Fixed lattices

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

Let us now turn our attention to liquid water. Just as in ice I, molecular motions may be divided into rapid vibrations and slower diffusional motions. In the liquid, however, vibrations are not centred on essentially fixed lattice sites, but around temporary equilibrium positions that are themselves subject to movement. Water at any instant may thus be considered to have an I-structure. An instant later, this I-structure will be modified as a result of vibrations, but not by any additional displacements of the molecules. This, together with the first I-structure, is one of the structures that may be averaged to allow for vibration, thereby contributing to the V-structure. Lastly, if we consider the structure around an individual water molecule over a long time-period, and realize that there is always some order in the arrangement of adjacent molecules in a liquid even over a reasonable duration, then we have the diffusionally averaged D-structure. 

Fig. 14.1 Model of a solid with cores at fixed lattice positions and valence electrons free to move throughout the crystalline solid.

A metal can be considered as a fixed lattice of positive ions permeated by a gas of free electrons. Positive ions are the atomic cores the electrons are the valence electrons. For example, copper has a configuration (electronic structure)... 

Table 1 b. Potential parameters for selected rare earths (hep structure assumed) and actinides (fee structure assumed) at fixed lattice parameters equivalent to an fee lattice parameters of 5.0 A and 4.8 A, respectively... 

The only contribution to primary pyroelectricity is the change in dipole oscillations with temperature at fixed lattice constants (strain). The calculated values for the primary and secondary pyroelectric coefficients are plotted in Figure... 

He concludes that the first (associative) mechanism gives values nearest the observed heat of adsorption determined by Beeck (30), and is therefore accepted as nearest the truth (34) (Qo (calculated) = 42 kcal./ mole Qo (observed) = 58 kcal./mole). Experiments on tungsten and nickel films (Beeck (35), Trapnell (36), and more recent work in Rideal s laboratory) have shown that when ethylene is added to a clean metal surface ethane appears in the gas phase. A self hydrogenation mechanism must be operative and at least in these cases dissociation of ethylene must occur on the catalyst. It is suggested that the calculations might be complicated by the energy of bond strain in the adsorption of an ethylene molecule to the fixed lattice distances of the metal. 

June et al. (11) also performed MD calculations to characterize the dynamics of Xe in silicalite. A fixed lattice was assumed with potential parameters close to those used in previous MD studies. The potential between zeolite and guest was determined prior to the calculation over a three-dimensional grid spanning the asymmetric unit. From these grid points, the potential at any point in the lattice could be found by interpolation. Temperatures of 200, 300, and 400 K were imposed during the simulations, which ran for 1 ns. 

The diffusion coefficients calculated from a simulation employing a flexible framework were all between 5 and 10 times larger than those calculated from fixed lattice simulations. A comparison between flexible framework results and NMR measurements (57) illustrated the influence of the cations in the experimental sample calculated diffusion coefficients from the cation-free (flexible) framework were approximately 5 times higher than the experimental results. The increase in diffusion coefficient as a function of loading found in experimental studies was reproduced by the simulations. 

The appearance of the IR spectrum of a compound depends somewhat on the sample s phase. Under high resolution, gas-phase IR bands consist of closely spaced lines—the rotational fine structure however, IR bands of liquids and solids very rarely show rotational fine structure. In most solids, the molecules are held in fixed lattice positions and are not free to rotate. In liquids, the high rate of intermolecular collisions and the substantial intermolecular interactions cause random shifts in the rotational energies, thereby broadening the rotational lines of a band sufficiently to merge them into one another, and eliminate the rotational fine structure. (Broadening of fine structure lines is also observed in gas-phase spectra when the pressure is increased.)... 

A metal can be considered as a fixed lattice of positive ions permeated by a gas of free electrons. Positive ions are the atomic cores, while the electrons are the valence electrons. For example, copper has a configuration (electronic structure) ls22s22p63s23p63dl04sl (superscripts designate number of electrons in the orbit) with one valence electron (4s). The atomic core of Cu+ is the configuration given above, less the one valence electron 4s1. The free electrons form an electron gas in the metal and move nearly freely through the volume of the metal. Each metal atom contributes its valence electrons to the electron gas in the metal. Interactions between the free electrons and the metal ions makes a large contribution to the metallic bond. 

One approach (4) is to calculate, for a certain zeolite structure, the Madelung and polarization energies for fixed lattice positions. The heat of formation due to ionic bonding is calculated both for the zeolitic aluminosilicate with varying amount of aluminum and... 

For a solid, the spin-lattice relaxation will be related to interactions with spins at fixed lattice positions This tends to be slow (T1 is large) the interactions between neighboring spin orientations occur very rapidly, so T2 tends to be much shorter, T2[Pg.712]

To analyse the complete pressure/volume curve simple statistical methods were applied The p/V curves of every single measurement are linear interpolated and then evaluated on a fixed lattice of points at the logarithmic pressure scale. 

D. H. Andrews, Leiden Commun. Suppl., No. 56 (1926) also assumes the existence of rotatory vibrations of large molecules in the fixed lattice in his explanation of specific heats (see also E. Schrodinger, Handbuch der Pfiysik, 10, 314, note 2). 

Systematic thermodynamic analyses, along the lines shown in sec. 3.4c, in combination with a comparative study of different equations of state (table 3.3) have, to the author s knowledge, not (yet) been carried out. This is a pity because it would be interesting to find out whether 2D equations of state, derived from partition functions for mobile adsorbates do function better for the G and L+G range than those based on a fixed lattice model. On the other hand, lattice models should satisfy for the condensed parts of the isotherm, although there the choice of the lattice size becomes critical. One of the reasons for the absence of such studies may be that experiments in the G-state are very susceptable to traces of contamination. 

Since, the surface energy curve is obtained for a pseudomorphic alloy on a fixed lattice of the substrate, its behavior can be related directly to the type of so-called effective interactions which are responsible for the ordering of A and B atoms on the surface. That is, if the multisite interactions are small in the system, which is usually the case for metallic alloys on a fixed lattice, the mixing energy can be written in terms of pair potentials between alloy... 

A Electronic conductors. Ionic conductivity is much lower as the ions are much larger. This means ions require more energy to move through the lattice and overcome energy losses, such as collision with the fixed lattice ions. 

Cooling Cgo down to very low temperatures causes the molecules to lock in a particular conformation. Below 90 K, the static solid state pattern broadens into the spectrum expected for a normal solid with fixed lattice sites. 

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