NUCLEAR INDUCTION / LONGITUDINAL RECOVERY

NMR Inversion Recovery Relaxometer

Invert the equilibrium magnetization, wait, then read it with a right-angle pulse. Scan independent delay preparations and infer longitudinal relaxation without assuming a perfect inversion or a known receiver amplitude.

One exponential longitudinal reservoir, ideal hard pulses and a crusher that removes residual transverse coherence. Each delay starts at equilibrium. The on-resonance, phase-calibrated receiver samples the same short time after readout; its attenuation is absorbed into an unknown scale. No direct observation of the longitudinal magnetization, no MRI image, and no historical fit.

Vertical static field; horizontal transmit/receive coil. A switch isolates the receiver during the pulse, then routes the induced voltage to a phase-referenced quadrature receiver. Cable motion is schematic. $B_0=0.35\,\mathrm{T},\quad f_{\rm ref}\simeq14.902\,\mathrm{MHz}$

PULSE SEQUENCE / AVERAGED RECEIVER VOLTAGES

Invert. Wait. Read the sign.

Equilibrium reset

\alpha_x

Ideal crusher

t_d

(\pi/2)_x

Receiver blankingSigned quadratures

Each of 41 delays uses a fresh equilibrium ensemble. After inversion and ideal transverse spoiling, a calibrated right-angle pulse maps the recovered longitudinal component into induction. Every signed sample is taken 0.4 milliseconds after readout. Repetitions are fully reset; playback is accelerated.

Ready. Predict, then acquire.

RAW VOLTAGE RECORD / SEPARATE SOURCE EXPECTATION

Two quadratures. Keep the sign.

$t_d/\mathrm{s},\quad I,Q\;(\mathrm{\mu V})$

Teal: in-phase voltage. Gold: quadrature voltage. Dots: retained observations. Thin lines: source expectations. Purple dashes: fit using records only. Gray: saved observations.

SIGNED RECOVERY / RECORD-ONLY INFERENCE

A zero crossing carries phase information.

$t_d/\mathrm{s},\quad I/\mathrm{\mu V}$

The deficit and equilibrium voltage are inferred together with recovery time. The quadrature noise channel is retained above; it does not directly measure longitudinal magnetization.

INFERENCE / SOURCE PARAMETERS EXCLUDED

Inferred longitudinal time

\text{—}

Inferred preparation deficit ratio

\text{—}

Inferred equilibrium voltage

\text{—}

Retained complex samples

0

Voltage residual RMS

\text{—}

Global bounded search with profiled amplitude. Weak-information and boundary checks are diagnostic heuristics, not parameter confidence intervals. A synthetic recovery does not independently establish the physical law.

Latest raw observations

Index	$t_d/\mathrm{s}$	$I/\mathrm{\mu V}$	$Q/\mathrm{\mu V}$

SAVED COMPARISON / MODEL BOUNDARY

Single noninteracting resonance; no molecular chemical shifts, coupling multiplets, diffusion, radiation damping, coil loading or realistic pulse bandwidth. Gain and phase are calibrated; no drift or digitizer quantization. Instantaneous hard-pulse approximation; the Cauchy static-offset model is not a computed field map.

Compare: free induction spectrum Read the magnetic-resonance evidence TeachSpin original experimental guide Manufacturer instrument manual Bruker transmit/receive probe manual NIST reference gyromagnetic ratio

Start with the essentials

Focus question: Keep the sign. Watch magnetization recover.
One-sentence intuition: Keep source assumptions separate from estimates made from records.

Core mathematical model

Recover a longitudinal component

\frac{M_y(t_d)}{M_0}=1-(1-\cos\alpha)e^{-t_d/T_1}

The vertical static field defines the longitudinal direction. An ideal crusher removes residual transverse coherence after preparation; each delay starts from a fresh equilibrium ensemble.

Read the sign through induction

\begin{aligned}\langle I(t_d)\rangle&=a-be^{-t_d/T_1}\\b/a&=1-\cos\alpha\end{aligned}

A calibrated right-angle readout maps longitudinal magnetization into the receiving plane. Fixed receiver dead time and gain enter the unknown scale. The receiver does not directly detect a static longitudinal component.

Only perfect inversion gives this zero

t_{\rm zero}=T_1\ln 2\quad(\alpha=\pi)

Fit the equilibrium scale, deficit and recovery time together. Imperfect inversion shifts the zero, so the zero alone is not a general lifetime estimator.

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
Keep a negative signal
Scan the full delay grid and inspect the initial negative in-phase voltage.
What to observe: The phase reverses as the recovered longitudinal component crosses zero. Taking the magnitude would fold the curve.
02
Change preparation, not relaxation
Save the run, then select imperfect inversion and acquire again.
What to observe: The inferred deficit changes while the recovery time stays compatible. No source angle is supplied to inference.
03
A flat record contains no clock
Select no preparation and repeat the same scan.
What to observe: A fitted exponential is insufficient evidence of a relaxation time. A constant-response comparison withholds weak estimates.