Crystal symmetry macroscopic

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. 

The symmetry of many molecules and especially of crystals is immediately obvious. Benzene has a six-fold symmetry axis and is planar, buckminsterfullerene (or just fullerene or footballene) contains 60 carbon atoms, regularly arranged in six- and five-membered rings with the same symmetry (point group //,) as that of the Platonic bodies pentagon dodecahedron and icosahedron (Fig. 2.7-1). Most crystals exhibit macroscopically visible symmetry axes and planes. In order to utilize the symmetry of molecules and crystals for vibrational spectroscopy, the symmetry properties have to be defined conveniently. 

What can we leaxn about the internal arrangement of atoms, ions, or molecules from the appearance and physical properties of a crystal The macroscopic physical properties of crystals are related to the arrangements of atoms within them. A careful examination of these physical properties leads to much useful information on the symmetry of the atomic arrangement, on the shape of the unit cell, and on the overall arrangement of molecules within the unit cell. Selected examples of such studies are described in this Chapter. 

In principle, any modification of the intra- or intermolecular relationships that break the averaged crystal symmetry on a macroscopic scale corresponds to a structural phase transition. Diffraction techniques (X-ray or neutron) are thus of primary importance to characterize the different phases. From the Bragg peaks that measure the long-range order of the mean crystal structure, the space groups, atomic positions and thermal parameters can be determined in each phases (Fig. 1). Moreover, in the most favorable cases, these methods can directly measure information on the order parameter and pretransitional ordering by following the superstructure at 7< Tc,... 

Quasi-crystals have macroscopic symmetries which are incompatible with a crystal lattice (Section 2.4.1). The first example was discovered in 1984 when the alloy AlMn is rapidly quenched, it forms quasi-crystals of icosahedral symmetry (Section 2.5.6). It is generally accepted that the structure of quasicrystals is derived from aperiodic space filling by several types of unit cell rather than one unique cell. In two-dimensional space, the best-known example is that of Penrose tiling. It is made up of two types of rhombus and has fivefold symmetry. We assume that the icosahedral structure of AlMn is derived from a three-dimensional stacking analogous to Penrose tiling. As is the case for incommensurate crystals, quasi-crystals can be described by perfectly periodic lattices in spaces of dimension higher than three in the case of AlMn, we require six-dimensional space. 

Note that the letter is different in front and back. The other symmetry elements of the letter A are also indicated. The six basic point-symmetry elements (1, 2, 3,4, 6, and i) can describe the crystal symmetry as it is macroscopically recognizable by inspection, if needed, helped by optical microscopy. 

It is known that the crystal symmetry defines point symmetry group of any macroscopic physical property, and this symmetry cannot be lower than corresponding point symmetry of a whole crystal. The simplest example is the spontaneous electric polarization that cannot exist in centrosymmetric lattice as the symmetry elements of polarization vector have no operation of inversion. We remind that inversion operation means that a system remains intact when coordinates x, y, z are substituted by —x, —y, —z. If the inversion center is lost under the phase transition in a ferroic at T < 7), Tc is the temperature of ferroelectric phase transition or, equivalently, the Curie temperature), the appearance of spontaneous electrical polarization is allowed. Spontaneous polarization P named order parameter appears smoothly... 

The formation of domain structures and the available variants in the low synunetry phase is dictated by the crystal symmetry of the high temperature phase. However, because domain patterns may produce new symmetries at the mesoscopic scale, it is the global symmetry, not the local symmetry, which controls the macroscopic functionality of the material. Therefore, at the macroscopic level, one can make composite structures of designed average symmetries to produce better functional properties. 

In real crystals of macroscopic sizes translation symmetry, strictly speaking, is absent because of the presence of borders. If, however, we consider the so-called bulk properties of a crystal (for example, distribution of electronic density in the volume of the crystal, determining the nature of a chemical bond) the influence of borders can not be taken into account (number of atoms near to the border is small, in comparison with the total number of atoms in a crystal) and we consider a crystal as a boundless system, [13]. 

Monoclinic System It has two Bravais lattices, i.e., primitive (P) and base-centered C, and three point groups 2, m, and 2/m. In detailed study of symmetry, the array of atoms that constitutes the structure of the crystal, a macroscopic mirror plane m, might be a glide plane c, while twofold rotation axis might be a screw axis as 2i. Considering these aspects of possible symmetry, the complete set is given as follows ... 

The same symmetry considerations, which here are stated for individual molecules, may of course also be applied to macroscopic systems, in particular to crystals. The determining aspect is here the overall crystal symmetry, and instead of the molecular... 

The maintenance of a connection to experiment is essential in that reliability is only measurable against experimental results. However, in practice, the computational cost of the most reliable conventional quantum chemical methods has tended to preclude their application to the large, low-symmetry molecules which form liquid crystals. There have however, been several recent steps forward in this area and here we will review some of these newest developments in predictive computer simulation of intramolecular properties of liquid crystals. In the next section we begin with a brief overview of important molecular properties which are the focus of much current computational effort and highlight some specific examples of cases where the molecular electronic origin of macroscopic properties is well established. 

Several kinds of intermediate states exist between the state of highest order in a crystal having translational symmetry in three dimensions and the disordered distribution of particles in a liquid. Liquid crystals are closest to the liquid state. They behave macroscopically like liquids, their molecules are in constant motion, but to a certain degree there exists a crystal-like order. 

The occurrence of twinned crystals is a widespread phenomenon. They may consist of individuals that can be depicted macroscopically as in the case of the dovetail twins of gypsum, where the two components are mirror-inverted (Fig. 18.8). There may also be numerous alternating components which sometimes cause a streaky appearance of the crystals (polysynthetic twin). One of the twin components is converted to the other by some symmetry operation (twinning operation), for example by a reflection in the case of the dovetail twins. Another example is the Dauphine twins of quartz which are intercon-verted by a twofold rotation axis (Fig. 18.8). Threefold or fourfold axes can also occur as symmetry elements between the components the domains then have three or four orientations. The twinning operation is not a symmetry operation of the space group of the structure, but it must be compatible with the given structural facts. 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

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