NMR Multiplet Animation

For the NMR-literate

Each cycle shows the first-order multiplet pattern of a spin coupled to 3–6 heteronuclear partners, each with \(I \in \{\tfrac{1}{2}, 1, \tfrac{3}{2}\}\) and \(J\) sampled uniformly from 5–50 Hz. The stick spectrum is built by iterative splitting, and each line is then broadened with a Gaussian lineshape.

For the NMR-illiterate

In NMR spectroscopy, atomic nuclei with a non-zero spin quantum number \(I\) possess a property called spin, and behave as tiny magnets. When placed in an external magnetic field, the nuclear spin precesses at a characteristic frequency called its resonance frequency. If a second, nearby nucleus also has a non-zero spin quantum number, the two interact through the bonding electrons in an interaction called scalar coupling. The effect of scalar coupling is to split a single resonance line up into multiple lines (a multiplet).

Splitting rules

Coupling to a spin-\(I\) nucleus produces \(2I + 1\) lines of equal intensity, symmetrically spaced by the coupling constant \(J\) (in Hz):

• Spin-\(\tfrac{1}{2}\) (e.g. ¹H, ¹³C, ³¹P): doublet — 2 lines
• Spin-\(1\) (e.g. ²H, ¹⁴N): triplet — 3 lines
• Spin-\(\tfrac{3}{2}\) (e.g. ¹¹B, ²³Na): quartet — 4 lines

When multiple coupling partners are present, the multiplet is built up iteratively: each new coupling splits every existing line into its own sub-multiplet. Lines that land on the same frequency (which occurs when coupling constants share a simple integer ratio) are summed.

The animation

Each cycle, the animation draws a new random multiplet: 3–6 coupling partners are generated, each is assigned a spin quantum number \(I \in \{\tfrac{1}{2}, 1, \tfrac{3}{2}\}\) and a coupling constant \(J\) drawn uniformly from 5–50 Hz. The resulting line spectrum is convolved with a Gaussian lineshape to simulate the appearence of peaks in a real NMR spectrum. Such coupling schemes are unlikely to be encountered in practice: it is chemically implausible for so many magnetically diverse nuclei to coexist within coupling range of a single site, and quadrupolar nuclei (\(I > \tfrac{1}{2}\)) typically undergo rapid relaxation that means that they often do not cause observable peak splittings (²H is a notable exception).

Real NMR signals

In practice, spectra are derived by performing a Fourier transform on a time-domain signal detected by the probe, called the free induction decay (FID). The FID is often manipulated using window functions before the Fourier transform to bestow desirable lineshapes on the peaks. The Lorentz-Gauss transform is commonly used to convert the naturally Lorentzian peaks to sharper Gaussian lines.