Plane spacing

Lattice plane spacing d Mean ionic activity coeffi- y ... 

Figure 5.7 Schematic edge dislocation after Peierls. Top part of crystal, T and bottom part B, are joined between planes a and P across a glide plane with an extra half-plane of atoms ending at c. The displacement along the glide plane is b, and the glide plane spacing is a.

Figure 5.8 Projection of the diamond structure so the (111) glide planes (AB) are perpendicular to the plane of the figure. Then the covalent bonds connecting the atoms in planes (AB) and (A ) are perpendicular to the (111) planes.The glide plane spacing, a of the figure corresponds to the bond length AA. The Burgers displacement, b corresponds to the atomic spacing along A or A. ...

Note 3 Depending on the plane space-group symmetries, three rectangular mesophases are distinguished (See Fig. 15a-c). 

Note 3 The plane space-group symmetry of a Colob mesophase is Pi (see Fig. 15d). 

Plane space group symmetries in the projection of the International System are ... 

In the shear stmctures discussed in the preceding sections, smaller compositional variations can be accommodated by changing the CS plane spacings. This, in... 

Moriguchi et al. (43) noted a correlation between the change in the basal plane spacing with H2S exposure of CdSt, and the ionic radius of sulfide, and used this as evidence for CdS monolayer formation. Measurements of high lateral conductivity in metal ion fatty acid films exposed to H2S (20,23) and of photoelectric properties (21) have also been used to invoke the concept of continuous sheets of metal sulfide forming. 

Operation (7) The mother structure (for example, TiO2 (rutile)) is divided into blocks with the dimension of a = x (n is an integer), where is a shear plane with plane indices (hkl) and is the plane spacing of H. (Fig. 2.2(a)). 

In a similar way, in a crystal exhibiting a threefold screw axis 3X or 32, identical atoms are repeated on planes spaced one-third the length of the axis therefore the reflections from the plane normal to the screw axis would, by themselves, appear to indicate a repeat distance only one-third the true axial length. The first of these reflections w ould be the third-order reflection in reference to the true repeat distance, wrhile the second would be actually the sixth-order reflection. In other words, a threefold screwr axis 3X or 32 causes the absence of the first and second orders of reflection from the plane perpendicular to the screw axis, as... 

Thus all points having the same 1 index lie at the same distance from the horizontal plane x y, and thus lie on a plane parallel to the plane x y moreover, the distance of each such plane of points from the plane x y is proportional to l. Therefore successive sets of points for the successive values of the index l lie on a set of equidistant planes, spaced Ajc apart. 

Fio. 12. STM image of ReS2 basal plane (57). Atomic spacing corresponds to actual ReS2 basal plane spacing. 

X-ray diffraction allows the identification of the crystalline phase of nanoparticles and, combined with HRTEM, permits the extraction of particular lattice plane spacings, estimate the lattice parameters, the crystallite size and the apparent diameter of the nanoscale particles. 

The particle diameter D is related to the full width at half maximum A by the Debye-Scherrer equation D = 0.9 XIA cos0, where 20 is the diffraction angle and X is the X-ray wavelength. Table 27.1 lists the particle size and lattice plane spacing calculated using the strongest (h,k,l) peak for the Fe, W, Mo carbides, nitrides, oxynitrides and oxycarbides. It is important to note that the calculated particle size using the Debye-... 

In X-ray crystallography, 2-A model" means that analysis included reflections out to a distance in the reciprocal lattice of 1/(2 A) from the center of the diffraction pattern. This means that the model takes into account diffraction from sets of equivalent, parallel planes spaced as closely as 2 A in the unit cell. (Presumably, data farther out than the stated resolution was unobtainable or was too weak to be reliable.) Although the final 2-A map, viewed as an empty contour surface, may indeed not allow us to discern adjacent atoms, structural constraints on the model greatly increase the precision of atom positions. The main constraint is that we know we can fit the map with groups of atoms — amino-acid residues — having known connectivities, bond lengths, bond angles, and stereochemistry. 

Atoms at solid surfaces have missing neighbors on one side. Driven by this asymmetry the topmost atoms often assume a structure different from the bulk. They might form dimers or more complex structures to saturate dangling bonds. In the case of a surface relaxation the lateral or in-plane spacing of the surface atoms remains unchanged but the distance between the topmost atomic layers is altered. In metals for example, we often find a reduced distance for the first layer (Table 8.1). The reason is the presence of a dipole layer at the metal surface that results from the distortion of the electron wavefunctions at the surface. 

Fig. 8. Generation of the form of the helical diffraction pattern. (A) shows that a continuous helical wire can be considered as a convolution of one turn of the helix and a set of points (actually three-dimensional delta-functions) aligned along the helix axis and separated axially by the pitch P. (B) shows that a discontinuous helix (i.e., a helical array of subunits) can be thought of as a product of the continuous helix in (A) and a set of horizontal density planes spaced h apart, where h is the subunit axial translation as in Fig. 7. This discontinuous set of points can then be convoluted with an atom (or a more complicated motif) to give a helical polymer. (C)-(F) represent helical objects and their computed diffraction patterns. (C) is half a turn of a helical wire. Its transform is a cross of intensity (high intensity is shown as white). (D) A full turn gives a similar cross with some substructure. A continuous helical wire has the transform of a complete helical turn, multiplied by the transform of the array of points in the middle of (A), namely, a set of planes of intensity a distance n/P apart (see Fig. 7). This means that in the transform in (E) the helix cross in (D) is only seen on the intensity planes, which are n/P apart. (F) shows the effect of making the helix in (E) discontinuous. The broken helix cross in (E) is now convoluted with the transform of the set of planes in (B), which are h apart. This transform is a set of points along the meridian of the diffraction pattern and separated by m/h. The resulting transform in (F) is therefore a series of helix crosses as in (E) but placed with their centers at the positions m/h from the pattern center. (Transforms calculated using MusLabel or FIELIX.)...

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