Unit Cell Notations

In many cases ordering is no longer observable in the presence of steps. Ordered carbonaceous layers form on the Ir(l 11) crystal face, for example, while ordering is absent on the stepped iridium surface. Ordering is absent on stepped Pt surfaces for most molecules that would order on the low Miller-Index (111) or (100) surfaces. 

In some cases, the step sites have different chemistry, i.e., they break chemical bonds, thereby producing new chemical species on the surface. This happens for example during NO adsorption on a stepped platinum surface l In this circumstance the step effect on ordering is through the new types of chemistry introduced by the presence of steps. Hydrocarbons for example dissociate readily at stepped surfaces of platinum or nickel while this occurs much more slowly on the low Miller-Index surfaces in the absence of a large concentration of steps As a result ordered hydrocarbon surface structures cannot be formed on the stepped surfaces of these metals while they can be produced on the low Miller-Index surfaces. 

In the majority of cases where adsorbates form ordered surface structures, the unit cells of those structures are larger than the unit cell of the substrate the surface lattice is then called a super lattice. The surface unit cell is the basic quantity in the description of the ordering of surfaces. It is necessary therefore to have a notation that allows the unique characterization of superlattices relative to the substrate lattice. 

Two common notations are used to relate superlattices to substrate lattices, one of these notations being a simplification of the other for simple cases. Let the substrate 

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In addition, the surface unit cell may be rotated with respect to the bulk cell. Such a rotated unit cell is notated as... 

These simple examples serve to show that instinctive ideas about symmetry are not going to get us very far. We must put symmetry classification on a much firmer footing if it is to be useful. In order to do this we need to define only five types of elements of symmetry - and one of these is almost trivial. In discussing these we refer only to the free molecule, realized in the gas phase at low pressure, and not, for example, to crystals which have additional elements of symmetry relating the positions of different molecules within the unit cell. We shall use, therefore, the Schdnflies notation rather than the Hermann-Mauguin notation favoured in crystallography. 

Fig. 2. Structures for the solid (a) fee Cco, (b) fee MCco, (c) fee M2C60 (d) fee MsCeo, (e) hypothetical bee Ceo, (0 bet M4C60, and two structures for MeCeo (g) bee MeCeo for (M= K, Rb, Cs), and (h) fee MeCeo which is appropriate for M = Na, using the notation of Ref [42]. The notation fee, bee, and bet refer, respectively, to face centered cubic, body centered cubic, and body centered tetragonal structures. The large spheres denote Ceo molecules and the small spheres denote alkali metal ions. For fee M3C60, which has four Ceo molecules per cubic unit cell, the M atoms can either be on octahedral or tetrahedral symmetry sites. Undoped solid Ceo also exhibits the fee crystal structure, but in this case all tetrahedral and octahedral sites are unoccupied. For (g) bcc MeCeo all the M atoms are on distorted tetrahedral sites. For (f) bet M4Ceo, the dopant is also found on distorted tetrahedral sites. For (c) pertaining to small alkali metal ions such as Na, only the tetrahedral sites are occupied. For (h) we see that four Na ions can occupy an octahedral site of this fee lattice.

The Wood notation, as this way of describing surface structures is called, is adequate for simple geometries. However, for more complicated structures it fails, and one uses a 2x2 matrix which expresses how the vectors al and a2 of the substrate unit cell transform into those of the overlayer. 

Vector notation is being used here because this is the easiest way to define the unit-cell. The reason for using both unit lattice vectors and translation vectors lies in the fact that we can now specify unit-cell parameters in terms of a, b, and c (which are the intercepts of the translation vectors on the lattice). These cell parameters are very useful since they specify the actual length eind size of the unit cell, usually in A., as we shall see. 

A special notation is used to describe surface reconstructions and surface overlayers and is described in books on surface crystallography (Clarke, 1985). The lattice vectors a and b of an overlayer are described in terms of the substrate lattice vectors a and b. If the lengths la I = mlal and Ib l = nibl, the overlayer is described as mXn. Thus, a commensurate layer in register with the underlying atoms is described as 1 X 1. The notation gives the dimension of the two-dimensional unit cell in terms of the dimensions of an ideally truncated surface unit cell. 

In the majority of cases where adsorbates form ordered structures, the unit cells of these structures are longer than that of the substrate they are referred to as superlattices. Two notations are used to describe the superlattice, the Wood notation and a matrix notation.18 Some examples of overlayer structures at an fcc(llO) surface are as follows ... 

Figure 6.10 illustrates LEED patterns of the clean Rh(lll) surface, and the surface after adsorption of 0.25 monolayers (ML) of NH3 [22]. The latter forms the primitive (2x2) overlayer structure (see Appendix I for the Wood notation). In the (2x2) overlayer, a new unit cell exists on the surface with twice the dimensions of the substrate unit cell. Hence the reciprocal unit cell of the adsorbate has half the size of that of the substrate and the LEED pattern shows four times as many spots. 

Adsorbates may form ordered overlayers, which can have their own periodicity. The adsorbate structure is given with respect to that of the substrate metal. For simple arrangements the Wood notation is used some examples are given in Fig. A.3. The notation Pt (110) - c(2x2) O means that oxygen atoms form an ordered overlayer with a unit cell that has twice the dimensions of the Pt (110) unit cell, and an additional O in the middle. Note that this abbreviation does not specify where the O is with respect to the Pt atoms. It may be on top of the Pt atoms but also in bridged or fourfold sites, or in principle anywhere as long as the periodic... 

We now introduce a Fourier transform procedure analogous to that employed in the solution theory, s 62 For the purposes of the present section a more detailed specification of defect positions than that so far employed must be introduced. Thus, defects i and j are in unit cells l and m respectively, the origins of the unit cells being specified by vectors R and Rm relative to the origin of the space lattice. The vectors from the origin of the unit cell to the defects i and j, which occupy positions number x and y within the cell, will be denoted X 0 and X for example, the sodium chloride lattice is built from a unit cell containing one cation site (0, 0, 0) and one anion site (a/2, 0, 0), and the translation group is that of the face-centred-cubic lattice. However, if we wish to specify the interstitial sites of the lattice, e.g. for a discussion of Frenkel disorder, then we must add two interstitial sites to the basis at (a/4, a/4, a]4) and (3a/4, a/4, a/4). (Note that there are twice as many interstitial sites as anion-cation pairs but that all interstitial sites have an identical environment.) In our present notation the distance between defects i and j is... 

Perhaps the most well known of the lithium intercalation compounds is Li jTiSj. Both Li (for x < 1) and Ti are octahedrally coordinated by S (Fig. 7.1) in the ABC notation, the structure is AbC(b)AbC, where the letter in parentheses denote lithium atoms. This structure is also called the IT form, because of its trigonal (T) symmetry and the single layer per unit cell. The electrochemical behaviour of Li in TiSj is described below in connection with staging. 

Fig. 4. Illustration of the idealized structures for type I through V of the zeolite-like 3D frameworks. Notations for atoms are the same to those in Fig. 3. The chains of solid circles and squares are at c = 0.5 if the c-axis is taken vertical to the sheet those of open ones are at c = 0 or 1. A selection of the unit cell, containing net two layers, is outlined by thin lines for each structure.

Fig. 1. The a notation indicates that the Tantalum atoms are aligned along the crystallographic c-axis. R indicates rhombohedral symmetry. The 3R phase has three trigonal prismatic slabs per unit cell, whereas the 6R phase has six slabs that are alternately trigonal prismatic and octahedral. F. Jellinek, J. Less-Common Met., 4, 9-15 (1962) F. J. DiSalvo et al., J. Phys. Chem. Solids, 34, 1357 (1973).

These chiral domains of tartaric acid are found to be formed by its bitartrate form. They follow a 2-dimensional ordered structure. In the case of the (R,K)-isomer the supramolecular assembly can be described by the following matrix notation, which defines the unit cell of the adlayer unambiguously in terms of the unit cell of the substrate ... 

Using the accepted notation, the S, site (16 per unit cell) is located at the... 

This is the matrix notation, in which the matrix M uniquely characterizes the relationship between the unit cells (note that the concept of unit cell is not unique — different unit cells can describe the same lattice — and so different matrices M can characterize the relationship between two given lattices). 

A non-matrix notation, called Wood notation can be used when the angles between the pairs of basis vectors are the same for the substrate and the superlattice, i.e., when the angle between and is the same as the angle between bj and b2-Then the unit cell relationship is given by, in general,... 

Discuss what is meant by a Bravais lattice and the Pearson notation. Sketch the unit cells for fee, bcc, hep and diamond giving the corresponding Pearson notation. 

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

Fig, 2.20 Ideal structure of M-NbjOj [4 x 4], belonging to the type of block structure shown in Fig. 2.19(c). The unit cell is outlined. For the notation, see text. 

Figure 5a shows the diffraction pattern associated with the clean (100) platinum surface. There are extra diffraction features in addition to those expected for this surface structure from the X-ray unit cell. This surface exhibits a so-called (5x1) surface structure (8). There are two perpendicular domains of this structure and there are 3, , f, and f order spots between the (00) and (10) diffraction beams. The surface structure is not quite as simple as the shorthand notation indicates, as shown by the splitting of the fractional order beams. The surface structure appears to be stable at all temperatures... 

SCHEME 22. MO-Schemee), PES data (eV), and structural parameters of the Group 14-decamethyl-metallocenes X-ray 6gED two independent molecules dtwo conformers in the unit cell in orbital notation for C2v symmetry... 

The determination of the eigenvalues wy(q)2 may be simplified by an orthogonal transformation to symmetry coordinates, which are linear combinations of the Cartesian displacements of the atoms which represent the actual displacements of the atoms in the unit cell. Simultaneously, the eigenvectors undergo the same orthogonal transformation (see Section 9.4, especially eqs. (9.4.4) and (9.4.6)). In matrix notation,... 

Let us start with the simple case of an ideal crystal with one atom per unit cell that is cut along a plane, and assume that the surface does not change. The resulting surface structure can then be described by specifying the bulk crystal structure and the relative orientation of the cutting plane. This ideal surface structure is called the substrate structure. The orientation of the cutting plane and thus of the surface is commonly notated by use of the so-called Miller indices. 

Figure 8.1 Notation of a cutting plane by Miller indices. The three-dimensional crystal is described by the three-dimensional unit cell vectors di, 02, and 03. The indicated plane intersects the crystal axes at the coordinates (3,1,2). The inverse is (, j, ). The smallest possible multiplicator to obtain integers is 6. This leads to the Miller indices (263).

Obviously, much of the development of crystallography predates the discovery of diffraction of X-rays by crystals. Early studies of crystal structures were concerned with external features of crystals and the angles between faces. Descriptions and notations used were based on these external features of crystals. Crystallographers using X-ray diffraction are concerned with the unit cells and use the notation based on the symmetry of the 230 space groups established earlier. 

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