Siegfried Haussuhl Physical Properties of Crystals An Introduction

Siegfried Haussühl Physical Properties of Crystals An Introduction WILEY-VCH Verlag GmbH & Co KGaA This Page Intentionally Left Blank Siegfried Haussühl Physical Properties of Crystals 1807–2007 Knowledge for Generations Each generation has its unique needs and aspirations When Charles Wiley first opened his small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity And we were there, helping to define a new American literary tradition Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused on building the future Once again, we were there, supplying the critical scientific, technical, and engineering knowledge that helped frame the world Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own borders and a new international community was born Wiley was there, expanding its operations around the world to enable a global exchange of ideas, opinions, and know-how For 200 years, Wiley has been an integral part of each generation’s journey, enabling the flow of information and understanding necessary to meet their needs and fulfill their aspirations Today, bold new technologies are changing the way we live and learn Wiley will be there, providing you the must-have knowledge you need to imagine new worlds, new possibilities, and new opportunities Generations come and go, but you can always count on Wiley to provide you the knowledge you need, when and where you need it! William J Pesce President and Chief Executive Officer Peter Booth Wiley Chairman of the Board Siegfried Haussühl Physical Properties of Crystals An Introduction WILEY-VCH Verlag GmbH & Co KGaA The Authors Prof Dr Siegfried Haussühl Institute of Crystallography University of Cologne Zülpicher Str 49b 50674 Cologne Germany All books published by Wiley-VCH are carefully produced Nevertheless, authors, editors, and publisher not warrant the information contained in these books, including this book, to be free of errors Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate Library of Congress Card No.: applied for Translation Peter Roman, Germany British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Nationalbibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.d-nb.de © 2007 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim All rights reserved (including those of translation into other languages) No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers Registered names, trademarks, etc used in this book, even when not specifically marked as such, are not to be considered unprotected by law Printed in the Federal Republic of Germany Printed on acid-free paper Printing: Strauss GmbH, Mörlenbach Bookbinding: Litges & Dopf Buchbinderei GmbH, Heppenheim Wiley Bicentennial Logo: Richard J Pacifico ISBN: 978-3-527-40543-5 v Contents Fundamentals 1.1 1.2 1.3 1.4 1.7.5 1.7.5.1 1.7.5.2 1.7.5.3 1.8 1.9 Ideal Crystals, Real Crystals The First Basic Law of Crystallography (Angular Constancy) Graphical Methods, Stereographic Projection The Second Basic Law of Crystallography (Law of Rational Indices) Vectors 10 Vector Addition 10 Scalar Product 13 Vector Product 14 Vector Triple Product 17 Transformations 18 Symmetry Properties 19 Symmetry Operations 19 Point Symmetry Groups 24 Theory of Forms 32 Morphological Symmetry, Determining the Point Symmetry Group 42 Symmetry of Space Lattices (Space Groups) 42 Bravais Types 42 Screw Axes and Glide Mirror Planes 45 The 230 Space Groups 46 Supplements to Crystal Geometry 47 The Determination of Orientation with Diffraction Methods 48 Sample Preparation 51 2.1 2.2 Crystal Preparation 51 Orientation 54 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.6 1.7 1.7.1 1.7.2 1.7.3 1.7.4 ¨ Physical Properties of Crystals Siegfried Haussuhl Copyright c 2007 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40543-5 vi 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.8.1 3.8.2 3.9 3.10 Definitions 57 Properties 57 Reference Surfaces and Reference Curves 59 Neumann’s Principle 60 Theorem on Extreme Values 61 Tensors 62 Theorem on Tensor Operations 65 Pseudo Tensors (Axial Tensors) 70 Symmetry Properties of Tensors 72 Mathematical and Physical Arguments: Inherent Symmetry 72 Symmetry of the Medium 74 Derived Tensors and Tensor Invariants 78 Longitudinal and Transverse Effects 80 Special Tensors 83 4.1 4.2 4.2.1 4.2.2 4.3 4.3.1 4.3.2 Zero-Rank Tensors 83 First-Rank Tensors 85 Symmetry Reduction 85 Pyroelectric and Related Effects 86 Second-Rank Tensors 89 Symmetry Reduction 89 Tensor Quadric, Poinsots Construction, Longitudinal Effects, Principal Axes’ Transformation 93 Dielectric Properties 99 Ferroelectricity 106 Magnetic Permeability 108 Optical Properties: Basic Laws of Crystal Optics 112 Reflection and Refraction 118 Determining Refractive Indices 127 Plane-Parallel Plate between Polarizers at Perpendicular Incidence 4.3.3 4.3.4 4.3.5 4.3.6 4.3.6.1 4.3.6.2 4.3.6.3 130 4.3.6.4 4.3.6.5 4.3.6.6 4.3.6.7 4.3.6.8 4.3.6.9 4.3.7 4.3.8 4.3.9 4.3.10 Directions of Optic Isotropy: Optic Axes, Optic Character 133 S´enarmont Compensator for the Analysis of Elliptically Polarized Light 136 Absorption 139 Optical Activity 141 Double refracting, optically active, and absorbing crystals 148 Dispersion 148 Electrical Conductivity 150 Thermal Conductivity 152 Mass Conductivity 153 Deformation Tensor 154 vii 4.3.11 4.3.12 4.3.13 4.4 4.4.1 4.4.1.1 4.4.1.2 4.4.1.3 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 4.5.7 4.5.7.1 4.5.8 4.5.9 4.5.9.1 4.5.9.2 4.5.10 4.5.11 4.5.12 4.6 4.6.1 4.6.2 4.6.3 5.1 5.2 5.3 5.4 Thermal Expansion 159 Linear Compressibility at Hydrostatic Pressure 164 Mechanical Stress Tensor 164 Third-Rank Tensors 168 Piezoelectric Tensor 173 Static and Quasistatic Methods of Measurement 174 Extreme Values 180 Converse Piezoelectric Effect (First-Order Electrostriction) 182 First-Order Electro-Optical Tensor 184 First-Order Nonlinear Electrical Conductivity (Deviation from Ohm’s Law) 194 Nonlinear Dielectric Susceptibilty 195 Faraday Effect 204 Hall Effect 205 Fourth-Rank Tensors 207 Elasticity Tensor 214 Elastostatics 217 Linear Compressibility Under Hydrostatic Pressure 220 Torsion Modulus 221 Elastodynamic 222 Dynamic Measurement Methods 231 Strategy for the Measurement of Elastic Constants 266 General Elastic Properties; Stability 267 The Dependence of Elastic Properties on Scalar Parameters (Temperature, Pressure) 270 Piezooptical and Elastooptical Tensors 271 Piezooptical Measurements 272 Elastooptical Measurements 273 Second-Order Electrostrictive and Electrooptical Effects 285 Electrogyration 286 Piezoconductivity 288 Higher Rank Tensors 288 Electroacoustical Effects 288 Acoustical Activity 289 Nonlinear Elasticity: Piezoacoustical Effects 290 Thermodynamic Relationships 297 Equations of State 297 Tensor Components Under Different Auxiliary Conditions 301 Time Reversal 305 Thermoelectrical Effect 307 12.6 Bibliography ¨ B W Mott Die Mikroh¨arteprufung Berliner Union, Stuttgart 1957 M J.P Musgrave Crystal acoustics Holden-Day, San Francisco 1970 T S Narasimhamurty Photoelastic and electro-optic properties of crystals Plenum Press, New York, London 1981 R E Newnham Structure-property relations Springer, Berlin 1975 A Nowick Crystal properties via group theory Cambridge University Press, Cambridge 1995 J F Nye Physical properties of crystals Oxford University Press, London, New York 1957 P Paufler Phasendiagramme Akademie-Verlag, Berlin 1981 P Paufler Physikalische Kristallographie Akademie-Verlag, Berlin 1986 ¨ P Paufler, G E.R Schulze Physikalische Grundlagen mechanischer Festkorpereigenschaften I u II Vieweg, Braunschweig 1978 F Pockels Lehrbuch der Kristalloptik Teubner, Leipzig 1906 H Poulet, J P Mathieu Vibration spectra and symmetry of crystals Gordon and Breach, New York 1976 H Rabin, C L Tang (eds.) Quantum electronics: a treatise Vol I, Nonlinear Optics, Parts A and B Academic Press, New York 1975 G N Ramachandran, S Ramaseshan Crystal optics In: Handbuch der Physik ¨ (S Flugge ed.) Vol XXV/1 Springer, Berlin 1967 R Rath Theoretische Grundlagen der allgemeinen Kristalldiagnose im durchfallenden Licht Springer, Berlin 1969 Lord Rayleigh The theory of sound Dover, New York 1945 C Z Rosen, B V Hiremath, R Newnham (eds.) Key papers in physics: Piezoelectricity American Institute of Physics, New York 1992 ¨ A Seeger Kristallplastizit¨at In: Handbuch der Physik (S Flugge ed.) Vol VII/2 Kristallphysik II, pp 1–210 Springer, Berlin 1958 J N Sherwood (ed.) The plastically crystalline state Wiley, Chichester 1979 L A Shuvalov (ed.) Modern Crystallography IV (Physical properties of crys¨ tals) Ubers aus d Russ Springer, Berlin 1983 425 426 12 Appendix G A Smolenskij, N N Krajnik Ferroelektrika und Antiferroelektrika Teubner, Leipzig 1972 U Stille Messen und Rechnen in der Physik Fr Vieweg & Sohn, Braunschweig 1955 G Szivessy Kristalloptik In: Handbuch der Physik Vol XX Springer, Berlin 1928 H Tertsch Die Festigkeitserscheinungen der Kristalle Springer, Wien 1947 J Tich´y, G Gautschi Piezoelektrische Meßtechnik Springer, Berlin 1980 H F Tiersten Linear piezoelectric plate vibrations Plenum Press, New York 1969 R Truell, C Elbaum, B B Chick Ultrasonic methods in solid state physics Academic Press, New York, London 1969 W Voigt Lehrbuch der Kristallphysik B Teubner, Leipzig 1910 (2 Aufl 1928) D C Wallace Thermoelastic theory of stressed crystals and higher-order elastic constants In: Solid State Physics, Vol 25, pp 301–404 Academic Press, New York, London 1970 D C Wallace Thermodynamics of crystals Wiley, New York 1972 L M Wasiliev Physikalische Kristallographie (in Russian) Metallurgia, Moscow 1981 W A Wooster A text-book on crystal physics Cambridge University Press, Cambridge 1949 W A Wooster Experimental crystal physics Oxford University Press, New York 1957 W A Wooster Diffuse X-ray reflections from crystals Clarendon Press, Oxford 1962 W A Wooster Tensors and group theory for the physical properties of crystals University Press, Oxford 1973 G S Zhdanov Crystal physics Oliver & Boyd, Edinburgh, London 1965 I S Zheludev Physics of crystalline dielectrics, I & II Plenum Press, New York, London 1971 ¨ ¨ J M Ziman Prinzipien der Festkorpertheorie Harri Deutsch, Zurich, Frankfurt a.M., 1975 12.6 Bibliography 12.6.2 Articles K Aizu Possible species of ”ferroelastic crystals” and of simultaneously ferroelectric and ferroelastic crystals J Phys Soc Japan 27 (1969) 387–396 C Andeen, J Fontanella, D Schuele Low-frequency dielectric constants of the alkaline earth fluorides by the method of substitution J Appl Phys 42 (1971) 2216–2219 R Bechmann Contour modes of square plates excited piezoelectrically and determination of elastic and piezoelectric coefficients Proc Phys Soc (London) B64 (1951) 323–337 P S Bechthold Quantitative Interpretation der Maker-Interferenzen bei der Frequenzverdopplung in optisch zweiachsigen Kristallen Diss Uni¨ 1976 versity Koln L Bohat´y, S Haussuhl ¨ Quadratic electrostrictive effects in LiF, NaF, NaCl, NaBr, KCl, KBr, KI, RbCl, RbBr and RbI Acta Crystallogr A33 (1977) 114–118 K Brugger Thermodynamic definition of higher order elastic coefficients Phys Rev A133 (1964) 1611–1612 M J Buerger Crystallographic aspects of phase transformations In: Phase Transformations in Solids, pp 183–209 Wiley, New York 1951 M J Buerger Phase transformations Sov Phys Crystallography 16 (1972) 959–968 C F Buhrer, L Ho, J Zucker Electrooptic effect in optically active crystals Appl Optics (1964) 517–521 S Chandra, M P Hemkar Evaluation of the elastic constants of 1,3,5-triphenylbenzene from thermal diffuse scattering of X-rays Acta Cryst A29 (1973) 25–28 H H Demarest Cube-resonance method to determine the elastic constants of solids J Acoust Soc Am 49 (1969) 768–775 R W Dixon Acoustic diffraction of light in anisotropic media IEEE J Quant Electron QE-3 (1967) 85–93 H J Eichler Forced light scattering at laser induced gratings–a method for investigation of optically excited solids Adv Solid State Phys XVIII (1978) 241–263 427 428 12 Appendix W.v Engelhardt, S Haussuhl ¨ Schleiffestigkeit und spezifische freie Grenzfl¨achenenergie der Alkalihalogenide vom NaCl-Typus Kolloid Zeitschr 173 (1960) 20–35 W.v Engelhardt, S Haussuhl ¨ Festigkeit und H¨arte von Kristallen Fortschr Miner 42 (1965) 5–49 P P Ewald, C Hermann Strukturbericht 1913–1928 Z Kristallogr Erg¨anzungsband 1931 Akad Verlagsgesellschaft, Leipzig 1931 R L Forgacs Improvements in the sing-around technique for ultrasonic velocity measurements J Acoust Soc Amer 32 (1960) 1697–1698 ¨ J Gahm Ein neuer Mikroh¨arteprufer Zeiss Informationen 14 Jahrgang (1966) 120–127 M W Guinan, A D Ritchie Evaluation of third-order elastic constants for cubic crystals J Appl Phys 41 (1970) 2256–2258 U Hauser, V Oestreich, M D Rohrweck On optical dispersion in transparent molecular systems Z Physik A280 (1977) 17–25, 125–130 et ibid A284 (1978) 9–19 S Haussuhl ¨ Physikalische Kristallographie von Calcium- und Cadmiumformiat Z Kristallogr 118 (1963) 33–50 S Haussuhl ¨ Die Abweichungen von den Cauchy-Relationen Phys kondens Materie (1967) 181–192 S Haussuhl ¨ Elastische und thermoelastische Konstanten von Benzil C6 H5 COCOC6 H5 , gemessen mit dem Schaefer-Bergmann-Verfahren Acta Cryst 23 (1967) 666–667 S Haussuhl ¨ Elastic and thermoelastic properties of 1,3,5-triphenyl-benzene derived from ultrasonic wave velocities Acta Cryst A30 (1974) 106–107 S Haussuhl ¨ Ferroelasticity and phase transformation in Rb2 Hg(CN)4 spinel Acta Cryst A34 (1978) 965–968 S Haussuhl ¨ Pyroelectric, dielectric, piezoelectric, elastic, and thermoelastic properties of triclinic lithium hydrogen oxalate hydrate, LiHC2 O4 · H2 O VII Intern Conf on Crystal Growth, Stuttgart 1983 ¨ S Haussuhl, ¨ W Muller ¨ Zur Auflosungsgeschwindigkeit von Kristallen in Wasser Kristall und Technik (1972) 533–554 12.6 Bibliography S Haussuhl, ¨ H Siegert Bestimmung des Elastizit¨atstensors trikliner Kristalle: Beispiel CuSO4 · 5H2 O Z Kristallogr 129 (1969) 142–246 S Haussuhl, ¨ K Uhl Die Bestimmung des piezooptischen Tensors monokliner Kristalle: Beispiel Taurin Z Kristallogr 128 (1969) 418–423 M V Hobden Phase-matched second-harmonic generation in biaxial crystals J Appl Phys 38 (1967) 4365–4372 A R Hutson, D L White Elastic wave propagation in piezoelectric semiconductors J Appl Phys 33 (1962) 40–47 J Jerphagnon A contribution to the study of certain non-linear optical phenomena Ann Telecommunications 23 (1968) 203–232 J Jerphagnon, S K Kurtz Maker Fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals J Appl Phys 41 (1970) 1667–1681 R C Jones A new calculus for the treatment of optical systems VII Properties of the N-matrices J Opt Soc Amer 38 (1948) 671–685 S K Joshi, B M.S Kashyap Determination of elastic constants of benzalazine from thermal diffuse scattering of X-rays Acta Cryst 17 (1964) 629–632 D A Kleinman Theory of second harmonic generation of light Phys Rev 128 (1962) 1761–1775 E Kr¨oner Calculation of the elastic constants of a polycrystalline material from the elastic constants of the single crystal Z Physik 151 (1958) 504– 518 H Kuppers ¨ Die Messung von Schallgschwindigkeiten in optisch anisotropen Medien mit dem Schaefer-Bergmann-Verfahren Acta Cryst A27 (1971) 316–322 J J Kyame Wave propagation in piezoelectric crystals J Acoust Soc Amer 21 (1949) 159–167 R G Leisure, F A Willis Resonant ultrasound spectroscopy J Phys Condens Matter (1997) 6001–6029 429 430 12 Appendix P D Maker, E W Terhune, M Nisenoff, C M Savage Effects of dispersion and focussing on the production of optical harmonics Phys Rev Letts (1962) 21–22 H J McSkimin Pulse superposition method for measuring ultrasonic wave velocities in solids J Acoust Soc Amer 33 (1961) 12–16 R Meier, K Schuster Zur Theorie der Schallausbreitung in piezoelektrischen Kristallen Ann Physik 11 (1953) 397–406 H Mueller Determination of elasto-optical constants with supersonic waves Z Kristallogr 99 (1938) 122–141 D F Nelson, M Lax New symmetry for acousto-optic scattering Phys Rev Letters 24 (1970) 379–380 D F Nelson, M Lax Theory of the photoelastic ineraction (1971) 2778–2794 Phys Rev B3 R Nitsche, V Kr¨amer, M Schuhmacher, A Bussmann Preparation and properties of ferroelastic/ferroelectric polytypes of antimony(III) oxide iodide Sb5 O7 I J Cryst Growth 42 (1977) 549–559 I Ohno Free vibration of a rectangular parallelepiped crystal and its application to determination of elastic constants of orthorhombic crystals J Phys Earth 24 (1976) 355-379 M Onoe, H F Tiersten, A H Meitzler Shift in the location of resonant frequencies caused by large electromechanical coupling in thickness-mode resonators J Acoust Soc Amer 35 (1963) 36–42 P Preu, S Haussuhl ¨ Dielectric properties and phase transition in Li2 Ge7 O15 Solid State Comm 41 (1982) 627–630 P Preu, S Haussuhl ¨ Quadratic electrostrictive effects in NaCl and KAl(SO4 )2 ·12H2 O derived from stress dependence of dielectric constants Solid State Comm 45 (1983) 619–623 N W Schubring, J P Nolta, R A Dork Ferroelectric hysteresis tracer featuring compensation and sample grounding J Sci Instr 35 (1964) 1517–1521 A Seeger, O Buck Die experimentelle Ermittlung der elastischen Konstanten ¨ hoherer Ordnung Z Naturforsch 15a (1960) 1056–1067 L A Shuvalov Symmetry aspects of ferroelectricity J Phys Soc Japan 28 (1970) 38–51 12.6 Bibliography 431 R N Thurston, K Brugger Third-order elastic constants and the velocity of small amplitude elastic waves in homogenously stressed media Phys Rev A133 (1964) 1604–1610 R W Whatmore New polar materials: their application to SAW (surface acoustic waves) and other devices J Cryst Growth 48 (1980) 530–547 ¨ H Wondratschek Uber Tensorsymmetrien in den einzelnen Kristallklassen N Jahrbuch Miner Jahrgang 1952, 217–234 I S Zheludev The change of complete symmetry of the crystals during the phase transitions in ferroelectrics and ferromagnetics Proc Ind Acad Sci LIX (1964) 168–184 12.6.3 Data Sources Th.J Ahrens (ed.) Mineral physics & crystallography: A handbook of physical constants American Geophysical Union, Washington 1995 S L Altmann, P Herzig Point-group theory tables Clarendon Press, Oxford 1994 B P Belikov, K S Alexandrov, T V Ryshova Elastische Eigenschaften von gesteinsbildenden Mineralen und Gesteinen In russ Sprache NAUK publishing, Moscow 1970 M G Graham, H E Hagy (eds.) Thermal Expansion—1971 AIP Conf Proceed No American Institute of Physics, New York 1972 P Groth Chemische Krystallographie Vol I-V V W Engelmann, Leipzig 1906–1919 Landolt-B¨ornstein, Aufl Springer, Berlin 1959–1971 ¨ Vol II/1 : Mechanisch-thermische Zustandsgroßen (1971) ¨ Vol II/4 : Kalorische Zustandsgroßen (1961) Vol II/6 : Elektrische Eigenschaften I (1959) Vol II/8 : Optische Konstanten (1962) Vol II/9 : Magnetische Eiegnschaften I (1962) Vol II/10 : Magnetische Eigenschaften II (1967) 432 12 Appendix Landolt-B¨ornstein, New Series Springer, Berlin 1966–2000 Vol III/1 : Elastic, piezoelectric, piezooptic and electrooptic constants of crystals (1966) Vol III/2 : Supplement and extension to III/1 (1969) Vol III/3 : Ferro and antiferroelectric substances 1969) Vol III/4a,b : Magnetic and other properties of oxides and related compounds (1970) Vol III/9 : Ferro and antiferroelectric substances (1975) Vol III/11 : Elastic, piezoelectric, pyroelectric, piezooptic, electrooptic constants, and nonlinear dielectric susceptibilities of crystals (1979) Vol III/12a,b,c : Magnetic and other properties of oxides and related compounds (1978-1981) Vol III/16a,b : Ferroelectrics and related substances (1981, 1982) Vol III/17a,b,c : Semiconductors (1981, 1982) Vol III/18a,b : Ferroelectrics and related substances (1990) Vol III/29a : Low frequency properties of dielectric crystals: Second and higher order elastic constants (1992) Vol III/29b : Low frequency properties of dielectric crystals: Piezoelectric, pyroelectric and related constants (1993) Vol III/30a : High frequency properties of dielectric crystals: Piezooptic and electrooptic coefficients and nonlinear dielectric susceptibilities (1996) Vol III/30b : High frequency properties of dielectric crystals: Nonlinear dielectric susceptibilities (2000) Vol IV/4 : High-pressure properties of matter (1980) Vol V/1 : Physical properties of minerals and rocks (1981) R E Taylor, G L Denman (eds.) Thermal expansion—1973 AIP Conf Proceed No 17 American Institute of Physics, New York 1974 Y S Touloukian, C Y Ho (eds.) Thermophysical properties of matter Vols 1– 6, 12 and 13 The TPRC Data Series IFI/Plenum, New York, Washington 1970 and 1972 A N Winchell The optical properties of organic compounds Academic Press, New York 1954 A N Winchell, H Winchell The microscopic character of artificial inorganic solid substances Academic Press, New York 1964 12.6 Bibliography 12.6.4 Journals Acta Crystallographica Crystal Research and Technology Journal of Applied Crystallography Journal of Crystal Growth Kristallografiya (Soviet Physics Crystallography) physica status solidi Solid State Communications ¨ Kristallographie Zeitschrift fur In addition, almost all Journals in physics and physical chemistry contain articles on crystal physics 433 This Page Intentionally Left Blank 435 Index α Quartz, 178, 271 abrasive hardness, 318 absolute measurement, 272 acoustical activity, 289 acute bisectrix, 134 adiabatic temperature increase, 301 adsorption energy, 321 angular constancy, anisotropic crystals, 334 anisotropy, 334 antiferroelectricity, 108 antiferromagnetism, 110 axial angle, 133 basic law of crystaloptics, 117 basis vector, beam bending, 219 black–white groups, 48 boundary surface energy, 318, 320 boundary surface energy, relative, 320 Bragg condition, 235 Bragg method, 49 Bravais indices, 47 Bravais types, 42, 44 – multiple-primitive, 43 – primitive, 43 Brillouin scattering, 231, 260 – stimulated, 262 capacitive dilatometer, 160, 219 Cartesian coordinate system, Cauchy relations, 217, 342 character, 360 Christoffel determinant, 224 class, 359 Clausius–Clapeyron relation, 350 Clausius–Mossotti formula, 106, 149, 335 color groups, 48 combination elastic wave, 225 conjugated element, 359 converse piezoelectric effect, 182 correlation of properties, 329 coupling factors, 228 crystal – ideal, – real, crystal class, 24 crystal systems, 32 crystallographic reference system, cubic crystal system, 31 Curie–Weiss law, 108 cyclic groups, 366 Debye–Waller factor, 265 deformation tensor, 155 deformation tensor, mechanoelectric, 301 determination of orientation, 48 ¨ Physical Properties of Crystals Siegfried Haussuhl Copyright c 2007 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40543-5 436 Index deviation from Ohm’s law, 194 diamagnetism, 110 dielectric tensor, 100, 113 – pressure dependence, 301 dipyramid, 36 direct matrix product, 364 dislocations, 314 dispersion, 148 disphenoid, 36 displacement tensor, 155 dissolution, 323 dome, 36 dynamic elasticity, 226 dynamic measurement methods, 231 elastic constants – measurement strategy, 266 elastic properties, 332, 341 elastic stability, 267 elasticity tensor, 214 elastodynamic basic equations, 223 elastodynamics, 222 elastooptical tensor, 271, 273 elastostatics, 217 electrical conductivity tensor, 150 electroacoustical effects, 288 electrooptic effect – second-order, 285 electrooptic tensor, 185 elektrogyration, 286 equations of state, 297 equivalent matrix, 360 Ewald construction, 275 Faraday effect, 204, 335 ferrimagnetism, 110 ferroelastics, 311 ferroelectricity, 108 ferromagnetism, 110 first basic law of crystallography, first-order electrostriction, 182 first-order transformations, 351, 352 forced vibrations of piezoelectric crystals, 244 frequency doubling, 196 Fresnel formula, first, 123 Fresnel formula, second, 123 Gibbs potential, 298 Gibbs’ condition, glide directions, 311 glide lines, 311 glide mirror planes, 45 glide plane, 46, 311 glide system, 311 ¨ Gruneisen constant, 333 ¨ Gruneisen relation, 333 ¨ Gruneisen tensor, 402 group, 357 – isomorphic, 359 – representations, 358 group algebra, 385 group table, 358 group theory, 357 gyration vector, 142 habit, Hall effect, 205 hardening, 314 hardness, 310 Hermann-Mauguin Symbole, 25 higher-rank tensors, 288 Hobden classes, 197 Hooke’s law, 214, 290 hysteresis, 352 ideal crystal, idempotents, 386 immersion method, 101 indentation hardness, 315 index equation, 115 index surface, 202 Index indicatrix, 116 inductive dilatometer, 159 inner Kronecker product, 364 interpretation of properties, 329 invariant linear subspace, 371 irreducible representation, 361 irreducible representation, construction, 364 Jamin interferometer, 187 Kleinman’s rule, 200, 301 Kusnezow method, 318 Lagrangian deformation tensor, 156 λ transformation, 354 Lambert’s absorption law, 139 Landau theory, 351 Laplace equation, 379 lateral contraction, 219 lattice energy, 336 Laue groups, 409 Laue method, 49 law of angular constancy, law of rational indices, 8, law of reflection, 119, 122 law of refraction, 122 Levi-Civit´a tensor, 217, 342 LiHSeO3 (PSG 222), 243 line groups, 47 linear compressibility, 220 linear compressibility tensor, 164 linear subspace, 387 linear vector space, 371 list of symbols, 405 lithium iodate, 179 lithium niobate, 178 longitudinal elastic wave, 225 Lorentz–Lorenz formula, 149, 284, 335 magnetic permeability tensor, 110, 113 magnetic point symmetry groups, 305 magnetic susceptibility, 335 magnetic susceptibility tensor, 109 Maker interferences, 197, 198 mass attenuation coefficient, 336 mass conductivity, 153 mass permeability tensor, 154 Maxwell relations, 115 – first-order, 300 – second–order, 300 Maxwell’s equations, 113 mechanical stress tensor, 164 meroedries, 34 metric, Michelson interferometer, 182 micro hardness tester, 317 Miller indices, 10 Mohs hardness, 323 mole polarization, 335 mole volume, 336 morphological symmetry, 42 Neutron scattering, 231, 263 non-tensorial properties, 309 nonadditive properties, 338 nonlinear dielectric susceptibility, 195 nonlinear elasticity, 290 nonlinear electrical conductivity, 194 obtuse bisectrix, 134 Onsager principle, 150 Onsager’s principle, 152 optic axes, 118, 133 optic character, 134 optical absorption, 139 optical activity, 141 437 438 Index optical interference dilatometer, 160 optically biaxial, 118 optically isotropic, 118 optically negative, 134 optically positive, 134 optically uniaxial, 118 optorotation, 277 order parameter, 351 order–disorder transformations, 352 potassium bromate, 178 precision measurement of lattice constants, 161 principal axes’ transformation, 95 principal refractive indices, 115 prism, 36 projection operators, 385, 388 pulse–echo methods, 231 pyramid, 36 pyroelectric effect, 86 pyroelectric groups, 86 paramagnetism, 110 parametric oscillator, 201 pedion, 36 phase matching, 196, 201 phase transformation, 313 phase transformations, 347, 348 phonon dispersion curve, 263 photochromatic effect, 355 physical properties, 332 piezoacoustical effect, 290 piezoconductivity, 288 piezodielectric tensor, 284 piezoelastic constants, 270 piezoelastic properties, 344 piezoelectric effect, 173 piezoelectric effect, extreme values, 180 piezoelectric effect, second-order, 301 piezooptical tensor, 271 pinacoid, 36 plane groups, 47 plane principal axis transformation, 98 Poinsot construction, 94 point symmetry group, 24, 29, 30, 408 – determination, 42 Poisson’s ratio, 220 polarization constants, 116 quasiadditive properties, 331 real crystals, reciprocal system, 15 refractive index, 115 regular representation, 360 resonances of plates and rods, 231, 239 resonant ultrasound spectroscopy (RUS), 255 Reuss case, 400 rotation, 21 rotation tensor, 155 rotoinversion, 21 sawing velocity, 324 scalar product, 13 scalar triple product, 15 scalenohedron, 36 Schaefer–Bergmann method, 231, 234 scratch hardness, 323 screw axes, 45 second basic law of crystallography, second harmonic generation, 196 second-order electrostriction, 285, 301 second-order optical activity, 148 second-order transformations, 351, 354 Index Sellmeier equation, 149 sing-around-method, 234 skalares Produkt, 13 Snellius’ law of refraction, 119 soft modes, 270 space groups, 42, 46 specific heat, 84 specific weight, 83 spectroscopic properties, 326 sphenoid, 36 spherical harmonics, 379 static and quasistatic methods of measurement, 174 stereographic projection, 4–6 strain gauge, 161, 219 strength, 317 strength properties, 309 streptohedron, 36 structure and properties, 329 substitution method, 102 superconductivity, 339 symmetry reduction – first-rank tensors, 85 – fourth-rank tensors, 207 – second-rank tensors, 89 – tensors of nonlinear elasticity, 291 – third-rank tensors, 168 tensor components under auxilliary conditions, 301 tensor representation, 370 theory of forms, 32 thermal conductivity, 152 thermal conductometer, 153 thermal expansion, 159, 339 thermal scattering, 231 thermoelastic constants, 270 thermoelastic properties, 344 thermoelectrical effect, 307 thickness resonator, 232 thiourea, 354 third basic law of crystallography, 25 time reversal, 305 torsion modulus, 221 total reflection, 124 trace, 359 tracht, transformation, dilative, 351 transformation, displacive, 351 transformation, reconstructive, 351 transformation, rotative, 351 translation, 20 transverse elastic wave, 225 twin formation, 47, 311 unitary matrix, 360 units, 407 vector addition, 10 vector product, 14 vector transformations, 18 vector triple product, 17 vectors, 10 Vegard’s rule, 337 Verdet constant, 335 Vickers hardness, 316 Voigt case, 400 Voigt notation, 216 volume compressibility, 220, 336 wedge method, 197, 198 whiskers, 222, 317 Wulff’s theorem, X-cut, 232 X-ray scattering, 231, 264 Y-cut, 232 Young’s modulus, 218 zero-cut, 271 zinc blende, 179 439