Diffraction vector

The Dark-Field symmetry is observed inside a hkl diffracted disk (often characterized by its diffraction vector g) which is exactly in Bragg orientation. This situation occurs when the hkl Bragg line goes through the center of its hkl diffracted disk. 

Figure 1.7 A Lang topograph (in transmission) of a silicon wafer. MoK i radiation (0.07 nm), conventional X-ray tube, Ilford L4 ultra-high resolution plate. Field size 3 nun by 3. 3 mm 022 reflection. Diffraction vector in direction up the page. The black lines are images of individual dislocations...

For symmetric reflections the peak search may now begin. For asymmetric reflections, the specimen must be rotated about its normal until the desired diffraction vector lies in the incidence plane of the beam conditioner. This is normally the diffractometer surface. An accurate knowledge of the orientation of the specimen in two axes is required to set asymmetric reflections this is usually taken from the position of the orientation flat or groove. 

As will become apparent, it is important to place the photographic plate as close to the specimen as possible. With rotating anode generators, care should be taken not to allow the full power of the beam to fall on the plate when stationary as this leads to an unsightly overexposed vertical line on the topograph. The presence of a horizontal stripe on the recorded topograph is often due to the presence of a second reciprocal lattice point lying on the Ewald sphere. It can be removed by a small rotation of the crystal about the diffraction vector as if to take a stereo pair. 

The black-white contrast reverses with the diffraction vector and also with the sense of the strain in the lattice. This is a useful means of determining the nature of a precipitate with a convenient rule of thumb . On the side of positive g, if the contrast is enhanced, the lattice is under compression, if reduced, it is under tension. 

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)... 

As seen in the last chapter, the image width is easy to quantify for a screw dislocation, where the diffraction vector is parallel to the dislocation line. Around a screw dislocation, the misorientation at a distance r from the core is given by... 

The contribution of harmonics is not normally large at first- and second-generation synchrotron sources, as the F i i value falls rapidly with increasing diffraction vector. A major exception is where weak quasi-forbidden reflections... 

The third difference arises from the continuons natnre of the radiation giving rise to orientation contrast only at the boundary between misoriented regions. Here the beams overlap or diverge depending on the sense of the misorientation with respect to the diffraction vector (Figure 10.9(a)). Best contrast occurs when... 

FIGURE 1 Cross-sectional TEM image of GaN on c-plane sapphire (grown by MOCVD) taken near the [1100] zone with diffraction vector g-2g (g = 1120). Threading dislocations extend from a highly defective low temperature GaN buffer layer to the film surface. The density of threading dislocations is 1010 cm 2. The majority of dislocations are edge defects with b = <1120>. 

Stacking faults are characterised by a fault plane and a fault displacement vector. On one side of the fault plane, the atoms that are located fer from the fault are displaced by a vector R in relation to the positions they would occupy in the absence of the fault. Strain fields emanating from any reconstructive bonding that is present near the fault plane will lead to additional displacements for atoms near the fault plane. Thus, the specification of R determines the positions of the atoms that are sufficiently distant so that the strain field generated by the fault is below some specified tolerance. For a planar fault, R may be determined experimentally by analysis of the diffraction contrast obtained with different diffraction vectors g. The positions of atoms near the fault may be determined theoretically by total energy minimisation calculations. Knowledge of these positions is essential to determine the electronic structure of the fault. 

Modulations is a perturbation of the crystal lattice which, unlike random perturbations due to thermal motion and static disorder, has regular character and therefore creates sharp diffraction peaks, usually as satellites of ordinary reflections. The diffraction vector can be then expressed as (cf Section 2.2.1)... 

Figure 7.16. Illustration of Ewald sphere construction, and diffraction from reciprocal lattice points. This holds for both electron and X-ray diffraction methods. The vectors AO, AB, and OB are designated as an incident beam, a diffracted beam, and a diffraction vector, respectively.

Figure 12.9 Bright-field image showing a typical platelet from the VPO-0.1 sample. Corresponding dark-field micrographs taken in (b) the g" ° = (024) reflection of (VO)2P207 and (c) the g = (022) reflection of 5-VOPO4 (g = diffraction vector) [94], (Reproduced with permission).

If parts I and II are twin-related and the operating diffraction vector g is normal to the twin plane (which is common to both parts), then the two parts will diffract with equal intensity and no contrast between I and II will be observed. For this reflection, the twin is said to be out-of-contrast. 

If an image is formed using a diffraction vector g normal to the twin plane, then the crystal on either side of the boundary will diffract with equal intensity and the twin will be out-of-contrast, as in Section 5.1. For this reflection, the 6-component is zero, and the twin boundary is also out-of-contrast (a) if R = 0 or (b) if R 0, but it lies in the twin plane, as is usually the case. 

In its simplest form, this matching of observation and theoretical prediction involves finding diffraction vectors g for which the dislocation is invisible (or very weak) and determining the direction of b by assuming that the condition g-b = 0 is satisfied. However, g b = 0 is a sufficient condition for invisibility only for a pure screw dislocation, and the invisibility of a pure edge dislocation also requires g b x u = 0. Thus, invisibility of a dislocation of mixed character is not expected for any reflection, even though the condition g b = 0 is satisfied, because g b x u 0. The... 

Figure 5.22. Imaging a spherical inclusion by its strain field in the surrounding isotropic crystal matrix, (a) Diagram illustrating the compressive strain around the inclusion, (b) Schematic diagram illustrating the nature of the image and, in particular, the line of no contrast CC normal to the diffraction vector g.

Three microtwins (A, B, and C) in Cazadero albite (Ano) imaged in DF with sbc different diffraction vectors g are shown in Figure 8.13. From... 

Figure 9.33. WBDF images of the dislocations and faults in a-spodumene observed with several different diffraction vectors.

In Ewald s construction, when the magnitude of the wavelength of the X rays is constant, the vectors describing the incident beam CO and the diffracted beam CP are equal in length and proportional to the reciprocal of the wavelength. The vectors CO and CP form an isosceles triangle in which the base OP is parallel to the diffraction vector. 

Diffraction vector A vector perpendicular to the lattice planes hkl causing a Bragg reflection. The diffraction vector bisects the directions of the incident and diffracted beams and lies in their plane. 

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