Springer Series in Materials Science

Volume 221

Series editors

Robert Hull, Charlottesville, USAChennupati Jagadish, Canberra, AustraliaRichard M. Osgood, New York, USAJürgen Parisi, Oldenburg, GermanyTae-Yeon Seong, Seoul, Korea, Republic of (South Korea)Shin-ichi Uchida, Tokyo, JapanZhiming M. Wang, Chengdu, China

The Springer Series in Materials Science covers the complete spectrum of materialsphysics, including fundamental principles, physical properties, materials theory anddesign. Recognizing the increasing importance of materials science in future devicetechnologies, the book titles in this series reflect the state-of-the-art in understand-ing and controlling the structure and properties of all important classes of materials.

More information about this series at http://www.springer.com/series/856

Juan F.R. Archilla • Noé JiménezVíctor J. Sánchez-MorcilloLuis M. García-RaffiEditors

Quodons in MicaNonlinear Localized TravellingExcitations in Crystals

123

EditorsJuan F.R. ArchillaDepartamento de Física Aplicada IGroup of Nonlinear Physics

Universidad de SevillaSevillaSpain

Noé JiménezFísica Aplicada, EPS GandíaUniversidad Politécnica de ValenciaGandiaSpain

Víctor J. Sánchez-MorcilloFísica Aplicada, EPS GandíaUniversidad Politécnica de ValenciaGandiaSpain

Luis M. García-RaffiMatemática AplicadaUniversidad Politécnica de ValenciaValenciaSpain

ISSN 0933-033X ISSN 2196-2812 (electronic)Springer Series in Materials ScienceISBN 978-3-319-21044-5 ISBN 978-3-319-21045-2 (eBook)DOI 10.1007/978-3-319-21045-2

Library of Congress Control Number: 2015944443

Springer Cham Heidelberg New York Dordrecht London© Springer International Publishing Switzerland 2015This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

Printed on acid-free paper

Springer International Publishing AG Switzerland is part of Springer Science+Business Media(www.springer.com)

To F.M. Russell, inspirator of all the bookand author of a substantial part of it.

Preface

This books commemorates the lifelong dedication of physicist Francis MichaelRussell to science, for a large part of which he has been fascinated with the darktracks in mica muscovite. Some of these lines he was able to explain as the fossilrecord of charged elementary particles in a series of amazing papers which startswith Tracks in mica caused by electron showers published in Nature in 1967.However, many of the tracks were produced by other causes and as they were alongthe primary lattice directions he suggested that they were the record of some kind ofquasi one-dimensional excitations of the lattice, to which he gave the name quo-dons. The theory was improved by his collaboration with nonlinear physicist Prof.Chris Eilbeck, which after many years had a high point in 2007 with the experi-mental verification of the transmission of localized energy and momentum in micaalong the lattice directions. Thereafter Russell, or Mike as we all know him, hascontinued doing research in mainly two directions: What many other things arerecorded in mica? What is the exact nature of quodons among the many nonlinearlocalized travelling excitations in crystals?

At present, Mike has just turned 83 and he is still extremely active. Evidence ofthis is apparent in the chapters he has written for this book, his nomination in 2013as honorary professor of the University of Pretoria, South Africa, his trek to theHimalayas in 2014 and last but not least the recent theory of charged quodons. For along time he has been worried that his ideas, dispersed in publications in manyjournals over many years may be forgotten. To prevent it he decided to write hislifelong research in such a way that it would be accessible to a wider audience. Theresult of his labour is in Chap. 20 of this book.

With the same objective, we conceived the idea of organizing a conferencewhere Mike could explain in detail his ideas to an interested audience of scientistsand they would contribute with the latest advances of science in this field. Finally,the conference took place in Altea in 2013. At the end of the conference it wasproposed that a book be written in a year’s time so that work could be finalized and

vii

ideas clarified. The objective of the book was to describe the phenomena in mica,the past and present research of Mike Russell and to show that phenomena similarto quodons appear in theory and experiments in many different systems, hence, thetitle Quodons in Mica and the subtitle Nonlinear Localized Travelling Excitationsin Crystals.

The conference had to be in a very special place, where the atmosphere wouldinspire the participants and leave them with lasting memories of what they havethought and felt. The region of Valencia was in prefence to Sevilla because of itsmany seaside locations. Soon, the enchanting village of Altea, overlooking awonderful bay and surrounded by impressive mountains was selected. The con-ference took place in September 2013 with participants from all over the world.

The book is the present volume. It is organized into several parts, although theclassification issomehow arbitrary and many chapters could also be included inother parts:

Part I Mica and Mica-Related Systems

• Tracks in mica, 50 years later. Review of evidence for recording the tracks ofcharged particles and mobile lattice excitations in muscovite mica by F.M.Russell.

• Numerical simulations of nonlinear modes in mica: past, present and future byJ. Bajars, J.C. Eilbeck and B. Leimkuhler.

• A supersonic crowdion in mica: Ultradiscrete kinks with energy between 40Krecoil and transmission sputtering by J.F.R. Archilla, Yu.A. Kosevich, N.Jiménez, V.J. Sánchez-Morcillo and L.M. Garca-Raffi.

Part II Two-dimensional Lattices

• Pattern formation by traveling localized modes in two-dimensional dissipativemedia with lattice potentials by V. Besse, H. Leblond D. Mihalache and B.Malomed .

• A numerical study of weak lateral dispersion in discrete and continuum modelsby L.A. Cisneros-Ake and A.A. Minzoni

• Breather mobility and the Peierls-Nabarro potential by M. Johansson andP. Jason.

• Asymptotic approximation of discrete breather modes in two-dimensional lat-tices by J.A.D. Wattis.

Part III Molecular Dynamics in Three Dimensions

• Moving discrete breathers in 2D and 3D crystals by S.V. Dmitriev, A.A.Kistanov and V.I. Dubinko.

• Standing and moving discrete breathers with frequencies above the phononspectrum by V. Hizhnyakov, M. Haas, A. Shelkan and M. Klopov.

• Phonon interference and energy transport in nonlinear lattices with resonancedefects by Yu.A. Kosevich, H. Han, L.G. Potyomina, A.N. Darinskii and S.Volz.

viii Preface

Part IV Electrons and Lattice Vibrations

• Electron transfer and tunneling from donor to acceptor in anharmonic crystallattices by A.P. Chetverikov, L. Cruzeiro, W. Ebeling and M.G. Velarde.

• Bound states of electrons in harmonic and anharmonic crystal lattices by L.S.Brizhik, A.P. Chetverikov, W. Ebeling, G. Röpke and M.G. Velarde.

• Solitons and charge transport in triangular and quadratic crystal lattices by A.P. Chetverikov, W. Ebeling and M.G. Velarde.

Part V Semiconductors

• Experimental observation of intrinsic localized modes in germanium by J.F.R.Archilla, S.M.M. Coelho, F.D. Auret, C. Nyamhere, V.I. Dubinko and V.Hizhnyakov.

• The origin of defects induced in ultra-pure germanium by Electron BeamDeposition by S.M.M. Coelho, J.F.R. Archilla, F.D. Auret and J.M. Nel.

• Rate theory of acceleration of defect annealing driven by discrete breathers byV.I. Dubinko, J.F.R. Archilla, S.V. Dmitriev and V. Hizhnyakov.

Part VI Other Systems

• The amide I band of crystalline acetanilide: old data under new light by L.Cruzeiro.

• Extreme waves and branching flows in optical media by M. Mattheakis and G.P. Tsironis.

• Discrete bright solitons in Bose-Einstein condensates and dimensional reductionin quantum field theory by L. Salasnich.

Part VII A Historical Perspective

• I saw a crystal: A historical account of the deciphering of the markings in micaby F.M. Russell.

The last part is not only interesting in this particular field, but also for showinghow the mind of a scientist works, how science is mixed with life, sometimespersonal events being an obstacle, but often science and life fertilizing each other.

I think it will also be useful for young students to show them that science needsnot be boring and how determination and persistence can lead to success.

Success in science is not money or social approval but the satisfaction of dis-covery. I think that will be the lasting legacy of Mike Russell.

Sevilla Juan F.R. ArchillaValencia Noé Jiménez

Víctor J. Sánchez-MorcilloLuis M. García-Raffi

Preface ix

Mike Russell in his laboratory in 2011 when he celebrated his 80th birthday

x Preface

Acknowledgments

I wish to express my deep gratitude to Juan F.R. Archilla for suggesting this book,for his unstinting encouragement and his constructive support of the mica studies.My thanks go also to Noé Jiménez, Víctor J. Sánchez-Morcillo and Luis M.García-Raffi and all the support team who have brought this book to completion.I would also like to express my delight and thanks to the many distinguished peoplewho have contributed articles to this book thereby illustrating the challenges andbroad spectrum of studies relating to nonlinear excitations with relevance to mica.I am grateful to the University of Seville for financial support in presenting thesestudies.

Abingdon F. Michael Russell

xi

Contents

Part I On Mica and Mica Related Systems

1 Tracks in Mica, 50 Years Later: Review of Evidencefor Recording the Tracks of Charged Particles and MobileLattice Excitations in Muscovite Mica . . . . . . . . . . . . . . . . . . . . . 3F. Michael Russell1.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Origin and Properties of Charged Particles

in Mica Underground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Measurement of the Sensitivity and Duration of Recording. . . 111.4 Interaction Between Theory and Experiment . . . . . . . . . . . . . 121.5 The Role of Atomic Chains in Propagation of Energy . . . . . . 131.6 Ejection of Atoms by Elastic Scattering of Quodons . . . . . . . 161.7 Thermal Stability of Lattice Excitations . . . . . . . . . . . . . . . . 181.8 Creation of Quodons by High Energy Particles . . . . . . . . . . . 201.9 Nuclear Scattering of Muons. . . . . . . . . . . . . . . . . . . . . . . . 201.10 Recording Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.11 Interactions of Mobile Lattice Excitations

with Stored Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.12 Confined Lattice Excitations . . . . . . . . . . . . . . . . . . . . . . . . 271.13 Internal Structure of Fans . . . . . . . . . . . . . . . . . . . . . . . . . . 281.14 The Position at the Beginning of 2015 . . . . . . . . . . . . . . . . . 311.15 The Puzzle Solved: Quodons Have Charge . . . . . . . . . . . . . 31References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

xiii

2 Numerical Simulations of Nonlinear Modes in Mica:Past, Present and Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Janis Bajars, J. Chris Eilbeck and Ben Leimkuhler2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.1 Solitons, Kinks and Breathers in Two Dimensions. . . 372.1.2 The Work of Marín, Eilbeck and Russell

on Breathers in the Potassium Layer of Mica . . . . . . 392.2 Preliminary Results from Numerical Experiments . . . . . . . . . 43

2.2.1 On-Site Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 442.2.2 Interaction Potential . . . . . . . . . . . . . . . . . . . . . . . . 442.2.3 Time Integration Method . . . . . . . . . . . . . . . . . . . . 472.2.4 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3 Conclusions and Future Plans . . . . . . . . . . . . . . . . . . . . . . . 64References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3 A Supersonic Crowdion in Mica . . . . . . . . . . . . . . . . . . . . . . . . . 69Juan F.R. Archilla, Yuriy A. Kosevich, Noé Jiménez,Víctor J. Sánchez-Morcillo and Luis M. García-Raffi3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2 Description of the System. . . . . . . . . . . . . . . . . . . . . . . . . . 733.3 The Magic Mode Revisited. . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.1 Basic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3.2 Fundamental Ansatz . . . . . . . . . . . . . . . . . . . . . . . 753.3.3 Phasors for the Magic Mode . . . . . . . . . . . . . . . . . . 78

3.4 Kinks with Substrate Potential: The Crowdion . . . . . . . . . . . 813.5 Phonons and Crowdions. . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5.1 Phonons in Presence of a Substrate Potential . . . . . . 833.5.2 Crowdion Phonon Tail . . . . . . . . . . . . . . . . . . . . . . 85

3.6 Some Numerical Simulations with Ultradiscrete Kinksor Crowdions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.6.1 Excess Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.6.2 Thermalized Medium . . . . . . . . . . . . . . . . . . . . . . . 88

3.7 Recoil Energy of 40K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.7.1 40K Decay Branches . . . . . . . . . . . . . . . . . . . . . . . 913.7.2 Secondary Processes . . . . . . . . . . . . . . . . . . . . . . . 94

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

xiv Contents

Part II Two-dimensional Lattices

4 Pattern Formation by Traveling Localized Modesin Two-Dimensional Dissipative Media with Lattice Potentials . . . 99Valentin Besse, Hervé Leblond, Dumitru Mihalacheand Boris A. Malomed4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.1.1 Dissipative Solitons: A Brief Overview . . . . . . . . . . 1004.1.2 The Subject of the Consideration

in the Present Chapter . . . . . . . . . . . . . . . . . . . . . . 1044.2 The Cubic-Quintic Complex Ginzburg-Landau Model

with the Cellular Potential . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3 The Pattern Formation by Kicked Dipoles . . . . . . . . . . . . . . 107

4.3.1 Generation of Multi-dipole Patternsby a Dipole Moving in the Transverse Direction . . . . 107

4.3.2 Dynamical Regimes Initiated by the LongitudinalKick Applied to the Dipole. . . . . . . . . . . . . . . . . . . 109

4.3.3 Collision Scenarios for Moving Dipoles in theSystem with Periodic Boundary Conditions . . . . . . . 110

4.4 The Pattern Formation by Kicked Quadrupoles . . . . . . . . . . . 1124.5 The Pattern Formation by Kicked Vortices . . . . . . . . . . . . . . 116

4.5.1 Chaotic Patterns Generated by Kicked Rhombic(Onsite-Centered) Vortices . . . . . . . . . . . . . . . . . . . 116

4.5.2 Kicked Square-Shaped (Offsite-Centered)Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 A Numerical Study of Weak Lateral Dispersion in Discreteand Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Luis A. Cisneros-Ake and Antonmaria A. Minzoni5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.2 Numerical Approximation to the Kadomtsev-Petviashvili

I Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.2.1 Continuous and Discrete Lump Propagation . . . . . . . 133

5.3 Lateral Motion and Interaction of Pulses with Obstacles. . . . . 1375.4 The Effect of Impurities in a Prestressed Lattice . . . . . . . . . . 1405.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6 Breather Mobility and the Peierls-Nabarro Potential:Brief Review and Recent Progress . . . . . . . . . . . . . . . . . . . . . . . . 147Magnus Johansson and Peter Jason6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Contents xv

6.2 PN-Barriers and Discrete Soliton Mobility in 1D. . . . . . . . . . 1506.3 Discrete Soliton (Breather) Mobility in 2D . . . . . . . . . . . . . . 153

6.3.1 Discrete Soliton Mobility in the 2D SaturableDNLS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.3.2 The Kagome Lattice . . . . . . . . . . . . . . . . . . . . . . . 1616.4 Travelling Discrete Dissipative Solitons

with Intrinsic Gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.5 Mobility of Quantum Lattice Compactons . . . . . . . . . . . . . . 1696.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7 Asymptotic Approximation of Discrete Breather Modesin Two-Dimensional Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Jonathan A.D. Wattis7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.1.2 The One-Dimensional FPU System . . . . . . . . . . . . . 1827.1.3 Generalisation to Two Dimensions . . . . . . . . . . . . . 183

7.2 The Asymptotic Reduction for the FPU Chain . . . . . . . . . . . 1847.3 Two-Dimensional Square Lattice . . . . . . . . . . . . . . . . . . . . . 187

7.3.1 Asymptotic Calculations. . . . . . . . . . . . . . . . . . . . . 1877.3.2 Case I: The Symmetric Potential (a ¼ 0) . . . . . . . . . 1897.3.3 Form and Stability of Soliton Solutions . . . . . . . . . . 1907.3.4 Higher Order Asymptotic Analysis

for Stationary Breathers . . . . . . . . . . . . . . . . . . . . . 1927.4 Honeycomb Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.4.1 OðeÞ—Dispersion Relationfor the Honeycomb Lattice . . . . . . . . . . . . . . . . . . . 194

7.4.2 General Approach for the Higher Order Terms . . . . . 1957.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.5.1 Future Directions. . . . . . . . . . . . . . . . . . . . . . . . . . 198References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Part III Molecular Dynamics

8 Moving Discrete Breathers in 2D and 3D Crystals . . . . . . . . . . . . 205Sergey V. Dmitriev, Andrei A. Kistanov and Vladimir I. Dubinko8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2068.2 Moving DB in 2D Hexagonal Lattice with Long-Range

Morse Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2078.2.1 Simulation Setup and Moving DB Ansatz . . . . . . . . 2088.2.2 Head-On Collision of Moving DB. . . . . . . . . . . . . . 211

xvi Contents

8.3 DB in Pure Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2128.3.1 Collision of Moving DB . . . . . . . . . . . . . . . . . . . . 2128.3.2 Application to Radiation Effects . . . . . . . . . . . . . . . 215

8.4 Wandering DB in an Ionic Crystal. . . . . . . . . . . . . . . . . . . . 2188.4.1 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . 2188.4.2 Pairs of Discrete Breathers . . . . . . . . . . . . . . . . . . . 219

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

9 Standing and Moving Discrete Breathers with FrequenciesAbove the Phonon Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Vladimir Hizhnyakov, Mati Haas, Alexander Shelkanand Mihhail Klopov9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2299.2 Mean Field Theory of Discrete Breathers . . . . . . . . . . . . . . 2309.3 Splitting of Discrete Breathers from the Top

of the Phonon Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 2339.4 Standing Discrete Breathers Above the Phonon Spectrum. . . . 235

9.4.1 Standing DBs in Ge and Diamond. . . . . . . . . . . . . . 2389.5 Moving Discrete Breathers with Frequencies

Above the Phonon Spectrum. . . . . . . . . . . . . . . . . . . . . . . . 2419.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

10 Phonon Interference and Energy Transport in NonlinearLattices with Resonance Defects . . . . . . . . . . . . . . . . . . . . . . . . . 247Yuriy A. Kosevich, Haoxue Han, Lyudmila G. Potyomina,Alexandre N. Darinskii and Sebastian Volz10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24810.2 Model Structures and Simulation Methodology . . . . . . . . . . . 249

10.2.1 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

10.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 25210.3.1 Interference Resonance Profile . . . . . . . . . . . . . . . . 25210.3.2 Isotopic Shift of Resonances . . . . . . . . . . . . . . . . . . 25410.3.3 Phonon Screening Effect . . . . . . . . . . . . . . . . . . . . 25510.3.4 Two-Path Phonon Interference in Si Crystal

with Ge Impurities. . . . . . . . . . . . . . . . . . . . . . . . . 25610.3.5 Random Distribution of Atoms . . . . . . . . . . . . . . . . 25710.3.6 Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . 25810.3.7 Wave Packet Coherence Length Determination . . . . . 259

10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Contents xvii

Part IV Electrons and Lattice Vibrations

11 Electron Transfer and Tunneling from Donor to Acceptorin Anharmonic Crystal Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 267Alexander P. Chetverikov, Leonor Cruzeiro, Werner Ebelingand Manuel G. Velarde11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26811.2 Hamiltonian and Equations of Motion

of the Electron-Lattice Dynamics. . . . . . . . . . . . . . . . . . . . . 27011.3 Free Electron Dynamics with One Bound State in a Lattice

with Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . 27211.4 Free Electron Dynamics in a Lattice with Fixed

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27511.5 Computer Simulations of ET from Donor to Acceptor

in a Lattice at Low Temperature . . . . . . . . . . . . . . . . . . . . . 28011.6 Dependence of Transition Time on a . . . . . . . . . . . . . . . . . . 28411.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 284References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

12 Bound States of Electrons in Harmonic and AnharmonicCrystal Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Larissa S. Brizhik, Alexander P. Chetverikov, Werner Ebeling,Gerd Röpke and Manuel G. Velarde12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29212.2 Hamiltonian of the System and Dynamic Equations . . . . . . . . 29512.3 Bisolitons in Harmonic Lattices. . . . . . . . . . . . . . . . . . . . . . 29712.4 Bisolectrons in Anharmonic Lattices . . . . . . . . . . . . . . . . . . 30012.5 Bisolectrons with Account of the Coulomb Repulsion . . . . . . 30712.6 Comparison with Numerical Simulations . . . . . . . . . . . . . . . 30912.7 Supersonic Bisolectrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 31212.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

13 Solitons and Charge Transport in Triangularand Quadratic Crystal Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 321A.P. Chetverikov, W. Ebeling and M.G. Velarde13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32113.2 Excitations in Square Lattices . . . . . . . . . . . . . . . . . . . . . . . 32213.3 Dispersion Relation for Two-Dimensional Excitations

and KP Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32813.4 Tight-Binding Dynamics of Charges Interacting

with the Lattice Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

xviii Contents

13.5 Control of Electrons and Losses-Free Transporton Longer Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

13.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Part V Semiconductors

14 Experimental Observation of Intrinsic Localized Modesin Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343Juan F.R. Archilla, Sergio M.M. Coelho, F. Danie Auret,Cloud Nyamhere, Vladimir I. Dubinko and Vladimir Hizhnyakov14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34414.2 Germanium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34514.3 Phonons in Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34614.4 Defects and Their Detection with DLTS. . . . . . . . . . . . . . . . 35114.5 Experiment of Plasma Induced Annealing. . . . . . . . . . . . . . . 35514.6 ILM Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35614.7 Thermal Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35814.8 Comparison of Thermal and Plasma-Induced Annealing . . . . . 35914.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

15 The Origin of Defects Induced in Ultra-Pure Germaniumby Electron Beam Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 363Sergio M.M. Coelho, Juan F.R. Archilla, F. Danie Auretand Jackie M. Nel15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36415.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36815.3 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36915.4 Intrinsic Localized Modes or Breathers . . . . . . . . . . . . . . . . 372

15.4.1 Limitations of Harmonicity . . . . . . . . . . . . . . . . . . 37315.4.2 Effect of Intrinsic Localized Modes . . . . . . . . . . . . . 376

15.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

16 Rate Theory of Acceleration of Defect Annealing Drivenby Discrete Breathers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381Vladimir I. Dubinko, Juan F.R. Archilla, Sergey V. Dmitrievand Vladimir Hizhnyakov16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38116.2 Discrete Breathers in Metals and Semiconductors . . . . . . . . . 382

16.2.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38316.2.2 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Contents xix

16.3 DB Excitation Under Thermal Equilibriumand External Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38916.3.1 Thermal Activation . . . . . . . . . . . . . . . . . . . . . . . . 39016.3.2 External Driving . . . . . . . . . . . . . . . . . . . . . . . . . . 391

16.4 Amplification of Sb-Vacancy Annealing Ratein Germanium by DBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

16.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Part VI Other Systems

17 The Amide I Band of Crystalline Acetanilide: Old DataUnder New Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401Leonor Cruzeiro17.1 Setting the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40117.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40417.3 The Acetanilide Crystal Structure and Dynamics . . . . . . . . . . 40817.4 No Orientational Influence on the Amide I Energy . . . . . . . . 41117.5 Weak Orientational Dependence of the Amide I Energy . . . . . 41617.6 Strong Orientational Dependence of the Amide I Energy . . . . 41717.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 419References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

18 Extreme Waves and Branching Flows in Optical Media . . . . . . . . 425Marios Mattheakis and George P. Tsironis18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42518.2 Mathematical Tools for Electromagnetic

Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42718.2.1 Quasi-two Dimensional Ray Solution. . . . . . . . . . . . 42818.2.2 Parametric Two Dimensional Ray Solution . . . . . . . . 43118.2.3 Helmholtz Wave Equation Approach . . . . . . . . . . . . 43318.2.4 Numerical Solution of Maxwell Equations . . . . . . . . 435

18.3 Branching Flow in Weakly Disordered Media. . . . . . . . . . . . 43718.3.1 Statistics of Caustics . . . . . . . . . . . . . . . . . . . . . . . 43818.3.2 Branching Flows in Physical Systems . . . . . . . . . . . 443

18.4 Rogue Wave Formation Through Strong ScatteringRandom Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44718.4.1 Rogue Waves in Optics . . . . . . . . . . . . . . . . . . . . . 448

18.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

xx Contents

19 Discrete Bright Solitons in Bose-Einstein Condensatesand Dimensional Reduction in Quantum Field Theory . . . . . . . . . 455Luca Salasnich19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45519.2 Bose-Einstein Condensate in a Quasi-1D Optical Lattice . . . . 457

19.2.1 Axial Discretization of the 3D Gross-PitaevskiiEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

19.2.2 Transverse Dimensional Reduction of the 3D DiscreteGross-Pitaevskii Equation . . . . . . . . . . . . . . . . . . . . 458

19.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 46019.2.4 Collapse of the Discrete Bright Soliton . . . . . . . . . . 462

19.3 Dimensional Reduction of a ContinuousQuantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46419.3.1 Dimensional Reduction of the Hamiltonian . . . . . . . . 46519.3.2 1D Nonpolynomial Heisenberg Equation . . . . . . . . . 46719.3.3 Generalized Lieb-Liniger Theory . . . . . . . . . . . . . . . 468

19.4 Dimensional Reduction for Bosons in a Quasi-1D Lattice. . . . 46919.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Part VII An Historical Perspective

20 I Saw a Crystal: An Historical Account of the Decipheringof the Markings in Mica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475F. Michael Russell20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47520.2 The Early Years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47720.3 Hunting for Mica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49220.4 The Discovery of Charged Particle Tracks . . . . . . . . . . . . . . 49720.5 Rebuttal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50420.6 Decay of Potassium Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . 51020.7 Lattice Excitations from Scattering of Muons . . . . . . . . . . . . 51820.8 Experimental Confirmation of Quodons . . . . . . . . . . . . . . . . 52820.9 Still Not Understood Tracks . . . . . . . . . . . . . . . . . . . . . . . . 53620.10 Quodons, Breathers and Extraterrestrials. . . . . . . . . . . . . . . . 54520.11 Question Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55020.12 After Dinner Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 55220.13 Recent Developments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

Contents xxi

Contributors

Juan F.R. Archilla Group of Nonlinear Physics, Departamento de FísicaAplicada I, Universidad de Sevilla, Sevilla, Spain

F. Danie Auret Department of Physics, University of Pretoria, Pretoria, SouthAfrica

Janis Bajars College of Arts and Science, School of Science & Technology,Nottingham Trent University, Nottingham, UK

Valentin Besse LUNAM Université, Université d’Angers, Laboratoire dePhotonique d’Angers, Angers, France

Larissa S. Brizhik Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine

Alexander P. Chetverikov Department of Physics, Saratov State University,Saratov, Russia

Luis A. Cisneros-Ake Department of Mathematics, ESFM, Instituto PolitécnicoNacional, México D.F., México

Sergio M.M. Coelho Department of Physics, University of Pretoria, Pretoria,South Africa

Leonor Cruzeiro CCMAR and Physics, FCT, Universidade do Algarve, Faro,Portugal

Alexandre N. Darinskii Institute of Crystallography, Russian Academy ofSciences, Moscow, Russia

Sergey V. Dmitriev Institute for Metals Superplasticity Problems, Ufa, Russia;National Research Tomsk State University, Tomsk, Russia

Vladimir I. Dubinko NSC Kharkov Institute of Physics and Technology,Kharkov, Ukraine

xxiii

Werner Ebeling Institut für Physik, Humboldt-Universität Berlin, Berlin,Germany

J. Chris Eilbeck Maxwell Institute and Department of Mathematics, Heriot-WattUniversity, Edinburgh, UK

Luis M. García-Raffi Instituto Universitario de Matemática Pura y Aplicada,Universidad Politécnica de Valencia, Valencia, Spain

Mati Haas Institute of Physics, University of Tartu, Tartu, Estonia

Haoxue Han CNRS, UPR 288 Laboratoire D’Energétique Moléculaire etMacroscopique, Combustion (EM2C) and Ecole Centrale Paris, Châtenay-Malabry,France

Vladimir Hizhnyakov Institute of Physics, University of Tartu, Tartu, Estonia

Peter Jason Department of Physics, Chemistry and Biology (IFM), LinköpingUniversity, Linköping, Sweden

Noé Jiménez Instituto de Investigación para la Gestión Integrada de las ZonasCosteras, Universidad Politécnica de Valencia, Grao de Gandia, Spain

Magnus Johansson Department of Physics, Chemistry and Biology (IFM),Linköping University, Linköping, Sweden

Andrei A. Kistanov Institute for Metals Superplasticity Problems, RAS, Ufa,Russia

Mihhail Klopov Institute of Physics, Tallinn University of Technology, Tallinn,Estonia

Yuriy A. Kosevich Semenov Institute of Chemical Physics, Russian Academy ofSciences, Moscow, Russia

Hervé Leblond LUNAM Université, Université d’Angers, Laboratoire dePhotonique d’Angers, Angers, France

Ben Leimkuhler Maxwell Institute and School of Mathematics, University ofEdinburgh, Edinburgh, UK

Boris A. Malomed Department of Physical Electronics, Faculty of Engineering,Tel Aviv University, Tel Aviv, Israel

Marios Mattheakis Crete Center of Quantum Complexity and Nanotechnology(CCQCN), Department of Physics, University of Crete, Heraklion, Greece; Instituteof Electronic Structure and Laser, Foundation for Research and Technology—Hellas (FORTH), Heraklion, Greece; Harvard University, Cambridge, MA, USA

xxiv Contributors

Dumitru Mihalache LUNAM Université, Université d’Angers, Laboratoire dePhotonique d’Angers, Angers, France; Horia Hulubei National Institute for Physicsand Nuclear Engineering, Magurele-Bucharest, Romania; Academy of RomanianScientists, Bucharest, Romania

Antonmaria A. Minzoni FENOMEC, Department of Mathematics andMechanics, IIMAS, Universidad Nacional Autónoma de México, México, D. F.,México

Jackie M. Nel Department of Physics, University of Pretoria, Pretoria, SouthAfrica

Cloud Nyamhere Physics Department, Midlands State University, Gweru,Zimbabwe

Lyudmila G. Potyomina Department of Physics and Technology, NationalTechnical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine

F. Michael Russell Department of Physics, University of Pretoria, Pretoria, SouthAfrica

Gerd Röpke Institut für Physik, Universität Rostock, Rostock, Germany

Luca Salasnich Department of Physics and Astronomy “Galileo Galilei” andCNISM, Universita di Padova, Padova, Italy; Istituto Nazionale di Ottica (INO) delConsiglio Nazionale delle Ricerche (CNR), Sezione di Sesto Fiorentino, SestoFiorentino, Italy

Alexander Shelkan Institute of Physics, University of Tartu, Tartu, Estonia

Víctor J. Sánchez-Morcillo Instituto de Investigación para la Gestión Integradade las Zonas Costeras, Universidad Politécnica de Valencia, Grao de Gandia, Spain

George P. Tsironis Crete Center of Quantum Complexity and Nanotechnology(CCQCN), Department of Physics, University of Crete, Heraklion, Greece; Instituteof Electronic Structure and Laser, Foundation for Research and Technology—Hellas (FORTH), Heraklion, Greece; Department of Physics, NazarbayevUniversity, Astana, Republic of Kazakhstan

Manuel G. Velarde Instituto Pluridisciplinar, Universidad Complutense, Madrid,Spain

Sebastian Volz CNRS, UPR 288 Laboratoire D’Energétique Moléculaire etMacroscopique, Combustion (EM2C) and Ecole Centrale Paris, Châtenay-Malabry,France

Jonathan A.D. Wattis School of Mathematical Sciences, University ofNottingham, Nottingham, UK

Contributors xxv