1. X-ray diffraction

2. Braggs' law = 2d hkl sin  hkl X-ray diffraction From this set of planes, only get reflection at one angle -  From this set of planes, only get reflection at one angle -  set of lattice planes d  X-rays d sin 

3. Braggs' law = 2d hkl sin  hkl X-ray diffraction another set of lattice planes X-rays

4. Braggs' law = 2d hkl sin  hkl X-ray diffraction another set of lattice planes X-rays Different sets of planes reflect at different  angles As d —> small,  —> large ( is constant) As d —> small,  —> large ( is constant)  

5. Ewald construction Think of set of planes reflecting in x-ray beam

6. Center sphere on specimen origin x-ray beam is a sphere diameter Think of set of planes reflecting in x-ray beam Center sphere on specimen origin x-ray beam is a sphere diameter Ewald construction

7. Think of set of planes reflecting in x-ray beam Center sphere on specimen origin x-ray beam is a sphere diameter Construct lines as below Think of set of planes reflecting in x-ray beam Center sphere on specimen origin x-ray beam is a sphere diameter Construct lines as below Ewald construction

8. Think of set of planes reflecting in x-ray beam Center sphere on specimen origin x-ray beam is a sphere diameter Construct lines as below Think of set of planes reflecting in x-ray beam Center sphere on specimen origin x-ray beam is a sphere diameter Construct lines as below Ewald construction

16. Most common in single crystal studies is to move (usually rotate) crystal Consider crystal placed at sphere center oriented w/ planes of points in reciprocal lattice as below Most common in single crystal studies is to move (usually rotate) crystal Consider crystal placed at sphere center oriented w/ planes of points in reciprocal lattice as below Ewald construction

17. Looking down on one plane of points.... the equatorial plane: Looking down on one plane of points.... the equatorial plane: Ewald construction

18. Looking down on one plane of points.... the equatorial plane No points on sphere (here, in 2-D, a circle); must rotate reciprocal lattice to observe reflections. Looking down on one plane of points.... the equatorial plane No points on sphere (here, in 2-D, a circle); must rotate reciprocal lattice to observe reflections. rotate around axis here, perpendicular to screen rotate around axis here, perpendicular to screen Ewald construction

19. Looking down on one plane of points.... the equatorial plane Must rotate reciprocal lattice to observe reflections. Looking down on one plane of points.... the equatorial plane Must rotate reciprocal lattice to observe reflections. rotate around axis here, perpendicular to screen rotate around axis here, perpendicular to screen Ewald construction

20. Looking down on one plane of points.... the equatorial plane Must rotate reciprocal lattice to observe reflections. Looking down on one plane of points.... the equatorial plane Must rotate reciprocal lattice to observe reflections. rotate around axis here, perpendicular to screen rotate around axis here, perpendicular to screen Ewald construction

21. Looking down on one plane of points.... the equatorial plane Must rotate reciprocal lattice to observe reflections. Looking down on one plane of points.... the equatorial plane Must rotate reciprocal lattice to observe reflections. rotate around axis here, perpendicular to screen rotate around axis here, perpendicular to screen Ewald construction

22. hk0 reflected rays all lie in the equatorial plane. Ewald construction

23. hk0 reflected rays all lie in the equatorial plane. hk1 reflected rays lie in a cone. hk0 reflected rays all lie in the equatorial plane. hk1 reflected rays lie in a cone. Ewald construction

24. hk0 reflected rays all lie in the equatorial plane. hk1 reflected rays lie in a cone. hk0 reflected rays all lie in the equatorial plane. hk1 reflected rays lie in a cone. Ewald construction

25. Sheet of film or image paper wrapped cylindrically around crystal.... Sheet of film or image paper wrapped cylindrically around crystal.... Ewald construction

26. Sheet of film or image paper wrapped cylindrically around crystal.... looks like this after x-ray exposure of oscillating crystal.....when flattened: Sheet of film or image paper wrapped cylindrically around crystal.... looks like this after x-ray exposure of oscillating crystal.....when flattened: Ewald construction

27. To see reflections: move sphere move crystal change sphere size To see reflections: move sphere move crystal change sphere size Ewald construction real spacereciprocal space only one set of planes - one (hkl) only one set of planes - one (hkl) use polycrystalline sample

28. To see reflections: move sphere move crystal change sphere size To see reflections: move sphere move crystal change sphere size Ewald construction real spacereciprocal space only one set of planes - one (hkl) only one set of planes - one (hkl) use polycrystalline sample

29. Ewald construction reciprocal lattice representation reciprocal lattice representation Ewald sphere

30. Ewald construction