Surface relaxation, lattice

FIGURE 27.8 Specular reflectivity for a clean Au(lOO) surface in vacuum at 310 K ( ). The solid line is calculated for an ideally terminated lattice. The dashed line is a fit to the data with a reconstmcted surface with a 25% increase in the surface density combined with a surface relaxation that increases the space between the top and next layers by 19%. In addition, the data indicate that the top layer is buckled or cormgated with a buckling amplitude of 20%. (From Gibbs et al., 1988, with permission from the American Physical Society.)... 

The T) and T2 dependence is described by Eqs. (3.4.3) and (3.4.4) [34] where Qi and q2 are spin-lattice and spin-spin surface relaxivity constants, and S/ Vis the surface-to-volume ratio of the pore. These equations provide the basis of a methodology for crack detection in cement paste specimens [13]. 

NMR relaxation of liquids such as water in porous solids has been studied extensively. In the fast exchange regime, the spin-lattice relaxation rate of water in pores is known to increase due to interactions with the solid matrix (so-called surface relaxation ). In this case, T) can be described by Eq. (3.5.6) ... 

The surface condition of a silicon crystal depends on the way the surface was prepared. Only a silicon crystal that is cleaved in ultra high vacuum (UHV) exhibits a surface free of other elements. However, on an atomistic scale this surface does not look like the surface of a diamond lattice as we might expect from macroscopic models. If such simple surfaces existed, each surface silicon atom would carry one or two free bonds. This high density of free bonds corresponds to a high surface energy and the surface relaxes to a thermodynamically more favorable state. Therefore, the surface of a real silicon crystal is either free of other elements but reconstructed, or a perfect crystal plane but passivated with other elements. The first case can be studied for silicon crystals cleaved in UHV [Sc4], while unreconstructed silicon (100) [Pi2, Ar5, Th9] or (111) [Hi9, Ha2, Bi5] surfaces have so far only been reported for a termination of surface bonds by hydrogen. 

The clean siuface of solids sustains not only surface relaxation but also surface reconstruction in which the displacement of surface atoms produces a two-dimensional superlattice overlapped with, but different from, the interior lattice structure. While the lattice planes in crystals are conventionally expressed in terms of Miller indices (e.g. (100) and (110) for low index planes in the face centered cubic lattice), but for the surface of solid crystals, we use an index of the form (1 X 1) to describe a two-dimensional surface lattice which is exactly the same as the interior lattice. An index (5 x 20) is used to express a surface plane in which a surface atom exactly overlaps an interior lattice atom at every five atomic distances in the x direction and at twenty atomic distances in the y direction. 

The calculation results indicate that a plane wave cutoff energy of 280 eV and Monkhorst-Paek k-point sampling density of 4 x 4 x 4 are sufficient for the lattice constant and total energy to converge to within 0.0005 A and lO" eV respectively. For surface relaxation, a plane wave cutoff of 280 eV and a 4 x 4 x 1 k-point mesh are sufficient to converge the surface geometry to within 0.001 A and relaxed surface energies to within 0.001 Vrc. ... 

In dynamic ETEM studies, to determine the nature of the high temperature CS defects formed due to the anion loss of catalysts at the operating temperature, the important g b criteria for analysing dislocation displacement vectors are used as outlined in chapter 2. (HRTEM lattice images under careful conditions may also be used.) They show that the defects are invisible in the = 002 reflection suggesting that b is normal to the dislocation line. Further sample tilting in the ETEM to analyse their habit plane suggests the displacement vector b = di aj2, b/1, 0) and the defects are in the (120) planes (as determined in vacuum studies by Bursill (1969) and in dynamic catalysis smdies by Gai (1981)). In simulations of CS defect contrast, surface relaxation effects and isotropic elasticity theory of dislocations (Friedel 1964) are incorporated (Gai 1981). 

Crystalline surfaces can be classified using the five two-dimensional Bravais lattices and a basis. Depending on the surfaces structure, the basis may include more than just the first surface layer. The substrate structure of a surface is given by the bulk structure of the material and the cutting plane. The surface structure may differ from the substrate structure due to surface relaxation or surface reconstruction. Adsorbates often form superlattices on top of the surface lattice. 

The structures of the three families of faces, terminating the polyhedra of well-sintered samples, as obtained by simply cutting the crystal lattices along the (0112), (2116), and (1120) planes taking into account surface relaxation have been discussed extensively (22) and are summarized only briefly here (Fig. 24). 

Early studies involving NMR include the work by Hanus and Gill is [6] in which spin-lattice relaxation decay constants were studied as a function of available surface area of colloidal silica suspended in water. Senturia and Robinson [7] and Loren and Robinson [8] used NMR to qualitatively correlate mean pore sizes and observed spin-lattice relaxation times. Schmidt, et. al. [9] have qualitatively measured pore size distributions in sandstones by assuming the value of the surface relaxation time. Brown, et. al. [10] obtained pore size distributions for silica, alumina, and sandstone samples by shifting the T, distribution until the best match was obtained between distributions obtained from porosimetry and NMR. More recently, low field (20 MHz) NMR spin-lattice relaxation measurements were successfully demonstrated by Gallegos and coworkers [11] as a method for quantitatively determining pore size distributions using porous media for which the "actual" pore size distribution is known apriori. Davis and co-workers have modified this approach to rapidly determine specific surface areas [12] of powders and porous solids. 

Recent investigations of surface properties of materials have revealed that many covalent (e.g. Si), ionic (e.g. LiF) and metallic (e.g. Pt) materials undergo crystallographic changes at the surface. The electronic structure near the surface is correspondingly modified (66,67) relative to that of the bulk. Ionic compounds with NaCl-structure have long been known to undergo a so-called surface relaxation 68, 69). This is a differential contraction (relaxation) of the surface cation (anion) lattice constant. 

In some sense this is similar to a uniaxial stress applied to the bulk crystal. For the semiconducting SmS (100) surface such surface relaxation, if large enough, is expected to have drastic influence on the electronic structure of the Sm ions, because the semiconductor-metal transition occurs when the bulk lattice constant has been... 

The result of such calculations show that, on a clean semiconductor, surface atomic sites in equilibrium always differ substantially from those of a semi-infinite lattice and there is an inward force on these surface atoms, since the presence of a dangling bond on a surface atom strengthens the back bonds to atoms in the second layer. This means that the back bonds assume some double bond character (i.e. the bond order becomes greater than unity). The consequent change in bond length leads to so-called surface relaxation (see Sect. 3.3). 

Fig. 10. Si(l 11 surface models after Levine et al. [141]. (a) Ideally terminated lattice (b) relaxation of surface atoms (c) relaxation of sub-surface atoms (d) periodic ripple (shown greatly exaggerated in amplitude) formed from residual stress in surface layers.

Figure 6. (A) A terminated crystal having surface lattice spacings, a and b, and layer spacing c along the surface normal direction. (B) The reciprocal-space structure for the structure in (A) note that every Bragg peak is intersected by a CTR. (C) Side view of the surface in (A) whose surface symmetry has been modified by adsorption resulting in a doubling of the unit cell dimension, 2a, and by a subsequent lateral surface relaxation in the outermost substrate layer. (D). A reciprocal space schematic of the surface in (C). The doubling of the surface unit cell results in new surface rods at Qx = n a, 3 / ...(shown as vertical dashed lines) that do not intersect any bulk Bragg peaks.

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