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Fluctuation X-ray Scattering
Fluctuation X-ray scattering (FXS) is an X-ray scattering technique similar to small-angle X-ray scattering (SAXS), but is performed using X-ray exposures below sample rotational diffusion times. This technique, ideally performed with an ultra-bright X-ray light source, such as a free electron laser, results in data containing significantly more information as compared to traditional scattering methods. FXS can be used for the determination of (large) macromolecular structures, but has also found applications in the characterization of metallic nanostructures, magnetic domains and colloids. The most general setup of FXS is a situation in which fast diffraction snapshots of models are taken which over a long time period undergo a full 3D rotation. A particularly interesting subclass of FXS is the 2D case where the sample can be viewed as a 2-dimensional system with particles exhibiting random in-plane rotations. In this case, an analytical solution exists relation the FXS data to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-ray Scattering Techniques
X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystal structure, chemical composition, and physical properties of materials and thin films. These techniques are based on observing the scattered intensity of an X-ray beam hitting a sample as a function of incident and scattered angle, polarization, and wavelength or energy. Note that X-ray diffraction is now often considered a sub-set of X-ray scattering, where the scattering is elastic and the scattering object is crystalline, so that the resulting pattern contains sharp spots analyzed by X-ray crystallography (as in the Figure). However, both scattering and diffraction are related general phenomena and the distinction has not always existed. Thus Guinier's classic text from 1963 is titled "X-ray diffraction in Crystals, Imperfect Crystals and Amorphous Bodies" so 'diffraction' was clearly not restricted to crystals at that time. Scattering techniques Ela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small-angle X-ray Scattering
Small-angle X-ray scattering (SAXS) is a small-angle scattering technique by which nanoscale density differences in a sample can be quantified. This means that it can determine nanoparticle size distributions, resolve the size and shape of (monodisperse) macromolecules, determine pore sizes, characteristic distances of partially ordered materials, and much more. This is achieved by analyzing the elastic scattering behaviour of X-rays when travelling through the material, recording their scattering at small angles (typically 0.1 – 10°, hence the "Small-angle" in its name). It belongs to the family of small-angle scattering (SAS) techniques along with small-angle neutron scattering, and is typically done using hard X-rays with a wavelength of 0.07 – 0.2 nm.. Depending on the angular range in which a clear scattering signal can be recorded, SAXS is capable of delivering structural information of dimensions between 1 and 100 nm, and of repeat distances in partially ordered s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotational Diffusion
Rotational diffusion is the rotational movement which acts upon any object such as particles, molecules, atoms when present in a fluid, by random changes in their orientations. Whilst the directions and intensities of these changes are statistically random, they do not arise randomly and are instead the result of interactions between particles. One example occurs in colloids, where relatively large insoluble particles are suspended in a greater amount of fluid. The changes in orientation occur from collisions between the particle and the many molecules forming the fluid surrounding the particle, which each transfer kinetic energy to the particle, and as such can be considered random due to the varied speeds and amounts of fluid molecules incident on each individual particle at any given time. The analogue to translational diffusion which determines the particle's position in space, rotational diffusion randomises the orientation of any particle it acts on. Anything in a solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free-electron Laser
A free-electron laser (FEL) is a (fourth generation) light source producing extremely brilliant and short pulses of radiation. An FEL functions and behaves in many ways like a laser, but instead of using stimulated emission from atomic or molecular excitations, it employs relativistic electrons as a gain medium. Radiation is generated by a ''bunch'' of electrons passing through a magnetic structure (called undulator or wiggler). In an FEL, this radiation is further amplified as the radiation re-interacts with the electron bunch such that the electrons start to emit coherently, thus allowing an exponential increase in overall radiation intensity. As electron kinetic energy and undulator parameters can be adapted as desired, free-electron lasers are tunable and can be built for a wider frequency range than any other type of laser, currently ranging in wavelength from microwaves, through terahertz radiation and infrared, to the visible spectrum, ultraviolet, and X-ray. The first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structure Determination From FXS Data
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as biological organisms, minerals and chemicals. Abstract structures include data structures in computer science and musical form. Types of structure include a hierarchy (a cascade of one-to-many relationships), a network featuring many-to-many links, or a lattice featuring connections between components that are neighbors in space. Load-bearing Buildings, aircraft, skeletons, anthills, beaver dams, bridges and salt domes are all examples of load-bearing structures. The results of construction are divided into buildings and non-building structures, and make up the infrastructure of a human society. Built structures are broadly divided by their varying design approaches and standards, into categories including building structures, archi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Legendre Transform
The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum. Conversely, the inverse fLT (ifLT) reconstructs the original function from the components of the Legendre spectrum and the Legendre polynomials, which are orthogonal on the interval ˆ’1,1 Specifically, assume a function ''x''(''t'') to be defined on an interval ˆ’1,1and discretized into ''N'' equidistant points on this interval. The fLT then yields the decomposition of ''x''(''t'') into its spectral Legendre components, :L_x (k) = \frac\sum_^x(t)P_k(t), where the factor (2''k'' + 1)/''N'' serves as normalization factor and ''L''''x''(''k'') gives the contribution of the ''k''-th Legendre polynomial to ''x''(''t'') such that (ifLT) :x(t) = \sum_k L_x(k) P_k(t). The fLT should not be confused with the Legendre transform or Legendre transformation used in thermodynamics and quantum physics. Legendre filter The fLT of a noisy experimental out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum Transfer
In particle physics, wave mechanics and optics, momentum transfer is the amount of momentum that one particle gives to another particle. It is also called the scattering vector as it describes the transfer of wavevector in wave mechanics. In the simplest example of scattering of two colliding particles with initial momenta \vec_,\vec_, resulting in final momenta \vec_,\vec_, the momentum transfer is given by : \vec q = \vec_ - \vec_ = \vec_ - \vec_ where the last identity expresses momentum conservation. Momentum transfer is an important quantity because \Delta x = \hbar / , q, is a better measure for the typical distance resolution of the reaction than the momenta themselves. Wave mechanics and optics A wave has a momentum p = \hbar k and is a vectorial quantity. The difference of the momentum of the scattered wave to the incident wave is called ''momentum transfer''. The wave number k is the absolute of the wave vector k = p / \hbar and is related to the wavelength k = 2\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Schematic Representation Of Mathematical Relations In Fluctuation X-ray Scattering
A, or a, is the first Letter (alphabet), letter and the first vowel of the Latin alphabet, Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Letter names, ''a'' (pronounced ), plural English alphabet#Letter names, ''aes''. It is similar in shape to the Greek alphabet#History, Ancient Greek letter alpha, from which it derives. The Letter case, uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version can be written in two forms: the double-storey a and single-storey É‘. The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English grammar, "English articles, a", and its variant "English articles#Indefinite article, an", are Article (grammar)#Indefinite article, indefinite arti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardization co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
