International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 3-33
doi: 10.1107/97809553602060000628

Chapter 1.1. Introduction to the properties of tensors

Contents

  • 1.1. Introduction to the properties of tensors  (pp. 3-33) | html | pdf | chapter contents |
    • 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  (p. 11) | 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  (p. 12) | 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  (p. 13) | 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  (pp. 14-15) | 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}], [3]; [4/m], [{\bar 4}], [4]; [6/m], [{\bar 6}], [6]; [(A_{\infty}/M)C], [A_{\infty}]  (p. 16) | html | pdf |
          • 1.1.4.7.4.2. Groups [{\bar 3}m], [32], [3m]; [4/mm], [422], [4mm], [{\bar {4}}{2}m]; [{6}/mm], [622], [6mm], [{\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 [Ox_2])  (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 [1]  (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 [2]  (p. 18) | html | pdf |
          • 1.1.4.8.2.2. Group m   (p. 18) | html | pdf |
          • 1.1.4.8.2.3. Group [2/m]  (p. 18) | html | pdf |
        • 1.1.4.8.3. Orthorhombic system  (p. 18) | html | pdf |
          • 1.1.4.8.3.1. Group [222]  (p. 18) | html | pdf |
          • 1.1.4.8.3.2. Group [mm2]  (p. 18) | html | pdf |
          • 1.1.4.8.3.3. Group [mmm]  (p. 18) | html | pdf |
        • 1.1.4.8.4. Trigonal system  (p. 18) | html | pdf |
          • 1.1.4.8.4.1. Group [3]  (p. 18) | html | pdf |
          • 1.1.4.8.4.2. Group [32] with a twofold axis parallel to [Ox_{1}]  (p. 18) | html | pdf |
          • 1.1.4.8.4.3. Group [3m] 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 [4]  (pp. 18-19) | html | pdf |
          • 1.1.4.8.5.2. Group [422]  (p. 19) | html | pdf |
          • 1.1.4.8.5.3. Group [4mm]  (p. 19) | html | pdf |
          • 1.1.4.8.5.4. Group [4/m]  (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 [4/mm]  (p. 19) | html | pdf |
        • 1.1.4.8.6. Hexagonal and cylindrical systems  (p. 19) | html | pdf |
          • 1.1.4.8.6.1. Groups [6], [A_{\infty}], [622], [A_{\infty} \infty A_{2}], [6mm] 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 [6/m], [(A_{\infty}/M)C], [6/mm] 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 [23]  (p. 19) | html | pdf |
          • 1.1.4.8.7.2. Groups [432] 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}], [1])  (p. 20) | html | pdf |
        • 1.1.4.9.2. Monoclinic system (groups [2/m], [2], m )  (p. 20) | html | pdf |
        • 1.1.4.9.3. Orthorhombic system (groups [mmm], [2mm], [222])  (p. 20) | html | pdf |
        • 1.1.4.9.4. Trigonal system  (pp. 20-21) | html | pdf |
          • 1.1.4.9.4.1. Groups [3] and [{\bar 3}]  (p. 20) | html | pdf |
          • 1.1.4.9.4.2. Groups [{\bar 3}m], [32], [3m], 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 [4/m], [4], [{\bar 4}]  (p. 21) | html | pdf |
          • 1.1.4.9.5.2. Groups [4/m m], [422], [4mm], [{\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 [6/m], [{\bar 6}], [6]; [(A_{\infty}/M)C], [A_{\infty}]  (p. 21) | html | pdf |
          • 1.1.4.9.6.2. Groups [6/mm], [622], [6mm], [{\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 [23], [{\bar 3}m]  (p. 21) | html | pdf |
          • 1.1.4.9.7.2. Groups [m{\bar 3}m], [432], [{\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}], [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 |