H. Schenk University of Amsterdam Nieuwe Achtergracht 166. Amsterdam The Netherlands
I I
I
Solving the Phase Problem in crystal Structure Determination A simple introduction to direct methods
T h e phase problem in X-ray crystal structure determination arises from the fact that whereas only the reflection intensities can be measured, both intensity and relative phases of all reflections must be known in order to image the electron density. A very important part of a structure determination will therefore be the solution of the phase problem and thus phasing methods form an essential constituent of crystallography courses for students specializing in the subject. Direct Methods try to solve the phase problem by calculating the phases directly from the magnitudes of the reflections. During the last decade the importance of the Direct Methods for daily practice has increased rapidly, so that for specialist students the subject is now an important component of the curriculum as counterpart of the Patterson methods. This paper presents a simple way to introduce Direct Methods to undergraduates, based on a primitive physical picture of the &-relation. Since elaborate theories about the %relation fail to give reliable estimates of the true prohability of three-phase structure invariants1 the use of a more simole exolanation is iustified. Moreover. the mathematical cotnplexi;y of the rttccnt Dtrevt Method literature together with the fact that ammuter ororrams for Dirrct Methods are mainly written as black boxes, not make it possible to base a simple introduction on the recent developments. A programmed instruction2 following the lines as described in this paper has been used successfully in our laboratory for teaching Direct Methods to undergraduate chemistry stu. , aenrs.
Figure 2. If thereflections Hand K both havea high intensity, lhen theelectron density will likely lie in theneighborhmddthe intersecting linesofthetwosets of equidistant planes definedby H and K.
db
Teaching the &-relation The &-relation is formulated as $H+~K+$-H-K~O
for large values of the product of the normalized structure It is this relation which is used prifactors IEHEKE-H-KI. marily in all Direct Method program systems to solve the phase problem. Although other relations are employed for
Presented in the course "Teaching Crystallography for Today's Sciences," Ettore Majorana Centre for Scientific Culture, Erice, Italy. September 1977. Schenk, H., Acto Cryst. A29,503 (1973). Schenk, H., "The symbolic addition method, programmed text" (in Dutch), 1973. Woolfsan, M. M., "Direct Methods in Crystallography," at the Clarendon Press, Oxford, 1961.
'
Figure 1. A reflected beam has a high intensity in case the atoms lie in the neighbahood 0 1 t h setat planes H (Fig. la)and a weak imensily when lheatms are spread out randomly wim respect to the planes H (Fig. Ib).
Volume 56. Number 6, June 1979 1 383
Figure 4. In an arbitrary triangle ABC an arbitrary point 0 has been chosen. Theorem: AOIAD + BOIBE + COICF = 2. Proof: AOIAD = APIAC: COICF = C R I A C BOIBE = BOIBC = ASIAC: because RP = SC. AP + CR + AS = 2AC. q.e.d.
Figure 3. In case three reflections H. K, and -H-K are all strong it Is mare likely that the planes of high electron density of -H-K run through the intersecting lines of the planes H and K (Fig. 3a)than just in between (Fig.3b). special puryx)it.i such as calculating f i p r r s of merit for finding the correct Y ,-relation. i t is heref fore I e r ~ t ~ m ato t eteach the Z2-relation only in a simple presentation for undergraduates. The process of understanding the &relation is built up around the following three questions: (1) In which regions is the electron density likely to be found if reflection H is a strong one? (2) In which regions is the electron density likely to he found if both reflections H and K are strong? (3) In which regions is the electron density likely to be found if all three reflections H, K, and -H-K are strong?
384 1 Journal of Chemical Education
T h e first question is illustrated in Figure 1. Knowing Bragg's law, i t will be easy to understand that the situation in Figure l a results in a large diffracted intensity. Similar arguments lead to the conclusion that Figure l h illustrates a weak reflection. In reverse, if reflectionH is strong the electron density will peak in planar regions of the crystal which are interplanar distances d~ apart. In case two, if reflections H and K are strong it is easy to see that the electron density will he found near the lines of intersection of the two sets of planes H and K. In projection this situation is indicated in Figure 2. If the reflection -H-K is also strong this implies that the electron density will also peak in planar regions which lie distances of ~ - H - K apart. Then it is much more likely that these planes of high electron density run through the lines of intersection of the strong reflections H and K as shown in Figure 3a (compare with Fig. 2), than that they run in between these lines, illustrated in Figure 3b. The last step to the &-phase relation is the introduction of an origin in the crystal, e.g. a t any place in Figure 3a. The sum of the phases of the three reflections is then given by $H
+ $K + &H-K
= 0 (modulo 2s)
which follows easily from a planimetric theorem (see Fig. 4). Since the choice of 0 is arbitrary, it follows that this phase relation is a structure invariant. A further exercise can be the use of this relation to solve a projection of a simple structure. In our programmed instruction we use the svmbolic addition method to solve the strurtt~reof tetraethyl-.diphosphit,e.disulphirle. the structure factors of which are given h y \Voolfson:'
