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

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 NOT!

OUTLINE I. What is the reciprocal lattice? 1. Bragg’s law. 2. Ewald sphere. 3. Reciprocal Lattice. II. How do you use it? 1. Types of scans: Longitudinal or θ-2θ, Rocking curve scan Arbitrary reciprocal space scan

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? Starting from Braggs’ law… Bragg’s Law: 2d sin  = n d d  A B  A’ B’  Good phenomenologically Good enough for a Nobel prize (1915)

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…

To show this, start again from diffracting planes… Define unit vectors s 0, s d d  A B  A’ B’  Notice that |s-s 0 | = 2Sinθ Substitute in Bragg’s law… 1/d = 2Sinθ/λ … Diffraction occurs when |s-s 0 |/λ = 1/d (Note, for those familiar with q… q = 2π|s-s 0 | Bragg’s law: q = 2π/d = 4πSinθ/ λ s0s0 s s – s 0 s0s0

To show this, start again from diffracting planes… Define a map point at the end of the scattering vector at Bragg condition d d  A B  A’ B’  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 s – s 0 λ

Families of planes become points! Single point now represents all planes in all unit cells of the crystal that are parallel to the crystal plane of interest and have same d value. d  A B A’ B’ s0/λs0/λ s/λs/λ d s – s 0 λ

Ewald Sphere A Diffraction occurs only when map point intersects circle. =1/d A’ s – s 0 λ s0/λs0/λ s/λs/λ Circumscribe circle with radius 2/λ around scattering vectors…

Origin s0s0 s Thus, the RECIPROCAL LATTICE is obtained 1/d 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! b* a* (110) (010) (200) s – s 0 λ

Reciprocal Lattice of γ-LiAlO 2 a* b* a* c* Projection along c: hk0 layer Note 4-fold symmetry Projection along b: h0 l layer a = b = 5.17 Å; c = 6.27 Å; P (tetragonal) a* = b* = 0.19 Å -1 ; c* = 0.16 Å -1 general systematic absences (00 l n; l ≠4), ([2n-1]00) c* a* (200) (400) (600) (020) (040)(060) (110) (200)(400)(600) (004) (008)

In a powder, orientational averaging produces rings instead of spots s0/λs0/λ s/λs/λ

OUTLINE I. What is the reciprocal lattice? 1. Bragg’s law. 2. Ewald sphere. 3. Reciprocal Lattice. II. How do you use it? 1. Types of scans: Longitudinal or θ-2θ, Rocking curve scan Arbitrary reciprocal space scan

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s0s0 s

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ Reciprocal lattice rotates by θ during scan

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ  s-s 0 /λ

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ Note scan is linear in units of Sinθ/λ - not θ! Provides information about relative arrangements, angles, and spacings between crystal planes. 

2. Rocking Curve scan Sample moves on θ, Detector fixed Provides information on sample mosaicity & quality of orientation  s-s 0 /λ First crystallite Second crystallite Third crystallite

2. Rocking Curve scan Sample moves on θ, Detector fixed Provides information on sample mosaicity & quality of orientation  s-s 0 /λ Reciprocal lattice rotates by θ during scan

3. Arbitrary Reciprocal Lattice scans Choose path through RL to satisfy experimental need, e.g., CTR measurements s-s 0 /λ 

A note about “q” In practice q is used instead of s-s 0 d d  A B  A’ B’  q |q| = |k’-k 0 | = 2π * |s-s 0 | |q| = 4πSinθ/λ k0k0 k’

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) What we haven’t talked about:

The End The Beginning…

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’ s – s 0 s0s0 s s0s0  = 1/d 2/λ