The dispersion surface

Equation (4.23) is the fundamental equation of the dispersion surface, which we now investigate in detail. 

For the point /I3, which lies on the line joining L and Q, we can see that 0 = g and thus 

We now show that a tie point on the dispersion surface also characterizes the ratio of the amplitudes of the waves inside the crystal associated with the tie point. From the first dispersion equation, Eq. (4.19a), we have 

Similarly, from the second dispersion equation, Eq. (4.19b), we have 

Note that because the Fourier coefficients t/g can be complex, so can and g. However, only the real parts are plotted in reciprocal space. 

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The analysis demonstrates the elegant use of a very specific type of column packing. As a result, there is no sample preparation, so after the serum has been filtered or centrifuged, which is a precautionary measure to protect the apparatus, 10 p.1 of serum is injected directly on to the column. The separation obtained is shown in figure 13. The stationary phase, as described by Supelco, was a silica based material with a polymeric surface containing dispersive areas surrounded by a polar network. Small molecules can penetrate the polar network and interact with the dispersive areas and be retained, whereas the larger molecules, such as proteins, cannot reach the interactive surface and are thus rapidly eluted from the column. The chemical nature of the material is not clear, but it can be assumed that the dispersive surface where interaction with the small molecules can take place probably contains hydrocarbon chains like a reversed phase. 

For completeness we note that the dispersion surface is only strictly hyperbohc when the circles are accurately represented by straight lines. In three... 

Figure 4.12 Magnification of the region near Lq in Figure 4.11. L is the Lorentz point (the Lane point corrected for the mean refractive index). A is one of the tie-points selected on the dispersion surface, and o and are the deviation parameters at that point (shown on branch 2)...

Any point on either branch of the dispersion surface is an equally good solution of the Maxwell equations. However, the only points that will be selected are... 

The above discussion has in effect been for materials with zero absorption, but this affects only the intensities. The construction of the dispersion surface and the wavevector matching are all performed on the real part of the wavevectors. When absorption is considered, the reflectivity in the Bragg case falls below 100% but it can still be over 99% for a low-absorption material such as silicon. 

Figure 4.14 The selection of tie-points on the dispersion surface for the transmission (Lane) case, using the construction of Figure 4.13 (branch 2 is on the left, branch 1 is on the right)...

Figure 4.17 The standing wavefields set up in symmetrical Laue-case reflections. The nodes of wavefields from branch 2 of the dispersion surface lie on the atomic planes and the wavefields experience low absorption. The antinodes of wavefields from branch 1 of the dispersion surface lie on the atomic planes and the wavefields experience high absorption...

Figure 4.19 The range of strong diffraction in (a) the Laue case, (b) the Bragg case. The heavy lining shows the region of the dispersion surface excited as the rocking curve is traversed...

In Fignre 4.19 the hyperbolic region is shown separately to clarify its dependence on the diameter of the dispersion surface. Since K o is large, we may write... 

Figure 4.20 The dependence of the range of strong diffraction on the diameter of the dispersion surface, (a) Symmetric reflection, (b) Asymmetric reflection...

For a perfect, uniform crystal, whether in bulk or as a thin layer, the Takagi-Taupin equations can be solved exactly as given in the next section. For the general case with multiple layers, however, it is necessary to integrate them numerically. The concepts of the dispersion surface are lost, and we cannot tell directly in which directions wavefields are propagating. They do give directly the intensities of the direct and diffracted beams emerging from the crystal, and all interference features are preserved. 

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... 

Let us consider two points F and G between which the local reciprocal lattice varies from g to + rfg. If the deformation is small the shape of the dispersion surface does not change and only a displacement of the hyperbolae results. We can consider this as a rotation about the origin of reciprocal space,... 

Instead of considering the dispersion surface as a variable and the reciprocal lattice as invariant, it is usually easier to consider the reciprocal lattice as the variable. Then equation (8.30) determines the variation of the amplitude ratio of the reflected and transmitted components as the wavefield propagates through the crystal. The ratio R characterises a particular tie-point on the dispersion surface and if R varies the tie-point must migrate along the dispersion surface branch. This results in a change in the intensity of the transmitted and diffracted... 

Figure 8.10 Rotation of the local reciprocal lattice vector in a distorted crystal, giving rise to a displacement of the dispersion surface hyperbola...

But 0 Sg are related by Eq. (4.23). Therefore, the selection of the point A must satisfy this equation. Because the spheres about O and G can be represented as planes in the neighborhood of Q, the locus of the points A are hyperbolic sheets with these planes as asymptotes. These hyperbolic sheets are called the dispersion surfaces. A more detailed view of the neighborhood around Q is shown in Figure 4.3. The two branches of the dispersion surface are called a and jS, the one closer to the L point being a. A point on the dispersion surface is called a tie point. The arbitrarily selected tie points (A and A2) and the directions of their associated wavevectors are shown. Note that for the a branch, 0 and Jg are both positive, but for the branch, they are both negative. 

Figure 4.3. Detailed view of the dispersion surfaces in the neighborhood of the point Q.

In the discussion of the dispersion surface in Section 4.3, tie points on the dispersion surface were arbitrarily selected in order to show how the tie points characterize the allowed waves in the crystal. We must now examine how the tie points are selected in an actual experiment, but first we consider the boundary conditions that exist at the boundary between vacuum and the entrance face of the crystal. 

Figure 4.4. Construction in reciprocal space showing the selection of tie points A and B on the two branches of the dispersion surface.

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