International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 24-31

Section 1.1.4.10. Reduced form of polar and axial tensors – matrix representation

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.4.10. Reduced form of polar and axial tensors – matrix representation

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1.1.4.10.1. Introduction

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Many tensors representing physical properties or physical quantities appear in relations involving symmetric tensors. Consider, for instance, the strain [S_{ij}] resulting from the application of an electric field E (the piezoelectric effect): [S_{ij} = d_{ijk}E_{k} + Q_{ijkl}E_{k}E_{l}, \eqno(1.1.4.4)]where the first-order terms [d_{ijk}] represent the components of the third-rank converse piezoelectric tensor and the second-order terms [Q_{ijkl}] represent the components of the fourth-rank electrostriction tensor. In a similar way, the direct piezoelectric effect corresponds to the appearance of an electric polarization P when a stress [T_{jk}] is applied to a crystal: [P_{i} = d_{ijk}T_{jk}. \eqno(1.1.4.5)]

Owing to the symmetry properties of the strain and stress tensors (see Sections 1.3.1[link] and 1.3.2[link] ) and of the tensor product [E_{k}E_{l}], there occurs a further reduction of the number of independent components of the tensors which are engaged in a contracted product with them, as is shown in Section 1.1.4.10.3[link] for third-rank tensors and in Section 1.1.4.10.5[link] for fourth-rank tensors.

1.1.4.10.2. Stress and strain tensors – Voigt matrices

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The stress and strain tensors are symmetric because body torques and rotations are not taken into account, respectively (see Sections 1.3.1[link] and 1.3.2[link] ). Their components are usually represented using Voigt's one-index notation.

  • (i) Strain tensor [\left. \matrix{S_{1} = S_{11}\semi\hfill &S_{2} = S_{22}\semi\hfill &S_{3} = S_{33}\semi\hfill \cr S_{4} = S_{23} + S_{32}\semi\hfill &S_{5} = S_{31} + S_{13}\semi\hfill &S_{6} = S_{12} + S_{21}\semi\hfill \cr S_{4} = 2S_{23} = 2S_{32}\semi &S_{5} = 2S_{31} = 2S_{13}\semi &S_{6} = 2S_{12} = 2S_{21}.\cr}\right\} \eqno(1.1.4.6)]The Voigt components [S_{\alpha}] form a Voigt matrix: [\pmatrix{S_{1} &S_{6} &S_{5}\cr &S_{2} &S_{4}\cr & &S_{3}\cr}.]The terms of the leading diagonal represent the elongations (see Section 1.3.1[link] ). It is important to note that the non-diagonal terms, which represent the shears, are here equal to twice the corresponding components of the strain tensor. The components [S_{\alpha}] of the Voigt strain matrix are therefore not the components of a tensor.

  • (ii) Stress tensor [\left. \matrix{T_{1}= T_{11}\semi\hfill &T_{2} = T_{22}\semi\hfill &T_{3} = T_{33}\semi\hfill\cr T_{4} = T_{23} = T_{32}\semi &T_{5} = T_{31} = T_{13}\semi &T_{6} = T_{12} =T_{21}.\cr}\right\}]The Voigt components [T_{\alpha}] form a Voigt matrix: [\pmatrix{T_{1} &T_{6} &T_{5}\cr &T_{2} &T_{4}\cr & & T_{3}\cr}.]The terms of the leading diagonal correspond to principal normal constraints and the non-diagonal terms to shears (see Section 1.3.2[link] ).

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

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Equation (1.1.4.5)[link] can be written [P_{i} = \textstyle\sum\limits_{j} d_{ijj}T_{jj} + \textstyle\sum\limits_{j\neq k} (d_{ijk}+ d_{ikj})T_{jk}.]

The sums [(d_{ijk} + d_{ikj})] for [j \neq k] have a definite physical meaning, but it is impossible to devise an experiment that permits [d_{ijk}] and [d_{ikj}] to be measured separately. It is therefore usual to set them equal: [d_{ijk} = d_{ikj}. \eqno(1.1.4.7)]

It was seen in Section 1.1.4.8.1[link] that the components of a third-rank tensor can be represented as a [9 \times 3] matrix which can be subdivided into three [3 \times 3] submatrices:[\pmatrix{&{\bf 1}&|&{\bf 2}&|&{\bf 3}&\cr}.]

Relation (1.1.4.7)[link] shows that submatrices 1 and 2 are identical.

One puts, introducing a two-index notation, [\left. \matrix{\hfill d_{ijj} = d_{i\alpha}\quad (\alpha = 1, 2, 3){\phantom.}\cr d_{ijk} + d_{ikj}\,\, (j\neq k) = d_{i\alpha}\quad (\alpha = 4, 5, 6).\cr}\right\}]Relation (1.1.4.7)[link] becomes [P_{i} = d_{i\alpha} T_{\alpha}.]

The coefficients [d_{i\alpha}] may be written as a [3 \times 6] matrix: [\left(\matrix{11 &12 &13 \cr 21 &22 &23\cr 31 &32 &33\cr}\right | \left. \matrix{14 &15 &16\cr 24 &25 &26\cr 34 &35 &36\cr}\right).]This matrix is constituted by two [3 \times 3] submatrices. The left-hand one is identical to the submatrix 1, and the right-hand one is equal to the sum of the two submatrices 2 and 3:[\pmatrix{&{\bf 1}&|&{\bf 2} + {\bf 3}&\cr}.]

The inverse piezoelectric effect expresses the strain in a crystal submitted to an applied electric field: [S_{ij} = d_{ijk}E_{k},]where the matrix associated with the coefficients [d_{ijk}] is a [9 \times 3] matrix which is the transpose of that of the coefficients used in equation (1.1.4.5)[link], as shown in Section 1.1.1.4[link].

The components of the Voigt strain matrix [S_{\alpha}] are then given by [\left. \matrix{S_{\alpha } = d_{iik}E_{k} \hfill &(\alpha = 1,2,3)\cr S_{\alpha} = S_{ij} + S_{ji} = (d_{ijk} + d_{jik})E_{k} &(\alpha = 4,5,6).\cr}\right\}]This relation can be written simply as [S_{\alpha} = d_{\alpha k}E_{k},]where the matrix of the coefficients [d_{\alpha k}] is a [6 \times 3] matrix which is the transpose of the [d_{i\alpha}] matrix.

There is another set of piezoelectric constants (see Section 1.1.5[link]) which relates the stress, [T_{ij}], and the electric field, [E_{k}], which are both intensive parameters: [T_{ij} = e_{ijk}E_{k}, \eqno(1.1.4.8)]where a new piezoelectric tensor is introduced, [e_{ijk}]. Its components can be represented as a [3 \times 9] matrix:[\pmatrix{{\bf 1}\cr-\cr{\bf 2}\cr-\cr{\bf 3}\cr}.]

Both sides of relation (1.1.4.8)[link] remain unchanged if the indices i and j are interchanged, on account of the symmetry of the stress tensor. This shows that [e_{ijk} = e_{jik}.]

Submatrices 2 and 3 are equal. One introduces here a two-index notation through the relation [e_{\alpha k} = e_{ijk}], and the [e_{\alpha k}] matrix can be written[\left({{\bf 1}\over{{\bf 2}+{\bf 3}}}\right).]

The relation between the full and the reduced matrix is therefore different for the [d_{ijk}] and the [e_{kij}] tensors. This is due to the particular property of the strain Voigt matrix (1.1.4.6)[link], and as a consequence the relations between nonzero components of the reduced matrices are different for certain point groups (3, 32, [3m], [\bar{6}], [\bar{6}2m]).

1.1.4.10.4. Independent components of the matrix associated with a third-rank polar tensor according to the following point groups

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1.1.4.10.4.1. Triclinic system

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  • (i) Group 1: all the components are independent. There are 18 components.

  • (ii) Group [\bar{1}]: all the components are equal to zero.

1.1.4.10.4.2. Monoclinic system

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  • (i) Group 2: twofold axis parallel to [Ox_{2}]: [Scheme scheme47]

    There are 8 independent components.

  • (ii) Group m: [Scheme scheme48]

    There are 10 independent components.

  • (iii) Group [2/m]: all the components are equal to zero.

1.1.4.10.4.3. Orthorhombic system

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  • (i) Group 222: [Scheme scheme49]

    There are 3 independent components.

  • (ii) Group [mm2]: [Scheme scheme50]

    There are 5 independent components.

  • (iii) Group [mmm]: all the components are equal to zero.

1.1.4.10.4.4. Trigonal system

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  • (i) Group 3: [Scheme scheme51] where the symbol ⊖ means that the corresponding component is equal to the opposite of that to which it is linked, ⊙ means that the component is equal to twice minus the value of the component to which it is linked for [d_{ijk}] and to minus the value of the component to which it is linked for [e_{ijk}]. There are 6 independent components.

  • (ii) Group 32, twofold axis parallel to [Ox_{1}]: [Scheme scheme52] with the same conventions. There are 4 independent components.

  • (iii) Group [3m], mirror perpendicular to [Ox_{1}]: [Scheme scheme53] with the same conventions. There are 4 independent components.

  • (iv) Groups [\bar{3}] and [\bar{3}m]: all the components are equal to zero.

1.1.4.10.4.5. Tetragonal, hexagonal and cylindrical systems

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  • (i) Groups 4, 6 and [A_{\infty}]: [Scheme scheme54]

    There are 4 independent components.

  • (ii) Groups 422, 622 and [A_{\infty}\infty A_{2}]: [Scheme scheme55]

    There is 1 independent component.

  • (iii) Groups [4mm], [6mm] and [A_{\infty }\infty M]: [Scheme scheme56]

    There are 3 independent components.

  • (iv) Groups [4/m], [6/m] and [(A_{\infty }/M)C]: all the components are equal to zero.

  • (v) Group [\bar{4}]: [Scheme scheme57]

    There are 4 independent components.

  • (vi) Group [\bar{6} = 3/m]: [Scheme scheme58] with the same conventions as for group 3. There are 2 independent components.

  • (vii) Group [\bar{4}2m] – twofold axis parallel to [Ox_{1}]: [Scheme scheme59]

    There are 2 independent components.

  • (viii) Group [\bar{4}2m] – mirror perpendicular to [Ox_{1}] (twofold axis at [45^{\circ}]): [Scheme scheme60]

    The number of independent components is of course the same.

  • (ix) Group [\bar{6}2m]: [Scheme scheme61] with the same conventions as for group 3. There is 1 independent component.

  • (x) Groups [4/mm], [6/mm] and [(A_{\infty }/M) \infty (A_{2}/M)C]: all the components are equal to zero.

1.1.4.10.4.6. Cubic and spherical systems

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  • (i) Groups 23 and [{\bar 4}3m]: [Scheme scheme62]

    There is 1 independent component.

  • (ii) Groups 432 and [\infty A_{\infty }]: it was seen in Section 1.1.4.8.6[link] that we have in this case [d_{123}= - d_{132}.]It follows that [d_{14} = 0], all the components are equal to zero.

  • (iii) Groups [m{\bar 3}], [m{\bar 3}m] and [\infty (A_{\infty }/M)C]: all the components are equal to zero.

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

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Let us consider five examples of fourth-rank tensors:

  • (i) Elastic compliances, [s_{ijkl }], relating the resulting strain tensor [S_{ij}] to an applied stress [T_{ij}] (see Section 1.3.3.2[link] ): [S_{ij} = s_{ijkl}T_{kl}, \eqno{(1.1.4.9)}]where the compliances [s_{ijkl}] are the components of a tensor of rank 4.

  • (ii) Elastic stiffnesses, [c_{ijkl }] (see Section 1.3.3.2[link] ): [T_{ij}= c_{ijkl}S_{kl}.]

  • (iii) Piezo-optic coefficients, [\pi _{ijkl}], relating the variation [\Delta \eta _{ij}] of the dielectric impermeability to an applied stress [T_{kl}] (photoelastic effect – see Section 1.6.7[link] ): [\Delta \eta_{ij} = \pi_{ijkl}T_{kl}.]

  • (iv) Elasto-optic coefficients, [p_{ijkl}], relating the variation [\Delta \eta_{ij}] of the dielectric impermeability to the strain [S_{kl}]: [\Delta \eta _{ij}= p_{ijkl}S_{kl}.]

  • (v) Electrostriction coefficients, [Q_{ijkl}], which appear in equation (1.1.4.4)[link]: [S_{ij} = Q_{ijkl}E_{k}E_{l}, \eqno{(1.1.4.10)}]where only the second-order terms are considered.

In each of the equations from (1.1.4.9)[link] to (1.1.4.10)[link], the contracted product of a fourth-rank tensor by a symmetric second-rank tensor is equal to a symmetric second-rank tensor. As in the case of the third-rank tensors, this results in a reduction of the number of independent components, but because of the properties of the strain Voigt matrix, and because two of the tensors are endowed with intrinsic symmetry (the elastic tensors), the reduction is different for each of the five tensors. The above relations can be written in matrix form: [Scheme scheme63] where the second-rank tensors are represented by [1 \times 9] column matrices, which can each be subdivided into three [1 \times 3] sub­matrices and the [9 \times 9] matrix associated with the fourth-rank tensors is subdivided into nine [3 \times 3] submatrices, as shown in Section 1.1.4.9.1[link]. The symmetry of the second-rank tensors means that submatrices 2 and 3 which are associated with them are equal.

Let us first consider the reduction of the tensor of elastic compliances. As in the case of the piezoelectric tensor, equation (1.1.4.9)[link] can be written [S_{ij}= \textstyle\sum\limits_{l} s_{ijll}T_{ll} + \sum\limits_{k\ne l}(s_{ijkl}+ s_{ijlk})T_{kl}. \eqno(1.1.4.11)]

The sums [(s_{ijkl}+ s_{ijlk})] for [k \neq l] have a definite physical meaning, but it is impossible to devise an experiment permitting [s_{ijkl}] and [s_{ijlk}] to be measured separately. It is therefore usual to set them equal in order to avoid an unnecessary constant: [s_{ijkl} = s_{ijlk}.]

Furthermore, the left-hand term of (1.1.4.11)[link] remains unchanged if we interchange the indices i and j. The terms on the right-hand side therefore also remain unchanged, whatever the value of [T_{ll}] or [T_{kl}]. It follows that [\eqalignno{s_{ijll} & = s_{jill}\cr s_{ijkl} & = s_{ijlk} = s_{jikl} = s_{jilk}.\cr}]Similar relations hold for [c_{ijkl}], [Q_{ijkl}], [p_{ijkl}] and [\pi_{ijkl}]: the submatrices 2 and 3, 4 and 7, 5, 6, 8 and 9, respectively, are equal.

Equation (1.4.1.11)[link] can be rewritten, introducing the coefficients of the Voigt strain matrix: [\eqalignno{S_{\alpha} = S_{ii} &= \textstyle\sum\limits_{l} s_{iill}T_{ll} + \sum\limits_{k\ne l}(s_{iikl} + s_{iilk})T_{kl} \quad (\alpha = 1,2,3)\cr S_{\alpha} = S_{ij} + S_{ji} &= \textstyle\sum\limits_{l} (s_{ijll} + s_{jill})T_{ll}\cr &\quad + \textstyle\sum\limits_{k\ne l}(s_{ijkl} + s_{ijlk} + s_{jikl} + s_{jilk})T_{kl} \quad (\alpha = 4,5,6).\cr}]We shall now introduce a two-index notation for the elastic compliances, according to the following conventions: [\left.\matrix{i = j\semi &k = l\semi &s_{\alpha \beta} = s_{iill}\hfill\cr i = j\semi &k\neq l\semi &s_{\alpha \beta} = s_{iikl}+s_{iilk}\hfill\cr i \neq j\semi &k = l\semi &s_{\alpha \beta} = s_{ijkk}+s_{jikk}\hfill\cr i \neq j\semi &k \neq l\semi &s_{\alpha \beta} = s_{ijkl}+ s_{jikl} + s_{ijlk}+ s_{jilk}.\hfill\cr}\right\} \eqno(1.1.4.12)]We have thus associated with the fourth-rank tensor a square [6 \times 6] matrix with 36 coefficients: [Scheme scheme115]

One can translate relation (1.1.4.12)[link] using the [9 \times 9] matrix representing [s_{ijkl}] by adding term by term the coefficients of submatrices 2 and 3, 4 and 7 and 5, 6, 8 and 9, respectively: [Scheme scheme64]

Using the two-index notation, equation (1.1.4.9)[link] becomes [S_{\alpha} = s_{\alpha \beta}T_{\beta}. \eqno(1.1.4.13)]

A similar development can be applied to the other fourth-rank tensors [\pi_{ijkl}], which will be replaced by [6 \times 6] matrices with 36 coefficients, according to the following rules.

  • (i) Elastic stiffnesses, [c_{ijkl}] and elasto-optic coefficients, [p_{ijkl}]: [Scheme scheme65] where [\eqalignno{c_{\alpha \beta} &=c_{ijkl}\cr p_{\alpha \beta} &=p_{ijkl}.\cr}]

  • (ii) Piezo-optic coefficients, [\pi_{ijkl}]: [Scheme scheme66] where[\left. \matrix{i = j\semi &k = l\semi &\pi_{\alpha \beta} = \pi_{iil l}\hfill \cr i = j\semi &k \neq l\semi &\pi_{\alpha \beta} = \pi_{iikl} + \pi_{iilk}\hfill \cr i \neq j\semi &k = l\semi &\pi_{\alpha \beta} = \pi_{ijkk} = \pi_{jikk}\hfill \cr i \neq j\semi &k \neq l\semi &\pi_{\alpha \beta} = \pi_{ijkl} + \pi_{jikl} = \pi_{ijl k} + \pi_{jilk}.\hfill\cr}\right\}]

  • (iii) Electrostriction coefficients, [Q_{ijkl}]: same relation as for the elastic compliances.

1.1.4.10.6. Independent components of the matrix associated with a fourth-rank tensor according to the following point groups

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1.1.4.10.6.1. Triclinic system, groups [{\bar 1}], [1]

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[Scheme scheme67]

1.1.4.10.6.2. Monoclinic system

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Groups [2/m], 2, m, twofold axis parallel to [Ox_{2}]: [Scheme scheme68]

1.1.4.10.6.3. Orthorhombic system

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Groups [mmm], [2mm], 222: [Scheme scheme69]

1.1.4.10.6.4. Trigonal system

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  • (i) Groups 3, [{\bar 3}]: [Scheme scheme70a] [Scheme scheme71] where ⊖ is a component numerically equal but opposite in sign to the heavy dot component to which it is linked; ⊕ is a component equal to twice the heavy dot component to which it is linked; ⊙ is a component equal to minus twice the heavy dot component to which it is linked; ⊗ is equal to [1/2(p_{11} - p_{12})], [(\pi_{11} - \pi_{12})], [2(Q_{11} - Q_{12})], [1/2(c_{11} - c_{12})] and [2(s_{11} - s_{12})], respectively.

  • (ii) Groups 32, [3m], [{\bar 3}m]: [Scheme scheme72] [Scheme scheme73] with the same conventions. The sign of [c_{14}] depends on the orientation of the [Ox_1] axis.

1.1.4.10.6.5. Tetragonal system

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  • (i) Groups 4, [{\bar 4}] and [4/m]: [Scheme scheme74]

  • (ii) Groups 422, [4mm], [{\bar 4}2m] and [4/mm]: [Scheme scheme75]

1.1.4.10.6.6. Hexagonal system

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  • (i) Groups 6, [{\bar 6}] and [6/m]: [Scheme scheme76] [Scheme scheme77]

  • (ii) Groups 622, [6mm], [{\bar 6}2m] and [6/mm]: [Scheme scheme78]

1.1.4.10.6.7. Cubic system

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  • (i) Groups 23 and [3m]: [Scheme scheme79]

  • (ii) Groups 432, [{\bar 4}3m] and [m{\bar 3}m]: [Scheme scheme80]

1.1.4.10.6.8. Spherical system

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For all tensors [Scheme scheme81]

1.1.4.10.7. Reduction of the number of independent components of axial tensors of rank 2

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It was shown in Section 1.1.4.5.3.2[link] that axial tensors of rank 2 are actually tensors of rank 3 antisymmetric with respect to two indices. The matrix of independent components of a tensor such that [g_{ijk}= - g_{jik}]is given by [\left(\matrix{&122 &133\cr -121 & &223\cr -131 &-232 &\cr}\left|\matrix{123 &131 &\cr &231 &-122\cr -233 & &-132\cr}\right|\matrix{132 & &121\cr 232 &-123 &\cr &-133 &-231\cr}\right).]The second-rank axial tensor [g_{kl}] associated with this tensor is defined by [g_{kl} = {\textstyle{1\over 2}} \varepsilon_{ijk} g_{ijl}.]

For instance, the piezomagnetic coefficients that give the magnetic moment [M_{i}] due to an applied stress [T_{\alpha }] are the components of a second-rank axial tensor, [\Lambda_{i\alpha }] (see Section 1.5.7.1[link] ): [M_{i}= \Lambda _{i\alpha} T_{\alpha}.]

1.1.4.10.7.1. Independent components according to the following point groups

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  • (i) Triclinic system

    • (a) Group 1: [Scheme scheme82]

    • (b) Group [{\bar 1}]: all components are equal to zero.

  • (ii) Monoclinic system

    • (a) Group 2: [Scheme scheme83]

    • (b) Group m: [Scheme scheme84]

    • (c) Group [2/m]: all components are equal to zero.

  • (iii) Orthorhombic system

    • (a) Group 222: [Scheme scheme85]

    • (b) Group [mm2]: [Scheme scheme86]

    • (c) Group [mmm]: all components are equal to zero.

  • (iv) Trigonal, tetragonal, hexagonal and cylindrical systems

    • (a) Groups 3, 4, 6 and [A_{\infty}]: [Scheme scheme87]

    • (b) Groups 32, 42, 62 and [A_{\infty}\infty A_2]: [Scheme scheme88]

    • (c) Groups [3m], [4m], [6m] and [A_{\infty}\infty M]: [Scheme scheme89]

    • (d) Group [{\bar 4}]: [Scheme scheme90]

    • (e) Group [\bar{4}2m]: [Scheme scheme91]

    • (f) Groups [\bar{3}], [4/m], [\bar{6}2m], [\bar{3}m], [4/mm] and [6/mm]: all components are equal to zero.

  • (v) Cubic and spherical systems

    • (a) Groups 23, 432 and [\infty A_{\infty}]: [Scheme scheme92]

      The axial tensor is reduced to a pseudoscalar.

    • (b) Groups [m\bar{3}], [\bar{4}3m], [m\bar{3}m] and [\infty (A_{\infty}/M)C]: all components are equal to zero.

1.1.4.10.7.2. Independent components of symmetric axial tensors according to the following point groups

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Some axial tensors are also symmetric. For instance, the optical rotatory power of a gyrotropic crystal in a given direction of direction cosines [\alpha_{1,} \alpha_{2}, \alpha_{3}] is proportional to a quantity G defined by (see Section 1.6.5.4[link] ) [G = g_{ij}\alpha_{i}\alpha_{j},]where the gyration tensor [g_{ij}] is an axial tensor. This expression shows that only the symmetric part of [g_{ij}] is relevant. This leads to a further reduction of the number of independent components:

  • (i) Triclinic system

    • (a) Group 1: [Scheme scheme93]

    • (b) Group [\bar{1}]: all components are equal to zero.

  • (ii) Monoclinic system

    • (a) Group 2: [Scheme scheme94]

    • (b) Group m: [Scheme scheme95]

    • (c) Group [2/m]: all components are equal to zero.

  • (iii) Orthorhombic system

    • (a) Group 222: [Scheme scheme96]

    • (b) Group [mm2]: [Scheme scheme97]

    • (c) Group [mmm]: all components are equal to zero.

  • (iv) Trigonal, tetragonal and hexagonal systems

    • (a) Groups 3, 32, 4, 42, 6, 62: [Scheme scheme98]

    • (b) Group [\bar{4}]: [Scheme scheme99]

    • (c) Group [\bar{4}2m]: [Scheme scheme100]

    • (d) Groups [\bar{3}], [3m], [\bar{3}m], [4/m], [4mm], [4/mm], [\bar{6}], [\bar{6}2m] and [6/mm]: all components are equal to zero.

  • (v) Cubic and spherical systems

    • (a) Groups 23, 432 and [A_{\infty} \infty A_{2}]: [Scheme scheme101]

    • (b) Groups [m\bar{3}], [\bar{4}3m], [m\bar{3}m] and [\infty (A_{\infty}/M)C]: all components are equal to zero.

In practice, gyrotropic crystals are only found among the enantiomorphic groups: 1, 2, 222, 3, 32, 4, 422, 6, 622, 23, 432. Pasteur (1848a[link],b[link]) was the first to establish the distinction between `molecular dissymmetry' and `crystalline dissymetry'.

References

Pasteur, L. (1848a). Recherches sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire. Ann. Chim. (Paris), 24, 442–459.
Pasteur, L. (1848b). Mémoire sur la relation entre la forme cristalline et la composition chimique, et sur la cause de la polarisation rotatoire. C. R. Acad Sci. 26, 535–538.








































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