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This 11 page Class Notes was uploaded by Clement Bernier on Wednesday October 21, 2015. The Class Notes belongs to PHY 300 at Syracuse University taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/225632/phy-300-syracuse-university in Physics 2 at Syracuse University.
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Date Created: 10/21/15
Lec6 a Phase transitions critical phenomena o Magnetic systems Ising model Phase transitions o Many systems composed of very many degrees of freedom exhibit phase transi tions o These are abrupt changes in the macro scopic state appearance properties etc of the system as some parameter is changed a Historically that parameter was often the temperature eg Solid liquid transition at some critical Tc Transition from magnetic to non magnetic material for some TC Critical Phenomena Close to the phase transition T N Tc the system exhibits critical behavior eg spe cific heat C N T Tera The critical exponent 04 is universal it is the same for many different materials The underlying reason for this universal ity is that the critical system exhibits very long range correlations between individual molecular constituents The distance over which these correlations take place is called the correlation length 5 gt 00 This washes out details on scale of lattice spacing Mag netic systems Many ferromagnetic materials may possess permanent magnetization Every atom contains circulating electrons These yield small magnetic fields Some times these can add to give a large macro scopic magnetic field it is said to be a permanent magnet However if the temperature is raised this will in general disappear the system goes from ferromagnetic to paramagnetic This is a phase transition Critical exponents Various thermodynamic quantities diverge or have singular power law behavior there Specific Heat 0 Magnetization M Mag netic susceptibility X C N T Tca M N T TCW X N T TC7 Model Simple model for these magnetic systems is the Ising model Place elementary magnets on sites of sim ple lattice representing crystalline struc ture of material Allow these elementary magnets si to point in just 2 possible directions up and down 5 l1 Allow the energy for the system to be given by E Z J Z 8163 ltijgt Dynamics Can writesolve dynamical equations but very many atoms in material too cumber some and not necessary Sufiices to have a theory which describes only the probability of finding the system in some state statistical mechanics Take as basic assumption of this theory that Probability of finding the system in some state with energy I at tem perature T is given by e W Observables computed by averaging over all possible states using this probability 7 Examples Mean magnetization M ltMgt Z M5eE8kT states State of system corresponds specifying the state of each elementary magnet or spin on some lattice Impossible to do this sum exactly even with a computer Resort to Monte Carlo methods Monte Carlo 0 Use a simple algorithm to move from state 7 to state j a Design that algorithm to ensure that after some iterations the probability of any state occuring is just e Ek T a Measure observables by simple averaging over this set of states Yields eg lt M gt2 Zcon gCMC with statistical error that varies as 1xN for N states Metropolis algorithm Simplest algorithmupdate procedure for Monte Carlo 0 Pick a site Try to flip the spin 5 gt 5 Compute change in energy under such a flip AE Local AE o Accept the move with probabiliy e W 0 Keep going 10 Phase transitions in Ising model Simplest case two dimensions c Find for T TC 2269 fluctuations in M have a peak oMOfOFTgtTC MOforTltTc a Close to Tc X N T TC1 875 in 2 dimen sions M N T TOW5 11
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