Layer lines reciprocal lattice

In direct space successive layers are sheared homogeneously along cylindrical surfaces, one relative to the adjacent one, as a consequence of the circumference increase for successive layers. In diffraction space the locus of the corresponding reciprocal lattice node is generated by a point on a straight line which is rolling without sliding on a circle in a plane perpendicular to the tube axis. Such a locus... 

If unit cell is orthogonal there are layers lines on oblique texture electron diffraction pattern. These lines occur when certain reciprocal lattice planes lie perpendicular to the texture axis. In this case period c may be more accurately determined by measuring the minor semi-axis R of any ellipse (in the presence of layer lines it is measured directly, since there is a zero line with /=0) and H of any reflection on that ellipse (preferably with a large l) ... 

Figure 11. (a) formation of layer lines in the reciprocal lattice of a texture for a orthogonal unit cell, (b) the doubling of number of circular scattering regions in the reciprocal lattice of a texture and therefore the number of reflections on an ellipse of a pattern for a non-orthogonal unit cell, (c) measurement of a values of 2r and 2 D on a texture pattern. 

Consider first the equatorial reflections. For a crystal rotated round its c axis, the equatorial reflections are those of hkO planes. To assign correct indices it is only necessary to make a diagram (Fig. 87 b) of the zero level of the reciprocal lattice (the dimensions being already known from layer-line spacings on other photographs), and to measure with a ruler the distance f of each point from the origin it is then obvious which reciprocal lattice point corresponds to each spot on the rotation diagram. 

As for the upper and lower layer lines of the rotation diagram, it is immediately obvious that the spots on them lie exactly above or below equatorial spots— the f values for spots on all layer lines are the same (except where certain spots are missing). The reason is that the 101 point of the reciprocal lattice is at the same distance from the axis of rotation as 100 (Fig. 80), and in general a point hkl is at the same... 

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. 

If a triclinic crystal is rotated round any axis of the real cell (Fig. 93), the photograph exhibits layer lines (since the various levels of the reciprocal lattice are normal to the axis of rotation), but not row lines, since none of the points on upper or lower levels are at the same distance from the axis of rotation as corresponding points on the zero level. The indices for points on the zero level are found in the same way as for photographs of monoclinic crystals rotated round the 6 axis for the zero level of a triclinic crystal rotated round c, a net with elements a, 6, and y is constructed (Fig. 94), and distances of points from the origin are measured. The other levels, projected on to the equator, are displaced with regard to the zero level in a direction which does not lie along an equatorial reciprocal axis the simplest way of measuring values is, as before, to use the zero level network,... 

Assuming that the equatorial reflections have been shown to fit a rectangular reciprocal lattice net, attention may be turned to the upper and lower layer lines. The values for all the spots are read off on Bernal s chart, and the reciprocal lattice rotation diagram is constructed from these values if the values for the upper and lower layer lines correspond with those of the equator—that is, row7 lines as well as layer lines are exhibited as in Fig. 80—then the unit cell must be orthorhombic. It should be noted that some spots may be missing from the equator, and it may be necessary to halve one or both of the reciprocal axes previously found to satisfy the equatorial reflections. The dimensions of the unit cell, and the indices of all the spots, follow immediately from the reciprocal lattice diagrams. 

This done, consider the other layer lines on the photograph. A reciprocal lattice rotation diagram is prepared as before from the and f values of all the spots. If row lines are exhibited, then the remaining axis of the reciprocal lattice is normal to the zero-level not, as in Fig. 89 in other words, the crystals are monoclinic with their b axes parallel to the fibre axis. It is again necessary to remember that one or both reciprocal axes of the zero-level net may have to he halved to account for all the points on other levels. 

A characteristic feature of these partially oxidized bis(oxalato)platinate salts of divalent cations is the coexistence of two modulations of the lattice over a wide temperature range (a) a one-dimensional modulation, as detected by the appearance of diffuse lines on X-ray films, perpendicular to the [Pt(C204)2] anion stacking direction and surrounding the even Bragg reflection layer lines of non-zero order (b) a three-dimensional modulation which gives rise to a complicated pattern of fine satellite spots in the neighbourhood of every reciprocal layer line. 

The relation of an individual image or diffraction pattern to the 3D structure factor set is important to understand. The 3D data for a single-layer crystal falls on a set of reciprocal lattice lines, where one line passes through each of the diffraction spots in the pattern for an untilted crystal such as shown in Fig. 2. The data from a single crystal falls on a central section of the 3D data, that is a plane which passes through the origin and whose orientation corresponds to the orientation of the crystal. Thus, a crystal tilted to any orientation contributes one data point to each lattice line within the resolution limit for that particular orientation. 

Fig. 3. (10 ) CTR or the a- Al203(0001)-(lxl) surface. Experimental (solid circles) and best-fit models for each possible termination single A1 layer (thick solid line), double A1 layer (dashed line) and oxygen terminated surfaces (dotted line). The logarithm of the structure factor is reported as a function of the out-of-plane momentum transfer in reciprocal lattice units of AI2O3.

Below Tp, the static ( locked ) CDW of the conduction electrons at 2 kp (or 4 kp), couple with the other atomic or molecular electrons in the lattice, cause a slight lattice distortion, and gives rise to extra X-ray reflections. Above Tp, these CDW are mobile, with no phase locking between excitations on nearby chains one sees X-ray diffuse reflections, similar to thermal diffuse scattering, which sharpen as T is lowered. Below Tp the static distortion produces new, usually weak, reflections between reciprocal-lattice layer lines. When the band filling is a rational fraction (1/4, 1/2, 2/3,1, etc.) then these reflections overlap with certain Bragg reflections of the background lattice, and are more difficult to detect. 

The Laue method involves a stationary crystal and polychromatic ( white ) X rays. In the other camera methods, monochromatic radiation is used. In these cases the crystal may be oscillated over a small angular range (oscillation method) or rotated 360° about an axis (rotation method). The layer lines so formed may be selected individually. In the Weissenberg method, the oscillation of the crystal is coupled with a movement of the photographic film. The Buerger precession method, by a more complex motion of the instrument, produces an undistorted and magnified picture of the reciprocal lattice. 

As pointed out in section 2.5.1, where the reciprocal lattice is defined, it is possible to determine the shape and dimensions of any real lattice from the corresponding information about the reciprocal lattice. The spots on the /th layer line correspond to reciprocal-lattice points lying on the reciprocal-lattice plane containing all points hkl) for different h and k. The fibre diagram is in fact similar to what would be obtained if the following imaginary experiment were performed. Place the origin of a suitably scaled version of the reciprocal-lattice at the point where the incident X-ray beam would strike the film, with the normal to the planes of constant / parallel to the fibre axis and therefore to the meridian of the fibre pattern. Rotate the reciprocal lattice around the meridian and mark a point on the film every time a lattice point intersects it. To understand the difference between this pattern and a real fibre pattern it is useful to consider fig. 4.13. 

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