Point lattices translational periodicity

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... 

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. 

Since a crystal stractrrre corrstitutes a regular repetition of a rrrrit of structure, the unit cell, we may say that a crystal stracture is periodic in three dimerrsions. The periodicity of a crystal stractrrre may be represented by a point lattice in three dimensions. This is an array of points that is invariant to all the translations that leave the crystal stractrrre invariant and to no others. We shall find the lattice useful in deriving the conditiorrs for x-ray diffraction. 

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 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... 

Note that the vectors a, b, c define, not only the unit cell, but also the whole point lattice through the translations provided by these vectors. In other words, the whole set of points in the lattice can be produced by repeated action of the vectors a, b, c on one lattice point located at the origin, or, stated alternatively, the vector coordinates of any point in the lattice are Pz, Qh, and Rc, where P, Q, and R are whole numbers. It follows that the arrangement of points in a point lattice is absolutely periodic in three dimensions, points being repeated at regular intervals along any line one chooses to drav/ through the lattice. 

In a further development of detail, one can take into account how the atoms of the solid are distributed spatially. The issue of symmetry in context with a fixed point in the crystal, and the symmetry of Bravais lattices, has been addressed, but in order to describe the entire crystal the effects of two new types of symmetry operation must be included. A space group determined in this way describes the spatial symmetry of the crystal. By definition, a crystallographic space group is the set of geometrical symmetry operations that take a three-dimensional periodic crystal into itself The set of operations that make up the space group must form a group in the mathematical sense and must include the primitive lattice translations as well as other symmetry operations. 

Let us suppose that a real crystal lattice consists of a set of points R that are determined by vectors r. Any property p of the crystal lattice is periodic, p r) = p r + R), where R is the lattice translation vector. 

It can be shown that in 3-D there are 14 different periodic ways of arranging identical points. These 14 3-D periodic point lattices are called the (translational) Bravais lattices and are shown in Fig. 1.3-6. Table 1.3-4 presents data related to some of the crystallographic terms used here. The 1-D and 2-D space groups can be classified analogously but are omitted here. 

Unfortunately, at this point, the actual complexities of UPtj became apparent. UPtj is a non-symmorphic lattice (two atoms per unit cell, separated by a nonprimitive translation vector). Because of this, the observed dynamic susceptibility is not invariant under reciprocal lattice translations (ignoring form factors, it has a periodicity of two reciprocal lattice vectors in the c direction and three in the basal direction, due to the non-primitive translation vector). Using this susceptibility in a gap equation, then, gives gap functions which are not properly lattice periodic. Thus both the solutions of Norman and of Putikka and Joynt are invalid. 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

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