While basic features of polarons were well recognized a long time ago and have been described in a number of review papers and textbooks, interest in the role of electron-phonon interactions and polaron dynamics in di?- ent materials has recently gone through a vigorous revival. Electron-phonon interactions have been shown to be relevant in many inorganic and organic semiconductors and polymers, colossal magnetoresistance oxides, and tra- port through nanowires and quantum dots also often depends on vibronic displacements of ions. These interactions presumably play a role in hi- temperature superconductors as well. The continued interest in polarons extends beyond the physical description of advanced materials. The ?eld has been a testing ground for analytical, semi-analytical, and numerical techniques, such as path integrals, strong-coupling perturbation expansion, advanced variational methods, exact diagonalization, Quantum Monte Carlo, and other techniques. This book reviews some recent developments in the ?eld of polarons, starting with the basics and covering a number of active directions of research. Single- and multipolaron theories have o?ered more insight into colossal magnetoresistance and in a broad spectrum of ph- ical properties of structures with reduced dimension and dimensionality such as transport, optical absorption, Raman scattering, photoluminescence, magneto-optics, etc. While nobody - at present - has a ?nal theory of hi- temperature superconductivity, we discuss one alternative (polaronic) route. We have bene?ted from discussions with many experts in the ?eld.

Monte Carlo computer simulations are now a standard tool in scientific fields such as condensed-matter physics, including surface-physics and applied-physics problems (metallurgy, diffusion, and segregation, etc. ), chemical physics, including studies of solutions, chemical reactions, polymer statistics, etc., and field theory. With the increasing ability of this method to deal with quantum-mechanical problems such as quantum spin systems or many-fermion problems, it will become useful for other questions in the fields of elementary-particle and nuclear physics as well. The large number of recent publications dealing either with applications or further development of some aspects of this method is a clear indication that the scientific community has realized the power and versatility of Monte Carlo simula tions, as well as of related simulation techniques such as "molecular dynamics" and "Langevin dynamics," which are only briefly mentioned in the present book. With the increasing availability of recent very-high-speed general-purpose computers, many problems become tractable which have so far escaped satisfactory treatment due to prac tical limitations (too small systems had to be chosen, or too short averaging times had to be used). While this approach is admittedly rather expensive, two cheaper alternatives have become available, too: (i) array or vector processors specifical ly suited for wide classes of simulation purposes; (ii) special purpose processors, which are built for a more specific class of problems or, in the extreme case, for the simulation of one single model system."

1. Lyotropic Liquid Crystals The class of compounds known as thermotropic liquid crystals has been widely utilized in basic research and industry during recent years. The properties of these materials are such that on heating from the solid to the isotropic liquid state, phase transitions occur with the formation of one or more intermediate anisotropic liquids. The unique and sometimes startling properties of these liquid crystals are the properties of pure compounds. However, there exists a second class of substances known as lyotropic liquid crystals which obtain their anisotropic properties from the mixing of two or more components. One of the components is amphiphilic, containing a polar head group (generally ionic or zwitterionic) attached to one or more long-chain hydrocarbons; the second component is usually water. Lyotropic liquid crystals occur abundantly in nature, particularly in all living systems. As a consequence, a bright future seems assured for studies on such systems. Even now, many of the properties of these systems are poorly understood. It is the purpose of this review to consolidate the results obtained from nuclear magnetic resonance studies of such systems and to provide a coherent picture of the field. Probably the most familiar example of a lyotropic liquid crystal is soap in water. A common soap is sodium dodecylsulphate where an ionic group (sulphate) is attached to a hydrocarbon chain containing twelve carbons.

High magnetic fields have been an important tool in semiconductor physics for a long time. The area has been growing very rapidly since quantum effects in silicon field-effect transistors have become of practical interest. Since the discovery of the quantum Hall effect by Klaus von Klitzing in 1980, this subject has grown exponentially. The book contains 42 invited papers and 37 contributed papers which were presented at the 7th of the traditional Wurzburg conferences. For the area of high magnetic fields applied in semiconductor physics recent results are discussed, and the state-of-the-art is reviewed. More than 50% of the papers concern two-dimensional electronic systems. Other subjects of current interest are magneto-optics and magneto transport in three-dimensional semiconductors. Special attention has been paid to the rapidly growing field of semimagnetic semiconductors."

The advances in the theory of diffraction gratings and the applications of these results certainly determine the progress in several areas of applied science and engineering. The polarization converters, phase shifters and filters, quantum and solid-state oscillators, open quasi optical dispersive resonators and power compressors, slow-wave structures and patter forming systems, accelerators and spectrometer; that is still far from being a complete list of devices exploiting the amazing ability of periodic structures to perform controlled frequency, spatial, and polarization selection of signals.

Diffraction gratings used to be and still are one of the most popular objects of analysis in electromagnetic theory. The further development of the theory of diffraction gratings, in spite of considerable achievements, is still very important presently. The requirements of applied optics and microwave engineering present the theory of diffraction gratings with many new problems which force us to search for new methods and tools for their resolution. Just in such way there appeared recently new fields, connected with the analysis, synthesis and definition of equivalent parameters of artificial materials layers and coatings, having periodic structure and possessing features, which can be found in natural materials only in extraordinary or exceptional situations.

In this book the authors present results of the electromagnetic theory of diffraction gratings that may constitute the base of further development of this theory which can meet the challenges provided by the most recent requirements of fundamental and applied science.

The following issues will be considered in the book

• Authentic methods of analytical regularization, that perfectly match the requirements of analysis of resonant scattering of electromagnetic waves by gratings;
• Spectral theory of gratings, providing a reliable foundation for the analysis of spatial frequency transformations of electromagnetic fields occurring in open periodic resonators and waveguides;
• Parametric Fourier method and C-method, that are oriented towards the efficient numerical analysis of transformation properties of fields in the case of arbitrary profile periodic boundary between dielectric media and multilayered conformal arrays;
• Rigorous methods for analysis of transient processes and time-spatial transformations of electromagnetic waves in resonant situations, based on development and incorporation in standard numerical routines of FDTD of so called explicit absorbing boundary conditions;
• New approaches to the solution of homogenization problems the key problem arising in construction of metamaterials and meta surfaces;
• New physical results about the resonance scattering of pulse and monochromatic waves by periodic structures, including structures with chiral or left-handed materials;
• Methods and the results of the solutions of several actual applied problems of analysis and synthesis of pattern creating gratings, power compressors, resonance radiators of high capacity short radio pulses, open electromagnetic structures for the systems of resonant quasi optics and absorbing coatings.

This book was originally published in Japanese in honour of Professor S. Chikazumi on the occasion of his retirement from the University of Tokyo in March 1982. Physicists who had been supervised by him or had closely col laborated with him wrote articles on recent developments in magnetism and its engineering applications. In the preface of his excellent textbook Physics of Magnetism (Wiley, 1964), Professor Chikazumi says that recent research in magnetism deals with fundamental physical problems and, at the same time, with more secondary magnetic phenomena, as well as with engineering applications of magnetic materials to electromagnetic machines, permanent magnets and electronic computers, and that the purpose of his textbook is to give a general view of these magnetic phenomena, focusing its main interest at the center of such a broad field. Always keeping such a viewpoint in mind, Professor Chikazumi has contributed a great deal to both fundamental physics and applications of magnetism. This is described in Chap. 1 of this book. Many books have been published on both the physics and applications of magnetism. However, no single book has a viewpoint covering both of them. The recent development of high technology needs such a broad viewpoint for scientists and engineers since it is a product of both fundamental science and technology. Research in magnetism is based on the response which materials show to the application of magnetic fields."

An international workshop on Elementary Excitations and Fluctuations in Magnetic Systems was held in Turin for five days beginning 25 May, 1987. The workshop followed much the same format as the one with the same title held in San Miniato in 1984 (proceedings: Springer Series in-Solid-State Sciences, Vol. 54), that most participants contributed talks and provided papers for the proceedings. While many of the participants had attended the first workshop, 15 of the 40 invited review papers were presented by scientists who had not. The majority of the talks reported theoretical work concerned with the introduction of new techniques. However, experimental work was also well represented, not least because many of the reported theoretical studies were motivated by experimental findings, and a highlight of the workshop was an extremely stimulating session devoted to recent neutron scattering measure- ments, on various systems, that exploited polarization analysis. The fine venue of the workshop, Villa Gualino, with its excellent facili- ties and spacious accommodation, helped to produce a delightful relaxed and friendly atmosphere. For the use of Villa Gualino and significant financial support we are indebted to our host organization, the Institute for Scien- tific Interchange (lSI). Additional financial support came from the Consiglio Nazionale delle Ricerche (CNR), Centro Interuniversitario di Struttura della Material del Ministero della Pubblica Istruzione (CISM-MPI) and Gruppo Nazionale di Struttura della Materia (GNSM-CNR).

Speech by Toyosaburo Taniguchi Welcome my friends to the Third International Symposium, Division on the Theory of Condensed Matter, of the Taniguchi Foundation. The need is now greater than ever for Japan to consider how to strengthen and foster international understanding between nations, peoples and societies, and how to contribute towards the establishment of peace and prosperity in the world. For more than twenty years, I have been supporting a symposium on mathe matics in which distinguished scholars from allover the world have engaged in free discussions. In this symposium, all the participants live together in community style. I have heard from members of some of these study groups that this type of setup has helped to strengthen their ties and relationships with their colleagues on a personal basis. What developed in the mathematics group led me to reorganize and strengthen the Taniguchi Foundation only a few years ago through additional funding. In order to effectively translate the objectives of the Foundation into action with the funds available, it becomes necessary to select those fields which are not necessarily in the limelight of popular interest, which means those fields which, I am afraid, are low in funding. I would rather choose from modest unimpressive academic fields than for the Foundation, projects those that stand out in gaudy, gorgeous popular acclaim."

Since the discovery by Bednorz and Muller of Cu-O alloys displaying high temperature superconductivity, great energy has been put into research in this field. One of the most important and interesting issues, and the subject of this volume, is the clarification of the microscopic origin and mechanism of high temperature superconductivity. This book discusses the latest experimental results on magnetic, optical, electrical, thermal and mechanical properties of the Cu-O and Bi-O superconductors, as well as proposed theoretical models of the mechanisms. The participants in the symposium agreed that for the high Tc Cu-O superconductors electron correlation effects are of central importance. For the Bi-O superconductors the main topic was whether the mechanism of superconductivity is the same as that of high Tc Cu-O superconductors. What was and what was not resolved at the symposium is summarized at the end of the volume.

An international workshop on Elementary Excitations and Fluctuations in Magnetic Systems was held in San Miniato, Italy, for five days beginning 28 May, 1984. The workshop comprised eight working sessions that contai- ned a total of 43 invited talks, and 58 scientists were in attendance from 14 countries. Our aim was to review some topics of current interest in the statistical physics of magnetic materials and models, with an emphasis on theoretical studies and confrontations between these and experimental and computer simulation data. book contains summary papers written by the invited speakers, and This the material will be of immediate interest to graduate students and resear- chers engaged in studies of magnetic properties. There is, perhaps, no ef- fective way to record and convey the benefit of the numerous discussions between the participants that are a significant integral feature of a work- shop. The magnificent .venue of the workshop, I Cappuccini, was made availa- ble to us by the.Cassa di Risparmio San Miniato. Financial support for the workshop was received from Consiglio Nazionale delle Ricerche, Universita degli Studi di Firenze and the Gruppo Nazionale Struttura,della Materia. Our administrative load and the burden of preparing the proceedings for publication was made light by the talents of Carla Pardini (CNR, Florence), and Caroline Monypenny and Jane Warren (Rutherford Appleton Laboratory). Fina 11y, we wish to thank all the participants for their attendance and individual contributions to the success of the workshop.

This volume contains the proceedings of the ffiM Japan International Sympo sium on Strong Correlation and Superconductivity, which was held in Keidan ren Guest House at the foot of Mt. Fuji, May 21-25, 1989. The purpose of the Symposium was to provide an opportunity for discus sions on the problem of strong correlation of electrons in the context of high-Tc superconductivity. Sixty-eight scientists were invited from seven countries and forty-three papers were presented in the Symposium. Soon after the discovery ofhigh-Tc superconducting oxides, Professor P. W. Anderson proposed that the essence of high-Tc superconductivity lies in the strong correlation among the electrons in these materials. This proposal has stimulated a wide range of theoretical investigations on this profound and dif ficult problem, which are expected to lead eventually to new concepts describ ing strong electron correlation. In the Symposium, Anderson himself started lively discussions by his talk entitled "Myth and Reality in High-Tc Supercon ductivity," which was followed by various reports on theoretical studies and experimental results. Concise and thoughtful summaries of experiment and theory were given by Professors H. R. Ott and P. A. Lee, respectively. It is our hope that this volume reflects the present status of the research activity on this outstanding problem from the viewpoint of the basic physics and that it will further stimulate the effort to understand these fascinating systems, the high-Tc oxides."

The sixth Taniguchi Symposium on the Theory of Condensed Matter was held between 14-18 November 1983 at Kashikojima. Japan. During the Symposium, about twenty participants lived together and discussed the magnetic super conductors and related problems in an active and friendly atmosphere. This volume contains the papers presented at this Symposium. A strong impetus for organizing a Symposium of this subject is afforded by recent intense interest and accumulated information on magnetic and other novel superconductors newly discovered, and indeed the Symposium has pro duced many excellent contributions to this very exciting field of condensed matter theory, as reported in this volume. In order to give the readers a general outline of the subject, a brief sketch of the problem is made in the Introduction. Then the remainder of this volume is divided into four Parts and an Appendix. Part I is devoted to di scuss ions on several aspects of ferromagnetic superconductors includ ing superconductivity in heavy fermion systems. Part II treats problems on anti ferromagnetic superconductors. In Part III three papers on organic supercon ductors are presented. Part IV includes discussions on the exotic supercon ductors. The Appendix is concerned with the new research project towards high Tc superconductors in Japan. The last but not least remark is to mention the activity of the Taniguchi Foundation whose support makes this Symposium possible. For many years Mr."

The Sixth Annual Conference of the Center for Nonlinear Studies at the Los Alamos National Laboratory was held May 5-9, 1986, on the topic "Nonlinearity in Condensed Matter: Lessons from the Past and Prospects for the Future. " As conference organizers, we felt that the study of non linear phenomena in condensed matter had matured to the point where it made sense to take stock of the numerous lessons to be learned from a variety of contexts where nonlinearity plays a fundamental role and to evaluate the prospects for the growth of this general discipline. The successful 1978 Oxford Symposium on nonlinear (soliton) struc ture and dynamics in condensed matter (Springer Ser. Solid-State Sci., Vol. 8) was held at a time when the ubiquity of solitons was just begin ning to be appreciated by the condensed matter community; in subsequent years the soliton paradigm has provided a rather useful framework for in vestigating a large number of phenomena, particularly in low-dimensional systems. Nevertheless, we felt that the importance of nonlinearity in wider arenas than "solitonics" merited a significant expansion in the scope of the conference over that of the 1978 symposium. Indeed, many of the lessons are quite general and their potential for cross-fertilization of otherwise poorly connected disciplines was certainly one of the prime motivations for this conference. Thus, while these proceedings contain many contribu tions pertaining to soliton behavior in different contexts, the reader will find much more as well, particularly in the later chapters."

Many novel cooperative phenomena found in a variety of systems studied by scientists can be treated using the uniting principles of synergetics. Examples are frustrated and random systems, polymers, spin glasses, neural networks, chemical and biological systems, and fluids. In this book attention is focused on two main problems. First, how local, topological constraints (frustrations) can cause macroscopic cooperative behavior: related ideas initially developed for spin glasses are shown to play key roles also for optimization and the modeling of neural networks. Second, the dynamical constraints that arise from the nonlinear dynamics of the systems: the discussion covers turbulence in fluids, pattern formation, and conventional 1/f noise. The volume will be of interest to anyone wishing to understand the current development of work on complex systems, which is presently one of the most challenging subjects in statistical and condensed matter physics.

This volume deals with physical properties of electrically one-dimensional conductors. It includes both a description of basic concepts and a review of recent progress in research. One-dimensional conductors are those materials in which an electric current flows easily in one specific crystal direction while the resistivity is very high in transverse directions. It was about 1973 when much attention began to be focussed on them and investigations started in earnest. The research was stimulated by the successful growth of crystals of the organic conductor TTF-TCNQ and of the inorganic conductor KCP. New concepts, characteristic of one dimension, were established in the in vestigations of their properties. Many new one-dimensional conductors were also found and synthesized. This field of research is attractive because of the discovery of new ma terials, phenomena and concepts which have only recently found a place in the framework of traditional solid-state physics and materials science. The relation of this topic to the wider field of solid-state sciences is therefore still uncertain. This situation is clearly reflected in the wide distribution of the fields of specialization of researchers. Due to this, and also to the rapid progress of research, no introductory book has been available which covers most of the important fields of research on one-dimensional conductors."

This tract is based on lecture notes for a course in mechanics that has been offered at Rensselaer Polytechnic Institute on and off for the past twenty years. The course is intended to provide graduate students in mechanics with an understanding of electromagnetism and prepare them for studies on the interaction of the electric and magnetic fields with deformable solid continua. As such, it is imperative that the distinction between particle and continuum descriptions of matter be carefully made and that the distinction between that which is inherently linear and that which is intrinsically nonlinear be clearly delineated. Every possible effort has been made on my part to achieve these ends. I wish to acknowledge the contributions of a number of students and faculty who attended the lectures over the years and who, by their questions and suggestions, significantly improved some of the sections. This preface would not be complete if I did not point out that my interest in electromagnetism was initiated and my attitude towards the development of the equations was influenced by lectures given by the late Professor R.D. Mindlin at Columbia University in the late nineteen fifties. I would like to thank Professor C. Truesdell for his helpful suggestions, which I feel significantly improved the clarity and readability of the Introduction, and Dr. M.G. Ancona for his comment concerning the clarity of an important point in Sec. 1.1.

For several years, the two parallel worlds of Molecular Conductors in one hand and Molecular Magnetism in the other have grown side by side, the former essentially based on radical organic molecules, the latter essentially based on the high spin properties of metal complexes. Over the last few years however, organometallic derivatives have started to play an increasingly important role in both worlds, and have in many ways contributed to open several passages between these two worlds. This volume recognizes this important emerging evolution of both research areas. It is not intended to give a comprehensive view of all possible organometallic materials, and polymers for example were not considered here. Rather we present a selection of the most recent research topics where organometallic derivatives were shown to play a crucial role in the setting of conducting and/or magnetic properties in crystalline materials. First, the role of organometallic anions in tet- thiafulvalenium-based molecular conductors is highlighted by Schlueter, while Kubo and Kato describe very recent ortho-metalated chelating ligands appended to the TTF core and their conducting salts. The combination of conducting and magnetic properties and the search for p-d interactions are analyzed in two comp- mentary contributions by Myazaki and Ouahab, while Valade focuses on the only class of metal bis(dithiolene) complexes to give rise to superconductive molecular materials, in association with organic as well as organometallic cations.

This volume is the proceedings of the Tsukuba Institute '87 on Fermi Surface Effects, which was held August 27-29, 1987, at Tsukuba Science City in Japan. The topic of the Institute, Fermi surface effects, is one of the fascinating subjects of solid-state physics. It has been known since Sommerfeld's work that the conduction electrons of metals constitute a degenerate Fermi system, and it has also been recognized that the occu pation number of the electron states has a discontinuity across the Fermi surface. Several basic properties of metal electrons stem from this fact. Furthermore, it gives rise to a singular response of the metal electrons to local and dynamical perturbations of low frequency. Such singular behav ior of the metal electrons is called a Fermi surface effect. In his opening address, printed as the Foreword, Professor R. Kubo described Fermi sur face effects as due to "wild" behavior of the metal electrons. The Institute co sisted of five invited lectures, each of which was two hours long and dealt with theoretical aspects of a subject related to Fermi surface effects. Each lecturer is an expert in the field, and gave an in tensive treatment of his own subject. The experiment of inviting only very few lecturers and allotting them ample time for both presentation and discussion seems to have been successful. This Institute, which was sponsored by the Japan Industrial Technology Association, will probably be followed by other institutes, forming a series."

This book explores the basic principles and methods of paleomagnetology and gives a systematic description of paleomagnetic phenomena such as geomagnetic reversals, paleosecular variations, long-term intensity changes and apparent polar wandering paths. Special emphasis is laid on results obtained from research work done in the Soviet Union. Together with data from other parts of the world they allow the critical discussion of aspects of magnetostratigraphy, plate tectonics and accretion tectonics. In diesem Buch werden grundlegende Fragen der Palaomagnetologie und des Palaomagnetismus abgehandelt. Durch die Einbeziehung zahlreicher Daten aus der Sowjetunion werden Probleme der Magnetostratigraphie sowie der Platten- und Akkretionstektonik kritisch beleuchtet.

The present volume contains the courses given at a Summer School on "Magne tic Phase Transitions" held at the Ettore Majorana Centre for Scientific Culture, at Erice (Trapani), Italy in July 1983 under the auspices of the Condensed Matter Division of the European Physical Society in their series on Materials Science and Technology. The student participants came from West Germany, Great Britain, Brazil, Greece, Switzerland, Sweden, Italy, USA and The Netherlands. The lecturers came from various European countries, Israel, USA and Canada. The atmosphere at the meeting was excellent and a good spirit of companion ship developed during two weeks of working together. The spread of interests among the lecturers and students was divers;jfied but balanced. The main lec turing contributions are reported in this volume. They represent up-to-date reviews in a pedagogical style. In addition, informal presentations on cur rent research interests were made which have not been included. The school attempted to summarize the current position on the properties of magnetic phase transitions from several points of view. The range and scope of the oretical techniques, and of particular aspects of materials or phenomena as observed experimentally were very well put forward by the lecturers. The grouping of manuscripts in chapters does not represent, however, the sched ule followed during the school. Contributions on mean-field approximations and renormalization-group methods either for static or dynamic phenomena can be found at various places in the following sections."

Magnetism in surfaces, heterostructures, small particles, and layered compounds has become an important subject in pure and applied science and a major component of high-technology industrial applications, for example, for magnetic recording and information storage and retrieval. This book presents relevant theoretical techniques and summarizes the optical, electronic, and nuclear experimental techniques used in studying the magnetic properties of such systems. Methods of sample preparation and characterization are also discussed. The book provides an up-to-date account of recent developments in this active interdisciplinary field.

Magnetic resonance has constantly been able to surprise with its ability to exhibit new phenomena. Just when it appears to be entering a quiet middle age it bursts into activity with some new manifestation of its versatility. This happened a few years ago, when observations on anomalous intensities were looked at more closely, and the pursuit of explanations and further evidence laid the foundations of the subjects treated in this volume. In organizing the NATO Advanced Study Institute we attempted to bring together a number, but by no means all, of those who had contributed significantly to the subject, and to obtain from them a comprehensive and detailed exposition of the subject. We were particularly anxious to avoid a set of lectures that dealt solely with the theory of the subject, because much of the interest in chemically induced magnetic polarization is due to its usefulness in applications to chemical problems: it is a real chemical tech nique, not just an amusing diversion for theoreticians. We set about organizing the course with the idea of making it useful to people who wanted to use the technique (for, after all, in the case of nuclear polarization, CIDNP, the technique can be used in any laboratory with minor modification of standard equipment)."

The quantum theory of magnetism is a well-developed part of contemporary solid-state physics. The basic concepts of this theory can be used to describe such important effects as ferromagnetic ordering oflocalized magnetic moments in crystals and ferromagnetism of metals produced by essentially delocalized electrons, as well as various types of mutual orientation of atomic magnetic moments in solids possessing different crystal lattices and compositions. In recent years, the spin-fluctuational approach has been developed, which can overcome some contradictions between "localized" and "itinerant" models in the quantum mechanics of magnetic crystals. These are only some of the principal achievements of quantum magnetic theory. Almost all of the known magnetic properties of solids can be qualitat ively explained on the basis of its concepts. Further developments should open up the possibility of reliable quantitative description of magnetic properties of solids. Unfortunately, such calculations based on model concepts appear to be very complicated and, quite often, not definite enough. The rather small number of parameters of qualitative models are usually not able to take into account the very different types of magnetic interactions that appear in crystals. Further development of magnetic theory requires quantitative information on electronic wave function in the crystal considered. This can be proved by electronic band structure and cluster calculations. In many cases the latter can be a starting point for quantitative calculations of parameters used in magnetic theory."

Starting in 1995 numerical modeling of the Earth's dynamo has ourished with remarkable success. Direct numerical simulation of convection-driven MHD- ow in a rotating spherical shell show magnetic elds that resemble the geomagnetic eld in many respects: they are dominated by the axial dipole of approximately the right strength, they show spatial power spectra similar to that of Earth, and the magnetic eld morphology and the temporal var- tion of the eld resembles that of the geomagnetic eld (Christensen and Wicht 2007). Some models show stochastic dipole reversals whose details agree with what has been inferred from paleomagnetic data (Glatzmaier and Roberts 1995; Kutzner and Christensen 2002; Wicht 2005). While these models represent direct numerical simulations of the fundamental MHD equations without parameterized induction effects, they do not match actual pla- tary conditions in a number of respects. Speci cally, they rotate too slowly, are much less turbulent, and use a viscosity and thermal diffusivity that is far too large in comparison to magnetic diffusivity. Because of these discrepancies, the success of geodynamo models may seem surprising. In order to better understand the extent to which the models are applicable to planetary dynamos, scaling laws that relate basic properties of the dynamo to the fundamental control parameters play an important role. In recent years rst attempts have been made to derive such scaling laws from a set of numerical simulations that span the accessible parameter space (Christensen and Tilgner 2004; Christensen and Aubert 2006).