Coordinate system indices

Figure 9. Contour of the 14a level and the geometry of Cu(S(CHj)2)(SCH )(NH.) with copper and thioether based coordinate systems indicated. In the contour, all nuclei indicated are in the plane of the diagram except for those of the amine nitrogens and thioether carbons. Values of the contours are the same as in Figure 7. Reproduced from Ref. 14. Copyright 1985, American Chemical Society.

Figure 44 Diffraction from a heiicoidaiiy twisted iameiiar crystai. (a) Views of the twisted crystai with the axes of the iocai coordinate systems indicated, (b) Schematic iiiustration showing the Ewaid sphere and the shape of the reflections which have the momentum transfer vectors given by (po, 0,0), (0, rfo, 0), and (0,0, t/o) in tbe iocai coordinate system. The twisting axis is paraiiei to the direction of the peak (0,0, Po)- The (%, 0,0) and (0,0, %) peaks iie in the piane of the crystai whiie the (0, go, 0) peak is normai to the crystai surface, it is noteworthy that for the chosen cubic iattice the axes of the direct and reciprocai iocai coordinate systems are coiiinear. The red regions highiightthe intersections of the (go, 0,0) and (0,0, tfo) peaks of the Ewaid sphere and thus correspond to the experimentaiiy observed diffraction peaks. With permission from Luchnikov, V. A. ivanov, D. A. J. Appl. Cryst. 2009, 42,67. ° ...

Eq. 4.1 can be obtained from several mechanics textbooks (e.g., [20, 21]) and is shown here in 3-dimensional vector notation under consideration of relative motion. J is the matrix of the probe mass s moment of inertia 0,2 and eo0,2 are the angular acceleration and velocity of the probe mass s coordinate system (index 2) with respect to the inertial frame (index 0) 0,i and (001 are the angular acceleration and velocity of the reference system (index 1) with respect to the inertial frame. The reference system is attached to the sensing element s substrate and therefore also to the vehicle whose motion is to be measured. 2 and oi12 belong to the probe mass with respect to the reference system. M is the torque applied to the probe mass and is composed of the driving stimulus as well as the stiffness of the suspension beams and the damping of the mechanical resonator. 

For standardization of validation procedure we suggested normalized coordinate system (NCS) X. = 100-C/C", Y. = 100-A/A", where C is a concentration, A - analytical response (absorbance, peak ai ea etc.), index st indicates reference solution, i - number of solution. In this coordinate system recuperation coefficient (findings in per cent to entry) is found as Z = IQQ-Y/X. As a result, coordinates of all methods ai e in the unified... 

Cartesian tensors, i.e., tensors in a Cartesian coordinate system, will be discussed. Three Independent quantities are required to describe the position of a point in Cartesian coordinates. This set of quantities is X where X is (x, X2, X3). The index i in X has values 1,2, and 3 because of the three coordinates in three-dimensional space. The indices i and j in a j mean, therefore, that a j has nine components. Similarly, byi has 27 components, Cp has 81 components, etc. The indices are part of what is called index notation. The number of subscripts on the symboi denotes the order of the tensor. For example, a is a zero-order tensor... 

The Jk-th wave function of the electrons in a chain of hydrogen atoms results in a similar way. From every atom we obtain a contribution 2 cosnka, i.e. the Is function %n of the n-th atom of the chain takes the place of A0. All atoms have the same function x, referred to the local coordinate system of the atom, and the index n designates the position of the atom in the chain. The k-th wave function is composed of contributions of all atoms ... 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

The shear-stress convention is a bit more complicated to explain. In a differential control volume, the shear stresses act as a couple that produces a torque on the volume. The sign of the torques defines the positive directions of the shear stresses. Assume a right-handed coordinate system, here defined by (z, r, 9). The shear-stress sign convention is related to ordering of the coordinate indexes as follows a positive shear xzr produces a torque in the direction, a positive xrg produces a torque in the z direction, and a positive x z produces a torque in the r direction. Note also, for example, that a positive xrz produces a torque in the negative 6 direction. 

Square planar coordination systems (SP-Jt The configuration index is a single digit which b the priority number Tor the ligating atom irons to the ligating atom of priority number l.33... 

It is an easy matter to write the correct expressions for a pair of equivalent hybrids with a given angle a between them. Hybridization index m is found immediately from Equation A1.17. A direction (0ly i) must be chosen for the first hybrid, and a direction (02, < 2) for the second found such that the angle between them will be a. The orientation of the hybrids with respect to the coordinate system is arbitrary it will be easiest to set up the orbitals if they are oriented so that the first points along one of the axes (say x) and the second lies in one of the coordinate planes (say x, y), or if the two are placed in one of the coordinate planes with a coordinate axis bisecting the angle a. 

The index 1 indicates that this representation refers to the position vectors expressed in the laboratory coordinate system. The last equation may be commented upon as follows ... 

We are now in a position to write down the Hamiltonian operator for all nuclear spin and quadrupole terms for a diatomic molecule we allow for the possibility that both nuclei are involved and therefore sum over the nuclear index a. The terms are expressed in a molecule-fixed rotating coordinate system with origin at the nuclear centre of mass, except that we retain Ia as being quantised in a space-fixed axis system. We number the terms sequentially and then describe their physical significance. 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

Often in science, one associates transformation properties under coordinate transformations with the sub-index labels of a matrix M = R. And, one can associate such behaviour separately for each member of the index pair. Thus, the human in charge has the freedom to construct relationships between two different coordinate systems, each linked to one of the indices. Generally, R is real orthogonal but not symmetric. 

Finally, there are extensive classes of heterocycles, which are intramolecularly coordinated 3-boraheteropropanes or 3-boraheteropropenes. For example, compound 25 would be 1,2-phosphoniaboratolane. Chemical Abstracts Service (CAS), which uses the kappa system to name (T-coordination compounds, indexes 25 as dihydro[3-(phos-phino-KP)propyl-KC]boron. Heterocyclic chemists may find the CAS system cumbersome. It will not be used in this chapter. 

The application of this concept in chromatography requires the introduction of some extra components with previously known (postulated) retention indices [RI = /(f )] into the samples being analyzed. Their peaks form a mobile coordinate system for the recalculation of tp. data of the target analytes. Hence, the establishment of any retention index system needs the following ... 

First, the apparent recovery or movement of dosed systems towards the reference case may be an artifact of our measurement systems that allow the n-dimensional data to be represented in a two-dimensional system. In an n-dimensional sense, the systems may be moving in opposite directions and simply bypass similar coordinates during certain time intervals. Positions can be similar but the n-dimensional vectors describing the movements of the systems can be very different. One-time sampling indexes are likely to miss these movements or incorrectly plot the system in an arbitrary coordinate system. 

Any reciprocal lattice vector, or reciprocal lattice point is uniquely specified by the set of three integers, hkl, which are the Miller indexes of the family of planes it represents in the crystal. Thus there is a one-to-one correspondence between reciprocal lattice points and families of planes in a crystal. It will be seen shortly that the reciprocal lattice is the Fourier transform of the real lattice, and vice versa. This was in fact demonstrated experimentally in Figure 1.7 of Chapter 1 by optical diffraction. As such, reciprocal space is intimately related to the distribution of diffracted rays and the positions at which they can be observed. Reciprocal space, in a sense, is the coordinate system of diffraction space. 

Remember further that each reciprocal lattice point represents a vector, which is normal to the particular family of planes hkl (and of length 1 /d u) drawn from the origin of reciprocal space. If we can identify the position in diffraction space of a reciprocal lattice point with respect to our laboratory coordinate system, then we have a defined relationship to its family of planes, and the reciprocal lattice point tells us the orientation of that family. In practice, we usually ignore families of planes during data collection and use the reciprocal lattice to orient, impart motion to, and record the three-dimensional diffraction pattern from a crystal. Note also that if we identify the positions of only three reciprocal lattice points, that is, we can assign hkl indexes to three reflections in diffraction space, then we have defined exactly the orientation of both the reciprocal lattice, and the real space crystal lattice. 

It is possible to differentiate between tensors of different levels. Zero level tensors are scalars. They do not change by transferring to another coordinate system. Examples of scalars are temperature , pressure p and density g. No index is necessary for their characterisation. 

As discussed in Sect. 2.1.2, the index of an anisotropic medium is described by the index ellipsoid (Eq. 22). If the coordinate system is chosen such that the axes do not match with the principal symmetry axes of the crystal, the index ellipsoid is described by the more general expression [11]... 

The k index is running over the atoms in a single length slice i. The vector Rik is of rank 3 [Rt = (x, y, z)], and it provides the Cartesian coordinates of a single atom mt is the mass of the k atom. The vector R k is a reference coordinate system (the coordinates of the middle intermediate structure) that is used to determine the absolute orientation. 

It is important to note that vector relations such as the identity that we have proven in this example are invariant to the coordinate system. So, even though we have proven this result by using Cartesian index notation, the result is valid in all coordinate systems (e.g., Cartesian, cylindrical, spherical, etc.)... 

---