**OUTLINE**

**I. What is the reciprocal lattice?**

**I. What is the reciprocal lattice?**

1. Bragg’s law.

2. Ewald sphere.

3. Reciprocal Lattice.

**II. How do you use it?**

**II. How do you use it?**

1. Types of scans:

Longitudinal or θ-2θ, Rocking curve scan

Arbitrary reciprocal space scan

**BUT…**

**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…**

**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…**

**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)

**Better approach…**

**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.

**• Such a map could provide a convenient way to describe**

**the relationships between planes in a crystal**

### In the end, we’ll show that the reciprocal lattice provides such

### a map…

**To show this, start again from diffracting ** **planes…**

**To show this, start again from diffracting**

**planes…**

**Define unit vectors s**

**Define unit vectors s**

_{0}**, s**

**, s**

d

d q

**A**

**B** q

**A’**

**B’**

2q

**• Notice that |s-s**

**• Notice that |s-s**

_{0}### | = 2Sinθ

### • Substitute in Bragg’s law…

### 1/d = 2Sinθ/λ …

*Diffraction occurs when*

**|s-s**

**|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**

**s**_{0}**s**

**s – s**_{0}

**s**_{0}**s – s**_{0}

**To show this, start again from diffracting ** **planes…**

**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 – s**_{0}

λ

**Families of planes become points!**

**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.

**Single point now represents all planes in all**

**unit cells of the crystal that are parallel to**

d q

**A**

**B**

**A’**

**B’**

**s***_{0}*/λ

**s/λ**d
**s – s**_{0}

λ

**Ewald Sphere**

**Ewald Sphere**

**A**

**Diffraction occurs only ** **when map point ** **intersects circle**

**Diffraction occurs only**

**when map point**

**intersects circle**

*.*

=1/d
**A’**

**s – s**_{0}**s***_{0}*/λ λ

**s/λ**

### Circumscribe circle with radius 2/λ around

### scattering vectors…

Origin
**s**_{0}

**s**

**Thus, the RECIPROCAL LATTICE is obtained**

**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**

**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; P4**_{1}**2**_{1}**2 (tetragonal)**
**a* = b* = 0.19 Å**^{-1}**; c* = 0.16 Å**^{-1}

**general systematic absences (00****ln; l≠4)****, ([2n-1]00)**
**c*** **a***

(200) (400) (600)

(110)

(004) (008)

**In a powder, orientational averaging ** **produces rings instead of spots **

**In a powder, orientational averaging**

**produces rings instead of spots**

**s***_{0}*/λ

**s/λ****OUTLINE**

**I. What is the reciprocal lattice?**

**I. What is the reciprocal lattice?**

1. Bragg’s law.

2. Ewald sphere.

3. Reciprocal Lattice.

**II. How do you use it?**

**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**

**1. Longitudinal or θ-2θ scan**

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

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

**s**_{0}**s**

0 10 20 30 40

**1. Longitudinal or θ-2θ scan**

**1. Longitudinal or θ-2θ scan**

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

**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**

**1. Longitudinal or θ-2θ scan**

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

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

**s-s***_{0}*/λ

2q

0 10 20 30 40

0 10 20 30 40

**1. Longitudinal or θ-2θ scan**

**1. Longitudinal or θ-2θ scan**

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

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

**s-s***_{0}*/λ

2q

**1. Longitudinal or θ-2θ scan**

**1. Longitudinal or θ-2θ scan**

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

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

2q

0 10 20 30 40

**s-s***_{0}*/λ

**1. Longitudinal or θ-2θ scan**

**1. Longitudinal or θ-2θ scan**

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

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

0 10 20 30 40

**s-s***_{0}*/λ
2q

0 10 20 30 40

**1. Longitudinal or θ-2θ scan**

**1. Longitudinal or θ-2θ scan**

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

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

0 10 20 30 40

**s-s***_{0}*/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. **

2q

0 10 20 30 40

**2. Rocking Curve scan**

**2. Rocking Curve scan**

**Sample moves on θ, Detector fixed**

**Sample moves on θ, Detector fixed**

**Provides information on sample mosaicity & **

**Provides information on sample mosaicity &**

**quality of orientation **

**quality of orientation**

2q
**s-s***_{0}*/λ

**First crystallite**
**Second crystallite**
**Third crystallite**

**2. Rocking Curve scan**

**2. Rocking Curve scan**

**Sample moves on θ, Detector fixed**

**Sample moves on θ, Detector fixed**

**Provides information on sample mosaicity & **

**Provides information on sample mosaicity &**

**quality of orientation **

**quality of orientation**

2q
**s-s***_{0}*/λ

Reciprocal lattice rotates by θ during

scan

**3. Arbitrary Reciprocal Lattice scans**

**3. Arbitrary Reciprocal Lattice scans**

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

**Choose path through RL to satisfy experimental need,**

**e.g., CTR measurements**

**s-s***_{0}*/λ
2q

**A note about “q”**

**A note about “q”**

**In practice q is used instead of s-s**

**In practice q is used instead of s-s**

_{0}d

d q

**A**

**B** q

**A’**

**B’**

2q
**q **

**|q| = **

|k’-k**|q| =**

*| =*

_{0}### 2π * |s-s

_{0}### |

**|q| = 4πSinθ/λ**

**|q| = 4πSinθ/λ**

**k**_{0}**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:**

**What we haven’t talked about:**