John Bargar 2nd Annual SSRL School on Hard X-ray Scattering Techniques in Materials and Environmental Sciences May 15-17

1. What use is Reciprocal Space? An Introduction x a* You are here b* John Bargar 2nd Annual SSRL School on Hard X-ray Scattering Techniques in Materials and Environmental Sciences May 15-17, 2007

2. NOT! The reciprocal lattice is the Fourier transform of the electron density distribution in a crystal. Scattering from a crystal occurs when the following expression evaluates to non-zero values: F

3. OUTLINE I. What is the reciprocal lattice? • Bragg’s law. • Ewald sphere. • Reciprocal Lattice. II. How do you use it? • Types of scans: Longitudinal or θ-2θ, Rocking curve scan Arbitrary reciprocal space scan

4. A’ A B’ q q B d 2q d Starting from Braggs’ law… Bragg’s Law: 2d sin q = n l • Good phenomenologically • Good enough for a Nobel prize (1915) • BUT… • There are a gabillion planes in a crystal. • How do we keep track of them? • How do we know where they will diffract (single xtals)? • What are their diffraction intensities? • BUT… • There are a gabillion planes in a crystal. • How do we keep track of them? • How do we know where they will diffract (single xtals)? • What are their diffraction intensities?

5. Better approach… • Make a “map” of the diffraction conditions of the crystal. • For example, define a map spot for each diffraction condition. • Each spot represents kajillions of parallel atomic planes. • Such a map could provide a convenient way to describe the relationships between planes in a crystal – a considerable simplification of a messy and redundant problem. • In the end, we’ll show that the reciprocal lattice provides such a map…

6. A’ s – s0 B’ q s0 2q To show this, start again from diffracting planes… Define unit vectorss0,s • Notice that |s-s0| = 2Sinθ • Substitute in Bragg’s law… • 1/d = 2Sinθ/λ… • Diffraction occurs when • |s-s0|/λ =1/d • (Note, for those familiar with q… • q = 2π|s-s0| • Bragg’s law: q= 2π/d = 4πSinθ/ λ A s – s0 s0 s q B d d

7. s – s0 λ A’ B’ q 2q To show this, start again from diffracting planes… Define a map point at the end of the scattering vector at Bragg condition Diffraction occurs when scattering vector connects to map point. Scattering vectors (s-s0/λ or q) have reciprocal lengths (1/λ). Diffraction points define a reciprocal lattice. Vector representation carries Bragg’s law into 3D. Map point A q B d d

8. s – s0 λ Families of planes become points! Single point now represents allplanes in all unit cells of the crystal that are parallel to the crystal plane of interest and have same d value. A’ A s0/λ s/λ q B B’ d d

9. Ewald Sphere Circumscribe circle with radius 2/λ around scattering vectors… A’ A Diffraction occurs only when map point intersects circle. s/λ s – s0 λ =1/d s0/λ

10. s – s0 λ Thus, the RECIPROCAL LATTICE is obtained Distances between origin and RL points give 1/d. Reciprocal Lattice Axes: a* normal to a-b plane b* normal to a-c plane c* normal to b-c plane Index RL points based upon axes Each point represents all parallel crystal planes. Eg., all planes parallel to the a-c plane are captured by (010) spot. Families of planes become points! s 1/d b* s0 (010) Origin (110) a* (200)

11. Reciprocal Lattice of γ-LiAlO2 (008) (600) (004) (400) (200) a* c* a* a* b* c* (200) (400) (600) (110) (020) (040) (060) Projection along c: hk0 layer Note 4-fold symmetry Projection along b: h0l layer a = b = 5.17 Å; c = 6.27 Å; P41212 (tetragonal) a* = b* = 0.19 Å-1; c* = 0.16 Å-1 general systematic absences (00ln; l≠4), ([2n-1]00)

12. In a powder, orientational averaging produces rings instead of spots s/λ s0/λ

13. OUTLINE I. What is the reciprocal lattice? • Bragg’s law. • Ewald sphere. • Reciprocal Lattice. II. How do you use it? • Types of scans: Longitudinal or θ-2θ, Rocking curve scan Arbitrary reciprocal space scan

14. 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s0 s

15. 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s0/λ Reciprocal lattice rotates by θ during scan

16. 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s0/λ 2q

17. 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s0/λ 2q

18. 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s0/λ 2q

19. 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s0/λ 2q

20. 1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s0/λ 2q • Note scan is linear in units of Sinθ/λ - not θ! • Provides information about relative arrangements, angles, and spacings between crystal planes.

21. 2. Rocking Curve scan Sample moves on θ, Detector fixed Provides information on sample mosaicity & quality of orientation First crystallite Second crystallite Third crystallite s-s0/λ 2q

22. 2. Rocking Curve scan Sample moves on θ, Detector fixed Provides information on sample mosaicity & quality of orientation Reciprocal lattice rotates by θ during scan s-s0/λ 2q

23. 3. Arbitrary Reciprocal Lattice scans Choose path through RL to satisfy experimental need, e.g., CTR measurements s-s0/λ 2q

24. A’ B’ q 2q A note about “q” In practice q is used instead of s-s0 |q| = |k’-k0| = 2π * |s-s0| |q| = 4πSinθ/λ q A k0 k’ q B d d

25. What we haven’t talked about: • Intensities of peaks (Vailionis) • Peak width & shape (Vailionis) • Scattering from non-crystalline materials (Huffnagel) • Scattering from whole particles or voids (Pople) • Scattering from interfaces (Trainor)

26. The End The Beginning…

27. q = 1/d 2/λ Graphical Representation of Bragg’s Law • Bragg’s law is obeyed for any triangle inscribed within the circle: Sinθ = (1/d)/(2/λ) A’ A q s0 s s – s0 s0