Terminal planes

We refer to the planes which close the box parallel to the boundary as the terminating planes. Between the terminating planes, the box includes one or more translational unit cells of the system parallel to the boundary. We therefore have to assume that the system is periodic parallel to the boundary, but this assumption is anyway necessary for practical schemes of calculation. The terminating planes should be far enough away from the interface to be locally in the undistorted bulk material. 

The second difficulty is where to locate the terminating planes. In the case of an ordered alloy the microscopic position of these planes with respect to the unit cell might make a difference to the result, and it has to be correctly chosen, as discussed in detail below. 

Two problems in the calculation of interfacial energies in crystals have been addressed, namely how to obtain the chemical potentials and how to define the terminating planes on the atomic scale for the purposes of calculating the excess free energy. 

Exactly the same procedure can be followed to define the free energy of a dislocation core. It should be surrounded by a box, the terminating planes of which can be dealt with exactly as above. Special attention has to be given to the atoms at the comers of the box, but this presents no particular problems their weights are simply a oroduct of the weights generated by the planar terminations which they share. 

In alkaline earth metal oxides, the (100) surface termination plane, which exposes equal numbers of anions and cations, is prevalent and, as illustrated in Figure 21.2, it can be anticipated that an entire family of different co-ordination sites, of different basicity, can be exhibited. Furthermore, it would be expected that this would lead to a dependence upon crystallite morphology and/or particle size. 

Figure 11.24. A [110] lattice-fringe image of a diamond grain that is tilted by 4° with respect to Si. Such small angular deviations arc compensated for by unequal numbers of iil and 111 terminating planes [310].

Probing on the separation of incident and reflected waves, a concern still remains two field quantities must be determined on the source plane and updated interactively. For waveguide discontinuities, the regularly implemented scheme of the third case places the source and the near-end terminal plane at the same position, inserts the excitation, and then applies the ABC to the source plane after the pulse has been fully propagated. Nonetheless, before any truncation process is allowed to initiate, DC distortions are induced near the incident waveform by the electric and magnetic boundary conditions. This is another reason why usual techniques cannot be located very close to the discontinuity. An efficient way to alleviate the prior weakness is to... 

Line defects are dislocations around which some of the atoms of the crystal lattice are misaligned. There are two types of dislocations the edge dislocation and the screw dislocation. Dislocations are caused by the termination of a plane of atoms in the middle of a crystal. In such a case, the surrounding planes are not straight, but instead bend around the edge of the terminating plane so that the crystal structure is perfectly ordered on either side. 

Fig. 42. The stacking layers used to describe oxide surfaces can be of three types, (a) T3 e I, with a neutral stacking layer. The 100 planes of MgO are shown as an example. (b) T5rpe II, with a choice of termination planes, only some of which yield nonpolar surfaces. The 0001 planes of AI2O3 are used to illustrate this A1 termination between the A1 layers forms a nondipolar stacking layer (black rectangle) but termination to give an oxygen-terminated surface results in a dipolar stacking layer (grey rectangle), (c)...

A form termed the right rkomMe prism is fenned by the oonr-bination of the three terminal planes, and is found in nature as Anhydrite, CaSO, By oontinning tiie vratioal axis of the rhombic octohedra to the distance oe, we get tiie vertical prisms as follows ... 

Figure 7.37 (b) shows the anisotropy growth of Fe (211) reflection. The anisotropy of the elemental iron can be seen from the deviations of the measured intensities of the Fe (211) reflection from the theoretical intensity calculated from the intensity of the Fe (110) base reflection. A similar difference was also observed for the Fe (200) reflection. These observations which were made with several batches of catalyst show that the crystallites of the catalyst are not isotropic and that an average deviation from e.g. iron octahedra which are the normal structme of low-temperature iron develops during activation. The small munber of reflections observed and the absence of the (111) peak preclude any meaningful prediction of the main dominating crystal termination plane. It may well be that several nonequilibrium orientations do coexist with the smn of aU anisotropies being observed in the data of Fig. 7.37. The detection of this integral anisotropy is a strong hint towards the texture of the iron being a major difference between normal (isotropic) iron and ammonia iron .

The absence of structurdly characterized dimeric lanthanide alkoxides [9] prompted us to determine the crystal-structure of 3 by single-crystal X-ray diffraction [10]. As Figure 1 indicates, 3 is a dimer with a pseudotetrahedral geometry about each Ce atom. A crystallographic inversion center constrains the Ce and the bridging O atoms to a single plane and a dihedral angle of 84.1° relates this plane to the 0(terminal)-Ce-0(terminal) plane. 

The characteristic feature of crystal surfaces is that the atoms on the surface assume positions different from those on a bulk-terminated plane. The differences can be small, which is referred to as surface relaxation , or large, producing a structure that differs drastically from what is encountered in the bulk, which is referred to as surface reconstruction . The changes in atomic positions can be such that the periodicity of the surface differs from the periodicity of atoms on a bulk-terminated plane of the same orientation. The standard way to describe the new periodicity of the surface is by multiples of the lattice vectors of the corresponding bulk-terminated plane. For instance, a i x 2 reconstruction on the (klm) plane is one in which the lattice vectors on the plane are and 2 times the primitive lattice vectors of the ideal, uiueconstructed, bulk-terminated (klm) plane. Simple integer multiples of the primitive lattice vectors in the bulk-terminated plane often are not adequate to describe the reconstruction. It is possible, for example, to have reconstructions of the form x /n2, or c( i x 2), where c stands for centered . 

Figure 11.7. The missing row reconstruction in close-packed surfaces, illustrated in a 2D example. Left the unreconstructed, bulk-terminated plane with surface atoms two-fold coordinated. The horizontal dashed line denotes the surface plane (average position of surface atoms) with surface unit cell vector ai. Right the reconstructed surface with every second atom missing and the remaining atoms having either two-fold or three-fold coordination. The horizontal dashed line denotes the surface plane with surface unit cell vector 2ai, while the inclined one indicates a plane of close-packed atoms. The labels of surface normal vectors denote the corresponding surfaces in the 3D FCC structure.

Figure 11.9. Stmcture of the GaAs( 110) surface Left top view of the bulk-terminated (110) plane, containing equal numbers of Ga and As atoms. Right side views of the surface before relaxation (below) and after relaxation (above). The surface unit cell remains unchanged after relaxation, the same as the unit cell in the bulk-terminated plane.

Now, if the adatom is of a chemical type that prefers to form exactly three covalent bonds like the group-III elements Al, Ga and In, then placing it at one of the two stable positions will result in a chemically passive and stable structure. This will be the case if the entire surface is covered by adatoms, which corresponds to a reconstruction with a unit cell containing one adatom and three surface atoms. The resulting periodicity is designated ( /3 x V3), shown in Fig. 11.12, since the new surface lattice vectors are larger by a factor of V3 compared with the original lattice vectors of the bulk-terminated plane. The new lattice vectors are also rotated... 

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