Holdings: Symmetry and physical properties of crystals

Symmetry and physical properties of crystals: [selected by Grenoble Sciences]

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Bibliographic Details
Main Authors:	Malgrange, Cécile (Author), Ricolleau, Christian (Author), Schlenker, Michel (Author)
Format:	Book
Language:	English
Published:	Dordrecht [u.a.] Springer 2014
Subjects:	Kristallphysik
Online Access:	Inhaltsverzeichnis Klappentext
Physical Description:	XXV, 522 S. Ill., graph. Darst.
ISBN:	9401789924 9789401789929

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adam_text	Contents Chapter 1 Introduction 1 1.1. Crystal order ..................................................................... 1 1.2. Order on a macroscopic scale ................................................ 3 1.3. Order on a microscopic scale ................................................ 4 1.4. Basic assumptions of geometrical crystallography ..................... 6 1.5. Amsotropy of physical properties .......................................... 6 1.6. Remarks on the scope of this book : topics left out ................... 7 Complement 1С. Crystal growth .................................................... 8 ICI. Natural growth of single crystals ................................. 8 1С.2. Crystal growth for scientific purposes ........................... 8 1C.3. Industrial crystal growth ........................................... 9 1С.4. Growth of single-crystal thin films ............................... 10 Further reading .................................................................. 10 References ................................................................................. 11 Chapter 2 — Symmetry operations 13 2.1. Isometries ......................................................................... 13 2.2. Symmetry operations. Symmetry elements .............................. 14 2.2.1. ті- fold rotations and axes ............................................ 14 2.2.2. η -fold rotoinversions and η -fold rotoinversion axes, denoted as ñ-fold axes ............................................... 15 2.2.3. Translations ............................................................ 16 2.2.4. Screw axes and glide mirrors ...................................... 17 2.2.5. Note ...................................................................... 18 2.3. Introduction to symmetry groups .......................................... 18 2.4. Exercises .......................................................................... 20 Chapter 3 - Crystal lattices 23 3.1. Direct lattice ..................................................................... 23 3.1.1. Unit cell, row .......................................................... 23 3.1.2. Lattice planes ....................................-.................... 28 XVI Symmetry and Physical Properties of Crystals 3.1.3. Wigner-Seitz ceil ...................................................... 32 3.2. Reciprocal lattice ............................................................... 32 3.2.1. Introduction based on diffraction ................................ 32 3.2.2. Alternative definition of the reciprocal lattice ................ 34 3.2.3. Properties of the reciprocal lattice ............................... 35 3.2.4. CrystallographAc calculations ...................................... 37 3.3. Properties of crystal lattices ................................................. 40 3.3.1. Centers of symmetry ................................................ 40 3.3.2. τι- fold and ñ-fold axes consistent with the crystalline state 40 3.3.3. The direct lattice and the reciprocal lattice have the same symmetry elements ...................................... 41 3.3.4. Geometrical relation between the symmetry axes and the crystal lattice ............................................... 42 3.3.5. The lattice is at least as symmetrical as the crystal ........ 43 3.4. Crystal systems ................................................................. 43 3.4.1. Two-dimensional crystal systems ................................. 43 3.4.2. Three-dimensional crystal systems ............................... 44 3.5. Examples of reciprocal lattices .............................................. 47 3.5.1. Mono clinic lattice .................................................... 47 3.5.2. Orthorhornbic, tetragonal and cubic lattices .................. 47 3.6. Hexagonal lattice and rhombohedral lattice ............................. 48 3.6.1. Hexagonal lattice ..................................................... 48 3.6.2. Rhombohedral lattice ............................................... 48 3.7. Bravais lattices .................................................................. 50 3.7.1. Why they are needed ................................................ 50 3.7.2. The fourteen Bravais lattices ....................................... 52 3.7.3. Reciprocal lattices of the non-primitive lattices .............. 53 3.8. Surface crystal lattice ......................................................... 54 3.8.1. Cut surface ............................................................. 55 3.8.2. Real surface ............................................................ 55 3.8.3. Notations ................................................................ 56 3.8.4. Reciprocal lattice .................................................... - 58 3.9. Exercises .......................................................................... 59 Appendix ЗА. The metric tensor ................................................... 63 3A.1. Definition ............................................................... 63 ЗА. 2. Volume of the unit cell .............................................. 63 3A.3. Product of the matrices associated to the direct and reciprocal metric tensors ___................................. 64 ЗА. 4. Calculation of lattice distances ................................... 65 ЗА. 5. Applications ............................................................ 65 Contents______________________________ XVI I Chapter 4 — Relationship between space groups and point groups 67 4.1. Introduction ...................................................................... 67 4.2. Symmetry operations of the crystal ....................................... 70 4.2.1. Change in origin ...................................................... 70 4.2.2. The operations (S, t) form a group .............................. 71 4.2.3. The lattice translations form an invariant subgroup of the group of symmetry operations of the crystal ......... 72 4.3. Space groups and point groups ............................................. 73 4.4. Exercises .......................................................................... 75 Complement 4C. Probes used for crystal structure determination ......... 76 4C . 1. Possible probes and criteria for choice .......................... 76 4C.2. X-rays .................................................................... 76 4C.3. Neutrons ................................................................ 78 4C.4. Electrons ................................................................ 79 Further reading .................................................................. 79 Appendix 4A. General features of groups ........................................ 80 Chapter 5 Point groups 83 5.1. Introduction ...................................................................... 83 5.2. Stereographic projection ...................................................... 84 5.2.1. Definition ............................................................... 84 5.2.2. Examples ............................................................... 86 5.2.3. Application to rotoinversion axes or ñ-fold axes ............. 88 5.2.4. Family of equivalent directions ................................... 89 5.3. Improper groups ................................................................ 89 5.3.1. Preliminary remark .................................................. 89 5.3.2. Properties of improper groups .................................... 90 5.4. Enumeration of the proper point groups ................................. 91 5.4.1. Preamble ................................................................ 91 5.4.2. Groups containing only the symmetry operations associated to an axis An (cyclic groups) ....................... 92 5.4.3. Groups containing the symmetry operations associated to an axis An and to a perpendicular axis A2, or dihedral groups .................................................... 92 5.4.4. Cubic proper groups ................................................. 93 5.5. Enumeration of the improper point groups .............................. 94 5.5.1. Improper groups containing inversion ........................... 94 55.2. Improper groups which do not feature inversion ............. 97 5.6. Classification of the point groups .......................................... 101 XVIII Symmetry and Physical Properties of Crystals 5.7. Laue classes ...................................................................... 106 5.8. Plane point groups ............................................................. 107 5.9. Isotropy point groups .......................................................... 107 5.10. Exercises .......................................................................... 109 Appendix 5A. Complements on the ster e o graphic projection ............... Ill öA.l. Stereogxaphic projection of the transform of a given direction through the symmetry operations associated to various symmetry elements .................................... Ill 5 A. 2- Ster eographie projections of the symmetry elements of a cube ................................................................ 112 Chapter 6 — Bravais lattices 117 6.1. Introduction ...................................................................... 117 6.2. Plane lattices .................................................................... 118 6.2.1. Group 1 ................................................................. 118 6.2.2. Group 3 ................................................................. 118 6.2.3. Group 4 ................................................................. 119 6.2.4. Group m ................................................................ 119 6.2.5. Conclusion .............................................................. 120 6.3. S-dimensional lattices .......................................................... 120 6.3.1. Group 1 ................................................................. 120 6.3.2. Group 2................................................................. 120 6.3.3. Group 3 ................................................................. 122 6.3.4. Group 4 ................................................................. 124 6.3.5. Group 222 .............................................................. 126 6.3.6. Group 23 ................................................................ 127 Chapter 7 — Space groups 129 7.1. Introduction ...................................................................... 129 7.2. Enumeration of the operations (S, t) ...................................... 130 7.2.1. S is a rotation. Definition of screw axes ........................ 131 7.2.2. S is a rotoinversion, noted Š. Definition of glide mirrors .. 134 7.2.3. Product of a symmetry operation and a translation ........ 136 7.3. Enumeration of space groups ................................................ 139 7.3.1- Symmorphic space groups .......................................... 139 7.3.2. Non-symmorphic space groups .................................... 142 7.3.3. International Tables for Crystallography ....................... 146 74. Nomenclature .................................................................... 149 7.5. Examples: space groups of some structures ............................. 151 7.5.1. TiCb-type structure (rutile) ....................................... 151 Contents ________________ XIX 7.5.2. Metals with close-packed hexagonal structure ................ 152 7.5.3. Structure of diamond ................................................ 154 7.6. Exercises .......................................................................... 155 Complement 7C. Structure determination: outline ............................ 158 Further reading .................................................................. 161 References ................................................................................. 1 G2 Chapter 8 — Chemical bonds and crystal structures 163 8.1. Introduction ...................................................................... 163 8.2. Ionic bonds ....................................................................... 165 8.2.1. Nature and properties ............................................... 165 8.2.2. Bonding energy ....................................................... 168 8.2.3. Ionic structures with formula AX ................................ 169 8.2.4. Other cubic structures .............................................. 171 8.3. Covalent bonds .................................................................. 173 8.3.1. Nature of covalent bonds ........................................... 173 8.3.2. Basic property ......................................................... 171 8.3.3. Examples ............................................................... 171 8.3.4. Conclusion .............................................................. 17f> 8.4. Van der Waals bonds, or molecular bonds ............................... 177 8.4.1. Nature and properties ............................................... 177 8.4.2. Example ................................................................ 177 8.5. Metallic bonds ................................................................... 178 8.5.1. Nature and properties ............................................... 178 8.5.2. Examples ............................................................... 178 8.6. Remarks and conclusions ..................................................... 180 8.7. Exercises .......................................................................... 181 Complement 8C. Magnetic structures ............................................. 183 Further reading .................................................................. 185 References ......................................................................-.......... 186 Chapter 9 - Crystal anisotropy and tensors 187 9.1. Introduction ...................................................................... 387 9.2. Anisotropie continuous medium —......................................· - 188 9.3. Representing a physical quantity by a tensor ........................... 189 9.3.1. Example: electrical conductivity ................................. 189 9.3.2. A refresher on orthonormal frame changes .................... 191 9.3.3. Application to electrical conductivity ........................... 193 9.4. Tensors ............................................................................ 194 9.4.1. Definition of a tensor ................................................ 194 XX Symmetry and Physical Properties of Crystals 9.4.2. An important property ............................................. 195 9.4.3. Field tensors and material tensors ............................... 196 9.5. Symmetry properties of tensors ............................................. 197 9.5.1. Internal symmetry. Symmetric and antisymmetric tensors 197 9.5.2. External symmetry of material tensors. Curie s and Neumann s principles ................................ 197 9.6. Reduction in the number of independent coefficients of a material tensor ............................................................ 200 9.6.1. Method using the transformation matrix ...................... 200 9.6.2. Direct inspection method ........................................ 201 9.6.3. Special case: central symmetry (inversion symmetry) ...... 202 9.7. Exercises .......................................................................... 203 References ................................................................................. 204 Chapter 10 — Second-rank tensors 205 10.1. Introduction ...................................................................... 205 10.1.1. Symmetric and antisymmetric tensors .......................... 205 10.1.2. Matrix form of second-rank tensors ............................. 206 10.1.3. Trace ..................................................................... 206 10.2. Representative quadric for a symmetric rank-2 tensor ............... 207 10.2.1. Characteristic surface ............................................... 207 10.2.2. Principal axes and principal coefficients ........................ 207 10.2.3. Shape of the quadric ................................................. 209 10.3. Properties of the quadric ..................................................... 210 10.3.1. Normal to the quadric .............................................. 210 10.3.2. Length and physical meaning of the radius vector .......... 211 10.3.3. Intensity of a physical property in a given direction ........ 211 10.4- Geometrical determination of the principal axes and principal coefficients: the Mohr circle construction ................................ 213 10.5. Effect of crystal symmetry ................................................... 215 10.5.1. Triciinic system ....................................................... 216 10.5.2. Monoclinic system .................................................... 216 10.5.3- Orthorhombic system ............-.................................. 216 10.5.4. Uniaxial systems: rhombohedral, trigonal and hexagonal . 217 10.5.5. Cubic system .......................................................... 217 10.6. Axial vectors, or antisymmetric rank-2 tensors ......................... 218 10.6.1. Polar vectors, axial vectors ........................................ 218 10.6.2. Example of an axial vector: the cross product ............... 220 10.7. Exercises .......................................................................... 222 Contents XXI Chapter 11 — The stress tensor 225 11.1. Introduction ...................................................................... 225 11.2. Stress tensor ..................................................................... 225 11.2.1. Introduction ........................................................... 225 11.2.2. Definition of the stress tensor ..................................... 227 11.2.3. Normal stress and shear stress .................................... 229 11.3. Basic relation .................................................................... 230 11.4. Symmetry of the stress tensor ............................................... 233 11.5. Examples of stress tensors ................................................... 234 11.5.1. Uniaxial stress ......................................................... 234 11.5.2. Pure shear .............................................................. 235 11.5.3. Hydrostatic pressure ................................................. 235 11.6. Effect of gravity ................................................................. 236 11.7. Exercises .......................................................................... 238 References ................................................................................. 240 Chapter 12 —Deformation of a solid. The strain tensor 241 12.1. Distortion tensor (displacement gradient tensor) ...................... 241 12.1.1. Definition ............................................................... 241 12.1.2. Physical meaning of components c¡} ............................ 243 12.2. Decomposition of the distortion tensor into a rotation and a strain .................................................. 244 12.2.1. Introduction using a simple example ............................ 244 12.2.2. Expressing the distortion associated to small rotations __ 245 12.2.3. Strain tensor ........................................................... 246 12.3. Elongation in a given direction ............................................. 247 12.4. Volume expansion (volumetric strain) .................................... 248 12.5. Special cases of strain ......................................................... 249 12.5.1. Simple elongation .................................................... 249 12.5.2. Pure shear strain ..................................................... 249 12.5.3. Simple shear strain ................................................... 250 12.6. Thermal expansion ............................................................. 250 12.7. Exercises .......................................................................... 254 Chapter 13 -Elasticity 259 13.1. Introduction ...................................................................... 259 13.2. Compliance and stiffness tensors ........................................... 262 13.2.1. Generalized Hooke s law ............................................ 262 13.2.2. Symmetry of the compliance and stiffness tensors ........... 263 XXII Symmetry and Physical Properties of Crystals 13.3. Contracted notation (Voigt s notation) ................................... 264 13.3.1. Stress tensor ........................................................... 264 13.3.2. Strain tensor ........................................................... 265 13.3.3. Compliance tensor and stiffness tensor ......................... 265 13.3.4. Relation between the compliance and stiffness tensors ..... 268 13.4. Energy of a strained solid .................................................... 268 13.5. Effect of crystal symmetry on the form of the elastic tensor ........ 271 13.5.1. Center of symmetry .................................................. 274 13.5.2. Groups 2, m and 2/m ............................................... 274 13.5.3. Groups 222, mmm aad mm2 ...................................... 274 13.5.4. Groups 422, 4mm and 4/mmm .................................. 275 13.5.5. Cubic system .......................................................... 276 13.6. Isotropie materials .............................................................. 276 13.6.1. Expressing components saß as a function of E and z/ ...... 278 13.6.2. Stiffness components. Lamé coefficients ........................ 279 13.7. Representative surface for Young s modulus ............................ 279 13.8. Compressibility .................................................................. 281 13.8.1. Bulk compressibility ................................................. 281 13.8.2. Linear compressibility of a bar .................................... 282 13.9. Notes on non-uniform stresses and strains ............................... 283. 13.10. Exercises .......................................................................... 285 Complement 13C. Plastic deformation and crystal defects .................. 288 Further reading .................................................................. 290 References ................................................................................. 291 Chapter 14 — Elastic waves in crystals 293 14.1. Introduction ....................................................................... 293 14.2. Plane elastic waves ........................................................,__ 294 14.3. Application to a cubic crystal ............................................... 297 14.3.1. Propagation of a plane wave along direction [100] ........... 298 14.3.2. Propagation along direction [110] ................................ 299 14.4. Isotropie solid case ............................................................. 300 14.5. Microscopic approach - Crystal lattice dynamics .................___ 301 14.5.1. Linear chain of identical atoms ................................... 301 14.5.2. Linear chain with two different atoms .......................... 305 14.5.3. Extension to the real crystal case ................................ 308 14-6. Exercises .............................................................................. 309 Contents__________________________________________ ________ XXIII Chapter 15 Crystal thermodynamics. Piezoelectricity 311 15.1. Crystal thermodynamics ...................................................... 312 15.1.1. Conjugate variables .................................................. 312 15.1-2- Independent variables ............................................... 314 15.1.3. Principal effects vs crossed effects ............................... 317 15.1.4. Summary of the various effects ................................... 318 15.1.5. Condensed representation of the physical property matrix 320 15.2. Pyroelectricity. Pyroelectric crystals ...................................... 321 15.3. Piezoelectric crystals ........................................................... 322 15.3.1. Direct effect and inverse effect .................................... 322 15.3.2. Piezoelectric tensor [d] and its two-subscript notation ..... 323 15.3.3. Effect of crystal symmetry on the form of the tensor ....... 324 15.3.4. Longitudinal piezoelectricity surface ............................ 328 15.3.5. Other forms of piezoelectric coefficients ........................ 329 15.3.6. Applications ........................................................... 331 15.4. Principal and crossed effects under various conditions ............... 332 15.5. Exercises .......................................................................... 334 Complement 15C. Electrostriction and magnetostriction .................... 337 Further reading .................................................................. 338 References ................................................................................. 339 Chapter 16 Light propagation in crystals 341 16.1. Maxwell s equations ............................................................ 341 16.2. Light propagation in an isotropie material .............................. 342 16.3. Sinusoidal waves that are solutions of Maxwell s equations ......... 343 16.4. Plane monochromatic wave in an anisotropic material ............... 345 16.4.1. Basic equation ......................................................... 345 16.4.2. Birefringence ........................................................... 346 16.4.3. Index surface .......................................................... 347 16.4.4. Index ellipsoid ......................................................... 349 16.4.5. Determining the induction vectors ............................... 350 16.4.6. Direction of energy propagation .................................. 353 16.5. Refraction of a plane wave at the boundary between two materials 354 16-51. The wave-vectors follow the Snell-Descartes law ............. 354 16.5-2. Application to uniaxial materials ................................ 357 16.6. Conclusion ........................................................................ 358 16.7. Exercises .......................................................................... 360 XXIV Symmetry and Physical Properties of Crystals Appendix 16A. Wave surface (or ray surface) and Huygens construction 364 16A.1 . Wave surface (от тау surface) .................................... 364 16A.2. Huygens construction ............................................. 367 Chapter 17 —Polarization of light by crystals 369 17.1. Polarization state of an electromagnetic wave .......................... 369 17.1.1. Linearly polarized wave ............................................. 370 17.1.2. Circularly polarized wave .......................................... 370 17.1.3. Elliptically polarized wave ......................................... 372 17.1.4. Natural light ........................................................... 372 17.2. Jones notation ................................................................... 373 17.3. Linear polarizers ................................................................ 374 17.4. Phase-shifting plates ........................................................... 376 17.4.1. Half-wave plates ...................................................... 377 17.4.2. Quarter-wave plates ................................................. 378 17.5. Partially polarized waves and Stokes parameters ...................... 379 17.6. Exercises .......................................................................... 382 References ................................................................................. 384 Chapter 18 —Rotatory power and optical activity 385 18.1. Definition of the rotatory power of a material .......................... 385 18.2. Fresnel s interpretation ........................................................ 386 18.3. Interpretation of rotatory power through the influence of the local environment ...................................................... 388 18.3.1. Effect of spatial dispersion ......................................... 388 18.3.2. Wave propagation in optically active crystals ................ 390 18.4. Effect of crystal symmetry on the gyrotropy tensor .................-- 393 18.4.1. Centrosymmetric groups ............................................ 393 18.4.2. Non-centrosymmetric groups ...................................... 394 18.5. Rotatory power and chirality ................................................. 396 18.6. Absorption and conclusion ................................................... 400 Complement 18C. Magnetic optical rotation .....-.............................. 401 Further reading .................................................................. 402 Appendix 18A. Axial tensors, or pseudo-tensors ............................... 403 ΙδΑ.Ι. Definition of axial tensors, or pseudo- tensors ................. 403 18A.2. Levi-Civita tensor, or permutation tensor ..............-....... 403 18A-3- The gyrotropy tensor [O] is a rank-2 axial tensor ............ 404 18A.4. Relation between tensors [G] and [β] ............................ 405 References ................................................................................. 406 Contents_______________________________________________________ XXV Chapter 19 - Electro-optical and elasto-optical effects 407 19.1. Introduction ...................................................................... 407 19.2. Electro-optical effects .......................................................... 408 19.2.1. Linear electro-optical effect, от Pockels effect ................. 408 19.2.2. Applications of the linear electro- opti cal effect ............... 415 19.2.3. Quadratic electro-optical effect, or Kerr effect ................ 420 19.3. Elasto-optical effects ........................................................... 424 19.3.1- Definition ............................................................... 424 19.3.2. Application to the acousto-optical effects ...................... 427 19.4. Exercises .......................................................................... 430 Complement 19C. Frequency doubling, or Second Harmonic Generation 433 Further reading .................................................................. 434 Chapter 20 — Solutions to the exercises 435 20.1. Exercises for Chapter 2 ....................................................... 435 20.2. Exercises for Chapter 3 ....................................................... 436 20.3. Exercises for Chapter 4 ....................................................... 444 20.4. Exercises for Chapter 5 ....................................................... 445 20.5. Exercises for Chapter 7 ....................................................... 450 20.6. Exercises for Chapter 8 ....................................................... 456 20.7. Exercises for Chapter 9 ....................................................... 458 20.8. Exercises for Chapter 10 ...................................................... 459 20.9. Exercises for Chapter 11 ...................................................... 461 20.10. Exercises for Chapter 12 ...................................................... 463 20.11. Exercises for Chapter 13 ...................................................... 470 20.12. Exercises for Chapter 14 ...................................................... 475 20.13. Exercises for Chapter 15 ...................................................... 479 20.14. Exercises for Chapter 16 ...................................................... 487 20.15. Exercises for Chapter 17 ...................................................... 492 20.16. Exercises for Chapter 19 ...................................................... 499 General references 507 Index 511 Field Theory of Non-Equilibrium Systems The physics of non-equilibrium many-body systems is one of the most rapidly expanding areas of theoretical physics. Traditionally used in the study of Laser physics and superconducting kinetics, these techniques have more recently found applications in the study of dynamics of cold atomic gases, mesoscopic, and nano-mechanicalsystems. The book gives a self-contained presentation of the modern functional approach to non-equilibrium field-theoretical methods. They are applied to examples ranging from biophysics to the kinetics of superfluids and superconductors. Its step-by-step treatment gives particular emphasis to the pedagogical aspects, making it ideal as a reference for advanced graduate students and researchers in condensed matter physics. ALEX KAM EN EV is a Professorin the WilLiam I. FineTheoreticaL Physics Institute (FTPI) and in the Department of Physics, University of Minnesota. Sincegaining his Ph.D. at the Weizmann Instituteof Sciencein 1996, he has held a postdoctoral position atthe University of California at Santa Barbara and the Assistant Professorship at the Israellnstitute of Technology (Technion).
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author	Malgrange, Cécile Ricolleau, Christian Schlenker, Michel
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author_facet	Malgrange, Cécile Ricolleau, Christian Schlenker, Michel
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discipline	Physik
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id	DE-604.BV042027545
illustrated	Illustrated
indexdate	2024-07-10T01:10:53Z
institution	BVB
isbn	9401789924 9789401789929
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-027469058
oclc_num	900413431
open_access_boolean
owner	DE-20 DE-355 DE-BY-UBR DE-29T DE-384
owner_facet	DE-20 DE-355 DE-BY-UBR DE-29T DE-384
physical	XXV, 522 S. Ill., graph. Darst.
publishDate	2014
publishDateSearch	2014
publishDateSort	2014
publisher	Springer
record_format	marc
spelling	Malgrange, Cécile Verfasser (DE-588)1065197322 aut Symétrie et propriétés physiques des cristaux Symmetry and physical properties of crystals [selected by Grenoble Sciences] Cécile Malgrange ; Christian Ricolleau ; Michel Schlenker Dordrecht [u.a.] Springer 2014 XXV, 522 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Kristallphysik (DE-588)4165768-8 gnd rswk-swf Kristallphysik (DE-588)4165768-8 s DE-604 Ricolleau, Christian Verfasser (DE-588)1065199368 aut Schlenker, Michel Verfasser aut Erscheint auch als Online-Ausgabe 978-94-017-8993-6 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027469058&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027469058&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext
spellingShingle	Malgrange, Cécile Ricolleau, Christian Schlenker, Michel Symmetry and physical properties of crystals [selected by Grenoble Sciences] Kristallphysik (DE-588)4165768-8 gnd
subject_GND	(DE-588)4165768-8
title	Symmetry and physical properties of crystals [selected by Grenoble Sciences]
title_alt	Symétrie et propriétés physiques des cristaux
title_auth	Symmetry and physical properties of crystals [selected by Grenoble Sciences]
title_exact_search	Symmetry and physical properties of crystals [selected by Grenoble Sciences]
title_full	Symmetry and physical properties of crystals [selected by Grenoble Sciences] Cécile Malgrange ; Christian Ricolleau ; Michel Schlenker
title_fullStr	Symmetry and physical properties of crystals [selected by Grenoble Sciences] Cécile Malgrange ; Christian Ricolleau ; Michel Schlenker
title_full_unstemmed	Symmetry and physical properties of crystals [selected by Grenoble Sciences] Cécile Malgrange ; Christian Ricolleau ; Michel Schlenker
title_short	Symmetry and physical properties of crystals
title_sort	symmetry and physical properties of crystals selected by grenoble sciences
title_sub	[selected by Grenoble Sciences]
topic	Kristallphysik (DE-588)4165768-8 gnd
topic_facet	Kristallphysik
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027469058&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027469058&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
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