Showing papers on "Partition function (quantum field theory) published in 1970"

Inversion of the Partition Function: The First‐Order Steepest‐Descent Method

Abstract: This paper explains and advocates the use of a Laplace‐transform steepest‐descent method for the computation of energy‐level densities g(E) by inversion of the partition function. The earlier Lin–Eyring approach, based on the Darwin–Fowler method, is shown to be unnecessarily obscure and in part redundant By use of a much‐simpler formulation due to Kubo, it is shown that the first‐order steepest‐descent approximation to the inversion integral follows straightforwardly from a knowledge of the relevant partition function, its energy, and specific‐heat relations. In this way several explicit equations for g(E) and its integral W(E) are derived, including two for anharmonic vibration and internal vibration–rotation. The relationship of the S–D method to Tolman's classical formula is discussed and some thermodynamic implications pointed out. Numerical results agree closely with those obtained by the direct counting of energy levels and, with certain exceptions, the indirect computations of other authors.

Influence of order on electron states in metals

Abstract: Using the model of independent pseudoatoms, we have obtained the partition function of a random array in terms of a Mayer function determined by the effective potential representing the pseudoatom. This calculation of the partition function is shown to be exact to second order in the scattering potential. It is also correct in the limit of strong, but slowly varying, potentials. To treat the electron states in a liquid metal, built from the same pseudoatom, assumptions have to be made to deal with the three-atom and higher-order correlation functions. Using a simplified form of the Kirkwood approximation the partition function may again be obtained in terms of the same Mayer function as for the random case, plus the two-body correlation function. Results for the partition function are presented for a model of liquid Be. The conclusion is that the partition function for the liquid metal lies quite close to that of the solid, and the short-range order (or excluded volume) appears to be a major factor in determining the electron states. When this order is lost, the partition function changes greatly and the present calculations lead in this case to a very substantial low-energy tail on the density of states. On the other hand, this tail is found to be very small in the liquid metal. Structure in the density of states near the Fermi level, arising from band overlap in the divalent metal, appears to be reduced, but not removed, on melting.

Exact Partition Function for the Two-Dimensional Ising Model

Abstract: We present a relatively simple derivation, based on Burgoyne's combinational method, of the Onsager formula for the partition function of the two-dimensional square Ising model.

Correlation of isotope effects with molecular forces, I. The diatomic molecule.

Abstract: The convergence properties of the individual terms in the finite orthogonal polynomial expansion of the partition function of a harmonic oscillator are examined as a function of the energy of the oscillator and the temperature of the system (u = e/kT). The convergence of each term depends on the number of terms in the polynomial and the variable u. The jth term approaches within 1% of its asymptotic value when the order of the polynomial, n, equals (u/π) + j + 1. These favorable convergence properties of the finite polynomials are used to relate the partition function of an oscillator and the difference in chemical properties of isotopes directly to molecular and intermolecular forces. A detailed discussion is given for the diatomic molecule. In this case it is possible to treat the quantum corrections of any order in terms of a single particle function multiplied by kinetic energy-coupling factors. The latter are formulated as simple mass ratios of the order of unity.

Domain Structure in a Second-Neighbour Ising Chain

Abstract: Two different methods of obtaining the partition function for a spin t open ended Ising chain with interactions between nearest and second nearest neighbour spins are presented. These are non-matrix methods, in that they do not require the diagonalization of a transfer matrix, and they provide a useful representation for studying the analytical structure of the partition function. The pair correlation fun13tions are also derived, using the second of these methods. These correlation functions are utilized in a study of the ground state and the finite temperature spin configurations. An interesting application of these results to the problem of out-of-phase domain structure in binary alloys is discussed.

Notizen: “Frozen” Properties and Cut-off Criteria of High Temperature Gases

Abstract: Once the cut-off criterion has been selected the procedure for the calculation of equilibrium compositions and thermodynamic properties is straightforward as already discussed in previous works5 ' 6. The purpose of the present work is to emphasize the importance of the contribution coming from electronic excitation in evaluating the \"frozen\" properties of a plasma and the strong dependence of the electronic contribution on the cut-off criterion adopted. Frozen properties are important quantities indeed for the analysis of heat transfer in plasmas where their values are needed for the evaluation of the adimensional PRANDTL a n d LEWIS n u m b e r s . The observation 5> 7 9 that thermodynamic properties and total specific heats of high temperature gases are practically insensitive to the cut-off criteria adopted has often led to the neglection 10' 11 of the contribution coming from electronic excitation in the evaluation of the equilibrium properties of plasmas. The results obtained according to this procedure are however accurate only as far as total thermodynamic quantities are concerned (i. e. properties which include ionization effects). The contribution of electronic excitation to these quantities is indeed small when compared with the contribution coming from ionization. However, when \"frozen\" properties are calculated, which do not include the contribution from ionization, the neglect of the electronic contribution can lead to very inaccurate results, as will be shown below. This point has been overlooked in recent calculations 11. Helium plasma (He, He+, e) in the temperature range 10,000 — 35,000 °K and in the pressure interval 0.1 — 10 atmospheres will be taken as an example. Results obtained for N2 plasma (N2, N2+, N, N+, N++, N+++, e) will also be presented. The conclusions reached from analysis of these systems should be considered of general validity for high temperature gases.

Coexistence curves of polymer solutions

Abstract: The asymptotic method was applied to polymer solutions to obtain their coexistence curves. The configurational partition function of the lattice model was much improved by introducing free volumes into consideration. Consequently, there were two kinds of coexistence curves which depend not only on the molecular weight but also on coordination number z, one corresponding to the upper critical solution temperature and the other to the lower critical solution temperature. The agreement of them with the experimental results is good. The comparison of the present theory with existing theories was also discussed.

The Physics of Nucleic Acids

Abstract: Prior to speaking of the structure and physical properties of nucleic acids, it would seem desirable to examine their biological role. In this book it has been remarked already several times that DNA is the fundamental genetic substance and nucleic acids are responsible for the synthesis of proteins. It would seem that the time has come to bring out the evidence for these statements.