Lattice chemical potential

Transfer matrix calculations of the adsorbate chemical potential have been done for up to four sites (ontop, bridge, hollow, etc.) or four states per unit cell, and for 2-, 3-, and 4-body interactions up to fifth neighbor on primitive lattices. Here the various states can correspond to quite different physical systems. Thus a 3-state, 1-site system may be a two-component adsorbate, e.g., atoms and their diatomic molecules on the surface, for which the occupations on a site are no particles, an atom, or a molecule. On the other hand, the three states could correspond to a molecular species with two bond orientations, perpendicular and tilted, with respect to the surface. An -state system could also be an ( - 1) layer system with ontop stacking. The construction of the transfer matrices and associated numerical procedures are essentially the same for these systems, and such calculations are done routinely [33]. If there are two or more non-reacting (but interacting) species on the surface then the partial coverages depend on the chemical potentials specified for each species. 

At each interface the interfacial potential will depend upon the chemical potentials of the species involved in the equilibrium. Thus at the Zn/Zn electrode there will be a tendency for zinc ions in the lattice to lose electrons and to pass across the interface and form hydrated ions in solution this tendency is given by the chemical potential of zinc which for pure zinc will be a constant. Similarly, there will be a tendency for hydrated Zn ions in solution to lose their hydration sheaths, to gain electrons and to enter the lattice of the metal this tendency is given by the chemical potential of the Zn ions, which is related to their activity. (See equation 20.155.) Thermodynamically... 

The curves were determined from Eqs. 24"—26" in order to apply these in a numerical calculation one first has to know the values of the following functions at — 3°C Afz, the difference in chemical potential between the "empty Structure II lattice and ice Cpg. the Langmuir constant for propane in the larger cavities of Structure II (Cpi = 0 for geometrical reasons) Cmi> Cm2> the Langmuir constants for methane in the two types of cavities of Structure II. 

Chemical potential of hydrate lattice, 22 Chlorine, frequency, 189 hydrate, 3... 

The smoothing terms have a thermodynamic basis, because they are related to surface gradients in chemical potential, and they are based on linear rate equations. The magnitude of the smoothing terms vary with different powers of a characteristic length, so that at large scales, the EW term should predominate, while at small scales, diffusion becomes important. The literature also contains non-linear models, with terms that may represent the lattice potential or account for step growth or diffusion bias, for example. 

Situations that depart from thermodynamic equilibrium in general do so in two ways the relative concentrations of different species that can interconvert are not equilibrated at a given position in space, and the various chemical potentials are spatially nonuniform. In this section we shall consider the first type of nonequilibrium by itself, and examine how the rates of the various possible reactions depend on the various concentrations and the lattice temperature. 

Fig. 29. Schematic diagram of the influence of hydrogen on the distribution of Si—Si and Si—H bonds during film growth, (a) shows the hydrogen chemical potential intersecting a possible distribution of bonds while (b) illustrates the expected result due to lattice relaxation (Street et al., 1988).

For historical reasons, the incompressible lattice-fluid system description is used, even if the distribution of one of the components is coupled to the distribution of vacant sites. Constant pressure SCF calculations are the same as constant chemical potential calculations for the vacant sites. These conditions are used below. 

Results of the ideal solution approach were found to be identical with those arrived at on the basis of a simple quasichemical method. Each defect and the various species occupying normal lattice positions may be considered as a separate species to which is assigned a chemical potential , p, and at equilibrium these are related through a set of stoichiometric equations corresponding to the chemical reactions which form the defects. For example, for Frenkel disorder the equation will be... 

In these equations gv is the change in Gibbs free energy on taking one atom from a normal lattice site to the surface of the crystal and (gt + gv) the change when an atom is taken from a normal lattice site to an interstitial site, both at constant temperature and pressure. cr denotes a site fraction of species r on its sublattice, and is the chemical potential of a normal lattice ion in the defect-free crystal. 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

By taking the minimum in Eq. (69) subject to the restrictions specified and using Lagrange undetermined multipliers (see, for example, Ref. 6), one finds a set of relationships satisfied by the defect chemical potentials. The results for the three basic types of intrinsic lattice disorder are as follows ... 

Non-perturbative results have been obtained from first principles by lattice QCD computations at zero chemical potential and temperatures up to a few times the transition temperature Tc. 

This makes worthwhile an approach which extrapolates, with as few assumptions as possible, lattice QCD data from zero chemical potential to /j > 0. 

The coupling g(T) obtained from available lattice QCD data with a given number of flavors will be an input for the extrapolation to non-zero chemical potential, as outlined in the following. 

Figure 5. The quark number susceptibility for Nf = 2, calculated from the quasiparticle model with the same parameters as in Figs. 3 and 4, for several chemical potentials compared to the lattice data [14] at /u = 0.

R possesses a spherical core of radius a consisting of quark matter with CFL condensate surrounded by a spherical shell of hadronic matter with thickness R — a containing neutron and proton superfluids. The triangular lattice of singly quantized neutron vortices with quantum of circulation irh/jj, forms in response to the rotation. Since the quark vortices carry SttTj/fi quantum of circulation, the three singly quantized neutron vortices connect at the spherical interface with one singly quantized quark vortex so that the baryon chemical potential is continuous across the interface [19]. 

Fig. 24. Contour plot of the structure factor (the kinematic LEED intensity) of a x y/i monolayer in a triangular lattice gas with nearest-neighbor repulsion, at a temperature k TIi = 0.355 (about 5% above T ) and a chemical potential // = 1.5 (0c = 0.336 at the transition temperature.) Contour increments are in a (common) logarithmic scale separated by 0.1, starting with 3.2 at the outermost contour. Center of the surface Brillouin zon is to the left k, and k the radial and azimuthal components of kH, are in units of nlXla, a being the lattice spacing. Data are based on averages over 2x10 Monte Carlo steps per site. (From Bartelt et...

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

If the energy to put an isolated ion and its electrons into the lattice is denoted s, then the chemical potential is... 

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