Unit cell transformation

Consider two different choices of unit cells a F-centered unit cell with axes (ax, b, ci) and a primitive one with axes (fl2, b2, C2), as shown in Fig. 9.2.4. We can write 

These relationships may be expressed in the form of square matrices  

The hkl indices of a reflection referred to these two unit cells are transformed in the same manner, i.e., /12 = V2/11 + Oki + lnl, etc. The volume ratio is equal to the modulus of the determinant of the transformation matrix if we take the left matrix, V2/V1 is equal to 1/4. 

As a concrete example, consider a rhombohedral lattice and the relationship between the primitive rhombohedral unit cell (in the conventional obverse setting) and the associated triple-sized hexagonal unit cell, as indicated in Fig. 9.2.5. 

The triply primitive hexagonal unit cell has lattice points at (0 0 0), (2/3, V3, ln), and (V3,2/3,2/3). From Fig. 9.2.5, it can be seen that the rhombohedral and hexagonal axes, labeled by subscripts r and h respectively, are related by vector addition  

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Because the ID unit cells for the symmorphic groups are relatively small in area, the number of phonon branches or the number of electronic energy bands associated with the ID dispersion relations is relatively small. Of course, for the chiral tubules the ID unit cells are very large, so that the number of phonon branches and electronic energy bands is also large. Using the transformation properties of the atoms within the unit cell transformation... 

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. 

Further information on the rhombohedral-hexagonal relationship and on unit cell transformations in general may be obtained from the International Tables for X-Ray Crystallography [G.l l], Vol. 1, pp. 15-21. 

The matrix must transform the cell into an equivalent cell, which means that all cell constants remain nearly unchanged by this transformation. This can be checked with XPREP (option U, unit cell transformation). 

Unit cell transformation and origin shift if needed... 

Kea.tlte, Keatite has been prepared (65) by the crystallisation of amorphous precipitated silica ia a hydrothermal bomb from dilute alkah hydroxide or carbonate solutions at 380—585°C and 35—120 MPa (345—1180 atm). The stmcture (66) is tetragonal. There are 12 Si02 units ia the unit cell ttg = 745 pm and Cg = 8604 pm the space group is P42. Keatite has a negative volumetric expansion coefficient from 20—550°C. It is unchanged by beating at 1100°C, but is transformed completely to cristobahte ia three hours at 1620°C. 

Fig. 8.12. The structure of 0.8% carbon martensite. During the transformation, the carbon atoms put themselves into the interstitial sites shown. To moke room for them the lattice stretches along one cube direction (and contracts slightly along the other two). This produces what is called a face-centred tetragonal unit cell. Note that only a small proportion of the labelled sites actually contain a carbon atom.

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

Fig. 2. Depiction of conformal mapping of graphene lattice to [4,3] nanotube. B denotes [4,3] lattice vector that transforms to circumference of nanotube, and H transforms into the helical operator yielding the minimum unit cell size under helical symmetry. The numerals indicate the ordering of the helical steps necessary to obtain one-dimensional translation periodicity.

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

Carbon, ring oxygen replacement by, 141-143 2-Carboxy-5-(2-hydroxymethyl)-4-methylthiazole, synthesis and transformations, 284-286 Carrageenans, 366-368,418-419 Cellotetraose hemihydrate, 331 Cellulose, 326, 329-332 alternate unit cells, 329-330 derivatives, 332... 

Moreover, a constant section can only contain integer n and I values separated by Al, An) = m+ —m - - 1), which means that the horizontal and vertical lattice vectors in Fig. 22 are given by x = 0,m + 1) and (1,0), respectively. Hence an alternative basis may be dehned with a unit cell having sides Av,APj = (l,m+ 1), in which case the argument in Section IIC3 shows that the two-dimensional monodromy matrix for the fixed km section transforms to... 

Figure 2.68 Grid model of a porous medium (left) and renormalization group transformation replacing a cluster of grid cells by a unit cell of larger scale (right).

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