Anisotropy, of crystals

The magnetic properties of crystals composed of aromatic polynuclear molecules may give information on the relative orientations of the benzene rings to each other. Thus, Clews and Lonsdale (1937) concluded from the magnetic anisotropy of crystals of o-diphenylbenzene that the planes of the o-phenyl groups are inclined at 50° to the plane of the main ring. 

Some diamagnetic crystals (graphite, bismuth, naphthalene and other aromatic substances) show prohounced diamagnetic anisotropy. The observed anisotropy of crystals of benzene derivatives correspond to the molar diamagnetic susceptibility —54 X 10 with the field direction perpendicular to the plane of the benzene ring and —37 X lO"6 with it in the plane. This molecular anisotropy has been found to be of some use in determining the orientation of the planes of aromatic molecules in crystals.1... 

Sclerometry finds ever wider uses. The cited example of the scratch method exploited in structural tests is just one of many. Among others, interesting studies of the anisotropy of crystals, chiefly semiconducting, were carried by this method by Soviet investigators, including Boyarskaya (1972, 1977), Pasko et al. (1977), Zhitaru et al. (1977), Lazarenko et al. (1978), Simashko et al. (1978). 

Hardness testing of minerals shows in most cases a considerable difference in hardness according to crystallographic direction. By analogy with the optical anisotropy of crystals, we call this phenomenon hardness anisotropy. Dmitrev (1949) differentiated three kinds of hardness anisotropy ... 

Fig. 7.1. Examples of deformed impressions due to hardness anisotropy of crystals. (After Yushkin, 1971)...

The tendency to form inclusions also appears fundamentally related to certain crystal/solution parameters affecting heat and mass transfer to the growing crystal. In particular, inclusions can be influenced by the heat of fusion of the crystalline material, viscosity and thermal conductivity of the solution, and anisotropy of crystal shape (Buckley 1951). Thus, careful control of operating conditions and the choice of a solvent itself may be used to improve purity through a reduced number of liquid inclusions. The latter is true in HMT crystals that readily form inclusions... 

LCs represent a fascinating state of matter, combining features of isotropic liquids to the characteristic anisotropy of crystals due to this combination, LCs possess interesting technical appHcations in displays, optoelectronic devices, and as sensors and new functional materials. 

We now turn briefly to the question to what extent it might make sense to deflne these quantities also for crystals. Because of the anisotropy of crystals it is evident that this will always be possible only for one or two specified directions. 

Composites may be characterized according to type of anisotropy. The structural anisotropy created by appropriate distribution of fibres or inclusions should be distinguished from anisotropy of crystals or natural organic materials like bone or wood. The anisotropy of composites is generated more or less purposefully, according to design and adequate technology. At various scales composites with random, unidirectional, bi-directional (laminates) and multidirectional anisotropy are produced. 

For the reasons developed in Chapters 2 and 5, we would not usually recommend the use of spherical or conical indenters for hardness measurements in materials with a marked tendency to brittle behavior because of the circumferential tensile stress, or where a significant amount of pile-up, controlled by discrete slip planes, may cause distorted indentations. Consequently, in this part of Chapter 3, we shall be concerned only with pyramidal indenters such as the Knoop, Vickers, and Berkovich indenters as well as the pentagonal indenter, which was designed with the advantages of the pyramidal indenters in mind but so as to offset the intrinsic anisotropy of crystals. Here we shall identify the orientation of a given indenter with respect to its facets, as sketched in Figures 2.3 and 2.5, rather than its diagonals. 

In die late nineteenth century, scientists quickly adopted flie seminal publications of the Curie brothers. Consequently, piezoelectricity and electrostriction were first discovered and investigated on inorganic, mono- or polycrystalline materials (Katzir 2006). Therefore, the theoretical treatment of tire relevant electromechanical properties has been based on the physics and in particular on the structure and the anisotropy of crystals (Newnham 2005 Tichy et al. 2010). Semicrystalline or amorphous polymers are usually less anisotropic flian crystals, and the symmetry... 

As an illustration, we consider the case of SFIG from the (111) surface of a cubic material (3m. syimnetry). More general treatments of rotational anisotropy in centrosymmetric crystals may be found in the literature [62. 63 and M]- For the case at hand, we may detennine the anisotropy of the radiated SFl field from equation Bl.5.32 in conjunction with the fonn of -)from table Bl.5.1. We fmd, for example, for the p-in/p-out and s-... 

The importance of unsaturation is illustrated by the fact that 2,4-nonadienoic acid [21643-39-0] forms a Hquid crystal phase, whereas the aHphatic carboxyHc acids do not. The two double bonds enhance the polarizabiHty of the molecule and bring iatermolecular attractions to a level that is suitable for mesophase formation. The overall linearity of the molecule must not be sacrificed ia poteatial Hquid crystal candidates. For example, whereas /n j - -aIkoxyciaaamic acids (5) are mesomorphic, the cis isomers (6) are not, a reflection of the greater anisotropy of the trans isomer. 

Results from magnetic susceptibiHty studies have been reported (50—53). Measurements (50) obtained by the Gouy method are shown in Figure 3. These are lower than those of other investigators. However, the temperature dependences of the magnetic susceptibiHties, for the various plutonium allotropes were similar. a-Plutonium single crystals show a slight anisotropy of (54). 

In the following development we consider a plane wave of infinite lateral extent traveling in the positive Xj direction (the wave front itself lies in the Xj, Xj plane). When discussing anisotropic materials we restrict discussion to those propagation directions which produce longitudinal particle motion only i.e., if u is the particle velocity, then Uj = Uj = 0. The <100>, <110>, and <111 > direction in cubic crystals have this property, for example. The derivations presented here are heuristic with emphasis on the essential qualitative features of plastic flow. References are provided for those interested in proper quantitative features of crystal anisotropy and nonlinear thermoelasticity. 

Crystallography is a very broad science, stretching from crystal-structure determination to crystal physics (especially the systematic study and mathematical analysis of anisotropy), crystal chemistry and the geometrical study of phase transitions in the solid state, and stretching to the prediction of crystal structures from first principles this last is very active nowadays and is entirely dependent on recent advances in the electron theory of solids. There is also a flourishing field of applied crystallography, encompassing such skills as the determination of preferred orientations, alias textures, in polycrystalline assemblies. It would be fair to say that... 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

Two approaches to the attainment of the oriented states of polymer solutions and melts can be distinguished. The first one consists in the orientational crystallization of flexible-chain polymers based on the fixation by subsequent crystallization of the chains obtained as a result of melt extension. This procedure ensures the formation of a highly oriented supramolecular structure in the crystallized material. The second approach is based on the use of solutions of rigid-chain polymers in which the transition to the liquid crystalline state occurs, due to a high anisometry of the macromolecules. This state is characterized by high one-dimensional chain orientation and, as a result, by the anisotropy of the main physical properties of the material. Only slight extensions are required to obtain highly oriented films and fibers from such solutions. 

Any extended part of a linear polymer molecule exhibits a strong anisotropy of many properties since its atoms and atomic groups are oriented and the macromolecule is actually a one-dimensional crystal . The parallel packing of these parts during the formation of a uniaxially oriented polymer substance imparts the anisotropie properties of a single molecule to the whole polymeric material. 

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