Planes indexing reflections from

Indexing rotation photographs. Preliminary consideration. The spots on the equator of a rotation photograph are obviously reflections from atomic planes which were vertical during the exposure. In Plate VII the equatorial spots are reflections from planes parallel to the c axis, that is, hkO planes the third or l index for these reflections is 0 by inspection. The other two indices, h and k, of all the equatorial reflections may be found from the spacings of the planes, which are worked out from the reflection angles 6 by the Bragg equation. 

If the crystal is rotated round its b axis (Fig. 89) the equatorial spots are reflections from hOl planes. The values for these spots are found as before by measuring the distance from the origin to each point of the (non-rectangular) hOl net plane (Fig. 88). Note that the indexing of equatorial reflections in this case cannot be done by a log d chart, since there are three variables, a, c, and / the reciprocal lattice method is essential. Once the indices for the equatorial reflections have been found, those of the reflections on upper and lower layer lines follow at once, since all reciprocal lattice points having the same h and l indices (such a set as 201, 211, 221, 231, and so on) are at the same distance from the axis of rotation and thus form row lines. 

To check this by a specific example, consider the 431 rhombohedral reflection. To find the lH index in terms of rhombohedral indices, we must travel from O to O by way of rhombohedral axes, that is, via OD, DK, and KO it is evident from Fig. 243 b that from O to D we cross four planes (hn), from D to K three planes (kR), and from AT to O one plane (lR), so that in all eight planes are crossed (hRA-kR- lR). For hll9 we have to go from O to A to get there by way of rhombohedral axial directions, the simplest course is via OD and DA from O to D four planes are crossed, while from D to A three planes are crossed, but these have to be subtracted because we are returning towards the original plane through 0 (or, in other... 

The specular reflectivity of neutrons, like the analogous light or X-ray reflectivity, from a surface or interface provides information about the neutron refractive index gradient or distribution in the surface region and in a direction orthogonal to the plane. This can often be simply related to a composition or concentration profile in the direction orthogonal to the surface, to provide directly information about adsorption and the structure of the adsorbed layer. 

Electroreflectance — The reflectance intensity of polarized light reflected from a smooth surface is a function of the refractive index. The basic equations, derived originally by Fresnel, for light polarized parallel to the plane of reflection take the form rp = "aC°sfb "bC0Sl a where... 

The polarization is specified by the plane in which the electric vector oscillates the index s stands for perpendicular to the plane of reflection, and the index p stands for parallel to it (note the traditional definition of the plane of polarization is perpendicular to the plane of the vector oscillations ). The condensed expressions on the very right side of these equations result from applying Snell s law. For optically active media and incident circularly polarized radiation see Sec. 6.3 compare also Secs. 3.2, 4.6.4 and 4.6.5. The square of an electric field strength E is measured as intensity the quotient of the reflected intensity and the incident one is the reflectance R... 

Consider the face-centered cubic lattice in Fig. 27.30. The 100 planes are interleaved at just half the spacing by the 200 planes, which contain only face-centered atoms. The reflected rays from this second set of planes are 180° out of phase with those from the 100 planes. The two reflections interfere destructively so that a first-order reflection does not appear from the 100 planes in this lattice. (Higher-order reflections appear from both sets, but the intensities are much weaker.) For the same reason the first-order reflection from the 110 plane does not appear, being destroyed by the first-order reflection from 220 planes. The 111 planes are not interleaved in this way, so the first-order reflection comes through loud and clear, especially because the 111 planes are close packed. The absence of certain lines helps enormously in the assignment of indices, the indexing, of the lines that do appear. 

Figure 1.36. (a) Schematic of the diftraction pattern of PPV film (b) intensity profiles (molecular transform) of the first six layer lines predicted for a perfectly oriented, randomly shifted PPV molecular system (c) reciprocal net for the (a, b ) equatorial plane ( ) observed reflection, (O) predicted reflection (d) indexing of the (a, b ) plane shown in (c). (Reproduced from ref 266 with kind permis.sion. Copyright (1986) John Wiley Sons, Inc., New York.)... 

An x-ray beam was reflected from the (1011) plane of quartz or, in some cases, from a cleavage plane of mica (fifth-order reflection). The radii of curvature of the bent crystals were 1002 and 1217 mm. The dispersion achieved using these crystals was 2.33 XU/mm (i.e., 4.01 eV/mm) and 3.59 XU/mm (6.01 eV/mm). This enabled us to resolve completely the doublet whose components were separated by 3.65 XU. A crystal was bent and the aperture was reduced until we achieved the lowest value of the half-width of the line of metallic titanium (AA = 0.93 XU). The asymmetry index of this line was a = 1.13. 

The semi-infinite structure is assumed, bordered at the front plane by a dielectric of the same refractive index as the average refractive index of the cholesteric . In such a case, we neglect the reflection from the front boundary. 

For a given model of the structure normal to the interface, no matter how complex, it is possible to calculate the neutron reflectivity exactly using the same formulae, apart from the difference in the refractive index, as for light polarized at rightangles to the plane of reflection. For a multilayer structure the optical matrix method [4] can then be used, in which the interface is divided into as many layers as are required to describe it with adequate resolution. This method lends itself especially well to machine calculations and is therefore the most widely used method of analysing neutron reflectivity. However, it does not reveal the relatively simple relation between reflectivity and interfacial structure, which can be done more clearly using the kinematic approximation. In the kinematic approximation the reflectivity profile is given by [5,6]... 

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