Reciprocal lattice rod

This intensity distribution means that each reciprocal lattice point is effectively extended into a rod that is normal to the thin crystal specimen. Due to this extension of the reciprocal lattice points and to the fact that the radius of the Ewald sphere is large compared with the spacing between reciprocal lattice points, the Ewald sphere usually intersects many reciprocal lattice rods, as shown in Figure 3.12(a). Thus, many diffracted... 

Figure 3.12. Facing page, (a) Diagram showing the Ewald sphere cutting the reciprocal lattice rods of zero and higher order Laue zones and (b) a schematic diagram of the corresponding diffraction pattern.

Fig. 3 Ewald construction. The white half-circle indicates the Ewald sphere in two dimensions. The points of intersection between the reciprocal lattice rods and the Ewald sphere form the set of reciprocal lattice points (bright) which obey Bragg s law and appear as diffraction spots in the diffraction pattern. Zero-, first- and second-order Laue zone are indicated. Eor electron diffraction in TEM, the ratio between the radius of the Ewald sphere and the reciprocal lattice unit is larger than visualized in the figure. (View this art in color at www.dekker. com.)...

Figure 8.2.9 Ewald construction for elastic and inelastic scattering of a particle with initial and final wave vectors kj and kf. Six crystal truncation rods perpendicular to the crystal surface are indicated as solid lines, (a) Five conditions of elastic Bragg reflection are marked as gray dashed arrows, which are defined via the intersection of the Ewald sphere with the reciprocal lattice rods. Spheres for different final energies are marked as dashed circles with a constant increase in energy between the circles. An enlargement is shown in (b), which emphasizes the relation between both the energy loss AE and the momentum transfer hAq (short solid arrows) due to phonon excitation or annihilation, to allow scattering into a defined detector direction given by the long dashed arrow.

In many cases the surface is reconstructed (see Section 10.8) which breaks the periodicity in the surface plane. For example, for the (2x1) surface reconstruction in Figure 10.13, the repeat unit for the surface is one atom spacing along the rows of dimers and two atom spacings perpendicular to the rows. Doubling the periodicity in the surface plane halves the reciprocal lattice rod spacing. This leads to the appearance of extra spots in the diffraction pattern. Counting the spots shows the surface reconstruction. Thus, a Si (111) surface that reconstructs with a 7x7 structure... 

Typically the substrate is not cut exactly on a low-index plane. When the diffraction spots are sharp enough, the spots may be split along the reciprocal lattice rod direction (vertically on the pattern) into two spots when the beam is aligned up or down the staircase of surface steps causing the miscut. The separation of these spots reflects the miscut of the sample surface relative to the low index planes. This is useful to know because miscut is used in some cases to enhance heteroepitaxy. 

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

For a given structure, the values of S at which in-phase scattering occurs can be plotted these values make up the reciprocal lattice. The separation of the diffraction maxima is inversely proportional to the separation of the scatterers. In one dimension, the reciprocal lattice is a series of planes, perpendicular to the line of scatterers, spaced 2Jl/ apart. In two dimensions, the lattice is a 2D array of infinite rods perpendicular to the 2D plane. The rod spacings are equal to 2Jl/(atomic row spacings). In three dimensions, the lattice is a 3D lattice of points whose separation is inversely related to the separation of crystal planes. 

Electrons having energies and incident angles typical of RHEED can be treated as nearly nonpenetrating. As a result, atoms in the outermost plane are responsible for most of the scattering, and the resulting reciprocal lattice will be an array of rods perpendicular to the surfrce plane. 

The number of units in an MQW will be much more limited than the number of atomic planes sampled by the X-ray beam in a standard reflection. The intensity will be low, but also the MQW will behave as a thin crystal —the reciprocal lattice points will be extended into rods perpendicular to the crystal surface. This will broaden the reflection, and thus the width of each satellite peak is determined by the number of units in the MQW. It has even been possible, by... 

Dynamic smdies of the alloy system in CO and H2 demonstrate that the morphology and chemical surfaces differ in the different gases and they influence chemisorption properties. Subnanometre layers of Pd observed in CO and in the synthesis gas have been confirmed by EDX analyses. The surfaces are primarily Pd-rich (100) surfaces generated during the syngas reaction and may be active structures in the methanol synthesis. Diffuse scattering is observed in perfect B2 catalyst particles. This is attributed to directional lattice vibrations, with the diffuse streaks resulting primarily from the intersections of 111 reciprocal lattice (rel) walls and (110) rel rods with the Ewald sphere. 

Among the most common surface X-ray scattering techniques used to probe mineral-fluid interface structure is the measurement of crystal truncation rods (CTRs). CTRs are diffuse streaks of intensity connecting bulk reciprocal lattice (Bragg) points in the direction perpendicular to a surface, and arise as a natural... 

The formalism of the reciprocal lattice and the Ewald construction can be applied to the diffraction at surfaces. As an example, we consider how the diffraction pattern of a LEED experiment (see Fig. 8.21) results from the surface structure. The most simple case is an experiment where the electron beam hits the crystal surface perpendicularly as shown in Fig. A.5. Since we do not have a Laue condition to fulfill in the direction normal to the surface, we get rods vertical to the surface instead of single points. All intersecting points between these rods and the Ewald sphere will lead to diffraction peaks. Therefore, we always observe diffraction... 

Figure A.5 Ewald construction for surface diffraction, a) a side view of the reciprocal lattice at the surface. Constructive interference occurs for all intersection points of the vertical rods with the Ewald sphere. This is equivalent to the condition when the component qj of the scattering vector parallel to the surface is identical to a reciprocal lattice vector of the surface lattice, b) the top view of the reciprocal surface lattice. The circle is the projection of the Ewald sphere. If we disregarding the radiation scattered into the crystal, the number of lattice points within the circle (corresponding to the intersections of the rods with the Ewald sphere) is identical to the maximum number of observed diffraction peaks.

It is of some interest to consider the situation where one of the ideahzed 2D systems that have been addressed can be followed in a layer-by-layer growth mode from a strictly 2D plane to one that is more 3D like. Such is the situation in the formation of multilayer molecular films adsorbed to uniform substrates or where epitaxial metal or soft matter growth is realized in chemical vapor deposition, molecular beam epitaxy or polymeric deposition systems. The hneshape discussion above has to be modified to account for the development of the third dimension of order in the system. Conceptually this is rather straightforward. Instead of considering, as Warren did, an ideal 2D reciprocal lattice composed of an ordered array of uniform rods, the reciprocal lattice for an idealized multilayer (e.g., two to five individual layers) system is characterized by... 

Consider now a perfect crystal truncated by a sharp surface (or semi-infinite crystal). It can be obtained by the product of a step function describing the electron density variation perpendicular to the surface, and an infinite lattice. The diffraction pattern is then the convolution of the 3D reciprocal lattice with the Fourier transform of the step function. An infinity of Fourier components is necessary to build this latter, so that there remains non zero intensity in between Bragg peaks as a function of / the reciprocal space is made of rods of intensity, called crystal truncation rods (CTR), extending perpendicular to the surface, and connecting bulk Bragg peaks [24, 25]. The intensity variation as a function of (or Qj or /) is found by stopping the summation at n3 = 0 in Eq. (1) and (2), yielding ... 

Figure 18 Diffraction pattern of KHCO3 at 15 K in the (a, c ) plane. The arrows point to the ridges of intensity. The insert visualizes the correspondence between the direct and reciprocal lattices. The rods of diffuse scattering lie along the (30l) direction and as such are perpendicular to the plane of the dimers, which contain the x and y directions defined in Fig. 1.

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