Cubic crystal lattices

Only body-centered cubic crystals, lattice constant 428.2 pm at 20°C, are reported for sodium (4). The atomic radius is 185 pm, the ionic radius 97 pm, and electronic configuration is lE2E2 3T (5). Physical properties of sodium are given ia Table 2. Greater detail and other properties are also available... 

Primitive cubic crystal lattice. One unit cell is marked... 

KEY TERMS face-centered cubic crystal lattice octahedral hole... 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

Soft silvery metal body-centered cubic crystal lattice density 5.24 g/cm melts at 822°C vaporizes at 1,596°C electrical resistivity 81 microhm-cm reacts with water soluble in liquid ammonia. 

YeUow metal face centered cubic crystals lattice constant, a at 25°C 4.0786A density 19.3 g/cm hardness 2.5-3.0 (Mohs), 18.5 (BrineU) melts at 1,064°C vaporizes at 2,856°C electrical resistivity 2.051 microhm-cm at 0°C and 2.255 microhm-cm at 25°C Young s modulus 11.2x10 psi at 20°C (static) Poisson s ratio 0.52 thermal neutron capture cross section 98.8 barns insoluble in almost all single acids or hydroxide solutions dissolves in aqua regia. 

Silvery-white metal close-packed cubic crystals lattice constant 3.8394A at 20°C density 22.42 g/cm (highest among metals) melts at 2410°C vaporizes at 4,130°C hardness 6-6.5 Mohs electrical resistivity 4.71 j,ohm-cm Young s modulus 3.75 x 10 tons/in magnetic susceptibility 0.133 x 10 cm3/g thermal neutron absorption cross section 440 barns. 

The aim of this work is to elucidate these problems. To this end, we calculate the effective spin Hamiltonian of the 5f2—5f2 superexchange interaction between the neighboring U4+ ions in the cubic crystal lattice of UO2 and we calculate T5 <%> eg, rs f2g(l) ancl r5 f2g(2) linear vibronic coupling constants. These data are then used to draw a more definite conclusion about the driving force of the phase transition and especially about the actual mechanism of the spin and orbital ordering in U02. 

Peculiarity of the fullerene molecule formation also reveals itself in a fullerite crystal structure. Cubic crystal lattices of fullerites and hydrofullerites behave like those of different metals and alloys. Fullerene molecules are distributed in the lattice sites while atoms of elements are distributed in the octa- and tetrahedral interstitial sites forming the interstitial solid solutions. Fullerene molecules substitute each other in the sites of lattice and form the substitution solid solutions. Forming exo- and endocompounds, fullerene molecules that are in the lattice sites can change considerably the properties of crystal, whereas its crystalline structure remain unchanged. 

Most of the suggested refined methods of treating the environment require considerable complication of the scheme of calculations and much computer time. Therefore they were mainly used in the case of sufficiently homogeneous systems of rather simple structures graphite, metals, and oxides with cubic crystal lattices. In contrast, the real surface of most oxides is characterized by... 

CuS04-5H20 The ratio of copper to sulfate is 1 1, so why does it not have a cubic crystal lattice like NaCl ... 

Scanning electron micrographs of biooxidized pyrite showed the formation of deep pits in crystal surfaces. The pores appear to be hexagonal in cross-section, consistent with screw dislocations in a cubic crystal lattice (43), and suggesting that the bacteria have attacked... 

Figure 2,3 Three-dimensional face-centered cubic crystal lattice structure. The atoms of every third and all consecutive third layers are positioned above an empty B or C place of the first layer. A crystal with cubic and octahedral faces results. The structure of the octahedral (111) faces is a hep structure. The smooth quadratic structure of the cubic (100) faces is also clearly seen, The unit cell containing 14 atoms is shown on the right-hand side. The (111) plane is given by the hatched surface.

Face-centered cubic crystal lattice. Burns when heated with a hot enough flame (over 800, oxygen torch), df 3.513. rt 2 4173. Hardness — 10 (Mohs scale), Sp heat at 100°K 0.606 cal/g-atom/ K. Entropy at 298.I6 K 0,5684 cal/g-atom/ K. Band gap energy 6.7 ev. Dielectric constant 5.7. Electron mobility —1800 cm1/v-sec. Hole mobility 1200 cmz/v-sec. Can be pulverized in a steel mortar. Attacked by laboratory -type cleaning soln (potassi um dichromate + coned HiSO ), In the jewelry trade the unit of weight for diamonds is one carat — 200 mg. Ref Wall Street J. 164, no. 36, p 10 (Aug 19, 1964),... 

Body-centered cubic crystal lattice d 5,244 mp 826°, Sol in liq ammonia. Shows two reduction potentials —0,710 and —2.510 v. (referred to a normal calomel electrode) Noddack, Brukl, Angew. Chem. 50, 362 (1937) gives two definite series of salts one in which the metal Is bivalent, and another in which it is trivalent. 

In the patent by Hill, an aUoy of titanium containing 13 wt%vanadium, 11 wt% chromium and 3 wt% aluminum was developed as a hydrogen transport membrane material [12]. In this alloy, the crystal structure of titanium, which is normally hexagonal below 1153 K (880 °C), is stabilized in its high-temperature body centered cubic allotropic form. The body centered cubic crystal lattice is preferred for hydrogen transport. This titanium alloy was found to have hydrogen permeability superior to that of pure palladium in the range 300-450 °C (573-723 K)... 

The formation of PbOn during the photo-electrochemical reaction can be considered to be a process similar to the chemical oxidation of tet-PbO by O2 at elevated temperatures (3(X)—350 °C). During the thermal process, non-stoichiometric PbO is formed with a pseudo-cubic crystal lattice and variable composition (Fig. 2.18) [36]. 

Fig. 1.8 (a) Body-centered cubic crystal lattice, (b) Face-centered cubic crystal lattice. 

Figure 4.8a, iQustrates the construction of the Wigner-Seitz cells for the face-centered and the body-centered cubic crystal lattices. An atom is located at the center of each polyhedron. 

Figure 4.13 First Brillouin zone of the body-centered cubic crystal lattice, k, ky,kz are the axes of a Cartesian coordinate system in fe-space. The symmetry points and symmetry lines are indicated. See Table 4.1 for details.

Figures 4.13 and 4.14 present the Brillouin zones for cubic crystal lattices. One can see the Brillouin zone for hexagonal close-packed crystal lattice in Figure 4.15.

Table 4.1 Points and directions of high symmetry in the first Brillouin zones, fee is the face-centered cubic crystal lattice bcc is the body-centered cubic crystal lattice hep is the hexagonal close-packed crystal lattice.

The Q matrix in case of the cubic crystal lattice is written as... 

Foreman and Lomer were the first to report that if the vibrations propagate in the cubic crystal lattice along the high symmetrical directions (111), (110), (100), the mathematical description of the process can be reduced to a linear chain of atoms (48). The problem of vibrations of parallel atomic planes is reduced to the problem of interacting oscillating points of the equal mass. Every oscillatory mode can be treated as related to a separate linear chain. 

Table 12.3 shows the values of the calculated interplanar force constants for five metals with the body-centered cubic crystal lattice. For simple metals sodium and potassium the first interplanar coefficients are of the order of several N m h These coefficients of niobium and molybdenum make tens of N m. The F3 coefficient is maximal. 

Dispersion curves of longitudinal and transverse osciQations along direction a type of that is L[lll] and T[lll] for potassium are presented in Figure 12.4. This direction is chosen because the distance between neighboring atoms is minimal along it in the body-centered cubic crystal lattice. Similar curves are typical for other alkali elements. 

Ice can crystallize forming 12 different structures. At atmospheric pressure the freezing water transforms into ice termed Ih that has a hexagonal crystal lattice. The Ic ice with the cubic crystal lattice is formed at low temperatures, that is at 143 K (-130 ° C). Phases of ice other than the Ih phases are produced by the application of high pressures. 

The organic crystals constructed from nonpolar molecules often form structures packed face-to-face in stacks. Other neutral organic molecules compose the face-centered cubic crystal lattice. Examples of these structures are illustrated in Figures 15.6 and 15.7. 

I describe the arrangement of atoms in the common cubic crystal lattices and calculate the packing efficiency for a lattice. 

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.

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