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

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(1)

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)

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 q = n l

d

d q

A

B q

A’

B’

2q

• Good phenomenologically

• Good enough for a Nobel

prize (1915)

(3)

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…

(4)

To show this, start again from diffracting planes…

Define unit vectors s

0

, s

d

d q

A

B q

A’

B’

2q

• 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-s0|

Bragg’s law: q = 2π/d = 4πSinθ/ λ

s0 s

s – s0

s0 s – s0

(5)

To show this, start again from diffracting planes…

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

(6)

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 q

A

B

A’

B’

s0s/λ

d s – s0

λ

(7)

Ewald Sphere

A

Diffraction occurs only when map point intersects circle

.

=1/d A’

s – s0 s0/λ λ

s/λ

Circumscribe circle with radius 2/λ around

scattering vectors…

(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

(9)

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)

In a powder, orientational averaging produces rings instead of spots

s0s/λ

(11)

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)

1. Longitudinal or θ-2θ scan

Sample moves on θ, Detector follows on 2θ

s0 s

0 10 20 30 40

(13)

1. Longitudinal or θ-2θ scan

Sample moves on θ, Detector follows on 2θ

s-s0

0 10 20 30 40

Reciprocal lattice rotates by θ during

scan

(14)

1. Longitudinal or θ-2θ scan

Sample moves on θ, Detector follows on 2θ

s-s0

2q

0 10 20 30 40

(15)

0 10 20 30 40

1. Longitudinal or θ-2θ scan

Sample moves on θ, Detector follows on 2θ

s-s0

2q

(16)

1. Longitudinal or θ-2θ scan

Sample moves on θ, Detector follows on 2θ

2q

0 10 20 30 40

s-s0

(17)

1. Longitudinal or θ-2θ scan

Sample moves on θ, Detector follows on 2θ

0 10 20 30 40

s-s0/λ 2q

0 10 20 30 40

(18)

1. Longitudinal or θ-2θ scan

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)

2. Rocking Curve scan

Sample moves on θ, Detector fixed

Provides information on sample mosaicity &

quality of orientation

2q s-s0

First crystallite Second crystallite Third crystallite

(20)

2. Rocking Curve scan

Sample moves on θ, Detector fixed

Provides information on sample mosaicity &

quality of orientation

2q s-s0

Reciprocal lattice rotates by θ during

scan

(21)

3. Arbitrary Reciprocal Lattice scans

Choose path through RL to satisfy experimental need, e.g., CTR measurements

s-s0/λ 2q

(22)

A note about “q”

In practice q is used instead of s-s

0

d

d q

A

B q

A’

B’

2q q

|q| =

|k’-k0| =

2π * |s-s

0

|

|q| = 4πSinθ/λ

k0 k’

(23)

• 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:

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