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As shown by von Laue's ingenious experiment, x-rays are electromagnetic waves of high energy (or short wavelength). Waves have both amplitude and phase, important concepts for understanding x-ray crystallography (Figure 3). X-rays are scattered by electrons; other diffraction phenomena important for the biosciences are neutron crystallography (neutrons are scattered by atomic nuclei, i.e., by neutrons and protons) and electron crystallography (electrons are scattered by the charge distribution, i.e., by electrons and protons). X-ray diffraction is by far the most established of these techniques, and many procedures in x-ray crystallography have become more or less routine.

When an x-ray photon impinges on an electron, the electron begins to oscillate and thereby acts as a secondary source of x-radiation. The ensuing waves disperse radially from the scattering center, with the same wavelength as the incoming wave (assuming elastic scattering, i.e., no energy loss) but an amplitude corresponding to the scattering power of an electron and a phase shift of 180° (Figure 4). A second electron nearby would also produce the same scatter, but due to its different position, the resulting waves will have a phase different to those from the first electron. If a detector is placed at some distance away, it will record the resultant diffraction pattern arising from addition of the individually scattered waves. Due to their different phases, the amplitude will be modulated as a function of position along the detector.

Of course, electrons are not found in isolation in macromolecules; rather, they are associated with atoms and bonds. An approximation used in biological crystallography is that the charge distribution is centered on the atomic nuclei; in this case, scattering by individual electrons is replaced by that of electron density distributions. From solution of the

Figure 2 Schematic diagram depicting a single layer of a crystal with trigonal symmetry; triangles represent threefold axes. (a) If the three colored molecules are identical, the threefold axis represents a crystallographic symmetry, as a 120° rotation superimposes individual unit cells onto one another. This restricts the cell axes to a = b, a = b = 90°, and g = 120°. Each threefold axis represents an alternative origin of equal validity. Note the large solvent channels between the molecules. (b) The unit cell corresponding to (a) contains three molecules in a trigonal P3 space group (note that two of the molecules of the complete lattice are made up of two 'halves' in the unit cell). Each equivalent molecule represents an alternative choice of asymmetric unit. The next layer of the crystal would stack directly on top at a distance c corresponding to the crystallographic c axis. (c, d) In contrast to (b), each orientation of the molecule represents a single layer of the crystal, each layer separated by one-third of the crystallographic c axis. The transformation between the colored molecules represents a screw axis, clockwise in (c) and anticlockwise in (d) which correspond to the space groups P3-, and P32, respectively. These arrangements exhibit a tighter packing density than (b); note that the crystal contacts in each are fundamentally different.

Figure 2 Schematic diagram depicting a single layer of a crystal with trigonal symmetry; triangles represent threefold axes. (a) If the three colored molecules are identical, the threefold axis represents a crystallographic symmetry, as a 120° rotation superimposes individual unit cells onto one another. This restricts the cell axes to a = b, a = b = 90°, and g = 120°. Each threefold axis represents an alternative origin of equal validity. Note the large solvent channels between the molecules. (b) The unit cell corresponding to (a) contains three molecules in a trigonal P3 space group (note that two of the molecules of the complete lattice are made up of two 'halves' in the unit cell). Each equivalent molecule represents an alternative choice of asymmetric unit. The next layer of the crystal would stack directly on top at a distance c corresponding to the crystallographic c axis. (c, d) In contrast to (b), each orientation of the molecule represents a single layer of the crystal, each layer separated by one-third of the crystallographic c axis. The transformation between the colored molecules represents a screw axis, clockwise in (c) and anticlockwise in (d) which correspond to the space groups P3-, and P32, respectively. These arrangements exhibit a tighter packing density than (b); note that the crystal contacts in each are fundamentally different.

Schrôdinger equation, it is possible to derive the electron density distribution and thereby the degree of scattering of any particular atom at a given wavelength (the so-called scattering or form factor). The atomic form factor f falls off with increasing angle of diffraction; maximum scatter is in the forward (undeflected) direction and is directly related to the atomic number Z of the scattering atom (Figure 4). This belies a major problem of biological crystallography: biomolecules are composed predominantly of atoms with low electron density - hydrogen, carbon, nitrogen and oxygen, with Z = 1,6, 7, and 8, respectively - and therefore scatter x-rays only weakly. This is compounded by the fact that atoms are in motion; this motion leads to an exponential reduction in the intensities at higher diffraction angle and thereby a reduction in resolution (see below). The degree of damping of the scattering factor is governed by the temperature- or B-factor - the higher the B-factor, the more rapid the falloff in intensity with scattering angle (Figure 4).

Addition of the diffracted waves from each atom in our object results in a defined diffraction pattern. Mathematically, the diffraction pattern is a Fourier transform of the diffracting object, each direction within the diffraction pattern associated with an amplitude F and a phase f. If we have an electron density distribution p(r), then the diffraction F in a given direction 9 is given by

Figure 3 The anatomy of a wave. (a) A wave is defined by its wavelength, its amplitude, and its phase. The red and the blue sine waves have the same wavelength and amplitude but differ in their phases. Addition of the two waves results in the pink wave, also of the same wavelength; due to the phase difference of the two waves, however, the resultant wave exhibits a new amplitude and phase. Addition of the waves is simplified by using the Argand representation in (b) amplitudes are given by the radii of the circles, while the phases are denoted by the angle subtended by the wave vector from the horizontal axis (straight lines). Every point of the sine wave from (a) is found as the projection of the height on the vertical axis of (b) (compare the blue and green elements of the wave cycle). The amplitude and phase of the resultant wave is then given by the simple vector addition of the two components (bottom).

where the magnitude of the diffraction vector | S | = 2sin9/1. A special property of the Fourier transform is that it can be inverted: the electron density is simply given by the inverse Fourier transform:

Note here however that F is a vector quantity, with magnitude and phase.

The phenomenon of diffraction is not restricted to x-radiation; the same occurs for example with visible light. With the latter, however, it is possible to recombine the diffracted waves to obtain an image of the diffracted object. This is achieved through the use of a lens; the lens ensures that both the phase and the amplitude information present in the diffraction pattern are maintained and recombined up to the point of image formation. For x-radiation, however, there exist at present no suitable lenses; an alternative means to image formation is therefore necessary. In principle, construction of the image is possible providing the phases of every part of the diffraction pattern are known - this is a property of Fourier transforms. Unfortunately, however, current diffraction experiments can measure accurately the intensities of the scattered rays, but all phase information is lost. While knowledge of the atomic positions within the molecule (i.e., the structure) can be used to completely reconstruct amplitudes, phases, and intensities in the diffraction pattern, the absence of experimental phase information means that the reverse process, from intensities to structure, is not possible directly. It is the burden of the crystallographer to determine these phases, i.e., to solve the phase problem.

A special situation arises when the object is a crystal. Under special conditions, all the unit cells in the crystal diffract in phase, leading to an enormous amplification in the scattering, so that individual reflections can be measured. While not strictly correct physically, Bragg provided a graphic illustration of the conditions for crystal diffraction (Figure 5).4 He considered that diffraction was caused by reflection of the incoming x-rays by imaginary planes within the crystal. For constructive interference to occur, all reflected waves from the crystal must be in phase. Bragg's law states that, for a wavelength l, a glancing angle 9, and an interplane spacing d, constructive interference occurs when where n is an integer. If the diffraction pattern fades away at an angle 29max, then this defines a minimum distance dmin = 1/2sin9max that can be resolved from the crystal - this is termed the resolution of the diffraction data.

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