This law states that, in any energy transfer or transformation, the total entropy of a system will always increase over time. This means that natural processes tend to move towards a state of greater disorder, and it is this principle that underlies the directionality of time and the irreversible nature of many processes.One of the most intriguing aspects of thermodynamics is its application in the field of heat engines. A heat engine is a device that converts heat into mechanical work. The efficiency of a heat engine is determined by its ability to convert heat into work, and this is governed by the Carnot cycle, which is an idealized thermodynamic cycle. The Carnot cycle provides an upper limit on the efficiency that any heat engine can achieve, and it is a testament to the power of thermodynamics in setting fundamental limits on the performance of machines.Another important application of thermodynamics is in the field of refrigeration and air conditioning. These systems work by transferring heat from a cooler region to a warmer region, which is the opposite of the natural flow of heat. This process is made possible by the use of refrigerants, which are substances that can absorb and release heat at different temperatures. The principles of thermodynamics are crucial in designing these systems to ensure that they are efficient and effective in cooling the spaces.In the realm of power generation, thermodynamics plays a vital role in the design and operation of various types of power plants. For example, in a steam power plant, water is heated to produce steam, which then drives a turbine to generate electricity. The efficiency of this process is governed by the principles of thermodynamics, and understanding these principles is essential for optimizing the performance of the power plant.Moreover, thermodynamics is also relevant in the field of environmental science. The study of thermodynamics helps us understand the energy flows in ecosystems and the impact of human activities on the environment. For instance, the concept of entropy can be used to analyze the sustainability of different energy sources and the environmental impact of energy production and consumption.In conclusion, thermodynamics is a subject of immense importance and relevance in our lives. It provides a deep understanding of the fundamental principles that govern the behavior of energy and its transformations. From powering our homes and industries to understanding the natural world, thermodynamics plays a crucial role in shaping our world and our future. As we continue to explore and harness the power of thermodynamics, we can look forward to a future in which energy is used more efficiently and sustainably, and in which our understanding of the universe continues to expand.。
MolecularThermodynamicsPDF：分子热力学的PDF
Molecular ThermodynamicsBy John SimonMolecular Thermodynamics Details:Molecular Thermodynamics: Donald A. McQuarrie, John D. Simon ...Evolved from McQuarrie and Simon's best-selling textbook, Physical Chemistry: A Molecular Approach, this text focuses in on the thermodynamics portion of the course. Thermodynamics - Wikipedia, the free encyclopediaThermodynamics is a branch of natural science concerned with heat and its relation to energy and work. It defines macroscopic variables (such as temperature, internal energy, entropy, and pressure) that characterize materials and radiation, and explains how they are related and by what laws they ...Molecular Thermodynamics - Johns Hopkins UniversityMolecular Thermodynamics. The explosion in structural data on bio-molecules provides new opportunities to understand their function in terms of their structure. 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Although many of the chapters are similar ...Introduction to Molecular ThermodynamicsIntroduction to Molecular Thermodynamics. by Robert M. Hanson and Susan Green, St. OlafCollege.enquiries:******************.Publisher'swebsiteforthisbookMolecular Thermodynamics (Physical chemistry monograph series ...Molecular Thermodynamics (Physical chemistry monograph series) [Richard E. Dickerson] on . *FREE* shipping on qualifying offers.Molecular Thermodynamics — ITQB - Universidade Nova de LisboaMolecular Thermodynamics of Liquids and Liquid Solutions, in particular, studies of Ionic Liquids and ionic liquid-containing systems constitute the main activity.Molecular thermodynamics of fluid-phase equilibria - J. M ...Molecular Thermodynamics of Fluid-Phase Equilibria, Third Edition is a systematic, practical guide to interpreting, correlating, and predicting thermodynamic properties used in mixture-related phase-equilibrium calculations. Completely updated, this edition reflects the growing maturity of ...Molecular thermodynamics - definition of Molecular ...ther·mo·chem·is·try (thûr m-k m-str) n. The chemistry of heat and heat-associated chemical phenomena. ther mo·chem i·cal (--k l) adj. ther mo·chem ist n.Prausnitz Group Home Page - University of California, BerkeleyMolecular Thermodynamics John M. Prausnitz. Professor, University of California, Berkeley Faculty Senior Scientist, Lawrence Berkeley National LaboratorymThermoWait for more... ... The website is going through major upgrade ,, wait for the new look soon :)Molecular thermodynamics - HmolpediaAn encyclopedia of topics on the thermodynamics, chemistry, and physics of human existence.Molecular Thermodynamics of Fluid-Phase Equilibria / Edition ...The classic guide to mixtures, completely updated with new models, theories, examples, and data. Efficient separation operations and many other chemical processes depend upon a thorough understanding of the properties of gaseous and liquid mixtures. Molecular Thermodynamics of Fluid-Phase ...Molecular Thermodynamics of Fluid-Phase EquilibriaThermodynamics Home - Chemical Engineering - Thermodynamics: Molecular Thermodynamics of Fluid-Phase Equilibria by Rudiger N. Lichtenthaler, Edmundo Gomes De Azevedo, John M. Prausnitz, Edmundo G. De AzevedoMolecular thermodynamics definition of Molecular ...thermochemistry [¦th?r·m??kem·?·str?] (physical chemistry) The measurement, interpretation, and analysis of heat changes accompanying chemical reactions and changes in state.Molecular Thermodynamics by Donald A. McQuarrie — Reviews ...Molecular Thermodynamics has 7 ratings and 1 review. Jacob said: This books is incredible. It's extremely comprehensive. If you love physical chemistry a...Introduction to Molecular Thermodynamics, by Robert M. Hanson ...Introduction to Molecular Thermodynamics, by Robert M. Hanson and Susan Green , University Science Books, copyright 2008Molecular Thermodynamics of Fluid Phase Equilibria - 3rd Edition2Molecular Thermodynamics of Fluid Phase Equilibria - 3rd Edition2 - Free ebook download as PDF File (.pdf), Text file (.txt) or read book online for free.Molecular thermodynamics (Book, 1999) []Get this from a library! Molecular thermodynamics. [Donald A McQuarrie; John D Simon]Molecular Thermodynamics - Download for freeMolecular Thermodynamics download from FileSnail - 3188189138905X pdf, Thermodynamics Processes an rar, FilePost DepositFiles and RapidShare files. Molecular Thermodynamics of Fluid Phase Equilibria (Prausnitz ...Molecular Thermodynamics of Fluid Phase Equilibria (Prausnitz- 2th Edition ) - Free ebook download as Text file (.txt), PDF File (.pdf) or read book online for free. k Molecular thermodynamics (Open Library)Molecular thermodynamics by Richard Earl Dickerson, 1969,W. A. Benjamin edition, in EnglishMolecular thermodynamics - definition of Molecular ...Molecular thermodynamics. Molecular Time Scale Generalized Langevin Equation Molecular Typing by Reverse Dot-Blotting Molecular Typing Working Group Molecular Unit Cell Approximation molecular velocity Molecular vibration Molecular Vibrational SpectrometerCenter for Molecular & Engineering ThermodynamicsThe University of Delaware’s Thomas H. Epps, III, is one of 30 early-career engineers nationwide invited to attend the 2013 European Union-United States Frontiers of Engineering Symposium to be held Nov. 21-23 in Chantilly, France.ChemE | Thermodynamics and Molecular ComputationsThermodynamics and Molecular Computations. Thermodynamics governs processes as diverse as chemical production, bioreaction, creation of advanced materials, protein separation, and environmental treatment.Molecular Thermodynamics Mcquarrie : ?g Rt K Q. 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Thermodynamics of superconducting lattice fermions
a r X i v :c o n d -m a t /9707225v 1 [c o n d -m a t .s u p r -c o n ] 22 J u l 1997Thermodynamics of superconducting lattice fermionsE.Otnes and A.SudbøDepartment of PhysicsNorwegian University of Science and Technology,N-7034Trondheim,NorwayWe consider the Cooper-problem on a lattice model including onsite and near-neighbor interac-tions.Expanding the interaction in basis functions for the irreducible representation for the point group C 4v yields a classification of the symmetry of the Cooper-pair wave function,which we calcu-late in real-space.A change of symmetry upon doping,from s-wave at low filling fractions,to d x 2−y 2at higher filling fractions,is found.Fermi-surface details are thus important for the symmetry of the superconducting wave function.Symmetry forbids mixing of s-wave and d-wave symmetry in the Cooper-pair wavefunction on a square lattice,unless accidental degeneracies occur.This conclusion also holds for the selfconsistent treatment of the many-body problem,at the critical temperature T c .Below T c ,we find temperatures which are not critical points,where new superconducting channels open up in the order parameter due to bifurcations in the solutions of the nonlinear gap-equation.We calculate the free energy,entropy,coherence length,critical magnetic fields,and Ginzburg-Landau parameter κ.The model is of the extreme type-II variety.At the temperatures where subdomi-nant channels condense,we find cusps in the internal energy and entropy,as well as as BCS-like discontinuities in the specific heat.The specific heat anomalies are however weaker than at the true superconducting critical point,and argued to be of a different nature.I.INTRODUCTION The non-perturbative effect on the ground state wave-function of an electron gas with arbitrarily weak effective attraction between the quasi-particles on the Fermi surface was first demonstrated by Cooper in his essentially exact solution of the corresponding two-body problem [1].Regardless of the strength of the interaction,two electrons interacting attractively on opposite sides of the Fermi-surface in an otherwise inert Fermi-gas form a bound state,the Cooper-pair.This simple calculation alone suffices to yield precisely the correct non-analytic dependence of the binding energy of the Cooper-pair on the dimensionless coupling constant λlater found by solving the full selfconsistent problem [2].Cooper solved the problem for the simple “jellium”model of a metal,where the Fermi-sea was taken to be spherical,and the interaction between the two extra electrons of s-wave symmetry.Of primary interest was the two-body spectrum,the Cooper-pair wave-function in k-space under such circumstances being a trivial constant.In this paper,we reconsider this simple problem on a square lattice.The tight-binding band structure includes nearest and next-nearest neighbor hopping,while the two-body term in the Hamiltonian includes an onsite repulsive Hubbard-term,a nearest neighbor effective electrostatic interaction,which may be taken to be attractive ,and also a next-nearest neighbor electrostatic interaction.The problem now includes two additional non-trivial features:i)The Fermi-surface is no longer spherical and one has to work directly in k-space rather than transforming the problem to energy-space.ii)The interaction between the quasiparticles no longer has simple s-wave symmetry,but rather may be expanded as a bilinear combination of basis functions for the irreducible representation of the point group of the 2D square lattice,C 4v .This leads to the possibility of a number of interesting effects.In addition,we use a selfconsistent scheme to calculate the superconducting gap,thermodynamic quantities,and temperature dependence of critical magnetic fields of this phenomenological lattice fermion model.This paper is organized as follows.In Section II,we define the model to be considered.The method of calculating the Cooper-pair wavefunction is presented in Section III,while specific numerical results pertaining to this quantity are presented in Section IV.More detailed analytical results on the binding energy of the Cooper-pair are given in Section V.In Section VI,we present the results for thermodynamic quantities and critical magnetic fields from a self-consistent scheme for a gap-function with several symmetry-channels,belonging to various s-wave and d-wave channels.In Section VII,we give a discussion of the specific heat anomalies one may expect in such a model.We emphasize thatthroughout this paper,the superconducting order parameter is not a vector order parameter,but assumed to be a spin-singlet scalar complex order parameter such as is believed to describe conventional low-temperature superconductors and high-T c cuprates.II.THE MODELThe model we consider is an extended Hubbard-model on a square lattice defined by the HamiltonianH =−ti,j ,σc +i,σc j,σ−t ′ i,j ,σc +i,σc j,σ−µ i,σc +i,σc i,σ+12 i,σn i,σn i,−σ+V i,j ,σ,σ′n i,σn j,σ′+Wi,j σ,σ′n i,σn j,σ′ .(1)Here i,j and i,j denote nearest neighbor and next-nearest neighbor couplings,respectively.t ,and t ′are corre-sponding hopping matrix elements,and µis the chemical potential.U is an onsite repulsion term,while V and W are effective electrostatic Coulomb matrix-elements between nearest and next-nearest neighbors,respectively.When viewed as an effective interaction term within a one-band model,as reduced from a multiband-band model,these terms may be attractive [3–6].In this paper we simply take them as effective attractions without further justification.After introducing a plane-wave basis and performing a standard BCS truncation of the interaction piece of the Hamiltonian [2,7],it takes the usual formH = k,σε k c + k,σc k,σ+k, k ′V k, k ′c + k,↑c +− k,↓c − k ′,↓c k ′,↑,(2)where we have defined,after an appropriate redefinition of the zero-point of energy and a rescaling of t ′and µε k =−2t cos(k x )+cos(k y )−2t ′cos(k x )cos(k y )−(2−2t ′−µ) .(3)The choice of such a quasiparticle dispersion is obviously motivated by its relevance as a simple means of modeling the quasi-particle band crossing the Fermi-surface of the high-T c cuprates [8].Note that in such a context,the inclusion of the t ′-term is crucial;a bipartite lattice with nearest-neighbor hopping only,is inconsistent with the observed Fermi-surfaces in the high-T c cuprates,where |t ′|≈|t |/2[9].(This also has motivated the choice t ′=0.45t in our numerical calculations).The importance of including this term in correctly interpreting experiments,has recently been strongly emphasized [10].In the above truncation of the interaction term,the interaction is assumed to be operative between fermion spin-singlets on opposite sides of the Fermi-surface,the inert and rigid Fermi-sea merely giving rise to Pauli-blocking factors.With the interactions given in Eq.1,it is readily shown that V k, k ′is given byV k, k ′=5 η=1ληB η( k )B η( k ′),(4)where λ1=U/2,and λ2=λ4=V ,λ3=λ5=W .We have also found it convenient to introduce the simple,but sufficient,subset of basis functions {B η( k )}for irreducible representations of the symmetry group C 4v of the square lattice,B 1( k )=1N ,B 2( k )=1N [cos(k x )+cos(k y )],B 3( k )=2N [cos(k x )cos(k y )],B 4( k )=1N[cos(k x )−cos(k y )],B 5( k )=2N[sin(k x )sin(k y )],where N is the number of lattice sites.Inclusion of longer ranged interactions will in general require an augmentation of this subset,but any finite ranged interaction will yield a separable potential.III.THE COOPER PAIR W A VE FUNCTION We define a two-particle state for the non-interacting case,i.e.U =V =W =0,by |k,σ;−k,−σ 0obeying the Schr¨o dinger equationH 0|k,σ;−k,−σ 0=2ε k |k,σ;−k,−σ 0,(5)where H 0denotes the Hamiltonian of the free particles,and ε k is given in Eq. 3.Note that in this notation,the hopping between next nearest neighbors is given by the matrix element,4t ∗t ′,and we limit ourselves to situations where 2|t ′|<1,such that the bottom of the band ε k is located at the Brillouin-zone center.The problem we will consider is a simplification of the one posed by H =H 0+H int .We imagine that we have a rigid Fermi-sea of non-interacting electrons with a spectrum given by Eq.3.To this inert Fermi-sea we add two electrons on opposite sides of the Fermi-sea which interact with the matrix element V k,k ′.This interaction term scatters a pair of electrons in the state |k,σ;−k,−σ 0to the state |k ′,σ;−k ′,−σ 0.The exact two-particle state for the two extraelectrons,for which k is no longer a good quantum number,is defined by |1,2 .This state is expanded in two-particle plane-wave states as follows|1,2 = k>k F ,σa k,σ|k,σ;−k,−σ 0,(6)such that the problem is reduced to one of determining the Fourier-coefficients a k,σ.The exact two-particle state for this problem obeys the Schr¨o dinger equation:(H 0+H int )|1,2 =E |1,2 .Upon inserting the expansion in plane-wave states,and projecting onto plane-wave states,we obtain upon using Eq.4,an integral equation for the expansion coefficientsηB η( k ′) k>k Fληa k B η( k ) A η=(E −2ε k′)a k ′,which immediately yields an expression for the expansion coefficientsa k ′= ηA ηB η( k′)E −2ε k D ηη′= η′A η′B η′( k ′).(8)The functions B η( k )are linearly independent,and a comparison of coefficients therefore yields the following set of equations for the amplitudes A ηηA ηλη′D ηη′=A η′.(9)The result can be written as a vector equation,(T −I ) A=0,where T η′η=λη′D ηη′and I denotes the identity matrix.A nontrivial solution exists if and only if the system determinant vanishes |T −I |=0,which determines the eigenvalue E ,and thus in turn the eigenvectors.Note that this analysis shows that at this level,in general one cannot obtain a Cooper-pair wave function with a mixed (s,d )-symmetry.This is a consequence of the fact that the matrix D ηη′is block-diagonal in the s-wave and d-wave sectors due to the fact that the “pair-susceptibility”χk =1/(E −2ε k )transforms as a function with s-wave symmetry expandable in the functions B 1,B 2and B 3.In Section VI,we show that this conclusion holds for the mean-field gap-equation that results from a self-consistent solution to the full problem at the critical point T c .The wave-function may thus have one of the following formsa k =3 η=1A ηB η( k )χ k ;εk >εk F ,a k =5η=4A ηB η( k )χ k ;εk >εk F .(10)All the amplitudes a k are zero for ε k <εk F .Since the D -matrix is block diagonal,we may find two different eigenvalues,one for each irreducible representation.The correct eigenvector corresponds to the lowest eigenvalue E .Furthermore,an immediate consequence of the fact that A η= k>k F ληa k B η(k )is that A η=0if and only ifλη=0.Specifying theλη’s thus immediately determines which of the basis functions Bη( k)that may contribute to the Cooper pair wave function.The Cooper pair wave function in real-space is calculated by applying the inverse lattice Fourier-transform to ak,σ,defined byψ( r)=1n k( k1,k2 i,j ),(11)where r= i,j is the relative distance between the electrons comprising the Cooper-pair, k= k1,k2 .IV.NUMERICAL RESULTSIn Fig.1(a)-(f)we have plotted|ψ(i,j)|2of the Cooper pair wave function given by Eq.(11)in the case of no onsite interaction,U=0,and nearest neighbor attraction,V=−0.75∗t.Throughout this discussion,we set W=0. The hopping matrix element t=0.10eV.At lowfilling fractions,the wave function displays s-wave symmetry which can be recognized by the peak present at the point(0,0).Electrons situated at the same site forming a pair cannot be in a d-wave state.An example of a d-wave state is shown in Fig.1(c).This state is in fact found when increasing thefilling fraction to n=0.17,for the same value of the Coulomb-parameters.The pairing state has changed its transformation properties due to doping.Further increasing thefilling does not change the symmetry,but Figs. 1(c)-(f)show that the pairing state broadens as doping is increased.The region where|ψ(i,j)|2=0can roughly be interpreted as the size of the Cooper pair,i.e.the coherence length,which is thus seen to increase with increasing filling fraction.This is verified directly by calculating the quantity i,j|i−j|2|ψ(i,j)|2/ i,j|ψ(i,j)|2.To further investigate at which doping level the symmetry change occurs,we have plotted the eigenvalue E of the two-particle Schr¨o dinger equation,in Fig.2as a function of doping.The eigenvalue E is determined by numerically solving|T−I|=0.The Fig.shows that for smallfilling fractions,i.e.n<0.1,the eigenvalue for s-wave pairing is lower than the corresponding value for d-wave pairing,and hence s-wave pairing is therefore energetically favorable. At n≈0.10,we see that the two energy curves intersect,and for n>0.10,d-wave pairing is thus expected to be energetically favorable.We have done similar eigenvalue calculations as above in the case of an onsite repulsion,U=1.0∗t.The results are shown in Fig. 3.Since an onsite interaction does not affect d-wave pairing,the curve that shows the d-wave eigenvalues is the same as shown in the scenario of no onsite interaction.The s-wave curve has however changed,and appears to be shifted to higher energies,which means that the change in the symmetry of the wave-function occurs at a lowerfilling fraction,in this case n≈0.08.The results presented in Fig.3indicate that increasing onsite repulsion favors d-wave pairing,as one would expect.The extended s-wave component B2of the Cooper-pair wave function will inevitably have afinite on-site component,although it avoids the hard core to a considerable extent.Figs.4(a)-(f)show the results for|ψ(i,j)|2as a function of doping for U=4.0∗t.This onsite repulsion is large enough to suppress pairing in any of the s-wave channels for allfillings that we have considered,in the range n∈[0.06,0.85].Asfilling is increased,one starts seeing a considerable broadening of the wavefunction with increasing filling fraction.Figs.5(a)-(f)show the Cooper-pair wave function with n=0.06at six different values for the onsite repulsion U.In Fig.5(a),the onsite repulsion U=0,and the s-wave solution is the energetically favorable(see Fig.2).In Fig.5(b)a weak onsite repulsion U=0.5∗t is applied.Note that for this case,we obtain the somewhat counter-intuitive result that,since|ψ(i,j)|2;|i−j|=0evidently has increased upon increasing the onsite repulsion,the weak increased repulsion effectively promotes attraction between electrons.Similar effects have been seen in exact many-body calculations on strongly correlated1D lattice fermion models[11].We will comment on this result later.Figs. 5(c)-(f)display the wave function when the onsite repulsion is increased in the range U/t=(0.7,1.0,1.30,1.45)and it is evident that between U=1.30∗t and U=1.45∗t,the wave function has changed its transformation properties to d-wave in order to completely eliminate the effect of the hard core.Increasing the onsite repulsion further will not affect the d-wave function,as already mentioned.To show in more detail for which values of U the wave function changes its transformation properties,we have calculated the energy eigenvalues as a function of onsite repulsion for afilling fractions n=0.06.The results are shown in Fig. 6.The trends in the results clearly show that in lattice models with four-fold symmetry,s-wave superconducting pairing is a low-filling effect,while d-wave pairing practically always wins in situations close to half-filling.V.ANALYTICAL RESULTSIt is instructive also to perform a simplified analytical treatment of the Cooper-problem.A major simplification results by rewriting D ηη′(E )as followsD ηη′(E )= ∞−∞dεkδ(ε−ε k )θ(ε k )B η( k )B η′( k )E −2εθ(ε)≈N ηη′ωc +µµdε1ωc ),(12)where we have introduced the binding-energy ∆≡2µ−E ,ωc is an upper band-cutoffand the function F is defined as F (x )≡1x ).In addition,we have introduced the projected densities of state N ηη′(ε)= k B η( k )B η′( k )δ(ε−ε k ).The constants N ηη′in Eq.12are chosen appropriately to get the correct value for the k -space sum.With our choice of coupling constants,the matrix D ηη′block-diagonalizes into a 3×3matrix in the s-wave sector,and a 2×2matrix in the d-wave sector.For simplicity,in the following we will set W =0.It will also be convenient to introduce the following definitionsν≡UV2(U|γ|2+|ν|22λs ;U >|V |N 22|γ|2+|ν|22λs ;U <|V |N 222λs ;∆s =2ωc 2λd ;∆d =2ωc2π2|t |11−ρa K 1−ρa ;|(a +ρ)/2|<1≈1√1−ρawhere we have definedρ=−2t′,a=ε/2t,and K(k)is a complete elliptic integral of thefirst kind,with modulus k [13].The result is valid for|ρ|<1,the requirement for having a band-minimum at the center of Brillouin-zone.(It is quite remarkable that the approximate logarithmic expression in Eq.15reproduces the exact expression excellently over the entire band.The reason is that the main effect of the elliptic integral K is to produce a narrow logarithmic singularity at a=ρ,while the remaining variation in N(ε)is due to the prefactor.)The density of states has a logarithmic singularity N(ε)=(1/2π2|t| 1−ρ2) atε=2tρ,afinite cusp at the lower band-edge,N(ε=2t(−2−ρ))=(1/4π|t|)/(1+ρ),while the density of states close to the upper band-edge has a smooth behavior with a value at the upper band-edge given by N(ε=2t(2−ρ))=(1/4π|t|)/(1−ρ).Were we to reverse the sign of t′,we would get a cusp at the upper band-edge,while the density of states at the lower edge would be smooth.The singularities that appear in the projected densities of states must come from the singularities in N(ε).We have not been able to reduce the integrals for the projected densities of states to useful expressions in terms of elliptic integrals,and have simply calculated them numerically using the tetrahedron algorithm[14].The peaks in N22(ε)originate from energies close to the bottom of the band,essentially identified with the cusps in N(ε),whereas the peaks in the N44(ε)come from energies identified with the logarithmic singularities in N(ε).As soon as thefilling fraction increases from the bottom of the band,the major singularities in N22(ε)are eliminated from the integrated projected density of states N22,and the contributions to N44coming from the logarithmic singularities in N44(ε)atε=2tρwill dominate.Hence,λd and thus d-wave pairing will always dominate close to half-filling, while s-wave pairing will win out close to the bottom of the band.Note for instance in the case where t′=0,that N22(ε)=0in the middle of the band,while N44(ε)has a weak logarithmic singularity.When t′=0the situation changes somewhat,and both projected densities of state arefinite at halffilling.Nonetheless,as long as t′<0.5,such that the bottom of the band is located at the zone-center,N44>N22forfilling fractions such that the Fermi-surface is close to the middle fo the band.For non-pathological situations,d-wave pairing will in general dominate close to half-filling.The conclusion holds for any pairing kernel with a predominant B4-symmetry.A somewhat counter-intuitive result obtains if we introduce a weak repulsive coupling U in the problem.In our notation,this means that in Eq.14,we may expand F in Eq.14in powers of|ν|/|γ|,in order tofind the corrections toλs to leading order in U/V.The leading order correction in U is given byλs−|γ|=|ν|N1222thereby reducing the indirect attractive effect of the onsite-repulsion.As a consequence,the pair-wave function will again spread out in real space,as observed in our numerics.Note that the above results are not sensitive to the basic pairing mechanism operative in the underlying microscopic theory.All that enters at this level,are competing channels in the pairing kernel with isotropic and extended s-wave symmetries,as well as d-wave symmetries.Hence,for instance anti-ferromagnetic spin-fluctuations [16],who some believe to be relevant in producing superconductivity in the high-T c cuprate CuO 2-planes,would fit into the scenario Eq.4for the pairing kernel.Nonetheless,the above “attraction from repulsion”effect within the isotropic and extended s-wave sector is not particularly relevant to the physics of the high-T c oxides,which show superconductivity only in the vicinity of half-filled bands,where we find that d x 2−y 2-wave pairing almost invariably wins out.VI.SELFCONSISTENT ANALYSISIn this Section,we consider the full nonlinear mean-field gap-equation.Starting from the full Hamiltonian,Eq.1,one performs a standard BCS-truncation of the interaction term,and a further anomalous mean-field decomposition to obtain the gap-equation∆ k =− k ′V k, k ′∆ k ′χ k ′V k, k ′=ηληB η( k )B η( k ′)χ k =12E k =this is equivalent to a bifurcation in the solution of the nonlinear gap-equation.Such bifurcations have recently been studied for the BCS-gap equation[17].We have solved the coupled mean-field gap equations numerically,to obtain the superconducting gap,and hencevarious thermodynamic quantities.We omit details of our calculations of the gap-function∆k itself,suffice it to saythat we have reproduced in detail the results of Ref.[15].The free energy,internal energy,entropy,specific heat,andcritical magneticfields are found by using the standard expressions for the free energy F and the entropy SF= k(ε k−χ k|∆ k|2)−12B c1. Our results exhibiting the quantities described above,are shown in Figs.8-10.As the temperature is lowered,onewill observe cusps in|∆k|,and hence the internal energy and entropy,at the temperatures where new channels are coupled into the gap.The specific heat will show BCS-discontinuities at all the temperatures where the new channelscondense,in addition to the BCS-discontinuity at the superconducting transition[17].However,the amplitude-fluctuations of the orderparameter are massive at these lower temperatures,and the superconducting correlation length isfinite.This is a generic feature of superconductivity in systems with competing pairing channels,irrespective of whether the dominant pairing is d-wave or s-wave.VII.SPECIFIC HEATFor the case of extreme type-II superconductors,criticalfluctuations will surely modify the results for the specific heat close to the critical temperature.The anomaly in the specific heat at the critical temperature is therefore not expected to be of the BCS-type at all,due to phase-fluctuations of the order parameter.As recently emphasized [18],in extreme type-II superconductors with a low superfluid density,the dominant contributions to the specific heat anomaly are expected to be phase-fluctuations in the order parameter,not amplitude-fluctuations as in BCS.This is easily seen by noting that the superfluid stiffness is related to the free energy F and a phase-twistδφin the order parameter across the system via the Fisher-Barber-Jasnow relation[19]ρs= ∂2FA large onsite repulsion U ultimately suppresses pairing in the isotropic and extended s-wave channels.This may explain why two anomalies in the specific heat so far have not been observed in zero magneticfield in Y Ba2Cu3O7[22].However one should note that an inclusion of the W-term in Eq.1will give pairing in another d-wave channel,d xy,which could compete with d x2−y2in the presence of a very large Hubbard-U,in the relevant doping regime,and hence also give two anomalies in the specific heat even in large-U compounds such as the high-T c cuprates.Recent experiments show intriguing features in microwave conductivity as well as London penetration depth well inside the superconducting phase,which may be consistent with the above picture[24].VIII.ACKNOWLEDGEMENTSSupport from the Research Council of Norway(Norges Forskningsr˚ad)Grants No.110566/410and110569/410,is gratefully acknowledged.The authors thank G.Angilella for discussions.LIST OF FIGURESFIG.1.The probability densityω(i,j)=|ψ(i,j)|2of the Cooper-pair wave function with U=0at differentfilling fractions. In Figs.(a)and(b)the wave function exhibits s-wave symmetry for n=0.06and n=0.12,respectively.Figs.(c)-(f)all exhibit d-wave symmetry of the wave function where n=0.14in(c),n=0.17in(d),n=0.51in(e),and n=0.85in(f).FIG.2.The binding energy of the Cooper-pair as a function of doping,at U=0.For n<0.11the binding energy of the s-wave pairing is energetically favorable compared to d-wave pairing.As doping increases,the d-wave pairing becomes favorable.FIG.3.The binding energy of the Cooper-pair as a function of doping,at U=1.0∗t.For n<0.11the binding energy of the s-wave pairing is energetically favorable to d-wave pairing.As doping increases,the d-wave pairing becomes favorable.FIG.4.The probability densityω(i,j)=|ψ(i,j)|2of the Cooper-pair wave function with U=4.0∗t at differentfilling fractions.Figs.(a)-(f)all exhibit d-wave symmetry of the wave function where n=0.06in(a),n=0.12in(b),n=0.14in (c),n=0.17in(d),n=0.51in(e),and n=0.85in(f).FIG.5.The probability densityω(i,j)=|ψ(i,j)|2of the Cooper-pair wave function with n=0.06at different onsite repulsions.For U=0,s-wave pairing is favorable(a).For the values U=0.50∗t,U=0.70∗t,U=1.0∗t and U=1.30∗t as shown in(b),(c),(d)and(e),respectively,the Figs.exhibit s-wave symmetry.For U=1.45∗t the Cooper pair wave function tranforms as a d-wave.FIG.6.The binding energy of the Cooper-pair with n=0.06as a function of onsite repulsion.Since the onsite potential couples to an s-wave,the d-wave binding energy is not affected by increasing U.For U<1.4∗t,s-wave paring is favored.For larger U’s,d-wave pairing is favored.FIG.7.The projected densities of states N22and N44along with the single particle density of states N(ε)= kδ(ε−ε k).FIG.8.Thermodynamic quantities in superconducting and normal states for n=0.25and U=0.Figs.(a)-(d)show the entropy,free energy,internal energy and specific heat,respectively.They all exhibit a critical behavior at T≈90K.In(d) it can easily be seen that another symmetry channel switches on at T≈60K,giving rise to cusps in the internal energy and entropy,and a discontinuity in the specific heat.Figs.(e)and(f)show the coherence length and the criticalfields,respectively. Note that the coherence length has singular behavior only at T≈90K.FIG.9.Thermodynamic quantities in superconducting and normal states for n=0.60and U=4.0∗t.Figs.(a)-(d)show the entropy,free energy,internal energy and specific heat,respectively.They all exhibit a critical temperature at T≈90K. Figs.(e)and(f)show the coherence length and the criticalfields,respectively.In this case,there are no additional features below T c,due to the complete suppression of the instability in the competing s-wave channel as a result of the large onsite U. FIG.10.The Ginzburg-Landau parameter,κ,as a function of temperature for the cases:(a)n=0.25and U=0and(b) n=0.60and U=4.0∗t.The result shows that the superconductor model is of the extreme type-II variety.Hence,one expects the dominant criticalfluctuations in the order parameter to be phase-fluctuations,not amplitudefluctuations.(a)(b)(c)(d)(e)(f)FIG.1.FIG.2.(a)(b)(c)(d)(e)(f)FIG.4.(a)(b)(c)(d)(e)(f) FIG.5.0.050.0100.0Temperature [K]0102030 [µe V /K ]S en S es(a)0.050.0100.0Temperature [K]-1.0-2.0-0.5-1.5[m e V ]F en F es(b)0.050.0100.0Temperature [K]01.02.0-1.0[m e V ]U en U es(c)0.050.0100.0Temperature [K]0.010.020.030.040.050.0[µe V /K ]C en C es(d)0.020.040.060.080.0100.0Temperature [K]0.020.040.060.080.0100.0120.0C o h e r e n c e l e n g t h [/l a t .](e)0.050.0100.0Temperature [K]0.010.020.030.040.0[T e s l a ]B c2B c1(f)。
作文未来超导技术450字的
作文未来超导技术450字的英文回答：Superconducting technology is poised to revolutionize various industries and sectors in the future. With itsability to conduct electricity with zero resistance, superconductors have the potential to greatly enhanceenergy efficiency, transportation systems, and even medical applications.In terms of energy efficiency, superconductingmaterials can significantly reduce energy loss during transmission and distribution. This means that electricity can be transmitted over long distances without any loss of power, leading to more efficient and cost-effective energy systems. This technology can also be applied to power grids, allowing for better management and stability.In the transportation sector, superconductors can revolutionize the way we travel. Magnetic levitation(maglev) trains, which use superconducting magnets to float above the tracks, can achieve incredible speeds and efficiency. This technology has the potential to transform long-distance travel, making it faster, safer, and more environmentally friendly.Furthermore, superconducting technology can greatly benefit the medical field. Magnetic resonance imaging (MRI) machines, which use superconducting magnets, provide detailed and accurate images of the human body. This technology has revolutionized medical diagnosis and treatment, allowing for earlier detection of diseases and more precise surgical procedures.In conclusion, the future of superconducting technology is bright. Its potential to enhance energy efficiency, revolutionize transportation, and improve medical applications cannot be overstated. As research and development in this field continue to advance, we can expect to see even more innovative and practical applications of superconductors in the near future.中文回答：超导技术有望在未来革新各个行业和领域。
new study roomtemperature superconductor
目前的研究表明，常温超导仍然是一个挑战性的目标。
尽管科学家们一直在不断寻找新的材料和方法，但实现真正的常温超导仍然存在许多技术上的难题。
在过去的几十年中，已经发现了一些高温超导材料，其中最具代表性的是铜氧化物和铁基超导体。
这些材料在相对较高的温度下展现出超导性质，但仍需冷却至较低温度来维持其超导状态。
尽管这些温度较传统的超导材料更高，但仍然需要复杂的冷却系统。
目前，科学家们正在寻找更多具有高温超导特性的材料，并尝试改变材料的结构和化学组成，以提高超导转变温度。
此外，研究人员还在探索新的物理机制和理论模型，以加深对高温超导机制的理解。
常温超导研究的难点之一是找到一种材料，它能够在常温（室温）下显示出相对较高的超导转变温度，而无需复杂的冷却。
这要求对超导机制有更深入的理解，并且需要在材料合成、结晶和处理方面取得进一步的突破。
此外，常温超导材料还需要具备良好的电流输运性能和稳定性，以满足实际应用的要求。
虽然现在的常温超导研究还面临许多挑战，但科学家们对于未来实现这一目标仍然持有乐观态度。
随着技术的不断进步和新的科学发现，我们可能会在未来看到常温超导的实现，并带来革命性的应用和发展。
The current status of room-temperature superconductivity remains a challenging goal. Despite ongoing efforts by scientists to search for new materials and approaches, achieving true room-temperature superconductivity still poses many technological obstacles.Over the past several decades, some high-temperature superconducting materials have been discovered, with cuprates and iron-based superconductors being the most notable examples. These materials exhibit superconducting properties at relatively higher temperatures, although they still require cooling to maintain their superconducting states. While these temperatures are higher than those of traditional superconductors, complex cooling systems are still necessary.Currently, scientists are actively searching for more materials with high-temperature superconducting properties and exploring ways to manipulate the structures and chemical compositions of materials to enhance the superconducting transition temperatures. Additionally, researchers are investigating new physical mechanisms and theoretical models to deepen our understanding of high-temperature superconductivity.One of the challenges in room-temperature superconductivity research is finding a material that can exhibit relatively high superconducting transition temperatures at ambient conditions without the need for complex cooling. This requires a deeper understanding of the underlying superconducting mechanisms and breakthroughs in material synthesis, crystallization, and processing. Furthermore, materials exhibiting room-temperature superconductivity need to possess good electrical transport properties and stability to meet the requirements of practical applications.While there are still significant challenges in achieving room-temperature superconductivity, scientists remain optimistic about the possibility in the future. With advancing technologies and new scientific discoveries, it is possible that we may witness the realization of room-temperature superconductivity, leading to revolutionary applications and developments.。
化学英语翻译课件Thermodynamics
Heat and enthalpy 热和焓
元素在标准态下 的焓值为零
Exothermic and endothermic reactions 放热和吸热反应
负焓变
Exothermic and endothermic reactions 放热和吸热反应
正焓变
Standard enthalpy changes 标准焓变
常用英语口语
• 11. I have no choice． 我别无选择。 12. I'll try my best． 我尽力而为。 13. I'm On your side． 我全力支持你。 14. Long time no see! 好久不见! 15. No pain，no gain． 不劳无获。 16. Well，it depends 噢，这得看情况。 17. We're all for it． 我们全都同意。 18. What a good deal! 真便宜! 19. You asked for it! 你自讨苦吃! 20. You have my word． 我保证.
键焓或键离解焓
First law of thermodynamics 热力学第一定律 能量转换定律
• Or the law of conservation of energy states that energy can neither be created nor destroyed, but only transferred from one form to another
在其他变量都保持常数时，为了使不同的焓变可以比 较，所有的标准焓变都由1mol物质在标准状态下反 应而测得。
Standard enthalpy changes