Parametral plane

Miller indiees are used to numerieally define the shape of erystals in terms of their faees. All the faees of a erystal ean be deseribed and numbered in terms of their axial intereepts (usually three though sometimes four are required). If, for example, three erystallographie axes have been deeided upon, a plane that is inelined to all three axes is ehosen as the standard or parametral plane. The intereepts Z, Y, Z of this plane on the axes x, y, and z are ealled parameters a, b and c. The ratios of the parameters and Irx are ealled the axial ratios, and by eonvention the values of the parameters are redueed so that the value of b is unity. 

For the eube (Figure 1.2) sinee no faee is ineluded none ean be ehosen as the parametral plane (111). The intereepts Y and Z of faee A on the axes y and z are at infinity, so the Miller indiees h, k and / for this faee will be a/a, bloa and c/oo. or (100). Similarly, faees B and C are designated (010) and (001), respeetively. Several eharaeteristie erystal forms of some eommon substanees are given in Mullin (2001). 

The domain of the stable flow is located to the right of the boundary PeL(i ) (the shaded region in the graph). To the left of this curve is the domain in which stable flows in a heated capillary cannot occur. From the relation between the parameters and Ja, the parametric plane Pep — may be subdivided into two domains (1) < Ja, and (2) > Ja. Within the first of these the stable flows cannot occur... 

The dependence of P (PeL) and g (PeL) is shown in Fig. 11.4. The parameter P (PeL) is a parabola with an axis of symmetry left of the line Pcl = 0. Since the Peclet number is positive, for any value of the operating parameters, the physical meaning is that only for the right branch of this parabola, which intersects the axis of the abscissa at some critical value of Peclet number, Pcl = Peer- The vertical line PeL = Peer subdivides the parametrical plane P - Pcl into two domains, corresponding to positive (PeL < Peer) or negative (PeL > Peer) values of the parameter P . The critical Peclet number is... 

Thus the mechanism formed by steps (l)-(4) can be called the simplest catalytic oscillator. [Detailed parametric analysis of model (35) was recently provided by Khibnik et al. [234]. The two-parametric plane (k2, k 4/k4) was divided into 23 regions which correspond to various types of phase portraits.] Its structure consists of the simplest catalytic trigger (8) and linear "buffer , step (4). The latter permits us to obtain in the three-dimensional phase space oscillations between two stable branches of the S-shaped kinetic characteristics z(q) for the adsorption mechanism (l)-(3). The reversible reaction (4) can be interpreted as a slow reversible poisoning (blocking) of... 

The faces of a crystal, irrespective of the overall shape of the crystal, could always be labelled with respect to the crystal axes. Each face was given a set of three integers, (h k l), called Miller indices. These are such that the crystal face in question made intercepts on the three axes of a/h, b/k and c/l. A crystal face that intersected the axes in exactly the axial ratios was given importance as the parametral plane, with indices (111). [Miller indices are now used to label any plane, internal or external, in a crystal, as described in Chapter 2, and the nomenclature is not just confined to the external faces of a crystal.]... 

Consider a rotating physical plane parametrized by the complex variable y e C for convenience we assume the fixed primaries of the restricted three-body problem to be situated at the points A, C given by the complex posititons y = — 1 and y = 1, respectively (see Figure 5). The complex variable of the parametric plane will be denoted by v and will be normalized in such a way that the primaries are mapped to v = -1 or v = 1, respectively. 

Figure 1.12 shows two simple crystals belonging to the regular system. As there is no inclined face in the cube, no face can be chosen as the parametral plane (111). The intercepts Y and Z of face A on the axes y and z are at infinity. 

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