TY - JOUR

T1 - Floquet theory in magnetic resonance

T2 - Formalism and applications

AU - Ivanov, Konstantin L.

AU - Mote, Kaustubh R.

AU - Ernst, Matthias

AU - Equbal, Asif

AU - Madhu, Perunthiruthy K.

N1 - Funding Information:
We thank Shimon Vega for his encouragement and introducing us to the various facets of Floquet theory. KLI acknowledges support from the Ministry of Science and Higher Education of the Russian Federation (Grant No. 075–15-2020–779). KRM and PKM acknowledge support of the Department of Atomic Energy, Government of India, under Project Identification No. RTI 4007. ME acknowledges support by the Swiss National Science Foundation (Grant No. ). Konstantin L. Ivanov (Kostya, as he was known to many of his friends and colleagues) passed away on March 5, 2021, during the revision of this manuscript. Kostya had varied interests in fields such as theory of reaction kinetics, hyperpolarisation, solid-state NMR, zero-field NMR, long-lived states, and level (anti) crossing. He was instrumental in initiating a web-based seminar series and conference series (ICONS) to disseminate topics in magnetic resonance.
Funding Information:
We thank Shimon Vega for his encouragement and introducing us to the various facets of Floquet theory. KLI acknowledges support from the Ministry of Science and Higher Education of the Russian Federation (Grant No. 075?15-2020?779). KRM and PKM acknowledge support of the Department of Atomic Energy, Government of India, under Project Identification No. RTI 4007. ME acknowledges support by the Swiss National Science Foundation (Grant No. 200020_188988). Konstantin L. Ivanov (Kostya, as he was known to many of his friends and colleagues) passed away on March 5, 2021, during the revision of this manuscript. Kostya had varied interests in fields such as theory of reaction kinetics, hyperpolarisation, solid-state NMR, zero-field NMR, long-lived states, and level (anti) crossing. He was instrumental in initiating a web-based seminar series and conference series (ICONS) to disseminate topics in magnetic resonance.
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/10/1

Y1 - 2021/10/1

N2 - 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.

AB - 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.

KW - Average Hamiltonian theory

KW - Dynamic nuclear polarisation

KW - Floquet theory

KW - Level crossing

KW - Magic-angle spinning

KW - NMR

KW - Solid-state NMR

KW - Spin chemistry

UR - http://www.scopus.com/inward/record.url?scp=85108859108&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85108859108&partnerID=8YFLogxK

U2 - 10.1016/j.pnmrs.2021.05.002

DO - 10.1016/j.pnmrs.2021.05.002

M3 - Review article

C2 - 34852924

AN - SCOPUS:85108859108

SN - 0079-6565

VL - 126-127

SP - 17

EP - 58

JO - Progress in Nuclear Magnetic Resonance Spectroscopy

JF - Progress in Nuclear Magnetic Resonance Spectroscopy

ER -