Crystal approximation

Temary and quaternary semiconductors are theoretically described by the virtual crystal approximation (VGA) [7], Within the VGA, ternary alloys with the composition AB are considered to contain two sublattices. One of them is occupied only by atoms A, the other is occupied by atoms B or G. The second sublattice consists of virtual atoms, represented by a weighted average of atoms B and G. Many physical properties of ternary alloys are then expressed as weighted linear combinations of the corresponding properties of the two binary compounds. For example, the lattice constant d dependence on composition is written as ... 

To further discuss the underlying mechanisms that forces the phase stabilities we also did calculations where the alloying effects were treated within the so-called virtual crystal approximation (VGA) where the real alloy constituents are replaced by an atom with an average (noninteger) atomic number. 

In a previous work we showed that we could reproduce qualitativlely the LMTO-CPA results for the Fe-Co system within a simple spin polarized canonical band model. The structural properties of the Fe-Co alloy can thus be explained from the filling of the d-band. In that work we presented the results in canonical units and we could of course not do any quantitative comparisons. To proceed that work we have here done calculations based on the virtual crystal approximation (VGA). In this approximation each atom in the alloy has the same surrounding neighbours, it is thus not possible to distinguish between random and ordered alloys, but one may analyze the energy difference between different crystal structures. 

Figure 2.Virtual crystal approximation calculations (solid line) compared with coherent potential approximation calculations for Fe-Co (longdashed line), Fe-Ni (dot-dashed line) and Fe-Cu (dashed line). The fcc-bcc energy difference is shown as a function of the atomic number.

Cathodic electrodeposition of microcrystalline cadmium-zinc selenide (Cdi i Zn i Se CZS) films has been reported from selenite and selenosulfate baths [125, 126]. When applied for CZS, the typical electrocrystallization process from acidic solutions involves the underpotential reduction of at least one of the metal ion species (the less noble zinc). However, the direct formation of the alloy in this manner is problematic, basically due to a large difference between the redox potentials of and Cd " couples [127]. In solutions containing both zinc and cadmium ions, Cd will deposit preferentially because of its more positive potential, thus leading to free CdSe phase. This is true even if the cations are complexed since the stability constants of cadmium and zinc with various complexants are similar. Notwithstanding, films electrodeposited from typical solutions have been used to study the molar fraction dependence of the CZS band gap energy in the light of photoelectrochemical measurements, along with considerations within the virtual crystal approximation [128]. 

In the virtual-crystal approximation (VCA) (Nordheim 1931), the site energy of an alloy atom is taken to be... 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

Crystal approximants. Several crystalline phases contain more or less closely packed atomic assemblies (polyhedra, clusters) which have been considered fundamental constituents of several quasicrystals, metal glasses and liquids. Such crystalline phases (crystal approximants), as reported in the previous paragraph, are often observed in the same (or similar) systems, as those corresponding to the formation of quasicrystals and under similar preparation conditions. Crystalline phases closely related to the quasicrystals (containing similar building blocks) have generally complex structures as approximants to the ico-quasicrystals we may, for instance, mention the Frank-Kasper phases (previously described in 3.9.3.1). 

To determine static properties of the SeO radical in KDP and DKDP, the temperature dependence of the hyperfine interaction between unpaired electron and Se (I = 1/2) nucleus was measured [53]. The hyperfine tensor component A, where the direction is along the c-axis, shows an isotope effect, because its value is higher in DKDP than in KDP. Furthermore, its value shows a jump at Tc for DKDP and a considerable temperature dependence in the PE phase of both crystals, approximated by the relation A (T) = A (0) - B coth(ro/T), where To 570 K for both crystals. It is interesting to note that A, similarly to the As NQR frequency and P isotropic chemical shift, should be constant in the PE phase if the two-state order-disorder mechanism of the corresponding tetrahedron holds. However, while the temperature dependencies of the As NQR frequency and P isotropic chemical shift in the PE phase were explained as originating from a six-state order-disorder mechanism [42] and additional displacive mechanism [46], respectively, here it was assumed that excitation of some extra lattice vibration mode with frequency Tq affects the hyperfine tensor components and causes the temperature dependence of A. ... 

The first contribution inside the curly brackets represents the change in the bond energy within the virtual crystal approximation, that is... 

At the Elsnig factory the crystallization of cyclonite is accomplished as follows. About 110 kg of cyclonite are introduced into a closed tank, with a capacity of 10001., equipped with stirrers and lined with a woollen filter cloth. Approximately 900 1. of acetone heated to 50°C are run into the tank to dissolve the cyclonite, after which the solution filtering through the filter cloth is drained down into a 30001. tank. (The filter cloth is changed every 10 hr). Here ahout 13501. of water is added over a period of 5 min, while the temperature is maintained at 25°C, and cyclonite is precipitated from the acetone solution in the form of fairly large crystals approximately 90% of the total are longer than 0.1 mm. The precipitated cyclonite is separated on a vacuum filter. 

Isophane insulin is a white suspension of rod-shaped crystals approximately 30- Jm long, and free from large aggregates of crystals after being subjected to moderate agitation. It contains either 1.4-1.8% glycerol, 0.15-0.17% meta-cresol, and 0.06-0.07% phenol on a wt/vol basis, or 1.4—1.8% glycerol and 0.20-0.25% phenol (wt/vol), at a pH of 7.1-7.4. It also contains 0.15-0.25% (wt/vol) of sodium phosphate, 0.01-0.04 mg of zinc, and 0.3—0.6 mg of protamine for each USP insulin unit. The insoluble matter in the suspension is crystalline and contains not more than traces of amorphous material. 

Note A/B implies A grown or strained to B and vice versa. A B implies no growth direction or explicit strain dependence, i.e. natural. ) T = theoretical E = experimental AVL = average lattice XPS = X-ray photoelectron spectroscopy PL = photoluminescence CL = cathodoluminescence UPS = ultraviolet photoelectron spectroscopy LMTO = linear muffin tin orbital method LAPW = linearised augmented plane wave method PWP = plane wave pseudopotential method VCA = virtual crystal approximation. 

FIGURE 1.3 Butylammonium vermiculite (Kenya) crystals before and after swelling in water (lateral dimensions of the crystals approximately 2.5 x 2.5 mm). (Reproduced with kind permission of the Clay Minerals Society, from Garrett, W.G. and Walker, G.F., Clays Clay Min., 9, 557, 1962.)... 

Impurity and Aperiodicity Effects in Polymers.—The presence of various impurity centres (cations and water in DNA, halogens in polyacetylenes, etc.) contributes basically to the physics of polymeric materials. Many polymers (like proteins or DNA) are, however, by their very nature aperiodic. The inclusion of these effects considerably complicates the electronic structure investigations both from the conceptual and computational points of view. We briefly mentioned earlier the theoretical possibilities of accounting for such effects. Apart from the simplest ones, periodic cluster calculations, virtual crystal approximation, and Dean s method in its simplest form, the application of these theoretical methods [the coherent potential approximation (CPA),103 Dean s method in its SCF form,51 the Hartree-Fock Green s matrix (resolvent) method, etc.] is a tedious work, usually necessitating more computational effort than the periodic calculations... 

The destabilization of the premlcellar aggregates at high water content may give rise to a) separation of liquid water, b) formation of Inverse micelles, or c) separation of a lamellar liquid crystal. Approximate calculations using the Tanford-Nlnham approach gave correct Information for a model system, but the critical ratio appeared too Insensitive to the alcohol/soap ratio to be useful. 

The oxide was identified by the occurrence of major peaks at 4.18, 2.69, and 2.44 A (Cu radiation with curved crystal monochrometer) in the X-ray diffraction pattern (6). The surface area was 48.5 + 0.2 m g l as determined by N2 adsorption by the B.E.T. method (21). SEM pictures of the oxide revealed needle-shaped crystals approximately 1 micron in length and 0.2 microns wide. The value for the total surface sites (FeO-p) was taken from Yates (22) work on aFeOOH. Yates (22) determined FeO to be equal to 27.8 ymol m by tritum exchange. Further details of the solid s preparation, identification, and basic surface characteristics can be found in Balistrieri (23). 

We make the ensemble average of Eq. (6.7) and suppose that t are decoupled so that each t is replaced by i. In reality, in Eq. (6.7) only immediately successive indices caimot repeat, and the first corrections are of third order in the I matrix this is an advantage with respect to the ensemble average of Eq. (6.4) in the virtual crystal approximation (VCA), because first corrections would be of second order in the w matrix. In the ATA we replace each by I in Eq. (6.7) the result corresponds to Eq. (6.4) with w = t/(l -F GooO- We obtain... 

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