Method of finite elements

Glatter, O.

doi:10.1107/97809553602060000581

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 104

Section 2.6.1.7.4. Method of finite elements

O. Glatter^a

2.6.1.7.4. Method of finite elements

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Models of arbitrary shape can be approximated by a large number of very small homogeneous elements of variable electron density. These elements have to be smaller than the smallest structural detail of interest.

Sphere method . In this method, the elements consist of spheres of equal size. The diameter of these spheres must be chosen independently of the distance between nearest neighbours, in such a way that the total volume of the model is represented correctly by the sum of all volume elements (which corresponds to a slight formal overlap between adjacent spheres). The scattering intensity is calculated using the Debye formula (2.6.1.67), with $[\Phi_i(h)=\Phi_k(h)=\Phi(h)]$ .

The computing time is mainly controlled by the number of mutual distances between the elements. The computing time can be lowered drastically by the use of approximate $[d_{ik}]$ values in (2.6.1.67). Negligible errors in I(h) result if $[d_{ik}]$ values are quantized to D_max/10000 (Glatter, 1980c). For the practical application (input operation), it is important that a certain number of elements can be combined to form so-called sub-structures that can be used in different positions with arbitrary weights and orientations to build the model.

The sphere method can also be used for the computation of scattering curves for macromolecules from a known crystal structure. The weights of the atoms are given by the effective number of electrons $[Z_{\rm eff}=Z-\rho _0V_{\rm eff},\eqno (2.6.1.68)]$ where $[V_{\rm eff}]$ is the apparent volume of the atom given by Langridge, Marvin, Seeds, Wilson, Cooper, Wilkins & Hamilton (1960).

Cube method . This method has been developed independently by Fedorov, Ptitsyn & Voronin (1972, 1974a,b) and by Ninio & Luzzati (1972) mainly for the computation of scattered intensities for macromolecules in solution whose crystal structure is known. In the cube method, the macromolecule is mentally placed in a parallelepiped, which is subdivided into small cubes (with edge lengths of 0.5–1.5 Å. Each cube is examined in order to decide whether it belongs to the molecule or to the solvent. Adjacent cubes in the z direction are joined to form parallelepipeds. The total scattering amplitude is the sum over the amplitudes from the parallelepipeds with different positions and lengths. The mathematical background is described by Fedorov, Ptitsyn & Voronin (1974a,b). The modified cube method of Fedorov & Denesyuk (1978) takes into account the possible penetration of the molecule by water molecules.

References

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 104