Fourier shape coefficients

A number of methods have been proposed for particle shape analysis, including shape coefficients, shape factors, verbal descriptions, curvature signatures, moment invariants, solid shape descriptors, and mathematical functions (Fourier series... 

In the most simplistic means of defining particle shape, measurements may be classified as either macroscopic or microscopic methods. Macroscopic methods typically determine particle shape using shape coefficients or shape factors, which are often calculated from characteristic properties of the particle such as volume, surface area, and mean particle diameter. Microscopic methods define particle texture using fractals or Fourier transforms. Additionally electron microscopy and X-ray diffraction analysis have proved useful for shape analysis of fine particles. 

A number of methods have been proposed for particle shape analysis these include verbal description, various shape coefficients and shape factors, curvature signatures, moment invariants, solid shape descriptors, the octal chain code and mathematical functions like Fourier series expansion or fractal dimensions. As in particle size analysis, here one can also detect intense preoccupation with very detailed and accurate description of particle shape, and yet efforts to relate the shape-describing parameters to powder bulk behaviour are relatively scarce.10... 

If, moreover, the general case of Fourier s coefficients with complex shape is considered ... 

Although most modem particle characterization methods are developed, validated and presumably used for spherical particles or equivalent spherical particles, real particles are rarely such ideal. In many instances, particle shape affects powder packing, bulk density, and many other macroscopic properties. Shape characterization of particulate systems only scatters in the literature [60], since there are hardly any universal methodologies available. Several methods exist that use shape coefficients, shape factors, Fourier analysis, or fractal analysis to semi-quantitatively describe shape [Ij. 

Instead of this methodology, we have chosen to use Fourier analysis of the entire peak shape. By this procedure all of the above problems are eliminated. In particular, we focus on the cosine coefficients of the Fourier series representing a peak. The instrumental effects are readily removed, and the remaining coefficient of harmonic number, (n), A, can be written as a product ... 

Many other filter functions can be designed, e.g. an exponential or a trapezoidal function, or a band pass filter. As a rule exponential and trapezoidal filters perform better than cut-off filters, because an abrupt truncation of the Fourier coefficients may introduce artifacts, such as the annoying appearance of periodicities on the signal. The problem of choosing filter shapes is discussed in more detail by Lam and Isenhour [11] with references to a more thorough mathematical treatment of the subject. The expression for a band-pass filter is H v) = 1 for v j < v < else... 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

There are problems in determining crystallite size from line broadening alone, since factors other than crystallite size contribute to the broadening, including local strain in the crystallites and shape anisotropy. Some of these problems can be overcome by the use of Fourier analysis of the peak shape. The cosine coefficients of the Fourier series can be used to determine a surface weighted average size for the crystallites. 

An image of the particle is obtained, the profile of that image is converted to a set of x,y pairs, a process known as digitizing. The x,y set is then converted to polar coordinates, (R,0). The curve in the R,0 space is converted to a Fourier equation, the coefficients of which are extracted, and then mathematically transformed to morphic terms which themselves constitute the shape features of the particle. A sample usually consists of 100 particles, upwards of 150 profile points are extracted from each particle, giving a total of 15,000 x,y points per analysis. Once the morphic terms are obtained, the data analysis can be carried out in order to facilitate the... 

Equation (2) defines the value of the size normalized mean radius of the particle. Equation (3) defines the size normalized sum of the squares of the Fourier coefficients. Equation (4) defines the sum and differences of the multiples. It has been shown that these size and shape descriptors can be used to regenerate the original particle profile. This Indicates that the descriptors together contain all of the size and shape Information contained In the original profile. 

Assumptions 1 The egg is spherical in shape with a radius of r, = 2.5 cm. 2 Heat conduction in the egg is one dimensional because or thermal symmetry about the midpoint. 3 The thermal properties of the egg and the heat transfer coefficient are constant, 4 The Fourier number is t > 0,2 so that the one-term approximate solutions are applicable. 

First, we need to know what is meant by a periodic function. The crystal contains a periodic arrangement - a regular array - of atoms but, as mentioned above, X-rays scatter electrons. Therefore it is more convenient to think about the crystal and thus the unit cell in terms of its electron density not fixy,z) where /describes the scattering factor of the atoms, but p(xy,z), where p(xy,z) is the electron density at point xy,z. As the atoms are periodically arranged, so also is their electron density p(xy,z) is a periodic function. We can therefore approximate it with a Fourier series just as above. If we know the electron density function, we can use a FT to calculate the individual coefficients Fbyy However this is completely useless the shape of the electron density, that is, the arrangement of the atoms in the crystal its structure - is precisely what we want to find out. In order to achieve this we need to do quite the opposite - calculate the electron density from the diffraction pattern. Before we consider how, we will try to build up a physical picture of what the FT of the electron density means. 

In 1949, however, Warren pointed out that there was important information about the state of a cold-worked metal in the shape of its diffraction lines, and that to base conclusions only on line width was to use only part of the experimental evidence. If the observed line profiles, corrected for instrumental broadening, are expressed as Fourier series, then an analysis of the Fourier coefficients discloses both particle size and strain, without the necessity for any prior assumption as to the existence of either [9,3, G.30, G.39]. Warren and Averbach [9.4] made the first measurements of this kind, on brass filings, and many similar studies followed [9.5]. Somewhat later, Paterson [9.6] showed that the Fourier coefficients of the line profile could also disclose the presence of stacking faults caused by cold work. (In FCC metals and alloys, for example, slip on 111 planes can here and there alter the normal stacking sequence ABCABC... of these planes to the faulted... 

The surprising result of the evaluation of Fourier coefficient signatures of homogeneous powders is that there are usually five or more different shapes and the question arises as to the relation between these predominant shapes. Research is still continuing in this area. 

Fig. 10.21. Series of equilibrium shapes for particles (adapted from Thompson et al. (1994)). The parameter L is a measure of the particle s size, while is one of the Fourier coefficients that defines...

If a homodyne method is used, the measured autocorrelation function (g,x) can be interpreted by using the Siegert relation. Equation 5.454. The translational and rotational diffusion coefficients for several specific shapes of the particles are given in Table 5.9. The respective power spectrum functions can be calculated by using the Fourier transform. Equation 5.449b. 

The coefficients of the truncated Fourier series of Eq. (2) or Eq. (3) are the molecular parameters for the hindering potential. The use of a limited number of those potential parameters implicitly means an assumption of the barrier shape. Usually, the first term (V3 or ) is the only term that is determined from the experiment. Using only one term of Eq. (2) means the assumption of a pure sinusoidal shape of the potential with the height of V3. If also higher terms V3n are included, the shape and height change. 

Each Fourier coefficient in a transform with apodisation represents a band of frequencies. The width of that band is controlled via the length of the signal that is transformed and the shape of the apodisation function. We can introduce the notion of frequency localisation as an extension of the previously introduced frequency resolution and in analogy to localisation in time. When the bands are wide, the frequency information returned by the transform is less localised than when the bands are narrow. In other words, when the time localisation is good, the frequency localisation is poor. 

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