计算物理英文讲义.pdf

Crystal Structure What is crystal?The material with long range order structure.The material with long range periodic structure.The material with macro-symmetrical structure.How to describe crystal structure?Crystal Structure=primitive unit of matter+latticeaaaaaPeriodic StructurexaxaxaxaxaWhat is needed to describe the periodic structure?The periodic unit vectors.),2,1,0(L=iiaxi),2,1,0()(),2,1,0(LL=+=ibaiaxoribiaxiix0b0Are they enough?The coordinate the matter in the unit.The type of the matter.Periodic unit cellPeriodic StructureTwo dimensional periodic structureTwo vectors in different direction are used to describe the shape of the periodic unit.Periodic StructureCu crystalThree vectors in different faces are used to describe the shape of the periodic unit.XYZCrystal StructureCu crystalAl crystalUnit Vectors:A1=aX,A2=aY,A3=aZPosition Vectors:B1=0B2=A2/2+A3/2B3=A1/2+A3/2B4=A1/2+A2/2CuCuCuCuAlAlAlAlA1=aY+aZA2=aX+aZA3=aX+aYUnit Vectors:Position Vectors:B1=0XYZCu AlLatticePrimitive unit cellCrystal unit cellCrystal Structure What is Lattice?The set of the lattice point.What is the lattice point?The point in the crystal with the same Geometrical Environment Chemic Environment Physical Environment How to construct the lattice?Select a point in the crystal arbitrarily as the origin.Find all the point in the crystal with the same property as the origin.Crystal StructureThe matter particle is not obliged to be on the lattice point.aaaaaaaaaaKey Mathematic Description for Crystal Structure Use Miller Indices to denote the orientations and the faces in the crystal structure.R1R1:0 1 0R2R2:1 1 0R3OABCDEFGR3:1 1 1Face ACEG:(1 1 0)Face BEG:(1 1 1)Heterogeneous Stacking in Crystal StructureFCC(110)(111)(001)Face Spacing Calculation Volume per atom()For the cell vectors selected in right hand rule Area per atom for face(hkl)(shkl)Face spacing for face(hkl)(dhkl)()N321AAA=N:number of atoms in the unit cellhklhklsd=Example:FCC calculation33111ad=2100ad=42110ad=Question:d111,d100,d110ExercisesBCCA1=aX,A2=aY,A3=aZXYZUnit VectorsQuestion:d001=?,d110=?,d111=?Answer of ExercisesConfiguration of face stacking for bcc(001)(110)(111)63111ad=2001ad=22110ad=Coordination Number of neighbors For cubic Structurehkl:hkl0;hkl:hk0 and l=0;hkl:h=k0 or l,l0;hkl:h=k 0,and l=0;hkl:h=k=l0;hkl:h=k=0 and l0;4824241286Coordination number(CN):the number of the neighbors with the same distance.CN Degeneration:5 0 0 and 3 4 0Super-Lattice Crystal What is superlattice?The lattice is composed of more than or equal to two sets of sub-lattice.FCCL12A1=aY+aZA2=aX+aZA3=aX+aYPrimitive Vectors:Basis Vectors:B1=0 CuPrimitive Vectors:A1=aX,A2=aY,A3=aZBasis Vectors:B1=0 (Al)B2=A2/2+A3/2 (Ni)B3=A1/2+A3/2 Ni)B4=A1/2+A2/2 (Ni)XYZSuper-Lattice CrystalBCCPrimitive Vectors:A1=-aX+aY+aZA2=aX-aY+aZA3=aX+aY-aZBasis Vectors:B1=0 (W)XYZA1A2A3Primitive Vectors:A1=aX,A2=aY,A3=aZBasis Vectors:B1=0 (Al)B2=A2/2+A2/2+A3/2 (Ni)B2Supper-Lattice CrystalL21DO3A1=aY+aZ A2=aX+aZ A3=aX+aYPrimitive Vectors:Basis Vectors:B1=0(4a)B2=-A1+A2+A3=aX(4b)B3=-A1-A2-A3=-a X-a Y-a Z(8c)B4=A1+A2+A3=a X+a Y+a Z(8c)aL21(A2BC)BCAADO3(A3B)BAAAXYZAtomic Site Preference for Crystal All solid phases are solutions.The study of solid solutions is an essential and very broad field of solid state physics.The atomic distribution of defects in solid.How to describe it?A statistical model is necessary.The focus of the problem is:In view of statistics theory:Question is:Question:How to set up a statistical model?Atomic Site Preference in Crystal It should be noticed:All experimentally measurable physics quantities should be macro state parameters.The macro state physics quantity is the statistical average of the micro physics quantity.The macro state parameters must be developed to describe the physics state.All the related quantities to describe the crystal structure are state parameters,not micro quantities.Question:How to distinguish the crystal from the non-crystal?Atomic Site Preference in Crystal Special Case for atomic site preference Two sets of sublattice.Two components.No composition vacancies.Stoichiometric composition Mathematic descriptions for the case.Ci(i=1 or 2):fraction of sublattice i.C(=A or B):atomic composition of atom .C1+C2=1 and CA+CB=1.C1=CAand C2=CBBragg-Williams(BW)model:The long range order parameter()is used to describe the atomic site preference.Atomic Site Preference in Crystal Order-disorder transformationL12FCC=1=0P1A=1P1B=0P2A=0P2B=1P1A=0.75P1B=0.25P2A=0.75P2B=0.25Pi:the possibility of the atom occupies the sublattice i.BW model:a linear relationship between and Pi Order state.Disorder stateAtomic Site Preference in Crystal Generalized case for atomic site preference n types of sublattice(n 2)m types of component No composition vacancies.Mathematic descriptions Ci(i=1,2,n):fraction of sublattice i.C(=t1,t2,tm):atomic composition of atom 11=niiC11=mttC i:fraction of atom distributed on the sublattice i.No composition vacancies.iiiCCP=11=nii11=mttiPAtomic Site Preference in Crystal The freedom of i:(n-1)(m-1)11=nii11=mttiiCC Procedure to produce possible configuration:One type atom then another,until complete.One sublattice then another,until complete.Question:How to design a program to produce the possible configuration?LoopWith a random generator to produce a i;Whether it is reasonable?Risk:no end loop.Atomic Site Preference in Crystal The value range of i The bottom limit.Not less than zero.The left sublattices is enough to hold the left atoms.The upper limit.Not larger than unit.Not larger than the sublattice fraction.iis in the range of i,i=CCCnijji,0max=iiCC,1minAtomic Site Preference in Crystal The efficient arithmetic for produce i.bottom limit for itk Not less than zero The left space of the sublattice is enough to hold the left atom.Upper limit for itk Not larger than the left atomic fraction.Not larger than the left space of the sublattice.Loop itk=aitk+(bitk-aitk)is a random number.=+=tknijtktjjijjtkitkCCCa111111,0max=tktktiiijjtkitkCCCb1111,1minThe left atom fractionThe left sublattice spaceDescription of Non-Crystal Structure Properties of non-crystal With direction homogenous physical and chemic properties(independent on direction)With short range order structure State parameter of non-crystal structure independent on direction,just depend on distance Radial distribution function(RDF):g(r)Description of Non-Crystal Definition of RDF:g(r)Radial Density Function:(r)()drrrdn24=dn:the number of matter particles in the neighbor shell of r to r+dr of the reference matter particle(RMP).r+drrRMP(r)is the average neighbor matter particles density.()()0rrg=RDF0:the density of ideal gas=N0N:total particle:total volumeDescription of Non-Crystala)ideal gas,b)liquid,c)non-crystal,d)crystalDescription of Non-CrystalMolecular Dynamics(MD)Calculated Result of Cu0246810120123456789g(r)r/10-1nmPerfect Crystal0246810120.00.51.01.52.02.53.0g(r)r/10-1nmRelaxed Crystal0246810120.00.51.01.52.02.5g(r)r/10-1nmNonCrystalAtomic Interaction Empirical method:Pair Potential Lennard-Jones(LJ)potential Born-Mayer potential Morse potential()612rBrArijijij=()6rCeArBr=()()()()0022rrBrrBeeAr=,:index of atom type,r:distance of atom pair()=NiNijijjitotrE1,121Total energy of the systemAtomic Interaction0r(r)r0()00=rrdrrd()0022=rrdrrdr0:the equilibrium distancerrdrdrr=FAtomic Interaction Semi-Empirical method:Multi-body potentialEmbedded atom method(EAM)()=Nijijjirf,1()()=+=NiNijijjiNiiitotrFE1,1121i:the environmental electron density at the site of the atom i embedded.F:the embedding energy function of atom.f:electron density function of atom.Atomic Interaction Rigid theoretic method(first principle calculation,or AB Initio calculation)Adiabatic approximation Average Field and Single electron approximation()()()rErrVm=+222hV(r):average potential field(r):wave function of electronBasic Concepts of Thermodynamics Statistic Ensemble Interaction between ensemble andenvironmentEnergy exchangeMatter exchange Type of ensembleMicro-Canonical Ensemble(isolate system)?No energy exchange,no matter exchange(NEV)Canonical Ensemble?Energy exchange,no matter exchange(NPT,NPV,etc.)Macro-Canonical Ensemble?Energy exchange,matter exchangeBasic Concepts of Thermodynamics Basic States of Matter Solid Liquid Gas Description of the State of the Ensemble Macro-State parameters Statistic Average of Micro-State parameters Equilibrium State The state spontaneously exists at the specified macroscopical conditions.P,T,and VBasic Concepts of Thermodynamics Statistic Assumptions At the specified macroscopic conditions,any possible micro-state can appear.The probability of a micro-state occurring depends on its micro-state energy.Existing Probability of MicroState(Pt)Geometrical Probability(Pg)Physic Probability(Pp):total number of possible microscopic states.TkpBeP=:energy of micro-state,T:temperaturekB:Boltzmann constant=1gP=TkgptBePPPBasic Concepts of Thermodynamics Existing Probability of Macro-State(PT)Geometrical Probability(PG)Physic ProbabilityPP)Z:number of possible microscopic states for the specified macroscopic state.TkEPBeP=E:energy of the macro-state=GPTkFZTkETkTkEGPTBBBBeeZePPP=ln1F:free energyTSEF=S:entropyBasic Concepts of Thermodynamics Entropy The disorder degree of the macroscopic state.Disorder degree Magnitude of the number of the microscopic states for the macroscopic state.Free Energy It is not an actual energy,it is the magnitude of the existing probability of the macroscopic state.Equilibrium state The macroscopic state with the maximum existing probability.Basic Concepts of Thermodynamics Question:Why can ice melt into water and water solidify into ice?For NEV ensemble,the equilibrium state is the state with the maximum entropy.Monte Carlo Method Profile of Monte Carlo A Classical Method.A Random Method.A Statistic Method.Famous and popular for its application in the Nuclear-Bomb design.Named by its place where it became famous.Monte Carlo Method A group of methods in which physical or mathematical problems are simulated by using random numbers.Monte Carlo Method One Dimension Monte Carlo Integration()=10dxxfP01xf(x)0 f(x)1,0 x 1Normal method:Discretization MethodMonte Carlo Method:Construct a unit Squre.Loop N0produce a pair of random numbers and in the range of 0,1.If f(),add 1 to count number N,Else if f(),nothing to do.P=N/N0(1,1)(,)Monte Carlo Method One Dimension Monte Carlo Integration()=badxxfP0 f(x)M,a x bNormalization:1)Set y=(x-a)/(b-a);2)Set F(y)=f(x)/M.0bxf(x)a01yF(y)()=10dyyFPP=PMonte Carlo Method One Dimension Monte Carlo Integration()=badxxfP0 f(x)f(),nothing to do.P=N/N0Monte Carlo Method Buffon modell2al 2aQuestion:What is the probability of the noodle intersection with anyone of the parallel lines?Monte Carlo Method Buffon modell2al 2aQuestion:What is the probability of the noodle intersection with anyone of the parallel lines?Answer:P=4a/l Monte Carlo Method Buffon modelxlx0l/2asin()()laldaP42sin0=How to describe the state of the noodle on the desk?Monte Carlo Method Metroplis Sample Boltzmann Distribution Function()TkBef=Thermodynamic average0f()Where is the energy of the microstates.=dedyeyTkTkBBMonte Carlo Method Metroplis Sample Thermodynamic average=TkTkiBiBieeyyWhere iis the energy of the system in state i.Detailed balance conditionijjjiiTpTp=Where piis the probability of finding the system in state i.Tijis the probability(or rate)that a system in state i will make a transition to state j.()TkijijjiBijeppTT=Monte Carlo Method Metroplis Sample()=)or p(if)or p(if1jijiijTkijijjipeorpppTBij Sample procedure1.Generate an arbitrary state as the initial sample state.2.Calculate its energy 0.3.Generate another arbitrary state4.Calculate its energy i+1(i=0,1,).5.If i+1 i,select the new state as state i+1.6.If i+1i,calculate Tii+1and produce a random number.If Tii+1,select the new state as state i+1,else select state I as state i+1.7.Iterate step 2 to step 6.Monte Carlo Method Ising Model of ferromagnet=jijiSSJH,Where J is a positive energy,and S=1/2For a system with 1010 electrons,the number of the spin state is 21001030.With a computer of a speed of 1THZ(1012/sec)and with the simplest run.The consumption time of 1030runs is about 3.2 1010years.Monte Carlo Method General Procedure of Monte Carlo1.Construct a Statistic distribution model that relates the expected value with the probability.2.Determine the probability by the experiment.3.Evaluate the expected value.Monte Carlo Method Characteristics of Monte Carlo1.Convergence:Not dependent on the dimensions and converge with a speed of N-1/2.2.The result has a statistic fluctuation.3.The result depend on the quality of pseudo random number generator(RNG).Integrals:discretizationmethod converges as N-2/d(d:dimensions of problem)1.Uncorrelated sequence.2.Long period.3.Uniformity.4.Efficiency.Monte Carlo Method Linear Congruential Generator(LCG)()mcaxxnnmod1+=+a,c:integer constantm:integer modulusx0:initial seed(0,1,2,3,4,5)mod 3(0,1,2,0,1,2)()()1,0=nnnRmfloatxR()()1,01=nnnRmfloatxRMonte Carlo Method Random Walk method(RWM)Sample of Distribution()()()=00001xxxxxx01x0 x(x)Sample procedure:1.Generate a random number 2.If-x0,finished,select as the solution.3.If-x0,generate another random number.Monte Carlo Method Solving Equation by RWM()axf=Sample procedure:1.Generate a random number 2.If f()-a,finished,select as the solution.3.If f()-a,generate another random number.Monte Carlo Method Minimizing by RWM()()xfminSample procedure:1.Select x0,arbitrarily and evaluate f(x0)as fmin.2.Generate a random number and evaluate fas.If fmin,replace x0by and fminby f().Else,nothing to do.3.Iterate step 2 until the desired loops.Molecular Dynamics What is Molecular Dynamics(MD)A computer simulation technique where the time evolution of a set of interacting particles is followed by integrating their equations of motion.iiimaFrr=Newtons Law22dtdiirarr=iiirFrr=Fi:force acting on particle i.ai:accelerator of particle i.i:inter-particle potential acting on particle i.Molecular Dynamics Micro-State of the system()is Denoted by 3N positions and 3N momenta.Macro-State of the system It is a micro-canonical system It is a system of (H()-E)It is NVE algebra General Procedure of MD Initialize(ri0,vi0)Start simulation and let the system reach equilibrium Continue simulation and store resultsMolecular Dynamics Canonical MD Algebra NVT Algebra(Extended system method)Thermal ReservoirsystemPerfect heat conductor boundaryTs:The instantaneous temperature of the systemTh:The temperature of Heat Pool.When TsTh,the system releases heat to the thermal reservoir without changing Th.When TsPrxpmrrxmmiijijiiiijijijijijiii92/1222prFprrrrP:the desired pressure