Unit cell defined

A crystal is a solid with a periodic lattice of microscopic components. This arrangement of atoms is determined primarily by X-ray structure analysis. The smallest unit, called the unit cell, defines the complete crystal, including its symmetry. Characteristic crystallographic 3D structures are available in the fields of inorganic, organic, and organometallic compounds, macromolecules, such as proteins and nucleic adds. 

Analytical solutions for the closure problem in particular unit cells made of two concentric circles have been developed by Chang [68,69] and extended by Hadden et al. [145], In order to use the solution of the potential equation in the determination of the effective transport parameters for the species continuity equation, the deviations of the potential in the unit cell, defined by... 

The challenge is to form compounds with structures that we design, not Mother Nature. Superlattices are examples of nano-structured materials [1-3], where the unit cell is artificially manipulated in one dimension. By alternately depositing thin-films of two compounds, a material is created with a new unit cell, defined by the superlattice period. 

On the basis of crystallochemistry consideration and taking into account electron microscopy observations of the surface of crystals upon which some polymer was formed,99 Arlman and Cossee13 concluded that the active sites are located on crystal surfaces different from the basal (001) ones. In particular, these authors considered in detail active sites located on crystal surfaces parallel in the direction a — b of the unit cell defined as in Ref. 98. Figure 1.13 illustrates that, if we cut a TiCl3 layer parallel to the direction defined above, which corresponds to the line connecting two bridged Ti atoms, electroneutrality conditions impose that each Ti atom at the surface of the cut be bonded... 

When 1/0 is a rational number m/n (m,n integer, undivisible), there are m substrate unit cells per set of n adsorbates a superlattice with unit cell area mS can form, with the superlattice unit cell containing n arbitrarily-positioned adsorbates. It may happen in this case that the adsorbates between themselves (ignoring the substrate) form a structure that has a smaller unit cell than the superlattice unit cell one must then distinguish between the overlayer unit cell (defined in the absence of a substrate) and... 

The situation for body-centred cubic metals (A2) is more complicated, but related to the ccp arrangement. As shown in Figure 5.14 a tetragonal face-centred unit cell can be constructed around the central axis of four contiguous body-centred cells. The interstitial points in the transformed unit cell define an equivalent face-centred cell, as before, and the same sites also define a body-centred lattice (shown in stipled outline) that interpenetrates the original A2 lattice. Each metal site is surrounded by six fee interstices at an average distance d6 - four of them at a distance a/s/2 and two more at a/2. 

The vector OP points along the [120] direction in the orthorhombic unit cell defined by abc. 

A more purist approach, which in principle ignores the accumulated structural data and associated structural trends, involves placing each independent molecule with a defined conformation at a certain location (X, K, Z) with a certain orientation (p, 6, (p) m a. unit cell defined by its six parameters (a, b, c, a, p, y). Each additional independent molecule in the asymmetric unit is defined with six additional location and orientation parameters. Each variation of conformation for one or all of the individual molecules constitutes a new starting model. 

In the case of alkali feldspars the unit cell is quite complicated. It contains four formula units, and 53 atoms in the asymmetric unit. As a result, the simulated sample of Tsatskis and Salje (1996) had the form of a very thin slab (or film) the computational unit cell defined for the whole slab contained slightly more than four formula units. In the simulation the slab had 101 orientation, which allowed the observation of only the... 

Corresponding to any crystal lattice, we can construct a reciprocal lattice, so called because many of its properties are reciprocal to those of the crystal lattice. Let the crystal lattice have a unit cell defined by the vectors ai, a2, and 83. Then the... 

Let (/) (r) be a periodic function in three dimensions, so that (r) = (r + R) with R = Mjai+/i2a2+ 3a3 witha/O = 1,2,3) being three vectors that characterize the three-dimensional periodicity and nj any integers (see Section 4.1). The function is therefore characterized by its values in one unit cell defined by the three a vectors. Then the integral (1.37) vanishes if the volume of integration is exactly one unit cell. 

Fig. 14.10 Two views of the unit cell (defined by the yellow lines) of the nickel arsenide (NiAs) lattice colour code Ni, green As, yellow. View (a) emphasizes the trigonal prismatic coordination environment of the As centres, while (b) (which views (a) from above) illustrates more clearly that the unit cell is not a cuboid.

The final factor to consider is that of the angle between the x, y, and z directions in the lattice. In our examples so far, angles were 90° in aU directions. If the angles are not 90°, then we have additional lattices to define. For a given unit-cell defined by the axes a, b and c, the corresponding angles are defined as a, p, y, where a is the angle in the x-direction, etc. 

The simplest model for the PAT structure, proposed by Gustafsson etal. [61], has an assumed orthorhombic unit cell defined by the values of the three lattice parameters a, b and c as just discussed. The model is shown is Figure 2.11. The unit cell will contain two... 

Figure 3.8 A cubic unit cell defined by three 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... 

L is the number of primitive unit cells in the large unit cell defining the cyclic cluster. The number m in CL.m defines the choice of the corresponding transformation matrix given at the bottom of each column. 

Figure 2.19c illustrates a hep unit cell defined by lattice species at (0, 0, 0) and (2/3, 1/3, 1/2). There are four tetrahedral sites and two octahedral sites per unit cell. The sizes of tetrahedral or octahedral holes within a hep and fee array are equivalent, respectively accommodating a sphere with dimensions of 0.225 or 0.414 times (or slightly larger) the size of a close-packed lattice atom/ion. 

Consider the 2D honeycomb lattice with two atoms per unit cell, defined by the lattice... 

Since translation is so important in crystals, consider first what happens if a molecular object is translated in two directions in space to form a periodic structure defined by two translational periods (Fig. 5.1). There are many different choices of the unit cell the cell defined by periods a and b is preferable over the cell defined by a and b because the cell angle (aOb) is closer to 90° than the cell angle (a Ob). The unit cell defined by a and b is called a non-primitive (centered) unit cell, because it encloses one molecule related by pure translation by a sub-multiple of the cell periodicity. The choice of such a cell introduces a quite unnecessary complication in this case. 

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