Die Wahrheit ist das Kind der Zeit, nicht der Autoritaet. Unsere Unwissenheit ist unendlich, tragen wir einen Kubikmillimeter ab! (from B. Brecht, in "Leben des Galilei")
There is no end to this wonderful world of experimental discovery and mental constructions of reality as new facts become known. That is why physicists have more fun than most people. (Miklos Gyulassy)
To Luca Lorenzo Strozzi Rigamonti, with hope (A.R.) To Gegia, Enri, Cate and Dario (P.C)
Attilio Rigamonti
Pietro Carretta
Structure of Matter An Introductory Course with Problems and Solutions
fl
Springer
ATTILIO RIGAMONTI PIETRO CARRETTA
Dipartimento di Fisica "A. Volta", Universita degli Studi di Pavia
Series Unitext Editorial board: G. Parisi, M. Cini, S. Forte, M. Inguscio, G. Montagna, O. Nicrosini, F. Pacini, L. Peliti, A. Rotondi
Library of Congress Control Number: 2007923018 ISBN 978-88-470-0559-4 Springer Milan Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole of part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in databanks. Duplication of this pubblication or parts thereof is only permitted under the provisions of the Italian Copyright Law in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the Italian Copyright Law. Springer is a part of Springer Science+ Business Media springer.com ©
Springer-Verlag Italia 2007
Printed in Italy Cover design: Simona Colombo, Milano Typeset by the authors using a Springer Macro package Printing and binding: Grafiche Porpora, Segrate, Milano Printed on acid-free paper
Preface
Intended Audience, Approach and Presentation This text is intended for a course of about fifteen weeks for undergraduate students. It arises from the adaptation and the amendments to a text for a full-year course in Structure of Matter, written by one of the authors (A.R.) about thirty years ago. At that time only a few (if any) textbooks having the suited form for introduction to basic quantum properties of atoms, molecules and crystals in a comprehensive and interrelated way, were available. Along the last twenty years many excellent books pursuing the aforementioned aim have been published (some of them are listed at the end of this preface). Still there are reasons, in our opinion, to attempt a further text devoted to the quantum roots of condensed matter properties. A practical aspect in this regard involves the organization of the studies in Physics, after the huge scientific outburst of the various topics of fundamental and technological character in recent decades. In most Universities there is now a first period of three or four years, common to all the students and devoted to elementary aspects, followed by a more advanced program in rather specialized fields of Physics. The difficult task is to provide a common and formative introduction in the first period still suitable as a basis for building up more advanced courses and to bridge the large area between elementary physics and the topics pertaining to research activities. The present attempt towards a readable book, hopefully presenting those desired characteristics, essentially is based on a mixture of simplified institutional theory with solved problems. The hope, in this way, is to provide physical insights, basic culture and motivation, without deteriorating the possibility of more advanced subsequent learning.
Organization Structure of Matter is such a wide field that a first task to undertake is how to confine an introductory text. The present status of that discipline
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Preface
represents a key construction of the scientific knowledge, possibly equated only by the unitary description of the electromagnetic phenomena. Even by limiting attention only to the conventional parts of the condensed matter, namely atoms, molecules and crystals, still we are left with an ample field. For instance, semiconductors or superconductors, the electric and magnetic properties of the matter and its interaction with the electromagnetic radiation, the microscopic mechanisms underlying solid-state devices as well as masers and lasers, are to be considered as belonging to the field of structure of matter (without mentioning the "artificial" matter involving systems such as nanostructures, photonic crystals or special materials obtained by subtle manipulations of atoms by means of special techniques). In this text the choice has been to limit the attention to key concepts and to the most typical aspects of atoms (Chapters 1-5) , molecules (Chapters 7-10) and of crystalline solids (Chapters 11-14) , looking at the basic "structural" aspects without dealing with the properties that originate from them. This choice is exemplified by referring to crystals: electronic states and quantum motions of the ions have been described without going into the details regarding the numerous and relevant properties related to these aspects. Only in a few particularly illustrative cases favoring better understanding or comprehensive view, derivation of some related properties has been given (examples are some thermodynamical properties due to nuclear motions in molecules and crystals or some of the electric or magnetic properties). Chapter 6 has the particular aim to lead the reader to an illustrative overview of quantum behaviors of angular momenta and magnetic moments, with an introduction to spin statistics, magnetic resonance and spin motions and a mention to spin thermodynamics, through the description of the adiabatic demagnetization used in order to approach the zero-temperature condition. All along the text emphasis is given to the role of spectroscopic experiments giving access to the quantum properties by means of electromagnetic radiation. In the spirit to limit the attention to key arguments, frequent referring is given to the electric dipole moment and to selection rules, rather than to other aspects of the many experiments of spectroscopic character used to explore the matter at microscopic level. Other unifying concepts present along the text are the ones embedded in statistical physics and thermal excitations, as it is necessary in view of the many-body character of condensed matter in equilibrium with a thermal reservoir.
Prerequisite, appendices and problems Along the text the use of quantum mechanics, although continuous, only involves the basic background that t he reader should have achieved in undergraduate courses. The knowledge in statistical physics is the one based on the Boltzmann, Fermi-Dirac and Bose-Einstein statistical distributions, with the relationships of thermodynamical quantities to the partition function (some of the problems work as proper recall, particularly for the statistical physics of
Preface
VII
paramagnets or for the black-body radiation). Finally the reader is assumed to have knowledge of classical electromagnetism and classical Hamiltonian mechanics. Appendices are intended to provide ad hoc recalls, in some cases applied to appropriate systems or to phenomena useful for illustration. The Gaussian cgs emu units are used. The problems should be considered entangled to the formal presentation of the arguments, being designed as an intrinsic part of the pathway the student should move by in order to grasp the key concepts. Some of the problems are simple applications of the equations and in these cases the solutions are only sketched. Other problems are basic building blocks and possibly expansions of the formal description. Then the various steps of the solution are presented in some detail. The aim of the melange intuition-theory-exercises pursued in the text is to favor the acquisition of the basic knowledge in the wide and wonderful field of the condensed matter, emphasizing how phenomenological properties originate from the microscopic, quantum features of the nature. It should be obvious that a book of this size can present only a minute fraction of the present knowledge in the field. If the reader could achieve even an elementary understanding of the atoms, the molecules and the crystals, how they are affected by electric and magnetic fields, how they interact with electromagnetic radiation and respond to thermal excitation, the book will have fulfilled its purpose. The fundamental blocks of the physical world are thought to be the subnuclear elementary particles. However the beauty of the natural world rather originates from the architectural construction of the blocks occurring in the matter. Ortega Y Gasset wrote "If you wish to admire the beauty of a cathedral you have to respect for distance. If you go too close, you just see a brick" . Furthermore, one could claim that the world of condensed matter more easily allows one to achieve a private discovery of phenomena. In this respect let us report what Edward Purcell wrote in his Nobel lecture: "To see the world for a moment as something rich and strange is the private reward of many a discovery" .
Acknowledgments The authors wish to acknowledge Giacomo Mauro D' Ariano , who has inspired and solved several problems and provided enlightening remarks with his collaboration to the former course "Structure of Matter" given by one of us (A.R.), along two decades. Acknowledgments for suggestions or indirect contributions through discussions or comments are due to A. Balzarotti, A. Barone, G. Benedek, G. Bonera, M. Bornatici, F. Borsa, G. Caglioti, R. Cantelli, L. Colombo, M. Corti, A. Debernardi, A. Lascialfari, D. Magnani, F. Miglietta, G. Onida, G. Pastori Parravicini, E. Reguzzoni, S. Romano, G. Senatore, J. Spalek, V. Tognetti, A.A. Varlamov.
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N. Papinutto is acknowledged for his help in preparing several figures. The problems have been revised by Dr. D. Magnani and Dr. G. Ventura, when students. The authors anticipate their gratitude to other students who, through vigilance and desire of learning will found errors and didactic mistakes. Dr. M. Medici is gratefully thanked for her c areful revision of the typed text. This book has been written while recelvmg inspiration from a number of text books dealing with particular items or from problems and exercises suggested or solved in them. The texts reported b elow are not recalled as a real "further-reading list " , since it would be too ample and possibly useless. The list is more an acknowledgment of t he suggestions received when seeking inspiration, information or advices.
A. Abragam, L'effet Mossbauer et ses applications a l'etude des champs internes, Gordon and Breach (1964). M. Alonso and E.J. Finn, Fundamental University Physics Vol.III- Quantum and Statistical Physics, Addison Wesley (1973). D.J. Amit and Y. Verbin, Statistical Physics - An Introductory course, World Scientific (1999).
N.W. Ashcroft and N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston (1976). P.W. Atkins and R.S. Friedman , Molecular Quantum Mechanics, Oxford University Press, Oxford (1997). A. Balzarotti, M. Cini, M. Fanfoni, Atomi, Molecole e Solidi. Esercizi risolti, Springer V erlag (2004). F. Bassani e U.M. Grassano , Fisica della Stato Solido , Bollati Boringhieri (2000).
J .S. Blakemore, Solid State Physics, W.B. Saunders Co. (1974). S. Blundell, Magn etism in Condensed Matter, Oxford Master S eries in Condensed Matter Physics, Oxford U.P. (2001). S. Boffi, Da Laplace a Heisenberg, La Goliardica P avese (1992).
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B.H. Bransden and C.J. Joachain, Physics of atoms and molecules, Prentice Hall (2002). D. Budker, D.F. Kimball and D.P. De Mille, Atomic Physics - An Exploration Through Problems and Solutions, Oxford University Press (2004). G. Burns, Solid State Physics, Academic Press, Inc. (1985). G. Caglioti, Introduzione alla Fisica dei Materiali, Zanichelli (1974). B. Cagnac and J .C. Pebay - Peyroula, Physique atomique, tome 2, Dunod Universit, Paris (1971). P. Caldirola, Istituzioni di Fisica Teorica, Editrice Viscontea, Milano (1960). M. Cini, Corso difisica atomica e molecolare, Edizioni Nuova Cultura (1992). L. Colombo, Elementi di Struttura della Materia, Hoepli (2002). E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, London (1959). C.A. Coulson, Valence, Oxford Clarendon Press (1953). J.A. Cronin, D.F. Greenberg, V.L. Telegdi, University of Chicago Graduate Problems in Physics, Addison-Wesley (1967). G.M. D ' Ariano , Esercizi di Struttura della Materia, La Goliardica Pavese (1989) . J.P. Dahl, Introduction to the Quantum World of Atoms and Molecules, World Scientific (2001).
w.
Demtroder, Molecular Physics, Wiley-VCH (2005).
W. Demtroder, Atoms, Molecules and Photons, Springer Verlag (2006). R.N. Dixon, Spectroscopy and Structure, Methuen and Co LTD London (1965) . R. Eisberg and R. Resnick , Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, J. Wiley and Sons (1985).
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H. Eyring, J. Walter and G.E. Kimball, Quantum Chemistry, J. Wiley, New York (1950). R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures an Physics Vol. III, Addison Wesley, Palo Alto (1965). R. Fieschi e R. De Renzi, Struttura della Materia, La Nuova Italia Scientifica, Roma (1995). A.P. French and E.F. Taylor, An Introduction to Quantum Physics, The M.LT. Introductory Physics Series, Van Nostrand Reinhold (UK)(1986). R. Gautreau and W. Savin, Theory and Problems of Modem Physics, (Schaum 's series in Science) Mc Graw-Hill Book Company (1978). H. Goldstein, Classical Mechanics, Addison-Wesley (1965). H.J. Goldsmid (Editor) , Problems in Solid State Physics, Pion Limited (London, 1972). D.L. Goodstein, States of Matter, Dover Publications Inc. (1985). G. Grosso and G. Pastori Parravicini , Solid State Physics, Academic Press (2000). A.P. Guimares, Magnetism and Magnetic Resonance in Solids, J. Wiley and Sons (1998). H. Haken and H.C. Wolf, Atomic and Quantum Physics, Springer V erlag Berlin (1987). H. Haken and H.C. Wolf, Molecular Physics and Elements of Quantum Chemistry, Springer Verlag Berlin (2004). G. Herzberg, Molecular Spectra and Molecular Structure, Vol. I, II and III, D. Van Nostrand, New York (1964-1966 , reprint 1988-1991). J.R. Hook and H.E. Hall, Solid State Physics, J. Wiley and Sons (1999).
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H. Ibach and H. Liith, Solid State Physics: an Introduction to Theory and Experiments, Springer Verlag (1990). C.S. Johnson and L.G. Pedersen, Quantum Chemistry and Physics, Addison - Wesley (1977). C. Kittel, Elementary Statistical Physics, J. Wiley and Sons (1958). C. Kittel, Introduction to Solid State Physics, J. Wiley and Sons (1956,1968). J.D. Mc Gervey, Quantum Mechanics - Concepts and Applications, Academic Press, New York (1995). L. Mih:ily and M.C. Martin, Solid State Physics - Problems and Solutions, J. Wiley (1996). M.A. Morrison, T.L. Estle and N.F. Lane, Quantum States of Atoms, Molecules and Solids, Prentice - Hall Inc. New Jersey (1976). E.M. Purcell, Electricity and Magnetism, Berkley Physics Course Vol.2, Mc Graw-Hill (1965). A. Rigamonti, Introduzione alla Struttura della Materia, La Goliardica Pavese (1977). M.N. Rudden and J. Wilson, Elements of Solid State Physics, J. Wiley and Sons (1996). M. Roncadelli, Aspetti Astrofisici della Materia (2004).
OSCUTa,
Bibliopolis, Napoli
H. Semat, Introduction to Atomic and Nuclear Physics, Chapman and H all LTD (1962).
J.C. Slater, Quantum Theory of Matter, Mc Graw-Hill, New York (1968). C.P. Slichter, Principles of Magnetic Resonance, Springer Verlag Berlin (1990) . S. Svanberg, Atomic and Molecular Spectroscopy, Springer Verlag Berlin (2003). D. Tabor, Gases, liquids and solids, Cambridge University Press (1993).
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P.L. Taylor and O. Heinonen, A Quantum Approach to Condensed Matter Physics, Cambridge University Press (2002). M.A. Wahab , Solid State Physics (Second Edition), Alpha Science International Ltd. (2005).
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Weinberg, The first three minutes: a modem view of the origin of the universe, Amazon (2005).
M. White, Quantum Theory of Magnetism, McGraw-Hill (1970). J .M. Ziman, Principles of the Theory of Solids, Cambridge University Press (1964).
Pavia, J anuary 2007
Attilio Rigamonti Pietro Carretta
Contents
1
2
3
Atoms: general aspects .................................... 1.1 Central field approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.2 Self-consistent construction of the effective potential .. . . . . . .. 1.3 Degeneracy from dynamical equivalence. . . . . . . . . . . . . . . . . . .. 1.4 Hydrogenic atoms: illustration of basic properties ........... Problems 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.5 Finite nuclear mass. Positron, Muonic and Rydberg atoms ... Problems 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.6 Orbital and spin magnetic moments and spin-orbit interaction Problems 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 7 Spectroscopic notation for multiplet states. . . . . . . . . . . . . . . . .. Appendix 1.1 Electromagnetic spectral ranges and useful numbers ................................................ Appendix 1.2 Perturbation effects in two-levels system. . . . . . .. Appendix 1.3 Transition probabilities and selection rules. . . . .. Problems F.I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Typical atoms ............................................. 2.1 Alkali atoms ............................................ Problems 11.1 ......................................... 2.2 Helium atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2.1 Generalities and ground state. . . . . . . . . . . . . . . . . . . . . .. 2.2.2 Excited states and the exchange interaction. . . . . . . . . .. Problems 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.3 P auli principle, determinantal eigenfunctions and superselection rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Problems F.II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
1 2 5 5 7 14 23 25 27 32 35 39 40 43 47 63 63 70 73 73 76 79 84 85
The shell vectorial model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91 3.1 Introductory aspects ..................................... 91 3.2 Coupling of angular momenta ............................. 93
XIV
Contents LS coupling model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. The effective magnetic moment ..................... Illustrative examples and the Hund rules for the ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Problems 111.2 ........................................ 3.3 jj coupling scheme ....................................... Problems 111.3 ........................................ 3.4 Quantum theory for multiplets. Slater radial wavefunctions ... 3.5 Selection rules .......................................... Problems F.II1 ........................................ 3.2.1 3.2.2 3.2.3
4
93 97 99 103 110 114 116 120 121
Atoms in electric and magnetic fields ...................... 4.1 Introductory aspects ..................................... 4.2 Stark effect and atomic polarizability ...................... Problems IV.2 ........................................ 4.3 Hamiltonian in magnetic field ............................. 4.3.1 Zeeman regime .................................... 4.3.2 Paschen-Back regime .............................. Problems IV.3 ........................................ 4.4 Paramagnetism of non-interacting atoms and mean field interaction .............................................. 4.5 Atomic diamagnetism .................................... Problems IV.5 ........................................ Appendix IV.1 Electromagnetic units and Gauss system ...... Problems F.IV ........................................
129 129 132 136 138 139 140 141
5
Nuclear moments and hyperfine interactions ............... 5.1 Introductory generalities ................................. 5.2 Magnetic hyperfine interaction - F states ................... Problems V.2 ......................................... 5.3 Electric quadrupole interaction ............................ Problems V.3 ......................................... Appendix V.1 Fine and hyperfine structure in Hydrogen ...... Problems F. V .........................................
167 167 169 174 177 180 184 187
6
Spin statistics, magnetic resonance, spin motion and echoes205 6.1 Spin statistics, spin-temperature and fluctuations ............ 205 Problems Vl.l ........................................ 210 6.2 The principle of magnetic resonance and the spin motion ......................................... 214 Problems VI.2 ........................................ 218 6.3 Spin and photon echoes .................................. 221 6.4 Ordering and disordering in spin systems: cooling by adiabatic demagnetization ...................... 223 Problems F. VI ........................................ 226
147 151 153 154 157
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7
Molecules: general aspects ................................. 237 7.1 Born-Oppenheimer separation and the adiabatic approximation238 7.2 Classification of the electronic states ....................... 242 7.2.1 Generalities ....................................... 242 7.2.2 Schrodinger equation in cylindrical symmetry ......... 243 7.2.3 Separated-atoms and united-atoms schemes and correlation diagram ................................ 245 Problems F.VII ....................................... 249
8
Electronic states in diatomic molecules .................... 251 8.1 Ht as prototype of MO approach ......................... 251 8.1.1 Eigenvalues and energy curves ...................... 251 Problems VIlLI ...................................... 258 8.1.2 Bonding mechanism and the exchange of the electron .. 260 8.2 Homonuclear molecules in the MO scenario ................. 262 Problems VIIL2 ...................................... 266 8.3 H2 as prototype of the VB approach ....................... 267 Problems VIIL3 ...................................... 271 8.4 Comparison of MO and VB scenarios in H 2: equivalence from configuration interaction ................................. 272 8.5 Heteronuclear molecules and the electric dipole moment ...... 275 Problems VIIL5 ...................................... 279 Problems F. VIII ...................................... 280
9
Electronic states in selected polyatomic molecules ......... 9.1 Qualitative aspects of NH3 and H 20 molecules .............. 9.2 Bonds due to hybrid atomic orbitals ....................... 9.3 Delocalization and the benzene molecule ................... Appendix IX.1 Ammonia molecule in electric field and the Ammonia maser ......................................... Problems F .IX ........................................
285 286 286 292 294 300
10 Nuclear motions in molecules and related properties ....... 303 10.1 Generalities and introductory aspects for diatomic molecules .. 303 10.2 Rotational motions ...................................... 305 10.2.1 Eigenfunctions and eigenvalues ...................... 305 10.2.2 Principles of rotational spectroscopy ................. 306 10.2.3 Thermodynamical energy from rotational motions ..... 309 10.2.4 Orientational electric polarizability .................. 310 10.2.5 Extension to polyatomic molecules and effect of the electronic motion in diatomic molecules .............. 311 Problems X.2 ......................................... 313 10.3 Vibrational motions ..................................... 316 10.3.1 Eigenfunctions and eigenvalues ...................... 316
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Contents
10.4 10.5 10.6 10.7 10.8 10.9
10.3.2 Principles of vibrational spectroscopy and anharmonicityeffects .............................. 318 Problems X.3 ......................................... 321 Morse potential ......................................... 323 Problems X.4 ......................................... 324 Roto-vibrational eigenvalues and coupling effects ............ 326 Problems X.5 ......................................... 328 Polyatomic molecules: normal modes ....................... 332 Principles of Raman spectroscopy ......................... 336 Problems X.7 ......................................... 340 Franck - Condon principle ................................ 340 Problems X.8 ......................................... 342 Effects of nuclear spin statistics in homonuclear diatomic molecules ............................................... 343 Problems X.9 ......................................... 347 Problems F.X ......................................... 348
11 Crystal structures ......................................... 353 11.1 Translational invariance, Bravais lattices and Wigner-Seitz cell 354 11.2 Reciprocal lattice and Brillouin cell ........................ 359 11.3 Typical crystal structures ................................. 362 Problems F .XI ........................................ 365 12 Electron states in crystals ................................. 369 12.1 Introductory aspects and the band concept ................. 369 12.2 Translational invariance and the Bloch orbital ............... 371 12.3 Role and properties of k .................................. 374 Problems XII.3 ....................................... 375 12.4 Periodic boundary conditions and reduction to the first Brillouin zone ........................................... 377 12.5 Density of states, dispersion relations and critical points ...... 379 12.6 The effective e el ctron mass ............................... 382 Problems XII.6 ....................................... 383 12.7 Models of crystals ....................................... 385 12.7.1 Electrons in empty lattice .......................... 385 12.7.2 Weakly bound electrons ............................ 389 12.7.3 Tightly bound electrons ............................ 393 Problems XII.7 ....................................... 398 Problems F .XII ....................................... 405 13 Miscellaneous aspects related to the electronic structure ... 13.1 Typology of crystals ..................................... 13.2 Bonding mechanisms and cohesive energies ................. 13.2.1 Ionic crystals ..................................... 13.2.2 Lennard-Jones interaction and molecular crystals ......
409 409 412 412 414
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XVII
Problems XIII.2 ...................................... 13.3 Electron states of magnetic ions in a crystal field ............. 13.4 Simple picture of the e lectric transport ..................... Appendix XIII. 1 M agnetism from itinerant e lectrons ......... Problems F.XIII ......................................
417 419 423 427 431
14 Vibrational motions of the ions and thermal effects ........ 435 14.1 Motions of the ions in the harmonic approximation .......... 435 14.2 Branches and dispersion relations .......................... 437 14.3 Models of lattice vibrations ............................... 437 14. 3.1 Monoatomic one-dimensional crystal ................. 438 14.3.2 Diatomic one-dimensional crystal .................... 440 14.3.3 Einst ein and Debye cryst als ......................... 443 14.4 Phonons ................................................ 447 14.5 Thermal properties related to lattice vibrations .............. 449 Problems XIV.5 ....................................... 451 14.6 The Mossbauer effect .................................... 453 Problems F .XIV ...................................... 458 Index .......................................................... 465
