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Introduction of the Floquet-Magnus Expansion in Solid-State Nuclear Magnetic Resonance Spectroscopy Eugène S Introduction of the Floquet-Magnus expansion in solid-state nuclear magnetic resonance spectroscopy Eugène S. Mananga and Thibault Charpentier Citation: J. Chem. Phys. 135, 044109 (2011); doi: 10.1063/1.3610943 View online: http://dx.doi.org/10.1063/1.3610943 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i4 Published by the American Institute of Physics. Related Articles Second-order dipolar order in magic-angle spinning nuclear magnetic resonance J. Chem. Phys. 135, 154507 (2011) Single crystal nuclear magnetic resonance in spinning powders J. Chem. Phys. 135, 144201 (2011) Resistive detection of optically pumped nuclear polarization with spin phase transition peak at Landau level filling factor 2/3 Appl. Phys. Lett. 99, 112106 (2011) High-resolution 13C nuclear magnetic resonance evidence of phase transition of Rb,Cs-intercalated single- walled nanotubes J. Appl. Phys. 110, 054306 (2011) Distribution of non-uniform demagnetization fields in paramagnetic bulk solids J. Appl. Phys. 110, 013902 (2011) Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 31 Jan 2012 to 128.103.149.52. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions THE JOURNAL OF CHEMICAL PHYSICS 135, 044109 (2011) Introduction of the Floquet-Magnus expansion in solid-state nuclear magnetic resonance spectroscopy Eugène S. Mananga1 and Thibault Charpentier2,a) 1CEA, Neurospin/I2BM, Laboratoire de Résonance Magnétique Nucléaire, F-91191 Gif-sur-Yvette cedex, France 2CEA, IRAMIS, Service Interdisciplinaire sur les Systèmes Moléculaires et Matériaux, LSDRM, UMR CEA/CNRS 3299, F-91191 Gif-sur-Yvette cedex, France (Received 20 August 2010; accepted 27 June 2011; published online 25 July 2011) In this article, we present an alternative expansion scheme called Floquet-Magnus expansion (FME) used to solve a time-dependent linear differential equation which is a central problem in quantum physics in general and solid-state nuclear magnetic resonance (NMR) in particular. The commonly used methods to treat theoretical problems in solid-state NMR are the average Hamiltonian theory (AHT) and the Floquet theory (FT), which have been successful for designing sophisticated pulse sequences and understanding of different experiments. To the best of our knowledge, this is the first report of the FME scheme in the context of solid state NMR and we compare this approach with other series expansions. We present a modified FME scheme highlighting the importance of the (time- periodic) boundary conditions. This modified scheme greatly simplifies the calculation of higher order terms and shown to be equivalent to the Floquet theory (single or multimode time-dependence) but allows one to derive the effective Hamiltonian in the Hilbert space. Basic applications of the FME scheme are described and compared to previous treatments based on AHT, FT, and static perturbation theory. We discuss also the convergence aspects of the three schemes (AHT, FT, and FME) and present the relevant references. © 2011 American Institute of Physics. [doi:10.1063/1.3610943] I. INTRODUCTION time-dependent interactions, was found to be less descriptive to rotating systems, such as sample-spinning experiments.21 Much progress has been made in the application of solid- However, these types of experiments were found to be state nuclear magnetic resonance (NMR) to elucidate molec- more conveniently described using Floquet theory.22–24 In ular structure and dynamics in systems not amenable to char- this work, we introduce the fusion of AHT and FT as pro- acterization by any other way. The importance of solid-state vided by the Floquet-Magnus expansion (FME) that can be nuclear magnetic resonance stands in its ability to determine very useful in simplifying calculations and also for pro- accurately intermolecular distances1, 2 and molecular torsion viding a more intuitive understanding of spin dynamics angles.3, 4 The methods have been used to systems including processes. both microscopically ordered preparations such as membrane The purpose of this article is to introduce the FME proteins,5–8 nanocrystalline proteins,9–11 amyloid fibrils,12–16 scheme to solid-state NMR, to give a general and coherent and also disordered or amorphous systems such as glasses.17 framework of the scheme, and to compare its use in solid- Site-specific resolution can be obtained either by uniaxial state NMR with other averaging approaches. Similarly to the orientation of the sample with respect to the static mag- AHT and FT theory, the primary aim of the FME approach is netic field18 or, through magic-angle sample spinning (MAS). to provide a scheme to build an approximation of the Hamil- Nowadays, MAS is widely used to obtain high resolution tonian describing the stroboscopic evolution of the system NMR spectroscopy of solids because of its effect of averag- over several periods. The FME approach is essentially dis- ing out the orientation-dependent component of nuclear spin tinguished from AHT with its function (t) which provides interactions, principally chemical shifts anisotropic and mag- an easy and alternative way for evaluating the spin behavior netic dipolar couplings. This technique can be combined with in between the stroboscopic observation points. The FME ap- cross polarization to increase the spectral sensitivity of rare proach is in fact the fusion of the two major methods used and low-gamma nuclei such as 13C, 15 N (Ref. 19) in biopoly- to describe the spin dynamics in solid-state NMR: the aver- mers or other organic solids. Therefore, MAS NMR tech- age Hamiltonian theory based on the Magnus expansion (ME) niques have improved to the point where complete structure and the Floquet theory based on the Fourier expansion. The determination is possible.10, 12, 13 first method (AHT) was developed by Haeberlen and Waugh As the technique of MAS spreads to the field of solid state in 196820 and is appropriate for stroboscopic sampling. The NMR, the concept of average Hamiltonian theory (AHT),20 AHT technique does not satisfactorily describe the case of which is the main theoretical tool to describe the effect of MAS spectra because in this case, the signal is usually ob- served continuously with a time resolution much shorter than a)Electronic mail: [email protected]. the rotor period.25, 26 Nevertheless, in a variety of cases, MAS 0021-9606/2011/135(4)/044109/11/$30.00135, 044109-1 © 2011 American Institute of Physics Downloaded 31 Jan 2012 to 128.103.149.52. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 044109-2 E. S. Mananga and T. Charpentier J. Chem. Phys. 135, 044109 (2011) experiments have been successfully analyzed using AHT,21, 27 However, this theory provides the option to evaluate the spin which yields an effective Hamiltonian averaged over some evolution between the time points of detection. In contrast cycle time of a periodic pulse sequence. From its natural to common approaches of AHT and FT, the main advan- formulation, AHT has been gradually applied to almost ev- tage of the FME scheme is to overcome the limitations of ery kind of situation, sometimes abusively.21 Some examples the stroboscopic detection schemes. In the Floquet-Magnus that lend to the application of AHT include the use of a pre- approach, even when the first and second order F1 and F2 requisite of the approach to design a frequency-modulated of the effective Hamiltonian are identical to their counter- analog of TPPM and explored other possibilities involv- parts in AHT and FT, the 1,2(t) functions provide an easy ing simultaneous phase and frequency modulation.28, 29 Sim- way for evaluating the spin evolution during ‘‘the time in be- ilarly, Eden and Levitt utilized symmetry arguments30 based tween’’ through the Magnus expansion of the operator con- on AHT to develop optimized heteronuclear decoupling se- nected to this part of the evolution. 1(t) and 2(t) are con- quences involving rotor-synchronized pulse sequences whose nected to the appearance of features like spinning sidebands fundamental element “C” is a 2π pulse. Therefore, despite in MAS. The evaluation of 1(t) and 2(t)isusefulespe- the emergence of alternative approaches such as the Floquet cially for the analysis of the non-stroboscopic evolution. For theory,22, 23 the exact effective Hamiltonian theory,31, 32 and example, in the case of C7,54–56 for non-stroboscopic detec- the Fer expansion,33 which have advantages in some circum- tion scheme, they can be used to estimate the intensity of stances, the average Hamiltonian theory still remains of cen- the spinning sidebands manifold in the double-quantum di- tral importance in theory of multiple-pulse NMR. All ap- mension. Higher order effects (F3, 3(t)) can also be eval- proaches are equivalent to first order. With the increase of uated using the FME approach, in a way easier than in the the level of sophistication of NMR experiments, second or- case of AHT or FT. The FME scheme can also be extended der terms are of increasing importance, such as in diffusion to multimode Hamiltonians for Hilbert space analysis espe- experiment.34 cially in the incommensurate case. To the best of our knowl- The second method (FT), which was first proposed by edge, we present here the first report highlighting the basics Shirley22 and introduced to NMR by Vega23, 24 and Maricq,35 of the FME scheme and compare this approach with the other provides a more universal approach for the description of series expansions. We present a generalized FME scheme, the full time dependence of the response of a periodically based on the importance of the boundary conditions (at the time-dependent system.36, 37 It allows the computation of the origin of time), which provides a natural choice for the op- full spinning sideband pattern that is of importance in many erators n(0) to simplify calculation of higher order terms MAS experimental circumstances to obtain information on and allows FT to be managed in the Hilbert space.
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