Origin of a unit cell

The third type of symmetry operation considered here involves inversion, the act of turning an object inside out. If a glove is taken as an example, the reader can verify that inversion will also convert a left hand to a right hand. The point about which inversion of an object occurs is called a center of symmetry (or a center of inversion. Figure 4.5). If an object lies at a distance r from a center of symmetry, its symmetry-related object lies at an equal distance (-r) in the opposite direction. This means that if a center of symmetry lies at the origin of a unit cell, it converts an object at x,y,z to one at -x,-y,-z. This type of symmetry operation will convert a left-handed molecule into a right-handed molecule, and vice versa. 

Origin of a unit cell The point in a unit cell (usually one corner) from which the X, y, and z axes originate. It is designated 0,0,0 (for its values of x, y, z). 

For the origin of the unit cell a geometrically unique point is selected, with priority given to an inversion center. 

An atom at the center of a unit cell would have a position specified as (, , ), irrespective of the type of unit cell. Atoms at the centers of the cell edges are specified at positions (, 0, 0), (0, i, 0), or (0, 0, ), for atoms on the a, b, and c axes. Stacking of the unit cells to build a structure will ensure that an atom at the unit cell origin will appear at every corner, and atoms on unit cell edges or faces will appear on all of the cell edges and faces. 

We now introduce a Fourier transform procedure analogous to that employed in the solution theory, s 62 For the purposes of the present section a more detailed specification of defect positions than that so far employed must be introduced. Thus, defects i and j are in unit cells l and m respectively, the origins of the unit cells being specified by vectors R and Rm relative to the origin of the space lattice. The vectors from the origin of the unit cell to the defects i and j, which occupy positions number x and y within the cell, will be denoted X 0 and X for example, the sodium chloride lattice is built from a unit cell containing one cation site (0, 0, 0) and one anion site (a/2, 0, 0), and the translation group is that of the face-centred-cubic lattice. However, if we wish to specify the interstitial sites of the lattice, e.g. for a discussion of Frenkel disorder, then we must add two interstitial sites to the basis at (a/4, a/4, a]4) and (3a/4, a/4, a/4). (Note that there are twice as many interstitial sites as anion-cation pairs but that all interstitial sites have an identical environment.) In our present notation the distance between defects i and j is... 

Figure 5. Changing the sign of the amplitude from minus to plus causes a phase shift of 180° in the Fourier map. Every cosine wave with a positive amplitude starts at the origin of the unit cell with a maximum (high potential) cosine waves with negative amplitudes on the other hand produce low (zero) potential at the origin.

Figure 6. Successive changes of the phase value of a Fourier wave with index h = 2 moves the region with high potential (black areas) from the origin at X = 0 in the top map towards X = 1/4 in the map at the bottom. This shows that the value of the phase (f) determines the positions with high potential within the unit cell, whereas the amplitude A just affects the intensity. Note, that the maps with a phase shift of (j) = 0° and (f) = 180° have a centre of symmetry at the origin of the unit cell, whereas the other maps have no symmetry centre. From this we can draw another important conclusion if we put the origin of the unit cell on a centre of symmetry we have only two choices for the phase value, = 0° or (j)= 180°. As we will see later, this feature is of great importance for solving centrosymmetric crystal structures.

Figure 14. Principle of direct methods using triplet relations (continued). In this case the origin of the unit cell was shifted half a unit cell along the a and 6-axis. Whereas the indices of the three waves remain unchanged, the relative phase values must change in order to...

Pi. The action of the lattice translations (i.e., the symmetry of the lattice itself) upon any one inversion center (1) that we introduce is to generate others (cf. the 2D group / 2). It is conventional to place one inversion center at the origin of the unit cell. The translational symmetry of the lattice then generates another one at the center of the cell (i,U), three more at face centers (e.g., 0,, i), and three at the midpoints of the edges (e.g., 2,0,0), for a total of eight inversion centers, none of which are equivalent. 

From crystallography, we obtain an image of the electron clouds that surround the molecules in the average unit cell in the crystal. We hope this image will allow us to locate all atoms in the unit cell. The location of an atom is usually given by a set of three-dimensional Cartesian coordinates, x, y, and z. One of the vertices (a lattice point or any other convenient point) is used as the origin of the unit cell s coordinate system and is assigned the coordinates x = 0, y = 0, and z = 0, usually written (0,0,0). See Fig. 2.4. 

The plumbides R(Ag, Pb)3 with R = Y, Sm, Gd-Tm crystallize with a very simple structure type, i.e. Q13AU, an ordered version of the cubic close packing. The rare earth atoms fill Wyckoff position la (the origin of the unit cell, see Figure 19), while the silver and lead atoms show random distribution on the 3c site. The phase analytical investigations reveal that up to 78% lead can occupy that site. Both sites have cuboctahedral coordination (CN 12). 

Consider the general equivalent positions of space group P2 /c as shown in Fig. 9.3.4(a). Let position 1 approach the origin of the unit cell in other words, let the coordinates x 0, y 0, and z 0. As this happens, position 4" also approaches the origin, while both 2 and 3 simultaneously approach the center of inversion at (0, 1 /2, 1 /2). When x = 0, y = 0, and z = 0,1 and 4 coalesce into one, and 2 and 3 likewise become the same position. There remain only two equivalent positions (0, 0, 0) and (0, 1/2, 1/2) that occupy sites of symmetry I, and they constitute the special equivalent position 2(a), which is designated as Wyckoff position 2(a). Other sets of special equivalent positions of site symmetry I are obtained by setting x = 1 /2, y = 0, z = 0 x = 0, y = 0,... 

When a translation is applied so that the origin of the unit cell now resides on the Ni atom, as shown in Fig. 10.2.3(b), the atomic coordinates become... 

To specify the position of a point within or on the surface of a unit cell, a system of coordinates similar to ordinary Cartesian coordinates is used. The corner of a cell is selected as the origin, and distances to the point are measured parallel to each axis these distances are expressed in terms of the identity distances along the respective axes. This is best shown in a two dimensional lattice (Fig. 20-3). Here there are four two-dimensional... 

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