John Bargar2nd Annual SSRL School on Hard X-ray Scattering Techniques in

Materials and Environmental SciencesMay 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!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 s0, s

d

d

A

B

A’

B’

• 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

s0 ss – s0

s0

s – s0

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 – s0

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/λ s/λ

d

s – s0

λ

Ewald Sphere

A

Diffraction occurs only when map point intersects circle.

=1/d

A’

s – s0

λ s0/λ

s/λ

Circumscribe circle with radius 2/λ around scattering vectors…

Origin

s0

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 planeb* normal to a-c planec* 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 – s0

λ

Reciprocal Lattice of γ-LiAlO2

a*b*

a*

c*

Projection along c: hk0 layerNote 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)

c*a*

(200)

(400)

(600)

(020

) (0

40)

(060

) (110)

(200

) (4

00)

(600

)

(004)

(008)

In a powder, orientational averaging produces rings instead of spots

s0/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 scanSample moves on θ, Detector follows on 2θ

s0 s

0 10 20 30 40

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

s-s0/λ

0 10 20 30 40

Reciprocal lattice rotates by θ during

scan

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

s-s0/λ

0 10 20 30 40

0 10 20 30 40

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

s-s0/λ

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

0 10 20 30 40

s-s0/λ

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

0 10 20 30 40

s-s0/λ

0 10 20 30 40

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

0 10 20 30 40

s-s0/λ

0 10 20 30 40

• Note scan is linear in units of Sinθ/λ - not θ!• Provides information about relative arrangements, angles, and

spacings between crystal planes.

0 10 20 30 40

2. Rocking Curve scanSample moves on θ, Detector fixed

Provides information on sample mosaicity & quality of orientation

s-s0/λFirst crystallite

Second crystallite

Third crystallite

2. Rocking Curve scanSample moves on θ, Detector fixed

Provides information on sample mosaicity & quality of orientation

s-s0/λ

Reciprocal lattice rotates by θ during

scan

3. Arbitrary Reciprocal Lattice scansChoose path through RL to satisfy experimental need,

e.g., CTR measurements

s-s0/λ

A note about “q”

In practice q is used instead of s-s0

d

d

A

B

A’

B’

q

|q| = |k’-k0| = 2π * |s-s0|

|q| = 4πSinθ/λ

k0 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 – s0 s0 s

s0

= 1/d

2/λ