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) 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) 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 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 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 r i - r j Suppose n random vacancies in monatomic solid w/ N atom sites Consider vectors r i - r j

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

7. Random vacancies Rearranging:

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

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

10. Random vacancies Rearranging: scattering power of ordered structure - no defects, reduced fcont s distrib of scatt power - decreases w/ u - approx proportional to n Rearranging: scattering power of ordered structure - no defects, reduced fcont s 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 fordeviation from ordered structure Now consider Patterson Suppose n random vacancies in monatomic solid w/ N electron density fordeviation from ordered structure

12. Random vacancies Now consider Patterson Suppose n random vacancies in monatomic solid w/ N electron density fordeviation from ordered structure Now consider Patterson Suppose n random vacancies in monatomic solid w/ N electron density fordeviation from 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 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 Einstein: monatomic, independent, harmonic, 1-D Spread electron density by Gaussian  

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

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

22. No average lattice Except for origin peak, all Patterson peaks spread by 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 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 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 Use Gaussian distrib w/ mean = c

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

27. No average lattice