CONTINUOUS NUCLEAR RESONANCE / LINE SHAPE

Continuous-wave NMR Spectrometer

Balance a radio-frequency bridge, step the drive frequency, and retain absorption and dispersion voltages. Estimate the resonance center and power-broadened line width from those records.

DRIVE / BALANCE / INFER

A resonance has two faces.

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.

One homogeneous resonance at fully settled Bloch steady state. The RF bridge ideally cancels direct drive pickup; phase and channel gains are calibrated. Independent Gaussian electronics noise, unknown constant baselines. Synthetic voltages, no material or historical data fit. A broadened line alone is not an 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 / SETTLED VOLTAGE EXPOSURES

Step the frequency. Keep 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 / SEPARATE SOURCE EXPECTATION

Absorption is even. 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 RETAINED VOLTAGES

What the fit leaves behind.

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

Only recorded coordinates appear. No residual is generated before a supported fit. A small residual shows consistency with this response family, not independent confirmation of the physical law.

INFERENCE / SOURCE PARAMETERS EXCLUDED

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. A lifetime is not inferred from this width. 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 / LIMITS & SOURCES

Single homogeneous, noninteracting resonance with exponential Bloch relaxation. No static field distribution, molecular shifts or coupling, heating, diffusion, radiation damping, finite settling, bridge imbalance, drive leakage, phase error or 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: A resonance has two faces.
One-sentence intuition: Keep source assumptions separate from estimates made from records.

Core mathematical model

Steady 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}

Normalized rotating-frame components. Detuning is sample frequency minus drive frequency; the bridge phase maps absorption to 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 height of baseline-subtracted absorption. An intrinsic time follows from inverse width only in the weak-drive homogeneous limit.

Fit what was recorded

\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}

Both channel amplitudes and constant baselines are unknown nuisance parameters. Center and half-width are found jointly, without source lifetimes or drive entering the fit.

Common difficulties

A simulation supplies its law

Typical misconceptionRecovering a source parameter independently confirms the law.

Better mental modelThese synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

Run the experiment

01
Read both channels
Scan the reference line; compare the absorption maximum and the baseline-subtracted dispersive sign change.
What to observe: Both responses share one center and width. Dots are retained voltages; thin source curves are a separate reference.
02
Broaden with drive
Save the run, select the strong-drive preset and scan again.
What to observe: The line broadens while the source transverse time is unchanged. The estimate reports the broadened width.
03
Remove the signal
Turn the RF drive off and acquire the same scan.
What to observe: A baseline and noise do not locate a resonance; weak-information diagnostics withhold the line estimate.