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Point-group symmetry and physical properties of crystals
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2, pp. 737-741 [ doi:10.1107/97809553602060000930 ]
... Authier (2014). A short resume is given by Klapper & Hahn (2005). As a consequence of the invariance of tensor ... non-centrosymmetry. J. Appl. Cryst. 9, 145-158. Klapper, H. & Hahn, Th. (2005). Point groups. In Encyclopedia of Condensed Matter ...

Pyroelectricity and ferroelectricity
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.5, p. 741 [ doi:10.1107/97809553602060000930 ]
Pyroelectricity and ferroelectricity 3.2.2.5. Pyroelectricity and ferroelectricity In principle, pyroelectricity can only exist in crystals with a permanent electric dipole moment. This moment is changed by heating and cooling, thus giving rise to electric charges on certain crystal faces, which can be detected by simple experimental procedures. An electric dipole moment ...

Second-harmonic generation (SHG)
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.4.3, p. 741 [ doi:10.1107/97809553602060000930 ]
Second-harmonic generation (SHG) 3.2.2.4.3. Second-harmonic generation (SHG) Light waves passing through a noncentrosymmetric crystal induce new waves of twice the incident frequency. This second-harmonic generation is due to the nonlinear optical susceptibility. The second-harmonic coefficients form a third-rank tensor, which is subject to the same symmetry ...

Optical activity
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.4.2, pp. 740-741 [ doi:10.1107/97809553602060000930 ]
Optical activity 3.2.2.4.2. Optical activity The symmetry information obtained from optical activity is quite different from that given by optical refraction. Optical activity is in principle confined to the 21 noncentrosymmetric classes but it can occur in only 15 of them (Table 3.2.2.1). In the 11 enantiomorphism classes, a single ...

Refraction
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.4.1, p. 740 [ doi:10.1107/97809553602060000930 ]
Refraction 3.2.2.4.1. Refraction The dependence of the refractive index on the vibration direction of a plane-polarized light wave travelling through the crystal can be obtained from the optical indicatrix. This surface is an ellipsoid, which can degenerate into a rotation ellipsoid or even into a sphere. Thus, the lowest symmetry ...

Optical properties
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.4, pp. 740-741 [ doi:10.1107/97809553602060000930 ]
Optical properties 3.2.2.4. Optical properties Optical studies provide good facilities with which to determine the symmetry of transparent crystals. The following optical properties may be used. 3.2.2.4.1. Refraction | | The dependence of the refractive index on the vibration direction of a plane-polarized light wave travelling through the crystal can be obtained ...

Etch figures
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.3, p. 740 [ doi:10.1107/97809553602060000930 ]
Etch figures 3.2.2.3. Etch figures Additional information on the point group of a crystal can be gained from the face symmetry, which is usually determined by observation of etch figures, striations and other face markings. Etch pits are produced by heating the crystal in vacuum (evaporation from the surface) or by ...

Morphology
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.2, pp. 739-740 [ doi:10.1107/97809553602060000930 ]
Morphology 3.2.2.2. Morphology If a crystal shows well developed faces, information on its symmetry may be derived from the external form of the crystal. By means of the optical goniometer, faces related by symmetry can be determined even for crystals far below 1 mm in diameter. By this procedure, however, only ...

Polar directions, polar axes, polar point groups
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.1.4, p. 739 [ doi:10.1107/97809553602060000930 ]
Polar directions, polar axes, polar point groups 3.2.2.1.4. Polar directions, polar axes, polar point groups A direction is called polar if its two directional senses are geometrically or physically different. A polar symmetry direction of a crystal is called a polar axis. Only proper rotation or screw axes can be polar. ...

Enantiomorphism, enantiomerism, chirality, dissymmetry
Klapper, H. and Hahn, Th.  International Tables for Crystallography (2016). Vol. A, Section 3.2.2.1.3, pp. 738-739 [ doi:10.1107/97809553602060000930 ]
Enantiomorphism, enantiomerism, chirality, dissymmetry 3.2.2.1.3. Enantiomorphism, enantiomerism, chirality, dissymmetry All these terms refer to the same symmetry restriction, the absence of improper rotations (rotoinversions, rotoreflections) in a crystal or in a molecule. This implies in particular the absence of a centre of symmetry, , and of a mirror plane, , but also of ...

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