Space lattice unit cell volume

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

Lattice parameters of Sm2Fel A.Co c)17C3, with various x and y values are listd in Table 10.2. The XRD lines observed on all the Sm2(Fe, xCox) 7Cy compounds were completely assignable to the Th2Zn]7-type crystal lattice of rhombohedral symmetry (space group = R3m), and the lattice parameters were in good agreement with reported values.3,35 The unit cell volume decreased with the concentration of Co but increased with the carbon content. This indicates that carbon atoms share the interstitial 9e sites.36... 

Independent of the lattice symmetry, a linear dependence of the lattice parameters (determined by the least-squares fit to the interplanar spacing of selected reflections in the XRD pattern [45] or by the more accurate full-profile fitting analysis [46]) on the Ti content has been found (Fig. 5). The equation relating the unit cell volume to the Ti content (Table 4) is particularly usefiil for determining the real framework composition directly from XRD analysis by comparing this with the Ti content resulting from elemental analysis, the amount of possible extra-framework Ti species can be estimated [46]. 

Fig. 10. Rietveld fits to the X-ray diffraction data in the orthorhombic Pnma space group for the Ndo.8 Sro.2 Mni-xCoxOs compounds. The labels "sg" and Td" indicates that samples were prepared using the sol gel and freeze-drying techniques, respectively. In each case, lattice parameters (a, b, c), unit cell volume (V) and theoretical density (p) data are inset.

However, the d 2 of the hcp(OOOl) surface of Be and Mg and the dimer bonds of the Il-b elements of Zn, Cd, and Hg have been reported to expand. With a reduction in Se grain size from 70 to 13 nm, the a lattice was found to expand by 0.3 %, but the c lattice spacing decreases shghtly, which expands the unit-cell volume by about 0.7 % at D = 13 nm [13]. The reported expansion appears off line with notations of Goldschmidt and Pauling who emphasized that the global bond contraction depends uniquely on the reduction in atomic CN, and it is independent of the bond namre or the particular constituent elements (Appendix A2) (Table 12.2). 

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

In cellulose II with a chain modulus of 88 GPa the likely shear planes are the 110 and 020 lattice planes, both with a spacing of dc=0.41 nm [26]. The periodic spacing of the force centres in the shear direction along the chain axis is the distance between the interchain hydrogen bonds p=c/2=0.51 nm (c chain axis). There are four monomers in the unit cell with a volume Vcen=68-10-30 m3. The activation energy for creep of rayon yarns has been determined by Halsey et al. [37]. They found at a relative humidity (RH) of 57% that Wa=86.6 kj mole-1, at an RH of 4% Wa =97.5 kj mole 1 and at an RH of <0.5% Wa= 102.5 kj mole-1. Extrapolation to an RH of 65% gives Wa=86 kj mole-1 (the molar volume of cellulose taken by Halsey in his model for creep is equal to the volume of the unit cell instead of one fourth thereof). 

Now that we know how to find the cell volume, we can use some previous information to calculate an important property of a material, namely, its density, which we represent with the lowercase Greek letter rho, p. For example, aluminum has an FCC space lattice. Recall that there are fonr atoms in the FCC unit cell. We know that each aluminum atom has an atomic weight of 27 g/mol. From Table 1.11, the cnbic lattice parameter for alnminnm is 4.05 A, or 0.405 nm (4.05 x 10 cm). This gives ns a volume of = 6.64 x 10 cm. You should confirm fhaf fhe fheorefical densify for aluminum is then ... 

Applying this criterion to. a lattice model, v corresponds roughly to the volume of a unit cell, and / to a lattice spacing. Hence the values of n 10 for exact enumeration quoted in Section IV seem quite reasonable. Certainly there is no support for the claim by Flory and Fisk31 that the 6/5 power law is attained only for n > 10.6... 

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