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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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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
Maslen, E. N., Fox, A. G. and O'Keefe, M. A.  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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