Basic crystallographyBasic crystallography

Paolo FornasiniDepartment of PhysicsUniversity of Trento, Italy

OverviewOverview

• X-rays• Crystals• Crystal lattices• Some relevant crystal structures• Crystal planes• Reciprocal lattice• Crystalline and non-crystalline materials

X-rays

1895 - Discovery of X-rays

Würzburg (Germany)November 8,1895

Wilhelm Konrad Röntgen(1845-1923)

PaoloFornasini

Univ. Trento

Electromagnetic waves

Speed (in vacuum):

€

c ≈ 3×108 m/s

PaoloFornasini

Univ. Trento

€

E = hν = hωPhoton energy

Frequency

Wavelength

€

ν = c /λ

€

λ

€

ω = 2πν

€

Electric field

Magnetic field

€

λ

€

c

1 Å = 10-10 m1 nm = 10-9 m

Wave-vector PaoloFornasini

Univ. Trento

PaoloFornasini

Univ. Trento

PaoloFornasini

Univ. Trento

Plane wave

€

r k

€

λ

€

k =2πλ

Plane wave PaoloFornasini

Univ. Trento

PaoloFornasini

Univ. Trento

PaoloFornasini

Univ. Trento

€

k =2πλ

€

r k

€

r r

€

r r

€

r E r r ( ) =

r E 0 cos

r k ⋅ r r ( ) =

r E 0 for

r k ⋅ r r = 2nπ

€

Re eir k ⋅r r { }

€

r E

Complex notation

Electromagnetic spectrum

€

E = hν = hc /λ

Photonenergy

Frequency

X-rays

€

λ ≈ 0.01÷10 Å

ν ≈ 1017 ÷1020Hz

E ≈ 0.4 ÷ 400 keV

Wavelength

PaoloFornasini

Univ. Trento

I R

Microwaves

E (eV)

108

1

10-12

10-8

10-4

104

λ (m)

10-14

10-6

106

102

10-2

10-10

ν (Hz)

1022

1014

102

106

1010

1018

X

UV

γ

ELF

Radiowaves

Interaction of x-rays with matter PaoloFornasini

Univ. Trento

Incoming beam(monochromatic)

€

hν

€

hν '

€

hν

€

hνF

€

E

€

EA

Elastic scattering

Inelastic scattering

Photo-electrons

Augerelectrons

FluorescencePhoto-electric

absorption

Scattering

Beam attenuation

X-RAYS and X-ray techniques PaoloFornasini

Univ. Trento

X-rays

10-12 10-10 10-8 10-6 10-4 10-2

1 Å 1 µm

λ (m)

U.V. I.R.

E[keV] = 12.4λ[Å]

Scattering

Bragg angle

Inte

nsity

λ = 2dsinθ

Spectroscopy

Abso

rptio

nEnergy

Imaging

Crystals

Crystals

Quartz crystal (SiO2)

Macroscopic regularities(e.g. constancy of angles)

Classification of crystals

Regular packing of microscopic structural unitsR.J. Haüy (1743-1822)

PaoloFornasini

Univ. Trento

Atoms and crystals

Structural units = atoms

Example: NaCl

Cubic structure1 cm3 m = 2.165 g N = 44.6 x 1021 atoms

HYPOTHESIS:

CONCLUSION: Inter-atomic distancesAtomic dimensions

Atomic masses: Na 38.12 x 10-24 gCl 58.85 x 10-24 g

0.28 nm = 2.8 Å

≈ X-ray wavelengths

PaoloFornasini

Univ. Trento

X-ray diffraction from crystals

Munich, 1912:• Max von Laue• W. Friedrich & P.Knipping

PaoloFornasini

Univ. Trento

Crystallography

William Henry Bragg (1862-1942)

William Lawrence Bragg (1890-1971)

Cambridge, 1912/13

Bragg spectrometer

PaoloFornasini

Univ. Trento

Crystal structure PaoloFornasini

Univ. Trento

1-D

3-D

2-D

Atom

Molecule

Protein

Bravais lattice + Basis

1-D

2-D

3-D

Bravais lattice + basis PaoloFornasini

Univ. Trento

1-D

2-D

3-D

Crystal lattices

Translation vectors (2D) PaoloFornasini

Univ. Trento

2-D

€

r a

€

r b

€

r R

Arbitrary origin

For every lattice point

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r R = n1

r a + n2r b

integers primitivevectors

Primitive vectors (2D) PaoloFornasini

Univ. Trento

€

r a

€

r b

€

r R

€

r R = n1

r a + n2r b

€

r R

€

r R

2-D

Different choices of primitive vectors

€

r a ,r b

Non-primitive vectors (2D) PaoloFornasini

Univ. Trento

€

r a

€

r b

€

r R

€

r R ≠ n1

r a + n2r b

Not all pairs are primitive

€

r a ,r b

€

r R

€

r R

2-D

Primitive vectors (3D) PaoloFornasini

Univ. Trento

€

r R = n1

r a + n2r b + n3

r c

Different choices of primitive vectors

€

r a ,r b , r c

3-D

€

r a

€

r b

€

r c

Primitive unit cells (2D) PaoloFornasini

Univ. Trento

2-D

Different choices of primitive unit cells

€

r a

€

r b

Primitive cell = 1 lattice point

Conventional unit cells (2D) PaoloFornasini

Univ. Trento

2-D

€

r a

€

r b

More than 1 lattice point per unit cell

Non-Bravais lattices PaoloFornasini

Univ. Trento

2-DAtoms

Un-equivalent sites

€

r R ≠ n1

r a + n2r b

Bravais lattices

Bravaislattice

Basis

PaoloFornasini

Univ. Trento

Classification of cells

2-D 3-D

€

r a

€

r b

€

γ

€

r a

€

r b

€

r c

€

γ

€

β

€

α

€

a b cα β γ

latin

greek

Surface Bravais lattice

Non-primitive unit cell

PaoloFornasini

Univ. Trento

2-D

3-D Bravais lattices

4 unit cells

P = primitiveI = body centeredF = face centeredC = side centered

7 crystal

systems

14 Bravaislattices

+ =

PaoloFornasini

Univ. Trento

3-D

Coordinates PaoloFornasini

Univ. Trento

€

14€

12

€

14

€

34

€

1

€

2

€

3€

1€

2

€

0

€

14, 14

€

34, 12

Inside cell: fractional coordinates

Lattice points:integer coordinates€

3,2( )

€

r R = n1

r a + n2r b

Some relevant crystalstructures

Simple cubic lattice PaoloFornasini

Univ. Trento

Primitive unit cell(1 lattice point per cell)

Coordination number = 6

lattice parameter

Bravais lattice

€

a

84-Po a=3.35 Å

Body centered cubic lattice (bcc) PaoloFornasini

Univ. Trento

conventional unit cell(2 lattice points per cell)

Bravais lattice

lattice parameter

€

a

Coordination number = 8

24-Cr a=2.88 Å26-Fe a=2.87 Å42-Mo a=3.15 Å

Face centered cubic lattice (fcc) PaoloFornasini

Univ. Trento

conventional unit cell(4 lattice points per cell)

Bravais lattice

lattice parameter

€

a

Coordination number = 12

29-Cu a=3.61 Å47-Ag a=4.09 Å79-Au a=4.08 Å

Diamond structure

conventional unit cell(8 atoms per cell)

PaoloFornasini

Univ. Trento

Coordination number = 4

€

a

Non-Bravais lattice

€

(0,0,0)

€

14, 14, 14

fcc Bravais lattice 2-atom basis+

6-C a=3.57 Å14-Si a=5.43 Å32-Ge a=5.66 Å

Zincblende (sphalerite) structure PaoloFornasini

Univ. Trento

conventional unit cell(8 atoms per cell)

Cordination number = 4

€

a

Non-Bravais lattice

€

(0,0,0)

€

14, 14, 14

fcc Bravais lattice 2-atom basis+

ZnS a=5.41 ÅGaAs a=5.65 ÅSiC a=4.35 Å

Rock-salt (NaCl) structure

Cordination number = 6

PaoloFornasini

Univ. Trento

conventional unit cell(8 atoms per cell)

NaCl a=5.64 ÅKBr a=6.60 ÅCaO a=4.81 Å

€

a

Non-Bravais lattice

€

(0,0,0)

€

12, 12, 12

fcc Bravais lattice 2-atom basis+

Simple hexagonal structure

€

a€

c

2 lattice parameters

PaoloFornasini

Univ. Trento

Primitive cellHexagonal symmetry

Top view

Primitive unit cell(1 lattice point per cell)

€

r a 1

€

r a 2

€

a1 = a2

Hexagonal close packed structure

€

a€

c

Cordination number = 12

4-Be a=2.29 Å12-Mg a=3.21 Å48-Cd a=2.98 Å

PaoloFornasini

Univ. Trento

Conventional unit cell(2 atoms per cell)€

c =83a

Non-Bravais lattice

€

(0,0,0)

€

12, 12, 12

primitive cell + 2-atom basis

Coordination number PaoloFornasini

Univ. Trento

N=4

diamond

N=6

cubic

N=8

bcc

N=12

fcc hcp

Close packing

Close-packing of spheres

1stlayer

A

A A A AAAAA

A

A A A AAAAA

A AAAA

2ndlayer

B B

B

BB B

B

B BB B

B B B

PaoloFornasini

Univ. Trento

3rdlayer

A

A A A AAAAA

A

A A A AAAAA

A AAAA

C C CC C

C CC

A B A A B C

hcp versus fcc

A

A A A AAAAA

A

A A A AAAAA

A AAAA

C C CC C

C CC

PaoloFornasini

Univ. Trento

A

B

A

AB

C

fcchcp

Crystal planes

Crystal planes PaoloFornasini

Univ. Trento

2-D

Miller indices, 2-D (a)

€

r a

€

r b

(hk) = (21) (hk) = (11)

2-D

PaoloFornasini

Univ. Trento

Miller indices, 2-D (b)

€

(hk) = (11 )

€

(hk) = (01)

€

(hk) = (10)

€

(hk) = (21 )

PaoloFornasini

Univ. Trento

Miller indices, 3-D PaoloFornasini

Univ. Trento

3-D

€

r c

€

r b

€

r a

€

r c

€

(hkl) = (100)

€

r b

€

r a

€

r c

€

(hkl) = (210)

€

r a

€

r c

€

r b €

(hkl) = (114)

€

r b

€

r a

€

(hkl) = (111)

Miller indices, cubic lattices PaoloFornasini

Univ. Trento

€

(111)

€

(110)

€

(100)

€

(200)

€

(110)

€

(222)

€

(200)

€

(111)

€

(220)

bcc

fcc

sc

Interplanar distance PaoloFornasini

Univ. Trento

€

dhkl

€

dhkl2 =

a2

h2 + k2 + l2

Cubic lattices

€

d200 =a2

=1.805 Å

€

d220 =a2 2

=1.276 Å

€

d111 =a3

= 2.084 Å

Copper, fcc, a=3.61 Å

Planes and directions

Family of planes

Perpendicular direction

€

[hkl]

€

(hkl)

Equivalent planes and directions

€

(010)

€

(100)

€

(001)€

[100]€

[010]

€

[001]

Equivalent directions

€

<100 >

Equivalent planes

€

{100}

Reciprocal lattice

X-ray diffraction pattern PaoloFornasini

Univ. Trento

Direct space3-dimensional lattice

Reciprocal space2-dimensional projection

Basic idea PaoloFornasini

Univ. Trento

€

dhkl

€

r K hkl

€

Khkl =2πdhkl

€

r K h'k 'l '

€

dh'k 'l '

A) Family of planes → wave-vector

B) Wave-vectors → set of points

C) Set of points → lattice

Reciprocal quantities PaoloFornasini

Univ. Trento

frequencytime

position wave-vector

€

λ

€

ω = 2π /T

€

k = 2π /λ€

T

Periodic behaviour

Harmonics PaoloFornasini

Univ. Trento

€

ω0

€

T

€

2ω0

Periodic behaviour

€

3ω0

Periodic behaviour

fundamental

2nd harmonic

3rd harmonic

time

frequency

1-D

€

a* =2πa

Direct space Reciprocal space

€

r a

€

r a

€

r a *

€

r a *

PaoloFornasini

Univ. Trento

2-D, rectangular lattice (a) PaoloFornasini

Univ. Trento

Direct space Reciprocal space

€

r a

€

r b

€

r R

€

r a *

€

r b *

€

r R *

€

r R = n1

r a + n2r b

€

r R * = m1

r a * + m2

r b *

€

a* =2πa

=2πbab

b* =2πb

=2πaab

€

r a *⊥r b

r b *⊥ r a

2-D, rectangular lattice (b)

Reciprocal spaceDirect space

PaoloFornasini

Univ. Trento

2-D, rectangular lattice (c)

Reciprocal spaceDirect space

PaoloFornasini

Univ. Trento

2-D, rectangular lattice (d)

Reciprocal spaceDirect space

PaoloFornasini

Univ. Trento

2-D, oblique lattice (a) PaoloFornasini

Univ. Trento

€

r a *

€

r b *

€

r a *⊥r b

r b *⊥ r a

€

r a

€

r b

€

r R

Reciprocal spaceDirect space

€

a* =2π

asinθ=

2πbabsinθ

b* =2π

bsinθ=

2πaabsinθ

2-D, oblique lattice (b) PaoloFornasini

Univ. Trento

Reciprocal spaceDirect space

Cubic lattices (a)

Reciprocal spaceDirect space

€

r b *

€

r c *

€

r a

€

r c

€

r b

€

r a *

PaoloFornasini

Univ. Trento

€

(111)

€

(110)

€

(100)

€

(111)

€

(110)

€

(100)

Cubic lattices (b)

Reciprocal spaceDirect space

PaoloFornasini

Univ. Trento

€

(200)

€

(110)

€

(222)

€

r a

€

r c

€

r b

€

r c *

€

r b *

€

r a *

€

(200)

Cubic lattices (c)

Reciprocal spaceDirect space

PaoloFornasini

Univ. Trento

€

(200)

€

(111)

€

(220)

€

r a

€

r c

€

r b

€

r b *

€

r c *

€

r a *

€

(200)

Primitive vectors: general rule

€

r a * = 2πr b × r c

r a ⋅r b × r c ( )

r b * = 2π

r c × r a r a ⋅

r b × r c ( )

r c * = 2πr a ×

r b

r a ⋅r b × r c ( )

€

r a

€

r b

€

r c

€

r a

€

r b

€

r c

€

r a *

€

r b *

€

r c *

€

r c *

€

r b *

€

r a *

Direct space Reciprocal space

€

r a ⋅r b × r c ( )

PaoloFornasini

Univ. Trento

Reciprocal lattice and lattice planes

For any family of lattice planes separated by a distance dthere are reciprocal lattice vectors perpendicular to the planes, the shortest of which have a length 2π/d.

For any reciprocal lattice vector R*, there is a family of lattice planes normal to R*

and separated by a distance d, where 2π/d is the length of the shortest reciprocal lattice vector

parallel to R*.

PaoloFornasini

Univ. Trento

Microscopic structureof materials

Macro and micro-crystals

Monocrystalline silicon, ∅ 13 cmCr, electron microscopy

Grain structure

PaoloFornasini

Univ. Trento

Effects of temperature PaoloFornasini

Univ. Trento

Temperature

Spread of atomic positionsThermal motion

Crystalline and non-crystalline materials PaoloFornasini

Univ. Trento

Crystalline solids Non-crystalline systems

Long-range order No long-range order

Cooling rate

High T:liquid

Low T:solid

Slow cooling Fast cooling

No thermodynamic equilibrium

Thermodynamic equilibrium

PaoloFornasini

Univ. Trento

Radial Distribution Function PaoloFornasini

Univ. Trento

r0

1

g(r)

€

RDF = 4πr2ρ r( ) = 4πr2ρ0 g r( )

PDF = Pair Distribution Function

average density

Short-range order

Summary

• Plane waves and wavevector• Crystal structure = Bravais lattice + basis• Bravais lattices: primitive vectors, unit cells

(primitive and conventional), classifications• Crystal structures (sc, bcc, fcc, hcp ...)• Crystal planes and Miller indices• Reciprocal lattice• Crystalline and non-crystalline materials

Dolomite mountains in winter