Dynamic-angle spinning of quadrupolar nuclei. 1990

In dynamic-angle spinning (DAS), a sample spins around an axis inclined at an angle θ(t) with respect to the magnetic field such that the averages of P_n(cosθ) are zero. The simplest case is where θ(t) assumes two discrete values θ₁ and θ₂ (complementary DAS angles) such that the averages of P₂(cosθ) and P₄(cosθ) are zero, thereby removing second-order quadrupolar (and dipolar) broadening. Examples of DAS complementary angles are θ₁=37.38° and θ₂=79.18°. Experimental details for DAS experiments are provided and applications to sodium-23 and oxygen-17 NMR illustrate the enhanced resolution achieved by removing the second-order broadening inherent in magic-angle spinning.     

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.     

Studying the effect of biaxial anisotropy on nonlinear magnetization oscillations accompanying the 90° pulsed magnetization process in ferrite–garnet films with easy-plane anisotropy

Time dependences of the azimuthal component of the torque T _φ(t) acting on magnetization are calculated to understand the nature of the delayed magnetization acceleration effect observed during the 90° pulsed magnetization of real ferrite–garnet films, in which biaxial anisotropy exists alongside with in-plane anisotropy. A calculation technique based on analyzing an operating point trajectory is used. Calculations show that if the effective anisotropy field H _K2 is comparable to the magnetizing pulse amplitude H _ma, abruptly ascending regions at characteristic times t* in curves T _φ(t) arise, in the limit of which nonlinear magnetization oscillations formed. The shape of these regions depends weakly on the magnetizing pulse front duration τ_f. This explains the reason of the weak dependence of the nonlinear magnetization oscillations on duration of the magnetizing pulse front. Calculations also show that the main features of the delayed acceleration effect are less clear upon an increase of the pulse amplitude: the behavior of curves T _φ(t) becomes smoother near times t*, and an increase in the pulse front duration is accompanied by a stronger drop in the intensity of magnetization oscillations.     

An ab initio study of vibration and rotation effects in isocyanatomethane,isothiocyanatomethane and azidomethane

The potential energy surfaces (PESs) for methyl group rotation accompanied by N-atom inversion in isocyanatomethane (MeNCO), isothiocyanatomethane (MeNCS) and azidomethane (MeN₃), have been investigated by Möller-Plesset (MP2), coupled cluster (CCD) and, in part, CCSD(T) ab initio calculations. Analytical forms for these surfaces are represented in terms of the (H₃)CNX (X=N, C) angle (θ°) and the HCNX torsion angle (φ°) using the expression E(θ, φ) = E₀(θ) + ½δE(θ)(1 + cos 3φ), where E₀(θ) and δE(θ) are even polynomials in (180—θ). In contrast to previous works of this nature, here it is shown explicitly that the fitting procedure is sufficiently flexible to provide accurate representations of the PES. It is found that the barrier to linearity and the barrier to internal methyl-group rotation both increase in the order MeNCS < MeNCO < MeN₃. The ratio of harmonic to quartic coefficients in the expansion of E₀(θ) also increases along this same series, and is largely responsible for the geometry of the CNX group becoming increasingly bent. The energetic and structural results compare favourably with data from previous experimental and theoretical studies.     

Extension to velocity dependent gauge transformations

In previous papers it was shown that in phase space a generator of gauge transformations for a singular Lagrangian \(L(q,\dot q)\) is given byG[ε^α]=ε _k ^α φ _α ^k (q,p,t) where φ _α ^k are first class constraints andG is subjected to a stationarity condition. A non-trivial extension from velocity (or momentum) independent gauge transformations to velocity dependent ones is realized, by replacing gauge functions ε^α(t) with momentum dependent functions ε^α(q,p,t), as long asG satisfies the stationarity condition. Inversely, it is proved in a classical framework that, within velocity dependent gauge transformations, all generators of gauge transformations can be expressed in terms of the linear combinations of φ _α ^k .     

A simple and direct periodic solution of the Korteweg-de vries equations

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrödinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw _t (x, t, λ)=φ _t (x + c, x, t, λ) withq _t (x, t). The functionφ(x, x ₀,t, λ) obeys the Schrödinger equation and the boundary conditionsφ(x ₀,x ₀,t, λ)=0,φ _x (x ₀,x ₀;t, λ)=1. The shiftingc is equal to the period. We differentiatew _t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq _t (x, t) by an expression of the KdV hiearchy in the relation betweenq _t (x, t) andw _t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.     

On the transfer of line radiation in random magnetic fields

An analysis is made of the influence of a magnetic field of constant magnitude but with random orientations, described by the spherical coordinates θ and φ, on the polarization transfer in spectral lines. The assumption that the statistical properties of the field are determined by the one- and two-point probability densities P₁(x, θ, φ) and P₂(x₁, θ₁, φ₁; x₂, θ₂, φ₂) where x is the coordinate along the ray, implies that the variations of θ and φ can be considered as being due to a Markovian process in x. The expectation values of the Stokes parameters 〈Iγ〉, γ = 1, …4, are calculated functions of x or the optical depth τ. The formalism includes the limiting cases of infinite and zero correlation length. Assuming an artificial triplet and simple conditions, the results suggest that isotropic magnetic fields of the type considered have no influence on the polarization characteristics during the transfer.     

Nonergodic dynamics of a system of nuclear spins 1/2 with identical spin-spin coupling constants

The exact solution for the evolution of nuclear spin polarizations in a system with spin-spin coupling constant g identical for all spin pairs is obtained on the condition that only one (first) spin is polarized at zero time. It is shown that the polarization P ₁(t) of the first spin has the form of periodic time pulsations with the period 4π/g. In every period, the function P ₁(t) changes from its initial value P(0)=1 to 1/3 during the time on the order of t≈4π/Ng, if the number of spins N?1, and remains in the state P ₁(t)=1/3 virtually during the entire period. A simple classical model within the framework of mean-field theory explains the physical nature of the nonergodic dynamics of the system.     

圆对称环域Josephson结中的零场台阶

对计及干涉项的描述圆对称环域Josephson结扰动sine-Gordon方程数值研究表明:在无外磁场时,对于内外半径分别为7.5λ_j和12.5λ_j的结,在它的Ⅰ-V特性曲线上呈现两支零场台阶,它们分别对应于单孤子和三孤子激发,双孤子的激发是不稳定的,它仅存在于暂态过程之中,三孤子激发相当于一维结中三孤子的对称模式,且伴随着包括不同振荡周期等的运动不规则性。 关键词：     

Barrier to internal rotation of the silyl group in cyclopropylsilane from combination band spectra

Combination band spectra arising in the region 2000–2300 cm^?1 of cyclopropylsilane can be attributed to the sum and difference bands of the internal rotation with an SiH stretching mode. Analysis of these spectra leads to a potential function for the internal rotation V(cm^?1) = 693.92(1 ? cos 3φ) ? 10.28(1 ? cos 6φ), where φ is the angle of internal rotation. The barrier to internal rotation is thus 1.98 kcal/mole.     

A simple relationship between chandrasekha's S and T functions

Chandrasekhar's equations for the functions S and T which represent the diffuse scattering and the diffuse transmission, respectively, of a beam of radiation by a planar atmosphere are invariant under the transformations S(τ₁; μ, φ - μ₀), φ₀) exp (-τ₁/μ₀) = T(τ₁; μ, φ; μ₀, φ₀) and T(τ₁; μ, φ; -μ₀, φ₀) exp(-τ_t/μ₀) = S(τ₁; μ, φ; μ₀, φ₀). This relationship reflects the physical symmetry of radiative transfer in a planar atmosphere. It offers a simple method for obtaining the expression for S (T) when an analytical expression for T (S) exists. However, it does not appear to ofter any simplification of numerical solutions.     

Asymptotic perturbation expansion in theP(φ)2 quantum field theory

In theP(φ)₂ model it is proved that the perturbation series for the infinite volume Schwinger functionsS(λ) are asymptotic in the limit as the coupling constant λ goes to zero. We also give conditions which imply smoothness ofS(λ) at arbitrary λ.     

Principles of generalized spin-echo spectroscopy

The concept is proposed for determining the total dynamic scattering function of an object under study, representing a sum of odd and even parts measured by the generalized neutron spin-echo method in the form of the signals S _odd(q, t) ～ ΣS(ω, q)sin(ωt)dω and S_even(q, t) ～ ΣS(ω,q)cos(ωt)dω as functions of the momentum q transferred to the neutron and the time t corresponding to the frequency ω and the transferred energy ?ω. The principle of the generalized spin echo and the results of mathematical modeling are confirmed in experiments on inelastic scattering on magnetic fluids and polymer solutions. The developed method makes it possible to study the features of the dynamics of atomic and molecular systems, e.g., to analyze soft vibrational spectra of nanoparticle ensembles against the background of intense relaxation processes, which is inaccessible for classical spin-echo spectrometry.     

How to measure the charm density in the proton

We study two experimental ways to measure the heavy-quark content of the proton: using the Callan-Gross ratio R(x, Q ²) = F _L /F _T and/or the azimuthal cos(2φ) asymmetry in DIS. Our approach is based on the following observations. First, the ratio R(x, Q ²) = F _L /F _T and azimuthal cos(2φ) asymmetry in heavy-quark leptoproduction are stable, both parametrically and perturbatively, within pQCD. Second, both these quantities are sensitive to resummation of the mass logarithms of the type α_sln(Q ²/m ²). We conclude that the heavy-quark densities in the nucleon can, in principle, be determined from high-Q ² data on the Callan-Gross ratio and/or the azimuthal asymmetry. In particular, the charm content of the proton can be measured in future studies at the proposed Large Hadron-Electron (LHeC) and Electron-Ion (EIC) Colliders.     

On the oscillons in the signum-Gordon model

New periodic solutions of signum-Gordon equation are presented. We first find solutions φ₀(x, t) defined for (x, t) ∈ ? × [0, T ] and satisfying the condition φ₀(x, 0) = φ₀(x, T ) = 0. Then these solutions are extended to the whole spacetime by using (2.4).     

Time vs. ensemble averages for nonstationary time series

We analyze whether sliding window time averages applied to stationary increment processes converge to a limit in probability. The question centers on averages, correlations, and densities constructed via time averages of the increment x(t,T)=x(t+T)−x(t), e.g. x(t,T)=ln(p(t+T)/p(t)) in finance and economics, where p(t) is a price, and the assumption is that the increment is distributed independently of t. We apply Tchebychev’s Theorem to the construction of statistical ensembles, and then show that the convergence in probability condition is not satisfied when applied to time averages of functions of stationary increments. We further show that Tchebychev’s Theorem provides the basis for constructing approximate ensemble averages and densities from a single, historic time series where, as in FX markets, the series shows a definite ‘statistical periodicity’. The convergence condition is not satisfied strongly enough for densities and certain averages, but is well-satisfied by specific averages of direct interest. Rates of convergence cannot be established independently of specific models, however. Our analysis shows how to decide which empirical averages to avoid, and which ones to construct.     

A remark on transition probability

Uhlmann's transition probability P(ψ, φ) of two normal states of a von Neumann algebra M, which is the supremum of |(Ψ, Φ)|² for all possible choices of representative vectors Ψ and Φ of ψ and φ, is shown to be the infimum of (∫d(μ_{ψ, e})^1/2)² for the induced measures μ_{ω, e}(B)=ω(e(B)) (B: Borel set in ℝ, ω=ψ, φ) for all possible projection-valued measures e belonging to M.     

A convergent expansion about mean field theory: I. The expansion

We give a convergent expansion for nearly Gaussian quantum field theory in the multiphase region. The expansion combines (1) an expansion in phase boundaries, (2) a cluster expansion, and (3) a perturbation expansion to isolate dominant behavior. We study in detail the ground state of the $\text{P}$(φ)₂ = (λφ⁴ ? φ² ? μφ)₂ model, with ∥ μ ∥ ? λ² ? 1. The ground state is close to the classical free field, obtained by replacing $\text{P}$(φ) by the quadratic mean field polynomial $\text{P}$_c(φ), tangent to $\text{P}$ at a global minimum. Selecting one minimum gives a pure phase (ergodic ground state) satisfying the Wightman-Osterwalder-Schrader axioms with a positive mass. We also establish analyticity in λ for μ = 0 in the sector ∥ Im λ ∥ < ? Re λ ? 1, for ? ? 1.