What is a crystal?

What is a crystal?

Originally from Greek: CRYSTAL – NATURAL ICE

Visit www.snowcrystals.com for your own pleasure

What is a crystal?

Originally from Greek: CRYSTAL – NATURAL ICE

Visit www.snowcrystals.com for your own pleasure

Minerals

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

http://webmineral.com

Mackayite, Fe ₃ Te ₂ O ₅ (OH)

http://webmineral.com

“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

a-Quartz crystals (SiO₂)

Bragg peaks

1 Dimensional crystal (1D periodic structures)

x

Unit cell

Crystal lattice

Lattice vectors

A=na

To obtain the whole crystal structure one has to

translate the UNIT CELL to each LATTICE POINT

Different choices of unit cell

Unit cell

Crystal lattice

+

Unit cell

Crystal lattice

+

2 Dimensional crystal (2D periodic structures)

+ Unit cell

Crystal lattice

b

A_uv=ua+vb = [uv]

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

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

a b

Building a lattice : choice of basis vectors 2

Building a lattice : choice of basis vectors 3

a b

Theorem about the choice of basis vectors

a b

A₁ A₂

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

Take two other lattice vectors A₁=[u₁v₁]=u₁a+v₁b

A₂=[u₂v₂]=u₂a+v₂b u₁,u₂,v₁ and v₂ 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 A₁ and A₂

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 A₁ and A₂ can be chosen as basis vectors for the SAME LATTICE

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

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

a

b

d

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

a b

d

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

a

b

d

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

a

d

b

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

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

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

B₁

B₂

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

The angle between faces a₁₂ = < (B₁,B₂)

1 2 12

1 2

( )

cos a  | B B || |

B B

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

2θ

1

d 3

4

5

Periodische Strukturen zeigen ähnliche

Beugungserscheinungen, wie eine Beugungsgitter.

6

Koennen KRISTALLEN als BEUGUNGSGITTER fuer Roentgenstrahlen benutzbar sein?

d ~ l

Lattice planes in crystals (described by three integer numbers)

1 2

2dsinθ=nλ 2θ

d

The BASIC physical principles of X-ray diffraction

X-rays are electromagnetic waves with the wavelength of the order l1 Å (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

r E¹

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

the scattered beam is described by the wave vector k

. The length of both wave vectors is 1/ l

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

k₀

E0

k₀

k₁ k₁ 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 )

k₀

k₁ H

2θ

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

and k

). Replacing two

separated scatters (see previous page) by distributed density:

   

( ) exp 2

A ^H    ^r  i ^Hr dV

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

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

- k

Scattering vector

Scattering angle

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

)

unit CELL uvw

A ^H    ^r  i ^Hr dV    i ^HA

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

 ( ) r

( ) (

)

 r   r  A

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

( ) exp(2

)

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

dot product between vectors H and A is reduced to

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

u v v

L ^H    ihu   ikv   ilw

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

N

and N

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

HA

= hu + kv + lw

Laue interference function

Laue interference function

Laue interference function

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

LAUE EQUATION for the diffraction by a crystal

k ₁ - k ₀ = h a* + k b* + l c*

h, k and l are indices of reflection

SCATTERING DIFFRACTION

1/λ k

H

k

-k

=H

Ewald construction

X-ray beam, l 2θ

Detector

k

k

2 θ

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

|=|k

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

-k

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

. This sphere is referred to as an Ewald sphere

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

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

H

1/λ

ω

2θ

Point detector

Ewald sphere

Rocking curve of (hkl) reflection k₀

k₁

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)

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

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

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

C

H

N

F

PClRu crystal (Guy Clarkson,

Department of Chemistry, University of Warwick)

Reciprocal space of a crystal: Example 2

a=b=c=3.8 Å a = b = g =90 deg

a*

b*

Na

Bi

TiO

crystal (Ferroelectrics and crystallography group,

Department of Physics, University of Warwick)

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

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

The brilliance of synchrotron radiation is approximately 10

times higher than the intensity, given by normal X-ray source

Synchrotron radiation is naturally

collimated in the vertical plane, and

has divergence 1/γ in the horizontal

plane

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

Didcot, UK www.diamond.ac.uk

The newest synchrotrons : PETRA III

Ring circumference : 2.3 km Energy: 6 GeV

www.hasylab.eu

The newest synchrotrons : SSRF

Shanghai, China

The newest synchrotrons : ALBA CELLS

Energy: 3 GeV

First phase beamlines: 7 Second phase beamlines: 8

Ewald construction for the Laue method

1/λ

Detector

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

contribute to the Laue pattern)

1/λ

Laue diffraction pattern

Laue diffraction pattern

λ₁

λ₂

λ₄ λ₃

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

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 sphere

Primary beam

Reciprocal lattice point Sphere

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

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

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

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

. 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

High symmetry (cubic) phase

Lower symmetry (tetragonal) phase

{200} {210} {211}

(200) (020)

(002) (012)

(021)

(210)

Some electron diffraction patterns

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

Some high resolution electron microscopy images

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