Continuous nuclear resonance / Saturation

NMR Power Saturation Analyzer

Why does the absorption signal of a nuclear resonance shrink when the radio-frequency drive gets too strong? The drive tips the nuclear magnetization faster than relaxation can rebuild it, so the magnetization along the field shrinks. Past a critical drive the absorption signal falls, yet the absorbed power keeps rising toward a ceiling set by the longitudinal relaxation time. Where the signal peaks tells you only the product of the two relaxation times, not each one.

Lock the drive to resonance and step the calibrated RF nutation rate. Record how the absorption voltage first rises and then falls, estimate the critical drive, and find out what the data cannot tell apart about the relaxation times.

Drive the sample, balance the bridge, then infer

Why does the absorption signal of a nuclear resonance shrink when the radio-frequency drive gets too strong?

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

Calibrated RF nutation frequency

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

1.6

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

The spins form one homogeneous Bloch reservoir, the drive is locked exactly on resonance, and every exposure is independent and fully settled. The calibrated nutation frequency is proportional to the co-rotating RF field, not to the power in watts. The voltage maximum fixes the product of the relaxation times; because the receiver scale is unknown, the two times cannot be recovered separately. This is a simulated bridge experiment, not a power calorimeter.

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

Find the drive strength where the curve turns over.

Set frequency and RF amplitudeReach exact steady stateAverage independent IQ voltages

61 independent settled exposures span calibrated nutation rates from zero to twelve hertz, at exact resonance. Rate is proportional to RF field; drive power is proportional to its square for an unchanged coil. No watt calibration or resonance-lock hardware is solved.

Ready. Predict, then acquire.

Raw voltages, with the source expectation shown separately

The voltage rises, then falls.

$r/\mathrm{Hz},\quad I,Q\;(\mathrm{\mu V})$

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 critical nutation rate

\text{—}

Inferred relaxation-time product

\text{—}

Inferred in-phase baseline

\text{—}

Retained complex exposures

0

Voltage residual RMS

\text{—}

Unknown amplitude and baseline are profiled out. Only the product of the relaxation times can be determined. Records that do not resolve the turnover are rejected by comparison with a linear, unsaturated response. Search bounds and weak-information thresholds are diagnostic heuristics, not confidence intervals.

Latest raw observations

Index	$r/\mathrm{Hz}$	$I/\mathrm{\mu V}$	$Q/\mathrm{\mu V}$

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.

Compare: absorption and dispersion 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: Why does the absorption signal of a nuclear resonance shrink when the radio-frequency drive gets too strong?
One-sentence intuition: Keep the assumptions built into the simulated source apart from what you estimate from the recorded data.

Core mathematical model

The critical RF amplitude

\begin{aligned}I(r)&=b+\frac{a r}{1+(r/r_c)^2}\\r_c&=\frac{1}{2\pi\sqrt{T_1T_2}}\end{aligned}

On exact resonance, the absorption voltage is largest when the calibrated nutation rate reaches its critical value. The RF field here is the co-rotating component, which is half the peak of the linearly oscillating field.

Only the product of the two times can be found

T_1T_2=\frac{1}{(2\pi r_c)^2}

The inference is not given the source scale. Two pairs of relaxation times with the same product give the same voltage curve once the unknown gain compensates.

Absorbed power and the measured voltage

\frac{P_{\rm abs}}{M_0B_0}=\Omega v=\frac{\Omega^2T_2}{1+\Omega^2T_1T_2}

The normalized absorbed power rises toward a plateau, while the voltage component can fall. The page gives no sample magnetization density, no volume and no watt calibration.

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
Scan past the turnover
Acquire over the full calibrated drive range, from zero drive out to the strong-drive tail.
What to observe: The fit finds the unknown baseline and scale together with the critical drive. The residuals are computed from the recorded voltages.
02
Test what the data cannot separate
Save the reference run, then choose the Same product, different times preset and acquire again.
What to observe: The source times are different, yet the mean voltage curves lie on top of each other. Neither time is reported on its own.
03
Separate the voltage from the absorbed energy
Compare the falling tail of the voltage curve with the absorbed-power formula.
What to observe: A smaller voltage does not prove that less energy is being absorbed. The analyzer is not a calorimeter.