Basic crystallography - UniTrentoalpha.science.unitn.it/~rx/Dakar_school/1_Fornasini_a_cryst.pdf ·...
Transcript of Basic crystallography - UniTrentoalpha.science.unitn.it/~rx/Dakar_school/1_Fornasini_a_cryst.pdf ·...
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)
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Univ. Trento
Electromagnetic waves
Speed (in vacuum):
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c ≈ 3×108 m/s
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E = hν = hωPhoton energy
Frequency
Wavelength
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ν = c /λ
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λ
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ω = 2πν
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Electric field
Magnetic field
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λ
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c
1 Å = 10-10 m1 nm = 10-9 m
Wave-vector PaoloFornasini
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PaoloFornasini
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PaoloFornasini
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Plane wave
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r k
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λ
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k =2πλ
Plane wave PaoloFornasini
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PaoloFornasini
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PaoloFornasini
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k =2πλ
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r k
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r r
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r r
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r E r r ( ) =
r E 0 cos
r k ⋅ r r ( ) =
r E 0 for
r k ⋅ r r = 2nπ
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Re eir k ⋅r r { }
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r E
Complex notation
Electromagnetic spectrum
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E = hν = hc /λ
Photonenergy
Frequency
X-rays
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λ ≈ 0.01÷10 Å
ν ≈ 1017 ÷1020Hz
E ≈ 0.4 ÷ 400 keV
Wavelength
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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
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Incoming beam(monochromatic)
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hν
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hν '
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hν
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hνF
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E
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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)
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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
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X-ray diffraction from crystals
Munich, 1912:• Max von Laue• W. Friedrich & P.Knipping
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Crystallography
William Henry Bragg (1862-1942)
William Lawrence Bragg (1890-1971)
Cambridge, 1912/13
Bragg spectrometer
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Crystal structure PaoloFornasini
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1-D
3-D
2-D
Atom
Molecule
Protein
Bravais lattice + Basis
1-D
2-D
3-D
Bravais lattice + basis PaoloFornasini
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1-D
2-D
3-D
Crystal lattices
Translation vectors (2D) PaoloFornasini
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2-D
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r a
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r b
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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
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r a
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r b
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r R
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r R = n1
r a + n2r b
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r R
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r R
2-D
Different choices of primitive vectors
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r a ,r b
Non-primitive vectors (2D) PaoloFornasini
Univ. Trento
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r a
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r b
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r R
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r R ≠ n1
r a + n2r b
Not all pairs are primitive
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r a ,r b
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r R
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r R
2-D
Primitive vectors (3D) PaoloFornasini
Univ. Trento
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r R = n1
r a + n2r b + n3
r c
Different choices of primitive vectors
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r a ,r b , r c
3-D
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r a
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r b
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r c
Primitive unit cells (2D) PaoloFornasini
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2-D
Different choices of primitive unit cells
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r a
€
r b
Primitive cell = 1 lattice point
Conventional unit cells (2D) PaoloFornasini
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2-D
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r a
€
r b
More than 1 lattice point per unit cell
Non-Bravais lattices PaoloFornasini
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2-DAtoms
Un-equivalent sites
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r R ≠ n1
r a + n2r b
Bravais lattices
Bravaislattice
Basis
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Classification of cells
2-D 3-D
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r a
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r b
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γ
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r a
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r b
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r c
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γ
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β
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α
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a b cα β γ
latin
greek
Surface Bravais lattice
Non-primitive unit cell
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2-D
3-D Bravais lattices
4 unit cells
P = primitiveI = body centeredF = face centeredC = side centered
7 crystal
systems
14 Bravaislattices
+ =
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3-D
Coordinates PaoloFornasini
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14€
12
€
14
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34
€
1
€
2
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3€
1€
2
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0
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14, 14
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34, 12
Inside cell: fractional coordinates
Lattice points:integer coordinates€
3,2( )
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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
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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
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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)
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Coordination number = 4
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a
Non-Bravais lattice
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(0,0,0)
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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
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a
Non-Bravais lattice
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(0,0,0)
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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
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Univ. Trento
conventional unit cell(8 atoms per cell)
NaCl a=5.64 ÅKBr a=6.60 ÅCaO a=4.81 Å
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a
Non-Bravais lattice
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(0,0,0)
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12, 12, 12
fcc Bravais lattice 2-atom basis+
Simple hexagonal structure
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a€
c
2 lattice parameters
PaoloFornasini
Univ. Trento
Primitive cellHexagonal symmetry
Top view
Primitive unit cell(1 lattice point per cell)
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r a 1
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r a 2
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a1 = a2
Hexagonal close packed structure
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a€
c
Cordination number = 12
4-Be a=2.29 Å12-Mg a=3.21 Å48-Cd a=2.98 Å
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Univ. Trento
Conventional unit cell(2 atoms per cell)€
c =83a
Non-Bravais lattice
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(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
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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
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A
B
A
AB
C
fcchcp
Crystal planes
Crystal planes PaoloFornasini
Univ. Trento
2-D
Miller indices, 2-D (a)
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r a
€
r b
(hk) = (21) (hk) = (11)
2-D
PaoloFornasini
Univ. Trento
Miller indices, 2-D (b)
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(hk) = (11 )
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(hk) = (01)
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(hk) = (10)
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(hk) = (21 )
PaoloFornasini
Univ. Trento
Miller indices, 3-D PaoloFornasini
Univ. Trento
3-D
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r c
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r b
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r a
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r c
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(hkl) = (100)
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r b
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r a
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r c
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(hkl) = (210)
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r a
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r c
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r b €
(hkl) = (114)
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r b
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r a
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(hkl) = (111)
Miller indices, cubic lattices PaoloFornasini
Univ. Trento
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(111)
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(110)
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(100)
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(200)
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(110)
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(222)
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(200)
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(111)
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(220)
bcc
fcc
sc
Interplanar distance PaoloFornasini
Univ. Trento
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dhkl
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dhkl2 =
a2
h2 + k2 + l2
Cubic lattices
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d200 =a2
=1.805 Å
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d220 =a2 2
=1.276 Å
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d111 =a3
= 2.084 Å
Copper, fcc, a=3.61 Å
Planes and directions
Family of planes
Perpendicular direction
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[hkl]
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(hkl)
Equivalent planes and directions
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(010)
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(100)
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(001)€
[100]€
[010]
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[001]
Equivalent directions
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<100 >
Equivalent planes
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{100}
Reciprocal lattice
X-ray diffraction pattern PaoloFornasini
Univ. Trento
Direct space3-dimensional lattice
Reciprocal space2-dimensional projection
Basic idea PaoloFornasini
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€
dhkl
€
r K hkl
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Khkl =2πdhkl
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r K h'k 'l '
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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
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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
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Effects of temperature PaoloFornasini
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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