Normal vibration frequencies of two- and three-dimensional arrays of alternating point masses

Normal vibration frequencies are calculated for alternating point-masses of mass M and m in two-dimensional rectangular and a three-dimensional parallelepiped arrays. The method of intermediate co-ordinate transformation is used in the analysis. In applying this method, the vibrating systems consisting of alternating masses are treated as the basic systems of linear chains of identical point-masses of mass m (the zero-order system) that are coupled together to form the two- and three-dimensional systems and then further modified by inclusion of an additional mass ?m at alternating sites, where M = (1 + ?)m. The zero-order solution for the linear array of identical particles of mass m is well known. The present calculation shows the advantage of the process of intermediate co-ordinate transformation in simplifying problems which are otherwise complicated and also illustrates how such a process can be applied in more than one successive step.     

Clusters of bound particles in the derivative δ-function Bose gas

In this paper we discuss a novel procedure for constructing clusters of bound particles in the case of a quantum integrable derivative δ-function Bose gas in one dimension. It is shown that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasi-momenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. We also establish a connection between these special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles associated with the derivative δ-function Bose gas and allows us to study various properties of these clusters like their size and their stability under the variation of the coupling constant.     

Correlations of particle orientation in monolayer films

Two-dimensional and three-dimensional models of an extended layer that consists of orientation-ally ordered particles are used to describe the ordering in monolayer films. In both models, when the distance between particles of the layer increases, the correlation function decreases to zero (for the parameter of adhesion to the substrate a = 0) or remains constant (for a ≠ 0, i.e., when the energy of interaction between the particles and the substrate is taken into account). In the latter case, this means that the layers have a long-range orientation order. The proposed models of an extended monolayer can be used to interpret data obtained by the light scattering and molecular dynamics methods for Langmuir-Blodgett films with the adhesion parameter a = 0.05 and the particle-particle interaction parameter b = 0.6 in the three-dimensional model or b = 1 in the two-dimensional model. The smaller value of b in the three-dimensional layer model can be explained by a stronger effect of cooperation of particle-particle interactions as compared to the two-dimensional layer model.     

Superlattice structures of the intercalation complex NbS2·(pyridine)12

Satellite spots were observed in NbS₂·(pyridine)_{$\text{1}\text{2}$} at room temperature by means of electron diffraction. The superlattice is composed of rectangular unit cell of 2√3a × 13a in the plane perpendicular to c-axis and appears with three-fold rotational overlapping in the diffraction pattern. In NMR measurements no evidence was observed of any chalcogenide lattice distortions. Hence, it is suggested that the superlattice has no relation with a charge-density-wave. Origins of the superlattice are discussed on a view point of pyridine molecule arrangement.     

The dependence of the radiative transition probability of donor-acceptor pairs on pair separation

The probability W(R) for radiative donor-acceptor pair transitions is generally assumed to decrease exponentially with increasing pair separation R, with a characteristic length a. When one centre in the pair is deep and strongly localized and the other one is shallow, a equals half the Bohr radius of the shallower centre. For pairs in GaP involving two shallow centres we have previously reported that a exceeds half this Bohr radius, and have suggested this to be due to the spatial extent of the deeper centre. In the present paper we investigate this point, both experimentally, and by considering Novotny's calculations — for direct transitions — on the R-dependence of W(R), in which the Bohr radii of both donor and acceptor are parameters. According to these calculations, W(R) is not exponential in the case of two shallow centres. In a limited range of R, W(R) is approximately exponential, however, and theoretical “effective values” of a can be obtained. These effective values are compared with experimental data on a for the pair series S_P-Si_P, S_P-Cd_Ga, S_P-Zn_Ga and S_P-C_P in the indirect semiconductor GaP; a amounts to ∽ 5.4, 7.0, 7.7 and 9.1 Å, respectively. The comparison shows that the difference found between the experimental a and half the Bohr radius of the less localized centre in the pair can indeed be explained by the influence on a of the spatial extent of the more localized centre. This findings implies a non-exponential W(R) for pairs with two shallow centres. This conclusion depends critically on the values used for the Bohr radii of the acceptors, however. These are not accurately known, due to insufficient knowledge of m¹_h, one of the quantities needed in calculating the radii. We used m¹_h = 0.36 m₀ to 0.40 m₀, corresponding to the normally used hydrogen-model energy of 40–45 meV. An alternative explanation for the experimental values of a is that in reality m¹_h is significantly lower than the 0.36 m₀ to 0.40 m₀ used, leading to larger acceptor Bohr radii and, in the extreme case, to no influence on a of the more localized centre and to an exponential W(R). In order to decide between the two possible explanations, additional experiments are presented. These include an accurate comparison of a between S_P-C_P and O_P-C_P as well as between S_P-Zn_Ga and O_P-Zn_Ga pairs. This is done by analysing the intergral band decay and time-resolved spectra of these pairs. The conclusions for all pairs mentioned are: (i) The Bohr radius of the acceptors involved is significantly larger than calculated from an effective-mass energy of 40–45 meV; they correspond to a hydrogen-model energy of 28 meV (m¹_h = 0.25 m₀). (ii) There is in first approximation no influence on the shape of W(R) of the spatial extent of the more localized centre. (iii) The experimental results are well described by an exponential W(R) with a equal to half the Bohr radius of the less localized centre of the pair.     

On the exact mean-field description of continuous quantum systems in equilibrium

The thermodynamic limit of free energy density is investigated for quantum systems of n particles obeying Boltzmann, Fermi and Bose statistics, interacting via spin-independent 2-body bounded separable potentials and confined to a bounded region Λ ? R^v. The technique used exploits the Feynman-Kac theorem in finite volume and the saddle-point method of Tindemans and Capel. It is shown that the limiting free energy density of such systems is equal to that of a system of noninteracting particles subject to a mean field which is equal to the averaged 2-body interaction. The equations for the mean field of n particles obeying Boltzmann, Fermi or Bose statistics represent self-consistent field problems and their forms comply with the well-known theorems on mean occupation numbers of single-particle eigenstates of ideal quantum gases at inverse temperature β.     

Bose–Einstein condensation of a relativistic Bose gas in a harmonic potential

Using semiclassical method, Bose–Einstein condensation (BEC) of a relativistic ideal Bose gas (RIBG) with and without antibosons in the three-dimensional (3D) harmonic potential is investigated. Analytical expressions for the BEC transition temperature, condensate fraction, specific heat and entropy of the system are obtained. Relativistic effects on the properties of the system are discussed and it is found that the relativistic effect decreases the transition temperature T_c but enlarges the gap of specific heat at T_c. We also study the influence of antibosons on a RIBG. Comparing with the system without antibosons, the system with antibosons has a higher transition temperature and a lower Helmholtz free energy. It implies that the system with antibosons is more stable.     

An SU(2) model for weak interactions

It is shown that it is possible to obtain neutral current weak interaction in agreement with experiments in a pure SU(2) model. The main experimental consequences are the relation mW₃? √2 mW± and the prediction that mW± ? 53 GeV and mW₃ ? 75GeV.     

Effects of long-range interaction on critical and multicritical behavior of compressible disordered systems

A field-theoretic approach is applied to describe behavior of weakly disordered, isotropic elastic compressible systems with long-range interactions directly in the three-dimensional space for various values of the long-range interaction parameter a. A renormalization-group procedure is applied separately for a > 2 and a ≤ 2 directly in the three-dimensional space. Renormalization-group equations are analyzed in the two-loop approximation, and critical and tricritical points are determined. It is shown that long-range effects are not important when a ≤ 2, whereas they play a key role in the opposite case of a > 2. Critical exponents characterizing the system are obtained for various values of the long-range interaction parameter. Behavior of homogeneous and disordered systems characterized by two fluctuating order parameters is also described.     

Perturbative calculation of critical exponents for the Bose–Hubbard model

We develop a strategy for calculating critical exponents for the Mott insulator-to-superfluid transition shown by the Bose–Hubbard model. Our approach is based on the field-theoretic concept of the effective potential, which provides a natural extension of the Landau theory of phase transitions to quantum critical phenomena. The coefficients of the Landau expansion of that effective potential are obtained by high-order perturbation theory. We counteract the divergency of the weak-coupling perturbation series by including the seldom considered Landau coefficient a ₆ into our analysis. Our preliminary results indicate that the critical exponents for both the condensate density and the superfluid density, as derived from the two-dimensional Bose–Hubbard model, deviate by less than 1 % from the best known estimates computed so far for the three-dimensional XY universality class.     

Universality of Three-Body Systems in 2D: Parametrization of the Bound States Energies

Universal properties of mass-imbalanced three-body systems in 2D are studied using zero-range interactions in momentum space. The dependence of the three-particle binding energy on the parameters (masses and two-body energies) is highly non-trivial even in the simplest case of two identical particles and a distinct one. This dependence is parametrized for ground and excited states in terms of supercircles functions in the most general case of three distinguishable particles.     

Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems

The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using two different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the average values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes. This allows the calculation of both nonlinear dynamical behaviour and of dynamical linear response and Green's functions. Practical approximation schemes are considered. In a SUB n approximation the m-body amplitudes are restricted to the order $\text{m ? n}$ and the energy functional is a finite-order multinomial in the amplitudes to be variationally determined. It is shown that the solution corresponds to summing well-defined subsets of Goldstone diagrams. These subsets are conveniently specificed in terms of tree structures, the normal or extended generalized time ordering g.t.o. trees. The extended exp S method is in the SUB n approximation able to sum, in addition to the normal SUB n diagrams, a set which contains m-body cluster amplitudes of arbitrarily high order (m > n) in the ordinary sense. The article also discusses how the SUB n truncation schemes must be modified to be able to treat a system with a strong repulsive core in the two-body interaction. The method is formulated for the general cases of Bose and Fermi systems which may or may not conserve total particle number. It is shown that the simplest approximation, SUB 1, in the extended exp S method agrees with the mean field theory, which is the coherent-state approximation in the boson case or the Hartree-Fock approximation in the fermion case. It is argued that the extended exp S method already in low-order approximations can realistically treat a great variety of diverse many-body problems, even including systems which may undergo ground-state phase transitions. A few applications are described in more detail. The Bose liquid is treated in the extended SUB 2 approximation. It is shown that the ground-state results in the uniform limit are exact and agree with the hypernetted-chain approximation. The modifications due to hard-core interactions and the non-linear equations of motion are also discussed in this case. For Fermi systems it is shown that the supercondictive phase transition of the BCS model Hamiltonian and the deformation phase transition of the Lipkin model are properly obtained by the extended exp S method in a low-order approximation.     

Ideal bose gas in an oscillator potential (exact solution)

Exact expressions for the statistical sum of the grand canonical ensemble and the one-particle density matrix are derived based on the definition of the density matrix for a system of N identical noninteracting Bose particles in an oscillator potential as a sum with respect to the symmetric exchange of the density matrix coordinates of distinguishable particles. A quasi-classical scenario is analyzed in detail.     

Critical Relaxation of a Three-Dimensional Fully Frustrated Ising Model

Critical relaxation from the low-temperature ordered state of the three-dimensional fully frustrated Ising model on a simple cubic lattice is studied by the short-time dynamics method. Cubic systems with periodic boundary conditions and linear sizes of L = 32, 64, 96, and 128 in each crystallographic direction are studied. Calculations were carried out by the Monte Carlo method using the standard Metropolis algorithm. The static critical exponents for the magnetization and correlation radius and the dynamic critical exponents are calculated.     

Partition functions and thermodynamic properties of paraboson and parafermion systems

New formulas are given for the grand partition function of paraboson systems of order p with n orbitals and parafermion systems of order p with m orbitals. These formulas allow the computation of statistical and thermodynamic functions for such systems. We analyze and discuss the average number of particles on an orbital, and the average number of particles in the system. For some special cases (identical orbital energies, or equidistant orbital energies) we can simplify the grand partition functions and describe thermodynamic properties in more detail. Some specific properties are also illustrated in plots of thermodynamic functions.     

Bose spectrum of superfluid solutions 3He4He

The Bose spectrum of superfluid solutions ³He⁴He is investigated in the case of p-pairing of Fermi particles. Both three-dimensional and two-dimensional models are considered. It is shown that the Bose-Fermi interaction changes the sound velocities in both subsystems. The other collective modes (17 in the 3D-model and 11 in the 2D-model) remain almost unchanged.     

“Solidification” in a soluble model of bosons on a one-dimensional lattice: The “Boson-Hubbard chain”

Lieb and Liniger's soluble model of the 1-D Bose gas is put on a lattice, becoming a “boson-Hubbard model”. It remains soluble by the Bethe ansatz. When the coupling exceeds a critical value ($\text{U}\text{2πT}\text{)>}\text{2√3}\text{π}$, a gap is present when the density n bosons per site is 1.