Unit cell primitive

Let us now see how these conditions for systematic absences are used. Suppose we have established from the X-ray diffraction data that a crystal is monoclinic. See Table 11.7 for the monoclinic space groups. We can next see if the unit cell is primitive or centered. If we choose the unique axis to be c, we look for absences indicative of A centering (hkl, k + 2 n) or B centering... 

As previously mentioned, the primitive unit cell is the smallest unit of a crystal that reproduces itself by translations. Figure 1-37 illustrates the difference between a primitive and a centered or nonprimitive cell. The primitive cell can be defined by the lines a and c. Alternatively, we could have defined it by the lines a and c. Choosing the cell defined by the lines a" and c" gives us a nonprimitive cell or centered cell, which has twice the volume and two repeat units. Table 1-11 illustrates the symbolism used for the various types of lattices and records the number of repeat units in the cell for a primitive and a nonprimitive lattice. The spectroscopist is concerned with the primitive (Bravais) unit cell in dealing with lattice vibrations. For factor group selection rules, it is necessary to convert the number of molecules per crystallographic unit cell Z to Z, discussed later, which is the number of molecules per primitive cell. For example,... 

The unit cell is 2(7 ). The two La atoms sit on a C3h site, and the six chlorine atoms are on a Cs site (see Appendix 4). Since the Hermann-Mauguin nomenclature cites that the unit cell is primitive (Pb6 /m) we need not reduce it. For the two La atoms there are six degrees of freedom (3n,Z ) = 3 x 1 x 2 = 6. The six Cl atoms possess 18 degrees of freedom (3 , Z ) = 3x3x2=18. Since all vibrational modes can be considered external modes, we need only correlate the site group to factor group. For the... 

The monoclinic crystals now are listed with the b axis as the unique axis, but prior to 1940, another popular "setting" used c as the unique axis. Of the 230 space groups, 7 have two choices of unit cell, a primitive rhombohedral one (R) and, for convenience, a nonprimitive hexagonal one (H), with three times the volume of the rhombohedral cell. The 3x3 transformation matrices from rhombohedral (obverse, or positive, or direct) cipbj, Cr to hexagonal axes aH, bur Ch and vice versa are shown in the caption to Fig. 7.17. 

The actual infinite lattices are obtained by parallel translations of the Bravais lattices as unit cells. Some Bravais cells are also primitive cells, others are not. For example, the body-centered cube is a unit cell but not a primitive cell. The primitive cell in this case is an oblique parallelepiped constructed by using as edges the three directed... 

Cubic unit cells P, primitive I, body-centered F, face-centered. 

The group theory analysis of the vibrational spectrum of the spinel stmeture [32] reveals that the crystal symmetry is cubic, correspondent to the space group 0 Fd3m) with eight formulas per unit cell. The primitive cell is rhombo-hedral with two formula units per cell. The vibrational spectrum of the spinel is... 

Single crystals of CgAsKs gave precession photographs showing hexagonal symmetry. There are no systematic absences, hence the unit cell Is primitive. The unit cell dimensions for C8AsF0 are a = 4.92(2) c =... 

Primitive, i.e. non-centered unit cell. A primitive unit cell is shown schematically in Figure 1.22, and it contains one lattice point per unit cell (lattice points are located in eight comers of the parallelepiped, but each comer is shared by eight neighboring unit cells in three dimensions). 

Based on the symmetry of the unit cell shape, the proper table (crystal system) must be selected. Only in one case, i.e. when the unit cell is primitive with a-b c,a = = 90° and y = 120°, both trigonal and hexagonal crystal systems should be analyzed. 

The solid state structure of a homologous series of mesogenic aromatic-aliphatic azomethine ether polymers (AZMEP-n), with n (=1-16) methylene spacers in the main chain, has been studied by X-ray fiber diffraction techniques. Unit cell parameters determined from the fiber patterns of heat-treated single filaments have been used to classify the polymers into distinct groups having different chain conformations and crystal systems. The even members of the. series, where n = 2 4, have C-centered monoclinic unit cells while the unit cells for polymers with n = 6 8 are I-centered monoclinic. When n is even and 10 the unit cells are (primitive) triclinic. The fiber repeat, in the triclinic cells correspond to one chemical moiety while in the monoclinic cells the fiber repeat corresponds to two chemical residues. In addition, the fiber repeat distances are consistent with the methylene spacers adopting an all-trans conformation for all even AZMEP-n polymers. 

Table 2 Crystal structure parameters of cyclo-S (standard deviations in parentheses). There are three (one) molecules per unit (primitive) cell ...

As an example, consider the diagrams appropriate to the space group number 75, PA, (Figure 5.2). The space group symbol reveals that the unit cell is primitive, and so contains only one lattice point. The principal symmetry element present is a tetrad axis parallel to the c-axis. The tetrad axis is always chosen to be parallel to the c-axis, and there is no need to specify the space group symbol as P 1 1 4. The origin of the unit cell coincides with the tetrad axis, and the projection is down this axis, and... 

The Wood notation can be generalized somewhat further, by adding either the prefix c for centred, or the prefix p for primitive. For instance, one may have a c (2 x 2) unit cell or a p(2 x 2) unit cell, the latter often abbreviated to (2 x 2) because it is identical to it. In a centred unit cell, the centre of the cell is an exact copy of the comers of the cell this makes the cell non-primitive, i.e. it is no longer the smallest cell that, when repeated periodically across the surface, generates the entire surface stmcture. Nonetheless, the centred notation is often used because it can be quite convenient, as the next example will illustrate. 

There are only 14 possible three-dimensional lattices, called Bravais lattices (Figure 5.1). Bravais lattices are sometimes called direct lattices. The smallest unit cell possible for any of the lattices, the one that contains just one lattice point, is called the primitive unit cell. A primitive unit cell, usually drawn with a lattice point at each comer, is labelled P. All other lattice unit cells contain more than one lattice point. A unit cell with a lattice point at each corner and one at the centre of the unit cell (thus containing two lattice points in total) is called a body-centred unit cell, and labelled I. A unit cell with a lattice point in the middle of each face, thus containing four lattice points, is called a face-centred unit cell, and labelled F. A unit cell that has just one of the faces of the unit cell centred, thus containing two lattice points, is labelled A-face-centred if the faces cut the a axis, B-face-centred if the faces cut the b axis and C-face-centred if the faces cut the c axis. 

The most basic unit cell, a primitive one, is similar to a square box, with each corner of the box holding one-eighth of an atom, so there is a total of one complete atom in the unit cell. The size, shape, and other characteristics of a unit cell are described as its crystal structure. For example, a unit cell of NaCl, or rock salt, has a different shape and chemical characteristic than a unit cell of silicon dioxide (Si02) or quartz. Although they are both crystals, they have different shapes and different properties. 

For the bcc ciystal structure, the (110) crystal plane is the most densely packed plane. The bcc(llO) surface can be pictured as a distorted hexagonal stracture. It is often convenient to describe this surface using the centered rectangular unit cell highlighted in Fig. 5a rather than the primitive unit cell the primitive unit cell can be transformed into the centered rectangular cell by multiplying the primitive lattice vectors by the matrix ... 

The choice of a unit primitive cell, that is, a ceU of the smallest possible volume for a given lattice structure is somewhat arbitrary. Instead of using a conventional... 

The space-lattice of Fig. 5.3 helps in the description of crystals. One must, however, observe that the lattice is only a mathematical abstraction and not the actual crystal. The lattice points may or may not coincide with actual motif positions. All motifs, however, have a fixed relationship to the lattice. The lattice descriptor is the unit cell, drawn in heavy lines. The three non-coplanar axes of the unit cell have the lengths a, b, c, and the angles a, p, y. One lattice point occupies the comers of the unit cell. Since for any given unit cell only 1/8 of its comers lies inside the unit cell, each primitive unit cell shown, contains only one lattice point. 

The distance between the two carbons C—C is 1.52 A the angle —CH2— is 114°. The unit cell is orthorhombic, with a = 7.40 A,b = 4.93 A, and c = 2.53 A. Taking its position from the c axis, the structure is illustrated in Figure 20.15. The motif, CH2, has the point symmetry mm2. The chain axis (c coordinate) is a 2j screw axis. The space group is Pnam, that is, the unit cell is primitive with an n ghde plane perpendicular to the a axis, an a glide plane perpendicular to the b axis, and a mirror plane perpendicular to the c axis. The foiu carbon atoms are at positions 5 T- + -y+ y + y + k i where x = 0.038a and... 

Fig. 17.1 Three centred lattices and associated primitive unit cells. The primitive is one-quarter of the volume of the face-centred cubic (a). For the body-centred tetragonal (b) and end-centred orthorhombic (c) it is one-half the volume.

Thus, it anticipates that a unit cell (or primitive cell) by multiplication following certain proportions and rules can generate a vdiole assembly. Symmetry means proportion and the proportion is measured by numbers. 

Fig.3.4. a) Unit cell and primitive translation vectors of the CsCl structure. b) Brillouin zone of the CsCl structure [3.5]... 

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