Unit cell and crystal lattices

I Explain how the unit cell and crystal lattice are related. 

For example, if one-third of the A (or B) crystal lattice sites are coincidence points belonging to both the A and B lattices, then E = 1 / = 3. The value of also gives the ratio between the areas enclosed by the CSL unit cell and crystal unit cell. The value of E is a function of the lattice types and grain misorientation. The two grains need not have the same crystal structure or unit cell parameters. Hence, they need not be related by a rigid body rotation. The boundary plane intersects the CSL and will have the same periodicity as that portion of the CSL along which the intersection occurs (Lalena and Cleary, 2005). 

The structure of a crystal is solved in three steps (Cullity, 1956). Firstly, the size and shape of the unit cell (a crystal lattice consists of identical unit cells) is found from the angular distribution of the diffraction beams. Secondly, the number of molecules per unit cell is computed from the size and shape of the unit cell, the chemical composition of the sample and the sample s measured density. Lastly, the positions of the molecules within the unit cell are deduced from the relative intensities of the diffraction beams. Data analysis, which is complex, is described by Woolfson and Fan (1995) and Clegg (2001). 

Figure 3.15. a) Scattering of waves by crystal planes. While the angle of the scattered waves will depend only on d, the intensity will depend on the nature of the scaiterers. as is clear when one compares (6) and (c). They both have the same lattice type, but quite different unit cells and crystal structures, which in turn would be reflected in the intensity of the scattered waves. 

The regular arrangement of the components of a crystalline solid at the microscopic level produces the beautiful, characteristic shapes of crystals, such as those shown in Fig. 10.8. The positions of the components in a crystalline solid are usually represented by a lattice, a three-dimensional system of points designating the positions of the components (atoms, ions, or molecules) that make up the substance. The smallest repeating unit of the lattice is called the unit cell. Thus a particular lattice can be generated by repeating the unit cell in all three dimensions to form the extended structure. Three common unit cells and their lattices are shown in Fig. 10.9. Note from Fig. 10.9 that the extended structure in each case can be viewed as a series of repeating unit cells that share common faces in the interior of the solid. 

SOLUTION As one of the cubic lattice systems, the fee lattice must belong to one of the cubic crystallographic point groups listed in Table 13.1 T, Ty O, or O. We may first examine the face-centered cubic unit cell for any of the symmetry elements of these point groups if they are symmetry elements of the unit cell, they will also be symmetry elements of the crystal. The face-centered cubic unit cell with identical spherical atoms at each lattice point has all the symmetry elements of the perfect cube E, I, six Q axes, four C3 axes, three Q axes, and nine mirror planes. These are sufficient to identify the point group of the unit cell, and the lattice, as O. ... 

The internal structure of a crystal is characterized by regularity in three dimensions. Let us consider an elemental crystal, in which all the atoms are alike. If we represent the center of each atom by a point corresponding to its position in space, the arrangement of such points is called a CRYSTAL lattice, as shown, for example, in Fig. 9.2. Note that the straight lines connecting the lattice points outline a block of unit shapes, called cells. All such cells are identical, and any one of them is called a UNIT CELL, The crystal lattice may be considered to consist of an indefinite number of unit cells, each one having sides in common with those of its nearest neighbors, and all similarly oriented in space. 

Unit cell and the lattice dimensions, while necessary for crystal structure determination, are by themselves valuable in polymer characterization. This is because in contrast to the lattice dimensions in crystals of small molecules, which remain essentially unchanged, these dimensions in polymeric materials vary over a relatively large range. The polymer needs to be obtained in at least a fiber or a biaxially oriented film in the initial identification of the unit cells, and in assigning the reflections to appropriate lattice planes. Once the reflections are indexed, cell dimensions can be calculated... 

We have already dlsussed structure factors and symmetry as they relate to the problem of defining a cubic unit-cell and find that still another factor exists if one is to completely define crystal structure of solids. This turns out to be that of the individual arrangement of atoms within the unit-cell. This then gives us a total of three (3) factors are needed to define a given lattice. These can be stipulated as follows ... 

The crystal structure of NiAl is the CsCl, or (B2) structure. This is bcc cubic with Ni, or A1 in the center of the unit cell and Al, or Ni at the eight comers. The lattice parameter is 2.88 A, and this is also the Burgers displacement. The unit cell volume is 23.9 A3 and the heat of formation is AHf = -71.6kJ/mole. When a kink on a dislocation line moves forward one-half burgers displacement, = b/2 = 1.44 A, the compound must dissociate locally, so AHf might be the barrier to motion. To overcome this barrier, the applied stress must do an amount of work equal to the barrier energy. If x is the applied stress, the work it does is approximately xb3 so x = 8.2 GPa. Then, if the conventional ratio of hardness to yield stress is used (i.e., 2x3 = 6) the hardness should be about 50 GPa. But according to Weaver, Stevenson and Bradt (2003) it is 2.2 GPa. Therefore, it is concluded that the hardness of NiAl is not intrinsic. Rather it is determined by an extrinsic factor namely, deformation hardening. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

Questions about such interactions could only be resolved by knowledge of the exact geometry of the atoms of a pigment molecule in its unit cell and the relative position of each individual molecule within the crystal lattice. This is elucidated through three dimensional X-ray diffraction analysis of single crystals [8]. 

In crystallography, the difiiraction of the individual atoms within the crystal interacts with the diffracted waves from the crystal, or reciprocal lattice. This lattice represents all the points in the crystal (x,y,z) as points in the reciprocal lattice (h,k,l). The result is that a crystal gives a diffraction pattern only at the lattice points of the crystal (actually the reciprocal lattice points) (O Figure 22-2). The positions of the spots or reflections on the image are determined hy the dimensions of the crystal lattice. The intensity of each spot is determined hy the nature and arrangement of the atoms with the smallest unit, the unit cell. Every diffracted beam that results in a reflection is made up of beams diffracted from all the atoms within the unit cell, and the intensity of each spot can be calculated from the sum of all the waves diffracted from all the atoms. Therefore, the intensity of each reflection contains information about the entire atomic structure within the unit cell. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes, a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry... 

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