Lattice translational period

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.

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

The structure of a vapor-quenched alloy may be either crystalline, in which the periodicity of the unit cell is repeated within the crystallites, or amorphous, in which there is no translational periodicity even over a distance of several lattice spacings. Mader (64) has given the following criteria for the formation of an amorphous structure the equilibrium diagram must show limited terminal solubilities of the two components, and a size difference of greater than 10% should exist between the component atoms. A ball model simulation experiment has been used to illustrate the effects of size difference and rate of deposition on the structure of quench-cooled alloy films (68). Concentrated alloys of Cu-Ag (35-65%... 

For a one-dimensional lattice with translational period a, it follows that... 

The situation with respect to the photobehavior of 7-chlorocoumarin is interesting (Fig. 11). There are two reaction pathways in this crystal one that favors the formation of the vyn-IIII isomer arising from reaction between the transla-tionally related molecules with a center-to-center distance of 4.54 A and another that would yield the anti-HT dimer, corresponding to the reaction between the centrosymmetrically related molecules, the center-to-center distance being 4.12 A. Experiment clearly shows that only the yy -HH dimer is obtained—not the one that would correspond to the path of least motion. This is supported by the results of the lattice energy calculations. The implication is that the shape of the free volume is anisotropic, with the larger volume or extension in the direction of the translational periodicity of 4.54 A. 

Till now, we have only considered a mathematical set of points. However, a material, in reality, is not merely an array of points, but the group of points is a lattice. A real crystalline material is constituted of atoms periodically arranged in the structure, where the condition of periodicity implies a translational invariance with respect to a translation operation, and where a lattice translation operation, T, is defined as a vector connecting two lattice points, given by Equation 1.1 as... 

Incommensurate structures have been known for a long time in minerals, whereas TTF-TCNQ is one of the very first organic material in which a incommensurate phase has been observed. There are two main types of incommensurate crystal structures. The first class is that of intergrowth or composite structures, where two (or more) mutually incommensurate substructures coexist, each with a different three-dimensional translational periodicity. As a result, the composite crystal consists of several modulated substructures, which penetrate each other and we cannot say which is the host substructure. The second class is that of a basic triperiodic structure which exhibits a periodic distortion either of the atomic positions (displa-cive modulation) and/or of the occupation probability of atoms (density modulation). When the distortion is commensurate with the translation period of the underlying lattice, the result is a superstructure otherwise, it is an incommensurately modulated structure (IMS) that has no three-dimensional lattice periodicity. 

Discrete translations on a lattice. A periodic lattice structure allows all possible translations to be understood as ending in a confined space known as the unit cell, exemplified in one dimension by the clock dial. In order to generate a three-dimensional lattice, parallel displacements of the unit cell in three dimensions must generate a space-filling object, commonly known as a crystal. To ensure that an arbitrary displacement starts and ends in the same unit cell it is necessary to identify opposite points in the surface of the cell. A general translation through the surface then re-enters the unit cell from the opposite side. 

Compliance with the octet rule in diamond could be shown simply by using a valence bond approach in which each carbon atom is assumed sp hybridized. However, using the MO method will more clearly establish the connection with band theory. In solids, the extended electron wave functions analogous to MOs ate called COs. Crystal orbitals must belong to an irreducible representation, not of a point group, but of the space group reflecting the translational periodicity of the lattice. 

The special points method depends upon retention of the translational periodicity of a lattice, which is lost if we consider defects, surfaces, or lattice vibrations. (Even the special vibrational mode with frequency listed in Table 8-1 entailed a halving of the translational symmetry.) It is therefore extremely desirable to seek an approximate description in terms of bond orbitals, so that the energy can be summed bond by bond as discussed in Chapter 3. We proceed to that now. 

The electron density in a crystal precisely fits the definition of a periodic function in which an exact repeat occurs at regularly fixed intervals in any direction (the crystal lattice translations). Therefore the electron density in a crystal with a periodicity d can be described by a Fourier synthesis in which each component cosine wave (which we will call an electron-density wave) has a periodicity (i.e., wavelength) d/n, and the amplitude of the rath-order Bragg reflection. 

The screw axis derives its name from its relation to the screw thread. Rotation about an axis combined with simultaneous translation parallel to the axis traces out a helix, which is left- or right-handed according to the sense of the rotation. Instead of a continuous line on the surface of a cylinder there could be a series of discrete points, one marked after each rotation through 360°/ . After n points we arrive back at one corresponding to the first but translated by x, the pitch of the helix, which in a 3-dimensional pattern corresponds to a lattice translation. The symbol for a screw axis indicates the value of n (rotation through 360°/n) and, as a subscript, the translation in units of xjn where x is the pitch. The translation associated with each rotation of an -fold screw axis may have any value from xjn to (n - l)x/rt, and therefore the possible types of screw axis in periodic 3D patterns are the following ... 

FIGURE 1.7 In (a) the object, again exposed to a parallel beam of light, is not a continuous object or an arbitrary set of points in space, but is a two-dimensional periodic array of points. That is, the relative x, y positions of the points are not arbitrary they bear the same fixed, repetitive relationship to all others. One need only define a starting point and two translation vectors along the horizontal and vertical directions to generate the entire array. We call such an array a lattice. The periodicity of the points in the lattice is its crucial property, and as a consequence of the periodicity, its transform, or diffraction pattern in (b) is also a periodic array of discrete points (i.e., a lattice). Notice, however, that the spacings between the spots, or intensities, in the diffraction pattern are different than in the object. We will see that there is a reciprocal relationship between distances in object space (which we also call real space), and in diffraction space (which we also call Fourier space, or sometimes, reciprocal space). 

The idea of a lattice, which expresses the translational periodicity within a crystal as the systematic repetition of the molecular contents of a unit cell, is a salient concept in X-ray diffraction analysis. A lattice, mathematically, is a discrete, discontinuous function. A lattice is absolutely zero everywhere except at very specific, predictable, periodically distributed points where it takes on a value of one. We can begin to see, from the discussion... 

The first sum is over cells, the of which is specified by the Bravais lattice vector Rj, while the second sum is over the set of orbitals a which are assumed to be centered on each site and participating in the formation of the solid. If there were more than one atom per unit cell, we would be required to introduce an additional index in eqn (4.68) and an attendant sum over the atoms within the unit cell. We leave these elaborations to the reader. Because of the translational periodicity, we know much more about the solution than we did in the case represented by eqn (4.58). This knowledge reveals itself when we imitate the procedure described earlier in that instead of having nN algebraic equations in the nN unknown... 

Note that we have presumed something further about the solution than we did in our earlier case which ignored the translational periodicity. Here we impose the idea that the solutions in different cells differ by a phase factor tied to the Bravais lattice vector R . The result is that the eigenvector Ua has no dependence upon the site i we need only find the eigenvectors associated with the contents of a single unit cell, and the motions of atoms within all remaining cells are unequivocally determined. Substitution of the expression given above into the equations of motion yields... 

What we have learned is that our solutions may be labeled in terms of the vector q, which is known as the wavevector of the mode of interest. Each such mode corresponds to a displacement wave with wavelength X = 27r/ q. This interpretation makes it clear that there is a maximum q above which the oscillations are associated with lengths smaller than the lattice parameter and hence imply no additional information. This insight leads to an alternative interpretation of the significance of the first Brillouin zone relative to that presented in chap. 4. As an aside, we also note that had we elected to ignore the translational periodicity, we could just as well have solved the problem using the normal mode idea developed earlier. If we had followed this route, we would have determined 3 A vibrational frequencies (as we have by invoking the translational periodicity), but our classification of the solutions would have been severely hindered. 

A specification of the actual form of Bloch functions in terms of atomic orbitals is needed. They can be written as a product of a plane wave and a function, 2(r, A), with the translational periodicity of the lattice. With the Schrodinger problem written in A-dependent form, the solutions take the form of a linear combination of Bloch functions, whose coefficients are to be determined. For all the atomic orbitals %j r- rj) in the unit cell, each centered at position r -, in a crystal with N unit cells, a combination which has the same translational periodicity as the lattice is the replacement of a single AO in the reference cell with a sum over all transla-tionally equivalent AOs in the extended crystal. The final form of the crystal orbitals is then ... 

Since the lattice is periodic, so is the external potential that enters the Schrodinger equation, and therefore we can state that because of the translational symmetry ... 

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