Crystal plane

The structure and geometry of a surface play a dominant role with respect to its reactivity in adsorption and catalysis. It is therefore always necessary to specify which structure we are dealing with and, hence, it is important to have a notation that describes the various surfaces in a unique manner. A crystal surface is described by a vector normal to it, given by 

The surface is then indicated by the set h, k, and I between parentheses (hkl), often in combination with the metals, such as in Cu(lOO) or Pt(lll). For a (100) surface, the atoms reside in a plane parallel to the y-z-plane. Negative numbers are indicated by a bar over the value as in Cu(OlO). Note that permutations such as Cu(lOO), Cu(OlO), Cu(OOl), Cu(OlO), and Cu(lOO) all describe surfaces of the same structure as shown in Fig. 5.2. 

The three basal planes of cubic crystals are (100), (110), and (111). The respective cross sections are shown in Fig. 5.2. The positions of the surface atoms that appear by applying these cuts on the unit cells of Fig. 5.1 are shown in Fig. 5.3. 

An important parameter for surface reactivity is the density of atoms in the surface. The general rule of thumb is that the more open the surface, the more reactive it is. We return to this effect in much more detail in Chapter 6. Note that (110) is the most open basal plane of an fee crystal, whereas (111) exhibits the closest packing. For bcc crystals the order is the opposite, i.e. (Ill) is the most open and (110) the most packed. 

The difference between the fee and hep structure is best seen if one considers the sequence of dose-packed layers. For fee lattices this is the (111) plane (see Figs. 5.1 and 5.3), for hep lattices the (001) plane. The geometry of the atoms in these planes is exactly the same. Both lattices can now be built up by stacking dose-packed layers on top of each other. If one places the atoms of the third layer directly above those of 

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Bragg scattering Coherent elastic scattering of monochromatic neutrons by a set of crystal planes. 

Returning to the complete calculation, is then given by u multiplied by the number of atoms per unit area in the particular crystal plane. 

Actual crystal planes tend to be incomplete and imperfect in many ways. Nonequilibrium surface stresses may be relieved by surface imperfections such as overgrowths, incomplete planes, steps, and dislocations (see below) as illustrated in Fig. VII-5 [98, 99]. The distribution of such features depends on the past history of the material, including the presence of adsorbing impurities [100]. Finally, for sufficiently small crystals (1-10 nm in dimension), quantum-mechanical effects may alter various physical (e.g., optical) properties [101]. 

We noted in Section VII-2B that, given the set of surface tension values for various crystal planes, the Wulff theorem allowed the construction of fhe equilibrium or minimum firee energy shape. This concept may be applied in reverse small crystals will gradually take on their equilibrium shape upon annealing near their melting point and likewise, small air pockets in a crystal will form equilibrium-shaped voids. The latter phenomenon offers the possible advantage that adventitious contamination of the solid-air interface is less likely. 

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

Measuring the electron emission intensity from a particular atom as a function of V provides the work function for that atom its change in the presence of an adsorbate can also be measured. For example, the work function for the (100) plane of tungsten decreases from 4.71 to 4.21 V on adsorption of nitrogen. For more details, see Refs. 66 and 67 and Chapter XVII. Information about the surface tensions of various crystal planes can also be obtained by observing the development of facets in field ion microscopy [68]. 

Surface heterogeneity may merely be a reflection of different types of chemisorption and chemisorption sites, as in the examples of Figs. XVIII-9 and XVIII-10. The presence of various crystal planes, as in powders, leads to heterogeneous adsorption behavior the effect may vary with particle size, as in the case of O2 on Pd [107]. Heterogeneity may be deliberate many catalysts consist of combinations of active surfaces, such as bimetallic alloys. In this last case, the surface properties may be intermediate between those of the pure metals (but one component may be in surface excess as with any solution) or they may be distinctly different. In this last case, one speaks of various effects ensemble, dilution, ligand, and kinetic (see Ref. 108 for details). 

Certain materials, most notably semiconductors, can be mechanically cleaved along a low-mdex crystal plane in situ in a UFIV chamber to produce an ordered surface without contamination. This is done using a sharp blade to slice tire sample along its preferred cleavage direction. For example. Si cleaves along the (111) plane, while III-V semiconductors cleave along the (110) plane. Note that the atomic structure of a cleaved surface is not necessarily the same as that of the same crystal face following treatment by IBA. 

If the detection system is an electronic, area detector, the crystal may be mounted with a convenient crystal direction parallel to an axis about which it may be rotated under tlie control of a computer that also records the diffracted intensities. Because tlie orientation of the crystal is known at the time an x-ray photon or neutron is detected at a particular point on the detector, the indices of the crystal planes causing the diffraction are uniquely detemiined. If... 

Etch Profiles. The final profile of a wet etch can be strongly influenced by the crystalline orientation of the semiconductor sample. Many wet etches have different etch rates for various exposed crystal planes. In contrast, several etches are available for specific materials which show Httle dependence on the crystal plane, resulting in a nearly perfect isotropic profile. The different profiles that can be achieved in GaAs etching, as well as InP-based materials, have been discussed (130—132). Similar behavior can be expected for other crystalline semiconductors. It can be important to control the etch profile if a subsequent metallisation step has to pass over the etched step. For reflable metal step coverage it is desirable to have a sloped etched step or at worst a vertical profile. If the profile is re-entrant (concave) then it is possible to have a break in the metal film, causing an open defect. 

Fig. 5.5. Miller indices for identifying crystal planes, showing how the (131) plane and the (110) planes are defined. The lower part of the figure shows the family of 100 and of 110) planes.

For a given structure, the values of S at which in-phase scattering occurs can be plotted these values make up the reciprocal lattice. The separation of the diffraction maxima is inversely proportional to the separation of the scatterers. In one dimension, the reciprocal lattice is a series of planes, perpendicular to the line of scatterers, spaced 2Jl/ apart. In two dimensions, the lattice is a 2D array of infinite rods perpendicular to the 2D plane. The rod spacings are equal to 2Jl/(atomic row spacings). In three dimensions, the lattice is a 3D lattice of points whose separation is inversely related to the separation of crystal planes. 

Figure 2 View looking down on the real-space mesh (a) and the corresponding view of the reciprocal-space mesh (b) for a crystal plane with a nonrectangular lattice. The reciprocal-space mesh resembles the real-space mesh, but rotated 90°. Note that the magnitude of the reciprocal lattice vectors is inversely related to the spacing of atomic rows.

There is considerable evidence in the thermoset literature that the fracture energy decreases with increasing crosslink density, consistent with the intuitive result that crosslinking inhibits flow. In the limit of very high crosslink density, where for example we approach the structure of diamond, fracture can occur on a single crystal plane such that... 

Another special case of weak heterogeneity is found in the systems with stepped surfaces [97,142-145], shown schematically in Fig. 3. Assuming that each terrace has the lattice structure of the exposed crystal plane, the potential field experienced by the adsorbate atom changes periodically across the terrace but exhibits nonuniformities close to the terrace edges [146,147]. Thus, we have here another example of geometrically induced energetical heterogeneity. Adsorption on stepped surfaces has been studied experimentally [95,97,148] as well as with the help of both Monte Carlo [92-94,98,99,149-152] and molecular dynamics [153,154] computer simulation methods. 

Erist chen, n. Uttle crystal. Kristall-chloroform, n. chloroform of crystallization. -druse, /. crystal druse, crystal cluster. -ebene, /. crystal plane, crystal face, -ecke,/. (solid) crystal angle, kristallelektrisch. a. piezoelectric, kristallen, a. crystalline. 

Figure 1.53 shows diagrammatically various types of pits that can range from hemispherical with a polished surface, in which crystallographic etching has been completely suppressed, to crystallographic pits whose sides are composed of the crystal planes that corrode at the slowest rate. Pits formed on Ni during anodic polarisation in an acetic acid-acetate buffer of pH 4-6 are shown in Fig. 1.54. 

The test operates at a potential above 2-00 V (vs. S.H.E.), and the ditch structure obtained with sensitised alloys must be due, therefore, to the high rate of dissolution of the sensitised areas as compared with the matrix. The step structure is due to the different rates of dissolution of different crystal planes, and the dual structure is obtained when chromium carbides are present at grain boundaries, but not as a continuous network. 

An account of the use of Miller indices to describe crystal planes and lattice directions is beyond the sco[>e of this article a very adequate treatment of this topic is, however, given in Reference 1. 

Fig. 4-13. Bent arid ground crystal spectrograph. (Johansson). The x-rays make the same angle with the crystal planes as in Fig. 4 12, but the grinding has brought these planes to lie on the focal circle.

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