Lattice functions, convolutions

The relationship between the crystal lattice, the unit cell contents, and the crystal structure is shown in two dimensions in Figure 2.16. This raises the concept of convolution, which is the equivalent of folding one function into another. It is a method for converting a unit of a pattern into many identical copies arranged on a lattice all that need be defined is the unit of a pattern and the lattice the convolution gives the rest. 

The best evidence for a superlattice is the observation of satellites around the Bragg reflection in x-ray diffraction (XRD)." The satellites are caused by the superperiodicity in the system, because the x-ray pattern is the Fourier transform of the product of the lattice and modulation functions convoluted with the basis. For composition waveforms that vary sinusoidally, only first order satellites are expected, because only one Fourier term is needed to describe a sine wave. As the waveform... 

Convolution Illustration. By convolution with the 5-function 5(r — r/) (cf. p. 25) we displace (translate) the particle by r. If, for the purpose of particle translation by convolution, we employ an abstract one-dimensional lattice, i.e.,... 

For instance, inaccurate positions of spherical hard-domains in their lattice of colloidal dimensions 2SIn real space there is a convolution of the ideal atom s position (a delta-function) with the real probability distribution to find it. 

After each peak has been described by the parameters of a model function, the convolution in Eq. (8.13) can be carried out analytically. As a result, equations are obtained that describe the effects of crystal size, lattice distortion, and instrumental broadening38 on the breadth of the observed peak. Impossible is in this case the separation of different kinds of lattice distortions. 

History. Starting from the ID point statistics of Zernike and Prins [116] J. J. Hermans [128] designs various ID statistics of black and white rods. He applies these models to the SAXS curves of cellulose. Polydispersity of rod lengths is introduced by distribution functions, / , (,r)108. Hermans describes the loss of correlation along the series of rods by a convolution polynomial . One of Hermans lattice statistics is namedparacrystalby Hosemann [5,117]. Hosemann shows that the field of distorted structure is concisely treated by the methods of complex analysis. A controversial subject is Hosemann s extension of ID statistics to 3D [63,131,227,228],... 

According to Eq. (1.16), the elastic coherent X-ray scattering amplitude is the Fourier transform of the electron density in the crystal. The crystal is a three-dimensional periodic function described by the convolution of the unit cell density and the periodic translation lattice. For an infinitely extended lattice,... 

Twenty-two atoms were placed randomly on a discrete lattice at positions Xi, 0 < Xi < 100. The concentration curves are continuous and have areas that are approximately equal to the number of atoms in the random sample. Each atom contributes a unit area to the coarsegrained c(x). Broader convolution functions (higher values of B) produce greater degrees of coarse graining. 

The specimen intensity transform X is a type of convolution product of the particle intensity transform Ip and the particle orientation density function ( 1,2). The procedure that we have used to simulate Ip involves firstly the calculation of the intensity transform for an infinite particle, with appropriate allowances for random fluctuations in atomic positions and for matrix scattering. A mapping of Xp is then carried out which includes the effects of finite particle dimensions and of intraparticle lattice disorder, if this is present. A mapping of Is is then obtained from Tp by incorporating the effects of imperfect particle orientation. 

In choosing beam optics to measure xrd-rsm, one must consider resolution function in the reciprocal space. The resolution function is defined by the incident beam divergence and the acceptance window of the diffracted beam side optic. Figure 6.3 schematically shows the definition of the resolution function in the reciprocal space. The X-ray detector is located at the tip of the scattering vector H in the reciprocal space. The incident beam divergence 5u> and the acceptance window of the diffracted beam optic 520 define the resolution function (gray area in Figure 6.3) in the reciprocal space. The form of the obtained diffracted intensity distribution of the crystal by xrd-rsm is defined by the convolution of the resolution function and the reciprocal lattice point of the crystal. Therefore, a resolution function smaller than... 

Figure 1. 205T1-NMR spectrum (solid line) at 5 K. The intensity is plotted in a linear scale. The thin solid line depict the histogram at particular local fields of the Readfield pattern. The dotted line represents the simulation spectrum convoluted with Lorentzian broadening function. The filled circles show the frequency dependence of 205 f,1 1 at the T1 site. The inset shows the image of the field distribution in the vortex square lattice center of vortex core (A), saddle point (B) and center of vortex lattice (C).

Consider now a perfect crystal truncated by a sharp surface (or semi-infinite crystal). It can be obtained by the product of a step function describing the electron density variation perpendicular to the surface, and an infinite lattice. The diffraction pattern is then the convolution of the 3D reciprocal lattice with the Fourier transform of the step function. An infinity of Fourier components is necessary to build this latter, so that there remains non zero intensity in between Bragg peaks as a function of / the reciprocal space is made of rods of intensity, called crystal truncation rods (CTR), extending perpendicular to the surface, and connecting bulk Bragg peaks [24, 25]. The intensity variation as a function of (or Qj or /) is found by stopping the summation at n3 = 0 in Eq. (1) and (2), yielding ... 

In the first three chapters of this book, we considered the fundamentals of crystallographic symmetry, the phenomenon of diffraction from a crystal lattice, and the basics of a powder diffraction experiment. Familiarity with these broad subjects is essential in understanding how waves are scattered by crystalline matter, how structural information is encoded into a three-dimensional distribution of discrete intensity maxima, and how it is convoluted with numerous instrumental and specimen-dependent functions when projected along one direction and measured as the scattered intensity V versus the Bragg angle 20. We already learned that this knowledge can be applied to the structural characterization of materials as it gives us the ability to decode a one-dimensional snapshot of a reciprocal lattice and therefore, to reconstruct a three-dimensional distribution of atoms in an infinite crystal lattice by means of a forward Fourier transformation. 

Ultimately we want to know how a crystal diffracts X rays and produces the diffraction pattern that it does, and conversely, how the diffraction pattern can be used to reconstruct the crystal. It will be found useful in this regard to consider the crystal as the combination, or product of two distinct components, or functions. The first of these is the contents of a unit cell, characterized mathematically by the coordinates of the atoms in an asymmetric unit along with their space group symmetry equivalent positions. The second is a point lattice that describes the periodic distribution of the unit cell contents, and is characterized by a, b, and c. A crystal may then be concisely defined as the first component, or function, repeated in identically the same way at every nonzero point of the second. This physical process of repetitive superposition is termed a convolution. It can be formulated mathematically as the product of the two components, or functions as... 

The concept of a repeated distribution is important because it can be shown (we will forego a painful formal proof here) that the Fourier transform (or diffraction pattern) of the convolution of two spatial functions is the product of their respective Fourier transforms. This was demonstrated physically using optical diffraction in Figure 1.8 of Chapter 1. In principle, this means that if we can formulate an expression for the Fourier transform of a single unit cell, and if we can do the same for a lattice, then if we multiply them together, we will have a mathematical statement for how a crystal diffracts waves, its Fourier transform. 

Treatment of the experimental data requires that the diffraction peak be fitted to a function that characterizes the peak shape convoluted with the instrumental function a background, which may depend on Q, must also be subtracted. The choices involved in these procedures are not unequivocal. Even with intense synchrotron sources, higher-order reflections are not usually observed, so the assignment of a structure is based on the position of only one peak, i.e., on only one lattice spacing. The interpretation of the structures is made plausible by knowledge of the molecular structure of the amphiphile and of the crystal structures of similar molecules. 

Figure 1.13 A one-dimensional example illustrating the mathematical operations represented by Equations (1.95) and (1.96). The convolution of the unit cell content pu(r) with the lattice z(r) produces an infinite repetition of the unit cell pattern, while the product of the structure factor F(s) with the reciprocal lattice Z(s) produces the discrete amplitude function whose magnitude at the reciprocal lattice point is modulated by F(s). Note that (a) and (A), (b) and (B), and (c) and (C) are, respectively, the Fourier transforms of each other.

The structure of a molecular or a liquid crystal is a result of convolution of two density functions, the density of a group of atoms in a molecule and periodic density function of a lattice. Let us look at the convolution procedure. By definition, the... 

Going back to solid or liquid crystals we can say that the convolution procedure distributes molecular density over the sites of the crystal lattice. On the left side of Fig. 5.14, the two functions, the electron density of a molecule pmoi(r) and discrete points of the lattice density piattice(r) =28(ri Fj) are shown separately (before convolution). On the right side we see the result of their convolution. Note that the convolution operation/i(x) /2(x) is dramatically different from the multiplication operation/i(x)/2(x). An example is illustrated by Fig. 5.15, in which function/2(x) is the same piattice(r) function as in the previous picture and/i(x) is the so called box-function. The latter is equal to 1 within its contour and 0 outside. The multiplication selects only few 8-functions from the whole lattice. On the contrary, the convolution translates pmoi into new functional space, namely the space of piattice-... 

In this appendix we describe a stencil algorithm which avoids many of the drawbacks of quadrature rules used in classical lattice models, while the extra computational cost is modest. The derivation consists of finding a unique and optimal set of stencil coefficients for a convolution with a Gaussian kernel, adapted to the special case of off-lattice density functional calculations. Stencil coefficients are the multipliers of the function values at corresponding grid points. 

From the summation in Section 3.4.2.2.3, we found that dif action rods are perpendicular to the surface, but what happens if the surface has a misorientation, either from cutting or from the growth A convenient way to arrive at a general answer to this question is to use the so-caUed crystal shape function s(r) instead of a specific summation [17]. In the general formula for calculating the scattered amplitude, Eq. (3.4.2.1), we can write the electron density as the convolution of the electron density in a unit cell /Ou(r) with all the lattice points... 

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