Introduction to the Physics of Matter - Basic Atomic, Molecular, and Solid-State Physics

298 Pages·2020·12.694 MB·English

by  Nicola Manini

Undergraduate Lecture Notes in Physics
Nicola Manini
Introduction
to the Physics
of Matter
Basic Atomic, Molecular, and
Solid-State Physics
Second Edition
Undergraduate Lecture Notes in Physics
Series Editors
Neil Ashby, University of Colorado, Boulder, CO, USA
WilliamBrantley,DepartmentofPhysics,FurmanUniversity,Greenville,SC,USA
MatthewDeady,PhysicsProgram,BardCollege,Annandale-on-Hudson,NY,USA
Michael Fowler, Department of Physics, University of Virginia, Charlottesville,
VA, USA
Morten Hjorth-Jensen, Department of Physics, University of Oslo, Oslo, Norway
Michael Inglis, Department of Physical Sciences, SUNY Suffolk County
Community College, Selden, NY, USA
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Nicola Manini
Introduction to the Physics
of Matter
Basic Atomic, Molecular, and Solid-State
Physics
Second Edition
123
NicolaManini
Department ofPhysics
University of Milan
Milan,Italy
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
ISBN978-3-030-57242-6 ISBN978-3-030-57243-3 (eBook)
https://doi.org/10.1007/978-3-030-57243-3
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Preface
This book fulfills a twofold purpose: to provide a pedagogic panorama of the
current microscopic understanding of the basic physics of matter and to help stu-
dentstoacquireaquantitativefeelingofthetypicalordersofmagnitudeofthemain
physical quantities (energy, time, temperature, length) involved in the specific
conditions relevant for “matter” in its atomic, molecular, and condensed forms.
Both tasks are favored by keeping structurally and conceptually well distinct the
analysis of the adiabatically separate motions of electrons and of atoms. This dis-
tinct treatment is organized in close parallel for molecules and for solids.
While keeping different degrees of freedom well distinct, formal likeness is
noted whenever useful, following the standard strategy of similar solutions for
similar equations. Noteworthy examples of this approach include the spherically
symmetricmotionofelectronsinatomsandofnucleiindiatomicmolecules,aswell
as applications of the Fermi-gas model to electrons in metals and to fluid 3He.
This book includes several detailed derivations, when they are useful to
understand the physical reasons of certain facts. These details deserve being
understood and then forgotten. In contrast, a number of mathematical relations
summarize essential physical information and results, and are therefore worth
retaining. As a guide for the reader, a gray background highlights these essential
equations.
Numbers and orders of magnitude are important, at least as much as mathe-
matical derivations, probably more. A broad selection of numerical problems
invites the reader to familiarize with the conceptually simple but often practically
intricatenumericalevaluationsandunitconversionsrequiredtoreachquantitatively
correct estimates in real-life or laboratory conditions.
At variance with many textbooks in this field, the present one adopts SI units
throughout.Thischoicedoesnotonlyfollowtheinternationalrecommendation,but
alsohelpstocompareallresultswiththeoutputofinstruments.Theoneindulgence
to non-SI units is a frequent quotation of energies in eV, which represents a
practical unit for most atomic-scale phenomena and converts easily to Joule.
v
vi Preface
As a basic textbook, this one focuses on what is currently well under control: a
selection of well-established systems, phenomena, experimental techniques, and
conceptual schemes. From this panorama, the reader should be warned against
gatheringthefalseimpressionthatthelastwordhasbeensaidaboutthephysicsof
matter. On the contrary, physicists, chemists, materials scientists, and biologists
currently investigate matter in its multi-faceted forms and collect a wealth of
experimental data, for which understanding is often only qualitative and partial.
Creativity and insight help scientists to develop novel approximate schemes, or
models, to interpret these data and achieve a better understanding of the intimate
structure and dynamics of matter.
Not everything is equally important. For a reader wishing to focus on a bare
minimumcontent,whilestillgraspingthebasicsofthephysicsofmatter,theauthor
suggests to skip the following non-essential topics:
(cid:129) the spectral-broadening mechanisms—Sect. 1.2,
(cid:129) the hyperfine structure of H—Sect. 2.1.8,
(cid:129) perturbation theory applied to the 2-electron atom—Sect. 2.2.4,
(cid:129) the details of the variational calculation for H+ in Sect. 3.2.1,
2
(cid:129) the electronic molecular transitions discussed in Sect. 3.3,
(cid:129) the density-operator formalism—Sect. 4.1.2,
(cid:129) the foundation of the canonical ensemble—Sect. 4.2 before Eq. (4.10),
(cid:129) the connection of entropy and statistics—Sect. 4.2.2,
(cid:129) the laser—Sect. 4.4.1,
(cid:129) the tight-binding and plane-waves models for the electron bands of crystals—
Sect. 5.2.1,
(cid:129) theextrinsicsemiconductorsandtheirtechnologicalapplications,frompage214
to the end of Sect. 5.2.2.
On the other hand, a reader who wishes to broaden her/his view of the field
beyondthetopicscoveredbythepresenttextbookcanfollowextendedintroductory
treatments, for example, in Refs. [1, 2, 3, 4]. Specialized texts focus on advanced
approachesclosertothefrontiersofresearch, coveringboththetheoreticalandthe
experimental side. The reader is invited to browse in particular Refs. [5, 6, 7] for
atomic physics, Refs. [6, 8, 9] for molecules, and Refs. [10, 11, 12] for solid-state
physics. Finally, this textbook focuses on the present-day understanding of the
physics of matter, omitting most of the fundamental experiments and conceptual
steps through which the scientific community reached this understanding.
Reference [13] provides an insider’s view of the historical evolution of the basic
concepts in this field.
The present volume draws its initial inspiration from Luciano Reatto’s course
delivered in the 1990s at the University of Milano. In the past, the bulk of this
volume was made available as lecture notes, first released on January 15, 2004.
Precious feedback and suggestions from Giovanni Onida, other colleagues, and
several students in the Milan Physics Diploma course prompted numerous cor-
rectionsandimprovementstothesecondedition.Comparedtothefirstedition,the
Preface vii
second edition also benefits from the addition of an Appendix about lighting
applications, a few new problems, a periodic table, and an index. The problems
includedinthisbook,plusmanymore,areavailableinItalianatthewebsitehttp://
materia.fisica.unimi.it/manini/dida/archive_exam.html.
The author acknowledges the warm feedback and interest from the students in
physicsattheUniversityofMilano;thisbookwasoriginallywrittenforthem.Last
but not least, the patient care and love of the author’s family was a primary
ingredient in nurturing the present textbook.
Milan, Italy Nicola Manini
June 2020
Contents
1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Typical Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Perspectives on the Structure of Matter
(Contains Spoilers). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Spectra and Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 One-Electron Atom/Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 The Energy Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 The Angular Wavefunction. . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 The Radial Wavefunction. . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.4 Orbital Angular Momentum and Magnetic Dipole
Moment| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.5 The Stern-Gerlach Experiment . . . . . . . . . . . . . . . . . . . . 22
2.1.6 Electron Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.7 Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.8 Nuclear Spin and Hyperfine Structure . . . . . . . . . . . . . . . 31
2.1.9 Electronic Transitions, Selection Rules . . . . . . . . . . . . . . 33
2.1.10 Spectra in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Many-Electron Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.1 Identical Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.2 The Independent-Particles Approximation . . . . . . . . . . . . 40
2.2.3 Independent Electrons in Atoms . . . . . . . . . . . . . . . . . . . 43
2.2.4 The 2-Electron Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.5 The Hartree-Fock Method. . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.6 Electronic Structure Across the Periodic Table. . . . . . . . . 52
2.2.7 Fundamentals of Spectroscopy . . . . . . . . . . . . . . . . . . . . 54
ix
x Contents
2.2.8 Core Levels and Spectra. . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.9 Optical Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2.10 Electric-Dipole Selection Rules. . . . . . . . . . . . . . . . . . . . 67
3 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1 The Adiabatic Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Chemical and Non-chemical Bonding . . . . . . . . . . . . . . . . . . . . . 75
þ
3.2.1 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2
3.2.2 Covalent and Ionic Bonding . . . . . . . . . . . . . . . . . . . . . . 80
3.2.3 Weak Non-chemical Bonds . . . . . . . . . . . . . . . . . . . . . . 84
3.2.4 A Classification of Bonding . . . . . . . . . . . . . . . . . . . . . . 88
3.3 Intramolecular Dynamics and Spectra . . . . . . . . . . . . . . . . . . . . . 89
3.3.1 Rotational and Rovibrational Spectra. . . . . . . . . . . . . . . . 90
3.3.2 Electronic Excitations. . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3.3 Zero-Point Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4 Statistical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1 Introductory Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.1 Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . 102
4.1.2 Quantum Statistics and the Density Operator. . . . . . . . . . 104
4.2 Equilibrium Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.1 Connection to Thermodynamics . . . . . . . . . . . . . . . . . . . 107
4.2.2 Entropy and the Second Principle . . . . . . . . . . . . . . . . . . 110
4.3 Ideal Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 The High-Temperature Limit . . . . . . . . . . . . . . . . . . . . . 113
4.3.2 Low-Temperature Fermi and Bose Gases. . . . . . . . . . . . . 125
4.4 Matter-Radiation Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.4.1 The Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1 The Microscopic Structure of Solids . . . . . . . . . . . . . . . . . . . . . . 150
5.1.1 Lattices and Crystal Structures . . . . . . . . . . . . . . . . . . . . 157
5.1.2 The Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.1.3 Diffraction Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2 Electrons in Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.2.1 Models of Bands in Crystals. . . . . . . . . . . . . . . . . . . . . . 186
5.2.2 Filling of the Bands: Metals and Insulators . . . . . . . . . . . 197
5.2.3 Spectra of Electrons in Solids. . . . . . . . . . . . . . . . . . . . . 223
5.3 The Vibrations of Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.3.1 The Normal Modes of Vibration. . . . . . . . . . . . . . . . . . . 228
5.3.2 Thermal Properties of Phonons. . . . . . . . . . . . . . . . . . . . 235
5.3.3 Other Phonon Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 240