**What is a crystal? **

**What is a crystal?**

**Originally from Greek: CRYSTAL – NATURAL ICE **

**Originally from Greek: CRYSTAL – NATURAL ICE**

Visit www.snowcrystals.com for your own pleasure

**What is a crystal? **

**What is a crystal?**

**Originally from Greek: CRYSTAL – NATURAL ICE **

**Originally from Greek: CRYSTAL – NATURAL ICE**

Visit www.snowcrystals.com for your own pleasure

**Minerals **

**Minerals**

**Minerals are natural solids formed as a result of the certain geological processes ** **Minerals ** **are** ** the largest source of naturally formed crystalline solids **

**Minerals are natural solids formed as a result of the certain geological processes**

**Minerals**

**are**

**the largest source of naturally formed crystalline solids**

**QUARTZ, SiO**

**QUARTZ, SiO**

_{2}http://webmineral.com

**Mackayite, Fe** _{3} **Te** _{2} **O** _{5} **(OH) **

**Mackayite, Fe**

_{3}**Te**

_{2}**O**

_{5}**(OH)**

http://webmineral.com

**“Life” example of anisotropic physical properties **

**“Life” example of anisotropic physical properties**

**Cutting a scarf is a typical example of the directional dependence **

**The reason for that is the special STRUCTURE made by the stitching **

**BONDS **

~ 1 mm=10^{-3} m

**Hypothesis of Pierre Curie – anisotropy of crystals is due to the ** **periodic structure **

6

**Laue diffraction patterns **

**Laue diffraction patterns**

a-Quartz crystals (SiO_{2})

**Bragg peaks **

**1 Dimensional crystal (1D periodic structures) **

**1 Dimensional crystal (1D periodic structures)**

x

**Unit cell **

**Crystal lattice **

**Lattice vectors **

**A=na **

**To obtain the whole crystal structure one has to **

**To obtain the whole crystal structure one has to**

**translate the UNIT CELL to each LATTICE POINT **

**translate the UNIT CELL to each LATTICE POINT**

**Different choices of unit cell **

**Different choices of unit cell**

**Unit cell **

**Crystal lattice **

**+ **

**Unit cell **

**Crystal lattice **

**+ **

**2 Dimensional crystal (2D periodic structures) **

**2 Dimensional crystal (2D periodic structures)**

+
**Unit cell **

**Crystal lattice **

**b **

**A**_{uv}=ua+vb = [uv]

**BASIS VECTORS and CRYSTAL LATTICE PARAMETERS **

**BASIS VECTORS and CRYSTAL LATTICE PARAMETERS**

**a ****b **

Lattice parameters for two dimensional case: a=|a|, b=|b|, a= (a,b) For the given example: a=1.5, b=1, a=80 deg

**Different choices of basis vectors and lattice ** **parameters **

**Different choices of basis vectors and lattice**

**parameters**

**a ****b **

**a ****b ****a **

**b **

There is a freedom of choice of the lattice basis vectors and therefore lattice parameters

**a ****b **

**Building a lattice : choice of basis vectors 1 **

**Building a lattice : choice of basis vectors 1**

**a ****b **

**Building a lattice : choice of basis vectors 2 **

**Building a lattice : choice of basis vectors 2**

**Building a lattice : choice of basis vectors 3 **

**Building a lattice : choice of basis vectors 3**

**a ****b **

**Theorem about the choice of basis vectors **

**Theorem about the choice of basis vectors**

**a ****b **

**A**_{1 }**A**_{2 }

Consider the lattice built with two basis vectors, a and b

Take two other lattice vectors
A_{1}=[u_{1}v_{1}]=u_{1}**a+v**_{1}**b **

A_{2}=[u_{2}v_{2}]=u_{2}**a+v**_{2}* b *
u

_{1},u

_{2},v

_{1}and v

_{2}are integer Does this new pair of vectors build the same lattice???

It is necessary to provide that the area, S of the parallelogram built on a and b is the same as
the area of parallelogram built on A_{1} and A_{2 }

1 1

1 2

2 2

( , ) *u* *v* ( , )

*S* *S*

*u* *v*

**A A** **a b** ^{1} ^{1}

2 2

1 (1)

*u* *v*

*u* *v*

If equation (1) is fulfilled the pair of vectors A_{1} and A_{2} can be chosen as basis vectors for
the SAME LATTICE

**Lattice rows (2D) / Lattice planes (3D) **

**Lattice rows (2D) / Lattice planes (3D)**

**Lattice rows (2D) / Lattice planes (3D) **

**Lattice rows (2D) / Lattice planes (3D)**

**a **

**b **

**d **

**Lattice rows (2D) / Lattice planes (3D) **

**Lattice rows (2D) / Lattice planes (3D)**

**a ****b **

**d **

**Lattice rows (2D) / Lattice planes (3D) **

**Lattice rows (2D) / Lattice planes (3D)**

**a **

**b **

**d **

**Lattice rows (2D) / Lattice planes (3D) **

**Lattice rows (2D) / Lattice planes (3D)**

**a **

**d **

**b **

**Examples. MILLER INDICES AND LATTICE PLANES **

**Examples. MILLER INDICES AND LATTICE PLANES**

**a ****b **

**a ****b ****a **

**b **

**a ****b **

**a ****b **

1. There is a plane (number N) intersecting the main axes a and b in points [1,0] and [0,2].

2. According to equation of this plane h=N and k=N/2.

3. The mutually prime h and k are obtained by taking N=2. We get h=2 and k=1.

**How do you calculate the Miller indices of the given set of lattice planes? **

**Examples. MILLER INDICES AND CRYSTAL MORPHOLOGY **

**Examples. MILLER INDICES AND CRYSTAL MORPHOLOGY**

### According to the original idea of Haüy the faces of a crystal are parallel to the lattice planes. Now we can characterize the crystal faces in terms of

### the Miller indices.

We take a lattice and construct a polyhedron from the different number of faces

**Examples. MILLER INDICES AND CRYSTAL MORPHOLOGY **

**Examples. MILLER INDICES AND CRYSTAL MORPHOLOGY**

**The angles between faces: how do they depend on the ** **crystal lattice **

**The angles between faces: how do they depend on the**

**crystal lattice**

**B**_{1 }

**B**_{2 }

The face with the Miller indices (hkl) is perpendicular to the reciprocal lattice vector B=[hkl]*=ha*+kb*+lc*

The angle between faces a_{12} = < (B_{1},B_{2})

1 2 12

1 2

### ( )

### cos a | **B B** || |

**B** **B**

**Example: the polyhedral shape of a 2D for two different ** **crystal lattices **

**Example: the polyhedral shape of a 2D for two different**

**crystal lattices**

### a=1, b=1, a=60 deg a=1, b=1, a=70 deg

### The angles between the natural faces of a crystals are defined by the crystal lattice parameters. This is the background for the law of

### constancy of the interfacial angles

**Entstehung von Beuigungsmaxima in periodischen Systemen **

**Entstehung von Beuigungsmaxima in periodischen Systemen**

**2θ **

2
**2θ**

1

*d * 3

4

5

### Periodische Strukturen zeigen ähnliche

### Beugungserscheinungen, wie eine Beugungsgitter.

6

**Koennen KRISTALLEN als BEUGUNGSGITTER ** **fuer Roentgenstrahlen benutzbar sein? **

**Koennen KRISTALLEN als BEUGUNGSGITTER**

**fuer Roentgenstrahlen benutzbar sein?**

*d ~ *l* *

Lattice planes in crystals (described by three integer numbers)

1 2

*2dsinθ=nλ * **2θ **

**2θ**

*d *

**The BASIC physical principles of X-ray diffraction **

**The BASIC physical principles of X-ray diffraction**

*X-rays are electromagnetic waves with the wavelength of the * *order * l1 *Å (10*

^{-10 }*m) *

*X-rays are scattered by electrons (Thomson scattering) *

*The waves scattered at different crystal sites interfere with each * *other. The sum of two waves depends on the phase difference *

*between them *

**The amplitude of X-ray scattering **

**The amplitude of X-ray scattering**

**r ****E**^{1}

2 1exp( 2 (*i* 1 0) )

**E** **E** **k** **k r**

### Suppose that a monochromatic X-ray beam with wavelength l hits a crystal. The primary beam is described by the wave vector **k**

**k**

_{0 }### the scattered beam is described by the wave vector k

_{1 }**. The length of both wave vectors is 1/** l

**. The length of both wave vectors is 1/**

### The sum of the electric field vectors from the wave scattered at electrons 1 and 2 is

###

1 0 1 1 0 1 1 0 1 0

( , ) 1 exp(2 ( *i* ) exp(2 (*i* ) exp(2 (*i* )

**A k k** **E** **k** **k r** **E** **k** **k 0** **k** **k r**

**k**_{0 }

**E**0

**k**_{0 }

**k**_{1 }**k***_{1 }*
2

Phase difference

between beams 1 and 2

the amplitude of scattering depends on the DIFFERENCE between wave vectors 1

**Scattering vector (k** _{1} **-k** _{0} **) and scattering angle (2** q **) **

**Scattering vector (k**

_{1}**-k**

_{0}**) and scattering angle (2**

**)**

**k**_{0 }

**k**_{1 }**H**

### 2θ

### The amplitude of X-ray scattering is usually described as a function of scattering vector H (instead of k

_{1}** and k**

**and k**

_{0}**). Replacing two **

**). Replacing two**

### separated scatters (see previous page) by distributed density:

###

### ( ) exp 2

*A* ^{H} ^{r} *i* ^{Hr} *dV*

^{H}

^{r}

^{Hr}

### With (r) is an ELECTRON DENSITY so that (r)dV is the amount of electron in the small volume dV. The intensity of the X-ray scattering is the square of the absolute value of electric field so

### that

_{2}

### ( ) ~| ( ) | *I* **H** *A* **H** **H = k**

**H = k**

_{1}* - k*

_{0 }Scattering vector

### Scattering angle

**Scattering by a crystal (periodic system) **

**Scattering by a crystal (periodic system)**

### Consider the *scattering amplitude, A(H), for the periodic crystal. For this * case electron density can be described by LATTICE and UNIT CELL

( )

### ( ) ( ) exp(2 ) exp(2

_{uvw}### )

*unit CELL* *uvw*

*A* ^{H} ^{r} *i* ^{Hr} *dV* *i* ^{HA}

^{H}

^{r}

^{Hr}

^{HA}

**Contribution of the single UNIT CELL (F) ****CRYSTAL LATTICE (L) **

### ( ) **r**

### ( ) (

_{uvw}### )

### **r** **r** **A**

**a ** **b **

**a**

**b**

**A**_{uvw }

Lattice translation

### The sum (integration) over all crystal can be reduced to the

### integration over the single unit cell only:

**Contribution of the crystal lattice **

**Contribution of the crystal lattice**

### ( ) exp(2

_{uvw}### )

*uvw*

*L* ^{H} *i* ^{HA} **The sum is carried out over all lattice points ** **It is convenient to decompose the scattering vector by the reciprocal lattice ** **basis, a* b* and c* so that H= h a* + k b* + l c*. In this case (see lecture 1) the **

^{H}

^{HA}

**dot product between vectors H and A is reduced to **

### ( ) exp(2 ) exp(2 ) exp(2 )

*u* *v* *v*

*L* ^{H} *ihu* *ikv* *ilw*

^{H}

### It can be easily shown that in the case of an X-ray beam that illuminates N

_{a }*N*

_{b }*and* *N*

_{c}* unit cells in the directions a b and c then*

2 2 2

2

2 2 2

### sin ( ) sin ( ) sin ( ) ( )

### sin ( ) sin ( ) sin ( )

*a* *b* *c*

*N h* *N k* *N l*

*L hkl*

*h* *k* *l*

###

###

###

**LAUE INTERFERENCE FUNCTION **

**LAUE INTERFERENCE FUNCTION**

**HA**

**HA**

_{uvw}* = hu + kv + lw*

**Laue interference function **

**Laue interference function**

**Laue interference function **

**Laue interference function**

**Laue interference function **

**Laue interference function**

**Conclusion: LAUE EQUATION **

**Conclusion: LAUE EQUATION**

### The intensity of scattering by a crystal is non-zero for the scattering vectors **H whose reciprocal lattice coordinates are integer. That ** **means, the scattering from a crystal (with a given lattice) is ** **described by the corresponding RECIPROCAL LATTICE **

**H whose reciprocal lattice coordinates are integer. That**

**means, the scattering from a crystal (with a given lattice) is**

**described by the corresponding RECIPROCAL LATTICE**

**LAUE EQUATION for the diffraction by a crystal **

**LAUE EQUATION for the diffraction by a crystal**

**k** _{1 } **- k** _{0 } **= h a* + k b* + l c* **

**k**

_{1 }**- k**

_{0 }**= h a* + k b* + l c***

### h, k and l are indices of reflection

### SCATTERING DIFFRACTION

### 1/λ **k**

_{0}**H **

**H**

**k**

_{1}**-k**

_{0}### =H

**Ewald construction **

**Ewald construction**

### X-ray beam, l 2θ

### Detector

**k**

_{1}**k**

_{1}### 2 θ

### The wavelength of the primary X-ray beam is the same the wavelength of the diffracted beam, therefore |k

_{0}

### |=|k

_{1}

### |=1/ l . Therefore the scattering vector *H=k*

_{1}

### -k

_{0 }

### always points to the surface of the sphere of radius 1/ l with the center at the point –k

_{0}

### . This sphere is referred to as an Ewald sphere

### CRYSTAL

**X-ray diffraction from a still CRYSTAL**

**X-ray diffraction from a still CRYSTAL**

### What happens if we take a still photograph of a crystal with a monochromatic X-ray beam while rotating a crystal? To get anything we need to have at least one reciprocal lattice point on the surface of the Ewald sphere.

### 1/λ

**Ewald sphere **

### X-ray beam

### Detector

**X-ray diffraction from a rotated crystal**

**X-ray diffraction from a rotated crystal**

### What happens if we rotate a crystal by a certain angle whilst collecting an image from the detector?

### 1/λ

### Detector

### X-ray beam

**Ewald sphere **

**Measurements of a reflection with a point detector**

**Measurements of a reflection with a point detector**

**H **

**H**

1/λ

### ω

### 2θ

Point detector

**Ewald sphere **

**Rocking curve of (hkl) reflection **
**k**_{0}

**k**_{1}

### X-ray beam

While crystal is ROCKED about REFLECTION POSITION, the hkl point of the reciprocal lattice PASSES THROUTH the EWALD sphere

**Rotation photographs (taken at single crystal diffractometer, ** **rotation angle 1 degree)**

**Rotation photographs (taken at single crystal diffractometer,**

**rotation angle 1 degree)**

a=b=c=3.8 Å a = b = g =90 deg a=10.2 Å b=15.4 Å c=18.3 Å

**Modern X-ray diffraction experiment**

**Modern X-ray diffraction experiment**

### Taking the number of ROTATION PHOTOGRAPHS with very small step at different orientations of a crystal. Taking images frame by

### frame

### Reconstructing reciprocal space

### Calculating distribution of intensities in the selected sections of

### the reciprocal space

**Reconstruction of reciprocal space: Example 1**

**Reconstruction of reciprocal space: Example 1**

**a=37.05 b=8.69 c=20.7 Å a=90 b=95.97 g=90 deg **

**C**

_{10}**H**

_{10}**N**

_{2}**F**

_{6}**PClRu crystal (Guy Clarkson, **

**Department of Chemistry, University of Warwick) **

**Reciprocal space of a crystal: Example 2**

**Reciprocal space of a crystal: Example 2**

* a=b=c=3.8 Å *a

*b*

**=***g*

**=**

**=90 deg**a*

b*

**Na**

_{0.5}**Bi**

_{0.5}**TiO**

_{3}** crystal (Ferroelectrics and crystallography group, **

**Department of Physics, University of Warwick) **

**Diffractometer**

**Diffractometer**

**X-ray SOURCE ** **Crystal holder **

**DETECTOR of ** **X-rays **

**Goniometer ** **with motors ** **Liquid **

**nitrogen jet ** **Observation **

**camera **

2
*n*

###

**a** **R**

**A charged particle moving along a ** **circular trajectory must be the source **

**of an electromagnetic wave **

**Generation of synchrotron radiation **

**Generation of synchrotron radiation**

### Such kind of ‘classical’ light is usually emitted by high-energy particles and is therefore not as common in nature as usual

### ‘quantum’ light. A few rare examples of natural generation of synchrotron radiation are known in astrophysics (for example radio frequencies emitted by some stars, some part of the spectrum of emitted by aurora in Jupiter.

### On the other hand synchrotron radiation is produced artificially in

### particle acceleration

** Wide photon energy range **

**High brilliance ** ** Extra high natural collimation **

**Full control of the radiation properties **

**Wavelength, ** **polarization ** **type, ** **spectral properties ** of the radiation are easily tuned for the specific purposes

**Wavelength,**

**polarization**

**type,**

**spectral properties**

### The brilliance of **synchrotron radiation ** is approximately 10

**synchrotron radiation**

^{6}** times higher than ** the intensity, given by **normal X-ray ** **source **

**times higher than**

**normal X-ray**

**source**

### Synchrotron radiation is naturally

**collimated in the vertical plane, and **

**collimated in the vertical plane, and**

### has divergence 1/γ in the horizontal

**plane **

**plane**

**The most famous and powerful SR sources **

**The most famous and powerful SR sources**

**European ** **Synchrotron ** **Radiation ** **Facility **

Grenoble, France

**Advanced ** **Photon ** **Source **

In Argonne national laboratory

Chicago, USA

**SPring 8 **

Harima, Japan

The energy of electrons in the storage ring is 8 GeV

**The newest synchrotrons : Diamond **

**The newest synchrotrons : Diamond**

**Didcot, UK www.diamond.ac.uk **

**Didcot, UK www.diamond.ac.uk**

**The newest synchrotrons : PETRA III **

**The newest synchrotrons : PETRA III**

**Ring circumference : 2.3 km ** **Energy: 6 GeV **

**www.hasylab.eu **

**The newest synchrotrons : SSRF **

**The newest synchrotrons : SSRF**

**Shanghai, China **

**The newest synchrotrons : ALBA CELLS **

**The newest synchrotrons : ALBA CELLS**

**Energy: 3 GeV **

**First phase beamlines: 7 ** **Second phase beamlines: 8 **

**Ewald construction for the Laue method**

**Ewald construction for the Laue method**

### 1/λ

### Detector

### Any reciprocal lattice point within the LARGEST EWALD sphere (of radius 1/l

_{min }

### contribute to the Laue pattern)

### 1/λ

_{min }

Laue diffraction pattern

**Laue diffraction pattern**

**Laue diffraction pattern**

**λ**_{1}

**λ**_{2}

**λ**_{4}**λ**_{3}

### Each Bragg reflection is diffracted at its own wavelength. Usually 2D detectors do not give the information about the wave length therefore each spot reconstructs the direction of reciprocal space

### only

**Different Laue diffraction patterns**

**Different Laue diffraction patterns**

**Diffraction from the powder**

**Diffraction from the powder**

### A powder sample is composed of small, single crystalline, randomly oriented grains. Each grain has the same structure, i.e. the same lattice, space group and fractional positions of atoms in the unit cell. As a result each point of the reciprocal lattice is represented by a sphere.

### Reciprocal space of a powder is the set of concentric spheres

### Powder diffraction is an important technique for the structure determination. The solution of a structure from

### powder diffraction is known as Rietveld refinement

**Ewald construction for the powder diffraction**

**Ewald construction for the powder diffraction**

**Ewald sphere **

**Ewald sphere**

Primary beam

### Reciprocal lattice point Sphere

**Typical powder diffraction pattern**

**Typical powder diffraction pattern**

### Powder diffraction is one dimensional data representing the spherically averaged

### reciprocal lattice of a crystal. The position of each peak corresponds to the

### LENGTH of a reciprocal lattice vector.

**Splitting of the peak positions: example 1**

**Splitting of the peak positions: example 1**

a*=1 b*=1 a=90 deg

Cubic crystal system

a*=1.02 b*=1 a=90 deg

Tetragonal crystal system

**Splitting of the peak positions: example 2**

**Splitting of the peak positions: example 2**

a*=1 b*=1 a=90 deg

Cubic crystal system

a*=1 b*=1 a=89 deg

‘Rhombohedral’ crystal system

**Powder diffraction showing splitting of peaks**

**Powder diffraction showing splitting of peaks**

### Cubic to tetragonal (paraelectric to ferroelectric) phase transition in BaTiO

_{3}

### . The temperature of the phase transition is known as Curie temperature

_{ }

### {200} cubic

### (200) tetragonal

### (020) tetragonal

### (002) tetragonal

**Change of the symmetry as a phase transition**

**Change of the symmetry as a phase transition**

High symmetry (cubic) phase

Lower symmetry (tetragonal) phase

### {200} {210} {211}

### (200) (020)

### (002) (012)

### (021)

### (210)

**Some electron diffraction patterns**

**Some electron diffraction patterns**

Courtesy to Dr R. Beanland, University of Warwick, UK

**Some high resolution electron microscopy images**

**Some high resolution electron microscopy images**

Courtesy to Dr R. Beanland, University of Warwick, UK