Unit cell lengths

The suppression of C60 crystallite formation in mixed LB films was attempted by mixing C60 and amphiphilic electron donor compounds [259]. Observation of the C60 LB film transferred horizontally by TEM clearly showed 10-40-nm-size crystallites. The diffraction pattern gave an fee lattice with unit cell length 1.410 nm. Examination of the mixed films with arachidic acid by TEM showed extensive crystallite formation. Mixed LB films of three different amphiphilic derivatives of electron donors with C60 were examined. One particular derivative showed very little formation of C60 crystallites when LB films were formed from monolayers of it mixed with C60 in a 1 2 ratio, while two others reduced C60 crystallite formation but did not eliminate it. 

Thus, these intercepts are given in terms of the actual unit-cell length found for the specific structure, and not the lattice itself. The Miller Indices are thus the indices of a stack of planes within the lattice. Planes are important in solids because, as we will see, they are used to locate atom positions within the lattice structure. 

It is sometimes important to specify a vector with a definite length, perhaps to indicate the displacement of one part of a crystal with respect to another part. In such a case, the direction of the vector is written as above, and a prefix is added to give the length. The prefix is usually expressed in terms of the unit cell dimensions. For example, in a cubic crystal, a displacement of two unit cell lengths parallel to the b axis would be written 2a [010]. 

The position of an atom or ion in a unit cell is described by its fractional coordinates these are simply the coordinates based on the unit cell axes (known as the crystallographic axes), but expressed as fractions of the unit cell lengths. It has the simplicity of a universal system which enables unit cell positions to be compared from structure to structure regardless of variation in unit cell size. 

FIGURE 2.14 The magnetic ordering in NiO. The Ni planes only are pictured, and alternate close-packed layers have opposing magnetic moments. Note that the magnetic unit cell length is double that of the normal unit cell. 

The sin 6 values for Cs2TeBr0 are listed in Table 2.4 for the observed reflections. To which cubic class does it belong Calculate its unit-cell length assuming Cu-Ka radiation of wavelength 154.2 pm. 

Use the data given in Question 5 to calculate a unit-cell length for the NaCl unit... 

If the unit cell length of NaCl is a=563.1 pm and the density of NaCl is measured to be 2.17x10 kg m calculate Z, the number of formula units in the unit cell. (The atomic masses of sodium and chlorine are 22.99 and 35.45, respectively.)... 

A powder diffraction pattern establishes that silver crystallizes in a face-centred cubic unit cell. The 111 reflection is observed at 0=19.1°, using Cu-Ka radiation. Determine the unit cell length, a. If the density of silver is 10.5x10 kg m and Z=4, calculate the value of the Avogadro constant. (The atomic mass of silver is 107.9.)... 

Cu crystallizes with a cubic close-packed structure. The Bragg angles of the first two reflections in the powder pattern collected using Cu-Ka radiation are 21.6° and 25.15°. Calculate the unit cell length a, and estimate a radius for the Cu atom. 

In summary, non-stoichiometric compounds are found to exist over a range of composition, and throughout that range the unit cell length varies smoothly with no change of symmetry. It is possible to determine whether the non-stoichiometry is accommodated by vacancy or interstitial defects using density measurements. 

In the sphalerite structure the anions form a cubic close packed array. The structure has a single adjustable parameter, the cubic cell edge. The 0 ions are too small for them to be in contact in this structure (see Fig. 6.4) so ZnO adopts the lower symmetry hexagonal wurtzite structure which has three adjustable parameters, the a and c unit cell lengths and the z coordinate of the 0 ion, allowing the environment around the Zn " ion to deviate from perfect tetrahedral symmetry. In the sphalerite structure the ZnX4 tetrahedron shares each of its faces with a vacant octahedral cavity (one is shown in Fig. 2.6(a)), while in the wurtzite structure one of these faces is shared with an empty tetrahedral cavity which places an anion directly over the shared face as seen in Fig. 2.6(b). The primary coordination number of Zn " in sphalerite is 4 and there are no tertiary bonds, but in wurtzite, which has the same primary coordination number, there is an additional tertiary bond with a flux of 0.02 vu through the face shared with the vacant tetrahedron. 

From the computer program included with this book, at 273.15 K the hydrate is predicted to have a dissociation pressure of 25.1 atm. The hydrate is predicted to be si, with a fractional occupancy of cavities as 0S = 0.870 and l = 0.973. There are 46 water molecules per si unit cell, with 2 small (512) and 6 large (51262) cavities, and the unit cell length is 12 A on one side. The density is then calculated from Equation E5.2.1 as... 

Translation Search. A translation search was done by using the P2 probe molecule oriented by the rotation function studies and refined by the Patterson correlation method. The translation search employed the standard linear correlation coefficient between the normalized observed structure factors and the normalized calculated structure factors (Funinaga Read, 1987 Brunger, 1990). X-ray diffraction data from 10-3 A resolution were used. Search was made in the range x = 0-0.5, y = 0—0.5, and z = 0-0.5, with the sampling interval 0.0125 of the unit cell length. 

Figure 8.14 Translational symmetry operations in crystals. A screw axis, symbol nm, involves a translation by a fraction of m/n of the unit cell length followed by a rotation of 360In degrees thus a twofold screw axis, 2[, involves a 180° rotation.

Screw axis nm (180° rotation followed by translation by of the unit cell length parallel to the rotation axis, e.g. 2, 180° rotation plus a half cell translation)... 

Potassium crystallizes in a body-centered cubic lattice (unit cell length a = 520 pm). 

Each rubidium halide (Group VIIA element) crystallizing in the NaCl-type lattice has a unit cell length 30 pm greater than that for the corresponding potassium salt of the same halogen. What is the ionic radius of Rb+ computed from these data ... 

ASTM has issued a recommended standard practice for testing fixed wavelength detectors in LC it is ASTM 685-79. In contrast to normal practice, they have defined noise in absorbance units (AU) per unit cell length. Otherwise the specifications follow the discussions presented in Chapter 7. 

---