Nutation Resonance Research Articles

Nonlinear response of magnetization obeying inertial dynamics to superimposed dc and ac fields and dynamic magnetic hysteresis

The nonlinear ac stationary responses of magnetization and the dynamic magnetic hysteresis of ferromagnetic particles with uniaxial anisotropy acted on by both external dc and ac magnetic fields are evaluated by employing the inertial Landau–Lifshitz–Gilbert equation. Analytical expressions are obtained for the components of the second- and third-order nonlinear susceptibility tensors using the perturbation method. The dynamic nonlinear susceptibility strongly depends on the dc field strength. The effect of inertia on the area and slope of the dynamic magnetic hysteresis loop is demonstrated. In addition to nutation resonance, the main nonlinear effects previously observed in the ferromagnetic resonance region (frequency doubling and appearance of weak subharmonic resonance peaks) are predicted in the nutation resonance frequency region.

Antiferromagnetic and nutation resonance frequencies of antiferromagnets at an arbitrary strength of the applied dc field

Nutation and precession resonances in an antiferromagnet subjected to a dc magnetic field are investigated by employing coupled linearized inertial Landau–Lifshitz–Gilbert equations describing the dynamics of magnetizations of antiferromagnet sublattices with uniaxial magnetocrystalline anisotropy. Analytical expressions for the eigenfrequencies of such an antiferromagnet are obtained for the longitudinal and transverse directions of the external dc field and for different ranges of its strength. The effect of inertia on the values of the resonant frequencies is shown for all possible states of the antiferromagnet in both the longitudinal and transverse directions of the external field. The estimated resonant frequencies are compared with those obtained from the numerical solution of the system of undamped inertial Landau–Lifshitz–Gilbert equations for closed trajectories of sublattice magnetizations. The good agreement of both independent estimations is demonstrated.

Inertial rotational diffusion and magnetic relaxation of the spin system in a strong magnetic field

Analytical expressions for the longitudinal and transverse correlation functions pertaining to inertial magnetic relaxation are obtained by expanding the deterministic magnetization trajectories into a Fourier series and averaging the result over all possible initial conditions applying the stationary Boltzmann distribution function. The longitudinal and transverse components of the magnetic susceptibility tensor are calculated for a system of noninteracting macrospins in a strong uniform external field using Bloch's phenomenological approach, which postulates the exponential evolution of the spin system towards equilibrium, and the Lorentz model of rotational diffusion of the magnetization vector. It is shown that the strength of the external field and the magnitude of the inertia parameters noticeably affect the shape of the susceptibility in the THz (nutation resonance) spectrum region. It is also demonstrated that the simple analytical Lorentz-type expression describes the main features of the complex susceptibility in the ferromagnetic resonance and nutation resonance regions.

Ferromagnetic and nutation resonance frequencies of nanomagnets with various magnetocrystalline anisotropies

Nutation and precession resonances are investigated for nanomagnets with uniaxial, biaxial, and cubic magnetocrystalline anisotropies employing the linearized inertial Landau–Lifshitz–Gilbert equation. Analytical expression analogous to the Smit–Beljers–Suhl formula for resonance frequencies is obtained. The estimated nutation resonance frequencies are compared with those obtained from the undamped inertial Landau–Lifshitz–Gilbert equation by determining numerically closed trajectories near the bottom of the deepest potential well. The good agreement of both independent estimations is demonstrated.

Spin pumping at terahertz nutation resonances

We investigate spin pumping current injected by the nutation resonances of a ferromagnet or an antiferromagnet into an adjacent metal. Comparing the dc spin pumping current between the normal precession and nutation resonances, we find that the ratio of spin pumping current at the nutation resonance to the precession resonance is more pronounced in antiferromagnets. We further show that the spin pumping current injected by the nutation resonance is opposite in sign as compared to the normal precession mode. This could offer a useful experimental signature for identifying such nutation resonances. Analyzing the nature of the nutational eigenmodes, we show that the sign change in spin current is rooted in a reversal of the precession sense for the nutation mode(s). Furthermore, the nutational modes in antiferromagnets are found to be dominated by precession of one of the two sublattices only.

Influence of intersublattice coupling on the terahertz nutation spin dynamics in antiferromagnets

Spin nutation resonance has been well-explored in one-sublattice ferromagnets. Here, we investigate the spin nutation in two-sublattice antiferromagnets as well as, for comparison, ferrimagnets with inter-and intra-sublattice nutation coupling. In particular, we derive the susceptibility of the two-sublattice magnetic system in response to an applied external magnetic field. To this end, the antiferromagnetic and ferrimagnetic (sub-THz) precession and THz nutation resonance frequencies are calculated. Our results show that the precession resonance frequencies and effective damping decrease with intra-sublattice nutation coupling, while they increase with inter -sublattice nutation in an antiferromagnet. However, we find that the THz nutation resonance frequencies decrease with both the intra-and inter-sublattice nutation couplings. For ferrimagnets, conversely, we calculate two nutation modes with distinct frequencies, unlike antiferromagnets. The exchange-like precession resonance frequency of ferrimagnets decreases with intra-sublattice nutation coupling and increases with inter-sublattice nutation coupling, like antiferromagnets, but the ferromagnetic-like precession frequency of ferrimagnets is practically invariant to the intra and inter-sublattice nutation couplings.

Theroy of magnetic inertial dynamics in two-sublattice ferromagnets

The magnetic inertial dynamics have previously been investigated for one sublattice ferromagnets. Here, we develop the magnetization dynamics in two-sublattice ferromagnets including the intra- and inter-sublattice inertial dynamics. First, we derive the magnetic susceptibility of such a ferromagnet. Next, by finding the poles of the susceptibility, we calculate the precession and nutation resonance frequencies. Our results suggest that while the resonance frequencies show decreasing behavior with the increasing intra-sublattice relaxation time, the effect of inter-sublattice inertial dynamics has an opposite effect.

Nutation in antiferromagnetic resonance

The effect of inertial spin dynamics is compared between ferromagnetic, antiferromagnetic and ferrimagnetic systems. The linear response to an oscillating external magnetic field is calculated within the framework of the inertial Landau--Lifshitz--Gilbert equation using analytical theory and computer simulations. Precession and nutation resonance peaks are identified, and it is demonstrated that the precession frequencies are reduced by the spin inertia, while the lifetime of the excitations is enhanced. The interplay between precession and nutation is found to be the most prominent in antiferromagnets, where the timescale of the exchange-driven sublattice dynamics is comparable to inertial relaxation times. Consequently, antiferromagnetic resonance techniques should be better suited for the search for intrinsical inertial spin dynamics on ultrafast timescales than ferromagnetic resonance.

Nutation resonance in ferromagnets

The inertial dynamics of magnetization in a ferromagnet is investigated theoretically. The analytically derived dynamic response upon microwave excitation shows two peaks: ferromagnetic and nutation resonances. The exact analytical expressions of frequency and linewidth of the magnetic nutation resonance are deduced from the frequency dependent susceptibility determined by the inertial Landau-Lifshitz-Gilbert equation. The study shows that the dependence of nutation linewidth on the Gilbert precession damping has a minimum, which becomes more expressive with increase of the applied magnetic field.

Detection of free core nutation resonance variation in Earth tide from global superconducting gravimeter observations

In order to verify the time variability of free core nutation (FCN) period, global superconducting gravimeter (SG) observations were analyzed based on synthetic test data. The gravity data series were synthesized to check the detectability of resonance variation caused by FCN period change. The tests indicate that the discrepancy between the FCN periods determined by SG and VLBI observations is caused by the high correlation between the FCN parameter and the amplitude factor of the ψ1 wave. The K1 wave is more sensitive to the FCN period change than other diurnal waves. The limit of the standard deviation of the K1 wave is found for more precisely observing the FCN period change. Tidal parameters of diurnal waves estimated from long series of 20 global SG stations were analyzed. A common variation trend is found in the amplitude factor of both K1 and ψ1 waves in all 8 stations above the limit, which indicates the FCN period may be not so stable in time. Furthermore, the variation in the K1 and ψ1 waves constrains the FCN period change to between 2.5 and 4 sidereal days, which also agrees with the possible variation from the current VLBI and SG observations.

Highly Efficient Tuning of Ferromagnetic Spin Interactions in High-Spin Arylamine Structures by Incorporation of Spin Bearing Carbazole Units.

Arylamine moieties oxidized to radical cations are considered promising spin bearing units in high-spin-type compounds. Here, we report the first use of carbazole-3,6-diamine units as efficient, rigid spin containing units. The use of rigid spin bearing units enhances significantly spin exchange interactions. The design using density functional theory calculations shows the progressive increase of the exchange coupling constant dependent on the considered model molecules. Two of the most representative molecules containing flexible (dimer 1) and rigid spin coupling unit (dimer 2) were synthesized. Electrochemical and pulsed-electron paramagnetic resonance nutation studies showed that both dimers can be oxidized to yield a majority of dicationic diradicals exhibiting S = 1 ground states. The high values of the dimer 2 exchange coupling constant obtained both computationally ( J/ kB = 145 K; HHeis. = - JS1 S2) and experimentally ( J/ kB = 90-100 K) indicate the beneficial role of the carbazole moiety incorporated into spin bearing units.

Extensometric observation of Earth tides and local tectonic processes at the Vyhne station, Slovakia

Abstract The Vyhne Tidal Station of the Earth Science Institute of the Slovak Academy of Sciences is located in the former mining gallery of St. Anthony of Padua in the Vyhne valley, Štiavnické vrchy Mts., Central Slovakia. It is equipped with a 20.5 metre long quartz-tube extensometer measuring Earth’s tides, and long-term tectonic deformations of the Earth’s crust. Data between 2001 and 2015 with some diverse gaps were digitally collected, processed and analysed. The effects of the local conditions, such as structure of the observatory, cavity effect, topography and geological features of the surrounding rocks, were investigated in detail and these effects were taken into consideration during the interpretation of the results of the data analysis. Tidal analysis of the extensometric data between 2005 and 2015 revealed that the measured tidal amplitudes are close to the theoretical values. The tidal transfer of the observatory was also investigated by coherence analysis between the theoretical and the measured extensometric data. The coherence is better than 0.9 both in the diurnal and semidiurnal band. The effect of the free core nutation resonance was also investigated in the case of the K1 and P1 tidal components. Since the K1/O1 ratio was about the theoretical value 0.8, than the P1/O1 was between 1.0 and 1.15 instead of the theoretical value of 0.9. The rate of the long-term strain rate was also investigated and the obtained −0.05 μstr/y shows a good agreement with the strain rate inferred from GPS measurements in the Central European GPS Reference Network.

Deviation from the Landau-Lifshitz-Gilbert equation in the inertial regime of the magnetization

We investigate in details the inertial dynamics of a uniform magnetization in the ferromagnetic resonance context. Analytical predictions and numerical simulations of the complete equations within the Inertial Landau-Lifshitz-Gilbert (ILLG) model are presented. In addition to the usual precession resonance, the inertial model gives a second resonance peak associated to the nutation dynamics provided that the damping is not too large. The analytical resolution of the equations of motion yields both the precession and nutation angular frequencies. They are function of the inertial dynamics characteristic time τ, the dimensionless damping α, and the static magnetic field H. A scaling function with respect to ατγH is found for the nutation angular frequency, also valid for the precession angular frequency when ατγH ≫ 1. Beyond the direct measurement of the nutation resonance peak, we show that the inertial dynamics of the magnetization has measurable effects on both the width and the angular frequency of the precession resonance peak when varying the applied static field. These predictions could be used to experimentally identify the inertial dynamics of the magnetization proposed in the ILLG model.

Study of the EFG tensor at 75As nuclei in Ge‐As‐Se chalcogenide glasses

Asymmetry parameters of the electric field gradient tensor at (75) As nuclei were determined for chalcogenide glassy semiconductors (CGS) of the Ge-As-Se system by comparing the experimental and simulated (75) As nuclear quadrupole resonance nutation interferograms. The electric field gradient asymmetry in CGS was analyzed, and it is believed that a structural change in these glassy semiconductors takes place at r¯ = 2.425. Electron paramagnetic resonance spectra of the Ge-As-Se system were obtained for the first time. A comparison was made between the results of analysis of the Ge-As-Se system by nuclear quadrupole resonance and electron paramagnetic resonance methods, and this allowed us to make the supposition that a structural phase transition occurs at r¯ = 2.4 from two-dimensional to three-dimensional CGS structure.

A novel method for coil efficiency estimation: Validation with a 13C birdcage

AbstractCoil efficiency, defined as the B1 magnetic field induced at a given point on the square root of supplied power P, is an important parameter that characterizes both the transmit and receive performance of the radiofrequency (RF) coil. Maximizing coil efficiency will maximize also the signal‐to‐noise ratio. In this work, we propose a novel method for RF coil efficiency estimation based on the use of a perturbing loop. The proposed method consists of loading the coil with a known resistor by inductive coupling and measuring the quality factor with and without the load. We tested the method by measuring the efficiency of a 13C birdcage coil tuned at 32.13 MHz and verified its accuracy by comparing the results with the nuclear magnetic resonance nutation experiment. The method allows coil performance characterization in a short time and with great accuracy, and it can be used both on the bench and inside the scanner. © 2012 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 41B: 139–143, 2012

Pulsed EPR of P1 centers in synthetic diamond under bichromatic excitation conditions

The dynamics of the interaction of P1 centers in synthetic diamond with a bichromatic radiation, representing microwave (MW) and radio frequency (RF) fields in a configuration characteristic of the stationary EPR spectroscopy with modulated magnetic field, has been studied using the transient nutation technique. It is demonstrated that a thermobaric treatment of the crystal leads to an increase in the phase relaxation time of P1 centers. Additional increase in this relaxation time is observed under the conditions of a nutation resonance, where the RF field frequency is close to the effective Rabi frequency in the MW field. These data are taken into account in considering the inversion of the EPR lines of P1 centers that was recently discovered in the stationary EPR.

Lander radioscience for obtaining the rotation and orientation of Mars

The paper presents the concept, the objectives, the approach used, and the expected performances and accuracies of a radioscience experiment based on a radio link between the Earth and the surface of Mars. This experiment involves radioscience equipment installed on a lander at the surface of Mars. The experiment with the generic name lander radioscience (LaRa) consists of an X-band transponder that has been designed to obtain, over at least one Martian year, two-way Doppler measurements from the radio link between the ExoMars lander and the Earth (ExoMars is an ESA mission to Mars due to launch in 2013). These Doppler measurements will be used to obtain Mars’ orientation in space and rotation (precession and nutations, and length-of-day variations). More specifically, the relative position of the lander on the surface of Mars with respect to the Earth ground stations allows reconstructing Mars’ time varying orientation and rotation in space. Precession will be determined with an accuracy better by a factor of 4 (better than the 0.1% level) with respect to the present-day accuracy after only a few months at the Martian surface. This precession determination will, in turn, improve the determination of the moment of inertia of the whole planet (mantle plus core) and the radius of the core: for a specific interior composition or even for a range of possible compositions, the core radius is expected to be determined with a precision decreasing to a few tens of kilometers. A fairly precise measurement of variations in the orientation of Mars’ spin axis will enable, in addition to the determination of the moment of inertia of the core, an even better determination of the size of the core via the core resonance in the nutation amplitudes. When the core is liquid, the free core nutation (FCN) resonance induces a change in the nutation amplitudes, with respect to their values for a solid planet, at the percent level in the large semi-annual prograde nutation amplitude and even more (a few percent, a few tens of percent or more, depending on the FCN period) for the retrograde ter-annual nutation amplitude. The resonance amplification depends on the size, moment of inertia, and flattening of the core. For a large core, the amplification can be very large, ensuring the detection of the FCN, and determination of the core moment of inertia. The measurement of variations in Mars’ rotation also determines variations of the angular momentum due to seasonal mass transfer between the atmosphere and ice caps. Observations even for a short period of 180 days at the surface of Mars will decrease the uncertainty by a factor of two with respect to the present knowledge of these quantities (at the 10% level). The ultimate objectives of the proposed experiment are to obtain information on Mars’ interior and on the sublimation/condensation of CO 2 in Mars’ atmosphere. Improved knowledge of the interior will help us to better understand the formation and evolution of Mars. Improved knowledge of the CO 2 sublimation/condensation cycle will enable better understanding of the circulation and dynamics of Mars’ atmosphere.

Effect of the Bloch-Siegert shift on the frequency responses of Rabi oscillations in the case of nutation resonance

The dynamics of a two-level spin system dressed by bichromatic radiation is studied under the conditions of double resonance when the frequency of one (microwave) field is equal to the Larmor frequency of the spin system and the frequency of the other (radio-frequency) field ωrf is close to the Rabi frequency ω1 in a microwave field. It is shown theoretically that Rabi oscillations between dressed-spin states with the frequency ɛ are accompanied by higher-frequency oscillations at frequencies nωrf and nωrf ± ɛ, where n = 1, 2,.... The most intense among these are the signals corresponding to n = 1. The counter-rotating (antiresonance) components of the RF field give rise to a shift of the dressed-state energy, i.e., to a frequency shift similar to the Bloch-Siegert shift. In particular, this shift is manifested as the dependence of the Rabi-oscillation frequency ɛ on the sign of the detuning ω1 − ωrf from resonance. In the case of double resonance, the oscillation amplitude is asymmetric; i.e., the amplitude at the sum frequency ωrf + ɛ increases, while the amplitude at the difference frequency ωrf − ɛ decreases. The predicted effects are confirmed by observations of the nutation signals of the electron paramagnetic resonance (EPR) of E′1 centers in quartz and should be taken into account to realize qubits with a low Rabi frequency in solids.

Mars nutation resonance due to Free Inner Core Nutation

The rotational normal modes of the planet Mars are computed using a numerical approach. We deduce the associated resonances in the nutations induced by the external gravitational forcing from the Sun, Phobos, and Deimos. In particular, the influence of a possible solid inner core inside a liquid core is investigated. A normal mode associated with the inner core (Free Inner Core Nutation (FICN)) is computed, as well as the associated resonance effects in the nutations. For a small inner core, before the eutectic composition of the outer core is reached, the FICN effect on nutation is negligible. For a large inner core it can be very important, and for some nutations it can greatly decrease the resonance effect of the classical Free Core Nutation. Future observations of Mars nutation resonances will therefore provide information about the inner core.

Polar motions equivalent to high frequency nutations for a nonrigid Earth with anelastic mantle

The coefficients of polar motions of the rigid/nonrigid Earth in frequency bands other than the retrograde diurnal one are systematically computed using general expressions, derived here for the first time, for the prograde and retrograde torques exerted on the Earth by lunisolar potentials of arbitrary spherical harmonic type. Taken together with the already known coefficients of low frequency nutations and UT1 variations, they provide a complete characterization, with high precision, of the motions of the pole of the terrestrial reference frame in space; this is needed for high precision studies in astronomy and space geodesy. The inputs used for our computations are a table of tidal amplitudes, and values of the geopotential coefficients of degrees up to 4 and of other relevant basic Earth parameters. General relations which connect the coefficients of high frequency nutations to those of the equivalent polar motions are established and used for deducing the former. The Chandler resonance plays a significant role in low frequency polar motions. In this context, the role of mantle anelasticity and the nature of the Earth's deformational response to zero frequency forcing are given special consideration. The free core nutation (FCN) resonance of low frequency nutations is shown to affect the prograde semidiurnal nutations through the coupling produced between the nutations in the two frequency bands by triaxiality terms in the angular momenta of the whole Earth and of its fluid core. It is shown in a transparent fashion that the effect of the core triaxiality arises almost exclusively from the huge FCN-related resonance in the wobble of the core. The magnitude of the effect is found to be a few times smaller than reported in a recent paper; it is also found, unlike in that paper, that the changes in the eigenfrequencies due to trixiality are only of the second order in the triaxiality parameter. Numerical results for the polar motions of the nonrigid Earth in different frequency bands, as well as for the elliptical nutations of the rigid Earth, are tabulated and compared with available numbers from earlier works.