Harker Planes or Sections

FIGURE 9.5 A molecule in (a) is composed of four atoms. In (b) we see the molecular image produced in the Patterson map when the vectors from atom 1 to all other atoms are drawn, then, in (c) when those from atom 2 are drawn, and finally in (d) the set of molecular images that appear in the Patterson map when all intramolecular vectors are compiled. 

We know that any symmetry operator, such as a twofold axis or a sixfold screw axis, is equivalent to a set of general fractional coordinates in the unit cell where, by symmetry, corresponding atoms on different asymmetric units are found (see Chapter 3). Thus to specify that there is a twofold axis along y (and it is always essential to specify the direction of the operator) is the same as saying that for an atom at any x, y, z, there must be an identical atom at —x, y, —z. To say there is a axis along z is equivalent to stating 

FIGURE 9.7 Two molecules of tRNA in (a) are related by a twofold symmetry axis along z in the crystal. A point x, y, z, which could be the site of a heavy atom in one molecule, has an identical corresponding site in the dyad-related molecule at — x, —y, z The vector that connects the two sites will be (x, y, z) — (—x, —y, z) = 2x, 2y, 0. This vector, a Harker vector, must appear on the ro = 0 section of the corresponding Patterson map computed from the intensities of the diffraction pattern. In (b) the heavy atom site on the protein molecule at x, y, z appears on the 2i screw axis (along z) related asymmetric unit at —x, —y, Z—. But —, the unit translation, is the same as +, so the difference vector is 2x, 2y,. This Harker vector would appear on the plane of the Patterson map containing points for which w =.  

Second, algebraic differences between the equivalent positions for the space group are formed. For each pair of equivalent positions, one coordinate difference will turn out to be a constant, namely 0, 5, 3, 5, depending on the symmetry operator. These define the Harker sections for that space group, which are the planes having one coordinate u,v, or w constant, and that will contain peaks corresponding to vectors between symmetry equivalent atoms. In focusing attention only on Harker sections, the Patterson coordinates u,v,w 

VECTORS BETWEEN EQUIVALENT POINTS FOURFOLD AXIS 

---