Lecture IX Crystals dr hab. Ewa Popko

The Schrödinger equation The hydrogen atom The potential energy in spherical coordinates (The potential energy function is spherically symmetric.) Partial differential equation with three independent variables

S-states probability

P-states probability

Why Solids?  most elements are solid at room temperature  atoms in ~fixed position “simple” case - crystalline solid  Crystal Structure Why study crystal structures?  description of solid  comparison with other similar materials - classification  correlation with physical properties

Early ideas Crystals are solid - but solids are not necessarily crystalline Crystals have symmetry (Kepler) and long range order Spheres and small shapes can be packed to produce regular shapes (Hooke, Hauy) ?

Kepler wondered why snowflakes have 6 corners, never 5 or 7. By considering the packing of polygons in 2 dimensions, it can be shown why pentagons and heptagons shouldn’t occur. Empty space not allowed

CRYSTAL TYPES Three types of solids, classified according to atomic arrangement: (a) crystalline and (b) amorphous materials are illustrated by microscopic views of the atoms, whereas (c) polycrystalline structure is illustrated by a more macroscopic view of adjacent single-crystalline regions, such as (a).

quartz Crystal structure Amorphous structure

Definitions 1. The unit cell “The smallest repeat unit of a crystal structure, in 3D, which shows the full symmetry of the structure” The unit cell is a box with: 3 sides - a, b, c 3 angles - , ,  14 possible crystal structures (Bravais lattices)

3D crystal lattice cubic a = b = c  =  =  tetragonal a = b  c  =  =  = 90 o monoclinic a  b  c  =  = 90 o   90 o

orthorhombic a  b  c  =  =  = 90 o hexagonal a = b  c  =  = 90 o ;  = 120 o triclinic a  b  c       90 o trigonal (rhombohedral) a = b = c  =  =   90 o

Chemical bonding Types: Ionic bonding Covalent bonding Metallic bonding Van der Walls bonding + -

Metallic bond Atoms in group IA-IIB let electrons to roam in a crystal. Free electrons glue the crystal Na+ e-e- e-e- Attract Repel Additional binding due to interaction of partially filled d – electron shells takes place in transitional metals: IIIB - VIIIB

Core and Valence Electrons Simple picture. Metal have CORE electrons that are bound to the nuclei, and VALENCE electrons that can move through the metal. Most metals are formed from atoms with partially filled atomic orbitals. e.g. Na, and Cu which have the electronic structure Na 1s 2 2s 2 2p 6 3s 1 Cu 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 Insulators are formed from atoms with closed (totally filled) shells e.g. Solid inert gases He 1s 2 Ne 1s 2 2s 2 2p 6 Or form close shells by covalent bonding i.e. Diamond Note orbital filling in Cu does not follow normal rule

sodium ion (Na + ) Ionic bonding Metal atoms with 1 electron to lose can form ionic bonds with non-metal atoms which need to gain 1 electron: –Eg. sodium reacts with fluorine to form sodium fluoride: sodium atom (Na) fluoride ion (F - ) fluorine atom (F) So the formula for sodium fluoride is NaF

Examples of ionic bonding:NaCl Each sodium atom is surrounded by its six nearest neighbor chlorine atoms (and vice versa) Electronically – sodium has one electron in its outer shell: [Ne]3s 1 and Chlorine has 7 (out of 8 “available” electron positions filled in its outer shell) [Ne]3s 2 3p 5 Sodium “gives up” one of its electrons to the chlorine atom to fill the shells resulting in [Ne] [Ar] cores with Na + and Cl - ions Coulombic attraction with tightly bound electron cores

Properties of the ionic crystals medium cohesive energy (2-4 eV/ atom). –low melting and boiling temp.. Low electrical conductivity. –(the lack of the free electrons). Transparent for VIS light –( energy separation between neighbouring levels > 3 eV) Easily dissolved in water. –Electrical dipoles of water molecules attract the ions

Covalent bonding: molecular orbitals Consider an electron in the ground, 1s, state of a hydrogen atom The Hamiltonian is The expectation value of the electron energy is This gives = E 1s = -13.6eV + E 1s V(r)  (r)

Hydrogen Molecular Ion Consider the H 2 + molecular ion in which one electron experiences the potential of two protons. The Hamiltonian is We approximate the electron wavefunctions as and p+ e- R r

Bonding and anti-bonding states Expectation values of the energy are: E = E 1s –  (R) for E = E 1s +  (R) for  (R) - a positive function Two atoms: original 1s state leads to two allowed electron states in molecule. Find for N atoms in a solid have N allowed energy states V(r)

1s 2s bonding Anti-bonding bonding covalent bonding – H 2 molecule

R0R nuclear separation (nm) energy(eV) parallel spin antiparallel spin system energy (H 2 )

Covalent bonding Atoms in group III, IV,V,&VI tend to form covalent bond Filling factor T. :0.34 F.C.C :0.74

Covalent bonding Crystals: C, Si, Ge Covalent bond is formed by two electrons, one from each atom, localised in the region between the atoms (spins of electrons are anti-parallel ) Example: Carbon 1S 2 2S 2 2p 2 C C Diamond: tetrahedron, cohesive energy 7.3eV 3D 2D

Covalent Bonding in Silicon Silicon [Ne]3s 2 3p 2 has four electrons in its outermost shell Outer electrons are shared with the surrounding nearest neighbor atoms in a silicon crystalline lattice Sharing results from quantum mechanical bonding – same QM state except for paired, opposite spins (+/- ½ ħ)

Properties of the covalent crystals Strong, localized bonding. High cohesive energy (4-7 eV/atom). –High melting and boiling temperature. Low conductivity.

ionic – covalent mixed