Magnetic Resonance

Magnetic resonance is the family of techniques that uses a static magnetic field to split magnetic energy levels and an oscillating field to drive transitions between them. It made spin and magnetic moments experimentally controllable: not only deflected in a field gradient, as in Stern–Gerlach Experiment, but flipped, precessed, measured, and refocused with timed pulses.

Helpful background. Magnetic moments connects angular momentum to field response. Rabi oscillations develops the two-level transition calculation used below.

Spin in a Static Field

A magnetic moment $\boldsymbol\mu$ in a magnetic field has interaction energy

$$H = - \boldsymbol\mu\cdot\mathbf B.$$

For an angular momentum $\mathbf S$ with gyromagnetic ratio $\gamma$, write

$$\boldsymbol\mu = \gamma\mathbf S.$$

If the static field is $\mathbf B_0=B_0\hat{\mathbf z}$, then

$$H_0 = - \gamma B_0 S_z.$$

For a spin-$1/2$ degree of freedom,

$$S_z = \frac{\hbar}{2}\sigma_z, \qquad H_0 = - \frac{\hbar\gamma B_0}{2}\sigma_z.$$

The two spin projections are split by an angular frequency

$$\omega_0 = \lvert\gamma\rvert B_0,$$

so the energy separation is

$$\Delta E = \hbar\omega_0.$$

Here $B_0>0$ and $\gamma$ carries its sign. If $\gamma>0$, the $S_z=+\hbar/2$ state has lower energy; for an electron, $\gamma<0$ and the order reverses. The positive resonance frequency uses $|\gamma|$.

A static field $B_0$ creates a spin transition frequency $\omega_0$. A transverse oscillating field $B_1(t)$ near $\omega_0$ drives coherent transitions; pulse duration and detuning determine the rotation in the effective two-level system.

Resonant Transitions

To drive the spin, apply a weak transverse oscillating field. A schematic field is

$$\mathbf B(t) = B_0\hat{\mathbf z} + B_1\cos(\omega t)\hat{\mathbf x}.$$

The transverse part couples the two spin projections because it contains $S_x$. The system responds strongly when

$$\omega \simeq \omega_0.$$

Near resonance, and under the usual weak-drive assumptions, the rotating-wave approximation turns the driven problem into an effective two-level Hamiltonian. Choose an energy-ordered basis so that the excited state has $\sigma_z=+1$, and choose the transverse phase to make the coupling positive. After transformation to a rotating frame,

$$H_{\mathrm{eff}} = \frac{\hbar}{2} \left( \Delta\sigma_z+\Omega\sigma_x \right),$$

where

$$\Delta=\omega_0-\omega.$$

For the linearly oscillating field written above, $\Omega=|\gamma|B_1/2$: its co-rotating component has half the amplitude. A circular field of constant transverse magnitude $B_1$, rotating in the resonant sense, instead gives $\Omega=|\gamma|B_1$. The weak-drive rotating-wave approximation requires $\Omega,|\Delta|\ll\omega_0$; see the rotating-wave approximation for the discarded counter-rotating term.

On resonance, a pulse of duration $t$ rotates the two-level state by an angle

$$\theta = \Omega t.$$

For an ideal resonant pulse in this isolated two-level model, a $\pi$ pulse exchanges the level populations. Starting from one energy eigenstate, a $\pi/2$ pulse creates an equal coherent superposition. A unitary pulse cannot turn a maximally mixed state into a pure superposition. Those pulse ideas became central in nuclear magnetic resonance, electron spin resonance, atomic clocks, quantum optics, and quantum information.

Molecular Beam Resonance

The earliest magnetic-resonance program was closely tied to molecular and atomic beams. In the molecular beam magnetic-resonance method, particles pass through magnetic-field regions that select, drive, and analyze internal magnetic states. When the applied radio-frequency field matches an internal transition frequency, the beam intensity at a detector changes.

This was a major step beyond passive magnetic deflection. A Stern–Gerlach apparatus separates components; a resonance apparatus can induce transitions between them. It therefore measures magnetic moments through frequencies rather than only through spatial deflections. Frequency measurement is extraordinarily precise, so magnetic resonance turned small magnetic moments and hyperfine splittings into precision observables.

Historically, this route led to accurate nuclear magnetic moments, hyperfine structure measurements, and later atomic-clock methods. It also made the connection between magnetic moments and quantum transitions operational:

$$\hbar\omega = E_f-E_i.$$

The same equation appears in spectroscopy, but magnetic resonance made the controlled transition itself the instrument.

Nuclear Magnetic Resonance

Nuclear magnetic resonance, or NMR, uses transitions of nuclear magnetic moments in a magnetic field. A nucleus with nonzero spin has magnetic sublevels. A radio-frequency field can drive transitions between them when the frequency matches the Larmor frequency.

For a spin-$1/2$ nucleus,

$$\frac{\omega_0}{2\pi} = \frac{|\gamma|}{2\pi} B_0.$$

For protons,

$$\frac{\gamma_p}{2\pi} \approx 42.58\,\mathrm{MHz/T}.$$

Real NMR is richer than the isolated-spin formula. Chemical shielding shifts the local field and therefore the resonance frequency. Spin-spin couplings split lines. Relaxation times $T_1$ and $T_2$ describe return to thermal equilibrium and loss of transverse coherence. Pulse sequences can refocus inhomogeneous dephasing, select coherences, and measure structure.

For quantum mechanics, NMR is important because it turns abstract spin precession into a controllable laboratory language. It also teaches a recurring lesson: spectra, dynamics, and environment must be modeled together.

Electron Spin Resonance

Electron spin resonance, also called electron paramagnetic resonance, drives transitions of unpaired electron spins. Because the electron magnetic moment is much larger than a nuclear magnetic moment, electron resonance frequencies are much higher at the same field.

A useful scale is

$$\frac{\lvert\gamma_e\rvert}{2\pi} \approx 28.0\,\mathrm{GHz/T}$$

for a nearly free electron with $g\simeq2$. Thus a microwave frequency near $9.5\,\mathrm{GHz}$ corresponds to a field near $0.34\,\mathrm T$.

Electron spin resonance is sensitive to $g$ factors, local crystal fields, hyperfine coupling to nearby nuclei, exchange interactions, and spin relaxation. It became a central tool in chemistry, condensed matter, radiation damage studies, defects in solids, and spin-based quantum technologies.

The same caution applies as in the Zeeman effect: there is no universal resonance formula without specifying the effective Hamiltonian. A free electron, an electron in a molecule, a defect center in a crystal, and an effective spin in a many-body material can have different $g$ tensors and couplings.

Measuring Magnetic Moments

Magnetic resonance measures magnetic moments by comparing a transition frequency with a calibrated magnetic field. The simplest proportionality is

$$\omega_0 = \lvert\gamma\rvert B_0.$$

If $B_0$ is known, measuring $\omega_0$ determines $\lvert\gamma\rvert$. With the angular momentum known, this gives a magnetic moment or $g$ factor.

In practice, precision work must control:

• magnetic-field calibration and homogeneity;
• line broadening from field gradients and interactions;
• power broadening from strong drives;
• systematic shifts from neighboring levels;
• relaxation and decoherence;
• sample geometry and environmental couplings.

That list is not a nuisance added after the “real” quantum mechanics. It is part of the quantum experiment. The frequency being measured belongs to an effective Hamiltonian embedded in a real apparatus.

Pulses as tools for control

Resonant pulses rotate a spin; free evolution accumulates phase. Combining the two gives Ramsey interferometry, while a spin-echo pulse can reverse dephasing from a static distribution of resonance frequencies. It does not reverse arbitrary relaxation or irreversible noise. These methods connect magnetic resonance to modern quantum control.

Common pitfalls

Distinguish the level spacing from its sign convention. $\omega_0=|\gamma|B_0$ is positive. Which spin projection is the ground state depends on the sign of $\gamma$.

Specify the drive amplitude. For a linear field $B_1\cos\omega t$, the resonant Rabi frequency is $|\gamma|B_1/2$ in the convention used here. A circular drive of magnitude $B_1$ has twice that Rabi frequency.

A linewidth need not measure irreversible coherence loss. Static field inhomogeneity and drive power can broaden a line. Echo measurements help distinguish reversible inhomogeneous dephasing from other decay processes.

Exercises

1. Estimate the proton NMR frequency in a $3.0\,\mathrm T$ magnetic field using $\gamma_p/(2\pi)=42.58\,\mathrm{MHz/T}$.

Solution

Use

$$\frac{\omega_0}{2\pi} = \left( 42.58\,\mathrm{MHz/T} \right) B_0.$$

For $B_0=3.0\,\mathrm T$,

$$\frac{\omega_0}{2\pi} \approx 127.7\,\mathrm{MHz}.$$

2. A nearly free electron has $\lvert\gamma_e\rvert/(2\pi)\approx28.0\,\mathrm{GHz/T}$. What field corresponds to an ESR frequency of $9.5\,\mathrm{GHz}$?

Solution

Solve $f=(\lvert\gamma_e\rvert/2\pi)B_0$:

$$B_0 = \frac{9.5\,\mathrm{GHz}}{28.0\,\mathrm{GHz/T}} \approx 0.34\,\mathrm T.$$

3. If an on-resonance drive has Rabi frequency $\Omega/(2\pi)=25\,\mathrm{kHz}$, what is the duration of a $\pi$ pulse?

Solution

A $\pi$ pulse satisfies $\Omega t_\pi=\pi$, so

$$t_\pi = \frac{\pi}{\Omega} = \frac{1}{2(25\,\mathrm{kHz})} = 20\,\mu\mathrm s.$$

4. Why must the resonant driving field have a transverse component?

Solution

For a static field along $z$, the energy eigenstates are eigenstates of $S_z$. For this isolated-spin Hamiltonian, a field parallel to $z$ is diagonal in that basis at every time. It changes phases but cannot change the two populations. A transverse field contains $S_x$ or $S_y$, which has off-diagonal matrix elements between the two spin projections and can drive transitions.

References

• A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, 1961.
• F. Bloch, “Nuclear Induction,” Physical Review 70, 460-474, 1946, DOI: 10.1103/PhysRev.70.460.
• M. H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic Resonance, 2nd ed., Wiley, 2008.
• C. P. Poole Jr., Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques, 2nd ed., Dover, 1996.
• E. M. Purcell, H. C. Torrey, and R. V. Pound, “Resonance Absorption by Nuclear Magnetic Moments in a Solid,” Physical Review 69, 37-38, 1946, DOI: 10.1103/PhysRev.69.37.
• I. I. Rabi, S. Millman, P. Kusch, and J. R. Zacharias, “The Molecular Beam Resonance Method for Measuring Nuclear Magnetic Moments,” Physical Review 55, 526-535, 1939, DOI: 10.1103/PhysRev.55.526.
• C. P. Slichter, Principles of Magnetic Resonance, 3rd ed., Springer, 1990.