Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop term
<p>A modified Anderson–Weiss approximation for describing double-quantum (DQ) NMR experiments in systems with many <span class="inline-formula"><i>I</i></span> <span class="inline-formula">=</span> <span class="inline-formula"...
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Copernicus Publications
2025-01-01
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| Series: | Magnetic Resonance |
| Online Access: | https://mr.copernicus.org/articles/6/1/2025/mr-6-1-2025.pdf |
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| author | N. Fatkullin I. Brekotkin K. Saalwächter |
| author_facet | N. Fatkullin I. Brekotkin K. Saalwächter |
| author_sort | N. Fatkullin |
| collection | DOAJ |
| description | <p>A modified Anderson–Weiss approximation for describing double-quantum (DQ) NMR experiments in systems with many <span class="inline-formula"><i>I</i></span> <span class="inline-formula">=</span> <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M3" display="inline" overflow="scroll" dspmath="mathml"><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="20pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="caca869f2f384264fc5bcd3ae2b8aeff"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mr-6-1-2025-ie00001.svg" width="20pt" height="14pt" src="mr-6-1-2025-ie00001.png"/></svg:svg></span></span> spins is proposed, taking inter-spin flip-flop processes into special consideration. In this way, an analytical result is derived for multi-spin systems for the first time. It is shown that in the initial stages of DQ intensity buildup, the probability of flip-flop processes in DQ experiments is half as large as in analogous Hahn echo or free induction decay experiments. Their influence on the experimentally observed DQ NMR signal becomes dominant at times <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M4" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>t</mi><mo>></mo><msup><mfenced close=")" open="("><mrow><mn mathvariant="normal">9</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></mfenced><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mi>T</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">eff</mi></msubsup><mo>≈</mo><mn mathvariant="normal">2.12</mn><msubsup><mi>T</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">eff</mi></msubsup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="127pt" height="19pt" class="svg-formula" dspmath="mathimg" md5hash="e3b567b0a4f093c2c564657ed3ae0a0d"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mr-6-1-2025-ie00002.svg" width="127pt" height="19pt" src="mr-6-1-2025-ie00002.png"/></svg:svg></span></span>, where <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M5" display="inline" overflow="scroll" dspmath="mathml"><mrow><msubsup><mi>T</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">eff</mi></msubsup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="18pt" height="17pt" class="svg-formula" dspmath="mathimg" md5hash="6c2ebd9bc9eb03d4a0d777b4936bd874"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mr-6-1-2025-ie00003.svg" width="18pt" height="17pt" src="mr-6-1-2025-ie00003.png"/></svg:svg></span></span> is the effective spin–spin relaxation time measured by the Hahn echo. Calculations and a comparison with spin dynamics simulations of small spin systems of up to eight spins reveal a satisfactory agreement.</p> |
| format | Article |
| id | doaj-art-bd4579c85bfb42d68aa0eb244706eb23 |
| institution | DOAJ |
| issn | 2699-0016 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Magnetic Resonance |
| spelling | doaj-art-bd4579c85bfb42d68aa0eb244706eb232025-08-20T02:40:56ZengCopernicus PublicationsMagnetic Resonance2699-00162025-01-01611410.5194/mr-6-1-2025Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop termN. Fatkullin0I. Brekotkin1K. Saalwächter2Institut für Physik – NMR, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), GermanyInstitute of Physics, Kazan Federal University, 420008 Kazan, Tatarstan, RussiaInstitut für Physik – NMR, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany<p>A modified Anderson–Weiss approximation for describing double-quantum (DQ) NMR experiments in systems with many <span class="inline-formula"><i>I</i></span> <span class="inline-formula">=</span> <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M3" display="inline" overflow="scroll" dspmath="mathml"><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="20pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="caca869f2f384264fc5bcd3ae2b8aeff"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mr-6-1-2025-ie00001.svg" width="20pt" height="14pt" src="mr-6-1-2025-ie00001.png"/></svg:svg></span></span> spins is proposed, taking inter-spin flip-flop processes into special consideration. In this way, an analytical result is derived for multi-spin systems for the first time. It is shown that in the initial stages of DQ intensity buildup, the probability of flip-flop processes in DQ experiments is half as large as in analogous Hahn echo or free induction decay experiments. Their influence on the experimentally observed DQ NMR signal becomes dominant at times <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M4" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>t</mi><mo>></mo><msup><mfenced close=")" open="("><mrow><mn mathvariant="normal">9</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></mfenced><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mi>T</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">eff</mi></msubsup><mo>≈</mo><mn mathvariant="normal">2.12</mn><msubsup><mi>T</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">eff</mi></msubsup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="127pt" height="19pt" class="svg-formula" dspmath="mathimg" md5hash="e3b567b0a4f093c2c564657ed3ae0a0d"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mr-6-1-2025-ie00002.svg" width="127pt" height="19pt" src="mr-6-1-2025-ie00002.png"/></svg:svg></span></span>, where <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M5" display="inline" overflow="scroll" dspmath="mathml"><mrow><msubsup><mi>T</mi><mn mathvariant="normal">2</mn><mi mathvariant="normal">eff</mi></msubsup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="18pt" height="17pt" class="svg-formula" dspmath="mathimg" md5hash="6c2ebd9bc9eb03d4a0d777b4936bd874"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="mr-6-1-2025-ie00003.svg" width="18pt" height="17pt" src="mr-6-1-2025-ie00003.png"/></svg:svg></span></span> is the effective spin–spin relaxation time measured by the Hahn echo. Calculations and a comparison with spin dynamics simulations of small spin systems of up to eight spins reveal a satisfactory agreement.</p>https://mr.copernicus.org/articles/6/1/2025/mr-6-1-2025.pdf |
| spellingShingle | N. Fatkullin I. Brekotkin K. Saalwächter Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop term Magnetic Resonance |
| title | Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop term |
| title_full | Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop term |
| title_fullStr | Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop term |
| title_full_unstemmed | Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop term |
| title_short | Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop term |
| title_sort | analytical treatment of proton double quantum nmr intensity buildup multi spin couplings and the flip flop term |
| url | https://mr.copernicus.org/articles/6/1/2025/mr-6-1-2025.pdf |
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