Crystal partition function

The special case where only rotators are present, Np = 0, is of particular interest for the analysis of molecular crystals and will be studied below. Here we note that in the other limit, where only spherical particles are present, Vf = 0, and where only symmetrical box elongations are considered with boxes of side length S, the corresponding measure in the partition function (X Qxp[—/3Ep S, r )], involving the random variable S, can be simplified considerably, resulting in the effective Hamiltonian... 

Let us consider a polymer chain with N->oo identical skeletal atoms, either in solution or in the melt, representing our polymer system. Our reference temperature is T0, i.e., the temperature above which no bundles may effectively contribute to crystallization. At T = T0 the chain is assumed to be unperturbed and its configurational partition function is ZN(T0) = kN (N -> oo) [107] for simplicity we use a reduced form Zn = Z /kN (henceforth simply the partition function) so that Zn(T0) = 1. Only at T < To effective bundles may form, see Fig. 1, and we have ZN(T) = 1 + AZN(T - T0) note that the unit term corresponds to the bundle-free infinite-chain configuration. Each bundle with n chain atoms in -c N) will contribute to AZn... 

The set of occupation numbers, ntl> which describe the state of every site in the crystal, is denoted by n and corresponds to a particular configuration N. Corresponding to each n there are N configurations differing only in the exchange of identical atoms between sites. Let 1 denote the coordinates of the atom at site l relative to the centre of the site and let L denote the set of all coordinates for every site of the crystal. If n0l = 1 then fictitious coordinates are introduced. The partition function can now be written... 

For the perfect crystal 0 reduces to Um L ) and the partition function is evaluated by expanding this potential energy in a... 

Crystals lack some of the dynamic complexity of solutions, but are still a challenging subject for theoretical modeling. Long-range order and forces in crystals cause their spectrum of vibrational frequencies to appear more like a continuum than a series of discrete modes. Reduced partition function ratios for a continuous vibrational spectrum can be calculated using an integral, rather than the hnite product used in Equation (3) (Kieffer 1982),... 

The theory developed for perfect gases could be extended to solids, if the partition functions of crystals could be expressed in terms of a set of vibrational frequencies that correspond to its various fundamental modes of vibration (O Neil 1986). By estimating thermodynamic properties from elastic, structural, and spectroscopic data, Kieffer (1982) and subsequently Clayton and Kieffer (1991) calculated oxygen isotope partition function ratios and from these calculations derived a set of fractionation factors for silicate minerals. The calculations have no inherent temperature limitations and can be applied to any phase for which adequate spectroscopic and mechanical data are available. They are, however, limited in accuracy as a consequence of the approximations needed to carry out the calculations and the limited accuracy of the spectroscopic data. 

With the introduction of the lattice structure and electroneutrality condition, one has to define two elementary SE units which do not refer to chemical species. These elementary units are l) the empty lattice site (vacancy) and 2) the elementary electrical charge. Both are definite (statistical) entities of their own in the lattice reference system and have to be taken into account in constructing the partition function of the crystal. Structure elements do not exist outside the crystal and thus do not have real chemical potentials. For example, vacancies do not possess a vapor pressure. Nevertheless, vacancies and other SE s of a crystal can, in principle, be seen , for example, as color centers through spectroscopic observations or otherwise. The electrical charges can be detected by electrical conductivity. 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

At that date, palladium hydride was regarded as a special case. Lacher s approach was subsequently developed by the author (1946) (I) and by Rees (1954) (34) into attempts to frame a general theory of the nature and existence of solid compounds. The one model starts with the idea of the crystal of a binary compound, of perfect stoichiometric composition, but with intrinsic lattice disorder —e.g., of Frenkel type. As the stoichiometry adjusts itself to higher or lower partial pressures of one or other component, by incorporating cation vacancies or interstitial cations, the relevant feature is the interaction of point defects located on adjacent sites. These interactions contribute to the partition function of the crystal and set a maximum attainable concentration of each type of defect. Conjugate with the maximum concentration of, for example, cation vacancies, Nh 9 and fixed by the intrinsic lattice disorder, is a minimum concentration of interstitials, N. The difference, Nh — Ni, measures the nonstoichiometry at the nonmetal-rich phase limit. The metal-rich limit is similarly determined by the maximum attainable concentration of interstitials. With the maximum concentrations of defects, so defined, may be compared the intrinsic disorder in the stoichiometric crystals, and from the several energies concerned there can be specified the conditions under which the stoichiometric crystal lies outside the stability limits. 

In order to evaluate the canonical partition function Q for a gas, we shall consider the system to be composed of an aggregate of essentially independent particles (molecules). As we shall see later, a crystal may be considered to a good approximation as an aggregate of independent harmonic oscillators. Each of these has its own microcanonical partition function ... 

Since many of these oscillators differ from each other in the values of their frequencies, energy levels, and partition functions, it is conveiuent to define a new quantity which is the geometric mean of all of the for the crystal ... 

Crystalline l. The partition function for the crystalline state of I2 consists solely of a vibrational part the crystal does not undergo any significant translation or rotation, and the electronic partition function is unity for the crystal as it is for the gas. 

The geometric mean partition function for the crystal can be expressed as... 

In a number of cases, when the heterogeneity is such that groups of sites may be considered as independent subsystems, statistical approaches may also be helpful. This may be so for surfaces with periodical structures like binary crystals or copolymers. Consider, by way of a simple example, a surface where pairs of sites (say A and B) form independent subsystems, and assume that molecules can adsorb on A and/or on B (site partition functions nd q. respectively) and that there is a lateral Interaction energy w if the two sites are both occupied. This leads to Isotherm (1.5.411. 

Let us consider the B cavities of a crystal of zeolite together with the adsorbed substance. Each cavity may contain not more than m adsorbed molecules. We assume that the contribution of the interaction energy of molecules which arq in different cavities to the total energy of the system may be neglected. Denoting by n the number of cavities which contain s adsorbed molecules (5 = 0, 1,. . . , m), we can write for this model the canonical partition function, Q (i, 4)... 

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