Lecture IV Crystals dr hab. Ewa Popko

1. Lecture IVCrystalsdr hab. Ewa Popko

2. 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

3. 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) ?

4. 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

5. 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).

6. Crystal structure Amorphous structure quartz

7. Definitions1. 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)

8. 3D crystal lattice monoclinic a  b  c  =  = 90o   90o cubic a = b = c  =  =  tetragonal a = b  c  =  =  = 90o

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

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

11. Bonding in Solids

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

13. Note orbital filling in Cu does not follow normal rule Core and Valence Electrons Most metals are formed from atoms with partially filled atomic orbitals. e.g. Na, and Cu which have the electronic structure Na 1s2 2s2 2p63s1 Cu 1s2 2s2 2p6 3s23p63d104s1 Insulators are formed from atoms with closed (totally filled) shells e.g. Solid inert gases He 1s2 Ne 1s2 2s2 2p6 Or form close shells by covalent bonding i.e. Diamond Simple picture. Metal have CORE electrons that are bound to the nuclei, and VALENCE electrons that can move through the metal.

14. Examples of 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: So the formula for sodium fluoride is NaF fluoride ion (F-) sodium atom (Na) sodium ion (Na+) fluorine atom (F)

15. Examples of ionic bonding

16. 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]3s1 and Chlorine has 7 (out of 8 “available” electron positions filled in its outer shell) [Ne]3s23p5 • 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

17. NaCl • Potential energy: • a- Madelung constant, m – integer number • for ro, the equilibrium positionbetween the ions: U0 is the cohesive energy, i.e. the energy per ion to remove the ion out of the crystal.

18. Repulsive potential 1/rm Total potential Attractive potential -1/r Ionic bonding

19. 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

20. E1s F(r) + V(r) 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> = E1s = -13.6eV

21. e- r R p+ p+ Hydrogen Molecular Ion Consider the H2+ molecular ion in which one electron experiences the potential of two protons. The Hamiltonian is We approximate the electron wavefunctions as and

22. V(r) Bonding andanti-bonding states Expectation values of the energy are: E = E1s – g(R) for E = E1s + g(R) for g(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

23. Anti-bonding 2s bonding Anti-bonding 1s bonding covalent bonding – H2 molecule

24. 8 6 energy(eV) 4 parallel spin 2 0 -2 antiparallel spin -4 -6 R0 0.1 0.2 0.3 0.4 nuclear separation(nm) system energy (H2)

25. 3D 2D 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 1S2 2S2 2p2 C C Diamond: tetrahedron, cohesive energy 7.3eV

26. Covalent Bonding in Silicon • Silicon [Ne]3s23p2 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)

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

28. diamond lattice

29. zinc blend crystals (ZnS, GaAs) As

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

31. ionic – covalent mixed