Spectroscopic Properties of Rare Earths in Optical Materials (Springer Series in Materials Science, Volume 83)

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Spectroscopic Properties of Rare Earths in Optical Materials (Springer Series in Materials Science, Volume 83)

Springer Series in MATERIALS SCIENCE 83 Springer Series in MATERIALS SCIENCE Editors: R. Hull R. M. Osgood, Jr. J

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Springer Series in



Springer Series in


R. M. Osgood, Jr.

J. Paris!

H. Warlimont

The Springer Series in Materials Science covers the complete spectrum of materials physics, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series reflect the state-of-the-art in understanding and controlling the structure and properties of all important classes of materials. 62 Epitaxy Physical Principles and Technical Implementation By M.A. Herman, W. Richter, and H. Sitter

72 Predictive Simulation of Semiconductor Processing Status and Challenges Editors: J. Dabrowski and E.R. Weber

63 Fundamentals of Ion-Irradiated Polymers By D. Fink

73 Sic Power Materials Devices and AppHcations Editor: Z.C. Feng

64 Morphology Control of Materials and Nanoparticles Advanced Materials Processing and Characterization Editors: Y. Waseda and A. Muramatsu

74 Plastic Deformation in Nanocrystalline Materials By M.Yu. Gutkin and LA. Ovid'ko

65 Transport Processes in Ion-Irradiated Polymers By D. Fink 66 Multiphased Ceramic Materials Processing and Potential Editors: W.-H. Tuan and J.-K. Guo 67 Nondestructive Materials Characterization With Applications to Aerospace Materials Editors: N.G.H. Meyendorf, PB. Nagy, and S.I. Rokhlin 68 Diffraction Analysis of the Microstructure of Materials Editors: E.J. Mittemeijer and P. Scardi 69 Chemical-Mechanical Planarization of Semiconductor Materials Editor: M.R. Oliver 70 Applications of the Isotopic Effect in Solids ByV.G. Plekhanov 71 Dissipative Phenomena in Condensed Matter Some Applications By S. Dattagupta and S. Puri

75 Wafer Bonding AppHcations and Technology Editors: M. Alexe and U. Gosele 76 Spirally Anisotropic Composites By G.E. Freger, V.N. Kestelman, and D.G. Freger 77 Impurities Confined in Quantum Structures By PO. Holtz and Q.X. Zhao 78 Macromolecular Nanostructured Materials Editors: N. Ueyama and A. Harada 79 Magnetism and Structure in Functional Materials Editors: A. Planes, L. Manosa, and A. Saxena 80 Ion Implantation and Synthesis of Materials By M. Nastasi and J.W. Mayer 81 Metallopolymer Nanocomposites By A.D. Pomogailo and V.N. Kestelman 82 Plastics for Corrosion Inhibition By V.A. Goldade, L.S. Pinchuk, A.V. Makarevich and V.N. Kestelman 83 Spectroscopic Properties of Rare Earths in Optical Materials Editors: G. Liu and B. Jacquier

Volumes 10-61 are listed at the end of the book.

Guokui Liu

Bernard Jacquier


Spectroscopic Properties of Rare Earths in Optical Materials With 194 Figures and 61 Tables


Dr. Guokui Liu

Professor Bernard Jacquier

Chemistry Division Argonne National Laboratory Argonne, IL 60439 USA E-mail: [email protected]

Universite Lyoni, CNRS 10 rue Andr^ Marie Ampere 69622 Villeurbanne France E-mail: [email protected]

Series Editors: Professor Robert Hull

Professor Jiirgen Parisi

University of Virginia Dept. of Materials Science and Engineering Thornton Hall Charlottesville, VA 22903-2442, USA

Universitat Oldenburg, Fachbereich Physik Abt. Energie- und Halbleiterforschung Carl-von-Ossietzky-Strafie 9-11 26129 Oldenburg, Germany

Professor R. M. Osgood, Jr.

Professor Hans Warlimont

Microelectronics Science Laboratory Department of Electrical Engineering Columbia University Seeley W. Mudd Building New York, NY 10027, USA

Institut fiir Festkorperund Werkstofforschung, Helmholtzstrafie 20 01069 Dresden, Germany

ISSN 0933-033X ISBN 3-540-23886-7 Springer Berlin Heidelberg New York ISBN 7-302-07409-7 Tsinghua University Press Library of Congress Control Number: 2004114853 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, and the Chinese Copyright Law of September 7,1999, in their current versions, and permission for use must always be obtained from Springer and Tsinghua University Press. Violations are liable to prosecution under the German and Chinese Copyright Law. Springer is a part of Springer Science+Business Media. springeronline.com © Tsinghua University Press and Springer-Verlag Berlin Heidelberg 2005 Printed in P.R. China The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover concept: eStudio Calamar Steinen Cover production: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 10926611



Spectroscopic Properties of Rare Earths in Optical Materials)


Decades of scientific research on spectroscopic properties of f-shell electrons has spawned an extensive array of applications for rare-earth activated luminescent and laser materials. From phosphors activated by Eu^^ , Eu^^ and Tb^^ ions for lighting and display to crystals and glasses doped with Er^^ and Yb^* for infirared-to-visible up-conversion, rare-earths represent a large share of the lighting materials market. Currently, Nd^^ doped crystals such as Nd^"^: YAG (Y3AI5O12) and Nd^^ : YVO4 are the dominant laser media for high power and compact solid-state lasers, while Er^^ doped phosphate and siUcate glasses prevalent in optical fiber amplifiers and microlasers used in optical teleconmiunications. Thanks also to materials science development, rare-earths will receive even wider use in the future, entering in the new world of nanotechnologies for instance. In addition to rare-earth activated phosphors and solid-state lasers, optical applications of rare-earths are already an integral part of internet and data conununications and are expected to be applied to information storage and processing, and organic and biological devices. Since the 1960s, several classic books on spectroscopy of rare-earth ions in solids have been published, including those by Wyboume (1965) and Hiifner (1978). More recent advances in spectroscopic theory and laser experiments involving rare-earth ions in solids were also reviewed, for example, in the book edited by Kaplyanskii and Macfarlane ( 1 9 8 7 ) . However, in the f-elements research community, we feel the need of a book that updates the information of recent progresses in the field and facilitates understanding and applications of the principles and concepts that are required in rare-earth optical materials characterization and development. The goal of this book is to provide a connection between basic research and materials science through analysis of fundamental spectroscopic properties of rare-earth activated luminescent and laser optical materials. In addition, this book will serve as an updated reference for materials research by covering a number of currently active topics in the field of rare-earth photo-physics and photo-chemistry. Fundamental topics of optical V

spectroscopy are addressed with an emphasis on the physical interactions that determine die primary optical properties, including energy level schemes, transitions intensities, line broadening, mechanisms of non-radiative relaxation and energy transfer. Topics of appUed research are selected from recent advances in concepts and techniques that created significant opportunities for present and future applications of rare-earth optical materials. An international collaboration, which includes contributions from authors with both experimental and theoretical expertise, enables us to offer the reader a systematic review of fundamental aspects and to provide a wide coverage on new applications that utilize the electronic transitions of rare-earth ions in solid-state materials. From free-ion and crystal-field energy level structures to transition intensities and line broadening induced by ion-ion and ion-phonon interactions, the first four chapters survey the fundamentals of felement photo-physics and spectroscopy, and provide a theoretical framework for the subjects that are discussed in the rest of the book. From Chapter 5 onwards, each chapter is devoted to a particular area of importance in new materials characterization or technology development. From up-conversion phenomena, to materials requirements for frequency-domain and time-domain optical memory, to current progress in rare-earth laser materials and phosphors, the concepts and principles discussed in this book were taken directly from the forefront of research in rare-earth optical materials. Moreover, illustration of current progress in fundamental aspects of quantum confinement and quantum electrodynamics is also discussed. To make it easier to read, and also to avoid cluttering up Chapters 1, 2 and 3 , the more theoretical derivations of these chapters are given in Appendices A and B. We hope that this book will provide useful information to researchers and students in the field of f-element spectroscopy and materials development. Although chapters in this book are written independently by individual authors, significant efforts have been made to achieve a coherent connection and systematic balance for the book as a whole. We are grateful to all authors of this book for their excellent contributions. Guokui Liu Argonne, Illinois, USA B. Jacquier Lyon, France




Electronic Energy Level Structure 1.1 Introduction 1.2 Electronic States and Coupling Schemes 1.2.1 Central Field Approximation 1.2.2 LS Coupling and Intermediate Coupling 1. 3 Free-ion Interactions 1.3.1 Coulomb Interaction 1.3.2 Spin-orbit Interaction 1.3.3 Corrections to Free-ion Hamiltonian 1.3.4 Reduced Matrices and Free-ion State Representation 1.3.5 Parameterization of the Free-ion Interactions 1.3.6 Energy Levels of 4f^ Configurations and Binding Energies Relative to Host Band 1. 4 Crystal-field Interaction 1.4.1 Crystal-field Hamiltonian and Matrix Element Evaluation 1.4.2 Symmetry Rules 1.4.3 Empirical Evaluation of Crystal-field Parameters 1.4.4 Theoretical Evaluation of Crystal-field Parameters 1.4.5 Corrections to the Crystal-field Hamiltonian 1. 5 Analysis of Crystal-field Spectra 1.5.1 Experimental Data 1.5.2 Computational Modeling 1.6 Modeling o f 4 f ' ' - 4 f ' ' " ' 5 d Spectra • 1.6.1 Energy Gaps Between the 4f^ and Excited Configurations 1.6.2 Hamiltonian for 4f^"^ 5d Configurations 1.6.3 Determination of Hamiltonian Parameters 1.7 Magnetic and Hyperfine Interactions 1.7.1 Zeeman Effect 1.7.2 Magnetic Hyperfine Interaction 1.7.3 Nuclear Electric Quadrupole Interaction 1.7.4 lon-ligand Hyperfine Interaction 1.7.5 Pr'^ 1.7.6 Eu'^

1 1 4 '4 7 11 11 14 15 17 18 23 26 28 32 37 39 46 50 51 54 57 57 59 60 64 65 67 69 72 73 76 VII

1.7.7 1.7.8 References

Tb'^ Other Ions

85 88 89

Transition Intensities 2 . 1 Introduction 2.2 Basic Equations 2 . 2 . 1 Electric and Magnetic Dipole Operators 2 . 2 . 2 Polarization Selection Rules 2 . 2 . 3 Vibronic Transitions 2 . 3 One Photon Transitions Within the 4f^ Configuration 2 . 3 . 1 General Theory 2 . 3 . 2 Transitions Between Crystal-field Levels 2 . 3 . 3 Transitions Between/Multiplets 2 . 3 . 4 Selection Rules 2 . 3 . 5 Vibronic Transitions 2 . 3 . 6 Circular Dichroism 2 . 3 . 7 Parameter Fits 2 . 3 . 8 Comparison with First-principles Calculations 2 . 3 . 9 Extensions to the Models 2.4 Higher-energy Transitions 2 . 4 . 1 4f''^ 4 f ' ' - ' 5 d Transitions 2 . 4 . 2 Charge Transfer 2 . 5 Two-photon Processes 2 . 5 . 1 f"^^ f^ Transitions 2 . 5 . 2 4f''^ 4 f ' ' - ' 5 d Transitions 2 . 6 Conclusions References

95 95 96 96 98 100 102 103 104 108 109 109 110 Ill 117 121 122 122 123 124 124 126 126 126

lon-phonon Interactions 3.1 Introduction 3.2 Basic Concepts: Hamiltonian of the lon-phonon Interaction, Adiabatic and Nonadiabatic Terms 3.3 Coupling Constants 3.4 Density Matrix Formalism in the Quantum Theory of Relaxation • 3.5 Thermal Shifts of Zero-phonon Line Positions 3.6 Non-radiative Transitions Between Crystal Field Energy Levels 3 . 6 . 1 One- and Two-phonon Transition Rates 3 . 6 . 2 Phonon Contributions to Linewidths 3 . 6 . 3 Multiphonon Relaxation

130 130


131 139 145 151 153 153 155 158

3.7 3.8

Vibrational Structure of the Optical Spectra Simulations of the lon-phonon Interaction Effects 3.8.1

166 170

Impurity Rare Earth Single Ion Centers in the LiYF4 Crystal



Relaxation Broadening of Optical Transitions in Pr^"^ Duner Centers in CsCdBrg 3.9 Conclusions References 4


Line Broadening Mechanisms and Their Measurement 4.1 Introduction 4.2 Inhomogeneous Broadening • 4.2.1 Zero-phonon Linewidths of 4 f - ^ f and 4f->5d Transitions 4.2.2 Line Broadening from Defects and Disorder 4.2.3 Site-selective Spectroscopy 4.2.4 Ultranarrow Inhomogeneous Linewidths 4.3 Non-inhomogeneous Sources of Line Structure 4.3.1 Hyperfine Structure and Isotope Shifts 4.3.2 Vibronic Sidebands 4.4 Homogeneous Broadening 4.4.1 lon-phonon Interactions 4.4.2 Ion-ion Interactions 4.4.3 Ion-nuclear Spin-spin Interactions 4.5 Measurement of Line Broadening and Examples 4.5.1 Fluorescence Line-narrowing 4.5.2 Spectral Hole Burning 4.5.3 Coherent Transient Spectroscopy 4.6 Disordered, Low-dimensional and Nanostructure Crystalline Materials 4.6.1 Disordered Materials 4.6.2 Low Dimensional Systems 4.6.3 Nanocrystalline Materials 4.7 Conclusions References Up-conversion in RE-doped Solids 5.1 Introduction and Historical Background 5.2 Energy Transfers Between RE Ions; Role of Energy Diffusion in Up- and Down-conversion • 5.2.1 Recall of Basics of Energy Transfer with

182 185 186

191 191 194 194 195 197 201 202 202 203 204 205 208 213 214 215 218 227 238 239 246 251 257 257 266 267 268 IX

Activator in Its Ground State Up-con version Processes by Sequential Energy Transfers; Comparison with ESA and Typical Examples 5. 3 Up-conversion in Single-ion and Pair-level Level Description; Theoretical and Experimental Discrimination 5. 3.1 Application of Cooperative Luminescence; Theory and Examples 5.3.2 Some Experimental Results for APTE Effect and Their Implications in Various Field 5.4 Cross-relaxation and the Photon Avalanche Effect 5.4.1 The Avalanche Process as a Positive Feedback System 5.4.2 Conditions for Observing an Avalanche Threshold 5 . 4 . 3 Er^ ^ : LiYF4 as an Avalanche Model Experiment 5 . 4 . 4 Photon Avalanche in Er^^ -doped Fluoride Glasses in Fibre and Bulk Shape 5 . 4 . 5 Avalanche in Co-doped Systems 5.4.6 Up-conversion Laser with Multiphonon-assisted Pumping Scheme and Photon Avalanche • 5. 5 Perspectives and Future Advances 5 . 5 . 1 Up-conversion UV Tunable Lasers 5.5.2 New Materials for Low-intensity IR Imaging 5 . 5 . 3 Up-conversion Material Intrinsic Bistability 5.6 Conclusions References


Current Topics in Rare-earth Lasers 6.1 Introduction 6.2 Spectroscopic and Laser Parameters 6 . 2 . 1 Basic Laser Parameters 6 . 2 . 2 Determination of Absorption Cross Sections 6 . 2 . 3 Determination of Emission Cross Sections 6. 2.4 Determination of Radiative Lifetimes and Branching Ratios 6. 3 UV-visible Laser Sources 6. 3.1 Compact and Tunable UV Lasers Based on Ce^"^ Doped Crystals: Prospects with Other Ions 6.3.2 Lasers Based on Nd^^ and Yb^^ Doped Nonlinear Crystals 6.4 Near-and Mid-infrared Laser Sources 6 . 4 . 1 High-power and Ultrafast Lasers Based on Yb^"^ Doped Materials

320 320 321 322 328 329



276 279 284 286 296 298 300 302 306 309 310 311 311 312 312 314 315

336 340 340 345 349 349


Rare-earth Doped Crystals for Telecommunications and Eyesafe Laser Applications 6 . 4 . 3 Low-frequency Phonon Materials for Mid-infrared Lasers 6.5 Conclusions Appendix • 6. A Laser Threshold Condition 6.B Minimimi Fraction of Excited Population jS^ 6 . C Energy Transfer Rates 6.D Einstein Coefficients 6 . E List of Acronyms References Rare Earth Materials in Optical Storage and Data Processing Applications 7.1 Introduction 7 . 1 . 1 Equivalence of Holebuming and Photon Echoes in Storage and Signal Processing AppUcations 7.1 2 Material Parameters for Optical Data Storage 7.1 3 Dephasing and Spectral Diffusion 7.2 Eu^ Materials 7.2 .1 Properties of YjSiOj 7.2 .2 Eu^ :Y,Si05 7.2 .3 Other Experiments in Eu' rY.SiOs 7.2.4 Eu'^Y^Os 7 . 2 . 5 Eu'^-YAIO, 7.3 Pr'* Materials 7 . 3 . 1 Pr^ Y^SiO^ 7.3.2 Pr' YAIO, 7 . 3 . 3 Pr' Y3AI3O1, 7 . 3 . 4 Pr' Lap, 7.4 Tm'"^ Materials 7 . 4 . 1 Tm*^ Y3AI5O,, 7.. 4 . 2 Tm'^ LU3AI5O12 7.. 4 . 3 Tm^" Y1.5LU1.5AI5O12 7.. 4 . 4 Tm'* YaGajO.^ 7.. 4 . 5 Tm'" LaF3 7.. 4 . 6 Tm^^ Y^SiOs •• 7.. 4 . 7 Tm^^ Y^Si^O, •• 7 . 4 . 8 Tm'* Y2O3 7 . 4 . 9 Tm'* YAIO3 7.4.10 Tm' Materials Summary

355 364 369 370 370 371 371 372 372 373

379 379 381 382 382 •• 384 385 386 391 391 393 394 394 397 398 399 399 401 404 405 406 407 407 408 409 409 410 XI


Er"^ Materials 7.5.1 Properties of E/^'rYaSiOs 7.5.2 Properties of Er'^iYgOa 7.5.3 Properties of Er^'^iLiNbOg 7.5.4 Properties of Er^'^iYAlOg 7.5.5 Properties of Er'"^ lY^Al^O,^ 7.5.6 Properties of Er^"^ Doped Tungstates 7.5.7 Er^^ Materials Summary 7.6 Other Materials 7.6.1 Eu'^ Materials 7.6.2 Deuterated Fluorides 7.6.3 Eu^" Persistence 7.6.4 Nd'^ Systems 7.7 Conclusions References Rare Earth Doped Confined Structures for Lasers and Amplifiers 8.1 Introduction 8.2 Propagation and Amplification: the Key Parameters 8.2.1 Opto-geometrical Parameters 8.2.2 Spectroscopic Parameters 8.2.3 Amplification into a Waveguide 8.2.4 Material Requirements for Fabrication of Waveguide 8.2.5 Other Specific Properties: Photosensitivity and Photorefractivity 8.3 Waveguide Amplifiers and Lasers 8.3.1 Erbium-doped Fiber Amplifier 8.3.2 Praseodymium-doped Fiber Amplifier 8.3.3 Thulium-doped Fiber AmpUfier 8.3.4 Fiber Lasers 8.3.5 Optical Integrated Amplifiers and Lasers 8.4 Optical Microcavities and Nanoconfinement 8.4.1 Optical Confinement 8.4.2 Experimental Evidences 8.4.3 Various Devices 8.5 Conclusions References Rare Earth Luminescent Centers in Organic and Biochemical Compounds 9.1 Introduction XII

411 412 416 416 417 419 420 420 421 422 423 424 424 426 426 430 430 432 432 433 438 441 442 443 444 449 450 452 453 454 455 456 457 458 458 462 462


Sensitizing the Luminescence of Trivalent Lanthanide Ions 9 . 2 . 1 Establishing the Importance of the Triplet State 9 . 2 . 2 Mechanisms of Energy Transfer 9.3 Preventing Nonradiative Deactivation of the Metal Ion Luminescent States 9 . 3 . 1 Vibrational Deactivation Processes 9 . 3 . 2 Electronic Deactivation Processes 9.4 Designing a Luminescent Probe 9 . 4 . 1 Qualitative Rules 9 . 4 . 2 Quantitative Estimates 9.5 Luminescent Lanthanide Complexes with Organic Ligands 9.6 AppUcations in Biomedical Analyses 9 . 6 . 1 Fluoroimmunoassays 9.6.2 Responsive Systems 9.7 Conclusions Appendix 9. A Chemical Formulae of Compounds Cited in the Text 9. B Glossary and Chemical Formulae References


Rare Earth Ions in Advanced X-ray Imaging Materials 10.1 X-ray Phosphors 10.2 X-ray Phosphors Used for Intensifying Screens • • • • 10.2.1 Calcium Tungstate 10.2.2 Rare Earth Tantalate Based Phosphors 10.2.3 Europium Activated Barium Fluoro-chloride Phosphors 10.2.4 Tb- and Tm-activated Lanthanum Oxybromides 10.2.5 Tb^^-activatedGadoUniumOxysulfide 10.3 X-ray Storage Phosphors and Their AppUcations 10.3.1 Physical Mechanism of Photostimulated Luminescence 10.4 X-ray Phosphors for Computed Tomography 10.4.1 Scintillators for X-ray Computed Tomography 10. 5 Scintillators for Electromagnetic Calorimetric Detection 10.5.1 Cerium Fluoride

464 464 467 469 469 472 476 476 477 481 486 486 489 491 491 491 496 497 500 500 502 503 504 509 511 513 514 515 518 521 524 525 XIII

10.5.2 Ce^^-activated Gd2Si05 and Lu2Si05 10.6 Conclusions References Appendix A Effective Operator Calculations A. 1 Effective Hamiltonians and Effective Operators A. 2 Perturbation Expansions A. 3 Symmetries and Selection Rules A. 4 Implications References

526 527 527 530 530 531 534 535 536

Appendix B Matrix Elements of Tensor Operators B. 1 Angular Momentum States and Operators B. 2 Clebsh-Gordan Coefficients and 3-j Symbols B. 3 Tensor Operators and the Wigner-Eckart Theorem B. 4 More Complex Situations References

537 537 537 538 539 540

Keywords Index


Materials Index



List of Contributors

Editors: Guokui Liu

Chemistry Division, Argonne National Laboratory, Argonne, lUinois 60439, USA Tel. 630 262 4630, Fax 630 252 4225 E-mail: [email protected]

B. Jacquier

Universite Claude Bernard Lyon 1, 43 bd du 11 Novembre 1918, Villeurbanne F-69622, France Tel. 33-(0)4 72 44 83 36, Fax 33-(0)4 72 43 11 30 E-mail:Jacquier@pcml. univ-lyonl. fr

Chapter Authors: Guokui Liu (Chapter 1) M. F. Reid

Department of Physics & Astronomy, University of

(Chapter 2)

Cantebury, Christchurch 8020, New Zealand Tel. 64 3 364 2548, Fax 64 3 364 2469 E-mail: m. reid@phys. canterbury, ac. nz

B. Z. Malkin

Physics Department, Kazan State University, 420008


Kazan, Russia Federation E-mail: [email protected]

R. S. Meltzer

Department of Physics and Astronomy,

(Chapter 4)

University of Georgia, Athens, GA 30602, USA Tel. 706 542 5515, Fax 706 542 2492 E-mail: rmeltzer@hal. physast. uga. edu


F. Auzel

Centre National de la Recherche Scientifique, GOTR,


UMR7574, France E-mail: [email protected]

R. Moncorge


(Chapter 6)

Universite de Caen, 6 Boulevard Marechal Juin, 14050 Caen, France E-mail: [email protected]

Y. C. Sun

Physics Department,


Bozeman, Montana 59717, USA




Tel. 406 994 6163, Fax 406 994 4452 E-mail: [email protected] B. Jacquier

B. Jacquier, L. Bigot, S.Guy, A. M. Jurdyc


University Claude Bernard Lyon 1 Villeurbanne F-69622, France

J. -C. G. Biinzli

Chemistry Department, University of Lausanne, BCH

(Chapter 9)

1402, Lausanne, CH-1015, Switzerland Tel. 41 21 692 3821, Fax 41 21 692 3825 E-mail: Jean-claude. bunzli@icma. unil. ch

M. Z. Su

College of Chemistry and Molecular Engineering,

(Chapter 10)

Peking University, Beijing, 100871, China Tel. 86 10 6275 1715, Fax 86 10 6275 1708 E-mail: [email protected]

W. Zhao

Department of Chemistry, University of

(Chapter 10)

Arkansas, 2801 South University Ave. Little Rock, AR 72204, USA E-mail: [email protected]



Electronic Energy Level Structure

Guokui Liu



From fluorescent lamps, solid-state lasers, to optical amplifiers in fiber optics, rare earth (RE) elements, or lanthanides, have been widely used to activate luminescent and photonic materials. A majority of applications involve electronic transitions between states within a 4f^configuration of trivalent (or divalent) RE ions doped into transparent host materials. The popular solid-state laser of Nd^^: YAG, for example, utiUzes the 1.05-jxm electronic transition from the "^Fg/g to ^Iii/2 multiplets within the 4f^configuration of the Nd^^ ion (see Chapter 6). Similarly, the Er^^ fluorescence at 1.5-|xm, emitted from the first excited multiplet '*Ii3/2 of the 4f^^ configuration, is important because of its high quantum efficiency and optimal wavelength for optical amplification in telecommunications. Green fluorescence of Er^^ from the "^83/2 state can be induced through upconversion excitation from several states at lower energies (see Chapter 5). These applications utilize the unique properties of the 4f electrons that have localized states and exhibit weak coupling to Ugand electrons and lattice vibrations. The 4f spectroscopic properties, including the energy level structure and the dynamics of the electronic transitions of RE ions in solids, thus primarily define the optical properties of an RE activated device. With energy levels at more than 30,000 cm"^ above the ground states of the 4f^ configurations, there are 5d, 6s, 6p orbitals in the RE ion electronic structure. The 5d states exhibit less locaUzed nature and stronger coupling to lattice vibrations, and since the inter-configuration 4f^ to 4f^"^5d transitions of the RE ions are parity-allowed, they have intensities up to 10,000 times stronger than the strongest 4f^ to 4f^ transitions (see Chapter 2). Due to these electronic properties, 4f^ to 4f^"^ 5d transitions have become increasingly important in recent years for applications in fast scintillators and ultraviolet laser sources. A fundamental understanding of the electronic energy levels of RE ions in the 4f^ and 4f^"^ 5d configurations is essential not only in spectroscopy but also in materials characterization. What appears to be unique in solid-state RE spectroscopy is that the electronic energy level structure is estabUshed primarily using the quantum theory of atomic spectroscopy, and all collective solid-state effects can be treated as a perturbation known as the crystal-field interaction (Stevens, 1952; Wyboume, 1965; Newman and Ng, 2000). Such a simple approximation works very well

Guokui Liu for RE ions in a 4f^ configuration in which the electrons in the partially occupied 4f shell are shielded by the electrons in the 5s and 5p shells from interacting with the ligands and, therefore, have Httle participation in chemical bond formation (Reisfeld and Jc|)rgensen, 1977). Figure 1.1 shows, with Nd^'^as an example, the radial distribution of electrons in different shells. It is clear that most of the 4f orbitals are much less extended than the 5 s and 5p orbitals. As a result, the electronic transitions between the 4f states are very sharp and have atomic-like spectral characteristics. It is based on the localized nature of the electronic properties that a general theoretical framework of solidstate RE spectroscopy was developed (Stevens, 1952; Judd, 1963a; Wyboume, 1965; Dieke, 1968).



Figure 1.1 Radial wave function /if/^ as a function of r in atomic radius for 4f^ electrons of Nd^ ^ in comparison with the charge distribution of its core (Xe) configurations

When a 4f electron is excited into a 5d orbital that extends beyond the 5s and 5p orbitals, the spectroscopic properties of RE ions in an electronic configuration such as 4f^"^ 5d are influenced more strongly by the lattice. Therefore, the electronic transitions between the 4f^ and 4f^~^5d states, through absorption or emission of photons, are expected to be characteristically very different from transitions within the 4f^ configuration. A modification of the crystal-field theory is necessary for modeling stronger ion-lattice interactions in analysis of the energy level splitting and the excited state dynamics. With restrictions on localized electronic interactions in a well-defined crystalline environment, the overall spectroscopic properties of an RE ion may be determined by evaluating interactions in various forms of mechanisms. Table 1.1 lists the scales of electronic energy levels in terms of different mechanisms of electronic interactions. Accordingly, theoretical analyses of the electronic interactions and their contribution to energy level splitting will be discussed in the

1 Electronic Energy Level Structure order of energy level scales. Historically, the development of a complete Hamiltonian for 4f^ configurations was approached in two stages. The first dealt with the fundamental electronic interactions, including electrostatic Coulomb interactions and spin-orbit coupling. The second dealt with the crystal field interaction that arises when the ion is in a condensed phase. Subsequently, additional effective operators dealing with higher order free-ion interactions and corrective crystal-filed interactions were introduced to reproduce more accurately the energy level structures observed in experiments. The theoretical framework, thus utiUzes well-estabUshed theories in two conventional fields: (l)the quantum theory of atomic spectroscopy that is the foundation for establishing free-ion energy level structures; (2) the point group theory that facilitates the determination of crystal-field splittings according to the symmetry properties of a crystalUne lattice. As Usted in Table 1.1, the scale of crystal-field splitting is smaller than that of the free-ion splitting, while the hyperfine energy level structures are even smaller, thus ensuring the legitimate application of perturbation theory to calculations of crystal-field spUtting and hyperfine structures. In the framework of crystal-field modeling, electronic energy levels are calculated by diagonalizing an effectiveoperator Hamiltonian with the basis free-ion wave functions, and the parameters of the effective operators are determined by fitting the experimentally observed energy levels to the calculated ones. As an en5)irical approach to modeUng the energy level structure of RE ions in soUds, crystal-field theory was very successful not only in predicting the exact number of crystal-field levels for an RE ion in a given host material, but also in accurately determining their energies. Table 1.1

Energy level scales of rare earth ions in crystals

Interaction Mechanism Configuration splitting (4f^ -4f^"^ 5d)

Energy (cm'^)

Optical Probe UV and VUV spectroscopy


Splitting within a 4f^ configuration Non-central electrostatic field Spin-orbit interaction Crystal field interaction Ion-ion interaction induced band structure (in stoichiometric compounds)

10^ 1 10' I 10^ 10"^-10

Hyperfine splitting Superhyperfine splitting (ion-ligand hyperfine interaction)

10 " - 1 0 10

Absorption, fluorescence or laser excitation spectroscopy

^ "I Selective and nonlinear laser spectroscopy

In this chapter, we discuss the electronic energy level structure of RE ions in crystalline solids. Given that the theoretical foundation has already be established and discussed comprehensively in several books (Judd, 1963 a; Wyboume, 1965; Dieke 1968; Abragam and Bleaney, 1986; Hiifner, 1978; Weissbluth, 1978; Cowan, 1981; Newman and Ng, 2000), we will not enter into details of 3

Guokui Liu the spectroscopic theory. Instead, an overview of modeling the electronic energy level structure of RE ions in dielectric crystals is provided. An effective operator Hamiltonian is introduced with tensor operators defined according to the nature of electronic interactions. The terms of Hamiltonian include free-ion interactions such as electrostatic Coulomb interactions and spin-orbit coupling, the ion-ligand interactions described in the framework of crystal-field theory, and the hyperfine interactions treated as a perturbation on electronic energy levels. Solutions to the Hamiltonian are obtained primarily through empirical approaches in which the phenomenological parameters of the effective operators of the Hamiltonian are determined. The calculations of the electronic energy level structures of RE ions require complicated tensor operation and the theory of angular momentum coupling (Judd, 1963a; Weissbluth, 1978; Cowan, 1981). In Appendices A and B of this book, properties of tensor operators, particularly the effectiveoperator Hamiltonian, and angular momentum coupUng are briefly discussed. Spectroscopic studies of magnetic and hyperfine interactions of RE ions in solids are an important part of solid-state RE spectroscopy and have gained significant applications in both fundamental understanding of physical interactions and materials characterization (see Chapters 4, 5 and 7). The advances in high resolution and nonlinear laser spectroscopic techniques have facilitated the measurements of RE hyperfine energy levels in great detail (Levenson, 1982; Macfarlane and Shelby, 1987). In Section 1.6, we discuss various mechanisms of hyperfine interactions and Zeeman effect. Their contributions to energy level splittings from the GHz to the kHz spectral range are evaluated on the same basis of crystal-field interaction. It is the author' s intention to outline the practical procedures of analyzing experimental spectra and, at the same time, to provide the reader a clear theoretical understanding of the electronic interactions, including crystal-field and hyperfine interactions of RE ions in crystals. In this regard, this chapter may stand alone as a useful guide to chemists and materials scientists in analysis of RE spectra. However, more importantly, this chapter serves as an introduction to other chapters, which assume the reader's understanding of the concepts of electronic interactions and the RE energy level structures.

1.2 1.2.1

Electronic States and Coupling Schemes Central Field Approximation

A conventional approach to solving Schrodinger equation for an A^-electron atomic system is to use the central field approximation and Hartree-Fock method (Hartree, 1957; Slater, 1960; Weissbluth, 1978). In the central field approximation, each electron is assumed to move independently in the field of the nucleus and a central field that is made up of the spherically averaged 4

1 Electronic Energy Level Structure potential fields of each of the other electrons. The quantum mechanical solution for such a central field system is the same as that for a single electron hydrogen atom. With the concept of central field approximation, the non-spherical part of the electronic interactions is treated as a perturbation to a spherically synmietric potential, so that the basis of the hydrogen atom wave functions can be used to construct wave functions for an A^-electron atom (ion). This method has been used to classify electronic states and evaluate energy levels of lanthanide and actinide ions (Judd, 1963a; Wyboume, 1965). The primary terms of the Hamiltonian for an A^-electron ion in the absence of external fields is commonly expressed as (1.1) where -^2




(1.2) N



1 ! e- . i5^,,.

(1. 16)


The energy levels of the free-ion states are independent of M, so they are ( 2 / + l)-fold degenerate. The new basis Eq. (1.15) in the intermediate coupling scheme describes the energy states of the Hamiltonian including Coulomb and spin-orbit interactions and is obtained from mixing all LS terms with the same J in a given 4f^ configuration. The transformation coefficients of a^su ^^ the components of the eigenvector pertaining to the basis state in the LS coupling (Judd, 1963).


Free-ion Interactions

In spectroscopy, a powerful method for evaluating atomic energy level structure (or RE ion energy level structure in our case) is to define and diagonalize an effective-operator Hamiltonian with the wave functions of the central field Hamiltonian. Racah (1949) used this method for calculating the energy matrix elements of the tensor operators of the electronic angular momentum. Since then, many developments have been made, particularly for applications of the effective operator method to rare earth spectroscopy (Judd, 1963a; Wyboume, 1965). In this section, we will review the primary results of the theory that are important for understanding the free-ion properties of RE ions in solids. The effective operator Hamiltonian and its reduced matrix elements for the Coulomb electrostatic interaction and spin-orbit coupling are discussed, while the effective operators for higher order free-ion interactions are presented without derivations. An essential part of the effective operator method is to determine the irreducible matrix elements of tensor operators using the Wigner-Eckart theorem (see B. 9 in Appendix B). For more comprehensive theories of tensor operators and atomic spectroscopy, one may read textbooks by Judd (1963a) and Weissbluth (1978).


Coulomb Interaction

In the central field approximation, orbital electronic wave functions of an RE ion 11

Guokui Liu are represented by products of radial and angular parts as shown in Eq. (1.8). The effective operator for Coulomb electrostatic intra-ion interaction may be expressed by expanding l/r^j into scalar products of the tensor operators of spherical harmonics as follows:

where r< indicates the distance from the nucleus to a near electron, and r> the distance from the nucleus to a further electron, and Y II



where {... } is a 6-j symbol. The values for the reduced matrix elements of the tensor operator V^"^ are tabulated in the books by Sobelman (1972) and Nielson and Koster (1963). They may be output from the SPECTRA program.


Corrections to Free-ion Hamiltonian

The electrostatic and spin-orbit interactions give the right order for the energy level splitting of the f^ configurations. However, these primary terms of the free ion Hamiltonian do not accurately reproduce the experimentally measured energy level structures. This is because the parameters F* and ^^f, which are associated with interactions within a f^ configuration, cannot absorb all the effects of additional mechanisms such as relativistic effects and configuration interactions. Introduction of new terms to the effective operator Hamiltonian is required to better interpret the experimental data. Judd and Crosswhite (1984) demonstrated that, in fitting the experimental free-ion energy levels of Pr^"^(f configuration), the standard deviation could be reduced from 733 cm~^ to 24 cm~^ by adding nine more parameterized corrective effective operators into the Hamiltonian. Among several corrective terms included in the effective operator Hamiltonian, a significant contribution to the f^ energy level structure is from configuration interactions between the configurations of the same parity, which can be taken into account by a set of three two-electron operators recommended by Wyboume (1965), ^ , 1 = a L ( L + l ) -\-pG{G^) +yG(Rj),

(1.29) 15

Guokui Liu where a, /? and y are the parameters associated with GiGz) and G(Rj) (Rajnak and Wyboume, 1963), the latter being eigenvalues of Casimir operators for the groups G2 and R^i Judd, 1963) For f^ configurations of N^S, a three-body interaction term was introduced by Judd (1966) and Crosswhite et al. (1968) as ^c2 =




i =2, 3,4,6, 7, 8

where T' are parameters associated with three-particle operators t^. This set of effective operators scaled with respect to the total spin S and total orbital angular momentum L are needed in the Hamiltonian in order to represent the coupling of the f^ states to those in the higher energy configurations (5d, 5p, 5s) via interelectron Coulomb interactions. It is common to include six three-electron operators ^,( / =2,3,4,6, 7,8). When perturbation is carried beyond the second order, an additional eight three-electron operators f.(ll ^ / ^ 19, with / = 13 excluded) are required (Judd and Lo, 1996). A complete table of matrix elements of the 14 three-electron operators for the f-shell were published by Hansen et al. (1996). In addition to the magnetic spin-orbit interaction parameterized by ^„^, relativistic effects including spin-spin and spin-other-orbit, both being parameterized by the Marvin integrals M°, M^, and A^( Marvin, 1947), are included as the third corrective term of the effective operator Hamiltonian (Judd etal., 1968). .^03 = S



i =0, 2,4

where m, is effective operator and M' is the radial parameter associated with m^. As demonstrated by Judd et al. (1968) and Camall et al. (1983), for improving the parametric fitting of the f-element spectra, two-body effective operators can be introduced to account for configuration interaction through electrostatically correlated magnetic interactions. This effect can be characterized by introducing three more effective operators as i^c4 = I



I =2, 4, 6

where /?, is the operator and P' is the parameter. In sunmiary, we have introduced 20 effective operators including those for two- and three-electron interactions. The total effective-operator Hamiltonian of free-ion interactions is ^Fi = Z k =0, 2, 4, 6


F% ^CnAsoinl) -^aUL+l)

+ pG{G,)



i =2, 3,4,6, 7, 8

i =0, 2,4

Electronic Energy Level Structure

i =2,4,6

This effective-operator Hamiltonian has been used as the most comprehensive free-ion Hamiltonian in previous spectroscopic analyses of f-element ions in soUds (Crosswhite, 1977; Crosswhite and Crosswhite, 1984; Camall et al., 1984, 1989). The 20 parameters associated with the free-ion operators are adjustable in fitting of experimental data.


Reduced Matrices and Free-ion State Representation

In ( 1 . 3 3 ) , all effective operators for the free-ion interactions have well-defined group-theoretical properties (Judd, 1963a; Wyboume, 1965). Within the intermediate coupling scheme, all matrix elements can be reduced, using the Wigner-Eckart theorem, to new forms that are independent of / , viz. .




(1.38) WithEqs. ( 1 . 20), ( 1 . 37) and ( 1 . 38), we may write the reduced matrix elements of the crystal-field Hamiltonian as

and l'5>.

It is obvious that truncation of LS terms in the / = 7/2 multiplets should affect the scale of the coupled matrix elements, and thus affect the crystal-field splitting. The same effect is expected for the off-diagonal matrix elements between different / , but this is less important because of the large energy gap between the ground state and the first excited state. In the literature, the crystal-field interaction is often characterized by quantitative comparison of the crystal-field strength defined as (Auzel, 1983) N^ = [ J _ y A ^ i ^ l



or as (Chang et al., 1982) 31

Guokui Liu r 1



Symmetry Rules

We now discuss the geometric properties of the crystal-field operators and parameters in more detail. In addition to the angular momentum of the RE ions that restricts k and q for a set of non-vanishing crystal-field operators, the site symmetry of RE ion in a crystalUne lattice also imposes Umits on crystal field operators, because the tensor operators for the crystal-field interaction must be invariant under the point group symmetry operations. Here our interest is to identify the non-vanishing components of crystal-field operators and their matrix elements. First, if we restrict our attention to the states of the same parity, namely l = l\ k must have even values. It is also required that BJ must be real in any synunetry group that contains a rotation operation about the F-axis by IT or a reflection through the X-Z plane; otherwise 5^(^7^0) is complex. In the later case, one of the BJ can be made real by a rotation of system of coordinates about the Z-axis. The B^ forq < 0 are related to the imaginary ones for ^ > 0 by B'_^ ={-iyB';,


Also under the invariant conditions of the point group theory, the crystallographic axis of lowest symmetry determines the values of q for the nonvanishing crystal-field operators. For example, at a site with C^^ synmietry, there is a threefold axis of rotational symmetry with a reflection plane that contains the Cg axis (Tinkham, 1964; Hiifner, 1978). The ligand electric field must exhibit this symmetry and hence if a 2Tr/3 rotation is made on the crystal field potential, followed by a reflection with regard to the plane, the potential is invariant only if ^ = 0 , ± 3 , and ± 6 . Thus, within an f^ configuration, the crystal field Hamiltonian may be written as

M C3.) = X [Blcf' (i) + Kef-:' (0 + K[ c6








^6» ^3h' ^ 6 h ' ^ 6 »


^6v' ^3h» ^6h


r, r,, r„ o, o, CeOg


D: 1/2,3/2

S: ±0, ±2, D:l

D: 1/2,3/2

S: ±0, ±3 D:l,2

D: 1/2,3/2, 5/2

S: ±0, ±4 D:2,6 T:l

D:/^ = l / 2 , 7 / 2 Q:/t=3/2


Re(5l),fi^ I>4


Re(B:,Bj) D2


S: ±0 D:l

Y2O2S p2

^ 4 » '^4» ^ 4 h



Trigonal ^3d

S: ±0, ±1 ^ • ^ » BQ, Re(52,


^3v» ^ 3 »




Re(B^) BQ.BQ,


(a) Morrison et al., 1982; (b) Hufner, 1978; (c) S: singlet, D: doublet. T: triplet, Q: quadruplet;


Guokiii Liu

Because of the 3-j symbols in Eq. ( 1 . 39), the degeneracy in M may be completely or partially removed by crystal-field coupling between states of different JM. In the crystal-field matrix, the terms for which M - M' = q aie mixed by C[^^. Otherwise the crystal-field matrix elements are zero. Based on this property, the crystal-field matrix may be reduced into a number of independent submatrices each of which is characterized by a crystal quantum number jm (or F). Each fi represents a group of M, such that M-M' =q (0,2, 3 , 4 , 6 ) belongs to the same submatrix (Hiifner, 1978). All matrix elements between the submatrices are zero. The crystal-field quantum number may be used to classify the crystal-field energy levels even when J and M are not good quantum numbers. To consider C^^iand D^^^) as an example, the JM and J'M' with M -M' =6 belong to the same crystal-field submatrix. For even number of f electrons, there are four independent submatrices, and for odd number of f electrons, there are three independent submatrices. The parameters of nonvanishing crystal-field terms for synmietries of common crystal hosts of f-element ions are given in Table 1. 7 along with the numbers of reduced crystal-field matrices. Without a magnetic field, the electrostatic crystal field alone does not completely remove the free-ion degeneracy for the odd-numbered electronic configurations. Known as Kramers' degeneracy (Kramers, 1930), all crystal field levels are at least doubly degenerate. The crystal quantum number and JM classification are given for Cg^, and D^^ in Table 1.8. In calculation of energy level structure for degenerate doublets, one may cut off a half of the submatrix elements. In many cases, calculations of crystal-field energy levels have been successful by using higher site symmetry than the real one so that a smaller number of parameters are required. First of all, this is because RE ions in many solids occupy a low site symmetry and the limited number of observed energy levels could not accurately determine a large number of crystal-field parameters. Secondly, many crystal lattices do not have mirror symmetries in their coordinates so that the crystal-field parameters with q 9^0 are complex ( see Table 1.7). If one makes an approximation by using only the real part of the crystal-field operators, the energy level calculation becomes much easier. Since the high synmietry approximation is equivalent to up-grading a lower synunetry site to a higher one within the same symmetry group, this is called the descentof-symmetry method ( Gorller-Walrand and Binnemans, 1996). This method may be applied to the groups of monocHnic, trigonal, and tetragonal structures listed in Table 1.7. For example, the S4 site synmietry of trivalent lanthanide ions in LiYF4 is often treated as Dg^iCEsterowitz, 1979; Gorller-Wakand, 1985, Liu et al., 1994a). Similarly, the actual C2 synmietry of LaFg is replaced by €2^ (Camall et al., 1989) and the Cg^ symmetry of LaClg by Dg^ (Morrison and Leavitt, 1982).



Electronic Energy Level Structure

Table 1.8 Classification of crystal field energy levels for Q^ and Dg^ symmetry (a) Even number of electrons M{fjL=0

/ 0



M(/x= ±2


No. (Levels) 1 2












3, - 3



0 0


±2, + 4 ±2, + 4

3, - 3

6 7


±1, +5


3, - 3



3, - 3



±1, +5 ±7, ± 1 , + 5

±2, + 4


±2, + 4

3, - 3




±7, ± 1 , + 5

±8, ±2, + 4

3, - 3


(b) Odd number of electrons MiiJL=±l2


M(fi= ±3/2 V3)

M{fjL= ± 5 / 2

No. (Levels)














±5/2, qF7/2


1 2



±3/2, +9/2

±5/2, HP7/2



±1/2, qhll/2

±3/2, +9/2

±5/2, +7/2



±13/2, ±1/2, +11/2

±3/2, +9/2

±5/2, + 7 / 2



±13/2, ±1/2, +11/2

±15/2, ±3/2, +9/2

±5/2, +7/2


In practical cases, the crystal field of Cg^ symmetry has been found to exhibit an effective potential appropriate to Dg^ synmietry because the imaginary term of ImBl in the C^^ potential is negligibly small. This is known because the nearest Ugands have D^^ synmietry. When the second shell of coordination is considered, the site symmetry reduces to Cg^ because the second nearest ligands add a smaller contribution to the crystal field at the lanthanide site. As illustrated in Fig. 1.6 on the X-Y plane, a rotation (Ac/)) of the next nearest hgand (NNL) coordinates about the Z-axis converts the structure of LaClg from C^^^ to Dg^ synmietry. For the actual structure of the lanthanide ethylsulfates and trichlorides, Ac/) may be less than 1°. Similarly for lanthanide ions in LiYF4, the value of ImBl in the ^4 potential is less determined. With zero InLB4, an effective site synmietry of Dgj has been used in practical cases (Morrison and Leavitt, 1982).


Guokui Liu Conversion of the LiYF4 structure from S^^ symmetry to Dg^ synunetry only involves the NNL fluorine ions connected in a tetrahedron. The actual crystallographic site synunetry is a slight distortion of Dgd- This implies a distortion of the (/)-coordinates from the dodecahedron D24 values (Blanchfield, 1983; Urland, 1981). In addition to a rotation of Ac^ about the Z-axis, a small change in the radial distance from the Y^ ^ center to the NNL is required in order to construct a perfect Dgd symmetry for the two tetrahedrons of the fluorine ions.

Figure 1. 6 Illustration of descent-of-symmetry operations for LaCla type of hexagonal crystals. A rotation of the next nearest lignds on the larger trigonal prism by a small angle A.


In the definitions of the overlap integrals (see Eq. ( 1 . 5 5 ) ) , wave functions of the lanthanide and ligand ions are considered in the reference systems S^ and S^ with the common quantization axis. Parameters of the electrostatic interaction of the valence electron with charge distributions localized at the crystal lattice sites are usually presented by a sum of contributions from the electric fields of point charges (B^^""^*), point dipole and quadrupole moments (B^^^^, B^p^^). In particular, parameters of the effective crystal-field Hamiltonian are related to the electrostatic field of point charges and averaged over radial wave functions of the lanthanide ion are


xi-l)'>C'!lie„1. Therefore, the additions to the free-ion Hamiltonian may be defined as (Reid et al., 2000) ^,(fcl) = J^F'(fd)Mfd)


+^(dd)Aso(dd), (1.64) 59

Guokiii Liu

with A: =2, 4, and7 = l, 3, 5. The F*^(fd) and G'(fd) are direct and exchange Slater parameters for the Coulomb interactions between the 4f and 5d electrons (Cowan, 1981). The ^ ( d d ) parameter is associated with the spin-orbit interaction of the 5d electron. The 5d electron is also affected by the crystal-field interaction ^cF(dd) = J^Bl{dd)Cl'\dd),


with k = 2 and 4, and the same restrictions applied to q as for the 4f electron crystal-field Hamiltonian //cpCff) expressed in Eq. (1.36). The 4f^"^5d configuration has a higher average energy than the 4f^ configuration. This energy difference contains contributions from several sources (Cowan, 1981), including kinetic energy. Coulomb, and (isotropic) crystalfield effects. These effects cannot be distinguished experimentally and can be considered to contribute to a single term AE (fd) Ssifd) in the Hamiltonian, where the operator ^^(fd) is diagonal, with unit matrix elements for 4f^"^5d and zero matrix elements for 4f^. Since the 4f^~^5d and 4f^ configurations have opposite parity, there is no Coulomb configuration interaction between them. However, the odd-parity crystal-field interaction can couple the 4f^"^5d and 4f^ configurations. The contribution to the Hamiltonian is

^cF(fd) = X^^(fd)C(fd),



withfc= 1, 3, 5. This mixing of configurations is one of the major contributions to electric dipole transitions within the 4f^ configuration ( see Chapter 2 ) . However, it has only a small effect on energy level splitting in the states of a 4f^"'5d configuration (Reid et al., 2000).


Determination of Hamiltonian Parameters

Though, spectra of 4f^ to 4f^"^ 5d transitions have long been experimentally investigated, it is only recently that excitation spectra of most ions of the RE series doped into LiYF4, YPO4, and CaFa have been systematically analyzed using extensions to the standard crystal-field model for the 4f^ configuration (van Pieterson et al., 2002a, b). Some previous studies on the 4f^"^5d configuration have tried to analyze the spectra by considering only the 5d crystal field and the free-ion splitting of the 4f^"^ core. However, this is not a very good approximation since the Coulomb interaction between the 4f and 5d electrons has a significant effect. As an example, this effect on the 4f^5d states of Tb^^ is illustrated in Fig. 1.9, which shows the spHtting of the 5d crystal-field states as 60

1 Electronic Energy Level Structure the f-d interaction parameters (F^(fd), G^(fd)) are increased from zero to the values predicted for the free ion according to Cowan' s program (Cowan, 1981). The parameter A used as the horizontal axis of the graph is a scaUng factor for the f-d interaction parameters. The energy difference between the first spin-forbidden and spin-allowed 4f^ 4f^"^ 5d transitions for Tb^^ is dominated by the exchange operators G^(fd) (van Pieterson et al., 2002b; Reid et al., 2002).

Figure 1.9 Splitting of the five 4>f 5d crystal-field states of Tb^^ in LiYF^. The parameter a represents the f-d interactions as explained in the text. On the left (A =0) the parameters for these interactions are set to zero and on the right (A = 1) they have the values calculated for the free ion (van Pieterson et al., 2002b)

In contrast to sharp zero-phonon Unes of electronic transitions within a 4f^ configuration, inter-configuration transitions to the 4f^~^ 5d configuration are dominated by broad vibronic bands, with a limited number of observable zerophonon lines. Fitting all of the free-ion and crystal-field parameters for 4f^~^5d is therefore not currently possible. It is crucial to begin with good estimates of the parameters and then use spectroscopic information from as many ions as possible to refine the calculation, since the effect of some parameters is more apparent in particular ions. For example, the 4f^"^5d configuration for Ce^^ is simply 5d, for which the only parameters are ^(dd), and the Bj(dd), making this ion particularly suitable for determining the 5d crystal-field parameters. As a general approach, the Hamiltonian parameters may be determined from scaling of initial estimates. The initial estimates may be from: (1) Parameters for the 4f^"^ core from parameters for the 4f^ configuration of the same ion; (2) 5d crystal-field parameters 5^(dd) from the Ce^^ spectrum; 61

Guokui Liu

(3) Free-ion parameters involving the 5d electron from ab initio calculations. The atomic calculation methods are described by Cowan (1981). The calculations are Hartree-Fock, with relativistic corrections. In Table 1. 14, we give a summary of Hamiltonian parameters for the supplementary terms of the Hamiltonian expressed in Eqs. ( 1 . 62) - ( 1 . 6 6 ) . For comparison with the ff parameters listed in Table 1.5, the F* ( ff) parameters increase dramatically across the series, as the 4f orbitals contract. The calculated F*(fd) and G^(fd) parameters decrease gradually, since the 5d orbitals do not contract as much as the 4f orbitals. Similarly, the ^ ( f f ) increase dramatically, and the ^ ( d d ) somewhat less. It was found that the calculated F^(fd) and G^(fd) parameters are too large, particularly from examining the spUtting between the high-spin and lowspin states of the 4f^~^5d configuration of the heavy lanthanides (van Pieterson et al., 2002a, b ) . Better agreement is obtained when the fd parameters are reduced to about 67% of their calculated value (Reid et al., 2002). This reduction is not purely induced by the crystalUne environment. The free-ion data indicate that the overestimation is mainly for the F^(fd) and G^(fd) parameters. In fact, such a reduction between the Hartree-Fock values and the experimental values for freeion parameters is similar to that for the free-ion parameters of the 4f^ configurations we discussed in Section 1.3. Though the 5d orbitals are perturbed much more by the host lattice than for the 4f orbitals, the F^(fd) and G^(fd) parameters are largely determined by the part of the 5d orbitals close to the nucleus where they interact with the 4f orbitals, and the ^( dd) parameters also have their largest contribution from close to the nucleus. Therefore, the same mechanisms whose effects are not absorbed into the Hartree-Fock values of F*(fd) and F\ff) may be considered. The crystal-field parameters associated with the 5d electron of the 4f^~^5d configuration are large, typically 10 to 20 times the parameters for the 4f electrons. For the host LiYF4, the crystal-field splitting of the 5d orbitals is approximately 20,000 cm "^ (van Pieterson et al., 2002 a, b ) . Crystal-field parameters for trivalent RE series in this host are shown in Table 1.14. Whereas the 4f parameters decrease dramatically across the lanthanide series, the 5d parameters decrease by a relatively small amount. Since the 4f orbitals contract somewhat in the 4f^"^ 5d configuration, their values may also decrease somewhat. For ions near the beginning or end of the lanthanide series the 5d crystalfield splitting dominates the spectra, with the splitting caused by atomic interactions giving a fine-structure built on top of the crystal-field spHtting. Towards the center of the series the atomic spHtting of the 4f^~^ core is quite large and the 5d crystal-field spHtting is more difficult to definitively identify. A notable exception is Tb^'', where the large spHtting between the lowest states of the 4 f core makes the identification of the 5d crystal-field splitting straightforward (van Pieterson et al., 2002b). 62

cd O



a o (U +-» 0)


'^ >



=3 t>0

o^ CO rH CM r-
(Nl O^

00 in CO CJ>





00 t> CO

00 CO

a> i>

00 CO


o 00


oa a>

•^ (Ji»>






o^ CO

in CO






i> (M in


CO l> CO

CO t> CO


!>• CO -^ CO ^ 00








Oi iH

j - \

in in


CO l> CO in


in CO 00

in CO iH


O^ in CO 00


o rH

CM 00


O^ Tt iH

CTi in CO 00


Oi CO 00



Electronic Energy Level Structure


Guokui Liu In summary, an extension to the standard crystal-field model for 4f^ provides a good basis for interpreting the spectroscopy of the 4f^"^ 5d configurations of lanthanide ions in crystals. However, the exact values for many of the Hamiltonian parameters are not yet known. More detailed data (including polarization measurements) are required to give definitive identification of the electronic origins and allow the accurate determination of more of the parameters and hence better calculations of various spectroscopic properties.


Magnetic and Hyperfine Interactions

The hyperfine energy levels of RE ions in solids range from MHz to GHz on the spectral scale. Because of localized electronic states, the homogeneous line width of transitions between 4f^ crystal-field levels in many cases is less than 1 MHz at liquid heUum temperatures. Therefore, not only can the electronic Zeeman effect be observed, but hyperfine splitting and even superhyperfine (ligand hyperfine) structures also can be measured when elimination of inhomogeneous line broadening is possible. Laser spectroscopic methods with a resolution of 10 "^ or higher have proven to be effective in eliminating inhomogeneous line broadening; therefore, they enable energy level spHttings on the MHz order to be measured in electronic transitions in the infrared and visible spectral regions (Macfarlane and Shelby, 1987). Spectroscopy in the MHz -GHz range reveals very rich physical information concerning electron nuclear interactions that impact the properties of RE ions as optical activators. In this section, we introduce the primary Hamiltonian of Zeeman and hyperfine interactions. Apart from elucidation of the physical origins for the various terms of the Hamiltonian, emphasis is given to the understanding of experimental spectra. A brief review of recent work on Pr^ ^, Eu^ ^, and Tb^ ^ is presented. The importance of studying the Zeeman effect and hyperfine interactions extends far beyond energy level structure. Many fundamental properties and applications of RE ions in solids involving hyperfine interactions are also discussed in other chapters of this book. In particular, since the homogeneous line width of optical transitions between hyperfine energy levels in the ground and excited crystal-field states are much narrower at low temperatures than the inhomogeneous broadening induced by disturbed sites and local field stress, measurements of the line shapes in a hyperfine spectrum are used to characterize line broadening mechanisms. Line broadening mechanisms and Une width measurements are discussed in detail in Chapters 3, 4 and 7 of this book. For RE ions in ground states, extensive studies of Zeeman and hyperfine energy level structures along with spin dynamics have been conducted using electron paramagnetic resonance (EPR) and electron-nuclear double resonance (ENDR) methods (Abragam and Bleaney, 1986). In this chapter, we restrict our attention to optical spectroscopy. To evaluate the Zeeman effect and hyperfine interactions of RE ions in 64

1 Electronic Energy Level Structure crystals, an effective Hamiltonian, in addition to that of the free ion and crystal field interactions as described in proceeding sections, may include terms such as

where the first two terms, existing only when an external magnetic field is appUed, represent electronic Zeeman effect and nuclear Zeeman effect, respectively. The third term represents the hyperfine interactions between the 4f electrons and the spin of the RE nucleus, and the fourth term is for the nuclear electric quardrupole interaction. In most cases, one can use a perturbation method in analysis of Zeeman and hyperfine spectra. It is always assumed that the wave functions of electronic states and the nuclear spin states are independent from each other. For the electronic states, the crystal-field eigenfunctions with /-mixing in the form of Eq. ( 1 . 41) are used as the zeroorder wave functions for obtaining the eigenvalues of the Hamiltonian expressed in Eq. ( 1 . 6 7 ) .


Zeeman Effect

In the presence of an external magnetic field, H, the degenerate crystal-field states split further, so that the degeneracy of the crystal-field states, including the Kramers' doublets for odd-A/^ configurations, is removed completely. In each / multiplet, there are a total of 2 / + 1 energy levels. For the electronic Zeeman effect of RE ions in an isotropic crystal, the interaction Hamiltonian has the simple form, ^z



and the effective magnetic moment is defined as At/ = - Mfig;/,


where in the LS coupling scheme


/ ( J + 1) + 5 ( S + 1) - L ( L + 1) 2 7 ( 7 + 1)



is the effective Lande g factor, and /Xg is the Bohr magneton. In the simple case of an isolated crystal-field doublet where the magnetic field is in parallel to the crystal c-axis, the splitting energy, ^z = /XBg,// = /XB^yi^X auM,

( 1 . 71)


Guokui Liu is linear in H and symmetric about the energy level at zero field. Usually, in comparison with a free electron Zeeman split, an effective g factor is defined as




A crystal-field singlet does not spUt but its energy level may shift due to coupling to other crystal-field levels in the same J multiplet. In the case of weak crystalfield splitting and a relatively strong magnetic field, the Zeeman effect should be evaluated by diagonalizing a matrix that includes the entire group of the crystalfield levels in the multiplet. Apart from the diagonal matrix elements (see Eq. ( 1 . 7 1 ) ) , the off-diagonal matrix elements are = fJiBgjH'Z a,^arM'M'8^M'^

(1. 73)


Depending on the crystal-field energy gaps at zero-field, one may reduce the size of the matrix by considering only those crystal-field levels that are not far separated from each other. As it was demonstrated in the case of ^D4 in Tb^^ in LiYF4(Liu et al., 1988) the calculation of the Zeeman effect for Tb^^ at an S^ site synunetry agrees very well with experimental observation. In other cases, where the density of RE electronic states is high, the Zeeman coupling between multiplets is strong and inclusion of states in different J multiplets is necessary. In general, when the external magnetic field is not parallel with the c-axis, the Zeeman interaction is anisotropic, but the (X, Y, Z) axes can still be chosen as the principal axes for the tensor operators of Hamiltonian, and Eq. ( 1 . 68) should be replaced by (Abragam and Bleaney, 1986)

In axial symmetry, gxx = gw^8±'^ gzz^g//It is easy to show that, for nonKramers' doublets, the diagonal matrix elements of the X and Y components vanish, and only the Z component needs be considered. Because the electrons shield the nucleus from interaction with an external field, calculations of nuclear Zeeman effect require consideration of the electronic shielding effect. Phenomenologically, an anisotropic tenser a( a^x^ ^YY> GLZZ) is usually introduced based on the consideration of strong shielding of the nuclear magnetic moment by the magnetic field. The Hamiltonian is of the following form: ^^


' (1 - a ) •/,


where y„ is the nuclear gyromagnetic ratio. The shielding parameters iaxx> «IT» azz) may absorb contributions from several mechanisms including coupling to 66

1 Electronic Energy Level Structure electronic angular momentum.


Magnetic Hyperfine Interaction

One of the leading electron-nucleus coupling mechanisms is conmionly expressed in the form of magnetic interaction. This is based on the understanding of the interaction between the nuclear spin momentum / and the effective magnetic field induced by the electronic angular momentum / . For rare earth ions in crystals of axial symmetry, magnetic hyperfine interaction can be described by the following effective spin Hamiltonian. - ^// Jzh

+ A^




and Ix,Y,z ^^ electronic angular and nuclear spin operators, repectively and A// and A j^ are the components of the hyperfine interaction constant parallel and perpendicular to the crystal c-axis, respectively. A theoretical calculation of A// and A^ is possible based on the assumption of isotropic electron nuclear interactions, so that A// = A, = Aj= Ifi.y^r-'


\\ j}.


where ir ^) is the mean inverse third power of the distance of the electron from the nucleus, and 2/XB ^r~^ ) A^ is the effective magnetic field induced by the f-electrons at the nucleus. For the ground states of trivalent RE ions, the values of the reduced matrix element {j \\ N \\ j} and the values of Aj are tabulated by Abragam and Bleaney (1986). Using the reduced matrix elements of the tensor V^^^^, one may calculate Aj for any multiplet from (Wyboume, 1965):

2-^^2xfMl^J^]^ L / ( / + 1) S

X (frSLWV''' Wfr'S'L'yll 1/



I' 2] J



After the value of Aj is determined, the matrix elements of the hyperfine Hamiltonian of Eq. (1.76) can be calculated using the crystal-field eigenfunction, \ iT} = ^ a^l JM}, and the independent nuclear wave M

function, I/, m ) . The same procedure as that in calculation of Zeeman effect is required for evaluating the hyperfine energy level spHttings. Given that hyperfine 67

Guokui Liu

splittings are much smaller than spin-orbit coupling and the crystal-field splitting, it is sufficient in most cases to consider the coupling only to the nearest crystalfield levels in the evaluation of the contributions from the off-diagonal matrix elements. As a parameter, the value of Aj may be obtained from analyzing experimental spectra. Knowing Aj from experiments enables one to determine the nuclear magnetic moment if the value of ( r ~ ^ ) can be calculated. For electronic singlets, including all crystal-field levels of non-Kramers' ions at low synunetry sites such as Cg in LaFg or Q in YAIO3, the diagonal matrix elements of the magnetic hyperfine interaction are quenched. Therefore, there is no first order hyperfine splitting in these electronic states. The contributions to hyperfine splitting are from the off-diagonal matrices between different crystal-field levels and nuclear electric quadrupole interactions. In a static magnetic field, H, the effects of hyperfine and electronic magnetic interactions on a singlet crystal-field level are evaluated using a second order perturbation theory. The effective Hamiltonian, including the nuclear Zeeman effect, can be expressed as (Abragam and Bleaney, 1983; Macfarlane and Shelby, 1987): ^HF ^-g'jfilH'A'H^^^^.


The first term in Eq. ( 1 . 79) is the quadratic electric Zeeman shift, where the tensor A is given by

with I D b e i n g the levels to which Eq. ( 1 . 79) appUes, ai/3) = X, 7, Z being the principle axes of the A tensor, and AE^ p, being the energy difference between level F and F'. The second term is the enhanced nuclear Zeeman Hamiltonian that can be expressed as ^ ^





where the effective nuclear gyromagnetic ratios are given by 7 a =yN+2gjjjL^AjA^yti.


Due to the second term, which arises from coupling to the second order hyperfine interaction, the nuclear Zeeman effect is enhanced. In comparison with Eq. ( 1 . 7 5 ) , this is also known as an anti-shielding effect arising from the 4f electrons.


1 Electronic Energy Level Structure


Nuclear Electric Quadrupole Interaction

Because the expectation value of the nuclear electric dipole moment vanishes, the nuclear electric contribution to the electronic energy levels is primarily from quadrupole interactions (Weissbluth, 1978). The nuclear electric quadrupole moment, Q, is defined by

Q = ilm


= />,


where the sum is taken over all Z protons. Using tensor operation, the quadrupole interaction Hamiltonian can be written as HQ =^7(^-^{[3/z - / ( / + !)] + < ^ « - V X n . + / i , ) } , (1.84) where

The expectation values of V^ in Eq. (1.84) are directly related to the components of the electric field gradient in terms of crystal-field parameters that we discussed With this convention, the in the previous section, Vz^oc B^ and V^x -VYY^^Ioff-diagonal components are eliminated by a suitable choice of quadrupolar tensor axes, while the diagonal components are ordered such as that \V^\^\Vyy\:^ Since V is subjected to the Laplace equation (V^y + Vjy + V^^ = 0), it is conventional to define an asymmetry parameter of the quadrupole Hamiltonian 77 by

T7 =

7i/ \




and a nuclear electric quadrupole interaction constant P by

^ " 4 4/(2/ / ( 2 / -- l 1) )'


so that a simplified nuclear electric quadrupole Hamiltonian can be written as

Guokui Liu

The value of P depends linearly on iv^}, the total local electric field gradient along the Z-axis. The nuclear electric quadrupole interaction constant may absorb contributions from more than one origin. For example, in the ^FQ ground state of a 4 4^ = 50 A "^ (Blok and Shirley, 1966). Placing these values into Eq. ( 1 . 9 2 ) , we obtain P[fCF,)

= 2 . 14 x l O - ' ( l -/?^)B^(MHz),


where Bl is in c m " \ For the ^D^ excited state, the eigenfunctions are obtained (M =0 terms only) as 'Do(17,284 c m " ' ) : -0.244113,300)-0.546112,200)+0.678122,200) -0.178132,200)-0.174111,100) + 0.217131,100)+0.250162,200) ' D 2 ( 2 4 , 6 4 5 c m " ' ) : -0.121113,320)-0.427112,220) + 0.535122,220)-0.140132,220) -0.445112,810-10)+0.44611,281,010) The eigenfunction of 'D2 is significantly mixed with 'K^O, and there are no crystal field levels of 'D2 with components of 7 = 2 , M = 0 at an energy level below 24,645 c m " \ These eigenfunctions result in K ' D O I C^'^ I ' D ^ > I '


and P^f(^Do) = 7 . 7 x l O - ' ( l -/?^)5^(MHz).


For^'^Eu^^ in LiYF4, similar calculations result in P^fC^o)

= 2.45 x l O - ' ( l


= 3.27 X 10-^(1 -/?^)B^(MHz),



and (1.105)

In comparison of the 4f electronic contribution to the quadrupole interactions of Eu^"^ in LiYF4 and LaFg, we notice that the values of P^^^ C^o) are 79

Guokui Liu approximately the same, but that of P^^\^DQ) differ by a factor of 4 . 8 . This is due to the much larger variation in the eigenfiinctions and energy levels of the ^DQ and ^Dg multiplets in the two systems. In general, PI^\^DQ) has a stronger dependence on hosts. Using Eq. (1.95) and accepted values (Erickson, 1986), ( 1 -y^) = 8 1 , and 4f = 2 . 5 x l 0 " ^ ^ c m ^ for ^^^Eu^\ the lattice contribution to the quadrupole interaction constant is approximately

Pu. = -

1.39 X IQ-^Bl (^ _ ^ ^ ) ' (MHz),

(1, 106)

with Bo in c m " \ The coefficient of 1. 39 x 10"' in Eq. (1.106) should be the same for Eu^^ in both the ^FQ and ^D^ states, and insensitive to crystalline environments. Comparing Eq. (1.106) with Eqs. (1.103) and (1.105), we conclude that in the ^DQ state, the nuclear quadrupole splitting is dominated by the lattice contribution. The ratio of ^4^ (^DQ) /P^^^^ is expected to be on the order of 10% or less. For all practical purposes, PCDQ) =Pi^^ is a good approximation. Although the nuclear electric quadrupole splittings in the ^¥Q and ^D^ states were measured using spectral hole burning and ODNMR methods and the were extracted from analysis of the absolute values of P ( ^ F Q ) and PCDQ) experimental data for Eu^ ^ in a variety of crystalline hosts, the shielding factors, (1 - RQ) and ( 1 - 0-2) have not been consistently determined. Using Eqs. (1.102) and ( 1 . 106), we can express the ground state quadrupole interaction constant for ^^^Eu^^ in LiYF^ as P('Fo) = [2.14(1 -R,)




with BQ in c m " \ and the ratio of lattice contribution to electronic contribution as ^latt


0. 6 5

(1 -a,){\



Practically, through experimental measurements of the quadrupole splitting in ^F^ and ^DQ, and using the value of Bl determined from crystal field analysis, (1 RQ) and (1 -0-2) can be determined from



= - ^ .


{I -RQ)




Electronic Energy Level Stracture

where S = 10' X




and R=



P'^HX) +^la«*

By definition, the value of (1 -RQ) and (1 -era) must be between 0 and 1. It is known that PC^O) ^ P\an is always negative and P^^^ C^o) is positive, but PC^O) and BI can be positive or negative. For a specific system, if Bl>0 and P(^Fo) N . "^< / x >\¥#s

^ O

^^ " ^ " - f c











^T*»^ vr»^ ^^^^^



^ ^ ^--^

^/ S L ^ I W -





60 80 100 120 Magnetic Field Angle (°)






Figure 1.14 Angular dependence of the observed (symbols) and calculated (curves) ODNMR frequencies of l^ = 1/2-^3/2 transitions for ^^^Eu^^ in LaFg crystal at two magnetically inequivalent sites. Four lines for each site are represented by the solid and dashed curves. The external magnetic field of 0.26 T was always in a plane containing the c-axis and perpendicular to an , 2

= ( ^ ) .


This equation is only strictly correct for high-symmetry sites (Henderson and Imbusch, 1989). It is conmion to define a ^ that is a product of our XL ^f Eq. 97

M. F. Reid (2.8) and the bulk refractive index correction 1/n (e. g., May et al., 1987b). Since the exact form of the local correction has a much more tenuous justification than the bulk correction it seems appropriate to keep them separate. For spontaneous emission the observable quantities are related to the spontaneous emission rates, i. e., the Einstein A coefficients. The electric dipole and magnetic dipole contributions to the A coefficients for a single polarization are given by .ED



1 ^ED



= -w z

7 nvj — S

(2. 9)

and MD



4a>^ 3 1 c^D


= ^-.

T—^n — ^



Note the 1/3 factor built into these definitions, which is essentially the polarization-averaging factor of Eq. (2.12) but must be included here to obtain the correct transition rates for polarized transitions. The sum of the A coefficients over all polarizations is the inverse of the radiative lifetime, i. e.,


= i ( < FI,q : + .


There are two extreme cases of interest. The first case applies to transitions within the 4f^ configuration. In this case the coupling to the lattice is small and, most importantly, the coupling is almost identical for the initial and final states. Moreover, vibronic transitions are weak, and generally only become important for centrosynmietric systems, where purely electronic electric-dipole transitions are forbidden. Transitions where there is a change of more than one vibrational quantum number are rare. Another case applies to transitions between the 4f^ configuration and the 4f^~^ 5d configuration or charge-transfer states. In this case the coupUng to the lattice is very different for the initial and final states and transitions may involve a change of several quanta of vibration. We now discuss these two cases in more detail.

Weak CoupUng: 4f ^ ^ 4f ^ Transitions

Since the coupling of 4f electrons to the lattice is small, vibronic 4f^4f^ transitions generally involve a change of only one quantum of vibration. Vibronic transitions have been studied most thoroughly for cubic systems, especially elpasolite systems (Faulker and Richardson, 1977; Reid and Richardson, 1984c; Crooks et al., 1997), where zero-phonon electric-dipole transitions are forbidden and the only possible electric-dipole transitions are those associated with oddparity vibrations. Our approach (Reid and Richardson, 1984c; Crooks et al., 1997) is to use a linear Herzberg-Teller vibronic coupUng model (Piepho and Schatz, 1983). In addition to the potential V the effective Hamiltonian is now supplemented by firstorder and higher-order corrections in the normal coordinate Q^, to become