Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 392   | 1 | 2 |

Section Density modification

D. L. Dorsete* Density modification

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Another method of phase determination, which is best suited to refining or extending a partial phase set, is the Hoppe–Gassmann density modification procedure (Hoppe & Gassmann, 1968[link]; Gassmann & Zechmeister, 1972[link]; Gassmann, 1976[link]). The procedure is very simple but also very computer-intensive. Starting with a small set of (phased) [F_{{\bf h}}], an initial potential map [\varphi ({\bf r})] is calculated by Fourier transformation. This map is then modified by some real-space function, which restricts peak sizes to a maximum value and removes all negative density regions. The modified map [\varphi'({\bf r})] is then Fourier-transformed to produce a set of phased structure factors. Phase values are accepted via another modification function in reciprocal space, e.g. [E_{\rm calc}/E_{\rm obs} \geq p], where p is a threshold quantity. The new set is then transformed to obtain a new [\varphi ({\bf r})] and the phase refinement continues iteratively until the phase solution converges (judged by lower crystallographic R values).

The application of density modification procedures to electron-crystallographic problems was assessed by Ishizuka et al. (1982)[link], who used simulated data from copper perchloro­phthalo­cyanine within the resolution of the electron-microscope image. The method was useful for finding phase values in reciprocal-space regions where the transfer function [|C(s)| \leq 0.2]. As a technique for phase extension, density modification was acceptable for test cases where the resolution was extended from 1.67 to 1.0 Å, or 2.01 to 1.21 Å, but it was not very satisfactory for a resolution enhancement from 2.5 to 1.67 Å. There appear to have been no tests of this method yet with experimental data. However, the philosophy of this technique will be met again below in the description of the maximum entropy and likelihood procedure.


Gassmann, J. (1976). Improvement and extension of approximate phase sets in structure determination. In Crystallographic Computing Techniques, edited by F. R. Ahmed, pp. 144–154. Copenhagen: Munksgaard.
Gassmann, J. & Zechmeister, K. (1972). Limits of phase expansion in direct methods. Acta Cryst. A28, 270–280.
Hoppe, W. & Gassmann, J. (1968). Phase correction, a new method to solve partially known structures. Acta Cryst. B24, 97–107.
Ishizuka, K., Miyazaki, M. & Uyeda, N. (1982). Improvement of electron microscope images by the direct phasing method. Acta Cryst. A38, 408–413.

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