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Approximate representations of shaped pulses using the homotopy analysis method
The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relaxation or coupling interactions can be described by three Euler angles. The Euler angles, in turn, can be obtained from the solution of a Riccati differential equation; however, analytic solutions exist o...
Autores principales: | , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Copernicus GmbH
2021
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8372782/ https://www.ncbi.nlm.nih.gov/pubmed/34414395 http://dx.doi.org/10.5194/mr-2-175-2021 |
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author | Crawley, Timothy Palmer III, Arthur G. |
author_facet | Crawley, Timothy Palmer III, Arthur G. |
author_sort | Crawley, Timothy |
collection | PubMed |
description | The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relaxation or coupling interactions can be described by three Euler angles. The Euler angles, in turn, can be obtained from the solution of a Riccati differential equation; however, analytic solutions exist only for rectangular and hyperbolic-secant pulses. The homotopy analysis method is used to obtain new approximate solutions to the Riccati equation for shaped radiofrequency pulses in nuclear magnetic resonance (NMR) spectroscopy. The results of even relatively low orders of approximation are highly accurate and can be calculated very efficiently. The results are extended in a second application of the homotopy analysis method to represent relaxation as a perturbation of the magnetization trajectory calculated in the absence of relaxation. The homotopy analysis method is powerful and flexible and is likely to have other applications in magnetic resonance. |
format | Online Article Text |
id | pubmed-8372782 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Copernicus GmbH |
record_format | MEDLINE/PubMed |
spelling | pubmed-83727822021-08-18 Approximate representations of shaped pulses using the homotopy analysis method Crawley, Timothy Palmer III, Arthur G. Magn Reson (Gott) Research Article The evolution of nuclear spin magnetization during a radiofrequency pulse in the absence of relaxation or coupling interactions can be described by three Euler angles. The Euler angles, in turn, can be obtained from the solution of a Riccati differential equation; however, analytic solutions exist only for rectangular and hyperbolic-secant pulses. The homotopy analysis method is used to obtain new approximate solutions to the Riccati equation for shaped radiofrequency pulses in nuclear magnetic resonance (NMR) spectroscopy. The results of even relatively low orders of approximation are highly accurate and can be calculated very efficiently. The results are extended in a second application of the homotopy analysis method to represent relaxation as a perturbation of the magnetization trajectory calculated in the absence of relaxation. The homotopy analysis method is powerful and flexible and is likely to have other applications in magnetic resonance. Copernicus GmbH 2021-04-16 /pmc/articles/PMC8372782/ /pubmed/34414395 http://dx.doi.org/10.5194/mr-2-175-2021 Text en Copyright: © 2021 Timothy Crawley and Arthur G. Palmer III https://creativecommons.org/licenses/by/4.0/This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/ |
spellingShingle | Research Article Crawley, Timothy Palmer III, Arthur G. Approximate representations of shaped pulses using the homotopy analysis method |
title | Approximate representations of shaped pulses using the homotopy analysis method |
title_full | Approximate representations of shaped pulses using the homotopy analysis method |
title_fullStr | Approximate representations of shaped pulses using the homotopy analysis method |
title_full_unstemmed | Approximate representations of shaped pulses using the homotopy analysis method |
title_short | Approximate representations of shaped pulses using the homotopy analysis method |
title_sort | approximate representations of shaped pulses using the homotopy analysis method |
topic | Research Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8372782/ https://www.ncbi.nlm.nih.gov/pubmed/34414395 http://dx.doi.org/10.5194/mr-2-175-2021 |
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