International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 73-74   | 1 | 2 |

## Section 1.3.4.2.2.6. Structure-factor calculation

G. Bricognea

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#### 1.3.4.2.2.6. Structure-factor calculation

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Structure factors may be calculated from a list of symmetry-unique atoms by Fourier transformation of the orbit decomposition formula for the motif given in Section 1.3.4.2.2.4:i.e. finally:

In the case of Gaussian atoms, the atomic transforms areor equivalently

Two common forms of equivalent temperature factors (incorporating both atomic form and thermal motion) are

 (i) isotropic B:so that , or ; (ii) anisotropic β's:so that , or .

In the first case, does not depend on , and therefore:In the second case, however, no such simplification can occur:These formulae, or special cases of them, were derived by Rollett & Davies (1955), Waser (1955b), and Trueblood (1956).

The computation of structure factors by applying the discrete Fourier transform to a set of electron-density values calculated on a grid will be examined in Section 1.3.4.4.5.

### References

Rollett, J. S. & Davies, D. R. (1955). The calculation of structure factors for centrosymmetric monoclinic systems with anisotropic atomic vibration. Acta Cryst. 8, 125–128.
Trueblood, K. N. (1956). Symmetry transformations of general anisotropic temperature factors. Acta Cryst. 9, 359–361.
Waser, J. (1955b). The anisotropic temperature factor in triclinic coordinates. Acta Cryst. 8, 731.