Introduction

1. Introduction From 1912 to 1980s, 1990s activities in x-ray diffraction & scattering mainly centered around determination of crystal structures through, usually, single crystal techniques

2. Introduction From 1912 to 1980s, 1990s activities in x-ray diffraction & scattering mainly centered around determination of crystal structures through, usually, single crystal techniques identification and quantitative analysis of unknowns, principally using powder diffraction techniques.

3. Introduction From 1912 to 1980s, 1990s activities in x-ray diffraction & scattering mainly centered around determination of crystal structures through, usually, single crystal techniques identification and quantitative analysis of unknowns, principally using powder diffraction techniques. still important & challenging activities

4. Introduction Recently x-ray diffraction techniques increasingly applied to more difficult & complex areas, likely because of a. need to solve new types of (non-routine) problems (e.g., disordered systems, interface structures, nano materials, quasiperiodicity)

5. Introduction Recently x-ray diffraction techniques increasingly applied to more difficult & complex areas, likely because of a. need to solve new types of (non-routine) problems (e.g., disordered systems, interface structures, nano materials, quasiperiodicity) b. need to examine structures new types of materials (e.g., incommensurate materials, polymers, quasicrystals)

6. Introduction Recently x-ray diffraction techniques increasingly applied to more difficult & complex areas, likely because of a. need to solve new types of (non-routine) problems (e.g., disordered systems, interface structures, nano materials, quasiperiodicity) b. need to examine structures new types of materials (e.g., incommensurate materials, polymers, quasicrystals) c. development of better instrumentation d. availability of highly intense sources (synchrotron) e. availability of immense computing power

7. Introduction To understand how to apply x-ray diffraction to these “new” areas (with, probably, many yet to come) need to examine physics & mathematics of diffraction from basic, rather general standpoint development of Cowley will be followed - Fourier series, Fourier transforms, & convolutions employed

8. Introduction Example calculation of crystallite size & microstrain from breadths of diffraction peaks (this one is "old") (see Warren - X-ray Diffraction)

9. Introduction Example calculation of crystallite size & microstrain from breadths of diffraction peaks (this one is "old") (see Warren - X-ray Diffraction) shape of diffraction peak determined by a. instrument source size and geometry b. small crystallite size c. non-homogeneous strain in the structure

10. Introduction • Example • shape of diffraction peak determined by • a. instrument source size and geometry • b. small crystallite size • c. non-homogeneous strain in the structure • before size/microstrain analysis, instrumental contribution to observed diffraction maxima must be removed by deconvolution

11. Introduction • Example • shape of diffraction peak determined by • a. instrument source size and geometry • b. small crystallite size • c. non-homogeneous strain in the structure • before size/microstrain analysis, instrumental contribution to observed diffraction maxima must be removed by deconvolution • in so-called Warren-Averbach method, deconvolution of observed maximum h(x) • h(x) = ∫(g(z) f(x-z) dz • performed by applying Stokes correction - results in representation of size/strain broadened peak by a series of Fourier coefficients

12. Introduction • Example (see Introduction to Quasicrystals, Jaric, ed.)