Axes, unit cell

FIGURE 1.22 Definition of axes, unit cell dimensions, and angles for a general unit cell. 

Structural details concerning coordination polyhedra are still best elucidated by single crystal X-ray and neutron diffraction studies, although it is clear that the newer techniques of NMR and ESR have apphcation. The structural variety displayed by the complexes between the actinide fluorides and the alkah fluorides has been amply demonstrated in the foregoing sections. Even with their different structures, such complexes display common characteristics which are independent of such packing details. Prominent among such general properties axe unit cell volume and refractivity volume, both of which are additive functions of composition. 

Macromolecule Crystal system Space group Mol. helix Unit cell axes Unit cell angles No. units Pc (gcm-3)... 

An experimental teclmique that is usefiil for structure studies of biological macromolecules and other crystals with large unit cells uses neither the broad, white , spectrum characteristic of Lane methods nor a sharp, monocliromatic spectrum, but rather a spectral band with AX/X 20%. Because of its relation to the Lane method, this teclmique is called quasi-Laue. It was believed for many years diat the Lane method was not usefiil for structure studies because reflections of different orders would be superposed on the same point of a film or an image plate. It was realized recently, however, that, if there is a definite minimum wavelengdi in the spectral band, more than 80% of all reflections would contain only a single order. Quasi-Laue methods are now used with both neutrons and x-rays, particularly x-rays from synclirotron sources, which give an intense, white spectrum. 

More quantitatively, it appears to a first approximation that the unit cell contraction of a compound containing Mn2+, Co2+, Ni2+, or Fe2+, relative to the isomorphous Mg+2 compound, is a linear function of the Ax of the metal-ligand bond if we neglect selenides and tellurides. Inclusion of these more covalent compounds indicates a greater dependence on Ax. 

The sodium chloride structure, AX systems. Cubic Fm3m (Space Group 225) The sodium chloride or rock salt, NaCl, structure has a simple face-centered cubic unit cell (Figure 8) with alternating cations-anions along the three cubic axes. 

The description of this structure is more complicated than that of Ba2YCu3Ox. There are six layers in the unit cell of this structural type and they can be viewed in two quite different ways. In the first interpretation, we divide the six layers into two blocks of three layers each, the first being (AX)0(BX2)C(AX)0 and the second (AX)C(BX2)0 (AX)C. These layers and these sequences are typical of perovskite and, therefore, in this description the structure is considered to be made of two perovskite blocks related to one another by a shift of origin of t = (l/2)(a + b). We may also regard the structure, however, as containing alternate blocks of perovskite (layers (BX2)0 c) and rock salt (layers (AX)co(AX)oc). As before, the unit cell is made of two... 

The corresponding relation between the host and guest crystals when evaluating the misfit ratio may be a one-to-one lattice relation in the same direction (a X b to a xb axes), or in different axial directions (aX b axes versus aX <110> axes), or on the basis of one unit cell versus a few unit cell sizes (see Fig. 7.13). Royer s misfit ratio is generally a two-dimensional correspondence, but Hartman [13] extended this relation to the misfit ratio in PBCs (see Section 4.2), which is a one-dimensional correspondence. Royer s epitaxial relations correspond to a relation between the F faces of the host and guest crystals containing more than two PBCs, and an epitaxial relation is not allowed between S faces or K faces. In Hartman s analysis, rela-... 

Consider two different choices of unit cells a F-centered unit cell with axes (ax, b, ci) and a primitive one with axes (fl2, b2, C2), as shown in Fig. 9.2.4. We can write... 

Figure 4-2. Energy conservation in CP-MD the potential energy (Ee, main axis), temperature (kinetic energy, T, auxiliary, right-hand side axis), physical energy (T + Ee, auxiliary axis), and conserved energy (Econs). The difference between Ec0 s and T + Ee is the fictitious kinetic energy of the wavefunction. The data from the simulation for the ethylene molecule with the CPMD program13 (Troullier-Martins pseudopotentials1415, time step of 4 a.u., fictitious mass 400 a.u., cut-off energy 70 Ry, unit cell 12 Ax 12 A xl2 A)...

Fig. 1. The ax projection of half a unit cell indicating packing and hydrogen bond scheme of pilocarpine trichlorogermanate(II), according to Fregerslev and Rasmussen (48).

Figure 7.22. Distributions of the nuclear density in the unit cell of NiMn02(0H) at jc = 0 a) the difference Fourier map calculated using AF = IFobs-F aid and phase angles calculated without H atom (pmi = -2.4 fm/A, p ,ax = 2.4 fm/A, Ap = 0.4 fin/A, p(H) = -2.4 fm/A ) b) the conventional Fourier map computed using Fobs and phase angles calculated including the H atom (p j = -12 fm/A, pn,ax = 18 fm/A (high p on Ni atoms is not shown), Ap = 2 fm/A, p(H) = -10 fm/A, p(Mn) = -12 fm/A ). Solid lines show positive values, thin dotted lines negative, and thick dotted lines indicate zero level.

It is natural to enquire why different AX compounds should possess different structures, and, in particular, why CsCl, CsBr and Csl should have a structure different from that of the other alkali halides. We can answer this question if we consider fig. 3.07a, which represents a section through the caesium chloride unit cell on a vertical diagonal plane. The ions in this diagram are shown in their correct relative sizes for Cs+ and Cl-, and anions and cations are seen to be in contact at the points P. Now let us suppose that the cations are replaced by others of... 

A co-ordination of 4 2 is found among AX% halides only in BeF2, which has the (idealized) / -cristobalite structure, named after one of the forms of Si02. The cubic unit cell of this structure is shown in... 

In Fig. 12 these three different nets are shown with the P points in the voids of the G-net and the N-net marked by squares. The axis of the unit cell has to be enlarged by the factor for the G-net and by a factor of 2 for the N-net with respect to the primitive lattice, formed by centering voids. The first case is indicated by the subscript V, the second case by the subscript 2, actually it should be 22 for ax, 2- Therefore the following equations can be written ... 

Then, we calculate the elements Fpq for all atomic orbitals p, g for unit cells 7 — 0,1,2,... Jmax- What is 7 iax The answer is certainly non-satisfactory ymax = 00. In practice, however, we often take 7 ,ax as being of the order of a few cells most often, we take jn,ax = 1. For each from the FEZ, we calculate the elements Fpq and Spq of Eqs. (9.56) and (9.58), and then solve the secular equations within the Haitiee-Fock-Roothaan procedure. This step requires diagonalization (see Appendix K available at booksite.elsevier.com/978-0-444-59436-5, p. el05). As a result, for each k we obtain a set of coefiicients c for the crystal orbitals and the energy eigenvalue Sn(k)-... 

Then, we calculate the elements for all atomic orbitals p, q for unit cells j = 0,1,2,..., /max- What is max The answer is certainly non-satisfactory max = oo. In practice, however, we often take jmax as being of the order of a few cells, most often we take" ,ax = 1. 

In the numerical calculation, a 3-D mesh with unit cell size (Ax) is used. The angle at the lattice sites i,j, k) is calculated by... 

From the X-ray diffraction data, the basal spacings of [B] and [D] were estimated to be 13.6 A in both cases, which can be interpreted as the sum of the van der Waals interval occupied by alkyl chains, 4.0 A, and of that of the silicate layer, 9.6 A, respectively. It shows that the intercalated cations are oriented with the flat-lying monolayer structure in the interlamellar space of the silicate. The interlayer surface area of an unit cell can be calculated as ax b = 5. 14 Ax9.00 A = 46.3 A for montmorillonite and 5.28 Ax9.18 A = 48.5 A for saponite respectively. Since the effective surface area of one -hexylammonium ion is 45.16 A [11], the silicate surface is covered by n-hexylammonium cations up to about 69% in the montmorillonite and to 75% in the saponite. 

C4v symmetry C.S.4 molecule aligned with four-fold axis parallel to unique c axis of tetragonal unit cell Ax corrected for diamagnetic anisotropy Ax, /C, ... 

Consider the following two-dimensional model the lattice is a square of side a with lattice vectors ai = ax and 82 = fly there are three atoms per unit cell, one Cu atom and two O atoms at distances ai/2 and 82/2, as illustrated in Fig. 4.13. We will assume that the atomic orbitals associated with these atoms are orthogonal. 

To calculate the lattice energy, let us consider a common ionic solid NaCl. To confirm that NaCl is a predominantly ionic solid, we can find its Ax value. Putting the values for the electronegativity of Cl and Na in the equation for calculating Ax, we get Ax = 3.16-0.93 = 2.13, which is very high, showing that NaCl is predominantly ionic bonded. Figure 8.1a shows the model of a NaCl unit cell. 

Perhaps the most common AX crystal structure is the sodium chloride (NaCl), or rock salt, type. The coordination number for both cations and anions is 6, and therefore the cation-anion radius ratio is between approximately 0.414 and 0.732. A unit cell for this crystal structure (Figure 12.2) is generated from an FCC arrangement of anions with one cation situated at the cube center and one at the center of each of the 12 cube edges. An equivalent crystal structure results from a face-centered arrangement of cations. Thus, the rock salt crystal structure may be thought of as two interpenetrating FCC lattices—one composed of the cations, the other of anions. Some common ceramic materials that form with this crystal structure are NaCl, MgO, MnS, LiF, and FeO. 

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