Formalized quantum mechanical Floquet theory and its application to sample spinning in nuclear magnetic resonance

A critique on the suitability of Fer expansion in time-evolution studies in quantum mechanics

The present report examines the utility and exactness of time-propagators derived from Fer expansion (FE). While the mathematical intricacies of the FE scheme are well established, the operational aspects of the same in time-evolution studies remain less explored and authenticated in physical problems of relevance. Through suitable examples, the operational inconsistencies observed in time-evolution studies based on the FE scheme are identified and corroborated through rigorous comparisons with simulations emerging from exact numerical methods. The limitations outlined seriously undermine the advantages associated with the FE scheme over other existing analytic methods.

A perspective on the relative merits/demerits of time-propagators based on Floquet theorem

The present report examines the nuances of analytic methods employed in the derivation of evolution operators in periodically driven quantum systems based on Floquet theorem. Specifically, time-propagators of the form, U(t) = P(t)e-iH̄t defined in the Hilbert space (of finite dimension), are derived through generalized multimodal expansion of the operators involved. While Floquet methods defined in the extended Hilbert space (of infinite dimension) have remained the method of choice for the description of time-evolution at non-stroboscopic time-intervals, the expansion schemes discussed do present an attractive option for similar studies in the standard Hilbert space. Nevertheless, the convergence criteria and suitability of such methods deserve formal validation in problems of experimental relevance. Employing examples comprising periodic Hamiltonians from magnetic resonance spectroscopy, the exactness of Floquet based time-propagators in the Schrödinger and interaction representation is discussed. Through rigorous comparisons between simulations emerging from analytic and exact numerical methods, the relative merits and demerits of different formulations of Floquet based methods are also discussed.

Theory of radio-frequency pulses on periodically driven three-level systems: challenges and perspectives

Nuances of multiple-quantum transitions in periodically driven systems is discussed through analytical methods based on time propagators derived from Floquet theory.

Floquet theory in magnetic resonance: Formalism and applications

Floquet theory is an elegant mathematical formalism originally developed to solve time-dependent differential equations. Besides other fields, it has found applications in optical spectroscopy and nuclear magnetic resonance (NMR). This review attempts to give a perspective of the Floquet formalism as applied in NMR and shows how it allows one to solve various problems with a focus on solid-state NMR. We include both matrix- and operator-based approaches. We discuss different problems where the Hamiltonian changes with time in a periodic way. Such situations occur, for example, in solid-state NMR experiments where the time dependence of the Hamiltonian originates either from magic-angle spinning or from the application of amplitude- or phase-modulated radiofrequency fields, or from both. Specific cases include multiple-quantum and multiple-frequency excitation schemes. In all these cases, Floquet analysis allows one to define an effective Hamiltonian and, moreover, to treat cases that cannot be described by the more popularly used and simpler-looking average Hamiltonian theory based on the Magnus expansion. An important example is given by spin dynamics originating from multiple-quantum phenomena (level crossings). We show that the Floquet formalism is a very general approach for solving diverse problems in spectroscopy.

On the equivalence between different averaging schemes in magnetic resonance

Evolution of quantum mechanical systems under time-dependent Hamiltonians has remained a challenging problem of interest across all disciplines. Through suitable approximations, different averaging methods have emerged in the past for modeling the time-evolution under time-dependent Hamiltonians. To this end, the development of analytic methods in the form of time-averaged effective Hamiltonians has gained prominence over other methods. In particular, the advancement of spectroscopic methods for probing molecular structures has benefited enormously from such theoretical pursuits. Nonetheless, the validity of the approximations and the exactness of the proposed effective Hamiltonians have always remained a contentious issue. Here, in this report, we reexamine the equivalence between the effective Hamiltonians derived from the Magnus formula and Floquet theory through suitable examples in magnetic resonance.

Dynamically Manipulating Topological Physics and Edge Modes in a Single Degenerate Optical Cavity

We propose a scheme to simulate topological physics within a single degenerate cavity, whose modes are mapped to lattice sites. A crucial ingredient of the scheme is to construct a sharp boundary so that the open boundary condition can be implemented for this effective lattice system. In doing so, the topological properties of the system can manifest themselves on the edge states, which can be probed from the spectrum of an output cavity field. We demonstrate this with two examples: a static Su-Schrieffer-Heeger chain and a periodically driven Floquet topological insulator. Our work opens up new avenues to explore exotic photonic topological phases inside a single optical cavity.

Fokker-Planck formalism in magnetic resonance simulations

This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library - the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use.

Progress in spin dynamics solid-state nuclear magnetic resonance with the application of Floquet-Magnus expansion to chemical shift anisotropy

The purpose of this article is to present an historical overview of theoretical approaches used for describing spin dynamics under static or rotating experiments in solid state nuclear magnetic resonance. The article gives a brief historical overview for major theories in nuclear magnetic resonance and the promising theories. We present the first application of Floquet-Magnus expansion to chemical shift anisotropy when irradiated by BABA pulse sequence.

Correlation chemical shift imaging with low-power adiabatic pulses and constant-density spiral trajectories

In this work we introduce the concept of correlation chemical shift imaging (CCSI). Novel CCSI pulse sequences are demonstrated on clinical scanners for two-dimensional Correlation Spectroscopy (COSY) and Total Correlation Spectroscopy (TOCSY) imaging experiments. To date there has been limited progress reported towards a feasible and robust multivoxel 2D COSY. Localized 2D TOCSY imaging is shown for the first time in this work. Excitation with adiabatic GOIA-W(16,4) pulses (Gradient Offset Independent Adiabaticity Wurst modulation) provides minimal chemical shift displacement error, reduced lipid contamination from subcutaneous fat, uniform optimal flip angles, and efficient mixing for coupled spins, while enabling short repetition times due to low power requirements. Constant-density spiral readout trajectories are used to acquire simultaneously two spatial dimensions and f(2) frequency dimension in (k(x),k(y),t(2)) space in order to speed up data collection, while f(1) frequency dimension is encoded by consecutive time increments of t(1) in (k(x),k(y),t(1),t(2)) space. The efficient spiral sampling of the k-space enables the acquisition of a single-slice 2D COSY dataset with an 8 × 8 matrix in 8:32  min on 3 T clinical scanners, which makes it feasible for in vivo studies on human subjects. Here we present the first results obtained on phantoms, human volunteers and patients with brain tumors. The patient data obtained by us represent the first clinical demonstration of a feasible and robust multivoxel 2D COSY. Compared to the 2D J-resolved method, 2D COSY and TOCSY provide increased spectral dispersion which scales up with increasing main magnetic field strength and may have improved ability to unambiguously identify overlapping metabolites. It is expected that the new developments presented in this work will facilitate in vivo application of 2D chemical shift correlation MRS in basic science and clinical studies.

Low-power adiabatic sequences for in vivo localized two-dimensional chemical shift correlated MR spectroscopy

Novel low-power adiabatic sequences are demonstrated for in vivo localized two-dimensional correlated MR spectroscopy, such as correlated spectroscopy and total correlated spectroscopy. The design is based on three new elements for in vivo two-dimensional MRS: the use of gradient modulated constant adiabaticity GOIA-W(16,4) pulses for (i) localization (correlated spectroscopy and total correlated spectroscopy) and (ii) mixing (total correlated spectroscopy), and (iii) the use of longitudinal mixing (z-filter) for magnetization transfer during total correlated spectroscopy. GOIA-W(16,4) provides accurate signal localization, and more importantly, lowers the SAR for both total correlated spectroscopy mixing and localization. Longitudinal mixing improves considerably (fivefolds) the efficiency of total correlated spectroscopy transfer. These are markedly different from previous 1D editing total correlated spectroscopy sequences using spatially nonselective pulses and transverse mixing. Fully adiabatic (adiabatic mixing with adiabatic localization) and semiadiabatic (adiabatic mixing with nonadiabatic localization) methods for two-dimensional total correlated spectroscopy are compared. Results are presented for simulations, phantoms, and in vivo two-dimensional spectra from healthy volunteers and patients with brain tumors obtained on 3T clinical platforms equipped with standard hardware. To the best of our knowledge, this is the first demonstration of in vivo adiabatic two-dimensional total correlated spectroscopy and fully adiabatic two-dimensional correlated spectroscopy. It is expected that these methodological developments will advance the in vivo applicability of multi(spectrally)dimensional MRS to reliably identify metabolic biomarkers.

Operator-based Floquet theory in solid-state NMR

This article reviews the application of operator-based Floquet theory in solid-state NMR. Basic expressions for calculating effective Hamiltonians based on van Vleck perturbation theory are reviewed for problems with a single frequency or multiple incommensurate frequencies. Such a treatment allows calculation of effective Hamiltonians for resonant and non-resonant problems. Examples from literature are given for single-mode to triple-mode Floquet problems, covering a wide range of applications in solid-state NMR under magic-angle spinning and radio-frequency irradiation of a single nucleus or multiple nuclei.

Symmetry-based recoupling in double-rotation NMR spectroscopy

In this contribution, we extend the theory of symmetry-based pulse sequences of types CN(n) (nu) and RN(n) (nu) in magic-angle-spinning nuclear resonance spectroscopy [M. H. Levitt, in Encyclopedia of Nuclear Magnetic Resonance, edited by D. M. Grant and R. K. Harris (Wiley, Chichester, 2002), Vol. 9]. to the case of rotating the sample simultaneously around two different angles with respect to the external magnetic field (double-rotation). We consider the case of spin-1/2 nuclei in general and the case of half-integer quadrupolar nuclei that are subjected to weak radio frequency pulses operating selectively on the central-transition polarizations. The transformation properties of the homonuclear dipolar interactions and J-couplings under central-transition-selective spin rotations are presented. We show that the pulse sequence R2(2) (1)R2(2) (-1) originally developed for homonuclear dipolar recoupling of half-integer quadrupolar nuclei under magic-angle-spinning conditions [M. Eden, D. Zhou, and J. Yu, Chem. Phys. Lett. 431, 397 (2006)] may be used for the same purpose in the case of double rotation, if the radio frequency pulses are synchronized with the outer rotation of the sample. We apply this sequence, sandwiched by central-transition selective 90 degrees pulses, to excite double-quantum coherences in homonuclear spin systems consisting of (23)Na and (27)Al nuclei.

Bimodal Floquet description of heteronuclear dipolar decoupling in solid-state nuclear magnetic resonance

A theoretical treatment of heteronuclear dipolar decoupling in solid-state nuclear magnetic resonance is presented here based on bimodal Floquet theory. The conditions necessary for good heteronuclear decoupling are derived. An analysis of a few of the decoupling schemes implemented until date is presented with regard to satisfying such decoupling conditions and efficiency of decoupling. Resonance conditions for efficient heteronuclear dipolar decoupling are derived with and without the homonuclear (1)H-(1)H dipolar couplings and their influence on heteronuclear dipolar decoupling is pointed out. The analysis points to the superior efficiency of the newly introduced swept two-pulse phase-modulation (SW(f)-TPPM) sequence. It is shown that the experimental robustness of SW(f)-TPPM as compared to the original TPPM sequence results from an adiabatic sweeping of the modulation frequencies. Based on this finding alternative strategies are compared here. The theoretical findings are corroborated by both numerical simulations and representative experiments.

Through-space R3-HETCOR experiments between spin-1/2 and half-integer quadrupolar nuclei in solid-state NMR

We present several new methods that allow to obtain through-space 2D HETCOR spectra between spin-1/2 and half-integer quadrupolar nuclei in the solid state. These methods use the rotary-resonance concept to create hetero-nuclear coherences through the dipolar interaction instead of scalar coupling into the HMQC and refocused INEPT experiments for spin n/2 (n>1). In opposite to those based on the cross-polarization transfer to quadrupolar nuclei, the methods are very robust and easy to set-up.

Proton line narrowing in solid-state nuclear magnetic resonance: New insights from windowed phase-modulated Lee-Goldburg sequence

We present here a bimodal Floquet analysis of the windowed phase-modulated Lee-Goldburg (wPMLG) sequence for homonuclear dipolar decoupling. One of the main criteria for an efficient homonuclear dipolar decoupling scheme is an effective z-rotation condition. This is brought about by the presence of radio-frequency imperfections in the pulse sequence together with a systematic manipulation of the wPMLG pulses. Additional improvement in the (1)H spectral resolution was obtained by a proper understanding of the off-resonance dependence of the wPMLG irradiation scheme based on bimodal Floquet theory. Numerical investigations further corroborate both theoretical and experimental findings. Theoretical analysis points to accidental degeneracies between the cycle time of the wPMLG sequence and the rotor period leading to the experimentally observed off-resonance dependence of the resolution. Two-dimensional (1)H-(1)H homonuclear single-quantum correlation spectra of model amino acids are also presented, highlighting the improved spectral resolution of wPMLG sequences.

Cross polarization from spins I=1∕2 to spins S=1 in nuclear magnetic resonance with magic angle sample spinning

Spin locking of the nuclear magnetization of a spin with S=1 such as deuterium in the presence of a radio-frequency field under magic angle spinning (MAS) is described in terms of adiabatic modulations of the energy levels. In a brief initial period, part of the initial density operator nutates about the Hamiltonian and is dephased. The remaining spin-locked state undergoes persistent oscillatory transfer processes between various coherences with a periodicity given by the rotation of the sample. While all crystallites in the powder undergo such periodic transfer processes, the phases of the oscillations depend on the angle gamma of the crystallites. The angle gamma is the azimuthal angle defining the orientation of the unique axis of the quadrupolar interaction tensor in a rotor-fixed frame. The theory is extended to describe cross-polarization between spins S=1 and I=12 under MAS. There are four distinct Hartmann-Hahn matching conditions that correspond to four zero-quantum matching conditions, all of which are shifted and broadened compared to their spin S=12 counterparts. These matching conditions are further split into a family of sideband conditions separated by the spinning frequency. The theory allows the calculation of both shifts and broadening factors of the matching conditions, as verified by simulations and experiments.

Floquet-van Vleck analysis of heteronuclear spin decoupling in solids: the effect of spinning and decoupling sidebands on the spectrum

We investigate the effect of magic angle spinning on heteronuclear spin decoupling in solids. We use an analytical Floquet-van Vleck formalism to derive expressions for the powder-averaged signal as a function of time. These expressions show that the spectrum consists of a centerband at the isotropic frequency of the observed spin, omega(0), and rotational decoupling sidebands at omega(0)+/-omega(1)+/-momega(r), where omega(1) is the decoupling field strength and omega(r) is the rotation frequency. Rotary resonance occurs when the rotational decoupling sidebands overlap with the centerband. Away from the rotary resonance conditions, the intensity of the centerband as a function of omega(r)/omega(1) is simply related to the total intensity of the rotational decoupling sidebands. Notably, in the absence of offset terms it is shown that as omega(1) decreases, the centerband intensity can decrease without any associated broadening. Furthermore, the centerband width is shown to be independent of spinning speed, to second order for the effects we consider. The effects of I spin chemical shift anisotropy and homonuclear dipolar couplings are also investigated. The analytical results are compared to simulations and experiments.

Amplitude-modulated decoupling in rotating solids: a bimodal Floquet approach

This paper centers on a theoretical study of amplitude-modulated heteronuclear decoupling in solid-state NMR under magic-angle spinning (MAS). A spin system with a single isolated rare spin coupled to a large number of abundant spins is used in the analysis. The phase-alternating decoupling scheme (XiX decoupling) is analyzed using bimodal Floquet theory and the operator-based perturbation method developed by van Vleck. An effective Hamiltonian correct to second order is calculated for the spin system under XiX decoupling. The results of these calculations indicate that under XiX decoupling the main contribution to the residual line width comes from a cross-term between the heteronuclear and the homonuclear dipolar couplings. This is in contrast to continuous-wave decoupling, where the residual line width is dominated by the cross-term between the heteronuclear dipolar coupling and the chemical-shielding tensor of the irradiated spin. For high-power decoupling the method results in very good decoupling provided that certain unfavorable recoupling conditions, imposed by specific ratios of the amplitude modulation frequency and the MAS frequency, are avoided. For low-power decoupling, the method leads to acceptable decoupling when the pulse length corresponds to an integer multiple of a 2pi rotation and the rf-field amplitude is less than a quarter of the MAS frequency. The performance of the XiX scheme is analyzed over a range of values of the rf power, and numerical results that agree well with the most recent experimental observations are presented.

Decoupling and recoupling using continuous-wave irradiation in magic-angle-spinning solid-state NMR: A unified description using bimodal Floquet theory

The application of two or more different time-dependent coherent perturbations with, in general, incommensurable frequencies occurs quite commonly in NMR experiments. Here we develop a unified description of the entire class of experiments using bimodal Floquet theory and van Vleck-Primas perturbation theory. This treatment leads to a time-independent effective Hamiltonian in Hilbert space and can be looked at as a generalization of average Hamiltonian theory to several incommensurate time dependencies. As a prototype experiment we treat the application of continuous-wave (cw) radio-frequency irradiation in combination with magic-angle sample spinning. Practically relevant examples of this type of experiments are heteronuclear spin decoupling and recoupling experiments using cw irradiation, e.g., rotary-resonance recoupling. Perturbations up to the third order must be taken into account to explain all experimentally observed resonance conditions.

Multipole-multimode Floquet theory in nuclear magnetic resonance

In this paper, we present a new analytical approach for describing the spin dynamics of synchronous and asynchronous time-dependent modulations in solid-state nuclear magnetic resonance experiments. The approach, based on multimode Floquet theory, employs the multipole operator basis of Sanctuary for spin description and illustrates the time evolution in the Floquet-Liouville space using the effective Hamiltonians obtained from the contact (or van Vleck) transformation procedure. Since the Hamiltonian and the density operator are expressed in terms of irreducible tensor operators, extensions to higher spin magnitudes (I>12) and multiple spins are quite straightforward and permit analytical treatments for many problems. We outline the general underlying principles involved in this approach with a brief mention of its potential application in other branches of spectroscopy.

Theory of heteronuclear decoupling in solid-state nuclear magnetic resonance using multipole-multimode Floquet theory

A formal theory for heteronuclear decoupling in solid-state magic angle spinning (MAS) nuclear magnetic resonance experiments is presented as a first application of multipole-multimode Floquet theory. The method permits a straightforward construction of the multispin basis and describes the spin dynamics via effective Floquet Hamiltonians obtained using the van Vleck transformation method in the Floquet-Liouville space. As a test case, we consider a model three-spin system (I2S) under asynchronous time modulations (both MAS and rf irradiation) and derive effective Hamiltonians for describing the spin dynamics in the Floquet-Liouville space during heteronuclear decoupling. Furthermore, we describe and evaluate the origin of cross terms between the various anisotropic interactions and illustrate their exact contributions to the spin dynamics. The theory presented herein should be applicable to the design and understanding of pulse sequences for heteronuclear and homonuclear recoupling and decoupling.

Probing membrane protein orientation and structure using fast magic-angle-spinning solid-state NMR

One and two-dimensional solid-state NMR experiments are discussed that permit probing local structure and overall molecular conformation of membrane-embedded polypeptides under Magic Angle Spinning. The functional dependence of a series of anisotropic recoupling schemes is analyzed using theoretical and numerical methods. These studies lead to the construction of a set of polarization dephasing or transfer units that probe local backbone conformation and overall molecular orientation within the same NMR experiment. Experimental results are shown for a randomly oriented peptide and for two model membrane-peptides reconstituted into lipid bilayers and oriented on polymer films according to a method proposed by Bechinger et al.

Identifying anisotropic constraints in multiply labeled bacteriorhodopsin by 15N MAOSS NMR: a general approach to structural studies of membrane proteins

Structural models of membrane proteins can be refined with sets of multiple orientation constraints derived from structural NMR studies of specifically labeled amino acids. The magic angle oriented sample spinning (MAOSS) NMR approach was used to determine a set of orientational constraints in bacteriorhodopsin (bR) in the purple membrane (PM). This method combines the benefits of magic angle spinning (MAS), i.e., improved sensitivity and resolution, with the ability to measure the orientation of anisotropic interactions, which provide important structural information. The nine methionine residues in bacteriorhodopsin were isotopically (15)N labeled and spectra simplified by deuterium exchange before cross-polarization magic angle spinning (CPMAS) experiments. The orientation of the principal axes of the (15)N chemical shift anisotropy (CSA) tensors was determined with respect to the membrane normal for five of six residual resonances by analysis of relative spinning sideband intensities. The applicability of this approach to large proteins embedded in a membrane environment is discussed in light of these results.

Multiple-spin analysis of chemical-shift-selective (13C, 13C) transfer in uniformly labeled biomolecules

Chemical-shift-selective (13C, 13C) polarization transfer is analyzed in uniformly labeled biomolecules. It is shown that the spin system dynamics remain sensitive to the distance of interest and can be well reproduced within a quantum-mechanical multiple-spin analysis. These results lead to a general approach on how to describe chemical-shift-selective transfer in uniformly labeled systems. As demonstrated in the case of ubiquitin, this methodology can be used to detect long-range distance constraints in uniformly labeled proteins.

Low-power XiX decoupling in MAS NMR experiments

Low-power XiX proton decoupling under fast magic-angle spinning is introduced. The method is applicable if the MAS frequency exceeds the proton-proton interactions. For rigid organic solids this is the case for MAS frequencies above approximately 40 kHz. It is shown that the quality of the decoupling as well as the sensitivity to frequency offsets can be improved compared to low-power continuous-wave decoupling. The decoupling efficiency is somewhat reduced compared to optimized high-power decoupling: in a peptide sample investigated at an MAS frequency of 50 kHz a loss of about 10% in signal intensity for CH3 and CH groups, and of about 40% for CH2 groups was observed. Taking into consideration, that the rf amplitude in the low-power XiX was about 15 times lower than in high-power XiX decoupling, such a reduction in line intensity is sometimes tolerable.

Heteronuclear spin decoupling in solid-state NMR under magic-angle sample spinning

Achieving high spectral resolution is an important prerequisite for the application of solid-state NMR to biological molecules. Higher spectral resolution allows to resolve a larger number of resonances and leads to higher sensitivity. Among other things, heteronuclear spin decoupling is one of the important factors which determine the resolution of a spectrum. The process of heteronuclear spin decoupling under magic-angle sample spinning is analyzed in detail. Continuous-wave RF irradiation leads only in a zeroth-order approximation to a full decoupling of heteronuclear spin systems in solids under magic-angle spinning (MAS). In a higher-order approximation, a cross-term between the dipolar-coupling tensor and the chemical-shielding tensor is reintroduced, providing a scaled coupling term between the heteronuclear spins. In strongly coupled spin systems this second-order recoupling term is partially averaged out by the proton spin-diffusion process, which leads to exchange-type narrowing of the line by proton spin flips. This process can be described by a spin-diffusion type superoperator, allowing the efficient simulation of strongly coupled spin systems under heteronuclear spin decoupling. Low-power continuous-wave decoupling at fast MAS frequencies offers an alternative to high-power irradiation by reversing the order of the averaging processes. At fast MAS frequencies low-power continuous-wave decoupling leads to significantly narrower lines than high-power continuous-wave decoupling while at the same time reducing the power dissipated in the sample by several orders of magnitude. The best decoupling is achieved by multiple-pulse sequences at high RF fields and under fast MAS. Two such sequences, two-pulse phase-modulated decoupling (TPPM) and X-inverse-X decoupling (XiX), are discussed and their properties analyzed and compared.

An analytical formulation of CPMAS

An exact expression for the cross polarization between two spin-1/2 particles is derived from the quantum Liouville equation. This is given in the form of two integrodifferential equations. These can be solved exactly in the static case (no sample spinning) and a powder average easily performed numerically. With magic-angle spinning, the neglect of certain interference terms simplifies the numerical calculation. A further assumption decoupling the calculation of the sidebands gives a very simple formula that is capable of giving a qualitative interpretation of all experimental observations. Examples are given illustrating typical buildup curves and CPMAS matching profiles.

The continuous wave electron paramagnetic resonance experiment revisited

When the modulation frequency used in continuous wave electron paramagnetic resonance (cw EPR) spectroscopy exceeds the linewidth, modulation sidebands appear in the spectrum. It is shown theoretically and experimentally that these sidebands are actually multiple photon transitions, sigma(+)+kxpi, where one microwave (mw) sigma(+) photon is absorbed from the mw radiation field and an arbitrary number k of radio frequency (rf) pi photons are absorbed from or emitted to the modulation rf field. Furthermore, it is demonstrated that both the derivative shape of the lines in standard cw EPR spectra and the distortions due to overmodulation are caused by the unresolved sideband pattern of these lines. The single-photon transition does not even give a contribution to the first-harmonic cw EPR signal. Multiple photon transitions are described semiclassically in a toggling frame and their existence is proven using second quantization. With the toggling frame approach and perturbation theory an effective Hamiltonian for an arbitrary sideband transition is derived. Based on the effective Hamiltonians an expression for the steady-state density operator in the singly rotating frame is derived, completely describing all sidebands in all modulation frequency harmonics of the cw EPR signal. The relative intensities of the sidebands are found to depend in a very sensitive way on the actual rf amplitude and the saturation of single sidebands is shown to depend strongly on the effective field amplitude of the multiple photon transitions. By comparison with the analogous solutions for frequency-modulation EPR it is shown that the field-modulation and the frequency-modulation technique are not equivalent. The experimental data fully verify the theoretical predictions with respect to intensities and lineshapes.

Solid-state NMR spectroscopy under periodic modulation by fast magic-angle sample spinning and pulses: a review

Fast magic-angle spinning (MAS) holds promise for new approaches to pulsed high-resolution NMR in solids where homogeneous interactions dominate. Prerequisite for developing new pulse methods is the understanding of signal encoding by spin interactions under MAS conditions and of interferences between MAS and pulses. This review discusses corresponding strategies and techniques in a coherent way with particular concentration on homonuclear decoupling techniques for line-narrowing in solids.