Fundamentals of the Physics of Solids / 00-front-matter

1

IA

18 VIIIA

1

2

1

H

Periodic table of elements

He

1s1

2 IIA

13 IIIA 14 IVA

15 VA

16 VIA

17 VIIA

1s2

Hydrogen

Helium

3

4

5

6

7

8

9

10

2

Li

Be

B

C

N

O

F

Ne

[He] 2s1

1s2

2s22p1

2s22p2

2s22p3

2s22p4

2s22p5

2s22p6

Lithium

Beryllium

Boron

Carbon

Nitrogen

Oxygen

Fluorine

Neon

11

12

13

14

15

16

17

18

3

Na

Mg

8

9

10

Al

Si

P

S

Cl

Ar

[Ne] 3s1

3s2

3s23p1

3s23p2

3s23p3

3s23p4

3s23p5

3s23p6

Sodium

Magnesium 3 IIIB 4 IVB

5 VB

6 VIB

7 VIIB

VIIIB

11

IB

12

IIB

Aluminum

Silicon

Phosphorus

Sulfur

Chlorine

Argon

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

4

K

Ca

Sc

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Zn

Ga

Ge

As

Se

Br

Kr

[Ar] 4s1

4s2

3d14s2

3d24s2

3d34s2

3d54s1

3d54s2

3d64s2

3d74s2

3d84s2

3d104s1

3d104s2

3d104s24p1

3d104s24p2

3d104s24p3

3d104s24p4

3d104s24p5

3d104s24p6

Potassium

Calcium

Scandium

Titanium

Vanadium

Chromium

Manganese Iron

Cobalt

Nickel

Copper

Zinc

Gallium

Germanium

Arsenic

Selenium

Bromine

Krypton

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

5

Rb

Sr

Y

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Cd

In

Sn

Sb

Te

I

Xe

[Kr] 5s1

5s2

4d15s2

4d25s2

4d45s1

4d54s1

4d55s2

4d75s1

4d85s1

4d10

4d105s1

4d105s2

4d105s25p1

4d105s25p2

4d105s25p3

4d105s25p4

4d105s25p5

4d105s25p6

Rubidium

Strontium

Yttrium

Zirconium

Niobium

Molybdenum Technetium Ruthenium Rhodium Palladium

Silver

Cadmium

Indium

Tin

Antimony

Tellurium

Iodine

Xenon

55

56

57

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

6

Cs

Ba

La

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

[Xe] 6s1

6s2

5d16s2

5d26s2

5d36s2

5d46s2

5d56s2

5d66s2

5d76s2

5d96s1

5d106s1

5d106s2

5d106s26p1

5d106s26p2

5d106s26p3

5d106s26p4

5d106s26p5

5d106s26p6

Cesium

Barium

Lanthanum

Hafnium

Tantalum

Tungsten

Rhenium

Osmium

Iridium

Platinum

Gold

Mercury

Thallium

Lead

Bismuth

Polonium

Astatine

Radon

87

88

89

104

105

106

107

108

109

110

111

7

Fr

Ra

Ac

Rf

Db

Sg

Bh

Hs

Mt

Ds

Rg

[Rn] 7s1

7s2

6d17s2

6d27s2

6d37s2

6d47s2

6d57s2

6d67s2

6d77s2

6d97s1

6d107s1

Francium

Radium

Actinium

Rutherfordium

Dubnium

Seaborgium Bohrium

Hassium

Meitnerium Darmstadtium Roentgenium

Lanthanoids

58

59

60

61

62

63

64

65

66

67

68

69

70

71

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

4f26s2

4f36s2

4f46s2

4f56s2

4f66s2

4f76s2

4f75d16s2

4f96s2

4f106s2

4f116s2

4f126s2

4f136s2

4f146s2

4f145d16s2

Cerium

Praseodymium Neodymium Promethium Samarium

Europium Gadolinium

Terbium

Dysprosium

Holmium

Erbium

Thulium

Ytterbium

Lutetium

90

91

92

93

94

95

96

97

98

99

100

101

102

103

Actinoids

Th

Pa

U

Np

Pu

Am

Cm

Bk

Cf

Es

Fm

Md

No

Lr

6d27s2

5f26d17s2

5f36d17s2

5f57s2

5f67s2

5f77s2

5f76d17s2

5f86d17s2

5f107s2

5f117s2

5f127s2

5f137s2

5f147s2

5f146d17s2

Thorium

Protactinium

Uranium

Neptunium

Plutonium Americium

Curium

Berkelium Californium Einsteinium

Fermium

Mendelevium

Nobelium

Lawrencium

Fundamentals of the Physics of Solids

Jeno˝ Sólyom

Fundamentals

of the Physics

of Solids

Volume I

Structure and Dynamics

Translated by Attila Piróth

With 240 Figures and 50 Tables

BCA

Author

Professor Jeno˝ Sólyom

Research Institute for Solid State Physics and Optics

P.O.Box 49 1525 Budapest Hungary

and

Department of Physics Eötvös Loránd University 1117 Budapest

Pázmány sétány 1/A Hungary [e-mail]

Translator

Attila Piróth

www.pirothattila.com

Title of the Hungarian edition: A Modern Szilárdtestfizika Alapjai I. A szilárd testek szerkezete és dinamikája, ISBN 963 463 538 5, c ELTE Eötvös Kiadó 2002

Library of Congress Control Number: 2007929725

ISBN 978-3-540-72599-2 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law.

Springer is a part of Springer Science+Business Media springer.com

c Springer-Verlag Berlin Heidelberg 2007

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting: by the author, the translator, and Integra, India using a Springer LATEX macro package Cover design: F. Steinen, eStudio Calamar, Girona/Spain.

Printed on acid-free paper

SPIN: 11429470

5 4 3 2 1 0

To Márta, Gyöngyvér, Tünde, and lringó

Preface

The reader is holding the first volume of a three-volume textbook on solidstate physics. This book is the outgrowth of the courses I have taught for many years at Eötvös University, Budapest, for undergraduate and graduate students under the titles Solid-State Physics and Modern Solid-State Physics.

The main motivation for the publication of my lecture notes as a book was that none of the truly numerous textbooks covered all those areas that I felt should be included in a multi-semester course. Especially, if the course strives to present solid-state physics in a unified structure, and aims at discussing not only classic chapters of the subject matter but also (in more or less detail) problems that are of great interest for today’s researcher as well. Besides, the book presents a much larger material than what can be covered in a twoor three-semester course. In the first part of the first volume the analysis of crystal symmetries and structure goes into details that certainly cannot be included in a usual course on solid-state physics. The same applies, among others, to the discussion of the methods used in the determination of band structure, the properties of Fermi liquids and non-Fermi liquids, and the theory of unconventional superconductors in the second and third volumes. These parts can be assigned as supplementary reading for interested students, or can be discussed in advanced courses.

The line of development and the order of the chapters are based on the prerequisites for understanding each part. Therefore a gradual shift can be observed in the style of the book. While the intermediate steps of calculations are presented in considerable detail and explanations are also more lengthy in the first and second volumes, they are much sparser and more concise in the third one, thus this volume relies more on the individual work of the students. On account of the prerequisites, certain topics have to be revisited. This is why magnetic properties are treated in three, and superconductivity in two parts. The magnetism of individual atoms is presented in an introductory chapter. The structure and dynamics of magnetically ordered systems built up of localized moments are best discussed after lattice vibrations, along the same lines. Magnetism is revisited in the third volume, where the role of electron–electron

VIII Preface

interactions is discussed in more detail. Similarly, the phenomenological description of superconductivity is presented after the analysis of the transport properties of normal metals, in contrast to them, while the microscopic theory is outlined later, when the e ects of interactions are discussed.

Separating the material into three similar-sized volumes is a necessity in view of the size of the material – but it also reflects the internal logical structure of the subject matter. At those universities where the basic course in solid-state physics runs for three semesters working through one volume per semester is a natural schedule. In this case the discussion of the electron gas – which is traditionally part of the introduction – is left for the second semester. This choice is particularly suited to curricula in which the course on solid-state physics is held parallel with quantum mechanics or statistical physics. If the lecturer feels more comfortable with the traditional approach, the discussion of the Drude model presented in the second volume can be moved to the beginning of the whole course. Nevertheless the discussion of the Sommerfeld model should be postponed until students have familiarized themselves with the fundamentals of statistical physics. For the same reason the lecturer may prefer to change the order of other chapters as well. Apart from the presentation of the consequences of translational symmetry, the topics discussed in Chapters 3, 4, and 6 can be deferred to a later time, when students have acquired a sound knowledge of quantum mechanics, atomic, and molecular physics. The consequences of translational symmetry can also be analyzed after the discussion of phonons. All this is, to a large extent, up to the personal preferences of the lecturer.

In presenting the field of solid-state physics, special emphasis has been laid on discussing the physical phenomena that can be observed in solids. Nevertheless I have tried to give – or at least outline – the theoretical interpretation for each phenomenon, too. As is common practice for textbooks, I have omitted precise references that would give the publication data of the discussed results. I have made exceptions only for figures taken directly from published articles. At the end of the first, introductory chapter I have given a list of textbooks and series on solid-state physics, while at the end of each subsequent chapter I have listed textbooks and review articles that present further details and references pertaining to the subject matter of the chapter in question.

Bulky as it might be, this three-volume treatise presents only the fundamentals of solid-state physics. Today, when articles about condensed matter physics fill tens of thousands of pages every year in Physical Review alone, it would be obviously overambitious to aim at more. Therefore, building on the foundations presented in this book students will have to acquire a substantial amount of extra knowledge before they can understand the subtleties of the topics in the forefront of today’s research. Nevertheless at the end of the third volume students will also appreciate the number of open questions and the necessity of further research.

Preface IX

A certain knowledge of quantum mechanics is a prerequisite for studying solid-state physics. Various techniques of quantum mechanics – above all fieldtheoretical methods and methods employed in solving many-body problems

– play an important role in present-day solid-state physics. Some essential details are listed in one of the appendices of the third volume, however, I have omitted more complicated calculations that would have required the application of the modern apparatus of many-body problems. This is especially true for the third volume, where central research topics of present-day solid-state physics are discussed, in which the theoretical interpretation of experimental results is often impossible without some extremely complex mathematical formulation.

The selection of topics obviously bears the stamp of the author’s own research interest, too. This explains why the discussion of certain important fields – such as the mechanical properties of solids, surface phenomena, amorphous systems or mesoscopic systems, to name but a few – have been omitted.

I have used the International System of Units (SI), and have given the equations of electromagnetism in rationalized form. Nonrationalized equations as well as gaussian CGS (and other) units are nevertheless still very much in use in the solid-state physics literature. This has been indicated at the appropriate places. On a few occasions I have also given the formulas obtained from nonrationalized equations. In addition to the fundamental physical constants used in solid-state physics, the commonest conversion factors are also listed in Appendix A. Only once have I deviated from standard practice, denoting Boltzmann’s constant by kB instead of k – reserving the latter for the wave number, which plays a central role in solid-state physics.

To give an impression of the usual values of the quantities occurring in solid-state physics, typical calculated values or measured data are often tabulated. To provide the most precise data available, I have relied on the Landolt– Börnstein series, the CRC Handbook of Chemistry and Physics, and other renowned sources. Since these data are for information only, I have not indicated either their error or in many cases the measurement temperature, and I have not mentioned when di erent measurement methods lead to slightly disparate results. As a rule of thumb, the error is usually smaller than or on the order of the last digit.

I would like to thank all my colleagues who read certain chapters and improved the text through their suggestions and criticism. Particular thanks go to professors György Mihály and Attila Virosztek for reading the whole manuscript. In spite of all e orts, some mistakes have certainly remained in the book. Obviously, the author alone bears the responsibility for them.

Special thanks are due to Károly Härtlein for his careful work in drawing the majority of the figures. The figures presenting experimental results are reproduced with the permission of the authors or the publishers. M. C. Escher’s drawings in Chapter 5 are reproduced with the permission of the copyright holder © 2006 The M. C. Escher Company-Holland. The challenge

XPreface

of translating the book from the Hungarian original was taken up by Attila Piróth. I acknowledge his work.

Finally, I am indebted to my family, to my wife and children, for their patience during all those years when I spent evenings and weekends with writing this book.

Budapest, May 2007

Jenő Sólyom

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 The Structure of Condensed Matter . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Characterization of the Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.1 Shortand Long-Range Order . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2Order in the Center-of-Mass Positions, Orientation,

and Chemical Composition . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Classification of Condensed Matter According to Structure . . . . 20 2.2.1 Solid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Liquid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.3 Mesomorphic Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 The Building Blocks of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Solids as Many-Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.1 The Hamiltonian of Many-Particle Systems . . . . . . . . . . 32 3.1.2 E ects of Applied Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1.3 Relativistic E ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 The State of Ion Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 Hund’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.2 Angular Momentum and Magnetic Moment . . . . . . . . . . 44 3.2.3 The Magnetic Hamiltonian of Atomic Electrons . . . . . . 46 3.2.4 Magnetization and Susceptibility . . . . . . . . . . . . . . . . . . . 47 3.2.5 Langevin or Larmor Diamagnetism . . . . . . . . . . . . . . . . . 49 3.2.6 Atomic Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.7 Van Vleck Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.8 Electron Spin Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 The Role of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3.1 Interaction with Nuclear Magnetic Moments . . . . . . . . . 68 3.3.2 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . 71 3.3.3 The Mössbauer e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72