non-Lorentzian Shape Research Articles

Measurements of the dynamic structure factor near the lambda temperature in liquid helium

The dynamic structure factor $S(q,\ensuremath{\omega})$ was measured by Brillouin scattering in liquid $^{4}\mathrm{He}$ at $q=1.79\ifmmode\times\else\texttimes\fi{}{10}^{5}$ ${\mathrm{cm}}^{\ensuremath{-}1}$. Results were obtained at two densities: $\ensuremath{\rho}=0.175$ g/${\mathrm{cm}}^{3}$ for which ${P}_{\ensuremath{\lambda}}=23.1$ bars and $\ensuremath{\rho}=0.179$ g/${\mathrm{cm}}^{3}$ for which ${P}_{\ensuremath{\lambda}}=28.5$ bars. Values of the reduced temperature $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{\ensuremath{\lambda}})}{{T}_{\ensuremath{\lambda}}}$ varied from -5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ to -5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ and from 1.3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$ to 5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$. This range involves both the hydrodynamic and the critical regions; values of $q\ensuremath{\xi}$ fall between 5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ and 21 below ${T}_{\ensuremath{\lambda}}$ and between 4.5 and 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ above ${T}_{\ensuremath{\lambda}}$. This article presents the data on the critical mode, which is second sound below the transition and heat diffusion above. The most striking result is that the damping of the critical mode is essentially independent of $\ensuremath{\epsilon}$ over the entire range of temperatures studied. This behavior is qualitatively different from the strong temperature dependence, consistent with dynamic scaling, which is observed at low frequencies. On the other hand, any dispersion in the velocity of high frequency (\ensuremath{\sim} several MHz) second sound, if present, is less than 2%. $S(q,\ensuremath{\omega})$ retains a two-peaked structure well into the critical region below ${T}_{\ensuremath{\lambda}}$; it still appears in the measured spectra (which contain the instrumental width as well) at $q\ensuremath{\xi}\ensuremath{\sim}4$. At ${T}_{\ensuremath{\lambda}}$ and in the critical region just above, there is evidence for a non-Lorentzian shape of $S(q,\ensuremath{\omega})$. The critical mode contribution to $S(q,\ensuremath{\omega})$ is compared with a theoretical calculation of Hohenberg, Siggia, and Halperin based on the planar-spin model of $^{4}\mathrm{He}$. The theory matches both the overall width and the general features of the shape of $S(q,\ensuremath{\omega})$ to within our spectral resolution at ${T}_{\ensuremath{\lambda}}$. However, closer to $q\ensuremath{\xi}=1$ in the critical region, and in the hydrodynamic regions both above and below ${T}_{\ensuremath{\lambda}}$, the theory predicts linewidths which are too small.

Weyl’s theory applied to predissociation by rotation. I. Mercury hydride

Weyl’s theory for singular ordinary second order differential equations is presented as a numerical method for analyzing the spectral properties of the Schrödinger operator associated with predissociation by rotation in diatomic molecules. It is pointed out that the poles of Weyl–Titchmarsh’s m function characterize both the discrete eigenvalues and the resonances in the continuous spectrum, thus allowing a unified treatment of the entire spectrum. The conventional hard core treatment of the origin is compared to two alternative approaches, both applicable to more general singular behavior of the potential. A detailed discussion of the other singularity, infinity, is given and Kodaira’s theorem is generalized to an expression involving Wronskian’s between the regular and the asymptotic solutions. The calculation of the Weyl–Titchmarsh m function is based on numerical Runge–Kutta integrations of both independent solutions, combined with asymptotic information derived from a Riccati expansion. Im(m) is shown to have its physical significance as flux of predissociating particles. The resonances in the continuous spectrum are obtained by numerical analytical continuation of Im(m) across the real energy axis. The method is applied to Stwalley’s isotopically combined (IC1) potential for HgH. All resonance energies and lifetimes are reported and compared to Stwalley’s phase shift results. Excellent agreement is found for sharp resonances whereas considerable discrepancies occur for the broader ones. This latter fact is attributed to the failure of the Breit–Wigner fit to account for asymmetric non-Lorentzian shapes.

Propagative spin relaxation in the Ising-like antiferromagnetic linear chain

It is shown that at low temperatures the spin dynamics of an antiferromagnetic linear chain can be governed by propagation of boundaries between antiferromagnetic one-dimensional domains. A double maximum results in the neutron-scattering function S( q, ω) if the one-dimensional momentum transfer q is not too close to a one-dimensional superlattice point. In contrast with ordinary magnons of an anisotropic system, the dispersion law corresponding to the maximum exhibits no gap. If q = π, S( q, ω) is centered at ω = 0 but has a nonlorentzian shape. At moderately low temperatures the neutron linewidth at q = π amd the spin-lattice relaxation time are both proportional to the inverse correlation length K, whereas at very low temperature T, they are proportional to K T 1 2 . Much smaller values of T 1 and of the neutron linewidth are expected if the spin dynamics is governed by thermally activated processes, as occurs if boundaries are trapped by lattice imperfection and cannot propagate. In the ferromagnetic Ising-like chain, spin dynamics is always governed by thermally activated processes.

Non-Lorentzian Shape of the Depolarized Rayleigh Line and the Correlation Function of the Tensor Polarization

The correlation function of the tensor polarization relevant for the depolarized Rayleigh line of a gas of rotating linear molecules is calculated for the pressure broadening regime. Point of depar­ture is the Waldmann-Snider equation for the distribution function of the gas. Due to the collisional coupling between the tensor polarization and other moments of the distribution function the cor­relation function turns out to be a sum of exponential functions. Consequently the depolarized Rayleigh line has a non-Lorentzian shape.

Generalized hydrodynamics for dilute gases: The application to light-scattering experiments

Abstract The application of generalized hydrodynamics to dilute gases is considered. A density expansion technique for treating k -nad ω-dependent current-density correlation functions is discussed and carried out for dilute gases. It is shown that to order k2, these correlatio functions can be reduced to single integral expressions involving only two or three Legendre components of the linearized Boltzmann collision operator. The expressions are evaluated analytically for a dilute gas of Maxwell molecules and exact numerical expressions for the generalized transport coefficients, consistent with the order of approximation considered, are obtained. From these expressions, the appropriate approximation to the density-density autocorrelation functions is determined and compared to forms obtained from the Navier-Stokes and the Burnett equations. The failure of the Burnett equations to predict cross-correlation functions which satisfy the appropriate symmetry properties is alos discussed. The detailed form for the light-scattering spectrum is presented and analyzed. It is shown that the spectrum does not consist of just three lorentzian lines. To the order of the first correction to the speed of sound, which exhibits a positive dispersion in this regime, a shift in the positions of the maxima of the Brillouin lines toward the Rayleigh line is shown to arise solely from the non-lorentzian shape of the Brillouin doublet and not from an overlap with the central line as has been reported previously, The results are presented in a form which can readily be compared to experiments done on gases. The interpretation of experiments on liquids is also discussed with an emphasis on how the importance of generalized hydrodynamic effects can be deduced from the data.

Anomalies in shape and width of the Mössbauer line measured in Bragg scattering of 14.4 keV γ-rays from the α-57Fe2O3 perfect single crystal

Abstract It has been found that the Mossbauer line measured in nuclear Bragg scattering of 14.4 keV 57 Fe γ-rays from planes (111) of the α-Fe 2 O 3 perfect single crystal has a non-lorentzian shape and an appreciable broadening. Analysis of the results shows that these anomalies stem from a collective character of the interaction between resonant γ-rays and nuclei in the perfect single crystal.

Mössbauer Resonance of 57Fe in Oxidic Spinels Containing Copper and Iron

The Mössbauer resonance of 57Fe in tetragonal and cubic phases of the spinel copper ferrite (CuFe2O4) has been investigated between 4.2° and 1000°K. Complementary experiments have been performed on the spinel systems Cu1−xZnxFe2O4 and Cu1−xCdxFe2O4, with O≤x≤0.5. At 4.2°K, at least two nonequivalent six-line patterns, I and II, are resolved in the spectra of all phases, exhibiting a combined magnetic and electric hyperfine interaction. The isomer shifts, internal magnetic fields, and nuclear quadrupole splittings have been determined as a function of temperature. At higher temperatures, below the Néel point, the cubic phases show an apparent six-line pattern with lines of non-Lorentzian shape and unusually broad half-widths. Above the Néel temperature, a distinct electric quadruple splitting was observed in the spectra of the tetragonal phases while the cubic phases show an apparent one-line pattern of a complex shape.

Theory of Linewidths in Electron Spin Resonance Spectra

A general theory of the linewidths in the electron spin resonance spectra of dilute solutions of free radicals has been developed in terms of the relaxation-matrix theory of Bloch, Redfield, and Ayant. In contrast to previous theories, it is shown that a composite line arising from a set of degenerate nuclear-spin states should, in general, consist of a sum of superimposed lines of Lorentzian shape with different widths rather than a single line with an over-all Lorentzian shape. A single Lorentzian line is still obtained, however, as a limiting case when the variation of the widths of the different components of a composite line is small compared to the average width. Although the non-Lorentzian shape of a composite line is often difficult to observe experimentally, a number of other observable properties are predicted by the present development that are outside the scope of the previous theories. For example, linewidth effects resulting from differences in the widths of the separate components of a composite line are predicted that explain the alternation in the linewidths from one hyperfine line to another recently observed in the ESR spectra in certain free radicals. The detailed form of the relaxation matrix is presented for intramolecular anisotropic and isotropic electron—nuclear dipolar interactions, quadrupole interactions, and g-tensor relaxations. Modulations of the spin density and hyperfine splittings are included, as are internal motions, and a number of cross terms between the different relaxation mechanisms arise. In general the relaxation matrix of a composite line contains significant off-diagonal elements, and the determination of the linewidths requires the evaluation of the eigenvalues of the matrix. Problems involving rapid chemical exchange, or modulation by jumps to a small number of sites, can be treated by the relaxation-matrix theory and, under special restrictions, by either the modified Bloch equations or the Anderson theory of motional narrowing. When applicable, these latter procedures can be used over the entire range of exchange rates, while the relaxation-matrix theory is limited to fast rates only.