Singularities in the lineshape of a second-order perturbed quadrupolar nucleus and their use in data fitting

Even for large quadrupolar interactions, the powder spectrum of the central transition for a half-integral spin is relatively narrow, because it is unperturbed to first order. However, the second-order perturbation is still orientation dependent, so it generates a characteristic lineshape. This lineshape has both finite step discontinuities and singularities where the spectrum is infinite, in theory. The relative positions of these features are well-known and they play an important role in fitting experimental data. However, there has been relatively little discussion of how high the steps are, so we present explicit formulae for these heights. This gives a full characterization of the features in this lineshape which can lead to an analysis of the spectrum without the usual laborious powder average.The transition frequency, as a function of the orientation angles, shows critical points: maxima, minima and saddle points. The maxima and minima correspond to the step discontinuities and the saddle points generate the singularities. Near a maximum, the contours are ellipses, whose dimensions are determined by the second derivatives of the frequency with respect to the polar and azimuthal angles. The density of points is smooth as the contour levels move up and down, but then drops to zero when a maximum is passed, giving a step. The height of the step is determined by the Hessian matrix—the matrix of all partial second derivatives. The points near the poles and the saddle points require a more detailed analysis, but this can still be done analytically. The resulting formulae are then compared to numerical simulations of the lineshape.We expand this calculation to include a relatively simple case where there is chemical shielding anisotropy and use this to fit experimental ¹³⁹La spectra of La₂O₃.     

Fano resonance and magneto-optical Kerr rotaion in periodic Co/Ni complex plasmonic nanostructure

we report a pure rerromagneuc metallic magnetopiasmonic structure consisting or two-dimensional oraerea Ni nanodisks array on Co film.With a sufficient height of the nanodisks,a steep and asymmetric Fano resonance can be excited in this structure.We attribute the fascinating spectral lineshape to the strong coupling between the excitation of surface plasmon polaritons at the interface and the localized surface plasmon resonance of nanodisks.The conclusion is fully confirmed by spectrum measurements in nanostructures with different heights.Furthermore,the enhancement and sign of the magneto-optical Kerr rotation in this structure are significantly modified by the Fano resonance.     

Dynamic properties of P4O6S and P4O7:31P spin-echo and 31P MAS-NMR investigations.

The rotational dynamics of P₄O₆S and P₄O₇ in the solid state were studied by means of ³¹P NMR spectra of spinning and static powder samples in the temperature range of 153–295 K and 295–388 K, respectively. All spectra were simulated to confirm the type of the motion and to extract the time scales as a function of the temperature. Good agreement between experimental and theoretical data was obtained on the basis of a three-site jump model. For P₄O₆S, the activation energy and the pre-exponential factor derived from the lineshape simulations amount to 51(2) kJ/mol and 6(3)·10¹⁵ s⁻¹. For P₄O₇, the spectral analysis yields an activation energy of 67(1) kJ/mol and a pre-exponential factor of 6(2)·10¹⁴ s⁻¹. The dynamic behavior was checked independently by lineshape analyses under both MAS and static conditions. Activation energies are consistent within the errors for the lineshape analyses. Additionally, we have analyzed spin–lattice relaxation measurements, which show the correct trends for the activation energies.     

Optical Fiber Excitation of Fano Resonances in a Silicon Microsphere

In this article, Fano lineshape whispering gallery modes were observed in the light scattering spectrum of a silicon microsphere in near-infrared telecommunication wavelengths. A simple model is presented to explain the transition from Lorentzian lineshape to the Fano lineshape resonances with the coupled-mode theory of multiple whispering gallery modes. Polar mode spacing of 0.23 nm is observed in the spectra, which correlates well with the calculated value. The quality factor of the Lorentzian and Fano resonances are on the order of 10⁵. By using an appropriate interface design for the microsphere coupling geometries, Fano lineshape optical resonances herald novel device applications for silicon volumetric lightwave circuits.     

Concentration of small ring structures in vitreous silica from a first-principles analysis of the Raman spectrum

Using a first-principles approach, we calculate Raman spectra for a model structure of vitreous silica. We develop a perturbational method for calculating the dielectric tensor in an ultrasoft pseudopotential scheme and obtain Raman coupling tensors by finite differences with respect to atomic displacements. For frequencies below 1000 cm(-1), the parallel-polarized Raman spectrum of vitreous silica is dominated by oxygen bending motions, showing a strong sensitivity to the intermediate range structure. By modeling the Raman coupling, we derive estimates for the concentrations of three- and four-membered rings from the experimental intensities of the Raman defect lines.     

晶体NiX2(X=Cl,Br,I)的光谱和电子顺磁共振谱研究

在强场耦合图像中,采用双自旋-轨道耦合(SO)参量模型建立了过渡族3d2(3d8)离子的三角对称下全组态光谱能级和电子顺磁共振(EPR)公式.与经典的晶体场理论(仅考虑中心金属离子的自旋-轨道耦合作用)相比较,该公式还包括了配体离子的自旋-轨道耦合作用的贡献,这一模型在应用于计算共价性较强的晶体光谱和电子顺磁共振谱可得到合理的结果.作为验证,用完全对角化方法研究了品体NiX2(X=Cl,Br,I)的光谱和电子顺磁共振谱,结果表明,理论与实验很好地符合.建立的全组态谱能级和电子顺磁共振公式为更精确地计算光谱和电子顺磁共振谱提供了一条可行方法.     

Simulations of molecular dynamics in solid-state NMR spectra of spin-1 nuclei including effects of CSA- and EFG-terms up to second order

By numerical simulations MAS and QCPMG methods for acquiring spectra of spin-1 nuclei were compared in order to determine the most sensitive experiment for analysis of molecular dynamics. To comply with the large quadrupolar constants for 14N and the CSA reported for 6Li both of these interactions are included up to second order. For 2H and 6Li both QCPMG and single-pulse MAS experiments were suitable for dynamics studies whereas the single-pulse MAS experiment were the method of choice for investigation of 14N dynamics for C(Q)'s larger than 750kHz at 14.1T. This property prohibits excitation of the 14N lineshape using either single hard or softer composite rf-pulses. Focusing on 14N it was demonstrated that the centerband lineshape is sensitive toward both off-MAS and CSA effects. In addition, excitation by real-time pulses showed that proper lineshapes corresponding to a site with a C(Q) of 3MHz may be excited by a very short pulse.     

Application of fast amplitude-modulated pulse trains for signal enhancement in static and magic-angle-spinning 47,49Ti-NMR spectra

It is demonstrated that the use of fast amplitude-modulated RF pulse trains with constant (FAM-I) and incremented pulse durations (SW-FAM) leads to considerable sensitivity enhancement for the central-transition signal (via spin population transfer from the satellite transitions) for solid-state NMR spectra of titanium, 47Ti (I = [Formula: see text] and 49Ti (I = [Formula: see text]. For the magic-angle spinning spectra of TiO2 and BaTiO3, the intensity of the 49Ti central-transition line was more than doubled compared to simple Hahn-echo acquisition, while for the static case, enhancement factors of 1.6 (TiO2) and 1.8 (BaTiO3) were obtained. No lineshape distortions are observed in either MAS or static spectra of both compounds. Employment of the FAM and SW-FAM sequences should be useful in the routine acquisition of 47,49Ti spectra, as the NMR signal can be detected much faster.     

Calculation of solid-state NMR lineshapes using contour analysis

Two new methods for calculating lineshapes in solid-state NMR spectra are described. The first method, which we refer to as semi-analytical, allows the rapid calculation of quadrupolar central-transition lineshapes in both static and magic-angle spinning cases. The second method, which is fully numerical, allows the calculation of lineshapes resulting from any combination of interactions, including quadrupolar, dipolar and chemical shift anisotropy, and is not restricted to cases in which the principal axis systems for the different interactions are aligned. Both methods are derived from consideration of the contour lines on a plot of the resonance frequency against the Euler angles, allowing the intensity of the lineshape to be calculated at each frequency. Consequently, highly accurate lineshapes can be calculated more rapidly than previously possible, since only orientations contributing to each specific frequency are considered. For our semi-analytical method, the intensity of each point in the lineshape can be directly calculated in tens of milliseconds on a standard PC. In contrast, established methods can take several hours to calculate the same lineshape.     

Dynamic properties of P4O6S and P4O7: P spin–echo and P MAS–NMR investigations

The rotational dynamics of P₄O₆S and P₄O₇ in the solid state were studied by means of ³¹P NMR spectra of spinning and static powder samples in the temperature range of 153–295 K and 295–388 K, respectively. All spectra were simulated to confirm the type of the motion and to extract the time scales as a function of the temperature. Good agreement between experimental and theoretical data was obtained on the basis of a three-site jump model. For P₄O₆S, the activation energy and the pre-exponential factor derived from the lineshape simulations amount to 51(2) kJ/mol and 6(3)·10¹⁵ s⁻¹. For P₄O₇, the spectral analysis yields an activation energy of 67(1) kJ/mol and a pre-exponential factor of 6(2)·10¹⁴ s⁻¹. The dynamic behavior was checked independently by lineshape analyses under both MAS and static conditions. Activation energies are consistent within the errors for the lineshape analyses. Additionally, we have analyzed spin–lattice relaxation measurements, which show the correct trends for the activation energies.     

Generalization of the lineshape useful in magnetic resonance spectroscopy

A new lineshape function is derived from the Tsallis distribution to describe electron paramagnetic resonance (EPR) spectra, and possibly nuclear magnetic resonance (NMR) spectra as well. This lineshape generalizes the Gaussian and Lorentzian lineshapes that are widely used in simulations. The main features of this lineshape function are presented: the normalization, moments, and first derivative. A number of experimental EPR spectra are compared with the results of simulations employing the new lineshape function. The results show that the new lineshape often provides a better approximation of the experimental spectrum. It is also shown that the new parameter of the lineshape function can be used to quantify the intermolecular spin-spin interactions.     

Slow motion lineshapes

A slow motion expansion about the characteristic features of the powder spectrum is presented. Analytic expressions for the lineshape function, modulated by slow rotational diffusion, are derived. It is shown that the slow motion limit is characterized by harmonic oscillator equations of motion, and the resulting spectrum is determined by harmonic oscillator eigenvalues. The essential features of the lineshape show up naturally, and in particular the axial lineshape diverges like τ^1/4 while there is only a weak motional correction to the logarithmic divergence of the non-axial lineshape. The dynamic frequency shifts converge to their static limits like τ^-1/2 for all cases.     

Vibronic Sidebands of Coherent Phonon States

Recently experiments have been reported about phonon sidebands in doped crystals, which may originate from coherent phonon states. The corresponding modes are either confined phonon modes in nanocrystals or localized phonon modes in bulk materials, both showing small damping due to phonon-phonon interaction. We present a theory of the lineshape of vibronic sideband spectra due to coherent phonon states using the conventional model of linear electron-phonon coupling and displaced equilibrium positions of the oscillators in the initial and final electronic states. Unlike in the conventional theory, the initial state of the oscillator is taken as a coherent phonon state and not as a thermalized one. Under these conditions we got an exact analytical solution for the lineshape of the vibronic sideband. The lineshape is determined by two parameters, the Huang-Rhys parameter S and the coherence parameter α of the phonon state. For α = 0 the lineshape converts into the standard Pekarian form for T = 0.     

Rotational tunnelling splitting of a chain of four coupled methyl groups

It is our goal to obtain a reliable prediction of the rotational tunnelling spectrum to be expected for a long chain of coupled one-dimensional quantum rotors. The problem is intractable by the simple methods used so far for up to three coupled methyl groups. Therefore, an efficient, nevertheless sufficiently precise method for solving the stationary Schrödinger equation of interacting methyl groups is developed first; it proves to be valid for a broad range of not too weak potential strengths. Then, three scenarios are investigated: they differ with respect to the relative strength of the single-rotor potential and the interaction potential. For each scenario, we illustrate the dependence of the energy level scheme on the number of coupled groups. For strong coupling and weak single-particle potential, the characteristic features of the energy level scheme of interacting methyls are most clearly observable: For as few as four coupled methyl groups we predict tunnelling spectra which are hardly distinguishable from single-methyl spectra. However, the collective behaviour is still important for the value of the tunnelling splitting. Therefore, the interpretation of such a spectrum in terms of single-methyl tunnelling is obvious but misleading with respect to the potential seen by a methyl group in the crystal.     

The lineshape of the L2,3 VV Auger spectrum of silicon

An attempt is presented to understand the details of the lineshape of the Si L_2,3 VV Auger spectrum from the (111) surface in the 7 × 7 superstructure. In the experiments we have followed the variation of the lineshape induced by adsorption of O₂, H₂O, CO and by bombardment with 3 keV Ar⁺ ions, over a range from a small perturbation of the surface to major changes in surface structure. For small perturbations from the clean surface we were able to resolve changes in the local density of states at surface silicon atoms. By unfolding the experimental spectra, effective transition densities of states result, which compare quite closely with calculated densities of states, apart from a certain enhancement of surface features in the experiments. All peaks in the experimental spectra can be explained, based on densities of states at the surface of pure Si(111) (7 × 7) (91.8 and 84.8 eV), Si(111) + adsorbed oxygen (70.6 eV), SiO₂, (78.9 and 64.5 eV) and plasmon losses, at 71.0 and 57.5 eV for the clean surface.     

Lineshape of inter-subband optical transitions in space charge layers

Broadening of the inter-subband optical spectra caused by static scatterers is studied both in the absence and presence of a magnetic field normal to space charge layers on semiconductor surfaces. The width of the spectrum is not given by a simple arithmetic average of the widths of the subbands between which the transition takes place. We have a correction term, which cancels a part of the broadening due to intra-subband scattering process in some cases. Under strong magnetic fields, the lineshape depends on the relative position of the two subbands because of the discreteness of the density of states.     

Powder MAS NMR lineshapes of quadrupolar nuclei in the presence of second-order quadrupole interaction

We derive a complete analytical solution for the powder magic angle spinning (MAS) nuclear magnetic resonance (NMR) lineshape in the presence of second-order quadrupole interaction, considering a radiofrequency (rf) pulse of finite width, a finite MAS frequency, and a non-zero asymmetry parameter. I_x is calculated using two approaches. The first applies time-dependent perturbation theory in the presence of the rf pulse and stationary perturbation theory (SPT) in its absence. The second is based on the Magnus expansion of the density matrix in the interaction representation during the pulse and SPT in its absence. We solve the problem in the laboratory frame using the properties of the Fourier transform and spin operators. Diagonalisation is not required. Both approaches agree well with each other under all conditions and also with the transition probability approach for the central transition. The Magnus expansion exists at all times and the effect of the non-secular terms is negligible. We describe an analytical method of averaging I_x over the Euler angles and simulate the ¹¹B MAS NMR lineshapes for crystalline and vitreous B₂O₃. A critical analysis is given of all earlier calculations of the MAS NMR lineshape.     

Spectrum evolutions of spontaneous and pump-depleted stimulated Brillouin scatterings in liquid media

A theoretical model for calculating spontaneous and stimulated Brillouin scattering(SBS) spectra is described. An empirical formula for the Stokes output spectral linewidth, a function of spontaneous Brillouin linewidth and the exponential gain coefficient, is obtained by the calculated data fitting. The formula holds true for two cases involving pump undepletion and depletion. The lineshape change from spontaneous to highly pump-depleted SBS spectra is also investigated. The result shows that for the pump power below the SBS threshold, the Stokes output spectral lineshape evolves from Lorentzian to approximately Gaussian as the pump power increases. For the pump power near or beyond the threshold, the SBS spectrum is in the form of a steady Gaussian profile, and the spectral linewidth comes to a certain value about 7 times narrower than the spontaneous one. The theoretical results are experimentally demonstrated by using several common liquid media.