Cubic cell

The free streaming tenn can be written as the difference between the number of particles entering and leaving the small region in time 5t. Consider, for example, a cubic cell and look at the faces perpendicular to the v-... 

The most widely used type of trap for the study of ion-molecule reactivity is the ion-cyclotron-resonance (ICR) [99] mass spectrometer and its successor, the Fourier-transfomi mass spectrometer (FTMS) [100, 101]. Figure A3.5.8 shows the cubic trapping cell used in many FTMS instmments [101]. Ions are created in or injected into a cubic cell in a vacuum of 10 Pa or lower. A magnetic field, B, confines the motion in the x-y... 

The truncated octahedron and the rhombic dodecahedron provide periodic cells that are approximately spherical and so may be more appropriate for simulations of spherical molecules. The distance between adjacent cells in the truncated octahedron or the rhombic df)decahedron is larger than the conventional cube for a system with a given number of particles and so a simulation using one of the spherical cells will require fewer particles than a comparable simulation using a cubic cell. Of the two approximately spherical cells, the truncated octahedron is often preferred as it is somewhat easier to program. The hexagonal prism can be used to simulate molecules with a cylindrical shape such as DNA. 

The period of the lamellar structures or the size of the cubic cell can be as large as 1000 A and much larger than the molecular size of the surfactant (25 A). Therefore mesoscopic models like a Landau-Ginzburg model are suitable for their study. In particular, one can address the question whether the bicontinuous microemulsion can undergo a transition to ordered bicontinuous phases. 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

Therefore the relationship between these interconvertible structures originates from a cubic anion lattice of 32 0 ions in the cell. With 32 Fe ions in the octahedral holes stoichiometric FeO is formed. Replacement of a number of Fe ions with two-thirds of their number of Fe ions maintains electrical neutrality but provides non-stoichiometric Fei 0. Continual replacement in this way to leave 24 Fe atoms in the cubic cell produces Fej04, and... 

Simple cubic cell (SC). This is a cube that consists of eight atoms whose centers are located at the comers of the cell Atoms at adjacent corners of the cube touch one another. 

Face-centered cubic cell (FCC). Here there is an atom at each comer of the cube and one in the center of each of the six faces of the cube. In this structure, atoms at the comers of the cube do not touch one another they are forced slightly apart Instead, contact occurs along a face diagonal The atom at the center of each face touches atoms at opposite comers of the face. 

Table 9.6 lists three other ways in which these types of cubic cells differ from one another. 

Beta radiation Electron emission from unstable nuclei, 26,30,528 Binary molecular compound, 41-42,190 Binding energy Energy equivalent of the mass defect measure of nuclear stability, 522,523 Bismuth (m) sulfide, 540 Blassie, Michael, 629 Blind staggers, 574 Blister copper, 539 Blood alcohol concentrations, 43t Body-centered cubic cell (BCC) A cubic unit cell with an atom at each comer and one at the center, 246 Bohrmodd Model of the hydrogen atom... 

Face-centered cubic cell (FCC) A cubic unit cell with atoms at each corner and one at the center of each face, 246 Fahrenheit, Daniel, 8 Fahrenheit temperature scales, 8... 

Significant figure A meaningful digit in a measured quantity, 9,20-2 lq ambiguity in, 10 in inverse logarithms, 645-647 in logarithms, 645-647 Silicate lattices, 243 Silicon, 242-243 Silver, 540-541 Silver chloride, 433,443-444 Simple cubic cell (SC) A unit cell in which there are atoms at each comer of a cube, 246... 

M3Ta02F4 and M3Nb02p4, where M = K, Rb, Cs, form ciystal structures similar to that of Rb3TiF6, which is described by Bode and Voss [189]. The structure of the compounds can be described as a tetragonally distorted cubic cell of the (NFL )3FeF6 type, the structure of which is described by Steward and Rockby [190]. 

Self-Test 5.4A How many atoms are there in a primitive cubic cell (see Fig. 5.33) ... 

FIGURE 5.37 The calculation of the net number of atoms in a face-centered cubic cell. 

Buckminsterfullerene is an allotrope of carbon in which the carbon atoms form spheres of 60 atoms each (see Section 14.16). In the pure compound the spheres pack in a cubic close-packed array, (a) The length of a side of the face-centered cubic cell formed by buckminsterfullerene is 142 pm. Use this information to calculate the radius of the buckminsterfullerene molecule treated as a hard sphere, (b) The compound K3C60 is a superconductor at low temperatures. In this compound the K+ ions lie in holes in the C60 face-centered cubic lattice. Considering the radius of the K+ ion and assuming that the radius of Q,0 is the same as for the Cft0 molecule, predict in what type of holes the K ions lie (tetrahedral, octahedral, or both) and indicate what percentage of those holes are filled. 

The structure of PuB,qq is not yet clear. A satisfactory indexing can be made for a primitive cubic cell with a = 23.42 A, but with 10 reflections not characteristic of space group Fm3c . ... 

The low-temperature (/1-)AE3(BN2)2 phases exhibit two distinct structures for AE = Ca and Sr that can be derived from the cation disordering in their respective high-temperature phases. For / -Ca3(BN2)2 an orthorhombic (Cmca) superstructure of the cubic cell with fi-a bo a, Cq ly l a was obtained, in which the former 8f sites are occupied by seven calcium ions in an ordered fashion. In contrast, the structure of / -Sr3(BN2)2 is simply the result of a transition from a cubic body-centered (Im3m) into a primitive structure (Pm3m), in which the former 2 a position (0, 0, 0 1/2, 1/2, 1/2) is split into two independent positions, of which only one is occupied by strontium (Fig. 8.6). 

Fig. 8.6 Crystal structures of -Ca3(BN2)2 (s) snd -Sr3(BN2)2 (b). Squares indicate cation vacancies. The related cubic cell of a-Ca3(BN2)2 is highlighted in the structure ofy -Ca3(BN2)2-...

Fig. 9.2 (a) The extended icosahedral symmetry in Ga,2 Li2o Cu,2 Ga6o and (b) packing of such units in the cubic cell (Im3) of Lin3Cu6Ga2i-... 

In this diagram, a series of hexagon-shaped planes are shown which are orthogonal, or 90 degrees, to each of the corners of the cubic cell. Each plane connects to cuiother plane (here shown as a rectangle) on each fiace of the unit-cell. Thus, the faces of the lattice unit-cell and those of the reciprocal unit-cell can be seen to lie on the same pltme while those at the corners lie at right angles to the corners. 

The basis for FTMS is ion-cyclotron motion. A simple experimental sequence in FTMS is composed of four events quench, ion formation, excitation and ion detection. Ions are created in or injected into a cubic cell where they are held by an electric trapping potential and a constant magnetic field B. Each ion assumes... 

Also known for some time is a phase transition at low temperature (111K), observed in studies with various methods (NQR, elasticity measurement by ultrasound, Raman spectrometry) 112 temperature-dependent neutron diffraction showed the phase transition to be caused by an antiphase rotation of adjacent anions around the threefold axis ([111] in the cubic cell) and to lower the symmetry from cubic to rhombohedral (Ric). As shown by inelastic neutron scattering, this phase transition is driven by a low-frequency rotatory soft mode (0.288 THz 9.61 cm / 298 K) 113 a more recent NQR study revealed a small hysteresis and hence first-order character of this transition.114 This rhombohedral structure is adopted by Rb2Hg(CN)4 already at room temperature (rav(Hg—C) 218.6, rav(C—N) 114.0 pm for two independent cyano groups), and the analogous phase transition to the cubic structure occurs at 398 K.115... 

Ca3 Al2 Si3 012) in which silica tetrahedra and aluminum octahedra are linked together so that sets of eight oxygen ions form quasi-cubic cells with calcium ions at their centers (Figure 11.8). The overall cubic unit-cell contains eight formula units (80 atoms) and has a cell parameter of 11.82 A. 

VTO, and that of the G structure (Ia3d symmetry, Fig. 4) index to /6 /8 /l4 VT6 V O V"22 /7A. Finally the measurements of the surface area inside the cubic cell can yield partial information about a topology of the structure. One can expect a small unit cell and the small surface area per side of... 

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