International Tables for Crystallography (2013). Vol. D. ch. 1.1, pp. 3-34
https://doi.org/10.1107/97809553602060000900

Chapter 1.1. Introduction to the properties of tensors

Contents

1.1. Introduction to the properties of tensors (pp. 3-34) | html | pdf | chapter contents |
A. Authier
- 1.1.1. The matrix of physical properties (pp. 3-5) | html | pdf |
  - 1.1.1.1. Notion of extensive and intensive quantities (p. 3) | html | pdf |
  - 1.1.1.2. Notion of tensor in physics (pp. 3-4) | html | pdf |
  - 1.1.1.3. The matrix of physical properties (p. 4) | html | pdf |
  - 1.1.1.4. Symmetry of the matrix of physical properties (pp. 4-5) | html | pdf |
  - 1.1.1.5. Onsager relations (p. 5) | html | pdf |
- 1.1.2. Basic properties of vector spaces (pp. 5-7) | html | pdf |
  - 1.1.2.1. Change of basis (p. 5) | html | pdf |
  - 1.1.2.2. Metric tensor (p. 5) | html | pdf |
  - 1.1.2.3. Orthonormal frames of coordinates – rotation matrix (pp. 5-6) | html | pdf |
  - 1.1.2.4. Covariant coordinates – dual or reciprocal space (pp. 6-7) | html | pdf |
    - 1.1.2.4.1. Covariant coordinates (p. 6) | html | pdf |
    - 1.1.2.4.2. Reciprocal space (p. 6) | html | pdf |
    - 1.1.2.4.3. Properties of the metric tensor (pp. 6-7) | html | pdf |
- 1.1.3. Mathematical notion of tensor (pp. 7-10) | html | pdf |
  - 1.1.3.1. Definition of a tensor (p. 7) | html | pdf |
    - 1.1.3.1.1. Linear forms (p. 7) | html | pdf |
    - 1.1.3.1.2. Tensor product (p. 7) | html | pdf |
  - 1.1.3.2. Behaviour under a change of basis (p. 7) | html | pdf |
  - 1.1.3.3. Operations on tensors (pp. 7-8) | html | pdf |
    - 1.1.3.3.1. Addition (p. 7) | html | pdf |
    - 1.1.3.3.2. Multiplication by a scalar (p. 8) | html | pdf |
    - 1.1.3.3.3. Contracted product, contraction (p. 8) | html | pdf |
  - 1.1.3.4. Tensor nature of physical quantities (p. 8) | html | pdf |
  - 1.1.3.5. Representation surface of a tensor (pp. 8-9) | html | pdf |
    - 1.1.3.5.1. Definition (p. 8) | html | pdf |
    - 1.1.3.5.2. Representation surfaces of second-rank tensors (pp. 8-9) | html | pdf |
    - 1.1.3.5.3. Representation surfaces of higher-rank tensors (p. 9) | html | pdf |
  - 1.1.3.6. Change of variance of the components of a tensor (p. 9) | html | pdf |
    - 1.1.3.6.1. Tensor nature of the metric tensor (p. 9) | html | pdf |
    - 1.1.3.6.2. How to change the variance of the components of a tensor (p. 9) | html | pdf |
    - 1.1.3.6.3. Examples of the use in physics of different representations of the same quantity (p. 9) | html | pdf |
  - 1.1.3.7. Outer product (pp. 9-10) | html | pdf |
    - 1.1.3.7.1. Definition (p. 9) | html | pdf |
    - 1.1.3.7.2. Vector product (p. 10) | html | pdf |
    - 1.1.3.7.3. Properties of the vector product (p. 10) | html | pdf |
  - 1.1.3.8. Tensor derivatives (p. 10) | html | pdf |
    - 1.1.3.8.1. Interpretation of the coefficients of the matrix – change of coordinates (p. 10) | html | pdf |
    - 1.1.3.8.2. Generalization (p. 10) | html | pdf |
    - 1.1.3.8.3. Differential operators (p. 10) | html | pdf |
    - 1.1.3.8.4. Development of a vector function in a Taylor series (p. 10) | html | pdf |
- 1.1.4. Symmetry properties (pp. 10-31) | html | pdf |
  - 1.1.4.1. Introduction – Neumann's principle (p. 11) | html | pdf |
  - 1.1.4.2. Curie laws (p. 11) | html | pdf |
  - 1.1.4.3. Symmetries associated with an electric field and with magnetic induction (flux density) (pp. 11-12) | html | pdf |
    - 1.1.4.3.1. Symmetry of an electric field (pp. 11-12) | html | pdf |
    - 1.1.4.3.2. Symmetry of magnetic induction (p. 12) | html | pdf |
  - 1.1.4.4. Superposition of several causes in the same medium – pyroelectricity and piezolectricity (p. 12) | html | pdf |
    - 1.1.4.4.1. Introduction (p. 12) | html | pdf |
    - 1.1.4.4.2. Pyroelectricity (p. 12) | html | pdf |
    - 1.1.4.4.3. Piezoelectricity (pp. 12-13) | html | pdf |
  - 1.1.4.5. Intrinsic symmetry of tensors (pp. 12-14) | html | pdf |
    - 1.1.4.5.1. Introduction (p. 13) | html | pdf |
    - 1.1.4.5.2. Symmetric tensors (p. 13) | html | pdf |
      - 1.1.4.5.2.1. Tensors of rank 2 (p. 13) | html | pdf |
      - 1.1.4.5.2.2. Tensors of higher rank (p. 13) | html | pdf |
    - 1.1.4.5.3. Antisymmetric tensors – axial tensors (pp. 13-14) | html | pdf |
      - 1.1.4.5.3.1. Tensors of rank 2 (pp. 13-14) | html | pdf |
      - 1.1.4.5.3.2. Tensors of higher rank (p. 14) | html | pdf |
      - 1.1.4.5.3.3. Properties of axial tensors (p. 14) | html | pdf |
  - 1.1.4.6. Symmetry of tensors imposed by the crystalline medium (pp. 14-16) | html | pdf |
    - 1.1.4.6.1. Matrix method – application of Neumann's principle (p. 15) | html | pdf |
    - 1.1.4.6.2. The operator A is in diagonal form (pp. 15-16) | html | pdf |
      - 1.1.4.6.2.1. Introduction (p. 15) | html | pdf |
      - 1.1.4.6.2.2. Case of a centre of symmetry (p. 15) | html | pdf |
      - 1.1.4.6.2.3. General case (pp. 15-16) | html | pdf |
    - 1.1.4.6.3. The method of direct inspection (p. 16) | html | pdf |
  - 1.1.4.7. Reduction of the components of a tensor of rank 2 (pp. 16-17) | html | pdf |
    - 1.1.4.7.1. Triclinic system (p. 16) | html | pdf |
    - 1.1.4.7.2. Monoclinic system (p. 16) | html | pdf |
    - 1.1.4.7.3. Orthorhombic system (p. 16) | html | pdf |
    - 1.1.4.7.4. Trigonal, tetragonal, hexagonal and cylindrical systems (pp. 16-17) | html | pdf |
      - 1.1.4.7.4.1. Groups $[{\bar 3}]$ , ; , $[{\bar 4}]$ , ; , $[{\bar 6}]$ , ; $[(A_{\infty}/M)C]$ , $[A_{\infty}]$ (pp. 16-17) | html | pdf |
      - 1.1.4.7.4.2. Groups $[{\bar 3}m]$ , , ; , , , $[{\bar {4}}{2}m]$ ; , , , $[{\bar{6}}{2}m]$ ; $[(A_{\infty}/M) \infty (A_{2}/M)C]$ , $[A_{\infty}\infty A_{2}]$ (p. 17) | html | pdf |
    - 1.1.4.7.5. Cubic and spherical systems (p. 17) | html | pdf |
    - 1.1.4.7.6. Symmetric tensors of rank 2 (p. 17) | html | pdf |
      - 1.1.4.7.6.1. Triclinic system (p. 17) | html | pdf |
      - 1.1.4.7.6.2. Monoclinic system (twofold axis parallel to ) (p. 17) | html | pdf |
      - 1.1.4.7.6.3. Orthorhombic system (p. 17) | html | pdf |
      - 1.1.4.7.6.4. Trigonal, tetragonal and hexagonal systems, isotropic groups (p. 17) | html | pdf |
      - 1.1.4.7.6.5. Cubic system (p. 17) | html | pdf |
  - 1.1.4.8. Reduction of the components of a tensor of rank 3 (pp. 17-20) | html | pdf |
    - 1.1.4.8.1. Triclinic system (p. 17) | html | pdf |
      - 1.1.4.8.1.1. Group (p. 17) | html | pdf |
      - 1.1.4.8.1.2. Group $[{\bar 1}]$ (p. 17) | html | pdf |
    - 1.1.4.8.2. Monoclinic system (p. 18) | html | pdf |
      - 1.1.4.8.2.1. Group (p. 18) | html | pdf |
      - 1.1.4.8.2.2. Group m (p. 18) | html | pdf |
      - 1.1.4.8.2.3. Group (p. 18) | html | pdf |
    - 1.1.4.8.3. Orthorhombic system (p. 18) | html | pdf |
      - 1.1.4.8.3.1. Group (p. 18) | html | pdf |
      - 1.1.4.8.3.2. Group (p. 18) | html | pdf |
      - 1.1.4.8.3.3. Group (p. 18) | html | pdf |
    - 1.1.4.8.4. Trigonal system (p. 18) | html | pdf |
      - 1.1.4.8.4.1. Group (p. 18) | html | pdf |
      - 1.1.4.8.4.2. Group with a twofold axis parallel to $[Ox_{1}]$ (p. 18) | html | pdf |
      - 1.1.4.8.4.3. Group with a mirror normal to $[Ox_{1}]$ (p. 18) | html | pdf |
      - 1.1.4.8.4.4. Groups $[\bar{3}]$ and $[\bar{3}m]$ (p. 18) | html | pdf |
    - 1.1.4.8.5. Tetragonal system (pp. 18-19) | html | pdf |
      - 1.1.4.8.5.1. Group (pp. 18-19) | html | pdf |
      - 1.1.4.8.5.2. Group (p. 19) | html | pdf |
      - 1.1.4.8.5.3. Group (p. 19) | html | pdf |
      - 1.1.4.8.5.4. Group (p. 19) | html | pdf |
      - 1.1.4.8.5.5. Group $[\bar{4}]$ (p. 19) | html | pdf |
      - 1.1.4.8.5.6. Group $[\bar{4}2m]$ (p. 19) | html | pdf |
      - 1.1.4.8.5.7. Group (p. 19) | html | pdf |
    - 1.1.4.8.6. Hexagonal and cylindrical systems (p. 19) | html | pdf |
      - 1.1.4.8.6.1. Groups , $[A_{\infty}]$ , , $[A_{\infty} \infty A_{2}]$ , and $[A_{\infty} \infty M]$ (p. 19) | html | pdf |
      - 1.1.4.8.6.2. Group $[\bar{6} = 3/m]$ (p. 19) | html | pdf |
      - 1.1.4.8.6.3. Group $[\bar{6}2m]$ (p. 19) | html | pdf |
      - 1.1.4.8.6.4. Groups , $[(A_{\infty}/M)C]$ , and $[(A_{\infty}/M) \infty (A_{2}/M)C]$ (p. 19) | html | pdf |
    - 1.1.4.8.7. Cubic and spherical systems (pp. 19-20) | html | pdf |
      - 1.1.4.8.7.1. Group (p. 19) | html | pdf |
      - 1.1.4.8.7.2. Groups and $[\infty A_{\infty}/M]$ (p. 20) | html | pdf |
      - 1.1.4.8.7.3. Group $[{\bar 4}3m]$ (p. 20) | html | pdf |
      - 1.1.4.8.7.4. Groups $[m{\bar 3}]$ , $[m{\bar 3}m]$ and $[\infty(A_{\infty}/M)C]$ (p. 20) | html | pdf |
  - 1.1.4.9. Reduction of the components of a tensor of rank 4 (pp. 20-24) | html | pdf |
    - 1.1.4.9.1. Triclinic system (groups $[{\bar 1}]$ , ) (p. 20) | html | pdf |
    - 1.1.4.9.2. Monoclinic system (groups , , m) (p. 20) | html | pdf |
    - 1.1.4.9.3. Orthorhombic system (groups , , ) (p. 20) | html | pdf |
    - 1.1.4.9.4. Trigonal system (pp. 20-21) | html | pdf |
      - 1.1.4.9.4.1. Groups and $[{\bar 3}]$ (p. 20) | html | pdf |
      - 1.1.4.9.4.2. Groups $[{\bar 3}m]$ , , , with the twofold axis parallel to $[Ox_{1}]$ (p. 21) | html | pdf |
    - 1.1.4.9.5. Tetragonal system (p. 21) | html | pdf |
      - 1.1.4.9.5.1. Groups , , $[{\bar 4}]$ (p. 21) | html | pdf |
      - 1.1.4.9.5.2. Groups , , , $[{\bar 4}2m]$ (p. 21) | html | pdf |
    - 1.1.4.9.6. Hexagonal and cylindrical systems (p. 21) | html | pdf |
      - 1.1.4.9.6.1. Groups , $[{\bar 6}]$ , ; $[(A_{\infty}/M)C]$ , $[A_{\infty}]$ (p. 21) | html | pdf |
      - 1.1.4.9.6.2. Groups , , , $[{\bar 6}2m]$ ; $[(A_{\infty}/M) \infty]$ ; $[(A_{2}/M)C]$ , $[A_{\infty}\infty A_{2}]$ (p. 21) | html | pdf |
    - 1.1.4.9.7. Cubic system (pp. 21-22) | html | pdf |
      - 1.1.4.9.7.1. Groups , $[{\bar 3}m]$ (p. 21) | html | pdf |
      - 1.1.4.9.7.2. Groups $[m{\bar 3}m]$ , , $[{\bar 4}3m]$ (p. 22) | html | pdf |
    - 1.1.4.9.8. Spherical system (p. 22) | html | pdf |
      - 1.1.4.9.8.1. Groups $[\infty (A_{\infty}/M)C]$ and $[\infty A_{\infty}]$ (p. 22) | html | pdf |
    - 1.1.4.9.9. Symmetric tensors of rank 4 (pp. 22-24) | html | pdf |
      - 1.1.4.9.9.1. Triclinic system (p. 22) | html | pdf |
      - 1.1.4.9.9.2. Monoclinic system (p. 22) | html | pdf |
      - 1.1.4.9.9.3. Orthorhombic system (p. 22) | html | pdf |
      - 1.1.4.9.9.4. Trigonal system (pp. 22-23) | html | pdf |
      - 1.1.4.9.9.5. Tetragonal system (p. 23) | html | pdf |
      - 1.1.4.9.9.6. Hexagonal and cylindrical systems (p. 23) | html | pdf |
      - 1.1.4.9.9.7. Cubic system (pp. 23-24) | html | pdf |
  - 1.1.4.10. Reduced form of polar and axial tensors – matrix representation (pp. 24-31) | html | pdf |
    - 1.1.4.10.1. Introduction (p. 24) | html | pdf |
    - 1.1.4.10.2. Stress and strain tensors – Voigt matrices (p. 24) | html | pdf |
    - 1.1.4.10.3. Reduction of the number of independent components of third-rank polar tensors due to the symmetry of the strain and stress tensors (pp. 24-25) | html | pdf |
    - 1.1.4.10.4. Independent components of the matrix associated with a third-rank polar tensor according to the following point groups (pp. 25-26) | html | pdf |
      - 1.1.4.10.4.1. Triclinic system (p. 25) | html | pdf |
      - 1.1.4.10.4.2. Monoclinic system (p. 25) | html | pdf |
      - 1.1.4.10.4.3. Orthorhombic system (p. 25) | html | pdf |
      - 1.1.4.10.4.4. Trigonal system (p. 25) | html | pdf |
      - 1.1.4.10.4.5. Tetragonal, hexagonal and cylindrical systems (pp. 25-26) | html | pdf |
      - 1.1.4.10.4.6. Cubic and spherical systems (p. 26) | html | pdf |
    - 1.1.4.10.5. Reduction of the number of independent components of fourth-rank polar tensors due to the symmetry of the strain and stress tensors (pp. 26-27) | html | pdf |
    - 1.1.4.10.6. Independent components of the matrix associated with a fourth-rank tensor according to the following point groups (pp. 27-29) | html | pdf |
      - 1.1.4.10.6.1. Triclinic system, groups $[{\bar 1}]$ , (p. 27) | html | pdf |
      - 1.1.4.10.6.2. Monoclinic system (p. 27) | html | pdf |
      - 1.1.4.10.6.3. Orthorhombic system (p. 27) | html | pdf |
      - 1.1.4.10.6.4. Trigonal system (p. 28) | html | pdf |
      - 1.1.4.10.6.5. Tetragonal system (p. 28) | html | pdf |
      - 1.1.4.10.6.6. Hexagonal system (pp. 28-29) | html | pdf |
      - 1.1.4.10.6.7. Cubic system (p. 29) | html | pdf |
      - 1.1.4.10.6.8. Spherical system (p. 29) | html | pdf |
    - 1.1.4.10.7. Reduction of the number of independent components of axial tensors of rank 2 (pp. 29-31) | html | pdf |
      - 1.1.4.10.7.1. Independent components according to the following point groups (pp. 29-30) | html | pdf |
      - 1.1.4.10.7.2. Independent components of symmetric axial tensors according to the following point groups (pp. 30-31) | html | pdf |
- 1.1.5. Thermodynamic functions and physical property tensors (pp. 31-32) | html | pdf |
  - 1.1.5.1. Isothermal study (p. 31) | html | pdf |
  - 1.1.5.2. Other forms of the piezoelectric constants (p. 32) | html | pdf |
  - 1.1.5.3. Relation between the pyroelectric coefficients at constant stress and at constant strain (p. 32) | html | pdf |
  - 1.1.5.4. Adiabatic study (p. 32) | html | pdf |
- 1.1.6. Glossary (pp. 32-33) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 1.1.4.1. Symmetry of an electric field (p. 11) | html | pdf |
  - Fig. 1.1.4.2. Symmetry of magnetic induction (p. 12) | html | pdf |
- Tables
  - Table 1.1.1.1. Extensive quantities and associated intensive parameters (p. 3) | html | pdf |