The stacking of layers

Now consider two stacked layers of thicknesses tq and x — To, the scattering properties of which are generally different. Let the radiation fields incident on either side of the composite, 7(0, —p, p) and I(x, p, p), be identically zero. By analogy with Eqs. (2.3.1) through (2.3.6), the upward and downward intensities at the common 

The first term on the right side of Eq. (2.3.9) is due to thermal emission from the layer defined by the boundaries t = 0 and t = tq. The second and third terms are due, respectively, to direct and diffuse transmission through the same layer. Analogous interpretations with respect to the layer defined by the boundaries T = ToandT = ti are valid forEq. (2.3.10). 

Equations (2.3.9) and (2.3.10) define the outgoing upward and downward radiation fields emitted from a composite inhomogeneous layer. Repeated applications of the procedure admit solutions for an atmosphere composed of any finite number of layers with individually different thermal and scattering properties. It is also 

Consider the two original individually homogeneous layers with thicknesses tq and n — To. Let an outside point source of intensity ttFo 8 pi — pio)8 4 — 4 o) irradiate the composite from above, and ignore all thermally emitted radiation. By analogy with Eqs. (2.1.43), (2.1.46), and (2.3.1), the intensity at tq in the direction pi, p), in the absence of thermal emission, is given by 

Finally, in the absence of thermal emission, the outgoing radiation fields at either boundary of the composite are given by 

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The familiar diamond structure, with each atom covalently bonded in a perfect tetrahedral fashion to its four neighbors, is adopted not only by C but also by Si and Ge. Silicon can also adopt a wurtzite structure (see below), an example of a polytype (one of several crystal structures possible for a substance having an identical chemical composition but differing in the stacking of layers, and which may exist in a metastable state after its formation at some different temperature or pressure). 

Figure 7.32. The 24PPc"PO< " structure of CuGaC>2. Cu atoms are black and the stacking of layers is along the c axis.

Fig. 2 Structure of [C4N3Hi2][NH,]2[Mnn,F,(SO,)d, I, showing the stacking of layers and the presence of piperazine in the interlayer space.

Crystals with the lower symmetries, typical of many organic compounds, are built by the stacking of layers of three-dimensional objects the postulated requirement is that these layers must be closely packed. The close stacking can be obtained... 

Polytypism. Polytypes are substances that occur in different structural modifications, each of which can be regarded as being built up by the stacking of layers of (nearly) identical structure and composition, and with the modifications differing only in their stacking sequence. 

Figure 7.21 The structures of graphite and boron nitride (a) a single layer of the graphite structure (b) the stacking of layers, represented as nets, in graphite (c) a single layer of the boron nitride structure (d) the stacking of layers, represented as nets, in boron nitride...

Figure 12. Two views of FEYQOQ [Pt2(C6H804)3(H20)2] 4H2O. (o) Side view showing the stacking of layers (or slabs) of hydrated PrOj and PrOio polyhedra. (b) A single slab wherein intraslab coordination of adipate molecules is observed and shown.

The most common two-dimensional technique employs a precession camera, but any technique giving two-dimensional undistorted images of the reciprocal lattice is suitable as well. From these undistorted images, the geometry of the diffraction pattern can be analyzed by simple visual inspection. In the case of a precession camera study, the crystal must be mounted so as to have the (001) plane perpendicular to the goniometer rotation axis. In fact, the stacking of layers in micas is along c and the periodicity in reciprocal... 

Figure 63. The ciystal structures of nacaphite and arctite (a) one layer of nacaphite projected onto (100) (b) two layers of nacaphite projected onto (100) (c) two layers of arctite projected onto (001) (d) the stacking of layers along [001] in arctite. ( Na,Ca ( )n) polyhedra are shadow-shaded, Ba are shown as dark circles.

Figure 4.1-8 Polyhedral unit cell representation of NaBaFe4 (HP04)3(P04)3"H20 showing the stacking of layers which run parallel to the a axis. The sodium (small cross-hatched circles) and barium (large cross-hatched circles) atoms lie between the layers along with the isolated water molecules which are shown as open circles. The FeOe polyhedra are lined and the PO4 tetrahedra are dotted [90].

Figure 13. (a) A ball-and-stick representation of the stacking of layers in Fi2(V02)(P04), viewed... 

Figure 16. The structure of Ag2(V02)(P04). (a) View parallel to the a axis showing the stacking of layers, (h) Polyhedral representation of the V/P/O network.

Figure 17. Views of the structure of Li2V0P04 O.5H2O showing the stacking of layers and the locations of the Li+ cations (a) and looking normal to the V/P/O plane (b).

Different wavelengths are reflected at different depths. An x-ray penetrates the stack of layers until it arrives at the depth where the period resonates with its wavelength and angle. In this manner, a very broad band of reflection can be obtained, therefore, we call the reflectors extremely broad band or EBB multilayers. This system is not as critically dependent as the uniform period multilayer upon adhering to the optimum prescription of thickness versus depth. They were originally conceived for the reflection of slow neutrons (Mezei, 1976, and Mezd and Dalgleish, 1977). Christensen... 

Fig. 1.19 Unit cells for fee (center), and hep (right) geometries. The coloring of atoms and dashed triangles are indicated to match the stacking of layers in Fig. 1.18...

The picture below shows part of the structure of a crystal of CdCh. The layers in the picture continue longways and sideways, and the stack of layers upwards and downwards. Each cadmium ion (green) is surrounded by six chloride ions (blue), but each chloride ion has three cadmium ions on one side, and three chloride ions from the next layer on the other side. 

The II phase is similar to the / phase and contains layers of trigonal prisms linked at the top. The Rh atoms are in the centers of them. But the stacking of layers is different in the two strucmres. 

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