Accurate First-Principle Equation of State for the One-Component Plasma

Accurate “first-principle” expressions for the excess free energy F_ex and internal energy of U_ex of the classical one-component plasma (OCP) are obtained. We use the Hubbard-Schofield transformation that maps the OCP Hamiltonian onto the Ising-like Hamiltonian, with coefficients expressed in terms of equilibrium correlation functions of a reference system. We use the ideal gas as a reference system for which all the correlation functions are known. Explicit calculations are performed with the high-order terms in the Ising-like Hamiltonian omitted. For small values of the plasma parameter Γ the Debye-Huckel result for F_ex and U_ex is recovered. For Γ ? 1, these depend linearly on Γ in accordance with the Monte Carlo findings for the OCP. The MC data for the internal energy are reproduced fairly well by the expression for U_ex obtained.     

The approximating Hamiltonian method for an infinite-mode Dicke maser model withA 2-term

A new Dicke-type maser model is proposed. In the thermodynamic limit it involves infinitly many modes and anA ²-term. The approximating Hamiltonian method (AHM) is shown to be valid for the exact calculation of the free energy per particle and some thermodynamic averages for this model in the thermodynamic limit. In the rotatingwave approximation the superradiant phase transition persists at the same temperature as in the model withoutA ²-term.     

Spreadsheets to calculate P–V–T relations,thermodynamic and thermoelastic properties of silicates in the MgSiO3–MgO system

Modified equations of state (EoS) of forsterite, wadsleyite, ringwoodite, akimotoite, bridgmanite and post-perovskite based on the Helmholtz free energy are described using Microsoft Excel spreadsheets. The equations of state were set up by joint analysis of reference experimental data and can be used to calculate thermodynamic and thermoelastic parameters and P–V–T properties of the Mg-silicates. We used Visual Basic for Applications module in Microsoft Excel and presented a simultaneous calculation of full set of thermodynamic and thermoelastic functions using only T–P and T–V data as input parameters. Phase transitions in the MgSiO₃–MgO system play an important role in the interpretation of the seismic boundaries of the upper Earth’s mantle and in the D″ layer. Therefore, proposed EoSes of silicates in the MgSiO₃–MgO system have clear geophysical implications. The developed software will be interesting to specialists who are engaged to study the mantle mineralogy and Earth’s interior.     

Quantum chemical studies of the solvation of the hydroxide ion

Abstract

To understand and model the solvation of the hydroxide ion, OH(H₂O)^? _n clusters, n = 1?5, are studied using ab initio quantum chemical techniques, largely at the MP2 level of theory using a double zeta plus polarization functions basis extended by diffuse functions. Energies and vibrational frequencies, together with thermodynamic quantities such as enthalpies, entropies and Gibbs free energies, are computed. This permits comparison with experimental estimates of the successive thermodynamic changes associated with the reaction OH(H₂O)^? _n + H₂O → OH(H₂O)^? _n+1. The theoretical values are in good agreement with experiment. The free energy of hydration of OH^? is modelled by a composite discrete-continuum method where the effects of the first hydration shell (n = 3) are obtained from the gas phase cluster calculation, while the long-range effects are modelled using self consistent reaction field theory, namely by calculating the solvation energy of OH(H₂O)^? _n in a dielectric continuum. The best estimate of the solvation (free) energy at 298 K is ?84·5 kcal mol^?1, compared to the experimental value of ?102·8 kcal mol^?1.     

Study of the 2-channel systems using the J-matrix method

ABSTRACT

The partial widths and the branching ratios have been calculated for 2-channel systems, using the diagonalisation of a 2-channel full Hamiltonian and the J-matrix quantities related to the free Hamiltonian, for resonance states with different orbital angular momentum ? (? = 0, 1, 2, 3) and different charge z (z = +1,?0,?? 1) for Noro and Taylor potential. Moreover, the 1- and 2-channel Morse potential have been investigated for the two diatomic molecules H₂ and LiH. The basis used is the Laguerre basis. Two approximate methods have been used to calculate the widths and the ratios associated with a spinless projectile?target system. Many results are reported here for the first time, including the results of ? ≠ 0 and z ≠ 0 in 2-channel Noro and Taylor potential, and the results for the two diatomic molecules. The comparison of our outcome with earlier studies, wherever possible, on energies, branching ratios and partial widths, is discussed.     

Operator method for calculating the spectrum of states in the framework of the extended dicke model

A system of N two-level atoms interacting with a resonant single-mode quantum field (the Dicke model) is described using the operator method for solving the Schrödinger equation. The spectrum of states is calculated without recourse to the rotating-wave approximation and the assumption regarding smallness of the linear sizes of the system as compared to the radiation wavelength. Analytical approximate expressions are obtained for the energy spectrum. These expressions approximate the energy spectrum over the entire range of Hamiltonian parameters in the normal and collective states of the system and make it possible to calculate the thermodynamic characteristics.     

Inequivalent Representations of Commutator or Anticommutator Rings of Field Operators and Their Applications

Hamiltonian of a system in quantum field theory can give rise to infinitely many partition functions which correspond to infinitely many inequivalent representations of the canonical commutator or anticommutator rings of field operators. This implies that the system can theoretically exist in infinitely many Gibbs states. The system resides in the Gibbs state which corresponds to its minimal Helmholtz free energy at a given range of the thermodynamic variables. Individual inequivalent representations are associated with different thermodynamic phases of the system. The BCS Hamiltonian of superconductivity is chosen to be an explicit example for the demonstration of the important role of inequivalent representations in practical applications. Its analysis from the inequivalent representations’ point of view has led to a recognition of a novel type of the superconducting phase transition. PACS: 03.70.+k, 05.30.−d, 11.10.−z, 74.20.Fg, 74.25.Bt, 74.78.Bz     

The su q(2) algebra in the off-diagonal basis and applications to quantum optics

We consider a new exactly solvable nonlinear quantum model as a Hamiltonian defined in terms of the generators of the su _q(2) algebra. The corresponding matrix elements of finite rotations (the q-deformed Wigner d functions) are introduced. It is shown that the quantum optical model of the three-wave interaction has an approximate su _q(2) dynamical symmetry given by this Hamiltonian. Such q symmetry allows us to investigate the spectral and dynamical properties of the three wave model through new perturbation techniques.     

One-Particle Distribution Functions and Thermodynamics of Crystals with Many-Body Forces. General Consideration and Investigation of the Properties of the Inert Gas Solids

A crystal is described by a probability density which is not symmetric regarding the interchange of phase-space coordinates between two molecules. The set of approximate equations for one-particle distribution functions with many-body forces is derived from the LIOUVILLE equation. The expression for the HELMHOLTZ free energy and the equation of state are obtained. The developed method is applied to the determination of the thermodynamic properties of crystalline argon, krypton, and xenon. The nearest-neighbor distances, internal energies, isothermal and adiabatic compressibilities, linear thermal expansion coefficients, specific heats C_v and C_p are calculated using the pair potentials of BARKER -POMPE , of BARKER -BOBETIC , and of LENNARD -JONES , together with the AXILROD -TELLER three-body potential. The obtained results are compared with the experimental data and discussed.     

Optimized t-expansion method for the Rabi Hamiltonian

A polemic arose recently about the applicability of the t-expansion method to the calculation of the ground state energy E₀ of the Rabi model. For specific choices of the trial function and very large number of involved connected moments, the t-expansion results are rather poor and exhibit considerable oscillations. In this Letter, we formulate the t-expansion method for trial functions containing two free parameters which capture two exactly solvable limits of the Rabi Hamiltonian. At each order of the t-series, E₀ is assumed to be stationary with respect to the free parameters. A high accuracy of E₀ estimates is achieved for small numbers (5 or 6) of involved connected moments, the relative error being smaller than ¹⁰−4 (0.01%) within the whole parameter space of the Rabi Hamiltonian. A special symmetrization of the trial function enables us to calculate also the first excited energy E₁, with the relative error smaller than ¹⁰−2 (1%).     

Statistical mechanics of cation ordering in PbSc<Subscript>1/2</Subscript>Ta<Subscript>1/2</Subscript>O<Subscript>3</Subscript> and PbSc<Subscript>1/2</Subscript>Nb<Subscript>1/2</Subscript>O<Subscript>3</Subscript> solid solutions

A model Hamiltonian for B cation ordering (Sc-Nb(Ta)) in PbSc_1/2Nb_1/2O₃ and PbSc_1/2Ta_1/2O₃ solid solutions is constructed. The parameters of the model Hamiltonian are determined from the ab initio calculation within the ionic crystal model with allowance made for the deformability and the dipole and quadrupole polarizabilities of the ions. The temperatures of the phase transition due to the ordering of the B cations are calculated by the Monte Carlo method in the mean-field and cluster approximations. The phase transition temperatures calculated by the Monte Carlo method (1920 K for PbSc_1/2Ta_1/2O₃ and 1810 K for PbSc_1/2Nb_1/2O₃) are consistent with the experimental data (1770 and 1450 K, respectively). The thermodynamic properties of the cation ordering are investigated using the Monte Carlo method.     

Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds

The quantum Maxwell theory at finite temperature at equilibrium is studied on compact and closed manifolds in both the functional integral and Hamiltonian formalism. The aim is to shed some light onto the interrelation between the topology of the spatial background and the thermodynamic properties of the system. The quantization is not unique and gives rise to inequivalent quantum theories which are classified by θ-vacua. Based on explicit parametrizations of the gauge orbit space in the functional integral approach and of the physical phase space in the canonical quantization scheme, the Gribov problem is resolved and the equivalence of both quantization schemes is elucidated. Using zeta-function regularization the free energy is determined and the effect of the topology of the spatial manifold on the vacuum energy and on the thermal gauge field excitations is clarified. The general results are then applied to a quantum Maxwell gas on an n-dimensional torus providing explicit formulae for the main thermodynamic functions in the low- and high-temperature regimes, respectively.     

On the effective cross-section for recombination of free carriers with dipole centres

An approximate calculation of the effective cross-section for recombination δ_eff of free charge carriers with dipole centres for small mean path lenght l of free carriers and not very small gas-kinetical cross-section of recombination σ_k is conducted. A formula for the σ_eff suitable also for larger l and small σ_k is suggested.     

Flow equations for the one-dimensional Kondo lattice model: static and dynamic ground state properties

The one-dimensional Kondo lattice model is investigated by means of Wegner's flow equation method. The renormalization procedure leads to an effective Hamiltonian which describes a free one-dimensional electron gas and a Heisenberg chain. The localised spins of the effective model are coupled by the well-known RKKY interaction. They are treated within a Schwinger boson mean field theory which permits the calculation of static and dynamic correlation functions. In the regime of small interaction strength static expectation values agree well with the expected Luttinger liquid behaviour. The parameter K_ρ of the Luttinger liquid theory is estimated and compared to recent results from density matrix renormalization group studies.     

Specific heat of LiB3O5 crystals in the temperature interval 80–300 K

The specific heat of LiB₃O₅ crystals is measured by adiabatic calorimetry. The experimental data on the specific heat are used to calculate the change in the thermodynamic functions (the entropy, enthalpy, and Gibbs free energy) and the Debye temperature Θ_D(T) of crystalline LiB₃O₅. Fiz. Tverd. Tela (St. Petersburg) 39, 624–625 (April 1997)     

Some approaches to polaron theory

Here, in our approximation of polaron theory, we examine the importance of introducing theT product, which turn out to be a very convenient theoretical approach for the calculation of thermodynamical averages.We focus attention on the investigation of the so-called linear polaron Hamiltonian and present in detail the calculation of the correlation function, spectral function, and Green function for such a linear system.It is shown that the linear polaron Hamiltonian provides an exactly solvable model of our system, and the result obtained with this approach holds true for an arbitrary coupling constant which describes the strength of interaction between the electron and the lattice vibrations. Then, with the help of a variational technique, we show the possibility of reducing the real polaron Hamiltonian to a socalled trial or approximate linear model Hamiltonian.We also consider the exact calculation of free energy with a special technique that reduces calculations with the help of the T product, which, in our opinion, works much better and is easier than other analogous considerations, for example, the path-integral or Feynman-integral method.^(1,2) Here we furthermore recall our own work,⁽⁴⁾ where it was shown that the results of Refs. 7 and 8 concerning the impedance calculation in the polaron model may be obtained directly without the use of the path-integral method.The study of the polaron system's thermodynamics is carried out by us in the framework of the functional method. A calculation of the free energy and the momentum distribution function is proposed.Note also that the polaron systems with strong coupling⁽⁹⁾ proved to be useful in different quantum field models in connection with the construction of dynamical models of composite particles. A rigorous solution of the special strong-coupling polaron problem, describing the interaction of a nonrelativistic particle with a quantum field, was given by Bogolubov.⁽³⁾ The works of Tavkhelidze, Fedyanin, Khrustalev, and others^(10–13) are dedicated to the further development and generalization of the Bogolubov method.Notice, too, that the electron-photon interaction effects play an important part in many problems of modern solid state theory (see, e.g., Refs. 7 and 14–19).The present paper summarizes a set of lectures delivered as a special course in the physics department of Moscow State University.     

Renormalization group calculation of the critical temperature dependence on longitudinal magnetic field for a disordered superconductor model

We describe a calculation ofT _c (P, H) in a model of a disordered superconductor which is based on the de Gennes-Skal-Shklovskii (dGSS) picture of the large cluster in a percolation system. The calculation is done by carrying out successive decimations on the Landau-Ginzburg Hamiltonian describing the links in the model. We calculateT _c (P, H) in the presence of longitudinal magnetic field by evaluating the renormalized Landau-Ginzburg coupling when the renormalized Landau-Ginzburg length equals the percolation link length in the dGSS picture and obtain the approximate analytic form ofT _c (P, H).Work supported by China Natural Science Foundation     

Statistical mechanics of cation ordering and lattice dynamics of PbZr<Subscript>x</Subscript>Ti<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>O<Subscript>3</Subscript> solid solutions

An effective Hamiltonian for Zr-Ti cation ordering in PbZr_xTi_1?xO₃ solid solutions is written out. To determine the parameters of the effective Hamiltonian, a nonempirical calculation is performed within an ionic-crystal model taking into account the deformation and dipole and quadrupole polarizabilities of ions. The thermodynamic properties of cation ordering are studied using the Monte Carlo method. The calculated phase transition temperatures (180 and 250 K for the concentrations x=1/3 and 1/2, respectively) are much lower than the melting temperature of the compound under study. At such temperatures, the ordering kinetics is frozen and, in reality, the phase transition to the ordered phase does not occur, in agreement with experimental observations. Within the same ionic-crystal model, we calculated the high-frequency permittivity, Born dynamic charges, and the lattice vibration spectrum for a completely disordered phase and certain ordered phases. It is shown that soft vibration modes, including ferroelectric ones, exist in the lattice vibration spectrum of both the completely disordered and the ordered phases.     

Magnetic interactions in Pr3+ : LiYF4 for quantum manipulation : search for an efficient three-level Λ system

The influence of an external magnetic field on the hyperfine structure of the ³ H₄(0) and ¹D₂(0) crystal field states of Pr³⁺ in LiYF₄ is studied in order to find an efficient three-level Λ system. Using an experimentally determined spin Hamiltonian, we show that three-level Λ systems can be obtained with equal strengths for the optically excited transitions under various magnetic field magnitudes and orientations. An analytical analysis based on two levels is proposed to find useful magnetic fields without extensive numerical calculations and to understand the general behaviour of the system. Pr³⁺ hyperfine structure has also been directly calculated using a complete Hamiltonian including free ion, crystal field and magnetic interactions. A good agreement with the spin Hamiltonian approach is found for the ground state whereas the excited state results poorly reproduce the experiment. This is attributed to the low accuracy of ¹D₂ crystal field wavefunctions. This suggests that transition strengths ratios could be calculated directly from the crystal field Hamiltonian with improved crystal field parameters.     

The Perturbation of the Quantum¶Calogero–Moser–Sutherland System¶and Related Results

The Hamiltonian of the trigonometric Calogero–Sutherland model coincides with a certain limit of the Hamiltonian of the elliptic Calogero–Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In this article we show the essential self-adjointness of the Hamiltonian of the elliptic Calogero–Moser model and the regularity (convergence) of the perturbation for the arbitrary root system. We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x ₁, …,x _N ) for the A _{N -1}-case. As a result, the algebraic calculation of the perturbation is justified. Received: 30 May 2001 / Accepted: 27 November 2001