1. 2/20/2014PHY 770 Spring 2014 -- Lecture 121 PHY 770 -- Statistical Mechanics 12:00-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage: http://www.wfu.edu/~natalie/s14phy770http://www.wfu.edu/~natalie/s14phy770 Lecture 12 -- Chapter 5 Equilibrium Statistical Mechanics Canonical ensemble  Magnetic effects  Ising model

2. 2/20/2014PHY 770 Spring 2014 -- Lecture 122

3. 2/20/2014PHY 770 Spring 2014 -- Lecture 123 Summary of results for the canonical ensemble

4. 2/20/2014PHY 770 Spring 2014 -- Lecture 124 Canonical ensemble – example including magnetic effects Consider a system of N particles in a box of volume V, treated in the semiclassical limit. Since we are in the semiclassical limit, we can use the relationship

5. 2/20/2014PHY 770 Spring 2014 -- Lecture 125 Canonical ensemble – example including magnetic effects -- continued

6. 2/20/2014PHY 770 Spring 2014 -- Lecture 126 Canonical ensemble – example including magnetic effects -- continued B=1 B=5 B=10

7. 2/20/2014PHY 770 Spring 2014 -- Lecture 127 Canonical ensemble – example including magnetic effects -- continued

8. 2/20/2014PHY 770 Spring 2014 -- Lecture 128 Statistical mechanics of the Ising model Spin ½ system with competing effects of nearest-neighbor interactions and spin alignment energy in a magnetic field

9. 2/20/2014PHY 770 Spring 2014 -- Lecture 129 Calculation of Z 1 Thermodynamic functions:

10. 2/20/2014PHY 770 Spring 2014 -- Lecture 1210 Statistical mechanics of the Ising model -- continued Spin ½ system with competing effects of nearest-neighbor interactions and spin alignment energy in a magnetic field 37 1 2 6 54            

11. 2/20/2014PHY 770 Spring 2014 -- Lecture 1211 Ising model – effects of interaction term alone for N=2;  ij = 

12. 2/20/2014PHY 770 Spring 2014 -- Lecture 1212 Ising model -- full model for N=2;  ij = 

13. 2/20/2014PHY 770 Spring 2014 -- Lecture 1213 Partition function for 1-dimensional Ising system of N spins with periodic boundary conditions (s N+1 =s 1 )

14. 2/20/2014PHY 770 Spring 2014 -- Lecture 1214 1-dimensional Ising system of N spins with periodic boundary conditions (s N+1 =s 1 ) (continued)

15. 2/20/2014PHY 770 Spring 2014 -- Lecture 1215 1-dimensional Ising system of N spins with periodic boundary conditions (s N+1 =s 1 ) (continued)

16. 2/20/2014PHY 770 Spring 2014 -- Lecture 1216 1-dimensional Ising system of N spins with periodic boundary conditions (s N+1 =s 1 ) (continued)

17. 2/20/2014PHY 770 Spring 2014 -- Lecture 1217 1-dimensional Ising system of N spins with periodic boundary conditions (s N+1 =s 1 ) (continued)       B 

18. 2/20/2014PHY 770 Spring 2014 -- Lecture 1218 1-dimensional Ising system of N spins with periodic boundary conditions (s N+1 =s 1 ) (continued)

19. 2/20/2014PHY 770 Spring 2014 -- Lecture 1219  B   =0  =1 

20. 2/20/2014PHY 770 Spring 2014 -- Lecture 1220 Mean field approximation for 1-dimensional Ising model

21. 2/20/2014PHY 770 Spring 2014 -- Lecture 1221 Mean field partition function and Free energy:

22. 2/20/2014PHY 770 Spring 2014 -- Lecture 1222 s

23. 2/20/2014PHY 770 Spring 2014 -- Lecture 1223  =1  =1 BB One dimensional Ising model with periodic boundary conditions:

24. 2/20/2014PHY 770 Spring 2014 -- Lecture 1224 Extension of mean field analysis to more complicated geometries

25. 2/20/2014PHY 770 Spring 2014 -- Lecture 1225 Extension of mean field analysis to more complicated geometries -- continued Mean field partition function and Free energy:

26. 2/20/2014PHY 770 Spring 2014 -- Lecture 1226 Extension of mean field analysis to more complicated geometries -- continued

27. 2/20/2014PHY 770 Spring 2014 -- Lecture 1227 Extension of mean field analysis to more complicated geometries -- continued

28. 2/20/2014PHY 770 Spring 2014 -- Lecture 1228 Extension of mean field analysis to more complicated geometries -- continued

29. 2/20/2014PHY 770 Spring 2014 -- Lecture 1229 Extension of mean field analysis to more complicated geometries -- continued

30. 2/20/2014PHY 770 Spring 2014 -- Lecture 1230 Extension of mean field analysis to more complicated geometries -- continued