Noncubic lattice

The idea of the proposed modification of the MP meshes can be also used to generate MMP meshes for all noncubic lattices. 

To find an approximate form of this modulating function a tight-binding descrip>-tion of a one-dimensional chain of atoms possessing one atom per primitive unit ceU and two orbitals per atom was studied [280]. The sinc ( ii jy) function found was simply extended to the three-dimensional cubic lattice case. Unfortunately, the form of this modulating function depends on the direct lattice of superceUs and its extension to noncubic lattices is nontrivial. 

Nevertheless, the calculation [683] demonstrates (1) the efficiency of the more detailed symmetry analysis for the supercell choice when the periodic defect calculations are made in the complicated crystaUine structures with the symmetry of a nonsym-morphic space group and noncubic lattice (2) the reality of the supercell model for the nonempirical LCAO calculations of the point defects in such a complicated crystalline structure as a rutile structure 3) higher efficiency of LCAO basis compared with LAPW basis in the snpercecell calculations of defective crystals. Moreover, the supercell model allows the dependence of the electronic properties of doped crys-... 

Here we shall just mention the so-called Miller indices, which will be denoted h, k, 1). These three indices indicate the intercepts of a plane with the crystal axes. Additional indices are needed in noncubic lattices and yet another notation (i jk) is used to characterize the direction in a lattice. For a discussion of this the reader is referred to more specialized literature. 

For larger systems N> 1000 or so, depending on the potential range) anotlier teclmique becomes preferable. The cubic simulation box (extension to noncubic cases is possible) is divided into a regular lattice of n x n x n cells see figure B3.3.7. These cells are chosen so that the side of the cell = L/n is greater than the... 

The EFG parameters Vzz and described by (4.42a) and (4.42b) do not represent the actual EFG felt by the Mossbauer nucleus. Instead, the electron shell of the Mossbauer atom will be distorted by electrostatic interaction with the noncubic distribution of the external charges, such that the EFG becomes amplified. This phenomenon has been treated by Stemheimer [54—58], who introduced an anti-shielding factor (1 —y 00) for computation of the so-called lattice contribution to the EFG, which arises from (point) charges located on the atoms surrounding the Mossbauer atom in a crystal lattice (or a molecule). In this approach,the actual lattice contribution is given by... 

Te intermetallics and compounds of noncubic symmetry exhibit quadrupole splitting, as expected in lattices of cubic symmetry... 

CoNiMn408, Ni2Mn40s and NisMnsOj. In the spectra of the noncubic Mn40 and the noncubic CoMn204, the extended fine structure is missing. It is apparent that the extended fine structure does pertain solely to the crystal lattice and is not dependent on the valence, coordination, or nature of the atom giving rise to the absorption spectrum. 

According to the relationship between the lattice volume and Tc as described, cubic CssCgo would be an ultimate candidate for a higher Tc superconductor, but the conventional vapor-solid reaction affords only the thermodynamically stable CsCso and CS4C60 phases. In 1995, noncubic CssCgo was obtained by a solution process in liquid ammonia, and the superconductivity was observed below 40 K under an applied hydrostatic pressure of 1.4 GPa [311]. 

In order to calculate the lattice parameter for a noncubic substance, a system of linear equations with the help of the relations reported in Table 4.3 is formed, and then this system is solved. It is easier to understand the procedure with an example hence, we use the BaCe0 95Yb0 05O3 5 perovskite that crystallizes in an orthorhombic unit cell [34] to illustrate the methodology [32],... 

For reasons to be discussed in Chap. 11, the observed values of sin 6 always contain small systematic errors. These errors are not large enough to cause any difficulty in indexing patterns of cubic crystals, but they can seriously interfere with the determination of some noncubic structures. The best method of removing such errors from the data is to calibrate the camera or diffractometer with a substance of known lattice parameter, mixed with the unknown. The difference between the observed and calculated values of sin 6 for the standard substance gives the error in sin 9, and this error can be plotted as a function of the observed values of sin 6. Figure 10-1 shows a correction curve of this kind, obtained with a particular specimen and a particular Debye-Scherrer camera. The errors represented by the ordinates of such a curve can then be applied to each of the observed values of sin 0 for the diffraction lines of the unknown substance. For the particular determination represented by Fig. 10-1, the errors shown are to be subtracted from the observed values. 

Powder patterns of cubic substances can usually be distinguished at a glance from those of noncubic substances, since the latter patterns normally contain many more lines. In addition, the Bravais lattice can usually be identified by inspection there is an almost regular sequence of lines in simple cubic and body-centered cubic patterns, but the former contains almost twice as many lines, while a face-centered cubic pattern is characterized by a pair of lines, followed by a single line, followed by a pair, another single line, etc. 

Cohen s method of determining lattice parameters is even more valuable when applied to noncubic substances, since, as we saw in Sec. 11 -2, straightforward graphical extrapolation cannot be used when there is more than one lattice parameter involved. Cohen s method, however, provides a direct means of determining these parameters, although the equations are naturally more complex than those needed for cubic substances. For example, suppose that the substance involved is hexagonal. Then... 

More recently, the generation of noncubic metal cyanide frameworks has led to the discovery of colossal uniaxial NTE in these systems. In Ag3[Co(CN)6], which consists of a 3D lattice of hexagonal symmetry, variation in temperature leads to a highly pronounced temperature-dependent hinging of the structure (see Eigure 1.22), resulting in colossal thermal contraction along the c-axis (a = —125 x 10 K ) and colossal expansion in the afe-plane ( a =+140 X 10 This property arises due... 

The mechanism of diffusion varies greatly depending on the crystalline structure and the nature of the solute. For crystals with lattices of cubic symmetry, the dif-fusivity is isotropic, but not so for noncubic crystals. In interstitial mechanisms of diffusion, small diffusing solute atoms pass through from one interstitial site to the next. The matrix atoms of the crystal lattice move apart temporarily to provide the necessary space. When there are vacancies where lattice sites are unoccupied, an atom in an adjacent site may jump into such a vacancy. This is called the vacancy mechanism. 

Charges on ions surrounding the Mossbauer probe atom in noncubic symmetry give rise to the so-called lattice contribution, (EFG)iat or v . (for axial symmetry). 

In noncubic sohds, the phonon mode frequencies of the polar lattice vibrations depend, in general, on the phonon mode propagation direction. Likewise, directionally dependent free-chargescattering rates and the anisotropic inverse effective freecarrier mass tensor will produce nonscalar free-charge-carrier contributions. The infrared dielectric function is then represented by a complex-valued second-rank tensor s, which can be expressed in Cartesian coordinates (x,y,z) as ... 

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