Real lattice

It is important to note that we assume the random fracture approximation (RPA) is applicable. This assumption has certain implications, the most important of which is that it bypasses the real evolutionary details of the highly complex process of the lattice bond stress distribution a) creating bond rupture events, which influence other bond rupture events, redistribution of 0(microvoid formation, propagation, coalescence, etc., and finally, macroscopic failure. We have made real lattice fracture calculations by computer simulations but typically, the lattice size is not large enough to be within percolation criteria before the calculations become excessive. However, the fractal nature of the distributed damage clusters is always evident and the RPA, while providing an easy solution to an extremely complex process, remains physically realistic. 

Fig. 31. Schematics of (a) real lattice and (b), (c) the (n,n) and (n,2n) diffraction features of incommensurate layers. SI - striped incommensurate, HI - hexagonal incommensurate, HIR -hexagonal incommensurate rotated. All phases are assumed to be fully relaxed. O denotes the (93 X 93)R30° commensurate and the incommensurate structures.

Various lattices are described in Chapter 16. Since there are about 10 atoms in 1 cm of a metal, one can expect that some atoms are not exactly in their right place. Thus, one can expect that a real lattice will contain defects (imperfections). 

The process of reflection by the real lattice cannot be visualized in terms of the reciprocal lattice but the condition for reflection by the real lattice (the Bragg equation) naturally has its jjrecise geometrical equivalent in terms of the reciprocal lattice. This is illustrated in Fig. 81, in which X Y represents the orientation of a set of crystal planes which we will suppose is in a reflecting position. Along the normal to this... 

Consider a crystal with its c axis OZ in Fig. 241) vertical. All reciprocal points corresponding with the vertical lik0 (real) lattice planes lie in the horizontal plane x y. ... 

Consider now any set of real lattice planes having indices hid. If one plane passes through the origin ()i the next plane JRST makes an intercept of cjl on the OZ axis. Draw a perpendicular to this plane, meeting it at N (ON = d, the spacing of the planes), and produce ON to P, where OP = Ajd. P is the reciprocal lattice point corresponding to the set of real lattice planes hid. 

Let the primitive translations in the real lattice be the vectors a, b, c, then these quantities multiplied by the integers u, v, w, respectively define the lattice points. Thus... 

Figure 2.8 Lattice of spheres (left) and its diffraction pattern (right). If you look at the pattern and blur your eyes, you will see the diffraction pattern of a sphere. The pattern is that of the average sphere in the real lattice, but it is sampled at the reciprocal lattice points.

Because the real lattice spacing is inversely proportional to the spacing of reflections, crystallographers can calculate the dimensions, in angstroms, of the unit cell of the crystalline material from the spacings of the reciprocal lattice on the X-ray film (Chapter 4). The simplicity of this relationship is a dramatic example of how the macroscopic dimensions of the diffraction pattern are connected to the submicroscopic dimensions of the crystal. 

Figure 4.8 a shows an ab section of lattice with an arbitrary lattice point O chosen as the origin of the reciprocal lattice I am about to define. This point is thus the origin for both the real and reciprocal lattices. Each + in the figure is a real lattice point. 

Figure 4.8 (a) Construction of reciprocal lattice. Real-lattice points are +s, and rec-... 

Now continue this operation for planes (210), (310), (410), and so on, defining reciprocal lattice points 210, 310,410, and so on (Fig. 4.8b). Note that the points defined by continuing these operations form a lattice, with the arbitrarily chosen real lattice point as the origin (indices 000). This new lattice is the reciprocal lattice. The planes hkO, hOk, and 0kl correspond, respectively, to the xy, xz, and yz planes. They intersect at the origin and are called the zero-level planes in this lattice. Other planes of reciprocal-lattice points parallel to the zero-level planes are called upper-level planes. 

FIGURE 8 Comparison of the modification of the Ewald construction when going from ideally translational symmetric lattices to real lattices in which planes degenerate into lattice slabs. The quantity P is explained in Equation (2). 

However, in a real lattice the electrons cannot assume the same energies as correspond to Bragg lattice reflections. The energy levels just below these reflections are considered to have been forced down in energy, while those just above are raised. The gaps break the curve into a series of S-shaped curves each corresponding to a Brillouin zone. 

The most persuasive proofs of NS storage can be obtained with a conservative approach, when the real lattice of SP is replaced with an infinite one, while such highly neutron absorbing fission products as Sm, Gd etc. are not taken into... 

The magnitude of the reciprocal lattice vectors are equal to the inverse of the lattice spacings of the associated planes in the real lattice (Eq. 7) ... 

It should be noted that for a three-dimensional reciprocal lattice, a third vector (c ) is used that is perpendicular to both a and b axes of the real lattice. 

It is often easier and more elegant to express the conditions for diffraction in terms of a mathematical transformation known as the reciprocal lattice. The reciprocal lattice vectors a, b, c are defined in terms of the real lattice vectors a, b, c by the relations ... 

---