Two-Dimensional NMR: Axes and Cross-Peaks

Introduction

A one-dimensional NMR spectrum lists frequencies, but it does not directly say which signals belong to neighbouring nuclei. Two-dimensional NMR spreads information over a plane with two frequency axes, so that a single spot can state "a nucleus resonating here passed magnetisation to a nucleus resonating there". The key idea is surprisingly simple: the second axis is not measured directly at all. It is built by repeating an experiment many times while stepping a delay, then Fourier transforming how the signal changes from one repetition to the next.

Core explanation

Almost every 2D pulse sequence can be divided into four periods. Preparation creates transverse coherence, usually with a 90° pulse after a relaxation delay. Evolution is a delay t₁ during which that coherence precesses at the characteristic frequency of its nucleus, including chemical shift and coupling effects. Mixing is a pulse or group of pulses and delays that transfers magnetisation from one nucleus to another, by scalar coupling, dipolar cross-relaxation or another mechanism. Detection records a free induction decay during t₂, exactly as in a 1D experiment.

The trick lies in t₁. The spectrometer records a first FID with t₁ close to zero, then a second with t₁ increased by a small increment Δt₁, then a third, and so on, often for 128 to 1024 increments. Each FID is Fourier transformed along t₂ to give an ordinary-looking spectrum. Now look down any single column of those spectra: the intensity of a given peak rises and falls from one increment to the next, because the phase acquired during t₁ depends on how fast the original coherence precessed. That oscillation in intensity (amplitude modulation) or phase is a time-domain signal in its own right. A second Fourier transform along t₁ converts it into a frequency, F1.

The result is a surface S(F1, F2), usually drawn as a contour plot. F2, the direct dimension, reports the frequency at which magnetisation was detected. F1, the indirect dimension, reports the frequency at which it evolved before mixing. If magnetisation evolves on nucleus A and is detected on nucleus A, the signal falls at (δA, δA). In a homonuclear experiment these points lie on the diagonal , and they reproduce the 1D spectrum. If magnetisation evolves on A but is transferred during mixing and detected on B, the signal falls at (δA, δB): an off-diagonal cross-peak . Cross-peaks carry the chemical information, because they exist only when the mixing mechanism links A and B.

In heteronuclear experiments the axes belong to different isotopes, for example ¹³C in F1 and ¹H in F2. There is then no diagonal in the homonuclear sense; each peak directly pairs a carbon frequency with a proton frequency.

Resolution in F1 is limited by the longest t₁ reached, so adding increments improves resolution but lengthens the experiment. The spectral width in F1 is set by the increment: sampling must be fast enough to capture the highest frequency present, otherwise signals fold (alias) into the wrong place. Phase cycling or pulsed field gradients select the wanted coherence pathway and suppress unwanted ones, and schemes such as States or echo–antiecho recording allow the sign of F1 frequencies to be distinguished.

Formulae

F1 spectral width (Hz) = 1 ÷ Δt₁ for complex (quadrature) sampling. Digital resolution in F1 ≈ spectral width ÷ number of complex increments. Total time ≈ number of increments × scans per increment × (acquisition time + relaxation delay).

Step-by-step reasoning

1. Prepare coherence with a pulse after full or partial relaxation. 2. Let it evolve for time t₁, so each nucleus picks up a phase set by its frequency. 3. Mix, transferring part of the magnetisation to coupled or nearby nuclei. 4. Detect the FID during t₂ and Fourier transform it. 5. Repeat with t₁ incremented, stacking the spectra. 6. Fourier transform along t₁ to generate F1, then plot contours.

Visual explanation

Imagine a square plot with the 1D proton spectrum along the top and down the left side. A diagonal line of spots runs from corner to corner. Any spot away from that line sits where a vertical line from one peak meets a horizontal line from another, pointing to a pair of connected nuclei.

Real-world analogy

A time-lapse photograph of a tide gauge records one height per day. From that series you can work out the tide's period even though no single photograph shows motion. The t₁ increments are like the daily photographs: each FID is a snapshot, and the series reveals a frequency. The analogy omits the mixing step, which is what makes the second axis chemically informative.

Real-world example

Pharmaceutical analytical laboratories routinely record 2D spectra to confirm the structure of each new batch of a drug substance or to identify an impurity at below 1% of the sample. Overlapping multiplets that are impossible to separate in 1D often become separate cross-peaks once spread across two axes.

Why?

Why does stepping t₁ produce a frequency axis? Coherence precessing at frequency ν gains a phase of 2πνt₁ before mixing. As t₁ grows, that phase advances steadily, so the detected amplitude oscillates at ν. Any oscillation sampled regularly in time can be Fourier transformed into its frequency.

Common misconception

"The spectrometer measures two frequencies at once." Only F2 is detected directly. F1 is reconstructed from many separate experiments, which is why 2D spectra take minutes to hours, and why instrument instability between increments creates streaks along F1.

Worked example

Question: A 2D proton experiment uses Δt₁ = 200 μs with complex sampling and 256 complex increments. What are the F1 spectral width and approximate digital resolution? On a 500 MHz spectrometer, what range of chemical shift does F1 cover?

Reasoning: Spectral width = 1 ÷ (200 × 10⁻⁶ s) = 5000 Hz. Resolution ≈ 5000 Hz ÷ 256 ≈ 19.5 Hz per point. At 500 MHz, 5000 Hz ÷ 500 Hz per ppm = 10 ppm.

Answer: 5000 Hz (10 ppm), with about 20 Hz per point, which is too coarse to resolve most proton couplings in F1.

Quick check

1. In a homonuclear 2D spectrum, where does a peak appear if magnetisation was not transferred during mixing? Answer: On the diagonal, at F1 = F2, matching the peak's position in the 1D spectrum.

Exam focus

Name the four periods of a 2D experiment and state which one is incremented. Explain that F1 comes from a second Fourier transform along t₁. Interpret a cross-peak at (δA, δB) as evidence of transfer between A and B by whatever mechanism the mixing step uses.

Advanced insight

Because F1 is sampled point by point, 2D experiments are often limited by time rather than sensitivity. Non-uniform sampling records only a selected subset of t₁ increments and reconstructs the spectrum with compressed-sensing or maximum-entropy algorithms, shortening experiments considerably. Extending the idea with a second incremented delay gives 3D and higher-dimensional NMR, essential for proteins.

Summary

A 2D experiment consists of preparation, evolution, mixing and detection. Incrementing the evolution time t₁ and Fourier transforming along it creates the indirect axis F1, while the detected FID gives F2. Diagonal peaks show untransferred magnetisation; cross-peaks show nuclei linked by the mixing mechanism. F1 resolution and spectral width are controlled by the number and size of t₁ increments.

Practice questions

1. Which period of a 2D experiment determines what kind of correlation appears? Answer: The mixing period, because its mechanism (scalar coupling, NOE or heteronuclear transfer) decides which nuclei can exchange magnetisation. 2. Why do 2D experiments take much longer than 1D experiments of similar sensitivity? Answer: Each t₁ increment requires a separate FID, often hundreds of them, each with its own scans and relaxation delay. 3. A signal appears at the wrong F1 position and moves when the spectral width is changed. What is the likely cause? Answer: Folding or aliasing, because the F1 spectral width is smaller than the frequency range of the signals. 4. Why is there no diagonal in a ¹H–¹³C heteronuclear spectrum? Answer: The two axes represent different nuclei, so no signal can evolve and be detected on the same nucleus at the same frequency on both axes.