---
title: "NMR Inversion Recovery Relaxometer Interactive Lab | Sandbox Physics"
description: "Keep the sign. Watch magnetization recover. 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."
canonical_url: "https://sandboxphysics.com/labs/nmr-inversion-recovery/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/nmr-inversion-recovery/"
last_modified: "2026-10-18"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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 blanking Signed 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](https://sandboxphysics.com/labs/nmr-free-induction-spectrometer/) [Read the magnetic-resonance evidence](https://sandboxphysics.com/stories/quantum-evidence/magnetic-resonance-beams-absorption-induction/) [TeachSpin original experimental guide](https://home.sandiego.edu/~severn/p480w/PNMR_Concept_Tour.pdf) [Manufacturer instrument manual](https://physics.wm.edu/~inovikova/phys352/PHYS352-Manuals-Instructions/Pulsed-NMR/nmr_company.pdf) [Bruker transmit/receive probe manual](https://2210pc.chem.uic.edu/nmr/downloads/bruker/en-US/pdf/z31339.pdf) [NIST reference gyromagnetic ratio](https://physics.nist.gov/cuu/Constants/Table/allascii.txt)

Physics tutorial

## NMR Inversion Recovery Relaxometer

**Background** 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.

**Why it matters** A constant is inferred through a calibrated response and retained observations.

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 misconception** Recovering a source parameter independently confirms the law.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

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.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)