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Calculation of the Properties of Vacancies and Interstitials, Skyland UNITED STATES DEPARTMENT OF COMMERCE Alexander B. Trowbridge, Secretary • NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director Calculation of the Properties of Vacancies and Interstitials Proceedings of a Conference Shenandoah National Park, Va. May 1-5, 1966 Supported in part by the Advanced Research Projects Agency National Bureau of Standards Miscellaneous Publication 287 Issued November 17, 1966 For sale by the Superintendent of Documents, U.S. Government Printing Office. Washington, D.C 20402 Price $2.50 Abstract This is the Proceedings of a Conference on the Calculation of the Properties of Vacancies and Interstitials, Skyland. Virginia, USA, held on May 1-4, 1966. The Conference dealt with the theory and techniques of calculation of the properties of point defects in metallic and nonmetallic crystals. The contributed and invited papers divided about evenly among three major topics: (1) static-lattice calculations of the energies and configurations of simple vacancies and interstitials in, mainly, metals and ionic crystals; (2) electronic states at and near point defects in metals, rare gas solids, and insula- tors (/-centers, electron traps); and (3) vibrational states at point defects. The report of a panel dis- cussion on each topic is also included. The emphasis is on the theory of the properties of isolated, simple defects rather than on the statistical properties of defect assemblies. The Conference attempted to examine the point defect theory and calculations critically, from the standpoint of general theory, rather than simply compare results with experiment. Key Words: Calculations, electronic states, energies of formation, energies of motion, interstitials, point defects, theory, vacancies, vibrational states. Library of Congress Catalog Card Number: 67-60032 Foreword The Institute for Materials Research of the National Bureau of Standards has as a major responsibility the task of ensuring that basic, urgently needed data on the properties of materials are available to meet the requirements of the Na- tions' scientists and engineers. Data in this sense is a rather broad term. Not only does it imply experimentally measured numbers, but it also includes theoretical data, such as the calculated properties of point defects in crystals, which formed the subject of this Conference. In an immediate sense, this responsibility is met in part through measurements done in Institute laboratories, in part through compilation and publication of critically evaluated data gathered from the literature. It is also necessary for the Institute to take a long-range view, to look further ahead, and to try to help in stimulating fields from which data of the future will come. The Conference on the Calculation of the Properties of Vacancies and Interstitials was part of that long-range effort. The field is a rapidly growing one, containing diverse elements, related to each other in subject and often in method, but being pursued to some extent independently. No conference bringing together these diverse elements had previously been held on this topic. Thus it appeared that the present Conference could perform an important function in providing an opportunity for a critical examination of the whole field, and a forum for interchange of ideas and discussion of problems at a most opportune time for such an interchange. The Institute presents these Proceedings of the Conference in order to make the results available to a wider audience. The Institute would like to express appreciation to the participants, whose time, energy, and thought are embodied in these Proceedings, and to the Advanced Research Projects Agency, whose sponsorship made the Conference possible. Gordon K. Teal Director, Institute for Materials Research iii Introduction The Conference on the Calculation of the Properties of Vacancies and Interstitials was held at Skyland, on the Skyline Drive in the Shenandoah National Park, Virginia, on May 1-4, 1966. It brought together approximately 70 scientists from the United States, Great Britain, Japan. West Germany, France, Italy, and the Netherlands, to discuss theoretical problems concerning isolated point defects in crystalline solids. Prior to this conference, the theory and calculation of the properties of point defects had not been the sole topic of such a gathering. The field, however, had become an important one, and seemed ready to benefit by the concentrated discus- sions a conference of this kind makes possible. Several apparently unrelated currents of research bearing on this topic have developed in recent years, and it appeared desirable to try to bring their practitioners together. Essentially static-lattice calculations for metals and nonmetals have been quite successful in obtaining estimates of defect formation energies, and to a lesser extent, migration energies as well. These calculations have passed the initial rough-approximation state, and rather sophisticated problems involving interatomic potentials, electron redistributions, and atomic configurations about the defects are now being attacked. Calculations of electronic states in the vicinity of a point defect are also making progress. Those working on color centers in ionic crystals are struggling to find ways of dealing with polarization effects and with problems presented by the overlap of the defect wave function with the neighboring ions. The problem in metals is the collective electron one in the presence of a sizable perturbation in the periodic crystal potential. Rather similar methods are being developed to handle vibrational and electronic states induced by the presence of point defects in otherwise periodic lattices; in both cases scattering theory is proving to be of great value. These three general topics, static-lattice energies and configurations, electronic states, and vibrational states, formed the program of the conference. The object was to try to assess and compare the reliability of calculations of these proper- ties from a theoretical point of view; not simply to compare the results with experiment, but to examine the calculations critically, from general theory. The conference concentrated on the theory of isolated defects rather than of statistical properties of assemblies of defects, so that topics like radiation damage or diffusion do not appear as such. As the Table of Contents shows, the conference program was actually subdivided into three sections corresponding to these three topics. Within each section, contributed papers on current research plus several invited talks were presented. Then a panel discussion was held on the topic at hand, in which an attempt was made to summarize and comment upon what the contributed papers and their discussion had revealed. It was hoped in this way to provide some synthesis, some overview, of the topic. These proceedings consist of the contributed papers, arranged by topic, and followed by a report on the panel discussion on the topic. These reports were prepared by the several panel chairmen, and each of course bears the stamp of its author's approach to the problem presented. We believe this use of panels to have been an interesting experiment in communication among scientists, and hope these reports will provide guidance and stimulation to others involved in similar experiments. All conferences require a good deal of work on the part of many p'eople. The editor would like to take this opportunity to call attention to the indispensable efforts of those whose work made the conference possible. The brunt was borne by R. E. Howard (Institute for Materials Research, NBS) and R. F. Wood (Oak Ridge National Laboratory), and important contributions were made by A. B. Lidiard (United Kingdom Atomic Energy Research Establishment, Harwell). C. McCombie (Reading University), A. Seeger (Max Planck Institut fur Metallforschung, Stuttgart), G. Vineyard (Brookhaven National Laboratory), and J. A. Krumhansl (Cornell University). Messrs Lidiard, McCombie, and Krumhansl served as panel chair- men as well. V Contents Page Foreword iii Realistic Fermi Surfaces and Scattering Mech- Introduction anisms 75 I. Energies and Configurations A. Seeger and E. Mann Interatomic Potentials and Defect Calculations in Ionic Electrical Resistivity of Impurities and Vacancies in Crystals* Aluminum 97 M. P. Tosi Y. Fukai The Energy of Formation of Schottky Defects in Ionic Structure Factor Approach to Energies of Migration 99 Crystals H. B. Huntington and M. D. Feit /. M. Boswarva and A. B. Lidiard Perturbed Bloch Functions Obtained From Koster- Recent Calculations Concerning Point Defects in Alkali Slater Perturbation Theory 101 B. Mozer .Halides , „ 11 Resistivities of Dilute Aluminum Alloys 109 G. ./. Dienes, R. D. Hatcher, R. Smolchowski, and Residual W. D. Wilson F. J. Blatt and H. R. Fankhauser Energy of Formation of Vacancy Pairs in KC1 15 Point Imperfections in Solid Rare Gases Ill K. Tharmalingam A. Gold in Vacancies and Monovalent Cation Impurities in the The Use and Misuse of Models The Calculation of Alkali Halides 17 Energy Levels in Insulators 115 M. Doyama and M. P. Tosi W. B. Fowler Methods for Calculation of The Electronic Structure Born — Model Defect Energies for CaF 2 19 A. D. Franklin of Defects in Insulators 119 F. Lattice Distortion Around Defects 21 U. Opik and R. Wood Model to Treat Lattice Distortions and The F- T. Kurosawa A Center Electron Consistently 123 Calculation of Lattice Distortion Around Point De- H. Bennett fects by Lattice Statics 23 S. H. Kanzaki /-Matrix Theory of Localized Electronic
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