NMR Pulse Sequences and Coherence

Introduction

A simple one-pulse NMR experiment tips magnetisation away from B₀ and records the free induction decay. More informative experiments string together several pulses and carefully timed delays. These pulse sequences decide which interactions are allowed to act, which are refocused, and how information is passed from one nucleus to another. Understanding their common architecture, and the idea of coherence that flows through it, turns a bewildering diagram of rectangles and delays into a readable story.

Core explanation

Almost every multipulse NMR experiment can be divided into four periods. The preparation period begins with a relaxation delay so that magnetisation returns towards equilibrium, then applies one or more pulses to create the starting state, most often transverse magnetisation. The evolution period is a delay, often called t₁ in two-dimensional experiments, during which that state precesses under chemical shift and scalar coupling. The mixing period uses pulses or delays to transfer the state to other spins, for example through scalar couplings, isotropic mixing or cross-relaxation. Finally, the detection period records the FID, called t₂ in a 2D experiment, while the receiver is open.

A one-pulse experiment has only preparation and detection. Adding a systematically incremented evolution delay creates a second time axis. Fourier transformation along both time axes then produces a two-dimensional spectrum, a topic developed in the next part of this unit. The mixing step determines which correlations appear: through-bond transfer for COSY and TOCSY, through-space cross-relaxation for NOESY, heteronuclear transfer for HSQC.

The vector model describes net magnetisation as an arrow rotating in the rotating frame. A 90° pulse along x turns z-magnetisation onto the y-axis; a 180° pulse inverts it. This picture works well for isolated spins and simple echoes, but it fails when coupled spins exchange information. Quantum-mechanical descriptions use the density operator, often expressed with product operators such as Iz, Ix and 2IxSz, which track coupled terms that have no simple vector picture.

Coherence is the key concept. For an ensemble of spins, coherence is a well-defined phase relationship between two spin states across many molecules. A coherence between states whose total magnetic quantum number differs by p is said to have coherence order p. Equilibrium z-magnetisation has p = 0. Observable transverse magnetisation corresponds to single-quantum coherence, p = ±1, and conventionally only p = −1 is detected by the receiver. Zero-quantum and double-quantum coherences can exist between coupled spins but cannot be observed directly; a later pulse must convert them back into single-quantum coherence.

Pulses can change coherence order, whereas free precession during delays changes only the phase of a coherence, at a rate proportional to its order and frequency. A coherence transfer pathway diagram lists the orders a signal passes through, for example 0 → +1 → −1 in a simple echo. Double-quantum filtered COSY deliberately passes through p = ±2, which removes signals from uncoupled singlets such as solvent.

Two tools select the desired pathway. Phase cycling repeats the experiment while changing pulse and receiver phases so that wanted signals add and unwanted signals cancel on co-addition. Pulsed field gradients briefly make B₀ vary across the sample, dephasing coherences in proportion to their order; a later gradient rephases only the chosen pathway. Gradients can select pathways in fewer scans, whereas phase cycling needs a complete cycle for full suppression.

The spin echo is the most important building block. After a 90° pulse, spins at slightly different frequencies fan out. A 180° pulse at time τ reverses their relative phases, so they realign at 2τ. Static field inhomogeneity and chemical-shift differences are refocused, while genuine T₂ decay and, for homonuclear pairs, scalar coupling evolution are not fully removed. This separation of interactions is the basis of T₂ measurement, heteronuclear editing and many transfer elements.

Step-by-step reasoning

To read an unfamiliar sequence, first locate the relaxation delay and first pulse. Next find any incremented delay, which marks evolution. Then identify the pulses or blocks that transfer magnetisation, and state what interaction they exploit. Finally, trace the coherence order through each step to acquisition, checking which pathway gradients or phase cycling select.

Visual explanation

Picture a timeline divided into four coloured bands: a wait and 90° pulse, a growing gap labelled t₁, a mixing block, and a decaying FID. Beneath it, a coherence ladder with levels −2 to +2 shows a line stepping between orders at each pulse and staying level during every delay.

Real-world analogy

A relay race has a start, a lap run by one athlete, a baton exchange and a finishing lap. The evolution period records how the first runner behaves, mixing is the baton exchange, and detection times the second runner. A dropped baton is like an unwanted pathway that phase cycling discards.

Real-world example

Protein NMR laboratories routinely use heteronuclear sequences that pass magnetisation from ¹H to ¹⁵N and back through INEPT transfer steps built from spin echoes. Gradient coherence selection suppresses the enormous water signal while retaining amide correlations, allowing spectra of proteins dissolved mainly in water.

Why?

Why can double-quantum coherence not be detected directly? The receiver coil responds to oscillating transverse magnetisation, which corresponds to single-quantum coherence. A double-quantum coherence produces no net observable transverse magnetisation, so a further pulse must convert it to p = −1 before acquisition.

Common misconception

A 180° pulse does not simply cancel everything that happened before it. It refocuses chemical-shift offsets and inhomogeneity but does not reverse true T₂ relaxation, and homonuclear scalar coupling continues to modulate the echo. Treating an echo as a perfect reset gives wrong predictions.

Worked example

A spin echo uses τ = 10 ms, so the echo forms at 2τ = 20 ms. The sample has T₂ = 0.20 s. Relative echo amplitude = exp(−2τ/T₂) = exp(−0.020/0.20) = exp(−0.10) ≈ 0.90. About 90% of the signal survives. Increasing τ to 100 ms gives exp(−1.0) ≈ 0.37, which is why a series of echo times can be fitted to measure T₂.

Quick check

1. What coherence order is detected directly by the receiver in the usual convention, and what does a delay do to a coherence? Answer: Single-quantum coherence with p = −1 is detected; a delay changes the coherence's phase but not its order.

Exam focus

Label the preparation, evolution, mixing and detection periods on any given sequence. Explain what a 180° pulse refocuses and what it does not. State that pulses change coherence order while delays do not, and describe how phase cycling or gradients select one pathway.

Advanced insight

Gradient selection that keeps only one coherence order can halve the available signal unless combined with echo–antiecho methods that record both p = +1 and p = −1 pathways in t₁. Sensitivity-enhanced schemes recover both orthogonal components, showing that pathway selection design directly controls signal-to-noise ratio.

Summary

Multipulse NMR experiments share a four-part architecture of preparation, evolution, mixing and detection. Coherence describes phase relationships between spin states, with only single-quantum coherence observable. Pulses change coherence order, delays change phase, spin echoes refocus offsets, and phase cycling or gradients keep the chosen transfer pathway.

Practice questions

1. Which period of a 2D sequence contains the incremented delay t₁? Answer: The evolution period, where the initial coherence precesses under chemical shift and coupling before mixing. 2. Why is double-quantum filtering useful in COSY? Answer: Uncoupled spins such as solvent singlets cannot form double-quantum coherence, so their strong signals are removed. 3. A 90°–τ–180°–τ sequence is run on a sample in an inhomogeneous field. Why does an echo still form? Answer: The 180° pulse reverses the phase spread from static field differences, so spins refocus at 2τ. 4. State one advantage of pulsed field gradients over phase cycling. Answer: Gradients select the desired coherence pathway within a single scan, reducing the number of scans needed and suppressing artefacts.