Scattering three-dimensional

The measurable quantity in a three dimensional scattering experiment is the differential cross section da (9)/dQ. This is defined as... 

These are plotted in Fig. 10.6, which shows the net intensity envelope in the xy plane as a solid line and represents the horizontally and vertically polarized contributions to the resultant by the broken lines. Since 0 is symmetrical with respect to the x axis, the three-dimensional scattering pattern is generated by rotating the solid contour around the x axis. 

The results show that DE-MS alone provides evidence of the presence of the most abundant components in samples. On account of the relatively greater difficulty in the interpretation of DE-MS mass spectra, the use of multivariate analysis by principal component analysis (PCA) of DE-MS mass spectral data was used to rapidly differentiate triterpene resinous materials and to compare reference samples with archaeological ones. This method classifies the spectra and indicates the level of similarity of the samples. The output is a two- or three-dimensional scatter plot in which the geometric distances among the various points, representing the samples, reflect the differences in the distribution of ion peaks in the mass spectra, which in turn point to differences in chemical composition of... 

Nonlinear mapping (NLM) as described by Sammon (1969) and others (Sharaf et al. 1986) has been popular in chemometrics. Aim of NLM is a two-(eventually a one- or three-) dimensional scatter plot with a point for each of the n objects preserving optimally the relative distances in the high-dimensional variable space. Starting point is a distance matrix for the m-dimensional space applying the Euclidean distance or any other monotonic distance measure this matrix contains the distances of all pairs of objects, due. A two-dimensional representation requires two map coordinates for each object in total 2n numbers have to be determined. The starting map coordinates can be chosen randomly or can be, for instance, PC A scores. The distances in the map are denoted by d t. A mapping error ( stress, loss function) NLm can be defined as... 

The aim of the graphics software, (tMod, threeD), is to enable the user to rapidly examine a large number of two- and three-dimensional scatter plots. At present the program is capable of handling up to 120 variables with up to 200 observations for each. 

This is the generalized three-dimensional scattering relationship for the response just above the surface at x to an oscillatory pressure just above the surface at x, due to Rayleigh wave excitation, in the case where the y component of the wavevector is constant. The three-dimensional scattering function can now be calculated. 

Figure 8.39 shows a three-dimensional scatter plot of the first three PC scores obtained from a PCA analysis of 987 calibration spectra that were collected for a specific on-line analyzer calibration project. In this case, cluster analysis was done using the first six PCs (all of which cannot be displayed in the plot ) in order to select a subset of 100 of these samples for calibration. The three-dimensional score plot shows that the selected samples are well distributed among the calibration samples, at least when the first three PCs are considered. 

Figure 8.39 Three-dimensional scatter plot of the first three PCA scores obtained from a set of original calibration data. The calibration samples selected by the cluster analysis method are marked with an x. ...

For bivariate data the simple scatter plot of variate x against variate y is popular and there are several ways in which this can be extended to accommodate fmther variables. Figure 9 iUustrates an example of a three-dimensional scatter plot. The data used are from Table 11, representing the results of the... 

Figure 16 A three-dimensional scatter plot of the polymer spectra projected on to the first three principal components...

Figure 7.20 Three-dimensional scatter plot of the first three principal components analysis. Subjects that are removed from the bulk of the data represent influential observations that may be influencing the parameter estimates overall. ...

For bivariate data the simple scatter plot of variable y against variable a is popular and there are several ways in which this can be extended to accommodate further variables. Figure 1.12 illustrates an example of a three-dimensional scatter plot. The data used are from Table 1.11, representing the results of the analysis of nine alloys for four elements. The concentration of three analytes, zinc, tin, and iron, are displayed. It is immediately apparent from the illustration that the samples fall into one of two groups, with one sample lying between the groups. This pattern in the data is more readily seen in the graphical display than from the tabulated data. 

This style of representation is limited to three variables and even then the diagrams can become confusing, particularly for a lot of points. One method for graphically representing multivariate data ascribes each variable to some characteristic of a cartoon face. These Chernoff faces have been used extensively in the social sciences and adaptations have appeared in the analytical chemistry literature. Figure 1.13 illustrates the use of Chernoff faces to represent the data from Table 1.11. The size of the forehead is proportional to tin concentration, the lower face to zinc level, mouth to nickel, and nose to iron concentration. As with the three-dimensional scatter plot, two groups can... 

If we had a data set with three variables it would still be possible to visualize the whole information by a three-dimensional scatter plot, in which the coordinates of each object are the values of the variables. But what to do if there are more than three variables What we need therefore is a technique permitting the visualization by simple bi- or tri-dimensional scatter plots of the majority of the information contained in a highly dimensional data set. This technique is Principal Component Analysis (PCA), one of the simplest and most used methods of multivariate analysis. PCA is very important especially in the preliminary steps of an elaboration, when one wants to perform an exploratory analysis in order to have an overview of the data. 

Fig. 3.8 Three dimensional scattering wave functions for 7 = 0 at the collision energy of 0.26 kcal/mol (a) and 0.46 kcal/mol (b). From [24], reprinted with permission from AAAS...

The plot of scattered field amplitude is shown in the upper part of Fig. 5.1 lb. It is the so-called sine-integral function. The scattering intensity is shown in the lower part of the figure. Integrating over the y and z co-ordinates we obtain the three-dimensional scattering amplitude F(q)= pFA AyA and intensity 7(q)= p"F2(A,AyA,)2. 

Just like structural keys, pharmacophore keys can be readily extended to account for multiple conformations. Three-point pharmacophore keys also lend themselves to visualization in the form of a three-dimensional scatter plot (see Section 5.4). A number of people have followed up Sheridan s work, most notably the groups at Chemical Design, Rhone-Poulenc, and Abbott. ... 

Pharmacophore plots are three-dimensional scatter diagrams that represent the three-point pharmacophores exhibited by a particular structure. The axes represent the distances between the three centers, and each point represents a distinct pharmacophore, labeled by a symbol that indicates the three centers, and color-coded according to whether it contains 1, 2, or 3 identical centers. A typical pharmacophore plot generated using Chemical Design s Chem-X suite is shown in Figure 6. 

As is the case for structural keys, pharmacophore keys can be readily extended to account for multiple conformations. Additionally, because pharmacophores are two- and three-dimensional objects, they are able to capture information on molecular shape and chirality. Three-point pharmacophore keys also lend themselves well to visualization via three-dimensional scatter plots (see section Visualization without Dimensionality Reduction below). Sheridan s original work has been extended by a number of groups, most notably those at Chemical Design [27], Rhone-Poulenc [30], and Abbott [10]. Davies and Briant [31] have employed pharmacophore keys for similarity/diversity selection using an iterative procedure that takes into account the flexibility of the compounds and the amount of overlap between their respective keys (see section Boolean Logic). 

Doicu, A., Wriedt, T., Formulation of the Extended Boundary Condition Method for Three-dimensional Scattering Using the Method of Discrete Sources,/. Mod Opt, 1998, 45, 199-213. 

Classical Ion Trajectory Simulations. Classical ion trajectory simulations were carried out by means of the three-dimensional scattering and recoiling imaging code (SARIC) developed in this laboratory. SARIC is based on the binary collision approximation, uses the ZBL universal potential to describe the interactions between atoms, and includes both out-of-plane and multiple scattering. Details of the simulation have been published elsewhere 11). 

Several other contributions have dealth with the quantum theory of triatomics scattered by thermal neutral atoms.The subject of ion scattering by solid surfaces has been recently reviewed for scattering regimes somewhat complementary to the present one. Classical trajectory studies have been carried out for coplanar scattering of Li" by the Ni(lOO) metal surfaceand more recently for three-dimensional scattering including the influence of surface... 

---