Effective crystallographic index

Different from the critical radius, Mori et al. [89] proposed a concept of effective crystallographic index to maximize the oxide ionic conductivity of doped ceria. The index, I, is defined as... 

Using single crystals it has been shown that different low-index crystal faces see Section 20) exhibit different corrosion rates. However, the relative corrosion rate of the different faces varies with the environment and these structural effects are of little practical significance. On the other hand, the fact that polycrystal grains of different crystallographic orientation may corrode at different rates, is of some importance. 

In non-polar, isotropic crystals or in glasses, there is no crystallographic direction distinguished and the linear electro-optic effect is absent. Nevertheless a static field may change the index by displacing ions with respect to their valence electrons. In this case the lowest non-vanishing coefficients are of the quadratic form, i.e. the refractive index changes proportionally to the square of the applied field Kerr effect . 

Toraya s WPPD approach is quite similar to the Rietveld method it requires knowledge of the chemical composition of the individual phases (mass absorption coefficients of phases of the sample), and their unit cell parameters from indexing. The benefit of this method is that it does not require the structural model required by the Rietveld method. Furthermore, if the quality of the crystallographic structure is poor and contains disordered pharmaceutical or poorly refined solvent molecules, quantification by the WPPD approach will be unbiased by an inadequate structural model, in contrast to the Rietveld method. If an appropriate internal standard of known quantity is introduced to the sample, the method can be applied to determine the amorphous phase composition as well as the crystalline components.9 The Rietveld method uses structural-based parameters such as atomic coordinates and atomic site occupancies are required for the calculation of the structure factor, in addition to the parameters refined by the WPPD method of Toraya. The additional complexity of the Rietveld method affords a greater amount of information to be extracted from the data set, due to the increased number of refinable parameters. Furthermore, the method is commonly referred to as a standardless method, since the structural model serves the role of a standard crystalline phase. It is generally best to minimize the effect of preferred orientation through sample preparation. In certain instances models of its influence on the powder pattern can be used to improve the refinement.12... 

Polycrystalline surfaces result from a mixing of all possible crystal orientations. From an energetic point of view, the low index faces discussed above predominate. The work function of the polycrystalline surface reflects a weighted average of the work functions for each crystallographic orientation. In the case of face-centered cubic systems, it falls between that of the (110) and (100) single crystal surfaces. Since polycrystalline metals are involved in most practical applications, the effect of surface structure on the work function is not discussed further here. More information on this topic is available in reviews by Trasatti [G3, 5]. 

The position of the vCO band depends on the Cu crystallographic plane [108]. The low-index Cu(lOO) and Cu(l 11) surfaces give bands at 2080 and 2076 cm", respectively, while the Cu(llO), Cu(311), Cu(211), and Cu(755) surfaces give bands between 2096 and 2110 cm", close to the band positions for evaporated polycrystalUne Cu and supported Cu (2100-2103 cm" ). This effect has been attributed [108] to a predominance of stepped or higher index planes at the surface of the evaporated films and snpported Cu. 

The stracture of CGZA crystal can be indexed to the cubic system. For the final acceptable refinement, which relates to a lower R factor, the Al " ions occupy the A sites, the larger Zr" ions occupy the D sites, and the largest Ca " and Gd " ions occupy the L sites with a ratio of 2 1 [16]. The crystallographic data and stracture refinement for the CGZA crystal are listed in Table 9.1 [16]. The 24c, 16a, and 24d WyckofF sites are fully occupied by Ca VGd, Zr", and Al, respectively. Based on the effective ionic radii (r) of the cations, Ce " (r = 1.14 A for CN = 8) is expected to preferably occupy the sites of Gd [16]. 

It is important to recogiuze that, as the expression derived above demonstrates, the effective mass of a crystal electron depends on the wave-vector k and band index n of its wavefunction, as well as on the wavefunctions and energies of all other crystal electrons with the same k-vector. This is a demonstration of the quasiparticle nature of electrons in a crystal. Since the effective mass involves complicated dependence on the direction of the k-vector and the momenta and energies of many states, it can have different magnitude and even different signs along different crystallographic directions ... 

The value of for ammonium alum has been determined and the points in Fig. 12 for ammonium alum can be replotted as log g versus the reciprocal of log p jp). Surprisingly a straight line is obtained (Fig. 13), in agreement with the form of Eq. (12) when r is taken as effectively independent of p. This agreement may mean that the growth of patches takes place by Volmer nucleation. Volmer nucleation would only be necessary if the interface between the patch and the parent crystal were flat and parallel to a low index plane. Acock et al [( ), p. 510] mention that patches on ammonium alum and other hydrates tend to have crystallographic interfaces. The patches on chrome.alum appear to be hemispherical in shape two-dimensional nucleation would not be necessary for curved interfaces. 

---