Convergence of spherical harmonic expansions for the evaluation of hard-sphere cluster integrals

ForN particles (N>2), by means of a spherical harmonic expansion of Silverstone and Moats, a 3N-dimensional cluster may be reduced to 2N+1 trivial integrals andN–1 interesting integrals. For hard spheres, theN–1 interesting integrals are products of polynomials integrated between binomial bounds. With simple clusters, closed forms are obtained; for more complex clusters, infinite series inl (ofY _lm ) appear. It is here shown for representative cases that these series converge exponentially rapidly, the leading pair of terms accounting for all but a few tenths of a percent of the total cluster integral.     

Functional differential equation approach to the large N expansion and mean field perturbation theory

An apparent difference between formulating mean field perturbation theory for λφ⁴ field theory via path integrals or via functional differential equations when there are external sources present is shown not to exist when mean field theory is considered as the N = 1 limit of the 0(N)λφ⁴ field theory. A simple method is given for determining the 1/N expansion for the Green's functions in the presence of external sources by directly solving the functional differential equations order by order in 1/N. The 1/N expansion for the effective action Γ(φ, χ) is obtained by directly integrating the functional differential equations for the fields φ and χ ( ) in the presence of two external sources j = −δΓ/δφ, S = −δΓ/δχ.     

ExactS-matrix of the adjoint SU(N) representation

We have calculated the exact factorisedS-matrices of the adjoint SU(N) representation in 1+1 space-time dimensions. Besides the trivial solution the only realised solution exhibits anO(N ²–1) symmetry.     

α-Decay calculations of medium mass nuclei within generalized density-dependent cluster model

The α-decay half-lives of even–even medium mass nuclei have been systematically investigated using a radial wave function within the generalized density-dependent cluster model. The α-decay width is calculated using the overlap integral of the quasi-bound initial state wave function, the scattering final state wave function, and the difference of potentials. The effective α–nucleus potential is constructed using the double-folded integral of realistic nucleon–nucleon interactions with the mass density distributions of α particles and daughter nuclei. For comparison calculations are also carried out for the Woods–Saxon shape potential. The present study is restricted to even–even nuclei with 82<N126, where the shell effect on the α-preformation factor has been taken into account for even–even N=126 isotones. The obtained α-decay half-lives are found to agree with the experimental data with a mean factor of less than 2 for both the double-folded potential and the Woods–Saxon shape potential.     

Electronic structure and spectrum of Cr3+ in LiCaAlF6

The electronic structure and spectrum of Cr³⁺ in LiCaAlF₆ are investigated by using the discrete variatitional-local density functional (DV-LDF) method with embedded cluster model. The clusters (CrF₆)^3– withC ₃,D _3d andO _h point group symmetries embedded in the crystal are treated. The one-electron energy levels, densities of states, orbital populations, spin polarization splittings and energies of some terms are calculated. The results show that the relaxation of F^– ions around the Cr³⁺ impurity is inevitable, and that theD _3d andO _h (CrF₆)^3– clusters, with an extended bond-lengthR(Cr–F) chosen to be equal to 1.88 Å can represent this relaxation in a much better way. All the ligand-field transition energies, which are obtained from the transition-state energy and the Griffith parameters, as yielded by a restricted one-electron DV-LDF calculation, compare well with the experimental ones.     

The first and second nearest neighbour Cr3+ pairs in ruby

The optical spectrum of the exchange coupled first nearest neighbour Cr³⁺ pairs in ruby was studied by absorption, fluorescence and excitation spectroscopy. It was analysed by means of the group theoretical considerations given in part I [1]. The analysis could be verified by measuring the Zeeman effect in fields up to 150 KG. The Zeeman splitting deviates considerably from that of pure spin states even for the orbital singulet of the pair ground state. The exchange integrals of the ground state are found to beJ=–115 cm^–1 andj=–0.7 cm^–1.This work was supported by the Deutsche Forschungsgemeinschaft, SFB 65     

Kinetic theory of dense fluids. IV. Entropy balance equation and irreversible thermodynamics

The viscosity coefficient obtained in a previous paper of this series is calculated as a function of density by developing the N-particle collision operator into a dynamic cluster expansion. The excess transport coefficient Δη is given in an exponential form, where η₀ is the two-body Chapman-Enskog result for the transport coefficient, n is the density, and β_l is a density-independent quantity consisting of connected cluster contributions of (l + 2) particles. Therefore, the leading term β₁ consists of connected three-body cluster contributions. The excess shear viscosity coefficient is calculated for a monatomic hard-sphere fluid by computing β_l up to the three-body contributions and the result is compared with the molecular dynamics result by Ashurst and Hoover and also with the experimental data on Ar at 75°C. In spite of the crudity of the potential model used and the approximations made the agreement is good. The result can be improved if l-body clusters (l 4) are included in the calculation. The thermal conductivity coefficient can be obtained in a similar form by using exactly the same procedure used for the viscosity coefficient.     

Vector-spinor space and field equations

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists a vector-spinor space with N_v vector dimensions and N_s spinor dimensions, where N_v=2k and N_s=2 ^k, k3. This space is decomposed into a tangent space with4 vector and4 spinor dimensions and an internal space with N_v–4 vector and N_s–4 spinor dimension. A variational principle leads to field equations for geometric quantities which can be identified with physical fields such as the electromagnetic field, Yang-Mills gauge fields, and wave functions of bosons and fermions.     

Nuclear spectral energy functions beyond the shell model

We predictl=0 nucleons in¹²C to have a negative (binding) energy centered around –22 MeV with a full width at half-maximum of 5.3 MeV. Thel=1 (P _3/2 nucleons) are predicted to have a much narrower spectral energy function centered around –10.6 MeV. A strongly correlated translational invariant wave function was used to describe the ground state nucleus. A central two-nucleon potential was utilized in the hyperspherical harmonic method to approximately solve the Schrödinger equation for the ground state wave function. Both confirmation and failings of the independent particle shell model are exposed.     

Analytical perfect fluid perturbation for the matter dominated epoch

We present the analytical solution to the linear evolution equation of a one component Friedmann perturbation using an equation of state of the form p = (1/3)μσ²(t), where μ is the mass density and σ(t) is the root mean square (rms) velocity in the matter dominated epoch. It is assumed that this rms velocity depends only on the time coordinate and decreases as 1/a, a being the expansion factor of the Friedmann background. The evolution equations are written for scales below the horizon using the longitudinal gauge. The general solution, in the coordinate space, of the evolution equation for the scalar mode is obtained. In the case of spherical symmetry, this solution is expressed in terms of unidimensional integrals of the initial conditions: the initial values of the Newtonian potential and its first time derivative. This perfect fluid solution is a good approximation to the evolution of warm dark matter perturbations obtained by solving the Vlasov’s equation for collisionless particles.     

Spherical harmonics and cartesian tensor scalar product expansions of the distribution function

On the basis of the expansion of the distribution function in a sum of the spherical harmonics, the distribution functionf(v, r, t) is expanded in a series of scalar products of two Cartesian tensors term by term, i.e. The tensors and ^(l) (l=2, 3) are constructed in dependence on the spherical harmonic expansion coefficients (the tensors and ^(l) (l=0, 1) have been constructed by Jancel and Kahan [3]). On the basis of the knowledge of the analytic form off ₂ andf ₃ the equations forf ₁ f ₂ andf ₃ for the case of the Boltzmann's equation are determined.Technická 2, Praha 6, Czechoslovakia.     

Mössbauer and infrared studies of the Cu-Cr ferrites

Ferrites of the system CuCr_xFe_2–x O₄, wherex=0,0.2,0.4, 0.6 and 0.8, have been studied by Mössbauer and IR absorption spectra. Mössbauer spectra were recorded at room temperature. The spectra of all samples showed two well defined Zeeman patterns corresponding to A and B sites. The effect of the variation of chromium substitution on the various hyperfine interactions has been discussed. The cationic distribution makes clear that all Cr³⁺ ions occupy octahedral sites. The IR spectra in the range 200–4000 cm^–1 showed the presence of four bands. The high and low frequency bandsv ₁ andv ₂ belong to the tetrahedral and octahedral sites, respectively. Small bandsv ₃ andv ₄are observed around v₂ and are assigned to the octahedral divalent metal-oxygen ion complexes and the lattice vibrations of the system, respectively.     

Scaling properties of ℤ k- 1 actions on the circle

We derive universal scaling properties for ^k–1 actions on the circle whose generators have rotation numbers algebraic of degreek. As fork=2 these properties can be explained for arbitraryk in terms of a renormalization group transformation. It has at least one trivial fixed point corresponding to an action whose generators are pure rotations. The spectrum of the linearized transformation in this fixed point is analyzed completely. The fixed point is hyperbolic with a (k–1)-dimensional unstable manifold. In the casek=2 the known results are therefore recovered.     

TheCP N−1 1/N-action by inverse scattering transformation in angular momentum

The effective action which generates 1/N expansion of theCP _N–1 model in two dimensions is studied here by inverse-problem methods. The action contains a functional determinant, in which auxiliary scalar and vector fields are assumed to have a spherical symmetry. This leads to the introduction, as an associated linear problem, of a radial Schrödinger equation with two potentialsv and , and a potential-dependent centrifugal term {(–r)²/r ²–1/4r ²}. The full inverse scattering formalism is developed here for this diffusion problem. It is formulated in terms of two-component Jost solutions, and leads to a matricial Gel'fand-Levitan-Marchenko equation. The scattering data associated to the potentials by this IST are then used to obtain a closed local form for the whole effective action. This is indeed possible for theCP _N–1 model, owing to the classical integrability. Moreover it is found that no spherically symmetric instanton exists in this case. However the absence of supplementary informations on the 1/N series, due to the non-integrability at quantum level, does not allow safe quantitative conclusions on the general behaviour of the 1/N series at large orders.Laboratoire associé au CNRS UA 280     

Contribution to the Analysis of theA2←XA1“Wulf” Transition of Ozone by High-Resolution Fourier Transform Spectrometry

The analysis of the rotational structure of the high-resolution Fourier transform 0⁰₀absorption spectrum of the³A₂←X¹A₁band system of the “Wulf” transition of the isotopomer¹⁶O₃of ozone is reported for the first time. With a near pure case (b) coupling model for the upper triplet state, we have assigned a significant portion of the spectrum, mainly theF₁(J=N+ 1) andF₂(J=N) spin components, primarily in the lower frequency region of the band. The lines corresponding to theF₃(J=N− 1) component are weak at lower frequencies and heavily congested in the central and higher frequency regions of the spectrum. Perturbations and predissociation phenomena have reduced the effective lifetime of the metastable³A₂state and have also limited the number of transitions included in the least-squares fit of the band. Approximately 100 lines have been assigned in the range from 9100–9550 cm⁻¹. Three rotational, three centrifugal distortion, three spin–rotation, and one spin–spin constant were varied. The geometry of the molecule in the³A₂state, as determined from these constants, isr= 1.345 Å and θ = 98.9°, in good agreement withab initioresults.     

Excess capacitance of ZnO-Au varactors

Doped ZnO single crystals were deposited with gold and indium in 1×10^–8 Torr vacuum. The lithium-doped ZnO single crystals and the gold interface revealed not only a Schottky diode but also varactor characteristics. TheI-V andC-V characteristics of ZnO:Li-Au devices were determined in the 0–140 mV and 0–1.5 V ranges.The frequency dependence of ZnO:Li-Au varactors was investigated in the 6–550 kHz range and the value of the most efficient varactor frequency was found to be 50 kHz for the lithium-doped samples prepared.To bring further insight into the matter the concept of excess capacitance was introduced and 1/C ²=f(–V) curves were rearranged between 0–150 mV where Schottky characteristics are non-linear. The excess capacitance values of lithium-doped varactors were determined at four different frequencies and ranged from 26 pF at 50 kHz to 70 pF at 6kHz.Finally, the bulk donor concentrations of the single crystals were calculated from the modifiedC-V curves to beN _D= 3×10²⁰ m^–3. On the other hand, the bulk donor concentration determined from the non-modifiedC-V curves wasN _D=1.02×10²² m^–3.     

Trapping of random walks on the line

Several features of the trapping of random walks on a one-dimensional lattice are analyzed. The results of this investigation are as follows: (1) The correction term to the known asymptotic form for the survival probability ton steps is O(( ₂n)^–1/3), where =–ln(1–c), andc is the trap concentration. (2) The short time form for the survival probability is found to be exp[–a(c)n ^1/2], wherea(c) is given in Eq. (21). (3) The mean-square displacement of a surviving random walker is found to go liken ^2/3for largen. (4) When the distribution of trap-free regions is changed so that very large regions are much rarer than for ideally random trap placement the asymptotic survival probability changes its dependence onn. One such model is studied.     

Lamb shifts for hydrogen,using the sachs elementary interaction theory

The Lamb shifts of theS _1/2 andP _1/2 states of hydrogen atoms are calculated using the Sachs elementary interaction theory. BothS _1/2 andP _1/2 levels are shifted with respect to the Dirac levels by energies of the same order of magnitude. Agreement with experiment is obtained for the² S _1/2 –² P _1/2 and³ S _1/2 –³ P _1/2 Lamb shifts, but the predicted¹ S _1/2 –² P _1/2 Lamb shift term of 12,164 MHz is in disagreement with the experimental value of 7860 ± 1140 MHz.     

Note on the interfacial tension of phase-separated polymer solutions

Earlier theoretical calculations of the interfacial tension of phase-separated polymer solutions as a function of the degree of polymerizationN and the temperatureT, based partly on the mean-field approximation, had led toN ^–1/4(1–T/T _c )^3/2 for fixedN1 andT approaching the critical solution temperatureT _c It is here remarked that the scaling procedure of de Gennes then modifies this toN ^–0.37(1–T/T _c )^1.26, which is in close accord with the experimentalN ^–0.44(1–T/T _c )^1.26. The simplest mean-field picture yieldsN ^–1/2(1–T/T _c )^3/2.     

Λ spin–orbit splitting in heavy hypernuclei as deduced from DWIA analyses of the reaction

By taking nuclear core excitations into account, detailed structure calculations of ⁸⁹ _ΛY have been performed as a function of the Λ spin–orbit splitting. The obtained wave functions have been used to estimate the (π⁺,K⁺) reaction cross sections within the distorted-wave impulse approximation (DWIA). A theoretical explanation is given quantitatively for the first time of how to understand the doublet substructure of each major peak observed in medium-heavy hypernuclear production. A small Λ spin–orbit splitting of around δ(0f^Λ)=0.2 MeV is deduced, a value consistent with the small ΛN two-body spin–orbit interactions deduced from γ-ray measurements in three light hypernuclei.