Scattering from imperfect crystals (see Cowley Sect. 7.1)

1. Scattering from imperfect crystals (see Cowley Sect. 7.1) Two types average lattice exists (point defects, dislocations, thermal vibrations) no average lattice (stacking faults, twinning)

2. Scattering from imperfect crystals (see Cowley Sect. 7.1) Two types average lattice exists (point defects, dislocations, thermal vibrations) no average lattice (stacking faults, twinning)

3. Scattering from imperfect crystals (see Cowley Sect. 7.1) Two types can't put in all atoms; consider average of atoms surrounding particular set of N atoms

4. Scattering from imperfect crystals Two types can't put in all atoms; consider average of atoms surrounding particular set of N atoms for a monatomic solid

5. Random vacancies (see Cowley Sect. 7.3) Suppose n random vacancies in monatomic solid w/ N atom sites Consider vectors ri- rj

6. Random vacancies (see Cowley Sect. 7.3) Suppose n random vacancies in monatomic solid w/ N atom sites Consider vectors ri- rj

7. Random vacancies Rearranging:

8. Random vacancies Rearranging: scattering power of ordered structure - no defects, reduced f

9. Random vacancies Rearranging: scattering power of ordered structure - no defects, reduced f conts distrib of scatt power - decreases w/ u - approx proportional to n

10. Random vacancies Rearranging: scattering power of ordered structure - no defects, reduced f conts distrib of scatt power - decreases w/ u - approx proportional to n

11. Random vacancies Now consider Patterson Suppose n random vacancies in monatomic solid w/ N electron density for deviation from ordered structure ordered structure

12. Random vacancies Now consider Patterson Suppose n random vacancies in monatomic solid w/ N electron density for deviation from ordered structure ordered structure

13. Random vacancies

14. Vacancy clusters Use Patterson approach again

15. Interstitials Assume n small interstitials w/ negligible scattering power Average structure is

16. Interstitials Assume n small interstitials w/ negligible scattering power

17. Interstitials Assume n small interstitials w/ negligible scattering power

18.  Thermal vibrations Einstein: monatomic, independent, harmonic, 1-D Spread electron density by Gaussian

19.  Thermal vibrations Except for origin peak, all Patterson peaks spread by

20. Thermal vibrations Except for origin peak, all Patterson peaks spread by Intensity is

21. Thermal vibrations Except for origin peak, all Patterson peaks spread by Intensity is

22. No average lattice Except for origin peak, all Patterson peaks spread by

23. No average lattice Except for origin peak, all Patterson peaks spread by d(z) is a set of fcns (r) considered periodic in x,y

24. No average lattice Except for origin peak, all Patterson peaks spread by d(z) is a set of fcns (r) considered periodic in x,y reciprocal lattice

25. No average lattice Use Gaussian distrib w/ mean = c

26. No average lattice Use Gaussian distrib w/ mean = c

27. No average lattice

28. No average lattice

29. No average lattice

30. No average lattice