Reciprocal coordinate

The respective reciprocal coordinate is called dual coordinate by the mathematician. 

Fig. 6. X-ray diffraction intensity (arbitrary units) as a function of reciprocal coordinate for prion peptides. 106-122 meridional scan of the pattern of SHal06-122 dried from 50% AcN H1(S/D) meridional scan of the pattern of SHal09-122 dried from 50% AcN H1(L) lyophilized HI A8A SHall3-120 dried from 50% AcN. SHal06-122 is from Fig. 2A in Inouye et al (2000). Diffraction patterns of A8A, HI (L), and HI (S/D) were previously reported (Nguyen et al., 1995). The strongest reflections (with Miller indices) are 4.56 A (201) in SHal06-122, 4.77 A (200) in HI (S/D), 4.44 A (201) in HI (L), and 4.33 A (201) in A8A. Low-angle reflections arising from the stacking of slabs are indicated by. ...

The indexing proeedure of any diffraction pattern requires the knowledge of positions of some reflections on the pattern. The relations between the position of each peak on a two-dimensional diffraction pattern and the corresponding point in reciprocal space can be established only after a successful indexing. The combined use of the reciprocal coordinates and the determined peak-shape are essential for extracting integrated intensities from diffraction data. 

Guggenheim, 1952), where z is the number of reciprocal coordinations of species 1 and 2 in the gaseous mixture. 

Fig. 13a and b. Intensity contour maps around the 5.9-nm and 5.1-nm actin layer lines (indicated by arrows) a resting state b contracting state. Z is the reciprocal-space axial coordinate from the equator. M5 to M9 are myosin meridional reflections indexed to the fifth to ninth orders of a 42.9-nm repeat, (c) intensity profiles (in arbitrary units) of the 5.9- and 5.1-nm actin reflections. Dashed curves, resting state solid curves, contracting state. Intensity distributions were measured by scanning the intensity data perpendicular to the layer lines at intervals of 0.4 mm. The area of the peak above the background was adopted as an integrated intensity and plotted as a function of the reciprocal coordinate (R) from the meridian... 

During CNS patterning, cells reciprocally coordinate their replication rate, fate, and relative position. These interactions are mediated by direct cell-to-cell contacts, by the local action of secreted polypeptides called morphogens, and by hormonal influences from the macroenvironment. 

Figure 4. Michaelis-Menten rate law expressed in double-reciprocal coordinates. In this plot, which is attributed to Lineweaver-Burk, 1 /V is given by the intercept on the //vaxis. The parameter K can be obtained from the slope of the straight line or the intercept on the negative i/S axis.

Resorting to the definition of the Fourier transform, Eq. (2.9), we notice that for the redundant coordinates the harmonic kernel degenerates and becomes exp (0) = 1. Thus for the redundant coordinates the Fourier transform turns into a simple integration with respect to the respective reciprocal coordinate - a projection . 

Nineteenth-century crystallography may be considered to be the mathematical branch of mineralogy. It is based on two empirical laws, the law of constancy of angle and the law of rational indices. These laws will be presented in the following pages after a discussion of some mathematical principles fundamental to crystallography, non-unitary coordinate systems and reciprocal coordinates. 

The direct base vectors and the reciprocal coordinates h, k, I transform in a covariant manner. The reciprocal base vectors and the direct coordinates u, v, w transform in a contravariant manner. 

The reciprocal coordinate system was defined above (1.2). If a, b, c are the reciprocal base vectors of the vectors a,b,c, the vector r = Ha + Kh + Lc (H, K, L being coprime integers) represents the normal to the plane Hu + Kv-h Lw-= 1. Its norm is r = l/(d//KL) where is the distance of the plane from the origin, and hence the distance between consecutive planes in the same family. The reciprocal lattice is the set of vectors... 

The experimental X-ray diffraction profiles of an uniaxially oriented PTFE sample are reported in Fig. 27 at different temperatures, as a function of the reciprocal coordinate redrawn from [246]. The value of reciprocal coordinate is fixed close to the position of intensity maximum on the 7 and... 

Fig. 27 Experimental X-ray fiber diffraction patterns of PTFE at = 0.20 A in the region of 7-th and 8-th layer line as a function of the reciprocal coordinate f. (Reprinted with permission from [246]. Copyright 1988 by the American Chemical Society)...

The reciprocal coordinate-momentum bra-ket representation transformation provides the formalism for Fourier transformation between the wave-function y/(x) and momentum function amplitude y/(p) ... 

Fig. 10. Dependence of the Volta potential measured in chain (8) on NADH concentration in reciprocal coordinates. Medium ...

Where Fis a matrix of layer form factors F Fis a matrix of (succession probabilities) multiplied by phase shift and f, are continuous reciprocal coordinates in the three directions of space. Nis the total number of layers. 

In conclusion, a 3D scattering scheme has to be applied for a correct analysis of off-specular scattering in terms of Yoneda scattering, Bra -sheet scattering, and GISANS combined into complete reflectometty. This implies that the reflected and scattered intensities have to be presented in the three reciprocal coordinates, a presentation that emerges as an indispensable tool for the data analysis. The theoretical descriptions for features of reflected and scattered intensities are added in text boxes. 

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