Lecture IX

Crystals

dr hab. Ewa Popko

zyxEzyxzyxVzyxm

,,,,,,2 2

2

2

2

2

22

zyxEzyxH ,,,,ˆ

The Schrödinger equationThe hydrogen atom

The potential energy in

spherical coordinates

(The potential energy function is spherically symmetric.)

Partial differential equation with three independent variables

r

erV

2

04

1)(

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

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)

3D crystal lattice

cubica = b = c = =

tetragonala = b c = = = 90o

monoclinica b c = = 90o

90o

orthorhombica b c = = = 90o

hexagonala = b c = = 90o; = 120o

triclinica b c 90o

trigonal (rhombohedral)a = b = c = = 90o

Chemical bonding

Types:

Ionic bonding

Covalent bonding

Metallic bonding

Van der Walls bonding + -+ -

Metallic bond

Atoms in group IA-IIB let electrons to roam ina crystal. Free electrons glue the crystal

Na+ Na+

e-

e-

Attract

Attract

Attract

AttractRepelRepel

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 1s2 2s2 2p6 3s1

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

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]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

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 orbitalsConsider 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

o

2

4e = where

RadiusBohr theis a where a 1

= (r) i.e. oo e ar/-3/2 o

r

- 2m

- = H

22

(r)dV H (r) = > E <

+

E1s

V(r)

(r)

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

|R - r|-

r -

2m

- = )rU( +

2m

- = H

2222

] + A[ |)] R - r(| + )r([ A = )r( 21

] B[ |)]R - r(| )r([ B = )r( 21

p+ p+

e-

R

r

Bonding andanti-bonding states Expectation values of the energy are:

E = E1s – (R) for

E = E1s + (R) for

(R) - a positive function

Two atoms: original 1s stateleads to two allowed electron states in molecule.

Find for N atoms in a solid have N allowed energy states

)r(

)r(

)r(

-6 -4 -2 0 2 4 6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

r

-6 -4 -2 0 2 4 6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

r

V(r)

2)r(

)r(

1s

2s

bonding

Anti-bonding

Anti-bonding

bonding

covalent bonding – H2 molecule

• 8

6

4

2

0

-2

-4

-6

R00.1 0.2 0.3 0.4

nuclear separation (nm)

ener

gy(e

V)

parallel spin

antiparallel spin

system energy (H2)

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 areanti-parallel )

Example: Carbon 1S2 2S2 2p2

C C

Diamond: tetrahedron, cohesive energy 7.3eV

3D 2D

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)

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