Unit cell deformation

The structural relationship of this monoclinic structure to the cubic one of the Prussian blue analogs is not very obvious. A comparison can be made if the dinuclear MnN6(H20)4 group is replaced by a hypothetical MNe octahedron and the monoclinic unit cell deformed to a cubic one. The resulting hypothetical structure is cubic face-centered with positions 4 a and 4 b alternatively occupied by Ru and M (56). 

An example of unit cell deformation is shown in Figure 4. Here the distribution of deformation on the side AD is drawn. The magnitude is normalized by the differences between the maximum deformation and minimum... 

Let us calculate the nanowire piezoelectric reaction to electric field Es applied along polar axes z. Using the calculated elastic field, one could find piezoelectric reaction as dkij = duij /dEk. One of fhe nonfrivial consequences of fhe flexoeffect is the local appearance of new piezoelectric tensor components, related to the unit cell deformation (see Fig. 4.23a), absent in the bulk system ... 

Fig. 34.7. Crystal structure of NaLuO and NaGdOz (schematic). To make the relation between the two structures clear, the unit cube of the rock-salt structure is drawn rather than the unit cells. Deformations of the ideal structure are not represented (from Blasse and Bril, 1970).

The two most widely accepted mechanisms for this phenomenon are defect inclusion in the crystalline phase and surface stresses related to reduced lamellar thickness [13,31,33]. In the second mechanism, defects are mostly rejected from the crystals, but preferentially reside in the interfacial layer, thereby exerting stress on the crystal surface [31,34,35]. As the counit concentration increases, the lamellar crystal becomes thinner. This leads to an increase in the crystal s surface-to-volume ratio, which in turn amplifies the magnitude of this stress-induced unit cell deformation. 

The a-rhombohedral form of boron has the simplest crystal stmcture with slightly deformed cubic close packing. At 1200°C a-rhombohedral boron degrades, and at 1500°C converts to P-rhombohedral boron, which is the most thermodynamically stable form. The unit cell has 104 boron atoms, a central B 2 icosahedron, and 12 pentagonal pyramids of boron atom directed outward. Twenty additional boron atoms complete a complex coordination (2). 

The studies show that the observed crystal volume is in fact composed of the fractional contributions from the unit cell volumes of the HS and LS isomers of the compound and a linear volume change with temperature as expressed in Eq. (128). Similarly, the observed lattice constants are formed from a deformation contribution proportional to the HS fraction and a contribution from thermal expansion following Eq. (131). This is a convincing demonstration that it is the internal variation of the molecular units occurring in the course of the spin-state transition which determines, at least in principle, the observed crystal properties. 

The crystal structure of NiAl is the CsCl, or (B2) structure. This is bcc cubic with Ni, or A1 in the center of the unit cell and Al, or Ni at the eight comers. The lattice parameter is 2.88 A, and this is also the Burgers displacement. The unit cell volume is 23.9 A3 and the heat of formation is AHf = -71.6kJ/mole. When a kink on a dislocation line moves forward one-half burgers displacement, = b/2 = 1.44 A, the compound must dissociate locally, so AHf might be the barrier to motion. To overcome this barrier, the applied stress must do an amount of work equal to the barrier energy. If x is the applied stress, the work it does is approximately xb3 so x = 8.2 GPa. Then, if the conventional ratio of hardness to yield stress is used (i.e., 2x3 = 6) the hardness should be about 50 GPa. But according to Weaver, Stevenson and Bradt (2003) it is 2.2 GPa. Therefore, it is concluded that the hardness of NiAl is not intrinsic. Rather it is determined by an extrinsic factor namely, deformation hardening. 

For mi = m2, the expression reduces to that obtained for a monoatomic chain (eq. 8.18). When q approaches zero, the amplitudes of the two types of atom become equal and the two types of atom vibrate in phase, as depicted in the upper part of Figure 8.10. Two neighbouring atoms vibrate together without an appreciable variation in their interatomic distance. The waves are termed acoustic vibrations, acoustic vibrational modes or acoustic phonons. When q is increased, the unit cell, which consists of one atom of each type, becomes increasingly deformed. At < max the heavier atoms vibrate in phase while the lighter atoms are stationary. 

Close to this limit the displacements of the two types of atom have opposite sign and the two types of atom vibrate out of phase, as illustrated in the lower part of Figure 8.10. Thus close to q = 0, the two atoms in the unit cell vibrate around their centre of mass which remains stationary. Each set of atoms vibrates in phase and the two sets with opposite phases. There is no propagation and no overall displacement of the unit cell, but a periodic deformation. These modes have frequencies corresponding to the optical region in the electromagnetic spectrum and since the atomic motions associated with these modes are similar to those formed as response to an electromagnetic field, they are termed optical modes. The optical branch has frequency maximum at q = 0. As q increases slowly decreases and... 

In general a crystal that contains n atoms per unit cell have a total of 3L n vibrational modes. Of these there are 3L acoustic modes in which the unit cell vibrates as an entity. The remaining 3L(n - 1) modes are optic and correspond to different deformations of the unit cell. At high temperatures where classical theory is valid each mode has an energy k T and the total heat capacity is 3R, in line with the Dulong-Petit law. 

Transformation (deformation) of a face-centred cubic unit cell into a body-centred cubic cell. 

Figure 7.36. Projections on characteristic planes of the unit cells of AlB2-derivative structures (binary deformation variants). Open circles represent atoms on the projection plane, dashed circles atoms on other parallel planes. For A1B2 compare with Fig. 7.5 and, for KHg2, with 7.33. (Adapted from Gladyshevskii et al. 1992)...

The model of Jayanthi etal. overcomes the phenomenological force-constant models and thus avoids the large number of hypothetical force constants, sometimes used in these calculations. Jayanthi et al. calculate the charge density in each unit cell by an expansion over many-body interactions, which arise from the coupling of the electronic deformations to... 

Also reported - is another polymorph a -Ca2Si04 which is orthorhombic, the unit cell volume of 344.8 being almost the same as that of P-Ca2Si04, 343.9 A. It seems likely that the a polymorph is only a small deformation of p-Ca2Si04. Contrast y-Ca2Si04 (Sect. 2.6.2) which is also orthorhombic, but with a much larger unit-cell volume, viz. 385.3 A ... 

Dumont and Bougeard (68, 69) reported MD calculations of the diffusion of n-alkanes up to propane as well as ethene and ethyne in silicalite. Thirteen independent sets of 4 molecules per unit cell were considered, to bolster the statistics of the results. The framework was held rigid, but the hydrocarbon molecules were flexible. The internal coordinates that were allowed to vary were as follows bond stretching, planar angular deformation, linear bending (ethyne), out-of-plane bending (ethene), and bond torsion. The potential parameters governing intermolecular interactions were optimized to reproduce infrared spectra (68). 

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