Crystal structure Miller indices

A.R. Siedle, 3M Central Research Laboratory If one of the extended structures described by Professor McCarley were truncated through a low Miller index plane, can one, following the appproach of Solomon, predict what metal orbitals would protrude from the surface so generated Have ultraviolet photoelectron spectra been obtained on single crystals of any of these materials ... 

If the vector defining the surface normal requires a negative sign, that component of the Miller index is denoted with an overbar. Using this notation, the surface defined by looking up at the bottom face of the cube in Fig. 4.4 is the (001) surface. You should confirm that the other four faces of the cube in Fig. 4.4 are the (100), (010), (100), and (010) surfaces. For many simple materials it is not necessary to distinguish between these six surfaces, which all have the same structure because of symmetry of the bulk crystal. ... 

This surface is therefore the (111) surface. This surface is an important one because it has the highest possible density of atoms in the surface layer of any possible Miller index surface of an fee material. Surfaces with the highest surface atom densities for a particular crystal structure are typically the most stable, and thus they play an important role in real crystals at equilibrium. This qualitative argument indicates that on a real polycrystal of Cu, the Cu(l 11) surface should represent a significant fraction of the crystal s surface total area. 

Figure 4.7 shows top-down views of the fee (001), (111), and (110) surfaces. These views highlight the different symmetry of each surface. The (001) surface has fourfold symmetry, the (111) surface has threefold symmetry, and the (110) has twofold symmetry. These three fee surfaces are all atomically flat in the sense that on each surface every atom on the surface has the same coordination and the same coordinate relative to the surface normal. Collectively, they are referred to as the low-index surfaces of fee materials. Other crystal structures also have low-index surfaces, but they can have different Miller indices than for the fee structure. For bcc materials, for example, the surface with the highest density of surface atoms is the (110) surface. 

The metal substrates used in the LEED experiments have either face centered cubic (fee), body centered cubic (bcc) or hexagonal closed packed (hep) crystal structures. For the cubic metals the (111), (100) and (110) planes are the low Miller index surfaces and they have threefold, fourfold and twofold rotational symmetry, respectively. 

In the last few years LEED studies of high Miller index or stepped surfaces have become more frequent. Almost all of these studies have been on fee metals, where the atomic structure of these surfaces consists of periodic arrays of terraces and steps. A nomenclature which is more descriptive of the actual surface configuration has been developed for these surfaces, as described in Section III. In Table 5.5 the stepped surface nomenclature for several high Miller index surfaces of fee crystals has been tabulated. In Fig. 5.1 the location of these high Miller index surfaces are shown on the... 

By the use of mainly LEED and lately ion scattering techniques the location of many atomic adsorbates, their bond distances and bond angles from their nearest neighbor atoms have been determined. The substrates utilized in these investigations were low Miller Index surfaces of fee, hep and bcc metals in most cases, and low Miller Index surfaces of semiconductors that crystallize in the diamond, zincblende and wurtzite structures in some cases that could be cleaned and ordered with good reproducibility. 

II. The Atomic Structure of Surfaces. Structures of Low and High Miller Index Crystal Surfaces. 5... 

Structures of Low and High Miller Index Crystal Surfaces... 

Platinum crystal surfaces that were prepared in the zones indicated by the arrows at the sides of the triangle are thermally unstable. These surfaces, on heating, will rearrange to yield the two surfaces that appear at the end of the arrows. There is reason to believe that the thermal stability exhibited by various low and high Miller index platinum surfaces is the same for other fee metals. There are, of course, differences expected for surfaces of bcc solids or for surfaces of solids with other crystal structures. 

One of the most exciting observations of LEED studies of adsorbed monolayers on low Miller index crystal surfaces is the predominance of ordering within these layers (18). These studies have detected a large number of surface structures formed upon adsorption of different atoms and molecules on a variety of solid surfaces. Conditions range from low temperature, inert gas physisorption to the chemisorption of reactive diatomic gas molecules and hydrocarbons at room temperature and above. A listing of over 200 adsorbed surface structures, mostly of small molecules, adsorbed on low Miller index surfaces can be found in a recent review (/). 

The chemisorption of over 25 hydrocarbons has been studied by LEED on four different stepped-crystal faces of platinum (5), the Pt(S)-[9(l 11) x (100)], Pt(S)-[6(l 11) x (100)], Pt(S)-[7(lll) x (310)], and Pt(S)-[4(l 11 x (100)] structures. These surface structures are shown in Fig. 7. The chemisorption of hydrocarbons produces carbonaceous deposits with characteristics that depend on the substrate structure, the type of hydrocarbon chemisorbed, the rate of adsorption, and the surface temperature. Thus, in contrast with the chemisorption behavior on low Miller index surfaces, breaking of C-H and C-C bonds can readily take place at stepped surfaces of platinum even at 300 K and at low adsorbate pressures (10 9-10-6 Torr). Hydrocarbons on the [9(100) x (100)] and [6(111) x (100)] crystal faces form mostly ordered, partially dehydrogenated carbonaceous deposits, while disordered carbonaceous layers are formed on the [7(111) x (310)] surface, which has a high concentration of kinks in the steps. The distinctly different chemisorption characteristics of these stepped-platinum surfaces can be explained by... 

The chemisorption of acetylene, ethylene, benzene, and cyclohexane were also studied on the Ir(lll) and stepped Ir[6(111) x (100)] crystal surfaces (30). Chemisorption characteristics of the Ir(lll) and Pt(lll) surface are markedly different. Also, the chemisorption characteristics of the low Miller index Ir(l 11) surface and the stepped Ir[6(l 11) x (100)] surface are markedly different for each of the molecules studied. The hydrocarbon molecules form only poorly ordered surface structures on either the Ir(l 11) or stepped iridium surfaces. Acetylene and ethylene (C2H2 and C2H4) form surface structures that are somewhat better ordered on the stepped iridium than on the low Miller index Ir(l 11) metal surface. The lack of ordering on iridium surfaces as compared to the excellent ordering characteristics of these molecules on... 

In a series of studies, the dehydrogenation and hydrogenolysis of cyclohexane was studied on both the stepped and low Miller index (111) crystal faces of platinum at a surface temperature of 300°C and a hydrogen to cyclohexane ratio of 20 1. While the rates on the stepped and low Miller index surfaces were not very different for the formation of benzene and hexane, the formation of cyclohexene was very structure sensitive, its rate being 100 times greater on the stepped surface than on the (111) crystal face. In Table III mrnnare the initial turnover numbers for the various reactions at low... 

A 3D crystal has its atoms arranged such that many different planes can be drawn through them. It is convenient to be able to describe these planes in a systematic way and Fig. 4 shows how this is done. It illustrates a 2D example, but the same principle applies to the third dimension. The crystal lattice can be defined in terms of vectors a and b, which have a defined length and angle between them (it is c in the third dimension). The box defined by a and b (and c for 3D) is known as the unit cell. The dashed lines in Fig. 4A show one set of lines that can be drawn through the 2D lattice (they would be planes in 3D). It can be seen that these lines chop a into 1 piece and b into 1 piece, so these are called the 11 lines. The lines in B, however, chop a into 2 pieces, but still chop b into 1 piece, so these are the 21 lines. If the lines are parallel to an axis as in C, then they do not chop that axis into any pieces so, in C, the lines chopping a into 1 piece and which are parallel to b are the 10 lines. This is a simple rule. The numbers that are generated are known as the Miller indices of the plane. Note that if the structure in Fig. 6.4 was a 3D crystal viewed down the c axis, the lines would be planes. In these cases, the third Miller index would be zero (i.e., the planes would be the 110 planes in A, the 210 planes in B, and the 100... 

One of the strengths of the approach employed here is that we have freedom over the choice of the transition metal for the tip and also the structure of the tip employed. Usually we use Pt and W tips and represent the tip apex as a pyramid-like cluster epitaxed on a substrate that is orientated along some low Miller index crystal plane (for example, 111, 110, or 100 surface planes). Generally we find that the structure of the tip has quite a big impact on the images obtained. Sharp tips, such as those constructed on Pt 100, Pt 111 or W 100 surfaces tend to yield higher resolution images than those obtained with more blunt tips (for example the 111 surface of (bcc) W as shown in Scheme II). However,... 

There are still a number of surface systems where the structure cannot be determined by LEED for theoretical and experimental reasons. High Miller-index surfaces, such as stepped or kinked surfaces, have layers separated by very small distances normal to the surface. The calculational tools normally used for LEED break down in this case, and no new approach has yet been developed to solve this problem. Experimental difficulties restrict the study of insulator surfaces, because of charging problems, and of molecular crystal surfaces, because of beam damage problems. 

In some structures, several planes and directions may be equivalent by symmetry. For example, this is the case for the (100), (010), (001), (100), (010), and (OOl) planes in the diamond cubic structure. Equivalent directions are denoted concisely as a group by using angular brackets. Thus, the (100) directions in a diamond cubic lattice include all of the directions that are perpendicular to the six planes noted above. The Miller index notation thus provides a concise designation for describing the surfaces of semiconductor crystals. 

Cr203 has the same bulk crystal structure as a-Al203, namely corundum. Of its several low Miller index surfaces only one, (0001), has been employed for adsorbate structural determinations so far. To overcome sample charging problems a thin film has been utilised for these studies, rather than a single crystal. The surface structure of this (0001) oriented thin film has been investigated by LEED-IV [112]. Simulations of the experimental data evidence a chromium terminated surface with large vertical interlayer relaxations, reaching down five or six layers. 

Fig. 3.2. Surface structure of platinum single crystal catalysts with the corresponding Miller index notations given in brackets. (Reproduced, with permission, from Ref. 14.)...

In this crystal lattice system, all surfaces with Miller indices, (hkl), satisfying the conditions h x k x 1 and h k l h are chiral [11]. Although such high Miller index surfaces have been studied for decades, it was not until recently that McFadden et al. specifically pointed out and demonstrated that their low synunetry structures render them chiral and, therefore, that they might have enantiospecific interactions with chiral adsorbates [12]. There has been a growing interest in the enantiospecific properties of naturally chiral metal surfaces and in the possibility of using such surfaces for enantioselective chemical processes. 

These difficulties have stimulated the development of defined model catalysts better suited for fundamental studies (Fig. 15.2). Single crystals are the most well-defined model systems, and studies of their structure and interaction with gas molecules have explained the elementary steps of catalytic reactions, including surface relaxation/reconstruction, adsorbate bonding, structure sensitivity, defect reactivity, surface dynamics, etc. [2, 5-7]. Single crystals were also modified by overlayers of oxides ( inverse catalysts ) [8], metals, alkali, and carbon (Fig. 15.2). However, macroscopic (cm size) single crystals cannot mimic catalyst properties that are related to nanosized metal particles. The structural difference between a single-crystal surface and supported metal nanoparticles ( 1-10 nm in diameter) is typically referred to as a materials gap. Provided that nanoparticles exhibit only low Miller index facets (such as the cuboctahedral particles in Fig. 15.1 and 15.2), and assuming that the support material is inert, one could assume that the catalytic properties of a... 

Similar to structure solution from first principles, the ab initio indexing implies that no prior knowledge about symmetry and approximate unit cell dimensions of the crystal lattice exists. Indexing from first principles, therefore, usually means that Miller indices are assigned based strictly on the relationships between the observed Bragg angles. 

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