Conventional unit cell

The FCC structure is illustrated in figure Al.3.2. Metallic elements such as calcium, nickel, and copper fonu in the FCC structure, as well as some of the inert gases. The conventional unit cell of the FCC structure is cubic with the lengdi of the edge given by the lattice parameter, a. There are four atoms in the conventional cell. In the primitive unit cell, there is only one atom. This atom coincides with the lattice pomts. The lattice vectors for the primitive cell are given by... 

The rocksalt stmcture is illustrated in figure Al.3.5. This stmcture represents one of the simplest compound stmctures. Numerous ionic crystals fonn in the rocksalt stmcture, such as sodium chloride (NaCl). The conventional unit cell of the rocksalt stmcture is cubic. There are eight atoms in the conventional cell. For the primitive unit cell, the lattice vectors are the same as FCC. The basis consists of two atoms one at the origin and one displaced by one-half the body diagonal of the conventional cell. 

Figure 4.5 shows a conventional unit cell of an fee crystal. It consists of atoms at the eight edges of a cube and at the centers of the six sides. The length a of the side of the cube is the lattice constant-, for our present purpose we may assume that it is unity. The lattice of an infinite, perfect solid is obtained by repeating this cubic cell periodically in all three directions of space. 

Figure 4.5 Conventional unit cell of a face-centered cubic crystal. The lattice contains the points at the corners of the cube and the points at the centers of the six sides.

Figure 4.6 Conventional unit cell of a face-centered cubic crystal and the principal lattice planes (a) (100), (b) (111), (c) (110).

The conventional unit cell of a body-centered cubic crystal (bcc) consists of the eight corners of a cube and the point in the center. Describe the structures of the (100), (111), and (110) planes. 

An alternative way of viewing this structure is to think of it as a cubic close-packed array of chloride Ions with sodium Ions filling all the octahedral holes. The conventional unit cell of a ccp array Is an F face-centred cube (hence the cubic in ccp) the close-packed layers lie at right angles to a cube diagonal (Figure 1.32). Filling all the... 

Figure 16.2. Conventional (non-primitive) unit cells of (a) the face-centered cubic and (b) the body-centered cubic lattices, showing the fundamental vectors a1 a2, and a3 of the primitive unit cells. (A conventional unit cell is one that displays the macroscopic symmetry of the crystal.)...

The neutron-diffraction pattern from cellulose II, prepared by treating cotton linters with 20% sodium hydroxide at 0°, showed prominent peaks at (sin 8/ ) = 0.0089 - 0.0094 nm-1 and 0.0075 nm-1, which can be indexed only by enlarging the conventional unit-cell. The proposed cell-dimensions are a = 1.57 nm, b (fiber axis) = 1.03 nm, c = 1.84 nm, and /3 = 63°. The validity of the P2i space group and the twofold screw axis along the chain was questioned. 

Neutron diffraction studies of cellulose I (cotton crystallites) showed that the a andc dimensions (b is the fiber axis) of the conventional unit-cell should be doubled. The dimensions deduced are a = 1.678 nm, b (fiber axis) = 1.03 nm, c = 1.588 nm, and /3 = 82°. These are the same as the values proposed by Honjo and Watanabe,33 except that the b dimension is less than the value of 1.058 nm proposed by them. It was found that, in the region of 101, 101, and 002 reflections, there are a number of additional reflections that cannot be indexed by using the Meyer-Misch unit cell, but they can be indexed as 102, 102, 211, 211, 203, 203, 121, and 121 by using the larger cell. 

Characteristics of Periodic Minimal Surfaces (Conventional Unit Cell)... 

Figure 2.26 Decoration of vertices in the diamond lattice to illustrate the possibility of growing the lattice by decoration about a single point using the orbits of the point group Tj (a) within the conventional unit cell (b) about the larger circle, O, in the extended lattice of interlocked cyclohexane chairs .

Figure 1.18(c) Computer image of 6x6x6 conventional unit cells of the P-surface (courtesy P. Pieruschka). 

Figure 1.27 A conventional unit cell of the Neovius surface.

The other crystal lattices can be generated by adding to some of the above-defined cells extra high-symmetry points by the so-called centering method. TableB.2 shows the new systems added to the simple crystal lattices (noted s, or P, for primitive) and the numbers of lattice points in each conventional unit cell. The body-centred lattices are noted be or I (for German Innenzentrierte), the face-centred, fc or F, and the side-centred or base-centred lattices are noted C (an extra atom at the Centre of the base). These 14 lattice systems are known as the Bravais lattices (noted here BLs). A representation of their unit cells can be found in the textbook by Kittel [7]. 

Figure B3.2.3. The muffin-tin spheres in the (110) plane of a zincblende crystal. The nuclei are surrounded by spheres of equal size, covering about 34% of the crystal volume. Unoccupied tetrahedral positions are indicated by crosses. The conventional unit cell is shown at the bottom the crystal directions are noted.

The conventional unit cells for the fourteen three-dimensional Bravais lattices are shown in Fig. 2.25. Each of these cells represents one lattice class. Some important supplementary information is presented in Section 1.4.1 to which may be added the following comments ... 

Y. Le Page, J, Appl. Crystallogr., 15, 255 (1982). The Derivation of the Axes of the Conventional Unit Cell from the Dimensions of the Buerger-Reduced Cell. 

Fig. 81. Possible stmcture of the Si(100)-c(4x4)Sb surface with the Sb dimers (0.25 ML) lying above Si dimer rows and oriented perpendicular to the direction of the Si dimers in the uppermost substrate layer. The Sb-Sb dimers are shown in black, with high and low-lying first-layer Si atoms indicated by shaded and open circles, respectively. Both primitive and conventional unit cells are shown. The model was chosen as having the minimal free energy and yielding the best fit to the CAICISS data as compared to the other candidate models (with Sb dimers substituting Si dimers and with Sb dimers sitting in between the underlying Si dimer rows) [98D1].

Mb stands for /(subgroups, obtained by enlarging of the conventional unit cells,... 

Fig. 15. Conventional unit cell of the ThCr2Si2 structure of CeM2X2 where M = Cu, Ni, Ru, Rh, Pd, Au,. .. and X = Si, Ge. This is also the unit cell of URu2Si2 (sect. 4.4).

Fig. 35. Conventional unit cell of UPd2Al3 a = 5.350 K, c = 4.185 A) and simple AF magnetic structure with propagation vector Q = (0,0, ). The large and small grey spheres in hexagonal planes correspond to the U and Pt atoms, respectively, intercalated by A1 atoms (small black spheres).

Coordinates of the crystallographic positions (in units of bottom) of all ions are given. The symbol Z stands for one conventional unit cell... 

The space groups are listed and described in the International Tables for X-Ray Crystallography (1962). For our purposes, we are interested only in the primitive unit cell, or the smallest unit cell that can be used to reproduce the crystal by means of translations only. The primitive unit cell often has lower symmetry than the conventional unit cell (Kittel, 1968). According to the international nomenclature for space groups, the letter P denotes a primitive lattice. If we deal with such a crystal, the conventional unit cell is the primitive cell. If, however, the group designation begins with another letter (e.g., I, f, C), then the primitive cell has to be determined and this is smaller and less symmetric than the conventional um t cell. The unit cell vectors of the primitive cell are called primitive translation vectors. 

A special unit cell of a crystal is the primitive unit cell, defined as the smallest unit cell from which the crystal can be built. As visualised in figure 1.7, the primitive unit cell is not uniquely defined but can be chosen in different ways. However, all possible primitive unit cells obviously have the same volume. One primitive unit cell of a body-centred cubic lattice is shown in figure 1.8. This cell is only part of the cube that one usually visualises when putting together the crystal lattice. As the crystal symmetries are less obvious when using this cell, frequently the cubic unit cell is used instead, called conventional unit cell. It is easy to determine whether a unit cell of a... 

Fig. 1.8. Body-centred cubic lattice, primitive unit cell (thick lines) and conventional unit cell (thin lines). Both cells are centred on one atom...

Bravais lattice is primitive If it contains only one atom, it is primitive if it contains more, it is not. While counting the atoms it has to be kept in mind to count only the appropriate fractions of those atoms occupying more than one cell. For instance, the conventional unit cell of the body-centred cubic lattice contains two atoms and is therefore not primitive, the conventional unit ceU of the face-centred cubic lattice contains four atoms and is thus not primitive either. 

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