Cubic crystals, definitions

Solids tend to crystallize in definite geometric forms that often can be seen by the naked eye. In ordinary table salt, cubic crystals of NaCl are clearly visible. Large, beautifully formed crystals of such minerals as fluorite, CaF2, are found in nature. It is possible to observe distinct crystal forms of many metals under a microscope. 

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]. 

When iodine is dissolved in hydriodic acid or a soln. of a metallic iodide, there is much evidence of chemical combination, with the formation of a periodide. A. Baudrimont objected to the polyiodide hypothesis of the increased solubility of iodine in soln. of potassium iodide, because he found that an extraction with carbon disulphide removed the iodine from the soln. but S. M. Jorgensen showed that this solvent failed to remove the iodine from an alcoholic soln. of potassium iodide and iodine in the proportion KI I2, and an alcoholic soln. of potassium iodide decolorized a soln. of iodine in carbon disulphide. The hypothesis seemed more probable when, in 1877, G. S. Johnson isolated cubic crystals of a substance with the empirical formula KI3 by the slow evaporation of an aqueous-alcoholic soln. of iodine and potassium iodide over sulphuric acid. There is also evidence of the formation of analogous compounds with the other halides. The perhalides or poly halides—usually polyiodides—are products of the additive combination of the metal halides, or the halides of other radicles with the halogen, so. that the positive acidic radicle consists of several halogen atoms. The polyiodides have been investigated more than the other polyhalides. The additive products have often a definite physical form, and definite physical properties. J. J. Berzelius appears to have made the first polyiodide—which he called ammonium bin-iodide A. Geuther called these compounds poly-iodides and S. M. Jorgensen, super-iodides. They have been classified 1 as... 

Note that, of course, the cubic crystal field splitting dzi, dx2 y2 (eg) at 6Dq and dxz, dyz, dxy at -4Dq is reproduced if Ds and Dt are both zero. Note also that the centre of gravity of neither of the eg nor of the t2g sets is maintained as the symmetry departs from cubic. This means that, in low symmetry, the concept of the cubic field splitting is not clearly defined. For small departures from cubic symmetry the lack of definition is not serious in practice, but to maintain the concept in, say, a square-planar complex, as is sometimes done, requires care. 

The aim of this work is to elucidate these problems. To this end, we calculate the effective spin Hamiltonian of the 5f2—5f2 superexchange interaction between the neighboring U4+ ions in the cubic crystal lattice of UO2 and we calculate T5 <%> eg, rs f2g(l) ancl r5 f2g(2) linear vibronic coupling constants. These data are then used to draw a more definite conclusion about the driving force of the phase transition and especially about the actual mechanism of the spin and orbital ordering in U02. 

Silver sulphide, Ag2S.—The sulphide occurs in nature as argentite, and also in combination with many other sulphides. It is formed by direct synthesis from its elements at ordinary temperatures,7 and also by immersion of the metal in solutions of alkali-metal poly sulphides. It can be prepared in cubic crystals by passing sulphur-vapour over heated silver, or by the action of hydrogen trisulphide, H2S3, on silver oxide. It is the only definite compound produced by fusion of mixtures of silver and sulphur in different proportions.9 The black, amorphous form is precipitated by the action of hydrogen sulphide or a soluble sulphide on a solution of a silver salt. It is also produced by the action of hydrogen sulphide on metallic silver, a reversible reaction.10... 

Body-centered cubic crystal lattice d 5,244 mp 826°, Sol in liq ammonia. Shows two reduction potentials —0,710 and —2.510 v. (referred to a normal calomel electrode) Noddack, Brukl, Angew. Chem. 50, 362 (1937) gives two definite series of salts one in which the metal Is bivalent, and another in which it is trivalent. 

A direction in a crystal is given as a set of three integers in square brackets [uvw] u, v, and w correspond to the above definition of the translation vector r. A direction in a cubic crystal can be described also by Miller indices, as a plane can be defined by its normal. The indices of a direction are expressed as the smallest integers which have the same ratio as the components of a vector (expressed in terms of the axis vectors a, b, c) in that direction. Thus the sets of integers 1, 1, 1 and 3, 3, 3 represent the same direction in a crystal, but the indices of the direction are [111] and not [333]. To give another example, the x axis of an orthogonal x, y, z coordinate system has Miller indices [ 100] the plane perpendicular to this direction has indices (100). 

Because the material properties are direction-dependent in a cubic crystal, they have to be stated together with the corresponding direction. According to the definition, the load direction has to be stated for Young s modulus Ei. Because the shear stress Tij and shear strain -y j have two indices, two indices are needed for the shear modulus Gij. Poisson s ratio relates strains in two directions. Here the second index j denotes the direction of the strain that causes the transversal contraction in the direction marked by the first index i eu = If the coordinate system is aligned with the axes... 

The characteristic time defined in (9.21) establishes a time scale for surface evolution of the kind discussed in the preceding section. Its definition depends on a number of parameter values that are not measurable and, therefore, are not known with any certainty. To get some idea of its magnitude, estimate the value of the characteristic time for the particular case of a Si surface with a mismatch strain of Cin = 0.008 at a temperature of T = 600 °C. Base the estimate on the unit cell dimension of a = 0.5431 nm for the diamond cubic crystal structure, and on the following values of macroscopic material parameters an elastic modulus of E = 130 GPa, a Poisson ratio of = 0.25, a mass density oi p = 2328kg/m, and the surface energy of 70 = 2J/m. Assume that 10% of the surface atoms are involved in the mass transport process at any instant so that = 0.1. 

We shall start with the so-called intermediate-field case. In many of the ionic complexes of the 3group elements, while the free ion energies due to the Coulomb repulsion between electrons have the order of magnitude of 10 to 10 cm, energies of the cubic crystal field are of the order of 10 cm. So we can treat the effect of the latter as the perturbation acting on the multiplet terms characterized by definite values of L and S. 

Patrykiejew et al. [329] have also simulated the behavior of 2D L-J fluids onto the (100) face of a face-centered-cubic crystal. Nevertheless, the Knight and Monson [225] work was not mentioned, so that no comparison of results was performed. To model the gas-surfaee potential, Patrykiejew et al. [329] used the first live terms of Steele s Fourier series [Eq. (14)] for a perpendicular reduced distance less than or equal to 2.5. The results show that at low temperature, the structure of the monolayer film depends strongly on the gas-surface potential corrugation, as well as on the size of the adsorbed atoms. Also, the influence of the corrugation on the melting transition is studied, indicating a different structure of the solid phase. Unfortunately, definitive conclusions about the nature and order of the observed phase transitions were not obtained. 

A unit, or perfect, dislocation is defined by a Burgers vector which regenerates the structure perfectly after passage along the slip plane. The dislocations defined above with respect to a simple cubic structure are perfect dislocations. Clearly, then, a unit dislocation is defined in terms of the crystal structure of the host crystal. Thus, there is no definition of a unit dislocation that applies across all structures, unlike the definitions of point defects, which generally can be given in terms of any structure. 

We now need to define a collection of atoms that can be used in a DFT calculation to represent a simple cubic material. Said more precisely, we need to specify a set of atoms so that when this set is repeated in every direction, it creates the full three-dimensional crystal stmcture. Although it is not really necessary for our initial example, it is useful to split this task into two parts. First, we define a volume that fills space when repeated in all directions. For the simple cubic metal, the obvious choice for this volume is a cube of side length a with a corner at (0,0,0) and edges pointing along the x, y, and z coordinates in three-dimensional space. Second, we define the position(s) of the atom(s) that are included in this volume. With the cubic volume we just chose, the volume will contain just one atom and we could locate it at (0,0,0). Together, these two choices have completely defined the crystal structure of an element with the simple cubic structure. The vectors that define the cell volume and the atom positions within the cell are collectively referred to as the supercell, and the definition of a supercell is the most basic input into a DFT calculation. 

An alternative definition of line breadth, first used by Laue (1926), is the area of the curve divided by its height—the integral breadth if this definition is used, the constant C in the Scherrer equation is rather larger Stokes and Wilson (1942) give values from 1 0 to 1 3 for Various reflecting planes of differently shaped crystals of cubic symmetry in practice, when the shape is not known, a value of 1 15 should be used. 

Definitive x-ray diffraction data on structure I was obtained by McMullan and Jeffrey (1965) for ethylene oxide (EO) hydrate, as presented in Table 2.2a. The crystal consists of a primitive cubic lattice, with parameters as given in Table 2.2a. The common pictorial view of structure I is presented in Figure 1.5a. In that figure, the front face of a 12 A cube is shown, with two complete 51262 (emphasizing hydrogen bonds) connecting four 512. 

After crystal structure II was deduced, a definitive x-ray diffraction study of tetrahydrofuran/hydrogen sulfide hydrate was undertaken by Mak and McMullan (1965), two of Jeffrey s colleagues. The crystal consists of a face-centered cubic lattice, which fits within a cube of 17.3 A on a side, with parameters as given in Table 2.2a and shown in Figure 1.5b. In direct contrast to the properties of structure I, this figure illustrates how a crystal structure may be completely defined by the vertices of the smaller 512 cavities. Because the 512 outnumber the 51264 cavities in the ratio 16 8, only 512 are clearly visible in Figure 1.5b. 

---