Crystal symmetry microscopic

However, its was found possible to infer all four microscopic tensor coefficients from macroscopic crystalline values and this impossibility could be related to the molecular unit anisotropy. It can be shown that the molecular unit anisotropy imposes structural relations between coefficients of macroscopic nonlinearities, in addition to the usual relations resulting from crystal symmetry. Such additional relations appear for crystal point group 2,ra and 3. For the monoclinic point group 2, this relation has been tested in the case of MAP crystals, and excellent agreement has been found, triten taking into account crystal structure data (24), and nonlinear optical measurements on single crystal (19). This approach has been extended to the electrooptic tensor (4) and should lead to similar relations, trtten the electrooptic effect is primarily of electronic origin. 

So called to distinguish them from certain microscopic symmetry operations with which we are not concerned here. The macroscopic elements can be deduced from the angles between the faces of a well-developed crystal, without any knowledge of the atom arrangement inside the crystal. The microscopic symmetry elements, on the other hand, depend entirely on atom arrangement, and their presence cannot be inferred from the external development of the crystal. 

The theory of crystal symmetry and of the periodicity of microscopic structures (translational symmetry) was developed during the 18th and 19th centuries from... 

We do not observe an image of the crystal in the focal plane of the objective of the microscope, but an interference pattern. Each point in this plane corresponds to the direction of a wave normal characterized by a birefringence, and hence, by an interference color or light intensity. The intensity goes to zero for all waves polarized according to the polarizer and the analyzer. The interference figures are characteristic of the crystal symmetry, the optical sign and the orientation of the indicatrix. 

In recent years, electron diffraction has been used to characterize fuel cell catalysts, as information about the crystal symmetry of active components can be obtained from electron diffraction. Most of the electron diffraction for fuel cell catalysts is performed in a Transmission Electron Microscope (TEM), where the electrons pass through a thin film of the samples being studied. The resulting diffraction pattern is then observed on a fluorescent screen and recorded on photographic film or with a CCD camera. 

There are more general problems of stability of materials and of phase transformations that are closely related to the tensile tests described above. Namely, the tensile test may be considered as a special case of so-called displacive phase transformation path. These paths are well known in studies of martensitic transformations. Such transformations play a major role in the theory of phase transitions. They proceed by means of cooperative displacements of atoms away from their lattice sites that alter crystal symmetry without changing the atomic order or composition. A microscopic understanding of the mechanisms of these transformations is vital since they occur prominently in many materials. 

A crystal is a solid with a periodic lattice of microscopic components. This arrangement of atoms is determined primarily by X-ray structure analysis. The smallest unit, called the unit cell, defines the complete crystal, including its symmetry. Characteristic crystallographic 3D structures are available in the fields of inorganic, organic, and organometallic compounds, macromolecules, such as proteins and nucleic adds. 

Figure 4.19 Scanning electron microscope picture of cuprous oxide crystals as shown in Fig. 4.18. Note the partial octahedral symmetry.

Electron Diffraction (CBED) and Large-Angle Convergent-Beam Electron Diffraction (LACBED) allow the identification of the crystal system, the Bravais lattice and the point and space groups. These crystallographic features are obtained at microscopic and nanoscopic scales from the observation of symmetry elements present on electron diffraction patterns. 

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