{"title":"1.4 Thermal expansion","authors":"H. Küppers","doi":"10.1107/97809553602060000903","DOIUrl":"https://doi.org/10.1107/97809553602060000903","url":null,"abstract":"This chapter discusses the reduction in the number of independent tensor components by crystal symmetry, representation surfaces, the quasiharmonic approximation and the Gruneisen relation. Experimental methods including diffraction, optical and electrical methods are presented. Finally, the relation between thermal expansion and crystal structure is discussed. \u0000 \u0000 \u0000Keywords: \u0000 \u0000Gruneisen relation; \u0000acoustic branches; \u0000anharmonicity; \u0000capacitance method; \u0000interferometry; \u0000pushrod dilatometry; \u0000thermal expansion","PeriodicalId":338076,"journal":{"name":"International Tables for Crystallography","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132499443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"3.2 Twinning and domain structures","authors":"V. Janovec, T. Hahn, H. Klapper","doi":"10.1107/97809553602060000916","DOIUrl":"https://doi.org/10.1107/97809553602060000916","url":null,"abstract":"This chapter forms the introduction to the treatment of twinning in Chapter 3.3 and of domain structures in Chapter 3.4 . It starts with a historical overview of twinning (beginning with a paper by Rome de l'Isle from 1783) and continues with the history of the various forms of domain structures: ferromagnetism, ferroelectricity and ferroelasticity, summarized as ferroic by Aizu in 1970. This historical survey is followed by a brief excursion into the rather new field of bicrystallography and grain boundaries. The major part of the chapter is concerned with an extended exposition of the mathematical tools needed in the subsequent parts, especially in Chapter 3.4 . One section introduces the basic concepts of set theory and explains the notion of unordered and ordered pairs, mappings of sets and the partition of a set into equivalence classes. The next section deals with basic group theory and is devoted mainly to group–subgroup relations and relevant notions, of which black-and-white and colour groups and coset decompositions of a group into left and double cosets are of central importance. In the final section, group theory is combined with set theory in the ‘action of a group on a set’ which represents an effective algebraic tool for the symmetry analysis of domain structures. The notions of stabilizer, orbit and stratum are explained and their significance in the analysis is illustrated by concrete examples. Keywords: bicrystallography; bicrystals; black and white symmetry groups; coincidence-site lattice; conjugate subgroups; cosets; daughter phase; dichromatic complexes; dichromatic groups; domain structures; domains; double cosets; equivalence classes; equivalence relation; ferroelectric domain structures; ferroic domains; mappings; normalizers; orbit; parent phases; partition; sets; stabilizers; twinning","PeriodicalId":338076,"journal":{"name":"International Tables for Crystallography","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125483126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Tolédano, V. Janovec, V. Kopský, J. Scott, P. Boček
{"title":"3.1 Structural phase transitions","authors":"J. Tolédano, V. Janovec, V. Kopský, J. Scott, P. Boček","doi":"10.1107/97809553602060000915","DOIUrl":"https://doi.org/10.1107/97809553602060000915","url":null,"abstract":"Aspects of phase transitions in crystals that are of interest to crystallographers are described in this chapter. The chapter starts with a brief introduction aimed at defining the field of structural transitions and the terminology used. The theory of structural phase transitions is then described. This theory relates the symmetry characteristics of the transitions to their physical characteristics. The application of the symmetry principles that derive from this theory is illustrated by the results contained in Tables 3.1.3.1 and 3.1.4.1. The first of these two tables concerns the simple but experimentally widespread situation in which a structural transition is not accompanied by a change in the number of atoms per primitive crystal cell. The second table concerns the general case, in which the number of atoms changes, and which corresponds to the onset of superlattice reflections at the phase transition. This table provides, for a set of hypothetical transformations, the various symmetry-based predictions of the theory. The important topic of soft modes, which is related to the microscopic mechanism of a structural transition, is then discussed. The final section of the chapter is an introduction to the accompanying software package Group Informatics. \u0000 \u0000 \u0000Keywords: \u0000 \u0000Curie temperature; \u0000Landau theory; \u0000Landau–Devonshire theory; \u0000domain states; \u0000enantiomorphism; \u0000equitranslational phase transitions; \u0000equitranslational subgroups; \u0000ferroelastic materials; \u0000ferroelastic phases; \u0000ferroelastic transitions; \u0000ferroelectric materials; \u0000ferroelectric phases; \u0000ferroelectric transitions; \u0000ferroic classes; \u0000ferroic domain states; \u0000ferroic phases; \u0000ferroic single-domain states; \u0000ferroic symmetry; \u0000ferroic transitions; \u0000free energy; \u0000high-symmetry phases; \u0000high-temperature superconductors; \u0000irreducible representations; \u0000low-symmetry phases; \u0000non-equitranslational phase transitions; \u0000order parameter; \u0000parent phases; \u0000parent symmetry; \u0000phase transitions; \u0000physical property tensors; \u0000prototype phases; \u0000soft modes; \u0000superconductors; \u0000tensor parameter","PeriodicalId":338076,"journal":{"name":"International Tables for Crystallography","volume":"481 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121330637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"1.3 Elastic properties","authors":"A. Authier, A. Zarembowitch","doi":"10.1107/97809553602060000902","DOIUrl":"https://doi.org/10.1107/97809553602060000902","url":null,"abstract":"In this chapter, the strain and stress tensors are defined and their main properties are derived. The elastic tensors, elastic stiffnesses and elastic compliances are then introduced. Their variation with orientation, depending on the crystal class, is given in the case of Young's modulus. The next part is devoted to the propagation of waves in continuous media (linear dynamic elasticity and the Christoffel matrix); the relation between the velocity and the elastic constants is given for the cubic, hexagonal and tetragonal classes. The experimental determination of elastic constants and their pressure and temperature dependence are discussed in separate sections. The last two sections of the chapter concerns nonlinear elasticity (second and higher-order elastic constants) and nonlinear dynamical elasticity. \u0000 \u0000 \u0000Keywords: \u0000 \u0000Eulerian description; \u0000Hooke's law; \u0000Lagrangian description; \u0000Poisson's ratio; \u0000Voigt notation; \u0000Young's modulus; \u0000bulk modulus; \u0000compressibility; \u0000cubic dilatation; \u0000dynamic elasticity; \u0000elastic compliances; \u0000elastic constants; \u0000elastic stiffnesses; \u0000elastic strain energy; \u0000elastic waves; \u0000elasticity; \u0000elongations; \u0000energy density; \u0000harmonic generation; \u0000homogeneous deformation; \u0000polarization; \u0000pulse-echo technique; \u0000pulse-superposition method; \u0000resonance technique; \u0000shear; \u0000spontaneous strain; \u0000strain field; \u0000strain tensor; \u0000stress tensor","PeriodicalId":338076,"journal":{"name":"International Tables for Crystallography","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130527804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"1.5 Magnetic properties","authors":"A. Borovik-romanov, H. Grimmer, M. Kenzelmann","doi":"10.1107/97809553602060000904","DOIUrl":"https://doi.org/10.1107/97809553602060000904","url":null,"abstract":"This chapter gives a short review of the structure and some properties of magnetic substances that depend mainly on the symmetry of these substances. Aspects related to the magnetic symmetry receive the most emphasis. The magnetic symmetry takes into account the fact that it is necessary to consider time inversion in addition to the usual spatial transformations in order to describe the invariance of the thermodynamic equilibrium states of a body. The first part of the chapter is devoted to a brief classification of disordered and ordered magnetics. The classification of ferromagnets according to the type of the magnetic structure is given in Section 1.5.1.2.1. In Section 1.5.1.2.2, the antiferromagnets are classified by the types of their magnetic structures: collinear, weakly non-collinear and strongly non-collinear antiferromagnets. Incommensurate structures are briefly mentioned in Section 1.5.1.2.3. Section 1.5.2 is devoted to magnetic symmetry. Different types of magnetic point (Section 1.5.2.1) and magnetic space (Section 1.5.2.3) groups are defined. The 22 magnetic Bravais lattices are displayed in Section 1.5.2.2. The transition from the paramagnetic state into the magnetically ordered state entails a transition from one magnetic group into another. These transitions are considered in Section 1.5.3. The domain structure of ferromagnets and antiferromagnets is considered in Section 1.5.4, where 180° and T-domains are described. Non-collinear antiferromagnetic structures (weakly ferromagnetic, non-collinear and non-coplanar antiferromagnetic structures) are described in Section 1.5.5. Besides the magnetic phase transition from the disordered into the ordered state, there exist transitions from one magnetic structure into another. Those of these that are obtained by a rotation of the ferromagnetic or antiferromagnetic vector relative to the crystallographic axis are called reorientation transitions and are analysed in Section 1.5.6. Sections 1.5.7 and 1.5.8 are devoted to phenomena that can be (and were) predicted only on the basis of magnetic symmetry. These are piezomagnetism (Section 1.5.7) and the magnetoelectric effect (Section 1.5.8). In Section 1.5.9, the magnetostriction in ferromagnets is briefly discussed.Keywords:Bravais lattices;Gaussian system of units;Landau theory;S-domains;SI units;angular phase;anisotropy energy;antiferromagnetic ferroelectrics;antiferromagnetic helical structures;antiferromagnetic phases;antiferromagnetic structures;antiferromagnetic vectors;antiferromagnets;diamagnets;domains;easy-axis magnetics;easy-plane magnetics;exchange energy;exchange symmetry;ferrimagnets;ferroelectric antiferromagnets;ferroelectric materials;ferroic domains;ferromagnetic ferroelectrics;ferromagnetic materials;ferromagnetic vectors;ferromagnetism;ferromagnets;helical structures;incommensurate structures;magnetic Bravais lattices;magnetic anisotropy energy;magnetic birefringence;magnetic fields;magnetic induction;magnetic lattices;m","PeriodicalId":338076,"journal":{"name":"International Tables for Crystallography","volume":"545 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116254935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"2.3 Raman scattering","authors":"I. Gregora","doi":"10.1107/97809553602060000913","DOIUrl":"https://doi.org/10.1107/97809553602060000913","url":null,"abstract":"Raman scattering is due to the inelastic scattering of photons by excitations in molecules, liquids or solids. This chapter deals specifically with the inelastic scattering of photons by lattice vibrations (phonons) in crystals, but the treatment is also applicable to other types of collective excitations in a crystal. The basic notions are introduced in the first section. First-order scattering by phonons is then described. A table summarizes the symmetry properties of Raman tensors in the 32 crystallographic classes. Further sections discuss morphic effects in Raman scattering, spatial-dispersion effects and higher-order scattering. \u0000 \u0000 \u0000Keywords: \u0000 \u0000Raman activity; \u0000Raman scattering; \u0000Raman spectral line shape; \u0000Raman tensor; \u0000dispersion; \u0000inelastic scattering; \u0000longitudinal optic mode; \u0000morphic effects; \u0000optic modes; \u0000phonon bands; \u0000Raman scattering cross section; \u0000selection rules; \u0000susceptibility derivatives","PeriodicalId":338076,"journal":{"name":"International Tables for Crystallography","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115056382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"1.8 Transport properties","authors":"G. D. Mahan","doi":"10.1107/97809553602060000907","DOIUrl":"https://doi.org/10.1107/97809553602060000907","url":null,"abstract":"The flow of either either electricity or heat is regarded as transport. These properties are of tensorial nature and are reviewed in this chapter. The topic is restricted to steady-state flows and to linear response. The three main transport coefficients are the electrical conductivity, the thermal conductivity and the Seebeck coefficient. Section 1.8.3 concerns the electrical resistivity of metals and semiconductors and the Hall effect. Section 1.8.4 concerns the thermal conductivity. Heat flow can be carried by two kinds of excitations: phonons and electrons. The cases of boundary scattering, impurity and defect scattering, isotope scattering and alloy scattering are distinguished as well as anharmonic interactions. The last section, Section 1.8.5, describes the properties of the Seebeck coefficient.","PeriodicalId":338076,"journal":{"name":"International Tables for Crystallography","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121295134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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