Stress free crystal

Show that regardless of the orientation of a straight dislocation line and its Burgers vector, there will exist a stress system that will convert the dislocation line into a helix whose axis is along the position of the original dislocation when the point-defect concentration is at the equilibrium value characteristic of the stress-free crystal. Use the simple line-tension approximation leading to Eq. 11.12. 

Since 4 orientation states co-exist in the ferroelastic phases of LSGMO, the condition of a stress-free crystal can be written ... 

Minimalization of the total crystal energy with respect to the lattice parameters yields a lattice constant which guarantees a stress-free crystal. It is found that in particular the TA-mode is very sensitive to compatibility between lattice constant and total energy. 

AH intrinsic germanium metal sold is specified to be N-type with a resistivity of at least 40 H-cm at 25°C or 50 H-cm at 20°C. Germanium metal prepared for use in infrared optics is usuaHy specified to be N-type with a resistivity of 4-40 Hem, to be stress-free and fine annealed, and to have certain minimum transmission (or maximum absorption) characteristics in the 3—5 or 8—12 pm wavelength ranges. Either polycrystaHine or single-crystal material is specified. 

For example, consider a binary alloy in which the stress-free molar volume is a function of concentration, V(cs). The linear expansion due to the composition change can be inferred from diffraction experiments under stress-free conditions (Vegard s effect) and is characterized by Vegard s parameter, ac [e.g., in cubic or isotropic crystals e ° = e°y 0 = = ac(c — c0)]. The assumption of coherency... 

Slip relies on chemical bond breaking and bond reformation as two planes of atoms pull apart. It is observed that the critical resolved shear stress required to cause plastic deformation in real materials is much lower (by several orders of magnitude) than the shear stress required in deforming perfect defect-free crystals, the so-called ideal shear stress. The latter is equivalent to the stress required for the simultaneous ghding motion (bond breaking and reformation) of aU the atoms in one plane, over another plane. 

Having derived the condition for crystal resonance allows the displacement profile at resonance to be calculated. When the crystal is operated in air or vacuum, the crystal faces experience no external restoring force and are oHisidered to be stress-free boundaries diis implies that dujdy = 0 at the upper and lower crystal faces. Applying this boundary condition to Equation 3.1 yields the shear displacement profile across the crystal ... 

In order to satisfy the stress-free boundary condition, coupled compressional and shear waves propagate together in a SAW such that surface traction forces are zero (i.e., T y = 0, where y is normal to the device surface). The generalized surface acoustic wave, propagating in the z-direction, has a displacement profile u(y) that varies with depth y into the crystal as... 

At equilibrium, the crystal must be stress free (dG/d(ei-e2) = 0), giving... 

Ferroelastic compounds are characterized by the buildup of a homogeneous spontaneous strain. As a consequence of the phase transition, the low-symmetry phase splits into different domains of identical free energy that are related by some of the lost symmetries of the high-symmetry phase, and that could be switched within each other by application of an external stress. Single-crystal NMR is uniquely suited to study such transitions, but also NQR to retrieve informations about chemical equivalence. 

X-rays through a suitable filter of solid material. Owing to the difficulties involved in obtaining stress-free single crystals the Laue method is no longer used. 

Fig. 2a shows that some calculated positions of reflections from symmetry allowed domains do not coincide with observed reflections of domain TR4. Therefore, we proceed to calculate the orientation matrix of domain TR3 (previously determined with respects to TRI), and taking this domain as a reference . Positions of reflections are given in Fig. 2b which shows that the domain TR3 is connected with domain TRI via (121), and it is also coimected with the domain TR4 via the plane (110). However, there is no stress-free wall between the domains TR3 and TR2. Based on the identification of domain walls between 4 observed orientation states we can now assume that the domain pattern of LSGMO crystal has a chevron-like configuration in the trigonal phase (Fig. 3). 

Consider a partially coherent thin p layer on an a single crystal. Given the distance between the misfit dislocations formed at the ajp interface to be d and the intrinsic (stress-free) lattice parameters of a and p to be a and a, respectively, derive the coherency strain s in the p layer. 

There are two electrical equivalent circuits in common usage, the transmission line model (TLM) and a lumped element model (LEM) commonly referred to as the Butterworth-van Dyke (BvD) model these are illustrated in Figs. 2(a and b), respectively. In the TLM, there are two acoustic ports that represent the two crystal faces one is exposed to air (i.e. is stress-free, indicated by the electrical short) and the other carries the mechanical loading (here, a film and the electrolyte solution, represented below by the mechanical loading Zs). These acoustic ports are coimected by a transmission line, which is in turn connected to the electrical circuitry by a transformer representing the piezoelectric coupling. For the TLM, one can show [18, 19] that the motional impedance (Zj ) associated with the surface loading can be related to the mechanical impedances of... 

At different stretches, different crystals are formed and these can have different stress free states. Crystallization may be different under conditions of varying stress and strain. Then the manner in which the crystallites melt is important. One can assume that the crystallites formed last melt first. 

---