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# Selected Topics PHY 300

Syracuse

GPA 3.64

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This 13 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 23 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

Lec8 a Phase transitions critical phenomena o Magnetic systems Ising model a Mean field theory correlations 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 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 Ising 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 Thermodynamic quantities a Take as basic assumption of this theory that Probability of finding the system in some state with energy at tem perature T is given by 6 album a Dbservat es conwputed by averaghwg over all possible states using this probability eg Estates Ee HkT lt E gt2 Estates 6 HkT Single spin in a magnetic field Hard to do sum explicitly for many spins sim plify Consider interaction of single spin with exter nal magnetic field E 2 H5 Using statistical mechanics mean spin is given by Zszil 86 HskT lt 8 gt2 8 e HskiT leading to lt 3 gt tanh HkT Mean field approximation Go back to original 2D Ising system Replace the sum over nearest neigbor spins by the in teraction with an effective magnetic field JZZS 83 gt 4J lt 8 gt 28139 j i 139 Now solve the equation lt 5 gt tanh 4J lt s gt kT Graphically see 2 solns o T gt Tc lt 3 gt 0 o T lt Tc 2 solns One with lt 5 gt72 0 Numerical Solution Use bisection algorithm to solve for lt s gt eg To solve equation x 2 0 find 2 points 121 and m2 at which f has opposite sign Root lies somewhere between Examine the midpoint x1 x22 and hence determine new interval to search Note interesting region close to T TC 4 Close to the transition For small lt s gt use approx tanthx x33 Find Power laws Implies X lt M2 gt lt M gt2 83 18 diverges Critical exponents While mean field theory is usually quantita tively wrong the basic idea of critical exponents describing behavior near phase transitions is right a Eg Specific Heat 0 Magnetization M Magnetic susceptibility X c7T nraa0 M T Tc OOli I 15 X n 77 39 10 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 correlation function lt 5051 gt 6 15 11 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 12 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 13

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