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Results for DC.creator="M." AND DC.creator="A." AND DC.creator="O'Keefe" in section 6.1.1 of volume C page 1 of 2 pages. |
X-ray scattering
International Tables for Crystallography (2006). Vol. C, Section 6.1.1, pp. 554-590 [ doi:10.1107/97809553602060000600 ]
... scattering 6.1.1.1. Coherent (Rayleigh) scattering | | An electromagnetic wave incident on a tightly bound electron is scattered coherently. For an incident wave ... beams (Fig. 6.1.1.1 ), the amplitude of the scattered wave at a distance r is where is the classical radius of the ... and k are the incident and scattered wavevectors, respectively. For a wave with the electric vector parallel to the plane ...
Structure factor
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.7, p. 590 [ doi:10.1107/97809553602060000600 ]
... coherent scattering from the contents of one unit cell in a crystalline material is the structure factor where the integration extends ...
The quasi-Gaussian approximation for curvilinear motion
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.6, p. 590 [ doi:10.1107/97809553602060000600 ]
... this limiting case, the p.d.f. assumes the same form as a one-dimensional rectilinear Gaussian density function except that the variable is the angle . A similar relation must exist between the p.d.f. on the sphere ... Gaussian function. This `quasi-Gaussian' approximation is the basis for a number of structure-factor equations for atoms with relatively ...
Model-based curvilinear density functions
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.5, pp. 589-590 [ doi:10.1107/97809553602060000600 ]
... Johnson & Levy (1974). The p.d.f. for an atom in a group of atoms undergoing large-amplitude rotational oscillation (libration) can sometimes be approximated satisfactory by a standard p.d.f. on the circle or on the sphere. The ... of their simpler forms. The p.d.f. for Brownian diffusion on a circle, also called the `wrapped normal' p.d.f. (Feller, 1966; ...
Curvilinear density functions
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.4, pp. 588-589 [ doi:10.1107/97809553602060000600 ]
... functions For groups of atoms moving on the surface of a circle or sphere, perturbation expansions in Cartesian coordinates may converge ... Hüller (1973). For atoms constrained to rotate about a single axis, where are cylindrical coordinates for the displacement u. ... and using yields For atoms moving on the surface and a sphere, the density function may be written where are ...
Cumulant expansion
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.3, p. 588 [ doi:10.1107/97809553602060000600 ]
Cumulant expansion 6.1.1.6.3. Cumulant expansion In a cumulant expansion (Johnson & Levy, 1974), the entire series is expressed ... for the generalized temperature factor is where the coefficient tensor , a symmetric tensor of order p, is the pth-order cumulant. ... and vice versa. The pth moment (if it exists) of a general p.d.f., [rho](x), is a symmetric tensor defined ...
Fourier-invariant expansions
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.2, pp. 586-588 [ doi:10.1107/97809553602060000600 ]
... expansions 6.1.1.6.2. Fourier-invariant expansions When truncated, an expression for a multipole expansion, p.d.f. or temperature factor must retain those terms ... desirable for the expansion to converge rapidly, and to have a form related to physical theory. In principle, the one-particle ... and anisotropy are small, the p.d.f. may be expressed as a rapidly converging expansion in spherical polar coordinates : for non- ...
Gram-Charlier series
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.1, p. 586 [ doi:10.1107/97809553602060000600 ]
... The operator is the pth partial (covariant) derivative , and is a contravariant component of the coefficient tensor. The quasi-moment coefficient ... of the Hermite polynomial expansion about the Gaussian p.d.f. is a power-series expansion about the Gaussian temperature factor with even ... the symmetry of the relationship between the Fourier transform of a real function and its inverse, the functional form of ...
The generalized temperature factor
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6, pp. 585-590 [ doi:10.1107/97809553602060000600 ]
... In the cases where the potential function V(u) is a close approximation to the Gaussian (harmonic) potential, series expansions based on a perturbation treatment of the anharmonic terms provide a satisfactory representation of the temperature factors. That is, if ...
The temperature factor
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.5, pp. 584-585 [ doi:10.1107/97809553602060000600 ]
The temperature factor 6.1.1.5. The temperature factor The atoms in a solid vibrate about their equilibrium positions, with an amplitude that increases with temperature. As a result of this vibration, the amplitude for coherent scattering is ... to (6.1.1.32) for C unity only if v coincides with a principal axis of the vibration ellipsoid. The probability that ...
International Tables for Crystallography (2006). Vol. C, Section 6.1.1, pp. 554-590 [ doi:10.1107/97809553602060000600 ]
... scattering 6.1.1.1. Coherent (Rayleigh) scattering | | An electromagnetic wave incident on a tightly bound electron is scattered coherently. For an incident wave ... beams (Fig. 6.1.1.1 ), the amplitude of the scattered wave at a distance r is where is the classical radius of the ... and k are the incident and scattered wavevectors, respectively. For a wave with the electric vector parallel to the plane ...
Structure factor
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.7, p. 590 [ doi:10.1107/97809553602060000600 ]
... coherent scattering from the contents of one unit cell in a crystalline material is the structure factor where the integration extends ...
The quasi-Gaussian approximation for curvilinear motion
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.6, p. 590 [ doi:10.1107/97809553602060000600 ]
... this limiting case, the p.d.f. assumes the same form as a one-dimensional rectilinear Gaussian density function except that the variable is the angle . A similar relation must exist between the p.d.f. on the sphere ... Gaussian function. This `quasi-Gaussian' approximation is the basis for a number of structure-factor equations for atoms with relatively ...
Model-based curvilinear density functions
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.5, pp. 589-590 [ doi:10.1107/97809553602060000600 ]
... Johnson & Levy (1974). The p.d.f. for an atom in a group of atoms undergoing large-amplitude rotational oscillation (libration) can sometimes be approximated satisfactory by a standard p.d.f. on the circle or on the sphere. The ... of their simpler forms. The p.d.f. for Brownian diffusion on a circle, also called the `wrapped normal' p.d.f. (Feller, 1966; ...
Curvilinear density functions
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.4, pp. 588-589 [ doi:10.1107/97809553602060000600 ]
... functions For groups of atoms moving on the surface of a circle or sphere, perturbation expansions in Cartesian coordinates may converge ... Hüller (1973). For atoms constrained to rotate about a single axis, where are cylindrical coordinates for the displacement u. ... and using yields For atoms moving on the surface and a sphere, the density function may be written where are ...
Cumulant expansion
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.3, p. 588 [ doi:10.1107/97809553602060000600 ]
Cumulant expansion 6.1.1.6.3. Cumulant expansion In a cumulant expansion (Johnson & Levy, 1974), the entire series is expressed ... for the generalized temperature factor is where the coefficient tensor , a symmetric tensor of order p, is the pth-order cumulant. ... and vice versa. The pth moment (if it exists) of a general p.d.f., [rho](x), is a symmetric tensor defined ...
Fourier-invariant expansions
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.2, pp. 586-588 [ doi:10.1107/97809553602060000600 ]
... expansions 6.1.1.6.2. Fourier-invariant expansions When truncated, an expression for a multipole expansion, p.d.f. or temperature factor must retain those terms ... desirable for the expansion to converge rapidly, and to have a form related to physical theory. In principle, the one-particle ... and anisotropy are small, the p.d.f. may be expressed as a rapidly converging expansion in spherical polar coordinates : for non- ...
Gram-Charlier series
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6.1, p. 586 [ doi:10.1107/97809553602060000600 ]
... The operator is the pth partial (covariant) derivative , and is a contravariant component of the coefficient tensor. The quasi-moment coefficient ... of the Hermite polynomial expansion about the Gaussian p.d.f. is a power-series expansion about the Gaussian temperature factor with even ... the symmetry of the relationship between the Fourier transform of a real function and its inverse, the functional form of ...
The generalized temperature factor
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.6, pp. 585-590 [ doi:10.1107/97809553602060000600 ]
... In the cases where the potential function V(u) is a close approximation to the Gaussian (harmonic) potential, series expansions based on a perturbation treatment of the anharmonic terms provide a satisfactory representation of the temperature factors. That is, if ...
The temperature factor
International Tables for Crystallography (2006). Vol. C, Section 6.1.1.5, pp. 584-585 [ doi:10.1107/97809553602060000600 ]
The temperature factor 6.1.1.5. The temperature factor The atoms in a solid vibrate about their equilibrium positions, with an amplitude that increases with temperature. As a result of this vibration, the amplitude for coherent scattering is ... to (6.1.1.32) for C unity only if v coincides with a principal axis of the vibration ellipsoid. The probability that ...
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