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# What is a crystal?

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### Originally from Greek: CRYSTAL – NATURAL ICE

Visit www.snowcrystals.com for your own pleasure

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### Originally from Greek: CRYSTAL – NATURAL ICE

Visit www.snowcrystals.com for your own pleasure

(3)

### QUARTZ, SiO

2

http://webmineral.com

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### Mackayite, Fe3Te2O5(OH)

http://webmineral.com

(5)

BONDS

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~ 1 mm=10-3 m

6

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### Laue diffraction patterns

a-Quartz crystals (SiO2)

Bragg peaks

(8)

x

Unit cell

Crystal lattice

Lattice vectors

A=na

(9)

Unit cell

Crystal lattice

+

Unit cell

Crystal lattice

+

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+ Unit cell

Crystal lattice

b

Auv=ua+vb = [uv]

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### 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

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### 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

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a b

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a b

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a b

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### 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

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a

b

d

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a b

d

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a

b

d

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a

d

b

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### Examples. MILLER INDICES AND LATTICE PLANES

a b

a b a

b

a b

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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?

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### the Miller indices.

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

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### 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

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2

1

d 3

4

5

6

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

d ~ l

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Lattice planes in crystals (described by three integer numbers)

1 2

d

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-10

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### The amplitude of X-ray scattering

r E1

2 1exp( 2 (i1 0) )

E E k k r

0

1

###    

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

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k0

k1 H

1

0

2

1

### - k

0

Scattering vector

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( )

uvw

unit CELL uvw

### AH    r  iHrdV    iHA

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

uvw

### a b

Auvw

Lattice translation

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uvw

uvw

u v v

a

b

c

2 2 2

2

2 2 2

a b c

uvw

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0

1

0

1

1

0

1

1

0

0

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Ewald sphere

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Ewald sphere

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1/λ

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

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### 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

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### Reconstruction of reciprocal space: Example 1

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

10

10

2

6

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### Reciprocal space of a crystal: Example 2

a=b=c=3.8 Å a = b = g =90 deg

a*

b*

0.5

0.5

3

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2 n

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6

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### The most famous and powerful SR sources

Grenoble, France

In Argonne national laboratory

Chicago, USA

### SPring 8

Harima, Japan

The energy of electrons in the storage ring is 8 GeV

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min

### 1/λ

min

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Laue diffraction pattern

λ1

λ2

λ4 λ3

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Primary beam

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### 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

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### 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

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3

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### Change of the symmetry as a phase transition

High symmetry (cubic) phase

Lower symmetry (tetragonal) phase

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### Some electron diffraction patterns

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

(69)

### Some high resolution electron microscopy images

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

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