Combination tones in the response of single degree of freedom systems with quadratic and cubic non-linearities

The method of multiple scales is used to analyze the response of a single-degree-of-freedom system to either the combination resonance of the additive type $\text{Ω}{}_2\text{+ Ω}{}_1\text{≈ ω}{}_0$ or the combination resonance of the difference type $\text{Ω}{}_2\text{? Ω}{}_1\text{≈ ω}{}_0$, where $\text{Ω}{}_1$ and $\text{Ω}{}_2$ are the frequencies of the excitation and ω₀ is the linear undamped natural frequency of the system. To the second approximation, the combination resonance of the additive type has three effects on the steady state response. First, it produces terms having the frequencies $\text{Ω}{}_1\text{, Ω}{}_2$ and $\text{Ω}{}_2\text{+ Ω}{}_1$ at first order and terms having the frequencies 0, $\text{2Ω}{}_1\text{, 2Ω}{}_2\text{, Ω}{}_2\text{? Ω}{}_1$, $\text{2(Ω}{}_2\text{+ Ω}{}_1\text{), Ω}{}_2\text{+ 2Ω}{}_1$ and $\text{2Ω}{}_2\text{+ Ω}{}_1$ at second order. Second, it produces a shift in the natural frequency of the system. Third, it produces a virtual primary-resonant excitation having the frequency $\text{Ω}{}_2\text{+ Ω}{}_1\text{≈ ω}{}_0$ that makes the component having the frequency $\text{Ω}{}_2\text{+ Ω}{}_1$ be of first rather than second order. Similar effects are produced by a combination resonance of the difference type or a superharmonic resonance of order two.     

Two-magnon Raman scattering in the two-dimensional antiferromagnet K2FeF4

Two-magnon Raman scattering in the planar quadratic antiferromagnet K₂FeF₄ is investigated. The temperature dependence of the energy shift is in good agreement with second-order Green-function theory, as is the linewidth at low temperature. Numerical results, including renormalization, are the Heisenberg exchange $\text{J}\text{k}{}_{\mathrm{B}}\text{= ?14.5 ± 0.7}\text{K}$ and the anisotropy $\text{Δ(T = 0) = gμ}{}_{\mathrm{B}}\text{H}{}_{\mathrm{A}}\text{4|J|S = 0.18 ± 0.05}$, but with $\text{J[1 + Δ(T = 0)]}\text{k}{}_{\mathrm{B}}\text{= ?17.06 ± 0.10}\text{K}$.     

Paзлoжeниe пo 1Z для чapтpи-фoкoвcкoй и кoppeляциoннoй энepгии, peлятивиcтcкич пoпpaвoк, дипoльнoгo мaтpичнoгo элeмeнтa

Energies and dipole matrix elements have been calculated for He, Li, Be, B, C, N, O, F, and Ne-like ions (configurations 1s²2sⁿ¹2pⁿ²?1s²2s^n1?12pⁿ²⁺¹). The Hartree-Fock energy, the correlation energy, and relativistic corrections were taken into account. Relativistic corrections were obtained by computing the entire quantity H_B. Numerical results are presented for energies of the terms in the form

$\text{E=E}{}_0\text{Z}{}^2\text{+ ΔE}{}_1\text{Z + ΔE}{}_2\text{+}\text{1}\text{Z}\text{ΔE}{}_3\text{+}\text{α}{}^2\text{4}\text{(E}{}_0{}^{\mathrm{p}}\text{Z}{}^4\text{+ ΔE}{}_1{}^{\mathrm{p}}\text{Z}{}^3\text{)}$

, and for the fine structure of the terms in the form

$\text{〈1s}{}^2\text{2s}{}^{\mathrm{n1}}\text{2p}{}^{\mathrm{n2}}\text{LSJ|H}{}_{\mathrm{Б}}\text{|1s}{}^2\text{2s}{}^{\mathrm{n1\prime }}\text{2p}{}^{\mathrm{n2\prime }}\text{L′S′J〉=(?1)}{}^{\mathrm{L+S\prime +J}}\text{LSJ}\text{S′L′1}\text{×}\text{α}{}^2\text{4}\text{(Z?A)}{}^3\text{[E}{}^{\mathrm{(0)}}\text{(Z ? B)+}\text{E}{}_{\mathrm{c0}}\text{]+(?1)}{}^{\mathrm{L\; +\; S\prime \; +\; J}}\text{LSJ}\text{S′L′2}\text{α}{}^2\text{4}\text{(Z?A)}{}^3\text{E}{}_{\mathrm{cc}}$

. Dipole matrix elements are required for calculation of oscillator strengths or transition probabilities. For the dipole matrix elements, two terms of the expansion in $\text{1}\text{Z}$ have been obtained. Numerical results are presented in the form P(a, a′) = (a/Z)[1 + (τ/Z)].     

Magnetic moments,E3 transitions and the structure of high-spin core excited states in 211Rn

The results of g-factor measurements of high-spin states in ²¹¹Rn are: $\text{E}{}_{\mathrm{<mtext>x</mtext>}}\text{= 8856 + Δ′}\text{keV}\text{(J}{}^{\mathrm{\pi }}\text{=}\text{63}\text{2}{}^{\mathrm{?}}\text{)}$, g = 0.626(7); $\text{6101 + Δ′}\text{KeV}\text{(}\text{49}\text{2}{}^{\mathrm{+}}\text{), 0.766(8)}$; $\text{5347 + Δ′}\text{KeV}\text{(}\text{43}\text{2}{}^{\mathrm{?}}\text{), 0.74(2)}$; $\text{3927 + Δ}\text{KeV}\text{(}\text{35}\text{2}{}^{\mathrm{+}}\text{), 1.017(12)}$; $\text{1578 + Δ}\text{KeV}\text{(}\text{17}\text{2}{}^{\mathrm{?}}\text{), 0.912(9)}$. These results together with measured E3 transition strengths and shell model calculations are used to assign configurations to the core excited states in ²¹¹Rn. Mixed configurations are required to explain the g-factors and enhanced E3 strengths simultaneously.     

Radiative decay of the 2 3Po0 level of beryllium-like ions

A Z-expansion technique is used to estimate the radiative rate for the $\text{2}{}^3\text{P}{}^{\mathrm{<mtext>o</mtext>}}{}_0\text{?2}{}^1\text{S}{}_0$ transition in berryllium-like ions. It is concluded that the transition rate is so slow as to be unobservable.     

A cellular model for the Mössbauer isomer shift of 99Ru, 193Ir, 195Pt and 197Au in transition metal alloys

It is demonstrated that the isomer shift of Mössbauer nuclei in transition metal alloys can be quantitatively described in terms of an atomic cell model. The isomer shift ΔE, relative to the pure metal as a reference, is derived from a change in boundary conditions for the atomic cell; $\text{ΔE = PΔφ}{}^1\text{+ QΔn}{}_{\mathrm{ws}}$, where ø¹ is the electronegativity parameter, n_ws the cell boundary electron density and P and Q are constants for a given Mössbauer nucleus. For solid solutions there is in addition a relatively small size mismatch term.     

Perturbation calculation of the susceptibility in the symmetric Anderson model

Perturbation theoretic expression for the longitudinal susceptibility has been derived taking into account that electron-electron correlations have the same amplitude as electron-hole correlations for the symmetric Anderson model. Numerical calculation of the susceptibility up to third order in the Coulomb interaction U gives: $\text{χ = (}\text{g}{}^2\text{μ}{}^2\text{2}\text{) χ}{}_0\text{(0) [1 + Uχ}{}_0\text{(0) + 0.54 (Uχ}{}_0\text{(0))}{}^2\text{+ 0.21 (Uχ}{}_0\text{(0))}{}^3$].     

The evolution of a cluster of particles and the invariants of a time-dependent oscillator

A simple physical realization is found for the time-dependent harmonic oscillator equation $\text{q}\text{?}\text{+ φ(t)q = 0}$, for its auxiliary equation $\text{g}\text{?}\text{9 + φ(t)? = h}{}^2\text{?}{}^{\mathrm{?3}}$, and for the Ermakov invariant. Through generalization of the model, a set of invariants is obtained for a system of equations of the form .     

The vibration frequency of an atom at the surface of a tight binding metal

The frequency of vibration ω⊥ of an atom normal to the surface of a tight binding metal is calculated in the Einstein model, and is compared with the bulk value ω₀. The value of $\text{(ohms}{}_0{}^2\text{?ohms}{}_{\mathrm{\perp }}{}^2\text{)}\text{ohms}{}_0{}^2$ is found to be greater than the value expected from a nearest neighbour central force model for a small number of electrons or holes in the band, whilst for a half filled band it is lower. Equality is obtained at the band filling for which there is zero surface relaxation. For the (001) surface of a simple cubic lattice this occurs for 0.5 and 1.5 electrons per atom in the band. These results appear to be consistent with the behaviour of Cr in the middle of the transition metal series, and of Ni, Pd and Pt at the end. The possible effect of the spill out of s electrons at the surface, not treated in the model, is briefly discussed.     

Investigation of the central component of SrTiO3 by neutron scattering with eV resolution

We studied the energy width and the width in reciprocal space Δq of the central mode of SrTiO₃ above T_c. At T_c+4° we observed an energy width of about 6×10^?7 eV. If the measured Δq is interpreted by a correlation length $\text{Δq}{}^{\mathrm{?1}}\text{= ξ = ξ}{}_0\text{?}{}^{\mathrm{<mtext>?</mtext><mtext>2</mtext><mtext>3</mtext>}}$ we obtain $\text{ξ}{}_0\text{= 75}\text{A}\text{?}$.     

Rotation-vibration analysis of the Coriolis-coupled ν5g, ν6g, ν8g and ν9 bands of H2CCO

Medium resolution infrared grating spectra of gaseous ketene, H₂CCO were recorded between 1000 and 400 cm^?1, both at instrument temperature (40°C) and with cooling (?40°C). Interferometric Fourier spectra were also measured at ?70°C with resolution 0.22 cm^?1 between 450 and 330 cm^?1. The K structure of the fundamentals ν₅, ν₆, ν₈, and ν₉ was assigned. These fundamentals are coupled by a-axis Coriolis interactions. These couplings were analysed on the symmetric top basis for setting up the perturbation matrix and by utilizing the K-dependent Coriolis shifts of levels. A preliminary analysis of the Coriolis intensity anomalies was also undertaken.Band center values from combination differences are $\text{ν}{}_5{}^0\text{= 587.30 (27)}$ and $\text{ν}{}_6{}^0\text{= 528.36 (39)}\text{cm}{}^{\mathrm{?1}}$. Synthetic spectra indicate the band origins of ν₈ and ν₉ to be close to 977.8 and 439.0 cm^?1, respectively. Estimates of Coriolis coupling constants obtained from synthetic spectra are $\text{ζ}{}_{58}{}^{\mathrm{a}}\text{= + 0.33 (5)}$, $\text{ζ}{}_{68}{}^{\mathrm{a}}\text{= + 0.714 (20)}$, $\text{ζ}{}_{59}{}^{\mathrm{a}}\text{= ? 0.774 (20)}$, and $\text{ζ}{}_{69}{}^{\mathrm{a}}\text{= ? 0.30 (2)}$. Approximate ratios of unperturbed vibrational transition moments obtained from spectral simulations are $\text{M}{}_8{}^0\text{:±iM}{}_5{}^0\text{:±iM}{}_6{}^0\text{:M}{}_9{}^0\text{≈ +2:?9:+10:+0.5}$.     

Lindemann''s criterion and the necessary condition for a melting curve maximum

Dielectric and differential microcalorimetry measurements have been realized on the $\text{Bi}{}_{\mathrm{1?x}}\text{Te}{}_{\mathrm{x}}\text{O}{}_{\mathrm{<mtext>3+x</mtext><mtext>2</mtext>}}$ (0,33 ? x ? 0,50) solid solution. A phase transition has been detected. The variation of the transition temperature with the composition has been determined. The low temperature phase, stable at 20°C, has piezoelectric and non-linear optical properties.     

A further contribution to the theory of the transient hot-wire technique for thermal conductivity measurements

We study the density of states of a one-dimensional tightbinding electron model with random hopping elements. The Hamiltonian is $\text{H = -∑}{}_{\mathrm{i}}\text{J}{}_{\mathrm{<mtext>i+</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(a}{}^{\mathrm{+}}{}_{\mathrm{i}}\text{a}{}_{\mathrm{i+1}}\text{+a}{}^{\mathrm{+}}{}_{\mathrm{i+1}}\text{a}{}_{\mathrm{i}}\text{)}$, where the $\text{J}{}_{\mathrm{<mtext>i+</mtext><mtext>1</mtext><mtext>2</mtext>}}$'s are independent identically distributed random variables. It is proved that the single particle density of states D(E) diverges near E = 0 as $\text{1}\text{|(E}\text{log}{}^3\text{|E|)|}$.     

The Miura transform and the existence of an infinite number of conservation laws of the cylindrical KdV equation

The cylindrical KdV equation, $\text{u}{}_{\mathrm{t}}\text{+ 6uu}{}_{\mathrm{x}}\text{+ u}{}_{\mathrm{xxx}}\text{+}\text{u}\text{2t}\text{= 0}$, is investigated. The Miura transform is obtained by a simple new method based on a “closedness” ansatz. By use of the obtained Miura transform, an infinite number of conservation laws are proved to exist.     

Dissociation activation barrier and heat of chemisorption: A morse-type analytical approach

The activation barrier ΔE¹_ABfor dissociation AB → A + B on transition-metal surfaces is analyzed within an additive Morse-type approach based on the bond-order conservation. It is shown that $\text{Δ}\text{E}{}^1{}_{\mathrm{AB}}\text{=}\text{D}{}_{\mathrm{AB}}\text{?(}\text{Q}{}_{\mathrm{A}}\text{+}\text{Q}{}_{\mathrm{B}}\text{+}\text{Q}{}_{\mathrm{A}}\text{Q}{}_{\mathrm{B}}\text{/(}\text{Q}{}_{\mathrm{A}}\text{+}\text{Q}{}_{\mathrm{B}}\text{)}$ where D_AB is the gas-phase dissociation energy and Q_A(Q_B) is the heat of atomic chemisorption. Estimates of $\text{Δ}\text{E}{}^1\text{for H}{}_2\text{, N}{}_2\text{, O}{}_2$, and NO are shown to be in reasonable agreement with experiment. The two-dimensional potential diagram of the metal-AB interactions is defined analytically and discussed in some detail.     

Hyperfine structure of CH3OH

Hyperfine structure of the (0, 0, 1) - (1, 0, 1) transition of methanol has been investigated by beam absorption and of the (J, 1, 3?) → (J, 1, 3+) transitions for J = 2, 3, and 6 by beam-maser spectroscopy. The best-fit results for the spin-rotation and spin-spin coupling constants $\text{C}{}_{\mathrm{JK\tau \pm }}{}^{\mathrm{(i)}}$ and $\text{D}{}_{\mathrm{JK\tau \pm }}{}^{\mathrm{(i)}}$, respectively, are in kHz¹: $\text{C}{}_{101}{}^{\mathrm{(1)}}\text{= 2.4(10)}$, $\text{C}{}_{101}{}^{\mathrm{(2)}}\text{= ?0.6(10)}$, $\text{D}{}_{101}{}^{\mathrm{(1)}}\text{= ?13.8(9)}$, $\text{D}{}_{101}{}^{\mathrm{(2)}}\text{= 7.0(9)}$, $\text{C}{}_{\mathrm{213?}}{}^{\mathrm{(1)}}\text{= ?5.0(10)}$, $\text{C}{}_{\mathrm{213?}}{}^{\mathrm{(2)}}\text{= ?5.5(10)}$ and ($\text{C}{}_{\mathrm{J13?}}{}^{\mathrm{(2)}}\text{- C}{}_{\mathrm{J13+}}{}^{\mathrm{(2)}}\text{) = 0.98(9)}$.     

Effect of constant mismatch on stimulated scattering in laser plasma interactions

The presence of a constant wave number mismatch (Δk) renders the asymptotic steady of the backward scattered wave time dependent. While the pump wave is unaffected by the mismatch, the excited waves (with group velocities v₂, v₃) are modulated at the frequency $\text{[}\text{v}{}_2\text{v}{}_3\text{(v}{}_2\text{+ v}{}_3\text{)}\text{] Δk}$.     

Pressure effect on the Curie temperature of rare-earth and transition metal compounds

A simple formula for the pressure dependence of the Curie temperature $\text{d}\text{T}{}_{\mathrm{c}}\text{d}\text{P}$ is derived in the s-d model for heavy rare-earth and transition metal (Fe, Co, and Ni) compounds. A qualitative agreement between the calculated and experimental results for $\text{d}\text{T}{}_{\mathrm{c}}\text{d}\text{P}$ is obtained for various intermetallic compounds.