Primitive cell, defined

Coincidence-IA (POL) p,q,r, and s are all rational numbers. A superceU, which in the case of Fig. 7(b) is a 2 x 2 array of primitive cells, defines the phase-coherent registry with the substrate. By convention, the supercell is defined by comers that coincide with substrate lattice points. If these sites are considered energetically preferred, this condition implies that the other overlayer lattice points on or within the perimeter of the supercell are less favorable. Consequently, if only the overlayer-substrate interface is considered, coincidence is less preferred than commensurism. Two alternative primitive unit cells are depicted here, constructed from different primitive lattice vectors. Though the matrix elements differ, the determinants and, therefore, the areas are identical. Note that the description of the unit cell with coinciding with [0,1] illustrates the reciprocal space criterion = ma [m = 1). 

The empirical pseiidopotential method can be illustrated by considering a specific semiconductor such as silicon. The crystal structure of Si is diamond. The structure is shown in figure Al.3.4. The lattice vectors and basis for a primitive cell have been defined in the section on crystal structures (ATS.4.1). In Cartesian coordinates, one can write G for the diamond structure as... 

Fig. 3. Translation vectors used to define the symmetry of a carbon nanotube (see text). The vectors a, and 82 define the 2D primitive cell.

Ni [182], V [183], and A1 [184]. SU-M [185] is a mesoporous germanium oxide with crystalline pore walls, possessing one of the largest primitive cells and the lowest framework density of any inorganic material. The channels are defined by 30-rings. Structural and thermal information show that there exists a mismatch between framework stability and template decomposition. The latter requires temperatures higher than 450 °C, while the structure is preserved only until 300 °C. 

An infinite three-dimensional crystal lattice is described by a primitive unit cell which generates the lattice by simple translations. The primitive cell can be represented by three basic lattice vectors such as and h defined above. They may or may not be mutually perpendicular, depending on the crystal... 

The research dealing with models for the first primitive cells has had one central topic for many years the minimal cell . According to Luisi et al. (2006a), this is defined as an artificial or semi-artificial cell which contains a minimal (but sufficient) number of components to keep the cell alive . The cell is considered to be living when three conditions are fulfilled ... 

The choices we made above to define a simple cubic supercell are not the only possible choices. For example, we could have defined the supercell as a cube with side length 2a containing four atoms located at (0,0,0), (0,0,a), (0,a,0), and (a,0,0). Repeating this larger volume in space defines a simple cubic structure just as well as the smaller volume we looked at above. There is clearly something special about our first choice, however, since it contains the minimum number of atoms that can be used to fully define the structure (in this case, 1). The supercell with this conceptually appealing property is called the primitive cell. 

The primitive cell for the fee metal can be defined by connecting the atom at the origin in the structure defined above with three atoms in the cube faces adjacent to that atom. That is, we define cell vectors... 

In Chapter 2 we mentioned that a simple cubic supercell can be defined with lattice vectors a, = a or alternatively with lattice vectors a = 2a. The first choice uses one atom per supercell and is the primitive cell for the simple cubic material, while the second choice uses eight atoms per supercell. Both choices define the same material. If we made the second choice, then... 

The three-dimensional shape defined by the reciprocal lattice vectors is not always the same as the shape of the supercell in real space. For the fee primitive cell, we showed in Chapter 2 that... 

We previously introduced the concept of a primitive cell as being the supercell that contains the minimum number of atoms necessary to fully define a periodic material with infinite extent. A more general way of thinking about the primitive cell is that it is a cell that is minimal in terms of volume but still contains all the information we need. This concept can be made more precise by considering the so-called Wigner-Seitz cell. We will not go into... 

Besides the conventionel, cubic cell, the BCC lattice can be build from a primitive cell. The primitive cell is akward for many purposes. First it is a parallelipiped and not cubic. Secondly, the crystallograhic directions are defined with respect to the conventional cell. 

If we wish to have a primitive cell, we must choose the one defined by the vector t3, and two others of equal length related to it by threefold rotation. We then obtain a rhombohedral cell bounded by six rhombuses. 

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,... 

It is necessary to define a factor group and to describe how it relates to a space group. In a crystal, one primitive cell or unit cell can be carried into another primitive cell or unit cell by a translation. The number of translations of unit cells then would seem to be infinite since a crystal is composed of many such units. If, however, one considers only one translation and consequently only two unit cells, and defines the translation that takes a point in one unit cell to an equivalent point in the other unit cell as the identity, one can define a finite group, which is called a factor group of the space group. 

Here d is the nearest-neighbor distance in the structure. Notice that this reduces to Eq. (13-2) for the case n, = 2 = 1 and that the value of a does not change if we define a molecular unit to be an integral number of primitive cells of a simpler structure for any definition that does not meet these requirements, the values of the corresponding constant will vary significantly from one structure to another. For this definition they do not, as can be seen in Table 13-1, where values obtained by Johnson and Templeton (1961) for a number of structures are tabulated. 

A primitive cell of a BL is a cell of minimum volume that contains only one lattice point, so that the whole lattice can be generated by all the translations of this cell. This definition allows for different primitive cells for the same BL, but their volumes must be the same. The parallelepiped defined by the three primitive vectors ai, a2, and a3 of a simple BL is a primitive cell of this lattice. 

In fee and bcc lattices, there are no cubic primitive cells whereas in simple cubic system, the reciprocal lattice is also simple cubic and the Miller indices of a family of lattice planes represent the coordinates of a vector normal to the planes in the usual Cartesian coordinates. As the lattice planes of a fee cubic lattice or a bcc cubic lattice are parallel to those of a sc lattice, it has then been fixed as a rule to define the lattice planes of the fee and bcc cubic lattices as if they were sc lattices with orthogonal primitive vectors of the reciprocal lattice. 

In the following we consider for simplicity a crystal with one atom per primitive cell, and within a single muffin-tin well (Fig.5.1) we define the potenti al... 

The next NL lines contain and the four standard potential parameters defined by (4.1). To be able to specify = 0 the fourth parameter must be given as < >2. If the primitive cell holds more than one atom, each extra atom should be.described by data analogous to the above NL + 2 lines. 

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