Diffraction space

Fadley C S et al 1997 Photoelectron diffraction space, time and spin dependence of surface structures Surf. Rev. Left 4 421-40... 

Fig. 8. Diffraction space according to the "disordered stacking model" (a) achiral (zigzag) tube (b) chiral tube. The parallel circles represent the inner rims of diffuse coronae, generated by streaked reflexions. The oo.l nodes generate sharp circles. In (a) two symmetry related 10.0 type nodes generate one circle. In the chiral case (b) each node generates a separate corona [9].

In direct space successive layers are sheared homogeneously along cylindrical surfaces, one relative to the adjacent one, as a consequence of the circumference increase for successive layers. In diffraction space the locus of the corresponding reciprocal lattice node is generated by a point on a straight line which is rolling without sliding on a circle in a plane perpendicular to the tube axis. Such a locus... 

Assuming kinematical diffraction theory to be applicable to the weakly scattering CNTs, the diffraction space of SWCNT can be obtained in closed analytical form by the direct stepwise summation of the complex amplitudes of the scattered waves extended to all seattering centres, taking the phase differenees due to position into aeeount. 

The diffraction space of MWCNTs can be computed by summing the complex amplitudes due to each of the constituent coaxial tubes. Taking into account possible differences in Hamada indices (Lj, Mj) as well as the relative stacking (described by zQ.j, can formally write... 

The diffraction space of ropes of parallel SWCNT can similarly be computed by summing the complex amplitudes of the individual SWCNTs taking into account the relative phase shifts resulting from the lattice arrangement at... 

A hexagonal lattice of identical SWCNT s leads in diffraction space to a 2D lattice of nodes at positions h +h2 2 A Bj = 27i5,y. Spots corresponding to such nodes are visible in Fig. 3. 

Fig. 11. Simulated diffraction space of a chiral (40, 5) SWCNT. (a) Normal incidence diffraction pattern with 2mm symmetry (b),(c),(d) and (e) four sections of diffraction space at the levels indicated by arrows. Note the absence of azimuthal dependence of the intensity. The radii of the dark circles are given by the zeros of the sums of Bessel functions [17].

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

Fig. 12. S imulated diffraction space for a (10, 10) armchair tube, (a) Normal incidence pattern, note the absence of oo.l spots, (b) Equatorial section. The pattern has 20-fold symmetry, (c) The section The pattern contains 20 radial...

Fig. 13. Simulated diffraction space of a 10-layer monochiral MWCNT with Hamada indices (40+8/ , 5+k) with / =0,...,9. In (a), (a ) and (02) the initial stacking at ( q was ABAB. whereas in (b), (b[) and (b2) the initial stacking was random, (a) The normal incidence pattern has a centre of symmetry only. (3 )(a2) The cusps are of two different types. The arc length separating the cusps is c (b) The normal incidence pattern now exhibits 2mm symmetry. (b )(b2) The cusps are distributed at random along the generating circles of the evolutes. These sections represent the diffuse coronae referred to in the "disordered stacking model" [17].

Figure 11.13 Changes of PTT ( ) and PBT ( ) fibers c-axis lattice strains measured from X-ray diffraction spacings as a function of applied external strains the dotted line represents affine deformation between lattice and applied strains...

Compound Diffraction Spacing in 20 Degrees Intensity in Percent... 

Figure 9. Molecular arrangements in aqueous precipitates of bovine serum albumin and lecitnin-cardiolipin (14). Center lipid bilayers and left ana right two alternative structures of the precipitates based on the x-ray diffraction spacings.

The best way to ascertain the structure of any compound is to determine its structure by X-ray or neutron diffraction. Indeed, as indicated in Table I, many mixed-metal clusters have had their structures examined by X-ray crystallography, and at least one by neutron diffraction. Space does not permit presentation of the structures of all of these clusters, and the reader is referred to the original articles. 

FIGURE 3.11. Relationships between a crystal (in crystal space) and its reciprocal lattice (in diffraction space). 

Fourier transforms (between crystal and diffraction space)... 

X-Ray diffraction powder patterns were developed by using a chromium source and a Phillips Debye-Sherrer camera. Due to their small size, the samples from the back of the halberd were irradiated for 12.5 h, and the green pseudomorph sample for 12.3 h. The black pseudomorph sample, larger, was irradiated for only 5 h. The exposed film was measured in accordance with standard procedures (8) and the resulting diffraction spacing values were compared to standard values of known materials (9). 

The interference function in Eq. 2.14 describes a discontinuous distribution of the scattered intensity in the diffraction space. Assuming an infinite number of points in a one-dimensional periodic structure N -> oo), the distribution of the scattered intensity is a periodic delta-function (see above) and therefore, diffraction peaks occur only in specific points, which establish a one-dimensional lattice in the diffraction space. Hence, diffracted intensity is only significant at certain points, which are determined (also see Eq. 2.13, Figure 2.23, wad Figure 2.25) from... 

In three dimensions, a total of three integers h, k and If are required to define the positions of intensity maxima in the diffraction space ... 

Diffraction space, in which diffraction peaks are arranged into a lattice, is identical to a reciprocal space. 

In the case of well-ordered crystals, It Is possible to deduce their atomic structures by appropriate manipulation of diffraction Intensities. In the case of x-ray scattering by liquids, direct use of measured intensities yields, at best, very limited structural Information (radial distribution functions). For ordered liquids, however, it is possible to posit structural models and to calculate what their scattering Intensities would be so that it is more productive to conduct the comparisons in diffraction space. To this end, it is possible to devise a point model to represent the spatial repetition of the constituent units in the ordered array and to compare its scattering maxima to the observed ones (6,9). More sophisticated analyses (10-12) make use of the complete electron densities (or projections onto the chain axis z), usually by calculating their Patterson functions P(z) since the scattering intensity function is its Fourier transform. 

FIGURE 1.7 In (a) the object, again exposed to a parallel beam of light, is not a continuous object or an arbitrary set of points in space, but is a two-dimensional periodic array of points. That is, the relative x, y positions of the points are not arbitrary they bear the same fixed, repetitive relationship to all others. One need only define a starting point and two translation vectors along the horizontal and vertical directions to generate the entire array. We call such an array a lattice. The periodicity of the points in the lattice is its crucial property, and as a consequence of the periodicity, its transform, or diffraction pattern in (b) is also a periodic array of discrete points (i.e., a lattice). Notice, however, that the spacings between the spots, or intensities, in the diffraction pattern are different than in the object. We will see that there is a reciprocal relationship between distances in object space (which we also call real space), and in diffraction space (which we also call Fourier space, or sometimes, reciprocal space). 

The complete diffraction pattern from a protein crystal is not limited to a single planar array of intensities like those seen in Figures 1.13 and 1.14. These images represent, in each case only a small part of the complete diffraction pattern. Each photo corresponds to only a limited set of orientations of the crystal with respect to the X-ray beam. In order to record the entire three-dimensional X-ray diffraction pattern, a crystal must be aligned with respect to the X-ray beam in all orientations, and the resultant patterns recorded for each. From many two-dimensional arrays of reflections, corresponding to cross sections through diffraction space, the entire three-dimensional diffraction pattern composed of ten to hundreds of thousands of reflections is compiled. 

FIGURE 1.14 Seen here is the hk0 zone diffraction pattern from a crystal of M4 dogfish lactate dehydrogenase obtained using a precession camera. It is based on a tetragonal crystal system and, therefore, exhibits a fourfold axis of symmetry. The hole at center represents the point where the primary X-ray beam would strike the film (but is blocked by a circular beamstop). Note the very predictable positions of the diffraction intensities. All the intensities, or reflections, fall at regular intervals on an orthogonal net, or lattice. This lattice in diffraction space is called the reciprocal lattice. 

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