🧲 NMR: Implementation of Bloch equations to visualise a standard 1D NMR experiment in 3D

An interactive JavaScript visualisation of the bulk magnetisation vector and detected signals during a standard 1D NMR (Nuclear Magnetic Resonance) experiment.

🔗 Live demo: tkimhofer.github.io/nmr_visualisation

Note: I've added a button top-left to apply RF pulse.

🎯 Overview

This project implements and visualises the fundamental physical processes in a standard 90° NMR experiment — from nuclear spin alignment to RF excitation, signal precession, relaxation, and data detection — using an animated 3D scene rendered with Three.js.

It is meant as an educational tool for students of spectroscopy or physics learning about spin dynamics and NMR signal formation.

🧩 Implementation

The magnetisation dynamics are computed from the analytical solutions to the Bloch equations ¹ for free precession and relaxation after a 90° pulse in the laboratory frame.

Key features

• Animated bulk magnetization vector with 3D axes.
• Real-time exponential T1 / T2 relaxation simulation
• Orthogonal detectors and complex signal traces
• Interactive orbit, zoom, and reset button (see upper left)

See src/code.js for more implementation details of the signal model and relaxation terms.

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🧠 NMR Spectroscopy in Essence

At its core, an NMR spectrometer comprises a powerful magnet, a probe with radio-frequency coils and transmit/receive electronics for signal excitation and detection.

NMR systems commonly found in research settings have a magnetic field strength of ~14.1 Tesla. Conventionally, this number is quoted by the corresponding 1H (Proton) resonance frequency: 14.1 T ≈ 600 MHz (nominal).

In comparison, clinical MRI systems have much lower field strengths, usually 1.5–3 T, with up to 7 T being considered as ultra-high-field MRI.

1. Sample placement

• Temperature & field stabilisation
• Locking, shimming, tuning & matching, RF pulse calibration, etc.

2. Nuclear spin alignment

• NMR-sensitive nuclei align parallel or antiparallel to magnetic field B0 (vertical/blue axis in visualisation)
• Parallel/antiparallel orientations correspond to low/high energy states (for 1H, where γ > 0 → antiparallel = low energy, see also reference #2)
• Slight excess population in the low-energy state → net bulk magnetization (M)
• Population difference proportional to magnetic field strength ³
• (higher magnetic field strength increases chemical shift dispersion, ie., less peak overlap)

3. Bulk magnetization in 3D space

• Bulk magnetization vector M (white) is the sum of individual nuclear magnetic moments
• Longitudinal axis: z along B0 (Mz component)
• Transverse axes: x (red) and y (green), forming the Mxy plane

4. RF excitation

• Exposed to radiofrequency (RF) pulse -> B1
• RF pulse calibrated to rotate M by 90° into xy-plane
• M starts precessing around z-axis at the Larmor frequency (omega0 in code)

5. Signal evolution and detection

A. Relaxation effects

• Magnitude of Mxy decays while Mz recovers, due to:
• T2: mutual magnetic interactions between spins → loss of phase coherence (spin-spin relax.)
• T1: interactions with lattice/environment → recovery along z-axis (spin-lattice relax.)
• Typically, T2 ≤ T1.

B. Signal detection

• The precessing Mxy induces voltage in two orthogonal receiver coils (90° phase difference)
• Detectors / recever channels shown as orange and green cones ⁴
• Resulting time-domain signals are drawn as orange and green traces

C. Complex signal representation

• Recorded time-domain signals are a complex-valued: s(t) = s_Re(t) + i* s_Im(t)
• The real (s_Re) and imaginary (s_Im) parts correspond to the two receiver channels
• After Fourier transformation:
  • Real → absorptive line shape
  • Imaginary → dispersive component

6. Recovery

• Longitudinal magnetization Mz grows until equilibrium re-established

7. Repetition

• Once equilibrium is reached, the acquisition cycle repeats
• The number of repetitions (Bruker parameter NS: Number of Scans) determines signal averaging
• Signal-to-noise ratio (S/N) increases proportionally to the square root of the number of scans

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Refs:

1: Bloch equations

2: Gyromagnetic ratio (γ)

3: Boltzmann distribution

4: Quadrature detection