A systematic extended iterative solution for quantum chromodynamics

An outline is given of an extended perturbative solution of Euclidean QCD which systematically accounts for a class of nonperturbative effects, while still allowing renormalization by the perturbative counterterms. Euclidean proper verticesΓ are approximated by a double sequenceΓ ^[r,p] , wherer denotes the degree of rational approximation with respect to the spontaneous mass scaleΛ _QCD, nonanalytic in the couplingg ², whilep represents the order of perturbative corrections ing ² calculated fromΓ ^[r,0]-rather than from the perturbative Feynman rulesΓ ^(0)pert-as a starting point. The mechanism allowing the nonperturbative terms to reproduce themselves in the Dyson-Schwinger equations preserves perturbative renormalizability and is intimately tied to the divergence structure of the theory. As a result, it restricts the self-consistency problem for theΓ ^[r,0] rigorously — i.e. without decoupling approximations — to the seven superficially divergent vertices. An interesting aspect of the solution is that rational-function sequences for the QCD propagators contain subsequences describing short-lived elementary excitations. The method is calculational, in that it allows the known techniques of loop computation to be used while dealing with integrands of truly nonperturative content.     

The inner charge defect method as an analog to the quantum defect method

A method is proposed for the calculation of one-electron wave functions for excited bound and free atomic states. For the interaction potential between the outer electron and the atomic core, we have adopted the following potential: V(r) = q₀/r for r < r₀ and V(r) = -1/r for r > r₀, where r₀ is approximately equal to the core radius, q₀(∈, 1) = Δq + 1, and Δq > 0 is the inner charge defect. It is shown, for atomic argon, that the method has about the same accuracy as those of Bates and Damgard and Brugess and Seaton.     

Asymptotic properties of the solutions of a differential equation appearing in QED and QCD

Stimulated by the study described in the preceding paper, we establish the asymptotic behaviour of the ratio h′(0)/h(0) for g → ∞, where h(r) is a solution, vanishing at infinity, of the differential equation h″(r) = igω(r)h(r) on the domain 0 ≤ r ≤ ∞ and ω(r) = (1 − √rK₁(√r))/r. Some results are valid for more general ω's.     

The velocity distribution and the density near a particle of an equilibrium one component plasma

The equation governing the conditional relative velocity distribution of an equilibrium one component plasma at distances smaller than the interparticle distance is derived from elementary principles of classical dynamics and probability, the latter defined from the temporal point of view. It becomes obvious that in accordance with todays accepted views the maxwellian velocity distribution satisfies the above equation. Using this result it is also shown that the conditional number density has the form n(r) = n₀exp(-e²/rk_BT).     

Effect of mean free path on the strength of the Ruderman-Kittel-Kasuya-Yoshida interaction in a dilute (Ag0.6Au0.4)Mn alloy

From measurements of magnetic susceptibility from 1.2 to 200 K and magnetization (0–50 kG) from 1.2 to 4.2 K, the strength of the RKKY interaction between Mn spins in a Ag_0.6Au_0.4 + 810 ppm Mn alloy has been determined to be V₀ = (2.4±0.5) x 10^-37 erg cm³. It is shown that this value for V₀ is consistent with an RKKY interaction strength which has has been reduced by mean free path effects by a factor exp (— 〈 r〉/l), where 〈 r 〉 is the average separation between Mn spins and l is the electron mean free path in the alloy.     

Wavefunctions of helium-like systems in limiting regions

We find an approximate analytic form for the solution ψ(r ₁, r ₂, r ₁₂) of the Schrödinger equation for a system of two electrons bound to a nucleus in the spatial regions r ₁ = r ₂ = 0 and r ₁₂ = 0, which are of great importance for a number of physical processes. The forms are based on the well-known behavior of ψ(r ₁, r ₂, r ₁₂) near the singular triple coalescence point. The approximate functions are compared to the locally exact ones obtained earlier by the correlation function hyperspherical harmonic (CFHH) method for the helium atom, light helium-like ions, and the negative ion of hydrogen H^?. The functions are shown to determine a natural basis for the expansion of CFHH functions in the considered spatial region. We demonstrate how these approximate functions simplify calculations of high-energy ionization processes.     

Integral representation for non-maxwell models and the approach to equilibrium

An integral representation is proved for the nonequilibrium distribution function of molecules interaction through potentials which are repulsive near r = 0 and diverge less rapidly than r^?4 as r→ 0. The asymptotic approach to the absolute-equilibrium distribution is studied for general initial states.     

Coulomb gauge vacua of (2+1) dimensional Yang-Mills theory

The vacuum structure of the (2+1) dimensional Yang-Mills theory is analysed. The non-trivial spherically symmetric vacuum fields in this theory can be calculated in closed form. It is shown that these non-trivial vacuum fields fall faster than 1/r at large r unlike the (3+1) dimensional case, where the vacuum is uniquely A_i = 0 if one requires lim_r→∞rA_i=0.     

An approach to radially symmetrical solutions of the sine-Gordon equation

The solution φ(r, t) of the radially symmetric sine-Gordon equation is considered in three and two spatial dimensions for initial curves, analogous to a 2π-kink, in the expanding and in the shrinking phase, for R(t)j? R(0). It is shown that the parameterization φ(r, t) = 4 arcian exp[γ(r?R(0)] + x(r, t), where R(t) describes the exact propagation of the maximum of φ,(r, t), is suitable. Using an appoximate differential equation, recently given for the propagation of the solitary ring wave, a rough analytic approximation for the correction function x(r = R(t), t) is found and tested numerically. A relationship between the fluctuations in x(r = R(t), t) and those in $\text{R}\text{?}\text{(t), t)}\text{and}\text{R(t)}$ explains why the solitary wave is almost stable. From x(r = R(t), t) and the supposition x(1, t) ≈ x(∞, t) ≈ 0 an assymetry in φ_r(r, t) with respect to r = R(t) is predicted. It also exhibits fluctuations corresponding to those in x(r = R(t), t). The condition for validity of this approximation apparently is also a limit for the stability of the solitary ring wave.     

A comprehensive approach to an equation of state for hard spheres and Lennard-Jones fluids

We present a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way.We will guess equations of state by using suitable axiomatic functional forms (n=1,2,3,4,5) for surface tension S n m (r),r ≥ d/2 with intermolecular separation r as a variable,where m is an arbitrary real number (pole).Among the equations of state obtained in this way are Percus-Yevick,scaled particle theory and Carnahan-Starling equations of state.In addition,we have found a simple equation of state for the hard sphere fluid in the region that represents the simulation data accurately.It is found that for both hard sphere fluids as well as Lennard-Jones fluids,with m=3/4 the derived equation of state (EOS) gives results which are in good agreement with computer simulation results.Furthermore,this equation of state gives the Percus-Yevick (pressure) EOS for the m=0,the Carnahan-Starling EOS for m=4/5,while for the value of m=1 it corresponds to a scaled particle theory EOS.     

Klein paradox in the Breit equation

We demonstrate that in the Breit equation with a central potentialV(r) having the propertyV(r ₀)=E there appears a Klein paradox atr=r ₀. This phenomenon, besides the previously found Klein paradox arr→∞ appearing ifV(r)→∞ atr→∞, seems to indicate that in the Breit equation valid in the singleparticle theory the sea of particle-antiparticle pairs is not well separated from the considered two-body configuration. We conjecture that both phenomena should be absent from the Salpeter equation which is consistent with the hole theory. We prove this conjecture in the limit ofm ⁽¹⁾→∞ andm ⁽²⁾→∞, where we neglect the terms ～1/m ⁽¹⁾ and 1/m ⁽²⁾. In Appendix I we show that in the Breit equation the oscillations accumulating atr=r ₀ in the case ofm ⁽¹⁾≠m ⁽²⁾ are normalizable to the Dirac δ-function. In Appendix II the analogical statement is justified for the nonoscillating singular behaviour appearing atr=r ₀ in the case ofm ⁽¹⁾=m ⁽²⁾.     

Kinetic perturbation theory for dilute gases

We prove, for two classes of smooth, repulsive interparticle potentials ø(r) = ø₀(r) + δø₁(r), that the collision integ rals of the linearized Boltzmann equation are analytic functions of λ in the neighborhood of λ = 0. It then follows, for example, that the first Enskog approximation for the transport coefficients can be represented by a power series in λ.     

Frustrations and phase transitions in the three-vertex Potts model with next-nearest-neighbor interactions on a triangular lattice

Phase transitions in two-dimensional arrays described by the three-vertex Potts model involving the interactions between magnetic moments located at the nearest-neighbor and next-nearest-neighbor sites of a triangular lattice are studied using the Monte Carlo method. The ratio of the next-nearest-neighbor and nearest-neighbor exchange constants r = J ₂/J ₁ is chosen within the 0–1 range. The analysis of the low temperature behavior of the entropy and density of states in the system, as well as of the fourth-order Binder cumulants, shows that in the range 0 ≤ r < 0.2, this model exhibits a first-order phase transition, whereas at r ≥ 0.2, frustrations arise in such a system.     

Growth of nonpolar a-plane GaN on nano-patterned r-plane sapphire substrates

Sapphire substrates were nano-patterned by inductive coupled plasma etching process. Nonpolar a-plane GaN films were grown on planar and nano-patterned r-plane sapphire substrates by metal organic chemical vapor deposition. The anisotropic characteristic and the crystalline quality of the a-plane GaN films were studied through XRD rocking curves. The cross section and surface morphologies of the a-plane GaN films were studied using SEM and AFM measurements, respectively. The crystal quality and surface flatness of the nonpolar a-plane GaN were greatly improved through the usage of the nano-patterned r-plane sapphire substrates.     

The quark mixing matrix in a composite model

The generalized mixing matrix of quarks is computed based on a composite model of quarks and leptons. Among potentials between constituent particles examined V(r) = Ar^s (As > 0) and A ln r (A > 0), it is shown that potentials with exponent 0 ? s ? 3 or ln r are consistent with current experimental data for the mixing matrix elements.     

Baryon- and lepton-number non-conserving processes and intermediate mass scales

An analysis of the possible mechanisms to mediate various baryon- and lepton-number non-conserving processes is presented. Processes considered include the Δ(B + L) = 0 proton decay, ΔB = 2 neutron-antineutron oscillations and neutrino Majorana masses. Among our results we find that, in the absence of elementary scalars and exotic fermions, all the renormalizable interactions of vector bosons and ordinary fermions conserve B - L. Therefore, the observation of Δ(B - L) ≠ 0 processes would imply the existence of elementary scalars and/or exotic fermions.     

On stability of the elliptical “orbits” of diamagnetic positronium

The stability of the elliptical orbits of diamagnetic positronium is studied and it is found to depend only on the eccentricity ε and the combination of parameters λ = B²r_0c³ (where B is the magnetic field strength and r_0c the conserved guiding center). For small and large values of λ the orbits remain stable while at approximately λ = 1 unstable behavior appears when the eccentricity becomes larger than ε = 0.25.     

Microwave l-type resonance transitions of the v4 = 1 state in PF3: Detailed interactions and molecular structure

Observation of the direct l-type resonance transitions in the microwave spectrum of the v₄ = 1 state of PF₃ has been extended to J = 36. The w-type interaction, (Δl = 0, ΔK = 6), has been found from measurements on the “forbidden” Stark trasitions in the $\text{K}\text{= 3}$ series. Also in this series a close accidental degeneracy was found between J = 30, $\text{K}\text{= 3}$ and 0, leading to new zero-field “forbidden” transitions through the r-type interaction (Δl = 2, ΔK = ?1) and to the determination of the C rotational constant. Nine spectroscopic parameters were determined using 140 observed frequencies including two “forbidden” trasitions. After suitable correction the B and C constants were used to determine the r₀, r_z, and r_e structures for PF₃. The equilibrium structure is estimated to be P-F = 1.561 ± 0.001 Å and ∠FPF = 97.7 ± 0.2°.     

On the ground state energy of a two-dimensional electron fluid

A new theory of the ground state energy of a two-dimensional electron fluid is presented. It is shown that the ring diagram contribution changes its analytical behavior atr _s =2^1/2, wherer _s is the usual density parameter defined by r_S = 1/a ₀(π n)^1/2,a ₀ being the Bohr radius andn is the electron density. For smallr _s , a high density series is obtained in agreement with the previous calculation. For larger _s , a hitherto unknown low density series is obtained. In the low density region, the first order exchange energy is completely cancelled out by a term from the ring contribution so that the ground state energy decreases in proportion tor _s ^?2/3 , followed byr _s ^/?4/3 and higher order terms. The energy is found to be minimum atr _s=1.4757, the minimum value being ?0.481915 Rydbergs.     

Frei zerfallende Magnetfelder von Kugelschalen

The decay constants of free decaying magnetic fields of sphere shells are calculated. In the regionr ₁<r<r ₀ the electrical conductivity isσ. For different values of the parameterγ=r ₀/(r ₀-r ₁) the following models are considered: The conductivity is zero forr>r ₀ andr<r ₁ (model 1). The conductivity is zero forr>r ₀, and forr<r ₁ the magnetic field is excluded by infinite conductivity (model 2). The magnetic field is excluded forr>r ₀ andr<r ₁ (model 3).