NMR Relaxation: T1 and T2

Introduction

An RF pulse pushes nuclear magnetisation away from thermal equilibrium. Longitudinal magnetisation then returns toward the field direction while transverse magnetisation loses coherent phase. These processes have different time constants, T1 and T2. Their values affect signal recovery, line width, quantitative integration and the design of multidimensional experiments. Calling all NMR decay “relaxation” without separating the directions hides important experimental consequences.

Core explanation

T1 describes longitudinal or spin–lattice relaxation: recovery of magnetisation M z toward its equilibrium value M₀. After an ideal inversion pulse in a simple single-exponential system, M z(t)=M₀[1−2exp(−t/T1)]. At t=0 it is −M₀, at long times it approaches +M₀, and T1 sets the recovery scale. The “lattice” is the molecular surroundings that accept or supply energy as spin populations return to their thermal imbalance. Magnetic-field fluctuations at relevant frequencies, produced by molecular motion and interactions, mediate this process.

T2 describes loss of coherent transverse magnetisation in the rotating frame. In a simple homogeneous system, transverse signal falls approximately as exp(−t/T2). Spins need not transfer net energy to the surroundings in every dephasing event; local magnetic interactions can cause their phases to spread so the vector sum shrinks. The homogeneous Lorentzian full width at half maximum in hertz is approximately 1/(πT2) for a simple exponentially decaying signal. Shorter T2 therefore gives broader lines. This relationship assumes the simple linewidth model; overlapping peaks and inhomogeneous broadening complicate measured widths.

T2 is the observed FID dephasing time when static magnetic-field inhomogeneity and other reversible frequency differences add to intrinsic transverse relaxation. It is usually no longer than T2. Spins at different positions precess at slightly different rates, causing their signals to cancel in the ensemble. A spin-echo pulse can refocus some static frequency offsets and allow a measurement closer to T2, but not remove truly irreversible dephasing or all exchange and diffusion effects.

T1 matters strongly for repeated scans. If a recycle delay is too short, some spins have not regained equilibrium longitudinal magnetisation before the next pulse. Different sites may have different T1 values and thus different apparent peak areas even if their proton numbers are known. Quantitative NMR requires suitable pulse angle, delay, calibration and attention to exchange or saturation. Repeating scans rapidly may increase scans per minute while biasing integration.

T2 affects how long coherent transfer can survive through a pulse sequence. In 2D NMR, magnetisation may need to persist during evolution and mixing delays. A very short T2 weakens cross-peaks and broadens direct or indirect dimensions. Paramagnetic centres often accelerate relaxation of nearby nuclei and may cause extreme broadening or signal loss. An absent NMR peak may therefore reflect relaxation rather than absence of an atom.

Relaxation depends on molecular motion, temperature, viscosity, magnetic field and interactions. A simple “faster tumbling always means longer T1” rule is unreliable because relaxation is strongest when fluctuation frequencies match transition frequencies in an appropriate spectral-density model. Macromolecules, small molecules and solids can occupy different motional regimes. T1 and T2 should be measured or justified for the specific experiment.

For an isolated spin-half system under common conditions, T2 is not greater than 2T1 in the usual Bloch-type relaxation framework, but equality is not generally expected. The precise relationship depends on mechanisms and model. More important in basic interpretation is the directional distinction: T1 controls population recovery; T2 controls coherence loss; T2 adds imperfect-field dephasing.

Step-by-step reasoning

Identify the magnetisation direction being observed. For M z recovery after perturbation, discuss T1 and recycle delay. For transverse FID decay, discuss T2 and T2 . Ask whether a spin echo is used to separate static inhomogeneity from intrinsic dephasing. Finally, connect measured times to line width, signal intensity or sequence timing under stated assumptions.

Visual explanation

Imagine an arrow initially tipped into the horizontal plane. Its vertical projection gradually returns upward on the T1 timescale. Meanwhile, many horizontal arrows fan out in angle, reducing their summed horizontal length on the T2 or T2 timescale. The two changes can occur simultaneously but describe different components of magnetisation.

Real-world analogy

Runners around a circular track can lose alignment as some move a little faster, even if their total energy is unchanged. That resembles transverse dephasing. Returning all runners to a preferred starting distribution after a disturbance resembles longitudinal recovery, though nuclear energy exchange is the real mechanism behind T1.

Real-world example

An analyst records a quantitative ¹H NMR spectrum of two non-exchanging sites. If one site has a much longer T1, a short recycle delay can underrepresent its integrated area. Extending the delay allows more complete longitudinal recovery and improves quantitative reliability, while improved shimming can narrow peaks by reducing the field-inhomogeneity part of T2 .

Why?

Why can an echo restore a decayed-looking signal? If spins dephase because each experiences a stable frequency offset, a refocusing pulse reverses their relative phase evolution so they re-align later. It cannot undo random interactions that irreversibly erase coherence during the interval; those contribute to true T2 decay.

Common misconception

T1 and T2 are not two names for the same exponential decay. T1 describes longitudinal population recovery, while T2 describes transverse coherence loss. T2 is also not merely another chemical property; it includes instrument and sample field homogeneity, so shimming can change it without changing intrinsic molecular T2.

Worked example

Assume a simple site has T2=0.20 s and homogeneous Lorentzian broadening only. Its approximate full width at half maximum is 1/(πT2)=1/(π×0.20)=1.59 Hz. If measured FID dephasing is T2 =0.05 s, the observed width may be closer to 1/(π×0.05)=6.37 Hz because of added inhomogeneity. A spin echo can refocus part of that additional width, but it does not change the site's intrinsic T2 under the assumed model.

Quick check

1. Which relaxation time primarily governs longitudinal signal recovery between repeated NMR scans? Answer: T1 governs recovery of M z toward equilibrium, so insufficient recycle delay can bias signal intensities.

Exam focus

Associate T1 with M z and spin–lattice energy exchange, T2 with transverse coherence and homogeneous linewidth, and T2 with observed FID dephasing including static inhomogeneity. Qualify linewidth formulas with their simple Lorentzian assumptions and avoid using peak area quantitatively without a relaxation-delay check.

Advanced insight

Relaxation rates are governed by spectral densities of fluctuating local fields at specific frequencies. Dipole–dipole interactions, chemical-shift anisotropy, quadrupolar coupling and paramagnetic effects contribute differently with field and motion. Pulse sequences can measure T1 by inversion recovery and T2 by echo trains, but model fitting may require multiple components when environments or exchange processes are heterogeneous.

Summary

T1 measures longitudinal return toward equilibrium, T2 intrinsic transverse coherence loss, and T2 additional observed dephasing including field imperfections. T1 controls quantitative recycle timing, while T2 and T2 influence line width and pulse-sequence sensitivity. Their values depend on molecular motion and interactions and should not be conflated.

Practice questions

1. What happens to quantitative peak areas if a long-T1 site is scanned with a very short recycle delay? Answer: It may be partly saturated and underrepresented relative to sites that recover more fully. 2. For a simple exponential transverse decay with T2=0.10 s, estimate homogeneous linewidth. Answer: About 1/(π×0.10)=3.18 Hz under the Lorentzian model. 3. Can improved field shimming change T2 without changing intrinsic T2? Answer: Yes. It reduces static field inhomogeneity and thus reversible ensemble dephasing. 4. What part of transverse signal loss can a spin echo refocus best? Answer: Dephasing from stable frequency offsets such as static field inhomogeneity, rather than irreversible T2 relaxation.