What use is Reciprocal Space? An Introduction

What use is Reciprocal Space? An Introduction a* b* x You are here John Bargar 5th Annual SSRL Workshop on Synchrotron X-ray Scattering Techniques in Materials and Environmental Sciences June 1-3, 2010

OUTLINE I. What is the reciprocal lattice? 1. Bragg s law. 2. Reciprocal Lattice. II. How do you use it? 1. Ewald sphere. 2. Types of scans: 3. Etc.

Starting from Braggs law Bragg s Law: nλ = 2d sinθ Good phenomenologically Good enough for a Nobel prize (1915) A B θ 2θ θ A B d d BUT There are a gabillion planes in a crystal. How do we keep track of them? How do we know where in space a crystal lattice will diffract? What are their diffraction intensities?

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. Objective of this talk: show that the reciprocal lattice provides such a map

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

Now, add the points A B Define a map point at the end of the scattering vector at Bragg condition θ s s 0 λ θ Map point A B Diffraction occurs when scattering vector connects to map point. Scattering vectors (s-s 0 /λ or q) have reciprocal lengths (1/λ). Diffraction points define a reciprocal lattice. 2θ d d Reciprocal cuz units are in inverse distance Vector representation carries Bragg s law into 3D.

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. A s s 0 λ A B θ s 0 /λ s/λ B d d

Families of planes become points on a line! Parallel planes with different d-spacings have reciprocal lattice points on a line. A A Diffraction occurs when s-s 0 /λ = 1/d = 2Sinθ/λ B θ s 0 /λ s/λ B Larger d = smaller s-s 0 d 2d d

Crystals have families of planes in real space

Differently oriented planes project into different directions in reciprocal space. Thus, the RECIPROCAL LATTICE is obtained s 0 s s 0 λ Origin 1/d b* (010) (110) a* (200) s Families of planes become points! Distances between origin and RL points give 1/d. Reciprocal Lattice Axes: a* normal to b-c plane b* normal to a-c plane c* normal to a-b plane Index RL points based upon axes Diffraction occurs when s-s 0 /λ = 1/d = 2Sinθ/λ

Symmetry and peaks The symmetry of the unit cell is reflected in the reciprocal lattice, which means s 0 s s 0 λ Origin 1/d a* b* (010) (110) s the locations and number of diffraction peaks is determined by unit cell symmetry (200) But, peak intensities are determined by atomic number and atomic position in unit cell (more on this from Apurva, Joanne, and Misra/Joanna)

Reciprocal Lattice of γ-lialo 2 (008) (600) (400) (004) (200) (110) a* b* c* a* a* c* Projection along c: hk0 layer Note 4-fold symmetry Projection along b: h0l layer a = b = 5.17 Å; c = 6.27 Å; P4 1 2 1 2 (tetragonal) a* = b* = 0.19 Å -1 ; c* = 0.16 Å -1 general systematic absences (00ln; l 4), ([2n-1]00)

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

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 θ s 0 θ 2/λ s s s 0 = 1/d s 0

The Ewald Sphere An elegant way to understand / rationalize diffraction phenomena A Circumscribe circle with radius 2/λ around scattering vectors A s 0 /λ s/λ s s 0 λ =1/d s-s 0 /λ = 1/d = 2Sinθ/λ Diffraction occurs only when map point intersects circle.

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s 0 s 0 10 20 30 40

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 0 10 20 30 40 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 2θ 0 10 20 30 40

1. Longitudinal or θ-2θ scan Sample moves on θ, Detector follows on 2θ s-s 0 /λ 2θ 0 10 20 30 40 The periodicity of peaks is linear in units of Sinθ/λ - not θ! q-scanning is more efficient than θ scanning

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

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-s 0 /λ 2θ

A related point: in a powder, orientational averaging produces rings instead of spots s 0 /λ s/λ

3. Arbitrary Reciprocal Lattice scans Choose path through RL to satisfy experimental need, (Arturas will give examples) s-s 0 /λ 2θ

A note about q In practice q is used instead of s-s 0 q = k -k 0 = 2π * s-s 0 q = 4πSinθ/λ A q A B θ k 0 k θ B d 2θ d

What we haven t talked about: Intensities of peaks (Mehta/Stubbs) Peak width & shape (Stubbs) Scattering from thin films (PM session)

QUIZ (summary): 1. Symmetry of the reciprocal lattice = symmetry of the unit cell. 2. Dimensions that are large in direct space are small in reciprocal space. 3. Which is a more useful parameter: q or 2θ? Answer: q is! 4. Why? Cuz q contains info about θ and λ! 5. Peak shape has information about what? (orientation, size, strain, etc Mehta talk).

The End The Beginning