Amplitude, diffracted beams

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.

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. 

Thermal properties of overlayer atoms. Measurement of the intensity of any diffracted beam with temperature and its angular profile can be interpreted in terms of a surface-atom Debye-Waller factor and phonon scattering. Mean-square vibrational amplitudes of surfece atoms can be extracted. The measurement must be made away from the parameter space at which phase transitions occur. 

Es is the electric field amplitude of the diffracted beam. Ec and Einc are the coherent and incoherent electric field amplitudes of the background intensity, respectively. 0 is the phase shift between the signal and the coherent background, and the phase of Ec is arbitrarily chosen to be zero. For convenience, the proportionality factor between E2 and / is set to unity. Sbom = E2 is the homodyne and Sbet = 2 ECES cos 0 the heterodyne signal. The total background is Jb=E2c + Efnc. 

Ideally, there is no phase shift between the reference and the diffracted beam (0=0), and, since TDFRS is completely nondestructive without dye bleaching, the signal can be accumulated over almost arbitrary times. In order to maximize the heterodyne signal amplitude, some means for phase adjustment and stabilization are needed. Without such active phase-tracking, 0 would have some arbitrary value and would slowly drift away due to the almost unavoidable slow thermal drift of the whole setup. 

Table I shows the calculated diffracted beam amplitudes for the (000), (002), (004) and (006) beams for both topologies, for a crystal 60 A thick. The principal difference between the intensities for the two structures lies in the intensity of the (004) beam relative to (006). [Note that in both cases the intensity of the central (000) beam is approximately the same and hence the intensities are approximately normalised with respect to each other thus allowing direct comparison]. In the case of the Imma structure the (004) beam is weak with respect to the (006) and hence the (006) must be included to resolve the satellite channels clearly. At a limited resolution of 3.5 A the (006) beam lies outside the objective aperture and hence the satellite channels are only very faint in the image. However, for the Cmcm model the relative intensities are reversed with the (004) beam stronger than the (006) and hence even if the (006) beam is excluded from the aperture the satellite channels will be clearly resolved.

At the entrance face of the crystal there is an incident (or transmitted) beam with amplitude Aq, but no diffracted beam. Hence,... 

It is clear from Eq. (4.50) and Figure 4.15 that the amplitudes Ag and y4g are both symmetrical about s = 0. Recall that in Section 4.8 it was shown that the waves with wavevectors originating on the 8 branch of the dispersion surface are preferentially absorbed, whereas the waves whose wavevectors originate on the a branch suffer little or no absorption. However, this effect does not give rise to any asymmetry about s = 0 of the intensity of the diffracted beam, which is shown in Figure 4.13. 

At P, the amplitude of the wave passing into the slab dz in the direction of the scattered (or diffracted) beam is... 

FIGURE 1.5. The experimental setup used by Friedrich and Knipping to measure X-ray diffraction intensities. The important components consisted of an X-ray source to provide a finely collimated X-ray beam, a crystal to scatter X rays, and a detection system, such as photographic film, to measure the directions and intensities of the diffracted beams. The intensities so measured are related to the squares of the amplitudes of the scattered beams, but information on the relative phases of these scattered beams is lost. This same general experimental setup is currently used, although the source of X rays and the detection system are now much more sophisticated-... 

Phase problem The problem of determining the phase angle (relative to a chosen origin) that is to be associated with each diffracted wave that is combined to give an electron-density map. The measured intensities of diffracted beams give only the squares of the amplitudes, but the relative phases cannot normally be determined experimentally (see Chapter 8). The determination of the relative phases of the Bragg reflections is crucial to the calculation of the correct electron density map. 

The amplitudes of diffracted beams are summed with a periodicity inversely proportional to their order (and with a relative phase, which needs to be determined) to give an electron density map. 

When a diffraction grating, such as a crystal, interacts with X rays, the electron density that causes this diffraction can be described by a Fourier series, as discussed in Chapter 6. The diffraction experiment effects a Fourier analysis, breaking down the Fourier series (of the electron density) into its components, that is, the diffracted beams with amplitudes, F[hkl). The relative phases a(hkl) are, however, lost in the process in all usual diffraction experiments. This loss of the phase information needed for the computation of an electron-density map is referred to as the phase problem. The aim of X-ray diffraction studies is to reverse this process, that is, to find the true relative phase and hence the true three-dimensional electron density. This is done by a Fourier synthesis of the components, but it is now necessary to know both the actual amplitude F[hkl) and the relative phase, a[hkl), in order to calculate a correct electron-density map (see Figure 8.1). We must be able to reconstruct the electron-density distribution in a systematic way by approximating, as far as possible, a correct [but so far unknown) set of phases In this way the crystallographer, aided by a computer, acts as a lens for X rays. 

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