Lattice strains

The presence of lattice strain seriously complicates the problem, and even raises some doubts about the validity of the definition for the size effect in the presence of distortions. 

The effects of lattice strain on powder line profiles are also developed in detail in Chapters 12 and 13. Again, here we give an introduction and overview. 

The most convincing lattice strain description is again provided by the Fourier analysis approach. A profile that would be affected by only distortion effects can be asymmetrical and has to be represented by sine as well as cosine terms  

The demonstration is usually made for an 001 reflection from a crystal having orthorhombic axes (Warren, Chapter 13.4), leading one to obtain the following definitions for and which are dependent on the harmonic number / of the reflection family considered  

This time, the problem is continuous in Z however, the integral is not supposed to extend from — go to + oo, since Z is expected to vary between —1 2 and +1 2 times the cell parameter (if it would be more, then we would have to consider Z +j, etc.). We could obtain a generalization for the whole 001 family (/= 1, 2, etc) that would enable us to calculate the profile shapes. 

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The breadth of the peaks in an x-ray diffractogram provide a detennination of the average crystallite domain size, assuming no lattice strain or defects, tlirough the Debye-Scherr fonnula ... 

For the same lattice strains, the larger the valency difference between solute and solvent, the greater the hardening. The strengthening influence of alloying elements persists to temperatures at least as high as 815°C. Valency effects may be explained by modulus differences between the various alloys... 

Raman Microspectroscopy. Raman spectra of small soflds or small regions of soflds can be obtained at a spatial resolution of about 1 p.m usiag a Raman microprobe. A widespread appHcation is ia the characterization of materials. For example, the Raman microprobe is used to measure lattice strain ia semiconductors (30) and polymers (31,32), and to identify graphitic regions ia diamond films (33). The microprobe has long been employed to identify fluid iaclusions ia minerals (34), and is iacreasiagly popular for identification of iaclusions ia glass (qv) (35). 

Physical Properties. Raman spectroscopy is an excellent tool for investigating stress and strain in many different materials (see Materlals reliability). Lattice strain distribution measurements in siUcon are a classic case. More recent examples of this include the characterization of thin films (56), and measurements of stress and relaxation in silicon—germanium layers (57). 

These techniques have very important applications to some of the micro-structural effects discussed previously in this chapter. For example, time-resolved measurements of the actual lattice strain at the impact surface will give direct information on rate of departure from ideal elastic impact conditions. Recall that the stress tensor depends on the elastic (lattice) strains (7.4). Measurements of the type described above give stress relaxation directly, without all of the interpretational assumptions required of elastic-precursor-decay studies. 

PRE-TRANSFORMATION LATTICE STRAIN ANISOTROPY AND CENTRAL PEAK SCATTERING... 

The observation of the departure from cubic symmetry above Tm co-incident with the appearance of the central peak scattering serves to resolve the conflict between dynamic and lattice strain models. The departure from cubic symmetry may be attributed to a shift in the atomic equilibrium position associated with the soft-mode anharmonicity. In such a picture, the central peak then becomes the precusor to a Bragg reflection for the new structure. 

Lattice gas Lattice parameter Lattice strain anisotropy Lic[uid alloys... 

A different melting point, and hence supercooling, is predicted for the strained sector. This is the basis for a different interpretation of the (200) growth rates a regime //// transition occurs on (110) but not on (200). This is despite the fact that the raw data [113] show a similar change in slope when plotted with respect to the equilibrium dissolution temperature (Fig. 3.15). It is questionable whether it is correct to extrapolate the melting point depression equation for finite crystals which is due to lattice strain caused by folds, to infinite crystal size while keeping the strain factor constant. 

The requirement I > 2 can be understood from the symmetry considerations. The case of no restoring force, 1=1, corresponds to a domain translation. Within our picture, this mode corresponds to the tunneling transition itself. The translation of the defects center of mass violates momentum conservation and thus must be accompanied by absorbing a phonon. Such resonant processes couple linearly to the lattice strain and contribute the most to the phonon absorption at the low temperatures, dominated by one-phonon processes. On the other hand, I = 0 corresponds to a uniform dilation of the shell. This mode is formally related to the domain growth at T>Tg and is described by the theory in Xia and Wolynes [ 1 ]. It is thus possible, in principle, to interpret our formalism as a multipole expansion of the interaction of the domain with the rest of the sample. Harmonics with I > 2 correspond to pure shape modulations of the membrane. 

It has been determined that there is a distribution coefficient for the impurities between crystal and melt which favors the melt. We can see how this arises when we reflect that impurities tend to cause formation of intrinsic defects within the crystal and lattice strain as a result of their presence. In the melt, no such restriction applies. Actually, each impurity has its own distribution coefficient. However, one can apply an average value to better approximate the behavior of the majority of impurities. 

The physical and chemical characteristics of zinc oxide powders are known to affect cement formation (Smith, 1958 Norman et al., 1964 Crisp, Ambersley Wilson, 1980 Prosser Wilson, 1982). The rate of reaction depends on the source, preparation, particle size and surface moisture of the powder. Crystallinity and lattice strain have also been suggested as factors that may change the reactivity of zinc oxide powders towards eugenol (Smith, 1958). 

Various other interactions have been considered as the driving force for spin-state transitions such as the Jahn-Teller coupling between the d electrons and a local distortion [73], the coupling between the metal ion and an intramolecular distortion [74, 75, 76] or the coupling between the d electrons and the lattice strain [77, 78]. At present, based on the available experimental evidence, the contribution of these interactions cannot be definitely assessed. Moreover, all these models are mathematically rather ambitious and do not show the intuitively simple structure inherent in the effect of a variation of molecular volume considered here. Their discussion has to be deferred to a more specialized study. 

Mineral-Melt Partitioning ofU-Series Nuclides Lattice strain modei... 

Figure 3. Cartoon illustrating the lattice strain model of trace element partitioning. For an isovalent series of ions with charge n+ and radius entering crystal lattice site M, the partition coefficient,, can be...

Figure 4. Fits of lattice strain model to experimental mineral-melt partition coefficients for (a) plagioclase (run 90-6 of Blundy and Wood 1994) and (b) elinopyroxene (ran DC23 of Blundy and Dalton 2000). Different valence cations, entering the large cation site of each mineral, are denoted by different symbols. The curves are non-linear least squares fits of Equation (1) to the data for each valence. Errors bars, when larger than symbol, are 1 s.d. Ionic radii in Vlll-fold coordination are taken from Shannon (1976).

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