Bragg planes

As one may infer from the quotation, W. L. Bragg realized that a crystal can act as an x-ray grating made up of equidistant parallel planes (Bragg planes) of atoms or ions from which unmodified scattering of x-rays can occur in such fashion that the waves from different planes are in phase and reinforce each other. When this happens, the x-rays are said to undergo Bragg reflection by the crystal and a diffraction pattern results. 

Although onty two Bragg planes have been considered, it is clear that planes below BBf will also contribute to the reflected beam, though to an exponentially decreasing extent. The relationship between X and sin will evidently be important in determining the useful wavelength range of a particular crystal... 

Mile. Cauchois20 used in transmission with a convergent beam a thin crystal bent cylindrically with the Bragg planes about as shown in Figure 4-11. Mica, gypsum, and quartz proved suitable materials. The action of the crystal is shown graphically in Figure 4-11, and it can be demonstrated analytically as well.22... 

With x-rays, however, one can have his cake and eat it too That the two conditions given above can both be met for a curved crystal was - appreciated first by Du Mond and Kirkpatrick18 and put into practice first by Johansson.21 This situation exists because-the crystal is a three-dimensional lattice of exceedingly small spacing. It is therefore possible to bend the crystal until the Bragg planes have the radius R,... 

Tilt of the Bragg plane of the specimen, so that its dispersion plane is not parallel... 

The theory of this aberration was worked out in the 1920s by Schwarzchild. For simphcity we shall discuss the case of a beam conditioner comprising a single crystal and an aperture as in the classic double-crystal arrangement. If the Bragg planes are tilted about an axis contained in the incidence plane and the... 

Bragg planes, then rays which are not contained in the incidence plane will not see equal angles with respect to the specimen and the reference. If we set the crystals so that the median ray (in the incidence plane) makes equal angles, then an inclined ray may make the Bragg angle for the reference crystal but will not be diffracted from the specimen (Figure 2.21). The result is that only a band of rays satisfies the Bragg conditions for both crystals. The band moves up (or down) as the crystals are rotated. The consequences are ... 

Figure 2.21 Bragg plane tilt aberration, (a) Diffracting planes parallel, diffraction occurs simultaneously over the whole height of the beam, (b) Diffracting planes skewed, diffraction only takes place over a narrow band, (c) As the crystal is rotated to measure the rocking curve, the band moves up or down the crystal. The integrated intensity remains approximately the same as in case (a) but the peak intensity decreases and the width increases...

Tilting about axis in incidence plane and Bragg plane... 

Two peaks should be visible, the K 2 being half of the intensity of the K 1 peak. If this intensity ratio is not 2 1 it implies that the Bragg planes are not vertical and the orientation of the crystal in the plane normal to the incident beam should be adjusted. 

For the Eikonal theory to be valid, the distortions must be suffieiently small that the eoneept of a ray is retained. The criterion is that the radius of curvature of the reflecting planes does not exceed a critical value R. approximately equivalent to an angular rotation of the Bragg planes by half the reflecting curve width in an extinction distance. The critical radius of curvature is thus... 

The width of the image can be deduced using this simple idea of contrast being formed when the misorientation around the defect exceeds the perfect crystal reflecting range. We consider the case of a screw dislocation nmning normal to the Bragg planes, where the line direction / coincides with the diffraction vector g. The effective misorientation at distance r from the core is =bH r (8.41)... 

Owing to the distance between somce and specimen, the divergence of the beam normal to the dispersion plane is small and with SR it is uimecessaty to perform tilt adjustments to bring the Bragg planes of specimen and beam conditioner exactly parallel. 

In the kinetic energy term, qz will equal (q - K)2 only when q = q G, that is, when q lies in a Bragg plane in that case... 

Brillouin zones are delimited by Bragg planes (see Fig. 8.12) The (n +1) th Brillouin zone is the set of points that are neither in the (n — l)th nor in the nth zone and that can be reached from the nth zone by crossing only one Bragg plane. 

Brillouin zones for a square-planar Bravais lattice. The small circles indicate reciprocal lattice points. The first three Brillouin zones lie entire-lywithinthesquare of side2fa each of them has area fa2. The first Brillouin zone, indicated by "1", is centered atthe origin and includes the origin point. The second Brillouin zone is indicated as "2", etc. the third as "3", etc. The diagonal and horizontal lines indicate Bragg "planes" (which must be lines in 2D). Zones 4, 5 (not shown), and 6 (not shown) lie partially outside the square of side 2b. Adapted from Ashcroft and Mermin [4]. 

Because the amplitude of the lattice distortion in the Peierls insulator is very small, the X-ray scattering associated with it is weak. Furthermore the one-dimensional nature of this distortion gives rise to diffuse Bragg planes instead of the usual well-defined Bragg reflections. These two facts have led to the development of a special diffuse X-ray photographic technique often referred to as the monochromatic Laue technique or XDS (for X-ray diffuse scattering) (64). 

This may be stated as the condition that an electron can only be diffracted if its wave number lies on a plane bisecting some lattice wave number. (We have used the fact that if q is a lattice wave number, then so is —q.) Such planes, called Bragg planes, are illustrated in Fig. 16-4 for the simple cubic lattice. 

Only quite small electric fields can be applied in a metal, because of the high conductivity in the time an electron moves before colliding with a defect or with the surface it can only be accelerated very slightly. Unless it happens to lie very close to a Bragg plane it will not be affected by the diffraction. Since most electrons are thus unaware of the lattice, conductivity can to a large extent be treated in the free-electron approximation, as we indicated earlier. 

The lattice wave numbers and Bragg planes for the system. shown in Fig. 16-3. We imagine, though, a much larger crystal, so the mesh of wave numbers allowed by periodic boundary conditions becomes very fine and is not shown. 

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