Analytical treatment of proton double-quantum NMR intensity buildup: multi-spin couplings and the flip-flop term - N. Fatkullin; I. Brekotkin; K. Saalwächter (Magnetic Resonance)
A modified Anderson–Weiss approximation for describing double-quantum (DQ) NMR experiments in systems with many I = 1/2<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> 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 t><mfenced close=")" open="(">9/2</mfenced>1/2<msubsup>T2eff</msubsup>≈2.12<msubsup>T2eff</msubsup><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>, where <msubsup>T2eff</msubsup><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> 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.