Rationalizing Unit Cells

Tetrafluoro-1,3,2-dithiazoldine (7) has an envelope conformation in the gas phase (electron diffraction study) and in the crystal (x-ray structural analysis) <93JPC9625). The S atom is located in the flap in the first case S and N play the part alternatively in the second case, both conformational varieties comprising the unit cell while being interconnected by N H—N bonds. A quantum chemical calculation was made to rationalize these structural features (Section 4.12.2). 

When 1/0 is a rational number m/n (m,n integer, undivisible), there are m substrate unit cells per set of n adsorbates a superlattice with unit cell area mS can form, with the superlattice unit cell containing n arbitrarily-positioned adsorbates. It may happen in this case that the adsorbates between themselves (ignoring the substrate) form a structure that has a smaller unit cell than the superlattice unit cell one must then distinguish between the overlayer unit cell (defined in the absence of a substrate) and... 

In practice the distinction between rational and irrational values of 1/6 is unimportant, because LEED cannot distinguish between unit cells larger than the coherence distance of the electron beam ( 100 A). It is customary to designate as incommensurate any overlayer that produces a coincidence unit cell larger than the LEED coherence distance. In fact, a truly incommensurate overlayer is impossible, since it could only occur in the limit of vanishing adsorbate-substrate forces parallel to the surface. 

Preferential sorption in the sinusoidal channels was confirmed by Nicholas et al. (67) in an MD study of methane and propane adsorption. This preference was most noticeable at infinite dilution at a loading of 12 molecules per unit cell the distribution of molecules over the channels was found to be close to that expected from the relative volumes of the channel segments. The propane molecules were predicted to spend more time in the intersections than the straight channel at infinite dilution. This result is rationalized by considering the slow motion of the molecules and the conformational changes necessary to move from one channel type to another via an intersection. The distribution of propane backbone bond angles was predicted to be similar to that of gas-phase propane, indicating the rather minor effect of the zeolite on the internal coordinates of propane. 

This formula rationalizes all of the experimentally discovered homologous phases. For example PrssOieo would have 88 modules and m = 8, which means it contains 16 oxygen vacancies in a crystallographic unit cell. Table 1 lists the values of n and m for all of the experimentally discovered phases in the lanthanide higher oxides. 

An alternative grid, shown in red, defines an alternative set of planes that also contains all equivalent atoms, and under different conditions, also satisfies Bragg s equation. Such planes, which may be constructed in an endless number of ways, all have one property in common - they make rational intercepts on the axes of the unit cell. The fractional intercepts are defined as a/h, h/k, c/l in terms of the unit cell constants and the integer Miller indices, hkl. 

R = rhombohedral (unit cell can be primitive or nonprimitive see notes to Table 1-11) Principal axis of ration given number n = order e.g., 2 = twofold axis of rotation... 

Studies on other high-temperature superconductors Positron annihilation measurements across Tc, coupled with the calculations of PDD have been carried out in a variety of hole-doped superconductors that include YBa2Cu40g [48], Bi-Sr-Ca-Cu-0 [49], and Tl-Ba-Ca-Cu-0 [50, 51] systems. We will not labor with the details here, except to state that a variety of temperature dependencies are seen and these can be rationalized when the results are analysed in terms of positron density distribution and the electron-positron overlap function [39]. These calculations show that the positron s sensitivity to the superconducting transition arises primarily from the ability to probe the Cu-O network in the Cu-0 layer. The different temperature dependencies of lifetime, i.e., both the increase and decrease, can be understood in terms of a model of local electron transfer from the planar oxygen atom to the apical oxygen atom, after taking into account the correct positron density distribution within the unit cell of the cuprate superconductor. 

It is important to describe each crystal face in a numerical way if data on different crystals or from different laboratories are to be compared. The method used to describe crystal faces is derived from the Law of Rational Indices, proposed by Haiiy and Arnould Carangeot. This Law states that each face of a crystal may be described, by reference to its intercepts on three noncollinear axes, by three small whole numbers (that is, by three rational indices)/ From this law, William Whewell introduced a specific way of designating crystal faces by such indices, and William Hallowes Miller popularized it. The integers that characterize crystal faces are called Miller indices h, k, and 1. When this method is used to describe crystal faces, it is rare to find h, k, or / larger than 6, even in crystals with complicated shapes. An example of the buildup of unit cells to give crystals with different faces is shown in Figure 2.11. 

This assumption is unrealistic using even the most powerful single processor PC available in late 2002. A more rational estimate is between 10 and 10 unit cells per second for a well optimized computer code. 

Below we will examine some practical applications of the theory of kinematical diffraction to solving crystal structures from powder diffraction data. When considering several rational examples in reciprocal space, we shall implicitly assume that the crystal structure of each sample is unknown and that it must be solved based solely on the information that can be obtained directly from a powder diffraction experiment and from a few other, quite basic properties of a polycrystalline material. The solution of a number of crystal structures in direct space will be based on the previously known structural data and supported by the results of powder diffraction analysis, such as unit cell dimensions and symmetry. 

---