Magnetization and Polarization of Coupled Nuclear Spins Ensembles at High Magnetic Fields

In nuclear magnetic resonance (NMR), the bulk magnetization of a sample is commonly assumed to be proportional to spin polarization, with each spin of the same type contributing equally to the measured signal. Herein, the high-field theorem for general spin-I systems (where I is the spin quantum number): the total measurable NMR signal remains unaffected by the grouping of spins into equivalent units (e.g., molecules) is proved, provided the system is at thermodynamic equilibrium in the high field limit ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>ℏ</mml:mi> <mml:msub><mml:mi>ω</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo>≫</mml:mo> <mml:mo>|</mml:mo> <mml:msub><mml:mi>H</mml:mi> <mml:mrow><mml:mtext>spin</mml:mtext> <mml:mo>-</mml:mo> <mml:mtext>spin</mml:mtext></mml:mrow> </mml:msub> <mml:mo>|</mml:mo></mml:mrow> <mml:annotation>$$ \hslash {\omega }_{0}\gg |{H}_{\text{spin}-\text{spin}}|$$</mml:annotation></mml:semantics> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>ω</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:mrow> <mml:annotation>$\left(\omega\right)_{0}$</mml:annotation></mml:semantics> </mml:math> is the Larmor frequency and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mrow><mml:msub><mml:mi>H</mml:mi> <mml:mrow><mml:mtext>spin</mml:mtext> <mml:mo>-</mml:mo> <mml:mtext>spin</mml:mtext></mml:mrow> </mml:msub> </mml:mrow> <mml:mo>|</mml:mo></mml:mrow> </mml:mrow> <mml:annotation>$\left|\right. H_{\text{spin} - \text{spin}} \left|\right.$</mml:annotation></mml:semantics> </mml:math> characterizes internal spin-spin interactions). The results are derived using both magnetization equations and density matrix formalism. The theorem, however, does not extend to conditions far from thermodynamic equilibrium (e.g., hyperpolarization), NMR of solids in the regime when quadrupolar or dipole-dipole interactions are not negligible, and zero- to ultralow-field NMR. Three educational problems designed to deepen understanding of the material in classroom settings are also presented. This work reinforces established principles in magnetic resonance but also highlights areas for further exploration.