Three-dimensional arrays of points

Let us replace each repeat unit in the crystal by a point (called a lattice point) placed at the same place in the unit. All such points have the same environment and are indistinguishable from one another. The resulting three-dimensional array of points is called a lattice. It is a simple but complete description of the way in which a crystal structure is built up. 

Crystal Lattice A three-dimensional array of points related by translational symmetry. The translation can occur in three independent directions giving three independent base vectors. We can fully describe such a lattice by three vectors, a, b, c, and three angles, a, (3, y. The special property of a crystal lattice is that the lattice points are identical if we have an atom at or near one point, there must be an identical atom at the same position relative to every other lattice point. 

Crystal Lattice Three-dimensional array of points, each of which represents a unit cell. 

A cr5 talline solid can be represented by a three-dimensional array of points, each of which represents an identical environment wilhin the crystal. Such an array of points is called a crystal lattice. We can imagine forming the entire crystal structure by arranging the contents of the unit cell repeatedly on the crystal lattice. 

Conclusion The Lattice is the sites of motifs where they can be placed to generate the pattern. It can be two-dimensional regular arrays of points for two-dimensional patterns or it can be three-dimensional arrays of points for three-dimensional patterns. Therefore, the lattice bears the knowledge of the scheme of repetition and when the motifs are placed in the lattice sites the entire pattern takes the shape and changes whenever the orientation of the motifs takes their role to play. If the order of this orientation of the motifs is maintained in some way or other, it retains the pattern characteristics of being geometrically symmetric otherwise not. 

When groundwater contaminant plumes are suspected of having significant depth as well as lateral distribution, a three-dimensional array of monitoring points is needed to identify and characterize such plumes. Thus, groundwater data must be obtained from a number of different locations and from a number of different depths at each location. As a result, either a large number of drillholes are required, each with separate instrumentation installed, or instruments must be combined and installed at multiple levels in each of a smaller number of drillholes. 

Consider a suspension composed of an ordered, repetitive, three-dimensional array of identical, rigid spheres immersed in an otherwise homogeneous fluid continuum and extending indefinitely in every direction. From a formal point of view, the lattice A, representing the group of translational self-coincidence symmetry operations of this spatially periodic medium, consists of the set of points... 

Compared to the unit cell, a macroscopic crystal can be regarded as infinitely large. Thus, a NaCl crystal of 1 imn contains lO unit cells. Ignoring its finiteness, each vertex of the unit cell and every one generated by translations are physically identical the atomic environments and all physical properties at these points are the same. An (infinite) three-dimensional array of such points in space is called the crystal lattice. Note that the latter is only a mathematical device to describe the crystal s regularity and is not synonymous with... 

Now if a, b, c are the primitive translation vectors that define the unit cell of the three-dimensional array of scattering points, then we have, using Eq. (3.6), the following conditions for diffraction maxima ... 

We have discussed the unit cell, which has imaginary boundaries. For simplification, each unit cell in a crystal can be represented by a single point. The result is called a crystal lattice, which is an imaginary three-dimensional arrangement of points such that the view in a given direction from each point in this lattice is identical to the view in the same direction from any other lattice point. In other words, it is an array of points that... 

Here gg, Fg, and Gg are, respectively, the first (gradient), second (force constants), and third energy derivatives evaluated at Xg. The square brackets indicate that the three-dimensional array of third derivatives is contracted with the vector of coordinate changes to yield a square matrix. If Xg is a stationary point, i.e., a minimum, then the usual theory of small vibrations applies the gradient term vanishes, and truncation after the second-order term leads to separable, harmonic normal modes of vibration. However, on the MEP, the gradient term generally is not zero. The second relevant expansion is the Taylor series representation of the path (of the solution to Eq. [8]) in the arc length parameter, s, about the same point, Xg ... 

Crystalline solids consist of atoms, ions, or molecules that are arrayed into a long-range, regularly ordered structure known as a crystalline lattice. A crystal consists of a pattern of objects that repeats itself periodically in three-dimensional space, so that it has the property of translational symmetry. A lattice is simply a three-dimensional array of lattice points, where the atoms, ions, or molecules are held together in the solid state by a balance of attractive and repulsive forces. Lattice points are geometrical constructs it is not a necessary condition for a physical entity, such as an atom or ion, to actually occupy the lattice point Indeed, many lattice points are simply empty space, around which a basis, or motif, of particles is centered. Two examples of two-dimensional lattices are shown in Figure 11.1. 

The ideal crystal is a rigid, three-dimensional array of molecules extending infinitely in all directions. This is the model used to evaluate the symmetry of a group of real atoms. The infinite extent of this array allows us to add new symmetry operations to our list of point group symmetry elements (Section 6.1). Previously, we counted only operations that leave the center of mass unchanged. However, the center of mass is not defined for an infinite number of atoms, so we can ignore that constraint now by adding translational symmetry elements to the list. 

A few solids make an almost infinite three-dimensional array of covalent bonds to neighboring atoms. Such solids are called covalent network solids. Diamond (a form of carbon), elemental silicon, elemental germanium, and silicon dioxide are examples (see Figure 21.3). Although few solids can be described this way, the ones that can have distinctive properties They are very hard with high melting points. It takes a lot of energy, either mechanical or thermal, to break the almost infinite network of covalent bonds. 

The ideal network structure can be envisaged as a three-dimensional array of crosslink points, each crosslink point being connected to at least three other crosslink points via linear polymer segments, which are called elastically active network chains. In practice non-ideal network elements are also present, such as loops or dangling ends (Figure 16.1). Network density, or crosslink density, is expressed as the concentration of either the crosslink joints or the elastically active network chains (those chains that are part of the infinite structure and attached to crosslink junctions at both ends) per unity of volume of the unswollen material. 

To specify a unit cell fuUy, we need to know not only its symmetry but its size, such as the lengths of its sides. There is a useful relation between the spacing of the planes passing through the lattice points, which (as we shcJl see) we can measure, and the lengths we need to know. Because two-dimensional arrays of points sue easier to visualize than three-dimensional arrays, we shall introduce the concepts we need by referring to two dimensions initially and then extend the conclusions to three dimensions. 

The system to be considered consists of a three-dimensional array of elements (the macromolecules) that are joined together by discrete crosslinks, or junction points, to form a network. The deformation of an element is described conveniently in terms of the vector that connects its two ends. Referring to Figure 1, the junction... 

Diffraction patterns can be described in terms of three-dimensional arrays called lattice points.33 The simplest array of points from which a crystal can be created is called a unit cell. In two dimensions, unit cells may be compared to tiles on a floor. A unit cell will have one of seven basic shapes (the seven crystal systems), all constructed from parallelepipeds with six sides in parallel pairs. They are defined ac-... 

Fig. 8. Generation of the form of the helical diffraction pattern. (A) shows that a continuous helical wire can be considered as a convolution of one turn of the helix and a set of points (actually three-dimensional delta-functions) aligned along the helix axis and separated axially by the pitch P. (B) shows that a discontinuous helix (i.e., a helical array of subunits) can be thought of as a product of the continuous helix in (A) and a set of horizontal density planes spaced h apart, where h is the subunit axial translation as in Fig. 7. This discontinuous set of points can then be convoluted with an atom (or a more complicated motif) to give a helical polymer. (C)-(F) represent helical objects and their computed diffraction patterns. (C) is half a turn of a helical wire. Its transform is a cross of intensity (high intensity is shown as white). (D) A full turn gives a similar cross with some substructure. A continuous helical wire has the transform of a complete helical turn, multiplied by the transform of the array of points in the middle of (A), namely, a set of planes of intensity a distance n/P apart (see Fig. 7). This means that in the transform in (E) the helix cross in (D) is only seen on the intensity planes, which are n/P apart. (F) shows the effect of making the helix in (E) discontinuous. The broken helix cross in (E) is now convoluted with the transform of the set of planes in (B), which are h apart. This transform is a set of points along the meridian of the diffraction pattern and separated by m/h. The resulting transform in (F) is therefore a series of helix crosses as in (E) but placed with their centers at the positions m/h from the pattern center. (Transforms calculated using MusLabel or FIELIX.)...

---