Magnetic Resonance: NMR and ESR Overview

Hold a person inside an MRI scanner and you are performing nuclear magnetic resonance on the protons in their tissue water. A superconducting magnet aligns billions of tiny nuclear compasses. A radio pulse tips them, and as they relax they broadcast a signal that a computer turns into an image. The same physics, applied to electrons instead of nuclei, lets chemists watch free radicals form and die in real time. Magnetic resonance matters because it is the most information-rich structural technique in chemistry. A single NMR spectrum reveals connectivity, stereochemistry, and dynamics without destroying the sample. Electron spin resonance (ESR, also called EPR) does the same for unpaired electrons, probing radicals, transition-metal complexes, and reaction intermediates invisible to ordinary optical spectroscopy. This article builds the theory from the ground up. You will see how a magnetic field splits spin states. Then you will see the resonance condition that a spectrometer must satisfy. Finally you will connect that condition to the two experimental observables that make spectra useful: chemical shift and spin-spin coupling.

Key takeaways

A magnetic field lifts the degeneracy of nuclear or electron spin states via the Zeeman effect.
Resonance occurs when photon energy $h\nu$ matches the Zeeman gap, $h\nu = g\mu B_0$ .
Electrons (ESR) split ~658 times more per tesla than protons (NMR), because $\gamma_e \gg \gamma_N$ .
Chemical shift reports the local electronic shielding around a nucleus, in parts per million, field-independent.
Spin-spin coupling (J, in hertz) transmits through bonds and reveals how many neighbors a nucleus has.
Key idea: Both techniques measure the same phenomenon, spin energy splitting in a field, read out through a resonance condition whose fine structure encodes chemical environment.

The core insight

Nuclear and electron spins behave as tiny magnets whose energy in an external field $B_0$ splits into discrete levels (Zeeman splitting); a photon of matching energy drives transitions between them, and the precise resonance frequency, shifted by local electrons (chemical shift) and split by neighboring spins (coupling), encodes molecular structure with exquisite detail.

The physical picture

A nucleus with nonzero spin quantum number $I$ , or an unpaired electron with spin $S = 1/2$ , possesses an intrinsic magnetic moment. In the absence of a field, all $2I+1$ (or $2S+1$ ) spin orientations have identical energy. Switch on a static magnetic field $B_0$ , conventionally along $z$ , and the moment's projection along the field determines its energy. This is the Zeeman effect, familiar from atomic spectroscopy but here applied to spin rather than orbital angular momentum. For a spin-1/2 nucleus (like $^1$ H or $^{13}$ C) there are exactly two levels: spin aligned with the field ( $m_I = +1/2$ , lower energy) and spin opposed ( $m_I = -1/2$ , higher energy). For an unpaired electron the pattern is the same but inverted in sign because the electron's magnetic moment points opposite to its spin. The energy gap between adjacent levels grows linearly with $B_0$ ; there is no splitting at all when the field is off, which is why magnetic resonance requires a strong magnet as an active ingredient of the experiment, unlike optical spectroscopy where the transition exists intrinsically. A photon of electromagnetic radiation drives transitions between these levels only when its frequency satisfies the resonance condition, and only when its magnetic component oscillates perpendicular to $B_0$ so it can torque the spin. For typical laboratory fields, this frequency falls in the radiofrequency range for nuclei (tens to hundreds of MHz) and the microwave range for electrons (~9-35 GHz), consistent with the electromagnetic spectrum hierarchy where spin transitions sit at the low-energy end, well below rotational and vibrational transitions. Populations of the two levels are governed by the Boltzmann distribution, exactly as in any spectroscopic technique. Because the Zeeman gap is minuscule compared with thermal energy $k_BT$ at room temperature, the population difference between spin-up and spin-down states is tiny, typically a few parts per million for protons in a strong magnet. This is the fundamental sensitivity limitation of NMR: unlike electronic or vibrational spectroscopy, magnetic resonance operates on a minute population imbalance, and the observed signal intensity scales directly with that imbalance. Two further phenomena convert a bare resonance condition into a structural tool. First, the electrons surrounding a nucleus generate small local magnetic fields that shield or deshield it from $B_0$ , shifting its exact resonance frequency, the chemical shift. Second, spins on neighboring atoms couple magnetically through the bonding electrons, splitting a single resonance into a multiplet, spin-spin coupling. Both effects are far smaller than the primary Zeeman splitting but are exactly what make spectra chemically informative rather than a single uninformative line per nucleus.

Figure

Zeeman splitting of a spin-1/2 nucleus as B0 increases, with the resonance photon shown

The core relations

E_{m} = -g\mu m B_0

where: $g$ is the nuclear or electron g-factor, $\mu$ the relevant magneton (nuclear magneton $\mu_N$ or Bohr magneton $\mu_B$ ), $m$ the magnetic spin quantum number, $B_0$ the applied field.

\Delta E = g\mu B_0

where: $\Delta E$ is the energy gap between adjacent spin states for $\Delta m = \pm 1$ , the only allowed transition.

h\nu = g\mu B_0

where: this is the resonance condition; $\nu$ is the frequency of radiation absorbed, $h$ Planck's constant.

\nu = \gamma B_0 / 2\pi

where: $\gamma$ is the gyromagnetic (magnetogyric) ratio of the spin, linking frequency directly to field strength.

\delta = \frac{\nu_{\text{sample}} - \nu_{\text{ref}}}{\nu_{\text{ref}}} \times 10^{6}

where: $\delta$ is the chemical shift in ppm, field-independent, referenced to a standard (TMS for $^1$ H, $^{13}$ C).

\frac{N_{\beta}}{N_{\alpha}} = e^{-\Delta E / k_BT}

where: $N_\alpha, N_\beta$ are populations of lower and upper spin states, $k_B$ Boltzmann's constant, $T$ temperature.

Derivation: From the Zeeman Hamiltonian to the NMR/ESR resonance condition

Goal: show that the observed resonance frequency is fixed by the spin's magnetic moment and the applied field, and quantify why electrons resonate at far higher frequency than nuclei at the same field.

Step 1. The magnetic moment operator. A spin angular momentum $\mathbf{S}$ (or $\mathbf{I}$ for a nucleus) has an associated magnetic moment

\boldsymbol{\mu} = g\gamma_0 \mathbf{S}

where $\gamma_0$ collects fundamental constants and $g$ is a dimensionless correction factor (2.0023 for a free electron, nucleus-specific for nuclei, e.g. 5.586 for the proton). This proportionality is an experimental fact traceable to the Dirac equation and to nuclear structure, but here it is simply the input.

Step 2. The Zeeman interaction energy. Placing the moment in a static field $B_0\hat{z}$ gives an interaction Hamiltonian

\hat{H} = -\boldsymbol{\mu}\cdot\mathbf{B}_0 = -g\mu B_0 \hat{S}_z / \hbar

using the appropriate magneton $\mu$ ( $\mu_B$ for electrons, $\mu_N$ for nuclei) so that $\hat{S}_z/\hbar$ has eigenvalues $m = -S, \ldots, +S$ in integer steps.

Step 3. Diagonalize. Because $\hat{S}_z$ is already diagonal in the $|S,m\rangle$ basis, the energy eigenvalues follow immediately:

E_m = -g\mu B_0 m

For spin-1/2 (proton, electron), $m = \pm 1/2$ , giving two levels separated by

\Delta E = E_{-1/2} - E_{+1/2} = g\mu B_0

Step 4. Selection rule. The oscillating magnetic field of electromagnetic radiation, if polarized perpendicular to $B_0$ , couples states differing by $\Delta m = \pm 1$ (it acts like a raising/lowering operator on spin). No other transitions are allowed to first order, so there is exactly one resonance frequency per chemically distinct spin-1/2 nucleus in a first approximation.

Step 5. Impose resonance. Absorption occurs when the photon energy matches the gap exactly:

h\nu_{\text{res}} = g\mu B_0 \quad\Longrightarrow\quad \nu_{\text{res}} = \frac{g\mu B_0}{h} = \frac{\gamma B_0}{2\pi}

where $\gamma = g\mu/\hbar$ is the gyromagnetic ratio, a spin-specific constant measured experimentally.

Step 6. Compare electron and proton. The Bohr magneton exceeds the nuclear magneton by the electron-to-proton mass ratio, $\mu_B/\mu_N = m_p/m_e \approx 1836$ . Combined with the g-factors ( $g_e \approx 2.0023$ , $g_p \approx 5.586$ ), the frequency ratio at fixed field is

\frac{\nu_e}{\nu_p} = \frac{g_e \mu_B}{g_p \mu_N} \approx \frac{2.0023 \times 1836}{5.586} \approx 658

This is why a 0.34 T field, trivial for ESR, drives electron resonance near 9.5 GHz (X-band microwave), while the same field would put proton NMR at only ~14.5 MHz, a very weak radiofrequency signal.

\boxed{\nu_{\text{res}} = \dfrac{g\mu B_0}{h}, \qquad \dfrac{\nu_e}{\nu_p} \approx 658 \text{ at fixed } B_0}

Worked examples

Example 1: Proton NMR frequency and population difference at 9.4 T. A 9.4 T magnet is a common "400 MHz" NMR spectrometer (the number names the proton frequency). Verify the frequency and find the population excess in the lower spin state at 298 K. Step 1 — frequency. For the proton, $\gamma_p/2\pi = 42.577$ MHz T $^{-1}$ (a tabulated, experimentally measured constant).

\nu = \gamma B_0/2\pi = 42.577 \text{ MHz T}^{-1} \times 9.4 \text{ T} = 400.2 \text{ MHz}

This matches the instrument's name, confirming the resonance relation. Step 2 — energy gap.

\Delta E = h\nu = (6.626\times10^{-34}\text{ J s})(400.2\times10^{6}\text{ s}^{-1}) = 2.652\times10^{-25}\text{ J}

Step 3 — population ratio via Boltzmann, per mole basis using $k_B T$ at 298 K:

k_BT = (1.381\times10^{-23}\text{ J K}^{-1})(298\text{ K}) = 4.116\times10^{-21}\text{ J}

\frac{N_\beta}{N_\alpha} = \exp\left(-\frac{\Delta E}{k_BT}\right) = \exp\left(-\frac{2.652\times10^{-25}}{4.116\times10^{-21}}\right) = \exp(-6.443\times10^{-5}) = 0.999936

Step 4 — fractional excess. The population difference relative to the total is approximately

\frac{N_\alpha - N_\beta}{N_\alpha + N_\beta} \approx \frac{\Delta E}{2k_BT} = 3.22\times10^{-5}

Only about 32 spins per million are "extra" in the lower state. This tiny excess, not the total number of spins, generates the entire measurable NMR signal — the physical reason NMR is inherently less sensitive than optical or even ESR spectroscopy, and why signal averaging over many scans is standard practice. Sanity check: doubling the field to 18.8 T would double both $\nu$ and the population excess, consistent with $\Delta E \propto B_0$ ; this is exactly why "higher field" magnets are prized for sensitivity as well as resolution.

Example 2: ESR of a methyl radical — hyperfine coupling and spectrum prediction. The methyl radical, $\cdot$ CH $_3$ , has one unpaired electron coupling to three equivalent protons with hyperfine coupling constant $a_H = 2.30$ mT. Predict the ESR spectrum and locate the resonance field at X-band (9.500 GHz), given $g = 2.0025$ for this radical. Step 1 — number of lines. For $n$ equivalent spin-1/2 nuclei, hyperfine coupling splits the ESR line into $n+1$ lines with binomial (Pascal's triangle) intensity ratios. For $n = 3$ protons:

n + 1 = 4 \text{ lines}, \qquad \text{intensity ratio } 1:3:3:1

This mirrors the $n+1$ rule for NMR multiplets, since both arise from the same angular-momentum coupling mathematics, only here nuclear spins split an electron resonance rather than the reverse. Step 2 — line spacing. Each pair of adjacent lines is separated by exactly the hyperfine coupling constant,

\Delta B_{\text{adjacent}} = a_H = 2.30\text{ mT}

so the four lines span a total width of $3 \times 2.30\text{ mT} = 6.90\text{ mT}$ from outermost to outermost. Step 3 — center field from the resonance condition. Rearranging $h\nu = g\mu_B B_0$ :

B_0 = \frac{h\nu}{g\mu_B} = \frac{(6.626\times10^{-34}\text{ J s})(9.500\times10^{9}\text{ s}^{-1})}{(2.0025)(9.274\times10^{-24}\text{ J T}^{-1})}

Numerator: $6.626\times10^{-34} \times 9.500\times10^{9} = 6.295\times10^{-24}$ J. Denominator: $2.0025 \times 9.274\times10^{-24} = 1.857\times10^{-23}$ J T $^{-1}$ .

B_0 = \frac{6.295\times10^{-24}}{1.857\times10^{-23}} = 0.339\text{ T} = 339\text{ mT}

Step 4 — assemble the spectrum. Four lines centered at 339 mT, spaced 2.30 mT apart, at approximately 335.85, 338.15, 340.45, 342.75 mT, intensities 1:3:3:1. Sanity check: 339 mT for X-band is squarely in the typical ESR range (free electron at 9.5 GHz resonates near 339 mT since $g\approx g_e$ ), and a hyperfine constant of order millitesla is typical for protons directly attached to a carbon bearing the unpaired electron. Both numbers are physically reasonable.

Limiting cases & common traps

Situation	What happens	Common error
$B_0 \to 0$	Zeeman splitting vanishes; no resonance possible	Forgetting NMR/ESR transitions do not exist without a field, unlike vibrational/rotational spectra
High field limit	Chemical shift dispersion (Hz) grows with $B_0$ ; ppm scale stays fixed	Confusing chemical shift in ppm (field-independent) with the Hz separation (field-dependent)
Many equivalent coupled spins	Multiplet follows $n+1$ rule with binomial intensities	Assuming coupling constant $J$ or $a$ changes with spectrometer field — it does not, it's in Hz/mT directly
Electron vs. nuclear spin	ESR gap ~658× larger than NMR at same field	Assuming ESR and NMR use comparable magnet strengths; ESR uses far smaller fields for similar frequency range
Fast chemical exchange	Distinct resonances coalesce into one averaged line	Misreading a broadened, averaged peak as a single "real" chemical environment

The chemist behind it

Felix Bloch and Edward Purcell independently detected nuclear magnetic resonance in bulk matter in 1945-46, sharing the 1952 Nobel Prize in Physics. Purcell's group at Harvard measured absorption in paraffin wax; Bloch's group at Stanford used induction in water, developing the classical vector picture (the Bloch equations) still taught today. Their discovery transformed a nuclear-physics curiosity into the analytical backbone of modern chemistry, culminating decades later in magnetic resonance imaging.

Origins · 1 minEdward Mills Purcell and how this was discovered

Edward Mills Purcell

1912-1997 · Co-discovered nuclear magnetic resonance in bulk matter in 1946, sharing the 1952 Nobel Prize in Physics with Felix Bloch

Ask the tutor: How do relaxation times $T_1$ and $T_2$ actually shape the linewidth and lineshape you see in a real NMR or ESR spectrum?

Why do NMR and ESR use such different frequency ranges for similar magnet strengths

The resonance frequency is proportional to the gyromagnetic ratio, which depends on the magnetic moment of the spin. Electrons carry a magnetic moment (Bohr magneton) about 658 times larger than a proton's (nuclear magneton), largely reflecting the electron-to-proton mass ratio. So at a typical laboratory field, electron transitions land in the microwave range (gigahertz) while proton transitions land in the radiofrequency range (megahertz). ESR spectrometers therefore use much weaker magnets, often under 1 T, while achieving comparably practical frequencies.

Does chemical shift depend on the strength of the spectrometer magnet

No, not when expressed in ppm. Chemical shift is defined as a ratio of frequency difference to the reference frequency, both of which scale together with field strength, so the ppm value is intrinsic to the molecule's electronic environment. However, the shift expressed in hertz does grow with field, which is why higher-field spectrometers give better peak resolution: the same ppm separation corresponds to a larger frequency gap in Hz.

What physically causes spin-spin coupling if it is not a direct magnetic dipole interaction through space

In solution, molecular tumbling averages direct through-space dipolar coupling to zero. The coupling observed in liquid-state spectra, called J-coupling or scalar coupling, is transmitted indirectly through the bonding electrons: one nuclear spin polarizes nearby electron spins via the Fermi contact interaction, and that polarization propagates through the bond network to influence a neighboring nucleus. Because it depends on bonding electrons, J is sensitive to the number of intervening bonds, bond angles, and hybridization.

Why is ESR so much rarer in undergraduate labs than NMR

ESR only detects species with unpaired electrons, meaning free radicals, many transition-metal complexes, and triplet states. Most stable organic molecules are closed-shell and completely ESR-silent, whereas essentially every organic molecule contains protons or carbons visible in NMR. ESR spectrometers are also less common instruments in synthetic teaching labs, reserved more for radical chemistry, catalysis, and materials research where paramagnetic species are the object of study.

How does linewidth affect what a magnetic resonance spectrum can tell us

As with any spectroscopic transition, natural, lifetime, and exchange broadening set a floor on how narrow a resonance line can be, following the same time-energy uncertainty relations covered for line broadening generally. In NMR, relaxation times $T_1$ and $T_2$ govern linewidth; short $T_2$ (common for large molecules or paramagnetic samples) broadens lines and can obscure fine coupling structure. In ESR, unpaired electrons typically relax so fast that lines are much broader than in NMR, another reason hyperfine structure must sometimes be resolved with great care.