= Nuclear magnetic resonance
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= NMR
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{synonym}
= NMR spectroscopy
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<Nuclear magnetic resonance> detects the response of <atomic nuclei> with a <magnetic moment> to a <radio frequency> <magnetic field> in a steady <magnetic field>. Its striking possibility is to measure a nuclear property through an ordinary electrical <signal> from a bulk sample: an enormous number of tiny <magnetic moments> can produce a measurable change in a receiving circuit.
For a nucleus with <spin angular momentum> $\mathbf J$, define its <gyromagnetic ratio> by $\boldsymbol\mu=\gamma\mathbf J$. Throughout this article $\gamma$ is angular, measured in radians per second per tesla; $\gamma/(2\pi)$ is the corresponding value in cycles per second per tesla. With $\mathbf B_0=B_0\hat{\mathbf z}$, the <Hamiltonian> is $H=-\gamma B_0J_z$. Adjacent <energy levels> differ by
$$
\Delta E=\hbar|\gamma|B_0,\qquad \omega_0=|\gamma|B_0,\qquad f_0=\frac{|\gamma|B_0}{2\pi}.
$$
Thus a measured resonance <frequency> and a calibrated <magnetic field> determine $|\gamma|$. Converting that into a <nuclear magnetic moment> also requires the <spin> quantum number: the maximum projection has magnitude $|\gamma|\hbar I$. This simple experiment does not by itself determine the sign of $\gamma$.
The <Rabi molecular beam magnetic resonance experiment> detects transitions through changes in a selected <molecular beam>. The bulk experiments below instead detect electrical consequences of nuclear <magnetization> in matter at ordinary density. A small <Boltzmann distribution> population imbalance is multiplied by a very large number of nuclei. For noninteracting <spin one-half> nuclei of number density $n$ and positive $\gamma$, their equilibrium <magnetization> is
$$
M_0=n\frac{\gamma\hbar}{2}\tanh\!\left(\frac{\gamma\hbar B_0}{2k_{\rm B}T}\right).
$$
This follows by weighting the two projections $\pm\gamma\hbar/2$ with their <Boltzmann distribution> probabilities. It explains both the smallness of the bulk polarization at ordinary <temperature> and the value of many nuclei contributing to one receiving circuit.
<NMR spectroscopy> measures resonant responses of <atomic nuclei> to applied <electromagnetic radiation> in a <magnetic field>. In <residual-group monitoring of polyurethane formation>, distinguishable signal <integrals> supply a relative residual-group readout; overlap or very weak signals can limit that inference.
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