Principal planes

Figure Bl.6.1 Equipotential surfaces have the shape of lenses in tlie field between two cylinders biased at different voltages. The focusing properties of the electron optical lens are specified by focal points located at focal lengthsandy2, measured relative to the principal planes, The two principal rays emanating...

Figure 2. The focal length is the distance between the principal plane and the focus, where the principal plane is defined as the surface where tlic input and out-pul light rays would intercept.

Fig. 35.3. Scatter plot of 16 olive oils scored by two sensory panels (Dutch panel lower case British panel upper case). The combined data are shown after Procrustes matching and projection onto the principal plane of the average configuration.

The turbine type of helicity in canals is a toroidal turbine, and can occur with proper rotation axes and with 2V 42, 63 screw axes, and depends on the object rotated. It is necessary that a distinctive principal plane or axis of the host molecule or molecular fragment be canted in the cylindrical surface of the canal so that it is neither parallel to or perpendicular to the axis of the canal. No unequivocal instance of this type of helical canal has been reported. The cyclodextrin unimolecular hosts 3 4) might be... 

Figure 3. SHG phase-matching configurations in the three principal planes of POM as a function of fundamental wavelength (-)for Type IIphase-matching and(—)for Type 1 phase-matching [( ) theoretical possibilities forbidden by symmetry].

Kuz min et al. (15) pointed out a standard result of classical mechanics If a configuration of particles has a plane of symmetry, then this plane is perpendicular to a principal axis (19). A principal axis is defined to be an eigenvector of the inertial tensor. Furthermore, if the configuration of particles possesses any axis of symmetry, then this axis is also a principal axis, and the plane perpendicular to this axis is a principal plane corresponding to a degenerate principal moment of inertia (19). 

In examining a list of X-ray reflections for this purpose, it is best to look first for evidence of the lattice type—whether it is simple (P) or compound systematic absences throughout the whole range of reflections indicate a compound lattice, and the types of absences show whether the cell is body-centred (/), side-centred ( 4, P, Or C), or face-centred (F). When this is settled, look for further absences systematic absences throughout a zone of reflections indicate a glide plane normal to the zone axis, while systematic absences of reflections from a single principal plane indicate a screw axis normal to the plane. The result... 

The plane of the nitrite ion can be defined it must lie in the only principal plane showing a normal decline of intensities—that is, 200. The nitrite ions must therefore lie as in Fig. 181 a (with c as the polar twofold axis) or as in Fig. 181 b (with b as the polar twofold axis). 

Reflections. If a plane of reflection is chosen to coincide with a principal Cartesian plane (i.e., an xy, xz, or yz plane), reflection of a general point has the effect of changing the sign of the coordinate measured perpendicular to the plane while leaving unchanged the two coordinates whose axes define the plane. Thus, for reflections in the three principal planes, we may write the following matrix equations ... 

Figure 6.5 Observed dHvA frequencies in three principal planes of WC grown by the flux method. Dotted lines are expected frequencies from the present model.

Figure 4.31 Data for the characterization of an electrostatic lens, (a) Positions of the focal and principal planes (left-hand and right-hand sides are indicated by the subscripts Y and r respectively) and their distances (optical sign conventions are disregarded, i.e., the distances are described only by their lengths). (b) Geometrical construction applied to image the arrow ye by means of characteristic asymptotic trajectories, (c) Geometrical construction for an asymptotic ray with a pencil angle a,e. The shaded areas are needed for the derivation of the linear and angular magnification factors of the lens. For details see main text.

A conical hopper of a half angle 20° and an angle of wall friction 25° is used to store a cohesionless material of bulk density 1,900 kg/m3 and an angle of internal friction 45°. The top surface of the material lies at a level 3.0 m above the apex and is free of loads. Apply Walker s method to determine the normal and shear stresses on the wall at a height of 1.2 m above the apex if the angle between the major principal plane at that height and the hopper wall is 30°. Assume a distribution factor of 1.1. 

The expression is easily coded, since T<0>, Eq. 41, and the r 1 are known. It simplifies for substitution on a principal plane or axis and for symmetrically equivalent multiple substitution, because several of the elements of the matrix T will then vanish. It is clear from Eq. 45 that the derivatives are nonvanishing only for those atoms a that have actually been substitued in the particular isotopomer s. Therefore, the Jacobian matrix X generated from these derivatives is, in general, a sparse matrix. 

For a least-squares solution of the system of Eqs. 40 for all s = 2, Ns, we have to identify the components of the vector of observations y, the components of the vector of variables p and the elements of the Jacobian matrix X as shown below (Eqs. 46—48). A left arrow has been used instead of a sign of equation to indicate that, in general, the dimensions of p, X, and y are preliminary and must be reduced before least-squares processing can take place some of the P/,ma = A/j 1 may not be independent because symmetrically equivalent atoms have been substituted. Other coordinates may be kept fixed intentionally (e.g., at zero when an atom is known to lie on a principal plane or axis). The respective component(s) must then be eliminated from the vector of variables. Also, one or more of the observations may have to be dropped in order to comply with the recommendations given for the Chutjian-type treatment of substitutions on a principal plane or axis [44],... 

The increment of a coordinate that is kept fixed, e.g. fixed at zero for the substitution on a principal plane or axis, must not be a component of the vector p of variables. The required elimination is most easily implemented by letting the matrix D of Eq. 56a be followed by a matrix E, which consists of a unit matrix where only the columns (not the rows) corresponding to any components of f) to be dropped have been eliminated ... 

The first and second moment conditions can be very easily introduced into the r5-fit method as least-squares constraints [7,54] if the number of isotopomers is sufficient for a complete restructure. The effect on the coordinates is not expected to be particularly unbalanced unless the moment conditions are required for the sole purpose of locating atoms that could not be substituted (e.g., fluorine or phosphorus) or that have a near-zero coordinate. While all coordinates may change, the small coordinates will, of course, change more. In the cases tested, the coordinate values of the rs-fit with constraints and those of the corresponding r/e-fit (not of the r0-fit), including errors and correlations, differed by only a small fraction of the respective errors, i.e., much less than reported above. This was true under the provision that all atoms could be substituted and that the planar moments that were excluded from the r -fit because of substitution on a principal plane or axis, were also omitted from the r/E-fit. With these modifications, the basic physical considerations and the input data are the same in both cases, and the results should be identical in the limit where the number of observations equals that of the variables. 

Note, however, that the Cartesian axes are not C4 axes (though they are C2 axes) and the principal planes (namely, xy, xz, yz) are not symmetry planes. Thus we have here an example of the existence of the 5 axis without C or crh having any independent existence. The ethane molecule in its staggered configuration has an S6 axis and provides another example. 

The locus of the middle points of a system of parallel chords of a conicoid is called the diametral plane, and if it is perpendicular to the chords it bisects it is called a principal plane. The roots of the quadratic equation (4), 12.VIII N giving the segments of a chord through (a, /S, y) are equal and opposite if ... 

For this purpose we view a craze as a lenticular region with its principal plane normal to the tensile axis z. Although actual shapes of isolated crazes differ significantly from regular shapes as we will discuss below, a craze can be considered as a very eccentric oblate spheroid with a radius a in its principal plane and a half thickness b parallel to the z axis. Furthermore, since the main microstructural feature of a craze is extended fibrils with a relatively constant extension ratio we can view the... 

Because of the complexity of Fig. 12.1, the detailed characteristics of binary VLB are usually depicted by two-dimensional graphs that display what is seen on various planes that cut the three-dimensional diagram. The three principal planes, each perpendicular to one of the coordinate axes, are illustrated in Fig. 12.1. Thus a vertical plane perpendicular to the temperature axis is outlined as ALBDEA. The lines on this plane represent a Pxy phase diagram at constant T, of which we have already seen examples in Figs. 10.1, 11.7, 11.9, and 11.11. If the lines from several such planes are projected on a single parallel plane, a diagram like Fig. 12.2 is obtained. It shows Pxy plots for three different temperatures. The one for represents the section of Fig. 12.1 indicated by ALBDEA. 

In this approach, the principal planes of the environment are taken as the best approximation of the potential symmetry planes, and the asymmetry of the environment as seen from the considered / th atom is defined as proportional to the distance from / to the symmetry plane. 

Figure S-1. Projections and embedding rectangles of toluene molecule in the three principal planes.

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