Equivalent crystal

In this circumstance, it is more reliable to measure the absolute integrated intensity of the substrate peak and compare this with the integrated intensity from an equivalent crystal of the substrate material on which no layer has been grown. In the angular position for diffraction from the substrate, the layer will not diffract and the substrate peak intensity will be simply reduced by normal photoelectric absorption. For a symmetric reflection, it is easy to see that the integrated intensity 1 of the substrate peak with the layer of thickness t present is related to the integrated intensity of the bare substrate / o by... 

The Malvern uses light scattering measurements to determine the weight percent of spherical equivalent crystals in each of 16 size classes and the mean size diameter. 

The host crystal of chrysoberyl has a hexagonal-close-packed structure. The space group is orthorhombic Pnma with four molecules per unit cell. The AP ions are octahedrally coordinated by the oxygen ions and occur in two not equivalent crystal field sites in the lattice. The AP" sites lying in the mirror-... 

Indices of a set of all symmetrically equivalent crystal faces or net planes are enclosed in braces, hkl is a form. 

Fig. 7.16. Illustration of the reduction of the density of states at in a disordered material compared to the equivalent crystal.

Fig. 7.17 estimates the parameter g for the actual conduction band density of states distribution of a-Si H in Fig. 3.16. The integral of the density of states up to energy E is plotted against N E). The equivalent ordered state is taken to be a parabolic band with the density of states of crystalline silicon. The parameter g decreases from the middle of the band to the band edge as expected and the results indicate that the mobility edge should occur near N E = 10 cm" eV", which is quite close to the value indicated by experiment. Unfortimately, this does not provide an accurate procedure for measuring E(, because there is not an exactly equivalent crystal with which to compare the density of states. Nevertheless it illustrates the principle.

Fig. 7.17. The calculated reduction in the density of states of a-Si H compared to the equivalent crystal. The mobility edge is expected to occur when g es 1/3.

American or bar-/i, etc., for British practice). Planes with these indices cut the unit cell axes at a/h, -b/k, -c/l, respectively. If the values of h, k, and / are small for all observed crystal faces, then a reasonable unit cell has probably been chosen. For reasons of symmetry, planes in hexagonal crystals are conveniently described by four axes, three in a plane at 120° to each other. This leads to four indices, hkil, where i — - h + k), and equivalent crystal faces will have similar indices. 

From a computational standpoint, the usefulness of the method relies on the simplicity of the calculations needed for the determination of the three equivalent crystals associated with each atom i. This is accomplished by building on the simple concepts of Equivalent Crystal Theory (ECT) [25,26], as will be discussed in detail below. The procedure involves the solution of one simple transcendental equation for the determination of the equilibrium Wigner-Seitz radius i WSE) of ch equivalent crystal. These equations are written in terms of a small number of parameters describing each element in its reference state, and a matrix of perturbative parameters Ay , which describe the changes in the electron density in the vicinity of atom / due to the presence of an atom j (of a different chemical species), in a neighboring site. The determination of parameters for each atom in... 

ECT is based on the concept that there exists for each atom i, a certain perfect equivalent crystal with its lattice parameter fixed at a value so that the energy of atom i in the equivalent crystal is the BPS strain energy contribution ef. This equivalent crystal differs from the actual ground state crystal only in that its lattice constant may be different from the ground state value. We compute ef via perturbation theory, where the perturbation arises from the difference in the ion core electronic potentials of the actual defect solid and those of the effective bulk single crystal. 

Four different contributions to the energy of atom i, which find their origin in four different perturbations, are singled out. The linear independence attributed between these four terms is consistent with the limit of small perturbations which is assumed in the formulation of ECT. Correspondingly, four different equivalent crystals have to be determined for each atom i [25]. 

The higher order terms are relevant for the case of anisotropic deformations [25]. The second term, [02(1,7)], is a two-body term which accounts for the increase in energy when N bonds are compressed below their equilibrium value. This effect is also modeled with an equivalent crystal, whose lattice parameter is obtained by solving a perturbation equation given by... 

Once the lattice parameter of the (strain) equivalent crystal, /, is determined, the BFS strain energy contribution is computed using the UBER [27], which contains all the relevant information concerning a single-component system ... 

To achieve bioequivalence for a given drug compound usually requires equivalent crystal structures in the drug substance, although exceptions are known to exist. 

CU2P7 VEC=. 4.111, thus Nnbo = 1/9 VECa = 5.286, thus AA = 19/7. One can formulate two equivalent crystal chemical formulae which differ only in the single non-bonding orbital being either on a P atom with two or on a P atom with three homonuclear bonds ... 

For the ethylsulphates, with only four independent terms, the equivalent crystal-field operators were derived by Elliott and Stevens (1952) and used by them to interpret the magnetic resonance results (Elliott and Stevens 1953). Only minor modifications of the numerical coefficients of the terms were needed to give a reasonable fit for each member of the series. Some further interactions are required, however, for two cerium compounds. In the hexagonal crystal field of the ethylsulphate, the energy levels of the ground manifold J = f are split into three... 

The interaction of the crystal with the substrate can be described in a more perceivable way by the adhesion energy /8, J cm , as introduced by Kaischew. " The adhesion energy is defined by = ctj + o-sub erf, taking into account that with the separation of the crystal from the substrate, crystal-solution and the substrate-solution interfaces are created, while the equivalent crystal-substrate interface disappears. 

An initial configuration of >1,000 atoms is required this can be a random configuration, output of an MD simulation, or at stoichiometric compositions the equivalent crystal structure. 

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