Continuous nuclear resonance / Line shape

Continuous-wave NMR Spectrometer

How do you find the center and width of a nuclear resonance with a continuous radio-frequency scan? Step the drive frequency and record two signals, one in phase with the drive and one a quarter cycle out of phase. Absorption peaks at resonance; dispersion changes sign there. Both share one center and width. Remove the baseline, then measure the width at half height. A strong drive saturates the spins and broadens the line, so only a weak drive reveals the intrinsic transverse relaxation time.

Balance a radio-frequency bridge, step the drive frequency, and record the absorption and dispersion voltages. From those readings, estimate the center of the resonance and its power-broadened line width.

Drive the sample, balance the bridge, then infer

How do you find the center and width of a nuclear resonance with a continuous radio-frequency scan?

Source longitudinal time · excluded from fit

T_1/\mathrm{s}

0.35

Source homogeneous transverse time · excluded from fit

T_2/\mathrm{ms}

25

Source receiver scale · excluded from fit

G/\mathrm{\mu V}

300

Source in-phase baseline · excluded from fit

b_I/\mathrm{\mu V}

2

Calibrated single-exposure noise

\sigma/\mathrm{\mu V}

1

Independent settled repeats

n

8

Source resonance offset · excluded from fit

f_0-f_{\rm ref}\;(\mathrm{Hz})

12

Calibrated RF nutation frequency

r=\Omega/(2\pi)\;(\mathrm{Hz})

1

Drive frequency offset

f-f_{\rm ref}\;(\mathrm{Hz})

0

Moving the frequency or drive probe retains records. Source changes clear current records; a saved comparison is retained.

There is one homogeneous resonance, and each point is taken after the Bloch steady state has fully settled. The RF bridge ideally cancels the drive signal picked up directly, and the phase and the gain of each channel are calibrated. The electronics noise is independent and Gaussian, and the baselines are constant but unknown. The voltages are simulated and are not fitted to any material or historical data. A broadened line by itself does not give the intrinsic transverse lifetime.

Nominal vertical static field, transverse sample coil. Continuous RF and a matched reference arm feed a balanced bridge; the residual reaches a phase-referenced receiver. Ideal cancellation and flat calibrated gain, not a solved RF circuit. Cable motion is schematic. $B_0=0.35\,\mathrm{T},\quad f_{\rm ref}\simeq14.902\,\mathrm{MHz}$

Continuous drive, voltages averaged at steady state

Step the frequency and record both phases.

Set frequency and RF amplitudeReach exact steady stateAverage independent IQ voltages

121 independent settled exposures span the drive offset from minus 180 to plus 180 hertz. Each point uses the selected fixed RF amplitude. This is a stepped steady-state scan; the accelerated playback does not model rapid passage or finite dwell.

Ready. Predict, then acquire.

Raw voltages, with the source expectation shown separately

Absorption is even about resonance, and dispersion changes sign.

$(f-f_{\rm ref})/\mathrm{Hz},\quad I,Q\;(\mathrm{\mu V})$

Teal: absorptive in-phase voltage. Gold: dispersive quadrature voltage. Dots: observations. Thin lines: source expectations. Purple dashes: record-only fit. Gray: exact saved records. Constant receiver baselines are retained and fitted.

Residuals computed from the retained voltages

What the fit leaves behind.

$(I-I_{\rm fit}),\;(Q-Q_{\rm fit})\quad(\mathrm{\mu V})$

Only recorded coordinates appear, and no residual is drawn until a supported fit exists. A small residual means the data are consistent with this family of responses. It does not independently confirm the physical law.

Inference without the source parameters

Inferred resonance offset

\text{—}

Inferred full width at half height

\text{—}

Inferred absorption peak above baseline

\text{—}

Retained complex exposures

0

Voltage residual RMS

\text{—}

Unknown channel amplitudes and baselines are profiled out. The line width includes drive saturation, so no lifetime is inferred from it. Search bounds and weak-information thresholds are diagnostic heuristics, not confidence intervals.

Latest raw observations

Index	$(f-f_{\rm ref})/\mathrm{Hz}$	$I/\mathrm{\mu V}$	$Q/\mathrm{\mu V}$

Saved comparison, model limits and sources

The model has a single homogeneous, noninteracting resonance with exponential Bloch relaxation. It leaves out static field distribution, molecular shifts or coupling, heating, diffusion, radiation damping, finite settling, bridge imbalance, drive leakage, phase error and RF circuit loading. Receiver gain is assumed flat over the narrow carrier span. The bridge measures response voltage, not absorbed watts or individual nuclear outcomes.

Continue: power saturation Compare pulsed free induction Magnetic-resonance evidence Bloch 1946 · equations and steady state Purcell, Torrey & Pound · original absorption report TeachSpin CW bridge manual NIST reference constant

Start with the essentials

Focus question: How do you find the center and width of a nuclear resonance with a continuous radio-frequency scan?
One-sentence intuition: Keep the assumptions built into the simulated source apart from what you estimate from the recorded data.

Core mathematical model

Steady-state absorption and dispersion

\begin{aligned}v&=\frac{\Omega T_2}{1+(\Delta T_2)^2+\Omega^2T_1T_2}\\u&=\Delta T_2v\end{aligned}

These are the normalized magnetization components in the rotating frame. The detuning is the sample frequency minus the drive frequency, and the bridge phase puts the absorption signal in the in-phase channel.

Power broadens the line

\Gamma_{\rm FWHM}=\frac{\sqrt{1+\Omega^2T_1T_2}}{\pi T_2}

This is the full width at half maximum of the absorption peak after the baseline is subtracted. Only for a weak drive and a homogeneous line does the inverse width give the intrinsic transverse time.

Fitting both recorded channels

\begin{aligned}I&=b_I+\frac{a}{1+x^2}\\Q&=b_Q+\frac{d x}{1+x^2},\quad x=\frac{c-f}{h}\end{aligned}

The amplitude and constant baseline of each channel are unknown and are fitted along the way. The center and half-width are searched together; neither the source relaxation times nor the drive enters the fit.

Common difficulties

The simulation already contains the law

Typical misconceptionGetting the source parameter back independently confirms the law.

Better mental modelThe data were generated from that law, so getting the parameter back tests the measurement and estimation steps. The data are not historical observations, they do not give a confidence interval for the parameter, and they are not external scientific certification.

Run the experiment

01
Read both channels
Choose the Reference line preset and scan. Compare where the absorption peaks with where the dispersion signal, after baseline subtraction, changes sign.
What to observe: Both signals share one center and one width. The dots are the recorded voltages; the thin curves show the source expectation separately for comparison.
02
Broaden the line with drive
Save the run, choose the Power broadened preset and scan again.
What to observe: The line gets wider even though the source transverse time has not changed. The estimate reports this broadened width.
03
Remove the signal
Turn the RF drive off and run the same scan.
What to observe: A baseline and noise alone cannot locate a resonance, so the weak-information check withholds the line estimate.

Continuous-wave NMR Spectrometer

Step the frequency and record both phases.

Absorption is even about resonance, and dispersion changes sign.

What the fit leaves behind.