1. Chemical bonding and crystal structure
1.1 Atoms
19
21
Hydrogen atom
Scanning electron microscopy
Cleaved surface
ZnO, TiO2, NiO, NaCl, Si, Ge, GaAs, InP
Ni surface
ª only for s-waves (l=0)
Rn0(r = 0) ≠ 0
ªCrystals are build by „small“
repeating units (= basis)
like atoms and molecules
Scanning tunneling microscopy
1. Chemical bonding and crystal structure
1.1 Atoms
20
22
1.1 Atoms
Polar plot
1.1 Atoms
Chemical bonding and Crystal structure
1.1 Atoms
Chemical bonding and Crystal structure
23
25
Ni (Z = 28), Zeff = 8, d = 2.49 Å
corresponds to relativistic corrections which
result from Dirac´s equation
i) Relativistic, kinetic energy
ii) Spin-orbit coupling (LS – coupling)
iIi) Darwin term
(see, e.g., p1/2 - p3/2 splitting in Nickel)
1.1 Atoms
Chemical bonding and Crystal structure
1.1 Atoms
Chemical bonding and Crystal structure
24
Hydrogen atom
26
i) Relativistic, kinetic energy
Estimate of (v/c)2 via uncertainty relation (a = Bohr radius):
10950 MHz = 0.045 meV, beachte aber ΔE/E ~ Z2α2
1.1 Atoms
Chemical bonding and Crystal structure
Chemical bonding and Crystal structure
1.1 Atoms
27
29
ii) Classical Hamiltonian for the spin-orbit interaction
B field from the proton in the electron's rest frame is
Valenzniveaus
perturbation Hamiltonian
Nicht die Größe der Bindungsenergie
sondern der Grad der Lokalisation
der Wellenfunktion entscheidet.
Rumpfniveaus
recall
1.1 Atoms
Chemical bonding and Crystal structure
Chemical bonding and Crystal structure
1.1 Atoms
28
iii) Darwin term
Flickering motion of electron in nucleus leads to average potential
30
Hund‘sche Regeln für teilweise gefüllte Schalen
(Valenzelektronen)
1.
2.
3.
4.
S maximal
L maximal
J = |L-S| für nicht mehr als halbgefüllte Schalen
J = L+S für mehr als halbgefüllte Schalen
Ni: Ar 3d8 4s2
contribution only for s-waves with finite amplitude at x = 0
1. S = 1
2. L = 3
3. J = 4
2S+1L
J
= 3F4
Chemical bonding and Crystal structure
1.1 Atoms
Chemical bonding and Crystal structure
1.2 Molecules
31
He 1s2 – excited states
2S+1L
33
r < r0
J
U(r)
r = r0
attractive Coulomb
interaction:
r > r0
„s-wave in core, p-wave not“
r
2s+1 = 1
2s+1 = 3
singulett state
triplett state
antisymmetric
symmetric
bonding (attraction) due to valence electrons
Pauli repulsion between neighbouring atoms
ª equilibrium distance r0 (related to lattice parameter)
Spatial part accordingly (Hund‘s rules !)
1.1 Atoms
Chemical bonding and Crystal structure
1.2 Molecules
32
Aufbauprinzip
Chemical bonding and Crystal structure
34
Hydrogen ion H2-
1.2 Molecules
Chemical bonding and Crystal structure
Chemical bonding and Crystal structure
1.3 Crystals
35
LCAO – linear combination of atomic orbitals
37
Covalent bonding (3-9 eV)
Si
Metalic bonding
(1-2 eV)
Ionic bonding (6 -10 eV)
NaCl
CsCl
Hydrogen-bridge
bonding (0.1 eV)
Van der Waals bonding (< 0.2 eV)
1.2 Molecules
Chemical bonding and Crystal structure
1.3 Crystals
Chemical bonding and Crystal structure
36
Molecular orbitals (e.g. benzene)
38
Covalent bonding
Electron density (contour plot)
LCAO
graphite: planar sp2 structure
diamond, silicon: tetrahedral sp3 structure
Chemical bonding and Crystal structure
1.3 Ion Crystals
Chemical bonding and Crystal structure
1.3 Ion Crystals
39
41
Ionic bonding - energetics
ionization electron
energy
affinity
(eV)
(eV)
Li
Na
K
Rb
5.39
5.14
4.34
4.18
0.62
0.55
0.5
ionization electron
energy
affinity
(eV)
(eV)
F
Cl
Br
I
17.4
13.0
11.8
10.5
3.40
3.61
3.36
3.06
Na + Cl → Na+ + Cl- + 1.53 eV
electrostatic interaction between ions
Na+ and Cl-: 5.1 eV (r0 ~ 2.8 Å)
total energy gain of 3.57 eV
Parameters ρ and B of the repulsive potential determined by
equilibrium distance r0 and compressibility κ
Different structures of ionic crystals:
Nearly spherical charge
distributions (closed shell)
stability depends on the ratio rA /rB of ionic radii:
CsCl ↔ NaCl ↔ ZnS
1.37 < rA /rB < 2.44
Electron density (contour plot)
1.3 Ion Crystals
Chemical bonding and Crystal structure
Chemical bonding and Crystal structure
1.3 Crystals - metals
40
Ionic bonding - electrostatic energy (Born-Mayer potential)
42
Metallic bonding
A Madelung constant, z coordination number
Overlapping wave functions form delocalized states (Bloch states)
NaCl (z = 6): A = 1.747565
- s – electrons of Alkali metals Li, Na, K, Cs, Rb (bcc)
- s,p – electrons of 3d metals Fe (bcc), Co (hcp), Ni (fcc), Cu (fcc)
(d-electrons add covalent character)
hexagonal
bcc
CsCl (z=8): A = 1.762675
fcc
hcp
Chemical bonding and Crystal structure
1.3 Crystals
Chemical bonding and Crystal structure
1.3 Van der Waals Crystals
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45
Hydrogen-bridge bonding
Hydrogen: 1s1, Ip = 15.6 eV, ‘ion core’ (proton) with radius ~10-15 m
- Electron transfer to strongly electronegative atoms (F, O, ...)
- Small size of proton leads to hydrogen bond A-H...B
between two negatively charged atoms (double well potential)
Origin: Interplay between attractive (van der Waals) and repulsive forces
- van der Waals interaction:
zero point fluctuations of electrons lead to induced dipole forces
- Short range repulsive interaction due to Pauli exclusion principle
δ+
o
104.5
H
μ
δ+
H
2δ-
potential energy
Water
´Model potential: Lennard-Jones potential
O
proton position (on A-B)
1.3 Crystals
Chemical bonding and Crystal structure
1.3 Rare gas crystals
Chemical bonding and Crystal structure
44
46
Hydrogen-bridge bonding
Ice
closed packed
fcc (ABC – stacking)
A12 = 12.13; A6 = 14.45
hcp (AB – stacking)
DNA
hcp
fcc
fcc
fcc
fcc
1.3 Crystal binding energy / atom
Chemical bonding and Crystal structure
1.4 2d - Bravais lattice
47
metals: ~1 - 2 eV/atom
covalent: ~3 - 9 eV/atom
ionic: ~6-10 eV/atom
1.4 Bravais lattice
Chemical bonding and Crystal structure
49
- choice of unit cell is not unique
- primitive unit cell contains only one point
van der Waals: 20-200 meV/atom
hydrogen: ~100 meV/bond
Chemical bonding and Crystal structure
48
fcc
bcc
diamond
1.4
14 Bravais lattices
Chemical bonding and Crystal structure
50
Chemical bonding and Crystal structure
1.4 Cubic Bravais lattices
1.4 Crystal lattice
Chemical bonding and Crystal structure
51
53
32 crystallographic point groups
C4
C3
simple cubic
Schönflies symbols
C2
body-centered cubic
face-centered cubic
Diamond point group Td ; fcc and bcc point group Oh
Chemical bonding and Crystal structure
1.4 Cubic Bravais lattices
1.5 Crystal structure
Chemical bonding and Crystal structure
52
2 - dim.
Wigner-Seitz cell
54
Hexagonal close-packed
stacking ABABAB...
hcp
C4
hexagonal
bcc
C3
hcp
fcc
C2
fcc
stacking ABCABC...
Wigner-Seitz cell reflects symmetry of point group
Chemical bonding and Crystal structure
1.5 Crystal structure
1.5 Crystal structure
Chemical bonding and Crystal structure
55
Diamond structure C, Si, Ge
Zink sulfid structure ZnS, GaAs, AgI
57
Substitutional binary alloys
Ge/Si
Two elements crystallizing
with the same structure
NaCl
30% Ag/ 70% Cu
CsCl, NiAl, CuBe
86 % Ag
95 % Cu
Rasterelektronenmikroskop
Chemical bonding and Crystal structure
1.5 Crystal structure
56
Quasicrystals
long-range orientational, but non-periodic order
Penrose tiles
Al65Cu20Fe15
produced by cooling with 106 K/s
fivefold symmetry