# II. How do you use it? I. What is the reciprocal lattice? OUTLINE

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## 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

(2)

### • 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?

### • 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?

d

d q

A

B q

A’

B’

2q

(3)

(4)

0

d

d q

A

B q

A’

B’

2q

0

0

### |/λ = 1/d

(Note, for those familiar with q…

q = 2π|s-s0|

Bragg’s law: q = 2π/d = 4πSinθ/ λ

s0 s

s – s0

s0 s – s0

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### Define a map point at the end of the scattering vector at Bragg condition

d

d q

A

B q

A’

B’

2q

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

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d q

A

B

A’

B’

s0s/λ

d s – s0

λ

(7)

A

.

=1/d A’

s – s0 s0/λ λ

s/λ

(8)

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 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 – s0

λ

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### Reciprocal Lattice of γ-LiAlO

2

a*

b*

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; 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)

(110)

(004) (008)

(10)

s0s/λ

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## 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

(12)

s0 s

0 10 20 30 40

(13)

### Sample moves on θ, Detector follows on 2θ

s-s0

0 10 20 30 40

Reciprocal lattice rotates by θ during

scan

(14)

s-s0

2q

0 10 20 30 40

(15)

0 10 20 30 40

s-s0

2q

(16)

2q

0 10 20 30 40

s-s0

(17)

0 10 20 30 40

s-s0/λ 2q

0 10 20 30 40

(18)

### Sample moves on θ, Detector follows on 2θ

0 10 20 30 40

s-s0

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.

2q

0 10 20 30 40

(19)

### quality of orientation

2q s-s0

First crystallite Second crystallite Third crystallite

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### quality of orientation

2q s-s0

Reciprocal lattice rotates by θ during

scan

(21)

s-s0/λ 2q

(22)

0

d

d q

A

B q

A’

B’

2q q

|k’-k0| =

0

k0 k’

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### What we haven’t talked about:

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