Diffracted beams

It is relatively straightforward to detemiine the size and shape of the three- or two-dimensional unit cell of a periodic bulk or surface structure, respectively. This infonnation follows from the exit directions of diffracted beams relative to an incident beam, for a given crystal orientation measuring those exit angles detennines the unit cell quite easily. But no relative positions of atoms within the unit cell can be obtained in this maimer. To achieve that, one must measure intensities of diffracted beams and then computationally analyse those intensities in tenns of atomic positions. 

One fiirther method for obtaining surface sensitivity in diffraction relies on the presence of two-dimensional superlattices on the surface. As we shall see fiirtlrer below, these correspond to periodicities that are different from those present in the bulk material. As a result, additional diffracted beams occur (often called fractional-order beams), which are uniquely created by and therefore sensitive to this kind of surface structure. XRD, in particular, makes frequent use of this property [4]. Transmission electron diffraction (TED) also has used this property, in conjunction with ultrathin samples to minimize bulk contributions [9]. 

The diffraction of low-energy electrons (and any other particles, like x-rays and neutrons) is governed by the translational syimnetry of the surface, i.e. the surface lattice. In particular, the directions of emergence of the diffracted beams are detemiined by conservation of the linear momentum parallel to the surface, bk,. Here k... 

Experimentally, this technique is very similar to the TDI technique described above. A laser beam is incident normally on a diffraction grating or a preferentially scratched mirror deposited on the surface to obtain the normally reflected beam and the diffracted beams as described above. Instead of recombining the two beams that are located symmetrically from the normally reflected beam, each individual beam at an angle d is monitored by a VISAR. Fringes Fg produced in the interferometers are proportional to a linear combination of both the longitudinal U(t) and shear components F(t) of the free surface velocity (Chhabildas et al., 1979), and are given by... 

In diffraction experiments a narrow and parallel beam of x-rays is taken out from the x-ray source and directed onto the crystal to produce diffracted beams (Figure 18.5a). The primary beam must strike the crystal from many different directions to produce all possible diffraction spots and so the crystal is rotated in the beam during the experiment. Rotating the crystal is much easier than rotating the x-ray source, especially when it is a synchrotron. 

Figure 18.5 Schematic view of a diffraction experiment, (a) A narrow beam of x-rays (red) is taken out from the x-ray source through a collimating device. When the primary beam hits the crystal, most of it passes straight through, but some is diffracted by the crystal. These diffracted beams, which leave the crystal in many different directions, are recorded on a detector, either a piece of x-ray film or an area detector, (b) A diffraction pattern from a crystal of the enzyme RuBisCo using monochromatic radiation (compare with Figure 18.2b, the pattern using polychromatic radiation). The crystal was rotated one degree while this pattern was recorded.

Figure 18.6 Diffraction of x-rays by a crystal, (a) When a beam of x-rays (red) shines on a crystal all atoms (green) in the crystal scatter x-rays in all directions. Most of these scattered x-rays cancel out, but in certain directions (blue arrow) they reinforce each other and add up to a diffracted beam, (b) Different sets of parallel planes can be arranged through the crystal so that each corner of all unit cells is on one of the planes of the set. The diagram shows in two dimensions three simple sets of parallel lines red, blue, and green. A similar effect is seen when driving past a plantation of regularly spaced trees. One sees the trees arranged in different sets of parallel rows.

Each diffracted beam, which is recorded as a spot on the film, is defined by three properties the amplitude, which we can measure from the intensity of the spot the wavelength, which is set by the x-ray source and the phase, which is lost in x-ray experiments (Figure 18.8). We need to know all three properties for all of the diffracted beams to determine the position of the atoms giving rise to the diffracted beams. How do we find the phases of the diffracted beams This is the so-called phase problem in x-ray crystallography. 

Figure 18.8 Two diffracted beams (purple and orange), each of which is defined by three properties amplitude, which is a measure of the strength of the beam and which is proportional to the intensity of the recorded spot phase, which is related to its interference, positive or negative, with other beams and wavelength, which is set by the x-ray source for monochromatic radiation.

Since such heavy metals contain many more electrons than the light atoms, H, N, C, O, and S, of the protein, they scatter x-rays more strongly. All diffracted beams would therefore increase in intensity after heavy-metal substitution if all interference were positive. In fact, however, some interference is negative consequently, following heavy-metal substitution, some spots measurably increase in intensity, others decrease, and many show no detectable difference. 

How do we find phase differences between diffracted spots from intensity changes following heavy-metal substitution We first use the intensity differences to deduce the positions of the heavy atoms in the crystal unit cell. Fourier summations of these intensity differences give maps of the vectors between the heavy atoms, the so-called Patterson maps (Figure 18.9). From these vector maps it is relatively easy to deduce the atomic arrangement of the heavy atoms, so long as there are not too many of them. From the positions of the heavy metals in the unit cell, one can calculate the amplitudes and phases of their contribution to the diffracted beams of the protein crystals containing heavy metals. 

The intensity differences obtained in the diffraction pattern by illuminating such a crystal by x-rays of different wavelengths can be used in a way similar to the method of multiple isomorphous replacement to obtain the phases of the diffracted beams. This method of phase determination which is called Multiwavelength Anomalous Diffraction, MAD, and which was pioneered by Wayne Hendrickson at Columbia University, US, is now increasingly used by protein cystallographers. 

In the early days of protein crystallography the determination of a protein structure was laborious and time consuming. The diffracted beams were obtained from weak x-ray sources and recorded on films that had to be manually scanned and measured. The available computers were far from adequate for the problem, with a computing power roughly equal to present-day pocket calculators. Computer graphics were not available, and models of the protein had to be built manually from pieces of steel rod. To determine the... 

Crystallization of proteins can be difficult to achieve and usually requires many different experiments varying a number of parameters, such as pH, temperature, protein concentration, and the nature of solvent and precipitant. Protein crystals contain large channels and holes filled with solvents, which can be used for diffusion of heavy metals into the crystals. The addition of heavy metals is necessary for the phase determination of the diffracted beams. 

In bright-field microscopy, a small objective aperture is used to block all diffracted beams and to pass only the transmitted (undiffracted) electron beam. In the... 

Any periodicity in the surfiice or overlayer different from that of the clean unreconstructed surface produces addidonal diffracted beams. A unit mesh latter than... 

Figure 4 Interference pettern created when regularly spaced atoms scatter an incident plane wave. A spherical wave emanates from each atom diffracted beams form at the directions of constructive interference between these waves. The mirror reflection—the (00) beam—and the first- and second-order diffracted beams are shown.

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