1.1 Atoms. 1.1 Atoms

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1. Chemical bonding and crystal structure 19 21 Hydrogen

ZnO, TiO 2, NiO, NaCl, Si, Ge, GaAs, InP Crystals are build

only for s-waves (l=0) R n0 (r = 0) 0 Scanning tunneling

1 1. Chemical bonding and crystal structure Hydrogen atom Scanning electron microscopy Ni surface Cleaved surface ZnO, TiO 2, NiO, NaCl, Si, Ge, GaAs, InP Crystals are build by small repeating units (= basis) like atoms and molecules only for s-waves (l=0) R n0 (r = 0) 0 Scanning tunneling microscopy 1. Chemical bonding and crystal structure Polar plot

2 23 25 Ni (Z = 28), Z eff = 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., p 1/2 -p 3/2 splitting in Nickel) Hydrogen atom i) Relativistic, kinetic energy Estimate of (v/c) 2 via uncertainty relation (a = Bohr radius): MHz = mev, beachte aber ΔE/E ~ Z 2 α 2

3 27 ii) Classical Hamiltonian for the spin-orbit interaction B field from the proton in the electron's rest frame is Valenzniveaus 29 perturbation Hamiltonian Nicht die Größe der Bindungsenergie sondern der Grad der Lokalisation der Wellenfunktion entscheidet. Rumpfniveaus recall iii) Darwin term Flickering motion of electron in nucleus leads to average potential 28 Hund sche Regeln für teilweise gefüllte Schalen (Valenzelektronen) 1. S maximal 2. L maximal 3. J = L-S für nicht mehr als halbgefüllte Schalen 4. J = L+S für mehr als halbgefüllte Schalen 30 contribution only for s-waves with finite amplitude at x = 0 Ni: Ar 3d 8 4s 2 1. S = 1 2. L = 3 3. J = 4 2S+1 L J = 3 F 4

4 Molecules 33 He 1s 2 excited states 2S+1 L J U(r) r < r 0 attractive Coulomb interaction: s-wave in core, p-wave not r = r 0 r > r 0 r 2s+1 = 1 singulett state antisymmetric 2s+1 = 3 triplett state symmetric Spatial part accordingly (Hund s rules!) bonding (attraction) due to valence electrons Pauli repulsion between neighbouring atoms equilibrium distance r 0 (related to lattice parameter) Molecules 34 Aufbauprinzip Hydrogen ion H 2 -

5 1.2 Molecules LCAO linear combination of atomic orbitals Crystals Covalent bonding (3-9 ev) Ionic bonding (6-10 ev) 37 Si NaCl CsCl Metalic bonding (1-2 ev) Hydrogen-bridge bonding (0.1 ev) Van der Waals bonding (< 0.2 ev) 1.2 Molecules Molecular orbitals (e.g. benzene) Crystals Covalent bonding 38 LCAO Electron density (contour plot) graphite: planar sp 2 structure diamond, silicon: tetrahedral sp 3 structure

1.3 Ion Crystals 39 1.3 Ion Crystals 41 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.

57 ev Parameters ρ and B of the repulsive potential determined by equilibrium distance r 0 and compressibility κ Nearly spherical charge distributions (closed shell) Different structures of

3 Ion Crystals Ionic bonding - electrostatic energy (Born-Mayer potential) 40 1.3 Crystals - metals Metallic bonding 42 A Madelung constant, z coordination number NaCl (z = 6): A = 1.

6 1.3 Ion Crystals Ion Crystals 41 Ionic bonding - energetics ionization energy (ev) electron affinity (ev) ionization energy (ev) electron affinity (ev) Li F Na Cl K Br Rb 4.18 I Na + Cl Na + + Cl ev electrostatic interaction between ions Na + and Cl - : 5.1 ev (r 0 ~ 2.8 Å) total energy gain of 3.57 ev Parameters ρ and B of the repulsive potential determined by equilibrium distance r 0 and compressibility κ Nearly spherical charge distributions (closed shell) Different structures of ionic crystals: stability depends on the ratio r A /r B of ionic radii: CsCl NaCl ZnS 1.37 < r A /r B < 2.44 Electron density (contour plot) 1.3 Ion Crystals Ionic bonding - electrostatic energy (Born-Mayer potential) Crystals - metals Metallic bonding 42 A Madelung constant, z coordination number NaCl (z = 6): A = 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 (), Cu () (d-electrons add covalent character) bcc hexagonal hcp CsCl (z=8): A =

1.3 Crystals Hydrogen-bridge bonding 43 1.3 Van der Waals Crystals 45 Hydrogen: 1s 1, I p = 15.

..) - Small size of proton leads to hydrogen bond A-H.

forces - van der Waals interaction: zero point fluctuations of electrons lead to induced dipole forces - Short range repulsive

7 1.3 Crystals Hydrogen-bridge bonding Van der Waals Crystals 45 Hydrogen: 1s 1, I p = 15.6 ev, ion core (proton) with radius ~10-15 m Water - 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 μ o H δ+ H δ+ O 2δ- potential energy Model potential: Lennard-Jones potential proton position (on A-B) 1.3 Crystals Hydrogen-bridge bonding Rare gas crystals 46 Ice closed packed (ABC stacking) A 12 = 12.13; A 6 = hcp (AB stacking) DNA hcp

8 1.3 Crystal binding energy / atom d - Bravais lattice 49 metals: ~1-2 ev/atom covalent: ~3-9 ev/atom ionic: ~6-10 ev/atom van der Waals: mev/atom hydrogen: ~100 mev/bond - choice of unit cell is not unique - primitive unit cell contains only one point 1.4 Bravais lattice Bravais lattices 50 bcc diamond

cubic face-centered cubic Diamond point group T d ; and bcc point group O h 1.4 Cubic Bravais lattices 52 1.

9 1.4 Cubic Bravais lattices Crystal lattice crystallographic point groups C 4 C 3 simple cubic Schönflies symbols C 2 body-centered cubic face-centered cubic Diamond point group T d ; and bcc point group O h 1.4 Cubic Bravais lattices Crystal structure 54 Wigner-Seitz cell 2 - dim. Hexagonal close-packed stacking ABABAB... C 4 hcp hexagonal bcc C 3 hcp C 2 stacking ABCABC... Wigner-Seitz cell reflects symmetry of point group

10 1.5 Crystal structure Crystal structure 57 Diamond structure C, Si, Ge Zink sulfid structure ZnS, GaAs, AgI Substitutional binary alloys Ge/Si Two elements crystallizing with the same structure NaCl CsCl, NiAl, CuBe 30% Ag/ 70% Cu 86 % Ag 95 % Cu Rasterelektronenmikroskop 1.5 Crystal structure Quasicrystals long-range orientational, but non-periodic order 56 Penrose tiles Al 65 Cu 20 Fe 15 produced by cooling with 10 6 K/s fivefold symmetry

Quantum Condensed Matter Physics Lecture 4

Quantum Condensed Matter Physics Lecture 4 David Ritchie QCMP Lent/Easter 2019 http://www.sp.phy.cam.ac.uk/drp2/home 4.1 Quantum Condensed Matter Physics 1. Classical and Semi-classical models for electrons