Solid State Physics Course - Full Course.pdf - kutas

Chapter 1: Chemical Bonding

Linus Pauling (1901{1994)

December 28, 2001

Contents

1 The development of Bands and their ¯lling

2 Di®erent Types of Bonds

2.1 Covalent Bonding . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Ionic Bonding . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Madelung Sums . . . . . . . . . . . . . . . . . . . 17

2.3 Metallic Bonding . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Van der Waals Bonds . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Van der Waals-London Interaction . . . . . . . . 21

Periodic Table

H 1

Hydrogen

1.008

He 2

Helium

4.003

Li 3

Lithium

6.941

Be 4

Beryllium

9.012

O 8

Oxygen

15.999

Ne 10

Neon

20.180

Boron

10.811

Carbon

12.011

Nitrogen

14.007

Fluorine

18.998

Na 11

Sodium

22.990

Mg 12

Magnesium

24.305

Al 13

Aluminum

26.982

Si 14

Silicon

28.086

P 15

Phosphorous

30.974

S 16

Sulfur

32.066

Cl 17

Chlorine

35.453

Ar 18

Argon

39.948

K 19

Potassium

39.098

Ca 20

Calcium

40.078

Sc 21

Scandium

44.956

Ti 22

Titanium

47.88

V 23

Vanadium

50.942

Cr 24

Chromium

51.996

Mn 25

Manganese

54.938

Fe 26

Iron

55.847

Co 27

Cobalt

58.933

Ni 28

Nickel

58.69

Cu 29

Copper

63.546

Zn 30

Zinc

65.39

Ga 31

Gallium

69.723

Ge 32

Germanium

72.61

As 33

Arsenic

74.922

Se 34

Selenium

78.96

Br 35

Bromine

79.904

Kr 36

Krypton

83.80

Rb 37

Rubidium

85.468

Sr 38

Strontium

87.62

Y 39

Yttrium

88.906

Zr 40

Zirconium

91.224

Nb 41

Niobium

92.906

Mo 42

Molybdenum

95.94

Tc 43

Technetium

(98)

Ru 44

Ruthenium

101.07

Rh 45

Rhodium

102.906

Pd 46

Palladium

106.42

Ag 47

Silver

107.868

Cd 48

Cadmium

112.411

In 49

Indium

114.82

Sn 50

Tin

118.71

Sb 51

Antimony

121.75

Te 52

Tellurium

127.60

Xe 54

Xenon

131.29

Iodine

126.905

Cs 55

Caesium

132.905

Ba 56

Barium

137.327

Lu 71

Lutetium

174.967

Hf 72

Halfnium

178.49

Ta 73

Tantalum

180.948

W 74

Tungsten

183.85

Re 75

Rhenium

186.207

Os 76

Osmium

190.2

Ir 77

Iridium

192.22

Pt 78

Platinum

195.08

Au 79

Gold

196.967

Hg 80

Mercury

200.59

Tl 81

Thallium

204.383

Pb 82

Lead

207.2

Bi 83

Bismuth

208.980

Po 84

Polonium

(209)

At 85

Astatine

(210)

Rn 86

Radon

(222)

Fr 87

Francium

(223)

Ra 88

Radium

226.025

Lr 103

Lawrencium

(260)

La 57

Lanthanum

138.906

Ce 58

Cerium

140.115

Pr 59

Praseodymium

140.908

Nd 60

Neodymium

144.24

Pm 61

Promethium

(145)

Sm 62

Samarium

150.36

Eu 63

Europium

151.965

Gd 64

Gadolinium

157.25

Tb 65

Terbium

158.925

Dy 66

Dysprosium

162.50

Ho 67

Holmium

164.93

Er 68

Erbium

167.26

Tm 69

Thulium

168.934

Yb 70

Ytterbium

173.04

Ac 89

Actinium

227.028

Th 90

Thorium

232.038

Pa 91

Protactinium

231.036

U 92

Uranium

238.029

Np 93

Neptunium

237.048

Pu 94

Plutonium

(244)

Am 95

Americium

(243)

Cm 96

Curium

(247)

Bk 97

Berkelium

(247)

Cf 98

Californium

(251)

Es 99

Einsteinium

(252)

Fm 100

Fermium

(257)

Md 101

Mendelevium

(258)

No 102

Nobelium

(259)

Solid state physics is the study of mainly periodic systems (or things

that are close to periodic) in the thermodynamic limit ¼ 10 21 atoms/cm 3 .

At ¯rst this would appear to be a hopeless task, to solve such a large

system.

Figure 1: The simplest model of a solid is a periodic array of valance orbitals embedded

in a matrix of atomic cores.

However, the self-similar, translationally invariant nature of the pe-

riodic solid and the fact that the core electrons are very tightly bound

at each site (so we may ignore their dynamics) makes approximate so-

lutions possible. Thus, the simplest model of a solid is a periodic array

of valance orbitals embedded in a matrix of atomic cores. Solving the

problem in one of the irreducible elements of the periodic solid (cf. one

of the spheres in Fig. 1), is often equivalent to solving the whole sys-

tem. For this reason we must study the periodicity and the mechanism

(chemical bonding) which binds the lattice into a periodic structure.

The latter is the emphasis of this chapter.

1 The development of Bands and their ¯lling

nl elemental solid

1s H,He

2s Li,Be

2p B!Ne

3s Na,Mg

3p Al!Ar

4s K,Ca

3d transition metals Sc!Zn

4p Ga!Kr

5s Rb,Sr

4d transition metals Y!Cd

5p In-Xe

6s Cs,Ba

4f Rare Earths (Lanthanides) Ce!Lu

5d Transition metals La!Hg

6p Tl!Rn

Table 1: Orbital ¯lling scheme for the ¯rst few atomic orbitals

We will imagine that each atom (cf. one of the spheres in Fig. 1)

is composed of Hydrogenic orbitals which we describe by a screened

Coulomb potential

¡ Z nl e 2

V (r) =

(1)

where Z nl describes the e®ective charge seen by each electron (in prin-

ciple, it will then be a function of n and l). As electrons are added to

the solid, they then ¯ll up the one-electron states 1s 2s 3s 3p 3d 4s 4p

4d 4f¢¢¢, where the correspondence between spdf and l is s ! l = 0,

p ! l = 1, etc. The elemental solids are then made up by ¯lling these

orbitals systematically (as shown in Table 1) starting with the lowest

energy states (where E nl = me 4 Z nl

tion of n as we would expect from a simple Hydrogenic model with

E n = mZ 2 e 4

2h 2 n 2

(with all electrons seeing the same nuclear charge Z). For

example, the 5s orbitals ¯ll before the 4d! This is because the situation

is complicated by atomic screening. I.e. s-electrons can sample the core

and so are not very well screened whereas d and f states face the an-

gular momentum barrier which keeps them away from the atomic core

so that they feel a potential that is screened by the electrons of smaller

n and l. To put is another way, the e®ective Z 5s is larger than Z 4d . A

schematic atomic level structure, accounting for screening, is shown in

Fig. 2.

Now let's consider the process of constructing a periodic solid. The

simplest model of a solid is a periodic array of valence orbitals embed-

ded in a matrix of atomic cores (Fig. 1). As a simple model of how

2h 2 n 2

Note that for large n, the orbitals do not ¯ll up simply as a func-

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