Experimental cubic lattice

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

The side chain separation varies in a range of 1 nm or slightly above. The network of aqueous domains exhibits a percolation threshold at a volume fraction of 10%, which is in line with the value determined from conductivity studies. This value is similar to the theoretical percolation threshold for bond percolation on a face-centered cubic lattice. It indicates a highly interconnected network of water nanochannels. Notably, this percolation threshold is markedly smaller, and thus more realistic, than those found in atomistic simulations, which were not able to reproduce experimental values. 

The original cluster-network model proposed by Gierke et al. (also referred to as the cluster-channel model) has been the most widely referenced model in the history of perfluorosulfonate ionomers. Despite the very large number of papers and reports that have strictly relied on this model to explain a wide variety of physical properties and other characteristics of Nafion, this model was never meant to be a definitive description of the actual morphology of Nafion, and the authors recognized that further experimental work would be required to completely define the nature of ionic clustering in these iono-mers. For example, the paracrystalline, cubic lattice... 

Experimental studies on the thermal decomposition and combushon processes of AP have been carried out and their detailed mechanisms have been reported.P-iil Fig. 5.1 shows the thermal decomposition of AP as measured by differential thermal analysis (DTA) and thermal gravimetry (TG) at a heating rate of 0.33 K s"f An endothermic peak is seen at 520 K, corresponding to an orthorhombic to cubic lattice crystal structure phase transition, the heat of reaction for which amounts to... 

Thermal Conductivities of Liquids. As was the case with viscosity, it is difficult to derive useful relationships that allow us to estimate thermal conductivities for liquids from molecular parameters. There is a theoretical development by Bridgman, the details of which are presented elsewhere [11], which assumes that the liquid molecules are arranged in a cubic lattice, in which energy is transferred from one lattice plane to the next at sonic velocity, v. This development is a reinterpretation of the kinetic theory model used in the last section, and with some minor modifications to improve the fit with experimental data, the following equation results ... 

The NFE behaviour has been observed experimentally in studies of the Fermi surface, the surface of constant energy, F, in space which separates filled states from empty states at the absolute zero of temperature. It is found that the Fermi surface of aluminium is indeed very close to that of a spherical free-electron Fermi surface that has been folded back into the Brillouin zone in a manner not too dissimilar to that discussed earlier for the simple cubic lattice. Moreover, just as illustrated in Fig. 5.7 for the latter case, aluminium is found to have a large second-zone pocket of holes but smaller third- and fourth-zone pockets of electrons. This accounts very beautifully for the fact that aluminium has a positive Hall coefficient rather than the negative value expected for a gas of negatively charged free carriers (see, for example, Kittel (1986)). 

We have performed such calculations for samples of non-functional polybutadienes 66) (Fig. 11) and, using the found Ax and X0 values, we calculated Kd0) for a cubic lattice model and a slit-like pore within the whole experimentally accessible eab range using Eq. (3.16). The result presented in Fig. 12 shows a good agreement of the experimental data with the calculated curves. Even such a crude model as the lattice-like model and a slit-like pore can be successfully applied to assess the change in the retention volume as a function of the composition of the mobile phase. 

The ground state behaviour observed for Jahn-Teller impurities in cubic lattices has often been explained a posteriori by fitting experimental data to quantities like the average random strains, the Jahn-Teller energy, E]T, or the % frequency [5]. In particular when EJT/iuov < 1 the observation at low temperatures of a cubic angular pattern is favoured, while if E /Hcoe > 1 a static EPR spectrum is expected [1-3]. 

The X-ray patterns taken from the two sections of the intermetallic layer adjacent to the Ni phase were also closely comparable, though this layer is visually seen in Fig. 3.13 to consist of three sublayers. One of these sections corresponded to a layer composition of 16.0 at.% Ni and 84.0 at.% Zn, while the other to 19.0 at.% Ni and 81.0 at.% Zn. The experimental interplanar distances were found to be in better agreement with the calculated values from the orthorhombic lattice parameters of =3.3326 nm, b=0.8869 nm and c=1.2499 nm reported by G. Nover and K. Schubert,279 rather than from the cubic lattice parameter =0.892 nm. Moreover, a few diffraction lines, including the strong line corresponding to the interplanar distance 0.207 nm, could not be indexed on the basis of the cubic structure. 

The theoretical model [130] was developed as an extension of the classical theory of dipolar broadening in dilute solid solutions in the absence of exchange interactions [16]. It was suggested in [130] how to determine the dipolar part in the line width by subtraction of the calculated input of Heisenberg exchange interaction of pairs of exchange-coupled ions. Equations were written for the three cubic lattices as a function of ion s concentration for various numbers of cationic sites included in a sphere of radius Rc with the assumption that clustering effects were absent. The results were compared well with experimental data on Cr3+ in MgO powders. 

Below the T0 temperature fullerite has simple cubic lattice (set), above this temperature it has face-centered cubic lattice (feel) and in the area of T0 temperature the first-kind phase transition from sc phase to fee phase occurs. The orientation ordering takes place in fullerite that was experimentally studied in papers [6-11] and at 260 K the fee lattice was formed from the simple cubic one due to this orientation ordering. The orientation ordering is defined not only by temperature, but by pressure as well [12]. 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

We will not deal here with the subject of EPR spectroscopy of the solid state. In this field of investigation a kind of delta-like approach such as that recently proposed to deal with molecular dynamics in the liquid state has developed naturally. According to the European Molecular Liquid Group (EMLG), the symbol A symbolizes the cooperative efforts of computer simulation, experiment, and theory. Knak Jensen and Hansen, for instance, carried out a computer simulation of the dynamics of N identical spins placed in a rigid simple cubic lattice subject to an external magnetic field Bq. a further example of numerical study is the paper of Sur and Lowe. Free-induction decay measurements,on the other hand, represent the experimental comer of this ideal triangle, the theoretical comer of which is, of course, expressed by the theoretical papers mentioned above. 

The wide variation of the radii of Table CL. Experimental Interionic Distances different ions has been considered as and Sums of Radn m C CX type Lattices a possible explanation of the formation of different lattices, possessing different coordination numbers. In the case of the close packing of equivalent spheres it is possible, as we have seen, to pack twelve spheres round a central sphere. If, however, the surrounding ions are larger than the central ion, it is not possible for it to be in contact with more than eight, thus replacing a close packed lattice by a body centred cubic lattice. The coordination number would thus... 

Experimental results of water vapor adsorption. Helium relative permeability, Pr, and water vapor permeability, Pe, for the two alumina pellets are presented in figures 6a and 6b, for water relative pressures up to unity. As the amount of water adsorbed starts to rapidly increase with P/Po, due to capillary condensation, a significant increase of its permeability may also be observed due to the resulting capillary enhancement of flow. At a certain value of P/Po where Vs is close to unity, all pores of the membrane are in the capillary condensation regime and thus follow the capillary enhanced type of flux. At this point water vapor permeability reaches its maximum value while, helium relative permeability decreases rapidly and falls to zero well below the point of saturation. This may be attributed, according to percolation theory, to the fact that in a simple cubic lattice, if -75% of the pores are blocked by capillary condensate, the system has reached its percolation threshold and helium... 

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