Theoretical Crystal Density

Titanium pyrophosphate [13470-09-2] TiP20y, a possible uv reflecting pigment, is a white powder that crystallizes ia the cubic system and has a theoretical density of 3106 kg/m. It is iasoluble ia water and can be prepared by heating a stoichiometric mixture of hydrous titania and phosphoric acid at 900°C. 

X-ray diffraction allows the dimensions of the unit cell to be accurately measured. If the structure type of the material is known, the ideal cell contents are also known. Thus, the unit cell of a crystal of composition M2O3 that adopts the corundum structure contains 12 M atoms and 18 O atoms (Supplementary Material, SI). This readily allows the theoretical density of a solid to be calculated. The weights of all of the atoms in the cell are added, and this is divided by the cell volume. 

Several issues remain to be addressed. The effect of the mutual penetration of the electron distributions should be analyzed, while the use of theoretical densities on isolated molecules does not take into account the induced polarization of the molecular charge distribution in a crystal. In the calculations by Coombes et al. (1996), the effect of electron correlation on the isolated molecule density is approximately accounted for by a scaling of the electrostatic contributions by a factor of 0.9. Some of these effects are in opposite directions and may roughly cancel. As pointed out by Price and coworkers, lattice energy calculations based on the average static structure ignore the dynamical aspects of the molecular crystal. However, the necessity to include electrostatic interactions in lattice energy calculations of molecular crystals is evident and has been established unequivocally. 

Electron Density Studies of Molecular Crystals 281 table 12.3 Topological Analysis of Theoretical Densities on Strained Ring Molecules... 

The single crystals thus far prepared are too small for many physical property measurements. Also, the two techniques described for preparation of powders result in fine grained (and not sintered) materials which are also not appropriate for many measurements. A different technique, however, was recently developed which produces near theoretical density polycrystalline pellets (56). Stoichiometric mixtures of BaO, K02 and Bi2Os are mixed and melted in N2 gas and quickly quenched onto a copper block under the N2 atmosphere. The process must be performed... 

The theoretical density of a crystal can be obtained from the volume of the unit cell and the mass of the unit cell contents. The results of an X-ray diffraction structure determination gives both of these data, as the unit cell dimensions are accurately measured and the type and number of formula units in the unit cell are also determined. An example of this type of calculation for FeO follows ... 

The cubic form resembles diamond in its crystal structure and is almost as hard. The theoretical density is 3.48 g/mL. It is colodess and a good electrical insulator when pure traces of impurities add color and make it semiconducting, eg, a few ppm of Be make it blue and />-type whereas small amounts of S, Si, or CN favor yellow, -type crystals. It is possible to makep—n junctions by growing -type material on j -type seed crystals (12). If this is done carefully in an alkaline-earth nitride bath using a temperature difference technique, as with large diamond crystals (see Diamond, SYNTHETIC), the resulting diodes are several mm in size and emit blue light when forward-biased (13,14). 

This value, based on the experimental density, is to be considered the more reliable, since it is based on a measured property of CsCl, while the ionic radii are based on averages over many different compounds. Unit cell dimensions can be measured accurately by x-ray diffraction and from them the theoretical density can be calculated. The measured density is usually lower because most samples that are large enough to measure are not perfect single crystals and contain empty spaces in the form of grain boundaries and various crystalline imperfections. 

Paschalis, E., and A. Weiss (1969). Hartree-Fock-Roothaan wave functions, electron density distributions, diamagnetic susceptibility, dipole polarizability and antishielding factor for ions in crystals. Theoret. Chim. Act. 13, 381-403. 

An important implication of the phase transformation and crystallization discussed above is that they are often accompanied by a volume change which can cause shrinkage or expansion and consequently cracking. This is attributed to the different theoretical densities of different phases. A notable example is the transition from the high-temperature tetragonal to the monoclinic phase upon cooling. The transition is... 

A. Volkov, Yu. Abramov, P. Coppens and C. Gatti, On the origin of topological differences between experimental and theoretical crystal charge densities, Acta Cryst. A56, 332-339 (2000). 

The theoretical density of a crystal can be found by calculating the mass of all the atoms in the unit... 

The measured density is 2165 kg m-3. The theoretical density is almost always slightly greater than the measured density because real crystals contain defects that act so as to reduce the total mass per unit volume. 

The theoretical density of a solid with a known crystal structure can be determined by dividing the mass of all the atoms in the unit cell by the unit cell volume, (Chapter 1, Section 1.12). This information, together with the measured density of the sample, can be used to determine the notional species of point defect present in a solid that has a variable composition. However, as both techniques are averaging techniques they say nothing about the real organisation of the point defects. The general procedure is ... 

The values of detonation velocity, D, of the polynitro arenes studied were calculated using the known relationships of Kamlet Jacobs [72] for the maximum theoretical densities of crystals (i. e. for monocrystal). When the heat of formation of the substance was not available, the D values were obtained by using the relationship of Rothstein Petersen [73]. For compounds with sulphur heteroatom in molecule the D values were taken from Ref. [40]. The D values used are summarised in Table 1. 

Polonium crystallizes in a simple cubic unit cell with an edge length of 3.36 A. (a) What is the mass of the unit cell (b) What is the volume of the unit cell (c) What is the theoretical density of Po ... 

The atomic contents of the unit cell give the composition of the material. The theoretical density of a crystal can be found by calculating the mass of all the atoms in the unit cell. (The mass of an atom is its molar mass divided by the Avogadro constant see Section Sl.l). The mass is divided by the unit cell volume. To count the number of atoms in a unit cell, we use the following information ... 

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