tructure - michael goerz

191. Chemical bonding and crystal structure

Scanning electron microscopy Cleaved surface

Scanning tunneling microscopy

Ni surfaceZnO, TiO2, NiO, NaCl, Si, Ge, GaAs, InP

Crystals are build by „small“repeating units (= basis) like atoms and molecules

201. Chemical bonding and crystal structure

1.1 Atoms

211.1 Atoms

only for s-waves (l=0) Rn0(r = 0) ≠ 0

Hydrogen atom

221.1 Atoms

Polar plot

231.1 Atoms Chemical bonding and Crystal structure

Ni (Z = 28), Zeff = 8, d = 2.49 Å


Hydrogen atom

10950 MHz = 0.045 meV, beachte aber ΔE/E ~ Z2α2


corresponds to relativistic corrections whichresult from Dirac´s equation

i) Relativistic, kinetic energyii) Spin-orbit coupling (LS – coupling)iIi) Darwin term

(see, e.g., p1/2 - p3/2 splitting in Nickel)


i) Relativistic, kinetic energy

Estimate of (v/c)2 via uncertainty relation (a = Bohr radius):


ii) Classical Hamiltonian for the spin-orbit interaction B field from the proton in the electron's rest frame is

perturbation Hamiltonian

recall


iii) Darwin term

Flickering motion of electron in nucleus leads to average potential

contribution only for s-waves with finite amplitude at x = 0


Rumpfniveaus

Valenzniveaus

Nicht die Größe der Bindungsenergiesondern der Grad der Lokalisationder Wellenfunktion entscheidet.


Hund‘sche Regeln für teilweise gefüllte Schalen (Valenzelektronen)

1. S maximal2. L maximal3. J = |L-S| für nicht mehr als halbgefüllte Schalen4. J = L+S für mehr als halbgefüllte Schalen

Ni: Ar 3d8 4s2

1. S = 12. L = 33. J = 4

2S+1LJ = 3F4


He 1s2 – excited states 2S+1LJ

2s+1 = 1 singulett state antisymmetric2s+1 = 3 triplett state symmetric

attractive Coulombinteraction:

„s-wave in core, p-wave not“

Spatial part accordingly (Hund‘s rules !)


Aufbauprinzip

331.2 Molecules

bonding (attraction) due to valence electronsPauli repulsion between neighbouring atoms

equilibrium distance r0 (related to lattice parameter)

r = r0r > r0

r

U(r)

r < r0

Chemical bonding and Crystal structure

341.2 Molecules

Hydrogen ion H2-


351.2 Molecules Chemical bonding and Crystal structure

LCAO – linear combination of atomic orbitals

361.2 Molecules Chemical bonding and Crystal structure

Molecular orbitals (e.g. benzene)

LCAO

371.3 Crystals Chemical bonding and Crystal structure

NaCl CsCl

Ionic bonding (6 -10 eV)Covalent bonding (3-9 eV)

Hydrogen-bridge bonding (0.1 eV)

Van der Waals bonding (< 0.2 eV)

Metalic bonding (1-2 eV)

Si

381.3 Crystals

graphite: planar sp2 structure

diamond, silicon: tetrahedral sp3 structure


Covalent bonding

Electron density (contour plot)

391.3 Ion Crystals Chemical bonding and Crystal structure

Ionic bonding - energetics

ionization energy

(eV)

electron affinity (eV)

ionization energy

(eV)

electron affinity (eV)

Li 5.39 0.62 F 17.4 3.40 Na 5.14 0.55 Cl 13.0 3.61 K 4.34 0.5 Br 11.8 3.36

Rb 4.18 I 10.5 3.06

Na + Cl → Na+ + Cl- + 1.53 eVelectrostatic interaction between ions

Na+ and Cl-: 5.1 eV (r0 ~ 2.8 Å)total energy gain of 3.57 eV

Nearly spherical charge distributions (closed shell)

Electron density (contour plot)


Ionic bonding - electrostatic energy (Born-Mayer potential)

A Madelung constant, z coordination number

NaCl (z = 6): A = 1.747565

CsCl (z=8): A = 1.762675


Parameters ρ and B of the repulsive potential determined by equilibrium distance r0 and compressibility κ

stability depends on the ratio rA /rB of ionic radii:CsCl ↔ NaCl ↔ ZnS

1.37 < rA /rB < 2.44

Different structures of ionic crystals:

421.3 Crystals - metals Chemical bonding and Crystal structure

Metallic bonding

Overlapping wave functions form delocalized states (Bloch states)

- 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)

bcc fcchexagonal

hcp

431.3 Crystals

Hydrogen-bridge bonding

proton position (on A-B)

pote

ntia

l en

ergy

μ

104.5o

Hδ+ O

δ-2

Hδ+

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)

Water


441.3 Crystals

Ice

DNA


Hydrogen-bridge bonding

451.3 Van der Waals Crystals

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

´Model potential: Lennard-Jones potential


461.3 Rare gas crystals

closed packedfcc (ABC – stacking)A12 = 12.13; A6 = 14.45hcp (AB – stacking)

hcpfccfccfccfcc


471.3 Crystal binding energy / atom

metals: ~1 - 2 eV/atomcovalent: ~3 - 9 eV/atomionic: ~6-10 eV/atom

van der Waals: 20-200 meV/atomhydrogen: ~100 meV/bond


481.4 Bravais lattice

fcc

bcc

diamond


491.4 2d - Bravais lattice Chemical bonding and Crystal structure

- choice of unit cell is not unique- primitive unit cell contains only one point

501.4 14 Bravais lattices Chemical bonding and Crystal structure

511.4 Cubic Bravais lattices Chemical bonding and Crystal structure

face-centered cubic

body-centered cubic

simple cubic

521.4 Cubic Bravais lattices Chemical bonding and Crystal structure

Wigner-Seitz cell2 - dim.

bcc fcc

Wigner-Seitz cell reflects symmetry of point group

C4

C3

C2

531.4 Crystal lattice Chemical bonding and Crystal structure

32 crystallographic point groups

Schönflies symbols

Diamond point group Td ; fcc and bcc point group Oh

C4

C3

C2

541.5 Crystal structure Chemical bonding and Crystal structure

Hexagonal close-packedstacking ABABAB...

stacking ABCABC...

fcc

hcp

hexagonal

hcp


Diamond structure C, Si, Ge Zink sulfid structure ZnS, GaAs, AgI

NaCl CsCl, NiAl, CuBe


Quasicrystalslong-range orientational, but non-periodic order

Penrose tiles

Al65Cu20Fe15produced by cooling with 106 K/s

fivefold symmetry


Substitutional binary alloysGe/Si

Two elements crystallizingwith the same structure

30% Ag/ 70% Cu

86 % Ag

Rasterelektronenmikroskop

95 % Cu

tructure - michael goerz

Documents