Simple cubic crystal

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

The easiest ciystal lattice to visualize is the simple cubic stracture. In a simple cubic crystal, layers of atoms stack one directly above another, so that all atoms lie along straight lines at right angles, as Figure 11-26 shows. Each atom in this structure touches six other atoms four within the same plane, one above the plane, and one below the plane. Within one layer of the crystal, any set of four atoms forms a square. Adding four atoms directly above or below the first four forms a cube, for which the lattice is named. The unit cell of the simple cubic lattice, shown in... 

Figure 4.1 Schematic dislocation line a simple cubic crystal structure. The line enters the crystal at the center of the left-front face. It emerges at the center of the right-front face. The shortest translation vector of the structure is the Burgers Vector, b. The line bounds the glided area of the glide plane (100) from the unglided area.

As the temperature is raised, the vibrational energy increases, because it is kBT in each direction. If we have a simple cubic crystal in which the intermolecular spacing is r then the molar volume is Nar3. The Young s modulus for the crystal is Y and we assume a Hooke s law spring. We can define the local stress as the applied force per molecule, Fm, divided by r2, giving a local strain of x/r where x is the extension caused by the oscillation. Hence ... 

Simple cubic crystal array indicating lattice points and unit cell. 

First attempts (14-161 described only the hardness of simple cubic crystals of the type AB as a function of their interatomic distances and the valencies of the atoms. This is shown in Equation 2 ... 

You now know how to define a supercell for a DFT calculation for a material with the simple cubic crystal structure. We also said at the outset that we assume for the purposes of this chapter that we have a DFT code that can give us the total energy of some collection of atoms. How can we use calculations of this type to determine the lattice constant of our simple cubic metal that would be observed in nature The sensible approach would be to calculate the total energy of our material as a function of the lattice constant, that is, tot(a). A typical result from doing this type of calculation is shown in Fig. 2.1. The details of how these calculations (and the other calculations described in the rest of the chapter) were done are listed in the Appendix at the end of the chapter. 

Figure 2.1 Total energy, L tot, of Cu in the simple cubic crystal structure as a function of the lattice parameter, a. The filled symbols show the results of DFT calculations, while the three curves show the fits of the DFT data described in the text.

The simple cubic crystal structure we discussed above is the simplest crystal structure to visualize, but it is of limited practical interest at least for elements in their bulk form because other than polonium no elements exist with this structure. A much more common crystal stmcture in the periodic table is the face-centered-cubic (fee) structure. We can form this structure by filling space with cubes of side length a that have atoms at the corners of each cube and also atoms in the center of each face of each cube. We can define a supercell for an fee material using the same cube of side length a that we used for the simple cubic material and placing atoms at (0,0,0), (0,g/2,g/2), (g/2,0,g/2), and (g/2,g/2,0). You should be able to check this statement for yourself by sketching the structure. 

Recently, the kinetics of faceting was studied using a Monte Carlo simulation (109). Evaporation-condensation and surface diffusion were considered for the simple cubic crystal with nearest- and next-nearest-neighbor interactions. It was... 

Estimate for a simple cubic crystal the relative energies of a 410 and 420 face. (Find /4io/7420-)... 

Atoms in the bulk simple cubic crystal have the six nearest neighbors and 12 second-nearest neighbors shown in the upper right figure for a total binding energy of 6cj>i + 122- Each bond is shared between two atoms. Hence, the mean sublimation energy of the crystal is one-half that value, or... 

PROBLEM 7.4.4. For a monoatomic cubic crystal consisting of spherical atoms packed as close as possible, given the choices of a simple cubic crystal (SCC atom at cell edges only this structure is rarely used in nature, but is found in a-Po), a body-centered cubic crystal (BCC, atom at comers and at center of body), and a face-centered cubic crystal (FCC body at face comers and at face centers), show that the density is largest (or the void volume is smallest) for the FCC structure (see Fig. 7.12). In particular, show that the packing density of spheres is (a) 52% in a simple cubic cell (b) 68% for a body-centered cell (c) 71% for a face-centered cubic cell. 

PROBLEM 8.6.1. A fictional simple cubic crystal has a lattice constant a = 4.21 A. Compute the four lowest free-electron energy levels along the wavevector k in the reduced zone scheme at the fc-space point (n/2a, 0, 0). 

Formation of the energy-band structure of KCl. We start with argon atoms, and then put them in a simple-cubic crystal structure. Protons arc then transferred between neighboring nuclei to form potassium and chlorine ions. 

FIGURE 20.7 Schematic illustrating the step growth mechanism, considering a 100 surface of a simple cubic crystal as an example and each atom as a cnbe with a coordination number of six (six chemical bonds) in bnUc crystal. 

FIGURE 20.9 Schematic illustrating the PBC theory. In a simple cubic crystal, 100 faces are flat surfaces (denoted as F-faces) with one PBC running through one such surface, 110 are stepped surfaces (S-faces) that have two PBCs, and 111 are kinked surfaces (K-faces) that have three PBCs. (From Hartman, R, and Perdok, W.G., Acta CrystL, 8, 49, 1955.)... 

FIGURE 21.11 Constructive Interference of x-rays scattered by atoms In lattice planes. Three beams of x-rays, scattered by atoms In three successive layers of a simple cubic crystal, are shown. Note that the phases of the waves are the same along the line CH, Indicating constructive Interference at this scattering angle 20. 

As stated in Sec. 3-6, when monochromatic radiation is incident on a single crystal rotated about one of its axes, the reflected beams lie on the surface of imaginary cones coaxial with the rotation axis. The way in which this reflection occurs may be shown very nicely by the Ewald construction. Suppose a simple cubic crystal is rotated about the axis [001]. This is equivalent to rotation of the reciprocal lattice about the bs axis. Figure A1-9 shows a portion of the reciprocal lattice oriented in this manner, together with the adjacent sphere of reflection. 

Figure 4-15 A screw dislocation in a simple cubic crystal. AB, BC are dislocations. The screw dislocation AD is parallel to BC (D is not visible).

Mn has a complicated (simple cubic) crystal structure with four inequivalent atomic positions and 58 atoms in the unit cell. See Table 25 3 of Ref. 191. "Extended x-ray absorption fine structure (EXAFS) in argon. 

Fig. 3. Periodic variation of energy of adsorption on a simple cubic crystal.

Figure 8.S I The three cubic crystal lattices are shown. In a simple cubic crystal, atoms are located at each of the corners of a cube. In a body-centered cubic crystal, an additional atom sits at the center of the cube, and in a face-centered cubic crystal, atoms are found at the center of each face of the cube. Each of these arrangements repeats throughout the crystal. All of the atoms in each of these structures are identical dififerent colors are used only to help you to see the different positions in the lattice.

Polonium is the only metal that forms a simple cubic crystal structure. Use the fact that the density of polonium is 9.32 g/cm to calculate its atomic radius. 

The orientation of an atomic plane in a crystal lattice is the reciprocal of the fractional intercepts (i.e. how far along the unit cell) which the plane makes with the crystallographic axes. Figure 12.18 depicts a simple cubic crystal, where the distance OG is taken as unity and the Miller Indices are calculated as shown in Table 12.1. The Zig-zag (101) and Armchair (112) faces of the graphite crystal are depicted as shown ... 

Table 12.1 Some Miller Indices of a simple cubic crystal (see Fig. 12.17)...

As indicated in Sect. 3.3.1, dislocations are line defects. The two basic types of dislocations are the edge and screw dislocations. A schematic three-dimensional (3D) illustration of an edge dislocation appears in Fig. 3.25. A (100) plane of Fig. 3.25 in a simple cubic crystal is illustrated schematically in Fig. 3.28. This illustration will help to define the Burgers vector later on. Figure 3.29 is a schematic view of edge and screw dislocations. 

At elevated temperatures, due to thermal fluctuations, singular faces acquire a surface roughness manifested by the formation of adatoms and surface vacancies, di- or polyatomic surface clusters, and vacancies and steps with kinks as shown in Figure 1. The surface roughness becomes appreciable only above a critical roughening temperature T,. The theoretical value of Tr for a simple cubic crystal lattice, T, = where i(fx denotes the... 

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