Bragg conditions

Fig. 1. Structures of (O) atoms and corresponding electron and x-ray diffraction patterns for (a) a periodic arrangement exhibiting translational symmetry where the bright dots and sharp peaks prove the periodic symmetry of the atoms by satisfying the Bragg condition, and (b) in a metallic glass where the atoms are nonperiodic and have no translational symmetry. The result of this stmcture is that the diffraction is diffuse.

The two-dimensional Bragg condition leads to the definition of reciprocal lattice vectors at and aj which fulfil the set of equations ... 

For materials which are available not in the form of substantial individual crystals but as powders, the technique pioneered by Debye and Scherrer is employed (Moore, 1972). The powder is placed into a thin-walled glass capillary or deposited as a thin film, and the sample is placed in the X-ray beam. Within the powder there are a very large number of small crystals of the substance under examination, and therefore all possible crystal orientations occur at random. Hence for each value of d some of the crystallites are correctly oriented to fulfil the Bragg condition. The reflections are recorded as lines by means of a film or detector from their positions, the d values are obtained (Mackay Mackay, 1972). 

This is the Bragg condition, or Bragg s law, the fundamental law of X-ray crystallography. 

The +/-g symmetries are obtained in the following way at first, a hkl reflection (g) is set to the Bragg condition as described in the previous paragraph. Then, the opposite reflection -h-k-1 (-g) is also set to the Bragg condition and the symmetry present in these two disks is observed and compared with the 12 possibilities shown on figure 5c. 

The Bragg condition defines a cone of angles,, normal to the (hkl) planes. Alternatively, we use the Laue condition to specify the Bragg diffraction ... 

The deviation of the electron beam from the Bragg condition is measured by the distance from the reciprocal lattice vector to the Ewald sphere along the zone axis direction, which approximately is defined by... 

Here 80 is the deviation angle from the Bragg condition. S is negative for positive A and positive for negative A. 

For high order reflections with a large g, the rapid increase in the excitation error away from the Bragg condition results a rapid decrease in diffraction intensity. Under the kinematical condition, the maximum intensity occurs at the Bragg condition, which appears as a straight line within a small convergence angle. 

Figure 5. A record Si CBED pattern with (111) and (222) at Bragg condition and the intensity profile (cross) and the theoretical fit (continuous curve). The structure factors of (111) and (222) reflections are obtained from the best fit.

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 ... 

This Laue condition is a little less restrictive than the Bragg law, in that we no longer have the condition that K g = K 1=1/, bnt we still expect strong diffraction only when we are near the Bragg condition. Ewald proposed, and Bloch showed that waves that exist in a crystal must have the periodicity of the lattice, that is, the solutions shonld look like... 

We should first correct the wavevector inside the crystal for the mean refractive index, by multiplying the wavevectors by the mean refractive index (1 + IT). This expression is derived from classical dispersion theory. Equation (4. 18) shows us that is negative, so the wavevector inside the crystal is shorter than that in vacuum (by a few parts in 10 ), in contrast to the behaviom of electrons or optical light. The locus of wavevectors that have this corrected value of k lie on spheres centred on the origin of the reciprocal lattice and at the end of the vector h, as shown in Figure 4.11 (only the circular sections of the spheres are seen in two dimensions). The spheres are in effect the kinematic dispersion surface, and indeed are perfectly correct when the wavevectors are far from the Bragg condition, since if D 0 then the deviation parameter y, 0 from... 

Thus we calculate the reflectivity of a whole layered material from the bottom up, using the amplitude ratio of the thick crystal as the input to the first lamella, the output of the first as the input to the second, and so on. At the top of the material the amplitude ratio is converted into intensity ratio. This calculation is repeated for each point on the rocking curve, corresponding to different deviations from the Bragg condition. This results in the plane wave reflectivity, appropriate for synchrotron radiation experiments and others with a highly collimated beam from the beam conditioner. 

Figure 7.4 Defect peak in GaAs due to polishing damage. 004 reflection with CuK Triple axis -2 scans with displacements of specimen from Bragg condition solid (upper) line, 10 displacement, next highest line, 20 displacement, lowest hne, 20 displacement...

Figure 7.9 Ewald constructions, (a) at the Bragg condition, (b) off the Bragg condition...

We see that very close to the Bragg condition, where the dispersion strrface is highly cttrved, R K and the crystal acts as a powerful angrtlar amplifier. A reaches 3.5xl0 in the centre of the dispersion surface for sihcon in the 220 reflection with MoK radiation. Far from the centre, the dispersion strrface becomes asymptotic to the spheres about the reciprocal lattice points and A approaches unity. Thus when the whole of the dispersion strrface is excited by a spherical wave, owing to the amplification close to the Bragg condition, the density of wavelields will be veiy low in the centre of the Borrmann fan and... 

Around defects, the scattering power differs from that in the perfect crystal because X-rays which do not satisly the Bragg condition in the perfect crystal may be diffracted in the deformed region arotmd the defect. Just as in the Lang projection topograph, these regions behave as small crystals which diffract kinematically and the net result is an increase in the intensity over that from the perfect crystal. 

Figure 10.15 Simulated image width as a function of deviation parameter in Bragg case weak beam topographs. Here, the specimen is set off the Bragg peak and an image of the defect occurs only when the lattice planes are locally rotated or dilated back into the Bragg condition. As this occurs only close to the dislocation core, the images are narrowed from those under strong beam conditions...

In solid-state physics the opening of a gap at the zone boundary is usually studied in the free electron approximation, where the application of e.g., a ID weak periodic potential V, with period a [V x) = V x + a)], opens an energy gap at 7r/a (Madelung, 1978 Zangwill, 1988). E k) splits up at the Brillouin zone boundaries, where Bragg conditions are satished. Let us consider the Bloch function from Eq. (1.28) in ID expressed as a linear combination of plane waves ... 

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