1. 9/28/2015PHY 711 Fall 2015 -- Lecture 151 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 15: Continue reading Chapter 4 1.Normal modes for extended one- dimensional systems 2.Normal modes for 2 and 3 dimensional systems

2. 9/28/2015PHY 711 Fall 2015 -- Lecture 152

3. 9/28/2015PHY 711 Fall 2015 -- Lecture 153 Example – linear molecule m1m1 m2m2 m3m3

4. 9/28/2015PHY 711 Fall 2015 -- Lecture 154

5. 9/28/2015PHY 711 Fall 2015 -- Lecture 155 m1m1 m2m2 m3m3 m1m1 m2m2 m3m3 m1m1 m2m2 m3m3

6. 9/28/2015PHY 711 Fall 2015 -- Lecture 156 m m m Consider an extended system of masses and springs:

7. 9/28/2015PHY 711 Fall 2015 -- Lecture 157

8. 9/28/2015PHY 711 Fall 2015 -- Lecture 158

9. 9/28/2015PHY 711 Fall 2015 -- Lecture 159

10. 9/28/2015PHY 711 Fall 2015 -- Lecture 1510

11. 9/28/2015PHY 711 Fall 2015 -- Lecture 1511

12. 9/28/2015PHY 711 Fall 2015 -- Lecture 1512 Example for N=4: Note that solution form remains correct for N  ∞

13. m m M 9/28/2015PHY 711 Fall 2015 -- Lecture 1513 m m M Consider an infinite system of masses and springs now with two kinds of masses:

14. 9/28/2015PHY 711 Fall 2015 -- Lecture 1514

15. 9/28/2015PHY 711 Fall 2015 -- Lecture 1515  qa/ 

16. 9/28/2015PHY 711 Fall 2015 -- Lecture 1516 Eigenvectors:

17. 9/28/2015PHY 711 Fall 2015 -- Lecture 1517 V(x,y) Potential in 2 and more dimensions

18. 9/28/2015PHY 711 Fall 2015 -- Lecture 1518 Example – normal modes of a system with the symmetry of an equilateral triangle 1 3 2 u1u1 u3u3 u2u2 k k k m m m

19. 9/28/2015PHY 711 Fall 2015 -- Lecture 1519 Example – normal modes of a system with the symmetry of an equilateral triangle -- continued 1 3 2 u1u1 u3u3 u2u2

20. 9/28/2015PHY 711 Fall 2015 -- Lecture 1520 Example – normal modes of a system with the symmetry of an equilateral triangle -- continued

21. 9/28/2015PHY 711 Fall 2015 -- Lecture 1521 Example – normal modes of a system with the symmetry of an equilateral triangle -- continued =  2

22. 9/28/2015PHY 711 Fall 2015 -- Lecture 1522 Example – normal modes of a system with the symmetry of an equilateral triangle -- continued 1 3 2 u1u1 u3u3 u2u2 k k k m m m

23. 9/28/2015PHY 711 Fall 2015 -- Lecture 1523 3-dimensional periodic lattices Example – face-centered-cubic unit cell (Al or Ni) Diagram of atom positions Diagram of q- space (q)

24. 9/28/2015PHY 711 Fall 2015 -- Lecture 1524 From: PRB 59 3395 (1999); Mishin et. al. (q)

25. 9/28/2015PHY 711 Fall 2015 -- Lecture 1525 Lattice vibrations for 3-dimensional lattice Example: diamond lattice Ref: http://phycomp.technion.ac.il/~nika/diamond_structure.html

26. 9/28/2015PHY 711 Fall 2015 -- Lecture 1526

27. 9/28/2015PHY 711 Fall 2015 -- Lecture 1527

28. 9/28/2015PHY 711 Fall 2015 -- Lecture 1528

29. 9/28/2015PHY 711 Fall 2015 -- Lecture 1529 B. P. Pandy and B. Dayal, J. Phys. C. Solid State Phys. 6 2943 (1973)