Density of a crystal

The presence of small concentrations of point defects changes the density of a crystal, and four values of the density can be calculated, depending on... 

Several formulations were proposed [65, 66], but the intuitive notation introduced by Hansen and Coppens [67] afterwards became the most popular. Within this method, the electron density of a crystal is expanded in atomic contributions. The expansion is better understood in terms of rigid pseudoatoms, i.e., atoms that behave stmcturally according to their electron charge distribution and rigidly follow the nuclear motion. A pseudoatom density is expanded according to its electronic stiucture, for simplicity reduced to the core and the valence electron densities (but in principle each atomic shell could be independently refined). Thus,... 

The crystal property perhaps most sensitive to structure is density and, conversely, the increase in density of a crystal with pressure is accompanied by significant structural changes. The latter can be of two types - either a smooth variation of the free parameters of the structure with pressure, or a first-order transition to a new structure type. 

An iron deficiency could be accommodated by a defect structure in two ways either iron vacancies, giving the formula Fe] /D, or alternatively, there could be an excess of oxygen in interstitial positions, with the formula FeOi+ f. A comparison of the theoretical and measured densities of the crystal distinguishes between the alternatives. The easiest method of measuring the density of a crystal is the flotation method. Liquids of differing densities which dissolve in each other, are mixed together until a mixture is found that will just suspend the crystal so that it neither floats nor sinks. The density of that liquid mixture must then be the same as that of the crystal, and it can be found by weighing an accurately measured volume. 

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 ... 

Keeping in mind that a crystal is composed of many of its unit cells and assuming there is no contamination, the density of a crystal can be computed from the properties of the unit cell. It is necessary to apportion the mass of the crystal among the various unit cells, and then to divide the mass apportioned to one unit cell by the volume of the unit cell. In computing the mass of a unit cell, it is important to assign to the cell only that fraction of... 

Each electron-density wave provides a component for summation to give the electron-density map, shown in Figure 6.11. If the electron density of a crystal could be described precisely by a single cosine wave that repeats three times in the unit cell dimension d, then the electron density has a periodicity of d/3 and the diffraction pattern will have intensity only in the third order (only one diffracted beam, 3 0 0). This is the electron-density wave that is used in the summation that gives an electron-density map if there is only one term because only one Bragg reflection is ob-... 

Measurement of the density of a crystal permits calculation of the weight of the contents of the unit cell. 

Will the presence of Frenkel defects change the measured density of a crystal ... 

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

Thus physical properties can be used as a probe of symmetry, and can reveal the crystallographic point group of the phase. Note that Neumann s principle states that the symmetry elements of a physical property must include those present in the point group, and not that the symmetry elements are identical with those of the point group. This means that a physical property may show more symmetry elements than the point group, and so not all properties are equally useful for revealing tme point group symmetry. For example, the density of a crystal is controlled by the unit cell size and contents, but the symmetry of the material is irrelevant, (see Chapter 1). Properties similar to density, which do not reveal symmetry are called non-directional. Directional properties, on the other hand, may reveal symmetry. 

The one-dimensional equation can be generalised to three dimensions and the electron density of a crystal at any point in the unit cell x, y, z, is given by ... 

The weight of all the atoms or ions in the unit cell can be calculated if the unit cell dimensions and the density of the crystal are known. This means that crystal density measurements are important in the early stages of a crystal structure analysis. A vertical column with a density gradient, calibrated with crystals of known density, is used to measure protein densities. These are determined by where along the column the crystal setdes and so that, by interpolation, its density can be estimated (39). This method must be done with care to ensure that the measurement is good. The density of a crystal is given by... 

The target bentonite, Kunigel VI, consists of about 50% of montmorillonite in weight and the rest of macro-grains, mainly quartz particles. The mass density of a crystal is 2.7g/cm, both for montmorillonite and for quartz, so one third (1/3) of the bentonite with its dry density 1.8g/cm is void (i.e., the void ratio e=0.5). Based on a... 

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 ... 

Will the density of a crystal go up or down if it contains (a) Schottky defects (b) Frenkel defects (c) vacancies (d) interstitials ... 

Figure 12.19 also summarizes the relationship between the atomic radius r and the edge length a of a simple cubic cell, a body-centered cubic cell, and a face-centered cubic cell. This relationship can be used to determine the density of a crystal, as Example 12.3 shows. 

Measurement of lattice defects. As the defect has little influence on the lattice sizes and crystal density, its measurement needs a rather high accuracy. If the defect concentration changes obviously with temperature, it is comparatively easy to distinguish the effect caused by defects and that by crystal itself. At higher temperatures, if the crystal sizes increase outstandingly, one can consider that it is the result of a formation of FYenkel defect. There will be excess amounts of interstitial metallic ions in lattice by measurement of the lattice constant accurately. According to comparison of the true density of a crystal with that of calculated by lattice constant obtained by XRD, it cannot only determine the defect concentration, but also can classify the types of defects. 

These observations form the basis of a new approach to nucleation theory built on density flmctional methods of statistical mechanics. The predictions and results of this nonclassical theory are described in greater detail in Ref 1. The starting point is the observation that the average density of a crystal can be written as a sum over Fourier components ... 

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