Laue method, diffraction

Laue Method for Macromolecule X-Ray Diffraction. As indicated above it is possible to determine the stmctures of macromolecules from x-ray diffraction however, it normally takes a relatively long period of data collection time (even at synchrotrons) to collect all of the data. A new technique, the Laue method, can be used to collect all of the data in a fraction of a second. Instead of using monochromated x-rays, a wide spectmm of incident x-rays is used. In this case, all of the reflections that ate diffracted on to an area detector are recorded at just one setting of the detector and the crystal. By collecting many complete data sets over a short period of time, the Laue method can be used to foUow the reaction of an enzyme with its substrate. This technique caimot be used with conventional x-ray sources. 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

Laudanosine, 2 87, 89 Laue method, for macromolecule X-ray diffraction, 26 442 Laundering... 

Macromolecule single-crystal structure determination, 26 426-427 Macromolecule structure, interactions related to, 13 742-743 Macromolecule X-ray diffraction, Laue method for, 26 442... 

FIGURE 2.2 X-ray diffraction by a crystal of beryl using tbe Laue method. 

The other source is the continuous wavelength spectrum of neutrons produced by stopping an accelerated beam of electrons, i.e., the spallation source . Since the electron beam is pulsed, so is the neutron beam [230]. The diffraction experiment uses the Laue method and the wavelengths are measured by their time of flight (TOF). In place of Bragg s law, dhk) = X/2 sin 0hk), the TOF relationship is... 

The Laue method was the first diffraction method ever used, and it reproduces von Laue s original experiment. A beam of white radiation, the continuous spectrum from an x-ray tube, is allowed to fall on a fixed single crystal. The Bragg angle 6 is therefore fixed for every set of planes in the crystal, and each set picks out and diffracts that particular wavelength which satisfies the Bragg law for the particular values of d and 9 involved. Each diffracted beam thus has a different wavelength. 

There are two variations of the Laue method, depending on the relative positions of source, crystal, and film (Fig. 3-5). In each, the film is flat and placed perpendicular to the incident beam. The film in the transmission Laue method (the original Laue method) is placed behind the crystal so as to record the beams diffracted in the forward direction. This method is so called because the diffracted beams are partially transmitted through the crystal. In the back-reflection Laue method the film is placed between the crystal and the x-ray source, the incident beam passing through a hole in the film, and the beams diffracted in a backward direction are recorded. 

Fig. 5-10 Focusing of diffracted beam in the transmission Laue method. 5 = source, C = crystal, F = focal point.

Fig. 8-11 Use of a stereographic ruler to plot the pole of a reflecting plane on a stereo-graphic projection in the back-reflection Laue method. Pole 1 is the pole of the plane causing diffraction spot 1.

In either Laue method, the diffraction spots on the film, due to the planes of a single zone in the crystal, always lie on a curve which is some kind of conic section. When the film is in the transmission position, this curve is a complete ellipse for sufficiently small values of 0, the angle between the zone axis and the transmitted beam (Fig. 8-12). For somewhat larger values of 0, the ellipse is incomplete because of the finite size of the film. When 0 = 45°, the curve becomes a parabola when 0 exceeds 45°, a hyperbola and when 0 = 90°, a straight line. In all cases, the curve passes through the central spot formed by the transmitted beam. 

Fig. 8-13 Relation between plane normal orientation and diffraction spot position in the transmission Laue method.

The sensitivity of the ordinary Laue method in the detection of crystal disorientation may be estimated as follows. Suppose the crystal-to-film distance is 5 cm, and assume that the minimum detectible broadening of a Laue spot is 1 mm. Then the diffracted beam has diverged by about 1/50 radian or 1°. This divergence corresponds to a disorientation of the reflecting planes of about 0.5°. This disorientation applies only to the area irradiated by the incident beam, which is typically 1 mm in diameter the irradiated area is therefore only 10" cm. ... 

The common methods of x-ray diffraction are differentiated by the methods used for bringing reciprocal-lattice points into contact with the surface of the sphere of reflection. The radius of the sphere may be varied by varying the incident wavelength (Laue method), or the position of the reciprocal lattice may be varied by changes in the orientation of the crystal (rotating-crystal and powder methods). 

There is another way of treating the Laue method which is more convenient for many purposes. The basic diffraction equation, Eq. (7), is rewritten in the form... 

This construction also shows why the diffracted beams from planes of a zone are arranged on a cone in the Laue method. All reciprocal-lattice lines representing the planes of one zone lie on a plane passing through the origin of the reciprocal lattice. This plane cuts the reflection sphere in a circle, and all the diffracted beam vectors S must end on this circle, thus producing a conical array of diffracted beams, the axis of the cone coinciding with the zone axis. 

That one can use both monochromatic and Laue methods, each to independently provide firm evidence for the photo-stmctural change in this test compound, is highly encouraging. One would expect that monochromatic methods are likely to yield real-space resolution better than Laue-based results, but this test shows that it is viable to observe photo-structural differences with Laue diffraction on small molecules and to undertake full data collections much more quickly than monochromatic methods would allow. Given that the Laue method is the only viable method in some cases, it is important to pursue the development of Laue diffraction in this area concurrent with the more major monochromatic developments. 

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