Isotropic lattice strain

Wood and Blundy (2001) developed an electrostatic model to describe this process. In essence this is a continuum approach, analogous to the lattice strain model, wherein the crystal lattice is viewed as an isotropic dielectric medium. For a series of ions with the optimum ionic radius at site M, (A(m))> partitioning is then controlled by the charge on the substituent (Z ) relative to the optimum charge at the site of interest, (Fig. 10) ... 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

One of the most frequently observed phenomena in epitaxial growth is the formation of strain relief patterns. These are caused by the mismatch of the unit cell size of the substrate and the deposited film. In many cases the strain or stress, which is imposed on the thin film by fhe subsfrafe lattice is relieved by reconstruction of the film. This reconsfrucfion can but must not necessarily lead to a nanopatterned film. An inferesfing example is the growth of Ag on Pt(l 11) (see Fig. 10) [41]. It has been shown for this particular system that the first Ag layer grows pseudomorphically exhibiting an isotropic compressive strain of 4.3% whereas in higher layers this strain is relieved by the formation of a dislocation network [42-47]. In order to improve the long-... 

Fig. 6.15 The relationship between experimental ORR activity (in units of kT ln(7a]j /7p,), T=298 K) of dealloyed Pt-Cu BMNCs after annealing at 800 °C red triangles) and 950 °C blue triangles) and the lattice strain of the Pt shell. The inset shows a scheme of a detilloyed Pt-Cu BMNC. The dashed line shows the DFT prediction of the ORR activity for a Pt(lll) singlecrystalline slab under isotropic strain. Modified with permission from ref [8]...

The thermal strain measurements described above have the common feature of anisotropic behaviour in a supposed isotropic state (cubic structure). These observations go well beyond the short-range, static strain fields associated with the lattice impurities responsible for Huang scattering. This then raises the question of the temperature at which the lattice symmetry changes and the implications of this for the central mode scattering. 

Incoherent Clusters. As described in Section B.l, for incoherent interfaces all of the lattice registry characteristic of the reference structure (usually taken as the crystal structure of the matrix in the case of phase transformations) is absent and the interface s core structure consists of all bad material. It is generally assumed that any shear stresses applied across such an interface can then be quickly relaxed by interface sliding (see Section 16.2) and that such an interface can therefore sustain only normal stresses. Material inside an enclosed, truly incoherent inclusion therefore behaves like a fluid under hydrostatic pressure. Nabarro used isotropic elasticity to find the elastic strain energy of an incoherent inclusion as a function of its shape [8]. The transformation strain was taken to be purely, dilational, the particle was assumed incompressible, and the shape was generalized to that of an... 

The Cu-Co system is a particularly simple precipitation system in which a Corich /3 phase precipitates in a Cu-rich terminal a phase. The f.c.c. lattices of both phases are well matched in three dimensions, so that the precipitate interfaces are coherent with respect to either lattice as a reference structure and the interfacial energy is sufficiently isotropic so that they are almost spherical, as in Fig. 19.2. Both the interfacial energy and strain energy are therefore relatively low and the nucleation of the f3 phase is therefore relatively easy and occurs homogeneously. This system has been used to test the applicability of the classical nucleation theory (Section 19.1.1) [11, 12]. In this work, the experimental conditions under which... 

The quantity e in Eq. (31) represents an isotropic average of the strain tensor, and F 1, F<2>,. .. describe orbital-lattice interactions. The transition probability of a direct process between states a > and b > is given by... 

In a single crystal, the excited X rays are Bragg diffracted at the lattice plane, and their isotropic angular characteristics show defect and excess Kossel lines of apex angle 90°-0b. which can be used for accurate measurement of lattice parameters and strains when the Kossel pattern is recorded on a photographic emulsion. 

Consider the substrate-film system shown in Figure 6.30. The bi-layer is constrained against bending, presumably by a relatively thick handle wafer, but it is unconstrained against extension or contraction in the plane of the interface. The isotropic elastic materials of the film and the substrate have the same shear modulus p and Poisson ratio u. The common biaxial modulus is then M = 2/i(l + v)/ l — v). The elastic biaxial extensional strains in the substrate and film are denoted by Cg and Cf, respectively. The mismatch in lattice parameter is denoted by Cm. 

The layer is flat for any uniform composition and it remains so if there is no lattice mismatch between constituents A and B. Assume that constituent B has an isotropic extensional mismatch strain ei with respect to constituent A. As a result, the midplane of the layer is curved, in general, for a nonuniform distribution From (2.58), it is known that the state of stress inducing this curvature is... 

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