Content of the unit cell

The amplitude and therefore the intensity, of the scattered radiation is detennined by extending the Fourier transfomi of equation (B 1.8.11 over the entire crystal and Bragg s law expresses die fact that this transfomi has values significantly different from zero only at the nodes of the reciprocal lattice. The amplitude varies, however, from node to node, depending on the transfomi of the contents of the unit cell. This leads to an expression for the structure amplitude, denoted by F(hld), of the fomi... 

The minimum amount of information needed to specify a crystal structure is the unit cell type, that is, cubic, tetragonal, and so on, the unit cell parameters, and the positions of all of the atoms in the unit cell. The atomic contents of the unit cell are a simple multiple, Z, of the composition of the material. The value of Z is equal to the number of formula units of the solid in the unit cell. Atom positions are expressed in terms of three coordinates, x, y, and z. These are taken as fractions of a, b, and c, the unit cell sides, say and j. The x, y, and z coordinates are plotted with respect to the unit cell axes, not to a Cartesian set of axes. The space group describes the symmetry of the unit cell, and although it is not mandatory when specifying a structure, its use considerably shortens the list of atomic positions that must be specified in order to built the structure. 

The crystallinity of organic pigment powders makes X-ray diffraction analysis the single most important technique to determine crystal modifications. The reflexions that are recorded at various angles from the direction of the incident beam are a function of the unit cell dimensions and are expected to reflect the symmetry and the geometry of the crystal lattice. The intensity of the reflected beam, on the other hand, is largely controlled by the content of the unit cell in other words, since it is indicative of the structural amplitudes and parameters and the electron density distribution, it provides the basis for true structural determination [32],... 

Notice that in the description of the content of the unit cell, the atomic coordinates are given for the first cell from the chosen origin, and are formulated modulo 1 thus, for instance, —x, —y, —z is written (x,y, z) rather than 1 — x, 1 — y, 1 — z and the following equivalences could be considered as typical examples ... 

To the extent that a crystal is a perfectly ordered structure, the specificity of a reaction therein is determined by the crystallographic symmetry. A crystal is built up by repeated translations, in three dimensions, of the contents of the unit cell. However, the space group usually contains elements additional to the pure translations, such as a center of inversion, rotation axis, and mirror plane. These elements can interrelate molecules within the unit cell. The smallest structural unit that can develop the whole crystal on repeated applications of all operations of the space group is called the asymmetric unit. This unit can consist of a fraction of a molecule, sometimes fractions of two or more molecules, a single whole molecule, or more than one molecule. If, for example, a molecule lies on a crystallographic center of inversion, the asymmetric unit will contain half... 

A crystal is made up of repeating units called unit cells. Each unit cell in the crystal has the same number of atoms or molecules arranged in a pattern that repeats regularly in three-dimensions (Fig. 4.6). It is the regularity or periodicity that makes a crystal diffract X-rays, while it is the content of the unit cell that gives a crystal its unique diffraction pattern. Furthermore, the degree to which all unit cells and their content have the same orientation in a crystal is directly proportional to its diffraction resolution. 

From the cubic unit cell dimension a, we can calculate the volume of the unit cell, Elf the density, p, of the crystals are known, then the mass of the contents of the unit cell, M, can also be calculated... 

The contents of the unit cell of any compound must contain an integral number of formula units. (Why ) Note that unit cell boundaries slice" atoms into fragments An atom on a face Will be split in half between two cells one on an edge will be splu into gunners among Jour cells, etc Identify the number of Na and Cl ions in the unit cell of sodium chloride illustrated in Fig. 4.1a and state how many formula units of NaCl the unit cell contains. Give a complete analysis. 

The chain conformation it was treated mathematically in terms of atoms regularly spaced along helices, an approach equivalent to defining the contents of the unit cell in terms of cylindrical coordinates. Packing of the chains was characterized by defining the distances between molecules as determined from the equatorial reflections on an x-ray fiber pattern. j ... 

The lines in the figure divide the crystal into identical unit cells. The array of points at the corners or vertices of unit cells is called the lattice. The unit cell is the smallest and simplest volume element that is completely representative of the whole crystal. If we know the exact contents of the unit cell, we can imagine the whole crystal as an efficiently packed array of many unit cells stacked beside and on top of each other, more or less like identical boxes in a warehouse. 

This model of diffraction implies that the directions of reflection, as well as the number of reflections, depend only on unit-cell dimensions and not upon the contents of the unit cell. The intensity of reflection hkl depends on the values of p(x,y,z) on planes (hkl). We will see (Chapter 5) that the intensities of the reflections give us the structural information we seek. 

Thinking again about the potential usefulness of computing IT s in crystallography, you will see that we can use the Fourier transform to obtain information about real space, f(x,y,z), from information about reciprocal space, F(h,k, 1)- Specifically, the diffraction pattern contains information whose Fourier transform is information about the contents of the unit cell. 

To understand the subtlety and electronic structure of the complex covalent network of /9-R105 boron, it is useful to view the contents of the unit cell [Fig. 13.4.11(b)] as assembled from Bi2fl (at the center of a B84 cluster) and B57 (lying on a C3 axis) fragments connected by 2c-2e bonds. 

Fig. 13. The contents of the unit cell of WSCI4 in the b projection (large open circles, tungsten small open circles, chlorine small closed circles, sulphur)...

Translational symmetry requires that any matter located at xea + yeb + zec must also be replicated exactly at the coordinates (x + l a ) ea + (y + m b ) eb + z + n c ) ecr where /, m, and n are integers (positive, negative, or zero). This translation symmetry defines the unit cell and the unit cell axes a, b, c. In the least symmetric case, the contents of the unit cell (atoms, ions, molecules, trapped solvents, proteins) may not have any symmetry at all. 

In all the applications of structure solution discussed here, we assume that the unit cell parameters a, b, c, a, /J, y and space group S are already known from prior analysis of the experimental powder diffraction pattern. We also assume that the contents of the unit cell (e.g. the types and number of atoms, ions or molecules) are known, at least to a sufficiently good approximation, but that the positions and structural arrangement of these constituents within the unit cell are not known. The structure is defined in terms of a structural fragment , which represents an appropriate collection of atoms within the asymmetric unit, and is coded using a set (denoted T) of variables that represents the positions of the atoms and/or molecules in the unit cell. For a collection of independent... 

Build the MO diagram associated with the contents of the unit cell (in the simplest case of an atom repeat unit these are its valence AOs). 

It should be recalled that the solution of crystal structures by diffraction techniques involves at least two kinds of averaging, the time average and the average content of the unit cell. Other techniques which observe on a shorter time scale may legitimately suggest a lower local symmetry in the same phase than that obtained by crystallography. 

In X-ray diffraction the envelope profile is the diffraction pattern of the atomic contents of the unit cell. 

Absolute scale When the structure factors are on an absolute scale, they represent the scattering of X rays by the total contents of the unit cell relative to the scattering of X rays by a single electron under the same condition. 

It is somewhat surprising that, in spite of what has been said in earlier chapters about the loss of phase information in the X-ray diffraction experiment, the intensity distribution contains information on whether the structure is centrosymmetric or noncentrosymmetric. This information is important when there is a spaoe group ambiguity in which one possibility is centrosymmetric, while the other is not (see Chapter 4. Arthur J. C. Wilson - noted that, while the intensities of Bragg reflections on the average depend only on the atomic contents of the unit cell, the distribution of intensities is different depending on whether the structure is centrosymmetric or noncentrosymmetric. The intensities... 

The lattice itself, including the shape of the unit cell, may be chosen in an infinite number of ways. As an example, a second alternative lattice with a different unit cell is shown in Figure 1.3. Both the origin of the lattice and the shape of the unit cell have been changed when compared to Figure 1.1, but the content of the unit cell has not - it encloses the same molecule. 

Figure 1.5. Illustration of the content of the unit cell. The coordinates of the center of each atom are given as doublets, i.e. Xi, Xi, y2 and Xf 73. In three dimensions, they become triplets, i.e. a ,-, yi, Zj.

An example of the unit cell in two dimensions and its content in terms of coordinates of all atoms is shown in Figure 1.5. Here, the three atoms ( large , medium , and small ) have coordinates Xu yu X2, and Xj, yj, respectively. Strictly speaking, the content of the unit cell should be described by specifying other relevant atomic parameters in addition to the position of each atom in the unit cell. These include types of atoms (i.e. their chemical symbols or sequential numbers in a periodic table instead of big , medium and small ), site occupancy and individual displacement parameters. All of these terms will be defined and explained later in this book. 

The entire content of the unit cell can be established from its asymmetric unit using the combination of symmetry operations present in the unit cell. Here, this operation is a rotation by 180° around the line perpendicular to the plane of the projection at the center of the unit cell. It is worth noting that the rotation axis shown in the upper left comer of Figure 1.6 is not the only axis present in this crystal lattice - identical axes are found at the beginning and in the middle of every unit cell edge as shown in one of the neighboring cells. ... 

So far, our discussion of symmetry of the lattice was limited to lattice points and symmetry of the unit cell. The next step is to think about symmetry of the lattice including the contents of the unit cell. This immediately brings translational symmetry into consideration to reflect the periodic nature of crystal lattices, which are continuous or infinite object. As... 

Using the International Tables for Crystallography describe every atom in terms of the multiplicities and Wyckoff letters of their site positions and establish the content of the unit cell, the simplest chemical formula and the number of formula units (Z) per unit cell. 

Once the content of the unit cell has been established, a model of the crystal structure should be created using either direct or reciprocal space techniques, or a combination of both. Direct space approaches do not mandate immediate use of the observed integrated intensities, while reciprocal space methods are based on them. 

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