Thermodynamics of Crystalline States (2nd edition)

Thermodynamics of Crystalline States

Minoru Fujimoto

Thermodynamics of Crystalline States Second Edition

Minoru Fujimoto Department of Physics University of Guelph Guelph, ON NIG 2W1 Canada

ISBN 978-1-4614-5084-9 ISBN 978-1-4614-5085-6 (eBook) DOI 10.1007/978-1-4614-5085-6 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012954672 # Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of 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 transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Originated from steam engines, in thermodynamics the energy exchange is formulated traditionally for homogeneous states of materials. Applying to condensed matters however, laws of thermodynamics have to be used with respect to the structural detail. Following Kirkwood’s Chemical Thermodynamics, this book is written for lattice dynamics in crystalline states under the laws of thermodynamics. It is noted that lattice symmetry remains implicit in thermodynamic functions of crystalline states, if assumed as homogeneous. In contrast, deformed crystals with disrupted symmetry are not stable and inhomogeneous, exhibiting mesoscopic properties, for which the lattice dynamics should be redefined under thermodynamic principles. I have selected the topics of structural changes, magnetic crystals, and superconducting transitions in this book to discuss these basic thermodynamic processes in crystalline states. Born and Huang have laid ground for thermodynamics of crystalline states in their book Dynamical Theory of Crystal Lattices. However, they considered that order–disorder phenomena was independent from the lattice dynamics, and so excluded it from their book. On the other hand, today new evidence indicates that the problem should be treated otherwise; in fact, the lattice plays a vital role in ordering processes. Accordingly, I was motivated to write about the physics of crystal lattice in the light of Born-Huang’s principles, which constitutes my primary objective in this book. In modern experiments, mesoscopic objects in crystals can be investigated within the timescale of observation, yielding results that appear somewhat unusual, if compared with macroscopic experiments. It is important that thermodynamic relations in mesoscopic states should be described with respect to the timescale of observation. Also significant is that mesoscopic quantities in crystals are driven by internal interactions in nonlinear character. I have therefore discussed thermodynamic quantities with regard to the timescale of observation, including elementary accounts of nonlinear physics to deal with long-range correlations. For the convenience of readers who are not particularly familiar with nonlinear dynamics, Appendix is attached for some useful formula of elliptic functions.

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Preface

I have rewritten this second edition for advanced students in physics and material science, assuming basic knowledge of traditional thermodynamics and quantum theories. This book is designed for serving as a useful textbook for classroom and seminar discussions. I hope that it will stimulate advanced studies of condensed matter. I should mention with my sincere appreciation that I have benefited for my writing from numerous discussions with and comments from my colleagues and students. Finally, I thank my wife Haruko for her continuous support and encouragements. Mississauga, Ontario

Minoru Fujimoto

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Crystalline Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Structural Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 New Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Modulated Phases and Renormalized Coordinates . . . 1.3.2 Thermodynamic Changes and Nonlinear Dynamics . . 1.3.3 Fields within Thermodynamic Boundaries . . . . . . . . . 1.3.4 Adiabatic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Sampling Modulated States . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 4 7 7 8 8 8 9 9 10

2

Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Normal Modes in a Simple Crystal . . . . . . . . . . . . . . . . . . . . 2.2 Quantized Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Phonon Field and Momentum . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Specific Heat of a Monatomic Crystal . . . . . . . . . . . . . . . . . . 2.6 Approximate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Einstein’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Debye’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Phonon Statistics Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Compressibility of a Crystal . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 11 14 16 18 20 22 22 23 25 28 30 31

3

Order Variables and the Adiabatic Potential . . . . . . . . . . . . . . . 3.1 A One-dimensional Ionic Chain . . . . . . . . . . . . . . . . . . . . . . 3.2 Displacive Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Born-Oppenheimer’s Approximation . . . . . . . . . . . . . . . . . .

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33 33 35 39

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3.4

Equilibrium Crystals and the Bloch Theorem . . . . . . . . . . . . 3.4.1 Reciprocal Lattice and Renormalized Coordinates . . . 3.4.2 The Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Lattice Symmetry and the Brillouin Zone . . . . . . . . . . 3.5 Phonon Scatterings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Phonon Scatterings in Crystals . . . . . . . . . . . . . . . . . 3.5.2 The Born-Huang Relaxation and Soft Modes . . . . . . . Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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45 45 47 49 51 51 52 53 54

4

Mean-Field Theories of Binary Ordering . . . . . . . . . . . . . . . . . . 4.1 Probabilities in Binary Alloys . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Bragg-Williams Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Becker’s Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Ferromagnetic Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Ferromagnetic Transition in Applied Magnetic Field . . . . . . . Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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55 55 57 60 62 64 66 66

5

Pseudospin Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Pseudospins for Binary Displacements . . . . . . . . . . . . . . . . . 5.2 A Tunneling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Pseudospin Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Pseudospin Clusters in Crystals . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Classical Pseudospins in Crystals . . . . . . . . . . . . . . . . 5.4.2 Clusters of Pseudospins . . . . . . . . . . . . . . . . . . . . . . . 5.5 Examples of Pseudospin Clusters . . . . . . . . . . . . . . . . . . . . . 5.5.1 Cubic to Tetragonal Transition in SrTiO3 . . . . . . . . . . 5.5.2 Monoclinic Crystals of Tris-Sarcosine Calcium Chloride (TSCC) . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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67 67 70 71 73 73 74 78 78

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80 82 82

Critical Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Landau Theory of Binary Transitions . . . . . . . . . . . . . . . 6.2 Internal Pinning of Adiabatic Fluctuations . . . . . . . . . . . . . . 6.3 Critical Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Observing Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Extrinsic Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Pinning by Point Defects . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Pinning by an Electric Field . . . . . . . . . . . . . . . . . . . 6.5.3 Surface Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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83 83 87 88 90 92 93 93 94 94 95

6

Contents

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7

Pseudospin Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Propagation of a Collective Pseudospin Mode . . . . . . . . . . . . 7.2 Transverse Components and the Cnoidal Potential . . . . . . . . 7.3 The Lifshitz Condition for Incommensurability . . . . . . . . . . . 7.4 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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97 97 101 103 105 108 109

8

Soliton Theory of Long-Range Order . . . . . . . . . . . . . . . . . . . . 8.1 A Longitudinal Dispersive Mode of Collective Pseudospins . . 8.2 The Korteweg–deVries Equation . . . . . . . . . . . . . . . . . . . . . 8.3 Solutions of the Korteweg–deVries Equation . . . . . . . . . . . . 8.4 Cnoidal Theorem and the Eckart Potential . . . . . . . . . . . . . . 8.5 Condensate Pinning by the Eckart Potentials . . . . . . . . . . . . . 8.6 Born–Huang’s Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Topological Mapping of Mesoscopic Fields . . . . . . . . . . . . . Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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111 111 115 117 120 122 126 129 132 135

9

Soft Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Lyddane–Sachs–Teller Relation . . . . . . . . . . . . . . . . . . . 9.2 Soft Lattice Modes of Condensates . . . . . . . . . . . . . . . . . . . . 9.2.1 The Lattice Response to Collective Pseudospins . . . . . 9.2.2 Temperature Dependence of Soft-Mode Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Cochran’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Symmetry Change at Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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137 137 141 141

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144 148 151 153 154

Experimental Studies on Critical Fluctuations . . . . . . . . . . . . . . 10.1 Diffuse X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Neutron Inelastic Scatterings . . . . . . . . . . . . . . . . . . . . . . . 10.3 Light Scattering Experiments . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Brillouin Scatterings . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Raman Scatterings . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Principles of Nuclear Magnetic Resonance and Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Paramagnetic Resonance of Impurity Probes . . . . . 10.4.3 The Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Hyperfine Interactions . . . . . . . . . . . . . . . . . . . . . . 10.5 Magnetic Resonance in Modulated Crystals . . . . . . . . . . . . 10.6 Examples of Transition Anomalies . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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155 155 160 164 165 169 171

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173 175 176 180 180 183 187

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Contents

Magnetic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Microscopic Magnetic Moments in Crystals . . . . . . . . . . . . 11.2 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Spin–Spin Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Spin Clusters and Magnetic Symmetry . . . . . . . . . . . . . . . . 11.5 Magnetic Weiss Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Magnetic Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Antiferromagnetic and Ferrimagnetic States . . . . . . . . . . . . 11.9 Fluctuations in Ferromagnetic and Antiferromagnetic States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9.1 Ferromagnetic Resonance . . . . . . . . . . . . . . . . . . . 11.9.2 Antiferromagnetic Resonance . . . . . . . . . . . . . . . . Exercise 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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189 189 193 194 198 200 203 204 208

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210 211 212 214 214

12

Phonon and Electron Statistics in Metals . . . . . . . . . . . . . . . . . . 12.1 Phonon Statistics Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Conduction Electrons in Metals . . . . . . . . . . . . . . . . . . . . . 12.2.1 The Pauli Principle . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 The Coulomb Interaction . . . . . . . . . . . . . . . . . . . . 12.2.3 The Bloch Theorem in Equilibrium Crystals . . . . . . 12.3 Many-Electron Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Fermi–Dirac Statistics for Conduction Electrons . . . . . . . . . Exercise 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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215 215 218 218 220 221 222 226 226 227

13

Superconducting Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Superconducting States . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Near-Zero Electrical Resistance . . . . . . . . . . . . . . . 13.1.2 The Meissner Effect . . . . . . . . . . . . . . . . . . . . . . . 13.1.3 Thermodynamic Equilibrium Between Normal and Superconducting Phases . . . . . . . . . . . . . . . . . 13.2 Long-Range Order in Superconducting States . . . . . . . . . . . 13.3 Electromagnetic Properties of Superconductors . . . . . . . . . . 13.4 The Ginzburg–Landau Equation and the Coherence Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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229 229 229 231

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233 235 237

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244 247 248

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249 249 252 255

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14

Theories of Superconducting Transitions . . . . . . . . . . . . . . . . . . 14.1 The Fro¨hlich’s Condensate . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 The Cooper Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Critical Anomalies and the Superconducting Ground State . . 14.3.1 Critical Anomalies and a Gap in a Superconducting Energy Spectrum . . . . . . . . . . . . .

Contents

14.3.2 Order Variables in Superconducting States . . . . . . . 14.3.3 BCS Ground States . . . . . . . . . . . . . . . . . . . . . . . . 14.3.4 Superconducting States at Finite Temperatures . . . . Exercise 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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257 262 266 267 268

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271

Chapter 1

Introduction

The role played by the lattice for symmetry changes in a crystal is a basic subject for discussion in this book. On the other hand, as related to lattice excitations, the concept of order variables needs to be established in thermodynamics of crystalline states. Normally inactive at elevated temperatures, order variables at lattice sites should be responsive to a symmetry change at a critical temperature. Exhibiting a singular behavior, an adiabatic potential should also emerge with order variables in finite magnitude with lowering temperature. Well documented experimentally, we should analyze their properties under thermodynamic principles. In reviewing thermodynamic principles, the present thermodynamics of crystalline states is overviewed in this chapter, emphasizing dynamic roles played by the lattice structure.

1.1

Crystalline Phases

Originated from early studies on steam engines, thermodynamics today is a well-established discipline of physics for thermal properties of matter. Specified by a uniform density, a thermodynamic state of a crystal can be described by state functions of temperature and pressure of the surroundings. In contrast, characterized by symmetry of the lattice structure, properties of a single crystal are attributed to masses and other physical properties of constituents at lattice points, while symmetry per se cannot be responsible for the physical properties. We realize that structural changes cannot be described by uniform state functions, unless these functions are defined as related with the structural detail. The structural transition is generally discontinuous at a critical temperature T c , below which even a chemically pure crystal becomes heterogeneous, composed of substructures in smaller volume, for example, domains in thermal equilibrium. Nevertheless, an external force or field can transform domains from one type to another while

M. Fujimoto, Thermodynamics of Crystalline States, DOI 10.1007/978-1-4614-5085-6_1, # Springer Science+Business Media New York 2013

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1 Introduction

maintaining structural stability. In addition, there are inhomogeneous crystals among various types, which are signified by a sinusoidal density, representing modulated structure. Following Kirkwood [1] in Chemical Thermodynamics we can consider state functions of continuous internal variables in crystalline states in equilibrium with the surroundings. In Dynamical Theory of Crystal Lattices, Born and Huang [2] proposed the principle for lattice stability, laying the foundation of thermodynamics of crystalline states. However, their principle was not quite verified with experimental results, which were not fully analyzed at that time. Today, in spite of many supporting results, their theory is still not considered as sufficiently substantiated. A crystalline state is very different from a gaseous state of a large number of free independent particles. A crystal structure packed with identical constituents is characterized by distinct symmetry; in contrast, fast molecular motion prevails in gaseous states. The internal energy of a gas is primarily kinetic energies of constituent particles in free motion, whereas for a crystal, it is intermolecular potential energies in the packed structure. The structural transformation in crystals is a dynamic process, whereas a gas is normally in a single phase, except in condensation process. In addition, a gas confined to a container has a finite volume, whereas external conditions determine a crystal volume. Simplifying by a mathematical conjecture, we can say that surfaces have a little contribution to bulk properties of a sizable crystal. However, a small volume change is inevitable in practical crystals under structural changes. In thermodynamics, a crystal must always be in contact with the surroundings. Assuming no chemical activity, surfaces are in physical contact with the surroundings, exchanging heat between them. Joule demonstrated that heat is nothing but energy, although it is of unique type microscopically. Boltzmann considered that heat originates from randomly distributed microscopic energies in the surroundings. On the other hand, excitations in a crystal are primarily lattice vibrations at a given temperature, which are quantum mechanically expressed as phonons in random motion. The phonon spectrum is virtually continuous; the internal energy of lattice is given by a statistical average of distributed phonon energies. In this context, the lattice symmetry is implicit in thermodynamic functions, as specified by a uniform density in a constant volume. Equilibrium between two bodies in thermal contact, as characterized by a common temperature T, signifies that there is no net flow of heat across the surfaces. For a combined system of a crystal and its surroundings, the total fluctuating heat is always dissipative, so that DQ is expressed as negative, that is, DQ
0, implying that any thermal process is fundamentally irreversible in nature, as stated by the second-law of thermodynamics. Depending on the nature of heat dissipation, such DQ cannot generally be expressed by a function of temperature and pressure; namely, the heat quantity Q is not a state function. Instead, a function DQ S, defined by T ¼ DS , can be employed as a state function if the integrating denominator T can be found for an infinitesimal S to be a total differential, representing a reversible contact between Hthe crystal and H surroundings. Clausius generalized this argument in integral form C Q
0 or C dS
0 around a closed T

1.1 Crystalline Phases

3

curve C in a phase diagram, where the equality sign indicates an idealized case of reversible process. Thermal equilibrium can thus be determined by maximizing the state function S, which is known as the entropy. Boltzmann interpreted the entropy S in terms of thermodynamic probability gðTÞ at constant pressure to represent a thermal average of randomly distributed microscopic energies in the surroundings, for which he wrote the relation S ¼ kB ln gðTÞ;

(1.1)

where kB is the Boltzmann constant. For crystalline states in equilibrium at temperature T, such a probability is a valid concept if random phonon energies are responsible for the statistical description under a constant volume V. However, if a small volume change DV is considered at constant T and p, a process at a constant crystal volume V is not an acceptable assumption for a crystal under constant p. Hence, Boltzmann’s statistical theory is valid only if V is constant; otherwise, the dynamical system in a crystal is statistically non-ergodic. In thermodynamics, physical properties of a crystal can be represented by the internal energy U ðp; T Þ, which is varied not only by heat DQ but also by external work W, as stated by the first law of thermodynamics. Namely, DU ¼ DQ þ W:

(1.2)

A mechanical work W can change the volume V, where a structural modification may be induced, for example, by an electric or magnetic field, straining crystals by their forces X. ConsideringP that such variablessm respond to applied X, the work W can be expressed as W ¼  sm X . It is noted that such X can be associated with m

internal interactions with sn at different sites n, signifying correlations between internal variables sm and sn . Such internal correlations can occur adiabatically in a crystal, independent of temperature. However, they are weakly temperature dependent in practice, via interactions with phonons. In contrast, the heat energy Q depends only on the temperature of the surroundings, flowing in and out of a crystal at a constant V. Correlated sm is generally in collective motion at long wavelength, propagating through an excited lattice. Such motion in a crystal should occur when driven by an adiabatic potential DU m at the lattice point m, as will be discussed in Chap. 7. In a modulated structure, the propagation can be detected in a standing wave whose period is not necessarily the same as in the lattice. On the other hand, if the correlation energy in short range is expressed in a form  J mn sm sn , where J mn is a function of the distance between sm and sn , we can write 

X

J mn sm sn ¼ sm Xm

(1.3)

n

which allows us to define the internal field Xm ¼

P n

J mn sn at a site m. Averaging Xm

and sm over all lattice sites, we can define the mean-field average hXm i and hsm i, respectively. For a nonzero hsm i, the expression (1.3) can be interpreted as work

4

1 Introduction

performed by a field hXm i on the hsm iin mean-field accuracy. On the other hand, by writing (1.3) as  hsm iXint, Xint can represent another kind of internal field, defined in better accuracy than the mean-field average hXm i. In this case, attributed to the adiabatic potential DU m , we can write Xint ¼ rm ðDU m Þ, corresponding to the Weiss field that was originally proposed for a magnetic crystal. In the presence of an external field X, the effective field can be defined as X þ Xint ¼ X0 , considering X0 as if applied externally. Such a field X0 is usually temperature dependent, as contributed by temperature-dependent Xint; hence, we can write X0 ¼ X0 ðTÞ under a constant p. The macroscopic energy relation can therefore be expressed as DU
DQ  DðsX0 Þ

or

DU  TDS þ DðsX0 Þ
0:

Writing the Gibbs potential as G ¼ U  TS þ sX0 , we can obtain the inequality DG
0;

(1.4)

indicating that the minimum of G determines the equilibrium at constant T and p. In practice, order variables sm should be identified experimentally; thereby, equilibrium can be determined from the Gibbs function at given T and p. On the other hand, the mean-field average hsm i ¼ s , called order parameter, can be utilized for simplifying description. In this case, the Gibbs function can be written as Gðp; T; sÞ, where the order parameter s is defined for the range 0
s
1, where 0 and 1 are designated to perfect disorder and order, respectively. In spontaneously ordered crystals, the internal field Xint can be related to spatially distributed sm. For a ferromagnetic crystal, Weiss (1907) postulated that Xint / ðmagnetizationÞ, which is traditionally called Weiss’ molecular field in a ferroelectric crystal.

1.2

Structural Changes

The structure of an equilibrium crystal is stable, which is however transformable from one stable state to another by applying an external field. In addition, we consider that the order variables sm emerging at T c in finite magnitude are in collective motion, leading to spontaneous phase transitions. Although assumed primarily as independent of the hosting lattice, the correlated order variables modulate translational lattice symmetry. Interpreting this by Newton’s action–reaction principle, correlated lattice displacements um should interact with correlated vectors sm . Born and Huang [2] proposed their principle for the lattice to maintain stability with minimum strains, which can be evaluated by minimizing the Gibbs potential. Such a minimizing process can nevertheless be observed as thermal relaxation. Regarding a transition from a phase 1 to another phase 2 at a critical point ðpc ; T c), the Gibbs potential may be written as Gðpc ; T c ; s1 ; s2 Þ, where s1 and s2 represent

1.2 Structural Changes

5

these phases 1 and 2 which coexist during the transition. Although somewhat imprecise, we write the whole Gibbs potential under an applied field X as Gtrans ¼ Gðpc ; T c ; s1 ; s2 Þ  ðs1 þ s2 ÞX: In the noncritical region below T c, we consider two Gibbs potentials G1 and G2, coexisting at p and T, which are expressed as G1 ¼ Gðp; T Þ  s1 X and G2 ¼ Gðp; T Þ  s2 X; respectively. For a binary transition characterized by inversion s1 ¼ s2 , we consider  P @Gtrans fluctuations Ds1 ¼ Ds2 , and hence, Dsi ¼ 0 . Therefore, @si p ;T i¼1;2 c

c

expanding with respect to Dsi , the Gibbs potential can be expressed as DGtrans ¼

  1 X @ 2 Gtrans Dsi Dsj þ . . . : 2 i¼1;2 @si @sj

(1.5)

pc ;T c

Neglecting higher than second-order terms for small Dsi, a nonvanishing DGtrans gives a discontinuity due to the leading term of the second-order in (1.5). On the other hand, if such an inversion does not apply, DGtrans is dominated by the finite first-order derivative terms; hence, the transition is called the first-order, according to the Ehrenfest’s classification. In the binary case, DGtrans is dominated by the second-order terms, and hence, the transition is called the second-order. It is noted from (1.5) that both the correlations Dsi Dsj and the second-order derivatives should not vanish for the leading term to come into effect; nonzero correlations are essential, as substantiated by transition anomalies observed for the specific heat. Such fluctuations Dsi should be quantum mechanical in nature, being responsible for the threshold anomalies, while si behave like a classical vector at and below T c. The transformation below T c is signified by the difference DG1;2 ¼ G2  G1 , which is finite at X 6¼ 0, however zero if X ¼ 0. Accordingly, DG1;2 ¼ ðs2  s1 ÞX:

(1.6)

For X ¼ 0 and X 6¼ 0 , the transformations are the second- and the first-order, respectively. In thermodynamics, both the Gibbs potential and order parameter are extensive variables by definition, as they are proportional to the corresponding volume. Denoting domain volumes by V 1 and V 2, the total volume is V 1 þ V 2 ¼ V, which can be assumed as invariant under a constant p. If so, (1.6) can be determined by either the volume ratio V 1 =V 2 or order-parameter ratio s1 =s2 ; both can be changed by applying X (see Chap. 4).

6

1 Introduction

Fig. 1.1 Phase equilibrium in the p–T diagram. Two curves of Gibbs functions. G1 ðp; T Þ and G2 ðp; T Þ cross at a point Pðpo ; T o Þ, representing thermal equilibrium between phase 1 and phase 2. If the transition is discontinuous at P, such Gibbs functions cannot specify the equilibrium sufficiently at P, requiring another variable such as s.

Fig. 1.2 (a) The secondorder phase transition is characterized by two functions G1and G2 that have a common tangent and different curvatures at P in the p–T diagram. (b) Equilibrium between binary domains specified by s1 and s2 , which are separated phases at temperatures below the critical temperature T c . Transition anomalies near T c are shown schematically by the shaded area.

Figure 1.1 shows a phase diagram, where two Gibbs functions of phases G1 and G2 are schematically plotted in a p–T plane. The crossing point Pðpo ; T o Þ indicates that a transition occurs between G1 and G2 at Pðpo ; T o Þ. The transition is generally first-order, because of discontinuity in the perpendicular direction to p–T plane, although implicit in the two-dimensional p–T diagram; there is a finite change of curvature at P as related to DGtrans. Nevertheless, the discontinuity vanishes, if we ignore fluctuations. Figure 1.2(a) illustrates a continuous binary transition G1 ! G2 at T c , assuming that there are no

1.3 New Concepts

7

fluctuations. In the diagram s–T in Fig. 1.2(b), a continuous transition is sketched by solid lines, which are broadened by fluctuations in the vicinity of T c . Below T c , the transformation s1 $ s2 can take place in the first order, if performed by an external X. It is noted that such a process under a constant p condition is not strictly adiabatic, because Xint is temperature dependent in practical crystals.

1.3

New Concepts

In this book, we discuss thermodynamics of crystalline states under the laws of thermodynamics, for which new concepts, such as renormalized coordinates, solitons, and fields, are added to express collective properties of order variables in periodic structure. These are listed in the following.

1.3.1

Modulated Phases and Renormalized Coordinates

We describe fluctuations in crystals as arising from quantum-mechanical space–time uncertainties for order variables si that represent classical vectors at all sites of a periodic lattice. Their correlated motion described as propagation in the continuum field suited particularly to low excitations of lattice under critical conditions. Associated with sinusoidal lattice displacements ui , the related variables si are signified by their displacements and expressed as a sinusoidal field si ¼ so exp iðk:r i  oti þ ji Þ; accompanying phase uncertainties Dji . Surfaces are significant for thermal properties of a crystal; however, by ignoring them, we can idealize a crystal as infinite periodic structure, where phases of order variables si fluctuate by Dji at a given T. Such sinusoidal fluctuations are detectable if their standing waves are pinned by the modulated lattice, visualizing mesoscopic fluctuations. (See Chap. 8 for the detail.) The order variable sðr; tÞ ¼ sðfÞ is continuous in space–time or phase f ¼ k:r  ot, for which one full angular period 0
f
2p is sufficient instead of a repeated expression. Hence, such a continuous variable sðfÞ can be utilized for expressing thermodynamic functions in a mesoscopic state. Expressed in the form s ¼ so f ðfÞ incorporated with periodic boundary conditions, so is a finite amplitude and f ¼ k:r  ot in the range 0
f
2p . The spatial phase fs ¼ k:r in standing waves visualizes distributed mesoscopic phases in the angular range 0
fs
2p in a whole crystal without referring to lattice sites. Noted that this definition is similar to a renormalization group of order variables in statistical mechanics [3], we shall call fs andðr; tÞ renormalized phase and coordinates, respectively. The Gibbs function in a mesoscopic phase can also be expressed as Gðso ; fs Þ in terms of renormalized variables.

8

1.3.2

1 Introduction

Thermodynamic Changes and Nonlinear Dynamics

Physical properties of matter can change with the responsible dynamical system in equilibrium under isothermal and adiabatic conditions. Small deviations from equilibrium are always restorable in stable crystals, where the dynamical system is dissipative and dispersive under these conditions. Accordingly, the equation of motion during thermodynamic changes can be nonlinear, leading to the modulated lattice structure. While not particularly familiar in traditional physics, we need to deal with the nonlinearity in thermodynamics of crystalline states. Soliton theory [4] in nonlinear physics provides with collective motion in an accurate approximation better than the mean-field approach.

1.3.3

Fields within Thermodynamic Boundaries

For collective motion in crystals, it is convenient to use the field concept [5]. The superconducting transition in metals can be discussed with the field concept on condensation of electrons in the reciprocal lattice space. Characterized by supercurrents and Meissner effects, superconducting phase transitions are more complex than other ordering phenomena, although exhibiting similar specific heat anomalies arising from condensing charge carriers. Fro¨hlich’s field-theoretical model of electron–phonon interaction provides a clear image of condensates to initiate a phase transition, on which the theory of Bardeen, Cooper, and Schrieffer can define order variables. The last three chapters in this book are for superconducting transitions in metals, which can be discussed by analogy of structural changes. Phase changes in crystals are generally associated with adiabatic potentials, where the nonlinearity arises from interactions with the lattice. In this book, dynamical variables in crystals are discussed within thermal boundaries at constant T and p, which can be described as isothermal and adiabatic processes in crystals. The former remains the same as in traditional theories, whereas the latter process cannot be separated from the former, showing thermal relaxation to stabilize structure.

1.3.4

Adiabatic Potentials

Introduced early by Boltzmann and Ehrenfest, the concept of adiabatic invariance should be extended explicitly to crystalline states to deal with long-range correlations, which can be discussed by means of the internal adiabatic potential in the lattice. Recognized in practical applications of thermodynamics, it was however not seriously dealt with in most available references. We therefore emphasized its significance for driving condensates for structural order.

Exercise 1

1.4

9

Sampling Modulated States

For critical fluctuations, the timescale t is a significant for sampling experiments of the spatial distribution of s. For that purpose, the timescale to of sampling should be shorter than the timescale t of fluctuations. A Gibbs potential, if sampled at a given p–T condition, should be expressed by a space–time average over to , namely,

Gp;T

1 ¼ to

ðto 0

1 dt V

!

ð

gðr; tÞd r ; 3

(1.7a)

V

where gðr; tÞ is the density of Gibbs potential in the sampling volume V. Writing the Rto time average as t1o gðr; tÞdt ¼ hgðr; tÞit , (1.7a) can be expressed by 0

Gp;T ¼

1 V

ð hgðr; tÞit d3 r:

(1.7b)

V

The average density hgðr; tÞit over to can be sampled from a specimen of volume V. It is noted that such a sampling results in a useful information only if t>to . Sampled mesoscopic hsðr; tÞit represents sðfs Þ , where 0
fs
2p and the distributed sðfs Þ is explicit, provided that t>to .

Exercise 1 1. What is the mean-field average? If a physical event at each lattice point is random and independent, the average should always be zero, should it not? There must be some kind of correlations among them in order to have a nonzero average. Discuss this issue. 2. Review Ehrenfest’s classification of phase transitions from a standard textbook on thermodynamics. In his definition of second-order transitions, vanishing firstorder derivatives and nonzero second-order derivatives of the Gibbs function are necessary. On the other hand, as we mentioned in Sect. 1.2, nonzero correlations Dsi Dsj among internal variables are required, in addition to Ehrenfest’s definition. Is this a conflict? This problem should be discussed to clear this issue before proceeding to the following chapters. 3. In traditional thermodynamics, the Gibbs potential was defined as a function of macroscopic variables. However, macroscopic heat and order parameter are associated with internal variables in a crystal, which are essentially related to thermal and adiabatic averages of fluctuating microscopic variables, respectively. In this context, the Gibbs function defined in this chapter is associated with all sites that may not be identical. Discuss this issue for thermodynamics of heterogeneous matters.

10

1 Introduction

References 1. J.G. Kirkwood, I. Oppenheim, Chemical Thermodynamics (McGraw-Hill, New York, 1961) 2. M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford, 1968) 3. D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, New York, 1987) 4. G.L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980) 5. H. Haken, Quantenfeldtheorie des Festko¨rpers (B. G. Teubner, Stuttgart, 1973)

Chapter 2

Phonons

Disregarding point symmetry, we can simplify the crystal structure by the space group [1, 2], representing the thermodynamic state in equilibrium with the surroundings at given values of p and T. In this approach, the restoring forces secure stability of the lattice, where the masses at lattice points are in harmonic motion. In this case, we realize that their directional correlations in the lattice are ignored so that a possible disarrangement in the lattice can cause structural instability. In this chapter, we discuss a harmonic lattice to deal with basic excitations in equilibrium structure. Lattice vibrations in periodic structure are in propagation, specified by frequencies and wavevectors in virtually continuous spectra. Quantum mechanically, on the other hand, the corresponding phonons signify the dynamical state in crystals. In strained crystals, as modulated by correlated constituents, lowfrequency excitations dominate over the distorted structure, which is however thermally unstable as discussed in this chapter.

2.1

Normal Modes in a Simple Crystal

A crystal of chemically identical constituent ions has a rigid periodic structure in equilibrium with the surroundings, which is characterized by translational symmetry. Referring to symmetry axes, physical properties can be attributed to the translational invariance, in consequence of energy and momentum conservations among constituents. Constituents are assumed to be bound together by restoring forces in the lattice structure. Considering a cubic lattice of N3 identical mass particles in a cubic crystal in sufficiently large size, we can solve the classical equation of motion with nearestneighbor interactions. Although such a problem should be solved quantum mechanically, classical solutions provide also a useful approximation. It is noted that the lattice symmetry is unchanged with the nearest-neighbor interactions, assuring

M. Fujimoto, Thermodynamics of Crystalline States, DOI 10.1007/978-1-4614-5085-6_2, # Springer Science+Business Media New York 2013

11

12

2 Phonons

Fig. 2.1 (a) Onedimensional monatomic chain of the lattice constant a. (b) A dispersion curve o vs: k of the chain lattice.

structural stability in this approach. In the harmonic approximation, we have linear differential equations, which can be separated into 3N independent equations; this one-dimensional equation describes normal modes of N constituents in collective motion along the symmetry axis x, y, or z [3]. Denoting the displacement by a vector qn from a site n, we write equations of motion for the components qx;n ; qy;n and qz;n independently, that is,
 q€x;n ¼ o2 qx;nþ1 þ qx;n1  2qx;n ;
 q€z;n ¼ o2 qz;nþ1 þ qz;n1  2qz;n ;


 q€y;n ¼ o2 qy;nþ1 þ qy;n1  2qy;n

and

where o2 ¼ k=m and k and m are the mass of a constituent particle and the force constant, respectively. As these equations are identical, we write the following equation for a representative component to name qn for brevity:
 q€n ¼ o2 qnþ1 þ qn1  2qn ;

(2.1)

which assures lattice stability along any symmetry direction. Defining the conjugate momentum by pn ¼ mq_ n, the Hamiltonian of a harmonic lattice can be expressed as H¼

N  2 X p

 2 mo2 mo2
2 þ qnþ1  qn þ ðqn  qn1 Þ : 2m 2 2 n

n¼0

(2.2)

Each term in the summation represents one-dimensional infinite chain of identical masses m, as illustrated in Fig. 2.1a. Normal coordinates and conjugate momenta, Qk and Pk , are defined with the Fourier expansions kN 1 X Qk expðiknaÞ and qn ¼ pffiffiffiffi N k¼0

pk ¼

kN X k¼0

Pk expðiknaÞ;

(2.3)

2.1 Normal Modes in a Simple Crystal

13

where a is the lattice constant. For each mode of qn and pn, the amplitudes Qk and Pk are related as Qk ¼ Qk  ; Pk ¼ Pk 

and

N X

exp iðk  k0 Þna ¼ Ndkk0 ;

(2.4)

n¼0

where dkk0 is Kronecker’s delta, that is, dkk0 ¼ 1 for k ¼ k0, otherwise zero for k ¼ k0. Using normal coordinates Qk and Pk , the Hamiltonian can be expressed as   2p=a  1 X  2 2 2 ka  H¼ Pk Pk þ Qk Qk m o sin ; 2m k¼0 2

(2.5)

from which the equation of motion for Qk is written as Q€k ¼ m2 o2 Qk ;

(2.6)

where ok ¼ 2o sin

ka ¼2 2

rffiffiffiffi k ka sin : m 2

(2.7)

As indicated by (2.7), the k-mode of coupled oscillators is dispersive, which are linearly independent from the other modes of k0 6¼ k . H is composed of N independent harmonic oscillators, each of which is determined by the normal coordinates Qk and conjugate momenta Pk. Applying Born–von Ka´rman’s boundary conditions to the periodic structure, k can take discrete values as given by k ¼ 2pn Na and n ¼ 0; 1; 2; . . . ; N. Figure 2.1b shows the dispersion relation (2.7) determined by the characteristic frequency ok : With initial values of Qk ð0Þ and Q_ k ð0Þ specified at t ¼ 0, the solution of (2.7) can be given by Qk ðtÞ ¼ Qk ð0Þ cos ok t þ

Q_ k ð0Þ sin ok t: ok

Accordingly,

kN X 1 X q_ n0 ð0Þ 0 0 p ffiffiffi ffi qn ðtÞ ¼ qn0 ð0Þ cosfkaðn  n Þ  ok tg þ sinfkaðn  n Þ  ok tg ; ok N k¼0 n0 ¼n;n1

(2.8) where aðn  n0 Þ represents distances between sites n and n0 so that we write it as x ¼ aðn  n0 Þ in the following. The crystal is assumed as consisting of a large number of the cubic volume L3 where L ¼ Na , if disregarding surfaces.

14

2 Phonons

The periodic boundary conditions can then be set as qn¼0 ðtÞ ¼ qn¼N ðtÞ at an arbitrary time t. At a lattice point x ¼ na between n ¼ 0 and N , (2.8) can be expressed as qðx; tÞ ¼

X

½Ak cosðkx  ok tÞ þ Bk sinðkx  ok tÞ;

k ð0Þ ffiffiffi and Bk ¼ q_ kp ffiffiffi , and x is virtually continuous in the range 0  x where Ak ¼ qpk ð0Þ N ok N  L, if L is taken as sufficiently long. Consisting of waves propagating in  x directions, we can write qðx; tÞ conveniently in complex exponential form, that is,

qðx; tÞ ¼

X

Ck exp iðkx  ok t þ jk Þ;

(2.9)

k

where C2k ¼ A2k þ B2k and tan jk ¼ BAkk . For a three-dimensional crystal, these onedimensional k-modes along the x-axis can be copied to other symmetry axes y and z; accordingly, there are 3N normal modes in total in a cubic crystal.

2.2

Quantized Normal Modes

The classical equation of motion of a harmonic crystal is separable to 3N independent normal propagation modes specified by kn ¼ 2pn aN along the symmetry axes. In @ quantum theory, the normal coordinate Qk and conjugate momentum Pk ¼ ih @Q k h are operators, where  h ¼ 2p and h is the Planck constant. For these normal and conjugate variables, there are commutation relations: ½Qk ; Qk0  ¼ 0; ½Pk ; Pk0  ¼ 0

and

½Pk ; Qk0  ¼ ihdkk0 ;

(2.10)

and the Hamiltonian operator is Hk ¼

1  y y Pk Pk þ m2 o2k Qk Qk : 2m

(2.11a)

y y Here, Pk and Qk express transposed matrix operators of the complex conjugates Pk   and Qk , respectively. Denoting the eigenvalues of Hk by ek , we have the equation Hk C k ¼ e k C k :

(2.11b)

For real eigenvalues ek , Pk and Qk should be Hermitian operators, which are y y characterized by the relations Pk ¼ Pk and Qk ¼ Qk , respectively. Defining operators

2.2 Quantized Normal Modes

bk ¼

15

y mok Qk þ iPk pffiffiffiffiffiffiffiffiffiffi 2mek

and

y y mok Qk  iPk pffiffiffiffiffiffiffiffiffiffi ; bk ¼ 2mek

(2.12)

we can write the relation io  1  2 2 y y y y k m ok Qk Qk þ Pk Pk þ Qk Pk  Pk Qk 2mek 2ek Hk iok ¼ þ ðQ Pk  Pk Qk Þ: ek 2ek k

y bk b k ¼

From this relation, we can be derive   1 y Hk ¼  hok bk bk þ ; 2

if

1 ek ¼ hok : 2

(2.13)

y Therefore, Hk are commutable with the operator bk bk , that is, h i y Hk ; bk bk ¼ 0; and from (2.12) h i y bk 0 ; bk ¼ dk 0 k ; ½ bk 0 ; bk  ¼ 0

and

h i y y bk0 ; bk ¼ 0:

Accordingly, we obtain h i y y Hk ; bk ¼ hok bk

and

½bk ; Hk  ¼ hok bk :

Combining with (2.11b), we can derive the relations   y y ho k Þ bk C k Hk bk C k ¼ ð e k þ 

and Hk ðbk Ck Þ ¼ ðek  hok Þðbk Ck Þ;

y indicating that bk Ck and bk Ck are eigenfunctions for the energies ek þ hok and y hok , respectively. In this context, bk and bk are referred to as creation and ek   annihilation operators for the energy quantum  hok to add and subtract in the energy ek ; hence, we can write y bk bk ¼ 1:

(2.14)

y Applying the creation operator bk on the ground state function Ck nk -times, the  nk
 y eigenvalue of the wavefunction bk Ck can be given by nk þ 12 hok, generating a

16

2 Phonons

state of nk quanta plus 12  hok . Considering a quantum hok like a particle, called a phonon, such an exited state with nk identical phonons is multiply degenerate by permutation nk ! Hence, the normalized wavefunction of nk phonons can be  nk y expressed by p1ffiffiffiffi bk Ck . The total lattice energy in an excited state of n1 ; n2 ; n ! k

::::: phonons in the normal modes 1, 2,. . ... can be expressed by X nk hok ; Uðn1 ; n2 ; :::::Þ ¼ U o þ

(2.15a)

k

where U o ¼

P hok k

2

is the total zero-point energy. The corresponding wavefunction

can be written as  n1  n2 y y b1 b2 ::::: Cðn1 ; n2 ; :::::Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC1 C2 :::::Þ; n1 ! n2 ! :::::

(2.15b)

which describes a state of n1 ; n2 ; :::: phonons of energies n1 hok1 ; n2 hok2 ; :::::. The total number N ¼n1 þ n2 þ ::::: cannot be evaluated by the dynamical theory; however, we can determine the value in thermodynamics, as related to the level of thermal excitation at a given temperature.

2.3

Phonon Field and Momentum

In a one-dimensional chain of identical mass particles, the displacement mode qk is independent from each other’s modes, and hence representing normal modes in a three-dimensional crystal. However, this model is only approximate, in that these normal modes arise from the one-dimension harmonic chain model, where mutual interactions between different normal modes are prohibited. For propagation in arbitrary direction, the vibrating field offers more appropriate approach than the normal modes, where quantized phonons move in any direction like free particles in the field space. Setting rectangular coordinates x; y; z along the symmetry axes of an orthorhombic crystal in classical theory, the lattice vibrations are described by a set of equations 2
2 o px;n1 2 k n
qx;n1  qx;n1 þ1 þ qx;n1  qx;n1 1 þ ¼ ex;n1 ; 2 2m 2
2 o py;n2 2 k n
qy;n2  qy;n2 þ1 þ qy;n2  qy;n2 1 þ ¼ ey;n2 ; 2 2m and 2
2 o pz;n3 2 k n
qz;n3  qz;n3 þ1 þ qz;n3  qz;n3 1 þ ¼ ez;n3 ; 2 2m

(2.16)

2.3 Phonon Field and Momentum

17

where ex;n1 þ ey;n2 þ ez;n3 ¼ en1 n2 n3 is the total propagation energy along the direction specified by the vector qðn1 ; n2 ; n3 Þ and k is the force constant. The variables qx;n1 ; qy;n2 ; qz;n3 in (2.16) are components of a classical vector qðn1 ; n2 ; n3 Þ , which can be interpreted quantum theoretically as probability amplitudes of components of the vector q in the vibration field. We can therefore write the wavefunction of the displacement field as Cðn1 ; n2 ; n3 Þ ¼ qx;n1 qy;n2 qz;n3 , for which these classical components are written as qðx; tÞ ¼

X


 Ck;x exp i kx x  okx t þ jkx ;

kx

qðy; tÞ ¼

X

 Cky exp i ky y  oky t þ jky ;

ky

qðz; tÞ ¼

X


 Cky exp i kz z  okz t þ jkz ;

kz

and hence, we have C ð n1 ; n2 ; n3 Þ ¼

X k

  n1 ex;n1 þ n2 ey;n2 þ n3 ez;n3 t þ jk : A k exp i k:r  h


 Here, A k ¼ Ckx Cky Ckz ; jk ¼ jkx þ jky þ jkz , and k ¼ kx ; ky ; kz are the amplitude, phase constant, and wavevector of Cðn1 ; n2 ; n3 Þ , respectively. Further writing n1 ex;n1 þ n2 ey;n2 þ n3 ez;n3 ¼ ok ðn1 ; n2 ; n3 Þ ¼ ok ; h

(2.17a)

the field propagating along the direction of a vector k can be expressed as Cðk; ok Þ ¼ A k exp iðk:r  ok t þ jk Þ;

(2.17b)

representing a phonon of energy hok and momentum  hk. For a small jkj, the propagation in a cubic lattice can be characterized by a constant speed v of propagation determined by ok ¼ vjkj, indicating no dispersion in this approximation. The phonon propagation can be described by the vector k, composing a reciprocal lattice space, as illustrated in two dimensions in Fig. 2.2 by kx ¼

2pn1 2pn2 ; ky ¼ L L

and

kz ¼

2pn3 ; L

18

2 Phonons

Fig. 2.2 Two-dimensional reciprocal lattice. A lattice
 point is indicated by kx ; ky . Two quarter-circles of radii k and k + dk show surfaces of constant ek and ekþdk for small jkj in kx ky plane.

ky =

2πny a

dk

4 3 2

k

1 0

1

2

3

4 kx =

2πnx a

in a cubic crystal, where aN ¼ L . A set of integers ðn1 ; n2 ; n3 Þ determines an energy and momentum of a phonon propagating in the direction of k, where ok qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2x þ n2y þ n2z . In the reciprocal space, all points on a ¼ ok ðn1 ; n2 ; n3 Þ and jkj ¼ 2p L spherical surface of radius jkjcorrespond to the same energy hok , representing a sphere of a constant radius jkj and energy ek. Quantum mechanically, we can write the phonon momentum as p ¼  hk to supplement the energy hok , characterizing a phonon particle.

2.4

Thermal Equilibrium

In thermodynamics, a crystal must always be in thermal contact with the surroundings. At constant external pressure p, the quantized vibration field can be in equilibrium with the surroundings at a given temperature T. A large number of phonons are in collision-free motion, traveling in all directions through the lattice, colliding with surfaces to exchange their energy and momentum with the surroundings. Assuming the crystal volume as unchanged, the average of phonon energies can be calculated with the Boltzmann probability at T. In equilibrium, the total energy of a crystal can be expressed as U þ U s, where U s is the contribution from the heat reservoir and Urepresents the energy of a stable crystal. In this case, the total energy U þ U s should be stationary with any thermodynamic variation around equilibrium. Using probabilities w and ws for keeping the crystal in equilibrium with the surroundings, the product wws should be calculated as maximizing U þ U s to determine the most probable value. Setting this variation problem as

2.4 Thermal Equilibrium

19

dðwws Þ ¼ 0

and

dð U þ U s Þ ¼ 0

for arbitrary variations dw and dws , these variations can be calculated as ws dw þ wdws ¼ 0

and

dU þ dUs ¼ 0;

respectively. We can therefore write dðln wÞ dðln ws Þ ¼ ; dU dU s which is a common quantity between U and Us . Writing it as equal to b ¼ kB1T , we can relate b to the conventional absolute temperature T. Therefore, dðln wÞ 1 ¼b¼ dU kB T

  U or w ¼ wo exp  : kB T

(2.18)

Here, w is called the Boltzmann probability; wo is the integration constant that can be determined by assuming U ¼ 0 at T ¼ 0 K, where kB is the Boltzmann constant. Quantum theoretically, however, T ¼ 0 is fundamentally unreachable, as stated in the third law of thermodynamics. Accordingly, we write U ¼ 0 þ Uo at 0 K, where hoo is the zero-point energy. U o ¼ 12 N Although we considered only vibrations so far, physical properties of a crystal are also contributed by other variables located at lattice points or at interstitial sites. Though primarily independent of lattice vibrations, these variables can interact with the lattice via phonon scatterings. If accessed by random collisions of phonons, energies ei of these variables are statistically available with Boltzmann’s probabilities wi , so we can write equations U¼

X

ei

and w ¼ Pi wi ;

i

where   ei wi ¼ wo exp  kB T

and

X

wi ¼ 1;

(2.19)

i

as these wi are for exclusive events. In this case, the function Z ¼

P i

 exp  keBiT ,

called the partition function, is useful for statistical calculation. In such a system as called microcanonical ensemble, thermal properties can be calculated directly with Z. The Boltzmann statistics is a valid assumption for a dynamical system under the ergodic hypothesis. Despite of the absence of rigorous proof, the Boltzmann statistics can usually be applied to phonon gas in a fixed volume. Thermodynamically,

20

2 Phonons

however, it is only valid for isothermal processes, because the volume is not always constant in adiabatic processes of internal origin. The anharmonic lattice cannot be ergodic in strict sense, whereas the harmonicity is essentially required for stable crystals at constant volume and pressure.

2.5

Specific Heat of a Monatomic Crystal


 The specific heat at a constant volume CV ¼ @U @T V is a quantity measurable with varying temperature under a constant external pressure p. The phonon theory is adequate for simple monatomic crystals, if characterized with no structural changes. For such a crystal, the
specific  heat and internal energy are given by quantized phonon energies ek ¼ nk þ 12  hok , for which the wavevector k is distributed virtually in all directions in the reciprocal space. Assuming 3N phonons in total, the energies ek are degenerate with the density of k-states written as gðkÞ, which is a large number as estimated from a spherical volume of radius jkj in the reciprocal space. In this case, the partition function can be expressed as     1   ek hok X  nk hok Zk ¼ gðkÞ exp  exp  ¼ gðkÞ exp  ; kB T 2kB T n ¼0 kB T k

where the infinite series on the right converges, if khBoTk 0. We define the variable V by writing V  V 3 ¼ g for the above algebraic equation, which can then be expressed as g2x ¼ 2gðV 3  V 1  gÞðV 3  V 2  gÞ: Further, introducing another variable x by g ¼ ðV 3  V 2 Þx2, this equation can be written for x as    1 x2x ¼ ðV 3  V 1 Þ 1  x2 1  k2 x2 ; 2 Expressing the phase by f ¼

where

k2 ¼

V3  V2 : V3  V1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 3  V 1 x, the equation can be further modified as

   2x2f ¼ 1  x2 1  k2 x2 :

8.3 Solutions of the Korteweg–deVries Equation

119

Integrating this expression, we obtain f p1ffiffiffi ¼ 2

xð1

0

dx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi; 1  x2 1  k 2 x2

(8.8a)

where the phase f1 is determined by the upper limit x1 of this integral. The reverse function of f1 , known as Jacobi’s sn-function, can be written as   f1 x1 ¼ sn pffiffiffi ; k : 2

(8.8b)

The potential V ðfÞ can then be expressed by V ðfÞ ¼ V 3  ðV 3  V 2 Þsn2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V 3  V 1 f; k

for

0 < k < 1;

(8.9a)

where the sn2 -function is periodic with the period ð1

dx 2K ðkÞ ¼ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi: 2 2 x2 1  x 1  k 0 ð kÞ ffiffiffiffiffiffiffiffiffiffi , On the other hand, the potential V ðfÞ in (8.9a) has a periodic interval of p2K V 3 V 2 corresponding a finite amplitude V 3  V 2 . If V 2 ! V 1 , we have the curve B in Fig. 8.1, for which k ! 1 and Kð1Þ ¼ 1. In this case the potential is

V ðfÞ ¼ V 3 þ ðV 3  V 2 Þsech2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V3  V1f :

(8.9b)

This shows p a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pulse-shaped potential with height V 3  V 2 , propagating with ffi effective phase V 3  V 1 f. Specifically if V 3  V 2 is infinitesimal, the modulus k is almost zero; (8.9a) exhibits a sinusoidal variation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V ðfÞ ¼ V 3 þ ðV 3  V 2 Þsin2 V3  V1f : It is noted that the coefficients in (8.9a) and (8.9b) are related with the modulus k. By definition, we have V 3  V 2 ¼ k2 ðV 3  V 1 Þ, and hence the amplitude V 3  V 2 is proportional to k2 . Adjusting values of V 1 ; V 2 ; and V3 , the potential V ðfÞ can be made as consistent with adiabatic potentials derived in Chap. 7. Thus, solutions of the Korteweg–deVries equation are expressed by elliptic functions of the propagating phase, which is either oscillatory or in a pulse shape, depending on the modulus k; a solitary potential of (8.9b) is known as the Eckart potential. The latter potential behaves like an independent particle, which can be proven rigorously, and hence called a soliton, but we shall not go into the detail.

120

8 Soliton Theory of Long-Range Order

Physically, the transverse density rz can be responsible for dispersion of the longitudinal s1 mode; mathematically the relation between rz ðfÞ and the soliton potential V(f) is a direct solution of the Korteweg–deVries equation. In a crystal, the adiabatic potential DU should reflect symmetry of the three-dimensional lattice. In addition to V ðfÞ of (8.9a) that corresponds to the longitudinal density rx ðfÞ, we should have a transverse potential: V ? ðf0 Þ ¼ V 2 þ ðV 3  V 2 Þcn2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

V 3  V 1 f0 ; k0

(8.10)

can be associated with the transverse density rðf0 Þ, where f0 ¼ f  K and k0 ¼ k. Judging from what we discussed in Sect. 8.1, this is an acceptable postulate. The total adiabatic potential for the density so  so should therefore be given by DU ¼ V ðfÞ þ V ? ðf  K Þ;

(8.11)

which acts on the total density s2x þ s2z . Jacobi’s elliptic functions can be expanded into power series as in Landau’s expansion, which can be truncated at a quartic term s4 (see Appendix). It was therefore logical to consider such expansions in the Landau theory. The corresponding potential can be expressed as DU ¼ a2 s2 þ b4 s4 in the adiabatic approximation. The above interpretation of Landau’s expansion owes the developing operator B3 that signifies the dispersive nature of the developing (8.4), as supplemented by Born–Huang’s principle in thermodynamic environment. Consequently, elliptic and quartic potentials are both useful approximations, for practical analysis of mesoscopic variables. It is notable that the soliton potential (8.9b) can describe a domain wall, in which the reversing energy is proportional to s2? that shows singularity at f ¼ 0, after a thermal process for k ! 1. The adiabatic potential DU ðs? Þ in (7.9b) represents therefore the domain-wall energy in the limit of k ! 1.

8.4

Cnoidal Theorem and the Eckart Potential

It is significant that adiabatic potentials are conformable with the order variables density in the Korteweg–deVries equation; both are signified by the phase f ¼ x vt, as they are in thermal equilibrium. Furthermore, by the coordinate transformation x  vt ! x0 , the potential V ðx0 Þ becomes always in phase with the order density sðx0 Þ2 in the system of x0 . Hence disregarding t, these V ðx; kÞ and sðx; kÞ2 can be expressed in the limit k ! 1 as proportional to sech2 x. The Eckart potential (8.9b) characterized by a solitary peak can be written in such a form as

8.4 Cnoidal Theorem and the Eckart Potential

121

Fig. 8.2 (a) Eckart’s potential of the half-width 2d. (b) A sketch of apcnoidal ffiffiffi  potential cn2 f 2k with periodic peaks.

VðxÞ ¼ V o sech2

x

d

for

k ¼ 1;

(8.12)

excluding the constant term. Here, d is introduced to indicate the width 2d of the symmetric peak around x ¼ 0, as shown in Fig. 8.2a. It is noted that  (8.12) is related to s1 in (7.8), whose transverse component can be written as s? dx that corresponds to VðxÞ. In contrast, the potential (8.9a) is oscillatory and periodic in space. Although expressed as sn2 for 0 < k < 1, such a potential is practically the same as cn2 function because of the relation sn2 þ cn2 ¼ 1. In Chap. 7, we have shown that the transverse component s? ðx; kÞ is in motion with an adiabatic cnoidal potential V ðx; kÞ ¼ V o dn2 uðx; kÞ;

(8.13a)

where V o is a proportionality factor. Because of the relation dn2 u ¼ 1  k2 sn2 u, (8.13a) gives an identical potential V ðx; kÞ ¼ V o k2 sn2 uðx; kÞ

(8.13b)

ignoring the additive constant. In any case, these squared elliptic functions represent a periodic potential, called a cnoidal curve, as sketched in Fig. 8.2b. For a physical application, omitting a trivial additive constant, the adiabatic potential DUðsÞ is expressed by (8.13a) or (8.13b).

122

8 Soliton Theory of Long-Range Order

At this point, we pay attention to a mathematical theorem for such a sn2 function, stating that peaks in the cnoidal expansion can be replaced by sech2 peaks. The rigorous proof is available in the advanced theory of elliptic functions, which is however too tedious to follow at the present level of discussion. Quoting the resulting formula, here we use the theorem 2k2 sn2 x ¼ 2a

m¼þ1 X

sech2

pffiffiffi  ax  cm þ const.;

(8.14)

m¼1

where a¼

p2 4K 0 ðkÞ2

c¼

pK ðkÞ and K 0 ðkÞ

K 0 ðkÞ ¼ K

pffiffiffiffiffiffiffiffiffiffiffiffiffi

1  k2 :

Here, the index m specifies the peak positions at xm for  1 < m < þ 1 along the x-axis; K ðkÞ is the complete elliptic integral (7.7) and K 0 ðkÞ is defined in the above. Nevertheless, those readers who are interested in the derivation of (8.14) can consult a standard reference book on elliptic functions. Owing to (8.14), the cnoidal potential can be replaced by a periodic array of Eckart potentials located at x ¼ pcffiffia m, where m are  integers. By virtue of this theorem, problems for the oscillatory cnoidal potential can be replaced by series of Eckart’s sech2 -peaks. On the other hand, physically such a transition to sech2 potentials should be interpreted by Born–Huang principle for sðx; tÞ to be placed in thermodynamic environment; taking k ! 1 , we use Kð1Þand K 0 ð1Þ ¼ Kð0Þ for (8.14). In this interpretation, pseudospins are ordered in small oppositely polarized domains of 4Kð1Þ in succession in the mesoscopic state.

8.5

Condensate Pinning by the Eckart Potentials

In this section we discuss the eigenvalue problem of sðx; kÞ in the Eckerd potential field V ðx; kÞ. Denoting the eigenvalue by e, the perturbed wave equation is written as

d2 sðxÞ 2x þ e þ V sech sðxÞ ¼ 0: o dx2 d For stable pinning, the eigenvalue should be negative, so we write e ¼ m2. Replacing dx by x, and setting V o d2 ¼ vo and m2 d 2 ¼ b2 , this equation can be expressed as  d2 sðxÞ  2 þ b þ vo sech2 x sðxÞ ¼ 0: 2 dx

(8.15)

8.5 Condensate Pinning by the Eckart Potentials

123

This is a differential equation familiar in Mathematical Physics, whose solution is expressed by hypergeometric series. Following Morse and Feshbach [2], we transform (8.15) to the hypergeometric equation in standard form. By defining the relation sðxÞ ¼ Af ðxÞ sechb x, the differential equation for the function f ðxÞ can be obtained as   d2 f df   2bðtanh xÞ þ vo  b2  b sech2 x f ðxÞ ¼ 0: 2 dx dx Further, this equation can be modified by another variable z ¼ 12 ð1  tanh xÞ as zð 1  zÞ

 d2 f df  þ ð1 þ bÞð1  2zÞ þ vo  b2  b f ðxÞ ¼ 0: 2 dz dz

We define such parameters a, b, and c that satisfy the relations a þ b ¼ 2c  1; where c¼1þb

and

ab ¼ vo þ b2  b;

that is, 1 a; b ¼ þ b  2

rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 vo þ : 4

Using a, b, and c, we obtain the hypergeometric equation expressed in standard form, i.e., zð 1  zÞ

d2 f df þ fc  ða þ b þ 1Þzg  abf ¼ 0: dz dz2

(8.16)

Expressing as f ðzÞ ¼ Fða; b; c; zÞ , the solution of (8.16) is called a hypergeometric function. The mesoscopic pseudospin variable sðx; kÞ can then be expressed as sðxÞ ¼ A sechb xFða; b; c; zÞ:

(8.17)

However, we are only interested in extreme cases that can be specified by z ! 0 and z ! 1, in order to use it physically at distant points x ! 1, for which we expand Fða; b; c; zÞ as Fða; b; c; zÞ ¼ 1 þ

ab aða þ 1Þbðb þ 1Þ 2 zþ z þ : 1!c 2!cðc þ 1Þ

(8.18)

124

8 Soliton Theory of Long-Range Order

Fig. 8.3 Gamma function G (z).

Corresponding to z ! 0, for x ! þ1 we have sðxÞx!1 ! A2b expðbxÞ; whereas for z ! 1, we consider x ! 1. For calculatingsðxÞ in these cases, it is convenient to use the following formula [3]: Fða; b; c; zÞ ¼

GðcÞGðc  a  bÞ Fða; b; a þ b  c þ 1; zÞ Gðc  aÞGðc  bÞ

þ ð1  zÞcab

GðcÞGða þ b  cÞ Fðc  a; c  b; c  a  b þ 1; 1  zÞ; GðaÞGðbÞ

where Gð. . .Þ are gamma functions. It is noted from (8.15) that the first term dominates Fða; b; c; zÞ , if z ! 0 . On the other hand, if z ! 1 we have ð1  zÞcab  2b expðbxÞ and Fð. . . ; 1  zÞ ! 1 in the second term, dominating Fða; b; c; zÞ. Considering a small b, sechb x  2b expðþbxÞ and Fða; b; 1 þ b; zÞ ! 1 in the first term. In this case, writing b ¼ ik to express propagating waves, we can write that sðxÞx!1 /

GðcÞGða þ b  cÞ GðcÞGðc  a  bÞ expðþikxÞ þ expðikxÞ: (8.19) GðaÞGðbÞ Gðc  aÞGðc  bÞ

Using definitions of a, b, and c, we notice that the numerators of these coefficients are Gð1 þ bÞGðbÞ and Gð1 þ bÞGðbÞ , respectively, whereas the denominators have singularities, depending on the value of vo. In Fig. 8.3 the curve

8.5 Condensate Pinning by the Eckart Potentials

125

of a gamma function GðzÞ plotted against the variable z is shown, where there are a number of poles at z ¼ 0; 1; 2; . . .. Hence, the first coefficient has singularities at rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 a; b ¼ þ b  vo þ ¼ m, m ¼ 0; 1; 2; . . . (i) 2 4 whereas the second coefficient can be specified by singularities for the denominator to become infinity, i.e., rffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 1 1 1 p Gðc  aÞGðc  bÞ ¼ G þ vo þ G  vo þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1; 2 4 2 4 cos p v þ 1 o

4

hence rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 vo þ ¼ n þ 4 2

or

vo ¼ nðn þ 1Þ;

where n can be any integer. Combining (i) and (ii), we obtain   1 1 b¼ nþ  m  ¼ n  m, 2 2

(ii)

(iii)

where m ¼ 0; 1; 2; . . . ; n  1 . Therefore the eigenvalues for b > 0 can be represented as bp ¼ n  m ¼ n; n  1; n  2; . . . ; 1 . Imposing conditions for no transmission and reflection on (8.19), the pseudospin wave sðxÞ can be in phase with the soliton potential, if rðxÞ and VðxÞ are conformal with integers specified by (i) and (ii). Therefore, by expressing the potential by eigenvalues p, the steady-state equation (8.15) for longitudinal wave functions sp ðxÞ can be written as  d2 sp ðxÞ  2 þ b þ pðp þ 1Þ sech2 x sp ðxÞ ¼ 0; 2 dx

(iv)

where discrete eigenvalues ep ¼ b2p can be determined by b2p ¼ p2 that are indexed by p ¼ n, n  1; n  2; . . . 2; 1. Writing the (iv) for two adjacent levels at n and n  1, we have  d2 sn  2 ¼ n þ nðn þ 1Þ sech2 x sn ¼ 0 2 dx and o d2 sn1 n 2 2 þ  ð n  1 Þ þ ð n  1 Þn sech x sn1 ¼ 0: dx2 Hence, for a change sn1 ! sn the potential value decreases by DV n1;n ¼ 2nsech2 x. We consider that such transitions can take place as a

126

8 Soliton Theory of Long-Range Order

Fig. 8.4 Soliton levels in cnoidal potentials.

thermal relaxation process with varying temperature. Dynamically inaccessible, but such transitions should be observed thermally as a consequence of phonon scatterings. Corresponding lattice displacements un should occur in a relaxation process between strain energies related to u2n1 ! u2n , corresponding to T n1 ! T n in virtue of the equipartition theorem. Figure 8.4 shows a change of the periodic potential (8.13b) with lowering temperature, where eigenvalues of the pseudospin density are indicated in each of periodic Eckart’s potentials. Not shown in the figure, but the density levels are practically in band structure whose widths are significantly broader at smaller n. Therefore, it is reasonable that stepwise thermal relaxation processes are responsible for these transitions toward the lowest level determined by n2 in each potential well.

8.6

Born–Huang’s Transitions

For two transverse components s1 ðxÞ and s2 ðxÞ for the same eigenvalue  b2, there is generally a finite potential difference between V 1 ðxÞ and V 2 ðxÞ, which can be shown in the following.

8.6 Born–Huang’s Transitions

127

Writing the wave equations as   D2 s1 ðxÞ ¼ b2 þ V 1 ðxÞ s1 ðxÞ

and

  D2 s2 ðxÞ ¼ b2 þ V 2 ðxÞ s2 ðxÞ

(i)

we consider that these components are related by s2 ðxÞ ¼ Aðx; bÞs1 ðxÞ þ B Ds1 ðxÞ

(ii)

which is adequate for the present analysis, although B is assumed as constant. Substituting (ii) into (i), and setting the front factors of s1 and Ds1 equal to zero, we obtain the relations Axx þ V 1x þ AðV 1  V 2 Þ ¼ 0 and 2Ax þ ðV 1  V 2 Þ ¼ 0;

(iii)

respectively. Eliminating V 1  V 2 in (iii), we derive the relation Ax Axx  Ax A  V 1x Ax ¼ 0, which is integrated as ~2 ; A2  Ax  V 1 ¼ b ~2 is a constant of integration. This equation, known as Riccati’s equation, where  b ~ ðxÞ, i.e., can be linearized by transforming A to s A¼

D~ s ~Þ; ¼ Dðln s ~ s

(iv)

resulting in   ~ðxÞ ¼ b2 þ V 1 ðxÞ s ~1 ðxÞ: D2 s ~ðxÞ satisfies (i), if This is identical to equations (i), indicating that the function s ~ b ¼ b. In this case, using (iv) in the second equation of (iii), we can write   ~Þ s2 ðxÞ D2 s2 ðxÞ ¼ b2 þ V 1 ðxÞ  2ðln s

(v)

~Þ; V 2 ðxÞ  V 1 ðxÞ ¼ 2ðln s

(vi)

and hence

representing a significant property of the adiabatic potential. Although consistent ~ ðxÞ is not immediately with equations (iv) in Sect. 8.5, the physical implication of s clear. In the above analysis, the variable x represents the phase of propagation, and the functions sðxÞ and VðxÞ are conformal with the transformation x ! x  at, where a is constant, and t is an arbitrary parameter. For thermodynamic applications, we

128

8 Soliton Theory of Long-Range Order

consider that the temperature T can be assigned to t for changing VðxÞ. Therefore, writing DV ðx  atÞ ¼

    @V @V Dx  a Dt; @x t @t x

(vii)

we can assume Dt / DT . In this case, an isothermal change can be specified by DT ¼ 0, where   ~0 @V s DV ðx  atÞ ¼ Dx ¼ 2 Dx: ~ @x t s For an adiabatic change, we have DV ðx  atÞ ¼ a

  @V 2a~ s0 Dt: Dt ¼  ~ @t x s

Subjecting to phonon scatterings, such DV can be calculated as proportional to hK þ kjDV jKi2 , where K is the phonon vector in a crystal, and then evaluated by phonon densities at K + k and K states.   Using the equipartition theorem in statisti@V cal mechanics, both of these rates, @V @x t and @t x for phonon scatterings, can be expressed as proportional to DT ¼ T 2  T 1 , except for DK ¼ 0 in the isothermal process. Therefore, the latter process cannot be totally adiabatic while involved in thermal relaxation. The expression (vii) may therefore be called Born–Huang transitions in thermodynamic environment. In real crystals, excited states of correlations can be attributed to soliton excitations of order variables, where the adiabatic potential is composed of sech2 potentials, emerging at a critical temperature T c , after the crystal is stabilized by Born–Huang’s transition. Accordingly, n ¼ 0 is logically for the ground state of ~ðxÞ ¼ cosh x , which is symmetrical combination of expðxÞ , whereas the s mesoscopic state at lower temperatures can be signified by increasing number n, ~ ðxÞ ¼ coshnþ1 x . Therefore, we can consider that the number n becoming to s represents an elemental decrement of the adiabatic potential. In this interpretation, coshnþ1 x is a multi-soliton function in the correlation field. Assuming that the transformation (iv) considered for s1 ! s2 represents a Born–Huang transition, we can explain experimental results. On the other hand, subject to phonon scatterings in crystals, a change Ds ¼ s1  s2 should accompany a lattice displacement Du ¼ u1  u2, which is proportional to a temperature change DT. The temperature-dependence in long tail of the specific heat anomaly below T c, the so-called l-transitions, can therefore be analyzed as illustrated in Fig. 8.5. It is instructive to see that the tail for T < T c can be schematically interpreted as stepwise isothermal and adiabatic changes, characterizing the latter processes by DT and Dn, as shown in the figure.

8.7 Topological Mapping of Mesoscopic Fields

129

Fig. 8.5 A schematic diagram for Cp  T curve. A fast phasing process of sk for T>T c, and a slow phasing of clustered pseudospins sk for T Þ  eo w >

and e < ð0Þ ¼ eo ð1 þ w < Þ  eo w < :

Considering that the Curie–Weiss laws (6.6a, b) determine the critical temperature T o , we can express o2oT ðT > T o Þ / T  T o

and

o2oT ðT < T o Þ / T o  T;

(9.5)

implying that frequencies of the transverse modes diminish or soften, if T o is approached from both sides. We may say that soft modes arise from the LST formula, if combined with Curie–Weiss laws.

9.2 Soft Lattice Modes of Condensates

9.2

141

Soft Lattice Modes of Condensates

In the above theory, amplitudes Po and uo as functions of a position r are distributed in crystal space. As these are proportional to the exponential factor exp ik  r , experimental studies with time-dependent electric field can be performed with no reference to lattice points, if the condition exp ik  r ¼ 1 is fulfilled, namely, either for k ¼ 0 at r ¼ R or for k ¼ G2 at r ¼ 2R. These cases correspond to the center or boundaries of first Brillouin zone in the reciprocal lattice. In the previous section, we actually discussed the response of a dielectric polarization to an applied field at the zone center G ¼ 0. Similar soft-mode studies can be carried out at the zone boundary k ¼ G2 in neutron inelastic scattering experiments, as discussed in Chap. 10. Dielectric properties of crystals originate from displaced charges at the lattice sites R; on the other hand, the neutron inelastic scatterings exhibit the response of mass displacements to a structural change associated with combing two unit cells in size 2R. In both of these cases, soft modes are due to condensates in thermodynamic environment.

9.2.1

The Lattice Response to Collective Pseudospins

In the above approximation, harmonic displacements uo were assumed as given by (9.2). In the presence of correlations however, pseudospins are in collective propagation, driven by the internal adiabatic potential in modulated lattice, shifting frequency and damping energy via phonon scatterings. Accordingly, shifting softmode frequencies is significantly related to symmetry and anharmonicity of the adiabatic potential. Considering longitudinal and transverse displacements uL k k and uT ?k from equilibrium positions, the adiabatic lattice potential in a condensate can be expressed by   X  X  uT þ V ð2Þ u2L þ u2T DU ¼ V ð1Þ uL þ T

T

n X  X o n X o þ V ð3Þ uL u2T þ u2L uT u2T þ V ð4Þ u4L þ u2L T

þ ;

T

T

(9.6)

P where the summation . . . is for transverse directions along the y and z axes, and V ð1Þ ; V ð2Þ ; V ð3Þ ; V ð4Þ ; :::::T are expansion coefficients of the corresponding powers 1, 2, 3, 4, . . ... Here uLand uT are finite but small displacements at temperatures T in the vicinity of T c. The energy DU expressed by (9.6) can then be truncated at the fourth

142

9 Soft Modes

order, composed of component terms of V ð1Þ . . . V ð4Þ , if uL and uL are sufficiently small. Expressing the phonon wavefunction by jq; oi , we can calculate the matrix elements hq0 ; o0 jDUjq; oi for the isothermal average hDU iT at T. The secular part, written as DU s , for brevity, can be written as DU s ¼ DU sL þ DU sT ; where the longitudinal component is   DU sL ¼ V ð2Þ þ V ð4Þ u2oL u2L þ V ð4Þ u4L

(9.7a)

and the transverse potential can be decomposed as DU sT ¼ DUs þ þ DU s ; where     DUsþ ¼ V ð2Þ þ V ð4Þ u2oL u2þ ; . . . ; DU s ¼ V ð2Þ þ V ð4Þ u2oL u2 ; . . . . . .

(9.7b)

P

u2T ¼ u2þ þ u2 . On the other hand, the time-dependent term arises from    matrix elements hq0 ; o0 juT jq; oi and q0 ; o0 u3T q; o for q0 6¼ q, which are subjected to inelastic phonon scatterings at T. The temperature-dependent terms consist of these nonzero elements for q0  q ¼ k and o0  o ¼ Do, related to inelastic phonon scatterings. Taking the term of V ð1Þ , for example, we have and

T

hq0 ; o0 jDU ð1Þ jq; oi ¼ V ð1Þ hq0 juL;T jqi exp iðtDoÞ; the scattering probability can thereby be expressed as the time average of  2 1  cosðtDoÞ P ¼ 2V ð1Þ  hq0 juL;T jqi2 Do2 integrated over phonon collision time to . Calculating hPit ¼ 2t1o  2 sinðto DoÞ ; hPit ¼ 2V ð1Þ  hq0 juL;T jqi2 to Do

þt Ðo to

Pdt, we obtain (9.8)

o DoÞ 1 where the factor sintðotDo is nearly equal to 1, if to < Do . For modes uL ¼ uoL exp ðiotÞ and uT ¼ uoT expðiotÞ, the probability at temperature T can be calculated with (9.8) for inelastic phonon scatterings q0  q ¼ k , while hðo0  oÞ ¼ hDo represents the energy transfer to the surroundings,  2 which is considered to be equal to kB T at a temperature T. Thus, the factor V ð1Þ  represents energy damping from

9.2 Soft Lattice Modes of Condensates

143

the uL mode. Damping is also contributed by the third-order term V ð3Þ u3T and higher terms, so that the overall damping can effectively be expressed by an empirical constant g as in the following analysis. Setting uL aside, the equations of motion for the transverse mode uT can be written as
2  d uT duT @U s 2 þ o ¼ Fs expðiotÞ; m þ g u (9.9) ¼ o T dt2 dt @x

 @u2 where Fs ¼  V ð2Þ þ V ð4Þ u2oL @x that is the driving force; oo is the characteristic frequency and g the damping constant of the unforced mode. Equation (9.9) represents a harmonic oscillator forced by an external force, which can be used for a zone-center transition at G ¼ 0 perturbed by an applied electric field; this also is applicable to a zone-boundary transition at 12 G, where the momentum loss of neutrons is equal to 12  hG, and q0  q ¼ 12 G þ k. Using uT ¼ uo expðiotÞin (9.9), we obtain the steady solution of (9.9)

 m o2  iog þ o2o uo ¼ Fs ; from which we can define the complex susceptibility as wðoÞ ¼

muo 1 : ¼ 2 oo  o2  iog Fs

(9.10)

Writing this in a complex form wðoÞ ¼ w0 ðoÞ  iw00 ðoÞ, the real and imaginary parts of (8.10) can be expressed as w0 ðoÞ ¼

o2o  o2 2 o2o  o2 þ o2 g2

and

w00 ðoÞ ¼

og ; 2 o2o  o2 þ o2 g2

(9.11)

respectively. Real and imaginary parts of the complex susceptibility are plotted against o in Fig. 9.2. The imaginary part w00 ðoÞ is a dumbbell-shaped symmetric jo2 o2 j 1 curve, if g < oo2  2Do, where Do ¼ o  oo . Otherwise, w00 ðoÞ  og for g > Do, showing no peak at o ¼ oo, which is overdamped; otherwise, the case for g < Do is called underdamped. Neutron experiments have revealed that the energy loss expressed as proportional to w00 ðoo Þ. Experimentally, the characteristic frequency oo of soft modes becomes close to zero in the vicinity of T c , where the equation (9.9) is dominated by the damping term. Therefore, we write g Writing

duT Fo þ o2o uT ¼ expðiotÞ: dt m

o2o 1 Fo ¼ and ¼ F0 , this equation is expressed as t g mg

144

9 Soft Modes

Fig. 9.2 A complex susceptibility wðoÞ ¼ w0 ðoÞ  iw00 ðoÞ. (a) Imaginary part w00 ðoÞ and (b) real part w0 ðoÞ.

duT uT þ ¼ F0 expðiotÞ; dt t

(9.12)

where t is a relaxation time. Equation (9.12) is known as Debye relaxation. Setting uT ¼ A expðiotÞ, we obtain the susceptibility for such a relaxator, that is, wD ðoÞ ¼

1 io þ

1 t

¼

t iot2 þ : 2 2 1þo t 1 þ o2 t2

(9.13)

Equations (9.12) and (9.13) are known as Debye relaxation, where t and F0are related to g and Fs in the overdamped soft mode. In practical crystals with lattice imperfections, such a relaxation formula are also valid, if ot  1, although o is not necessarily low. Nevertheless, the soft mode near zero frequencies looks like Debye relaxator. Figure 9.3 shows an example of dielectric functions from ferroelectric crystals of BaTiO3 obtained by Petzelt and his coworkers [2], showing the presence of soft modes in the paraelectric phase, as characterized by dispersive e0 ðoÞ of Debye’s type and absorptive e00 ðoÞ of dumbbell shape. In observed squared frequencies o2o versus T c  T shown in Fig. 9.4 [3], there is a noticeable deviation from the straight line predicted by the mean-field theory.

9.2.2

Temperature Dependence of Soft-Mode Frequencies

Equation (9.9) represents a dynamical oscillator in thermodynamic environment. Experimentally, we know that the damping constant g and characteristic frequency

9.2 Soft Lattice Modes of Condensates

145

Fig. 9.3 Dielectric softmode spectra eðoÞ observed from BaTiO3 crystals by backward-wave techniques. Curves for 1, 2, 3, and 4 were obtained at temperatures 474, 535, 582, and 667 K, respectively. From J. Petzelt, G.V. Kozlow, and A.A. Volkov, Ferroelectrics 73, 101 (987).

Fig. 9.4 A plot of squared soft-mode frequencies versus T  T c obtained by neutron scatterings from SrTiO3 . From R. Currat, K.A. Mu¨ller, W. Berlinger, and F. Desnoyer, Phys. Rev. B17, 2937 (1978).

oo are temperature dependent, which are attributed to phonon scatterings in crystals. Under a constant pressure, the crystal structure is strained by the energy DU , resulting in a small volume change. In this case, we need to confirm if such an adiabatic potential can cause a frequency change in the phonon spectra. Assuming that a strained

146

9 Soft Modes

lattice is an isotropic and continuous medium, the Gibbs potential of the lattice can be written as GðkÞ ¼ kB T ln

hoðkÞ ðDV Þ2 þ ; kB T 2kV

where k is the compressibility, and here DV strains V expresses volume

 that may be temperature dependent. From the equilibrium condition, we set @G @V T ¼ 0 to obtain kB T @oðkÞ the relation DV ¼ k in the first order. On the other hand, from (2.36) V oðkÞ @V combined with the equipartition theorem U vib ¼ kB T, we can write DV kg kgkB T ¼ kDp ¼  U vib ¼  ; V V V where g ¼  d dlnlnoVðkÞ is Gru¨neisen’s constant, which is normally assumed as temperature independent. If so, the frequency oðkÞ may change only adiabatically, implying no thermal shift of oðkÞ is expected in isotropic crystals; hence, a thermal frequency shift should be attributed to time-dependent DUt . Disregarding the volume change, we consider DUt that can be written as  

 DU t ¼ V ð3Þ u2L uy  iuz þ V ð4Þ u2L u2y þ u2z ; for which

P

uT in (9.6) is replaced by a complex displacement uy  iuz in a yz

T

plane, for calculating convenience. Cowley [4] showed that inelastic phonon scatterings by the term of V ð3Þ in DU t include uy  iuz , whereas the second term of V ð4Þ is responsible for elastic scatterings via u2y þ u2z . In (9.6), the damping term

ð1Þ P V þ V ð3Þ u2L uT is also complex, so the total damping factor can be expressed T

in a complex form as g ¼ G  iF. In his theory, the secular frequency is written as oðk; DoÞ , where Do represents energy changes  hDo during the inelastic phonon process; G and F are also functions of k and Do. In Cowley’s theory, the steady-state solution of (9.9) can be written as uo ðk; DoÞ ¼

Fs =m 2

oðkÞ þ

o2o

 i oðkÞðG  iFÞ

;

for which the squared characteristic frequency can be expressed effectively as oðk; DoÞ2 ¼ o2o þ oðkÞFðk; DoÞ: Here, the factor Fðk; DoÞ is contributed by three parts, namely, Fðk; DoÞ ¼ Fo þ F1 ðkÞ þ F2 ðk; DoÞ; where

(9.14)

9.2 Soft Lattice Modes of Condensates

Fo ¼

147

@oðkÞ DV @V

is negligible for DV ¼ 0 as discussed earlier in this section. The second term is expressed as

 n þ 12 V ð4Þ u2L  h F1 ðkÞ ¼ hK; Kju2T jK0 ; K0 i; NoðkÞ o0 where n is the number of scattered phonons ðK0 ; o0 Þ, giving rise to a temperaturedependent frequency

 shift with F1 ðkÞ / T , if combining with the equipartition theorem  ho0 n þ 12 ¼ kB T. The expressions for F2 ðk; DoÞ and for Gðk; DoÞ are similar, thereby the effective damping parameter becomes   X V ð3Þ u2L hkjuT jK1 ; K2 i2 p h 2Gðk; DoÞ ¼ 16NoðkÞ 1;2 o1 o2  ½ðn1 þ n2 þ 1Þfdðo þ o1 þ o2 Þ þ dðo  o1  o2 Þg  ðn1  n2 Þfdðo  o1 þ o2 Þ þ dðo þ o1  o2 Þg : Considering two temperatures T 1 ¼ hokðBkÞ ðn1 þ n2 þ 1Þ and T 2 ¼ hokðBkÞ ðn1  n2 Þ, this damping term indicates that the phonon energy hoð2n2 þ 1Þ is transferred to the heat reservoir, which is equal to kB ðT 1  T 2 Þ. It was controversial, if the soft-mode frequency converges to zero. Experimentally it is a matter of timescale of observation; however, no answer can be given to this question, because the transition threshold is always obscured by unavoidable fluctuations. Considering that oðkÞ is determined at k ¼ 0 and signified by Do for a soft mode, that is, oð0Þ ¼ oo  Do, we can write (9.14) for the effective characteristic frequency oð0; DoÞ as oð0; DoÞ2 ¼ o2o þ F1 ð0Þðoo  DoÞ: This can be made equal to zero if we choose Do ¼  F14ð0Þ , in which case we 2 0 obtain oo ¼  Do 2 . Assuming that Do emerges at T o , we may write oo ¼ A T o . 0 Further writing the second term in the above as F1 ð0Þðoo  DoÞ ¼ A T, we obtain oð0; DoÞ2 ¼ A0 ðT o  T Þ, which agree with Landau’s hypothesis (9.5). For a zone-boundary transition, a soft mode can be studied with intensities of inelastically scattered neutrons in the scattering geometry K þ k ! K0 , where K and K0 are wavevectors of neutrons. The timescale of neutron collision is significantly short, so that the critical anomalies can be studied by scanning scattering angles with varying temperature of the sample crystal. In this case, (9.14) can be expanded for small jkj as

148

9 Soft Modes

oðk; DoÞ2 ¼ oð0; DoÞ2 þ kk2 þ . . . ; with a small constant k, so that we can write approximate relations oðk; DoÞ2 ¼ A0 > ðT  T o Þ þ k > k2

for

T > To

oðk; DoÞ2 ¼ A0 < ðT o  T Þ þ k < k2

for

T < To;

and (9.15)

which are useful formula for the intensity analysis of scattered neutrons in a fixed geometry, as shown by Fig. 9.4 [5].

9.2.3

Cochran’s Model

Temperature-dependent lattice modes were observed in early spectroscopic studies on structural transitions, but it was after Cochran’ theory (1690) [6] that such modes are called soft modes. Based on a simplified model of an ionic crystal, he showed that the frequency softening can occur as a consequence of counteracting short- and long-range interactions. In this section, we review Cochran’s theory but modifying that ionic displacements for a condensate in a crystal, assuming displaceable charges with effective masses that are considerably lighter than ions. If a particle of mass m and charge +e is displaced by u from its site, a hole of mass m0 and charge  e is left behind, creating a dipole moment s ¼ eu that is regarded as the order variable. Dynamically, such a dipole s is in motion with respect to the center of mass, mM characterized by the reduced mass m ¼ mþM , where m and M are masses of the charge and the ion with a hole. Nevertheless, as m  m if m < M, we can disregard the lattice structure in Cochran’s model. In his model, one-dimensional chain of mass particles m and m0 is considered in an applied field E ¼ Eo exp iot. We assume that these dipoles si ¼ eui are correlated between neighboring sites, so that the crystal as a whole is polarized as expressed by the polarization P along the direction of the chain. Accordingly, we consider a depolarizing field as expressed by  ePo along the direction of the chain. Further, taking dipolar interactions from all other chains into account, Cochran included the Lorentz field  3ePo , assuming a uniform crystal. Acceptable in the mean-field accuracy, his theory can bring a temperature dependence into dynamic equations through the temperature-dependent polarization P.  Denoting the transverse displacements of charges +e and  e by uþ T and uT , respectively, we can write equations of motion as

9.2 Soft Lattice Modes of Condensates




þ  d2 uþ P  T m 2 ¼ C uT  uT þ e E þ dt 3eo

149




  d2 u P þ T m ; ¼ C uT  uT  e E þ dt2 3eo 0

and

where C is a restoring constant that represents a short-range interaction. Assuming eðuþ u Þ P ¼ T v T , where v is the specific volume of a crystal, these equations can be combined as m


 d2 P e 2 P E þ ¼  CP: dt2 3eo v

(9.16a)

For a given field E ¼ Eo exp ioT t, we set P ¼ Po exp ioT t in (9.16a) and obtain the expression for the susceptibility

Po wðoT Þ ¼ ¼ eo E o

e2 eo v ; e2 2  moT C 3eo v 2

2

which shows a singularity if mo2T ¼ C  3eeo v . Therefore, if the condition C ¼ 3eeo v is met, oT can be zero, for which wðoT Þ or PðoT Þ can be regarded to represent a soft mode. On the other hand, if charges e; e are signified by longitudinal displacements  eðuþ þ L uL Þ uL and u , we have the equation L , respectively, by defining P ¼ v
 d2 P e 2 P P m 2 ¼ E þ  CP; dt eo 3eo v

(9.16b)

from which the longitudinal susceptibility can be written as e2 eo v : wðoL Þ ¼ 2e2 2  moL Cþ 3eo v 2

2e Since C is a positive constant, the frequency determined by mo2L ¼ C þ 3e ov cannot vanish. In Cochran’s theory, the soft mode should be a transverse mode, and both oT and oL can be temperature dependent. For a binary displacive structural change, we consider that pseudospins emerged at T c and then clustered in periodic lattice at T c . In the process of condensate formation, the corresponding lattice displacement uT of clustered pseudospins should be different from unclustered ones dynamically. Therefore, in the following, we specify these displacements by uT and vT , respectively, which are pinned and observed in practical experiments.

150

9 Soft Modes

We can assume that these two modes are coupled in such a way as vT ¼ cuT , where c is temperature dependent. For the coupled system, the equations of motion can be written as d2 uT duT dvT Fs þ g0 þ o2o uT ¼ expðiotÞ þg dt2 dt dt m and dvT vT þ ¼ F0 expðiotÞ; dt t for which the steady-state solutions can be expressed as

 Fs uo o2  iog þ o2o  iog0 vo ¼ m

and


 1 vo io þ ¼ F0 : t

We can therefore derive the susceptibility function for the coupled oscillator, wðoÞ ¼

muo 1 ¼ Fs 2 2 oo  o  iog  icg0 F0

ot : 1  iot

Writing d2 ¼ cg0 F0 , this susceptibility formula is simplified as 1

wðoÞ ¼ o2o



o2

d2 ot  iog  1  iot

:

(9.17)

If these two modes u and v are independent, that is, d ¼ 0 or c ¼ 0, (9.17) is characterized by a simple oscillator formula for the u-mode. If the observed susceptibility contains such an additional independent response as given by (9.13) in the vicinity of o ¼ 0, we have wðoÞ ¼

1 t : þ o2o  o2  iog 1  iot

(9.18)

Nevertheless, at oo  0 , these two terms at a very low frequency become overlapped and indistinguishable unless g > 1t . Experimentally however, such a response of Debye type happens to be a distinctively sharp peak and observed distinctively in zone-boundary transitions at near-zero frequency, which is called a central peak, as shown in Fig. 9.5 [7].

9.3 Symmetry Change at Tc

151

Fig. 9.5 Soft-mode spectra from SrTiO3 and KMnF3, consisting of a soft-mode and a central peak. From S.M. Shapiro, J.D. Axe, G. Shirane, and T. Riste, Phys. Rev. B6, 4332 (1972).

9.3

Symmetry Change at Tc

The presence of an additional mode vT near o  0 is hypothetical in the foregoing one-dimensional theory. However, such a transverse mode is logical to be included in the response function at o ¼ 0 . In fact, the mode vT is consistent with the observed soft-mode spectra in TSCC crystals, hence indicating symmetry change of the lattice at T c . A structural transformation is characterized macroscopically by a symmetry change in a crystal. We consider that a symmetry change occurs at T c as induced by emerging adiabatic potentials. Considering such a potential as given by (9.7a) and (9.7b) becomes significant at the outset of pseudospin ordering along the x-axis, we write the potential expressed as     DU tr ¼ V ð3Þ u2y þ u2z þ V ð4Þ u2x u2y þ u2z

(9.19)

for lowering symmetry via phonon scatterings by transverse displacements uy and uz. Noted that (9.7a), (9.7b), and (9.19) belong to the same class of a cubic symmetry group, we consider that a transition may take place for the lattice structure to deform at minimal energy. Figure 9.6a, b [8, 9] illustrate symmetry changes at transition temperatures T c , as interpreted by observed soft-mode spectra.

152

9 Soft Modes

Soft modes observed by optical experiments on TSCC crystals and identified as B2u modes for T > T c and A1 modes for T < T c , and propagating along x and z directions, respectively. Except for the critical region, the adiabatic potentials are written as 1 1 DUBu ¼ Au2x þ Bu4x 2 4

and

1 0 1 0 DU A1 ¼ Auz2 þ Buz4 : 2 4

For the symmetry change between B2u and A1 , we postulate the coordinate transformation from  ux to  uy 0  uz 0 , corresponding to phonon scatterings K x $ K x and K y ; K z $ K y ; K z , respectively. Then, the process can be determined by a linear combination u0 ¼ cx u0 x þ cy u0 y

where

c2x þ c2y ¼ 1;

with which the potential in the critical region is expressed by 1 1 2 2 DUcr ¼ Au0 þ Bu0 u2x ; 2 4

(9.20)

where the second quartic term represents a coupling between DUB2u and DU A1 . Therefore, we reexpress (9.20) as DUcr ¼ c2x



 A 0 2 B 0 4 A 0 2 B 0 2 0 2 u x þ u x þ c2y uy þ ux uy : 2 4 2 4

We assume that the critical state can change by c2x ! 0 and c2y ! 1 to establish 02 the normal state, where we should have ux ¼  AB in thermal equilibrium. Therefore, A limcy !1 DU cr ! þ u2y : 4

(9.21)

This indicates that the A1 mode below Tc is characterized by oð0Þ2 ¼ 2AðT o  T Þ, which appears to be substantiated by the curves in Fig. 9.6. It is notable at this stage that the potential (9.20) is theoretically predictable by the soliton theory as calculated by the method of reverse scattering [10]. For phonon-energy curves for SrTiO3 below 105 K in Fig. 9.6b, the quadratic potential should also be related effectively to anisotropic quartic potentials, depending on k. These curves are determined by o < ð0; kÞ2 ¼ AðkÞðT o  T Þ þ kk2 where the coefficient AðkÞdepends on the direction of k. Experimentally, near-linear relations between o < ð0; 0Þ2 and T o  T as shown in Fig. 9.4 were obtained for many structural transitions. Consistent with the mean-field theory however, we have no significant reason to support this linear relation. Hence, we usually write o < ð0; 0Þ / ðT o  T Þb , where b is called a critical exponent that can be determined from the curves, but close to b ¼ 12 in mean-field approximation.

Exercise 9

153

Fig. 9.6 Soft-mode spectra as a function of temperature: (a) from TSCC, (b) SrTiO3 . From J. F. Scott, Raman Spectroscopy of Structural Phase Transitions in Light Scattering Near Phase Transitions, ed. H.Z. Cummins and A.P. Levanyuk, North Holland, Amsterdam (1983). J. Feder and E. Pytte, Phys. Rev. B1, 4803 (1970).

Exercise 9 1. In Cochran’s theory, the polarization P is a macroscopic variable. How his theory can be modified for a mesoscopic polarization? 2. In Landau’s theory, he postulated that A ¼ A0 ðT  T o Þ where A0 is a positive constant; hence, A is positive for T > T o but negative for T < T o . In practical crystals, A0 is not necessarily a constant, depending on the wavevector k of the order parameter. Why? Discuss the mechanism for variable A0 . Representing primarily a harmonic potential, but often including a quartic potential effectively, the soft-mode frequency exhibits therefore a different temperature dependence, as shown in Fig. 9.6a, b. Is higher-order potentials the acceptable explanation for different A0 ?

154

9 Soft Modes

3. When deriving the LST relation in Sect. 9.1, the adiabatic potential was not considered for calculating eðoÞ. Is it an acceptable assumption? If not, discuss why the potential was disregarded. 4. Review Cochran’s model to justify his concepts of short- and long-range interactions. 5. Why the soft mode can be considered as representing a Born–Huang transition? Discuss the origin for a temperature-dependent mode in a crystal.

References 1. R.J. Elliott, A.F. Gibson, An Introduction to Solid State Physics and Its Applications (MacMillan, London, 1974) 2. J. Petzelt, G.V. Kozlov, V.V. Volkov, Ferroelectrics 73, 101 (1987) 3. R. Currat, K.A. Mu¨ller, W. Berlinger, F. Desnoyer, Phys. Rev. B17, 2937 (1978) 4. A. Cowley, Prog. Phys. 31, 123 (1968) 5. M. Iizumi, J.D. Axe, G. Shirane, K. Shimaoka, Phys. Rev. B15, 4392 (1977) 6. W. Cochran, The Dynamics of Atoms in Crystals (Edward Arnold, London, 1973) 7. S.M. Shapiro, J.D. Axe, G. Shirane, T. Riste, Phys. Rev. B6, 4332 (1972) 8. J.F. Scott, Raman spectroscopy of structural phase transitions in light scattering near phase transition (North-Holland, Amsterdam, 1983) 9. J. Feder, E. Pytte, Phys. Rev. B1, 4803 (1070) 10. Lamb GL Jr (1980) Elements of soliton theory Chap. 3, (Wiley, New York)

Chapter 10

Experimental Studies on Critical Fluctuations

In mesoscopic states of a crystal, properties of the condensates can be studied experimentally with renormalized variables, which are observable if they are pinned by intrinsic and extrinsic adiabatic potentials. Condensates in crystals are primarily longitudinal, propagating along the direction of wavevector, where the pseudospin arrangement and frequency dispersion can be sampled with varying temperature for T < T c . With appropriate probes, we can obtain useful information about critical anomalies; photons are sensitive to the temporal variation, neutrons are for studying displacements in responsible constituents, and magnetic resonance probes yield information about structural changes. Such sampling experiments as X-ray diffraction, dielectric measurements, light and neutron inelastic scatterings, and magnetic resonances are outlined in this chapter, with relevant results to substantiate theoretical arguments.

10.1

Diffuse X-Ray Diffraction

Normally, the crystal structure is determined by X-ray diffraction studies. A collimated X-ray beam is scattered from a crystal, exhibiting a diffraction pattern on a photographic plate placed perpendicular to the beam direction. The diffraction pattern arises from elastic scatterings of X-ray photons by distributed electron densities in the crystal, keeping nuclear masses intact. In the critical region, on the other hand, the structure is modulated along symmetry directions, resulting in diffraction spots broadened in anomalous shape, which are called diffuse diffraction. In early crystallographic studies, Bragg established the concept of crystal planes of constituents, like optical reflection planes, signifying periodic lattice structure in normal crystals. A collimated X-ray beam reflects from a large number of parallel crystal planes, showing an interference pattern on a photographic plate placed nearby. Figure 10.1 illustrates a practical arrangement commonly used for analyzing diffraction patterns. M. Fujimoto, Thermodynamics of Crystalline States, 155 DOI 10.1007/978-1-4614-5085-6_10, # Springer Science+Business Media New York 2013

156

10

Experimental Studies on Critical Fluctuations

Fig. 10.1 X-ray diffraction setup. Incident X-ray beam Ko, diffracted beam K on a conical surface. A modulated axis is indicated by Goka.

Considering that a collimated beam of monochromatic X-ray of a wavevector Ko and frequency oo is incident on crystal planes characterized by the normal vector G, scattered wavevectors K and frequency o are determined by Bragg’s law Ko
K ¼  G;

(10.1a)

for which jKo j ¼ jKj and

hoo ¼
ho


(10.1b)

express the conservation laws of momentum and energy for elastic scatterings. Here, G ¼ na is the lattice translation vector along the normal direction denoted by a, where n is integer, representing a group of parallel crystal planes perpendicular to a. We notice from (10.1b) that only the magnitude jGj is significant for the elastically scattered beam K; diffractions occur in all directions satisfying the relation (10.1a). All of these diffracted vectors K are on conical surfaces with respect to the a-axis, as shown in Fig. 10.1. Such conical diffractions constitute Laue’s law, providing a useful method of constructing diffraction patterns. In this

10.1

Diffuse X-Ray Diffraction

157

figure how the lattice modulation can occur along the a-axis for a given wavevector k is illustrated. Considering a charge element rðr o Þd3 r that scatters an incident X-ray photon field Eo exp iðKo :r o
oto Þ at a point r o and time to , the scattered field is spherical and expressed as ð Aðr; tÞ /

d3 r o rðr o ÞEo exp iðKo :r o
otÞ V ðr o Þ

exp ifK:ðr
r o Þ
oðt
to Þg : jr
r o j

Hence, the scattered amplitude at a distance r  r o is determined by an Þ approximate form Aðr; tÞ / exp iðK:r
ot ¼ Ao , which can be reexpressed by r Ao / Eo

ð d3 r o frðr o Þ exp iðKo
KÞ:r o g:

(10.2)

V ðr o Þ

Elastic scatterings from a rigid crystal are specified by the relation exp ið G:r o Þ ¼ 1; hence the scattered intensity ratio (10.2) is determined by maximum scattered amplitude, which is proportional to the net charge at the lattice point r o . In practice, however, a collimated X-ray beam strikes a finite area on crystal planes, so that the reflected beam arises from all scatterers in the impact area. Letting positions of scatterers as r om where m ¼ 1; 2; . . . in the impact area, (10.2) can be reexpressed as Ao X ¼ f m ðGÞ expð
iG:r om Þ; Eo m

(10.3)

where ð f m ðGÞ ¼

rðr om Þd3 r om : V ðrom Þ

This is called the atomic form factor. However, in crystals electronic densities overlap significantly between neighboring atoms, so (10.3) overestimates individual contributions. In this context, we redefine the factors that are independent of overlaps of constituents. Replacing r om by a continuous variable s
r om , we can write ð

ð rðsÞ expð
iG:sÞd3 s ¼ V ðsÞ

rðs
r om Þ expf
iG:ðs
r om Þgd3 ðs
r om Þ  expð
iG:r om Þ V ðsÞ

¼ f m ðGÞ expð
iG:r om Þ;

158

10

Experimental Studies on Critical Fluctuations

where ð f m ðGÞ ¼

rðs
r om Þ expf
iðs
r om Þgd3 ðs
r om Þ:

(10.4)

V ðsÞ

Using (10.4), the formula (10.3) can be expressed as SðGÞ ¼

X

f m ðGÞ expð
iG:r om Þ ¼

X

m

m



 2pi f m ðGÞ exp ðxm h þ ym k þ zm lÞ ; O

which is called the structural form factor; thereby the field of a scattered beam at a distant position r can be expressed as EG ð r Þ Sð G Þ exp iðK:r
otÞ: ¼ Eo r Using the structural factor, the intensity of the scattered beam is given by I ðGÞ 1  ¼ 2 S ðGÞSðGÞ: Io r

(10.5)

In a crystal is modulated in a direction of the a-axis, the periodicity of pseudospins is modified by the wavevectors G  k and the energies  De ¼
h ðo
oo Þ ¼ 
hDo, so that the scatterings should basically be inelastic. We therefore consider conservation laws ko
k ¼ k  k

and

eðko Þ
eðkÞ ¼ DeðkÞ;

by which (10.2) can be revised for inelastic scatterings as ð Ao exp ifK:ðr
r o Þ
oðt
to Þg / d3 r o rðr o Þ exp iðKo :r o
oo to Þ Eo jr
r o j ð r ð r Þ exp i ð K:r
ot Þ o exp ifðKo
KÞ:r
ðoo
oÞto g:  d3 r o r Using (10.6) in this expression, we obtain EG ðrÞ exp iðK:r
otÞ X / ff ðG þ kÞ exp iðk:r m
Do to Þ Eo r m þ f ðG
kÞ exp ið
k:r m þ Do to Þg;

(10.6)

10.1

Diffuse X-Ray Diffraction

159

where ð f ðG  kÞ ¼

d3 r o rðr o Þ exp iðG  kÞ:r o : V ðr o Þ

We consider jkj < jGj for a weak modulation, in which case f ðG  kÞ  f ðGÞ and phase fluctuations exp iðG  kÞ:r o can be separated to amplitude and phase modes, similar to the critical condition discussed in Chapter 6. Taking these symmetric and antisymmetric modes into account, we can derive the corresponding intensity at r, i.e., I ðG  kÞ I ðGÞ 2jf ðGÞj2 X ½cosfk:ðr m
r n Þ ¼ þ Io Io r2
Doðtm
tn Þg þ sinfk:ðr m
r n Þ
Doðtm
tn Þg :

(10.7a)

Here, the first term on the right represents elastic scatterings, while phase fluctuations in the impact area S are explicit in the second inelastic term. Considering timescale of observation to , the latter represents the intensity anomaly due to diffuse diffraction, i.e., hI ðG  kÞit
hI ðGÞit 2jf ðGÞj2 1 ¼ S Io r2

ð



 hcos fit þ hsin fit dS;

(10.7b)

S

where f ¼ k:ðr m
r n Þ
Doðtm
tn Þ represents fluctuating phases. Here, it is convenient to consider that r m
r n ¼ r and tm
tn ¼ t are continuous space–time, i.e., f ¼ k:r
Do:t; (10.7b) expresses that time averages hcos fit and hsin fit are integrated over the area S. Assuming a rectangular area S ¼ Lx Ly and continuous f ¼ kx
Do:t , the symmetric spatial fluctuations can be calculated as 1 S

ð hcos fit dS ¼ S

 1 Lðx Lx

0

hcos fit dx

L  1 ðy

Ly

0

D

f D 1 ð2 E dy : ¼ cos fdf kLx



2 f
f1 f þ f1 E cos 2 ¼ sin 2 : f2
f1 2 2 t

f1

t

Here, f1 ; f2 indicate limits of the space integration at x1 ; x2 ; hence f2
f1 ¼ Df Ð 1 ¼ kLx , and f2 þf ¼ kx
Do:t. After a similar calculation for S1 hsin fit dS, we 2 S

obtain the formula for the net intensity anomaly:

160

10

Experimental Studies on Critical Fluctuations

 hDI ðGÞit sin kLx sinðto DoÞ  ¼ hcos kxit þ hsin kxit : to Do Io kLx

(10.7c)

This formula explains diffused intensities of X-ray diffraction; however such broadened spots and lines are hardly analyzed in the timescale to of photochemical processes in X-ray experiments. For neutron diffraction, we can use (10.7c) for intensity anomalies of neutron scatterings. The intensity of a neutron beam is determined by counting technique characterized by a much shorter timescale than in X-ray case, but still anomalous intensities (10.7c) for neutrons cannot be analyzed.

10.2

Neutron Inelastic Scatterings

In contrast to X-ray experiments, neutron scatterers are normally associated with heavy nuclei, so symmetric and antisymmetric contributions to elastic scatterings DK ¼ G cannot be resolved, but can be resolved for inelastic scatterings for DK?G within timescale of neutron counters, if scattered neutrons are specified by a small wavevector k ¼ DK and energy De ¼
hDo. For one-dimensional modulation, if collimated X-ray beam is parallel to the modulated axis, we obtain K ¼ 0 from (10.1a) if Ko ¼ G , meaning no elastic scatterings. For neutrons, in contrast, inelastic scatterings can occur at the zone boundaries  G2 , if K
Ko ¼  12 G, without elastic diffraction in all directions. Figure 10.2 illustrates such an arrangement that inelastic scatterings can be detected efficiently, provided that the incident beam Ko is in high-enough intensity. Otherwise, as there are some unavoidable elastic scatterings in practice, feeble inelastic intensities are undetected. Nevertheless, the latter can be detected by minimizing elastic scatterings, if the wavevector is scanned in the vicinity of DK ¼ 0. Using thermal neutrons from a nuclear reactor, as characterized by their de Broglie’s wavelengths comparable with lattice spacing, inelastic scatterings can reveal anomalies resolved as expressed like (10.7c). An intense thermal neutron source was available for such experiments performed at leading nuclear reactor facilities. In this section, we only discuss the principle of the scattering experiment with representative results. In Fig. 10.3 a triple axis spectrometer is sketched, consisting of monochrometer, goniometer, and analyzer, which are all rotatable individually around their axes installed in parallel. The monochrometer and analyzer are diffraction devices using crystals of known lattice spacing; thereby the neutron wavelength can be selected and measured by adjusting angles y and y0, respectively, as indicated in the figure. A sample crystal is mounted on the goniometer, which can be rotated around a symmetry axis to scan jKj in the vicinity of Ko jjG, as shown in Fig. 10.2. The wavevector and energy conservation laws can be applied to the impact between a neutron and a scatterer, i.e.,

10.2

Neutron Inelastic Scatterings

161

Fig. 10.2 Experimental arrangement for neutron inelastic scatterings. Incident beam Ko . In perpendicular directions K, only inelastic scatterings can be observed with no diffraction. The fluctuation spectrum can be recorded by scanning the scattering angle.

K o
K ¼ Gi  k

and eðKo Þ
eðKÞ ¼ DeðkÞ ¼ 
hDo;

(10.8)

where DeðkÞ represents an amount of energy transfer. Here, we have written the lattice vector as Gi due to distributed collisions, where the phase transition occurs near the Brillouin zone boundaries at  G2 . The zone-boundary transition is signified by the scatterer of two unit cells, and hence called a cell-doubling transition. Approaching the phase translation, the vector Gi is not exactly equal to  G2 , but shifting from the boundary by a perturbation attributed to Born–Huang’s principle. Expressing incident neutrons by the wave function C  exp iðKo :r o
oo to Þ, the scattered neutron from a scatterer at a lattice point r o can be described by the wavefunction ð d3 r o rðr o Þ

Cðr; tÞ  V ðr o Þ

exp ifK:ðr
r o Þ
oðt
to Þg jr
r o j

 exp ifðK
Ko Þ:r o
ðo
oo Þto g; where r o and to represent the space–time of the impact. In the direction perpendicular to Ko , we require the condition for no elastic scatterings, i.e., K ¼  k and Ko ¼
Gi , for which case Cðr; tÞ can be modified as

162

10

Experimental Studies on Critical Fluctuations

Fig. 10.3 Triple-axis spectrometer for neutron scattering experiments. The monochrometer, goniometer, and analyzer are all rotatable around their axes in parallel.

ð Cðr; tÞ 

d3 r o rðr o Þ V ðr o Þ

exp ifk:ðr
r o Þ
oðt
to Þg jr
r o j

 exp ifðGi  kÞ:r o  Do:to g: The structural factor can be defined for a finite area that includes many scatterers at r om as ð f ðGi  kÞ ¼

d3 r o rðr o Þ exp iðGi  kÞ:r o ; V ðr o Þ

and express the wavefunction, by assuming jr
r om j  r and tom  to , as exp ifk:r  oðt
to Þg CðGi  kÞ / r X  ff ðGi þ kÞ exp iðk:r om
Do:tom Þ þ f ðGi
kÞ exp ið
k:r om þ Do:tom Þg: m

10.2

Neutron Inelastic Scatterings

163

Writing nm ðGi  kÞ ¼ f ðGi  kÞ exp iðk:r om  Do:tom Þ , the intensity of scattered neutrons can be given by D

IðGi  kÞ ¼ jCðGi  kÞj ¼

* X

2

E t

/

jnm ðGi  kÞj

m

þ

* X

* X +

  ; n nm ðGi þ kÞnn ðGi þ kÞ þ nm ðGi
kÞnn ðGi
kÞ

m

+ t

2 t

  6¼ n nm ðGi þ kÞnn ðGi þ kÞ þ nm ðGi
kÞnn ðGi
kÞ

m

+ : t

The first term represents elastic scatterings at  k , whereas the second one indicates inelastic contributions. The latter can be expressed as * X exp iGi :ðr om
r on Þ h DI ðGi  kÞit / h

m6¼n

 j f ðGi þ kÞj2 exp ifk:ðr om
r on Þ
Doðtom
ton Þg 2

þ j f ðGi
kÞj exp i
k:ðr om
r on Þ þ Doðtom
ton Þg

+ i

: t

Considering that r om
r on ¼ r and tom
ton ¼ t are continuous space–time, the intensity anomaly hDI ðGi  kÞit is signified by phase fluctuations f ¼ k:r
Do:t and
f ¼
k:r þ Do:t. If these scattering factors f ðGi  kÞ can be approximated for a small k as jf ðGi  kÞj2 ¼ jf ðGi Þj2  2ik:rk jf ðGi Þj2 ; we can express the intensity anomaly as ðDI ðGi  kÞÞt ¼ Ahcos fit þ Phsin fit ;

(10.9)

where ð

ð 2

A/

j f ðGi Þj cosðGi :r Þd r V ðr Þ

3

and

2k:rk jf ðGi Þj2 cosðGi :r Þd3 r:

P/ Vðr Þ

We have considered that the fluctuations are pinned at the zone boundary. It is noted that the anomalies (10.9) are identical to (10.7c) for X-ray diffraction. Expression (10.9) indicates that the intensity anomaly is contributed by two modes of phase fluctuations hcos fit and hsin fit at intensities A and P. In Chapter 6, we have discussed such distributed intensities with respect to the phase f ¼ k:r þ fo ðto Þ where fo ðto Þ is an arbitrary phase depending on to, as shown in Fig. 6.3. Figure 10.4 [1] shows examples of anomalous intensities of scattered

164

10

Experimental Studies on Critical Fluctuations

Fig. 10.4 Typical intensity distributions observed from a magnetic MnF2 crystal at the Nee´l temperature T N , showing A and P absorption peaks. Here, the experimental result is quoted as an example of neutron inelastic scattering experiments, assuming that neutron–nucleus collisions constitute the scattering mechanism. For magnetic crystals such as MnF2 , we normally consider neutron–magnetic spin collisions, which are nonetheless unidentifiable in neutron experiments; the difference is however insignificant. From M.P. Schulhof, P. Heller, R. Nathans and A. Linz, Phys. Rev. B1, 2403 (1970).

neutrons, which were observed from magnetic MnF2 crystals near the critical Nee´l temperature T N . These anomalies observed in different scattering geometries show clearly anisotropic intensities composed of A and P modes. For observing such fluctuations, it is noted that the timescale to in the scattering process should be comparable to Do
1 for the energy transfer; otherwise the scattering anomalies vanish for to >Do
1 .

10.3

Light Scattering Experiments

Using intense coherent light waves from a laser oscillator, we can normally detect feeble inelastic scatterings in enhanced intensity from a crystal, permitting detailed studies of scattered photon spectra. Incident light induces dielectric fluctuations due

10.3

Light Scattering Experiments

165

to random impacts with constituents, producing the Rayleigh radiation that is attributed to elastic collisions, while inelastic scatterings also occur in detectable intensity, yielding information about structural changes in dielectric crystals. Capable to study inelastic scatterings, laser spectroscopy provides useful results for studying modulated structures.

10.3.1 Brillouin Scatterings When sound waves are standing in one dimension inside liquid, a monochromatic light beam incident to it in perpendicular direction generates a diffraction pattern that is characterized by modulated liquid densities. In dielectric crystals, induced dielectric fluctuations are responsible for similar phenomena, known as the Brillouin scatterings. A dielectric crystal exhibits such scatterings in the presence of sound waves, for which a photo-elastic mechanism is responsible. In this case, inelastic collisions between photons and constituent ions generate such Brillouin scatterings, yielding useful information about a modulated structure analyzable with Born–Huang’ principle. In Fig. 10.5a, we consider that an intense monochromatic light of wavevector Ko polarized along the y-direction is incident upon the (101) plane of a sample crystal, and that the scattered light of K is detected in the perpendicular direction ½
101 . In this case, phonons of wavevectors  Q should participate in the scattering process as shown in Fig. 10.5b, illustrating a scattering geometry for K, Ko , and  Q. It is noted that owing to the photoelastic properties determined by the lattice symmetry, the phonon waves are not necessarily longitudinal but transversal as well; consequently the scattered light K can be polarized in two directions v–v and h–h, as indicated in Fig. 10.5a. The symmetry relation between incident and scattered lights should be indicated by group operations; however we simply consider that the scattered components fluctuate in vertical and horizontal directions. Namely, we can write Dpv ¼ av E and Dph ¼ ah E, where av and ah are polarizabilities, where the amplitude of light is E. Denoting the polarized laser light by the electric field Eo exp iðKo :r o
oo to Þ at the impact position and time ðr o ; to Þ, the induced polarization is Dpðr o ; to Þ ¼ ðaÞ Eo exp iðKo :r o
oo to Þ, where ðaÞ represents a tensor with components av ; ah . As given in Sect. 10.1, the scattered field can be expressed for a distant point jr
r o j  r as ðð exp iðK:r
otÞ Dpðr o ; to Þ:Eo Eðr; tÞ  r  exp ifðKo
KÞ:r o
ðoo
oÞto gd3 r o dto :

(10.10)

If o ¼ oo, (10.9) is maximum at all wavevectors for jKj ¼ jKo j, representing elastic collisions known as the Rayleigh scatterings. Defining the scattering angle j by

166

10

Experimental Studies on Critical Fluctuations

Fig. 10.5 (a) A light scattering arrangement for Brillouin scatterings. Incident light is polarized in the y-direction as marked v. . .v, while scattered light is depolarized to v. . .v and h. . .h directions. (b) A general Brillouin scattering geometry with phonon excitations  Q.

Ko :r o ¼ 2pro cos j, where l is the wavelength, the Rayleigh intensity I R ðKÞ can be l expressed for r o 0, n ¼ 1, 2, 3, 4 correspond to the lowest spin–orbit coupling, whereas for l < 0, n ¼ 6, 7, 8, 9. The latter case can be rephrased by holes n0 ¼ 10 – n ¼ 4, 3, 2, 1 to be consistent with l > 0. If n ¼ 5, in particular, the lowest energy state is 6 S; hence HLS ¼ 0, signifying L ¼ 0 and S ¼ 12. Table 11.1 lists multiplet states of transition group ions and experimental values of l. It is noted that 3d electrons of transition ions orbit closely to the outer shell so that their orbital momentum is strongly quenched in a crystal field. On the other hand, the 4f shell of rare-earth ions is relatively deep inside ions so that the quenching of L is weak and hence insignificant. Table 11.2 shows multiplet states for rare-earth elements. Table 11.1 Incomplete 3d shells in transition group ions

Ion Ti3+ V3+ Cr3+ Mn2+ V2+ Cr2+ Fe3+, Mn2+ Fe2+ Co2+ Ni2+ Cu2+

n 1 2 3 4 3 4 5 6 7 8 9

State D 3 F 4 F 5 D 4 F 5 D 6 S 5 D 4 F 3 F 2 D 2

lcm1 154 104 87 85 55 57 0 100 180 335 828

11.2

Paramagnetism

193

Table 11.2 Spin–orbit coupling constants Ti3+ d1 2 D 154

Ion d-shell J-state l(cm1)

11.2

V3+ d2 3 F 104

Cr3+ d3 4 F 87

Mn3+ d4 5 D 85

V2+ d3 4 F 55

Cr2+ d4 5 D 57

Fe2+ d6 5 D 100

Co2+ d7 4 F 180

Ni2+ d8 3 F 335

Cu2+ d9 2 D 828

Paramagnetism

In a magnetic crystal, magnetic ions at lattice sites cannot be independent from each other, as weak magnetic dipole–dipole interactions should also be considered in principle. Nevertheless, assuming the latter as negligible at elevated temperatures, magnetic ions are considered in paramagnetic states, where their electronic energy splits into multiplet levels by an applied magnetic field B, namely, eðJ; MJ Þ ¼ gJ bBMJ ; MJ ¼ J; J  1; . . . ; J: These levels are populated thermodynamically as proportional to the Boltzmann factors expfeðJ; MJ Þ=kB T g: For the paramagnetic state of a crystal of N ions, the macroscopic magnetization can be calculated as þJ P

M ðB; T Þ ¼ N

J

g bM B ðgJ bMJ Þ exp J k TJ B   : þJ P g bM B exp  J k TJ J

þJ P

B

gJ bBMJ is the partition function of magnetic ions. Writing kB T J x¼ gJ bB=kB T for convenience,

Here, Z ¼

Z¼

þJ P

exp

exp xMJ ¼ expðxJ Þ þ    þ exp xJ ¼ J  sinh J þ 12 x ¼ : sinh 12x

expðxJ Þ  exp xðJ þ 1Þ 1  exp x

Therefore, @ ln Z @ ln Z @x ¼ NkB T M ðB; T Þ ¼ NkB T @B @x @B ( )  J þ 12 cosh J þ 12 x cosh 12x   ¼ gJ b ; 2 sinh 12x sinh J þ 12 x

194

11

Magnetic Crystals

or M ðB; T Þ ¼ gJ bJBJ ðxÞ;

(11.5a)

where BJ ðxÞ ¼

2J þ 1 ð2J þ 1Þx 1 x coth  coth : 2J 2J 2J 2J

(11.5b)

The function BJ ðxÞis known as the Brillouin function. Experimentally, (11.5ab) were found to explain measured results of M =M o versus B=T from some rare-earth crystals, in particular, as illustrated in Fig. 11.2; here, M o is the value of M at B ¼ 0. In the limit of x ! 0, we can show that BJ ðxÞ ffi Jþ1 3J x approximately, with which we can derive the Curie law. The response function of a paramagnetic crystal, called susceptibility, is defined as   M ðB; T Þ gJ bJB Ng2J b2 J ðJ þ 1Þ ¼ lim NgJ bJBJ wm ¼ : ¼ B!0 B kB T 3kB T This susceptibility represents the Curie law expressed as C Ng2J b2 J ðJ þ 1Þ w ¼ ; where C ¼ T 3k

(11.6a)

B

is called Curie’s constant; the law is correctly applicable in the limit of B ! 0. Experimentally, however, the formula wm ¼

C þ aJ T

and

C¼

Nb2 P 2 ; P 2 ¼ gJ J ð J þ 1Þ 3kB T

(11.6b)

can be used for data analysis, where aJ is constant and P is the effective Bohr magneton number. Table 11.2 lists rare-earth ions, showing their ground states, comparing observed values of P with those calculated by (11.6b). If x ! 1, on the other hand, we obtain BJ ðxÞ ! 1, indicating that all magnetic moments are aligned in parallel with B, which is called saturated magnetization.

11.3

Spin–Spin Correlations

Magnetic moments of constituent ions are primarily independent from the lattice, where the magnetic and the crystal symmetries coexist. Inversion symmetry mn !mn holds by reversing its direction of an applied field, that is, B ! B . Characterized by fast motion at elevated temperatures, the average hmn i vanishes in

11.3

Spin–Spin Correlations

195

Fig. 11.2 M z =M o versus B=T of paramagnetic ions Cr3þ ; F e3þ ; and Gd3þ .

the paramagnetic phase. However, if the motion is slowed down on lowering temperature, these mn are observed like classical vectors. We consider a potential DU n at the site n that emerges in the direction of magnetization to drive mn at and below the critical temperature T c , similar to binary pseudospins. Owing to the relation mn ¼ gS bSn, inversion symmetry Sn ! Sn holds by reversing B, where mn and Sn are both in precession around the direction of B. Signified by precession, magnetic moments should be in relative motion with respect to the ions, where the mutual interactions are attributed to electron exchange between mn . In early physics, mutual magnetic dipole–dipole interactions were considered as too weak for magnetic ordering, but, instead, Heisenberg proposed in 1929 that electron exchange between ions in short distance can be responsible for ferromagnetic interactions, referring to its correct order of magnitude. Following his theory, we consider the exchange interaction for two Fe3þ ions in iron and similar cases in other magnetic crystals, in analogy of the covalently bonded hydrogen molecule H2. Figure 11.3a, b show two 1s orbitals in H2 and two adjacent orbitals of dð3z2  r 2 Þ in an iron crystal, respectively, showing that the magnitude of overlapped wavefunctions is essential for magnetic interactions. Denoting interactingatoms by A and B, the wavefunctions of two electrons can be written as jA;B ðr i ÞwA;B si;j , where i ¼ 1; 2 and j ¼ þ12; 12. The total wavefunction can be expressed in determinant form to satisfy Pauli’s principle, namely,

196

11

Magnetic Crystals

Fig. 11.3 Electron exchange between two ions (a) two 1 s orbitals in H2 , (b) two 3d ð3z2  r 2 Þ orbital of Fe3þ , (c) one 3d orbital of magnetic ion and p orbital of a negative ion.



jA ðr 1 ÞwA ðs1 Þ jB ðr 1 ÞwB ðs1 Þ



; c/

jA ðr 2 ÞwA ðs2 Þ jB ðr 2 ÞwB ðs2 Þ

indicating that the sign of c changes by exchanging  1 electrons  1 between r 1 and  r12 . ¼ 1; w  ¼ 0 and of w þ2 Also noted is that combined spin functions of w þ 2 2  ¼ 0; w 12 ¼ 1 signify ms ¼ þ12 and ms ¼ 12 states, respectively, which are denoted by a and b in the following discussion. Considering four states caa ; cab ; cba and cbb as the basis functions, we calculate the matrix of Hamiltonian H for two electrons, including their kinetic energies and all Coulomb potentials. The diagonal element Haa , for instance, is written as Haa ¼

Xð ð

Caa  HCaa d3 r 1 d3 r 2

s1 ;s2

ðð

’A  ðr 1 Þ’B  ðr 2 ÞH’A ðr 1 Þ’B ðr 2 Þd3 r 1 d3 r 2 ðð  ’A  ðr 1 Þ’B  ðr 2 ÞH’B ðr 1 Þ’A ðr 2 Þd3 r 1 d3 r 2 ¼

11.3

Spin–Spin Correlations

197

where jA and jB are orthogonal functions for Haa . Writing these terms in Haa as K AB and  J AB , the interaction HðA; BÞ can be expressed by a determinant

K AB  J AB

0 HðA; BÞ ¼

0

0

0 K AB J AB 0

0 J AB K AB 0



0



0

:

0

K AB  J AB

(11.7a)

Here, J AB is referred to as the exchange integral. Transforming this determinant to a diagonal form, we obtain two eigenvalues: e3 ¼ K AB  J AB and e1 ¼ K AB þ J AB , whose eigenfunctions are Caa ; and

p1ffiffi 2



p1ffiffi 2



Cab þ Cba ;

Cab  Cba



for

Cbb

for

s1 þ s2 ¼ 1

s1 þ s2 ¼ 0;

(11.7b)

representing parallel and antiparallel spins, respectively. Referring to the total spin S ¼ s1 þ s2 ¼ 1 and 0, the first three functions and the second function in (11.7b) are referred to as triplet and singlet states, respectively, whose energy gap is given by 2jJ AB j. The exchange integral can be calculated as ðð J AB ¼

jA  ðr 1 ÞjB ðr 2 Þ

e2 j ðr 1 ÞjA ðr 2 Þd3 r 1 d3 r 2 ; jr 1  r 2 j B

(11.8)

which is numerically evaluated as positive for s orbitals in H2 and negative for p orbitals in O2. The sign of J AB determines if the ground state is singlet or triplet. For the ferromagnetic state of iron crystals, Heisenberg’s theory predicted that the exchange integral estimated in Fig. 11.3b is positive for parallel spins, giving rise to ferroelectric spin arrangement in crystals. Exchanging unpaired electrons between neighboring magnetic ions binds them covalently together with parallel spins, whereas negative J AB gives rise to antiparallel spins. Such an exchange can occur indirectly in other compounds via an ion inbetween. In crystals of perovskite KMF3, where the intervening ion M can be either one of Mn2þ ; Fe2þ ; Co2þ ; Ni2þ , exhibiting antiferromagnetism. These crystals show antiferromagnetic phases below critical temperatures T N , called the Nee´l temperature; Fig. 11.4 illustrates the arrangement of magnetic spins in these crystals. In KMnF3 , for example, below T N ¼ 88K , unpaired electrons are exchanged through Mn2þ . . . F . . . Mn2þ , as shown in Fig. 11.3c. In this case, the integral J AB was considered as negative for antiferromagnetic spin arrangement. Spin–spin correlations associated with exchanging unpaired electrons can be indicated by sAz sBz ¼ 14, referring the AB axis to as z, for the two spin states to be triplet and singlet. Nonetheless, this relation can be generalized to the scalar

198

11

Magnetic Crystals

Fig. 11.4 Structure of a KM F3 crystal. M represents Mn, Fe, Co, or Ni.

product sA :sB ¼ 14 of twoclassical vectors sA and sB, which is valid if both spins are in rapid precession, so sAx sBx þ sAy sBy ¼ 0 in the first order. Therefore, the exchange integral J AB can be replaced conveniently by 12 ð1  4sA :sB ÞJ AB , for which sA :sB ¼ 14 and  14 specify for triplet and singlet states, respectively. Hence, the correlation energy can be written as Hex ¼ 2J AB sA :sB

(11.9)

excluding the constant term. This formula (11.9) for spin–spin correlations is identical to previous one (5.10) for the nonmagnetic correlation, except for the factor 2. Noted from (11.8), the integral J AB is a function of distance jr A  r B j and highly directional in crystal space via an intervening ion, if any, between magnetic ions. The latter indirect exchange is called superexchange. For thermodynamics, the sign of J AB in any case is significant to determine the spin–spin coupling to lower energy by Hex .

11.4

Spin Clusters and Magnetic Symmetry

Evidenced by easy magnetization phenomena, no appreciable adiabatic potentials can be expected in magnetic crystals for modulating mn, unless an external magnetic field B is applied in parallel to the easy axis. We therefore consider that B should be applied in the direction of magnetization axis; otherwise, magnetic symmetry is independent of crystalline symmetry. In this context, such an easy direction can be identified as an axis for electron exchanges in the lattice, which were substantiated by neutron diffraction experiments. In an axis for electron exchanges, collective motion of mn occurs as described by Fourier series, showing their propagation in the periodic lattice. In the condensate model, it is a valid assumption that such a propagation is driven by an adiabatic potential that synchronize collective mn in this direction. Considering the moment mn as relative displacements from the site n, the collective mn should be in phase

11.4

Spin Clusters and Magnetic Symmetry

199

with sinusoidal lattice excitations un . Disregarding uncertainties in the transition region, such a collective mn mode can be written as mn ¼ mo exp iðq:r n  otn Þ or directly by the Fourier amplitude mo ¼ mn expfiðq; r n  otn Þg: Including space–time uncertainties Dr n and Dtn at sites n, it is convenient to express these modes by the phase variables fn ¼ k:r n  otn with uncertainties D fn ¼ k:Dr n  oDtn . In an applied field B, the magnetic moments mn are in precession around its direction, where the angles of precession are considered to synchronize with lattice translation, as sketched in Fig. 11.5. Writing these moments as classical vectors mm ¼ mem and mn ¼ men , where jem j ¼ jen j ¼ 1 for unit vectors, the short-range interaction energy at the site n is given by Hn ¼ 

X

2J mn mm :mn ¼ m2

m

X

2J mn em :en ;

(11.10)

m

where J mn is an exchange integral between m and n sites. Using this for short-range correlations for a magnetic cluster, the summation over m covers the nearest-neighbor sites. Assuming m as constant, we write for these unit vectors to be real ones, that is, em ¼ eþq exp ifm þ eq expðifm Þ and

en ¼ eþq exp ifn þ eq expðifn Þ;

where fm ¼ q:r m  otm and fn ¼ q:r n  otn are phases at sites m and n, respectively. In Chap. 5, we showed that the average of Hn over the timescale of observation to is given by hHn it ¼ 2m2 Gt eþq :eq J ðqÞ;

(11.11a)

where J ð qÞ ¼

X

J mn exp iq:ðr m  r n Þ

(11.11b)

m

and 1 Gt ¼ 2to

þt ðo

expfiðtm  tn Þgdðtm  tn Þ ¼ to

sin oto : oto

(11.11c)

200

11

Magnetic Crystals

Fig. 11.5 One-dimensional array of spins in Larmor precession in applied magnetic field B k k.

Equations (11.11a) and (11.11b) represent such a cluster centered at mn with limited number of interacting neighbors, which can be considered as a seed for magnetic condensation. For J ðqÞ, however, we consider electron exchanges only with nearest neighbors to keep local symmetry unchanged, as next-nearest-neighbor interactions J 0 disrupt local symmetry. In the presence of a magnetic field B, we should have collective motion of magnetic moments mn in propagation mode. Minimizing JðqÞ with respect to q, we can determine the axis of propagation k for B ¼ 0, which should be the same axis for minimum strains in the lattice. If B 6¼ 0, on the other hand, the collective motion is described in terms of phases fn ; all mn waves are synchronized with Dfn ¼ 0 to ensure stability of the crystal. This process can be observed as thermal relaxation for Bkk. Further, such a mode of mn below T c is commensurate with the lattice, as signified by either k ¼ 0 or k ¼ G=2, corresponding to ferromagnetic or antiferromagnetic arrangements.

11.5

Magnetic Weiss Field

Implied by Larmor precession P of mn around an applied field B, the magnetic moment of a cluster mn ¼ mi can propagate as mn exp ifn in the direction i;cluster

of B. The space–time reference for mn can be reset for motion in this direction by replacing fn by a continuous phase angle in the range 0  f